Soil-Structure Interaction for foundations on High-Speed...

i

Soil-Structure Interaction for foundations on High-Speed Railway Bridges

JOHAN LIND ÖSTLUND, ANDREAS ANDERSSON, MAHIR ÜLKER KAUSTELL, JEAN-MARC BATTINI

TRITA-BKN, Report 166 ISSN 1103-4289 ISRN KTH/BKN/R--166--SE

Structural Engineering and Bridges, 2017 Civil and Architectural Engineering

KTH, SE-100 44 Stockholm www.byv.kth.se

Soil-Structure Interaction for foundations on High-Speed

Railway Bridges

JOHAN LIND ÖSTLUND, ANDREAS ANDERSSON,

MAHIR ÜLKER-KAUSTELL, JEAN-MARC BATTINI

Report Stockholm, Sweden 2017

TRITA-BKN. Report 166, 2017 ISSN 1103-4289 ISRN KTH/BKN/R--166--SE

KTH School of ABE SE-100 44 Stockholm

SWEDEN

Royal Institute of Technology (KTH) Department of Civil and Architectural Engineering Division of Structural Engineering and Bridges

i

Abstract

This report contains a parametric study on the dynamic response of railway bridges on flexible supports. The results are based on simulations using 2D and 3D models. The dynamic stiffness of the supports is described by separate models of the foundation, including relevant stress and strain dependent soil properties from permanent loading that is linearized in a subsequent dynamic analysis. The complex-valued dynamic stiffness constitutes the boundary conditions in a separate analysis of the bridge superstructure that is solved in frequency domain.

Two different foundation types are studied; shallow slab foundation with relatively good ground conditions, and pile group foundations with relatively poor ground conditions. In both cases, the foundation slab and the pile group have fixed geometry. In the parametric study, the corresponding vertical static foundation stiffness range from 2 – 20 GN/m for the slab foundation and 5 – 25 GN/m for the pile group foundation.

For the slab foundations, both the stiffness and damping highly depends on the properties of the soil, foundation depth and geometry of the foundation slab. For the pile group foundations, the stiffness is mainly governed by the pile group and the damping by the soil.

Based on the simulations, the additional damping from the slab foundation is in most cases negligible. Only for relatively soft foundations and short-span bridges significant additional damping is seen. For the pile group foundations, the additional damping is in some cases significant, especially for deeper foundations and short-span bridges. Considering a lower bound of the parametric study does however result in a negligible contribution.

The dynamic response from passing trains show that the assumption of fixed supports in most cases is conservative. However, the flexible supports may result in a lower natural frequency that should be accounted for in order to not underestimate the resonance speed of the train.

If flexible supports are included in a dynamic analysis, both the stiffness and damping component needs to be included. The frequency-domain approach presented in this report is a viable solution technique but is not implemented in most commercial software used in the industry.

Keywords: Dynamic soil-structure interaction, impedance, foundation stiffness, railway bridge, high-speed trains.

iii

Sammanfattning

I denna rapport redovisas en parameterstudie avseende den dynamiska responsen hos järnvägsbroar på flexibla upplag. Resultaten baseras på simuleringar med 2D och 3D modeller. Den dynamiska styvheten från upplagen beskrivs i separata modeller av grundläggningen, där jordens spännings- och töjningsberoende egenskaper från permanent last linjäriseras i en efterföljande dynamisk analys. Den komplexa dynamiska styvheten utgör randvillkor i en separat analys av broöverbyggnaden och löses i frekvens-domän.

Två olika grundläggningstyper undersöks, plattgrundläggning med lågt grundläggnings-djup och relativt goda grundförhållanden samt pålgrundläggning med relativt dåliga grundläggningsförhållanden. I båda fallen har platt- respektive pålgrundläggningen konstant geometri. I parameterstudien är den motsvarande vertikala statiska grundläggningsstyvheten i storleksordning 2 – 20 GN/m för plattgrundläggning och 5 – 25 GN/m för pålgrundläggning.

För plattgrundläggningar beror både styvhet och dämpning i hög grad på jordens egenskaper, grundläggningsdjup och bottenplattans storlek. För Pålgrundläggningar beror styvheten i första hand på pålgruppens styvhet och dämpningen på jordens egenskaper.

Baserat på simuleringarna visas att tilläggsdämpningen för plattgrundläggningen i de flesta fall är försumbar. Bara för några fall med relativt mjuk grundläggning och korta broar erhålls betydande tilläggsdämpning. För pålgrundläggningarna är tilläggs-dämpningen i vissa fall högre, särskilt för större grundläggningsdjup och kortare broar. Från ett undre intervall av parameterstudien är dock tilläggsdämpningen försumbar.

Den dynamiska responsen från passerande tåg visar fasta upplag i de flesta fall är ett antagande på säker sida. Eftergivliga upplag kan medföra en lägre egenfrekvens vilket bör beaktas för att inte underskatta tågets resonansfart.

Om flexibla upplag beaktas i en dynamisk analys bör dess bidrag från både styvhet och dämpning inkluderas. De frekvensdomänbaserade metoder som redovisas i denna rapport är en lämplig lösningsmetod, men är inte implementerad i de flesta kommersiella programvaror som används i industrin.

Nyckelord: Dynamisk jord-strukturinteraktion, impedans, grundläggningsstyvhet, järnvägsbro, höghastighetståg.

v

Preface

The research presented in this report has been funded by the Swedish Transport Administration via the research project TRV 2016/56775, ”Brodynamik på höghastighetsbanor – inverkan av upplagens eftergivlighet”.

The project consists of theoretical simulations of the dynamic response of non-ballasted railway bridges from passing high-speed trains. The resulting deck vertical acceleration and vertical displacement is studied for the case of different support flexibility and compared to the case of non-flexible supports.

Stockholm, November 2017

The authors

vii

Contents

Abstract i

Sammanfattning iii

Preface v

1 Introduction 1

Background . . . . . . . . . . . . . . . . . . . . . . . 1

Aim and scope . . . . . . . . . . . . . . . . . . . . . . 2

Limitations . . . . . . . . . . . . . . . . . . . . . . . 2

Outline of the report . . . . . . . . . . . . . . . . . . . . 3

2 Dynamic soil-structure interaction 5

Basic concepts . . . . . . . . . . . . . . . . . . . . . . 5

Dynamic properties of soil . . . . . . . . . . . . . . . . . . 6

Impedance functions . . . . . . . . . . . . . . . . . . . . 14

3 Slab foundations 17

Material properties . . . . . . . . . . . . . . . . . . . . . 17

2D models . . . . . . . . . . . . . . . . . . . . . . . . 17

3D models . . . . . . . . . . . . . . . . . . . . . . . . 20

Verification of impedance functions . . . . . . . . . . . . . . 22

Verification of the bridge-soil system . . . . . . . . . . . . . . 24

4 Pile group foundations 25

Material properties . . . . . . . . . . . . . . . . . . . . . 25

Pile group layout . . . . . . . . . . . . . . . . . . . . . 25

2D models . . . . . . . . . . . . . . . . . . . . . . . . 26

3D models . . . . . . . . . . . . . . . . . . . . . . . . 30

viii

5 Bridge models 33

Basis of design . . . . . . . . . . . . . . . . . . . . . . 33

Methods of analysis . . . . . . . . . . . . . . . . . . . . 34

2D models . . . . . . . . . . . . . . . . . . . . . . . . 35

3D models . . . . . . . . . . . . . . . . . . . . . . . . 36

Damping from SSI, 2D-model . . . . . . . . . . . . . . . . 38

Damping from SSI, 3D-model . . . . . . . . . . . . . . . . 41

Response from passing trains, 2D model . . . . . . . . . . . . 42

Response from passing trains, 3D model . . . . . . . . . . . . 43

6 Conclusions 45

General remarks . . . . . . . . . . . . . . . . . . . . . . 45

Slab foundations . . . . . . . . . . . . . . . . . . . . . 45

Pile group foundations . . . . . . . . . . . . . . . . . . . 46

Further research . . . . . . . . . . . . . . . . . . . . . 46

Bibliography 47

A Impedance functions 49

A.1 Slab foundation, 3D-model . . . . . . . . . . . . . . . . . . 49

A.2 Pile group foundation, 3D-model . . . . . . . . . . . . . . . 52

A.3 Darendeli constants . . . . . . . . . . . . . . . . . . . . 54

B Response from passing trains 55

B.1 Slab foundation, 2D-model . . . . . . . . . . . . . . . . . . 55

B.2 Pile group foundation, 2D-model . . . . . . . . . . . . . . . 60

B.3 Slab foundation, 3D-model . . . . . . . . . . . . . . . . . . 64

B.4 Pile group foundation, 3D-model . . . . . . . . . . . . . . . 65

1

1

Introduction

Background

For railway bridges on high-speed lines, the dynamic performance during train passages need to be verified. Eurocode EN 1990 stipulates a set of design criteria for the dynamic response, where the vertical deck acceleration and vertical deck displacement often are most important. To show that a bridge will fulfil the requirements according to EN 1990, time consuming dynamic simulations are often needed. In a previous study, so-called design diagrams were developed where the required mass and stiffness of the bridge deck could be determined by simple charts (Svedholm and Andersson, 2016). The design charts are based on a semi-analytical 2D-model on fixed supports. Dynamic simulations of a large set of bridges were also performed using a 3D-model on fixed supports, generally showing similar results as the 2D-model counterpart but with lower natural frequency due to 3D effects. (Andersson and Svedholm, 2016).

In both studies mentioned above, the flexibility of the supports was neglected. Modelling the supports with only a constant stiffness may result in unrealistic estimates of the dynamic response, especially for lower-bound static values. One of the reasons is that too flexible supports may induce impact-like loading when the train enters the bridge. This may partly be overcome by modelling the load distribution from the train to the bridge, especially for short-span bridges. A more realistic model of the support conditions during dynamic loading is however preferred but may result in even more time-consuming simulations. It also requires more input in terms of geotechnical parameters that are rarely known at the early design phase of the bridge.

