SECOND CYCLE, 30 CREDITS ,
STOCKHOLM SWEDEN 2018
Train Induced Vibration
Analysis of an End-frame
Bridge
- Numerical Analysis on Sidensjövägen
JASMIN HALILOVIC
NIKLAS WIBERG
Department of Civil and Architectural Engineering Division of Structural Engineering and Bridges
AF222X Degree Project in Structural Design and Bridges TRITA-ABE-MBT-18302
ISBN: 978-91-7729-885-4
Higher speeds and higher capacity will cause the Swedish rail network to be exposed to disturbing dynamic effects. Higher speeds cause higher vertical acceleration levels of the bridge deck. In this thesis, a numerical analysis of a three span end-frame bridge subjected to train induced vibrations is performed. The aim is to identify which structural components and boundary conditions that affect the dynamic behavior of the bridge. Furthermore, the influence of soil structure interaction (SSI) will be investigated as it may have contribution to the stiffness and damping of the structural system.
In order to capture the dynamic response of the bridge, an analysis in the frequency domain was preformed where frequency response functions (FRF) and acceleration envelopes were obtained. For this purpose, a detailed FE-model in 3D was created. Three different cases were studied, model subjected to ballast, model subjected to soil and model subjected to both ballast and soil in coherence. A high speed load model (HSLM) was used to create simulation of train passages at different speeds and applied to all cases so that the bridge deck accelerations could be studied. A simplified 2D-model with impedance functions representing the soil-structure interaction was created to validate the results from the detailed 3D-model and for practical design purposes.
The result of this numerical analysis showed that the vertical accelerations were within acceptable levels of the maximum allowed limits given in governing publications. Considering the surround-ing soil, the results revealed an increase of the dynamic response in the midspan at resonant frequency. However, it was identified that this behavior is not explained by the influence of soil structure interaction but rather the change in boundary conditions of the end-shields. The same dynamic behavior was identified for the simplified 2D-model, with a slight underestimation of the vertical accelerations at resonance.
Keywords
Högre hastigheter gör att det svenska järnvägsnätet utsätts för störande dynamiska effekter. Ökade hastigheter orsakar högre vertikala accelerationsnivåer av brodäcket. I detta examensar-bete utförs en numerisk analys av en ändskärms bro utformad med tre span som utsätts för tåg inducerade vibrationer. Syftet är att identifiera vilka strukturella komponenter och randvillkor som påverkar brons dynamiska beteende. Dessutom kommer påverkan av jord-struktur interak-tion (SSI) att undersökas, som kan bidra till ändringar i styvhet och dämpning av strukturen.
För att undersöka den dynamiska responsen av bron gjordes en analys i frekvensdomänen där frekvenssvarsfunktioner (FRF) erhölls och accelerationsenvelopper beräknades sedan. För detta ändamål skapades en detaljerad FE-modell i 3D. Tre olika fall studerades, en modell med endast ballast, en modell med endast jord och en modell med både ballast och jord i samverkan. En höghastighetslastmodell (HSLM) användes för att skapa simuleringar av tågpassager med olika hastigheter och applicerades i alla scenarion så att de vertikal accelerationer i brodäcket kunde undersökas. En förenklad 2D-modell med impedansfunktioner som representerar jord-struktur interactionen skapades för att validera resultaten från den detaljerade 3D-modellen och för praktiska konstruktionsändamål.
Resultatet av den numeriska analysen visade att de vertikala accelerationsnivåerna hölls under de maximalt tillåtna värden. Vid undersökningen av jord-struktur interaktion visade resul-taten en ökning av den dynamiska responsen i mittenspannet vid resonansfrekvensen. Det visade sig att detta beteende inte förklaras av jord-struktur interaktion utan snarare genom ändringen av randvillkoren för ändskärmarna. Samma dynamiska beteende identifierades för den förenklade 2D-modellen, med en viss underskattning av de vertikala accelerationerna vid resonans.
Nyckelord
This master thesis was initiated by the Department of Civil and Architectural Engineering at The Royal Institute of Technology at KTH Royal Institute of Science and Technology in cooperation with ELU Consulting AB.
We would like to express our sincere gratitude towards our supervisor Abbas Zangeneh
Ka-mali, Ph.D at KTH and ELU Consulting AB for his guidance, dedication and encouragement
throughout the work with this master thesis. His wisdom and patience in our discussions has helped us to develop a pleasant working environment. We would also like to thank Adjunct Prof.
Costin Pacoste for the opportunity to write our master thesis at ELU Consulting AB and for
enlightening discussions regarding dynamic problems and phenomena. A special thanks to all the master thesis colleges that contributed to a memorable spring.
