NumericalAnalysisofTwoCaseStudyBridges Soil-StructureInteractionAnalysisofPortalFrameRailwayBridges

STOCKHOLM, SWEDEN 2020

Soil-Structure

Interaction Analysis of Portal Frame

Railway Bridges

Numerical Analysis of Two Case Study Bridges

Nils Sandqvist and Marko Milicevic

KTH ROYAL INSTITUTE OF TECHNOLOGY

Analysis of Portal Frame Railway Bridges

Numerical Analysis of Two Case Study Bridges

Nils Sandqvist and Marko Milicevic

Master Thesis, 2020

KTH Royal Institute of Technology

School of Architecture and the Built Environment Department of Civil and Architectural Engineering Division of Structural Design and Bridges

AF223X Degree Project in Structural Engineering and Bridges SE-100 44, Stockholm, Sweden

TRITA-ABE-MBT-20360

© Nils Sandqvist and Marko Milicevic

This thesis concerns dynamic Soil-Structure Interaction (SSI) analysis of portal frame railway bridges. Dynamic problems are common for bridges used for high speed railway traffic. The passing trains induce harmonic loads on the bridges causing vibration amplitudes that may cause damage to the bridge structures and user discomfort.

Previous studies have shown that the effects of SSI are substantial for short span portal frame bridges. The damping ratio of the system is greatly increased due to the energy dissipation properties of the surrounding soil causing significant changes in the dynamic response of the structure. Therefore, it is of interest to investigate the effects of SSI for portal frame bridges with longer spans.

Two case study bridges with span lengths of approximately 16m have been investigated in detail in this study. Dynamic analyses of the bridges and train passage simulations have been performed. The results show that SSI significantly increases the damping ratio which leads to lower vibration amplitudes. It is also possible to draw the conclusion that more accurate results are achieved when modeling fixed foundations rather than using static spring foundations to replicate the stiffness of the subsoil.

Moreover, a simplified modeling approach accounting for the effects of SSI is proposed.

The proposed method provides satisfactory results, but more future work may increase the quality of the results further. To validate the conclusions from this study, a proposal for experimental validation is presented. Performing full-scale dynamic tests on the studied bridges would enable further comparison and validation of the results.

Keywords

Soil-structure Interaction, Portal Frame Bridge, Dynamic Analysis, Perfectly Matched Layer, Train Passage Analysis, Railway Bridge, High Speed Train

Denna studie behandlar dynamisk jord-struktur interaktionsanalys för plattrambroar.

Dynamiska problem är vanliga för järnvägsbroar som används för höghastighetståg.

De passerande tågen inducerar laster på broarna som orsakar vibrationer vilka kan skada brostrukturen och skapa obehag för tågpassagerare.

Tidigare studier har visat att effekterna av jord-struktur interaktion är väsentliga för plattrambroar med kort spännvidd. Systemets dämpning blir betydligt högre när jordens egenskaper beaktas vilket orsakar förändringar i brons dynamiska respons.

Det är därför av intresse att även undersöka effekterna av jord-struktur interaktion för plattrambroar med medellång spännvidd.

Två fallstudiebroar med spännvidder om 16m har undersökts närmre i denna studie.

Dynamiska analyser och simuleringar av tågpassager har genomförts. Resultaten visar att jord-struktur interaktion ökar systemets dämpningsförmga avsevärt vilket leder till lägre vibrationsamplituder. Dessutom går det att dra slutsatsen att mer exakta resultat erhålls genom att modellera fast inspända brofundament jämfört med att modellera statiska fjädrar för att imitera den underliggande jordens styvhet.

Vidare har en förenklad modelleringsmetod som tar hänsyn till jord-bro interaktion föreslagits. Den föreslagna metoden ger tillfredsställande resultat men kan förbättras ytterligare genom mer framtida utveckling. För att validera de slutsatser som dras i uppsatsen föreslås en hur en experimentell validering kan genomföras. Genom att utföra fullskaliga dynamiska tester på de undersökta broarna blir det möjligt att jämföra resultaten och validera resultaten ytterligare.

Nyckelord

Jord-struktur interaktion, Plattrambro, Dynamisk analys, Tågpassageanalys, Järnvägsbro, Höghastighetståg

3D Three Dimensional DOFs Degrees Of Freedom FE Finite Element

FEM Finite Element Methods FFT Fast Fourier Transform

FRF Frequency Response Function IFFT Inverse Fast Fourier Transform PMLs Perfectly Matched Layers SSI Soil-Structure Interaction VBM Viscous Boundary Method

1 Introduction

1.1 Background . . . 1

1.2 Previous studies . . . 2

1.3 Aims and scope . . . 3

1.4 Research contribution . . . 4

1.5 Outline of the thesis . . . 4

2 Theoretical Background

2.1.1 Natural frequency . . . 7

2.1.2 Damping . . . 8

2.2 Dynamic soil-structure interaction . . . 9

2.3 Wave propagation . . . 9

2.3.1 Body waves . . . 10

2.3.2 Surface waves . . . 12

2.4 Methods of analysis . . . 12

2.4.1 Direct method . . . 13

2.4.2 Substructure method . . . 24

3 Methods and Methodologies

3.2 Methodologies . . . 30

4 Finite Element Models

4.1.1 Bridge model . . . 32

4.1.2 Soil model . . . 36

4.1.3 Simplified model . . . 39

4.2 Södra Kungsvägen. . . 43

4.2.1 Bridge model . . . 43

4.2.2 Soil model . . . 47

4.2.3 Simplified model . . . 49

4.3 Train passages . . . 54

4.3.1 Modeling description . . . 54

4.3.2 Boundary conditions . . . 55

4.3.3 High Speed Load Model . . . 57

4.3.4 Load Formulation . . . 58

4.3.5 Analysis procedure . . . 58

5 Results

5.1.1 Bridge model only . . . 62

5.1.2 Full bridge-soil system . . . 64

5.1.3 Simplified model . . . 65

5.1.4 Summary . . . 68

5.2 Södra Kungsvägen. . . 69

5.2.1 Bridge model only . . . 69

5.2.2 Full bridge-soil system . . . 71

5.2.3 Simplified model . . . 72

5.2.4 Summary . . . 75

5.3 Train passages . . . 76

5.3.1 Norra Kungsvägen. . . 76

5.3.2 Södra Kungsvägen . . . 82

6 Proposal for Experimental Validation

6.2 Accelerometer placement . . . 89

6.3 Force level . . . 90

6.4 Frequency range . . . 90

6.5 Train passages . . . 91

7 Conclusions and Discussion

7.2 Discussion . . . 93

7.2.1 Soil-Structure Interaction Analysis . . . 93

7.2.2 Train Passage Analysis . . . 96

7.2.3 Future Work . . . 96

7.2.4 Final Words . . . 97

References

Introduction

This thesis concerns dynamic soil-structure interaction analysis of portal frame bridges. Firstly, a short background to the subject will be given in section 1.1, followed by a presentation of some key results from previous studies in section 1.2.