The dynamic soil-structure interaction (SSI) between the foundation and the bridge superstructure may be modelled in different ways. One way is to model the complete structure including the subgrade, foundation and bridge superstructure. Since a large set of dynamic simulations usually are required, this is likely to be too time consuming for practical design application and commonly available computer resources. A more feasible way is to determine the impedance of the bridge support in a separate model that can be condensed to a set of frequency dependent spring- and dashpot elements in another model of the bridge superstructure and supports. These parameters can be obtained from the complex-valued impedance function of the foundation, after which the dynamic response from passing trains is solved in frequency domain.

Chapter

CHAPTER 1. INTRODUCTION

2

Aim and scope

The aim of this report is to study the influence of dynamic soil-structure interaction for railway bridges with slab- or beam like behaviour and give recommendations for suitable methods of analysis. The objective is further to investigate if the dynamic response from bridges with fixed supports are conservative compared to cases with flexible supports.

Limitations

It should be emphasised that dynamic soil-structure interaction of railway bridges is a vast research field including both geotechnical engineering, geodynamics, bridge dynamics and different aspects of computational methods. The scope of this report is limited to a relatively small theoretical parametric study of different foundations and the resulting dynamic characteristics. The report includes the following:

- impedance functions for shallow slab foundations, - impedance functions for a case study pile group, - bridge models with flexible supports described by the impedance functions.

The subgrade for slab foundations are based on a single layer homogenous well-compacted granular soil material down to bedrock. A stress and strain dependent soil modulus is assumed for permanent loading but linear elastic for dynamic loading. A constant value for the material damping is assumed.

The layout of the pile group is based on the case of an existing railway bridge. The subgrade is assumed similar to soft clay, modelled as linear elastic with a rather low soil modulus that increase parabolic with depth. The piles are assumed fully fixed to the surrounding soil and at the bottom cap.

The bridges are modelled both in 2D and 3D with cross-sections according to (Svedholm and Andersson, 2016). The bridges are slab- and beam like structures and have previously been optimised to reach the dynamic design limits in EN 1990 for the case of non-ballasted bridges. The bridges have not been verified by any static design and it is likely that some of the cross-sections are unrealistically thin. The SSI is included by impedance functions corresponding to typical layout of intermediate supports. End-supports typically experience additional SSI due to interaction with the wing walls and back wall. This is however not included in the analysis. The train is modelled as a constant moving force applied directly onto the bridge deck, without interaction dynamic between the vehicle and the track.

Finally, it should be emphasized that the results in this report is a purely theoretical study with the aim of illustrating the influence of SSI on bridge dynamics. Many assumptions are afflicted with great uncertainties and simplifications, especially regarding the geotechnical parameters.

1.4. OUTLINE OF THE REPORT

3

Outline of the report

Chapter 2 presents some relevant dynamic properties of soil and basic concepts of dynamic soil-structure interaction (SSI). The concepts of using impedance functions for the supports and application using the finite element method is presented.

The slab foundation model is presented in Chapter 3, based on frictional soil models. Impedance functions are calculated using both a 2D axisymmetric and a full 3D model. The results are compared with semi-analytical solutions found in the literature.

The pile group foundation is presented in Chapter 4. Given the large variation of possible pile group layouts, it is deemed near impossible to cover all relevant cases. Instead, a parametric study is performed based on an existing pile group for a representative bridge. The conditions correspond to relatively soft clay and the soil modulus is based on dynamic CPT-testing at two locations.

The coupled bridge-soil models are presented in Chapter 5. Since the calculated impedance functions are complex-valued and frequency dependent, the analysis of the coupled system is performed in frequency domain. Simulations are performed using both 2D and 3D models, based on the impedance functions presented in Chapter 3 and Chapter 4. The bridge cross-sections are based on the design diagram approach descried in (Svedholm and Andersson, 2016). The results consist of both additional damping from SSI as well as simulated train passages. Further results from passing trains are found in Appendix B.

Conclusions from the study is presented in Chapter 6.

5

2

Dynamic soil-structure interaction

Basic concepts

Within the context of this report, dynamic soil-structure interaction (SSI) is referred to as the influence from the soil on the bridge deck dynamic response. SSI may alter the bridge response due to a change in stiffness, damping and mass. For bridges on conventional supports, the main part of SSI is expected from the foundation and transferred to the supports. For other types of bridges, e.g. integral- and portal frame bridges as well as bridges with integrated wing walls, significantly larger influence of SSI may be expected. This study focuses however on slab bridges on conventional supports, as illustrated in Figure 2.1.

Bridges with slab foundations are expected only for relatively good ground conditions of well compacted frictional soil and likely for moderate foundation depths. For larger depths of poor soil conditions, the ground conditions either has to be improved or excavated or substituted with pile group foundations.

Figure 2.1: Conceptual sketch of a continuous railway bridge on slab foundation.

Chapter

CHAPTER 2. DYNAMIC SOIL-STRUCTURE INTERACTION

6

Dynamic properties of soil

In the following section a brief general overview of different aspects of the soil parameters to be used in dynamic analysis within the current context is presented. The specific parameters used in the analyses are presented in section 3.1 for the slab foundation and in section 4.1 for the pile group foundation.

Elastic wave propagation

In order to give accurate and converged results in dynamic soil-structure interaction calculations wave motion needs to be considered. There are different types of waves in elastic continuum media, of which three of the more important types are presented here. First, the P-wave, or primary wave, is a dilatational wave with the soil motion parallel to the wave direction. This is the fastest of the waves and will travel via the water media in saturated soils. Secondly, the S-wave, also known as shear wave or secondary wave, is the second fastest of the waves and has its soil motion perpendicular to the wave direction. The S-wave travels through the soil grain structure and is not affected by saturation. Finally, the Rayleigh wave is a surface wave where the soil motion is circular in the direction of the wave. The amplitude of this wave decreases logarithmically with depth. The wave speeds are dependent on Poisson’s ratio, the density and the modulus of the soil medium and may be calculated as in Eq. (2.1), where vp is the P-wave speed, vs is the S-wave speed and vR is the Rayleigh-wave speed.

ρG

v =s , ν

νvv

21

22sp −

−= , ν

ν+=+R s

0.862 1.14

1v v (2.1)

S-wave speed is usually recorded in seismic field measurements, as it is not affected by soil saturation. By also knowing the density of the soil the shear modulus may be back-calculated, as in Eq. (2.2). To gain the E-modulus, Poisson’s ratio is required. The shear wave speed in gravel ranges from 200 to 400 m/s. The shear wave speed in clay and silt ranges from 0 to 250 m/s. In moraine, the shear wave speed may range from 200 to 700 m/s. (SGI, 2000)

ρ= 2max sG v (2.2)

Maximum shear modulus

The linear elastic shear modulus of soil is stress dependent. There are many empirical formulas to calculate the linear elastic shear modulus, see summaries of these formulas in e.g. Bodare (1997) and SGI (2000). Below is a general expression on affecting factors.

( ) nP σIOCRefG mmax ,, ′= (2.3)

2.2. DYNAMIC PROPERTIES OF SOIL

7

Here, e is void ratio representing the compaction rate, OCR is over-consolidation ratio, IP is plasticity index, σm' is effective confining pressure and n is a soil dependent factor between 0.3 to 0.6, in many cases put to 0.5. The factors within the function mark and n are internal material properties of the soil. The mean effective stress is dependent on external loading. The self-weight of the soil represents external loading for in-situ conditions. From this formula it would seem that a higher level of loading would increase the modulus with a higher stress level. However, as the loading is increased there is also a decrease of modulus from strain level. The effect of loading yields thus both increase and reduction of the modulus at the same time.

Material damping

The material damping ratio represents the energy dissipation in the soil. Among others, friction between particles in the soil, strain rate effect and non-linear soil behaviour are mechanisms that contribute to material damping. The hysteretic damping D is described in Figure 2.2, where WD is the energy dissipating in one cycle of loading and WS is the maximum strain energy stored during the cycle. The area within the hysteresis loop is WD and the triangular area represents WS. In theory, there would be no dissipation of energy in the linear elastic domain in the hysteresis model. However, there is energy dissipation even at very small strain levels evident from laboratory tests, which corresponds to the small-strain damping ratio, Dmin, and is a constant value. (Zhang, et al., 2005) The small-strain damping ratio depends on soil type. From laboratory tests, the material damping ratio of friction soils seems to be between 0.5-2%, while cohesion soils may have a somewhat higher soil damping ratio at around 1-4 %. These values depend on a number of factors.

Figure 2.2: The hysteresis loop of one loading cycle. Includes the maximum shear modulus Gmax, shear modulus G and damping ratio D. Reproduced from Zhang, et al (2005).

It should be noted that a non-disturbed in-situ soil may have different properties than that of a disturbed sample tested in laboratory environment. A study by Zeghal, et al. (1995) suggested that the in-situ damping ratio may be as high as double, or even more, of the value of a laboratory test for low shear strains. However, the accuracy of in-situ seismic test methods to determine damping ratio used within that and other studies give questionable results and there is no robust method today. It seems laboratory tests give conservative values regarding damping ratios and shear modulus (Darendeli, 2001)

τG

max

γ

1 1G = τ/γ

WD

4πWS

D =W

D

WS


8

The damping D is increased at strain levels greater than the linear elastic domain, adding damping to Dmin. This increase in damping, DMasing and the non-linear variation of the hysteresis loop may be described by the Masing’s rule theory (Masing, 1926) and is further described in the following section.

Non-linearity of soil

Soil is a non-linear material. The reduction of shear modulus and the determination of damping ratio of soils has been thoroughly investigated throughout the years. Studies on friction soils include authors such as Rollins, et al. (1998), Seed & Idriss (1970), & Massarch (2004) among many others. Studies considering cohesion soils include Zhang, et al. (2005), Vucetic & Dobry (1991) and (Hardin & Drnevich (1972).

Table 2.1 presents a summary of affecting parameters regarding the impact on non-linear modulus reduction and damping ratio determination on clean sand and cohesive soils. The primary affecting parameters are strain level, mean effective confining pressure and void ratio. The number of loading cycles is an important factor regarding damping ratio determination, but not regarding modulus. For cohesive soils, such as clay and silt, the dependency on degree of saturation is also important. Most affecting parameters are internal material properties such as void ratio and degree of saturation. Some of the more affecting parameters are however the results of loading, such as stress, strain, loading frequency and number of loading cycles. Figure 2.3 presents a study of the influence of each of these four parameters on shear modulus decrease and damping ratio determination compared to a laboratory study on gravelly soil samples. The shear strain level is applied on the x-axis.

