Guidebook-2 Design of Bridges

Guidebook 2

DESIGN of BRIDGESPietro Croce et al.

Guidebook 2

DESIGN of BRIDGESPietro Croce et al.

Editor:

Pietro Croce, University of Pisa, Department of Civil Engineering, Structural Division

Authors:

Pietro Croce, University of Pisa, Department of Civil Engineering, Structural Division

Milan Holický, Czech Technical University in Prague, Klokner Institute

Jana Marková, Czech Technical University in Prague, Klokner Institute

Angel Arteaga, E. Torroja Institute of Construction Sciences, CSIC, Madrid

Ana de Diego, E. Torroja Institute of Construction Sciences, CSIC, Madrid

Peter Tanner, E. Torroja Institute of Construction Sciences, CSIC, Madrid

Carlos Lara, E. Torroja Institute of Construction Sciences, CSIC, Madrid

Dimitris Diamantidis, University of Applied Sciences in Regensburg, Faculty of Civil Engineering

Ton Vrouwenvelder, Netherlands Organisation for Applied Scientific Research, Delft

Guidebook 2 Design of Bridges

ISBN: 978-80-01-04617-3

Edited by : Pietro Croce, Univers ity of Pisa

Publ ished by : Czech Technica l Univers ity in Prague, Klokner Inst ituteŠ ol ínova 7 , 166 08 Prague 6 , Czech Republ ic

Pages : 230

1 st edit ion

Foreword

3

FOREWORD

The Leonardo da Vinci Project CZ/08/LLP-LdV/TOI/134020 “Transfer of Innovations

Provided in Eurocodes, addresses the urgent need to implement the new system of European

documents related to design and construction work and products.

These documents, called Eurocodes, are systematically based on the recently

developed Council Directive 89/106/EEC “The Construction Product Directive” and its

Interpretative Documents ID 1 and ID 2. Implementation of Eurocodes in each Member State

is a demanding task as each country has its own long-term tradition in design and

construction.

The project should enable an effective implementation and application of the new

methods for designing and verification of buildings and civil engineering works in all the

partner countries (CZ, DE, ES, IT, NL) and in other Member States.

The need to explain and effectively use the latest principles specified in Eurocodes

standards is apparent from enterprises, undertakings and public national authorities involved

in construction industry and also from university and colleges. Training materials, manuals

and software programmes for education are urgently required.

The submitted Guidebook 2 completes the set of two guidebooks intended to provide

required manuals and software products for training, education and effective implementation

of Eurocodes:

Guidebook 1: Load Effects on Buildings

Guidebook 2: Design of bridges.

It is expected that the Guidebooks will address the following intents in further

harmonization o f European construction industry:

- reliability improvement and unification of the process of design,

- development of a single market for product and for construction services,

- improvement of the competitiveness of the European industries in the global world

market;

- new opportunities for trained primary target groups in the labour market.

The Guidebook 2 is focused on determining load effects on road, railway and

pedestrian bridges and special civil structures. The following main topics are discussed in

particular:

- basic requirements on bridges,

- basis of structural design,

- traffic loads for static and fatigue assessment and climatic actions,

- accidental actions,

- combination rules for bridges,

- examples and case studies.

Annex A to Guidebook 2 concerns new traffic trends in European countries and their

consequences on load models and on assessment of existing bridges; Annex B provides basic

information about action and combination rules for special structures, like cranes, masts,

towers and pipelines.

Foreword

4

The Guidebook 2 is written in a user-friendly way employing only basic mathematical

tools, supplemented by examples and case studies developed in detail.

A wide range of potential users of the Guidebooks and other training materials

includes practising engineers, designers, technicians, experts of public authorities, young

people – high school and university students. The target groups come from all territorial

regions of the partner countries. However, the dissemination of the project results is foreseen

to be spread into all Member States of CEN and other interested countries.

Pisa 2010

Contents

5

GUIDEBOOK 2 – DESIGN OF BRIDGES

CONTENTS

Foreword 3

Contents 5

Chapter 1: Basic requirements 7

Chapter 2: Basis of design – methodological aspects 15

Appendix A to Chapter 2 – Principles of probabilistic optimization 27

Chapter 3: Static loads due to traffic 33

Appendix A to Chapter 3 – Development of static load traffic models

for road bridges of EN 1991-2 63

Chapter 4: Fatigue loads due to traffic 79

Appendix A to Chapter 4 – Vehicle interactions and fatigue assessments 99

Chapter 5: Non traffic actions 105

Chapter 6: Accidental actions 119

Chapter 7: Combination rules for bridges in Eurocodes 134

Chapter 8: Case study - Design of a concrete bridge 147

Chapter 9: Case study – Design of a steel bridge 165

Chapter 10: Case study - Design of a composite bridge 191

Annex A: Effects of LHVs on road bridges and EN1991-2 load models 209

Annex B: Actions and combination rules for cranes, masts, towers and pipelines 221

Contents

6

Chapter 1: Basic requirements

7

CHAPTER 1: BASIC REQUIREMENTS

Angel Arteaga1, and Ana de Diego

1

1E. Torroja Institute of Construction Sciences, CSIC. Madrid. Spain

Summary

The Eurocode system establishes a series of basic requirements that must be met by all

structures to ensure their suitability for their intended use and durability. Those requirements,

based on European Commission Directives and other construction standards in place, are

reviewed and explained in Chapter 1 of Guidebook 1. This first chapter of Guidebook 2

describes the specific requirements applicable to bridges.

1 INTRODUCTION

1.1 Background documents All the essential requirements to be met by any construction work, bridges included,

are laid down in Eurocode EN 1990 [1]. The serviceability limits specifically applicable to

bridges, in which user safety and comfort are the prime concerns, are set out in EN 1990/A1

(EN 1990 Annex 2) [2]. That standard also specifies the combinations of actions to be

considered when verifying ultimate and serviceability limit states in bridges.

1.2 General principles

Chapter 1 of Guidebook 1 on buildings [3] provides a detailed account of the

requirements that, pursuant to the Construction Products Directive (CPD) [4], must be met by

construction products for their free circulation on the European construction products market.

The provisions of the CPD are applicable not only to buildings, but to construction

works in general. The definition of construction works contained in Interpretative document

No 1: Mechanical resistance and stability [5], expressly includes bridges. Hence, the entire

content of that Chapter 1 is relevant to bridges. Readers who wish to consult the general

requirements for bridges are referred to the aforementioned chapter of Guidebook 1: the

present chapter of Guidebook 2 is limited to questions exclusively pertinent to bridges.

In its Annex I [4], the CPD lists six essential requirements that must be met by all

construction products and works, as follows:

1. Mechanical resistance and stability

2. Safety in case of fire

3. Hygiene, health and the environment

4. Safety in use

5. Protection against noise

6. Energy economy and heat retention

Of these, only the first two are generally related to structural behaviour and

consequently only they are covered by structural Eurocodes.


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The provisions of essential requirement 2, safety in case of fire, include a number of

structural behaviour-related issues. Indeed, while Part 1.2 of all the Eurocodes, from EN 1991

on actions in general to EN 1992 to EN 1999 on structural materials, deal with structural

behaviour in the event of fire, this question was not addressed in Guidebook 1 nor is it

included in this Guidebook 2.

Consequently, the present publication and specifically this chapter cover only the first

of the essential requirements, mechanical resistance and stability. Inasmuch as the third

through the sixth requirements do not involve structural behaviour, compliance therewith

cannot, generally speaking, be ensured under the provisions of the structural Eurocodes. For

that reason, they are not considered in either Guidebook 1 [3] or the present text. That

notwithstanding, Annex 2 of EN 1990 [2] lists user safety and comfort requirements that

bridges must meet in connection with the fourth essential requirement, safety in use. Since [2]

relates these requirements to structural response, they are dealt with in this Guidebook as part

of the discussion on serviceability criteria.

2 BASIC REQUIREMENTS

2.1 General

The scope of EN 1990 Annex A2 [2] and EN 1991 Part 2 [6 ] covers road, rail and

foot bridges. By contrast, certain special kinds of bridges, such as moveable bridges,

aqueducts and combination road and railway bridges are excluded.

EN 1990 Annex A2 [2] also lists criteria for the combination of actions to be applied

to verify ultimate limit states (ULS), serviceability limit states (SLS), partial factors (γ values)

and combination coefficients (ψ values). These issues are addressed in detail in Chapter 6

hereunder.

Annex A2 also lays down procedures and methods for verifying SLS when such limit

states are not related to the structural materials.

2.2 Permanent design situation requirements

All the provisions of [3] on ULS and SLS requirements are applicable to bridges and

nothing more is needed to say. Only to highlight that, in fact, the indication on the Guidebook

1 on the greater influence, today, of SLS than ULS requirements is particularly pertinent to

bridges, whose longer spans and stronger and lighter materials intensify their susceptibility to

deformation and vibrations. As a result, serviceability limit states may be more quickly

reached in such structures. In addition to that technological challenge, designers are faced

with an aesthetic issue: in bridges, beauty is essentially a result of the structure itself, which

means that possible flaws cannot be cloaked for want of a superstructure.

2.3 Transient situation requirements

In general, the load conditions affecting bridges during construction (see figure 1) can

vary in an important way from the conditions prevailing during normal use. The structural

strength scheme may also differ. Piers and decks that are designed to form portal frames in the

finished structure, for instance, may work like cantilevered beams with wholly different stress

distributions during construction. Therefore construction stage conditions must be specially

taken into account and the construction phases planned with particular care.

One condition, whose verification is normally very important in bridges, but much less

so in buildings, is the static equilibrium during construction. In certain stages of the

construction, actions or limit states may be of greater influence than in the finished structure.

The EQU limit states in these transient situations are often relevant and meticulous planning


9

is required to prevent EQU failure. In such cases, the possible position and values of self-

weight and loads at different construction times must be taken into consideration, in the full

understanding that the values of characteristic actions, partial factors and combination

coefficients may differ from the permanent situation values. In this phase, the weight of

concrete cast on decks, for instance, is regarded to be not a permanent action as in the finished

structure, but a variable action. As a result, the γ-factor applicable will be γQ equal to 1.5

rather than γG equal to 1.35.

The values of these parameters applicable to transient situations are given in Eurocode

EN 1991-1-6: Actions during execution [7].

Figure 1. Bridge under construction

2.4 Accidental situation requirements

Of the accidental actions listed in EN 1991-1-7 [8], only impacts and gross errors are

generally applicable to bridges.

Impact may be the result of collision either under or on the bridge. These situations are

dealt with separately in the Eurocodes: the former in [8] and the latter in [6] as traffic loads.

In bridges, clearly, other strategies besides design can be implemented to protect the

structure from major impact damage. Such alternative strategies include:

- the use of low sensitivity, highly robust structural typologies, such as redundancies

- the use of structural systems that warn of collapse i.e., ductile members

- the prevention or reduction of possible hazards by providing suitable clearance, ample

distance between lane centrelines and bridge members liable to be impacted

(bollards...) and so on.

The effect of impact on lightweight structures, such as some footbridges, is not

covered by [7]. In these cases, the aforementioned alternative strategies would be ever more

relevant.

Accidental actions affecting bridges are discussed in depth in Chapter 5 hereunder.

3 SERVICEABILITY REQUIREMENTS

3.1 General

As specified in [3], most serviceability criteria defined in terms of structural materials

are equally applicable to bridges and buildings.


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In addition to general serviceability criteria, however, certain specific criteria are in

place for bridges in connection with user safety and comfort, more specifically to avoid

excessive deck vibration or deformation.

3.2 Serviceability criteria for road bridges

Table 1 gives the design values for the combinations of actions to be considered in

SLS verification. While the Eurocode recommends partial factors, γ, equal to 1,0 when

calculating the design values for actions from their characteristic values, this criterion may be

modified in national annexes.

Table 1. Design values for use in combinations of actions

Combination Permanent actions Gd Prestress Variable actions Qd

Unfavourable Favourable Leading Others

Characteristic

Frequent

Quasi-permanent

Gkj,sup

Gkj,sup

Gkj,sup

Gkj,inf

Gkj,inf

Gkj,inf

P

P

P

Qk,1

ψ1,1 Qk,1

ψ2,1 Qk,1

ψ0,i Qk,i

ψ2,i Qk,i

ψ2,i Qk,i

Generally speaking, EN 1990 [1] recommends the use of characteristic and frequent

combinations for irreversible and reversible SLS, respectively, and the quasi-permanent

combination for long-term effects and structural aesthetics. Certain specific SLS for road

bridges address durability or user comfort and safety.

Damage to structural load bearings before the end of their design working life must be

prevented by limiting the amplitude of deck vibration over the supports. Another solution is to

adopt for these elements, if replaceable, a shorter service life than for other members: 15-25

years, instead of the 100 years normally established for bridges.

There are SLS directly related to user comfort and safety, such as uplift of the deck in

the supports (and, which would be, as noted, also cause damage to the bearings). Since these

limit states are related to human safety may be demanding higher safety levels than other

SLS.

Wind- or traffic-induced deck vibrations may also have to be limited to ensure user

comfort.

3.3 Serviceability criteria for footbridges

The pursuit of ever lighter weight and more engaging bridge designs has led to some

well-known cases of footbridges subject to excessive vibrations, causing extreme user

discomfort and forcing to take important measures to improve its conditions.

Depending of the different situations, Annex 2 [2] recommends considering different

load values:

- for persistent design situations, the presence of a group of about 8 to 15 people

(depending on the deck area) walking at a normal pace

- for other traffic categories, depending on the design situations: permanent,

transient or accidental, or the deck area or part of it in consideration, specific load

cases should be considered, when relevant: presence of streams of pedestrians

(significantly more than 15 persons), or occasional festive or choreographic events

- EN 1990 Annex 2 [3] recommends the following maximum deck acceleration

values:

0.7 m/s2 for vertical vibrations


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0.2 m/s2 for horizontal vibrations due to normal use

0.4 m/s2 for exceptional crowd conditions.

Such verifications are generally necessary only in footbridges with low natural

frequencies, i.e., under 5 Hz for vertical and under 2.5 Hz for horizontal (lateral) and torsional

vibrations. These low natural frequencies often occur in light footbridges.

3.4 Serviceability criteria for railway bridges

As in road bridges, the SLS for railway bridges are related to durability and user

comfort or safety, all in the context of excessive deck deformation or vibration and ultimately

of bridge stiffness.

Traffic may be compromised by excessive bridge deformation, which generates

unacceptably large vertical and horizontal variations in track geometry and vibrations in

bridge members. Such excessive vibrations may also cause ballast instability, inadmissible

reductions in wheel-rail contact forces or rail fatigue, not to mention passenger discomfort.

Given that user safety is highly sensitive to track conditions, the following checks

should be performed:

- vertical acceleration in the deck, to prevent ballast instability;

- vertical deflection in the deck throughout each span;

- unrestrained uplift at the bearings, to avoid premature bearing failure;

- twist of the deck measured along the centre line of each track on the approaches to and

passage over bridges to minimise the risk of train derailment;

- horizontal transverse deflection;

- limits to the first natural frequency of lateral span vibration, to prevent resonance

between the bridge and the lateral motion of vehicles on their suspension.

Some of the above phenomena also affect passenger comfort due to excessive vertical

or horizontal acceleration.

To determine the effect of the actions on the bridge could be necessary to perform a

dynamic analysis, the EN 1991-2 [6] gives the conditions when this analysis is needed. In

general is needed in bridges serving lines with Maximum Line Speed in site bigger than 200

km/h, with no simple structure, spanning more than 40 m and with first natural torsional

frequency more than 1,2 times the first natural bending frequency. That document indicates

also the way to perform a dynamic analysis, but this is out of the scope of this Guidebook.

If this dynamic analysis is not needed, static load effects are enhanced by a dynamic

factor Φ. This factor assume the value Φ2 or Φ3 depending on the track conditions, it results

82.02.0

44,1

Φ

2 +−

=L

Φ 1.00 ≤ Φ2 ≤ 1.67 (1)

for carefully maintained track and

73.02.0

16,2

Φ

3 +−

=L

Φ 1.00 ≤ Φ3 ≤ 2.0, (2)

being LΦ the determinant length (length associated withΦ), which is given in EN 1991-2 [6]

paragraph 6.4.5.3 depending on kind and dimensions of the elements.

EN 1991/A1 [2] gives the criteria regarding the traffic safety limiting vertical

acceleration in deck, deck twist and vertical deformation of the deck:

- Vertical acceleration in deck: acceleration is limited to prevent track instability and

thereby ensure traffic safety. The maximum design values specified for frequencies of


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up to 3.5 Hz or 1.5 times the frequency of the fundamental mode of vibration of the

member considered are 3.5 m/s2 in ballasted track or 5 m/s

2 for decks in which the

elements supporting the track are secured directly.

- Deck twist: maximum track twist must be limited. For tracks with gauge s [m] of 1.435

m, t [mm/3m] measured over a length of 3 m (see figure 2) should not exceed the values

given in Table 2.

Figure 2 – Definition of deck twist

Table 2. Recommended maximum values for deck twist

- Vertical deformation of the deck: Passenger comfort depends on vertical acceleration

inside the coach during travel on, approach to, passage over and departure from the

bridge. Vertical acceleration must be limited, therefore, to ensure acceptable comfort

levels, according to Table 3.

Table 3. Recommended comfort levels

Comfort level Vertical acceleration bv (m/s2)

Very high 1.0

High 1.3

Acceptable 2.0

4 DESIGN WORKING LIFE AND RELIABILITY MANAGEMENT

One of the first things a designer must know when designing a structure is its

projected working life. The indicative design working life categories listed in Eurocode EN

1990 [2] and given in Table 4 below may be modified in the national annexes.

In bridges, as noted earlier, not all members need be designed to the same working

life: some of them, which are more or less readily replaceable, like bearings, may be classified

under category 2 and designed for shorter working lives than the main members.

In Table 4 bridges are included under category 5, with a design working life of 100

years. These values are indicative only, and in each specific case subject to an agreement

between the owner (usually the authorities) and the designer, in which bridge characteristics

play a significant role: traffic density, accessibility, existence of alternative routes and so on.

For instance, for a bridge in a main road the 100 years design working life appears adequate,

but for a minor bridge serving an area with little traffic with alternative paths a 25 or 50 years

would be more adequate design working life.

Speed range V (km/h) Maximum twist t (mm/3m)

V ≤ 120 t ≤ 4,5

120 < V ≤ 200 t ≤ 3,0

V > 200 t ≤ 1,5


13

Table 4. Design working life. Indicative values

Design working

life category

Indicative design

working life (years)

Examples

1 10 Temporary structures (1)

2 10-25 Replaceable structural parts, e.g. gantry

girders, bearings

3 15-30 Agricultural and similar structures

4 50 Building structures and other common

structures

5 100 Monumental building structures, bridges,

and other civil engineering structures

(1) Structures or parts of structures that can be dismantled with a view to being re-used

should not be considered as temporary.

Closely reliability level and its attainment are associated with design working life. In

the Eurocodes, partial factors and characteristic values are based on a design working life of

50 years and a reliability index of β = 3,8, assuming normal consequences.

In bridges, the β value and consequently the partial factors and characteristic values

listed in the Eurocodes derived from there would seemingly have to be revised to adapt them

to a 100-year design working life and potentially sizeable economic consequences.

This option shall be necessary in works of major importance. In most common

bridges, however, this approach to raising the reliability index is not always justified.

Rather, greater reliability should be attained via suitable quality control during

construction and satisfactory inspection and maintenance policies throughout the working life.

The aim of this approach is to reduce the dispersion of material strength values, lower the

probability of gross errors and raise the likelihood of detecting minor flaws. These

requirements are more commonly met and more readily assumed in bridges than buildings.

5 DURABILITY

Durability is a major issue in bridges. They normally have a fairly long design

working life (100 years) and are directly exposed to environmental conditions, for they have

not protective superstructure. Furthermore, in cold climates de-icing salts, which cause

aggressive corrosion in steel, are frequently strewn over bridges. In any event, the presence of

water always intensifies durability problems in any material.

The requirements for long durability can be summarised as follows:

- appropriate choice of materials, in keeping with the environmental conditions

- careful design from the standpoint of durability: speedy evacuation of rainwater, for

instance

- quality control measures

- inspection and maintenance programme tailored to the prevailing conditions.

6 REFERENCES

[1] EN 1990 Eurocode - Basis of structural design. CEN, Brussels, 2002.

[2] EN 1990/A1 – Application for bridges. CEN, Brussels, 2002.


14

[3] Milan Holický et al., Guidebook1: Load Effects on Buildings. Leonardo da Vinci Project,

CTU, Klokner Institute, Prague, 2009

[4] Construction Products Directive (Council Directive 89/106/EEC). European

Commission, Enterprise Directorate-General, 2003

http://ec.europa.eu/enterprise/construction/internal/cpd/cpd.htm

[5] Interpretative document No. 1: Mechanical resistance and stability. European

Commission, Enterprise Directorate-General, 2004

http://ec.europa.eu/enterprise/construction/internal/intdoc/idoc1.htm

[6] EN 1991-2 – Eurocode 1: Actions on structures – Part 2: Traffic loads on bridges. CEN,

Brussels, 2003.

[7] EN 1991-1-6 – Eurocode 1: Actions on structures – Part 6: Actions during execution.

CEN, Brussels, 2003.

[8] EN 1991-1-7 – Eurocode 1: Actions on structures – Part 7: Accidental actions. CEN,

Brussels, 2006.

Chapter 2: Basis of design – methodological aspects

15

CHAPTER 2: BASIS OF DESIGN – METHODOLOGICAL ASPECTS

Milan Holický1 and Dimitris Diamantidis

2

1Klokner Institute, Czech Technical University in Prague, Czech Republic

2University of Applied Sciences, Regensburg, Germany

Summary

Uncertainties affecting structural performance can never be entirely eliminated and

must be taken into account when designing any construction work. Various design methods

and operational techniques for verification of structural reliability have been developed and

worldwide accepted in the past. The most advanced operational method of partial factors is

based on probabilistic concepts of structural reliability and risk assessment. General principles

of structural reliability and risk assessment can be used to specify and further calibrate partial

factors and other reliability elements. Moreover, developed calculation procedures and

convenient software products can be used directly for verification of structural reliability

using probabilistic concepts and available experimental data.

1 INTRODUCTION

1.1 Background materials

Basic concepts of structural reliability are codified in a number of national standards,

in the new European document EN 1990 [1] and the International Standard ISO 2394 [2].

Additional information may be found in the background document developed by JCSS [3]

and in recently published handbook to EN 1990 [4]. Guidance for application of probabilistic

methods of structural reliability may be found in working materials provided by JCSS [5] and

in relevant literature listed in [4] and [5].


General principles of structural reliability are described in both the international

documents EN 1990 [1] and ISO 2394 [2]. Basic requirements on structures are specified in

Section 2 of EN 1990 [2]: a structure shall be designed and executed in such a way that it will,

during its intended life, with appropriate degrees of reliability and in an economic way

- sustain all actions and influences likely to occur during execution and use;

- remain fit for the use for which it is required.

It should be noted that two aspects are explicitly mentioned: reliability and economy

(see also Guidebook 1 [6]). However, this Guidebook shall be primarily concerned with

reliability of bridge structures, which include

- structural resistance;

- serviceability;

- durability.


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Additional requirements may concern fire safety of structures (see Handbook 5) or

other accidental design situations. In particular it is required by EN 1990 [1] that in the case

of fire, the structural resistance shall be adequate for the required period of time.

To verify all the aspects of structural reliability implied by the above-mentioned basic

requirements, an appropriate design lifetime, design situations and limit states should be

considered. Note that the basic lifetime for a common building is 50 years and that, in general,

four design situations are identified: permanent, transient, accidental and seismic. Two types

of limit states are normally verified: ultimate limit states and serviceability limit states.

2 UNCERTAINTIES

2.1 Classification of uncertainties

It is well recognised that construction works including bridges are complicated

technical systems suffering from a number of significant uncertainties in all stages of

execution and use. Depending on the nature of a structure, environmental conditions and

applied actions, various types of uncertainties become more significant than the others. The

following types of uncertainties can be identified in general:

- natural randomness of actions, material properties and geometric data;

- statistical uncertainties due to a limited size of available data;

- uncertainties of the resistance and load effect models due to simplifications of actual

conditions;

- vagueness due to inaccurate definitions of performance requirements;

- gross errors in design, during execution and use;

- lack of knowledge concerning behaviour of new materials and actions in actual

conditions.

The order of the listed uncertainties corresponds approximately to the decreasing level

of current knowledge and available theoretical tools for their description and consideration in

design (see following sections). It should be emphasized that most of the above listed

uncertainties (randomness, statistical and model uncertainties) can never be eliminated

absolutely and must be taken into account when designing any construction work.

2.2 Available tools to describe uncertainties

Natural randomness and statistical uncertainties may be relatively well described by

available methods provided by the theory of probability and mathematical statistics. In fact

the EN 1990 [1] gives some guidance on available techniques. However, lack of credible

experimental data (e.g. for new materials, some actions including environmental influences

and also for some geometrical properties) causes significant problems. In some cases the

available data are inhomogeneous, obtained under different conditions (e.g. for material

resistance, imposed loads, environmental influences, for inner dimensions of reinforced

concrete cross-sections). Then it may be difficult, if not impossible, to analyse and use them

in design.

The uncertainties of computational models may be to a certain extent assessed on the

basis of theoretical and experimental research. EN 1990 [1] and materials of JCSS [5] provide

some guidance. The vagueness caused by inaccurate definitions (in particular of serviceability

and other performance requirements) may be partially described by the means of the theory of

fuzzy sets. However, these methods have a little practical significance, as suitable

experimental data are rarely available. The knowledge of the behaviour of new materials and


17

structures may be gradually increased through theoretical analyses verified by experimental

research.

The lack of available theoretical tools is obvious in the case of gross errors and lack of

knowledge, which are nevertheless often the decisive causes of structural failures. To limit

gross errors due to human activity, a quality management system including the methods of

statistical inspection and control may be effectively applied.

Various design methods and operational techniques, which take these uncertainties

into account, have been developed and worldwide used. The theory of structural reliability

provides background concept techniques and theoretical bases for description and analysis of

the above-mentioned uncertainties concerning structural reliability.

3 RELIABILITY

3.1 General

The term "reliability" is often used very vaguely and deserves some clarification.

Often the concept of reliability is conceived in an absolute (black and white) way – the

structure either is or isn’t reliable. In accordance with this approach the positive statement is

understood in the sense that “a failure of the structure will never occur“. This interpretation is

unfortunately an oversimplification. Although it may be unpleasant and for many people

perhaps unacceptable, the hypothetical area of “absolute reliability” for most structures (apart

from exceptional cases) simply does not exist. Generally speaking, any structure may fail

(although with a small or negligible probability) even when it is declared as reliable.

The interpretation of the complementary (negative) statement is usually understood

more correctly: failures are accepted as a part of the real world and the probability or

frequency of their occurrence is then discussed. In fact in the design it is necessary to admit a

certain small probability that a failure may occur within the intended life of the structure.

Otherwise designing of civil structures would not be possible at all. What is then the correct

interpretation of the keyword “reliability” and what sense does the generally used statement

“the structure is reliable or safe” have?

Several bridge failures have been occurred in the past. Frequent causes of bridge

failures are floods, collisions and fatigue problems. Figure 1 shows the failure of the bridge

over the Mississippi River in central Minneapolis, which collapsed in 2007. The bridge had an

age of 40 years, not many compared to the 100 years desired lifetime of bridge. Therefore the

reliability of a bridge should be focussed through all phases i.e. design, construction,

operation, maintenance and upgrading.

Under this basic consideration it appears even more important to implement reliability

concepts from the very initial design stage and consequently to use modern reliability based

design elements. Therefore a scientific definition of reliability and an associated derivation of

the reliability based design elements are necessary. Such steps have been implemented in the

Eurocodes and are explained and illustrated next.

3.2 Definition of reliability

A number of definitions of the term “reliability” are used in literature and in national

and international documents. ISO 2394 [2] provides a definition of reliability, which is similar

to the approach of national standards used in some European countries: reliability is the

ability of a structure to comply with given requirements under specified conditions during the

intended life, for which it was designed. In quantitative sense reliability may be defined as the

complement of the probability of failure.


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Figure 1. Failure of the Mississippi River bridge in Minneapolis, August 2007.

Note that the above definition of reliability includes four important elements:

- given (performance) requirements – definition of the structural failure,

- time period – assessment of the required service-life T,

- reliability level – assessment of the probability of failure Pf,

- conditions of use – limiting input uncertainties.

An accurate determination of performance requirements and thus an accurate

specification of the term failure are of uttermost importance. In many cases, when considering

the requirements for stability and collapse of a structure, the specification of the failure is not

very complicated. In many other cases, in particular when dealing with various requirements

of occupants’ comfort, appearance and characteristics of the environment, the appropriate

definitions of failure are dependent on several vaguenesses and inaccuracies. The

transformation of these occupants' requirements into appropriate technical quantities and

precise criteria is very hard and often leads to considerably different conditions.

In the following the term failure is being used in a very general sense denoting simply

any undesirable state of a structure (e.g. collapse or excessive deformation), which is

unambiguously given by structural conditions.

The same definition as in ISO 2394 is provided. In Eurocode EN 1990 [1] including

note that the reliability covers the load-bearing capacity, serviceability as well as the

durability of a structure. Fundamental requirements include the statement (as already

mentioned) that ”a structure shall be designed and executed in such a way that it will, during

its intended life with appropriate degrees of reliability and in an economic way sustain all

actions and influences likely to occur during execution and use, and remain fit for the use for

which it is required”. Generally a different level of reliability for load-bearing capacity and

for serviceability may be accepted for a structure or its parts. In the documents [1] and [2] the


19

probability of failure Pf (and reliability index β) are indicated with regard to failure

consequences (see Guidebook 1 [6]).

3.3 Probability of failure

The most important term used above (and in the theory of structural reliability) is

evidently the probability of failure Pf. In order to defined Pf properly it is assumed that

structural behaviour may be described by a set of basic variables X = [X1, X2, ... , Xn]

characterizing actions, mechanical properties, geometrical data and model uncertainties.

Furthermore it is assumed that the limit state (ultimate, serviceability, durability or fatigue) of

a structure is defined by the limit state function (or the performance function), usually written

in an implicit form as

Z(X) = 0 (1)

The limit state function Z(X) should be defined in such a way that for a favourable (safe) state

of a structure the function is positive, Z(X) ≥ 0, and for a unfavourable state (failure) of the

structure the limit state function is negative, Z(X) < 0 (a more detailed explanation is given in

the following Chapters of this Guidebook 2).

For most limit states (ultimate, serviceability, durability and fatigue) the probability of

failure can be expressed as

Pf = P{Z(X) < 0} (2)

The failure probability Pf can be assessed if basic variables X = [X1, X2, ... , Xn] are

described by appropriate probabilistic (numerical or analytical) models. Assuming that the

basic variables X = [X1, X2, ... , Xn] are described by time independent joint probability density

function ϕX(x) then the probability Pf can be determined using the integral

xxX d)(

0)(Z

f ∫<

=X

P ϕ (3)

More complicated procedures need to be used when some of the basic variables are

time-dependent. Some details concerning theoretical models for time-dependent quantities

(mainly actions) and their use for the structural reliability analysis are given in other Chapters

of this Guidebook 2. However, in many cases the problem may be transformed to a time-

independent one, for example by considering in equation (2) or (3) a minimum of the function

Z(X) over the reference period T.

Note that a number of different methods [2] and software products [8, 9, 11] are

available to calculate failure probability Pf defined by equation (2) or (3).

3.4 Reliability index

An equivalent term to the failure probability is the reliability index β, formally defined

as a negative value of a standardized normal variable corresponding to the probability of

failure Pf. Thus, the following relationship may be considered as a definition

)(Φ f1

PU−−=β (4)

Here )(1 pu fΦ− denotes the inverse standardised normal distribution function. At present the

reliability index β defined by equation (4) is a commonly used measure of structural

reliability in several international documents [1], [2], [5].

It should be emphasized that the failure probability Pf and the reliability index β

represent fully equivalent reliability measures with one to one mutual correspondence given

by equation (4) and numerically illustrated in Table 1.


20

In EN 1990 [1] and ISO 2394 [2] the basic recommendation concerning required

reliability level is often formulated in terms of the reliability index β related to a certain

design working life.

Table 1. Relationship between the failure probability Pf and the reliability index ββββ.

Pf 10−1

10−2

10−3

10−4

10−5

10−6

10−7

β 1.3 2.3 3.1 3.7 4.2 4.7 5.2

3.5 Time variance of failure probability

When the vector of basic variables X = X1, X2, ... , Xm is time variant, then the failure

probability p is also time variant and should be always related to a certain reference period T,

which may be generally different from the design working life Td. Considering a structure of a

given reliability level, the design failure probability pd = pn related to a general reference

period Tn = n T1 can be approximately assessed as pd ~ n p1 corresponding to the period T1

where T1 denotes the basic period, for example 1 year.

4 RELIABILITY TARGETS

4.1 General on risk acceptance

Risk acceptance criteria are introduced in the EN 1990 [1] in terms of target and

acceptable (i.e. design) failure probabilities and associated reliability indices. They are used in

order to obtain safety factors for design purposes. The values have been derived through long

studies by combining the various approaches reviewed in the previous paragraph. The values

reflect the possible failure consequences, the reference time period and are valid for

component failures. Special attention must be given to global failure conditions and to target

reliability criteria for existing structures.

4.2 Reliability classes

Design failure probabilities pd are usually indicated in relation to the expected social

and economical consequences in order to reflect the aforementioned risk acceptance criteria.

Table 2 shows classification of target reliability levels provided in EN 1990 [1]. Reliability

indexes β are given for two reference periods T (1 year and 50 years) only, without any

explicit link to the design working life Td. The values are based on calibration and

optimization and reflect results from several studies. It is noted that similar β-values as in

Table 1 are given in other national and international guidelines (see for example [2], [3]).

It should be underlined that a couple of β values (βa and βd) specified in Table 2 for

each reliability class (for 1 year and 50 years) corresponds to the same reliability level.

Practical application of these values depends on the time period Ta considered in the

verification, which may be connected with available statistical information concerning time

variant actions (wind, earthquake, etc.) and the related vector of basic variables X = X1, X2, ...,

Xn.

By considering for example a 50 years design working period then the reliability index

βd = 4.3 should be used in the verification of structural reliability. The same reliability level

corresponding to class 3 is achieved when the time period Ta = 1 year and βa = 5.2 is used,

again in case of limit states dominated by time varying actions. If the working life is 100

years as it usual for bridges then in case of time dependent actions the lifetime target

reliability index (i.e. for T= 100years) can be obtained from the formulae discussed in §3.3 of

Guidebook 1 [6] and results approximately as βd = 4,1 (for T = 100 years)


21

Table 2. Reliability classification in accordance with EN [1]

Reliability

classes

Consequences for

loss of human life,

economical, social

and environmental

consequences

Reliability index β Examples of buildings and

civil engineering works

βa for Ta=

1 year

βd for Td=

50 years

RC3 – high High 5.2 4.3 Important bridges, public

buildings

RC2 – normal Medium 4.7 3.8 Residential and office

buildings

RC1 – low Low 4.2 3.3 Agricultural buildings,

greenhouses

Important bridges are in general important structures with considerable consequences

of failure and should be in general RC3 (reliability class 3) structures. However bridges can

be also classified in the three categories of Table 2 according to the consequences of failure.

A small bridge in a rural area with limited traffic can be for example associated to RC1. The

target reliability can be obtained also on the basis of an optimisation procedure as illustrated

in Appendix A of this chapter.

4.3 Global failure – robustness

Structures are composed of various elements such as columns, beams, plates etc. The

aforementioned target reliability values are valid for components, since structural design is

based on design of components. The global reliability i.e. the reliability against collapse of the

entire bridge system or a major part of it is a function of the reliability of all the elements

against local failure but also of the system response to local failure. The assumption that a

consistent level of reliability of a structural system is reached by an adequate reliability of its

members is not generally valid. Especially for bridges structures subjected to extreme

environmental loads such as earthquake or wind or accidental loads such as explosion or

impact is not sufficient.

Therefore structural codes have additional requirements regarding global failure or

progressive failure of the structure. The associated requirements in the Eurocodes are

discussed in Chapter 5 of Guidebook 1 [6].

5 DESIGN METHODS IN PRACTICE

5.1 General

During their historical development the design methods have been closely linked to

the available empirical, experimental as well as theoretical knowledge of mechanics and the

theory of probability. The development of various empirical methods for structural design

gradually crystallized in the twentieth century in three generally used methods, which are, in

various modifications, still applied in standards for structural design until today: the

permissible stresses method, the global factor and partial factor methods. All these methods

are often discussed and sometimes reviewed or updated.

The following short review of historical development illustrates general formats of

above mentioned design methods and indicates relevant measures that are applied to take into

account various uncertainties of basic variables and to control resulting structural reliability.

In addition a short description of probabilistic methods of structural reliability and their role


22

in further development of design procedures is provided. Detailed description of probabilistic

methods of structural reliability is given in Chapter 2, Chapter 3 and in Annex B of

Guidebook 1 [6].

5.2 Permissible stresses

The first of the worldwide-accepted design methods for structural design is the method

of permissible stresses that is based on linear elasticity theory. The basic design condition of

this method can be written in the form

σmax < σper, where σper = σcrit / k (6)

The coefficient k (greater than 1) is the only explicit measure supposed to take into account all

types of uncertainties (some implicit measures may be hidden). Moreover, only a local effect

(a stress) σmax is compared with the permissible stress σper and, therefore, a local (elastic)

behaviour of a structure is used to guarantee its reliability. No proper way is provided for

treating geometric non-linearity, stress distribution and ductility of structural materials and

members. For that reasons the permissible stress method leads usually to conservative and

uneconomical design.

However, the main insufficiency of the permissible stress method is lack of possibility

to consider uncertainties of individual basic variables and computational models used to

assess load effects and structural resistances. Consequently, reliability level of structures

exposed to different actions and made of different material may be not only conservative

(uneconomical) but also considerably different.

5.3 Global safety factor

The second widespread method of structural design is the method of global safety

factor. Essentially it is based on a condition relating the standard or nominal values of the

structural resistance R and load effect E. It may be written as

s = R / E > s0. (7)

Thus the calculated safety factor must be greater than its specified value s0 (for

example s0=1,9 is commonly required for bending resistance of reinforced concrete

members). The global safety factor method attempts to take into account realistic assumptions

concerning structural behaviour of members and their cross-sections, geometric non-linearity,

stress distribution and ductility; in particular through the resulting quantities of structural

resistance R and action effect E.

However, as in the case of the permissible stresses method the main insufficiency of

this method remains a lack of possibility to consider the uncertainties of particular basic

quantities and theoretical models. The probability of failure can, again, be controlled by one

explicit quantity only, by the global safety factor s. Obviously, harmonisation of reliability

degree of different structural members made of different materials is limited.

5.4 Partial factor method

At present, the most advanced operational method of structural design [1, 2] accepts

the partial factor format (sometimes incorrectly called the limit states method) usually applied

in conjunction with the concept of limit states (ultimate, serviceability or fatigue). This

method can be generally characterised by the inequality

Ed (Fd, fd, ad, θd) < Rd (Fd, fd, ad, θd) (8)

where the design values of action effect Ed and structural resistance Rd are assessed

considering the design values of basic variables describing the actions Fd = ψ γF Fk, material


23

properties fd = fk /γm, dimensions ad + ∆a and model uncertainties θd. The design values of

these quantities are determined (taking into account various uncertainties) using their

characteristic values (Fk, fk, ak, θk), partial factors γ, reduction factors ψ and other measures of

reliability [1, 2, 3, 4], Thus the whole system of partial factors and other reliability elements

may be used to control the level of structural reliability. Detailed description of the partial

factor methods used in Eurocodes method is provided in Guidebook 1.

Compared with previous design methods the partial factor format obviously offers the

greatest possibility to harmonise reliability of various types of structures made of different

materials. Note, however, that in any of the above listed design methods the failure

probability is not applied directly. Consequently, the failure probability of different structures

made of different materials may still considerably vary even though sophisticated calibration

procedures were applied. Further desired calibrations of reliability elements on probabilistic

bases are needed; it can be done using the guidance provided in the International standard ISO

2394 [2] and European document EN 1990 [1].

5.5 Probabilistic methods

The probabilistic design methods introduced in the International Standard [2] are

based on a requirement that during the service life of a structure T the probability of failure Pf

does not exceed the design value pd or the reliability index β is greater than its design value βd

Pf ≤ Pd or β > βd (9)

In EN 1990 [1] the basic recommended reliability index for ultimate limit states

βd = 3.8 corresponds to the design failure probability Pd = 7.2 × 10-5

, for serviceability limit

states βd = 1.5 corresponds to Pd = 6.7 × 10-2

. These values are related to the design working

life of 50 years that is considered for building structures and common structures. In general

greater β - values should be used when a short reference period (one or five years) will be

used for verification of structural reliability.

It should be mentioned that probabilistic methods are not yet commonly used in

design praxis. However, the developed calculation procedures and software products (for

example [8, 9] and [11]) already enable the direct verification of structural reliability using

probabilistic concepts and available experimental data. Recently developed software product

CalCode [11] is primarily intended for calibration of codes based on the partial factor method.

5.6 Risk assessment

The risk assessment of a system consists of the use of all available information to

estimate the risk to individuals or populations, property or the environment, from identified

hazards. The risk assessment further includes risk evaluation (acceptance or treatment). The

whole procedure of risk assessment is typically an iterative process as indicated in Figure 2.

The first step in the risk analysis involves the context (scope) definition related to the system

and the subsequent identification of hazards.

The system is understood as a bounded group of interrelated, interdependent or

interacting elements forming an entity that achieves in its environment a defined objective

through the interaction of its parts. In the case of technological hazards related to civil

engineering works, a system is normally formed from a physical subsystem, a human

subsystem, their management, and the environment. Note that the risk analysis of civil

engineering systems (similar to the analysis of most systems) usually involves several

interdependent components (for example human life, injuries, and economic loss).

Any technical system may be exposed to a multitude of possible hazard situations. In

the case of civil engineering facilities, hazard situations may include both environmental

effects (wind, temperature, snow, avalanches, rock falls, ground effects, water and ground


24

water, chemical or physical attacks, etc.) and human activities (usage, chemical or physical

attacks, fire, explosion, etc.). As a rule, hazard situations due to human errors are more

significant than hazards due to environmental effects.

Figure 2. Flowchart of iterative procedure for risk assessment.

6 CONCLUDING REMARKS

The basic concepts of the probabilistic theory of structural reliability are characterized

by two equivalent terms, the probability of failure Pf and the reliability index β. Although

they provide limited information on the actual frequency of failures, they remain the most

important and commonly used measures of structural reliability. Using these measures the

theory of structural reliability may be effectively applied for further harmonisation of

reliability elements and for extensions of the general methodology for new, innovative

structures and materials.

Historical review of the design methods worldwide accepted for verification of

structural members indicates different approaches to considering uncertainties of basic

variables and computational models. The permissible stresses method proves to be rather

conservative (and uneconomical). The global safety factor and partial factor methods lead to

similar results. Obviously, the partial factor method, accepted in the recent EN documents,

represents the most advanced design format leading to a suitable reliability level that is

S tart

Definition of the system

Ha za rd ide ntifica tion

Probability analysis Conse que nce a na lysis

Ris k es tima tion

Risk e va lua tion

Acce pta ble risk?Ris k tre a tm e nt

S top

No

Yes

Ris

k a

na

lysi

s

Ris

k a

ssess

men

t


25

relatively close to the level recommended in EN 1990 (β = 3.8). The most important

advantage of the partial factor method is the possibility to take into account uncertainty of

individual basic variables by adjusting (calibrating) the relevant partial factors and other

reliability elements.

Various reliability measures (characteristic values, partial and reduction factors) in the

new structural design codes using the partial factor format are partly based on probabilistic

methods of structural reliability, partly (to a great extent) on past empirical experiences.

Obviously the past experience depends on local conditions concerning climatic actions and

traditionally used construction materials. These aspects may be considerably different in

different countries. That is why a number of reliability elements and parameters in the present

suite of European standards are open for national choice.

It appears that further harmonisation of current design methods will be based on

calibration procedures, optimisation methods and other rational approaches including the use

of methods of the theory of probability, mathematical statistics and the theory of reliability

and risk assessment. The probabilistic methods of structural reliability provide the most

important tool for gradual improvement and harmonisation of the partial factor method for

various structures from different materials. Moreover, developed software products enable

direct application of reliability methods for verification of structures using probabilistic

concepts and available data.

Design of a structure assisted by risk assessment may be effectively used when there is

a need to consider failure consequences of a system containing the structure and costs of

safety measures. Probabilistic optimisation of the system utility may provide valuable

information concerning the optimum target reliability level. Finally it should be mentioned

that many famous structures have been designed according to the Eurocodes [12], and a

typical case is the famous bridge in Millau, France, which is illustrated in Figure 3.

Figure 3. Millau bridge in France


26

7. REFERENCES


[2] ISO 2394 General principles on reliability for structures, ISO, 1998.

[3] JCSS: Background documentation, Part 1 of EC 1 Basis of design, 1996.

[4] Gulvanessian, H. – Calgaro, J.-A. – Holický, M.: Designer's Guide to EN 1990,

Eurocode: Basis of Structural Design; Thomas Telford, London, 2002, ISBN: 07277

3011 8, 192 pp.

[5] JCSS: Probabilistic model code. JCSS working materials, http://www.jcss.ethz.ch/, 2001.

[6] Milan Holický et al., Guidebook1: Load Effects on Buildings. Leonardo da Vinci Project,

CTU, Klokner Institute, Prague, 2009

[7] EN 1991-1-1 Eurocode 1 Actions on structures. Part 1-1 General actions. Densities, self-

weight, imposed loads for buildings, CEN, Brussels, 2002

[8] VaP, Variable Processor, version 2.3, Petschacher Software and Project, Feldkirchen,

2009.

[9] COMREL, version 7.10, Reliability Consulting Programs, RCP MUNICH, 1999.

[10] ISO 13822. Basis for design of structures - Assessment of existing structures, ISO

2001.

[11] CodeCal, Excel sheet developed by JCSS, http://www.jcss.ethz.ch/.

[12] de Ville de Goyet, V. Important Structures designed Using the Eurocodes, Workshop

EU-Russia cooperation on standardisation for construction, Moscow, October 2008.

[13] Diamantidis, D. - P. Bazzurro, Target Safety Criteria for Existing Structures, Workshop

on Risk Acceptance and Risk Communication, Stanford University, CA, USA, March

2007.

[14] Joint Committee on Structural Safety (JCSS), Assessment of Existing Structures, RILEM

Publications S.A.R.L., 2000.

[15] Allen, D.E., 1993, Safety Criteria for the Evaluation of Existing Structures, Proceedings

IABSE Colloquium on Remaining Structural Capacity, Copenhagen, Denmark.

[16] CSA S6-1990, 1990, Design of Highway Bridges: Supplement No. 1-Existing Bridge

Evaluation, Canadian Standards Association, Ottawa, Ontario.


27

Appendix A to Chapter 2 – Principles of probabilistic optimization

A.1 General principles

Principles of probabilistic optimization are illustrated considering the objective

function in a basic form of the total cost Ctot(x,q,n) as

Ctot(x,q,n) = Cf∑=

n

i

q,iQx,iP1

f )( )( + C0 + x C1 (A.1)

Here x is a decision parameter of the optimization (a parameter of structure resistance), q is

annual discount rate (e.g. 0.03, an average long run value of the real discount rate in European

countries), n the number of years of a considered design working life (e.g. 50, 100), Pf(x,i)

failure probability at the year i, Cf malfunctioning costs (due to loss of structural utility),

Q(q,i) discount factor dependent on the annual discount rate q and number of years i, C0

initial cost independent of decision parameter x, and C1 cost per unit of the decision parameter

x.

Note that the design working life is considered here as a given determistic quantity. In

reality the working life for a given design is a random quantity depending on social and

physical factors. The design itself may aim at some optimum. This, option, however, is

neglected in this appendix.

Assuming independence, the annual probability of failure Pf(x,i) at the year i is given

by the geometric sequence

Pf(x,i) = p(x) (1 − p(x))i−1

(A.2)

where p(x) denotes the initial probability of failure that is dependent on the decisive parameter

of structural resistance x. Then the failure probability Pfn(x) during n years can be estimated

by the sum of the sequence Pf(x,i) given as

Pfn(x) = 1 – (1 – p(x))n ≈ n p(x) (A.3)

Where the approximation indicated in equation (A.3) is acceptable for small

probability p(x) < 10−3

.

The discount factor of the expected future costs at the year i is considered in a usual

form as

Q(q,i) = 1/ (1+q)i (A.4)

Here q denotes discount rate. Thus, the cost of malfunctioning Cf is discounted by the factor

Q(q,i) depending on the discount rate q and the point in time (number of year i) when the loss

of structural utility occurs.