Furthermore, the aims and scope of the work will be presented in section 1.3 and the research contribution described in section 1.4. Finally, the outline of the thesis is introduced in the end of the chapter in section 1.5.

1.1 Background

Dynamic problems can occur in bridges used for high-speed railway traffic due to the periodic loads that the passing trains induce on the bridge superstructure. The vibrations of the structure can reach very high levels under resonance conditions which occurs when the frequency of the load is the same as one of the natural frequencies of the bridge [8]. For train induced loads, the critical speed which causes resonant conditions can be evaluated using Eq. 1.1 [10].

vcr = f₀ · d

i (1.1)

where f₀ is the fundamental frequency of the bridge, d is the axle distance and i is an integer multiple.

The vibrations that occur during resonance conditions may cause multiple negative effects. Structural breakdown and instability of the bridge superstructure, ballast and

tracks caused by the vibrations will increase the maintenance costs and decrease the service life of the bridge. Furthermore, vibrations at high levels also cause discomfort for the train passengers. Vibration amplitudes should be kept below certain threshold values to ensure that structural damage does not occur and that the comfort criteria are fulfilled. For ballasted tracks the accelerations of the bridge deck should be kept lower than 3.5 m/s², for ballast free tracks it should be kept below 5 m/s² [7].

The dynamic properties of structures are mainly described by the natural frequency and damping ratio. Thus, it is necessary to correctly evaluate these factors in order to be able to predict the dynamic response of railway bridges in a realistic way [5].

The effects of Soil-Structure Interaction (SSI) are often neglected during design of bridges due to the lack of simple and reliable models that correctly account for the interaction between the structure and surrounding soil. Classical boundary conditions are instead usually implemented [29]. Eurocode 1991-2 [7] suggests that the maximum damping ratio of 2.5% should be considered for the dynamic analysis of railway bridges.

Previous studies, presented in the following section, have shown that the damping ratio may be significantly larger when the effects of SSI are taken into account. In the case of short span portal frame bridges, energy dissipation due to the backfill soil and subsoil increases the damping of the system which reduces the vibration amplitudes [29].

Neglecting the influence of SSI in dynamic analyses could therefore lead to unrealistic results, especially for frequencies close to resonance conditions. It would consequently lead to wrong maximum allowable speed of the trains passing over the bridge.

1.2 Previous studies

The effects of soil-structure interaction on portal frame bridges are not yet well investigated. However, some presented research in the area exists. Zangeneh et al.

[31] conducted a full scale dynamic testing on a short-span portal frame bridge with span length 8m. Numerical and experimental results from the dynamic analyses of the same bridge were presented by Zangeneh Kamali in 2018 [29]. The results showed that the effects of SSI were significant on the modal properties of the tested bridge.

Similarly to Zangeneh et al., Ikzer [13] has also been able to conclude that the effects of SSI are substantial in a study of short span portal frame bridges. The soil significantly

increases the damping ratio and the stiffness of the bridge-soil system. The changes in the modal properties of the bridge results in lower acceleration amplitudes and shifts in natural frequencies of the bridge.

Furthermore, the effects of SSI on the behavior of portal frame walls has been analyzed in a study performed by Malm [19]. The results showed that the soil significantly increased damping and stiffness of the bridge walls. Consequently SSI is thought to greatly influence the damping ratio and stiffness of the entire bridge-soil systems for portal frame bridges.

There have been limited published results about the effects of SSI on medium range portal frame bridges. However, Turguay et al. [26] studied the general effects of SSI on portal frame bridges for four different span lengths. The results from the study showed the importance of including SSI in the dynamic analysis of portal frame bridges as the damping and stiffness of the system increased. The effects of the backfill soil proved to have the biggest influence on short span bridges while the subsoil was more significant for longer span portal frame bridges. However, more research is needed concerning the effects of SSI on portal frame bridges with medium range span lengths.

1.3 Aims and scope

The aim of this report is to study the effects of Soil-Structure Interaction on the dynamic response of medium range portal frame bridges. The main focus is on how SSI affects the modal properties of the system. In further detail, the aims are listed below:

• Study the dynamic properties of reinforced concrete portal frame bridges with span length of approximately 16m.

• Clarify how SSI affects the natural frequency and damping ratio of the bridge-soil system for different foundation types and properties of the underlying soil.

• Propose a simplified method of modeling portal frame bridges that takes SSI for both backfill and subsoil into account which can be used for design purposes.

• Simulate train passages on the simplified bridge models to study the dynamic response of medium range portal frame bridges for train induced loads.

• Propose experimental validation procedures through a full-scale dynamic testing

of existing portal frame bridges.

In order to limit the scope of the work the following limitations apply:

• Focus will be solely on medium range portal frame bridges with the span length of approximately 16m. The dynamic response may vary substantially for different span lengths but this is beyond the scope of the study.

• Only the effects of SSI on the stiffness and damping ratio of the bridge-soil system will be studied.

• The study considers only linear time-invariant systems. Plastic behavior of the soil and cracking of the concrete are beyond the scope of this study.

1.4 Research contribution

The investigation that has been carried out in this study have resulted in the following research contributions:

• Identification of the effects of soil-structure interaction on the modal properties of medium range portal frame bridges.

• A proposal of a simplified modeling method that accounts for the effects of soil- structure interaction which is possible to use for dynamic analyses of medium range portal frame bridges.

• An estimation of the dynamic response of medium range portal frame bridges induced by passing trains.

1.5 Outline of the thesis

The rest of this thesis is structured in the following way. A theoretical background to the research subject is provided in Chapter 2. It will touch upon some of the fundamental theory of structural dynamics, soil-structure interaction, different modeling approaches and theory of absorbing boundary layers. In chapter 3, the methods and methodologies that have been implemented to achieve the aims of the study in a satisfactory way are described. Chapter 4 is a further description of the methods of the study. The chapter presents the Finite Element (FE)- models of the bridge-soil systems of two existing bridges that have been chosen for

further investigation. The geometry, materials, loads, boundary conditions, modeling assumptions and more is explained in great detail in this chapter. A description of the analysis of train passages is also be included in Chapter 4. The results from the study are presented in Chapter 5. In Chapter 6, a proposal for experimental validation of the results from the study is presented. Measurement preparation and optimal location for the excitation points are presented among other details about how to successfully perform full-scale dynamic tests on the studied bridges. Moreover, conclusions from the study are drawn in Chapter 7 along with a discussion of the results. Finally, suggestions for future research is presented.