The effective shear strain γeff is calculated according to Eq. (2.4), (Halabian & Naggar, 2002) and (Lysmer, et al., 1974). In the formula, α is a parameter in the interval 0.5-0.7, commonly chosen to 0.65 in seismic analysis.

( ) ( ) ( ) ( )2yz

2xz

2xy

2zy

2zx

2yxeff 6

3γγγεεεεεε

αγ +++−+−+−= (2.4)

Gmax is valid for γ <10-5, where the material modulus can be considered linear.


9

Table 2.1: Parameters that control non-linear soil behaviour and their relative importance of affecting shear modulus reduction and material damping determination of clean sand and cohesive soils. Adapted from Darendeli (2001), based on Hardin & Drnevich (1972).

Parameter

Impact on modulus

Impact on Damping

Clean Sands

Cohesive Soils

Clean Sands

Cohesive Soils

strain amplitude *** *** *** *** mean effective confining pressure *** *** *** *** void ratio *** *** *** *** number of loading cycles + * *** *** degree of saturation * *** ** -over consolidation ratio * ** * **effective strength envelope ** ** ** **octahedral shear stress ** ** ** **frequency of loading (above 0.1 Hz) * * * **other time effects (Thixotropt) * ** * **grain characteristics, size, shape, gradation, mineralogy * * * *

soil structure * * * * volume change due to shear strain below 0.5% - * - **** very important ** less important * relatively unimportant+ relatively unimportant except for sand- unknown


10

Figure 2.3: Shear strain vs. normalised shear modulus and damping as function of the mean effective stress σm′, load cycles and load frequency. Measured data from (Rollins, 1998) and theoretical from (Darendeli, 2001).


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Friction soil

Three empirical relationships regarding linear shear modulus distributions by depth of friction soil, from sand to gravelly soils, are compared in Figure 2.4. Also, the shear moduli calculated from the upper and lower bound shear wave speeds for gravel are presented. The formula suggested by TR Geo 13 (2014), Eq. (2.5a), is a simplified formula appreciating the upper and lower boundaries of friction soil, where KTRGeo is a user-chosen factor from 15·103 to 30·103, the lower value corresponding to sand and the higher to crushed rock material.

The TR Geo formula was validated against two empirical studies. Prange (1981) suggested the formula in Eq. (2.5b), valid for coarse, angular ballast material of fractions<70 mm. The void ratio, e, was chosen to 0.15 to 0.6. For friction soil, the void ratio may be between 0.15 and 0.9 according to SGI (2008), however, a void ratio of 0.9 means that almost half of the volume consists of voids which is assessed unlikely in the general case of in-situ soil. Seed et al (1984) studied gravelly soils, see Eq. (2.5c), where KSeed is a factor between 75 and 135 corresponding to the minimum and maximum values of gravel in the study. Both formulas by Seed et al. (1984) and TR Geo formula for determining the limits of the shear modulus dependency on stress include the effect of void ratio implicitly and do not give an option to include it explicitly.

Figure 2.4: Gmax as function of depth according to (2.2) and Eq. (2.5).

5.0mTRGeomax σKG ′= (MPa) (2.5a)

( ) 38.0m

26

max 1

97.21023.7 σ

e

eG ′

+−⋅= (MPa) (2.5b)

5.0mSeedmax 1000 σKG ′= (psf) (2.5c)

H (

m)


12

The initial shear modulus G/Gmax and the material damping D depend on the shear strain γ according to Eq. (2.6) and Eq. (2.7), (Rollins, 1998). The mean effective stress σm′ is described by Eq. (2.8). Note that there is a misprint in (Rollins, 1998) that is

adjusted in Eq (2.6) so that G/Gmax = 1 at γ = 0.

( )γγG

G2000

max 10116001

1−++

= , 0 ≤ γ ≤ 10-2 (2.6)

( )( ) 75.09.010015.01188.0−−++= γD (2.7)

( ) 3'z

'y

'x

'm σσσσ ++= (2.8)

Within this project, the reduction of modulus and determination of damping ratio have been calculated using the method presented in Darendeli (2001). Darendeli worked with a statistical approach using 18 parameters, ϕn, evaluated from laboratory tests and dependent on soil type, and given as mean values and variances. See further Appendix A.3.

Cohesive soil

In addition to void ratio, the empirical relationships of linear shear modulus are dependent on plasticity and over-consolidation ratio. There are several suggestions on empirical relationships of the linear shear modulus of cohesion soils. Hardin & Black (1968) suggested the formula in Eq. (2.9a) for low-plastic or stiff clays. This formula does neither consider OCR nor plasticity index. The void ratio, e, for clays ranges from 0.3 to 3.0 for clays (SGI, 2008). The formula in Eq. (2.9b) by Hardin (1978) is valid for low plastic clays and considers both plasticity index and OCR. Here, k is a factor dependent on the plasticity index and pa is the atmospheric pressure (100 kPa).

( ) 5.0m

2

max 197.2

3270 σee

G ′+−= (2.9a)

( ) 5.0am2max

7.03.0

625pσ

e

OCRG

k

′+⋅= (2.9b)

The non-linear reduction of Gmax and the damping ratio determination is in addition to stress and strain levels also dependent on the degree of saturation and void ratio, represented by the plasticity index. Figure 2.5 presents a comparison of modulus reduction and damping determination between calculated Darendeli curves against empirically measured results from Vucetic & Dobry (1991).


13

Figure 2.5: Comparison of non-linear material property change of cohesion soil between empirical distributions from Vucetic & Dobry (1991) against calculated results from Darendeli constants. The Darendeli constant results are varied in plasticity index.

In this report, the shear modulus is based on experimental CPT-testing at two locations with relatively poor ground conditions, denoted Location A and Location B in Figure 2.6. Both locations are situated in Södermanland/Östergötland in East of Sweden and consists of clay and silt. At Location A, the measured shear wave speed range between 60 to 160 m/s and the average density was about 1600 kg/m3. At Location B, the measured shear wave speed range between 80 to 180 m/s and the average density was about 1800 kg/m3. The data from Location A is approximated with a 2nd order fit and the data from Location B with a linear fit. Extrapolation outside the experimental data range may yield unreliable results.

Figure 2.6: Gmax as function of depth according to Eq. (2.9) and experimental CPT-testing at location A and B. The limit from (SGI i7, 2008) is based on vs = 250 m/s.


14

Impedance functions

Most of the simulations in this report is based on a finite element method approach. Modelling the complete system of the bridge, foundation and subgrade with a continuum approach will result in relatively large and computer-intensive analysis. In dynamic analysis, the soil often needs to be modelled with both a fine mesh and large geometry to accurately represent the propagating waves in the required frequency content and to mitigate reflecting waves due to model geometry. In addition, a large set of dynamic simulations of passing trains are usually required to verify the dynamic performance.

The simulation time can be drastically reduced by decoupling the foundation and the bridge superstructure. The resulting stiffness and damping can be calculated from the separate model of a single foundation and expressed as the impedance. The concept is illustrated for a single-degree of freedom system in Figure 2.7.

The equation of motion in time domain is presented in Eq. (2.10). The complex-valued steady-state displacement x(ω) due to a harmonic load F(ω) can be obtained by solving the equation of motion in frequency domain, defined by Eq. (2.11). The resulting impedance Z(ω) is given by Eq. (2.12), where the real part corresponds to the stiffness and the imaginary part to the damping, as in Eq. (2.13). The static stiffness is given by K(0). The result is illustrated in Figure 2.8.

Figure 2.7: SDOF-system subjected to a harmonic load F(ω), a) system with mass M and fixed parameters C and K, b) equivalent system with frequency-dependent parameters C(ω) and K(ω).

+ + = ( ) ( ) ( ) ( )Mx t Cx t Kx t F t (2.10)

( )ω ω ω ω− + + =2 i ( ) ( )M C K x F (2.11)

ω ω ω ω ω= = − + +2( ) ( ) ( ) iZ F x M C K (2.12)

ω ω= = − 2( ) Re( )K Z K M (2.13a)

ω ω= =( ) Im( )C Z C (2.13b)

M

C K C(ω) K(ω)

F(ω)F(ω)

b)a)

x

2.3. IMPEDANCE FUNCTIONS

15

Figure 2.8: Illustration of the frequency-dependent stiffness K(ω) and damping C(ω).

The full impedance matrix for a general system consists of all degrees of freedom at the load point, in 3D with 3 translations and 3 rotations represented by a 6×6×ω matrix. The main contributions are often found in the diagonal terms and for the present application especially the vertical translation component.

Analytical solutions of impedances for fundamental cases are found in the literature, e.g. (Kobori et al., 1971), (Kausel, 1974), (Kausel & Ushijama, 1979) and (Gazetas, 1983). The handbook of impedances (Sieffert & Cevaer, 1992) is a collection of some of these analytical solutions.

Figure 2.9 shows the impedance obtained from the 3D-model illustrated in Figure 2.10. As comparison, an SDOF-system is included, with parameters fitted to the peak amplitude and resonance frequency from the 3D-model. Before resonance, the 3D-model mainly undergoes quasi-static deformation with relatively low resulting damping. After resonance however, the response is dominated by wave propagation and larger resulting damping.

Figure 2.9: Illustration of a foundation impedance, comparison between a 3D FE-model and a SDOF-system.

0 10

1

0 10

1

ω/ω1

ω/ω1

Re(

Z)/

K

Im(Z

)/ω

C

|Z(

)-1| (

m/G

N)

K(

) (G

N/m

)

C(

) (M

Ns/

m)


16

a) b)

Figure 2.10: Contour plots of displacement amplitudes of the 3D model, a) before resonance, b) after resonance.

17

3

Slab foundations

Material properties

The slab foundations are assumed to consist of a single layer of well compacted granular soil down to bedrock. In practice, the geotechnical parameters can vary significantly and the material properties may be described with many different theories depending on the application. For the present application, a realistic resulting vertical support stiffness of the footing is of main interest.