The necessary conditions for the minimum of the total cost follow from equation (A.1)

as

0),(

),(),,(

1

1

ff

tot =+∂

∂=

∂∂

∑=

Cx

ixPiqQC

x

nqxC n

i

(A.5)

thus

f

1

1

f ),(),(

C

C

x

ixPiqQ

n

i

−=∂

∂∑

=

(A.6)

Equation (A.6) represents a general form of the necessary condition for the minimum

of total cost Ctot(x,q,n) and the optimum value xopt of the parameter x. It generates also the


28

optimum (target) probability of failure and corresponding target value of the reliability index

β.

Considering equation (2) and (4) the total costs Ctot(x,q,n) described by equation (A.1)

may be written as

Ctot(x,q,n) = Cf

)1(

))(1(1

)1(

))(1(1

)(

q

xp

q

xp

xp

n

+−−

+−−

+ C0 + x C1 (A.7)

Note that the total sum of expected malfunction costs during the period of n years is

dependent on the product of the one-time malfunction Cf , initial probability p(x) and a sum of

the geometric sequence having the quotient (1− p(x)/(1+ q).

Thus the total malfunction cost Ctot(x,q,n) depends on the annual probability of failure

p(x), discount rate q and on number of years n. For small probabilities of failure p(x), the cost

given by equation (A.7) may be well approximated as

Ctot(x,q,n) ≈ Cf p(x) PQ(x,q,n) + C0 + x C1 (A.8)

where the time factor PQ(x,q,n) depends primarily on the number of years n and discount rate

q. It is almost independent of the structural parameter x and, for small probability p(x) may be

approximate as

PQ (x,q,n)

)1(

11

)1(

11

)1(

))(1(1

)1(

))(1(1

q

q

q

xp

q

xpnn

+−

+−

≈

+−−

+−−

= = PQ(q,n) (A.9)

For given q and n the simplified time factor PQ(q,n) is independent of x; for q = 0.03 and n =

50, the simplified time factor PQ(q,n) ~ 26.5, for q = 0.03 and n = 100 PQ(q,n) ~ 32,5. If the

discount rate is small, q ~ 0, then the simplified time factor converts to number of years n,

PQ(q,n) ~ n.

The necessary condition for the minimum of the total costs then follows from

equations (A.8) and (A.9) as

),(d

)(d 1

nqPQC

C

x

xp

f

−= (A.10)

Equation (A.10) can be used for assessing the target (optimum) value pt(x) of the initial

annual probability p(x).

A.2 A special case

A special case concerns structures when a failure is dominated by a load like wind

with an exponential distribution. Then the failure probability p(x) may be expressed as

p(x) = exp{-x/a } (A.11)

where a is an appropriate statistical parameter. Then the target (optimum) probability pt(q,n)

follows from equation (A.10) as

),(

),(f

1t

nqPQC

Caxqp = (A.12)


29

The corresponding optimum structural parameter xopt(q,n) may be then written in an

explicit form as

),(

ln),(f

1opt

nqPQC

Caanqx −= (A.13)

Note that the simplified time factor PQ(q,n) is independent of x. Obviously the

optimum structural parameter xopt(q,n) and the target probability pt(q,n) depends on the cost

ratio C1/ C, number of years n and discount rate q.

Given the optimum value for x we may find the optimum value for the annual failure

probability as:

),(

)( 1

nqPQC

aCxp

f

opt = (A.14)

The optimum probability for the total design working life Td = n years is then:

f

optoptfnC

aC

nqPQ

nxnpxP 1

),()()( =≈ (A.15)

The value of n/PQ(q,n) runs for q = 0.03 from 1.1 for n = 1 to 3.0 for n=100. Note that for

small discount rates q ~ 0, the value n/PQ(q,n) = 1. This means that the optimum failure

probability is almost independent of n and, thus, of the design working life.

Consider for instance the case that for q = 0.03 and a certain value of aC1/Cf, the

optimum failure probability for a design life time of 50 years is 7.2 10-5

, which corresponds to

β = 3.8. If the design life time is 100 years, the optimum failure probability decreases to 10-4

,

or β = 3.7; if the design life time is 10 years, the optimum β increases to 3.9. The difference

between 3.7 and 3.9 is small enough to be neglected in practical design.

A.3 An example

The following example illustrates the general principles and special case of

probabilistic optimization described above. To simplify the analysis the total costs Ctot(x,q,n)

given by equation (A.1) is transformed to the standardized form κtot(x,q,n) as

κtot(x,q,n) = ),,( )(),,(

f

0 nqxPQxpC

CnqxCtot =− +x C1/ Cf (A.16)

Obviously, both costs Ctot(x,q,n) and κtot(x,q,n) achieve the minimum for the same

parameter xopt.

The exponential expression for the probability p(x) in equation (A.11) is simplified

assuming a = 1. Further it is considered that the discount rate q = 0.03 and the total period of

time is n = 50 years. Under this assumptions Figure A.1 shows variation of the total

standardized costs κtot(x,q,n) (given by equation (A.14)), and the reliability index β

corresponding to the probability Pfn(x) (given by equation (A.3)), with structural parameter x

for selected costs ratio C1/Cf. The optimum values xopt(q,n) of the structural parameter x are

indicated by the dotted vertical lines.

Figure A.2 shows variation of the optimum structural parameter xopt(q,n) with the costs

ratio C1/Cf , again for q =0.03, n =50. The optimum parameter xopt(q,n) may be obtained from

general condition (A.6) or from simplified expression (A.13) for the simplified time factor

Q(q,n) ~ 26.5.


30

Figure A.1. Variation of the total standardized costs κκκκtot(x,q,n) and the reliability index ββββ

with structural parameter x for q =0.03, n = 50 and selected costs ratios C1/Cf.

Figure A.2. Variation of the optimum structural parameter xopt obtained from equation

(4) or (13) with the costs ratio C1/Cf for q =0.03, n =50.

The optimum reliability index βopt is generally a function of a number of basic

variables. In the fundamental case considered above, the optimum reliability index βopt =

βopt(q,n,C1/Cf) depends particularly on the discount rate q, design working life n and the cost

ratio C1/Cf. However, the index βopt is primarily dependent on the cost ratio C1/Cf and its

dependence on the discount rate q and the design working life n seems to be insignificant.

This is well illustrated by Figure A.3 that shows variation of the optimum reliability index βopt

with the cost ratio C1/Cf for selected design working life n = 10, 50, 100, and the discount rate

q = 0.03.

It follows from Figure A.3 that with increasing working life n (and increasing discount

rate q) the optimum reliability index βopt slightly decreases. For very small discount rate q ∼ 0,

the value n/PQ(q,n) = 1 and the index βopt is independent of n.

6 8 10 12 14 16 18 200

0.01

0.02

0.03

0.04

1

2

3

4

5

6

β κtot(x,q,n)

x

C1/ Cf =0.002

C1/ Cf =0.001

C1/ Cf =0.0001

C1/ Cf =0.00001

β

1 .105

1 .104

1 .103

0.01

6

8

10

12

14

16

C1/ Cf

xopt(q,n) for q =0.03, n =50


31

Figure A.3. Variation of the optimum reliability index ββββopt with the cost ratio C1/Cf for

selected design working life n = 10, 50, 100, and the discount rate q = 0.03.

A.4 Concluding remarks

Probabilistic optimization may provide valuable background information concerning

reliability differentiation by assessing the target (optimum) probability of failure or the

reliability index. It appears that the target reliability index and corresponding resistance

parameter depends on

- the ratio of cost per unit of structural parameter and cost of structural failure

(malfunctioning costs),

- the statistical parameter of failure probability,

- discount rate and design working life.

Results obtained from analyzed example indicate more specific conclusions, validity

of which should be conditioned by the accepted assumptions concerning the objective

function and annual failure probability. It appears that with increasing malfunctioning cost,

the target reliability index and the optimum structural resistance increase (Figure A.1 and

A.2). The design working life seems to have a very limited influence on the optimum life time

reliability, particularly for small discount rates (Figure 3). For practical purposes the optimum

target reliability index and the corresponding structural parameter can be well assessed

considering reasonable lower bounds for the design working life (say 50 years) and the

discount rate (say 0.02).

Available experience indicates that applications of the optimization approach in

practice should be primarily based on properly formulated objective functions, and on

credible estimates for the cost per unit of structural parameter and cost of structural failure

(malfunctioning costs).

1 .106

1 .105

1 .104

1 .103

0.01

2

3

4

5

C1/Cf

βopt

n = 10

50

100


32

Chapter 3: Static loads due to traffic

33

CHAPTER 3: STATIC LOADS DUE TO TRAFFIC

Pietro Croce1

1Department of Civil Engineering, Structural Division - University of Pisa

Summary

In contemporary codes for bridges, load traffic models for static verification aim to

reproduce the real values of the effects induced in the bridges by the real traffic, i.e. the

effects having specified return periods; therefore they are artificial models, generally not

representing real vehicles. Static traffic load models of EN1991-2 are illustrated and their

origin is discussed.

1 INTRODUCTION

Whilst in traditional bridge codes static loads were represented by real vehicles, in

modern codes, static verifications are performed through artificial models, resulting in the

same values of the effects induced in the bridges by the real traffic.

Static traffic load models for road, pedestrian and railway bridges of the new

Eurocode EN 1991-2 [1] are illustrated, stressing the background philosophy and the applied

methodological criteria.

Calibration of traffic models for road bridges was based on real traffic data recorded in

two experimental campaign performed in Europe between 1980 and 1994 and mainly on the

traffic recorded in may 1986 in Auxerre (F) on the motorway Paris- Lyon. The Auxerre traffic

was identified, on the basis of the available data, as the most representative European

continental traffic in terms of composition and severity, also taking into account the expected

traffic trends. This conclusion was confirmed by more recent studies [2].

The calibration is discussed in much more detail in Appendix A to the present chapter.

2 THE EN 1991-2 LOAD TRAFFIC MODELS FOR ROAD BRIDGES

The static load model for road bridges of the EN 1991-2 is illustrated in the following.

As the load model is calibrated for road bridges having carriageway width smaller

than 42 m and span length up to 200 m, it cannot be used, in principle, outside the above

mentioned field. Anyhow, it results generally safe-sided for bigger spans.

2.1 Division of the carriageway and numbering of notional lanes

The carriageway is defined as the part of the roadway surface sustained by a single

structure (deck, pier etc.). It includes all the physical lanes (marked on the roadway surface),

the hard shoulders, the hard strips and the marker strips. Its width w should be measured

between the kerbs, if their height is greater than 100 mm, or between the inner limits of the

safety barriers, in all other cases.

The carriageway width does not include, in general, the distance between fixed safety

barriers or kerbs of a central reservation nor the widths of these barriers.


34

The carriageway is divided in notional lanes, generally 3 m wide, and in the remaining

area, according to Table 1, as reported, for example, in figure 1. If the carriageway is

physically divided in two parts by a central reservation, then:

- each part, including all hard shoulder or strips, should be separately divided in

notional lanes, if the parts are separated by a fixed safety barrier;

- the whole carriageway, central reservation included, should be divided in notional

lanes, if the parts are separated by demountable safety barriers or another road

restraint system. Table 1. Subdivision of the carriageway in notional lanes

Carriageway

width w

Number of

notional lanes nl

Width of a

notional lane

Width of the

remaining area

w<5.4 m 1 3 m w-3 m

5.4 m ≤w<6 m 2 0.5 w 0

6 m ≤w Int(w/3) 3 m w-3×nl

Notional lane n. 1

Remaining area

Remaining area

Remaining area

Remaining area

Notional lane n. 2

Notional lane n. 33.0

3.0

3.0w

Figure 1. Example of lane numbering

The location of the notional lanes is not linked with their numbering, so that number

and location of the notional lanes should be chosen each time in order to maximize the

considered effect. In particular cases, for example for some serviceability limit states or for

fatigue verifications, it is possible to derogate from this rule and to consider less severe

locations of the notional lanes. In general, the notional lane that gives the most severe effect is

numbered lane n. 1 and so on, in decreasing order of severity.

The numbering of the carriageway depends on the element under consideration.

When the carriageway is made by two separate supported by a unique deck, the lane

numbering should regard the entire carriageway, considering, obviously, that lane n. 1 can be

alternatively on the two parts (figure 2).

When, instead, carriageway consists of two separate parts on two independent decks

supported by the same abutments or the same piers, it needs to distinguish two cases: for deck

design purposes, each part is considered and numbered independently, while, on the contrary,

for abutment or pier design the two parts are considered and numbered together (figure 3).


35

Deck design: onenotional lane numbering

Pier design: onenotional lane numbering

Figure 2. Lane numbering in case the entire carriageway is supported by a single deck

Deck design: two separate

notional lane numbering

Pier design: one

notional lane numbering

Figure 3. Lane numbering in case the carriageway consists of two separate parts

supported by two separate decks

2.2 Load models for vertical loads

Load models representing vertical loads are intended for the evaluation of road traffic

effects associated with ULS verifications and with particular serviceability verifications.

Four different load models are considered:

- load model n. 1 (LM1) generally reproduces traffic effects to be taken into account

for global and local verifications; it is composed by concentrated and uniformly

distributed loads;

- load model n. 2 (LM2) reproduces traffic effects on short structural members; it is

composed by a single axle load on specific rectangular tire contact areas;

- load model n. 3 (LM3), special vehicles, should be considered only when

requested, in a transient design situation; it represents abnormal vehicles not

complying with national regulations on weight and dimension of vehicles. The

geometry and the axle loads of the special vehicles to be considered in the bridge

design should be assigned by the bridge owner;

- load model n. 4 (LM4), a crowd loading.

2.3 Load model n. 1

Load model n. 1 consists of two subsystems:

- a system of two concentrated axle loads, representing a tandem system weighing

2⋅αQ⋅Qk (see Table 2), whose geometry is shown diagrammatically in figure 4;

- a system of distributed loads having a weight density per square meter of αq⋅qk

(see Table 2).

The adjustment factors αQ and αq depend on the class of the route and on the expected

traffic type: in absence of specific indications, they are assumed equal to 1. The characteristic

loads values on the notional i-th lane are indicated αQi⋅Qki and αqi⋅qki while on the remaining

area the weight density of the uniformly distributed load is expressed as αqr⋅qkr.


36

Longitudinal axisof the bridge

0.4

0.4

2

0.4

1.6

0.4

0.4 0.8 0.4

1.2

Figure 4. Tandem system of LM1

Table 2. Load model n. 1 – characteristic values

Position Tandem system – Axle

load Qik [kN]

Uniformly distributed

load qik [kN/m2]

Notional lane n. 1 300 9.0



Other notional lanes 0 2.5

Remaining area 0 2.5

For bridges without road signs restricting vehicle weights, it should be assumed

αQ1≥0.8 for the tandem system on the first notional lane, while for i≥2, αqi≥1.0 except for the

remaining area.

The load model n. 1 should apply according to the following rules (see figure 5):

- in each notional lane only one tandem system should be considered, situated in the

most unfavourable position;

- the tandem system should be considered travelling in the direction of the

longitudinal axis of the bridge, centred on the axis of the notional lane;

- when present, the tandem system should be considered in full, i.e. with all its four

wheels;

- the uniformly distributed loads apply, longitudinally and transversally, only on the

unfavourable parts of the influence surface;

- the two load systems can insist on the same area;

- the impact factor is included in the load values αQi⋅Qki and αqi⋅qki;

- when static verification is governed by combination of local and global effects, the

same load arrangement should be considered for calculation of local and global

effects;

- when relevant, and only for local verifications, the transverse distance between

adjacent tandem systems should be reduced, up to a minimum of 0.4 m.


37

Lane n. 1Q =300 kN

q =9.0 kN/m

QikQik

qik

Lane n. 2

0.5

2.0

0.5

Q =200 kN2k

q =2.5 kN/m2k2

Lane n. 3Q =100 kN3k

q =2.5 kN/m3k2

Remaining area q =2.5 kN/mrk2

1k

1k

0.5

2.0

0.5

0.5

2.0

0.5

w

Figure 5. Example of application of load model n.1

2.4 Load model n. 2

The local load model n. 2, LM2 (figure 6), consists of a single axle load βQ⋅Qak with

Qak=400 kN, dynamic amplification included. Unless otherwise specified βQ=αQ1.


2

0.6

1.4

0.6

0.35

Figure 6. Load model n. 2 (single axle)

The load model, which is intended only for local verifications, should be considered

alone on the bridge, travelling in the direction of the longitudinal axis of the bridge.

The model should be applied in any location on the carriageway and, if necessary,

only one wheel load of βQ⋅200 kN should be considered. If not otherwise specified, the

contact surface of each wheel is rectangle, whose dimensions are 0.35 m×0.6 m.

2.5 Load model n. 3 - Special vehicles

Besides the above mentioned load models, the Eurocode also foresees the possibility

to consider special vehicles, whose transit on the road network and in particular on the bridges


38

is subject to special authorisation, because they exceed the legal limits in length, in width

and/or in mass.

These special vehicles are represented by a set of standardised arrangements of axle

loads, where the bridge owner can pick-up, according to his specific necessities, one or more

vehicles to be taken into account in the bridge design.

The load model should be considered only if expressly required and its application

should regard only the selected special vehicles.

A useful reference is represented by the set of standardized special lorries given in the

informative Appendix A of EN 1991-2, which is reported in Tables 3.a and 3.b.

The nominal values of the axle loads of the special lorries are associated exclusively to

transient design situations.

Each axle load is considered uniformly distributed over two or three narrow

rectangular surfaces 1.20 m long and 0.15 m wide. Axles weighing 150 or 200 kN are

considered distributed on two surfaces, axles weighing 240 kN are considered distributed on

three surfaces, as illustrated in figure 7.

Special vehicles characterised by axle loads in the interval 150 to 200 kN occupy the

notional lane n. 1, while special vehicles characterized by 240 kN axle loads occupy two

adjacent notional lane, lanes n. 1 and n. 2 (figure 8). The lanes are situated in the most

unfavourable position, at most excluding hard shoulders, hard strips and marker strips. More

favourable positions can be considered, if transit is allowed only under special limitations.

Table 3.a. Special vehicles with axle weighing 150 and 200 kN

150 kN axle loads 200 kN axle laods

Vehicle

weight Geometry Axle loads Vehicle

type Geometry Axle loads Vehicle type

600 kN 3×1.5 m 4×150 kN 600/150

900 kN 5×1.5 m 4×150 kN 900/150

1200 kN 7×1.5 m 4×150 kN 1200/150 5×1.5 m 6×200 kN 1200/200

1500 kN 9×1.5 m 4×150 kN 1500/150 7×1.5 m 1×100+7× 200 kN 1500/200

1800 kN 11×1.5 m 4×150 kN 1800/150 8×1.5 m 9×200 kN 1800/200

2400 kN 11×1.5 m 12×200 kN 2400/200

2400 kN 5×1.5+12+5×1.5 m 12×200 kN 2400/200/200

3000 kN 14×1.5 m 15×200 kN 3000/200

3000 kN 7×1.5+12+6×1.5 m 15×200 kN 3000/200/200

3600 kN 17×1.5 m 18×200 kN 3600/200

Table 3.b. Special vehicles with axles weighing 240 kN

240 kN axle loads

Vehicle weight Geometry Axle loads Vehicle type

2400 kN 8×1.5 m 10×240 kN 2400/240

3000 kN 12×1.5 m 1×120+12×200 kN 3000/240

3600 kN 14×1.5 m 15×240 kN 3600/240

3600 kN 7×1.5+12+6×1.5 m 15×200 kN 3600/240/240


39

Lo

ng

itud

ina

l axis

of th

e b

rid

ge

1.2

0.3

1.2

150 kN or 200 kN axle weight

1.2

0.3

1.2

240 kN axle weight

1.2

0.3

Figure 7. Axle lines and wheel contact areas for special vehicles

Figure 8. Arrangement of special vehicle on the carriageway

Since special vehicles are assumed to move at low speed (5 km/h), dynamic effects are

not significant; therefore dynamic magnification is considered included in the nominal values

of the axle loads.

As a rule, concomitance of the special vehicles with the load model n. 1 is taken into

account considering that the lane (lane n. 1) or the two adjacent lanes (lanes n. 1 and 2),

occupied by the standardized special vehicle, are not subjected to additional traffic loads in a

range of 25 m each side from the front axle and the rear axle of the special vehicle itself,

measured in the longitudinal direction as shown in figure 9.

According to the aforementioned general rules, the remaining parts of the notional

lanes and of the carriageway are loaded with the frequent values of load model n. 1.

2.6 Load model n. 4 – Crowd loading

The uniformly distributed load model n. 4, the crowd loading, is particularly

significant for bridges situated in urban areas and it should be considered only when expressly

required.

4.20


40

Figure 9. Simultaneity of special vehicles and load model n. 1

The nominal value of the load, including dynamic amplification, is equal to 5.0

kN/m2, while the combination value is reduced to 3.0 kN/m

2, even if it seems that calculations

are considerably simplified adopting a value of 2.5 kN/m2, like in lanes 2 and 3 and in the

remaining area.

The crowd loading should be applied on all the relevant parts of the length and width

of the bridge deck, including the central reservation, if necessary.

2.7 Characteristic values of horizontal actions

2.7.1 Braking and acceleration forces

The braking or acceleration force, denoted by Qlk, should be taken as a longitudinal

force acting at finished carriageway level.

The characteristic values of Qlk depends on the total maximum vertical load induced

by LM1 on notional lane n. 1, as follows

( ) kN 90010.026.0QkN 180 11lk 111≤⋅⋅⋅⋅+⋅⋅⋅=≤⋅ LwqQ lkqkQQ ααα , (1)

being w1 is the lane width and L the length of the loaded area.

This force, that includes dynamic magnification, should be considered located along

the axis of any lane. When the eccentricity is not significant, the force may be considered

applied along the carriageway axis and uniformly distributed over the loaded length.


41

2.7.2 Centrifugal force

The centrifugal force Qtk is a transverse force acting at the finished carriageway level

and perpendicularly to the axis of the carriageway. Unless otherwise specified, Qtk should be

considered as a point load at any deck cross section.

The characteristic value of Qtk, including dynamic magnification, depends on the

horizontal radius r [m] of the carriageway centreline and on the total maximum weight of the

vertical concentrated loads of the tandem systems of the main loading system Qv

)2( iki Qv QQi

⋅⋅=∑ α , (2)

and it is given by

vtk QQ ⋅= 2.0 [kN], r<200 m; r

QQ v

tk ⋅= 40 [kN], 200 m≤r≤1500 m;

0=tkQ , r>1500 m.

(3)

2.8 Groups of traffic loads on road bridges

When simultaneity of traffic actions with non-traffic actions is significant, the

characteristic values of the traffic actions can be determined considering the five different,

and mutually exclusive, group of loads reported in table 4, where the dominant component

action is underlined.

Each groups of loads indicated in the table should be considered as defining a

characteristic action for combination with non-traffic loads, but they can be used also to

evaluate infrequent and frequent values.

To obtain infrequent combination values it is sufficient to replace in table 4

characteristic values with the infrequent ones, leaving unchanged the others, while frequent

combination values are obtained replacing characteristic values with the frequent ones and

setting to zero all the others. The ψ-factors for traffic loads on road bridges are reported in

table 5.

Table 4. Assessment of characteristic values of multi-component actions for traffic loads

on road bridges

Carriageway

Footways and

cycle tracks

Vertical loads Horizontal loads Vertical loads only

Group of

loads

Main load

model

Special

vehicles

Crowd

loading

Braking force Centrifugal

force

Uniformly

distributed loads

1 Characteristic

values

Combination value

2 Frequent

values

Characteristic

values

Characteristic

values

3 Characteristic

values

4 Characteristic

values

Characteristic

values

5 see §2.5 and

figure 9

Characteristic

values


42

Table 5. Recommended values of ψψψψ- factors for traffic loads on road bridges

Action Symbol ψ0 ψ1infq ψ1 ψ2

gr1a (LM1)

Tandem System

UDL

0.75

0.40

0.80

0.80

0.75

0.40

0

0

gr1b (single axle) 0 0.80 0.75 0

Traffic loads gr2 (Horizontal Forces) 0 0 0 0

(see table 9) gr3 (Pedestrian loads) 0 0.80 0 0

gr4 (LM4 – Crowd loading)) 0 0.80 0.75 0

gr5 (LM3 – Special vehicles)) 0 1.0 0 0

The values of ψ0, ψ1, ψ2 for gr1a, referring to load model n.1 are assigned for routes

with traffic corresponding to adjusting factors αQi, αqi, αqr and βQ equal to 1, while those

relating to UDL correspond to the most common traffic scenarios, in which an accumulation

of lorries can occur, but not frequently. Other values may be envisaged for other classes of

routes or of other classes of expected traffic, according to the relevant α factors.

For example, for traffic situations characterised by severe presence of continuous

traffic, like for bridges in urban areas, a value of ψ2 other than zero may be envisaged for the

UDL system of LM1 only.

The factors for the UDL, given in table 5, apply not only to the distributed part of

LM1, but also to the combination value of the pedestrian load mentioned in table 5.

3 ACTIONS ON FOOTBRIDGES

The section of EN1991-2 concerning actions on footbridges covers explicitly actions

on footways, cycle tracks and footbridges and it is specifically devoted only to footbridges.

The uniformly distributed load qfk and the concentrated load Qfwk given in the following,

where relevant, can be also used for parts of road and railway bridges accessible to pedestrian.

Load models and their representative values include dynamic amplification effects and

should be used for all serviceability and ultimate limit state static calculations, excluding

fatigue limit states.

When vibration assessments based on specific dynamic analysis are necessary, ad hoc

studies should be performed. Some guidance about vibration check of footbridges is given in

EN1990-A2 [3] as summarized in the following §3.6

3.1 Vertical load models

Three different vertical load models can be envisaged for footbridges:

1. an uniformly distributed load representing the static effects of a dense crowd;

2. one concentrated load, representing the effect of a maintenance load;

3. one or more, mutually exclusive, standard vehicles, to be taken into account when

maintenance or emergency vehicles are expected to cross the footbridge itself.

3.1.1 Uniformly distributed load

The crowd effect on the bridge is represented by a uniformly distributed load.

When risk of dense crowd exists or if required, Load Model 4 for road bridges should

be considered also for footbridges.


43

On the contrary, if the application of the aforesaid Load Model 4 is not required, a

uniformly distributed load qfk, to be applied to the unfavourable parts of the influence surface,

should be considered.

Value of qfk depends on the loaded length L [m] and it is given by

22 kN/m0.530

1200.2kN/m5,2 ≤

++=≤

Lq fk .

(4)

In road bridges supporting footways or cycle tracks, the characteristic value 5 kN/m2

or the combination value (2.5 kN/m2) should be considered, according to figure 10.

q =5.0 kN/mfk2

Figure 10. Characteristic load on a footway (or cycle track) of a road bridge

3.1.2 Concentrated load

For local assessments, a 10 kN concentrated load Qfwk, representing a maintenance

load should be considered on the bridge, acting on a square surface of sides 0.1 m. The

concentrated load will not be combined with other variable non-traffic loads.

Obviously, when the service vehicle described in §3.1.3 is taken into account, Qfwk

should be disregarded.

3.1.3 Service vehicle

Service vehicles for maintenance, emergencies (e.g. ambulance, fire) or other services

can be assigned when necessary, depending on the particular situation. When no special

information is available and no permanent obstacle prevents the transit of vehicles on the

bridge deck, the special service vehicle defined in figure 11 should be considered in transient

design situations.

When consideration of the service vehicle is not required, the transit of the vehicle

shown in figure 11 should be considered as accidental.


3

0.2

0.2

1.3

0.2

0.2

0.2

0.2

Q =40 kNsv2Q =80 kNsv1

Figure 11. Service or accidental vehicle


44

3.2 Characteristic values of horizontal forces

For footbridges, it should be considered a horizontal force Qflk, acting simultaneously

with the corresponding vertical load, whose characteristic value is equal to the greater of:

- 10% of the total load corresponding to the uniformly distributed load or

- 60% of the total weight of the service vehicle, when relevant.

The horizontal force, which does not coexists with the concentrated load Qfwk, acts

along the bridge deck axis at the pavement level on a square surface of sides 0.1 m and it is

normally sufficient to ensure the horizontal longitudinal stability of the footbridge.

3.3 Groups of traffic loads on footbridges

Vertical loads and horizontal forces due to traffic should be combined, when relevant,

considering the groups of loads defined in table 6. Each one of these mutually exclusive

groups defines a characteristic action for combination with non – traffic loads.

When combinations of traffic loads with actions of different nature need to be

considered, any group of loads in table 6 should be considered as one action.

Table 6. Definition of groups of loads (characteristic values)

Load type Vertical forces Horizontal forces

Load system Uniformly

distributed load

Service

vehicle

Groups gr1 Fk 0 Fk

of loads gr2 0 Fk Fk

Wind and snow are not considered to act simultaneously with traffic loads on

footbridges, except on roofed bridges, which are considered according to the appropriate rules

given in EN 1991-1-3.

Wind and thermal actions should not be considered as simultaneous.

3.4 Application of the load models

The traffic models described in the previous paragraphs, with the exception of the

service vehicle model, may also be used for pedestrian and cycle traffic areas of the deck of

road bridges limited by parapets and not included in the carriageway, as well as on footpaths

of railway bridges.

These actions are free, so that the vertical loads should be applied anywhere within the

relevant areas, in order to obtain the most adverse effects.

3.5 Verifications of traffic induced deformations and vibrations for footbridges

As known, traffic induced deformations and vibrations of footbridges strongly

influence the serviceability level.

The main structure is generally affected by vertical and horizontal vibrations, as well

as torsional vibrations, either alone or coupled with vertical and/or horizontal vibrations

The design situations to be studied depend on the type of pedestrian traffic admitted

on individual footbridge during its design working life and on the level of regulation,

authorisation and control of the traffic itself.

In general, the following design situations could be taken into account:


45

1. A persistent design situation considering the simultaneous presence of a group of

about 8 to 15 persons walking normally;

2. the simultaneous presence of streams of pedestrians (significantly more than 15

persons), which could be persistent, transient or accidental depending on boundary

conditions, like location of the footbridge in urban or rural areas, the possibility of

crowding due to the vicinity of railway and bus stations, schools, important building

with public admittance, relevance of the footbridge itself and so on;

3. occasional sports, festive or choreographic events, requiring ad hoc investigations.

3.5.1. Bridge-traffic interaction

A pedestrian normally walking exert on the bridge vertical periodic forces, with a

frequency ranging between 1 and 3 Hz, and horizontal periodic forces, perfectly synchronised

with the vertical ones, with a frequency ranging between 0.5 and 1.5 Hz, but forces exerted by

several persons are generally not synchronised and characterised by different frequencies.

When the frequency of the periodic forces normally exerted by pedestrians is close to

a natural frequency of the deck, it commonly happens that the subjective perception of the

bridge oscillation induces the pedestrian to synchronise their steps with the vibrations of the

bridge, so that resonance occurs, increasing considerably the response of the bridge.

The number of pedestrians participating to the resonance is highly random and

depends on the number of persons on the bridge: when the number of persons on the bridge is

bigger than 10, the number of the participating persons is a decreasing function of their

number, but correlation between forces exerted by pedestrians themselves may increase with

movements.

3.5.2. Dynamic models of pedestrian loads

The studies about dynamic modelling of pedestrian loads for footbridge design are still

in progress, also in consideration of the serviceability failures produced by vibrations in

recently built footbridges (Millennium bridge in London, Solferino bridge in Paris), so that

what given in this chapter should be considered as merely informative.

Dynamic pedestrian loads could be defined by means of two separate models

consisting of:

1. a concentrated force (Fn), representing the excitation by a limited group of pedestrians,

which should be systematically used for the verification of the comfort criteria;

2. a uniformly distributed load (Fs), representing the excitation caused by a continuous

stream of pedestrians, which should be used separately from Fn where specified.

Load model Fn, which should be placed in the most adverse position on the bridge

deck, is represented by one pulsating force with a vertical component Fn,v

) 2sin()( 280, tffkF vvvvn π= [N] (5)

and an horizontal component Fn,h

) 2sin()( 70, tffkF hhhhn π= [N], (6)

to be considered separately.

In equations (5) and (6) fv is the natural vertical frequency of the bridge closest to 2

Hz, fh is the natural horizontal frequency of the bridge closest to 1 Hz, t is the time in s and

kv(fv) and kh(fh) are suitable coefficients, depending on the frequency according to figure 12.

For the evaluation of fv, fh and of the inertia effects, Fn should be associated, if

unfavourable, with a static mass equal to 800 kg, applied at the same location.


46

The uniformly distributed load model Fs, to be applied on the whole deck of the

bridge, consists in a uniformly distributed pulsating load with vertical component

) 2sin()( 15, tffkF vvvvs π= [N/m2], (7)

and horizontal component

) 2sin()( 4, tffkF hhhhs π= [N/m2], (8)

to be considered separately.

For the evaluation of fv, fh and of the inertia effects, Fs should be associated, if

unfavourable, with a static mass equal to 400 kg/m2, applied at the same area.

In special cases, likes relevant footbridges, it may be possible to increase the reliability

degree of the assessments, by specifying to apply Fs on limited unfavourable areas (e.g. span

by span) or with an opposition of phases on successive spans.

0 1 2 3 4 5 fv [Hz]

kv(f

v)

1

2

3

Vertical vibrations

0 1 2 3 4 5 fh [Hz]

kh(f

h)

1

2

3

Horizontal vibrations

Figure 12. Relationships between coefficients kv(fv), kh(fh) and frequencies fv, fh

3.5.3. Comfort criteria

In order to ensure pedestrian comfort, the maximum acceleration of any part of the

deck should not exceed

- 0,7 m/s2 for vertical vibrations; or

- 0,15 m/s2 for horizontal vibrations.

The assessment of comfort criteria should be performed for natural vertical frequency

of the footbridge up to 5 Hz or horizontal and torsional natural frequencies up to 2.5 Hz.

In the evaluation of natural frequencies fv or fh the mass of any permanent load should

be taken into account and the stiffness parameters of the deck should be calculated using the

short term dynamic elastic properties of the structural material and, if significant, of the

parapets. It must be noted that generally the mass of pedestrians is relevant only for very light

decks.

If comfort criteria cannot be satisfied with a significant margin, the possible

installation of dampers in the structure after its completion should be envisaged in the design.


47

Evaluation of accelerations shall take into account the damping of the footbridge,

through the damping factor ζ referring to the critical damping, or the logarithmic decrementδ,

which is equal to 2πζ.

For rather short spans, when calculations are performed using the groups of

pedestrians given before, the effect of the damping on the acceleration can be considered

through the reduction factors:

- ) 2exp(1, ζπ nk vn −−= for vertical vibrations or

- ) exp(1, ζπ nk hn −−= for horizontal vibrations,

where n is the number of steps necessary to cross the span under consideration.

For a simply supported bridge, the design value of the vertical acceleration a1d due to

the group of pedestrians may then be assumed as:

ζζπ

) 2exp(1)( 1651

M

nfka vvd

−−= [m/s2], (9)

where M is the total mass of the bridge, f is the relevant, i.e. the determining, fundamental

frequency, and kv(fv) is given in figure 12.

4 ACTIONS ON RAILWAY BRIDGES

Like road bridge load models, also railway bridges load models of EN 1991-2 do not

describe actual loads, although weight and geometry of trains are often exactly known.

Load models for railway bridges have been set-up in such a way that their effects,

amplified by the dynamic coefficients, which in this case are given separately, represent the

characteristic effects of the most severe train traffic expected on the European railways

network.

The rail traffic within the scope in EN1991-2 concerns standard track gauge and wide

track gauge of the European mainline network. In general, the load models given here are not

applicable to narrow-gauge railways, tramways and other light railways, preservation

railways, rack and pinion railways, funicular railways and so on, that require specific loading

models, to be specifically defined.

Of course, when other traffic conditions need to be considered, which are outside the

scope of the load models specified in EN 1991-2, specific alternative load models and

combination rules should be defined for the particular case under consideration.

4.1 Representation of actions and nature of rail traffic loads

Actions due to normal railway operations are usually represented by:

- vertical loads,

- vertical loading for earthworks,

- dynamic effects,

- centrifugal forces,

- nosing forces,

- traction and braking forces,

- combined response of a structure and track to variable actions,

- aerodynamic effects from passing trains,


48

- actions due to overhead line equipment and other railway infrastructure and

equipment.

4.2 Vertical loads

In EN 1991-2 five load models are given for railway loading:

1. Load Model 71, representing normal rail traffic on mainline railways;

2. Load Model SW/0, which could be relevant for continuous bridges;

3. Load Model SW/2, representing heavy trains;

4. Load Model HSLM, representing high speed (>200 km/h) passenger trains;

5. Load Model “unloaded train”, representing the effect of an unloaded train.

The loadings can vary depending on the nature, the volume and maximum weight of rail

traffic on different railways, as well as on different qualities of track.

4.2.1 Load Model 71

Load Model 71, representing the vertical static effect of normal rail traffic, is composed

by a 4-axles vehicle weighing 1000 kN and by a uniformly distributed load equal to 80 kN/m,

not limited in extension, as illustrated in figure 13.

Q =250 kNvk 250 kN 250 kN 250 kNq =80 kN/mvkq =80 kN/mvk

0.8 1.6 1.6 1.6 0.8

no limitation in extension no limitation in extension

Figure 13. Load Model 71

Heavier or lighter rail traffics can be taken into account multiplying the characteristic

values of loads given in figure 13 by a factor α, which should assume following values: 0.75;

0.83; 0.91; 1.10; 1.21; 1.33; 1.46, being α=1.0 the α factor for normal traffic.

In any case, the same factor α should be considered to evaluate equivalent vertical loading

for earthworks and earth pressure effects, centrifugal, traction and braking forces, combined response

of structure and track to variable actions, accidental actions and Load Model SW/0 for continuous

span bridges.

4.2.2 Load Models SW/0 and SW/2

Load Model SW/0 represents the static effect of normal traffic on continuous beams,

while Load Model SW/2, which represents the static effect of heavy rail traffic, should be taken

into account only where heavy rail traffic is foreseen.

LM SW/0 and SW/2 are represented by two uniformly distributed loads of length a,

spaced by c, as reported in table 7 and illustrated in figure 14.

Table 7. Characteristic values of vertical loads for Load Models SW/0 and SW/2

Load

Model

qvk

[kN/m]

a

[m]

c

[m]

SW/0

SW/2

133

150

15.0

25.0

5.3

7.0


49

qvk qvk

a c a

Figure 14. Load Models SW/0 and SW/2

4.2.3 Load Model “unloaded train”

The so called unloaded train is a particular load model consisting of a vertical uniformly

distributed load with a characteristic value of 10.0 kN/m, which could be used for some

particular verifications.

4.2.4 Eccentricity of vertical load models 71 and SW/0

The eccentricity of vertical load due to lateral displacement to be considered for static

verifications assessments can be taken into account considering the ratio of wheel loads on all

axles as up to 1.25:1.00 on each track, so that it results the eccentricity e shown in figure 15,

which should be not greater than r/18, being r the transverse distance between the wheel loads.

The loads to be taken into account are the appropriate uniformly distributed and

concentrated loads pertaining to LM71 and SW0 when required.

The load eccentricity e may be neglected in fatigue verifications

r

qv1

Qv1

qv2

Qv2

q +qv1

Q +Qv1

v2

v2

e

qv2

qv1

≤1.25

Qv2

Qv1

≤1.25

e≤r

18

Figure 15. Eccentricity of vertical loads

4.2.5 Distribution of axle loads

Distribution of axle loads by the rails, sleepers and ballast, for all kind of trains and

verifications, including fatigue, can be taken into account

- in the longitudinal direction considering that a point force or an axle load is

distributed by the rail over three adjacent sleepers, being the loaded one subjected to

the 50% of the load and each of the two adjacent one subjected to the 25% of the load

as indicated in figure 16; for local verifications a load dispersal with 4:1 slope


50

through the ballast can be considered according to figure 17, where a represents the

sleeper’s spacing;

- in the transverse direction depending on the track configuration: the actions should

be distributed transversely according to figure 18 for bridges with ballasted track

without cant, according to figure 19 for bridges with ballasted tracks with cant and

for full length sleepers, where the ballast is only consolidated under the rails, or for

duo-block sleepers; according to figure 20 for bridges with ballasted tracks with cant

and full length sleepers; finally, in case of bridges with ballasted track and cant and

for full length sleepers, where the ballast is only consolidated under the rails, or for

duo-block sleepers, figure 20 should be suitably modified to take into account the

transverse load distribution under each rail shown in figure 19.

Qvi

Qvi

aa a a

4

Qvi

2

Qvi

4

Figure 16. Longitudinal distribution of concentrated loads

Qvisleeper

reference plane

4:1

Figure 17. Longitudinal dispersal of sleeper loads through the ballast

4.2.6 Equivalent vertical loading for earthworks and earth pressure effects

To determine earth pressure effects or to design earthworks under or adjacent to the

track, an equivalent vertical loading due to rail traffic actions can be considered for the

evaluation of global effects, represented by the appropriate load of model LM71 or SW/2


51

uniformly distributed over a width of 3.0 m at a level 0.70 m below the running surface of the

track. Dynamic effects can be disregarded.

For local elements close to a track (e.g. ballast retention walls and so on), the maximum

local vertical, longitudinal and transverse loadings on the element due to rail traffic actions

should be evaluated.

4.2.7 Footpaths and general maintenance loading

For pedestrian and cycle paths and for general maintenance loads a uniformly

distributed load with a characteristic value qfk=5 kN/m² should be taken into account, while

for design of local elements a concentrated load Qk=2.0 kN acting alone should be applied on

a square surface with a 200 mm side.

4:1

Qv

Qh

Qr

MRA B

reference plane

h

Aσ B

σ

Mσ

Figure 18. Transverse distribution of action for ballasted tracks without cant

4:1

Qv

Qh

Qr

MRA B

reference plane

h

Aσ B

σ

running plane

0.6

4:1

0.6

Figure 19. Transverse distribution of action for duo-block sleepers


52

4:1

Qv

Qh

Qr

MRA B

reference plane

h

Aσ B

σ

Mσ

u

running plane

Figure 20. Transverse distribution of action for ballasted tracks with cant

4.3 Dynamic magnification factors ΦΦΦΦ (ΦΦΦΦ2, ΦΦΦΦ3)

Provided that risks of resonance effects and excessive vibrations of the bridge are

negligible, dynamic magnification of stresses and vibration effects can be taken into account

through the dynamic factor Φ.

On the contrary, when risks of resonance or excessive vibrations exist, a suitable

dynamic analysis is necessary. In these cases static load effects multiplied by the dynamic

factor Φ are unable to predict resonance effects from high speed trains, therefore dynamic

analysis techniques, taking into account the time dependant nature of the loading from the

High Speed Load Model (HSLM) and Real Trains (e.g. by solving equations of motion) are

required for predicting dynamic effects at resonance.

The dynamic factors can be applied also to structures with more than one track.

4.3.1. Definition of the dynamic factor Φ

The dynamic factor Φ which increases the static load effects induced by Load Models

71, SW/0 and SW/2 depends on the level of maintenance of tracks.

For carefully maintained track, it is

1.00≤ 82.02.0

44,12 +

−=

ΦLΦ ≤1.67,

(10)

while for standard maintained track, it is

1.00≤ 73.02.0

16,23 +

−=

ΦLΦ ≤ 2.00,

(11)

being LΦ the “determinant” length associated with Φ in [m].

For the most common and practically relevant cases, LΦ is defined in tables 8.a, 8.b

and 8.c. For cases not covered by the tables, a satisfactory estimate of LΦ can be obtained

evaluating LΦ itself as the base length of the influence line for the deflection of the member

under consideration.

The dynamic factor Φ shall not be used with the loading due to Real Trains, Fatigue

Trains, Load Model HSLM and load model “unloaded train”.

When the resultant stress in a structural member depends on several effects, each of

which relates to a separate structural behaviour, each effect should be calculated using the

appropriate determinant length.


53

Table 8.a. Determinant length LΦΦΦΦ

Case Structural element Determinant length LΦ

Steel deck plate: closed deck with ballast bed (orthotropic deck plate) (global and local transverse

stresses)

1.1

1.2

1.3

1.4

Deck with cross girders and continuous

longitudinal ribs

Deck plate (for both directions)

Continuous longitudinal ribs (including

small cantilevers up to 0.5 m –

cantilevers greater than 0.5 m require ad

hoc studies)

Cross girders

End cross girders

3 times cross girder spacing


Twice the length of the cross girder

3.6 m (it is recommended adoption of Φ3)

Deck plate with cross girder only

2.1

2.2

2.3

Deck plate for both directions

Cross girders

End cross girders

Twice the cross girder spacing + 3 m

Twice the cross girder spacing + 3 m


Steel grillage: closed deck with ballast bed (orthotropic deck plate) (global and local transverse

stresses)

3.1

3.2

3.3

3.4

Rail bearers:

- as an element of continuous

grillage

- simply supported

Cantilever of the rail bearers

Cross girders (as part of cross

girder/continuous rail bearers grillage)

End cross girders


Cross girder spacing + 3 m




For arch bridges and concrete bridges of all types with a cover of more than 1.00 m, Φ2

and Φ3 may be reduced according to the formula

1.0 10

1.00 -3,23,2

h - Φ = Φ red ≥ ,

(12)

being h [m] the height of cover from the top of the deck to the top of the sleeper, including the

ballast, or, in case of arch bridges, from the crown to the extrados.

Rail traffic actions on columns with a slenderness <30, abutments, foundations, retaining

walls and ground pressures may be calculated disregarding dynamic effects.

Bridges sensitive to dynamic effects and in any case bridge on high speed lines

(V≥200 km/h) require specific dynamic analysis considering Real trains or High Speed Load


54

Model, according to the specific application rules. The question is outside the scope of the

present Guidebook and it will not be discussed here.

Table 8.b. Determinant length LΦΦΦΦ


Concrete deck slab with ballast (global and local transverse stresses)

4.1

4.2

4.3

4.4

4.5

4.6

Deck slab as part of box girder or

upper flange of the main beam:

- spanning transversally to the

main girders

- spanning in longitudinal

direction

- cross girders

- transverse cantilever

supporting railway loadings

Deck slab continuous (in main girder

direction) over the cross girders

Deck slab for half through and through

bridges :

- spanning perpendicular to the

main girders


direction

Deck slab spanning transversely

between longitudinal steel beams in

filler bridge decks

Longitudinal cantilevers of deck slab

End cross girders or trimmer beams

3 times span of deck plate

3 times span of deck plate


e

- If e≤0.5 m, three times the distance

between the webs

- If e>0.5 m ad hoc studies are

necessary

Twice the cross girder spacing

Twice span of deck slab + 3 m

Twice span of deck slab

Twice the determinant length in the

longitudinal direction

- If e≤0.5 m, 3.6 m (it is recommended

adoption of Φ3)

- If e>0.5 m ad hoc studies are

necessary


Note: For cases 1.1 to 4.6 inclusive LΦ cannot exceed the determinant length of the main girders


55

Table 8.c. Determinant length LΦΦΦΦ


Main girders

5.1

5.2

5.3

5.4

5.5

5.6

Simply supported girders and slabs

(including steel beams embedded in

concrete)

Girders and slabs continuous over n

spans with Lm=(L1+ L2+....+ Ln)/n

Portal frames and closed frames or

boxes:

- single span


direction

Single arch, arch rib, stiffened girder of

bowstring

Series of arches with solid spandrels

retaining fills

Suspension bars (in conjunction with

stiffening girders)

Span in the main girder direction

LΦ=min[(1+0.1⋅n)⋅Lm; Li,max]

Consider as three span continuous beam (use

expression in 5.2 with horizontal and vertical

lengths of members of frame or box)

Consider as multi span continuous beam

(use expression in 5.2 with lengths of end

vertical members and members)

Half span of the bridge

Twice the clear opening

4 times the longitudinal spacing of

suspension bars

Structural supports

6

Columns, trestles, bearings, uplift

bearings, tension anchors and for

calculation of contact pressure under

bearings

Determinant length of the supported member

4.4. Application of traffic loads on railway bridges

In designing a railway bridge, the number and the position of the loaded tracks should

determined first, according to the relevant influence surfaces.

In each verification the greatest number of tracks geometrically and structurally

possible should be considered in the less favourable position, irrespective of the effective

positions of the intended tracks, according to the given minimum spacing between centre-lines

of adjacent tracks.

Actions of different nature should be combined considering traffic loads and forces

placed in the most unfavourable positions.