We would also like to thank Prof. Raid Karoumi and co-workers at the Division of Structural Engineering and Bridges for developing our interest in structural dynamics. The quality of the courses in structural dynamics and bridge design is a major reason why we chose to write our master thesis in this subject.
Finally we would like to thank our families for their encouragement, love and support through our five years at KTH.
Stockholm, Wednesday 4th July, 2018
Nomenclature xi
List of figures xv
List of tables xvii
1 Introduction 1
1.1 Background . . . 1
1.2 Botnia line & Sidensjövägen . . . 2
1.3 Governing Publications . . . 3
1.4 Aim and Scope . . . 4
2 Theoretical background 5 2.1 Structural Dynamics . . . 5
2.1.1 Equation of Motion . . . 5
2.1.2 Steady State and Transient Response . . . 6
2.2 Frequency Domain Method . . . 7
2.2.1 Fourier Series Representation . . . 7
2.2.2 Discrete Fourier Transform . . . 9
2.2.3 Fast Fourier Transform . . . 9
2.2.4 Sampling . . . 9
2.3 Frequency Response Function . . . 12
2.4 Solution Methods . . . 13
2.4.1 Direct Analysis . . . 13
2.4.2 Mode Superposition Method . . . 13
2.5 Soil Dynamics . . . 15
2.5.1 Waves Propagation in Elastic Solids . . . 15
2.5.2 The Linear Viscoelastic Model . . . 17
2.5.3 Impedance Functions of Foundations . . . 22
2.5.4 Transmitting Boundaries . . . 24
3 Method 25 3.1 Used Software . . . 26
3.2 Modeling . . . 27
3.2.2 Modeling of Soil in 2D . . . 32 3.2.3 Complete 2D-model . . . 34 3.2.4 3D Bridge Model . . . 35 3.2.5 Modeling of Soil in 3D . . . 38 3.2.6 Complete 3D-model . . . 39 3.3 Method Procedure . . . 40 3.4 Quality Assurance . . . 41 3.4.1 Convergence Analysis . . . 41 3.4.2 Static Deformations . . . 43
3.4.3 Natural Frequencies and Modes of Vibration . . . 44
3.5 Model Refinement . . . 45
3.5.1 Stiffness Modifications . . . 45
3.5.2 Mass Modifications . . . 46
3.6 Method of Analysis . . . 47
3.6.1 Calculation of FRF:s . . . 47
3.6.2 High Speed Load Model . . . 48
3.6.3 Calculation of Acceleration Envelopes . . . 50
3.7 Model Comparison . . . 51
3.7.1 Simplified Models . . . 51
3.7.2 Detailed Models . . . 52
3.7.3 Main Differences Between 2D & 3D Models . . . 53
3.8 Soil Analysis . . . 55
3.8.1 Effect of Subsoil in 3D . . . 55
3.8.2 Effect of Surrounding Soil in 3D . . . 57
3.8.3 Soil Sensitivity Analysis . . . 59
3.9 Cases subjected to Analysis . . . 60
4 Result 61 4.1 Case Analysis . . . 62
4.1.1 Case 1: Influence of Modeled Ballast . . . 62
4.1.2 Case 2: Soil Structure Interaction . . . 64
4.1.3 Case 3: Soil Structure Interaction & Modeled Ballast . . . 65
4.1.4 Different Load Positions and Output Points . . . 66
4.2 Comparison with 2D-model . . . 70
4.2.1 2D-model with Backfill . . . 70
5 Discussion & Conclusions 73 5.1 Conclusions . . . 73
5.1.1 Effect of Ballast . . . 74
5.1.2 Soil Structure Interaction . . . 74
5.1.3 2D-Model . . . 75
5.2 Further Investigations . . . 76
Appendix A Modes Shapes 79 Appendix B FRF:s and Acceleration Envelopes 83
B.1 Acceleration Envelopes for Train Passages HSLM A1-10 . . . 83
B.2 Influence of Transmitting Boundaries on Ballast . . . 84
Greek Symbols
η Loss factor −
γ Shear strain −
γbt Bridge deck acceleration limit for ballasted track m/s2
γdf Bridge deck acceleration limit for un-ballasted track m/s2
λ Wave length m
µ Elastic modulus P a
µ′ Loss modulus P a
µ∗ Complex modulus P a
ω Angular Frequency rad/s
ω0 Frequency of the fundamental harmonic excitation rad/s
ωD Damped natural frequency rad/s
ωn Natural frequency rad/s
τ Shear stress P a
ϕ Modal matrix −
ξ Damping Ratio −
ζ Structural damping Eurocode −
ν Poisson’s ratio P a
i unit imaginary number√−1
Latin Letters
a0 Dimensionless coefficient −
A Acceleration in frequency domain m/s2
Atrib Contributing area per node m2
Ampp Amplitude function in time domain −
bj Fourier series coefficient for sine expansion −
c Damping coefficient N s/m
ccr Critical Damping coefficient N s/m
C Damping matrix N s/m
C Wave velocity m/s
Cp Compression wave velocity m/s
CR Rayleigh wave velocity m/s
Cs Shear wave velocity m/s
Dn Normal damper coefficient N s/m
Dt Tangent damper coefficient N s/m
E Young modulus P a
fnyquist Nyquist frequency Hz
fs Sampling frequency Hz
∆f Frequency resolution Hz
Fp Axle load in time domain N
g Gravitational acceleration m/s2
G Shear modulus P a
H Complex frequency response function m/N s2