2D models

In the present study, the vertical translation component of the impedance functions is of largest importance. In 2D, this is simplified by using an axi-symmetric model of the shallow slab foundation as illustrated in Figure 3.1. The slab is assumed as rigid and massless with radius Rs. Reflecting waves are mitigated by viscous dampers cp and cs according to Eq. (3.1), An is the equivalent area that each spring is distributed on.

The soil is considered by an initial static analysis of permanent loads, consisting of the gravity load of the soil and a distributed stress at the footing. The resulting stress and strain dependent E-modulus using Eq.(2.5a) and Eq. (2.6) is updated in an iterative process. The subsequent steady-state analysis is assumed linearized at this stress state. The soil is modelled with 8-noded plane axisymmetric elements. The average element size is H/8 but is decreased in the vicinity of the slab.

The resulting impedance functions from a parametric study is presented in Figure 3.3. Fixed parameters are Rs = 4 m, R = 100 m, ρ = 2000 kg/m3, ν = 0.26 (friction soil) and damping D = 1.5%. The depth H is varied from 2 to 8 m and the vertical stress under the footing σbot is varied from 0 to 150 kPa. The case of KTRGeo = 15·103 and KTRGeo = 30·103 are studied. The resulting static stiffness is illustrated in Figure 3.2.

It shall be noted that zero stress at the footing may result in unrealistically low stiffness, as shown by the difference between σbot = 0 kPa and 50 kPa in Figure 3.3.

Chapter

CHAPTER 3. SLAB FOUNDATIONS

18

Figure 3.1: 2D axi-symmetric model of a slab foundation.

ρEAρc np = , ρGAρc ns = (3.1)

Figure 3.2: Static vertical foundation stiffness from the parametric study.

Rs

R

H

F(ω)

cp

cs

E(σ,γ), ρ, ν, ζ

2 4 6 8

H (m)

0

5

10

15

20

25

3.2. 2D MODELS

19

Figure 3.3: Impedance functions for vertical translation, based on the axi-symmetric model for slab foundation. K1 = 15·103 (black) and K1 = 30·103 (red), σbot is the vertical stress on the footing.


20

3D models

The geometry of the slab foundation model is shown in Figure 3.4. The radius R = 200 m and the slab is imbedded 1.6 m below the top surface corresponding to the frost-free depth. The geometry of the foundation slab is 9.4×3.3×1.0 m in all analyses. The mass of the foundation slab was not included when calculating the impedance functions and the contact with the adjacent soil was instead modelled using a coupling/constraint to a reference point with a rigid link.

The 3D model of the slab foundation is modelled in Abaqus using quadratic tetrahedral elements (C3D10). Linear infinite elements (CIN3D8) were used at the boundaries to mitigate reflecting waves. The mesh size in the vicinity of the foundation slab was set to 0.25 m and then varied from 0.5 m to 20 m at the boundaries.

The stress and strain dependent modulus of the soil is considered similar to the 2D model. In the case of the 3D model, Darendeli constants in Table A.1 were applied to calculate the modulus reduction instead of (Rollins et al., 1998). In this case, the modulus reduction curve also depends on the stress state, as illustrated in Figure 2.3. The lower bound of the E-modulus of the soil is limited to 100 MPa. Further, a damping D = 2%, ρ = 2000 kg/m3 and ν = 0.25 was used.

Figure 3.4: 3D model of the slab foundation.

Impedance functions are calculated for 8 different models, with parameters according to Table 3.1. The vertical translation component is presented in Figure 3.6. The static vertical stiffness is presented in Figure 3.5. All main impedance components for translation and rotation is presented in Appendix A.1.

Table 3.1: Parameters for the 3D model of the slab foundations.

model KTRGeo·103 H (m) σbot (kPa)

1 15 2 110 2 15 2 210 3 15 4 110 4 15 4 210 5 30 2 110 6 30 2 210 7 30 4 110 8 30 4 210

foundation slab

soil domain

infinite elements

R

a) b)

x

z

y

3.3. 3D MODELS

21

Figure 3.5: Static vertical foundation stiffness from the 3D-model.

Figure 3.6: Impedance functions for vertical translation, based on the 3D model for slab foundation. K1 = 15·103 (black) and K1 = 30·103 (red), σbot is the vertical stress on the footing.

Kst

atic

(G

N/m

)


22

Verification of impedance functions

The finite element approach for calculating the impedance functions from the slab foundation is verified using theoretical cases found in the literature. Three cases are illustrated in Figure 3.7, all consisting of a homogenous elastic soil with a constant stress-independent E-modulus.

Figure 3.7: Models for verification, a) flexible square slab on a viscoelastic soil, b) rigid square slab, viscoelastic or structurally damped soil, c) rigid circular slab on a structurally damped soil.

Flexible square slab

A semi-analytical solution to the flexible square slab is reported in (Kobori, 1971). The results are compared with the FEM-solution in Figure 3.8. Input to the analysis is B = 1 m, H = 4 m, E = 400 MPa, ρ = 2000 kg/m3, ν = 0.25. The viscosity μ′ is calculated according to Eq. (3.2). The results are presented in Figure 3.8 where a0 is a dimensionless frequency parameter according to Eq. (3.3).

Figure 3.8: Vertical impedance for a flexible square slab, comparison between 3D

FEM and semi-analytical solution by (Kobori et al, 1971).

μ′ = η G B / vs (3.2)

s0 vBωa = (3.3)

a)

z E,ρ,ν,η

2R

z E,ρ,ν,η/μ′

2B

2B

Hz E,ρ,ν,μ′

2B

2B

b) c)

3.4. VERIFICATION OF IMPEDANCE FUNCTIONS

23

Rigid square slab

A rigid slab is modelled with similar conditions as the flexible slab. Solutions the Boundary Element Method (BEM) is compared with FEM in Figure 3.9. The results have been normalised to the static stiffness for comparison.

Figure 3.9: Vertical impedance for a rigid square slab, comparison between 3D FEM using either viscoelastic or structural damping and the Boundary Element Method (BEM), η = 0.04 in all cases.

Circular rigid slab

A circular rigid slab is modelled using both FEM and BEM. In FEM, both the axisymmetric and the full 3D-model is compared. The input is R = 1 m, H = 2 m, E = 400 MPa, ρ = 2000 kg/m3, ν = 0.33.

Figure 3.10: Vertical impedance for a rigid circular slab, comparison between FEM in 3D/2D and BEM, η = 0.1 in all cases.

Re(

Z)

Im(Z

)


24

Verification of the bridge-soil system

A full 3D-model of the bridge and the slab foundation model is analysed and verified with the corresponding bridge-soil system modelled with impedance functions. The full 3D-model of a 10 m simply supported slab bridge is illustrated in Figure 3.11. The input for the soil is based on soil model no. 3 in Table 3.1.

In the reduced model where the foundation is described by impedance functions, no interaction between the two supports are included. Further, only the diagonal terms in the impedance matrix is included.

The frequency response function (FRF) for the models are presented in Figure 3.12. Good agreement is found between the two models.

Figure 3.11: 3D FE-model of the coupled bridge and slab foundation model, displacement field at 20 Hz steady-state response for a single wheel load on the bridge deck, located at the position of the outer rail.

Figure 3.12: Frequency response function for the vertical displacement at midspan.

|z|

|z|

25

4

Pile group foundations

Material properties

Pile group foundations are often used for poor ground conditions. Given the large variation in geotechnical conditions and the unique layout of individual pile groups makes it more difficult to make general simplifications compared to slab foundations. The main aim is therefore to conceptually show the dynamic behaviour of a typical pile group under relatively poor ground conditions. This will only represent a very small set of samples for plausible pile group foundations. The soil is based on the dynamic CPT-testing presented in Figure 2.6. In the 2D model, only data from Location A is used. In the 3D model, data from both Location A and B is used.

Pile group layout

The layout of pile groups is often site-specific depending both on the ground conditions and the loads from the superstructure. Within the scope of this report, a pile group from an existing bridge is selected for reference. The bridge Ullbrobäcken is a two-span double track railway bridge along Mälarbanan between Enköping and Lundby, Sweden. The bridge deck is designed as a continuous concrete slab with 24 m in both spans and was built in 1993. The layout of the pile group for the mid-support is illustrated in Figure 4.1. The dimension of the slab is 11.6×6.0×1.6 m. The pile group consists of a total of 40 piles, having a cross-section of 0.27×0.27 m, average length of 4.5 m and an inclination of 6:1 and 4:1.

[m] Figure 4.1: Pile group layout for the mid-support at Ullbrobäcken.

1.11.1 1.1 1.1 1.1 1.11.5 1.2 1.5

1.21.2

1.2

Chapter

CHAPTER 4. PILE GROUP FOUNDATIONS

26

2D models

Similar to the slab foundation, the pile group foundation is simplified using a 2D axi-symmetric model as illustrated in Figure 4.2. This is a rather crude simplification since the pile group in Figure 4.1 is not axi-symmetric. However, since the vertical translation component is of main interest, this is judged as a reasonable first approach.

The piles are modelled with Euler-Bernoulli beam elements that are pinned at the bottom at fixed at the top. Full interaction with the surrounding soil is assumed. The axi-symmetric model describes 1 rad of the full model. The corresponding stiffness from the total number of piles is compensated for by a factor α according to Eq. (4.1), where Npiles is the total number of piles and Npilerows the number of pile rows as illustrated in Figure 4.2.

pilerows

piles

π2

1

N

Nα = (4.1)

The E-modulus of the soil follows Figure 2.6, Location A. Further, ρ = 1600 kg/m3, ν = 0.48 and ζ = 3% is used for the soil and Ec = 34 GPa and ρc = 2400 kg/m3 for the piles. The slab is assumed rigid but with a lumped mass at the load point. The slab is only interacting with the piles and not the adjacent soil. Sufficient mitigation of reflecting waves is usually obtained for R>10H, but since the model is rather computationally efficient, R = 20H is used in further analysis. 8-noded plane axi-symmetric elements with an average size of 0.5 m is used.