Theoretical Background

This chapter will present a theoretical background to the current field of research to provide a theoretical base for the subject of the thesis. Firstly, a short introduction to structural dynamics and how it applies to the thesis is presented in section 2.1.

Following is the theory of soil-structure interaction in section 2.2. Wave propagation theory is then introduced in section 2.3. To conclude the chapter, theory concerning two different methods of analysis is given in section 2.4.

2.1 Structural dynamics

Structural dynamics concerns the behavior of structures that are subjected to dynamic loads, i.e. loads that vary with time. Such dynamic loading can for example appear from pedestrian or car traffic, winds, earthquakes or explosions.

The dynamic properties of a structure need to be considered during the design phase.

Even though the amplitude of the dynamic force would not harm the structure, if the force is applied statically, it could cause damage, discomfort or in some extreme cases it may even cause the structure to fail when the force is dynamic. This phenomenon should be taken into consideration especially in the cases when the frequency of the applied force is close to one of the natural frequencies of the structure. The ratio between the dynamic and static response is described by the dynamic amplification factor which can be calculated according to Eq. 2.1 [5].

R_d= u₀

u_st,0 = 1

√(1− (ω/ωn)²)²+ (2ζ(ω/ω_n))² (2.1)

where ω and ω_nare frequency of the external force and natural frequency of the system respectively while ζ is damping coefficient.

For railway bridges, structural dynamics is particularly important topic, especially for high-speed railway traffic. When trains travel across a bridge, the axles of the train impose periodic loads on the structure that sets the bridge in motion [8]. As mentioned before, the vibrations of the structure need to be kept below certain limits to ensure structural integrity and comfort for the train passengers [7].

As previously stated, the natural frequency and damping are the two most important factors that describe the dynamic response of structures [5]. A more detailed description of these factors is provided below.

2.1.1 Natural frequency

All structures have frequencies at which they tend to vibrate. These frequencies are called natural frequencies or eigenfrequencies. A natural frequency is a property of the system which depends on the stiffness, mass and damping of the structure. For undamped structures, the natural frequency can be evaluated according to Eq. 2.2 [5].

ω_n =

…k

m (2.2)

where k and m are stiffness and mass of the system respectively.

If an external dynamic force acts upon the system with the same frequency as one of the natural frequencies of the system, resonance will occur. At resonance, the amplitudes of vibration can become very large which could potentially cause damage to the structure. The most critical dynamic response will thus appear close to resonance conditions [5].

In the case of railway bridges, large vibration amplitudes may appear if the frequency of the load induced by the train is the same as a multiple of the natural frequency of the bridge [8]. The critical speeds can be evaluated according to Eq. 1.1. The maximum dynamic response appears for resonance conditions and close to resonance which is why it is important to study the dynamic response of bridges close to the natural frequencies of the structure.

2.1.2 Damping

Damping is the process which causes free vibrations to decrease in amplitude due to dissipation of energy [25]. Damping is caused by friction and hysteresis within the material (material damping) and due to the spreading of vibrations over an increasing area (geometrical damping or radiation damping) [29].

Material damping in railway bridges occurs due to a number of factors. For example opening and closing of cracks within the concrete, friction at supports and bearings, friction within the ballast and bending of materials. These factors all cause energy dissipation within the structure [8, 21].

Damping in the soil material partly occurs due to energy dissipation within the material. The soil medium is not perfectly elastic even for small strains, so the material will dissipate some energy under periodic loading [29]. The small strain material damping varies for different types of soils but is usually in the range of 2-7% [27].

Radiation damping within the soil is usually considerably larger than the material damping. Estimation of the material damping factor of the soil will thus not have a significant influence on the dynamic response in most of the cases [29].

Modal damping ratios of a structure can be estimated using the Half-power bandwidth method. This method has been implemented for the dynamic analyses performed in this study. By plotting a Frequency Response Function (FRF), the damping ratio can be estimated for the eigenmodes of the structure according to Eq. 2.3 [5].

ζ = f_b− fa

2· fn

(2.3)

where f_nis the natural frequency and f_aand f_b are the frequencies where the response amplitude is equal to u0. The response amplitude u0 is defined according to Eq.

2.4.

u₀ = u_max

√2 (2.4)

The influence of the damping ratio on the acceleration amplitudes is substantial, especially for frequencies near resonance conditions. Low damping ratios will often lead to an unacceptably high dynamic response. For high damping ratios the response

amplitudes will be significantly lower [5]. Underestimating the damping ratio in dynamic analyses of railway bridges leads to an overestimation of the vibration amplitudes of the structure. For obvious reasons, it is therefore very important to correctly estimate the damping ratio in order to accurately predict the dynamic behavior of structures [29]. From Eq. 2.1, it can be seen that the damping ratio is in fact the only limiting factor of the response in the case of resonance, i.e. when ω = ω_n.

2.2 Dynamic soil-structure interaction

During the design phase, a structure is often analyzed in isolation, neglecting the influence of the soil surrounding the structure. The principles of dynamic soil- structure interaction regards the influence of the surrounding soil and how the interaction between the structure and soil affects the dynamic response. Studying the structure in isolation may be adequate in some cases, but it has been shown that the influence of SSI often needs to be accounted for in order to achieve accurate results [23, 31].

When the structure vibrates, the surrounding soil will be influenced and it will start to vibrate along with the structure. In this way the stiffness and energy dissipation properties of the soil will affect the modal characteristics of the system [31].

Previous studies have shown that there are two main effects of SSI on railway bridges [30, 31]:

• The damping of the system is increased which lowers the dynamic response in the vicinity of resonance conditions.

• The stiffness of the system may be influenced causing changes in the natural frequency and consequently also the critical speeds.

2.3 Wave propagation

There are two main types of waves categorized based on how they propagate through soil - body waves and surface waves. The following subsections will provide a basic theoretical background about these types of waves.

2.3.1 Body waves

Body waves propagate through the inner parts of the soil. This group of waves can be categorized further into primary waves and secondary waves based on the motion of the particles.

Primary waves

Primary waves, or P-waves, cause movement of the particles in the same direction as the direction that the wave is traveling in. These waves are also known as compression waves and have high propagation speed [22]. An illustration of primary waves is provided in Fig. 2.3.1.

Figure 2.3.1: An illustration of a primary body wave [29].

Secondary waves

Secondary waves, or S-waves, produce displacement of the particles perpendicular to the axis of the wave propagation. These so called shear waves travel at lower speeds through the soil material compared to primary waves [22]. Fig. 2.3.2 shows an illustration of how secondary waves propagate through soil.

Figure 2.3.2: An illustration of an secondary body wave [29].