Iy Moment of intertia m4 k Stiffness N/m K Stiffness matrix N/m m Mass Kg M Mass matrix Kg Ms Compression modulus P a
pn IDFT of signal N
P Force in frequency domain N
Pj DFT of signal N
q Modal coordinate vector m
Q Point force N
S Dynamic impedance N/m
t Time s
T0 Time period s
u Displacement in time domain m
U Displacement in frequency domain m
˙
u Velocity in time domain m/s
¨
u Acceleration in time domain m/s2
W Maximum elastic energy J
Abbreviations
2D Two-dimensional
3D Three-dimensional
BC Boundary condition
DAF Dynamic Amplification Factor
DFT Discrete Fourier Transform
DOF Degrees of Freedom
EC Eurocode
FE Finite Element
FEA Finite Element Analysis
FEM Finite Element Method
FFT Fast Fourier Transform
FRF Frequency Response Function
HSLM High Speed Load Model
IDFT Inverse Discrete Fourier Transform IFFT Inverse Fast Fourier Transform SDOF Single Degree of Freedom
SGI Swedish Geotechnical Institute
1.1 Bridge location . . . 2
1.2 Drawing of bridge at Sidensjövägen . . . 2
2.1 Equation of motion . . . 5
2.2 Relationship between the frequency and time domains . . . 8
2.3 Leakage phenomena . . . 10
2.4 Compression wave . . . 16
2.5 Shear wave . . . 16
2.6 Rayleigh wave . . . 17
2.7 Shear strain behavior . . . 18
2.8 Hysteretic stress-strain curve . . . 19
2.9 Rate-dependent Kelvin model . . . 20
2.10 Rate-independent Kelvin model . . . 21
2.11 Vertically vibrating foundation block . . . 22
2.12 Viscous boundaries . . . 24
3.1 Bridge boundary conditions . . . 28
3.2 Bridge model in 2D . . . 29
3.3 Ballast as non structural mass . . . 30
3.4 Support conditions for 2D-model . . . 30
3.5 Explanation of movable directions of supports . . . 30
3.6 Modeled constraints of supports in 2D-model . . . 31
3.7 Soil structure system of 2D-model . . . 32
3.8 Soil consideration in 2D-model . . . 32
3.9 Spring coefficients for individual parts . . . 33
3.10 Complete 2D-model . . . 34
3.11 3D bridge model . . . 35
3.12 Modeled ballast & sleepers instance . . . 36
3.13 Support conditions for 3D-model . . . 36
3.14 Explanation of movable direction in supports . . . 36
3.15 Modeled constraints of supports in 3D-model . . . 37
3.16 Applied dashpot illustration . . . 38
3.17 Complete 3D-model . . . 39
3.19 Load cases for static deformation . . . 43
3.20 Static deformation . . . 43
3.21 Vertical bending modes for 2D-model and 3D-model . . . 44
3.22 Stiffness modification of bridge deck . . . 45
3.23 Stiffness modification through selection of constraint . . . 45
3.24 Mass modification of pillars . . . 46
3.25 Method of analysis . . . 47
3.26 Load position at sleepers . . . 48
3.27 HSLM-A load model . . . 48
3.28 Longitudinal distribution of a point force or wheel load by the rail . . . 49
3.29 Time history . . . 49
3.30 Simplified models . . . 51
3.31 Comparison between simplified models . . . 51
3.32 Detailed models . . . 52
3.33 Comparison between detailed models . . . 52
3.34 Different modeling approaches regarding varying thickness . . . 53
3.35 Different modeling approaches regarding the end shield . . . 53
3.36 Pillar constraints in 3D-model . . . 54
3.37 Representation of subsoil . . . 55
3.38 Influnce of subsoil . . . 56
3.39 Soil backfill conditions for 3D-model . . . 57
3.40 Convergence analysis of soil backfill . . . 57
3.41 Models for comparison of additional surrounding soil . . . 58
3.42 Influence of surrounding side soil . . . 58
3.43 Soil stiffness sensitivity of 2D-model with simplified backfill . . . 59
4.1 Models in Case 1 . . . 62 4.2 Results case 1 . . . 63 4.3 Models in Case 2 . . . 64 4.4 Result Case 2 . . . 64 4.5 Models in Case 3 . . . 65 4.6 Result Case 3 . . . 65
4.7 Influence of backfill at midpoint of first span . . . 66
4.8 Different load positions along the bridge . . . 66
4.9 FRFs at midspan for different load positions along the bridge . . . 67
4.10 Phase angle and magnitude at different load positions at midspan . . . 69
4.11 Phase angle and magnitude at different load positions at offset span . . . 69
4.12 Complete 2D-model . . . 70
4.13 Effect of backfill in 2D-model . . . 70
2.1 Comparison of different solution methods . . . 14
3.1 Minimum values of structural damping . . . 27
3.2 Material properties for bridge and soil . . . 27
3.3 Element types and mesh sizes tested for 2D-Model . . . 41
3.4 Assigned element types and mesh sizes in 2D-model . . . 41