Figure 4.2: 2D axi-symmetric model of a pile group foundation, two rows of piles, x1 = 0.6 m, x2 = 1.8 m.

To illustrate the conceptual behaviour of the pile groups, different parameters are varied in the results below. Figure 4.3 show that the total number of piles mainly influence the vertical stiffness and not the damping. Increased number of pile rows result in increased interaction between the soil and the piles and therefore increased damping. According to Figure 4.4, the pile inclination has a moderate influence on the response at low frequencies but may change the peaks of resonance.

Figure 4.5 shows the influence of the soil E-modulus. Increased soil modulus results in higher resonance frequencies and increased damping at resonance. The response at non-resonant loading, in this case below about 15 Hz, the soil has low impact on the vertical impedance. A constant E-modulus representing the average depth results in similar resonance frequency but somewhat higher damping.

x2

R

H

F(ω)

cp

cs

Es(H), ρ, ν, ζz

αEc, A

p, I

p, ρ

c

n

x1

1

4.3. 2D MODELS

27

Figure 4.3: Impedance function for vertical translation, H = 6 m, pile inclination 6:1, variation of the number of piles, x3 = x2 for 3 pile rows.

Figure 4.4: Impedance function for vertical translation, H = 6 m, 40 piles in 2 rows, variation of pile inclination.

Figure 4.5: Impedance function for vertical translation, H = 6 m, 40 piles in 2 rows, pile inclination 6:1, variation of the E-modulus of the soil.

0 10 20 30

f (Hz)

0

5

10

15

20

25

30

0 10 20 30

f (Hz)

0

5

10

15

20

25

3020 piles in 2 rows40 piles in 2 rows60 piles in 2 rows40 piles in 3 rows

K (

GN

/m)

C (

MN

s/m

)

K (

GN

/m)

C (

MN

s/m

)


28

Finally, the result from a parametric study is presented in Figure 4.7. Based on the reference pile group with 40 piles in two rows and a pile inclination 6:1, different depths H = 6, 12, 18 and 24 m is studied. For each depth, a high and low value for both the soil E-modulus Es and concrete pile E-modulus Ec is used. In total 4 impedance functions for each depth.

Figure 4.6 shows the static stiffness for all cases in the parametric study. The global vertical stiffness is mainly governed by the piles and decrease with increasing pile length.

Figure 4.6: Static stiffness for the pile groups in the parametric study.

4.3. 2D MODELS

29

Figure 4.7: Impedance function for vertical translation, based on the reference model with 40 piles in 2 rows and pile inclination 6:1.

0 10 20 30 40 50-10

0

10

20

30K

(G

N/m

)H = 6 m

0 10 20 30 40 500

50

100

150

C (

MN

s/m

)

H = 6 m

0 10 20 30 40 50-10

0

10

20

30

K (

GN

/m)

H = 12 m

0 10 20 30 40 500

50

100

150

C (

MN

s/m

)

H = 12 m

0 10 20 30 40 50-10

0

10

20

30

K (

GN

/m)

H = 18 m

0 10 20 30 40 500

50

100

150

C (

MN

s/m

)

H = 18 m

0 10 20 30 40 50f (Hz)

-10

0

10

20

30

K (

GN

/m)

H = 24 m

0 10 20 30 40 50f (Hz)

0

50

100

150

C (

MN

s/m

)

H = 24 m

1Es, 1E

c

2Es, 1E

c

1Es, 1.5E

c

2Es, 1.5E

c


30

3D models

The 3D-model for the pile group foundation is illustrated in Figure 4.8. The piles are modelled with Timoshenko beam elements (B31) coupled to the adjacent soil using the Embedded Element option in Abaqus. The bottom of the piles is pinned and the top is rigidly connected to the foundation slab. The slab is submerged 1.6 m below the top surface. In the model, R = 100 m was found sufficient. In the analysis, linear tetrahedral (C3D4) elements were used.

Figure 4.8: 3D model of the pile group foundation, a) pile group embedded in the soil, b) pile group layout.

The E-modulus of the soil vary with depth. Apart from the experimental results obtained at location A with poor soil conditions, location B with moderate soil conditions have also been analysed. The density was set to 1620 kg/m3 for Location A and 1788 kg/m3 for Location B. In both cases, the damping D = 2% and ν = 0.49 was used.

The experimental data scatters significantly and extrapolation at greater depths may be unreliable. For the present study however, the conceptual behaviour between poor and moderate soil is of main interest.

A parametric study of the pile groups is based on the data in Table 4.1. The vertical translation component of the impedance functions is presented in Figure 4.10. The static vertical stiffness is presented in Figure 4.9. All main impedance components for translation and rotation is presented in Appendix A.2.

pile group

soil domain

infinite elements

R

a) b)

x

z

y

4.4. 3D MODELS

31

Table 4.1: Parameters for the 3D model of the pile group foundations.

model location H (m) 1 A 6 2 A 12 3 A 18 4 A 24 5 B 6 6 B 12 7 B 18 8 B 24

Figure 4.9: Static stiffness for the pile groups in the 3D-model.

6 12 18 24H (m)

0

10

20

30Location ALocation B


32

Figure 4.10: Impedance function for vertical translation, based on reference 3D model with soil according to Location A and Location B.

0 10 20 30-10

0

10

20

30K

(G

N/m

)H = 6 m

0 10 20 300

50

100

150

C (

MN

s/m

)

H = 6 m

0 10 20 30-10

0

10

20

30

K (

GN

/m)

H = 12 m

0 10 20 300

50

100

150

C (

MN

s/m

)

H = 12 m

0 10 20 30-10

0

10

20

30

K (

GN

/m)

H = 18 m

0 10 20 300

50

100

150

C (

MN

s/m

)

H = 18 m

0 10 20 30

f (Hz)

-10

0

10

20

30

K (

GN

/m)

H = 24 m

0 10 20 30

f (Hz)

0

50

100

150

C (

MN

s/m

)

H = 24 m

Location ALocation B

33

5

Bridge models

Basis of design

The bridges analysed in this report are essentially based on the theoretical cross-sections presented in (Svedholm and Andersson, 2016). These are based solely on a dynamic design optimization with the objective to reach the design limits for either vertical deck acceleration (5 m/s2 for non-ballasted bridges) or vertical deck displacement (EN 1990, Figure A2.3). The train load HSLM-A is used and the design speed is 320 km/h. The cross-sections are based on a 2D model with fixed supports.

Flexible supports are generally expected to have a larger influence for short-span bridges due to the relatively high natural frequency. Therefore, the study focuses on short and medium size slab bridges. The cross-section is illustrated in Figure 5.1 and parameters are given in Table 5.1. Note that n0,eff is an effective natural frequency based on a 2D-model with fixed supports and no eccentricity, as used in (Svedholm and Andersson, 2016).

Figure 5.1: Cross-section of a two-track concrete slab bridge, the deck thickness t depends on the span length and number of spans. (Svedholm and Andersson, 2016)

12.0

7.0

0.30.4

0.4

0.5t

Chapter

CHAPTER 5. BRIDGE MODELS

34

Table 5.1: Cross-sectional data for the slab bridges. For continuous bridges in 3 or 4 spans, the outer span is 0.8L.

1 span simply supported 2 span continuous L (m) 10 15 20 25 30 40 10 15 20 25 30 40

m (ton/m) 23.0 29.5 35.0 41.4 44.7 35.3 19.7 24.6 29.3 30.8 33.5 35.6 EI (GNm2) 15.9 45.6 89.4 166.0 220.7 91.9 8.3 21.2 44.4 53.9 75.5 94.6

t (m) 0.8 1.2 1.5 1.9 2.0 1.5 0.6 0.9 1.2 1.2 1.4 1.5 n0,eff (Hz) 13.0 8.7 6.3 5.0 3.9 1.6 10.2 6.5 4.8 3.3 2.6 1.6

3 span continuous 4 span continuous

L (m) 10 15 20 25 30 40 10 15 20 25 30 40 m (ton/m) 18.6 22.7 26.0 29.8 33.5 33.1 18.5 22.5 26.0 28.0 30.6 35.0 EI (GNm2) 6.6 15.1 27.0 46.9 75.5 71.4 6.4 14.3 27.0 36.5 52.4 89.4

t (m) 0.5 0.8 1.0 1.2 1.4 1.4 0.5 0.8 1.0 1.1 1.2 1.5 n0,eff (Hz) 9.3 5.7 4.0 3.2 2.6 1.4 9.3 5.6 4.0 2.9 2.3 1.6

Methods of analysis

In this report, both the bridge and the foundation are modelled using the finite element method. The models consist of a stiffness matrix K, mass matrix M, damping matrix C and load vector F.

Eigenvalue analysis

The natural frequencies and mode shapes are obtained from an eigenvalue analysis. By solving the 1st order eigenvalue problem according to Eq. (5.1), the undamped natural circular frequency ωu = λ1/2. The mode shapes are given by the corresponding eigenvectors.

For a damped system, the 2nd order eigenvalue problem according to Eq. (5.2) needs to be solved, from which the damped natural circular frequencies are given by ωd = Im(λ) and the modal damping by ζ = -Re(λ)/ωu, and ωu = |λ|.

0=+ MλK (5.1)

02 =++ MλCλK (5.2)

5.3. 2D MODELS

35

Steady-state response

The complex-valued steady-state displacement x(ω) due to a harmonic load F(ω) can be obtained by solving the equation of motion in frequency domain, defined by Eq. (2.11). This method is performed both to calculate the impedance of the foundations and to simulate passing trains on the bridges. In the separate foundation models, F(ω) represent a unit harmonic load at the connection point to the bridge.

Since the foundation is expressed as a frequency dependent impedance, the simulation of passing trains are also solved in frequency domain. A unit harmonic load F(ω) is applied in each of the load points N on the bridge model from which the displacement matrix x(ω) with size N×ω is calculated. In the analysis, both the input load and output response are calculated at node 1 to N. The train load vector Ft(t) describes the load as function of time for a specific train speed v. The corresponding load Ft(ω) in frequency domain is obtained by a Fourier transformation. Finally, the displacement in time domain x(t) is obtained by Eq. (5.3). The acceleration can be calculated by -x(ω)/ω2, but since both displacement and accelerations are needed as output, it is more computationally efficient to calculate a(t) as the 2nd derivative of x(t).

