1

Abstract

The purpose of this work was to create a

computer program capable of performing

security verifications on reinforced concrete

columns subjected to seismic actions

according to Eurocode 8[2].

Ultimate Limit State of bending with axial force

(section 6.1 of EC2[1]) and shear (Section 6.2

of EC2) verifications are computed and two

methods of computation are developed for

verification and detailing for ductility

requirements of critical regions (sections

5.4.2.3 and 5.4. 3.2 of EC8). The work focuses

on ductility provision, since this is the

verification most directly linked to Eurocode 8

and the one that most differs from previous

Portuguese legislation, RSA and REBAP.

This work is developed on rectangular

columns, which together with circular ones are

the only ones that the EC8[2] presents a

simplified method that avoids explicit

calculation of curvature ductility by calculating

the moment curvature relation. In this study

two ways of ductility verification are compared:

by using the requirements presented on

EC8[2] (mostly about hoops pattern) and by

calculating the moment-curvature relationship.

Keywords:

Computing; Eurocode 8; reinforced concrete; columns; ductility; confinement.

Contents

1 Introdution ...............................................1

2 Materials .................................................1

2.1 Concrete .........................................1

2.2 Reinforcing steel .............................2

3 Ultimate limit states – Resistance ............2

3.1 Bending with axial force (uniaxial

and biaxial) .................................................3

3.1.1 Computing ..................................3

3.2 Shear..............................................4

4 Ductility Provision ....................................4

4.1 Detailing and ductility classes .........5

4.2 Computing ......................................5

4.2.1 Computation of the expression (4)

– 5.15 of EC8[2] .....................................5

4.2.2 Explicit (M-C) ductility verification 6

5 Example of Application ............................7

5.1 Problem ..........................................7

5.2 Flexural resistance ..........................7

5.3 Shear..............................................8

5.4 Ductility ...........................................8

5.4.1 Verification by expression (14) –

5.15 of EC8 .............................................8

5.4.2 Explicit (M-C) ductility verification 9

6 Conclusion ..............................................9

Symbols ....................................................... 10

References .................................................. 10

1

1 Introdution

The Eurocodes, specifically EC2[1] and

EC8[2], add to the Portuguese codes RSA[6]

and REBAP[5] more safety verifications in

structural elements and modifications in

verifications already existent, mostly in

detailing for ductility requirements. These

verifications are, mostly too long and repetitive.

Computer technology evolution permits solving

these calculations with a much higher rate than

it would be possible without its application.

Some aspects of the program that require

numerical methods are not viable in a non-

automatic procedure. At the same time

computation of routine processes can help

reduce human error.

Once it is not possible to cover all the

verifications present in the Eurocodes, this

work deals with the verification of columns

subjected to seismic actions according to

EC2[1] and EC8[2]. This article focuses on the

verification of the ultimate limit state of flexural

strength (section 6.2 of EC2[1]), shear (Section

6.2 of EC2[1]) and verification and detailing for

ductility requirements (sections 5.4.2.3 and

5.4.3.2 of EC8[2]).

The structure of this extended abstract is

similar to the one of the dissertation full version

in Portuguese language and it is divided into

three parts. The first one (Chapter 2) briefly

describes the properties of materials, namely

concrete and reinforcing steel. The second part

describes the verification to be performed:

shear and bending with axial force resistance

(Chapter 3) and ductility verifications and

provisions (Chapter 4). In each chapter is firstly

presented a theoretical introduction describing

which verifications are done, the methods

allowed by the Eurocodes and which of them

are automated. Then it is explained the way

the program addresses these issues: what

algorithms are chosen and how they work. The

third part (Chapter 5) presents an annotated

example of application, with special emphasis

on ductility provision, which is the most specific

verification of Eurocode 8[2].

The program is annexed in the CD that

includes the written part.

2 Materials

This chapter describes the stress-strain

relations of concrete and reinforcing steel used

for analyses and design of cross sections in

the following chapters.