3.5 Element types and mesh sizes tested for 3D-model . . . 42
3.6 Assigned element types and mesh sizes in 3D-model . . . 42
3.7 Natural frequencies for 2D-model and 3D-model . . . 44
3.8 Mass modification of pillars . . . 46
3.9 HSLM-A geometrical properties . . . 48
3.10 Comparison simplified models . . . 51
3.11 Comparison of dynamic properties for detailed models . . . 52
4.1 Dynamic properties of 3D - Models . . . 68
Introduction
1.1
Background
High-speed railway is a frequent discussion in the work with Swedish infrastructure. More countries over the world invest in high-speed railway hoping for a better alternative to air travel. To shorten the travel time for commuters, reduce environmental impact and contribute to regional growth through the construction of high-speed enticements a lot. However, previous research has shown that end-frame bridges gives rise to high acceleration levels that do not fulfill the criteria specified in the Eurocode (Andersson, 2010). Therefore, this type of bridge will be further investigated in detail to determine if the surrounding soil has more dynamic effect than suggested.
1.2
Botnia line & Sidensjövägen
Botniabanan is a 190 km single track railway line between Nyland and Umeå i the north of Sweden, see Figure 1.1. The maximum allowed speed is 250 km/h for passenger trains and 120 km/h for freight trains with a maximum axle weight of 25 tons. Previous field investigations has shown that a majority of train bridges on Botniabanan do not fulfill the dynamic requirements according to governing publications at 250 km/h. The problems concerns all bridge types, but specially for end-frame bridges (Andersson, 2010). The end-frame bridge located at Sidensjövägen is a bridge of this type, its location is marked in Figure 1.1.
Figure 1.1 Bridge location
The bridge located at Sidensjövägen is a slab bridge designed with end-frames. The total span of the bridge is 49 m and is supported on roller bearings at the ends and fixed bearings. The roller bearings are resting on two support walls and four pillars, see Figure 1.2 for illustration of the bridge.
1.3
Governing Publications
At present there are no recommendations in the Eurocodes regarding the consideration of soil-structure interaction in design of end-frame bridges. This is because the subject is on the up spring and ongoing research on the subject is performed. The work in this thesis has therefore been based on the dynamic requirements imposed on bridge structures exposed to train loads. The following publications have been followed:
• TRVK Bro 11 Swedish Transport Administration technical requirements - Bridge TRV publ nr 2011:085
• SS-EN 1991-2 Eurocode 1: Actions on structures – Part 2: Traffic loads on bridges • SS-EN 1990 Eurocode – Basis of structural design
The requirements according to the Swedish Transport Administration states that for railway bridges that may be exposed to traffic with velocities above 200 km/h, a dynamic analysis should be performed (TRV, 2011). The dynamic analysis should follow Eurocode 1991-2 section 6.4. In SS-EN 1991-2, the universal train load model and bridge parameters that should be used is given (CEN, 2003). Finally, the bridge deck acceleration limits that should be followed are given by SS-EN 1990 A2.4.4.2.1 (CEN, 2002):
• γbt ≤ 3.5 m/s2 (for ballasted tracks)
1.4
Aim and Scope
The aims and scope of this thesis can be summarized into following:
• Perform numerical analysis of train induced vibrations subjected on an end-frame bridge with excluded and included ballast and surrounding soil
• Analyze results from numerical analysis and draw conclusion about the dynamic behavior of the bridge and the effect of SSI
• Create a simplified model for practical design purposes The following limitations apply:
• The structures were modeled according to the theory of linear elasticity
• The soil medium was modeled according to the theory of linear elasticity and the interaction with the structure was assumed to be fully tied
Theoretical background
The following Chapter describes briefly the basic concepts of structural- and soil dynamics. The authors believe that following theory is needed in order to better understand the aspects of this thesis. If no other references are given in 2.1, 2.2, 2.3 and 2.4 the sections are referred from (Chopra et al., 1995).