( ) ( ) ( )( )( )=

−−−=N

n

xxtωt

newFωxIFFTtx1

)(i1 1Re (5.3)

2D models

In the 2D-model depicted in Figure 5.2, the bridge deck is modelled as a Euler-Bernoulli beam with constant flexural rigidity EI and mass m and damping ζ. The foundation is modelled as frequency dependent springs k(ω) and dashpots c(ω), based on impedance functions from a separate model. The bridge piers are not included in the foundation model and is instead represented by a lumped mass m1 = 200 ton at each support. All supports are assumed to have the same properties and only the vertical component is considered.

Material damping of the bridge deck is included as structural damping according to Eq. (5.4), where Kb is the stiffness matrix of the bridge deck.

Figure 5.2: Simply supported 2D beam on flexible supports.

( )ζKK i21b* += (5.4)

L

k(ω)

EI, m, ζm1

c(ω)


36

3D models

The 3D-model consists of the bridge deck, piers and foundation slab. The model is created in Abaqus and is illustrated in Figure 5.3. The cross-section of the deck follows Figure 5.1 and Table 5.1. The geometry of the piers and foundation slab is illustrated in Figure 5.4 and are the same in all analyses. For the pile group foundations however, the geometry of the foundation slab is modelled according to Figure 4.1. The height of the bearings was chosen to 0.15 m and the width to 0.6 m, equally applied to all bridges bearings.

The 3D-model consists of 4-noded shell elements (S4R) except for the edge beams that are modelled with beam elements. The maximum mesh size is 0.3 m for the bridge deck and 1.0 m for the flanges and the edge beams.

Figure 5.3: View of the 3D model of the bridge, L = 25 m.

Figure 5.4: Geometry of the bridge pier and the foundation slab.

z

yx

0.950.6

3.3 9.4

7.00.9

1.6

5.0

1.0

soil surface

[m]

5.4. 3D MODELS

37

Interactions and constraints

The interactions and constraints between the bridge deck, piers and foundation slab is illustrated in Figure 5.5.

In Abaqus, impedance functions may be applied as frequency dependent springs and dashpots using connector elements. The connector elements spans between two reference points with the same coordinates, at the bottom of the slab and at the position corresponding to where the impedances are calculated. The connection category of the connector section is chosen to Basic, with the translational connection type as cartesian and the rotational type as rotation. It is not possible to include both frequency dependency and coupled terms of the impedance matrices in Abaqus. Therefore, only the main directions of the impedances are included, excluding the coupled motion.

Roller and pinned support conditions of the bearings are applied between the two reference points at the same coordinates, between the bottom of the bridge and the top of the bearings.

The loads at train passages are applied as point loads at reference points coupled to square surfaces on the shell deck. The reference points correspond to the bottom of one sleeper and the square surfaces correspond to the load distribution of sleepers down to the bridge slab (0.2×0.2 m2).

Figure 5.5: Constraints used in the 3D-model.

support coupling

flange offset

edge beambearing

pier/slab connection

impedance connection point impedance to slab coupling

load pointload to deck coupling


38

Damping from SSI, 2D-model

Flexible supports are expected to change the dynamic properties of the bridge in terms of lower natural frequency, increased damping and change in mode shape. These changes can be assessed by solving the 2nd order eigenvalue problem in Eq. (5.2) for the combined system of the bridge with impedance functions.

Eurocode EN 1991-2 gives a set of lower bound values for damping to be used in dynamic analysis of passing trains, see Table 5.2. These values are based on the work summarized in (ERRI, 1999), consisting of experimental testing of a large number of bridges, see Figure 5.6. Estimating damping of railway bridges from experimental data is often afflicted with large uncertainties since it often depends on the load amplitude. Further, there are many sources of damping, depending on bridge type, boundary conditions and support conditions.

Table 5.2: Lower limit for critical damping (%) to be assumed in design according to EN 1991-2, Table 6.6.

Bridge type L < 20 m L ≥ 20 m Steel and composite 0.5 + 0.125(20 - L) 0.5 Prestressed concrete 1.0 + 0.070(20 - L) 1.0 Filler beam and reinforced concrete 1.5 + 0.070(20 - L) 1.5

Figure 5.6: Total damping in bridges, experimental data from ERRI D214/RP3. Many of the short-span concrete bridges are filler beam bridges.

For the simulations in this report, the total damping is assumed to consist of ζbridge and ζSSI according to Eq. (5.5). Damping can be non-additive, making it difficult to separate damping from different sources. Assuming that the lower-bound values in EN 1991-2 correspond to bridges with relatively low influence from SSI, these values are assigned to the bridge deck, i.e. ζbridge = ζEurocode. This may overestimate the damping in some cases but are judged to have a moderate effect compared to other uncertainties.

ζtot = ζbridge + ζSSI (5.5)

0 10 20 30 40 50 60 70

L (m)

0

2

4

6

8

10

concretesteelcomposite

5.5. DAMPING FROM SSI, 2D-MODEL

39

The 2D beam model in Figure 5.2 with vertical impedance functions from Figure 3.3 is used for calculating the additional damping from SSI. For each depth H, a total of 8 different impedances are studied, using different stress σbot at the footing and different values for KTRGeo. The cross-section of the bridge is based on the slab bridge depicted in Figure 5.1 based on the design diagrams in (Svedholm and Andersson, 2016) with fixed supports.

The total damping is estimated by solving Eq. (5.2) with k(ω) and c(ω) linearized at the first bending mode. Iterations are required since K and C depend on the support conditions at the natural frequency of the coupled bridge-foundation system. The corresponding eigenvalue is selected based on the largest MAC-value and that the damped natural frequency fd < f, where f is the undamped first natural frequency of the bridge with fixed supports. The results are presented in Figure 5.7, where Δζ = ζtot – ζEurode.

Figure 5.7: Additional modal damping for the first bending mode due to a flexible

slab foundation. H = 2 m (black), H = 4 m (blue), H = 6 m (red) and H = 8 m (green).

The largest damping from SSI is found for short-span simply supported bridges on soft foundations. The damping is rapidly decreasing with increased span length.

Some of the slab foundations used in the analysis may be unrealistically soft. If only including the case of H = 2 m and 4 m with σbot = 100 kPa and 150 kPa, the resulting damping according to Figure 5.8 is obtained. The two values at about 1% damping is for KTRGeo = 15·103 and H = 4 m.

0.7 0.8 0.9 1fd/f

0

2

4

6L = 10 m

0.85 0.9 0.95 1fd/f

0

0.2

0.4

0.6L = 20 m

0.9 0.95 1fd/f

0

0.05

0.1L = 30 m

0.98 0.985 0.99 0.995 1fd/f

0

0.01

0.02L = 40 m

1 span2 span3 span4 span


40


slab foundation, L = 10 m, σbot = 100 kPa and 150 kPa, H = 2 m and 4 m.

The first bending mode generally have similar shape for the case of fixed and flexible supports. In most cases MAC>0.95, only for a few cases of the 10 m bridge on soft foundation, 0.7<MAC<0.9 is obtained.

The similar procedure is used for estimating the additional damping from pile group foundations, using the impedance functions presented in Figure 4.7. The results are presented in Figure 5.9. Except a few outliers for the 10 m bridge, MAC>0.95. The resulting damping is generally higher compared to the slab foundation but show the same trend with decreased damping for longer spans.

Figure 5.9: Additional modal damping for the first bending mode due to a flexible pile group foundation. H = 6 m (black), H = 12 m (blue), H = 18 m (red) and H = 24 m (green).

0.85 0.9 0.95 1fd/f

0

2

4

6

8L = 10 m

0.9 0.95 1fd/f

0

0.5

1

1.5L = 20 m

0.96 0.98 1fd/f

0

0.1

0.2

0.3L = 30 m

0.99 0.995 1fd/f

-0.05

0

0.05

0.1

L = 40 m


5.6. DAMPING FROM SSI, 3D-MODEL

41

Damping from SSI, 3D-model

For the 3D-models, the complex eigenvalue-analysis available in Abaqus was used when estimating the damping from SSI. The natural frequency is estimated based on the FRF for vertical displacement at midspan/outer rail in a previous analysis. The selected frequency is then used as reference in a first order eigenvalue analysis and then a subsequent complex eigenvalue analysis. The damping obtained by the complex eigenvalue analysis was verified using the Half-Power Bandwidth method on the previously calculated FRFs. The bridge deck was assigned damping according to EN 1991-2, similar to the 2D-model.

The results for the bridges with slab foundation are presented in Figure 5.10. The results show that the additional damping due to SSI is negligible, generally less than 0.1%. It shall be noted that the results refer to impedance functions with H = 2 m and 4 m with σbot = 110 kPa and 210 kPa.

The results for the bridges with pile group foundation are presented in Figure 5.11. One can see that for short span bridges and high depths, the additional damping from SSI can be important.

Figure 5.10: Additional modal damping for the first bending mode due to a flexible slab foundation. H = 2 m (black), H = 4 m (blue).

0.92 0.94 0.96 0.98 1

0

0.05

0.1L = 10 m

0.92 0.94 0.96 0.98 1

0

0.05

0.1L = 15 m


0.92 0.94 0.96 0.98 1

0

0.05

0.1L = 20 m

0.92 0.94 0.96 0.98 1f/f

d

0

0.05

0.1L = 25 m


42


pile group foundation. H = 6 m (black), H = 12 m (blue), H = 18 m (red) and H = 24 m (green).

Response from passing trains, 2D model

The response from passing trains are simulated with the main objective to investigate if any model with flexible support may yield larger response compared with fixed supports. The same models are used as for the damping estimation in section 5.5. The response from HSLM A1-A10 is solved in frequency domain using Eq. (2.11) and Eq. (5.3). A comparison with time domain methods is shown in Figure 5.12 for a 10 m bridge on fixed supports. The results are found to be in sufficient agreement for further analysis.

Results for the same bridge on flexible supports are presented in Figure 5.13. It is found that the upper bound response for the model with flexible supports correspond well with the case of fixed supports. The obtained lower bound response generally corresponds to soft foundation.