Wave propagation speed

The speeds of primary waves, C_p, and secondary waves, C_s, can be calculated according to Eq. 2.5 and Eq. 2.6 respectively [29].

C_p =

 λ + 2G ρ = C_s

…2− 2ν

1− 2ν (2.5)

C_s =  G

ρ (2.6)

where λ is Lame’s first constant, ν is Poisson’s ratio, ρ density of the medium and G is the shear modulus of the media. Lame’s constant is calculated according to Eq. 2.7 [29].

λ = 2Gν

1− 2ν (2.7)

The wave speeds for different soil types are given in Table 2.3.1.

Table 2.3.1: Primary and secondary wave speed through different types of soil [12].

Soil type P-wave velocity S-wave velocity

[m/s] [m/s]

Ice 3 000 - 3 500 1 500 - 1 600

Water 1 480 - 1 520 0

Granite 4 500 - 5 500 3 000 - 3 500

Sandstone, Shale 2 300 - 3 800 1 200 - 1 600 Fractured Rock 2 000 - 2 500 800 - 1 400

Moraine 1 400 - 2 000 300 - 600

Saturated sand and Gravel 1 400 - 1 800 100 - 300 Dry sand and Gravel 500 - 800 150 - 350 Clay below GW level 1 480 - 1 520 40 - 100

Organic soils 1 480 - 1 520 30 - 50

2.3.2 Surface waves

Opposed to body waves, surface waves propagate along the surface or very close to the surface of the medium [20]. Rayleigh wave is one type of surface wave. For Rayleigh waves the soil particles rotate around the axis perpendicular to the direction of the traveling wave [20]. An illustration of a Rayleigh wave is given in Fig. 2.3.3.

Figure 2.3.3: An illustration of a Rayleigh wave [29].

The propagation speed of Rayleigh waves can be calculated according to Eq. 2.8 [29].

C_R= 0.862 + 1.14ν

1 + ν C_s (2.8)

2.4 Methods of analysis

There are several methods of analysis available to approach problems of dynamic SSI.

Among these methods, one possibility is to model the bridge-soil system in its entirety using Finite Elements (FE). This method is called the direct method. Another option is to decompose the soil-structure system and analyze the structure and soil separately.

This method is called the substructure method [29].

In the following sections the direct method and the substructure method are presented in more detail as ways of approaching dynamic soil-structure interaction problems.

Both methods have been used in the dynamic analysis of the bridge-soil systems within this study. The results of both approaches will be presented and compared in the following chapters of the report.

2.4.1 Direct method

The direct method accounts for the effects of SSI by modeling the entire soil-structure system using finite elements [28]. Even though this method has advantages such as accurate results and theoretical simplicity, the main disadvantage of this method is the extensive computational effort that is required. The computation time for solving such massive models is very large even for relatively small bridges which unfortunately makes this method less attractive to be used in the design phase.

When modeling soil-structure systems in their entirety, one should be aware of possible interruption of the dynamic response due to wave reflection at the boundaries of the FE-model. In order to prevent unwanted reflection at the boundaries, the soil domain size has to be well chosen and boundaries that absorb outward propagating waves have to be implemented [29]. Further information about absorbing boundary conditions is provided below.

Absorbing boundary conditions

To reduce size of the soil model and the number of degrees of freedom in the model, engineers use different techniques to simulate an infinite domain. By applying absorbing boundaries, it is possible to avoid wave reflection at the boundaries of the model and thus it is possible to reduce the model size. One approach to absorb wave propagation is to apply viscous dashpots at the boundaries of the model. This method is called the Viscous Boundary Method (VBM) [17].

The viscous boundary method uses dashpots to absorb outgoing propagating waves.

At each node on the boundaries of the soil model, three dashpots are placed, one perpendicular to the boundary and two tangential with the purpose of absorbing dilatational and shear waves. This method perfectly absorbs all waves propagating in the direction perpendicular to the boundary, but fails to completely absorb waves that reach that boundaries at angles other than the right [29].

An another fairly new approach is to implement Perfectly Matched Layers (PMLs) on the model boundaries. PMLs is an efficient method to absorb waves at the boundaries and can simply be implemented in commercial software COMSOL Multiphysics^®. This method uses artificial stretching of coordinates to absorb outward propagating waves [6].

Perfectly Matched Layers are boundary layers that are designed to absorb all waves at the boundaries of a model without any reflection. This makes it possible to create a model of a finite size with no wave reflection at the boundaries of the model. By using these layers, the computational costs of performing dynamic soil-structure interaction analyses can be reduced greatly because the size of the model can be decreased significantly [6].

PMLs were first introduced by Berenger in 1993 as a modification of previously used matched layers [3]. The major weakness of matched layers at the time was their inefficiency of absorbing waves propagating in directions other than perpendicular to the boundaries. By introducing PMLs, Berenger introduced layers that were able to absorb all waves regardless of their incidence angle [3]. Even though the primary purpose of PMLs was to enable absorption of electromagnetic waves in modeling, modifications of the layer properties allow expansion of their use to other research fields including structural dynamics.

The fundamental concept of PMLs is the principle of coordinate stretching functions.

Instead of handling a layer with an infinite thickness to completely absorb wave propagation, this technique allows having boundaries in a relatively short distance from the elastic domain. This enables reduction of the model size while still achieving complete wave absorption.

The waves are theoretically perfectly absorbed at the boundaries using PMLs, but due to the discretization of the model, some reflection may occur. However, if the PMLs are incorporated properly, the amplitudes of the reflected waves will be negligible [1].

Complex coordinate stretching is used to achieve a perfect match at the intersection between the computational domain and the attenuating layer. The stretching function λ(s)is used to stretch the coordinates in the complex plane within the PML domain.

The coordinates in this region are modified according to Eq. 2.9 [2].

˜ x =

∫ st

s0

λ(s)ds (2.9)

where s₀ and s_tis the inner and outer coordinate of the PML domain respectively as showed in Fig. 2.4.1.

Figure 2.4.1: An illustration of a PML domain [14].

There are numerous different proposals for stretching functions that can be implemented in PMLs. One of the most widely used is the stretch function proposed by Basu and Chopra [2] which is given in Eq. 2.10.

λ(x) = 1 + f_x^e(x)

a₀ − if_x^p(x)

a₀ (2.10)

where a₀ = ωL_s/C_s is a dimensionless frequency in which L_s is the characteristic wavelength and C_s secondary wave velocity. Additionally, f_x^e(x) and f_x^p(x) are attenuation functions of evanescent and propagating waves respectively.

As described by François et al. [9], the real part of the attenuation functions artificially stretches the thickness of the PML domain. As a consequence, the wavelength inside the PMLs changes according to Eq. 2.11. This imposes high requirements on the mesh size for high frequencies so that the shape of the waves is correctly captured.