For the determination of the most adverse load effects, the following rules apply: when

LM 71 is considered,

- any number of uniformly distributed loads qvk should be applied on the track and up

to four concentrated loads Qvk should be applied once per track,


56

- for elements carrying two tracks, Load Model 71 shall be applied to either track or

both tracks,

- for bridges carrying three or more tracks, LM 71 should be applied if loaded tracks

are less than three, while 0.75 times LM71 should be applied for three or more

loaded tracks;

when LM SW/0 is considered,

- the loading SW/0 should be applied once to a track;

- for elements carrying two tracks, LM SW/0 should be applied to either track or

both tracks;

- for bridges carrying three or more tracks, LM SW/0 should be applied if loaded

tracks are less than three, while 0.75 times LM SW/0 should be applied for three or

more loaded tracks;

- continuous beam bridges designed for LM71 must be checked for LM SW/0 too;

finally, when LMSW/2 is considered,

- the loading SW/2 shall be applied once to a track,

- for elements carrying more than one track, LM SW/2 shall be applied to any one

track only, with LM71 or LMSW/0 applied to the other tracks, in accordance with

the specific application rules.

When relevant, the following rules apply for the Load Model “unloaded train”:

- the Load Model “unloaded train” shall only be considered in the design of

structures carrying one track.

- any number of lengths of the uniformly distributed load qvk shall be applied to a

track.

For serviceability assessments concerning deformations, LM 71 and, if relevant, Load

Models SW/0 and SW/2 increased by the dynamic factor Φ should be applied.

For vibrations or resonance assessments, the above mentioned Load Models for High

Speed trains (HSLM) or Real Trains (RT) should be also taken into account.

The number of loaded tracks to be loaded when checking the limits of deflections and

vibration is given in table 9.

4.5 Horizontal forces - characteristic values

4.5.1 Centrifugal forces

When the track is curved over the whole or part of the length of the bridge, the

centrifugal force and the track cant should be considered.

The centrifugal forces act outwards in a horizontal direction and are applied 1.80 m

above the running surface, considering the Maximum Line Speed V allowed on the bridge,

except for Load Model SW/2, for which a maximum speed of 80 km/h may be assumed.

Said Qvk and qvk the characteristic values of the vertical load components of the

railway load models, LM 71, SW/0, SW/2 and “unloaded train”, any enhancement for

dynamic effects is disregarded, the characteristic values Qtk, qtk of the corresponding centrifugal

forces are given by


57

) (127

) (

22

vkvktk Qfr

VQf

rg

vQ

⋅== [kN]

) (127

) (

22

vkvktk qfr

Vqf

rg

vq

⋅== [kN/m],

(13)

being f a reduction factor, given in the following.

Table 9. Number of tracks to be loaded for checking limits of deflection and vibration

Limit States Checks and

associated acceptance criteria

Number of tracks

1 2 ≥≥≥≥ 3 Traffic Safety Checks:

– Deck twist (EN 1990 Annex 2 A2.4.4.2.2) 1 1 or 2 *)

1 or 2 or 3 or

more *)

– Deformation of the deck (EN 1990 Annex 2 A2.4.4.2.3) 1 1 or 2 *)

1 or 2 or 3 or

more *)

– Horizontal deflection of the deck (EN 1990 Annex 2

A2.4.4.2.4) 1 1 or 2

*)

1 or 2 or 3 or

more *)

– Combined response of structure and track to

variable actions including limits to vertical and

longitudinal displacement of the end of a deck (

6.5.4)

1 1 or 2 *) 1 or 2

*)

– Vertical acceleration of the deck (EN1991-2 6.4.6

and EN 1990 Annex 2 A2.4.4.2.1) 1 1 1

SLS Checks:

– Passenger comfort criteria (EN 1990 Annex 2

A2.4.4.3)

1 1 1

ULS Checks

– Avoidance of unrestrained uplift at bearings 1 1 or 2 *) 1 or 2 or 3 or

more *)

*) whichever is critical (see multi-component actions in §4.6)

In equations (13), v in m/s and V in km/h are the maximum line speed, g is the gravity

acceleration and r is the radius of curvature in m. In case of varying radius, r could suitably set

to its mean value.

Centrifugal forces should be combined with the pertinent vertical traffic load.

The centrifugal force shall not be multiplied by the dynamic factor Φ2 or Φ3.

The factor f takes into account the reduced mass of higher speed trains, therefore, as for

short loaded lengths the magnitude of centrifugal forces is dictated by faster light vehicles, for

Load Model 71 (and where significant Load Model SW/0) the cases considered in table 10 shall

be considered, depending on the line speed V and on the adjustment factor α.

For Load Model 71 (and LM SW/0, if relevant) the reduction factor f is given by:

−

+−−=fLV

Vf

88.2175.1

814

1000

1201 ≤1.0, (14)

where V is the maximum line speed in km/h and Lf is the influence length in m of the loaded part

of curved track on the bridge, which is most unfavourable for the design of the structural element


58

under consideration.

For Load Model 71 (and where significant Load Model SW/0) and V>120 km/h, two

cases should be taken into account:

- case a: in this case Load Model 71 (and where relevant Load Model SW/0) is taken

into account with its dynamic factor and the centrifugal force is evaluated according

to equations (13) setting V=120 km/h, so that the latter is not reduced (f = 1);

- case b: in this case Load Model 71 (and where relevant Load Model SW/0) is taken

into account with its dynamic factor and the centrifugal force is evaluated according

to equations (13) considering the maximum speed V=120 km/h and the related

reduction factor f, evaluated according (14).

Table 10. Load cases for centrifugal force evaluation

Value

of α

Max

Line

Speed

[km/h]

Centrifugal force based on Associated vertical traffic

based on V

[km/h] α f

α<1 >120 V 1 f 1⋅f⋅(LM71”+”SW/0) Φ⋅α⋅1⋅(LM71”+”SW/0) or

Φ⋅α⋅0.5⋅(LM71”+”SW/0)

when vertical traffic actions

are favourable

120 α 1 α⋅1 (LM71”+”SW/0) Φ⋅α⋅1⋅(LM71”+”SW/0) or

Φ⋅α⋅0.5⋅(LM71”+”SW/0)


are favourable

0 --- --- ---

≤120 V α 1 α⋅1 (LM71”+”SW/0)

0 --- --- ---

α =1 >120 V 1 f 1⋅f⋅(LM71”+”SW/0) Φ⋅1⋅1⋅(LM71”+”SW/0)

orΦ⋅1⋅0.5⋅(LM71”+”SW/0)


are favourable

120 1 1 1⋅1 (LM71”+”SW/0) Φ⋅1⋅1⋅(LM71”+”SW/0)

orΦ⋅1⋅0.5⋅(LM71”+”SW/0)


are favourable

0 --- --- ---

≤120 V 1 1 1⋅1 (LM71”+”SW/0)

0 --- --- ---

α >1 >120*) V 1 f 1⋅f⋅(LM71”+”SW/0) Φ⋅1⋅1⋅(LM71”+”SW/0)

orΦ⋅1⋅0.5⋅(LM71”+”SW/0)


are favourable

120 α 1 α⋅1 (LM71”+”SW/0) Φ⋅α⋅1⋅(LM71”+”SW/0) or

Φ⋅α⋅0.5⋅(LM71”+”SW/0)


are favourable

0 --- --- ---

≤120 V α 1 α⋅1 (LM71”+”SW/0)

0 --- --- ---

*) valid only if maximum speed of heavy freight traffic limited to 120 km/h


59

When the line speed V is bigger than 300 km/h and the influence length Lf is bigger

than 2.88 m a lower bound exists for f, f (V=300 km/h).

For Load Models SW2 and unloaded train f=1.0.

4.5.2 Nosing force

The nosing force Qsk, to be always combined with the vertical traffic load, is

represented by a concentrated force acting horizontally, applied at the top of the rails,

normally to the centre-line of track, on both straight and curved track. For rail traffic with

maximum axle load of 250 kN, the characteristic value should be taken as Qsk=100 kN and it

should be multiplied by the dynamic magnification factor Qsk.

Nosing force should by multiplied by the adjustment factor α only when α>1.

4.5.3 Actions due to traction and braking

Traction and braking forces are commonly considered as uniformly distributed over the

corresponding influence length La,b for traction and braking effects related to the structural

element considered.

Traction force is indicated with Qlak and braking force is indicated with Qlbk.

Traction and braking forces, applied the top of the rails in the longitudinal direction of

the track, should be determine according to the following expressions, which are applicable to

all types of track construction, e.g. continuous welded rails or jointed rails, with or without

expansion devices, being La,b in m:

Traction force: Qlak = 33 La,b [kN]≤1000 [kN] (15)

for Load Models 71, SW/0, SW/2, “unloaded train” and HSLM;

Braking force: Qlbk = 20 La,b [kN]≤6000 [kN]

for Load Models 71, SW/0 and Load Model HSLM

Qlbk = 35 La,b [kN] (16)

for Load Model SW/2

Qlbk = 2.5 La,b [kN]

for Load Model “unloaded train”.

In some special case, like for lines carrying special traffic (restricted to high speed

passenger traffic for example) the traction and braking forces may be taken as equal to 25% of

the sum of the axle-loads of the Real Train acting on the influence length of the action effect of

the structural element considered, with an upper limits of 1000 kN for Qlak and 6000 kN for Qlbk.

4.6 Multicomponent traffic actions

4.6.1 Characteristic values of multicomponent actions

The simultaneity of the above mentioned traffic loadings may be taken into account by

considering the mutually exclusive groups of loads given in EN 1991-2 and reported in table 11.

Each groups of loads reported in the table defines a single variable characteristic action,

that should be taken into account for combination with non-traffic loads and should be

considered as a single variable traffic action.

The number of group loads is high; therefore it implies a considerable increase of the

number of the load combinations to be considered in the analyses.

Anyhow, the number of the load groups can be considerably reduced, making some little

safe-sided simplification, so that only four relevant group of loads need to be taken in account,

according to table 12.


60

Table 11. Groups of traffic loads (characteristic values of multicomponent actions)

Groups of loads Vertical forces Horizontal forces

n = Reference EN 1991-2 6.3.2/6.3.3 6.3.3 6.3.4 6.5.3 6.5.1 6.5.2 Comment

1

2 ≥

3

Number

of tracks

Load

group

Loaded

track

LM 71(1)

SW/0 (1), (2)

HSLM(6)(7)

SW/2 (1),(3)

Unload

ed train

Traction,

Braking(1) Centrifugal

force(1) Nosing

force

1 11 T1 1 1 (5)

0.5 (5)

0.5 (5)

Max. vertical

1 with max.

longitudinal

1 12 T1 1 0.5 (5)

1 (5)

1 (5)

Max. vertical

2 with max.

transverse

1 13 T1 1 (4)

1 0.5 (5)

0.5 (5)

Max.

longitudinal

1 14 T1 1 (4)

0.5 (5)

1 1 Max. lateral

1 15 T1 1 0.5 (5)

1 (5)

1 (5

Lateral

stability with

“unloaded

train”

1 16 T1 1 1 (5)

0.5 (5)

0.5 (5)

SW/2

1 17 T1 1 0.5 (5)

1 (5)

1 (5)

SW/2

2 21 T1

T2

1

1

1 (5)

1 (5)

0.5 (5)

0.5 (5)

0.5 (5)

0.5 (5)

Max. vertical

1 with max

longitudinal

2 22 T1

T2

1

1

0.5 (5)

0.5 (5)

1 (5)

1 (5)

1 (5)

1 (5)

Max. vertical

2 with max.

transverse

2 23 T1

T2

1 (4)

1 (4)

1

1

0.5 (5)

0.5 (5)

0.5 (5)

0.5 (5)

Max.

longitudinal

2 24 T1

T2

1 (4)

1 (4)

0.5 (5)

0.5 (5)

1

1

1

1

Max. lateral

2 26 T1

T2

1

1

1 (5)

1 (5)

0.5 (5)

0.5 (5)

0.5 (5)

0.,5 (5)

SW/2

2 27 T1

T2

1

1 0.5 (5)

0.5 (5)

1 (5)

1 (5)

1 (5)

1 (5)

SW/2

≥3 31 Ti

0.75 0.75 (5)

0.75 (5)

0.75 (5)

Additional

load case

(1) All relevant factors (α, Φ, f, ...) shall be taken into account.

(2) SW/0 shall only be taken into account for continuous span bridges.

(3) SW/2 needs to be taken into account only if it is stipulated for the line.

(4) Factor may be reduced to 0.5 if favourable effect, it cannot be zero.

(5) In favourable cases these non dominant values shall be taken equal to zero.

(6) HSLM and Real Trains where required in accordance with 6.4.4 and 6.4.6.1.1. of EN 1991-2

(7) If a dynamic analysis is required in accordance with 6.4.4 of EN 1991-2 also see 6.4.6.5(3) of EN 1991-2.

Leading component action as appropriate

to be considered in designing a structure supporting one track (Load Groups 11-17)

to be considered in designing a structure supporting two tracks (Load Groups 11-27 except 15).

Each of the two tracks shall be considered as either T1 (Track one) or T2 (Track 2)

to be considered in designing a structure supporting three or more tracks;

(Load Groups 11 to 31 except 15. Any one track shall be taken as T1, any other track as T2 with all other

tracks unloaded. In addition the Load Group 31 has to be considered as an additional load case where all

unfavourable lengths of track Ti are loaded.


61

Table 12. Assessment of simplified groups of traffic loads

Load

type

Vertical load Horizontal load Notes

Group of

loads

Vertical load

(1)

Unloaded

train (1)

Traction

Braking

Centrifugal

force

Nosing force

Group 1

(2)

1.00 ---- 0.5 (0) 1.00 (0) 1.00 (0) Maximum

vertical and

lateral

action

Group 2

(2)

---- 1.00 0 1.00 (0) 1.00 (0) Lateral

stability

Group 3

(2)

1.00 (0.5) ---- 1.00 0.5 (0) 0.5 (0) Maximum

longitudinal

action

Group 4

(3)

0.8 (0.6, 0.4) 0.8 (0.6, 0.4) 0.8 (0.6, 0.4) 0.8 (0.6, 0.4) Concrete

cracking

Leading action

(1) Including all his coefficients φ, α and so on

(2) Simultaneity of several actions with their characteristic values, even if improbable, is considered for

groups 1, 2 and 3, considering that the consequences of this assumptions are not relevant

(3) Group 4 should be considered only for concrete cracking (values in parentheses are to be considered

when more than 1 track is loaded: 0.6 for two loaded tracks and 0.4 for three or more loaded tracks.

4.6.2 Other representative values of the multicomponent actions

The frequent values of multicomponent actions can be determined using the same rules

given above: for each group of loads, instead of characteristic values, the factors given in table

11 affect the frequent values of the relevant actions considered.

When the reduced number of groups of loads is considered, according to table 12, only

group 4 should be taken into account.

Quasi-permanent traffic actions shall be taken as zero.


In recent years, bridge codes have considerably evolved. This evolution is the result of

the positive interaction of several factors, namely

- recent progresses of the probabilistic theory of structural reliability;

- improved knowledge of the statistical parameters governing the extreme value

distributions of climatic actions; and,

- finally, availability of large databases regarding in situ traffic measurements all

around the world.

The most evident outcome of the modern code development is the “artificial” nature of

traffic load models.

Since they aim to reproduce to real traffic effects characterised by specified return

period or by given probability to be exceed in the design working life of the bridge, traffic

load models can differ considerably by real vehicles, in terms of silhouette and axle’s

arrangement as well as in terms of axle and vehicle weights.


62

Amongst the contemporary codes for bridge design, Eurocode emerges for its primary

importance, profiting of out to date background and ad hoc studies.

It must be highlighted that Eurocode has been widely checked and it is successfully

applied in current design practice in Europe.

In the present chapter the static traffic load models for road, pedestrian and railway

bridges of EN 1991-2 have been discussed, highlighting the application rules and the group of

traffic loads to be considered in combination with non-traffic actions.

When relevant, particular attention has been devoted to background information, also

aiming to suggest safe-sided simplified assumptions.

In the appendix A to this chapter, calibration study of the traffic load models for road

bridges are illustrated in details, while in the Annex A to the present Guidebook, the future

traffic trends of the lorry traffic in European road network is discussed and their consequences

on EN 1991-2 load models are analysed.

Non traffic actions are illustrated in chapter 5 and load combinations in chapter 7.

The practical application of the load models for road bridges is better detailed in

chapters 8, 9 and 10, where three case studies are developed.

7 REFERENCES

[1] EN1991-2, Eurocode 1: Actions on structures - Part 2: Traffic loads on bridges. CEN,

Brussels, 2003.

[2] O’Connor, A.J. et al., Effects of traffic loads on road bridges – Preliminary studies for the

re-assessment of the Eurocode 1, Part 3. Proceedings of the 2nd

European Conference on

Weigh-in-motion of road vehicles. Lisbon, 1998

[3] EN1990-A2, Eurocode: Basis of structural design – Annex A2: Applications for bridges.

CEN, Brussels, 2005

[4] Croce, P. & Sanpaolesi, L., Design of bridges. Pisa: TEP, 2004.

[5] O’Brien, E.J. et al., Bridge applications of weigh-in-motion. Paris: Laboratoire Central des

Ponts et Chaussées, 1998.

[6] Bruls, A et al., ENV1991 Part 3: The main model of traffic loads on road bridges.

Background studies. Proceedings of IABSE Colloquium on Basis of Design and Actions

on Structures. Background and Application of Eurocode 1. Delft, 1996.

[7] Croce, P. & Salvatore, W., Stochastic model for multilane traffic effects on bridges.

Journal of Bridge Engineering, ASCE, 6(2): 136-143, 2001

[8] Tschemmernegg, F. & al, Verbreiterung und Sanierung von Stahlbrücken. Stahlbau n. 9,

1989.

[9] Sedlacek, G. & al. 1991. Eurocode 1 - Part 12. Traffic loads on road bridges. Definition

of dynamic impact factors. Report of subgroup 5.


63

Appendix A to Chapter 3 – Development of static load traffic models for road bridges

of EN 1991-2


Static load models for road bridges of EN 1991-2 have been developed considering

that an up to date structural code should

- be easy to use;

- be applicable independently on the static scheme and on the span length of the

bridge;

- reproduce as accurately as possible the real load effects induced on the bridge by

all possible flowing and jammed traffic scenarios, that can occur on the bridge

during its design working life; the real load effects are characterised by a specified

return period or by a given probability to be exceeded during the design working

life;

- include the dynamic magnification due to the road-vehicle and to the bridge-

vehicle interactions in load values;

- allow combinations of local and global effects of actions;

- be unambiguous, covering all the cases that could occur in the design practice.

Obviously, as load models are to be defined and calibrated referring to traffic effects

having specified return periods, pre-normative studies had to deal with complicated

theoretical and methodological problems. Among these, especially significant were those

concerned with the extrapolation to very long time periods of effects due to flowing traffics,

recorded on the slow lane for few days or few weeks, taking into account the most severe

flowing and/or congested traffic scenarios that could happen on one or on several lanes.

A.2 Static load model philosophy

As a rule, the evaluation of real traffic effects and the subsequent drafting and

calibration of the load model can be carried out by analytical and numerical methodologies

consisting of:

- identification of the most significant real traffic records;

- choice of the static schemes and spans of the relevant bridges;

- specification of the influence surfaces the most significant effects;

- elaboration of the traffic data and their manipulation to obtain jammed, slowed

down and flowing traffic types;

- determination of the histograms of the extreme values of the effects induced by the

transit of the different traffic typologies on the influence surfaces;

- simulation of extreme scenarios for multilane traffic;

- elaboration and extrapolation of the histograms of the extreme values of the effects

to evaluate the reference values, characterized by specified return period;

- correction of the reference values to take into account the dynamic effects due to

road-vehicle and to vehicle-structure interactions;

- drafting and calibration of the load model;

- applicative trials;


64

- model refinement.

A.3 Statistical analysis of European traffic data

As said, the first phase of the study was devoted to statistic analysis of European

traffic data, in order to select the most representative traffics, in terms of the expected flow

and composition.

Available European traffic’s data were mainly the result of two large measurement

campaigns performed, respectively, within 1977 and 1982 on bridges situated in France,

Germany, Great Britain, Italy and Holland and within 1984 and 1988 on several roads all

around the Europe. Recorded daily flows on the slow lane were varying between 1000 and

8000 lorries on motorways, and between 600 and 1500 lorries on main roads, while fast lane

daily flows on motorway and slow lane daily flows on secondary roads resulted drastically

reduce to 100-200 lorries. [4, 5].

Statistical analyses, that allowed to know the distributions of the most significant

traffic parameters, like traffic composition, inter-vehicle distances, axle’s spacing, weight,

length and speed of each lorry, essentially was limited to data recorded in Italy, France and

Germany; in fact, UK data appeared poorly representative of the continental situation, while

Spanish and Dutch data seemed excessively influenced by the respective road systems

peculiarities.

Significant data, derived from long distance motorway traffics (Auxerre (F), Garonor

(F), Brohltal (D), Fiano Romano (I), Sasso Marconi (I) and Piacenza (I)), are summarized in

tables A.1÷A.5. Table A.1 shows the daily flows of cars and lorries per lane and the

percentage of inter-vehicular distances smaller than 100 meters; table A.2 illustrates the traffic

compositions in terms of standardized lorries, while table A.3 illustrates the composition of

the entire fleets of circulating commercial vehicles in the three above mentioned Countries.

Finally, daily flows of axles heavier than 10 kN and lorries, together with the respective

values of statistical parameters are shown in table A.4 and A.5, respectively.

Generally, the analysis of the European traffic data shows that

- mean values of axle-loads and total weight of heavy vehicles strongly depend on

the traffic typology, i.e. on the road classification; they are usually very scattered:

- the statistical distribution of the axle-load is generally unimodal, with the mode

around 60 kN, while the statistical distribution of the total weight is bimodal with

the first mode around 150 kN and the second mode around 400 kN;

Table A.1. Daily traffic flows per lane

Cars Lorries % intervehicle distance<100 m

Brohltal (D) 11126 4793 26.7

Garonor (F – 1982) -- 2570 32.6

Garonor (F – 1984) -- 3686 32.3

Auxerre (slow lane) (F) 8158 2630 18

Auxerre (slow lane) (F) 1664 153 8.5

Fiano R. (I) 8500 4000 26.1

Piacenza (I) 8500 5000 30.9

Sasso M. (I) 7500 3500 24.3


65

Table A.2. Composition of the commercial traffic

Lorries (%) (2 Axles)

Lorries (%) (>2 Axles)

Articulated lorries (%)

Lorries with trailer (%)

Brohltal (D) 16.6 1.6 40.2 41.6

Garonor (F - 1982) 38.6 2.6 47.6 11.2

Garonor (F - 1984) 47.5 2.2 44.3 6.0

Auxerre (slow lane) (F) 22.7 1.3 65.2 10.8

Auxerre (fast lane) (F) 27.6 3.5 58.4 10.5

Fiano R. (I) 41.4 7.0 29.0 22.6

Piacenza (I) 35.3 7.5 35.8 21.4

Sasso M. (I) 40.1 10.0 30.2 19.7

Table A.3. Composition of the circulating lorry fleets

Germany France Italy

2 axles 17.0 32.0 38.67

3 axles 5.0 5.8 9.0

4 axles 25.0 25.0 10.0

5 axles 52.0 33.2 33.0

6 axles 1.0 4.0 8.0

> 6 axles -- -- 1.33

Table A.4. Daily flows and statistical parameters of axles heavier than 10 kN and lorries

ALL AXLES TANDEM AXLES TRIDEM AXLES

Flow Pmean

[kN] σ

[kN] Pmax [kN]

Flow Pmean [kN]

σ [kN]

Pmax [kN]

Flow Pmean [kN]

σ [kN]

Pmax [kN]

Brohltal (D) 19970 59.0 28.4 165.0 1977 116.5 54.6 260.0 1035 60.0 230.0 355.0

Garonor

(F) 1982 8470 57.6 27.6 180.0 712 126.3 49.3 340.0 303 90.0 200.0 295.0

Garonor

(F) 1984 11593 59.3 30.0 195.0 1016 132.1 58.1 290.0 489 90.0 200.0 320.0

Auxerre (F)

slow lane 10442 82.5 35.2 195.0 844 165.6 54.0 305.0 961 130.0 250.0 390.0

Auxerre (F)

fast lane 581 73.1 41.2 200.0 47 141.2 63.9 275.0 51 120.0 250.0 390.0

Fiano R. (I) 15000 56.8 32.9 142.0 2000 115.2 45.5 245.0 900 80.0 260.0 360.0

Piacenza

(I) 20000 61.8 31.0 135.0 2500 127.0 44.1 260.0 1500 100.0 220.0 365.0

Sasso M. (I) 13000 61.9 30.8 135.0 1600 136.4 49.5 260.0 800 110.0 250.0 375.0


66

Table A.5. Daily flow and total weights of commercial vehicles

Flow Pmean [kN]

σ [kN]

Pmax [kN]

Brohltal (D) 4793 245.8 127.3 650.0

Garonor (F) 1982 2570 189.8 107.5 550.0

Garonor (F) 1984 3686 186.5 118.0 560.0

Auxerre (F) slow lane 2630 326.7 144.9 630.0

Auxerre (F) fast lane 153 277.2 163.6 670.0

Fiano R. (I) 4000 204.5 130.3 590.0

Piacenza (I) 5000 235.2 140.0 630.0

Sasso M. (I) 3500 224.9 149.0 620.0

- on the contrary, daily maxima are much less sensitive to traffic composition and

they vary between 130 and 210 kN for single axles, between 240 and 340 kN for

two axles in tandem, between 220 and 390 kN for three axles in tridem, and

between 400 and 650 kN for the total lorry weight;

- daily maxima of axle-loads and of total weight of the vehicle largely exceed the

legal values;

- in consequence of industrial choices of lorry manufacturers, vehicle geometries

have remained practically unchanged since the 1980’s: the inter-axle distance

distribution strongly results trimodal: the first mode, a little scattered, is located

around 1.30 m, corresponding to the usual axle’s spacing for tandem and tridem

arrangements of axles, the second mode, also characterized by low scattering, is

located around 3.20 m, a typical value for tractors of articulated lorries, while the

third one, located around 5.40 m, is much more dispersed;

- long distance continental Europe traffic data are sufficiently homogeneous;

- the heavy traffic composition evolved in a very straightforward way during the

1980’s: the percentage of articulated lorries stepped up despite a strong reduction

in the less commercially profitable trailer trucks, in conjunction with a contraction

of the number of single lorries, whose use is increasingly limited to local routes;

- in consequence of a better and more rational management of the lorry fleets, the

number of empty lorry passages has been strongly reduced and often limited to the

sole tractor unit in case of articulated lorries, , so raising the mean vehicle loads;

- long distance traffics are much more aggressive than local traffics;

- generally lorry flows tend to increase, even if the absolute maximum flow was

recorded in 1980 in Germany on the Limburger Bahn (8600 lorries per day on the

slow lane).

On the basis of the above mentioned considerations, the studies for calibration of EN

1991-2 load models for road bridges were based on the traffic recorded in Auxerre (France),

on the motorway A6 Paris-Lyon.

The Auxerre traffic is very severe and summarizes effectively the main characteristics

of the long distance European traffic, especially in terms of composition.

Other traffic data have been used only for checking the reliability of the results

obtained with Auxerre data.

The most relevant parameters of the slow lane Auxerre traffic are summarized in

figures A.1÷A.6. More precisely, in figures A.1, A.2 and A.3 are shown the histograms of


67

vehicle speeds, inter-vehicle distances and axle loads, respectively, referring to the total

vehicle’s flow (lorries plus cars), while in figures A.4, A.5 and A.6 are reported the analogous

histograms referring only to the lorry flow.

The statistical analyses allow to conclude that speed and length of vehicles are poorly

correlated and, from the probabilistic point of view, practically independent on the axle-loads

and on the total weight of the vehicles.

It must be stressed, finally, that European traffics exist which are more aggressive than

the Auxerre traffic, like the one recorded in Paris on the Boulevard Périférique. Such traffics,

nevertheless, are not very significant, since they depend on local situations and are hard to

generalize.

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

6 11 16 21 26 31 36 41 46 51

speed - [m/s]

Slow lane

Auxerre (F)

Figure A.1. Histogram of the vehicle speed frequency – Auxerre – total flow

0.00

0.10

0.20

0.30

0.40

1.36 102 5.07 10

3 1.00 10

4 1.50 10

4 2.00 10

4

Inter-vehicle distance - [m]

Slow lane

Auxerre (F)

Figure A.2. Histogram of the inter-vehicle distance frequency – Auxerre – total flow


68

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

12 25 38 51 64 77 90 103 116 129 142 155 168 181

Axle load - [kN]

Slow lane

Auxerre (F)

Figure A.3. Histogram of the axle load frequency – Auxerre – total flow

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

6 11 16 21 26 31 36 41 46 51

speed - [m/s]

Slow lane

Auxerre (F)

Figure A.4. Histogram of the vehicle speed frequency – Auxerre – lorries

A.4 Traffic scenarios

The evaluation of the reference values of the real traffic effects induced on the bridge

by the recorded traffic is not trivial.

Traffic records generally refer to normal flowing situations; they are often inadequate

to represent the most severe situations, which can happen in disturbed traffic scenarios. For

this reason, in order to consider extreme traffic situations as well, traffic data have been

opportunely manipulated, considering deterministic traffic scenarios being representative of

some relevant real situation [6], [7].

Concerning the single lane, four different types of traffic models have been developed

as follow: flowing, slowed down, and congested with/or without cars.


69

0.00

0.05

0.10

0.15

0.20

0.25

0.30

8.9 101 5.83 103 1.16 104

Intervehicle -distance - [m]

Slow lane

Auxerre (F)

Figure A.5. Histogram of the inter-vehicle distance frequency – Auxerre – lorries

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

10 37 64 91 118 145 172

Axle load - [kN]

Slow lane

Auxerre (F)

Figure A.6. Histogram of the axle load frequency – Auxerre – lorries

The flowing traffic is represented by the traffic as recorded. Flowing traffic, to which a

suitable dynamic coefficient must be associated, is particularly important for bridges spanning

up to 30 to 40 m, when characteristic values are sought. If frequent values are wanted, flowing

traffic is relevant in a much wider span range.

Slowed down traffic is significant when infrequent loads are sought. It can be easily

obtained considering the vehicles in the recorded order and reducing the distance among

adjacent axles of two consecutive vehicles to a suitable minimum value to simulate vehicle

convoys in braking phase. The minimum distance can be generally set to 20 m.

The congested traffic, which is relevant when the bridge span is greater than 50 m, can

be finally extracted from the recorded traffic reducing to 5 m the distance between the

adjacent axles of two consecutive vehicles, so reproducing a traffic column in slow (stop and

go) motion.


70

The traffic scenarios are particularly influenced by the driver behaviours, therefore,

among the congested traffic configurations, it is particularly meaningful that one

characterized by the presence on the slow lane of lorries only. In fact, when the traffic slows

down, the drivers of lighter and faster vehicles tend to change lane to overtake heaviest

vehicles.

This effect is very well represented in classical photos, like that reported by

Tschemmenerg & al. [8], relative to traffic jams on the Europa bridge (figure A.7). Obviously,

the congested traffic without cars can be simply obtained disregarding light vehicles, below

35 kN.

Figure A.7. Traffic jam on the Europa Bridge (from [8])

In defining the target values of traffic effects, extreme situations characterised by

flowing or jammed traffic on one or several lanes have been modelled considering several

deterministic traffic scenarios, as synthesized below, considering Auxerre traffic.

Flowing multilane traffics were obtained considering the following effects:

- for the most loaded lane, the first lane, the extrapolated effect induced by the slow

lane traffic as recorded;

- for the second lane, the daily maximum effect (not extrapolated), induced by the

slow lane traffic as recorded;

- for the third lane, the mean daily effect induced by the slow lane traffic as

recorded;


71

- for the fourth lane, the mean daily effect induced by the fast lane traffic as

recorded.

Jammed traffic scenarios took into account:

- for the most loaded lane, the first lane, the extrapolated effect induced by the

congested traffic without cars, deduced from the slow lane traffic;

- for the second lane, the daily maximum effect induced by the congested traffic

with cars, deduced from the slow lane traffic;

- for the third lane, the daily maximum effect induced by the slow lane traffic as

recorded;

- for the fourth lane, the mean daily effect induced by the slow lane traffic as

recorded.

Target values have then been evaluated referring to a considerable number of bridge

spans and influence surfaces. In particular, nine cylindrical influence surfaces have been

considered for simply supported as well as continuous bridges, spanning between 5 and 200

m.

A.5 Extrapolation methods

As mentioned above, the choice of the main load model and its calibration requires the

preliminary knowledge of the reference values, which are the relevant values of the effects -

characteristic, infrequent, frequent, and quasi-permanent - to be reproduced through the load

model itself.

Obviously, even considering deterministic traffic scenarios, the methodology to

evaluate the reference values cannot be taken for granted, so that suitable numerical

procedures, based on appropriate extrapolation methods of the histograms of the traffic

induced effects must be set-up.

The relationship between the return period and the distribution fractile can be easily

determined, assuming a uniform flow of lorries on the bridge. Under this hypothesis the

distance amongst two vehicles can be considered as equivalent to unit time interval, so that

the vehicles are described by a stationary time series X1, X2.…, Xi,…,Xn, being Xi the weight

of the i-th vehicle entering the bridge at time i.

If the weights Xi are independent and distributed according to the same cumulative

distribution function F(x), the return period Rx of the x value of Xi, which is defined as

[ ]xx NER = , where { }xXxXxXxXnN nnx ≥<<<= − ,,....,,|inf 121 , (A.1)

can be derived from

[ ] 1)(1

−−= xFRx . (A.2)

If the time series is replaced by a stationary stochastic process {Xt, t>0}, then

[ ]xx TER = , where { }tuxXxXtT utx <∀<∧≥= , |inf . (A.3)

If YN is the maximum value of Xi and N is the total number of vehicles crossing the

bridge during Rx, then it is

{ }NiXY iN ≤<= 0,max . (A.4)


72

As the Xi are independent and identically distributed, the cumulative distribution

function F[YN] of YN itself is given by

[ ] ( )[ ]n

N xFYF = . (A.5)

In conclusion, said yp the upper p-fractile of YN, p is

( )pyFp −= 1 , (A.6)

and, for N→∞ and T→∞, R results

( ) ( ) p

T

p

TyRR p ≅

−−==

1ln, where 10 <<< p . (A.7)

The expression (A.7), which does not depends on yp and on the distribution of X,

associates the return period and the fractile. For example, when the design life is 50 year, the

5% fractile (p=0.05) matches the value having a return period R=974.78≈1000 years.

In general, to evaluate the extreme values of the effects induced by the traffic, three

different methods of extrapolation have been employed in the framework of EN 1991-2

works.

The methods are based on the half-normal distribution, on the Gumbel distribution and

the Monte Carlo method, respectively. It is important to stress, however, that the characteristic

values of real traffic effects resulted practically independent on the extrapolation method.

A.5.1 Half-normal distribution extrapolation method

In the half-normal distribution extrapolation method the upper tail of the extreme

values distribution of the stochastic variable X is approximated by a normal distribution,

through an opportune choice of two parameters of the normal distribution.

The value xR, having a return period R, is given then by

RR zxx ⋅+= σ0 , (A.8)

where x0 is the last mode of the distribution and zR is the upper p-fractile of the standardized

normal variable Z,

( ) 12

−⋅= Np , (A.9)

being N the total number of lorries during the reference period R.

A.5.2 Gumbel distribution extrapolation method

Under hypotheses similar to those illustrated in the previous paragraph, the extreme

values distribution can be represented through a two parameters Gumbel (type I) distribution.

The parameters u and α’, which represent, respectively, the mode and the scattering of

the distribution, can be derived from the extreme values histogram as

σ⋅−= 45.0mu , ( ) 17797.0'

−⋅= σα , (A.10)

where m and σ are the mean value and the standard deviation of the histogram itself.

The value xR is then

'α⋅+= yuxR , (A.11)

being

( )[ ]11lnln

−−−−= Ry , (A.12)


73

the reduced variable of the Gumbel distribution.

An example of application of this extrapolation method on a Gumbel chart for a

generic effect T(y) is synthetically illustrated in figure A.8.

Gumbel chart

150

160

170

180

190

200

-2 0 2 4 6 8

Reduced variable y

Effect

T(y)

Figure A.8. Example of data extrapolation on a Gumbel chart

A.5.3 Monte Carlo extrapolation method

Numerical extrapolation procedures based on the Monte Carlo method make use of

automatic generations of a suitable set of extreme traffic situations, derived from the recorded

traffic data, in such a way that a suitable population of extreme values is obtained. This

population is elaborated in turn with an appropriate extrapolation method.

The population can be produced in several ways.

The simplest and intuitive procedure consists in the repeated application of the Monte

Carlo method.

The vehicles crossing the bridge are randomly selected amongst a complete set of

standard vehicles, representing the most common real lorries. Lorry silhouettes, axle loads,

axle spacings and the inter-vehicle distances are generated according to the relevant statistic

parameters of the recorded traffic.

An alternative procedure, more complicated but also much more effective, has been

proposed and adopted for calibrating the reference values of real traffic effects [7]. In this

methodology the crude Monte Carlo method is employed to obtain representative statistics of

the effects, which are the input data for the calculation of the statistical parameters of the

Gumbel distribution.

This latter method allowed also to underline that reference values are poorly

dependent upon the traffic jam frequency, at least for the most loaded lane [7].

A.6 Definition of dynamic magnification factors

In addition to the extrapolated static effects, the target values evaluation requires also

specific knowledge about the dynamic effects, due to vehicle-bridge interactions, to be

considered in calibration studies [9].


74

A.6.1 The inherent impact factors

Since the recorded traffic data refer to flowing traffic, they contain some dynamic

effects and they must be corrected through the so-called inherent impact factor, which is

intrinsic in the measurements and can be evaluated simulating numerically the weighing in

motion measurements.

For the purposes EN 1991-2, the Auxerre weighing in motion device has been

simulated considering the lorries, represented by a sequence of axles with shock-absorbers

having suitable dynamic characteristics, running on good roughness pavement resting on rigid

foundation.

In this way it was stated that the characteristic values, which are relevant for the

ultimate limit states, are affected by an inherent impact factor ϕin=1.10. When serviceability

limit states and fatigue are considered, instead, and the range between the 10% and the 90%

fractiles is taken into account, static and dynamic effects practically coincide and ϕin=1.00.

A.6.2 The impact factor

The impact factor depends on several parameters, like type, static scheme, span,

natural frequency and damping coefficient of the bridge, dynamic characteristics and speed of

the lorries, roughness of the road pavement and so on.

Generally, it results greater when the natural frequency of the bridge is close to the

natural frequencies of axles (10÷12 Hz) and lorries (1÷2 Hz).

In the framework of EN 1991-2 pre-normative studies, in order to determine global

local impact factors, a number of numerical simulations have been performed considering

several bridge schemes with varying traffic scenarios.

Concerning global effects, medium or good road pavement roughness has been

considered in turn. Local dynamic effects have been studied taking into account the presence

of a stepped irregularity, 30 mm height and 500 mm wide, simulating a road surface

discontinuity, like that caused by a damaged expansion joint, a pothole or an ice sheet.

The result of each numerical simulation was a time history of the considered effect,

like the one reported in figure A.9, from which the so-called physical impact factor ϕ can be

derived. Physical impact factor is the ratio between the maximum dynamic response and the

maximum static response of the bridge

st

dyn

max

max=ϕ . (A.13)

The physical impact factor refers to a well precise load configuration and it depends

on such a quantity of parameters that cannot to be directly employed for load model

calibration. Besides, heaviest vehicles, which mainly influence the extreme values of the

dynamic distribution of traffic effects, are generally slow and are characterised by small value

of ϕ factors.

For calibration purposes, two alternative approaches can be adopted:

- the first approach takes into account dynamic effects referring directly to the

dynamic effects distribution;

- an alternative method takes into account the static effect distribution multiplied by

a suitable calibration value of the impact factor, ϕcal. ϕcal can be defined as the

ratio between the dynamic value Edyn(p-fractile) and the static value Est(p-fractile)

corresponding to the same assigned p-fractile

)(

)(

fractilepst

fractilepdyn

calE

E

−

−=ϕ . (A.14)


75

Static oscillogram

Effect

Dynamic increment

N 0

Dynamic oscillogram

t

t

Figure A.9. Definition of the physical impact factor ϕϕϕϕ

Obviously, due to its conventional nature, ϕcal doesn't have a precise physical

meaning; in fact the static and dynamic x-fractiles don’t correspond to the same load

configuration.

The characteristic values of the calibration impact factors ϕcal, derived from Auxerre

traffic and employed in EN1991-2, are synthesized in figure A.10, depending on the span

length L.

The target dynamic values Edyn(x-fractile) can be finally evaluated through the expression

)()( fractilexst

in

localcalfractilexdyn EE −− ⋅

⋅=

ϕϕϕ

. (A.15)

where ϕlocal represents the local impact factor, when relevant.

Local impact factor ϕlocal takes into account concentrated irregularities of the roadway

surface.


76

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0 100 200

L [m]

φcal

Bending 1lane

Bending 2lanes

Bending 4lanes

Shear

Figure A.10. Calibration value of the impact factors ϕϕϕϕcal (EN 1991-2).


Traffic load models for road bridges of EN 1991-2 have been defined and calibrated

step by step balancing demand for accuracy and demands for ease of use.

Preliminary calibrations highlighted that load models best fitting the target values

should consist of concentrated loads and distributed loads:

- at least two concentrated loads should be considered in each relevant lane;

- Introduction of more than two concentrated loads doesn’t affect the precision of

the results.

- the intensity of the uniformly distributed loads result slowly decreasing functions

of the loaded length L.

The preliminary outcome has been successively modified to simplify the structure and

the application rules of the load model, mainly to eliminate any reason for ambiguity, finally

arriving to a load model characterised by:

- load values independent from the loaded length;

- dynamic effects include in load values;

- coexistence of concentrated and distributed loads on the same loaded area;

- aptitude to evaluate local and global, even simultaneous, effects;

- width of the notional lane equal to 3.0 m.

For the sake of model coherence, it has been established that, when relevant, the entire

carriageway width can be loaded, i.e. not only the part occupied by the physical lanes, but

also that one remaining.

In order to reproduce the real traffic effects in secondary elements, characterized by

influence surfaces with very small base length, it has been also introduced a local load model,

constituted by a single axle, which should be considered alone on the bridge.

Once opportunely calibrated, the so defined load model constitutes the load model of

EN 1991 - 2, which is illustrated more precisely in §2 of chapter 3.


77

Besides characteristic loads, having a probability of about 2% to be exceed in 50 years

design working life, i.e. about 1000 years return period, other relevant values of real traffic

effects exist, like infrequent, frequent and quasi-permanent values, which are particularly

relevant for SLS assessments.

Infrequent and frequent can be identified by one year or one week return period,

respectively. Quasi-permanent values result generally negligible and can be set zero, except

for particular cases, like, for example, bridges in the urban zone.

Frequent and infrequent values of traffic effects can be determined resorting to

methods substantially analogous to those used for the evaluation of characteristic values.

Their detailed illustration is omitted here, stressing only some relevant conclusion, on which

the relevant parts of EN 1991-2 are based:

- characteristic values of traffic effects increase slowly with the return period, in fact

- taking into account a medium roadway roughness, infrequent values of traffic

effects are about 90% of the corresponding characteristic values;

- taking into account a good roadway roughness, infrequent values reduce a little

and become about 80% of the corresponding characteristic values;

- frequent values of traffic effects are 70%÷80% of the corresponding characteristic

values;

- since the frequent values of traffic effects depend substantially on flowing traffic,

as the span increase frequent values tend to precise lower limits, which are

approximately 40%÷50% of the corresponding characteristic values.


78

Chapter 4: Fatigue loads due to traffic

79

CHAPTER 4: FATIGUE LOADS DUE TO TRAFFIC

Pietro Croce1


Summary

Fatigue performance of bridges is becoming more and more important due to the

growth of traffic flows, the increase of mean weight of the Heavy Good Vehicles, the

advances in conception and plan of bridges, the refinement of stress analysis techniques and

the deeper knowledge of the mechanical properties of the materials. In many cases fatigue

assessments strongly influence the design of new bridges and the assessment of existing

bridges. Fatigue traffic load models of EN1991-2 [1] for road and railway bridges are

illustrated and their background is discussed.

1 INTRODUCTION

In the last years the advances in conception and designing of bridges, the refinement

of stress analysis techniques and the deeper knowledge of the mechanical properties of the

materials have determined a relevant improvement of the bridge design as well as of the

bridge performances. At the same time, as the traffic flows and the mean weight of the Heavy

Good Vehicles have increased considerably, the fatigue resistance demand of modern bridges

have so significantly risen, that in many cases fatigue assessments strongly influence the

design.

On the base of the aforesaid considerations, in modern bridge codes sophisticated

fatigue load models should be given aiming to reproduce as well as possible the fatigue

induced by real traffic.

In the following fatigue traffic load models of EN1991-2 for road and railway bridges

are illustrated and their background is discussed.

The knowledge of the actual road traffic is affected by high uncertainties, on the

contrary, railway traffic is not only better known, but can be managed much more easily. For

this reason the calibration of fatigue traffic models for road bridges, mainly based on the real

traffic recorded in 1986 in Auxerre (F) on the motorway Paris- Lyon, is discussed in much

more detail.

2 DEVELOPMENT OF THE FATIGUE ROAD LOAD MODELS OF EN 1991-2

In the following the pre-normative background studies which have been carried out in

the framework of EN 1991-2 to define fatigue loads models for road traffic is discussed,

together with the main features of the models themselves.

2.1 Modelling of fatigue loads

As known, the ISO definition states that fatigue is the progressive, localised and

permanent structural change occurring in a material subjected to conditions that produce


80

fluctuating stresses and strains at some point or points and that may culminate in cracks or

complete fracture after a sufficient number of fluctuations.

In engineering structures, fatigue is induced by actions and loads varying with time

and/or space and/or by random vibrations. Thus fatigue can be originated by natural events,

like waves, wind and so on, or by loads deriving from the normal service of the structure

itself.

Among the civil structures exposed to fatigue, bridges occupy a prominent position, as

they are subjected to the fluctuating action of lorries or trains crossing the bridges themselves.

The assignment of appropriate fatigue load models is therefore a key topic in contemporary

bridge design codes of practice.

In principle, modelling of fatigue loads asks for the complete knowledge of the so-

called load spectrum, expressing the load variations or the number of recurrences of each load

level during the design working life of the structure. Load spectrum is generally given in

terms of an appropriate function, graph, histogram or table.

The load spectrum is often deduced from recorded data, referring to relatively short

time intervals. In this case, additional problems must be faced regarding the statistical

processing, the reliability over longer periods of the available data and the future trends of

traffic.

Whenever the real load spectrum results so complicated that cannot be directly used

for fatigue checks, as it happens for bridge, it is replaced by some conventional load

spectrum, aimed to reproduce the fatigue induced by the real one.

The evaluation of conventional load spectra is particularly problematic, because it

requires to consider the actions also from the resistance point of view. In fact, fatigue depends

on the nature of the varying actions and loads, and additionally on structural material details,

through the shape and the properties of the relevant S-N curves.

Problems become even tougher when details exhibit endurance (fatigue) limit. As

fatigue limits under constant amplitude represents a threshold value for the damaging stress

range, it needs to distinguish between equivalent load spectra, aiming to reproduce the actual

fatigue damage, and frequent load spectra, aiming to reproducing the maximum load range to

be taken into account for fatigue assessments.

Since fatigue verifications are performed in different ways, depending on the necessity

to assess fatigue damage or boundless fatigue life, the distinction between equivalent and

frequent spectra appears quite obvious.

Moreover, the powerful methods of the stochastic process theory, often used in

defining fatigue load spectra in other engineering structures, cannot be applied to bridges, as

road traffic loads induce broad band stress histories. All that implies that the link between the

action and the effect cannot be expressed by simple formulae, while further difficulties arise

when vehicle interactions, whether due to simultaneity or not, become significant.

Nevertheless, provided that vehicle interaction problems can be solved in some way,

as shown in the Appendix A to the present chapter, it is intuitive enough to think that fatigue

load spectra for bridges are composed by suitable sets of standardized lorries, where each

lorry is identified by its own relevant properties, i.e. relative frequency, number of axles, axle

loads, axle’s spacing, deduced processing appropriate traffic records.

At this stage, it appears quite evident that definition of load spectra for bridges

requires careful consideration of fatigue assessment methodology, to assure that assessments

based on conventional spectra or on real spectra lead to the same results.

2.1.1 Fatigue verification methods

The preliminary explanation of fatigue assessment methodology based on

conventional load spectra is a crucial question in studying fatigue load models.


81

It can be easily recognised that fatigue verification methods goes along with a well-

defined procedure, characterised by the following steps

1 assignment of fatigue load spectra, discriminating, if necessary, equivalent ones

from frequent ones;

2 detection and classification of structural details most vulnerable to fatigue

cracking and selection of the appropriate S-N curves;

3 choice of the pertinent partial factors γM;

4 evaluation, for each detail, of the appropriate influence surface.