2.1
Structural Dynamics
2.1.1 Equation of Motion
Figure 2.1 shows a single degree of freedom (SDOF) system subjected to a external dynamic force p(t). The system consists of three components, spring, damper and mass which relates to
the displacement u(t), velocity ˙u(t), and acceleration ¨u(t), respectively.
(a) SDOF system (b) Free-body diagram
Figure 2.1 Equation of motion
p(t) = m¨u(t) + c ˙u(t) + ku(t) (2.1)
Structural dynamic problems are often systems with a large number of degrees of freedom and therefore it’s convenient to manage the equation of motion in matrix form instead
P(t) = M¨u(t) + C ˙u(t) + Ku(t) (2.2) where M, C and K are the mass, damping and stiffness matrices, respectively. The displacement and force are described as vectors u(t) and P(t).
2.1.2 Steady State and Transient Response
If we assume the displacement to be a harmonic motion u(t) = u0sin(ωt + α) and the external
force to be harmonic p(t) = p0sinω the equation of motion 2.1 can be rewritten by dividing with
the mass as
po
msin(ωt) = ¨u(t) + 2ξωnu(t) + ω˙ n
2u(t) (2.3)
where ωn=qmk is the natural frequency of the system and
ξ = c
2mωn =
c ccr
(2.4)
is the damping ratio of the system and ccr is called the critical damping coefficient. The oscillation
of the system is dependent of the damping ratio of the system. If ξ ≥ 1 the system does not oscillate and will return to its equilibrium state in a faster rate.
The solution to the differential equation 2.3 is given by
u(t) = e−ξωnt(Acosω
Dt + BcosωDt) + Csinωt + Dcosωt (2.5)
where ωD = ωn
p
1 − ξ2 is the damped natural frequency of the system.
The first part of the equation containing ωD terms is called the transient response and is the free
2.2
Frequency Domain Method
The frequency domain method of dynamic analysis is alternative method to solve linear differential systems such as the equation of motion. It’s a useful method when dealing with large systems and structures interacting with unbounded boundaries like underlaying soil. The method uses discrete Fourier transforms to solve the equation of motion. The Fourier transform allows us to go from the time domain to the frequency domain. The Fourier transform of the equation of motion 2.2 derived earlier is given by
[−ω2M + iωC + K]U(ω) = P(ω) (2.6) which gives us the equation of motion as representation in the frequency domain. In dynamic analysis of soil-structure interaction where the stiffness matrix K and the damping matrix C are frequency dependent the equation becomes
[−ω2M + iωC(ω) + K(ω)]U(ω) = P(ω) (2.7) allowing the equation of motion to be solved for each frequency of interest.
2.2.1 Fourier Series Representation
The theory behind the discrete Fourier transform is based on Fourier series. Any arbitrary periodic function can be described by an infinite sum of cosine and sine functions. If p(t) is a
periodic function with the period T0 it can be represented using Fourier series as
p(t) = a0+ ∞ X j=1 ajcosjω0t + ∞ X j=1 bjsinjω0t (2.8) where ω0 = 2πT
0 is the frequency of the fundamental harmonic excitation.
The coefficient a0, aj and bj can be expressed as
a0 = 1 T0 Z T0 0 p(t)dt (2.9) aj = 2 T0 Z T0 0 p(t)cosω0tdt j = 1, 2, 3, ... (2.10) bj = 2 T0 Z T0 0 p(t)sinω0tdt j = 1, 2, 3, ... (2.11)
where a0is the average value of the excitation force p(t). The coefficients aj and bj are amplitudes
of the jth harmonics of frequency jω0.
Fourier Transform
The Fourier series as mention above can be used to represent periodic functions. If the function
p(t) = 1 2π Z ∞ −∞P (ω)e iωtdω (2.12) where P (ω) = Z ∞ −∞ p(t)eiωtdt (2.13)
The frequency function P (ω) is the Fourier transform (also called direct Fourier transform) of the time function p(t). Equation 2.12 is the inverse Fourier transform of P (ω) and the two equation are together called a Fourier transform pair. They can be derived using complex Fourier series
by letting the period T0 approach infinity.