λ˜_s(x) = λ(s)

1 + f_x^e(x) (2.11)

Since high values of both the real and imaginary attenuation parameters requires a fine mesh size in the FE-model, one should choose these values with great accuracy [9].

Basu and Chopra [2] recommend values for both the real and the imaginary attenuation parameters to be set to:

f_x^e(x) = 20 f_x^p(x) = 20

Similarly to the definitions of the stretching functions above, it is possible to define the

functions in a similar way for other directions.

To apply the previously described theory of PMLs, a user defined stretch function will be implemented in FE-software COMSOL Multiphysics^®. The user defined stretch function that will be used is presented in Eq. 2.12 [14].

˜

s = s₀+ λ(h^e_s(ξ, λ) + i· h^ps(ξ, λ)) (2.12)

where h^e_sis defined according to Eq. 2.13 and h^p_s is defined according to Eq. 2.14.

h^e_s(ξ, λ) = ξL_s

λ + ξ^(q+1)· f₀^e

2π(q + 1) (2.13)

h^p_s(ξ, λ) =−ξ^(q+1)· f₀^p

2π(q + 1) (2.14)

In the equations above, Lsis the thickness of the PML domain, λ is the stretch function and q is an attenuation profile parameter. f₀^e and f₀^p are the previously described attenuation parameters. The parameter ξ is the relative coordinate which is defined according to Eq. 2.15.

ξ = s− s0

L_s (2.15)

Verification of Perfectly Matched Layers

To verify the value of the real stretching parameter f₀^ethat is used to define the PMLs, a verification study may be conducted. The verification can be performed both for a stratum and for a half-space using a simplified 2D model and a 3D model. The model used for the verification is a cube of soil loaded with a vertical point load at the center of the upper surface. The deformation is then measured in another point, 1m from the load. The results are compared to the reference solutions available in literature to ensure that the boundary layers work as intended. In the following, the procedures of the verification is presented for the case of a half-space followed by the case of a stratum.

In cases where the subsoil is a thick homogeneous layer, it is reasonable to model it as a half-space because negligible wave reflection will occur within the soil. The verification

of the PMLs using a 2D and a 3D verification model of a half-space is presented below.

The verification model consists of a cube of soil with 2m thick PMLs modeled on the outsides and on the bottom of the soil to ensure that no wave reflection occurs at the boundaries. The model is loaded with a vertical force in the midpoint of the top surface of the soil. Deformation is measured in another point, 1m away from the load. Fixed supports are applied on the outsides of the PMLs on the sides and on the bottom of the model. The 2D verification model is presented in Fig. 2.4.2.

Figure 2.4.2: A schematic view of the FE-model used to verify PML parameters in 2D for a half-space.

Different values of the real stretching parameter f₀^eare tested to identify which value should be used for the analysis while f₀^p = 20 and q = 2 are held constant. The results from the verification models are compared to reference results to ensure that the PMLs work as intended. The reference solutions are calculated using the Direct Stiffness Method [15] and the Boundary Element Method [4] which are implemented in an elastodynamic toolbox (EDT) [18] in MATLAB. Fig. 2.4.3 shows the results from the 2D verification.

(a) (b)

Figure 2.4.3: The figure shows the a) real part and b) imaginary part of the deformation 1m from the concentrated force in the 2D verification model.

A similar comparison for a 3D case is presented below. The configuration of the model is the same as for the 2D case described above. The 3D model is presented schematically in Fig. 2.4.4.

(a) (b)

Figure 2.4.4: 3D FE-model used to verify PML parameters for a half-space. a) Top view b) Bottom view.

In the 3D-model the vertical load is also applied in the midpoint of the top surface of the soil and the deformation is measured in another point 1m away from the load. Fig.

2.4.5 shows the results for the 3D case.

(a) (b)

Figure 2.4.5: The figure shows the a) real part and b) imaginary part of the deformation 1m from the concentrated force in the 3D verification model.

The results from the verification study show that the parameter f₀^e has very little influence on the quality of the results in the case of a half-space subsoil model.

It follows that for such analyses the parameters that are recommended to be implemented for the case of a half-space are summarized in Table 2.4.1.

Table 2.4.1: PML parameter values that should be used for half-space soil models.

PML parameters

f₀^e 20

f₀^p 20

q 2

In the case of a soft soil layer on top of a much stiffer layer of soil or rock, the soil system may be modeled as a stratum. Dynamic waves will propagate outwards to the sides without reflection. However, wave reflection will occur at the bottom interface between the soft soil and the stiffer material.

The verification model is similar to the previously described model of a half-space. It consists of a cube of soil with 2m thick PMLs applied on the outsides of the model. A point load is applied in the middle of the top surface of the model and the deformation is measured in a second point 1m away from the load. The outsides of the model and the bottom is fixed. Fig. 2.4.6 shows an illustration of the 2D model that was used for the stratum case.

Figure 2.4.6: A schematic view of the FE-model used to verify PML parameters in 2D for a stratum.

For the verification, the parameters f₀^p = 20 and q = 2 are held constant to isolate the influence of the real stretching parameter f₀^e. Fig. 2.4.7 shows the results from the parameter study for the 2D case.

(a) (b)

Figure 2.4.7: The figure shows a) the real part of the deformation and b) the imaginary part of the deformation.

A similar verification may be conducted for a 3D case. The 3D FE-model used for the verification is presented schematically in Fig. 2.4.8.

(a) (b)

Figure 2.4.8: 3D FE-model used to verify PML parameters for a stratum. a) Top view b) Bottom view.

Fig. 2.4.9 show the results for the 3D verification model.

(a) (b)

Figure 2.4.9: The figure shows a) the real part of the deformation and b) the imaginary part of the deformation.

The results from the verification study show that the parameter f₀^ehas a large influence on the quality of the results in the case of subsoil modeled as a stratum. For the analyses of bridge-soil systems that are modeled in this way, the recommended parameters are summarized in Table 2.4.2.

Table 2.4.2: PML parameter values that should be used for stratum soil systems.

PML parameters

f₀^e 40

f₀^p 20

q 2

Pile group verification

To verify the impedance model for foundations with pile groups, a simplified pile group can be modeled and compared to reference solutions provided by Padrón et al.

[24]. The reference solution for a 2x2 pile group is compared to an impedance model created in COMSOL Multiphysics^® to ensure that no problems appear and the pile group impedance model works as intended. Properties of the impedance model are summarized in Table 2.4.3.

Table 2.4.3: Properties of the reference impedance model that was created to be compared to a reference solution.