At this stage, the assessment methodology splits up in two different branches,

accordingly as fatigue verification is devoted to compute fatigue damage or to assess

boundless fatigue life.

Damage computation procedure

5.a calculation of the design stress history σ=σ(t) produced in the detail by the

equivalent load spectrum travelling over the influence surface;

6.a analysis of the stress history by means of a suitable cycle counting method, like

the reservoir method or the rainflow method, to obtain the stress spectrum, where

the number of occurrences of each stress range in the design working life is

associated with the stress range itself;

7.a computation of the cumulative damage D using the Palmgren-Miner rule: if D≤1

the fatigue check is satisfied, otherwise, it is necessary to raise the fatigue strength

of the detail. Fatigue resistance can be enhanced both reducing the stress range,

i.e. enlarging the dimensions, or increasing fatigue category, i.e. adopting more

refined workmanship or details.

Boundless fatigue life assessment

5.b calculation of the design stress history σ=σ(t) produced in the detail by the

frequent load spectrum transiting over the influence surface;

6.b computation of the maximum stress range ∆σmax=σmax-σmin, being σmax and σmin,

respectively, the absolute maximum and the absolute minimum of the stress

history;

7.b boundless fatigue life assessment. If the verification is not satisfied, it is possible

to improve fatigue resistance using the provisions described in 7.a, or to attempt

to go through fatigue damage computation.

Obviously, in bridges exposed to high-density traffic, concrete slabs and orthotropic

steel deck details are subject to such a huge number of stress cycles, that boundless fatigue

life assessment using frequent load spectra becomes quite obligatory.

2.1.2 Reference traffic measurements

Also the fatigue load models of Eurocode 1 have been defined and calibrated on the

basis of the two large traffic measurement campaigns carried out in several European

countries in years 1977 to 1982 and 1984 to 1988, which have been discussed in Annex A to

chapter 3.

Unlike static loads, which depend only on the upper tail of the traffic load distribution,

fatigue loads are influenced by the entire distribution.


82

For this reason, fatigue models have been refined, supplementing the main calibration,

based on Auxerre traffic data, with supplementary studies, based on different traffic data, in

order to enlarge their field of application,.

These supplementary calibrations regarded not only motorway traffics - Brothal (D),

Piacenza, Fiano Romano, Sasso Marconi (I) – but also local traffic on secondary roads (Epone

(F)). In effect, long distance traffics, typical of motorways and main roads, are characterised

by high percentage of heavy vehicles, while local traffics, typical of secondary roads, are

lighter and composed mostly by two axle lorries. Besides, it should be considered that, as

confirmed by recent traffic data, European traffics show a trend characterised by

- marked increase of the number of articulated lorries vis-à-vis the simultaneous

reduction of the number lorries with trailer;

- reduction of the number of three axle lorries for the benefit of two axle lorries;

- increase of the average load per lorry.

2.2 The fatigue load models of EN 1991-2

The calibration method, the underlying philosophy, the methodological approach and

the main features of the fatigue models of Eurocode 1 are summarised in the following.

2.2.1 Calibration method

Fatigue load models have been calibrated referring to reference influence surfaces

relative to simply supported and continuous bridges, spanning in the range 3 m to 200 m.

In agreement with the general fatigue assessment procedures, calibration has been set-

up according to the following scheme,

- choice of the most significant European traffic data;

- selection of appropriate S-N curves;

- evaluation of the stress histories in reference bridges;

- cycle counting and stress spectra computation;

- first identification of fatigue models;

- definition of standardised lorry silhouettes;

- calibration of frequent load models, best fitting the maximum stress range ∆σmax

induced by real traffic;

- calibration of equivalent load models, best fitting the fatigue damage D induced by

the traffic.

2.2.2 Reference S-N curves

Reference S-N curves pertain to steel details, characterised by endurance limit.

As known, in the logarithmic S-N chart these curves are represented by a bilinear

curve, characterised by a sloping branch of constant slope, m=3, (figure 1), or by a trilinear

curve, characterised by two sloping branches, m=3 and m=5, (figure 2), according as

boundless fatigue life or fatigue damage is to be assessed.

Since steel details exhibit fatigue limit ∆σD, two cases can be envisaged, according to

the maximum stress range ∆σmax of the real stress history is higher than ∆σD or not.

To be significant for fatigue, ∆σmax must be exceeded several times during the design

working life of the bridge and its definition is not trivial. Two different approaches have been

proposed, leading to similar results: in the former, ∆σmax is defined as the stress range such

that the 99% of the total fatigue damage results from all stress ranges below ∆σmax; in the

latter ∆σmax is the stress range exceeded approximately 5⋅104 times during the bridge life.


83

O N

S

m=3

? σ 5⋅106

∆σD

Figure 1. Bilinear S-N curve

O N

S

m=3

?σ 5⋅106

D m=5

L

108

∆σ

∆σ

Figure 2. Trilinear S-N curve

This last definition implies that the return period of ∆σmax is about half a day, giving

so direct explanation of frequent load spectrum denomination.

In EN 1991-2 studies, to derive equivalent load spectra independently from the fatigue

classification of details, cumulative damage has been computed referring generally to

simplified S-N curves with unique slope, in turn m=3 (figure 3) or m=5 (figure 4). S-N curves

with double slope (figure 5) and without endurance limit have been used for some additional

calculations.

Some comparisons show that load spectra obtained using the simplified curve m=5 are

free from significant errors and reproduce generally well the actual fatigue damage.

O N

S

m=3

Figure 3. Single slope S-N curve (m=3)

O N

S

m=5

Figure 4. Single slope S-N curve (m=5)

O N

S

m=3

? σ 5⋅106

D m=5

∆σ

Figure 5. Double slope m=3- m=5 S-N curve without endurance limit


84

2.2.3 Fatigue load models

From the above-mentioned considerations, it derives that at least two conventional

fatigue load models must be considered: the one for boundless fatigue life assessments, the

other for fatigue damage calculations. Besides, since an adequate fitting of the effects induced

by the real traffic requires very sophisticated load models, whose application is often difficult,

the introduction of simplified and safe-sided models, to be used when sophisticated checks are

unnecessary, seems very opportune.

For this reason in EN 1991-2 two fatigue load models are foreseen for each kind of

fatigue verification: the former is essential, safe-sided and easy to use, the latter is more

refined and accurate, but also more complicated. In conclusion, four conventional models are

given:

- models 1 and 2 for boundless fatigue checks;

- models 3 and 4 for damage calculations.

Fatigue load model 1 is extremely simple and generally very safe-sided. It directly

derives from the main load model used for assessing static resistance, where the load values

are simply reduced to the frequent ones (figure 6.a), multiplying the tandem axle loads Qik by

0.7 and the weight density of the uniformly distributed loads qik by 0.3.

Obviously, for local verifications, the fatigue load model n. 1 is constituted by the

isolated concentrated axle weighing Q=280 kN (frequent value - figure. 6.b).

Lane n. 1Q =210 kN

q =2.7 kN/m

QikQik

qik

Lane n. 2

0.5

2.0

0.5

Q =140 kN2k

q =0.75 kN/m2k2

Lane n. 3Q =70 kN3k

q =0.75 kN/m3k2

Remaining area q =0.75 kN/mrk2

1k

1k

0.5

2.0

0.5

0.5

2.0

0.5

w

2

Figure 6.a. Fatigue load model n. 1

Figure 6.b. Fatigue load

model n. 1 for local

verifications

The verification consists of checking that the maximum stress range ∆σmax induced by

the model is smaller of the fatigue limit ∆σD. The application rules for the load model n. 1

agree exactly with those given for the main load model, so that the absolute minimum and

maximum stresses correspond as rule to different load configurations. The model allows

making “coarse” verifications also in multi-lane configurations, generally resulting extremely

safe-sided.

The simplified fatigue model n. 3, conceived for damage computation, is constituted

by a symmetrical conventional four axle vehicle, also said fatigue vehicle (figure 7). The

equivalent load of each axle is 120 kN. This model is accurate enough for spans bigger than

10 m, while for smaller spans it results safe-sided.


85

Figure7. Fatigue load model n. 3

Fatigue load models n. 2 and n. 4 are the most refined one and they are load spectra

constituted by five standardised vehicles, representative of the most common European

lorries.

Fatigue load model n. 2, which is a set of lorries with frequent values of axle loads,

and fatigue model n. 4, which is a set of lorries with equivalent values of the axle loads, are

illustrated in tables 1 and 2, respectively. They allow to perform very precise and

sophisticated verifications, provided that the interactions amongst vehicles simultaneously

crossing the bridge are negligible or opportunely considered.

Table 1. Fatigue load model n. 2 – frequent set of lorries

LORRY

SILHOUETTE

Axle spacing

[m]

Frequent

axle loads

[kN]

Wheel type

(see table 3)

4.5 90

190

A

B

4.20

1.30

80

140

140

A

B

B

3.20

5,20

1.30

1.30

90

180

120

120

120

A

B

C

C

C

3.40

6.00

1.80

90

190

140

140

A

B

B

B

4.80

3.60

4.40

1.30

90

180

120

110

110

A

B

C

C

C


86

Table 2. Fatigue load model n. 4 – equivalent set of lorries

LORRY SILHOUETTE TRAFFIC TYPE

Long

distance

Medium

distance

Local

traffic

LORRY

Axle

spacing

[m]

Equivalent

Axle loads

[kN]

Lorry

percentage

Lorry

percentage

Lorry

percentage

4.5 70

130

20.0 40.0 80.0

4.20

1.30

70

120

120

5.0 10.0 5.0

3.20

5.20

1.30

1.30

70

150

90

90

90

50.0 30.0 5.0

3.40

6.00

1.80

70

140

90

90

15.0 15.0 5.0

4.80

3.60

4.40

1.30

70

130

90

80

80

10.0 5.0 5.0

The types of wheels pertaining to each standardised lorries of fatigue load models n. 2

and n. 4 are indicated in table 1, referring to table 3.

The number of lorries to be taken into account for damage assessments depends on the

traffic category: indicative values of Nobs, representing the number of lorries of year per slow

lane, are given in table 4. The additional traffic on the fast lane can be assumed to be 10% of

the slow lane traffic.

In fact, in EN 1991-2 a further general purpose fatigue model is anticipated too,

denominated fatigue model n. 5. This model is constituted by a sequence of consecutive axle

loads, directly derived from recorded traffic, duly supplemented to take into account vehicle

interactions, where relevant.

Fatigue model n. 5 is aimed to allow accurate fatigue verifications in particular

situations, like suspended or cable-stayed bridges, important existing bridges or bridges

carrying unusual traffics, whose relevance justifies ad hoc investigations [2].

2.2.4 Accuracy of fatigue load models

In the following, some significant results obtained using the fatigue load models are

compared with those pertaining to the reference traffic, allowing to highlight the accuracy and

the field of application of the each conventional model.


87

Table 3. Definition of wheels and axles of standard lorries

Wheel axle

type Geometrical definition

A

Longitu

din

al axis

of th

e b

ridge

2

1.78

0.220.3

2

0.22 0.3

2

B

Lon

gitud

inal axis

of

the

bridg

e

2 0.22

0.3

2

0.22

0.3

2

0.220.22

0.54 0.54

C

Lo

ngitu

din

al axis

of th

e b

rid

ge

2

1.73

0.270.3

2

0.27 0.3

2

Table 4. Indicative number of lorries expected per year and for a slow lane

Traffic categories Nobs per year and per slow lane

1 Roads and motorways with 2 or more lanes per direction

with high flow rates of lorries 2.0⋅10

6

2 Roads and motorways with medium flow rates of lorries 0.5⋅106

3 Main roads with low flow rates of lorries 0.125⋅106

4 Local roads with low flow rates of lorries 0.05⋅106

Essentially, the comparison concerns the four influence surfaces shown in figure 8, for

bridges span L varying between 3 m and 100 m. The influence surfaces pertain to bending

moment M0 at midspan of simply supported beams, bending moments M1 and M2 at midspan

and on the support, respectively, of two span continuous beams and bending moment M3 at

midspan of three span continuous beams.


88

Figure 8. Reference influence lines

In figures 9 to 14 the outcomes of different fatigue load models for the aforesaid

influence lines are compared with the real traffic (Auxerre traffic) effects, in function of the

span length L, considering only one notional lane.

For unlimited fatigue life assessments, the ratios between the maximum bending

moment ranges due to fatigue load model 1, ∆Mmax,LM1, and fatigue load model 2, ∆Mmax,LM2,

and the maximum bending moment ranges due to real traffic, ∆Mmax,real, are diagrammatically

reported in figures 9 and 10, respectively.

For fatigue damage assessments reference can be made to the equivalent bending

moment range corresponding to 2⋅106 cycles.

The ratios between the equivalent bending moment ranges due to fatigue load model

3, ∆Meq,LM3, and the equivalent bending moment ranges due to real traffic, ∆Meq,real, are shown

in figures 11 and 12, considering one slope S-N curves characterised by m=3 and m=5,

respectively. Analogous diagrams for equivalent bending moment ranges due to fatigue load

model 4, ∆Meq,LM4, are illustrated in figure 13, for m=3 S-N curve.

Figure 9. Accuracy of fatigue load model n. 1


89


Figure 11. Accuracy of fatigue load model n. 3 – m=3

Figure 12. Accuracy of fatigue load model n. 3 – m=5


90


Analysis of the diagrams shows that:

- as just said, fatigue load model n. 1 appears very safe-sided, especially for short

spans;

- load model n. 2 results much more reliable; values little below the actual ones are

obtained for M2 in the span range 20 to 50 m, but this depends on the particular

shape of the influence line;

- as expected, model n. 4 fits very good actual results for short influence lines;

- fatigue model n. 3 looks unsafe for M2 influence lines when spans are above 30 m,

in particular for higher m values. To solve the problem it has been proposed to

modify the model n. 3 taking into account an additional fatigue vehicle each time

that the influence surface exhibits two contiguous areas of the same sign. This

additional fatigue vehicle, having equivalent axle loads set to 40 kN, should run on

the same lane of the basic fatigue vehicle, 40 m away from it. The adoption of such

an additional vehicle should mitigate the error in computation of ∆M2,eq, as it

appears evident in figure 14, where damage calculations for M2 influence line and

for m=3 and m=9 S-N curves, considering additional fatigue vehicle.

Figure 14. Accuracy of improved fatigue load model n. 3 – 2 vehicles


91

2.3 The λλλλ-coefficient method

Besides the usual damage computations based on Palmgren-Miner rule, EN 1991-2

also foresees a conventional simplified fatigue assessment method, said λ-coefficient method,

based on λ damage equivalent factors, which are dependent on the material.

The method, derived originally for railway bridges, is based for road bridges on

fatigue model n. 3 (fatigue vehicle) and it is aimed to bring back fatigue verifications to

conventional resistance checks, comparing a conventional equivalent stress range, ∆σeq,

depending on appropriate λ-coefficients, with the detail category [3], [4].

The equivalent stress range ∆σeq is given by

pfatpfateq σϕλσϕλλλλσ ∆⋅⋅=∆⋅⋅⋅⋅⋅=∆ 4321 , (1)

where

- min,max, ppp σσσ −=∆ is the maximum stress range induced by fatigue model n. 3;

- λ1 is a coefficient depending on the shape and on the base length of the influence

surface, i.e. on the number of secondary cycles in the stress history;

- λ2 is a coefficient allowing to pass from reference traffic, used in fatigue model

calibration, to expected traffic;

- λ3 depends on the design life of the bridge;

- λ4 takes into account vehicle interactions amongst lorries simultaneously crossing

the bridge;

- ϕfat is the equivalent dynamic magnification factor for fatigue verifications.

λ1, represented in graphical or tabular form, is calculated in the calibration phase,

comparing the damage due to the fatigue vehicle with the damage produced by a single stress

cycle having the maximum stress range ∆σp.

If m is the slope of S-N curve, it is

m

mp

i

miin

1

1

∆

∆⋅= ∑

σσ

λ .

(2)

λ2 depends on the annual lorry flow and on traffic composition.

In general, said N1 and Qm1 the flow and the equivalent weight of the actual traffic,

m

i

i

i

m

ii

mn

Qn

Q∑

∑ ⋅=1 ,

(3)

and N0 and Q0 the flow and the equivalent vehicle weight of the reference traffic, it results

mm

N

N

Q

Qk

1

0

1

0

12

⋅⋅=λ .

(4)

In equation (4) k represents a conversion parameter, given by

1

0

mv

ef

Q

Q

D

Dk ⋅= ,

(5)


92

where Dv is the damage produced by N0 fatigue vehicles and Def is the damage produced by

N0 real lorries.

For Auxerre traffic it ensues 4800 =Q kN and 0N = 6102 ⋅ lorries per year.

λ3 is given by

m

RT

T=3λ ,

(6)

where TR is the reference design working life (TR=100 years) and T is the actual design

working life.

λ4, which, as said, takes into account vehicle interactions, can be expressed as

m

i i

m

combcomb

m

ii

N

N

N

N

N

NNl ∑ ∑

⋅+

⋅+=

1111

*

1

*1

14 ),(η

ηηηλ ,

(7)

where N1 is the lorry flow (number of the lorries) on the main lane, Ni the lorry flow on the i-

th lane, ηi the max ordinate of the influence surface corresponding to i-th lane, *

iN the

lonely, i.e. not interacting, lorry flow on the i-th lane, Ncomb the number of interacting lorries

and ηcomb the overall ordinate of the influence surface for the ”interacting” lanes, being the

second summation extended to all relevant combinations of lorries on several lanes.

An appropriate closed form expression for λ4 can be theoretically derived for two

simultaneously loaded lanes, as shown in the Appendix A to the present chapter.

The equivalent impact factor ϕfat, finally, is the ratio between the damage due to the

dynamic stress history and the damage due to the corresponding static stress history

( )( )m

m

statistati

m

dymidymi

fatn

n

∑∑

∆⋅

∆⋅=

,,

,,

σσ

ϕ .

(8)

In conclusion, said ∆σc the detail category, the fatigue assessment reduces to check

that expression

cpfateq σσϕλσ ∆≤∆⋅⋅=∆ (9)

is satisfied.

2.4 Partial factors γγγγM

The partial factors γf, regarding the action aspect, and γm, regarding the fatigue

resistance aspect, cover uncertainties in the evaluation of loads and stresses as well as fatigue

strength scattering.

According to the experience from steel structures, these partial factors affecting stress

ranges are generally combined in a unique factor mfM γγγ ⋅= . Beside the material, the

numerical values of γM depend on the possibility to detect and repair fatigue cracks and on the

consequences of fatigue failure and are given ion the relevant Eurocodes.

2.5 Effects of future traffic trends

In the next future, the traffic composition could be sensibly modified.

In fact, in order to improve the organization of the European transportation network,

the European Commission enacted the 96/53/EC Directive limiting the total mass of Heavy

Good Vehicles (HGV) to 44 t, but admitting, on a parity basis, the possibility to allow the


93

circulation of Long and Heavy Vehicles (LHV), characterised by mass up to 60 t and length

till 25 m. This possibility has been admitted in Germany and in northern Countries, in

particular Sweden, Finland, Denmark and The Netherlands. For this reason in these Countries

a significant increase of the number of LHVs in long distance traffic has been experienced.

Despite of their effectiveness in terms of decrease of pollutant emissions and cost

reduction, LHVs could result too much demanding for existing infrastructures, in particular

for bridges, so that their impact requires careful examination.

In order to evaluate the aptness of EN 1991-2 fatigue load models to cover also the

effects of LHVs, some additional studies has been performed on relevant bridge schemes and

spans comparing the Auxerre traffic effects with those induced by a more recent one,

containing a relevant number of LHVs, recorded with a WIM device at the Moerdijk site in

the Netherlands in April 2007. These studies are illustrated in the Annex A to the present

Guidebook.

3 FATIGUE LOAD MODELS FOR RAILWAY BRIDGES OF EN 1991-2

For fatigue damage assessments of railway bridges subjected to normal railway traffic

based on characteristic values of Load Model 71, including the dynamic factor Φ, EN 1991-2

assigns three different fatigue load spectra.

These load spectra refer to three different traffic mixes, usual traffic mix, traffic with 250

kN-axles mix or light traffic mix, depending on whether the structure carries mixed (usual) traffic,

predominantly heavy freight traffic or lightweight passenger traffic.

The above mentioned load spectra are based on an annual traffic tonnage of 25 Mt per

each track.

If not otherwise specified, fatigue damage should be assessed taking into account that:

- in structures carrying multiple tracks, fatigue loadings shall be applied to a maximum

of two tracks, which should be the most unfavourable;

- the structural design working life of the bridge is 100 years;

- when dynamic effects can be considered through static dynamic factors, static

dynamic factors should be determined according to the method illustrated in § 3.2;

- bridges requiring dynamic analysis could necessitate of additional investigations

and/or recommendations.

3.1 Train types for fatigue

As just said, three different load spectra are given on the basis of three traffic mixes,

standard, heavy and light, characterized by illustrated respectively in tables 5.a, 5.b and 5.5 and

in figures 15.a, 15.b and 15.b, where also train type are indicated.

The load spectra refer to adjustment factors α=1.0. When relevant, the axle loads should

be multiplied by the appropriate adjustment factor α.

3.2 Dynamic magnification factors for fatigue assessments

The static dynamic factors Φ2 and Φ3 are applied to load models LM 71, SW/0 and

SW/2.

Since the static dynamic factors are intended for static assessments of bridge members,

they are evaluated considering extreme loading cases that can occur in the design working life.

For this reason, they would be excessively onerous if applied to Real Trains included in

fatigue load spectra.


94

Table 5.a. Standard traffic mix with axles ≤225 kN

Train type Number of

trains/day

Mass of

train [t]

Traffic volume

[Mt/year]

1 12 663 2.90

2 12 530 2.32

3 5 940 1.72

4 5 510 0.93

5 7 2160 5.52

6 12 1431 6.27

7 8 1035 3.02

8 8 1035 2.27

Total 67 24.95

Table 5.b. Heavy traffic mix with 250 kN axles


trains/day

Mass of

train [t]

Traffic volume

[Mt/year]

5 6 2160 4.73

6 13 1431 6.79

11 16 1135 6.63

12 16 1135 6.63

Total 51 24.78

Table 5.c. Light traffic mix with axles ≤225 kN


trains/day

Mass of

train [t]

Traffic volume

[Mt/year]

1 10 663 2.4

2 5 530 1.0

5 2 2160 1.4

9 190 296 20.5

Total 207 25.3

In reality, in fatigue verifications only the equivalent dynamic effect over the assumed

100 years design working life needs to be considered, therefore the dynamic enhancement for

each Real Train can be reduced, for Maximum Permitted Vehicle Speeds up to 200 km/h, to

++ "2

1'

2

11 ϕϕ (10)

where

41'

KK

K

+−=ϕ and 100

2

56,0"

L

e−

=ϕ (11)


95

being v the Maximum Permitted Vehicle Speed in m/s, L the determinant length LΦ in m of

the structural member and 160

vK = for L≤20 m and

408,016,47 L

vK = for L>20 m.

3.0

4x20 t 13x(4x150 kN)

Train type 3 - High speed train - V=250 km/h - P=9400 kN - n=5/d

8.4

6

16

.5

4x20 t

3.0

8.4

6

3.0

4.4

5

2.5

4.9

5

3.0

2.5

2.5

4.4

5

3.0

4x170 kN 8x(2x170 kN)

Train type 4 - High speed train - V=250 km/h - P=5100 kN - n=5/d

11

.0

3.0

3.3

15

.7

3.0

3x170 kN

15

.7

3.0

3.0

11

.0

3.0

4x170 kN

3.3

3.0

3x170 kN

15

.7

3.0

3.0

2.1

6x225 kN

Train type 5 - Freight train - V=80 km/h - P=21600 kN - n=7/d

4.0

4.4

2.1

15x(6x225 kN)

2.1

2.1

1.8

5.0

5.7

1.8

1.8

1.8

5.7

1.8

1.8

1.8

1.8

2.1

6x225 kN


3.9

4.4

2.1

2.1

2.1

6.5

3.8

1.8

1.8

2x70 kN

A

2x70 kN

A

6.5

3.7

12

.8

4x225 kN

B A

6.5

3.7

1.8

1.8

8.0

4x225 kN

C

3.5

3.2

CABBBAAC+

CABCAACCB

2.2

6x225 kN

10x(4x225 kN)


3.0

1.8

1.8

3.2

11

.0

6.9

2.2

2.2

2.2

1.8

11

.0

1.8

2.2

6x225 kN

20x(2x225 kN)

Train type 8 - Freigth train - V=100 km/h - P=10350 kN - n=6/d

3.5

4.2

5.5

6.9

2.2

2.2

2.2

5.5

4.2

5.5

2.2

6x225 kN

12x(4x110 kN)

Train type 1 - Passenger train - V=200 km/h - P=6630 kN - n=12/d

3.2

2.6

2.6

3.6

11

.5

6.9

2.2

2.2

2.2

2.6

11

.5

2.6

3.3

4x225 kN 10x(4x110 kN)


2.9

2.5

2.5

6.7

5.0

3.3

16

.5

2.5

16

.5

2.5

Figure 15.a. Fatigue load spectrum for normal traffic mix


96

2.1

6x225 kN

4.0

4.4

2.1

15x(6x225 kN)

2.1

2.1

1.8 5.0

5.7

1.8

1.8

1.8 5.7

1.8

1.8

1.8

1.8

2.1

6x225 kN

3.9

4.4

2.1

2.1

2.1

6.5

3.8

1.8

1.8

2x70 kN

A

2x70 kN

A

6.5

3.7

12.8

4x225 kN

B A

6.5

3.7

1.8

1.8 8.0

4x225 kN

C

3.5

3.2

CABBBAAC+

CABCAACCB

2.2

6x225 kN

10x(4x250 kN)


3.0

1.8

1.8

3.2

11.0

6.9

2.2

2.2

2.2

1.8

11.0

1.8

2.2

6x225 kN

20x(2x250 kN)


3.5

4.2

5.5

6.9

2.2

2.2

2.2

5.5

4.2

5.5



Figure 15.b. Fatigue load spectrum for heavy traffic mix

2.2

6x225 kN

12x(4x110 kN)


3.2

2.6

2.6

3.6

11

.5

6.9

2.2

2.2

2.2

2.6

11

.5

2.6

3.3

4x225 kN 10x(4x110 kN)


2.9

2.5

2.5

6.7

5.0

3.3

16.5

2.5

16.5

2.5

2.1

6x225 kN

4.0

4.4

2.1

15x(6x225 kN)

2.1

2.1

1.8

5.0

5.7

1.8

1.8

1.8

5.7

1.8

1.8

1.8

1.8


2.5

4x130 kN

Train type 9 - Suburban train - V=120 km/h - P=2960 kN - n=190/d

14

.0

2.5

4.3

2.5

4x110 kN

11

.5

2.5

2.5

2x(4x130 kN)

14

.0

4.3

2.5

4.3

2.5

4x110 kN

11

.5

2.5

4.3

2.5

4x130 kN

14

.0

2.5

Figure 15.c. Fatigue load spectrum for light traffic mix


97

3.3 The λλλλ-factor design method

Also for railway bridges, the fatigue assessment can be simplified and reduced to an

equivalent stress range verification, determined according to the relevant Eurocodes (EN 1992-2,

EN 1993-2, EN 1994-2), on the basis of the so-called λ-method.

For example, for steel bridges, the safety verification shall be carried out by ensuring that

the following condition is satisfied:

Mf

cFf γ

σσλγ ∆≤∆Φ 712 (12)

where γFf is the partial factor for fatigue loading, usually set to 1.00, λ is the damage

equivalent factor for fatigue, which takes account the expected railway traffic on the bridge and

the span of the member, Φ2 is the dynamic factor, ∆σ71 is the maximum stress range due to the

Load Model 71 (and where required SW/0), ∆σC is the reference value of the fatigue strength,

i.e. the class of the detail, and γMf is the partial factor for fatigue strength.

Load model should be placed in the most unfavourable position for the element under

consideration, disregarding any adjustment factor.


Fatigue verifications are decisive for designing new road and railway bridges as well

for assessing existing ones.

In up to date structural codes fatigue loads are given through suitable load spectra,

deduced from real traffic data, recorded using weighing in motion devices.

In theory, load spectrum can be directly deduced from real traffic data, provided that

they are representative of the traffic concerning the bridge, during its design working life. In

practice, the management of real load spectrum is very complicated and it requires a huge

amount of calculations; therefore, its application is justified only for particularly important

bridges.

Usually, in structural codes fatigue loads are assigned through conventional load

spectra, which reproduce the fatigue effects induced by the real traffic. Since fatigue effects

depend not only on the actions but also on the material properties, through the appropriate S-N

curve, the definition and the use of conventional load spectra is not trivial.

Duly taking into account the theoretical differences that exist between equivalent load

spectra, intended to reproduce fatigue damage, and frequent load spectra, intended to

reproduce the maximum load range for fatigue assessments, in EN 1991-2 five load spectra

are assigned for road bridge assessments and three load spectra are assigned for railway

bridges assessments.

In addition to the usual damage computations, based on Palmgren-Miner rule, EN

1991-2 allows to adopt also a conventional simplified fatigue assessment method, based on λ

damage equivalent factors, which are dependent on the material. This method brings back

fatigue verification to conventional resistance check, where an appropriate equivalent stress

range, ∆σeq, is compared with the detail category.

In the present chapter, background information and main features of EN 1991-2 load

spectra have been illustrated, discussing their possibilities and their fields of application and

highlighting the results of pre-normative calibration studies.

When vehicle’s interactions are significant, EN 1991-2 fatigue load models cannot be

used, unless additional information is available. Vehicle’s interactions problems are tackled


98

from the theoretical point of view in Appendix A to the present chapter, where simplified

formulae are also given for the evaluation of the pertinent damage equivalent factor λ4.

Finally, in Annex A to the present Guidebook, the aptness of EN 1991-2 fatigue load

models to face the actual trends of road traffic and in particular the effects of Long and Heavy

Vehicles, allowed by the 96/53/EC Directive is discussed and additional studies are

illustrated.

5 REFERENCES

[1] EN1991-2, Eurocode 1: Actions on structures - Part 2: Traffic loads on bridges.

Brussels: CEN, 2003

[2] Caramelli, S. & Croce, P., Messina bridge: testing assisted deck fatigue design,

International Institute of Welding Conference on Welded Constructions: Achievements

and perspectives for the new millennium. Florence, 2000.

[3] Croce, P., Background to Fatigue Load Models for Eurocode 1: Part 2 Traffic Loads.

Progress in Structural Engineering and Materials 1(3:4): 250-263, 2001

[4] Bruls A et al., Part 3: Traffic loads on bridges. Calibration of road load models for road

bridges. Proceedings of IABSE Colloquium on Basis of Design and Actions on Structures.

Background and Application of Eurocode 1. Delft, 1996.

[5] Ventsel E.S., Probability theory. Moscow: MIR, 1983.


99

Appendix A to Chapter 4 – Vehicle interactions and fatigue assessments


When vehicle interaction is relevant, stress histories cannot be determined using

conventional fatigue models or recorded traffic data, unless appropriate additional information

are available.

The achievement of general theoretical results in modelling vehicle interactions could

sensibly enlarge the field of application of the fatigue load models and it represents a main

objective in the improvement of EN 1991-2.

A.2 Modelling of vehicle interactions

The probability that several vehicles are running simultaneously on the bridge, on the

same lane or on several lanes, can be found theoretically in the framework of the queuing

theory.

The bridge can be assimilated to a service system, with or without waiting queue, and

the stochastic processes can be modelled as Markov processes. This allows to arrive to a

suitably modified load spectrum, composed by single vehicles or vehicle convoys travelling

alone on the bridge so that the complete stress history results a random assembly of their

individual stress histories.

A.2.1 Basic assumptions

Let the load spectrum consisting in a set of q types of lorries and be Nij the number of

i-th vehicle per year (annual flow) on the j-th lane. The total flow on the j-th lane is then

∑=

=q

i

ijj NN1

.

(A.1)

Obviously, the probability that several lorries are simultaneously travelling on the

bridges, which is negligible for L<40 m, becomes more and more relevant as the characteristic

length L of the influence line increases.

Basic hypotheses of the theory are that the vehicle arrivals are distributed according a

Poisson law and that the transit time Θ on L is exponentially distributed.

A.2.2 Interaction between lorries simultaneously travelling on one lane

The probability Pn that n lorries are simultaneously travelling on L can be calculated

considering the bridge as a single channel system with a waiting queue, in which the waiting

time, depending on the number of requests in the queue, and the number of the request in the

queue itself are limited.

In fact, as there is a minimum value for the time interval Ts between two consecutive

lorries, the waiting time for the i-th vehicle in the queue is given by si TiT ⋅−Θ= and the

number of requests in the queue is limited to ( ) 1int 1 −⋅Θ= −sTw .

Under the assumption that each Ti is distributed with an exponential law whose

parameter is 1−= ii Tϕ , the problem can be solved in a closed form [5]. The probability Pn to

have n vehicles on the lane, i.e. n-1 requests in queue, is then given by


100

1

2

11

1 1

1

−

=

−−

= =

+⋅⋅++⋅

= ∑ ∏ ∑w

i

i

s

s

j

ji

n

nP ϕααδαδ

αδ

n=0, 1,

(A.2)

and by

1

2

11

1 1

11

1 1

1

−

=

−−

= =

−−

= =

+++⋅

+= ∑ ∏ ∑∏ ∑

w

i

i

s

s

j

ji

n

s

s

j

j

n

nP ϕααδαδϕα

αδ

2≤n≤w, (A.3)

where δ represents the lorry flow density and 1−Θ=α . The annual number of interactions

between n vehicles i1, .., in on the j-th lane can be then obtained substituting these formulae in

the general equation,

∑ ∏

∏

⋅⋅

−=

=

=

nst

k

q

n

s

ji

n

k

ijn

i

N

N

n

N

P

PN

1

1

j

0

j ),i ...., ,i , (1n 21

(A.4)

where ∑2q

indicates the sum over all the possible choices with repetitions of n elements

among q.

In the practice, the problem is reduced to consider the simultaneous presence of two

lorries r and t only, so that it results

1

1

0 11

−

++⋅+=

ϕαδ

αδ

P , ( )

1

11

2

2 11

−

++⋅+⋅

+⋅=

ϕαδ

αδ

ϕααδ

P

(A.5)

and the annual number of interactions becomes

( ) 2

2

2

1

1

j t),,(

j

q s

ji

tjrj

r

N

N

NNN

st

⋅

⋅++

⋅⋅=

∑ ∏=

ϕαδ

δ.

(A.6)

When a single vehicle model is given, equation (A.6) simplifies further into

( )1

j 1), 1,(2 ϕαδ

δ++⋅

⋅= jN

N .

(A.7)

A.2.3 Interaction between lorries simultaneously travelling on several lanes

Under the aforementioned hypotheses, interactions between lorries simultaneously

travelling on several lanes can be tackled in analogous way.

The bridge is considered as a multiple channel system without waiting queue, where

new requests are refused if all channels are occupied. In this case the probability Pk to have

simultaneously vehicles on k lanes, i.e. k occupied channels, can be deduced solving an

Erlang type system.

Said µ the density of the total flow N* and recalling that 1−Θ=α , it results


101

1

0 !!

−

=

⋅⋅

⋅= ∑

m

ii

i

k

k

kik

Pα

µα

µ 0≤k≤m.

(A.8)

Substituting (A.8) in the general expression

∑ ∏

∏∏

=

=

=

⋅⋅

⋅

−=

km

k

s

h

k

j

hk

j h

hikhi

st

j

j

jj

N

N

k

N

N

N

P

PN

1

1*

10

h i ...., ,i ,h 1kk2 211

(A.9)

where ∑

km

represents the sum over all the possible choices of k elements among m, it is

possible to derive the annual number of interactions of k lorries, i1 on the h1-th lane,....., ik on

the hk-th lane,

∑ ∏

∏∏

∑

=

=

=

=

⋅⋅

⋅

⋅

⋅=

km

k

s

h

k

j

hk

j h

hi

m

jj

j

k

k

hi

st

j

j

jj

N

N

k

N

N

N

j

kN

1

1*

1

1

h i ...., ,i ,h

!

!kk2 211

αµ

αµ

.

(A.10)

As said before, usually only the case in which two lorries r and t are simultaneously

present on the h-th and the j-th lane is relevant, so that it results

12

02

2

2!2

−

=

⋅⋅

⋅= ∑

ii

i

iP

αµ

αµ

and

2!2

12

12

2

j t ,

jh

ii

i

jh

tjrh

hr

NN

iNN

NNN

+⋅

⋅⋅

⋅⋅

⋅⋅

=−

=∑α

µα

µ,

(A.11)

or, simply, when a single type of vehicle is considered,

2!2

12

12

2

j ,

jh

ii

i

h

NN

iN

+⋅

⋅⋅

⋅=

−

=∑α

µα

µ.

(A.12)

A.2.4 The time independent load spectrum

The procedures described above allow to obtain the so-called lonely vehicles

spectrum, which is time-independent being composed by individual vehicles and by vehicle

convoys travelling alone on the bridge.

Generally, the evaluation of the lonely vehicles spectrum requires to resort to both the

above mentioned procedure:

- the simultaneous transit on the same lane is considered first, obtaining a new load

spectrum, composed by individual vehicles and by vehicle convoys travelling alone

on the lane;

- new load spectra are then applied on different lanes to solve the multilane case.


102

A.2.5 Time independent interactions

Once the lonely vehicle spectrum is known, the complete stress history can be derived

as a random assembly of the individual stress histories.

Unfortunately, the stress spectrum cannot be determined, in general, as a pure and

simple sum of the individual stress spectra. In fact when maximum and minimum stresses are

given by different members of the spectrum, the individual stress histories can combine,

depending on the cycle counting method adopted, originating some kind of time independent

interaction.

If cycles are identified using the reservoir method or the rainflow method, the problem

can be solved in the general case. The demonstration is out of the scope of this Guidebook

and it will be shown only the main results.

Two individual stress histories iAσ and

jAσ interact if and only if

ji AA σσ maxmax ≤ ∧ ji AA σσ minmin ≤ (A.13)

or

ij AA σσ maxmax ≤ ∧ ij AA σσ minmin ≤ (A.14)

If the couples of interacting histories are sorted in such a way that the corresponding

maxσ∆ are in descending order, the number of the combined stress histories as well as the

residual numbers of each individual stress history can be computed in a very simple recursive

way, as follows.

In general, an individual stress history can interact with several others; therefore the

number of combined stress histories Ncij, obtained as h-th combination of the stress history

iAσ and as k-th combination of the stress history jAσ is given by

jk

ih

jk

ih

cijNN

NNN

)1()1(

)1()1(

−−

−−

+

⋅= ,

(A.15)

where ih

N)1( − and j

kN

)1( − are the number of the individual stress histories iAσ and

jAσ

which are not yet combined and being iAi NN =)0( and

jAj NN =)0( the number of

repetitions of iAσ and

jAσ in the lonely vehicle spectrum.

The actual number of individual stress histories iAσ , which do not combine with other stress

histories, is given by

( )∑≠

+−=ik

kiikiip

NNNN)0()( ,

(A.16)

being the sum extended to all the stress histories kAσ , which combine with

iAσ itself.

In conclusion, a new modified load spectrum is obtained, whose members, represented

by the lonely individual vehicles and convoys and by their time independent combinations,

are interaction free, so that it can be defined as interaction-free vehicle spectrum.


The above mentioned methods allow the derivation of some important general results.


103

The methods can be used to tackle relevant questions concerning the calculation of the

maximum length of the influence line for which lorry interaction on the same lane can be

disregarded or with the calibration of damage equivalent λ4-factor accounting for multilane

effect in λ-coefficient method.

The analysis, shortly illustrated below, can be performed taking into account:

- S-N curve in characterised by on slope m=5;

- four different annual lorry flow rates, N1=2.5×105; N2=5.0×105; N3=1.0×106;

N4=2.0×106, distributed over 280 working days;

- constant lorry speed v=13.89 m/sec.

Assuming an inter-vehicle interval Ts=1.5 s, application of (A.7) allows to calculate,

for example, how many vehicles per years are travelling simultaneously on the same lane, in

function of the annual vehicle flow and on the considered length L, as summarised in table

A.1.

Table A.1. Number of yearly interacting vehicles on one lane

L (m) N1 N2 N3 N4

40 1190 4729 18566 71605

50 1690 6670 25987 98813

60 2165 8515 32940 123618

75 2858 11177 42796 157689

100 3978 15423 58110 208240

These theoretical results, which are in good agreement with numerical simulations,

confirm that simultaneous presence of several lorries on the same lane is generally not

relevant for spans below 75 m. On the contrary, when bending moment on support of two

span continuous beams is considered under high traffic flows, simultaneity results significant

starting from 30 m span.

Closed form expression for calculation of λ4 coefficients can be obtained resorting to

equation (A.12) referring to two lanes carrying equal lorry flows per year, which is the most

relevant case for practical applications. The results are summarized in table A.2 for different

traffic flows and span length.

Table A.2. Number of yearly interacting vehicles on two lanes carrying equal lorry flows

L (m) N1 N2 N3 N4

10 1846 7331 28901 112358

20 3666 14450 56179 212764

30 5458 21367 81966 303028

50 8967 34626 129532 458712

75 13213 50200 182480 617280

100 17312 64766 229356 746264

150 25100 91240 308640 943390

200 32383 114678 373132 1086953


104

Taking into account lorry interactions in all possible relative positions of the two

lorries, equivalent stress ranges ∆σeq, can be easily evaluated from table A.2, provided that

influence coefficient pertaining to each lane is known.

Obviously, said ∆σ1 the equivalent stress range corresponding to one lane flow only,

the required λ4 coefficient is simply given

1

4 σσ

λ∆∆

= eq.

(A.17)

If the two lanes have the same influence coefficient, i.e. the influence surface is

cylindrical, λ4 values result those indicated in table A.3, being 5 2149.1 ≈ (m=5) the λ4 basic

value, corresponding to zero interactions.

Table A.3. λλλλ4-factors for two lanes carrying lanes carrying equal lorry flows

L (m) N1 N2 N3 N4

10 1.156 1.162 1.174 1.197

20 1.162 1.174 1.197 1.234

30 1.168 1.186 1.217 1.264

50 1.180 1.207 1.250 1.310

75 1.194 1.230 1.283 1.351

100 1.207 1.250 1.310 1.381

150 1.230 1.283 1.351 1.423

200 1.250 1.310 1.381 1.450

These results demonstrate that λ4, which takes into account globally vehicle

interactions, is a quasi-linear function of N⋅Θ , which can be expressed in closed form as

⋅⋅⋅+⋅

+=

65

1

214

1001.003.1

v

NL

ηηηλ .

(A.18)

where L is in m and v in m/s, being η1 and η2, η1≥η2, the influence coefficients related to the

two interacting lanes.

Chapter 5: Non traffic actions

105

CHAPTER 5: NON TRAFFIC ACTIONS

Pietro Croce1


Summary

In bridge design, climatic, geotechnical and environmental actions should be

considered, like wind, snow, temperature, earth pressure, water actions, uneven settlements

and so on. Variable climatic actions for bridges are discussed, stressing the peculiarities of the

relevant load models.

1 INTRODUCTION

Besides dead loads and imposed loads, other climatic, geotechnical and environmental

actions should be considered in bridge design, like wind, snow, temperature, earth pressure,

water actions, uneven settlements and so on.

In this chapter, the climatic actions are discussed, devoting special attention to

peculiarities of the relevant load models for bridges.

2 WIND ACTIONS

Wind actions on bridges are specified in EN1991-1-4 [1]; here only peculiarities

concerning bridges themselves will be taken into account, as general information are just

given in the chapter 4 of Guidebook 1 [2].

Strictly speaking, EN1991-1-4 specifications are applicable only to girder bridges

spanning up to 200 m with a constant cross section and one or more spans . Cross section can

be boxed, mono or multi-cell, or open with two or more longitudinal beams, which can be

made, in turn, by open or box sections or by truss, with a single deck (upper or lower).

In any case, it must be stressed that EN1991-1-4 rules can be easily extended to

variable cross sections, to double deck bridges as well as to other bridge types, provided that

wind-structure interactions are not relevant.

Bridges characterised by multiple or significantly curved decks, roofed bridges and

movable bridges could require some additional studies.

Lower or intermediate deck arch bridges or suspended and cable stayed bridges call

for specific studies, since for them wind-structure interactions cannot be disregarded.

In general, wind is considered blowing in two horizontal directions, x and y, being y

the longitudinal axis of the bridge and x the transversal axis (figure 1), originating forces in x,

y and z direction. Forces induced by wind blowing in direction x can be considered not

simultaneous with forces induced by wind blowing in direction y and vice versa; on the

contrary, wind forces acting in z direction should be considered acting simultaneously with

the corresponding x or y force. In some of particular orography, it could be necessary to

consider some inclination of the wind directions, out of the horizontal plane.


106

x

y

z

L

B

D

Figure 1. Reference system for wind actions on bridges.

2.1 Wind forces on bridges

For the evaluation of wind actions on bridges, two different load scenarios are taken

into account, depending on the compatibility of strong wind with traffic on bridge. In fact, if

the traffic is not protected against the wind, for example by wind shields, road, pedestrian or

railway traffic is not possible when the fundamental value of basic wind velocity vb,0 attains a

particular threshold value, equal to

m/s 23*0, =bv , (1)

for road bridges, and to

m/s 25**0, =bv , (2)

for railway bridges.

When bridge in unloaded, the wind forces Fwk, evaluated as indicated in the following,

should be considered.

If the bridge is not protected against the wind, the combination value of the wind

actions is given by

wkwk FF 0' ψ= , (3)

for road bridges, where ψ0=0.6 for persistent design situations and ψ0=0.8 for actions during

execution,

wkwk FF 0

'' ψ= , (4)

for railway bridges, where ψ0=0.75, and

wkwk FF 0

''' ψ= , (5)

for footbridges, where ψ0=0.30.

If the bridge is protected against the wind, i.e. wind shields are foreseen on the bridge

or the bridge is covered, the combination value of the wind should not exceed

( )*

0,

*

bwkwk vFF = , (6)

for road bridges, and

( )**

0,

**

bwkwk vFF = , (7)


107

for railway bridges, while for footbridges no additional limitation is necessary as ψ0 is low

enough.

If dynamic analysis is not required, as it happens in normal bridges, for example

bridge spanning up to 40 m, whatever the construction material used, the structural factor

cs⋅cd, can be assumed cs⋅cd=1.0, being cs the size factor and cd the dynamic factor.

2.1.1 Wind forces on the deck in the x-direction

Wind forces in the x-direction can be evaluated using the expression

xrefbwk ACvF ,

2 2

1 ρ= , (8)

where ρ=1.25 kg/m3 is the air density, vb is the basic wind speed for the site under

consideration, Aref,x is the reference area and C is the wind load factor for bridges.

In absence of traffic, reference area Aref,x should be evaluated taking into account:

- in case of plain (web) beams, the total height d of the projection on a vertical plane

of all the main beams, including the part of one cornice or footway or ballasted

track projecting over the front main girder, (see figure 2), plus the sum d1 of the

heights of solid parapets, noise barriers, wind shields and open safety barriers

installed on the bridge;

- in case of truss beams, the total height d of the projection on a vertical plane of all

the trusses, including the part of one cornice or footway or ballasted track

projecting over the front main girder, or the projection of the contour of the solid

section, whichever is less, plus the sum d1 of the heights of solid parapets, noise

barriers, wind shields, and open safety barriers installed on the bridge.

The height of open safety barrier is set to 0.3 m, so that the reference heights to be

considered in same relevant case can be derived from table 1.

d

0.3 d1

Op

en

pa

rap

et

Op

en

safe

ty b

arr

ier

Solid

safe

ty b

arr

ier

or

pa

rap

et o

r no

ise b

arr

ier

Fig. 2. Depth to be used in evaluation of Aref,w.

Table 1. Depth to be considered in evaluation of Aref,w

Road restraint systems and shields On one side On both sides

Open parapet or open safety barrier d+0.3 m d+0.6 m

Solid parapet or solid safety barrier d+d1 d+2 d1

Open parapet and open safety barrier d+0.6 m d+1.2 m


108

During execution, finishing can be disregarded in the evaluation of Aref,x and, prior of

the placement of the carriageway, the surface of two main beams should be considered.

In presence of traffic, reference area Aref,x should be assumed as the larger between the

area evaluated considering absence of traffic and the area obtained taking into account the

presence of traffic. Lateral surface of vehicles exposed to wind is represented

- in road bridges, with a rectangular area, 2 m in height, starting from the

carriageway level, on the most unfavourable position, independently of the location

of the vertical traffic loads;

- in railway bridges, with a rectangular area, 4 m in height, starting from the top of

the rail, on the whole length of the bridge.