A parametric study has been undertaken, for bridges with 1-4 span and a maximum span length 10 – 40 m. For each bridge, all impedances in Figure 3.3 and Figure 4.7 have been analysed, for train HSLM A1-A10 in the range 100 – 450 km/h in increments of 2 km/h. The results, comprising about 1.4 million train passages, are presented in Appendix B.1 and Appendix B.2.

The results show that bridge on flexible supports in most cases result in lower peak response compared to fixed supports. Flexible supports on soft foundation may however show a lower natural frequency and therefore lower resonance speed. In a few cases the results with flexible supports exceeds the case of fixed supports, in particular for the 30 m bridge in 2 spans on slab foundation.

0.94 0.96 0.98 1fd/f

0

2

4

6

(%

)

L = 10 m

0.94 0.96 0.98 1fd/f

0

2

4

6

(%

)

L = 15 m


0.94 0.96 0.98 1fd/f

0

0.5

1

1.5

2

(%

)

L = 20 m

0.94 0.96 0.98 1fd/f

0

0.5

1

1.5

2

(%

)

L = 25 m

5.8. RESPONSE FROM PASSING TRAINS, 3D MODEL

43

Figure 5.12: Simply supported beam with L = 10 m on fixed supports. Envelope of vertical acceleration and displacement due to HSLM A1-A10.

Figure 5.13: Simply supported beam with L = 10 m on flexible supports using the 2D model, envelope from HSLM A1-A10.

Response from passing trains, 3D model

A similar approach for simulating passing trains was made with the 3D-model. The same model as in section 5.6 was used and the response was solved in frequency domain. In this case, each HSLM point load was distributed to the two rail attachment points of each sleeper, with half of the amplitude of the force.

The results for a 10 m simply supported bridge is shown in Figure 5.14. Flexible supports results in a slightly lower resonance speed, especially for the slab foundation.

100 200 300 400

v (km/h)

0

2

4

6

8

10

100 200 300 400

v (km/h)

0

1

2

3

4

5

frequencydirectmodal

a max

(m

/s2 )

dm

ax (

mm

)


44

Figure 5.14: Simply supported beam with L = 10 m on flexible supports using the 3D model, envelope from HSLM A1-A10.

A parametric study was made for the single-spanned bridges, where SSI is expected to have its largest effect, with span lengths 10, 15, 20, 25 m. For each bridge, all impedances in Appendix A.1 and Appendix A.2 have been analysed for trains HSLM A1-A10 in the range 100-400 km/h with an increment of 2 km/h. The results are presented in Appendix B.3 and Appendix B.4.

The results from the parametric study show that the case of fixed supports is generally conservative but that the potential decrease in resonance speed due to the decrease of the natural frequency needs to be accounted for. The decrease in natural frequency may depend on both the support conditions and 3D effects of the bridge deck.

a max

(m

/s2 )

dm

ax (

mm

)

45

6

Conclusions

General remarks

The large natural variation of both geotechnical parameters and geometry of bridge foundations makes it difficult to derive general conclusion. The conclusions presented herein are based solely on theoretical results from a limited number of parametric studies. The parameter range and modelling techniques are somewhat different in the 2D and 3D models, but generally similar conclusions are obtained.

For most studied cases, bridges on flexible supports result in lower dynamic response compared to fixed supports. If flexible supports need to be accounted for, then it is important to include both the stiffness and damping components. Flexible supports may result in a lower natural frequency and therefore a lower resonance speed. In this report the decrease is generally less than 5% but depends both on the foundation stiffness, mass of the bridge pier and mass of the bridge superstructure.

Slab foundations

The impedance of the slab foundations mainly depends on the soil modulus, foundation depth and geometry of the slab. In this report, the stress and strain dependent soil modulus is considered. It is important to use a realistic range of the soil parameters, as conventional design values for long-term loading may result in unrealistically soft foundation stiffness. In this report, the resulting vertical static foundation stiffness range between 2 – 20 GN/m in the 2D-model and 4 – 16 GN/m in the 3D-model but may differ significantly in a real design.

The increase in modal damping is in most cases negligible, especially for higher foundation stiffness and longer bridge span. Only for the lower ranges of foundation stiffness in the 2D model and 10 m span bridges, the damping may be significant. It is difficult to assess if this correspond to realistic cases without experimental validation.

The simulations from passing trains generally show that bridges on fixed supports give conservative results. Again, soft foundation on short-span bridges may result in reduced dynamic response due to increased damping.

Chapter

CHAPTER 6. CONCLUSIONS

46

Pile group foundations

The vertical stiffness K(ω) of pile group foundations highly depends on the axial stiffness of the piles. The damping component C(ω) is mainly governed by the properties of the soil and the interaction with the piles. In this report, the resulting vertical static foundation stiffness range between 5 – 25 GN/m and is relatively similar between the 2D and 3D models since the same number of piles was used.

The results show that the increase in modal damping may be significant in some cases, especially for deeper foundation depths and short-span bridges. The lower bound of the result does however result in negligible additional damping.

The response from passing trains is generally higher for the case of fixed supports compared to flexible supports.

Further research

Based on the findings in this report, it is recommended that further research on dynamic soil-structure interaction on railway bridge dynamic focus on the following topics.

- Impedance of end-abutments. In this report all supports are generalized supportconditions valid for intermediate supports consisting of a foundation slab and apier. At end-abutments, additional interaction with the soil in contact with thewing walls and back walls is expected, that may have a larger impact especiallyon simply supported bridges.

- Bridges with integrated wing walls. Bridges designed with integrated end-shieldsmay experience significant contribution from dynamic soil-structure interaction,resulting in both additional damping and a change in boundary conditions. A toosimplified analysis of this bridge type may result in unrealistically high dynamicresponse, often owing to impact-loading on the cantilevering end of the bridgedeck. Including a more realistic soil-structure interaction approach may insteadshow that this bridge type has good dynamic performance.

- Validation by experimental testing. The results in this report is solely based ontheoretical simulations. To fully rely on the results, experimental testing isdeemed necessary. A combination of traditional geotechnical testing, includingseismic methods, together with controlled excitation of the bridge foundationsand bridge superstructure is seen as a viable solution.

In the presented study, the properties of the soil were linearized in the dynamic analysis, assuming that the increase in shear strain is small. This may need to be studied further, since larger shear strain generally results in a lower shear modulus.

47

Bibliography

Andersson, A. & Svedholm, C., 2016. Dynamisk kontroll av järnvägsbroar, inverkan av 3D-effekter (swedish), Stockholm: KTH Division of Structural Engineering and Bridges, Report KTH/BKN/R-158.

Bodare, A., 1997. Jord- och Bergdynamik (swedish), Stockholm: KTH Avdelningen för jord- och bergmekanik.

Darendeli, M. B., 2001. Development of a New Family of Normalized Modulus Reduction and Material Damping Curves, Austin: The University of Texas.

ERRI, 1999. Rail bridge for speed > 200 km/h, Recommendation for calculating damping in rail bridge decks. ERRI D 214/RP3.

Gazetas, G., 1983. Analysis of machine foundation vibrations: state of the Art. Soil Dynamics and Earthquake Engineering, 2(1), pp. 2-42.

Halabian, A. & Naggar, M., 2002. Effect of non-linear soil–structure interaction on seismic response of tall slender structures. Soil Dynamics and Earthquake, Volume 22, pp. 639-658.

Hardin, B. & Drnevich, V., 1972. Shear Modulus and Damping in Soils: Design Equations and Curves. Journal of Soil Mechanics and Foundations, 98(7), pp. 667-692.

Kausel, E., 1974. Forced Vibrations of Circular Foundations on Layered Media, Boston: Massachusetts Institute of Technology, MIT.

Kausel, E. & Ushijama, R., 1979. Vertical and Torsional Stiffnesses of cylindrical Footings, Research Rep. R 76- 6, Cambridge, Massachusetts: MIT.

Kobori, T., Minai, R. & Suzuki, T., 1971. The Dynamical Ground Compliance of a Rectangular Foundation on a Viscoelastic Stratum. 20(4), pp. 289-329.

Lysmer, J., Udaka, T., Tsai, C. & Seed, H. B., 1974. FLUSH - a computer program for approximate 3-D analysis of soil-structure interaction problems, s.l.: R.E.75-30, Berkeley: University of California.

Masing, G., 1926. Eigenspannugen und Verfestigung beim Messing (german). Proceedings of the 2nd International Congress on Applied Mechanics, pp. 332-335.

Massarch, K. R., 2004. Deformation properties of fine-grained soils from seismic tests. International Conference on Site Characterization, p. 14.

Prange, B., 1981. Resonant column testing of rail road ballast, s.l.: ISSMGE.

Rollins, K. M., Evans, M. D., Diehl, N. B. & Daily, W. D., 1998. Shear Modulus and Damping Relationships for Gravels. Journal of Geotechnical and Environmental Engineering, pp. 396-405.

BIBLIOGRAPHY

48

Seed, B. H., Wong, R. T., Idriss, I. M. & Tokimatsu, K., 1984. Moduli and Damoing Factors for Dynamic Analyses of Cohesionless Soils, Berkely: Berkely: University of California.

Seed, H. B. & Idriss, I. M., 1970. Soil Moduli and Damping Factors for Dynamic Response Analyses, Report EERC 70-10, Berkely: Berkely University of California.

SGI, 2000. SGI Information 17, Geodynamik i Praktien (swedish), Linköping: Sveriges Geotekniska Institut (Swedish Geotechnical Institute).

SGI, 2008. SGI Information 1, Jords egenskaper, Linköping: Statens geotekniska institut (Swedish Geotechnical Institute).

Sieffert, J.-G. & Cevaer, F., 1992. Handbook of Impedance Functions, Surface Foundations, s.l.: Ouest Éditions Presses Académiques.

Svedholm, C. & Andersson, A., 2016. Designdiagram för förenklad kontroll av järnvägsbroar (swedish), Stockholm: KTH Division of Structural Engineering and Bridges, Report KTH/BKN/R-157.