Model properties

Pile-to-pile separation ratio, s/d 5 Pile-to-soil stiffness ratio, E_p/Es 100 Soil-to-pile density, ρ_s/ρ_p 0.7

Stratum depth, H 15· d

Soil internal damping coefficient, β_s 0.05 Pile internal damping coefficient, β_p 0.00

Soil Poisson ratio, ν_s 0.4

Pile Poisson ratio, ν_p 0.25

The stiffness and damping properties from the impedance model is presented in Fig.

2.4.10. The results have been compared to the reference solutions [24].

(a) (b)

Figure 2.4.10: Results show the a) stiffness and b) damping properties of the impedance model for a 2x2 pile group foundation.

Element size

To capture the wave motion in the soil, it is of importance that the mesh size of the soil medium is well chosen. The highest frequency of interest determines the appropriate element size. Eq.2.16 provides a guideline for the choice of the smallest possible element size if linear elements are used. The minimum element size is based on the shortest wavelength that aims to be captured [16].

L_E,min = 1

8 · λmin (2.16)

where L_E,min is the minimum element size and λ_min is the shortest wavelength. The shortest wavelength is determined from Eq.2.17.

λ_min = Cs

f_max (2.17)

The previous equations are guidelines for linear elements. If quadratic elements are used, an element size twice as large is sufficient. If the guidelines are followed, the shortest wavelengths will be described by eight linear elements and four quadratic elements for the shortest wavelengths. Fig. 2.4.11 shows how the wave motion is

described by eight linear elements or four quadratic elements per wave length.

Figure 2.4.11: The description of a wave length using 8 linear elements or 4 quadratic elements.

2.4.2 Substructure method

When it comes to large structures, the number of Degrees Of Freedom (DOFs) quickly becomes very large when the direct approach is applied resulting in a very high computational cost to solve dynamic SSI problems. The substructure method is an alternative approach that requires less computational effort. The basic principle of the substructure method is to disintegrate the full soil-structure model into smaller components and to analyze each of them separately. Usually, the bridge model and the soil model are analyzed independently when this method is implemented. This method is very efficient when it comes to linear frequency dependent problems [29].

The aim of creating a simplified model using the substructure method is to achieve a model of a reasonable computational size that can correctly capture the dynamic response of the complete bridge-soil system. To achieve this, the full soil model is replaced by springs and dashpots.

A simplified FE-model of a bridge-soil system consists only of the studied bridge modeled in solid elements. The soil around the bridge is replaced by springs and dashpots that are placed at the bridge-soil interfaces. A schematic illustration of a simplified model is presented in Fig. 2.4.12.

Figure 2.4.12: A schematic figure of a simplified model [29].

Frequency dependent stiffness and damping matrices of the soil model are evaluated at the interface nodes between the bridge and the soil using impedance functions which are determined using FE-models in combination with empirical equations. The stiffness and damping matrices of the soil are then introduced in the equations of motion of the structure at the interface nodes [29].

Wall-backfill interaction

Distributed springs and dashpots are applied on the back walls and wing walls of the bridge to replicate the soil around the bridge. The values of the spring- and dashpot coefficients are evaluated using empirical formulas [31].

The normal and tangential spring stiffness are calculated according to Eq. 2.18 and Eq.

2.19 respectively.

ks,n = π

√2(1− ν) · G

H (2.18)

where ν is the Poisson ratio of the soil, G is the shear modulus and H is the height of the backfill soil.

k_s,t = ( Cs

C_La)· ks,n (2.19)

where C_sis the shear wave velocity of the soil and C_Lais Lysmer’s wave velocity which can be calculated according to Eq. 2.20.

C_La = 3.4· Cs

π· (1 − ν) (2.20)

The dashpot coefficients in the normal and tangential direction are evaluated using Eq.

2.21 and Eq. 2.22 respectively.

d_s,n = ρ· CLa· g(f) (2.21)

d_s,t = ( Cs

C_La)· ds,n (2.22)

where ρ is the density of the backfill soil and g(f ) is a factor that takes into account different behavior of the soil for different frequencies. This factor can be evaluated according to Eq. 2.23.

g(f ) =







0 if f < f_si

»1− (^f_f^si)² if f ≥ fsi

(2.23)

where f_si is the resonance frequency of the soil which can be evaluated according to Eq. 2.24.

f_si = C_s

4· H (2.24)

Foundation-subsoil interaction

To account for the subsoil, translational and rotational stiffness and damping values are evaluated using impedance functions. Using those functions, springs and dashpots are later applied at the center of the bottom surfaces of the foundations of the bridge.

In this case, foundations are assumed to be rigid.

The FE-model that is created to evaluate the impedance functions includes one bottom foundation of the bridge and the underlying soil. The soil is modeled with the same properties as the underlying ground. PMLs are also applied on the outside boundaries of the model to ensure that no wave reflection occurs.

Both the stiffness and damping properties of the soil are frequency dependent which means that dynamic equilibrium equations need to be solved. The equilibrium condition is presented in Eq. 2.25 [11].

m¨u_z(t) +κzu_z(t) + F_z(t) = 0 (2.25)

whereκzis the impedance function to be determined.

Due to the fact that applied force and corresponding displacement are out of phase, impedance functions are complex numbers. The impedance function is defined according to Eq. 2.26 [11].

κz(ω) = K_z(ω) + iωC_z(ω) (2.26)

where K_z is the stiffness of the springs in the direction of the applied force and C_z is the dashpot coefficient in the same direction. The real part of the impedance function thus gives the frequency dependent stiffness coefficients. The frequency dependent damping coefficients are on the other hand given by the imaginary part of the impedance function. It should be noted that the damping coefficients includes both radiation damping and material damping [11].

The well-known relationship between applied dynamic force and corresponding

displacement presented in Eq. 2.27, is used to evaluate the stiffness and damping coefficients.

Fz(ω) = (Kz(ω) + iωCz(ω))· Uz(ω) (2.27)

Two possible approaches for evaluating the impedances thus appear. One possibility is to first model the bottom foundation surface and the underlying soil with finite elements. The modeled soil in this case is significantly smaller comparing to the model of the whole structure. A prescribed harmonic load, F_z, is then applied to the foundation and the corresponding displacements, Uz, evaluated.

The same results would be obtained if the displacement U_z is prescribed and the corresponding reaction force, F_z, is evaluated which is the other available solution method.

Once one of the previously described methods have been applied the impedance functions can be calculated according to Eq. 2.28.

K_z(ω) + iωC_z(ω) = F_z(ω)

U_z(ω) (2.28)

As previously described, the stiffness coefficients are then evaluated from the real part and the damping coefficients from the imaginary part of the impedance function.