Calculation of wind load factor C

The wind load factor C is given by

xfe ccC , = , (9)

where ce is the exposure coefficient for kinetic pressure and cf,x is the force coefficient, which

is equal to cf,x0, being cf,x0 the force coefficient or drag coefficient without free end flow.

The exposure coefficient could be evaluated considering a reference height ze given by

the distance from the lowest point of the ground and the centre of the bridge beck,

disregarding additional parts, parapets, barriers and so on, included in the reference area.

The force coefficient cf,x can be assumed equal to 1.30 for normal bridges, or can be

determined using the expression

−= 1 ;3.05.2max ;4.2min,

tot

xfd

bc , (10)

for bridge with solid parapets and/or solid barriers and/or traffic, and using the expression

−= 3.1 ;3.05.2max ;4.2min,

tot

xfd

bc , (11)

for the construction phase and/or bridges with open parapets.

In expressions (10) and (11) b represents the total width of the bridge and dtot the

height considered in the evaluation of Aref,x, Aref,x=dtot⋅L, except for truss girder where dtot does

not include the truss height, so that truss girder should be considered separately.

Two generally similar decks, being at the same level and separated transversally by a

gap not significantly greater than 1 m, can be considered as a unique structure, when

windward structure forces are to be calculated. In other cases special studies are necessary.

If particular orography determines an incoming wind inclined more than 10° in the

vertical plane, the drag coefficient may be derived from special investigation.

When the windward face of the section is inclined to the vertical of an angle α1, the

drag coefficient cf,x0 may be decreased by a factor η1,

( ).70 ; 005.01max 11 αη −= , (12)

but this reduction does not affect Fw.

If the bridge is sloped transversely by an angle α2, the drag coefficient cf,x0 should be

increased by a factor η2,

( ).251 ; 03.01max 12 αη += . (13)


109

Provided that dynamic analysis is not necessary, the wind load factor C can be also

evaluated in a more simple way using table 2. In the table the force factors C refer to terrain

category II, Area with low vegetation such as grass and isolated obstacles (trees, buildings)

with separations of at least 20 obstacle height. For intermediate values of ze and/or b/dtot,

linear interpolation is permitted.

Table 2. Force factors C for bridges

b/dtot ze≤20 m ze=50 m

≤0.5 6.7 8.3

≥4.0 3.6 4.5

2.1.2 Wind forces on deck in the y-direction

When relevant, longitudinal wind forces in y-direction can be assumed equal to 25%

of wind forces in x-direction for plated bridges and to 50% of wind forces in x-direction for

truss bridges.

2.1.3 Wind forces on deck in the z-direction

Upward and downward vertical wind forces in z-direction can be determined using the

lift force coefficients cf,z, which should not be used to calculate vertical vibrations of bridge

deck.

If suitable wind tunnel tests are not available, it can be assumed cf,z=±0.9, so

considering globally the influence of a possible transverse slope of the deck, of the slope of

terrain and of fluctuations of the angle of the wind direction with the deck due to turbulence.

Considering the symbols indicated in figure 3, alternative values of cf,z, can be derived

from the diagrams reported in figure 4. These diagrams are valid in the range -10°≤θ≤10°,

where θ is the sum of α, which is the inclination of the wind direction in the vertical plane,

and β, which is the superelevation of the deck. When using the diagrams, the total depth dtot

may be limited to the depth of the deck structure, disregarding the presence of traffic and any

bridge equipment.

The reference height can be assumed be ze, as in previous cases.

Fze

b

dtot

β

αθ

wind direction

Aref,z=bL

β=superelevation

α=angle of the wind with the horizontal

θ=α+β

Figure 3. Transversal slope and wind inclination for z-direction wind forces

For hilly terrain, when the bridge deck is at least 30 m above ground, and in every case

for flat and horizontal terrain, the angle α of the wind with the horizontal, due to turbulence,

may be taken as ±5°.


110

Figure 4. Force coefficients cf,z for bridges with transversal slope and wind nclination

The reference area Aref,z should be set equal to the planar area of the bridge, Aref,z=b⋅L,

being b the total width and L the length of the bridge.

The vertical force Fz, which is relevant only if it is of the same order of magnitude of

the dead load, can be considered applied with an eccentricity e=b/4, if not otherwise specified.

2.2 Wind forces on piers

Wind actions for the entire bridge should be evaluated considering the most

unfavourable wind direction for bridge and supporting piers.

Wind forces on piers should be determined using the general formulae and the relevant

force and pressure coefficients.

During construction phases, intermediate situations can occur where horizontal

transmission or redistribution of wind actions by the deck are not granted: these transient

design situations could result the most severe for the piers and for some particular type of

deck. For this reason, besides persistent design situations, also transient design situations for

wind actions during execution should be assessed.


111

3 SNOW LOADS

Snow loads on bridges should be determined according the general procedure of

Eurocode EN 1991-1-3 [3], just described in Guidebook 1, therefore only some additional

information about simultaneity of snow load with other actions is given here.

In general, except for roofed bridges and for bridges situated in particular geographic

area, snow loads should not be combined with traffic actions

4 THERMAL ACTIONS

As known, in EN 1991-1-5 [4] a generic temperature profile in the cross section of a

structural element can be obtained by superposition of four essential components, as

illustrated in figure 5.

Figure 5. Essential constituents of a temperature profile

These components correspond to

- a uniform temperature component ∆Tu;

- a linearly variable component about the z-axis ∆TMy;

- a linearly variable component about the y-axis ∆TMz;

- a non linear component ∆TE, which results in a self-equilibrated system of stesses.

4.1 Temperature changes on bridges

The temperature changes on bridges are given in terms of uniform temperature

component, vertical difference component, which includes non linear component also, and,

when relevant, a horizontal difference component, which can be assumed linearly varying.

The temperatures in the bridge depend not only by the shade air temperature and solar

radiation, but also on the scheme, on the cross section, on the mass and on the material.

Therefore, bridges can be classified in terms of categories and subcategories as

follows:

1. Steel bridge: steel box girder

steel truss or plate girder;

2. Composite bridge

3. Concrete bridge: concrete slab

concrete beam

concrete box girder.


112

4.1.1 Uniform temperature component

The uniform temperature component depends on the maximum and minimum

temperature, Te,max and Te,min, that the bridge can attain during its working life.

Once determined the maximum and minimum shade air temperatures of the site

characterized by 50 years return period, Tmax and Tmin, the uniform temperature components

Te,max and Te,min can be determined according to the diagrams of figure 6, where Te,max and

Te,min, in °C, are expressed in terms of Tmax and Tmin, in °C, for each bridge category recalled

before. Values of Te,max values for truss or plated steel bridges (category 1) can be reduced by

3 °C.

-50 -40 -30 -20 -10 0 10 20 30 40 50-50

-40

-30

-20

-10

0

10

20

30

40

50

60

70

1

2

3

3

2

1

Tmin Tmax

Te

,min

T

e,m

ax

Figure 6. Correlation between shade air temperature (Tmin, Tmax) and uniform

components of the bridge temperature (Te,min, Te,max)

If T0 is the initial bridge temperature, i.e. the temperature of the bridge at the time

when it is restrained, the variation of the uniform bridge temperature ∆Tu is given by

conNespNeeu TTTTT ,,min,max, - ∆+∆==∆ , (14)

where

∆TN,exp=Te,max-T0 and ∆TN,con=T0-Te,min (15)

are the temperature variations to be considered when the bridge expands or contracts,

respectively.

Assessing bearing displacements it can be assumed ∆TN,exp=Te,max-T0+20°C and

∆TN,con=T0-Te,min+20°C.


113

4.1.2 Vertical temperature varying component

In consequence of the different heating and cooling of the top and bottom surfaces of

the bridge, vertical temperature variations can occur. These variations correspond to

maximum heating, when the top surface is warmer than bottom surface, and maximum

cooling, when the bottom surface is warmer than the top surface.

The vertical temperature profiles can be defined under two different hypothesis,

according as non-linear temperature profile ∆TE is disregarded or not: in the former case a

simplified equivalent linear profile can be considered, while in the latter one a non linear

profile, including ∆TE, is taken into account.

Equivalent linear vertical temperature profile

When equivalent linear vertical temperature profile is adopted, the maximum

temperature differences corresponding to maximum heating or maximum cooling, denoted as

∆TM,heat and ∆TM,cool, for the different bridge categories can be deduced by table 3.

The values given in table 3 represent upper bound values of the temperature

differences for road and railways bridges carrying a 50 mm surfacing. For different thickness

of the surfacing, values of table 3 should be multiplied by the adjustment factors ksur given in

table 4.

Table 3. Equivalent linear vertical temperature variations for bridges

Type of deck Top warmer than

bottom ∆TM,heat [°C]

Bottom warmer than

top ∆TM,cool [°C]

Type 1:

Steel deck

18

13

Type 2:

Composite deck

15

18

Type 3:

Concrete deck

- concrete box girder

- concrete beam

- concrete slab

10

15

15

5

8

8

Table 4. Adjustment factors ksur for road, foot and railway bridges

Surface thickness

[mm]

Type 1 Type 2 Type 3

Top

warmer

than bottom

Bottom

warmer

than top

Top

warmer

than bottom

Bottom

warmer

than top

Top

warmer

than bottom

Bottom

warmer

than top

insurfaced 0.7 0.9 0.9 1.0 0.8 1.1

water-proofed 1.6 0.6 1.1 0.9 1.5 1.0

50 1.0 1.0 1.0 1.0 1.0 1.0

100 0.7 1.2 1.0 1.0 0.7 1.0

150 0.7 1.2 1.0 1.0 0.5 1.0

ballast (750 mm) 0.6 1.4 0.8 1.2 0.6 1.0


114

Non linear vertical temperature profiles

When more refined analyses are necessary, the vertical temperature profile can be

assumed as non linear, considering in the heating and cooling conditions the temperature

profiles given in tables 5, 6 and 7 for steel bridges, concrete bridges and composite bridges,

respectively.

For composite bridges two alternative profiles, normal and simplified, are given in

tables 7.a and 7.b. The simplified profile is generally safe-sided.

The temperature differences ∆T given in tables 5 to 7 include both the vertical

temperature component ∆TM and the non linear temperature component ∆TE, together with a

little part of uniform component ∆TN, just considered in the uniform temperature component

∆TN, given in §4.1.1.

The temperatures for other surfacing depths of bridge decks of type 1 to 3 are given in

Tables B.1 to B.3 of EN 1991-1-5, Annex B.

4.1.3 Horizontal temperature varying component

Horizontal temperature differences in bridges can be generally disregarded, except in

special cases, for example when one side of the bridge is much more exposed to the sunlight

of the other one.

When horizontal component must be taken into account, a linear variation of 5 °C can

be assumed.

Table 5. Non linear vertical temperature differences for steel bridges

Type of construction Temperature difference (∆T)

Heating Cooling

h

Steel deck with 40 mm surfacing

0.1

m

0.2

m

0.3

m

24 °C

14 °C8 °C

4 °C

h

-6 °C0.5

m

h

h

Steel deck on truss or plate girder with 40 mm surfacing

21 °C

0.5

m

h

-5 °C

0.1

m

h


115

Table 6. Non linear vertical temperature differences for concrete bridges


Heating Cooling

h

100 mm surfacing

Concrete slab

100 mm surfacing

h

Concrete beam

h

100 mm surfacing

Concrete box girder

Table 7.a. Non linear vertical temperature differences for composite bridges, normal

profile


Heating Cooling

h

100 mm surfacing

Concrete deck on box, truss or plate girder


116

Table 7.b. Non linear vertical temperature differences for composite bridges, simplified

profile


Heating Cooling

h

100 mm surfacing

Concrete deck on box, truss or plate girder

On the contrary, special attention should be paid for concrete multicell box girder

where temperature of the inner webs can differ significantly (around 15°C) from the

temperature of the outer ones.

5. CONCLUSIONS

Effects of variable climatic actions, wind, snow and temperature given in Eurocodes

EN 1991-1-x have been illustrated, with special emphasis on their application on bridges,

discussing peculiarities, application rules and possible simplification of the relevant load

models.

Wind specifications are applicable only to girder bridges spanning up to 200 m with a

constant cross section and one or more spans, but they can be extended variable cross

sections, to double deck bridges as well as to other bridge types, provided that wind-structure

interactions are not relevant.

Bridge types which are sensitive to wind-structure interactions, like lower or

intermediate deck arch bridges or suspended and cable stayed bridges, call for specific

studies, duly supported by wind tunnel tests.

Simultaneity of snow loads with traffic actions is generally not significant, except in

very particular cases, as roofed bridge, and can be disregarded.

Air shade temperature variations and solar radiation result in temperature fields in the

bridge, depending on the bridge location and on the structural material. These temperature

fields are typically non linear and have been described in detail, but often it is possible to refer

to simplified and safe-sided linear distributions.

Seismic actions have been not considered here, as their illustration is beyond the scope

of the present Guidebook.

6. REFERENCES

[1] EN1991-1-4, Eurocode 1: Actions on structures - Part 1-4: General actions – Wind

actions. Brussels: CEN, 2005


117

[2] Holicky, M et al., GB1: Basis of design and actions on structures, Leonardo Project

number: CZ/08/LLP-LdV/TOI/134020, Prague, CTU, 2010

[3] EN1991-1-3, Eurocode 1: Actions on structures - Part 1-3: General actions – Snow

loads. Brussels: CEN, 2004

[4] EN1991-1-5, Eurocode 1: Actions on structures - Part 1-5: General actions – Thermal

actions. Brussels: CEN, 2004


118

Chapter 6: Accidental actions

119

CHAPTER 6: ACCIDENTAL ACTIONS

Ton Vrouwenvelder1, Dimitris Diamantidis

2

1TNO, Delft, Netherlands

2University of Applied Sciences, Regensburg, Germany

Summary

The accidental actions covered by Part 1.7 of EN 1991 are discussed and guidance for

their application in design calculations is given. A short summary is presented of the main clauses

in the code for collisions due to trucks. After the presentation of the clauses an example is given

in order to get some idea of the design procedure and the design consequences.

1 INTRODUCTION

1.1 General

General principles for classification of actions on structures, including accidental

actions and their modelling in verification of structural reliability, are introduced in EN 1990

Basis of Design. In particular EN 1990 defines the various design values and combination rules to

be used in the design calculations. A detailed description of individual actions is then given in

various parts of Eurocode 1, EN 1991 [2]. Part 1.7 of EN 1991 covers accidental actions and

gives rules and values for the following topics:

- -impact loads due to road traffic

- -impact loads due to train traffic

- -impact loads due to ships

It should be kept in mind that the loads in the main text are rather conventional. More

advanced models are presented in annex C of EN 1991-1-7. Apart from design values and other

detailed information for the loads mentioned above, the document EN 1991, Part 1-7 also gives

guidelines how to handle accidental loads in general. In many cases structural measures alone

cannot be considered as very efficient.

1.2 Background Documents

Part 1.7 of EN 1991 is partly based on the requirements put forward in the Eurocode on

traffic loads (ENV 1991-3) and some ISO-documents. For the more theoretical parts use has been

made of prenormative work performed in IABSE [5] and CIB [6]. Specific backgrounds

information can be found in [7] and [8].


120

2 BASIC FRAMEWORK

In order to reduce the risk involved in accidental type of load one might, as basic

strategies, consider probability reducing as well as consequence reducing measures, including

contingency plans in the event of an accident. Risk reducing measures should be given high

priority in design for accidental actions, and also be taken into account in design. Design with

respect to accidental actions may therefore pursue one or more as appropriate of the following

strategies, which may be mixed in the same design:

1. preventing the action occurring or reducing the probability and/or magnitude of the

action to a reasonable level. (The limited effect of this strategy must be recognised; it

depends on factors which, over the life span of the structure, are normally outside the

control of the structural design process)

2. protecting the structure against the action (e.g. by traffic bollards)

3. designing in such a way that neither the whole structure nor an important part thereof

will collapse if a local failure (single element failure) should occur

4. designing key elements, on which the structure would be particularly reliant, with

special care, and in relevant cases for appropriate accidental actions

5. applying prescriptive design/detailing rules which provide in normal circumstances

an acceptably robust structure (e. g. tri-orthogonal tying for resistance to explosions,

or minimum level of ductility of structural elements subject to impact). For

prescriptive rules Part 1.7 refers to the relevant ENV 1992 to ENV 1999.

The design philosophy necessitates that accidental actions are treated in a special manner

with respect to load factors and load combinations. Partial load factors to be applied in analysis

according to strategy no. 3 are defined in Eurocode, Basis of Design, to be 1.0 for all loads

(permanent, variable and accidental) with the following qualification in: "Combinations for

accidental design situations either involve an explicit accidental action A (e.g. fire or impact) or

refer to a situation after an accidental event (A = 0)". After an accidental event the structure will

normally not have the required strength in persistent and transient design situations and will have

to be strengthened for a possible continued application. In temporary phases there may be reasons

for a relaxation of the requirements e.g. by allowing wind or wave loads for shorter return periods

to be applied in the analysis after an accidental event. As an example Norwegian rules for

offshore structures are referred to.

The typical difference between permanent, variable and accidental loads is shown in

Figure 1 depending on time. Permanent loads are always present (e.g permanent weight of the

construction). Variable loads are nearly always present even its value may be small for a

considerable part of time. However values which are nonzero will occur many times during the

design life of the structure (traffic, snow, wind). Accidental loads, on the contrary, usually never

occur during the lifetime of a structure. But if they are present, it takes only a short time. The

duration depends on the manner of load. For example: Explosions take a shorter time (seconds)

than floods (some days).


121

Figure 1. Typical time characteristics of (a) accidental, (b) variable and (c) permanent loads

3 IMPACT DUE TO ROAD TRAFFIC (VEHICLE COLLISIONS)

3.1 Impact on substructure

Impacts on the substructure of bridges by road vehicles are a relatively frequent

occurrence and may have considerable consequences. Specific provisions are consequently

specified in EN 1991 Part 1-7. In the case of soft impact, design values for the horizontal actions

due to impact on vertical structural elements (e.g. columns, walls) in the vicinity of various types

of internal or external roads may be obtained from Table 1. Soft impact means that the impacting

body consumes most of the available kinetic energy. The forces Fdx and Fdy denote respectively

the forces in the driving direction and perpendicular to it. There is no need to consider them

simultaneously. The collision forces are supposed to act at 1.25 m above the level of the driving

surface (0.5 m for cars). The force application area may be taken as 0.25 m (height) by 1.50 m

(width) or the member width, whichever is the smallest.

In addition to the values in this Table 1 the code specifies more advanced models for

nonlinear and dynamic analysis in an informative annex. The values of design impact forces

given in Table 1 are left open for the national choice as Nationally Determined Parameters.

time

Force

(a)

time

Force

(b)

time

Force

(c)


122

Table 1. Horizontal static equivalent design forces due to impact on supporting

substructures of structures over roadways

Type of road Type of vehicle Force Fd,x [kN] Force Fd,y [kN]

Motorway

Country road

Urban area

Courtyards/garages

Courtyards/garages

Truck

Truck

Truck

Passengers cars only

Trucks

1000

750

500

50

150

500

375

250

25

75

The probabilistic methods of the reliability theory have been used in [9] to determine the

impact forces due to vehicle impact. Two alternative procedures given in EN 1991-1-7 [1], Annexes B and C have been analysed. The following assumptions have been taken:

1. The probability of a structural member being impacted by a lorry leaving its traffic

lane is 0.01 per year.

2. The target failure probability for a structural member, given a lorry hits the

substructure of the bridge is 10-4

/10-2

= 0.01 [3, 4].

3. The probabilistic models given by the working documents of JCSS [10] have been

implemented

The values of accidental impact forces have been computed in [9] and are shown here in

Table 5 for the three assumed distances d of the substructure from the road. The resulting impact

forces determined on the basis of above introduced alternative probabilistic procedures (see Table

2) are considerably higher than the minimum (indicative) requirement for impact forces given

Section 4 of EN 1991-1-7 [1] (see Table 1). This is mainly due to the rather high probability of a

structural member being impacted by a lorry leaving its traffic lane, i.e. 0.01 per annum, which

represents a conservative assumption. For roadways, the impact forces are in a range from 2.9 to

2.8 MN, for roads in urban areas, the impact forces are in a broader range from 1.9 to 1.4 MN

(depending on the applied probabilistic approach) for three study cases of distances d from 3 to 9

m. The study [9] indicates that for the design of structural members located nearby the traffic

routes the upper bound of the accidental impact forces should be recommended in the National

annex of EN 1991-1-7 [1] provided that no other safety measures are implemented. However it is

stated here that the values of Table 2 are relatively high and that they may be recomputed based

on recorded statistics in a certain region or for certain road types.

Table 2: Design values of impact forces based on the probabilistic approach.

Type of road Design impact force Fd,x [kN]

d = 3 m d = 6 m d = 9 m

Roadways 2910 2850 2810

Urban areas 1580 1500 1430

Design Example

Consider the reinforced concrete bridge pier of Figure 2. The cross sectional dimensions

are b = 0.50 m and h = 1.00 m. The column height h = 5 m and it is assumed to be hinged to both

the bridge deck as to the foundation structure. The reinforcement ratio ρ is 0.01 for all four


123

groups of bars as indicated in figure 4.1, right hand side. Let the steel yield stress be equal to 300

MPa and the concrete strength 50 MPa. The column will be checked for impact by a truck under

motorway conditions.

x

H h

y

Fdy

a b

Figure 2. Bridge pier under impact loading

According to the code, the forces Fdx and Fdy should be taken as 1000 kN and 500 kN

respectively and act at a height of a = 1.25 m. The design value of the bending moments and

shear forces resulting from the static force in longitudinal direction can be calculated as follows:

Mdx = H

)aH(a −Fdx =

00.5

)25.100.5(25.1 −1000 = 940 kNm

Qdx = H

aH −Fdx =

00.5

25.100.5 −1000 = 750 kN

Similar for the direction perpendicular to the diving direction:

Mdy= H

)aH(a −Fdy =

00.5

)25.100.5(25.1 − 500 = 470 kNm

Qxy = H

aH −Fdy =

00.5

25.100.5 −500 = 375 kN

Other loads are not relevant in this case. The self-weight of the bridge deck and traffic

loads on the bridge only lead to a normal force in the column. Normally this will increase the load

bearing capacity of the column. So we may confine ourselves to the accidental load only.

Using a simplified model, the bending moment capacity can conservatively be estimated

from:

MRdx = 0.8 ρ h2 b fy = 0.8 0.01 1.00

2 0.50 300 000 = 1200 kNm


124

MRdy = 0.8 ρ h b2 fy = 0.8 0.01 1.00 0.502 300 000 = 600 kNm

As no partial factor on the resistance need to be used in the case of accidental loading, the

bending moment capacities can be considered as sufficient. The shear capacity of the column,

based on the concrete tensile part (say fctk = 1200 kN/m2) only is approximately equal to:

QRd = .0.3 bh fctk = 0.3 1.00 0.50 1200 = 360 kN.

This is almost sufficient for the loading in y-direction, but not for the x-direction.

Additional shear force reinforcement is necessary.

3.2 Impact on superstructure

Design values for actions due to impact from lorries and/or loads carried by the lorries on

members of the superstructure should be defined unless adequate clearances or suitable protection

measures to avoid impact are provided. The recommended value for adequate clearance,

excluding future re-surfacing of the roadway under the bridge, to avoid impact is in the range 5.0

m to 6.0 m. The following scenarios are considered:

a) impact on restraint system on the superstructure (see Figure 3)

Indicative equivalent static design forces are given in the Eurocode 1 part 1.7 for that

scenario and are reported here in Table 3.

The forces apply perpendicular to the direction of normal travel.

Figure 3. Vehicle impact on restraint system


125

Table 3 - Indicative equivalent static design forces due to impact on superstructures.

Category of traffic Equivalent static design force

Fdx a [kN]

Motorways and country national and main roads 500

Country roads in rural area 375

Roads in urban area 250

Courtyards and parking garages 75 a x = direction of normal travel.

b) impact forces on the underside surfaces of bridge decks

The same impact forces as given in Table 3 with an upward inclination are considered as

also shown in Figure 4.

F(h)10°

F(h')10°

F(h)

h

h'

hdrivigdirection

Figure 4. Impact forces on underside surfaces of superstructure

4 LOADS DUE TO RAILWAY COLLISION

EN 1991 1-7 classifies structures that maybe subject to impact from derailed railway

traffic according to Table 4. Bridges belong consequently to class B. For that class each

requirement should be specified. To some extent it is questionable whether an analysis for

horizontal impact should be made at all since the probability of such an event is very small. The

probability depends on:

- Likelihood of train derailment

- Likelihood of train colliding with the bridge given train derailment

The likelihood of train derailment depends on the derailment rate, the number of trains per

day and the critical distance. The likelihood of collision depends on the lateral distance from the

structure and on the train velocity [11]. A risk analysis approach can be found in [12].


126

Table 4. Classes of structures subject to impact from derailed railway traffic.

Class A Structures that span across or near to the operational railway that are either permanently

occupied or serve as a temporary gathering place for people or consist of more than one

storey.

Class B

Massive structures that span across or near the operational railway such as bridges carrying

vehicular traffic or single storey buildings that are not permanently occupied or do not

serve as a temporary gathering place for people.

Recent studies in Switzerland have investigated the impact force on bridges after train

derailment. The impact force is a function of the speed and direction of impact, which depends

mainly on the train velocity vE and the distance from the point of derailment to the point of

impact, as well as on the dynamic friction coefficients. Before the engine or the rest of the train

impacts on a structure after derailment, the train, or a part of it, travels a certain distance across

the ballast and the platform. Thus, part of the kinetic energy is dissipated before impact. Thereby

a number of different cases have been treated. Some of the more important ones are shown in

Figure 5 and are:

1. Derailment of the engine at the front end of the train.

2. Derailment of a carriage at the front end of the train (engine at rear)

3. Derailment of two carriages at an arbitrary position of the train.

x

vE

y

Lw θ0

1)

vehicle on the railway track

derailed vehicle

L = enginew = carriage

v = velocity at moment

of derailment

x

vE

y

L w θ0

2)

derailed vehicle

xθ0θ0

vE

3)

derailed vehicles

Cases of derailmentengine Re 6/6: length=19.3 m; width=3.0 m;

height=4.5 m; mass=120 t

carriage EW IV: length=26.4 m; width=2.8 m;

height=4.1 m; mass=41/50 t

Figure 5. Cases of train derailment


127

The results of the studies have shown that the impact forces depend on the scenario and

can reach values up to 30 MN. Impact functions of train engines on stiff structures are reported in

Figure 6 as function of the engine velocity at moment of derailment.

It is recommended here to perform a risk analysis to define the impact forces in case that

this accidental load is considered, i.e. has an annual probability of occurrence greater than 10-6

.

Protection measures are also recommended to reduce the risk.

t [s/100]

F [MN]

O 2 4 6 8 10

10

20

30

v ≤8 m/sE

t [s/100]

F [MN]

O 2 4 6 8

10

20

30

8 m/s<v ≤12 m/sE

t [s/100]

F [MN]

O 2 4 6 8

10

20

30

v >12 m/sE

Figure 6. Impact functions of train engines on stiff structures

5 LOADS DUE TO SHIP IMPACT

Ship impact accidents (see Figure 7) have occurred several times in the past with

considerable consequences. Table 5 shows the most important casualties of accidents involving

ships and bridges, [13].

Table 5. Fatalities in ship-bridge collisions (1960-2002)


128

Figure 7. Ship impact on bridges

The probability of a ship colliding with a particular object in the water (here bridge deck

or bridge piers) depends on the intended course of the ship relative to the object and the

possibilities of navigation or mechanical errors. In order to find the total probability of an object

being hit, the total number of ships should be taken into account. Finally, the probability of

having some degree of structural damage also depends on the mass, the velocity at impact, the

place and direction of the impact and the geometrical and mechanical properties of ship and

structure.

When discussing ship collisions, it is essential to make a distinction between rivers and

canals on the one side and open water areas like lakes and seas on the other. On rivers and canals

the ship traffic patterns can be compared to road traffic. On open water, shipping routes have no

strict definitions, although there is a tendency for ships to follow more or less similar routes when

having the same destination.

A typical possible model for the ship distribution within a traffic lane is presented in

Figure 8. In general it will be possible to model the position of a ship in a lane as a part with some

probability density function. Details will of course depend on the local circumstances. It should

be noted that sometimes the object under consideration might be the destination of the ship, as for

instance a supply vessel for an offshore structure.

Navigation errors are especially important for collisions at sea. Initial navigation errors

may result from inadequate charts, instrumentation errors and human errors. The probabilistic

description of these errors depends on the type of ship and the equipment on board, the number of

the crew and the navigation systems in the sea area under consideration. Given a ship on collision

course, the actual occurrence of a collision depends on the visibility (day or night, weather

conditions, failing of object illumination, and so on) and on possible radar and warning systems

on the structure itself.

Mechanical failures may result from the machinery, rudder systems or fire, very often in

connection with bad weather conditions. The course of the ship after the mechanical failure is

governed by its initial position and velocity, the state of the (blocked) rudder angle, the current

and wind forces, and the possibility of controlling the ship by anchors or tugs. These parameters


129

together with the mass and dimensions of the ship should be considered as random. Given these

data, it is possible to set up a calculation model from which the course of the ship can be

estimated and the probability of a collision can be found. Such models have been applied several

times in ship collision specific studies for important bridges.

Figure 8. Ingredients for a ship collision model

The ship bridge accidents can be in general divided into three cases (Figure 9):

A. bow collision with bridge pillar,

B. side collision with bridge pillar,

C. deckhouse (superstructure) collision with bridge span.

Figure 9. Frequent types of ship bridge accidents: A) bow collision with bridge pillar; B)

side collision with bridge pillar; C) deckhouse (superstructure) collision with bridge span

The most important and frequent in scope of energy distributed during collision are bow

collisions.

The occurrence of a mechanical or navigation error, leading to a possible collision with a

structural object, can be modelled as an (inhomogeneous) Poison process. Given this Poison


130

failure process with intensity λ(x), the probability that the structure is hit at least once in a period

T can be expressed as:

(y) dx dyf(x,y) P(x) ) P-(T) = nT(P scac λλ∫∫1 (1)

whereas:

T = period of time under consideration

n = number of ships per time unit (traffic intensity)

λ(x) = probability of a failure per unit travelling distance

Pc(x,y) = conditional probability of collision, given initial position (x,y)

fs(y) = distribution of initial ship position in y direction

Pa = the probability that a collision is avoided by human intervention.

For the evaluation in practical cases, it may be necessary to evaluate Pc for various

individual object types and traffic lanes, and add the results in a proper way at the end of the

analysis. To give some indication for λ, in the Nieuwe Waterweg near Rotterdam in the

Netherlands, 28 ships were observed to hit the river bank in a period of 8 years and over a

distance of 10 km. Per year 80 000 ships pass this point, leading to λ=28/(10×8×80000) = 10-6 per

ship per km.

For practical applications mechanical models rules have been developed to calculate the

part of the total energy that is transferred into the structure. Some of these rules are based on

empirical models, others on a static approximation, starting from so-called load indentation

curves (F-u diagrams) for both the object and the structure. According to this model the

interaction force during collapse is assumed to raise form zero up to the value where the sum of

the energy absorption of both ship and structure equal the available kinetic energy at the

beginning of the impact.

Design values can be then defined then from the collision model. The occurrence of a

mechanical or a navigation error, leading to a possible collision with a structural object, can be

modelled as a Poisson process. If data about types of ships, traffic intensities, error probability

rates and sailing velocities are known, a design force could be found from:

(y) dx dyf] Fkm) > (x,y) λλ(x) P[v ) -p) = nT(P(F>Fsdad ∫∫1 (2)

Given target reliability and estimates for the various parameters in (2) design values for

impact forces may be derived. The values in Tables 4.5 and 4.6 of EN 1991 Part 1.7, however,

have not been derived on the basis of explicit target reliability.

For inland ships the values in Table 4.5 in [2] have been chosen in accordance with ISO

DIS 10252. For a particular design it should be estimated which size of ships on the average

might be expected, and on the basis of those estimates, design values for the impact forces can be

found. Table 6 shows a comparison between:

- the values in Table 4.5 of EN 1991-1-7;

- the values based on Annex C of EN 1991-1-7, equation (C1);

- the values based on Annex C of EN 1991-1-7, equation (C9).

The masses for the inland waterways ships should been taken in the middle of the class.

The velocity used is 3 m/s and the equivalent stiffness k = 5 MN/m.


131

Table 6. Design forces Fd for inland ships

m

[ton]

v

[m/s]

k

[MN/m]

Fd

[MN] Fd [MN]

Fd

[MN]

Table 4.5 of

EN 1991-1-7

eq (C.1) of EN

1991-1-7

eq (C.9) of EN

1991-1-7

300 3 5 2 4 5

1250 3 5 5 8 7

4500 3 5 10 14 9

20000 3 5 20 30 18

For sea going vessels values in Table 7 are based on equation (2), with v=3 m/s and

k0=15 MN/m for the smallest ship category and 60 MN/m for the heaviest category.

Table 7. Design forces Fd for seagoing vessels

m

[ton] v [m/s]

k

[MN/m]

Fd

[MN]

Fd

[MN]

Fd

[MN]

Table 4.6 of

EN 1991-1-7

eq(C.1) of EN

1991-1-7

eq (C.11) of

EN 1991-1-7

3000 5 15 50 34 33

10000 5 30 80 87 84

40000 5 45 240 212 238

100000 5 60 460 387 460

6 DISCUSSION ON ANNEX C

The informative Annex C of EN 1991 Part 1-7 gives the designer information on

background information for dynamic calculations in the case of impact loading.

A correct impact assessment s requires a nonlinear dynamic analysis of a model that

comprises both the structure as the impacting body.

The annex demonstrates the principles of such an analysis using simple empirical

models. It should be noted that more advanced models might be appropriate in special cases or

background studies.

In the assumption that the structure is rigid and immovable and the colliding object

deforms linearly during the impact phase and remains rigid during unloading, the maximum

resulting dynamic interaction force is given by:

mkvF r= (3)

where vr is the object velocity at impact; k is the equivalent elastic stiffness of the object (i.e. the

ratio between force F and total deformation); m is the mass of the colliding object. The stiffness,

of course, is some kind of an averaged equivalent value, incorporating all kind of geometrical and

physical nonlinearities in the mechanics of the collision process.

Some reasonable estimates for these quantities are shown in Table 8:


132

Table 8. Statistical parameters for input values

mean

value

standard

deviation

m mass 20 ton 12 ton

v velocity 80 km/hr 10 km/hr

k equivalent stiffness 300 kN/m

As we are considering a loading situation conditional upon the accidental event of

collision, there is no need to use extreme fractiles of these distributions. In many cases one

chooses to use the mean plus one standard deviation. In this case this leads to m=32 ton and v= 90

km/hr=25 m/s and from there we find:

F = 25 (300 ×32)0.5 = 2400 kN

Compared to the F = 1000 kN in table 3.1 this is a large number. However, we should

keep in mind that the load in table 3.1 is intended as a static value, where the force acts only over

a short period of time. The shape of the force due to impact can usually be assumed as a

rectangular pulse and the duration of the pulse is then given by:

kmt /=∆ (4)

In the given example the duration would be 0.3 s. Another point is that the vehicle

usually looses speed between the point where it leaves the track and the point where it hits the

structure (see Figure 10).

road

ϕ

v0

road

structure

s

d d

d

d

road

road structure

structure

structure

Figure 10. Situation sketch for impact by vehicles (top view and cross sections for upward

slope, flat terrain and downward slope)

For a given deceleration a, the velocity vr after a distance s from the critical point is:

vr = (v02– 2 a s )0.5 (5)


133

Using a = 4 m/s2 we arrive at a distance s = 80 m. This means that the force will be zero if

the distance between the centre line of the track and the structural element is about 20 m. Here it has

been assumed that the angle ϕ = 15o. For intermediate distances one may use the expression:

F = Fo bdd /1− (for d < db). (6)

Note that the value of db may be adjusted because of the terrain characteristics.

The force of eq. (3) is the force at the impact surface between the structure and the

impacting vehicle. Inside the structure this load will lead to dynamic effects. As long as the

structure behaves elastically there may be some the dynamic amplification (one may think of 40

percent). However, due to elastic-plastic effects stresses may be reduced.

7 RISK ANALYSIS

For important bridges risk analyses are performed in order to compute the risk associated

to impact forces and especially due to ship collision. Site specific data of the type of traffic are

combined with probabilistic analyses as mentioned above in order to define:

- Design impact force associated to a target probability level

- Protection measures

- Robustness measures to avoid global failure (see for example [14])

The risk analysis scheme given in the Eurocodes 1991 Part 1.7 can be useful in order to

analyse the aforementioned aspects. It is reported here in Figure 11. Decision measures are taken

based on such a procedure.

Figure 10. Risk analysis procedure in the Eurocodes


134

8 CONCLUSIONS

Accidental actions on bridges as specified in the Eurocode 1991 Part 1.7 have been

reviewed in this chapter. The following topics have been covered:

- -impact loads due to road traffic

- -impact loads due to train traffic

- -impact loads due to ships

The models presented in Annex C of EN 1991-1-7 have been discussed. Apart from

design values and other detailed information for the loads mentioned above, the document EN

1991, Part 1-7 also gives guidelines how to handle accidental loads in general. In many cases

structural measures alone cannot be considered as very efficient.

9 REFERENCES

[1] EN 1990 Eurocode - Basis of structural design. European Comittee for Standardisation,

04/2002.

[2] EN 1991-1-1 Eurocode 1: Actions on structures – Part 1-1: General actions – Densities, self

weight, imposed loads for buildings. European Comittee for Standardisation, 04/2002.

[3] ISO 2394, General principles on reliability for structures. 1998.

[4] I SO 3898, Bases for design of structures – Notations - General Symbols, 1997.

[5] Larsen, O.D.: Structural Engineering Documents “Ship Collision with bridges”, The

interaction between Vessel traffic and Bridge structures

[6] CIB: Actions on structures impact, CIB Report, Publication 167, CIB, Rotterdam 1992

[7] Vrouwenvelder, T.: Stochastic modelling of extreme action events in structural engineering,

Probabilistic Engineering Mechanics 15 (2000) 109-117

[8] Vrouwenvelder, T.:”Design for ship impact according to Eurocode 1, Part 2.7”, Ship

collision analysis, Gluver and Olson, 1998 Balkema, ISBN 9054109629

[9] Markova, J. and K. Jung: “Alternative procedures for impact forces in Eurocodes” Journal of

KONBIN, In: Proceeding of the 4th International Conference on Safety and Reliability,

Wydawnictwo Instytutu Technicznego Wojsk Lotniczych, 30 May-2 June 2006, ISSN 1895-

8281, pp 175-182

[10] Joint Committee on Structural Safety (JCSS), Probabilistic Model Code, www.jcss.ethz.ch

[11] UIC Code 777-2: Structures Built over Railway lines, Paris, 2003.

[12] Jung, K. and J. Markova: “Risk assessment of structures exposed to impact by trains" In:

Walraven, Blaauwendraad, Scarpas & Snijder (eds.), Proceedings of 5th International PhD

Symposium in Civil Engineering; 16-19 June 2004, Delft, The Netherlands, A.A. Balkema

Publishers, ISBN 90 5809 676 9; pp. 1057-1063

[13] Proske,D.:Ein Beitrag zur Risikobeurteilung von alten Brücken unter Schiffsanprall,

Dissertation, TU Dresden, 2003.

[14] Starossek, U., Progressive Collapse of Structures: Nomenclature and Procedure, Structural

Engineering International Vol. 2, 2006.

Chapter 7: Combination rules for bridges in Eurocodes

135

CHAPTER 7: COMBINATION RULES FOR BRIDGES IN EUROCODES

Milan Holický1, Jana Marková

1


Summary

The combination rules for bridges introduced in this Chapter are based on Eurocode

EN 1990/A1. The combinations of traffic loads with non-traffic actions and alternative

procedures for load combinations are presented here. An example of verification of the bridge

cantilever for the limit state of static equilibrium is included. Furthermore, selected results of

application of alternative combination rules for the design of prestressed concrete highway

bridge in the Bohemia are presented. Comparison of obtained action effects indicates that

alternative combination rules may lead to considerably diverse load effects. Further

harmonisation based on calibrations of partial factors and other safety elements is needed.

1 INTRODUCTION

1.1 Background documents EN 1990/A1 [2] provides basis for determination of combinations of actions for

ultimate and serviceability limit state verifications of bridges. The aim of this Chapter is to

describe principles of load combinations. Permanent actions, traffic loads and climatic actions

due to wind, snow and temperature are considered in accordance with relevant Parts of EN

1991. Supplementary information on the traffic load models provided in EN 1991-2 [3] is

given in the Background document [4] which is expected to be available on the JRC web site.


EN 1990/A1 [2] gives rules focused on the application of basis provided in EN 1990

[1] for the bridge design. The critical load cases should be determined for the selected design

situations and identified limit states.

Similarly as for buildings (see Annex A1 in [1]) the alternative combination rules for

the ultimate limit states provided in EN 1990/A1 [2] may lead to significantly diverse load

effects as illustrated in Section 5.

2 COMBINATION OF ACTIONS

2.1 General

The effects of actions that cannot occur simultaneously due to physical or functional

reasons should not be considered together in combinations of actions. In case when specific

measures are provided preventing some actions to act simultaneously, these combinations

need not be considered in analysis (e.g. when it is assured that some construction loads are not

simultaneously acting during a specific construction phase).

The expressions 6.9a to 6.12b in EN 1990 [1] are applied for the verification of

ultimate limit states and the expressions 6.14a to 6.16b are applied for the verification of the


136

serviceability limit states in bridge design.

2.2 Fundamental combination of actions

Three alternatives procedures for fundamental combination of actions is provided for

bridges, similarly as for buildings. The choice of combination is the National Determined

Parameter (NDP) which may be nationally selected.

For example the fundamental combination of actions in bridge design in the Czech

Republic is based on the twin expressions (6.10a), (6.10b) given as

iki

i

iQkQ

j

PjkjG QQPG ,,0

1

,1,1,01,

1

,, """""" ψγψγγγ ∑∑>≥

+++ (1)

∑∑≥≥

+++1

,,0,1,1,

1

,, """"""j

ikiiQkQ

j

PjkjG QQPG ψγγγγξ (2)

where the less favourable expression needs to be considered. Where relevant, the favourable

or unfavourable design values of permanent actions Gd,sup or Gd,inf should be considered.

The application of the combination of actions according to the twin expressions

(6.10a), (6.10b) gives in common cases more uniform reliability level of bridges for various

ratios of the characteristic values of variable loads and permanent loads. Furthermore, it was

also decided in the Czech Republic to allow application of the unique expression (6.10)

∑+∑ ++≥≥ 1

0111 j

i,ki,i,Q,k,Qj

Pj,kj,G Q""Q""P""G ψγγγγ (3)

Selected results of application of alternative combination rules for a prestressed

concrete road bridge is shown in the example 5.2 at the end of this Chapter.

The rules for simultaneous combinations of individual traffic load models and their

arrangements are provided in EN 1991-2 [3]. Five different and mutually exclusive groups of

traffic loads as given in [3] are shown in Table 1. Any group of traffic loads should be taken

into account as one variable action which is acting in combination with other variable actions.

Table 1. Assessment of groups of traffic loads (characteristic values of the multi-

component action)

Carriageway Footways and

cycle tracks

Vertical loads Horizontal loads Vertical loads

only

Groups

of

loads

Main load

model LM1

Special

vehicles LM3

Crowd

loading LM4

Braking,

acceleration

forces

Centrifugal.

transverse

force

Uniformly

distributed loads

gr1 Characteristic

values

gr2 Frequent

values

Characteristic

values

Characteristic

values

gr3 Characteristic

values

gr4 Characteristic

values

Characteristic

values

gr5 See EN 1991-

2

Characteristic

values

The rules for combinations of construction loads during execution for bridges were

transferred from EN 1991-1-6 [6] to EN 1990/A1 [2]. However, some rules for the


137

verification of ultimate and serviceability limit states during execution and for combination of

construction loads with other variable loads remain till now in Annex A2 of EN 1991-1-6 [5].

It should be taken into account in appropriate cases that construction loads Qc act

simultaneously with other types of actions. Different construction loads (Qca to Qce) should be

considered according to the project conditions as one single action, or also as several

individual construction loads that are combined with other variable actions. In some cases it

need not be considered in one combination some variable actions. For example, it is rather

unlikely the simultaneous occurrence of construction load Qca due to working personnel with

small site equipment together with maximum wind or snow actions. For individual project

however, it may need be considered in combination snow and wind simultaneously with other

types of construction loads, e.g. with cranes. The characteristic values of climatic actions may

be reduced for short-time construction periods on the basis of EN 1991-1-6 [6].

Where relevant, the thermal actions and water loads should be considered

simultaneously with construction loads. The various parameters governing water actions and

components of thermal actions should be taken into account when identifying appropriate

combinations with construction loads. The selection of actions to load combinations need to

be considered according to the conditions of individual project.

2.3 Combinations of actions for road bridges

Suplementary rules are introduced for load combinations on road bridges. The snow

load and wind actions need not to be combined with

- the braking and acceleration forces or the centrifugal forces or the associated group

of loads gr2

- the loads on footways and cycle tracks or with the associated group of loads gr3,

- the crowd loading (LM 4) or the associated group of loads gr4.

In common cases it is not necessary to consider the snow loads together with models

LM1 and LM2. However, in some mountain areas it may be also necessary to consider the

combination of snow with traffic.

LM1 or group of loads gr1a need not to be considered with wind actions greater

than *

WF or WkF0ψ .

For certain serviceability limit states of concrete bridges the infrequent combination of

actions is also recommended in EN 1990/A1 [2] given as

{ } 1;1;;;E ,,11,infq,1, >≥= ijQQPGE ikikjkd ψψ (4)

where the combination of actions in braquet may be expressed as

ik,

1

,1k,1infq1,

1

, "+""+""+" QQPGi

i

j

jk ∑∑>≥

ψψ (5)

The infrequent value of the traffic load corresponds to the mean return period of one

year which is based on the product of the characteristic value of variable load and factor

ψ1,infq. The recommended value of the factor for traffic loads ψ1,infq = 0.8.

2.4 Combination rules for footbridges

Infrequent combination of variable actions is not considered in footbridge design. The

concentrated load (wheel load) Qfwk need not to be combined with any other variable actions

than those due to traffic.

In general, wind loads and thermal actions need not be taken into account

simultaneously in common cases.


138

Snow loads need not be combined with groups of loads gr1 and gr2 for footbridges

unless otherwise specified for particular geographical areas and certain types of footbridges.

In case that fotbridges provide protection of the pedestrians and cyclists against all kinds of

unfavourable weather, the specific load combinations may be determined. The combination

similar to actions on buildings may be applied in which instead of relevant category of

imposed load the specific group of traffic loads is applied.

2.5 Combinations of actions for accidental design situations

One accidental action should be considered only in accidental load combination which

need not be combined with snow load or wind actions.

Models of accidental actions on bridges are given in EN 1991-2 [3], models of impact

forces due to cars, heavy vehicles and vessels on bridge substructures in EN 1991-1-7 [7].

Additional combinations of actions for other accidental design situations (e.g. combination of

road or rail traffic actions with avalanche, flood or scour effects) may be agreed for the

individual project.

2.6 Values of ψ factors

ψ factors for traffic loads, wind, thermal actions and construction loads are given for

road bridges in the following Table 2. The recommended values of ψ factors are given for the

road traffic corresponding to adjusting factors αQi, αqi, αqr and βQ equal to 1. The

recommended ψ0 value for thermal actions may in most cases be reduced to 0 for ultimate

limit states EQU, STR and GEO.

The characteristic values of wind actions and snow loads during execution are defined

in EN 1991-1-6 [6]. Representative values of water loads are not given in EN 1990/A1 [2].

They may be defined as NDP in the National Annex or for the individual project.

Table 2. Recommended values of ψψψψ factors for road bridges.

Type of action ψ0 ψ1,infq ψ1 ψ2

Traffic loads

gr1a

(LM1 +

pedestrian or

cycle-track

loads)1)

TS (tandem system) 0.75 0.8 0.75 0

UDL (uniform) 0.40 0.8 0.40 0

pedestrian or cycle-

track loads 2)

0.40 0.8 0.40 0

gr1b (single axle) 0 0.8 0.75 0

gr2 (horizontal forces) 0 0 0 0

gr3 (pedestrian loads) 0 0.8 0.40 0

gr4 (LM4 – crowd loading) 0 0.8 0.75 0

gr5 (LM3 – special vehicles) 0 0 0 0

Wind actions Fw – persistent design situation

– transient design situation

0.6

0.8

0.6

-

0.2

-

0

0

Fw* 1.0 1 - -

Thermal actions Tk 0.6 0.8 0.6 0.5

Snow loads QSn,k – transient design situation 0.8 - - -

Construction loads Qc 1.0 - - 1.0


139

The recommended values of ψ0, ψ1 and ψ2 for gr1a and gr1b are given for road traffic

corresponding to adjusting factors αQi, αqi, αqr and Qβ equal to 1. Those relating to UDL

correspond to common traffic scenarios, in which a rare accumulation of lorries can occur.