The Fourier transform is a powerful tool that can be used to calculate the response of a linear system subjected to an excitation force p(t). This is done by combining the response to individual
harmonic excitation terms in the Fourier integral 2.12. If we assume the excitation to be P (ω)eiωt
then the response of the system is given by
U (ω) = H(ω)P (ω)eiωt (2.14)
where H(ω) is the complex frequency response function. By superposing each individual response the total response in the time domain can be expressed as
u(t) = 1
2π
Z ∞
−∞
U (ω)dω (2.15)
which is the inverse Fourier transform of U (ω). The procedure is know as the frequency domain-method and can used to analyze the response of a structure in the frequency and time domain. Figure 2.2 shows the relationship between the frequency and time domains.
Figure 2.2 Relationship between the frequency and time domains (Hewlett-Packard, 1994)
2.2.2 Discrete Fourier Transform
The Fourier transform is valid for for both periodic and non periodic responses like we find in the real world. However, it requires that both the direct and inverse Fourier transform of the excitation and the frequency response are determined. This is not possible for large structures and complex excitations functions. In order to analytically evaluate these integrals we must perform a numerical integration to get a true approximation of the Fourier transforms. The method is called discrete Fourier transform and is useful to determine the dynamic response of a structure.
The discrete Fourier transform of a N -periodic excitation p(t) can be expressed as
Pj = 1 T0 N −1 X n=0 pne−i(jω0tn)∆t = 1 N N −1 X n=0 pnei−2πnj/N (2.16)
where pn describes the discretized forcing function and is the inverse discrete Fourier transform
of the sequence Pj. It can be expressed as
pn= N −1 X j=0 Pjei(jω0tn)= N −1 X j=0 Pjei(2πnj/N ) (2.17)
The two equations 2.16 and 2.17 represents a discrete Fourier transform pair.
2.2.3 Fast Fourier Transform
The discrete Fourier transform requires calculation of the direct DFT of the excitation sequence and the inverse DFT of the response sequence this leads to excessive amounts of computation time, especially when a large number of samples N is needed for the analysis. The fast Fourier transform is a fast and accurate algorithm to calculate the DFT which reduces the computational time drastically. It’s based on the Cooley-Tukey algorithm developed in 1965 (Cooley and Tukey, 1965) and is just another way of calculating the DFT. The FFT algorithm is often used in computer software like MatLab.
2.2.4 Sampling
The Sampling Frequency
The fast Fourier transform helps us transfer data from the time domain to the frequency domain or spectrum which is a favorable way of handling and analyzing data. However, since it’s not possible to transfer the data continuously, sampling is needed and because of this the transfered data is not an exact representation in either domain. To get an ideal representation the sampled data must be as close as possible. The sampling frequency can be expressed as
fs= 1
∆t =
N
T (2.18)
The Sampling Theorem
The sampling theorem says that a continuous signal may be reconstructed if and only if it is sampled at a frequency greater than its Nyquist frequency. Meaning that in order to have good approximation of the signal it must be sampled more than twice per period. The Nyquist frequency is defined as fnyquist= fmax = ∆f N 2 = fs 2 (2.19) where ∆f = fs
N is called the frequency resolution. The sampling theorem is important to fulfill
in order to prevent distortion of the signal like aliasing/folding.
Anti-Alias Filter
Aliasing occurs when under sampling is an issue. Frequency components that are higher than
half of the sampling frequency fs
2 will not be accurately represented. Frequencies above this limit
will be reflected over the limit and appear as lower frequency components and cause distortion of the signal. In real world situations it might be difficult to restrict the frequency range of the signal. Therefore, anti-aliasing or low pass filters are used to to restrict the frequency range and remove all frequencies above the Nyquist frequency before the sampler and A/D converter.
Windowing
The fast Fourier transform is computed through sampled blocks in the time history under the assumption that the waves are periodic (Hewlett-Packard, 1994). The Fourier replication of a sampled record that is not periodic leads to the phenomena called leakage. The reconstruction of the incorrectly sampled record leads to an incorrect frequency spectrum containing leakage. Figure 2.3 show how the phenomena arises through a incorrectly sampled time record.
Figure 2.3 Input signal not periodic in time record (Hewlett-Packard, 1994)
2.3
Frequency Response Function
This section will describe the steady state response of a system with rate-independent linear damping which is used in this thesis. The concept is explained more thoroughly in Section 2.5.2 by the rate-independent Kelvin model. The rate-independent damping is very useful in the frequency-domain method of analysis when the steady state response of the system is of great interest.