It should be noted that in the previously written equations, the vertical component of the impedance model is examined since the vertical force and vertical displacement, F_z and U_z, are considered. As for all 3D bodies, the number of DOFs is six (three translations and three rotations) so the procedure described above should be conducted six times to obtain all of the stiffness and damping coefficients for all DOFs.

Applying a translational force in three directions and bending moments along three axes returns all frequency dependent impedance functions [11].

Once the frequency dependent stiffness and damping properties of the soil have been evaluated according to the procedure described above, they can be introduced in the equations of motion of the structure using springs and dashpots. In this way it is possible to account for the effects of SSI without having to model the soil as a full Three Dimensional (3D) domain.

Methods and Methodologies

3.1 Methods

To achieve the desired aims of the study, two existing portal frame bridges located in Umeå, Sweden have been selected for further analysis. Dynamic analyses of the bridges will be performed using numerical modeling approaches.

The chosen bridges have very similar material and geometrical properties. Both bridges have a span length of 15.7m. What sets the bridges apart is the geological properties of the ground below them. One of the bridges, which is built on Norra Kungsvägen, is built on pile foundations in a relatively soft and thick layer of soil. The other bridge, on Södra Kungsvägen, is founded on foundations on packed fill on top of a thick layer of stiff moraine.

The bridges on Norra Kungsvägen and Södra Kungsvägen will be modeled with Finite Element Methods (FEM) using commercial software COMSOL Multiphysics^®. This software is primarily chosen because of the ability to simply implement absorbing boundary conditions with PMLs. The dynamic response of the bridges with and without the effects of SSI will be analyzed and compared. Due to the different soil properties at the sites, the influence of the underlying soil will also be investigated.

The effects of SSI on the damping ratio and natural frequency will be the main focus of the analyses with the aim to clarify how these factors are influenced by SSI.

A detailed description of the FE-models, modeling techniques and modeling assumptions for the bridges on Norra Kungsvägen and Södra Kungsvägen is given in

Chapter 4.1 and Chapter 4.2 respectively.

Once the dynamic analyses have been performed for the two bridges, a simplified modeling approach using springs and dashpots to replicate the effects of the surrounding soil will also be proposed. Impedance functions and empirical formulas for the damping and stiffness values will be used to create simplified models of the bridge-soil systems on both Norra and Södra Kungsvägen. The aim is to propose a modeling method that is possible to implement during the design phase of medium range portal frame bridges.

The results of the dynamic analyses for the bridges in isolation, the complete bridge- soil systems and for the simplified models will be presented and compared. Analyzing the results will hopefully enable clarification of the effects of SSI.

Simulations of train passages will then be conducted on both the bridges. Ten different variations of high speed load models will travel across the bridge models in varying speeds. The frequency response and acceleration of the bridges will be monitored.

For the train passage simulations the simplified models of the bridge-soil systems on Norra Kungsvägen and Södra Kungsvägen will be used. A more detailed description of the method used for the train passage analysis is provided in Chapter 4.3.

Once the results from the models of the bridges have been achieved, a method for experimental validation will be proposed. Performing a full-scale dynamic testing on the bridges that are studied is a great way of validating the results of the study.

Furthermore, any flaws in the modeling techniques can be detected by making this comparison. By applying periodic forces on the bridges using a bridge exciter and measuring the dynamic response with accelerometers it is possible to compare the results from the FE-models with the results from in situ testing.

3.2 Methodologies

The method choice is a combination of a case study and creating models. Two portal frame bridges with similar geometric properties but different foundation properties will be investigated. This method has been chosen in order to obtain the research goal in the best possible way. There are obviously both positive and negative aspects of the chosen methods which will be discussed further in the following.

The decision to focus the study on two existing bridges is twofold. To start with, focusing on real bridges creates the possibility of verifying theory with reality.

Secondly, the fact that the bridges have very similar structural and geometrical properties but with different properties of the underlying ground provides an opportunity to identify in which ways the subsoil and foundation types influence the modal properties of the bridge-soil system. If only one bridge was studied, one would lose the valuable opportunity to compare, relate and verify the results from one bridge to the results of another, in many aspects, similar bridge.

The bridges will be modeled using Finite Elements. A FE-model is a great way to accurately model structures of different kinds. It is a commonly used tool to investigate the properties of structures with great detail. It may require a significant amount of time to create the models with sufficient accuracy. However, it gives the most exact results compared to any alternative method.

A dynamic analysis of the structure could in theory be conducted using theory of structural dynamics in combinations with hand calculations, but the level of complexity of the structures with several hundred thousands of degrees of freedom would lead to enormous computational effort, especially considering the fact that the ground beneath the structures must be included in the analysis to answer the research question.

Ordinary structural calculations would therefore require far more computational time than a FE-model. Therefore, it seems rather straightforward to choose to analyze the damping properties of the structures using FE-models.

The FE-models that take soil-structure interaction into account are compared to the models which do not. All other factors except the surrounding soil are identical for both models. By creating models in this way, it is possible to ensure that all other factors are held constant and the only difference is the influence of SSI which is the influence of interest. Thus, in this way the influence of SSI on the dynamic effects can be identified which is the goal of the study.

Finite Element Models

4.1 Norra Kungsvägen

One of the bridges that has been a subject of the analysis in this study is a portal frame bridge situated on Norra Kungsvägen in Umeå, Sweden. In this subchapter the complete Finite Element model of the bridge and soil will be described thoroughly.

Initially the model of the bridge will be described (section 4.1.1). Secondly, the full soil model will be described (section 4.1.2). This will include all relevant information about how the soil has been modeled including the absorbing boundary conditions. Lastly, a description of the simplified model that has been proposed to account for the effects of SSI without modeling the soil in its entirety is explained (section 4.1.3). Detailed drawings of the bridge are given in Appendix A.

4.1.1 Bridge model

A complete model description of the bridge on Norra Kungsvägen is provided below.

The geometry of the bridge, material properties, loads, boundary conditions, element types and mesh size as well as modeling assumptions are all presented.

Geometry

The bridge is a portal frame bridge. Total height of the bridge is 6.6m measured from the top of the foundation to the deck surface. The span is 15.7m. Wing walls are perpendicular to the back walls and including the length of the wing walls, the total length of the bridge structure is 32.4m. The foundations of the bridge are connected

with two beams parallel to the longitudinal axis of the bridge. A figure of the bridge model is provided in Fig. 4.1.1.

Figure 4.1.1: The FE-model of the portal frame bridge on Norra Kungsvägen.

Thicknesses of the structural elements of the bridge are given in Table 4.1.1.

Table 4.1.1: Norra Kungsvägen geometry.