Other values may be expected for other classes of routes, or expected traffic, related to the

choice of the corresponding α factors.

Recommended ψ factors for footbridges are given in Table 3. The combination value

of the pedestrian and cycle-track load, mentioned in Table 4.4a of EN 1991-2 [3] is a reduced

value to which the factors ψ0 and ψ1 may be used.

The recommended ψ0 value for thermal actions may in most cases be reduced to 0 for

ultimate limit states EQU, STR and GEO.

Table 3. Recommended values of ψψψψ factors for footbridges.

Action ψ0 ψ1 ψ2

gr1 0.40 0.40 0

Traffic loads Qfw 0 0 0

gr2 0 0 0

Wind forces Fw 0.3 0.2 0

Thermal actions T 0.6 0.6 0.5

Snow loads QSn (transient design situation) 0.8 - 0

Construction loads Qc 1.0 - 1.0

3 ULTIMATE LIMIT STATES

3.1 Design values of actions in persistent and transient design situations

The design values of actions and recommended partial factors given in EN 1990/A1 [2]

for the ultimate limit states (EQU) in the persistent and transient design situations are given in

Table 4.


for the ultimate limit states (STR) in the persistent and transient design situations are given in

Table 5.


for the ultimate limit states (STR/GEO) in the persistent and transient design situations are

given in Table 6.

For the design of geotechnical structures (e.g. footings, piles, piers, abutments) where

geotechnical actions and the resistance of the ground are involved the following three

alternative approaches recommended in EN 1990/A1 [2]

- Approach 1: Applying in separate calculations design values from Table 6(C) and

Table 5(B) to the geotechnical actions as well as the actions from the structure;

- Approach 2: Applying design values of actions from Table 6(B) to the geotechnical

actions as well as the actions from the structure;

- Approach 3: Applying design values of actions from Table 6(C) to the geotechnical

actions and, simultaneously, applying design values of actions from 5(B) to the

actions from the structure.


140

Table 4. Design values of actions (EQU) (Set A)

Persistent

and transient

design

situation

Permanent actions

unfavourable favourable

Prestress Leading

variable

action

Accompanying variable

actions

main others

(if any)

Eq. (6.10) γGj,sup Gkj,sup γGj,inf Gkj,inf γP P γQ,1 Qk,1 γQ,iψ0,iQk,i

The recommended set of partial factors:

γGj,sup=1.05 for unfavourable actions and γGj,inf =0.95 for favourable permanent actions, γQi=1.35 for

unfavourable actions due to road and pedestrian traffic, γQ=1.45 for unfavourable actions due to

railway traffic, reduced to γQ=1.20 for load models SW/2 and unloaded train, γQi=1.5 for other

variable actions, 0 for favourable variable actions.

The values of partial factors for prestressing γP are given in the relevant design Eurocodes.

For unfavourable construction loads in transient design situations γQ=1.35.

When a counterweight is used, partial factor for its weight can be assumed γG,inf =0.8.

For the verification of uplift of bearings of continuous bridges or in cases where the verification of

static equilibrium also involves the resistance of structural members (e.g. where the loss of static

equilibrium is prevented by stabilising systems or devices, e.g. anchors, stays or auxiliary columns),

as an alternative to two separate verifications based on Tables 4 (Set A) and 5 (Set B), a combined

verification, based on Table 4 (Set A), may be applied.

The following values of γ factors are recommended in the combined verification:

γG,sup=1.35, γG,inf=1.25 for permanent actions, γQ =1.35 for unfavourable actions due to road traffic

and pedestrians, γQ= 1.45 for unfavourable actions due to railway traffic, reduced to γQ=1.20 for load

models SW/2 and unloaded train; γQ =1.50 for other variable actions and γQ=1.35 for construction

loads. For favourable variable actions, γQ=0, provided that applying γG,inf =1,00 both to the favourable

and unfavourable part of permanent actions does not give more unfavourable effect.

Table 5. Design values of actions (STR) (Set B)

Persistent

and transient

design

situation

Permanent actions


Prestress Leading

variable

action


actions

main others

(if any)

Exp. (6.10) γGj,supGkj,sup γGj,infGkj,inf γPP γQ,1 Qk,1 γQ,iψ0,iQk,i

Exp. (6.10a) γGj,supGkj,sup γGj,infGkj,inf γPP γQ,1ψ0,1Qk,1 γQ,iψ0,iQk,i

Exp. (6.10b) ξγGj,supGkj,sup γGj,infGkj,inf γPP γQ,1 Qk,1 γQ,iψ0,iQk,i

The choice between exp. 6.10, or 6.10a and 6.10b may be decided in the National Annex.

Recommended values:

γGj,sup = 1.35 for unfavourable and γGj,inf = 1.0 for favourable permanent actions;

γQ = 1.35 for unfavourable actions due to road or pedestrian traffic;

γQ = 1.45 for unfavourable actions due to railway traffic, reduced to γQ=1.20 for load models SW/2

and unloaded train;

For favourable variable actions, γQ = 0.

The characteristic values of all permanent actions from one source may be multiplied by γG,sup if

the total resulting action effect is unfavourable and γG,inf if the total resulting action effect is

favourable.

For particular verifications, the values for γG and γQ may be subdivided into γg and γq and the

model uncertainty factor γSd (a value of γSd in recommended in the range 1.0 to 1.15).


141

Table 6. Design values of actions (STR/GEO) (Set C)

Persistent and

transient

design

situation

Permanent actions


Prestress Leading

variable

action

Accompanying variable actions

main others

(if any)

Exp. (6.10)) γGj,supGkj,sup γGj,infGkj,inf γP P γQ,1 Qk,1 γQ,iψ0,iQk,i

Recommended set of partial factors for actions:

γGj,sup=γGj,inf =γGset=1.0 for permanent actions and actions due to settlements,

γQ =1.15 for unfavourable actions due to road and pedestrian traffic, γQ =1.15 for unfavourable actions

due to railway traffic, γQ = 1,30 for the variable part of horizontal earth pressure from soil, ground

water, free water and ballast, γQ = 1.30 for all other unfavourable variable actions (0 for favourable).

The selection of the geotechnical approach is a NDP which may be given the National

Annex. For example it was decided in the Czech Republic to recommend the Approach 2 for

footings, piles, anchors, underground walls etc., and the Approach 3 is recommended to be

applied for the stability of the slopes.

The design values of actions for the ultimate limit states in the accidental and seismic

design situations are given in Table 7.

Table 7. Design values of actions for use in accidental and seismic combinations

Persistent and

transient

design

situation

Permanent actions


Prestress Accidental

or seismic

action


actions

main others

(if any)

Accidental

exp. (6.11a/b)

Gkj,sup Gkj,inf P Ad ψ11 Qk1 or ψ21Qk1 ψ2,i Qk,i

Seismic

Exp. (6.12a/b)

Gkj,sup Gkj,inf P γIAEk or

AEd

ψ2,i Qk,i

In the case of accidental design situations, the main variable action may be taken with its

frequent or quasi-permanent values. The choice is given in the National Annex depending on the

accidental action under consideration.

For execution phases during which there is a risk of loss of static equilibrium, the

combination of actions is given as

kc,2

11

"+""+""""" dinfkj,supkj, QAPGGjj

ψ∑∑≥≥

++ (6)

where kcQ , is the characteristic value of construction loads as defined in EN 1991-1-6 [6], i.e.

the characteristic value of the relevant combination of groups Qca, Qcb, Qcc, Qcd, Qce and Qcf.

4 SERVICEABILITY LIMIT STATES

4.1 Design values of actions for serviceability verifications

Three combinations of actions are recommended for verification of serviceability

criteria. The design values of actions for the serviceability limit states are given in Table 8.


142

Several serviceability criteria are recommended in EN 1992 to EN 1999 including

cracking in concrete structures.

Table 8 Design values of actions for use in serviceability limit states



Characteristic

Frequent

Quasi-permanent

Gk,j,sup

Gk,j,sup

Gk,j,sup

Gk,j,inf

Gk,j,inf

Gk,j,inf

P

P

P

Qk,1

ψ1,1Qk,1

ψ2,1Qk,1

ψ0,iQk,i

ψ2,iQk,i

ψ2,iQk,i

5 EXAMPLES

Example 5.1 Verification of the ultimate limit states (EQU) during execution

The stability of a bridge cantilever during execution should be verified. The self-weight

G, construction loads Qc and wind actions W are acting on a bridge cantilever. The scheme is

illustrated in Figure 1.

a b

gd,inf

gd,sup Qcc,dqca,d qcb,d

wd

wd

Figure 1. Actions on a bridge cantilever during an execution phase

It is assumed that the coefficient of variation of self-weight g is very small and

therefore, the mean value of the self-weight may be applied. Three construction loads are

present on the right-side of the bridge cantilever: the uniformly distributed load due to

working personnel with small site equipment qca, the movable storage of materials qcb and

moveable heavy devices Qcc. Moreover, it is assumed a non-favourable effect of the wind

forces w acting on the bridge cantilever (on the right side in the downwards direction, on the

left side in the upward direction.

The following condition given in (6.7) of EN 1990 [1] is applied for the verification of

static equilibrium

Ed, dst ≤ Ed, stb (7)

The destabilizing effects of actions are determined on the basis Table 4

Ed,dst = 1.05 gk b2/2 + 1.35 (qca,k b

2/2 + qcb,k b

2/2 + Qcc,k b

2) + 1,5 ψ0 wk b

2/2

and stabilizing effects of actions are given as

Ed,dst = 0.95 gk a2/2 –1.5 wk b

2/2


143

where the lengths of bridge cantilevers are a = 24 m and b = 27 m. The self-weight of the

prestressed concrete cantilever is determined on the basis of nominal dimensions of the box

girder cross-section considering the mean value of density. The bridge cross-sectional area is

A = 7.6 m2 and the density of prestressed concrete γc = 25 kN/m

3. The self-weight is

determined as

gk = Ac γc = 7.6 × 25 = 190 kN/m

The heavy construction device Qk,cc (50 kN), construction loads due to working

personnel qca (1 kN/m2) and movable storage of material qcb (0.2 kN/m

2) are applied

considering the recommended values given in ČSN EN 1991-1-6 [6].

The wind action per 1 metre of bridge length is determined as wk = ±6.9 kN/m (the

procedure for specification of wind actions is not included here).

The destabilising effects in case that the leading construction load (dominant) is present

may be determined

Ed,dst=1.05×190×272/2+1.35×(50×27

2+1×27

2/2+0.2×27

2/2)+1.5×0.8×6.9×27

2/2=117.4 MNm

and in case that the wind is a leading variable action

Ed,dst = 1.05×190×272/2+1.5×6.9×27

2/2+1.35×(50×27

2+1×27

2/2+0,2×27

2/2) = 118.2 MNm

and the stabilizing effects of actions are given as

Ed,stb = 0.95 × 190 × 242/2 – 1.5 × 6.9 × 24

2/2 = 101 kNm.

The condition given by expression (7) is not satisfied and therefore, for assurance of the

cantilever stability it is necessary to accept additional measures. In case that the contra-weight

is applied, the uncertainties in the position of the contra-weight need to be considered or the

recommended partial factor γg,inf = 0.8 applied according to Table 4.

Example 5.2 Selected results of application of alternative combination rules

Selected results of the application of alternative load combinations given by expressions

(6.10) or twin expressions (6.10a), (6.10b) for the analysis of internal forces of the highway

prestressed concrete bridge in Čekanice are presented here. The bridge is a 13 span

continuous beam, length of spans from 22 m to 40.5 m. The first part of bridge is built of

beams (3 spans above the railway), characterised by the cross sections reported in Figure 2.a,

the second part is built of box girder, see Figure 2.b.

The self-weight, permanent loads, the group of traffic loads gr1 and thermal actions are

taken into account.

The two alternative approaches for the vertical difference component of thermal actions

provided in EN 1991-1-5[5] are also considered here (linear approach 1 and non-linear

approach 2). The approach 2 is selected in the National Annex of the Czech Republic.

Figure 2.a. Cross-sections of the Čekanice bridge on highway D3 (Prague - Tábor).


144

Figure 2.b. Box girder of the Čekanice bridge on highway D3 (Prague - Tábor).

Selected results of comparative studies of the highway bridge in Čekanice are given in

Tables 9 to 12.

Table 9. Moments based on alternative procedures given in EN 1990/A1[2] and EN

1991-1-5 [5], hogging cross-sections.

Type of cross-

section, EN 1991-

1-5 approach No.

Moments in MNm

(6.10) Q1 (6.10) T1 (6.10a) (6.10b) Q1 (6.10b) T1

Beam - 2 -12.47 -9.15 -7.68 -8.67 -5.36

Box girder - 1 -36.26 -32.67 -27.88 -28.92 -25.32

Box girder - 2 -31.39 -24.55 -23.01 -24.05 -17.20

Table 10. Moments based on alternative procedures given in EN 1990/A1 [2] and EN

1991-1-5 [5], sagging cross-sections.

Type of cross-

section,

EN 1991-1-5

approach No.

Moments in MNm

(6.10) Q1 (6.10) T1 (6.10a) (6.10b) Q1 (6.10b) T1

Beam - 2 14.48 13.42 10.83 12.59 11.54

Box girder - 1 34.97 34.65 27.61 30.60 30.28

Box girder - 2 31.99 29.69 24.64 27.63 25.32

Table 11. Upper stresses based on alternative procedures given in EN 1990/A1 [2] and

EN 1991-1-5 [5], hogging cross-sections

Type of cross-

section, EN

1991-1-5

approach No.

Stresses in MPa

(6.10) Q1 (6.10) T1 (6.10a) (6.10b) Q1 (6.10b) T1

Beam - 2 1.20 1.08 -0.51 -0.16 -0.28

Box girder - 1 1.23 0.85 0.34 0.45 0.07

Box girder - 2 2.85 3.55 1.96 2.07 2.77


145

Hogging and sagging moments based on alternative combination rules given by

expression (6.10) and twin expressions (6.10a,6.10b) of EN 1990/A1 [2] considered for two

types of bridge cross-sections are shown in Tables 9 and 10, for stresses in Tables 11 and 12.

The leading variable action is noted as Q1 for traffic loads or T1 for thermal actions.

Table 12 Lower stresses based on alternative procedures given in EN 1990/A1 [2] and

EN 1991-1-5 [5], hogging cross-sections

Type of cross-

section, EN

1991-1-5

approach No.

Stresses in MPa

(6.10) Q1 (6.10) T1 (6.10a) (6.10b) Q1 (6.10b) T1

Beam - 2 6.34 3.26 2.82 4.52 1.44

Box girder - 1 3.54 3.48 2.27 2.78 2.73

Box girder - 2 1.81 0.57 0.53 1.05 -0.16

6 REFERENCES


[2] EN 1990/A1 – Application for bridges. CEN, Brussels, 2002.

[3] EN 1992-1 – Eurocode 1: Actions on structures – Part 2: Traffic loads on bridges. CEN,

Brussels, 2003.

[4] Sedlacek G. et al., Background document to EN 1991-2 Traffic loads for road bridges –

and consequences for the design, JRC Report, 2009

[5] EN 1991-1-5 – Eurocode 1: Actions on structures – Part 5: Thermal actions. CEN,

Brussels, 2003

[6] EN 1991-1-6 – Eurocode 1: Actions on structures – Part 1: Actions during execution.

CEN, Brussels, 2005

[7] EN 1991-1-7 – Eurocode 1: Actions on structures – Part 1: Accidental actions. CEN,

Brussels, 2006.


146

Chapter 8: Case study - Design of a concrete bridge

147

CHAPTER 8: CASE STUDY - DESIGN OF A CONCRETE BRIDGE

Pietro Croce1

1Department of Civil Engineering, Structural Division, University of Pisa, Italy

Summary

In this chapter an example of a simply supported prestressed concrete road bridge with

open cross section. The load analysis is performed according to the provisions of EN 1990

and EN 1991, with special emphasis on traffic loads given in EN 1991-2. Aim of the case

study is to clarify load application and load combinations, taking into account their influence

on the local and global behaviour of the bridge members. Static and fatigue assessments are

out of the scope of the present paper and are not considered here.

1 INTRODUCTION

In the present case study, the design of a prestressed concrete road bridge is discussed,

with special emphasis on loads and load combinations.

Loads are determined according to EN 1991-1-1 [1], EN 1991-1-4 [2], EN 1991-1-5

[3], EN 1991-2 [4], and load combinations are derived from EN 1990 [5].

The simply supported bridge, which covers an effective span of 45.0 m (figure 1), is

located in a urban area and it is characterised by an open cross section composed by four

precast pre-stressed concrete longitudinal beams set at constant spacing of 2.95 m (figure 2),

connected by four stiff transverse beams. The transverse beam spacing is 15.0 m.

The upper flanges of the precast longitudinal beams are duly connected to a 0.30 m-

thick concrete slab, cast in situ in a second phase. The concrete slab is not prestressed.

Only end transverse beams (diaphragms) are connected to the concrete slab.

1515 15

45

Transverse beamsEnd transverse beam End transverse beam

Figure 1. Static scheme of the bridge

2 THE SIMPLY SUPPORTED BRIDGE

2.1 General

The total width of the bridge is 11.8 m. The carriageway, 7.50 m wide, is separated

from the two walkways, each one 1.50 m wide, by means of two fixed safety barriers.

The height of the longitudinal beams, whose geometry is represented in figure 3, is

2.76 m; therefore the total height of the cross section is 3.06 m.

The distance between the bridge’s intrados and an underlying roadway is 6.0 m.

The surfacing is made by a 60 mm thick asphalt layer.


148

7.51.5 0.5

11.8

1.50.5

Figure 2. Cross section of the bridge

0.8

2.95 2.95 2.95

11.8

2.7

6

3.0

6

0.3

0.6

1.8

10.3

5

0.3

0.9 0.2

0.3

50

.6

0.98

0.82

Figure 3. Geometry of the cross section

2.2 Mass properties of the longitudinal beams

The area A of each precast longitudinal beam is 1.27 m2.

The centroid of the precast beam is located 1.24 m away from its intrados and the

principal moments of inertia of the beam about its centroidal axes are Jx=1.099 m4 about the

horizontal axis x and Jy=0.045 m4 about the vertical axis y.

Once cast the slab, the mass properties of each beam modify as follows: area A=2.153

m2; centroid located 1.93 m away from the intrados; principal moments of inertia of the beam

about its centroidal axes Jx=2.564 m4 about the horizontal axis x and Jy=0.687 m

4 about the

vertical axis y.

The moment of inertia Jy of the whole cross section about its centroidal vertical axis y

is finally 96.453 m4.

2.3 Structural materials

Materials are chosen according to EN 1992-1-1 [6] and EN 1992-2 [7]

The strength class of structural normal weight concrete is C50/60.

Reinforcing steel is B450C.

Prestress is obtained using post-tensioned prestressing tendons.


149

3 LOAD ANALYSIS

3.1 Structural self-weight

Considering a density γ =25.0 kN/m3 for reinforced concrete, the nominal self-weight

of each precast beam is:

kN/m 0.31m 24.1 kN/m 0.52 231, =×== Ag bk γ . (1)

The self-weight of the cast-in-situ r.c. slab pertaining to each longitudinal beam is

kN/m 13.22m 2.95 m 3.0kN/m 0.52 31, =××== Ag sk γ . (2)

The total self-weight of each transverse beam is

kN 72.11m 69.4kN/m 0.52 331, =×== ttk VG γ , (3)

being Vt its volume.

3.2 Self weight of non structural elements

Self weights of non structural elements to be considered are those due to the

waterproofing, to the 60 mm asphalt surfacing, to the safety barriers and to the parapets. For

global verifications it is a reasonable approximation to consider these loads distributed per

unit surface, by “spreading out” the weight of safety barriers and parapets.

All things considered, the equivalent uniformly distributed loads corresponding to the

self-weight of non structural elements gk,1p is about 2.2 kN/m2, so that the load per unit length

of the bridge gk,2 is

kN/m 96.25m 80.11kN/m 2.2 32, =×=kg . (4)

3.3 Traffic loads

According to EN 1991-2 [4], traffic loads should be applied on the carriageway,

longitudinally and transversally, in the most adverse position, according to the shape of the

influence surface, in order to maximize or minimize the considered load effects.

The first operation to be performed consists of determining the width w of the

carriageway and the number of notional lanes.

The carriageway width w depends, first of all, on whether the walkways are accessible

to vehicular traffic or not. In the present case, as walkways are protected by fixed safety

barriers, only crowd loading needs to be considered on them.

The carriageway width w is given by the clear distance between the safety barriers,

therefore w=7.50 m.

As w>6.0 m, each notional lane is 3.0 m wide and the maximum number of notional

lanes nl which can be considered is given by

23

50.7Int

m 3Int =

=

= wnl ,

(5)

and the maximum width of remaining area wr is 1.50 m (figure 4),

m 50.1m 0.32m 50.7m 0.3 =⋅−=⋅−= lr nww . (6)

As known, EN 1991-2 calls for four separate static load models, being the single axle

load model n. 2 (LM2) devoted only to local verifications.


150

notional lane n. 1

notional lane n. 2

remaining area

7.5

3.0

3.0

1.5

Figure 4. Notional lane arrangement on the carriageway

For global verifications of the urban bridge in question, only load model n. 1 (LM1)

and the expressly required crowd loading, load model n. 4 (LM4), are relevant.

Load model n. 3 corresponding to special vehicles (LM3) is not considered, as the

bridge is not concerned with special vehicle transit.

On the i th notional lane, the main load model LM1 provides for a tandem system of

axles weighing αQi Qik, accompanied by a uniformly distributed load αqi qik, being αQi and αqi

the adjustment factors. In the present work it has been assumed αQi=αqi =1.0 for each lane,

while the values Qik and qik are summarized in table 1.

Table 1. Characteristic values for load model n. 1 (LM1)

Notional lane Qk [kN] qk [kN/m2]

Lane 1 300 9.0

Lane 2 200 2.5


Regarding the crowd loading, EN 1991-2 prescribes a nominal value of 5.00 kN/m2,

and a combination value of 3.0 kN/m2 (2.5 kN/m

2 in the Italian National Annex).

LM1 and LM4 loads must be distributed in the most unfavourable way (both

transversally and longitudinally) for the effect under consideration, bearing in mind, however,

that a single lane cannot hold more than one tandem system, and that the tandem system, if

present, must be considered in full, that is to say, with all the four wheels.

3.4 Wind actions

Wind actions can be represented by vertical and horizontal equivalent static forces.

Vertical force is orthogonal to the roadway plane, while horizontal forces are

represented by two components, parallel and orthogonal to the bridge’s longitudinal axis,

respectively.

The equivalent pressure exerted by the wind can be calculated through the expression

( ) ( ) 2

2beeep vzczq

ρ= , (7)

in which ρ is the air density, which is assumed to be constant and equal to 1.25 kg/m3, vb is

the basic wind velocity and ce(ze) is the so-called exposure coefficient, given by

( ) ( ) ( ) ( )[ ]eveeree zIzczczc ⋅+⋅⋅= 7120

2 , (8)

where ze stands for the reference altitude over the ground.

Expression (8) depends on the roughness coefficient cr, on the orography factor c0 and

on the turbulence intensity Iv. The orography factor, taking into account any significant local


151

variations in the site’s orography, can usually be assumed equal to 1.0. Iv and cr are defined by

the following expressions

( ) ( )

( )

≥

≤<

⋅=

zzzI

zz

z

zzc

k

zI

v

e

i

ev

minmin

min

0

0

if

m 200 if

ln ,

(9)

( ) ( )

( )

≥

≤<

⋅

=zzzc

zzz

zzk

zc

r

er

er

minmin

min

0

if

m 200 if ln,

(10)

where ki is the turbulence factor, usually set to 1.0. In formulae (9) and (10), the terrain factor

kr, the roughness length z0 and the minimum height zmin depend on the terrain category.

As said, the bridge in question is located in an urban area which can be classified in

terrain category IV, that is an area in which at least 15 % of the surface is covered with

buildings whose average height exceeds 15 m. For terrain category IV it results z0=1.0 m,

zmin=10.0 m, kr=0.234.

The reference height ze represents the distance between the lowest ground level to the

centre of the bridge deck structure, disregarding other parts (e.g. parapets) of the reference

areas Recalling that the intrados of the structure is 6.0 m above ground level, it is

ze=(6.0+0.5⋅3.06) m=7.53 m (figure 5).

3.0

6

1.5

36

ze

= 7

.53

Figure 5. Evaluation of the reference height ze for wind actions


152

As ze <zmin, it holds

( ) ( ) 434.0

m 0.1

m 10ln0.1

1min =

⋅== zIzI vev ,

(11)

( ) ( ) 539.0m 0.1

m 10ln234.0min =

⋅== zczc rer ,

(12)

so that

( ) ( ) [ ] 176.1434.0710.1539.0 22min =⋅+⋅⋅== zczc eee .

(13)

Basic wind velocity vb is function of geographic site. Here we assume vb=27 m/s,

obtaining an equivalent static pressure

( ) 222 kN/m 54.0N/m 9.5350.272

25.1176.1 ≅=⋅⋅=ep zq .

(14)

Said x the horizontal direction orthogonal to the bridge’s axis, the force Fwk,x is

( ) xrefxfepxwk AczqF ,,, ⋅⋅= , (15)

where the coefficient cf,x depends on the ratio between the deck’s width b and the total deck’s

height dtot exposed to wind.

When the bridge is unloaded the exposed height is 4.26 m, as the presence of two open

safety barriers and two open parapets is equivalent to an increase of 1.2 m in the exposed

height.

For unloaded bridge, the coefficient cf,x is

669.126.4

8.113.05.23.1 ;3.05.2max ;4.2min, =−=

−=

tot

xfd

bc . (16)

If must be noted that the simplified approach proposed in EN1991-1-4 [2] allowing to

set cf,x=1.3 is generally unsafe-sided.

Using (16), the force Fwk,x for unloaded bridge is then (figure 6)

( ) kN/m 84.326.4669.154.0,,, =⋅⋅== xrefxfepxwk AczqF . (17)

1.2

Fwk,x Figure 6. Equivalent static force Fwk,x (unloaded bridge)


153

When the bridge is loaded, the exposed height increases by 2.0 m (3.0 m in the Italian

National Annex), so it becomes 5.06 m (6.06 m).

In that case the coefficient cf,x is then

80.106.5

8.113.05.20.1 ;3.05.2max ;4.2min, =−=

−=

tot

xfd

bc , (18)

or, according to the Italian National Annex,

916.106.6

8.113.05.20.1 ;3.05.2max ;4.2min, =−=

−=

tot

xfd

bc , (19)

and also for loaded bridge the simplification cf,x=1.3 results unsafe-sided.

Since ψ0w=0.6, the combination values ψ0wFwk,x for the equivalent wind force for

loaded bridge result (figure 7)

( ) kN/m 95.206.58.154.06.0,,0,0 =⋅⋅⋅== xrefxfepwxwkw AczqF ψψ , (20)

( ) kN/m 76.306.6916.154.06.0,,0,0 =⋅⋅⋅== xrefxfepwxwkw AczqF ψψ , (21)

respectively.

2 m EN 1991-1-43 m Italian National Annex

wind action ψ0 Fwk,x

Figure 7. Equivalent static force ψψψψ0wFwk,x (loaded bridge)

Regarding the vertical action, lacking more precise data from wind tunnel tests, the

coefficient cf,z can be set to

9.0 , ±=zfc , (22)

where the sign is determined by the most unfavourable situation. As in that case the reference

area is the horizontal projection of the bridge deck, Fwk,z is

( ) ( ) kN/m 74.5m 80.119.054.0,,, ±=⋅±⋅=⋅⋅= zrefzfepzwk AczqF , (23)

applied with an eccentricity, e, with respect to the longitudinal axis of the bridge

m 95.24

m 80.11

4=== d

e . (24)


154

The vertical force Fwk,z is much smaller than the permanent loads, therefore, according

to EN 1991-1-4, it could be disregarded.

Finally, in the longitudinal direction y, the action to be considered is 25% of that for

the direction orthogonal to the axis, but it is not relevant in this case.

3.5 Thermal actions

As we are dealing with a statically determined structure, uniform and linear thermal

variations should be taken into account only to calculate deformations of the structure and, if

relevant, effects of friction forces in bearings.

Non linear thermal variations cause stresses, but they are not particularly relevant in

the present example.

Assuming that in the site under consideration the maximum and minimum air shade

temperatures with an annual probability of being exceeded of 0.02 are Tmax=40 °C and Tmin=-

10 °C, respectively, the uniform bridge temperature components for a concrete bridge result

Te,max=41.7 °C and Te,min=-1.8 °C (figure 8).

-50 -40 -30 -20 -10 0 10 20 30 40 50-50

-40

-30

-20

-10

0

10

20

30

40

50

60

70

1

2

3

3

2

1

Tmin Tmax

Te,m

in

Te,m

ax

Figure 8. Evaluation of Te,max and Te,min

Setting the initial bridge temperature T0 to 20 °C, the characteristic values of the

maximum expansion and contraction ranges, ∆TN,exp and ∆TN,con, result so

C 7.21207.410max,exp °=−=−=∆ TTT eN, , (25)

C 8.21)8.1(20min,0con °=−−=−=∆ eN, TTT . (26)

Linearly varying vertical temperature difference components can be set to

∆TM,heat=+15 °C for top warmer than bottom and to ∆TM,cool=+8 °C for bottom warmer than

top (figure 9).

Deeper investigations are out of the scope of the present example.


155

15 °C

8 °C

Figure 9. Simplified temperature distributions along the height of the cross section

4 STRESS CALCULATION

4.1 Global behaviour of the structure

Since transverse beams are much stiffer than longitudinal beams, transverse load

distribution can be studied through the classical Courbon-Engesser theory, which assumes

that transverse beams can be modelled as rigid beams resting on linear elastic supports,

corresponding to the longitudinal beams.

When the cross section is made of n longitudinal beams, the reaction Ri induced in the

i-th longitudinal beam by a concentrated load P, applied with an eccentricity e1 with respect to

the vertical centroidal axis of the deck, is given by

+=∑∑ ==

n

j jj

i

n

j j

ii

dJ

ed

JJPR

1

2

1

1

1 ,

(27)

where di is the distance of the i-th longitudinal beam from the vertical centroidal axis and Ji is

the moment of inertia of the i-th longitudinal beam.

If the n longitudinal beams are identical, expression (27) simplifies in

+=∑ =

n

j j

ii

d

ed

nPR

1

2

1 1,

(28)

that clearly indicates that the most heavily stressed beams are the external ones.

When, as in the present example, the longitudinal beams are also equally spaced,

expression (28) can be further simplified and for the external beams it becomes

( )

∆++=

1

61 11

nn

e

nPR , (29)

said ∆ the spacing of the longitudinal beams.

Setting P=1, the above mentioned formulae allow to determine the influence lines of

the load pertaining to each beam. In figures 10 and 11 are illustrated the influence lines

concerning loads on the external beam n. 1 and on the internal beam n. 2, respectively.

From these influence lines it can be easily derived the influence lines for shear forces

and bending moments acting on relevant cross sections of the transverse beam. For example,

in figure 12 it is illustrated the influence line for bending moment in the cross section A-A of

the transverse beam.


156

0.7 0

.4

0.1

-0.2

-0.3

50.8

5

1 2 3 4

Figure 10. Influence line for load on the external longitudinal beam (n. 1)

0.4

0.3 0.2

0.4

5

1 2 3 4

0.1

0.0

5

Figure 11. Influence line for load on the internal longitudinal beam (n. 2)

4.2 Local effects on the slab

Since the concrete slab behaves like a continuous plate supported by the longitudinal

beams, main stresses in it are in the transverse direction, except that in the neighbourhood of

end diaphragms, where the slab is connected to end diaphragms themselves.

For the evaluation of the local effects, the wheel contact pressures can be determined

resorting to the well known hypothesis of 45° diffusion of the loads through the surfacing and

the slab. In this way, recalling that the surfacing thickness is 60 mm and the slab thickness is

300 mm, the equivalent contact area dimensions result 820×820 mm2 for the wheels of the

tandem axle systems of LM1 (figure 13) and 1020×770 mm2 for the wheels of the isolated

axle of LM2 (figures 14), respectively.


157

1 2 3 4

-0.3

0 ∆

0.4

∆ 0.1

∆

-0.2

0 ∆

-0.6

5 ∆

-0.3

5 ∆∆ ∆ ∆

A

A

Figure 12. Influence line for bending moment in section A-A of the transverse beam

400

820

45°

60

300

Figure 13. Load dispersal for the wheel of

the tandem system of LM 1

600

1020

45°

60

300


the single axle of LM 2

Under these hypotheses, the relative contact pressures result

21 kN/m 1.233

82.0 82.0

150 =⋅

=kQp , (30)

for a single wheel of the heaviest tandem system of LM1 and

2kN/m 6.25477.0 02.1

200 =⋅

=Qakp , (31)

for a single wheel of the isolated axle load of LM 2.


158

4.3 Calculation of the main structure under traffic actions

4.3.1 Effects of the self-weight and dead loads

Recalling expressions (1) and (2), the characteristic value of the uniformly distributed

self-weight is

kN/m 13.53,1, =+= skbkk ggG , (32)

while the effect of the weight of each transverse beam on each longitudinal beam is a

concentrated load of 2.93 kN, placed 15.0 m away from the support.

From expression (4) the dead load of non structural parts pertaining to each

longitudinal beam is 6.49 kN/m.

Under permanent loads, the maximum bending moment occurs at midspan (cross

section C) and it is given

( ) kNm 3.15135kNm 0.159320.4562.598

1

348

1 2,12 =

⋅+⋅⋅=⋅+= .LG

LGCMtk

kkg

,

(33)

while the maximum shear force occurs at support (cross section A) and it results

( ) kN 4.1344kN 9320.4562.592

1

42

1 ,1 =

+⋅⋅=+= .G

LGAVtk

kkg . (34)

4.3.2 Effects of traffic loads on main beams

The influence lines to be considered for transversal distribution of traffic loads have

those previously discussed, shown in figures 10 and 11.

For an external beam, that is the most heavily loaded, the most unfavourable load

arrangements are the one represented in figure 15 for traffic loads and the one represented in

figure 16 for crowd loading.

0.7 0

.4

0.1

-0.2

-0.3

50.8

5

1 2 3 4

Q =300 kN

q =9 kN/m2

q =2.5 kN/m2

1k Q =300 kN1k

1k

Q =200 kN2k Q =200 kN2k

q =3.0 kN/m

2

2

0.7

6

0.4

8

0.1

7

2kfk

2

q =2.5 kN/m2

rk

0.0

1

Figure 15. Most unfavourable traffic load arrangement for beam n. 1


159

0.7 0

.4

0.1

-0.2

-0.3

50.8

5

1 2 3 4

q =5.0 kN/m2

q =5.0 kN/m

0.7

6

fkfk

0.3

1

2

Figure 16. Most unfavourable LM4 (crowd loading) arrangement for beam n. 1

When traffic loads are taken into account (figure 15), to beam n. 1 pertain a

concentrated load

kN 35617.0200248.03002 =⋅⋅+⋅⋅=kQ , (35)

and a uniformly distributed load

kN/m 66.1701.05.221.017.05.20.348.00.90.376.00.35.1 =⋅⋅+⋅⋅+⋅⋅+⋅⋅=kq . (36)

Considering the tandem system as a unique concentrated load (knife load), it is

( ) kNm 2.8475 4

0.453560.4566.17

8

1

48

1 22max =

⋅+⋅⋅=+=LQ

LqCM kkkq , (37)

and

( ) kN 3.753 3560.4566.172

1

2

1max =

+⋅⋅=+= kkkq QLqAV . (38)

When crowd loading (LM4) is considered instead (figure 16), to beam n. 1 pertains

only a uniformly distributed load

kN/m 31.1531.00.52.676.00.55.1 =⋅⋅+⋅⋅=kq , (39)

whose effects are, in the present example, less severe than those caused by lorry traffic.

4.3.3 Effects of traffic loads on transverse beams

Stresses in transverse beams are caused only by traffic loads.

Considering the just mentioned influence line (figure 12), the load arrangements that

determine maximum and minimum bending moment in cross section A-A of the transverse

beam are those illustrated in figures 17 and 18.

With the load arrangement of figure 17, the bending moment in section A-A results

( ) ( ) ( )( ) kNm 2.922

11.089.007.03155.207.020015

29.02.2305.08.0923.033.030095.2max =

⋅+⋅⋅⋅+⋅+⋅⋅⋅+⋅⋅++

=kq AAM , (40)


160

while with the crowd loading arrangement of figure 18 it is

( ) ( )kNm 7.25415

06.027.107.0

57.026.043.05.1595.2min −=

⋅

⋅+⋅⋅++⋅

⋅−=kq AAM . (41)

1 2 3 4

-0.3

0 ∆

0.4

∆ 0.1

∆

-0.2

0 ∆

-0.6

5 ∆

-0.3

5 ∆∆ ∆ ∆

A

A

Q =300 kN

q =9 kN/m

q =2.5 kN/m2

1k Q =300 kN1k

1k

2k

Q =200 kN2k Q =200 kN2k2

2

2

0.0

2 ∆

0.2

1 ∆

0.3

3 ∆

0.2

3 ∆

0.1

8 ∆

q =2.5 kN/m2

rk

0.1

1 ∆

Figure 17. Load arrangement maximizing M(AA) in the transverse beam

1 2 3 4

-0.3

0 ∆

0.4

∆ 0.1

∆

-0.2

0 ∆

-0.6

5 ∆

-0.3

5 ∆∆ ∆ ∆

A

Aq =5.0 kN/mfk

2q =5.0 kN/mfk

2

q =5.0 kN/mfk

2q =5.0 kN/mfk

2

-0.2

6 ∆

-0.0

6 ∆

-0.0

7 ∆

-0.4

3 ∆

Figure 18. Load arrangement minimizing M(AA) in the transverse beam


161

4.3.4 ULS combinations for permanent and traffic loads

Adopting expression (6.10) of EN 1990 and recalling that in ULS structural

verifications (STR) partial factor for permanent loads, γG, is set to 1.35 for unfavourable

effects and to 1.0 for favourable effects, and partial factor for traffic loads, γQ, is set to 1.35

for unfavourable effects and to 0 for favourable effects, design values of stresses can be

obtained.

Maximum bending moment at midspan M(C)dmax and maximum shear force at end

support V(A)dmax are

( ) kNm 2.31874)( 35.1)( 35.1max =+= kqkgd CMCMCM . (42)

and

( ) kN 9.2831)( 35.1)( 35.1max =+= kqkgd QVAVAV . (43)

Design value of braking or acceleration forces depends on the vertical loads applied on

notional lane n. 1 and it results

kN 0.65045)2.7360 ( 35.1) 1.0 2(0.6 35.1 111max =⋅+=+⋅= LwqQQ kkld . (44)

This value must be combined with an appropriate combination value of vertical traffic

load, corresponding to its frequent value. Recalling that ψ1=0.75 for the tandem systems of

LM1 and ψ1=0.4 for the uniformly distributed load component of LM1, loads pertaining to

beam n. 1 (see expressions (35) and (36)) become

( ) kN 26717.0200248.03002 75.0 =⋅⋅+⋅⋅=kQ , (45)

and

( ) kN/m 06.701.05.221.017.05.2348.00.90.376.035.1 4.0 =⋅⋅+⋅⋅+⋅⋅+⋅⋅=kq , (46)

so that maximum bending moment in C and maximum shear force in A result

( ) kNm 0.6469 4

45276

8

4506.7 1.35

48

1 35.1

22

max =

⋅+⋅=

+=LQ

LqCM kkdq (47)

and

( ) kN 0.575 2672

0.4506.7 35.1

2

1 35.1max =

+⋅=

+= kkdq QLqAV . (48)

When the leading traffic loads are vertical ones, the accompanying value of the

braking and acceleration forces are to be defined in National Annex and can be set to zero.

4.3.5 SLS combinations for permanent and traffic loads

Concerning SLS verifications, combinations of permanent and traffic load are

generally relevant only for characteristic and frequent load combinations, as quasi-permanent

values of traffic loads are zero, except in very particular cases.

Significant load arrangements to be considered look very similar to the ones just

illustrated regarding ULS verifications, so they will not be further discussed.

As just recalled, it is necessary to stress that frequent values of traffic loads are

obtained via the ψ1 factors, which depend on the nature of the load: in fact ψ1=0.75 for the


162

tandem systems of LM1, for the isolated single axle (LM2) and for crowd loading (LM4),

while ψ1=0.40 for the uniformly distributed load component of LM1.

4.4 Wind effects

Partial factor γQ for wind actions in ULS combinations is γQ=1.50 if unfavourable and

γQ=0 if favourable.

4.4.1 Design effects of vertical wind actions

The vertical component of the pressure exerted by the wind, as discussed in §3.4, is a

uniformly distributed load acting on the entire length of the bridge with an eccentricity e=2.95

m.

For unloaded bridge it results

kN/m 61.8kN/m 74.55.1,, ±=⋅±== zwkQzwd FF γ , (49)

so that the design wind actions on the external beams become

kN/m 74.495.220

95.261.86

4

61.8 ±=

⋅⋅⋅+±=wdq .

(50)

For loaded bridge, the combination value ψ0wqwk =±2.84 kN/m should be considered,

in place of qwd.

4.4.2 Design effects of horizontal wind actions

Design values of horizontal wind action should derived from expressions (17) for

unloaded bridge and from expressions (20) or (21) for loaded bridge.

For unloaded bridge, the design load is

kN/m 76.5kN/m 84.35.1,, =⋅== xwkQxwd FF γ , (51)

applied with an eccentricity of ez=2.13 m with respect to the bearing’s plane.

For loaded bridge, instead, the design load is

kN/m 43.4kN/m 95.25.1,0,0 =⋅== xwkwQxwdw FF ψγψ , (52)

applied with an eccentricity of ez=2.53 m, according to EN 1991-2, or

kN/m 64.5kN/m 76.35.1,0,0 =⋅== xwkwQxwdw FF ψγψ , (53)

applied with an eccentricity of ez=3.03 m, according to Italian National Annex.

Horizontal actions determine, as well as bending moments in the horizontal plane Mz,

total horizontal support reaction at each end given by

LFR xwkxd ⋅⋅= ,, 5.0 or LFR xwkwxd ,0, 5.0 ψ⋅= , (54)

and vertical reactions on the supports of the external beams given by

⋅⋅⋅

±=95.220

5.06 ,

,

zxwk

zd

eLFR or

⋅⋅⋅

±=95.220

5.06 ,0

,

zxwkw

zd

eLFR

ψ.

(55)

4.5 Effects of thermal variations

As said, uniform and linearly varying temperature variations induce only

displacements and secondary stress states due to friction in bearings, which can be determined


163

considering that the coefficient of thermal expansion for prestressed concrete is αT=10⋅10-6

°C.

5 FINAL REMARKS

In the present chapter, effects of loads and load combinations on a simply supported

concrete bridge with open cross section are discussed.

The road bridge is located in an urban area, so that also crowd loading needs to be

explicitly taken into account.

Application of permanent, climatic and traffic actions, derived from the relevant parts

of Eurocode 1, is illustrated in detail, paying special attention to traffic loads.

As the transverse beams are much stiffer than the longitudinal beams, transverse load

distribution has been studied resorting to the Courbon-Engesser theory.

Load combinations for ultimate and serviceability limit state assessments are

determined according to EN 1990 rules, highlighting specific features of local or global

behaviour and their consequences as well as possible simplifications.

The example confirms that Eurocodes are very appropriate for bridge design.

6 REFERENCES


weight, imposed loads for buildings, CEN, Brussels, 2002.

[2] EN 1991-1-4 Eurocode 1 Actions on structures. Part 1-4 General actions. Wind actions,


[3] EN 1991-1-5 Eurocode 1 Actions on structures. Part 1-5 General actions. Thermal

actions, CEN, Brussels 2004.

[4] EN 1991-2 Eurocode 1 Actions on structures. Part 2 Traffic loads on bridges, CEN,

Brussels, 2003.


[6] EN 1992-1-1 Eurocode 2 Design of concrete structures. Part 1-1 General rules and rules

for buildings, CEN, Brussels, 2004.

[7] EN 1992-2 Eurocode 2 Design of concrete structures. Part 2 Concrete bridges. Design

and detailing rule, CEN, Brussels, 2005.


164

Chapter 9: Case study – Design of a steel bridge

165

CHAPTER 9: CASE STUDY – DESIGN OF A STEEL BRIDGE

Pietro Croce1

1Department of Civil Engineering, Structural Division, University of Pisa, Italy

Summary

In the present chapter it is discussed the design of an orthotropic steel deck bridge

according to Eurocodes EN 1990 and EN 1991, in particular referring to traffic loads given in

EN 1991-2. The case study refers to a three span continuous bridge with box cross section.

Aim of the case study is to clarify the influence of load application and load combinations on

the local and global behaviour of bridge members.

1 INTRODUCTION

In the present case study, the design of an orthotropic steel deck bridge is discussed,

with special emphasis on loads and load combinations.

The steel bridge considered here is a three span continuous bridge on four supports.

The clear length of each span is 120 m, so that the length of the bridge is 360 m (figure 1).

Loads are determined according to EN 1991-1-1 [1], EN 1991-1-4 [2], EN 1991-1-5

[3], EN 1991-2 [4], and load combinations are derived from EN 1990 [5].

Fatigue aspects are out of the scope of the present paper; therefore they are not

discussed here.

2 THE THREE SPAN CONTINUOUS BRIDGE

2.1 General

The bridge’s structure is made up of an orthotropic steel deck, with closed trapezoidal

longitudinal stiffeners, sustained by a box girder, 3800 mm height (figure 2), whose current

geometry is described below.

The deck is made up by an 18 mm upper flange stiffened by 8 mm thick trapezoidal

stiffeners.

The trapezoidal stiffeners, which have a lower flange 200 mm wide and are 270 mm in

height, are characterised by a spacing of 600 mm (300+300 mm) and are continuous through

I-shaped transverse beams, 750 mm height. The span of the longitudinal stiffeners is 3000

mm, i.e. the transverse beam’s spacing.

The webs of the box girder are 12 mm thick, while the lower flange is 34 mm thick.

The webs and the lower flange are stiffened by L-shaped 200x100x14 longitudinal stiffeners.

The carriageway is composed by two physical lanes, each one 3.75 m wide, and by

two walkways, each one 1.50 m wide, so that the total width of the carriageway between the

safety barriers is 10.50 m and the overall width of the bridge is 11.60 m. The walkways are at

the same level of the physical lanes, from which they are separated only by the road signs.

The bridge is located in an extra-urban area and the distance between the bridge

intrados and the underlying ground is 20.0 m.


166

120 120 120

Figure 1. Static scheme of the bridge

Figure 2. Cross section of the bridge

2.2 Mass properties of the cross section

The total area A of the cross section is 0.584 m2 and its centroid is 1.54 m from the

deck extrados.

The principal moments of inertia about the centroidal axis are Jx=1.665 m4 about the

horizontal axis x and Jy=3.711 m4 about the vertical axis y.

The strength moduli are then W’x=-1.081 m3 and W”x=0.737 m

3, about the x-axis, and

W’y=-0.639 m3 and W”y=0.639 m

3, about the y-axis.

2.3 Material

Materials are chosen according to EN 1993-1-1 [6] and EN 1993-2 [7]

The structural steel is S355J2 grade.

3 LOAD ANALYSIS

3.1 Structural self-weight

Since the steel density is γ =78.5 kN/m3, the nominal self-weight of the bridge is:

kN/m 84.45m 584.0kN/m 78.5 231, =×== Ag bk γ . (1)

To take into account the weight of other structural parts (transverse beams, bracings

and so on), the value of gk,1b, is increased of about 6%, so that the self-weight gk,1 results

kN/m 6.48 06.1 1,1, ≅= bkk gg . (2)


167

3.2 Self weight of non structural elements

Self weights of non structural elements to be considered are those due to the

waterproofing, to the 60 mm thick asphalt surfacing and to the safety barriers. For global

verifications it is a reasonable approximation to consider these loads distributed per unit

surface, by “spreading out” the weight of safety barriers.

All told, the equivalent uniformly distributed loads corresponding to the self-weight of

non structural elements gk,1p is about 2.2 kN/m2, so that the load per unit length of the bridge

gk,2 is

kN/m 52.25m 60.11kN/m 2.2 32, =×=kg . (3)

3.3 Traffic loads

According to EN 1991-2 [1], traffic loads should be applied on the carriageway,

longitudinally and transversally, in the most adverse position according to the shape of the

influence surface, in order to maximize or minimize the considered load effect.