TR Geo 13, 2014. Trafikverkets Tekniska Råd för Geokonstruktioner -TR Geo 13 (swedish), s.l.: Trafikverket (Swedish Transport Administration).

Vucetic, M. & Dobry, R., 1991. Effect of Soil Plasticity on Cyclic Response. Journal of Geotechnical Engineering, 117(1), pp. 89-107.

Zeghal, M., Elgamal, A. E., T., T. H. & Stepp, J. C., 1995. Lotung Downhole Array II: Evaluation of Soil Nonlinear Properties. Journal of Geotechnical Engineering, 4(121), pp. 363-378.

Zhang, J., Andrus, R. D. & Juang, C. H., 2005. Normalized Shear Modulus and Material Damping Ratio Relationships. Journal of Geotechnical and Geoenvironmental Engineering, 131(4), pp. 453-464.

49

A

Impedance functions

The following appendix present the impedance functions from the 3D-model. The complete impedance matrix is 6×6×ω, but the off-diagonal terms are often several magnitudes lower than the main components. The directions follow Figure 5.3 where z is the vertical direction and ry the rotation along the axis transverse to the bridge.

A.1 Slab foundation, 3D-model

Appendix

Appendix A. Impedance functions

50

Figure A.1: Impedance functions for translation, based on the 3D model for slab foundation. K1 = 15·103 (black) and K1 = 30·103 (red), σbot is the vertical stress on the footing.

0 10 20 30f (Hz)

0

5

10

15

20

0 10 20 30f (Hz)

0

10

20

30

40

0 10 20 30f (Hz)

0

5

10

15

20

0 10 20 30f (Hz)

0

10

20

30

40

0 10 20 30f (Hz)

0

5

10

15

20

0 10 20 30f (Hz)

0

10

20

30

40

H = 2 m, bot = 110 kPa

H = 2 m, bot = 220 kPa

H = 4 m, bot = 110 kPa

H = 4 m, bot = 220 kPa

A.1. Slab foundation, 3D-model

51

Figure A.2: Impedance functions for rotation, based on the 3D model for slab foundation. K1 = 15·103 (black) and K1 = 30·103 (red), σbot is the vertical stress on the footing.


52

A.2 Pile group foundation, 3D-model

Figure A.3: Impedance functions for translation, based on the 3D model for pile group foundation, Location A (black) and Location B (red).

0 10 20 30f (Hz)

-10

0

10

20

30

0 10 20 30f (Hz)

0

50

100

150

200

0 10 20 30f (Hz)

-10

0

10

20

30

0 10 20 30f (Hz)

0

50

100

150

200

0 10 20 30f (Hz)

-10

0

10

20

30

0 10 20 30f (Hz)

0

50

100

150

200

H = 6 mH = 12 mH = 18 mH = 24 m

A.2. Pile group foundation, 3D-model

53

Figure A.4: Impedance functions for rotation, based on the 3D model for pile group foundation, Location A (black) and Location B (red).

0 10 20 30f (Hz)

-100

0

100

200

300

0 10 20 30f (Hz)

0

500

1000

1500

2000

0 10 20 30

f (Hz)

-20

0

20

40

60

0 10 20 30

f (Hz)

0

50

100

150

200

0 10 20 30f (Hz)

-20

0

20

40

60

0 10 20 30f (Hz)

0

200

400

600

H = 6 mH = 12 mH = 18 mH = 24 m


54

A.3 Darendeli constants

In the 3D-model of the slab foundation, stress and strain dependent modulus of the soil is considered using the Darendeli constants in Table A.1 and Eq.(A.1) to (A.5). Darendeli worked with a statistical approach, where 18 model parameters φ1 - φ18 evaluated from laboratory tests define the reduction of modulus/increase of damping ratio for different soil conditions. For further reading, see (Darendeli, 2001).

Table A.1: Darendeli constants.

( )( ) 1

rmax

−+= aγγaGG (A.1)

( ) 43021rφφ σOCRPIφφγ ′⋅⋅+= (A.2)

( ) minMasing1.0

maxadjusted DDGGbD += (A.3)

( )( )( ) 9807610min frqln1 φφ σOCRPIφφφD ′⋅⋅++= (A.4)

5φa = , ( )Nφφb ln1211 += (A.5)

G/Gmax is the normalised shear modulus, Dadjusted is the actual damping ratio considering linear and nonlinear effects, a is a curvature coefficient, b is a scaling coefficient, γr is the reference strain of a soil corresponding to the strain amplitude when the shear modulus is reduced to half of Gmax and DMasing is a Masing-behaviour damping related to the hysteresis loop and nonlinear damping.

55

B

Response from passing trains

The following appendix present envelopes of the bridge deck vertical acceleration and vertical displacement, evaluated for train model HSLM A1-A10. The results include the dynamic factor 1+0.5φ′′ according to EN 1991-2 Annex C.

Results from the 2D-models are based on the 2D beam model with support impedance calculated from the axisymmetric models. The results from the 3D models are based on the 3D shell model of the bridge and impedance functions from the 3D solid models.

B.1 Slab foundation, 2D-model

Appendix

Appendix B. Response from passing trains

56

Figure B.1: Results from the 2D-model with flexible supports from slab foundations, envelope from HSLM A1-A10, L = 10 m.

B.1. Slab foundation, 2D-model

57


100 200 300 400v (km/h)

0

5

10

15L = 20 m, 1 span

100 200 300 400v (km/h)

0

5

10

15L = 20 m, 1 span

100 200 300 400v (km/h)

0

5

10

15L = 20 m, 2 span

100 200 300 400v (km/h)

0

5

10

15L = 20 m, 2 span

100 200 300 400v (km/h)

0

5

10

15L = 20 m, 3 span

100 200 300 400v (km/h)

0

5

10

15L = 20 m, 3 span

100 200 300 400v (km/h)

0

5

10

15L = 20 m, 4 span

100 200 300 400v (km/h)

0

5

10

15L = 20 m, 4 span

flexible supportsfixed supports


58


B.1. Slab foundation, 2D-model

59


100 200 300 400v (km/h)

0

5

10a m

ax (

m/s

2 )L = 40 m, 1 span

100 200 300 400v (km/h)

0

20

40

60

80

dm

ax (

mm

)

L = 40 m, 1 span

100 200 300 400v (km/h)

0

5

10

a max

(m

/s2 )

L = 40 m, 2 span

100 200 300 400v (km/h)

0

20

40

60

80

dm

ax (

mm

)

L = 40 m, 2 span

100 200 300 400v (km/h)

0

5

10

a max

(m

/s2 )

L = 40 m, 3 span

100 200 300 400v (km/h)

0

20

40

60

80

dm

ax (

mm

)

L = 40 m, 3 span

100 200 300 400v (km/h)

0

5

10

a max

(m

/s2 )

L = 40 m, 4 span

100 200 300 400v (km/h)

0

20

40

60

80

dm

ax (

mm

)

L = 40 m, 4 span



60

B.2 Pile group foundation, 2D-model

Figure B.5: Results from the 2D-model with flexible supports from pile group foundations, envelope from HSLM A1-A10, L = 10 m.

a max

(m

/s2 )

d max

(m

m)

a max

(m

/s2 )

d max

(m

m)

a max

(m

/s2 )

d max

(m

m)

a max

(m

/s2 )

d max

(m

m)

B.2. Pile group foundation, 2D-model

61


a max

(m

/s2 )

dm

ax (

mm

)

a max

(m

/s2 )

dm

ax (

mm

)

a max

(m

/s2 )

dm

ax (

mm

)

a max

(m

/s2 )

dm

ax (

mm

)


62


100 200 300 400v (km/h)

0

5

10L = 30 m, 1 span

100 200 300 400v (km/h)

0

5

10

15

20L = 30 m, 1 span

100 200 300 400v (km/h)

0

5

10L = 30 m, 2 span

100 200 300 400v (km/h)

0

5

10

15

20L = 30 m, 2 span

100 200 300 400v (km/h)

0

5

10L = 30 m, 3 span

100 200 300 400v (km/h)

0

5

10

15

20L = 30 m, 3 span

100 200 300 400v (km/h)

0

5

10L = 30 m, 4 span

100 200 300 400v (km/h)

0

5

10

15

20L = 30 m, 4 span



63


100 200 300 400v (km/h)

0

2

4

6

8a m

ax (

m/s

2 )L = 40 m, 1 span

100 200 300 400v (km/h)

0

20

40

60

80

dm

ax (

mm

)

L = 40 m, 1 span

100 200 300 400v (km/h)

0

2

4

6

8

a max

(m

/s2 )

L = 40 m, 2 span

100 200 300 400v (km/h)

0

20

40

60

80

dm

ax (

mm

)

L = 40 m, 2 span

100 200 300 400v (km/h)

0

2

4

6

8

a max

(m

/s2 )

L = 40 m, 3 span

100 200 300 400v (km/h)

0

20

40

60

80

dm

ax (

mm

)

L = 40 m, 3 span

100 200 300 400v (km/h)

0

2

4

6

8

a max

(m

/s2 )

L = 40 m, 4 span

100 200 300 400v (km/h)

0

20

40

60

80

dm

ax (

mm

)

L = 40 m, 4 span



64

B.3 Slab foundation, 3D-model

Figure B.9: Results from the 3D-model with flexible supports from slab foundations, envelope from HSLM A1-A10, simply supported bridge in 10 – 25 m.

a max

(m

/s2 )

dm

ax (

mm

)

a max

(m

/s2 )

dm

ax (

mm

)

a max

(m

/s2 )

dm

ax (

mm

)

a max

(m

/s2 )

dm

ax (

mm

)


65

B.4 Pile group foundation, 3D-model

Figure B.10: Results from the 3D-model with flexible supports from pile group foundations, envelope from HSLM A1-A10, simply supported bridge in 10 – 25 m.

a max

(m

/s2 )

dm

ax (

mm

)

a max

(m

/s2 )

dm

ax (

mm

)

a max

(m

/s2 )

dm

ax (

mm

)

a max

(m

/s2 )

dm

ax (

mm

)

67

TRITA-BKN. Report 166, 2017

ISSN 1103-4289

ISRN KTH/BKN/R--166--SE

www.kth.se

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