2.1 Concrete

The stress-strain relations presented in Figure

1 a) and b), which are presented in EC2[1],

refer to non-confined concrete. The first one

refers to design of cross sections, the second

one to structural analysis. The relations in

Figure 1 c) (found in EC2[1]) and d) (MC90[3])

are modifications of a) and b) respectively to

take into account the effect of confinement.

Confinement of concrete results in a

modification (see equations (1) to (5)) of the

effective stress-strain relation: higher strength

and more significantly higher critical strains are

achieved.

2

Figure 1- Concrete stress-strains relations

a) parabola-rectangle diagram for non-confined concrete [1] b) diagram for structural analysis for non-confined concrete [1] c) parabola-rectangle diagram for confined concrete [1] d) diagram for structural analysis for confined concrete [1]

(1)

(2)

(3)

(4)

(5)

2.2 Reinforcing steel

EC2[1] considers two stress-strains diagrams

for reinforcing steel (see Figure 2):

One with an inclined top branch with a

strain limit of

Another with a horizontal top branch

without the need to check the strain

limit.

Figure 2 – stress-strains diagrams for reinforcing steel [1]

udfyd/ Es

fyk

kfyk

fyd = fyk/ s

kfyk

A

B

uk

kfyk/ s

a) c)

b) d)

3

3 Ultimate limit states –

Resistance

3.1 Bending with axial force

(uniaxial and biaxial)

Biaxial bending with axial force resistance can

be calculated according to:

Criterion 1 - Explicitly calculating the

stresses and strains for an axial force

and a moment not acting in a principal

direction;

Criterion 2 – According to condition

5.4.3.2.1(2)P of EC8[2], that permits

carrying out verification separately in

each principal direction, with the

uniaxial moment of resistance reduced

by 30%

Criterion 3 - Following the expression

(5.39) of EC2[1]:

(6)

In the last two cases it is only necessary to

calculate flexural resistance in the two

orthogonal directions. Only in the first case the

biaxial bending should be checked together.

The following hypotheses are considered:

plane sections remain plane; the strain in

bonded reinforcement, whether in tension or in

compression, is the same as that in the

surrounding concrete; the tensile strength of

concrete is ignored.

The possible range of strain distributions is

shown in Figure 3.

3.1.1 Computing

In this extended abstract only uniaxial bending

with axial force computation is presented.

The user inserts the cross-sectional geometry,

materials characteristics (for concrete the user

should choose the class and for steel the user

should choose the yield strength), the stress-

strains relationships and the reinforcement

pattern.

The software considers the longitudinal bars as

concentrated forces at their geometrical center

and integrates concrete stresses through the

following numerical method. The section is

discretized in areas parallel to the neutral axis

and each element has a uniform extension,

corresponding to the extension of its geometric

center (Figure 4). One million fibers are

considered in parable zone (extensions

between 0 and 0.2%) and only 1 fiber to the

rectangular area.

In order to draw the diagram of extensions

completely, the extensions are set up at both

ends of the section according to Figure 3.

Then tensions are determined through the

stress-strains relations described in Figures 1

and 2 and the axial forces and momentums by

the following equilibrium equations (7) and (8):

(7)

(8)

4

Figure 3 – extension fields admitted by the program

Figure 4 – discretization of cross section for axial force and bending resistance

3.2 Shear

The verification is based on the truss model

prescribed in EC2[1]. The user enters the

shear force and chooses the angle θ. The

distance z can be set to 0.9 d or may be

chosen by the user.

The program returns the cross-sectional area

of the shear reinforcement per unit length

(trough equation (9)) and checks the tension

introduced in the concrete according to

equation (10):

(9)

(10)

Where is equal to for and is

calculated according to (11):

( em MPa) (11)

4 Ductility Provision

The behavior factor used in linear elastic

analysis performed on a structure subjected to

seismic action, has implied a certain ductility of

the structure. In order to guarantee that the

designed resistance is sufficient it is necessary

to ascertain whether the ductility is consistent

with the choice of the behavior factor.