For example, to determine the maximum sag moments in the spans and the hog

moment at the supports, the relevant influence surfaces, illustrated in figures 3, 4 and 5, must

be considered.

According to EN 1991-2, it is necessary first to determine the total width of the

carriageway w and the number of conventional lanes. The width w depends on whether the

walkways are isolated from vehicular traffic by fixed safety barriers or by kerbs of sufficient

height (>100 mm) or not. In the present case, walkways are potentially interested by vehicle

traffic, as they are separated from the physical lanes only by road signs. For this reason, the

width w is represented by the inner distance between the safety barriers, and therefore

w=10.50 m.

-24.5898

51.36

-25

-20

-15

-10

-5

0

5

0 60 120 180 240 300 360

Figure 3. Influence line for max sag moment in span 1 [m]

-21

180

-25

-20

-15

-10

-5

0

5

0 60 120 180 240 300 360

Figure 4. Influence line for max sag moment in span 2 [m]


168

12.3168

69.24

-5

0

5

10

15

0 60 120 180 240 300 360

Influence Line for Hog Moment at Support 2

Figure 5. Influence line for hog moment at support 2 [m]

As the influence surfaces for bending in box girders are cylindrical, i.e. they have

rectangular cross section, to maximize bending moments the entire carriageway should be

loaded. The number of notional lanes nl is then given by

33

50.10Int

m 3Int =

=

= wnl ,

(4)

and the remaining area is 1.50 m wide,

m 50.1m 0.33m 50.10m 0.3 =⋅−=⋅−= lr nww . (5)

Obviously, to maximize the torque coexisting with the maximum bending moment, it

is necessary to maximize the load eccentricity, so obtaining the notional lane arrangement

illustrated in figure 6.

Clearly, when load conditions maximizing the torque are explored, load eccentricities

should be maximized and different lane arrangement should be considered, like the one

illustrated in figure 7, where only two notional lanes need to be loaded.

notional lane n. 1

notional lane n. 2

notional lane n. 3

remaining area

10.5

3.0

3.0

3.0

1.5

Figure 6. Notional lane arrangement on the carriageway (bending moment calculation)

notional lane n. 1

notional lane n. 2

10

.5

3.0

3.0

Figure 7. Notional lane arrangement on the carriageway (torque calculation)


169

As known, EN 1991-2 calls for four separate static load models, being the single axle

load model n. 2 (LM2) devoted only to local verifications.

For global verifications of the bridge in question, only load model n. 1 (LM1) is

relevant.

In fact, load model n. 3 corresponding to special vehicles (LM3) and load model n. 4,

crowd loading (LM4), are not accounted for, as the bridge is not interested by special vehicle

transit and it is located in an extra-urban area. In this regard, it must be recalled that load

models LM4 and LM3 need to be considered only when expressly required.

On the i th notional lane, the main load model LM1 provides for a tandem system of

axles weighing αQi Qik, accompanied by a uniformly distributed load αqi qik, being αQi and αqi

the adjustment factors. In the present work it has been assumed αQi=αqi =1.0 for each lane,

while the values Qik and qik are summarized in table 1. Only one tandem system should be

considered per lane, placed in the most unfavourable position.

Table 1. Characteristic values for load model n. 1 (LM1)

Notional lane Qk [kN] qk [kN/m2]

Lane 1 300 9.0

Lane 2 200 2.5

Lane 3 100 2.5


As said, when seeking a determined effect on the bridge, the LM1 must obviously be

arranged in the most unfavourable position and the tandem systems, when present, need to be

considered in full, that is, with all their four wheels.

By way of example, possible arrangements of the static traffic loads are represented in

figures 8 and 9, corresponding to notional lane numberings discussed below and illustrated in

figures 6 and 7, respectively.

2000

Q =300 kN

q =9 kN/m2

q =2.5 kN/m2

q =2.5 kN/m2

1k Q =300 kN1k

1k

2k 3kq =2.5 kN/m

2rk

2000

Q =200 kN2k Q =200 kN2k

2000

Q =100 kN3k Q =100 kN3k

Figure 8. Load condition corresponding to notional lane numbering in figure 6


170

2000

Q =300 kN

q =9 kN/m2

q =2.5 kN/m2

1k Q =300 kN1k

1k

2k

2000

Q =200 kN2k Q =200 kN2k

Figure 9. Load condition corresponding to notional lane numbering in figure 7

3.4 Wind actions

Wind actions can be represented by vertical and horizontal equivalent static forces.

Vertical force is orthogonal to the roadway plane, while the horizontal forces can be

represented by two components, parallel and orthogonal to the bridge’s longitudinal axis,

respectively.

The equivalent pressure exerted by the wind can be calculated through the expression

( ) ( ) 2

2beeep vzczq

ρ= , (6)

in which ρ is the air density, which is assumed to be constant and equal to 1.25 kg/m3, vb is

the basic wind velocity and ce(ze) is the so-called exposure coefficient, depending on the

reference altitude over the ground, ze, and given by

( ) ( ) ( ) ( )[ ]eveeree zIzczczc ⋅+⋅⋅= 7120

2 . (7)

Expression (7) depends on the roughness coefficient cr, on the orography factor c0 and

on the turbulence intensity Iv. The orography factor, taking into account any significant local

variations in the site’s orography, can usually be assumed equal to 1.0. Iv and cr, instead, are

defined by the following expressions

( ) ( )

( )

≥

≤<

⋅=

zzzI

zz

z

zzc

k

zI

v

e

i

ev

minmin

min

0

0

if

m 200 if

ln ,

(8)

( ) ( )

( )

≥

≤<

⋅

=zzzc

zzz

zzk

zc

r

er

er

minmin

min

0

if

m 200 if ln,

(9)


171

where ki is the turbulence factor, usually set to 1.0. The terrain factor kr, the roughness length

z0 and the minimum height zmin depend on the terrain category.

As said, the bridge in question is located in an extra-urban area which can be classified

in terrain category II, i.e. an area with low vegetation, such as grass, and isolated obstacles

(trees, buildings) with separations of at least 20 obstacle heights. For terrain category II it

results z0=0.050 m, zmin=2.0 m, kr=0.19.

The reference height ze represents the distance between the lowest ground level to the

centre of the bridge deck structure, disregarding other parts (e.g. parapets) of the reference

areas Recalling that the intrados of the structure is 20.0 m above ground level, it is

ze=20.0+(3.80/2) m= 21.90 (figure 10).

Figure 10. Evaluation of the reference height ze for wind actions

As ze>zmin, it results

( ) 164.0

m 05.0

m 90.21ln0.1

1 =

⋅=ev zI , (10)

( ) 156.1m 05.0

m 90.21ln19.0 =

⋅=er zc , (11)

whence

( ) [ ] 869.2164.0710.1156.1 22 =⋅+⋅⋅=ee zc . (12)

Considering for the site a basic wind velocity vb=27 m/s, qp(ze) results

( ) .N/m 13070.272

25.1869.2 22 =⋅⋅=ep zq

(13)


172

According to EN 1991-2, the y-axis is assumed parallel to the bridge axis, x-axis is

assumed horizontal and perpendicular to the y-axis, while the z-axis lies in the vertical plane

containing the y-axis.

The equivalent static force Fwk,x in the x direction is given by

( ) xrefxfepxwk AczqF ,,, ⋅⋅= , (14)

where the coefficient cf,x is a function of the ratio between the total deck’s width b and the

total deck’s height dtot exposed to wind.

When the bridge is unloaded the exposed height is 4.4 m, as the presence of two open

safety barriers determines an increase of 0.6 m in the exposed height.

When the bridge is loaded the exposed height increases by 2.0 m, so becoming 5.8 m.

For unloaded bridge, the coefficient cf,x is

709.14.4

6.113.05.23.1 ;3.05.2max ;4.2min, =−=

−=

tot

xfd

bc . (15)

If must be noted that the simplified approach proposed in EN1991-1-4, suggesting to

assume cf,x=1.3, is generally unsafe-sided.

Since the webs of the box girder are inclined by the angle α=10° with respect to the

vertical (figure 11), the coefficient cf,x calculated in (15) can be reduced by the factor η1

( ) 95.0.70 ; 005.01max 11 =−= αη , (16)

obtaining (figure 12)

( ) kN/m 34.94.495.0709.1307.1,1,, =⋅⋅⋅== xrefxfepxwk AczqF η . (17)

Figure 11. Web inclination allowing a reduction of the coefficient cf,x

For loaded bridge, the coefficient cf,x is

90.18.5

6.113.05.20.1 ;3.05.2max ;4.2min, =−=

−=

tot

xfd

bc ,

(18)

and also for loaded bridge the simplification cf,x=1.3 proposed in EN1991-1-4 is unsafe-sided.

Considering the reduction factor η1 calculated before, the combination value ψ0wFwk,x

for the equivalent wind force for loaded bridge results (figure 18)

( ) kN/m 21.88.595.09.1307.16.0,1,0,0 =⋅⋅⋅⋅== xrefxfepwxwkw AczqF ηψψ , (19)


173

being ψ0w=0.6.

It is interesting to note that in the Italian National Annex the exposed height of lorries

it has been set equal to 3.0 m, instead of 2.0 m. In this case, as dtot=6.8 m, cf,x=1.988 and

( ) kN/m 07.108.695.0988.1307.16.0,1,0,0 =⋅⋅⋅⋅== xrefxfepwxwkw AczqF ηψψ . (20)

Figure 12. Equivalent static force Fwk,x (unloaded bridge)

Figure 13. Equivalent static force ψψψψ0wFwk,x (loaded bridge)

Concerning the vertical action, lacking more precise data from wind tunnel tests, a

value of ±0.9 can be assumed for the force coefficient cf,z. Since the reference area Aref,z is the

horizontal projection of the bridge deck, Aref,z=b=11.6 m2/m, the equivalent static force for

unit length Fwk,z is then

( ) ( ) kN/m 64.13m 6.119.0kN/m 307.1 2,,, ±=⋅±⋅=⋅⋅= zrefzfepzwk AczqF . (21)

to be applied with eccentricity e=0.25 b=0.25⋅11.6 m=2.9 m with respect to the longitudinal

axis of the bridge.

According to EN1991-2, as Fwk,z is much lower than the permanent load (65.52 kN/m),

it could be disregarded.

Finally, when relevant, equivalent static forces in the longitudinal y-direction (the

bridge’s longitudinal axis) should be considered, which can be set equal to 25% of the forces

in the x-direction.


174

3.5 Thermal actions

As the structure under consideration is a continuous beam resting on four supports,

thermal actions induce displacements and stresses.

In order to account for thermal variations, two different contributions must be

distinguished. A uniform thermal variation along the cross section and a non-uniform

temperature variation along the section’s height, corresponding to situations where the top and

the bottom of the bridge are at different temperatures, due to differential heating or cooling

effects.

The former contribution does not provoke any stresses as long as the bridge can slide

horizontally in correspondence to its supports and it will cause only a shortening or elongation

of the structure’s line of axis (i.e. it is relevant only for design of bearings and expansion

joints).

A typical uniform temperature variation ∆TU can be derived from EN 1991-1-5.

Assuming that in the site under consideration the maximum and minimum air shade

temperatures with an annual probability of being exceeded of 0.02 are Tmax=40 °C and Tmin=-

10 °C, respectively, the uniform bridge temperature components for a steel bridge result

Te,max=56.6 °C and Te,min=-13.3 °C (figure 14).

Setting the initial bridge temperature T0 to 20 °C, the characteristic values of the

maximum expansion and contraction ranges, ∆TN,exp and ∆TN,con, result so

C 6.36206.560max,exp °=−=−=∆ TTT eN, , (22)

C 3.33)3.13(20min,0con °=−−=−=∆ eN, TTT . (23)

-50 -40 -30 -20 -10 0 10 20 30 40 50-50

-40

-30

-20

-10

0

10

20

30

40

50

60

70

1

2

3

3

2

1

Tmin Tmax

Te,m

in

T

e,m

ax

Figure 14. Evaluation of Te,max and Te,min


175

The second non uniform contribution is clearly very significant for the bridge under

consideration. EN 1991-1-5 offers two possible procedures to deal with it, provided that the

surfacing thickness is not less than 40 mm.

The first, more accurate one, calls for applying rather complex thermal variation laws

along the cross section’s height (figure 15), while the second instead makes use of simpler

linear variations. Consequently, while the first variation laws require employing dedicated

software for the structural analysis, simplified linear variations enable even manual

calculations, at least up to a certain degree. In case of steel deck structures, simplified linear

variations correspond to a raise in temperature of 18 °C for top warmer than bottom,

∆TM,heat=+18 °C, and to an increase of 13 °C for bottom warmer than top, ∆TM,cool=+13 °C,

(figure 16).

Figure 15. Accurate temperature distributions along the height of the cross section

Figure 16. Simplified temperature distributions along the height of the cross section

4 STRESS CALCULATION

4.1 Behaviour of the structure

The structural behaviour of the orthotropic steel deck bridges is usually described

taken into account three separate static functions for the stiffened plate. In effect, the stiffened

plate participates to three different resisting systems, each one characterised by precise stress

and deformation patterns.

In the local resisting system the deck plate directly sustains the traffic loads and

transfers them to the orthotropic plate system, composed by the deck plate, the longitudinal

stiffeners and the transverse beams.

In the intermediate resisting system the orthotropic system transmits the loads to the

main system.


176

Finally, in the global resisting system, the stiffened plate represents the upper flange of

the main girders.

The resultant stress pattern in the deck plate can be so obtained by the superposition of

the individual stress patterns induced by each one of the three above mentioned static systems.

4.2 Local effects on the deck plate

The stress pattern of the local resisting system is often not considered in the static

checks, as the combined effect of local plastic deformations and membrane-type behaviour

limits the stress peaks. In fact, when it collapses, the deck plate behaves mainly like a tensile

cylindrical membrane restrained by two longitudinal hinges located at the connections with

the stiffener’s webs: in this mechanism normal stresses in longitudinal direction are not

relevant.

However, if an evaluation of such a stresses is required, the wheel contact pressure can

be determined resorting to the well known hypothesis of 45° diffusion of the loads through

the surfacing and the deck plate. In this way, recalling that the surfacing thickness is 60 mm

and the deck plate thickness is 16 mm, the contact area dimensions result 536×536 mm2 for

the wheels of the tandem axle systems of LM1 (figure 17) and 736×486 mm2 for the wheels

of the isolated axle of LM2 (figures 18), respectively.


the tandem system of LM 1


the single axle of LM 2

Thus, the relative contact pressure for a single wheel of the heaviest tandem system of

LM1 is

21 kN/m 1.522

536.0 536.0

150 =⋅

=kQp (24)

and for a single wheel of the isolated axle load of LM 2 is

2kN/m 6.523486.0 736.0

200 =⋅

=Qakp . (25)

Local stresses are very important when fatigue assessments are concerned. As known,

fatigue assessments can be determinant in designing deck plates, ribs and transverse beams

details, but they are outside the scope of the present example.


177

4.3 Orthotropic deck plate analysis

The analysis of the orthotropic deck plate, whose function is to transfer the traffic

loads to the main girder, can be performed through analytical or numerical methods.

The analytical methods are based on the Hüber equation for the orthotropic plate.

Although all the calculation procedures available in the literature are based on defining an

equivalent orthotropic deck, they can be divided into two classical families, depending upon

whether both longitudinal ribs and transverse beams are spread out in calculating the

equivalent deck’s stiffness, or only longitudinal ribs are considered. The former methods

include, for instance, the Cornelius method, while the Pelikan-Esslinger method represents a

refined example of the second approach, largely applied, especially in studying influence

surfaces.

A detailed illustration of the analytical methods is beyond the scope of the present

example, but it is important to stress that in both cases local plate behaviour cannot be

determined, while the effective state of stress and deformation in the longitudinal stiffeners

can be derived only adopting the latter ones.

A very refined approach, able also to reproduce the local behaviour, consists in the

implementation of a finite-element model, made up essentially of isotropic shell elements,

which reproduces a significant portion of the deck. In the FE model the upper plate, the

stiffeners and the transverse beams are appropriately meshed as shown in figures 19 and 20.

Figure 19. FE mesh of an orthotropic deck

Figure 20. FE mesh detail

In order to reduce the degrees of freedom of the problem, alternative simplified FE

models can be implemented, using equivalent orthotropic shell elements, based on one of the

two analytical approaches illustrated before.

Since the wheel loads influence only few longitudinal stiffeners (three to five), it is

necessary to model only a limited deck width, not bigger than the width of a notional lane,

regardless of the calculation method adopted (FEM or numerical analysis), so limiting the

problem’s complexity.

The uniformly distributed loads to be considered can be determined, considering their

dispersal through the surfacing and the deck plate, as illustrated in §4.2.

4.4 Calculation of the main girder under vertical loads

The main girder is calculated simply as a continuous beam resting on four supports,

characterised by constant flexural stiffness EJx (Jx=1.151 m4).

4.4.1 Effects of the self-weight and dead loads

Applying the single source principle of EN 1990 to the structural self-weight and to

dead loads, their effects can be calculated considering a uniformly distributed load


178

kN/m 12.742,1, =+= kbkk ggG . (26)

Said A the cross section of the first span where bending moment attains its local

maximum, and denoted with B and C the intermediate supports, Gk yields the symmetrical

diagram of bending moments shown in figure 21, where

( ) kNm 8.85381kNm0.12012.7425

2

25

2 22 =⋅== LGAM kkg , (27)

( ) kNm 4.106727kNm0.12012.7410

1

10

1 22 −=⋅−=−= LGBM kkg , (28)

( ) kNm 8.26681kNm0.12012.7440

1

40

1 22 =⋅== LGCM kkg . (29)

Figure 21. Bending moment diagrams for permanent loads.

4.4.2 Effects of traffic loads

As said, since the bending moment influence surfaces of the box girder are cylindrical,

the most severe transversal load arrangement for bending moment is the one previously

illustrated in figure 8, where all three notional lanes as well as the residual area of the

carriageway are loaded.

For this reason, the total uniformly distributed load of LM1 to be applied on the bridge

is given by

( ) kN/m 75.45m 5.1kN/m 5.2m 0.3kN/m 5.25.29 22 =×+×++=kq , (30)

while the global effects of the tandem systems of LM 1 can be determined considering in the

worst longitudinal position a single concentrated load (knife load)

( ) kN 1200kN 200400600222 321 =++=++= kkktk QQQQ ; (31)

this is the sum of the axle loads of the three tandem systems applied on the three notional

lanes.

Referring to the bending moment in the side spans, in the central span and at the

intermediate supports, the most unfavourable load arrangements should be determined

according to the influence lines illustrated in figures 3, 4 and 5, respectively.

For example, to maximize the bending moment in the first span due to traffic loads

MQ(A’)kmax, the uniformly distributed load should be applied on the first and on the third span

as indicated in figure 22, while the concentrated load should be applied on the section A’,

located around 51.4 m from the first support, so that

( ) ( ) ( ) kNm 6.960567.29507 9.66548''' maxmaxmax =+=+= kQkqkQ AMAMAMt

. (32)

Sec. A

Sec. B Sec. C


179

Figure 22. Arrangement of UDL to maximize bending moment in lateral spans

To maximize the bending moment at midspan (section C) due to traffic loads,

MQ(C)kmax, the central span should be loaded with uniformly distributed traffic load (figure

23), being the concentrated load located at midspan

( ) ( ) ( ) kNm 0.746100.25200 0.49410maxmaxmax =+=+= kQkqkQ CMCMCMt

. (33)

Figure 23. Arrangement of UDL to maximize bending moment in the central span

To minimize the bending moment MQ(B)kmin due to traffic loads on cross section B,

corresponding to the second support, the first and the second span should be loaded with

uniformly distributed load (figure 24), while the concentrated load should be placed 69.24 m

away from the first support

( ) ( ) ( ) kNm 2.916402.14780 0.78860minminmin −=−−=+= kQkqkQ BMBMBMt

. (34)

Figure 24. Arrangement of UDL to minimize bending moment on the second support

When minimum traffic load effects are investigated, different load conditions are to be

considered, as indicated in the following.

The minimum bending moment in the section A’ of the first span, MQ(A’)kmin, is

obtained applying the uniformly distributed load on the central span, as just indicated in

figure 22, and the concentrated load around 49.9 m away from the second support, so that

( ) ( ) ( ) kNm 6.190243.4915 3.14109''' minminmin −=−+−=+= kQkqkQ AMAMAMt

. (35)

The minimum bending moment at midspan (section C), MQ(C)kmin, is obtained

applying the uniformly distributed load on the lateral spans, as indicated in figure 22, and the

concentrated load around 70.7 m away from the first or the last support, so that


180

( ) ( ) ( ) kNm 1.384791.5539 0.32940minminmin −=−+−=+= kQkqkQ CMCMCMt

. (36)

The maximum bending moment at intermediate support (section B), MQ(B)kmax, is

obtained applying the uniformly distributed load on the third span, and the concentrated load

around 50.5 m away from the third support, so that

( ) ( ) ( ) kNm 0.146750.3695 0.10980maxmaxmax =+=+= kQkqkQ BMBMBMt

. (37)

4.4.3 ULS combinations for permanent and traffic loads

Applying the single source principle for permanent loads and recalling that in ULS

structural verifications (STR) partial factor for permanent loads, γG, is set to 1.35 for

unfavourable effects and to 1.0 for favourable effects, and partial factor for traffic loads, γQ, is

set to 1.35 for unfavourable effects and to 0 for favourable effects, design values of bending

moments are obtained as follows, considering expression (6.10) of EN 1990.

The maximum bending moment in the first span M(A’’)dmax is obtained applying the

permanent design load Gd=1.35 Gk =100.06 kN/m on each span, the uniformly distributed

design traffic load qd=1.35 qk =61.76 kN/m on the lateral spans and the design concentrated

traffic load Qd=1.35 Qk =1620 kN on the section A’’, around 51.0 m away from the first

support, so that

( ) kNm 0.244421'' max =dAM . (38)

The corresponding bending moment and shear force diagrams are illustrated in figures

25 and 26, respectively.

Figure 25. Bending moment diagram corresponding to M(A’’)dmax in the first span

Even if, in general, cross section A’’ differs from A’, in the present example the

difference is so little that it could be disregarded. In fact, considering A’, it would result

( ) kNm 0.244364' max =dAM . (39)

Figure 26. Shear force diagram corresponding to Mdmax in the first span


181

The maximum bending moment at midspan M(C)dmax is obtained applying the


design traffic load qd=1.35 qk =61.76 kN/m on the central span and the design concentrated

traffic load Qd=1.35 Qk =1620 kN on section C, so that

( ) kNm 0.136742max =dCM . (40)

The corresponding bending moment and shear forces diagrams are illustrated in


Figure 27. Bending moment diagram corresponding to Mdmax at midspan

Figure 28. Shear force diagram corresponding to Mdmax ad midspan

The minimum bending moment on the second support M(B)dmax is obtained applying

the permanent design load Gd=1.35 Gk =100.06 kN/m on each span, the uniformly distributed

design traffic load qd=1.35 qk =61.76 kN/m on the first and the second span and the design

concentrated traffic load Qd=1.35 Qk =1620 kN approximately around 69.24 m away from the

first support, obtaining

( ) kNm 0.267796max −=dBM . (41)



Figure 29. Bending moment diagram corresponding to Mdmin on the second support


182

Figure 30. Shear force diagram corresponding to Mdmin on the second support

The minimum bending moment in the first span M(A’)dmin is obtained applying the



traffic load Qd=1.35 Qk =1620 kN on the second span, approximately around 49.9 m away

from the second support, so that

( ) kNm 4.59275' min −=dAM . (42)

The corresponding bending moment and shear force diagrams are illustrated in figures

31 and 32, respectively.

The minimum bending moment at midspan M (C)dmin is obtained applying the


design traffic load qd=1.35 qk =61.76 kN/m on the lateral spans and the design concentrated

traffic load Qd=1.35 Qk =1620 kN approximately 70.7 m away from the first or the last

support, so

( ) kNm 7.25261min −=dCM . (43)

Figure 31. Bending moment diagram corresponding to Mdmin in section A’

Figure 32. Shear force diagram corresponding to Mdmin in section A’


183



Figure 33. Bending moment diagram corresponding to Mdmin in section C

Figure 34. Shear force diagram corresponding to Mdmin in section C

The maximum bending moment on the second support M(B)dmax is obtained applying

the permanent design load Gd=1.0 Gk =74.12 kN/m on each span, the uniformly distributed

design traffic load qd=1.35 qk =61.76 kN/m on the third span and the design concentrated

traffic load Qd=1.35 Qk =1620 kN approximately around 50.5 m away from the third support,

obtaining

( ) kNm 2.86922max −=dBM . (44)



Figure 35. Bending moment diagram corresponding to Mdmax in section B

Particularly relevant cases concern minimum design shear force at the right hand end

of first span (section B-), maximum design shear force at the left hand end of the central span

(section B+) and maximum and minimum shear forces at midspan (sections C

- and C

+).

Minimum design shear force in section B- V(B

-)dmin is obtained applying the permanent

design load Gd=1.35 Gk =100.06 kN/m on each span, the uniformly distributed design traffic


184

load qd=1.35 qk =61.76 kN/m on the first and the third span and the design concentrated

traffic load Qd=1.35 Qk =1620 kN immediately left hand of section B-, obtaining the shear

force diagram illustrated in figure 37, where

( ) kN 0.12890min −=−dBV . (45)

Figure 36. Shear force diagram corresponding to Mdmax in section B

Figure 37. Shear force diagram corresponding to Vdmin in section B

-

Maximum design shear force in section B+ V(B

+)dmax is obtained applying the



traffic load Qd=1.35 Qk =1620 kN immediately right hand of section B+, obtaining the shear

force diagram illustrated in figure 38, where

( ) kN 2.11329max =+dBV . (46)

Figure 38. Shear force diagram corresponding to Vdmax in section B

+

Minimum design shear force in section C-, V(C

-)dmin is obtained applying the

uniformly distributed design traffic load qd=1.35 qk =61.76 kN/m on the left hand half of the

central span and the design concentrated traffic load Qd=1.35 Qk =1620 kN immediately left

hand of section C-, obtaining the shear force diagram illustrated in figure 39, where


185

( ) kN 0.2276min −=−dCV . (47)

-4

-3

-2

-1

0

1

2

3

4

5

0 120 240 360

V [M

N]

x [m]

Figure 39. Shear force diagram corresponding to Vdmin in section C-

Finally, maximum design shear force in section C-, V(C

-)dmax is obtained applying the

uniformly distributed design traffic load qd=1.35 qk =61.76 kN/m on the right hand half of the

central span and the design concentrated traffic load Qd=1.35 Qk =1620 kN immediately right

hand of section C-, obtaining the shear force diagram illustrated in figure 40, where

( ) kN 0.2276max =−dCV . (48)

Figure 40. Shear force diagram corresponding to Vdmax in section C

-

4.4.4 SLS combinations for permanent and traffic loads

Concerning SLS verifications, combinations of permanent and traffic load are

generally relevant only for characteristic and frequent load combinations, as quasi-permanent

values of traffic loads are zero, except in very particular cases.

Significant load arrangements to be considered look very similar to those illustrated

before regarding ULS verifications, so they will not be discussed in detail.

It is just necessary to recall that frequent values of traffic loads are obtained via the ψ1

factors, which depend on the nature of the load: in fact, ψ1=0.75 for the tandem systems of

LM1, for the isolated single axle of LM2 and for crowd loading (LM4), while ψ1=0.40 for the

uniformly distributed loads of LM1.

4.4.5 Wind effects

When relevant, the vertical component of the pressure exerted by the wind should be

considered as uniformly distributed load acting on the entire length of the bridge.

The bending moment diagram is then analogous to that due to dead loads and

therefore, recalling expression (21), for unloaded bridge it results


186

( ) kNm 3.15713kNm0.12064.1325

2

25

2 22, ±=⋅±== LFAM zwkkwz , (49)

( ) kNm 6.19641kNm0.12064.1310

1

10

1 22, mm =⋅=−= LFBM zwkkwz , (50)

( ) kNm 4.4910kNm0.12064.1340

1

40

1 22, ±=⋅±== LFCM zwkkwz , (51)

while for loaded bridge, the combination value ψ0wFwk,z=±8.18 kN/m, ψ0w=0.6, should be

considered, in place of Fwk,z.

Partial factor γQ for wind actions in ULS combinations is γQ=1.50 if unfavourable and

γQ=0 if favourable.

4.5 Design values of horizontal wind actions

The design values of horizontal wind action are derived from expressions given in

§3.4, recalling that γQ=1.50 if the effect is unfavourable and γQ=0 if the effect is favourable.

4.6 Effects of thermal variations

As said, in a continuous bridge the uniform temperature variation ∆TU induces only

longitudinal displacements. Considering that the coefficient of thermal expansion for steel is

αT=12⋅10-6

°C-1

, the longitudinal displacements uy associated with the uniform temperature

variations ∆TN,exp=36.6 °C and ∆TN,exp=33.3 °C result

mm/m 439.0100010126.36 6exp, =⋅⋅⋅= −

yu , (52)

mm/m 40.0100010123.33 6, =⋅⋅⋅= −conyu . (53)

The vertical temperature differences, assumed to be linear through the cross section’s

height, produce a bending moment diagram that is linear in the two side spans and constant in

the central one. In figure 41 is reported the bending moment diagram for heating (top warmer

then bottom).

120 120 120

Figure 41. Bending moment diagram for vertical temperature differences (heating)

Recalling that, for the continuous beams considered in the present example, the

extreme values of bending moments are given by

h

TEJM T

T

∆−=∆α

5

6, (54)

it results


187

kNm 8.238498.3

181012665.1101.2

5

6 68 −=⋅⋅⋅⋅⋅⋅−=

−

∆TM , (55)

in case of heating (top warmer then bottom), and

( )kNm 9.17224

8.3

131012665.1101.2

5

6 68 =−⋅⋅⋅⋅⋅⋅−=

−

∆TM . (56)

in case of cooling (bottom warmer then top).

When, depending on the local site conditions, the adoption of a more precise law of

variation for vertical temperature differences is required, as described in §3.5, more refined

analyses should be performed.

4.7 Assessment of the members of the cross section

To avoid use of complicated models, the assessment of members of the cross section

of the box girder can be performed in a very simplified way considering the classical

superposition of two structural systems.

In the first structural system, where member loads are considered, the cross section is

suitably modified so that it results kinematically determinate or restrained. In this restrained

structure the member-end forces due to member applied loads are calculated and the

equivalent fixed-end structure forces are determined.

In the second structural system, the static equilibrium in the kinematically released

structure is restored by applying the negative of the equivalent fixed-end structure forces on

the actual structure.

The application of this method to the relevant cases discussed in §3.3 and illustrated in

figures 8 and 9 is summarized in figures 42 and 43, respectively.

R1

M1

M1 M2

M2

R1 R2

R2

Figure 42. Structural systems for assessment of cross section members (ref. figure 8)


188

R1

M1

M1 M2

M2

R1 R2

R2

Figure 43. Structural systems for assessment of cross section members (ref. figure 9)

5 FINAL REMARKS

In the present chapter, a relevant example is developed concerning steel bridges.

The design of a three spans continuous bridge with orthotropic steel deck and box

cross section is illustrated, paying special attention to the evaluation of extreme effects

induced by permanent, climatic and traffic actions according to EN 1991.

The road bridge, whose total length is 360 m, is located in an extra-urban area, so that

also crowd loading can be disregarded.

Starting from the usual distinction between local and global static behaviour, different

methodological approaches for orthotropic deck bridges analysis are illustrated, stressing their

theoretical bases.

Discussion of load combinations according to EN 1990 rules demonstrates also in this

case the soundness of Eurocode system for bridge design.

6 REFERENCES


weight, imposed loads for buildings, CEN, Brussels, 2002.

[2] EN 1991-1-4 Eurocode 1 Actions on structures. Part 1-4 General actions. Wind actions,


[3] EN 1991-1-5 Eurocode 1 Actions on structures. Part 1-5 General actions. Thermal

actions, CEN, Brussels 2004.

[4] EN 1991-2 Eurocode 1 Actions on structures. Part 2 Traffic loads on bridges, CEN,

Brussels, 2003.


189


[6] EN 1993-1-1 Eurocode 3 Design of steel structures. Part 1-1 General rules and rules for

buildings, CEN, Brussels, 2005.

[7] EN 1993-2 Eurocode 3 Design of steel structures. Part 2 Steel bridges, CEN, Brussels,

2006.


190

Chapter 10: Case study - Design of a composite bridge

191

CHAPTER 10: CASE STUDY - DESIGN OF A COMPOSITE BRIDGE

Peter Tanner1, Carlos Lara

1 and Angel Arteaga

1

1Institute of Construction Science, IETcc – CSIC, Madrid, Spain

Summary

The identification of all the actions and action effects likely to arise during

construction and future use is a crucial step in bridge design. Actions and effects that go

unrecognised in this stage and are consequently ignored in further analyses may result in

structural designs with an unacceptably low level of reliability. With proper detection, on the

contrary, suitable safety measures can be readily adopted to ensure the required levels of

reliability are reached. This chapter deals with the application of the structural Eurocodes,

particularly EN-1991, “Actions on Structures”, to the analysis of composite decks on road

bridges. The effects of actions and combinations of actions relevant to the verification of the

ultimate and serviceability limit states in bridge decks are also studied.

1 INTRODUCTION

The primary purpose of the present chapter is to illustrate how Eurocode EN 1991,

“Actions on Structures” and other structural Eurocodes can be applied to the analysis of

composite steel and concrete decks on road bridges. The specific aims sought are:

1. to define the site-specific actions and environmental effects included in the EN

1991 models, as well as the combinations of actions that must be taken into

account when verifying conformity to structural safety and serviceability

requirements.

2. to identify the action effects to be used to verify bridge deck conformity to

structural safety and serviceability requirements for each relevant combination of

actions.

EN 1990/A1 [2] provides basis for determination of combinations of actions for

ultimate and serviceability limit state verifications of bridges. The aim of this Chapter is to

describe principles of load combinations. Permanent actions, traffic loads and climatic actions

due to wind, snow and temperature are considered in accordance with relevant Parts of EN

1991. Supplementary information on the traffic load models provided in EN 1991-2 [3] is

given in the Background document [4] which is expected to be available on the JRC web site.

1.1 Scope

Although bridge deck design normally covers serviceability, structural safety, fatigue

resistance and durability of all structural members, i.e., the bridge deck, piers, abutments and

foundations, the present chapter deals with the structural safety and serviceability of the

bridge deck only. The verification of ultimate, serviceability and fatigue limit states is not

explicitly covered. The example used illustrates only the questions relating to overall

structural analysis for determining the relevant action effects.


192

In the example introduced in this chapter, adapted from [1], the bridge deck consists of

two steel girders and a concrete slab. This example was chosen because as a result of the

ample coverage of this type of deck in Eurocode 4, Part 2 on the design of composite steel

and concrete bridges [2], Eurocode provisions can be applied rather straightforwardly.

For the intents and purposes of this example, certain simplifying assumptions have

been made, particularly respecting construction stages (§2.4) and the design situations

considered in the verification of ultimate limit states (§3.1). Nonetheless, the deck

dimensions, materials used and assumptions hereunder are realistic and the deck in the

example meets all the applicable structural safety, serviceability, fatigue and durability

requirements laid down in the relevant structural Eurocodes.

1.2 Chapter organisation

The introduction, the first of this chapter, describes the aims and scope of the example.

It is followed by a review of the bridge deck, including geometric characteristics, material

properties and the construction sequence.

Section 3 identifies all the actions and their effects likely to arise during bridge deck

construction and future use, along with the specific load models for the actions that should be

considered in analysis and design calculations.

A number of the features of the structural model used for the analysis of the composite

bridge deck are described in Section 4.

The values to be used for internal forces and moments and the displacements found

with bridge deck analysis are summarised in section 5.

Section 6 reviews the partial factors used to calculate the design values of the effects

of actions and resistance thereto, while Section 7 discusses the combinations of actions

applied to verify structural conformity to safety and serviceability standards.

The chapter concludes with an eighth section containing general remarks on

composite bridge deck analysis as stipulated in structural Eurocodes. 2 BRIDGE DECK DESCRIPTION

2.1 Geometry

The solution adopted is a continuous composite bridge deck carrying three lanes of

road traffic, with a constant cross-sectional height of 2,15 m and a total width of 10 m. The

deck cross-section comprises an in situ concrete slab 0,25 m deep and two 1,9-m deep welded

steel girders, set at a distance of 5,0 m (Figure 1). The bridge has a total length of 103,5 m

and three spans: 30,0 +43,5+30,0 m (Figure 2).

0,5 m 0,5 m

0,2

5 m

1,9

0 m

2,5 m 5,0 m 2,5 m

10,0 m

Figure 1. Composite deck cross-section for a road bridge


193

30,0 m 43,5 m 30,0 m

103,5 m

Figure 2. Static system

2.2 Material properties

Structural steel:

As per EN 1993-1-1 [3], §3.2:

Steel grade Nominal thickness

t [mm]

Yield strength

fy [N/mm2]

Modulus of

elasticity Ea

[kN/mm2]

S355 t ≤ 40 mm 355 210

S460 40 < t ≤ 80 mm 430

Concrete:

As per EN 1992-1-1 [4], §3.1:

Strength class C30

Characteristic cylinder strength fck = 30 N/mm2

Secant modulus of elasticity Ecm = 33 kN/mm2

Reinforcing steel 1:

As per EN 1992-1-1 [4], §3.2 and Annex C:

Steel grade B 500

Specified yield strength fsk = 500 N/mm2

Modulus of elasticity Es = 200 kN/mm2

In composite structures, the design value of the modulus of elasticity may be taken to

be equal to the value for structural steel: Es = 210 kN/mm2 ([2], §3.2.2).

Stud connectors 2:

Nominal ultimate strength fu = 450 N/mm2

Diameter φ = 19 mm

Height h = 125 mm.

1 While in EN 1992-1-1 [4] the yield strength of reinforcing steel is symbolized as fyk,, in EN 1994-1-1 [5] it is

shown fsk to distinguish it from structural steel.

2 In the context of the material recommended for stud connectors, reference is made in [2], §3.4.2.1, to EN-

13918.


194

2.3 Steel plate thickness, stiffeners and diaphragms

According to the distribution of stiffeners and diaphragms and the steel plate thickness

set out in Figure 3, referred to the bridge deck surface, a total of 100 kg/m2 of structural steel

is needed.

2,08 2,08 2,08 3,125 3,1253,125 3,35 3,35 3,35

3,35 6,70 6,70 5,00 5,00 6,25 6,25 6,25

450x25

tw = 12 mm

600x40 600x40 600x25 600x60 (S460M)

tw = 15 mm

6,25

3,125 3,00

30,00 m 21,75 m

3,00 3,35 3,35 2,50 2,50 2,50 2,50

ST

ST: TRANSVERSE STIFFENER; DI: INTERMEDIATE DIAPHRAGM; DB: BEARING DIAPHRAGM; SL: LONGITUDINAL STIFFENER

DI ST ST DI ST DI ST DI ST ST

DB SL SL

ST STDI DI

tw = 12 mm

BOTTOM FLANGE

DI

WEB

TOP FLANGE

Figure 3. Distribution of stiffeners and diaphragms

2.4 Construction

For the purpose of structural analysis, bridge construction is assumed to be divided

into the following stages:

- Erection of the steel structure.

- Casting of the in situ concrete in a single lift across the entire length of the bridge

without temporary supports.

- Simultaneous application of all dead loads, in particular the vehicle restraint system

and the asphalt layer, two weeks after the in situ concrete is poured.

3 ACTIONS

3.1 Introduction Structural reliability is closely related to the recognition of the actions and effects to

which the structure may be exposed during construction and use. The goal is to identify all

actions and effects likely to arise. Only then can a solution be found that meets the basic

requirements laid down in EN 1990 [6], §2.1. In light of the importance of this step, the

actions and action effects that might be relevant to the bridge deck in the example are

described below. Only persistent and transient design situations are considered for ultimate

limit state verification. In other words, the analysis does not address the effects of either

accidental or seismic actions. Loads affecting only one half of the bridge deck are considered,

however.

3.2 Permanent actions

Self-weight of steel structure

The self-weight of the steel structure is sustained by the steel structure alone without

temporary supports:

21,0 kN/m 5,0 m 5,0 kN/ms

g = ⋅ =


195

Self-weight of in situ concrete slab

Like the self-weight of the steel structure, the weight of the in situ concrete is borne by

the steel structure alone [7]:

30,25 m 25,0 kN/m 5,0 m 31, 25 kN/mc

g = ⋅ ⋅ =

Dead loads

The dead loads consist primarily of the vehicle restraint system and the asphalt layer

and are borne by the composite structure.

21,6 kN/m 5,0 m 8,0 kN/mdl

g = ⋅ =

3.2 Creep and shrinkage

Shrinkage Pursuant to EN 1994-2 [2] §3.1(3) and EN 1992-1-1 [4] §3.1.4 and Annex B, total

shrinkage strain has two components, drying and autogenous shrinkage strain. Taking

ambient relative humidity to be 70 % and assuming that the concrete is manufactured with

Class N cement:

3

70%

701,55 1 1,018

100β

= ⋅ − = .

The basic drying shrinkage strain, εcd,0, is calculated as:

( ) 6 6

,0

38 0,85 220 110 4 exp 0,12 10 1,018 362 10

10cdε − − = + ⋅ ⋅ − ⋅ ⋅ ⋅ = ⋅

The notional height is: 0

2 2 2,50, 244 m

20,5

cAh

u

⋅= = =

where Ac is the cross-sectional area and u the perimeter of the member in contact with the

atmosphere.

3

10000( , ) 0,985

10000 0,04 244ds st tβ = =

+ ⋅

Drying shrinkage strain develops over time as:

6 6( ) 0,985 0,85 362 10 303 10cd tε − −= ⋅ ⋅ ⋅ = ⋅

( )0.5( ) 1 exp 0,2 10000 1,0as

tβ = − − ⋅ =

6 6( ) 2,5 (30 10) 10 50 10ca

ε − −∞ = ⋅ − ⋅ = ⋅

The autogenus shrinkage strain is:

6 6( ) 1,0 50 10 50 10ca

ε − −∞ = ⋅ ⋅ = ⋅

and the total shrinkage strain at t = ∞ is:


196

6 6(303 50) 10 353 10cs

ε − −∞ = + ⋅ = ⋅

Creep

Given that all dead loads are applied 15 days after the in situ concrete is poured and

that the ambient relative humidity is 70 %, it follows that ([2], §2.3.3 and §3.1; [4], §3.1.4 and

Annex B):

70% 3

1 70 /1001 1,48

0,1 244ϕ −= + =

⋅

16,8 16,8( ) 2,73

30 8cm

cm

ff

β = = =+

0 0,2 0,2

0

1 1( ) 0,55

0,1 0,1 15t

tβ = = =

+ +

where t0 is the age of concrete at first loading in days. The notional creep coefficient 0ϕ is:

0 1,48 2,73 0,55 2, 22ϕ = ⋅ ⋅ =

( )181,5 244 1 0,012 70 250 632 1500Hβ = ⋅ ⋅ + ⋅ + = ≤

( )0.3

0

10000, 0,982

632 10000c

tβ ∞ = = +

The final creep coefficient is:

( ) ( )0 0 0, , 2, 22 0,982 2,18t tϕ ϕ β∞ = ⋅ ∞ = ⋅ =

3.3 Variable actions

Traffic loads. Further to EN 1991-2 [8], §4.3.2, only Load Model 1 is applied. In this model, used

for general verification calculations, both concentrated two-axle tandem loading and

uniformly distributed loads are considered.

The carriageway width w is assumed to be equal to the distance between the inner

limits of the vehicle restraint system, therefore w=9,0 m. As w>6,0 m, the number of

conventional lanes, each of which is wl=3,0 m wide, is afforded by the relation (Figure 4):

9,0int int 3

3 3l

wn

= = =

.

Since the span length is greater than 10 m, each of the three tandem load systems may

be replaced by a one-axle load equal to the total load exerted by the two axles constituting the

system [8], §4.3.2 (6).

Uniformly distributed loads: Lane 1 q1k = 9,0 kN/m2

other notional lanes qi,k = 2,5 kN/m2


197

remaining area qrk = 2,5 kN/m2

Tandem system Lane 1 Q1k = 300 kN (1 axle of 600 kN)

Lane 2 Q2k = 200 kN (1 axle of 400 kN)

Lane 3 Q3k = 1 00 kN (1 axle of 200 kN)

other notional lanes Qik = 0.

Figure 4. Traffic loads on the bridge deck

Due to the absence of specific indications related to the expected traffic, the

adjustment factors αQ and αq are assumed to be equal to 1,0.

Temperature

Linear temperature variation from the upper to the lower face of the bridge deck (see

§6.1.4.1 of [9] and §4 of chapter 5):

Top surface warmer (heating) ∆T = +15 ºC (7 ºC/m)

Bottom surface warmer (cooling) ∆T = −18 ºC (-8,4 ºC/m)

The same linear thermal expansion coefficient is assumed for the steel and the

concrete, namely α = 10·10-6

ºC-1

([2], §3.1(1) and §5.4.2.5(3); [4], §3.1.3(5)).

The value specified in EN 1991-2 [8], section 5, for uniform thermal variation is

disregarded because it is irrelevant to the present study.

Construction loads during the casting of concrete

Pursuant to EN 1991-1-6 [10], §4.11.2, the following construction loads are taken into

account simultaneously in the calculations to verify steel structure conformity. These loads

are intended to be positioned to cause the maximum effects, which may be symmetrical or

not.

- Actual area:

Self-weight of the formwork: qcc,k = 0,5 kN/m2

Q3k = 100 kN Q3k = 100 kN

2,0 m q1k = 9,0 kN/m

2 q2,3k = 2,5 kN/m

2

Q1k = 300 kN Q1k = 300 kN

2,0 m

Q2k = 200 kN Q2k = 200 kN

2,0 m

Lane 1 Lane 2 Lane 3

wl = 3,0 m wl = 3,0 m wl = 3,0 m

w = 9,0 m


198

Weight of the fresh concrete (density 26 kN/m3): qcf,k=0,25 26=6,5 kN/m

2

- Outside the working area:

Working personnel with small site equipment: qca,k = 0,75 kN/m2

- Inside the working area (3 m×3m):

10 % of the self-weight of fresh in situ concrete, but 0,75 ≤ qca,3·3 ≤ 1,5 kN/m2:

qca,3·3=0,1 6,5= 0,65 kN/m2<0,75 kN/m

2

therefore, in the present case qca,3·3 = 0,75 kN/m2.

4 STRUCTURAL ANALYSIS

4.1 Effective width

4.1.1 General remarks

The effective widths calculated in the items below, which are valid for verifying the

ultimate and serviceability limit states, refer to the concrete slab only. The reduction set out in

EN 1993-1-5 [11] should be applied to the steel flanges, in the present case ψel = 1.

4.1.2 Global bridge deck analysis

Each span is assumed to have a constant effective width across its entire length, which

is taken to be the mid-span value calculated as described in item 4.1.2 ([2], §5.4.1.2(4)).

4.1.3 Effective width for verifying cross-sections

The effective width of the concrete slab for verifying cross-sectional conformity to

requirements is determined as specified in EN 1994-2 [2], §5.4.1.2.

At mid-span or over an internal support, the total effective width beff is determined

from.

0eff eib b b= + Σ

where: / 8ei e i

b L b= ≤

bi is the actual width;

Le is the equivalent span

(approximate distance between

points of zero bending moment);

b0 is the distance between the

centres of the outstand shear

connectors.

In the present case it is assumed b0 = 0,20 m.

The effective width at the end supports is corrected by applying the βi factor:

,0 0eff i eib b bβ= + Σ ⋅

where:

(0,55 0,025 / ) 1,0i e ei

L bβ = + ⋅ ≤ ,

Consequently:

(0,55 0,025 25,5 / (2,5 0,1) 0,82i

β = + ⋅ − =

b

b

be1

eff

e2bo


199

,0 0,2 2 0,82 2,5 4,3 meff

b = + ⋅ ⋅ = .

The resulting distribution of equivalent spans and effective widths is shown in Figure

5.

Figure 5. Distribution of equivalent spans and effective widths

4.1.4 Construction

Given the construction sequence described in §3.4, the procedure to verify steel

structure conformity with the legislation must cover the actions generated by its self-weight,

the weight of the in situ concrete and the construction loads ([2], §5.4.2.4). The construction

sequence is also used as a factor in composite deck verification.

4.1.5 Creep

According to [2], §5.4.2.2, modular ratios nL for the concrete may be used to calculate

the effects of creep. Depending on the type of loading, the modular ratios are given by:

0 (1 )L L t

n n ψ ϕ= +

where: n0 is the modular ratio Ea /Ecm for short-term loading;

ψL is the creep multiplier depending on the type of loading;

φt is the creep coefficient;

Ecm is the secant modulus of elasticity of the concrete for short-term

loading.