Structural element Thickness [mm]

Back walls 700

Wing walls 800

Deck 520

Edge beams 580

Foundations 1000

Foundation beams 700

The bridge on Norra Kungsvägen is founded on pile groups which are described further in section 4.1.2 where the soil model is presented in detail. For further details about the measurements of the bridge the structural drawings are presented in Appendix A.

The following simplifications of the geometry have been made in order to simplify the model:

• The longitudinal inclination of the deck (1:50) is neglected i.e. the vertical coordinate of all points of the bridge deck is the same.

• The cross-section of the edge beams is simplified as rectangular along the whole bridge. The cross-section area is however identical to the drawings.

Material properties

The bridge is made of reinforced concrete. The material properties that have been used in the model are summarized in Table 4.1.2.

Table 4.1.2: Material properties of the concrete bridge on Norra Kungsvägen.

Material properties of concrete

Young’s modulus, E 35 GPa

Poisson’s ratio, ν 0

Density, ρ 2400 kg/m³

Damping ratio, ζ 1.5 %

To account for the ballast an added mass has been applied to the bridge deck. In Table 4.1.3 the properties of the ballast are presented.

Table 4.1.3: Material properties of the ballast on the bridge on Norra Kungsvägen.

Material properties of ballast

Young’s modulus, E 0.11 GPa

Density, ρ 2400 kg/m³

Thickness of ballast layer, t 0.85 m

Loads

For the dynamic analysis of the bridge a frequency domain study has been carried out. Two different positions of a concentrated load have been used to excite the bridge model:

• The load is placed at the longitudinal quarter point of the bridge along the center line at the bottom of the deck.

• The load is placed at the longitudinal quarter point of the bridge at the edge and at the bottom of the deck.

The excitation points on the bridge on Norra Kungsvägen are showed in Fig.

4.1.2.

Figure 4.1.2: The excitation points on the bridge on Norra Kungsvägen.

These excitation points were chosen to clearly excite the studied modes and because of the possibility to excite the bridge in the same points during the full scale dynamic testing. This will provide the possibility to compare the results from the model with the experimental results.

The amplitude of the load used in the frequency domain study is 1N . The load is directed upwards.

Boundary conditions

For the analysis of the bridge in isolation, fixed constraints have been applied at the bottom surfaces of the foundation slabs. There are no constraints on the back walls or wing walls.

Element types and mesh size

Solid elements have been used to model the bridge. The choice of solid elements was made to achieve higher accuracy compared to the accuracy that would have been achieved with shell elements. The full model of the bridge-soil system has close to 2 million DOFs, so to save a few thousand DOFs and potentially a couple of minutes of computational time by modeling the bridge in shell elements instead of solid elements

was ruled out.

Quadratic elements with mesh size 1m were used for the discretization of the FE-model.

A mesh convergence analysis presented in Appendix D resulted in the choice of the mesh size.

4.1.2 Soil model

This section describes the properties of the soil model that has been used to analyze the full bridge-soil system on Norra Kungsvägen. Details of how the absorbing boundary conditions have been implemented in the model using PMLs are also explained.

Geometry

The geometry of the soil model surrounding the bridge on Norra Kungsvägen consists of backfill soil behind the back walls and sloping soil on the sides of the wing walls.

As previously mentioned, the bridge on Norra Kungsvägen is founded on pile groups because of the soft soil conditions at the site. Concrete piles are modeled from the bottom of the foundations to the rock layer. The average length of the piles is approximately 16m. The subsoil is modeled as a cuboid with the base area of approximately 40 x 40 m and height 16m. In the model, the bottom surface of the foundations are in contact with the subsoil. Fig. 4.1.3 shows the FE-model of the full bridge-soil model on Norra Kungsvägen.

Figure 4.1.3: The FE-model of the portal frame bridge on Norra Kungsvägen including the full soil model.

Material properties

The soil conditions on Norra Kungsvägen are fairly complex. The subsoil is soft and consists of several layers with different properties. The properties of the soil have been estimated using the information provided by the geotechnical report from the site.

The backfill soil is made of crushed rock. The subsoil consists of a thin layer of crushed rock, layers of sand, clay and silt on top of a deeper moraine layer above the bedrock. Pile tips rest on the bedrock. Fig. 4.1.4 shows the soil conditions at the site schematically. The dynamic material properties of the soil surrounding the bridge are summarized in Table 4.1.4.

Figure 4.1.4: A schematic view of the subsoil on Norra Kungsvägen.

Table 4.1.4: Material properties of the soil on Norra Kungsvägen.

Backfill soil - crushed rock

Shear wave speed, C_s 300 m/s

Poisson’s ratio, ν 0.2

Density, ρ 1800 kg/m³

Young’s modulus, E 390 MPa

Damping ratio, ζ 2.5 %

Subsoil layer 1 - crushed rock Shear wave speed, C_s 250 m/s Poisson’s ratio, ν 0.49

Density, ρ 1800 kg/m³

Young’s modulus, E 335 MPa

Damping ratio, ζ 2.5 %

Subsoil layer 2 - sand/silty sand Shear wave speed, C_s 220 m/s Poisson’s ratio, ν 0.49

Density, ρ 1900 kg/m³

Young’s modulus, E 274 MPa

Damping ratio, ζ 5.0 %

Subsoil layer 3 - silty clay

Shear wave speed, C_s 100 m/s Poisson’s ratio, ν 0.49

Density, ρ 1700 kg/m³

Young’s modulus, E 51 MPa

Damping ratio, ζ 5.0 %

Subsoil layer 4 - silt

Shear wave speed, C_s 200 m/s Poisson’s ratio, ν 0.48

Density, ρ 1800 kg/m³

Young’s modulus, E 213 MPa

Damping ratio, ζ 5.0 %

Subsoil layer 5 - moraine

Shear wave speed, C_s 450 m/s Poisson’s ratio, ν 0.45

Density, ρ 2000 kg/m³

Young’s modulus, E 1175 MPa

Damping ratio, ζ 5.0 %

Boundary conditions

The bedrock layer beneath the soil on Norra Kungsvägen can be considered as infinitely stiff. Due to the thick layer of softer soil above the stiff bedrock, the dynamic waves will be reflected at the interface of the soil and bedrock. Waves traveling outwards to the sides will however propagate freely without reflection. It follows that the soil system surrounding the bridge on Norra Kungsvägen should be modeled as a stratum.

The backfill soil and the subsoil is modeled with 2m thick PMLs on the outsides of the model in order to simulate infinite soil in this direction and to ensure that no reflection occurs at the sides. The PMLs have the same material properties as the soil but enables the layers to be artificially stretched. Fixed constraints are applied on all outside boundaries and at the bottom of the model.

For the analysis of the bridge on Norra Kungsvägen the parameters used for the PMLs that have been implemented are summarized in Table 4.1.5.