In order to ensure an overall ductile behavior, it

is important to prevent brittle failure or other

undesired mechanisms (such as shear of

structural elements) and favor modes with

more ductile rupture such as bending.

Following the capacity design rule, that is

possible by deriving the design shear force

from equilibrium conditions, assuming that

plastic hinges with their possible overstrengths

have been formed in column ends.

For the required overall ductility of the

structured to be achieved, once the privileged

rupture mechanism is bending, the potential

regions for plastic hinge formation (critical

5

regions) shall possess sufficient curvature

ductility. This condition is deemed to be

satisfied if the curvature ductility factor of

critical regions is at least equal to the following

values1:

(12)

(13)

As the structure ductility depends on the

ductility of the materials that constitute it, the

"strategy" to increase structural ductility is to

increase materials ductility, more specifically

concrete. As noted in 2.1 the ultimate strain of

concrete can be greatly increased if it is

guaranteed a certain level of confinement

stress.

The value can be checked in two ways:

Explicitly. is “defined as the ratio of

post ultimate strength curvature at

85% of the moment of resistance, to

the curvature at yield, provided that the

limit strains and are not

exceeded” It is necessary to check the

resistance taking into account the loss

of section due to spalling of concrete,

but confinement effect on resistance

and ductility of concrete can be taken

into account.

According to expression (12) equal to

expression 5.15 of EC8[2]. This

expression is intended primarily to

ensure good confinement and prevent

buckling of reinforcing bars.

1 In case the longitudinal reinforcing steel is class B,

should be multiplied by 1,5.

(14)

Where:

is the mechanical volumetric ratio of

confining hoops within the critical regions

is the confinement effectiveness factor.

It takes into account the non-uniformity of

distribution of the confining stresses, both

“horizontally” and “vertically”.

4.1 Detailing and ductility classes

The EC8[2] defines 3 different classes of

ductility and energy dissipation for structures:

DCL (low), DCM (middle) and DCH (high).

DCL is only recommended for low seismicity

cases. To each category different behavior

factors and different constructive

displacements (that must always be met) are

assigned. These constructive displacements

include minimum and maximum reinforcement

and hoops ratio and distance between bars

and hoops.

The program checks these provisions through

the function DCM or DCH.

4.2 Computing

4.2.1 Computation of the expression (4) –

5.15 of EC8[2]

4.2.1.1 Data input

The user must insert the following data in order

that the program can solve the equation (14):

The concrete strength classes, steel

yield strength, cross section and

6

longitudinal reinforcement geometry

and axial force. All of these values are

expected to have already been defined

by the user when calculating bending

resistance.

The seismic parameters q0, Tc e Ti for

the calculations in both considered

directions.

The diameter, spacing and layout of

hoops. The user must define the

hoops patterns, denoting in which

longitudinal bars the hoops engage.

4.2.1.2 Data interpretation

By the information on hoop and reinforcement

pattern the program calculates all the values of

(distance between consecutive engaged

bars). In order to achieve this, the software

needs to know which reinforcement bars are

engaged and which ones of those are

consecutive. The user needs then to introduce

the longitudinal bars sequence.

4.2.2 Explicit (M-C) ductility verification

The calculation is done in accordance with

5.2.3.4. of EC8, in which a minimum value of

curvature ductility factor μφ has to be

ensured(see 4). This requires calculating the

moment-curvature relationship of critical

regions.

4.2.2.1 M-C calculation

The moment-curvature relationship is

calculated for a given value of axial force,

obtained in structural analysis and introduced

by the user. Thus, it is necessary to set a

curvature (independent variable) in order to

define the extension field compatible with

that satisfies the axial force equilibrium

condition, trough the chosen constitutive

relations. From the obtained extensions field,

the bending moment value is again obtained

by stress-strains relations and equilibrium of

bending moment.