The modular ratio n0 for analysing the structure when exposed to traffic loads and

temperature (and dead loads, where the analysis is performed for t = 0) is:

0 / 210 / 33 6, 4a cm

n E E= = =

The ratio for permanent loads3 is:

3 In this case, the dead load only, because the construction entails no shoring.

7,5 m

30,0 m

Le = 25,5 m

Le = 18,4 m

eff b 4,3 m 5,0 m

15,0 m 7,5 m 15,0 m

30,0 m

21,7 m

43,5 m

4,8 m 5,0 m

10,9 m

4,8 m 5,0m

7,5 m10,9 m

4,3 m

7,5 m

Le = 25,5 mLe = 30,5 m

Le = 18,4 m


200

6.4 (1 1,1 2,18) 21,7n = ⋅ + ⋅ =

The ratio for primary and secondary shrinkage effects is:

6,4 (1 0,55 2,18) 14n = ⋅ + ⋅ = .

4.2 Concrete cracking

Given that the ratio between lengths of adjacent continuous spans is

30,0 / 43,5 0,7 0,6= > , the effect of cracking is calculated by applying the flexural stiffness

of the cracked section to 15% of the span on both sides of each internal support ([2],

§5.4.2.3). This simplification is acceptable for all calculations with the exception of

connection design, where the result might err on the unsafe side.

Further to the previous assumption, the length affected by cracking is 0,15×30,0 = 4,5

m in the lateral spans and 0,15×43,5=6.5 m in the central span, both adjacent to each support.

Since the thickness of the web and the bottom flange is designed to change at a

distance of 5,0 m from the support on each side, for the sake of simplification, this is the

distance used as the region subject to cracking in the three spans.

4.3 Mechanical properties of cross-sections

As discussed in §4.1.2 and §4.1.3, a constant effective width of beff = 5 m along the

entire deck is assumed for structural analysis, obtaining the section properties reported below.

Type 1 section: over piers (5 m on both sides)

= 450 x 25

1,9

0 m

= 600 x 60 (S 460)

= 1815 x 15

5,0 m

0,2

5 m

Ø 20 a 20 (s)

Ø 16 a 20 (i)

Figure 6. Type 1 section

Table 1. Type 1 section properties

Structural steel section

Transformed section Cracked Section

n = 6.4 n = 14 n = 21,7

Area [m2] 0,074 0,270 0,164 0,132 0,087

Inertia [m4] 0,041 0,143 0,118 0,102 0,063

v [m] 1,247 0,504 0,749 0,898 1,291

v’ [m] 0,653 1,646 1,401 1,252 0,859

NOTE 1: v is the distance between the uppermost fibre in the section and the centroid.

v’ is the distance between the centroid and the lowermost fibre in the section.

In the cracked section, the upper face of the concrete slab is regarded to be the uppermost

fibre.

NOTE 2: The contribution of the reinforcing steel is disregarded in the calculations to determine the

characteristics of the transformed section.


201

Type 2 section: over abutments (5 m from bearing)

5,0 m

1,9

0 m

0,2

5 m

= 450 x 25

= 600 x 25

= 1850 x 12




Transformed section

n = 6.4 n = 14 n = 21,7

Area [m2] 0,048 0,244 0,138 0,106

Inertia [m4] 0,029 0,082 0,071 0,065

v [m] 1,023 0,353 0,529 0,649

v’ [m] 0,877 1,797 1,621 1,501

Type 3 section: span

= 600 x 40

= 1835 x 12

5,0 m

= 450 x 25

1,9

0 m

0,2

5 m




Transformed section

n = 6.4 n = 14 n = 21,7

Area [m2] 0,057 0,253 0,147 0,115

Inertia [m4] 0,034 0,108 0,092 0,082

v [m] 1,153 0,415 0,624 0,762

v’ [m] 0,747 1,735 1,526 1,388


202

4.4 Structural model

The structural model is a continuous beam whose characteristics are shown in Figure

9.

2

5,0 m

103,5 m

30,0 m

20,0 m 10,0 m 33,5 m 10,0 m 20,0 m 5,0 m

43,5 m

2

30,0 m

Type of section 3 1 3 1 3

Figure 9. Structural model adopted

The characteristics considered for each section type are given in Table 4, depending

on the applied action.

Table 4. Characteristics by section type and applied action

TYPE 1 SECTION TYPE 2 SECTION TYPE 3 SECTION

Self-weight of steel structure Structural steel Structural steel Structural steel

Self-weight of in situ concrete Structural steel Structural steel Structural steel

Dead load t = 0 Cracked section Transformed n=6,4 Transformed n=6,4

t = ∞ Cracked section Transformed n=21,7 Transformed n=21,7

Traffic load Cracked section Transformed n=6,4 Transformed n=6,4

Temperature Cracked section Transformed n=6,4 Transformed n=6,4

Shrinkage Cracked section Transformed n=14,0 Transformed n=14,0

5 ACTION EFFECTS

5.1 Internal forces and moments

Elastic analysis is used to calculate the action effects in the sections over the pier and

at mid-span in the intermediate bay. The findings are given in Table 5. The values for shear

refer to the section over the pier in the intermediate span.

The effect of thermal action is calculated by entering the gradient set out in §3.4 above

in the structural model.

Primary and secondary shrinkage-induced effects are estimated as follows ([2],

§5.4.2.2):

- in light of the properties of the sections where n = 14, the forces and moments

applied at the two ends of the deck are as follows (figure 10)

66 33 10

5 0.25 353 10 6649 kN1 0.55 2.18 2.19

cmc cs

EN A ε −

∞ ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ = + ⋅

( )0.256649 0.529 0.125 2686 kN m

2M N v

= ⋅ − = ⋅ − = ⋅

*


203

where the value of v is that pertaining to the section over the abutment;

- in addition to the effects discussed above, tensile stresses equal to N/Ac (using the

aforementioned values of N and Ac) must be considered to estimate the shrinkage-

induced stress in the slab.

Table 5. Internal forces and moments

Over pier At mid-span

M (kN·m) V (kN) M (kN·m) V (kN)

Self-weight of steel structure -744 +109 +439 0

Self-weight of in situ concrete -4648 +680 +2744 0

Dead load t = 0

t = ∞

-1011

-1087

+174

+174

+881

+805

0

0

Uniformly distributed load -4214 +707 +4380 0

Tandem system Mmax = -2760

Mconc = 0

Vconc = +519

Vmax = +800 +5938 +400

Temperature Cooling

Heating

-1709

+1424

0

0

-1709

+1424

0

0

Shrinkage -394 0 -394 0

103,5 m

30,0 m43,5 m30,0 m

N

M

N

M

Figure 9. Forces and moments to be applied to estimate shrinkage effects

5.2 Vertical displacements

The following mid-span displacements in the intermediate bay are obtained from the

model using the properties of the transformed sections where the effective width of the

concrete is reduced, according to the modular ratio related with the type of loading.

The mechanical property values used to calculate the deflection induced by distributed

traffic loads and tandem system are the values for the short-term loads at t = 0 and t = ∞.

Table 6. Displacements at mid-span in the central bay

Load state f [mm] (t = 0) f [mm] (t = ∞)

Self-weight of steel structure 8 8

Self-weight of concrete 51 51

Dead load 6 7

Shrinkage 0 5

Total traffic load 22 22

Distributed load on the central span 34 34

Tandem system 33 33


204

6 DESIGN VALUES

6.1 Effects of actions

In serviceability limit state verifications, the partial factors for the actions are assumed

to be equal to 1,0 ([6], §A1.4.1 (1)).

In ultimate limit state verifications, the partial factors used are given in [12] (§A.2.3)

and listed in Table 7 below:

Table 7. Partial factors for actions, ultimate limit state calculations

Type of action Partial factor

Permanent loads γG = 1,35

Traffic loads γQ = 1,35

Thermal action γQ = 1,5

Shrinkage γGsc = 1,0 ([2] §2.3.3)

6.2 Material properties

In serviceability limit state verifications, the partial factors for material properties are

assumed to be 1,0 ([6], section 6.5.4(1)).

In ultimate limit state verifications, the partial factors used are given in [2] (§2.4) and

listed in Table 8 below:

Table 8. Partial factors for material properties, ultimate limit state calculations

Material Partial factor

Structural steel γM0 = 1,0

γM1 = 1,1

Concrete γc = 1,5

Reinforcing steel γs = 1,15

The values used for partial factors γM0 and γM1 are as recommended in EN 1993-2

[13].

7 COMBINATIONS OF ACTIONS

7.1 ψ factor values

The ψ factors for the action combinations considered in this study, given in the

following table 9, are the factors recommended in EN 1990/A1 (Annex A2, on application to

bridges) [12].

7.2 Combinations for the verification of ultimate limit states4 (ULS)

The combinations of actions used for ULS verification, applying the above action

combination and partial factors, are:

4 Only combinations for t = ∞ are used in the present example.


205

Table 9. Combination factors used for ultimate limit state calculations

Type of action Ψ0 Ψ1,infq Ψ1 Ψ2

Uniformly distributed load 0,4 0,8 0,4 0,0

Tandem system 0,75 0,8 0,75 0,0

Thermal action 0,0 (ULS) 0,6 (SLS)

0,8 0,6 0,5

a) 1.35 1.35p scG G Q⋅ + + ⋅

b) 0 01.35 1.5 1.35 ( )p sc TS UDLG G T Q Qψ ψ⋅ + + ⋅ + ⋅ ⋅ + ⋅

where: ψ0 = 0,75 for tandem system (TS)

ψ0 = 0,40 for uniformly distributed loads (UDL)

Gp permanent loads

Gsc shrinkage

Q traffic loads

QTS traffic loads due to tandem system

QUDL uniformly distributed traffic loads

T temperature

7.2.1 Mid-span section

Where traffic loads are assumed to be the predominant variable action:

1.35 1.35p scG G Q⋅ + + ⋅

( ), 1.35 · 439 2744 805 394 1.35 (4380 5938) 18919 kN mEd maxM = + + − + ⋅ + = ⋅

1.35 · 400 540 kNEdV = =

Where thermal action is regarded to be the predominant variable action:

0 01.35 1.5 1.35 ( )p sc TS UDLG G T Q Qψ ψ⋅ + + ⋅ + ⋅ ⋅ + ⋅

1.35 (439 2744 805) 394 1.5 1424 1.35 (0.4 4380 0.75 5938)

15503 kN m

Ed,maxM = ⋅ + + − + ⋅ + ⋅ ⋅ + ⋅

= ⋅

1.35 · 0.75 ·400 405 kNEdV = =

7.2.2 Support section

The two combinations studied are: minimum bending moment and concomitant shear,

and maximum shear and concomitant bending moment.

- 1st combination: MEd,min - VEd,conc


1.35 1.35p sc

G G Q⋅ + + ⋅


206

1.35 (-744 4648 1087) 394 1.35 (-4214 2760) 18556 kN mEd,min

M = ⋅ − − − + ⋅ − = − ⋅

, 1.35 (109 680 174) 1.35 (707 519) 2955 kNEd conc

V = ⋅ + + + ⋅ + =


0 01.35 1.5 1.35 ( )p sc TS UDL

G G T Q Qψ ψ⋅ + + ⋅ + ⋅ ⋅ + ⋅

1.35 (-744 4648 1087) 394 1.5 (-1709) 1.35 (0.4 (-4214) 0.75 (-2760))

-16774 kN m

Ed,minM = ⋅ − − − + ⋅ + ⋅ ⋅ + ⋅

= ⋅

, 1.35 (109 680 174) 1.35 (0.4 707 0.75 519) 2207 kNEd concV = ⋅ + + + ⋅ ⋅ + ⋅ = .

- 2nd

combination: MEd,conc - VEd,max


1.35 1.35p scG G Q⋅ + + ⋅

, 1.35 (-744 4648 1087) 394 1.35 (-4214) -14829 kN mEd concM = ⋅ − − − + ⋅ = ⋅

, 1.35 (109 680 174) 1.35 (707 800) 3334 kNEd maxV = ⋅ + + + ⋅ + =


This combination does not condition the results.

Table 10 summarises the action effects found for the mid-span and support sections.

Table 10. Moments and shear forces acting on the mid-span and support sections

Type of section Action effects M [kN·m] V [kN]

Mid-span MEd,max - VEd,con 18919 540

Support MEd,min - VEd,con -18556 2955

MEd,conc – VEd,max -14829 3334

7.3 Combinations for verifying serviceability limit states (SLS)

The following combinations of actions are established for verifying SLS ([6], [8] and

[13]):

- Characteristic combination: (a) 0.6p scG G Q T+ + + ⋅

(b) 0 0p sc TS UDLG G T Q Qψ ψ+ + + ⋅ + ⋅

Frequent combination: (a) 1 1 0.5p sc TS UDLG G Q Q Tψ ψ+ + ⋅ + ⋅ + ⋅

(b) 0.6p scG G T+ + ⋅

Quasi-permanent combination: 0.5p scG G T+ + ⋅

where: ψ0 = ψ1 = 0,75 for tandem system (TS)

ψ0 = ψ1 = 0,40 for uniformly distributed loads (UDL).


207

8 FINAL REMARKS

In this chapter, the provisions of a number of items in EN-1991, “Actions on

Structures”, along with specific stipulations for composite steel and concrete bridges and

other structural Eurocodes are applied to analyse a composite steel and concrete bridge deck

for vehicle traffic. The following concluding remarks are in order:

- The study discussed hereunder focuses on the application, as reliably as possible, of

the rules laid down in the structural Eurocodes referenced.

- In keeping with the purpose and scope of this chapter, all the relevant actions and

their effects likely to arise during bridge deck construction and future use are

identified. The specific load models for the actions that should be considered in

analysis and design calculations are also determined.

- Further to the requirements for safety and serviceability of composite steel and

concrete structures, the relevant action effects are calculated and the combinations

of actions for verifying ultimate and serviceability limit states are determined for

the example chosen.

9 REFERENCES

[1] Monografía M-10. Comprobación de un Tablero Mixto. Comisión 5, Grupo de Trabajo

5/3 “Puentes Mixtos”, Asociación Científico-técnica del Hormigón Estructural, Madrid,

2006, ISBN 84-89670-47-1.

[2] EN 1994-2 Eurocode 4: Design of composite steel and concrete structures. Part 2: Rules

for bridges. CEN, Brussels, 2005.

[3] EN 1993-1-1 Eurocode 3: Design of steel structures. Part 1-1: General rules and rules

for buildings. CEN, Brussels, 2005.

[4] EN 1992-1-1 Eurocode 2: Design of concrete structures. Part 1-1: General rules and

rules for buildings. CEN, Brussels, 2004.

[5] EN 1994-1-1 Eurocode 4: Design of composite steel and concrete structures. Part 1-1:

General rules and rules for buildings. CEN, Brussels, 2004.

[6] EN 1990 Eurocode - Basis of structural design, CEN, Brussels, 2002.

[7] EN 1991-1-1 Eurocode 1 - Actions on structures. Part 1-1: General actions – Densities,

self-weight, imposed loads for buildings, CEN, Brussels, 2002.

[8] EN 1991-2 Eurocode 1: Actions on structures. Part 2: Traffic loads on bridges. CEN,

Brussels, 2003.

[9] EN 1991-1-5 Eurocode 1: Actions on structures. Part 1-5: General actions – Thermal

actions. CEN, Brussels, 2003.

[10] EN 1991-1-6 Eurocode 1: Actions on structures. Part 1-6: General actions – Actions

during execution. CEN, Brussels, 2005.

[11] EN 1993-1-5 Eurocode 3: Design of steel structures. Part 1-5: Plated structural

elements. CEN, Brussels, 2006.

[12] EN 1990:2002/A1:2005 Eurocode - Basis of structural design. Annex A2: Application for

bridges. CEN, Brussels, 2005.

[13] EN 1993-2 Eurocode 3: Design of steel structures. Part 2: Steel bridges. CEN, Brussels,

2006.


208

Annex A: Effects of LHVs on road bridges and EN1991-2 load models

209

ANNEX A: EFFECTS OF LHVs ON ROAD BRIDGES AND EN1991-2

LOAD MODELS

Pietro Croce1


Summary

To improve the European transportation network, the European Commission issued the

96/53/EC directive, that besides limiting the total mass of Heavy Good Vehicles (HGV) to 44 t,

allowed, on a parity basis, the possibility to permit the circulation of Long and Heavy Vehicles

(LHV), with total mass up to 60 t and length till 25 m. Northern European countries took

advantage of this possibility, experiencing a significant increase of LHVs on long distance traffic.

Since the static and fatigue models for road bridges of EN1991-2 have been calibrated on the

traffic recorded in Auxerre (F) in 1986, where LHVs were not included, the effect of the

introduction of LHVs could be excessively demanding, with disproportionate increase of costs.

Recent studies concerning this relevant question are discussed.

1 INTRODUCTION

In order to improve the organization of the European transportation network, the

European Commission issued the 96/53/EC Directive [1] limiting the total weight of Heavy Good

Vehicles (HGV) to 44 t, but admitting, on a parity and not discriminating basis, the possibility to

allow the circulation of Long and Heavy Vehicles (LHV), with a total mass up to 60 t and with

length till 25 m.

Some northern European countries, in particular Sweden, Finland, The Netherlands and

Germany, permitted the LHVs, so experiencing a significant increase of the number of LHVs in

long distance traffic.

The introduction of LHVs opens new prospects for traffic management thanks to of their

effectiveness in terms of decrease of pollutant emissions and cost reduction. However, it could

lead on the other hand to excessive surcharge of bridges, with disproportionate increase of costs.

Since the load models for static and fatigue assessments of road bridges given in EN1991-

2 [2] have been calibrated using the traffic recorded in Auxerre (F) in 1986, obviously not

including LHVs,, the impact of LHVs on existing infrastructures as well as on the load models

themselves could be very relevant.

In order to clarify this problem and to draw some preliminary conclusion, additional

studies have been performed on relevant bridge schemes and spans comparing the Auxerre traffic

effects with those induced by the traffic recorded in April 2007 in Moerdijk, the Netherlands,

characterised by an high percentage of LHVs.


210

2 LONG AND HEAVY VEHICLE TRAFFIC MEASUREMENT

As just said, at present the results of a wide campaign of in-situ measurements concerning

typical LHV traffic in the Netherlands are available, which have been used in the present study.

The above mentioned measurements have been performed in the first week of April 2007

in Moerdijk (NL), using a state of art weighing in motion (WIM) device, in the framework of the

studies concerning the assessment of equivalent fatigue loads for bridge decks, made by van

Bentum and Dijkstra [3].

In the records vehicles travelling with speed greater than 33 m/s were disregarded,

considering that in this case the reliability is not granted.

2.1 Characteristics of Moerdijk traffic

Analysis of Moerdijk traffic allows classifying commercial lorries in 53 relevant

subclasses, characterized by axle number varying between 2 and 9, described in figure 1. It must

be noted that lorries with up to 13 axles appear in the records and that the maximum recorded

lorry load was about 1140 kN, pertaining to a ten axle lorry type O13411, 19.5 m long.

Preliminary analysis of the records revealed that in few cases measurements were

inaccurate, so that it was necessary to eliminate wrong data from traffic database. In fact, the four

records summarized in table 1 and relative to two axle lorries appear largely unrealistic, also due

to excessive speed or length.

Table 1. Inaccurate data for 2 axle lorries in Moerdijk records

Speed

[m/s]

Length

[m]

Weight

[kN]

1st axle

Load

[kN]

2nd

axle

load

[kN]

30.6 21.02 707 398 309

3.3 13.05 613 208 405

3.3 62.02 684 318 366

27.2 11.32 689 353 336

Disregarding wrong data, the maximum recorded axle load results about 292 kN,

pertaining to the 3rd

axle of a T12O3 lorry, whose total weight is 636 kN, while the maximum

uniformly distributed load is about 63 kN/m, pertaining to a T12O21 silhouette weighing 813 kN

in total.

Moerdijk traffic measurements, amended according to the aforesaid considerations, were

taken into account for the evaluation of static and fatigue effects on reference bridge schemes and

spans, to be compared with those induced by the Auxerre traffic, as well as with those induced by

EN1991-2 load models, as described in the following.

3 HGV TRAFFIC AND LHV TRAFFIC COMPARISON

3.1 Axle and lorry loads

Static effects can be compared in a very simple way in terms of load spectra.

This is highlighted in particular in figures 2, 3, 4 and 5, which refer to comparison of

single, tandem, tridem axle’s weight spectra and total lorry weight spectra of Auxerre and

Moerdijk, respectively.


211

Figure 1. Lorry subclasses and symbols for Moerdijk (NL) traffic


212

Figure 2. Comparison of single axle load spectra for Auxerre and Moerdijk traffics.

Figure 3. Comparison of tandem axle load spectra for Auxerre and Moerdijk traffics

Figure 4. Comparison of tridem axle load spectra for Auxerre and Moerdijk traffics


213

Figure 5. Comparison of total lorry weight spectra for Auxerre and Moerdijk traffics.

3.2 Preliminary conclusions about static loads

The examination of the traffic records and the load spectra comparison shown above

highlight that:

- in consequence of the new traffic trend, the average axle number of the commercial

vehicle tends to increase significantly;

- total weight of LHVs could raise very high level, but usually this level is associated

with axle loads close to the legal limits;

- despite that occasionally axle loads can reach about 300 kN, Moerdijk traffic appears,

in general, less severe than the Auxerre traffic;

- it seems, therefore, that EN 1991-2 load models for static verifications cover also

Moerdijk traffic effects, so confirming, at this stage, its effectiveness;

- clearly, to draw more definitive conclusions it is necessary to enlarge the field of

investigation, also considering different traffic records.

It must be stressed, moreover, that Auxerre traffic data, obtained with less refined

WIM devices, probably are affected by a systematic overestimate.

4 FATIGUE DAMAGE

4.1 Reference traffics and equivalent load spectra

Besides in terms of static assessment, it is necessary to ascertain the aggressiveness of

traffic in terms of fatigue damage.

For this purpose four different traffic samples, composed by 10 000 vehicle each, have

been taken into account: two real ones, directly derived from Auxerre traffic measurements

used for calibration of EN 1991-2 models and from the new Moerdijk traffic measurements,

respectively, and two conventional ones, suitably derived from the fatigue load model n. 4 for

long distance traffic.

The aforesaid conventional traffic models were obtained taking into account an annual

flow of 2×106 standard LM 4 lorry silhouettes on the slow lane. These models, obtained


214

through a Monte Carlo simulation, differ on the inter-vehicle distances, that in the former case

are as simulated, in order to consider also interaction between vehicles simultaneously present

on the bridge, and in the latter case they are suitably increased in such a way that only isolated

lorries can cross the bridge, so avoiding interaction. It must be also stressed that EN 1991-2

states that, as rule, fatigue load models cannot be used directly when vehicle interactions

become significant, unless adequate additional ad hoc studies are performed.

The choice of these four reference traffics is particularly appropriate, because it allows

to compare the fatigue damage induced by the Auxerre traffic not only with the damage

induced by Moerdijk one, but also with those induced by the equivalent load spectra of EN

1991-2.

4.2 Reference influence lines and spans for bridges

Preliminarily, five influence surfaces for simply supported and continuous beams have

been selected to perform the fatigue damage assessment. The influence surfaces, illustrated in

figures 6 to 10, refer to the bending moment at midspan of a simply supported beam (Mo), to

the bending moment at intermediate support (M1) and at section located 0.432·L from the first

support, where the bending moment attains the maximum value(M2), in a two span continuous

beam and to the to the bending moment at the third support (M3) and at midspan (M4) of a five

span continuous beam.

0 0.5 1

Figure 6. Influence line for bending moment M0 at midspan of simply supported bridge

0 1 2

Figure 7. Influence line for hog moment M1 at intermediate support of two span

continuous bridge

0.4320 1 2

Figure 8. Influence line for max sag moment M2 in the section located 0.432·L from the

support in two span continuous bridge


215

0 1 2 3 4 5

Figure 9. Influence line for hog moment M3 at the third support of five span continuous

bridge

0 1 2 3 4 5

Figure 10. Influence line for max sag moment M4 at midspan of five span continuous

beam bridge

4.3 Comparison of fatigue damage

For each influence line, nine different span lengths have been considered, varying

from 3 m to 100 m (3 m, 5 m, 10 m, 20 m, 30 m, 50 m, 70 m, 100 m), and the bending

moment histories induced in each of them, in turn, by the four relevant traffics have been

determined.

Stress spectra have been derived from the above mentioned time histories using the

rainflow cycle counting method. Finally, fatigue damage has been evaluated using the

Palmgren Miner rule.

In the damage assessment, consistently with the assumptions made in EN1991-2

background studies, simplified single slope S-N curves, without fatigue limit, have been taken

into account, assuming, in turn, a slope m=3 and m=5. These statements are fully justified, as

they simplify significantly fatigue assessments, introducing negligible errors.

The fatigue damage induced by each relevant traffic is then compared with the one

induced by the Auxerre traffic, both for m=3 or m=5, and the aggressiveness of each traffic is

finally derived in terms of the equivalence factor Keq,t for the actual traffic, given by

eqAux

teq

Aux

t m

D

DteqK

σσ

∆∆

==

1

, (1)

where Dt is the fatigue damage induced by the actual traffic, DAux is the fatigue damage

induced by the Auxerre traffic, m is the slope of the S-N curve adopted for the evaluation of

Dt and DAux, ∆σeq,t is the equivalent value of the stress range for the actual traffic and ∆σeq,Aux

is the equivalent value of the stress range for the Auxerre traffic. Obviously, higher values of

Keq,t correspond to more aggressive traffics.

Keq,t is a characteristic traffic parameter and it represents a concise way to compare

different traffics: in fact, it can be interpreted as an adjustment factor for which the axle load


216

values of Auxerre traffic must be multiplied to reproduce the fatigue damage induced by the

actual traffic.

The Keq,t curves, pertaining to Moerdijk traffic as well as to conventional traffics

derived from fatigue load model LM4 of EN 1991-2, with and without vehicle interaction, are

plotted for each relevant influence line, in terms of span length, in figures 11 to 15 for m=3

and in figures 16 to 20 for m=5. More precisely figures 11 and 16 refer to M0, figures 12 and

17 to M1, figures 13 and 18 to M2, figures 14 and 19 to M3 and figures 15 and 20 to M4.

Figure 11. Keq curves for bending moment M0 (m=3)




217





218





219


Critical examination of these Keq,t curves allows to conclude that:

- load model LM 4 represents very satisfactorily the actual fatigue damage induced by

the Auxerre traffic and is generally safe-sided;

- despite of EN 1991-2 statement that establishes that isolated standard lorries cannot be

used when vehicle interactions are relevant (i.e. when L > 30 m), the use of

conventional often allows to approximate real traffic effects better than resorting to

improved load model LM4, taking into account interactions;

- conventional load model LM4, disregarding vehicle interactions, results unsafe-sided

for bending moment at intermediate supports of continuous beams: in this case, on the

contrary, the use of improved load model LM4 leads to significant overestimates of

fatigue damage, with Keq values raising up 1.25;

- the aforesaid phenomena can be explained considering that influence line of

intermediate support is characterized by two adjacent zones where the ordinates of the

influence line are comparable, so that from one side the stress range induced by

isolated vehicles, which affect only one zone, is too low, while, from the other side,

interacting LM4 vehicles, which affect both adjacent zones, determine too high stress

range, as their equivalent axle loads values, calibrated considering short and medium

span bridges, are similar;

- the Moerdijk traffic is characterized by Keq factors generally ranging between 0.8 and

0.85 except the cases M1 and M3 for span length L > 50 m and m=5. 5 CONCLUSIONS

The impact of LHVs on design of bridges in terms of static and fatigue assessments

has been discussed, comparing the effects induced by the Moerdijk (NL) traffic, characterized

by high percentage of LHVs, with those induced by the Auxerre traffic, used as reference

traffic in background of EN1991-2. The fatigue assessments have been supplemented also

considering two conventional traffics, deduced by the fatigue load model 4 of EN1991-2,

constituted by equivalent lorries.


220

Results of the study demonstrate that EN 1991-2 load models adequately cover the

effect induced by the LHVs, as included in Moerdijk measurements. This can be explained

considering, on the one hand, that overloads of single axles of LHVs are usually not so

relevant as for HGVs, on the other hand, that Auxerre data, obtained in 1986 with a less

refined WIM device, could be affected by some systematic overestimate.

Clearly, these results need to be supplemented and improved as they concern specific

traffic measurement; therefore further studies are necessary enlarging the field of

investigation and considering various traffic measurements.

6. REFERENCES

[1] Directive 96/53/EEC, OJ L.235, 17/09/1996.

[2] EN1991-2, Eurocode 1: Actions on structures - Part 2: Traffic loads on bridges. Brussels:

CEN 2003

[3] van Bentum, C.A. & Dijkstra, O.D. Process description of equivalent fatigue load on

bridge decks. TNO report 366 B UK. Delft: TNO, 2008.

Annex B: Actions and combination rules for cranes, masts, towers and pipelines

221

ANNEX B: ACTIONS AND COMBINATION RULES FOR CRANES,

MASTS, TOWERS AND PIPELINES

Milan Holický1, Jana Marková

1


Summary

Relevant loads due to cranes and machinery need to be considered in the design of

towers, masts and pipelines. For the selected design situations and identified limit states, the

critical load cases should be assessed. The combination rules for actions, partial factors and

other reliability elements provided in EN 1990 are supplemented for the purposes of cranes

and machinery in EN 1991-3, for towers and masts in EN 1993-3-1 and for pipelines in EN

1993-4-3. It is foreseen that during the planned revision of EN 1990 the basis of design for

these structures will be transferred to a new Annexes of EN 1990.

1 INTRODUCTION

1.1 Background documents The aim of this Chapter is to introduce an overview of actions and supplementary

rules for their combinations needed for the design of masts, towers, pipelines and crane

supporting structures. The basic procedures for the determination of characteristic and design

values of actions are given in EN 1990 [1] and in various Parts of EN 1991. Some Parts of

EN 1993 provide rules for specification of actions and their combinations for masts, towers

and pipelines.

1.2 Basic principles

EN 1990 [1] gives basic principles for the design of construction works. EN 1991-3

[6] provides supplementary rules for the design of cranes and machinery, EN 1993-3-1 [7] for

masts and towers and EN 1993-4-3 [8] for pipelines.

2 ACTIONS AND COMBINATION RULES FOR CRANES AND MACHINERY

2.1 General

EN 1991-3 [6] give guidance for specification of actions induced by cranes on runway

beams and by rotating machines on supporting structures.

2.2 Actions due to cranes

Actions due to cranes include hoist loads, inertial forces caused by

acceleration/deceleration and by other dynamic effects. Following vertical and horizontal

components are distinguished:

– variable vertical crane actions caused by the self-weight of the crane and the hoist load


222

– variable horizontal crane actions caused by acceleration/deceleration or by skewing or

by other dynamic effects.

The various representative values of variable crane actions are characteristic values

composed of a static and a dynamic component.

Dynamic components (induced by vibration due to inertial and damping forces) are in

general taken into account by dynamic factors ϕ to be applied to the static values of actions

Fϕ,k = ϕ i Fk (1)

Where Fϕ,k is the characteristic value of a crane action, ϕ i is the dynamic factor and Fk is the

characteristic static component of a crane action. The various dynamic factors and their

application are listed in Table 2.1 of EN 1991-3 [6].

The simultaneity of crane load components may be taken into account by considering

groups of loads. Each of these groups should be considered as defining one characteristic

crane action for the combination with non-crane loads.

Cranes can also evoke accidental actions due to collision with buffers or collision of

lifting attachments with obstacles. These actions should be considered for the structural

design where appropriate protection is not provided. They are represented by various load

models defining design values in the form of equivalent static loads.

2.3 Actions due to machinery

Structures supporting rotating machines are commonly loaded by permanent actions

due to the self-weight of all fixed or movable parts and static actions due to operations.

Variable actions due to machinery during normal service are dynamic actions caused by

accelerated masses including

– periodic frequency-dependent bearing forces due to eccentricities of rotating masses in

all directions, mainly perpendicular to the axis of the rotors

– periodic actions due to service depending on the type of machine that are transmitted

by the hull or bearings to the foundations

– free mass forces or mass moments

– forces or moments due to switching on/off or other transient procedures such as

synchronisations.

Moreover, accidental actions may occur due to e.g. accidental magnification of the

eccentricity of masses (for instance by fracture of brakes), by short circuit or out of

synchronisation of the generators and machines.

2.4 Combinations of actions

Effects of actions that cannot exist simultaneously due to physical or functional

reasons should not be considered together in combination of actions.

When combining a group of crane loads together with other actions, the group of crane

loads should be considered as one action. For runways outside buildings the wind actions,

snow and temperature should be considered. The maximum wind force *WF compatible with

crane operations need to be considered (the recommended value is *WF = 20 m/s).

The combination of actions for ultimate limit states is based on expressions (6.9a) to

(6.12b) and for serviceability limit states on expressions (6.14a) to (6.16b) according to EN

1990 [1].


223

Where an accidental action is to be considered, no other accidental action nor wind nor

snow actions need be considered to occur simultaneously.

Recommended values of ψ-factors are given in Table 1.

Table 1. Recommended values of ψψψψ-factors

Action ψ0 ψ1 ψ2

Single crane or groups of loads

induced by cranes

1,0 0,9 (*)

(*) The recommended value is the ratio between the permanent crane action and the

total crane action.

2.5 Ultimate limit states

For each critical load case, the design values of the effects of actions should be

determined by combining the values of actions which occur simultaneously in accordance

with EN 1990 [1].

Static equilibrium (EQU) and uplift of structural bearings for structures supporting

cranes and machinery should be verified using the design values of actions given in Table 2.

Table 2. Design values of actions (EQU) (Set A)

P/T

situations

Permanent actions Leading

variable

action (*)

Accompanying variable actions

Unfavourable Favourable Main (if any) Others

Eq. (6.10) γGj,supGkj,sup γGj,infGkj,inf γQ,1 Qk,1 γQ,iψ0,iQk,i

NOTE 1 The recommended set of values for γ are

γGj,sup = 1,05

γGj,inf = 0,95

γQ = 1,35 for unfavourable variable crane actions

γQ = 1,00 for favourable variable crane actions, where crane is present

γQ = 0 for favourable variable crane actions, where crane is not present

γQ = 1,5 for other unfavourable variable actions (0 where favourable)

NOTE 2 For verification of uplift of structural bearings or in cases where the verification of static

equilibrium also involves the resistance of structural members (e.g. where loss of static equilibrium is

prevented by stabilizing systems or devices, e.g. anchors, stays or auxiliary columns) as an alternative

to two separate verifications based on Tables 2.2 and 2.3 a combined verification based on Table 2.2

may be adopted with the following recommended values

γGj,sup = 1,35

γGj,inf = 1,25

γQ,1 = 1,35 for unfavourable variable crane actions

γQ,1 = 1,00 for favourable variable crane actions where crane is present

γQ,1 = 0 for favourable variable crane actions where crane is not present

γQ = 1,5 for other unfavourable variable actions (0 where favourable)

provided that applying γGj,inf = 1,00 both to the favourable part and to the unfavourable part of

permanent actions does not give a more unfavourable effect.


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The design values of actions and recommended partial factors based on EN 1990 [1]

and EN 1991-3 [6] for the ultimate limit states (STR) in the persistent and transient design

situations are given in Table 2.3.

Table 3. Design values of actions (STR) (Set B)

P/T

situations

Permanent actions


Prestress Leading

variable

action


actions

Main (if any) others

Exp. (6.10) γGj,supGkj,sup γGj,infGkj,inf γP Pk γQ,1 Qk,1 γQ,iψ0,iQk,i

Exp. (6.10a) γGj,supGkj,sup γGj,infGkj,inf γP Pk γQ,1ψ0,1Qk,1 γQ,iψ0,iQk,i

Exp. (6.10b) ξγGj,supGkj,sup γGj,infGkj,inf γP Pk γQ,1 Qk,1 γQ,iψ0,iQk,i

Recommended values:

γGj,sup = 1,35 for unfavourable and γGj,inf = 1,0 for favourable permanent actions

γQ = 1,35 for unfavourable crane actions, γQ = 1,00 for favourable crane actions, where crane is

present, γQ = 0 for favourable crane actions, where crane is not present.

γQ = 1,5 for other unfavourable variable actions, γQ = 0 for favourable variable actions where

crane is not present

ξ = 0,85 (so that ξγGj,sup = 0,85×1,35 = 1,15)

γP = recommended values are specified in the relevant design Eurocode.

The characteristic values of all permanent actions from one source may be multiplied by γG,sup if

the total resulting action effect is unfavourable and γG,inf if the total action effect is favourable.

For particular verifications, the values for γG and γQ may be subdivided into γg and γq and the

model uncertainty factor γSd (a value of γSd in recommended in the range 1,0 to 1,15).

The design values of actions for the ultimate limit states in the accidental and seismic

design situations are given in Table 4.

Table 4 Design values of actions for use in accidental and seismic combinations

P/T situations Permanent actions


Prestress Accidental

or seismic

action


actions

main (if any) others

Accidental

Exp.

(6.11a/b)

Gkj,sup Gkj,inf Pk Ad ψ11 Qk1 or ψ21Qk1 ψ2,i Qk,i

Seismic

Exp.

(6.12a/b)

Gkj,sup Gkj,inf Pk γIAEk or

AEd

ψ2,i Qk,i

In case of accidental design situations, the main variable action may be taken with its frequent or

quasi-permanent values. The choice is given in the National Annex depending on the accidental

action under consideration.

2.6 Serviceability limit states

Three combinations of actions are recommended for verification of serviceability

criteria. The design values of actions for the serviceability limit states are given in Table 2.5.

The serviceability criteria should be defined in relation to the serviceability

requirements in accordance with EN 1992 to EN 1999. Deformations should be calculated in

accordance with relevant EN 1991 to EN 1999 by using the appropriate combinations of


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actions according to expressions (6.14a) to (6.16b) taking into account the serviceability

requirements and the distinction between reversible and irreversible limit states.

Table 5. Design values of actions for use in serviceability limit states



Characteristic

Frequent

Quasi-permanent

Gk,j,sup

Gk,j,sup

Gk,j,sup

Gk,j,inf

Gk,j,inf

Gk,j,inf

Pk

Pk

Pk

Qk,1

ψ1,1Qk,1

ψ2,1Qk,1

ψ0,iQk,i

ψ2,iQk,i

ψ2,iQk,i

3 ACTIONS AND COMBINATION RULES FOR MASTS AND TOWERS

3.1 General

The lattice towers and guyed masts which are within the scope of EN 1993-3-1 [8]

should be designed in accordance with general rules of EN 1990 [1] for the basis of design

and EN 1993-1-1 [7] for design of steel structures.

Three reliability levels are distinguished for the ultimate limit state verifications of

structures, depending on the possible economic and social consequences of their collapse.

3.2 Actions and environmental influences

Models of actions are given in relevant Parts of EN 1991. Self-weight should be

determined in accordance with EN 1991-1-1 [2]. Imposed load is recommended to be

considered on members inclined within 30° to horizontal to carry the weight of a workman

making possible the maintenance. Wind loads need to be specified according to EN 1991-1-4

[4] however, using the wind force coefficients and supplementary rules for wind actions given

in Annex B of EN 1993-1-1 [7]. Basic guidance for modelling of climatic actions [3,4] are

provided in Annex B and for ice loads in Annex C [7], more details for icing are given in

ISO 12494 [10]. Hazard situations, proposed strategies for risk reduction, accidental actions

and guidance for their application are provided in EN 1991-1-7 [5].


Permanent, transient and accidental design situations are distinguished. The basis for

specification of characteristic and design values of actions are given in EN 1990 [1]. The

values of partial factors and ψ-factors for actions in the ultimate limit states are recommended

in Annex A to EN 1993-1-1 [7].


The serviceability limit states which need to be considered within the scope of

structures given in EN 1993-3-1 [8] include

– deflections or rotations that may adversely affect the use of the structure

– vibration, oscillation or sway that cause loss of transmitted signals

– deformations, deflections, vibration, oscillation or sway leading to damage to non-

structural elements.

3.5 Reliability differentiation

Three reliability classes 1 to 3 are distinguished in EN 1993-3-1 [8] related to the

consequences of structural failure as illustrated in Table 6.

The values of partial factors γG and γQ recommended in EN 1993-3-1 [8] for the

ultimate limit state verifications considering three reliability classes are introduced in Table 7.


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Table 6. Reliability differentiation for towers and masts.

Reliability

Class Examples of structures

3 towers and masts erected in urban locations, or where their failure is likely to cause

injury or loss of life; towers and masts used for vital telecommunication facilities;

other major structures where the consequences of failure would be likely to be very

high

2 all towers and masts that cannot be defined as class 1 or 3

1 towers and masts built on unmanned sites in open countryside; towers and masts,

the failure of which would not be likely to cause injury to people

Table 7. Partial factors for actions

Type of action

effect Reliability Class

γG for permanent

actions

γQ for variable

actions

unfavourable

3 1,2 1,6

2 1,1 1,4

1 1,0 1,2

favourable all classes 1,0 0,0

However, the values of partial factors of actions appear to be considerably lower than

those recommended for structures in the basic Eurocode EN 1990 [1], Annex A1. Presently,

the target values of reliability index βt are not recommended in EN 1993-3-1 [8].

3.6 Ice load and combinations with wind

The basis for specification of ice load is given in ISO 12494 [10] to which EN 1993-3-

1 [8] make reference. The ice loads are based on Ice Classes IC for rime (IR) and glaze (IG).

The specification of relevant Ice Class for the site location is left for national decision.

Combination of ice with wind can often govern the design of masts and towers. Two

combinations of wind and snow need to be considered for both symmetrical icing and

asymmetrical icing:

– leading ice load and accompanying wind

γG Gk + γice Qk,ice + γW k ψW Qk,w (2)

– leading wind and accompanying ice

γG Gk + γW k Qk,w + γice ψice Qk,ice (3)

where k is given in ISO 12494 [10] dependent on Ice Class and the values of reduction

coefficients ψI = ψW = 0,5 are recommended in EN 1993-3-1 [8].

4 ACTIONS AND COMBINATION RULES FOR PIPELINES

4.1 General

EN 1993-4-3 [9] provides principles and application rules for the structural design of

buried cylindrical steel pipelines for the transport of liquids or gases or mixtures of liquids

and gases at ambient temperatures. The design of pipelines should be in accordance with

provisions of EN 1990 [1] for the basis of design and EN 1991 for actions.


227

Different levels of reliability may be adopted for different types of pipelines,

depending on possible economic and social consequences of their collapse. Each CEN

Member State may define the minimum level of reliability. The recommended values given in

EN 1993-4-3 [9] are intended for medium safety requirements.

4.2 Actions and environmental influences

Actions which should be considered include

– internal and external pressure

– self weight of the pipeline and its content (transported product)

– soil loads, imposed deformation (due to settlements, landslides)

– traffic loads

– temperature variations

– construction loads

– seismic loads.


The basis for specification of characteristic and design values of actions are provided

in EN 1990 [1]. The values of partial factors shall be based on the required reliability level.

The following combinations of actions for ultimate limit states should be considered

a) Internal pressure: The difference between the maximum internal pressure and the

smallest external pressure. This limit state is generally used first for the

determination of the wall thickness.

b) Internal pressure plus other relevant loads: The internal and external pressure

conditions defined in (a), with the other relevant design loads added. This limit state

is generally used to check critical strains

c) External pressure plus other relevant loads: The difference between the maximum

external pressure and the smallest internal pressure, with the other relevant design

loads added. This limit state is generally used to check ovalisation, critical strains,

local buckling etc.

d) Temporal variations in pressure plus other relevant design loads. This case is

concerned with cyclic actions on the pipe. This limit state is generally used last to

check for fatigue.


For serviceability limit states the following combinations should be considered

a) Internal pressure plus other relevant loads: The difference between the maximum

internal pressure and the smallest external pressure with the other relevant design

loads.

b) External pressure plus other relevant loads: The difference between the maximum

external pressure and the smallest internal pressure, with the other relevant design

loads added.

4.5 Reliability differentiation

Pipelines usually comprise several associated facilities such as pumping stations,

operation centres, maintenance stations, etc., each of them housing different sorts of

mechanical and electrical equipment. Since these facilities have a considerable influence on


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the continued operation of the system, it is necessary to give them adequate consideration in

the design process aimed at satisfying the overall reliability requirements.

Different levels of reliability may be adopted for different types of pipelines,

depending on possible economic and social consequences of their collapse. The choice of the

target reliability should be agreed between the designer, the client and the relevant authority.

5 AN EXAMPLE OF SPECIFICATION OF ACTIONS

A simply supported steel beam span of 3 m of a transmitting tower is located at high

of 14 m above the terrain of category 2 where the basic wind velocity is vb0 = 27,5 m/s, see

Fig. 1. The steel beam (cross-sectional shape I, in mutual distances of 1 m) is loaded by the

self-weight of a steel deck thickness of 0,01 m and by average permanent load 3 kN/m2 due to

the transmitting facilities. The tower is located in region with Ice Class IR5.

The self-weight of a beam is 0,1 kN/m, the permanent load g = 3 kN/m2, wind

pressure w at high 14 m w = 0,6 kN/m2

(or suction -1,8 kN/m2) and icing i = 0,5 kN/m

2.

g

w

i

L = 3 m

Figure 1. Scheme of a structural member and its loading.

The fundamental combination and reliability elements given in EN 1990 [1] is

considered for a common structural reliability class RC2. The maximum moment for the

simple beam is determined for an icing considered as a leading action

M = 1/8 L2 (1,35 g + 1,5 i + 1,5 ψW k w)

M = 1/8×32 (1,35×3,1 + 1,5×0,5 + 1,5×0,6×0,6×0,6) = 5,9 kNm

where ψW = 0,6, k = 0,6 [10], and for the wind considered as a leading action

M = 1/8 L2 (1,35g + 1,5 w + 1,5 ψI i)

M = 1/8×32 (1,35×3,1 + 1,5×0,6 + 1,5×0,5×0,5) = 6,1 kNm

where ψI = 0,5. The values of reliability elements (partial factors and reduction factors) are

based on EN 1990 [1].

However, when the values of reliability elements given in EN 1993-3-1 [8] are

considered then the following result will be obtained if icing is a leading action

M = 1/8×32 (1,1×3,1 + 1,4×0,5 + 1,4×0,6×0,6×0,6) = 5 kNm

and for the wind considered as a leading action

M = 1/8×32 (1,1×3,1 + 1,4×0,6 + 1,4×0,5×0,5) = 5,2 kNm


229

Thus, the application of the reliability elements given in EN 1993-3-1 [8] leads to

about 15 % reduction of internal moment.

In case that the reliability elements recommended in the CENELEC standard EN

50341-1 [11] should be applied then the resulting moments decrease about 30 % in

comparison with results based on EN 1990 [1]. The reliability of structural members designed

considering reliability elements according to EN 50341-1 [11] appears to be significantly low.


The design of structures supporting cranes and machinery, towers and masts and

pipelines are based on the load combinations provided in EN 1990 [1]. Supplementary rules

for the specification of loads and load effects are given in relevant Parts of EN 1991 and EN

1993.

It is expected that within the revision of EN 1990 similar Annexes A for cranes and

machinery, masts and towers and pipelines will be further developed. However, proposed

values of partial factors and other reliability elements should be further calibrated and

harmonised.

9 REFERENCES

[1] EN 1990 Eurocode – Basis of design. CEN, Brussels, 2006.

[2] EN 1991-1-1 Eurocode 1: Actions on structures – Part 1-1: General actions – Densities,

self weight, imposed loads for buildings. CEN, Brussels, 2002.

[3] EN 1991-1-3 Eurocode 1: Actions on structures – Part 1-3: Snow loads. CEN, Brussels,

2003.

[4] EN 1991-1-4 Eurocode 1: Actions on structures – Part 1-4: Wind loads. CEN, Brussels,

2005.

[5] EN 1991-1-7 Eurocode 1: Actions on structures – Part 1-7: Accidental actions. CEN,

Brussels, 2006.

[6] EN 1991-3 Eurocode 1 – Actions on structures Part 3: Actions induced by cranes and

machinery. CEN, Brussels, 2006.

[7] EN 1993-1-1 Eurocode 3: Design of steel structures – Part 1.1: General rules and rules

for buildings. CEN, Brussels, 2005.

[8] EN 1993-3-1 Eurocode 3: Design of steel structures – Part 3.1: Towers, masts and

chimneys – Towers and masts. CEN, Brussels, 2006.

[9] EN 1993-4-3 Eurocode 3: Design of steel structures – Part 4.3: Pipelines. CEN,

Brussels, 2007.

[10] ISO 12494 Atmospheric icing on structures. ISO/TC98, 2001.

[11] EN 50341. Overhead electrical lines exceeding AC 45 kV – Part 1: General requirements

– Common specifications, CENELEC, Brussels, 2001.


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