The equation of motion for a rate-independent SDF system subjected to a external force p(t) is expressed as
m¨u +ηk
ω u + ku = p(t)˙ (2.20)
where the term ηkωu is the rate independent damping force at frequency ω. The factor η is a˙
damping coefficient and k is the stiffness of the structure.
If we assume the external force to be a harmonic unit load
p(t) = 1eiωt (2.21)
The steady-state response of the system will be a harmonic motion at the forcing frequency ω and can be expressed as
u(t) = Hu(ω)eiωt (2.22)
where Hu(ω) is the frequency-response function. By substituting equations 2.21 and 2.22 in
equation 2.20 and rewriting it gives
Hu(ω)eiωt− ω2m + k(1 + iη)= eiωt (2.23) The complex term k(1 + iη) is often called the complex stiffness of the system and considers both
the damping and elastic forces. If we cancel the eiωt on both sides and rearrange the equation a
expression for the frequency-response function is given by
Hu(ω) =
1
2.4
Solution Methods
2.4.1 Direct Analysis
Direct analysis is a solution method where the DOF of the whole model is set to be a variable. By integrating with time the dynamic equilibrium equation for the defined DOF can be solved. The solutions is dependent on each time stage and since the form to the equilibrium equation does not change it is possible to apply various integration methods. The analysis is performed for all time stages and the analysis time depends on the number of time stages. Example of direct solution method to a SDOF system can be found under Section 2.1.1.
2.4.2 Mode Superposition Method
The mode superposition method, also called modal analysis, is a solution method where the structural displacement is taken into consideration. The structural displacement is assumed as a linear combination of orthogonal displacements. This assumption allows for simplified calculation of the dynamic response for a selected mode. This is favorable when performing linear dynamic analysis of large models since it requires smaller amount computational cost. These assumptions and simplifications may affect the accuracy of the solution since the analysis is dependent on the number of modes used. If the number of modes is set to the same as number of DOF, we obtain accurate results. Otherwise, it is seen as approximations
Consider following undamped equation of motion for free vibration:
M¨u + Ku = 0 (2.25)
where M is the modal mass matrix and K is the modal stiffness matrix. This can be rewritten to following eigenvalue problem:
(K − ω2nM)u = 0 (2.26)
where ωn is the n:th eigenfrequency and u is the displacement vector. The displacement vector
u is expressed as the linear combination (or the sum of modal contributions):
u =
N
X
i=1
[ϕiqi] = Φq (2.27)
where Φ is the modal matrix containing all the eigenmodes and q is the modal coordinate vector. Thus the problem becomes:
ΦTMΦ¨q + ΦTKΦq = 0 (2.28)
Where the diagonal generalized mass and stiffness matrices are given by:
k = ΦTKΦ (2.30)
Table 2.1 Comparison of different solution methods
Direct method Mode superposition method Computational time More time Less time
Of importance Definitions of time step Number of modes
Size of models Smaller models Larger models
2.5
Soil Dynamics
2.5.1 Waves Propagation in Elastic Solids
The following section will briefly describe the concept of wave propagation in infinite and half-space solids. The wave propagation is part of the damping in the soil called radiation damping. The phenomena can be explained by energy losses throughout the soil when the waves propagate further and further away from its source. Following is based on concepts stated by (Bodare, 1997) and (Ülker Kaustell, 2009).
Waves in infinite solids
In an infinite linear elastic solid two type of waves exist if we only consider propagation in one direction. The first waves are associated with volumetric change or dilatation and its movement is parallel with the wave propagation. They are often referred to as ’Primary’, P waves. The other wave known as ’Secondary’, S waves are coupled to shear deformation and its movement is perpendicular to the wave propagation.
The two waves are governed by the wave equation that is given by
∂2u
∂t2 − C 2∂2u
∂x2 = 0 (2.31)
where C is the wave velocity, u is the displacement, x is the inference direction and t is the time.