The extension field of the cross section can be

defined by its curvature and by its center of

mass extension (see Figure 5).

(15)

Figure 5 – cross section extension field

Therefore set the curvature, it is needed to

determinate to what is the value of the center

of mass extension that solves the equilibrium

equation (7), in accordance to the chosen

value of axial force.

Stresses are calculated respecting the

assumptions:

Steel with horizontal top branch

(perfect plasticity) as recommended in

Designers Guide[4].

Confined concrete according to one of

the following hypotheses:

o Stress-strain relation c (see 2.1)

o Stress-strain relation d (see 2.1)

Cover concrete is considered as non-

confined concrete. The stress-strain

relation chosen for the cover concrete

is the parabola-rectangle relation a)

described in 2.1. Whenever the

concrete fibers reach the maximum

compression extension, they are no

longer considered (σ = 0).

7

The cross section is discretized as in the

calculation of bending and axial resistance

(see 3.1.1). The only difference lies on the

separation of concrete section in two: the

confined concrete core and cover concrete as

shown in Figure 6.

Figure 6 – cross section discretizaton: confined concrete and non-confined concrete

Iteratively solving the equation (7)2 with an

error of 0.1% in the axial force

, the extension and consequently the

extensions field are obtained.

The value of momentum is obtained by the

equation (8), using the same stress-strains

relations used for the calculation of the axial

force.

Values of curvature are arbitrated until one of

the following conditions is met:

(16)

(17)

5 Example of Application

Just part of the needed verifications is

presented in this extended abstract (major axis

and higher axial force). The verifications are

done both by equation (14) and explicitly (M-C

diagram).

5.1 Problem

2 Equation (15) is implicit in equation (7).

It must be made the safety verification of the 3

meters column with the following seismic

parameters and cross-section (30X60):

Class DCM

A400NR Class C

C25/30

Figure 7 – example: cross section and reinforcement pattern

5.2 Flexural resistance

In this example only the major axis resistance

and ductility are verified, thus no biaxial

bending function is executed.

The program checks (does not design) the

flexural strength. So it is first necessary to

arbitrate the reinforcement pattern (Figure 7).

Running the function of bending with axial

force, with the two types of stress-strains

diagrams for steel (presented in 2.2) the

following results are obtained (Figures 8 and

9):

8

Figure 8 – stress-strains diagram for steel and options for inclined top branch and material partial factors

Figure 9 – bending with axial force verification.

5.3 Shear

The shear force is calculated in accordance

with clause 5.4.2.3.(1)P and the capacity

design rule, on the basis of the equilibrium of

the column under end moments corresponding

to plastic hinge formation with an increase

factor of 1,1 for DCM.

Figure 10 – shear verification

5.4 Ductility

Longitudinal bars are already defined (Figure

7), the user must insert the hoops pattern, their

diameter and vertical spacing. To assist hoops

tracing, the program draws the section and

numbers the longitudinal bars, as shown in

Figure 11(the pattern adopted).

Figure 11 – hoops pattern

5.4.1 Verification by expression (14) – 5.15

of EC8

Inserting N = 1000KN and pressing the DCM

button the following result (Figure 12) is

obtained.

-3.000

-2.000

-1.000

0

1.000

2.000

3.000

4.000

5.000

6.000

0 200 400 600

N(KN)

M(KNm)

Msdy

not hardened

hardened

0,7 * not hardened

0,7 * hardened

9

0,0075; 536,30,007; 529,5

0,0155; 612,00,0155; 613,7

0,1475; 550,70,1365; 528,4

0

100

200

300

400

500

600

700

0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16

M(KNm)

φ(m-1)

M-C Rc)

M-C Rd)

My

Mmax

Mu

Figure 12 – DCM function output

5.4.2 Explicit (M-C) ductility verification

The ductility of the cross section represented in

Figure 11 is verified explicitly by the two

constitutive relations that take confinement into

account (c and d) described in 2.1. In both

.

It is necessary to run the function of DCM

ductility because the constructive

displacements, such as the limits of , ,

and , still have to be checked.

The M-C diagram is shown in Figure 13.

It is easily noticed that, in this example, there is

no significant difference between the results

obtained by relation c) and d). The major

difference is that after the spalling of cover

concrete, if the relation assumed is d), the

cross section still lightly decreases its

resistance when the curvature increases.

Curvature ductility factors are obtained by

and

, both values well above

the required value of 5,2. It can be noticed that

the result of the ductility verification made by

expression (14) was , implying

that the coefficient assumed by this method is

very similar to 5,2.

6 Conclusion

This work is developed on rectangular

columns, which together with circular ones are

the only ones for which the EC8[2] presents a

simplified method that avoids explicit

calculation of curvature ductility. That explains

why it has been given some emphasis to the

verification of ductility of columns by a method

other than the resolution of the equations

Figure 13 – M-C relation

10

proposed by EC8[2] for verification of critical

regions (in this case equation (14) of the

extended abstract or 5.15 in EC8[2])

Symbols

Area

Cross-sectional area of the shear

reinforcement per unit length

Width of confined concrete

Gross cross-sectional width

Secant modulus of elasticity of

concrete

Design value of modulus of elasticity

of reinforcing steel

Compressive strength of concrete

Compressive strength of confined

concrete

Design value of concrete

compressive strength

Characteristic value of concrete

compressive strength

Characteristic value of confined

concrete compressive strength

Design yield strength of

reinforcement steel

Characteristic yield strength of

reinforcement steel

Coefficient ; Factor

Length; Span

Length of critical region

Length of hoops

Bending moment

Design value of the bending moment

Flexural resistance

Axial Force

Behaviour factor

Basic value of ductility factor

Spacing of the hoops in critical

region

Fundamental period of the building

Period at the upper limit of the

constant acceleration region of the

spectrum

Shear force

z Lever arm of internal forces

Confinement effectiveness factor

Horizontal confinement effectiveness

factor

Vertical confinement effectiveness

factor

Partial factor for concrete

Partial factor for reinforcing steel

Extension

Compressive strain in the concrete

Compressive strain in the concrete

at the peak stress fc

Compressive strain in the confined

concrete at the peak stress fc

Ultimate compressive strain in the

concrete

Ultimate compressive strain in the

confined concrete

Extension in the steel at yield

Design ultimate extension of

reinforcement steel

Characteristic ultimate extension of

reinforcement steel

Truss angle

Ductility coefficient factor

Normalized design axial force

Reinforcement volumetric ratio

Stress

Lateral stress on concrete

Curvature

Mechanical volumetric ratio of confining hoops within the critical regions

References

[1] EN 1992-1:2004, Eurocode 2: Design

of concrete structures – Part 1:

General rules and rules for buildings.

CEN, Brussels, 2004

[2] EN 1998-1:2004, Eurocode 8: Design

of structures for earthquake resistance

– Part 1: General rules and rules for

buildings. CEN, Brussels, 2004

[3] CEB – FIB Model Code 90, CEB –

FIB. Thomas Telford 1993

[4] Faccioli, Ezio; Fardis, Michael; Pinto,

Paolo entre outros. Designers Guide to

En 1998-1 and 1998-5. Eurocode 8:

Design Provisions for Earthquake

Resistant Structures. Thomas Telford

2005.

11

[5] REBAP, Regulamento de Estruturas

de Betão Armado e Pré-Esforçado;

Decreto-Lei nº349-C/83, Imprensa

Nacional – Casa da Moeda, Lisbon,

1984.

[6] RSA, Regulamento de Segurança e

Acções para Estruturas de Edifícios e

Pontes; Decreto-Lei nº235/83,

Imprensa Nacional – Casa da Moeda,

Lisbon, 1983.