The wave velocity Cp for the P-waves can be derived to
Cp = s Mc ρ = s E ρ (1 − ν) (1 − 2ν)(1 + ν) (2.32)
where Mc is the compression modulus, E is the Young’s modulus, ρ is the mass density and ν is
the Poisson’s ratio. The P waves velocity is controlled by the compression modulus and is the faster of the two waves. In contrast to the S waves, that only can travel through solid materials,
the P waves can travel through any type of material. The shear wave velocity Cs can be derived
as Cs = s G ρ = s E ρ 1 2(1 + ν) (2.33)
where the velocity is controlled by the shear modulus G. The dilatation waves and shear waves exist independently from each other and are as mention the only two waves that can propagate in an unbounded homogeneous body. The relationship between the velocities can be expressed as
Cp Cs = s 2 − 2ν 1 − 2ν (2.34)
Figure 2.4 P-wave movement and particle motion
Figure 2.5 S-wave movement and particle motion
Waves in half-spaces
In a half-space elastic medium additional waves arises. The shear and dilatation waves will reflect against the free surface which can be seen as a ground surface. In addition to these reflection waves another type of wave is introduced called surface wave. There are several types of surface waves and they propagate along the free surface. The most important surface wave is called Rayleigh waves which is comparable to a wave that arises on water surface. The particle motion is elliptic and is retrograde in relation to the direction of the wave. The amplitude of the waves decreases exponentially with depth and are frequency dependent. Waves with low frequency penetrate further down in the material compared to waves with high frequency. The speed of the Rayleigh waves are a bit slower than the shear waves and in a homogeneous elastic half-space it can derived approximately to
CR=
0.87 + 1.12ν
1 + ν Cs (2.35)
Figure 2.6 Rayleigh wave movement and particle motion
2.5.2 The Linear Viscoelastic Model
Determining the soil properties and to accurately capture the response of the soil behavior is of great importance in order to understand soil structure interaction. Following section will describe basic theory and modeling in soil medias. The material used in this section can be found in (Ishihara, 1996) and (Kramer, 1996) whereas the linear viscoelastic model is based on.
Soil materials are not perfectly elastic and dissipates energy during cyclic loading. The energy loss is referred to as material damping. In the following section the concept will be described using linear viscoelastic theory. The theory is based on wave propagation and takes into account the cyclic behavior of the soil. The shear stress and the shear strain are two important parameters that must represented in a truthful way. The soil behavior is highly dependent on the strain level and therefore, obviously the shear modulus is a key factor to model the soil properly. If
the strain range is assumed to be small (< 10−5) a linear elastic model is a good representation.
However, if we reach medium strain levels at approximately (< 10−3) the soil behavior becomes
elasto-plastic. Due to the fact that the shear modulus seems to decrease while the shear strain increases. The soil material also experience energy losses during each load cycle. The energy dissipation properties of the soil material under cyclic stress can be represented by the damping ratio. The soil has energy absorbing properties when exposed to cyclic loading and for the most part the energy loss is rate independent and of hysteric nature. On the other hand the strains at this level are still small enough to not cause any major changes in the soil properties. Meaning that the soil behavior can be approximated quite accurately by the use of linear viscoelastic
theory. During strain levels larger than (10−2) the soil properties start to change with both the
Figure 2.7 Soil behavior in compliance with strain-deformation characteristics and appropriate model method (Ishihara, 1996)
Stress-strain relationship in general
The general expressions for the stress τ and strain γ in complex variables can expressed as
τ = τaeiω (2.36)
γ = γaei(ωt−δ) (2.37)
where γa and τarepresents the strain and stress amplitudes,respectively, ω is the frequency and δ
is the phase angle difference considering the time lag in the responses. The general ratio between the stress and strain can then be expressed as
τ γ = τa γa eiδ = τa γa (cos δ + i sin δ) (2.38)
by introducing the variables
µ = τa γa cos δ (2.39) µ′ = τa γa sin δ (2.40)
the equation 2.38 can be rewritten to
µ∗= τ
γ = µ + iµ
′ (2.41)
where µ and µ′ are referred to as the elastic modulus and the loss modulus and µ∗ is called the
complex modulus. The relationship can also be expressed as
η = tan δ = µ
′
µ (2.42)
Hysteretic stress-strain
The loss coefficient can also be derived from the hysteretic stress-strain curve. The damping is related to the loss of energy during each load cycle ∆W , which is equal to the area enclosed by the hysteresis loop shown in Figure 2.8. It can be calculated using the integral for an ellipse
∆W =
Z
τ dγ = µ′πγa2 (2.43)
It can be shown that the maximum elastic energy W that can be stored in a unit volume of a vicsoelastic body can be expressed as
W = 1
2µγ
2
a (2.44)
The ratio between the loss of energy per cycle ∆W and the maximum elastic energy W stored gives the relation
∆W W = µ′πγa2 1 2µγa2 = 2πµ ′ µ (2.45)
which is related the loss coefficient η derived in equation 2.42 in the following way
η = 1 2π ∆W W = µ′ µ = tan δ (2.46)
The rate-dependent Kelvin Model
To get a deeper understanding of how a linear viscoelstic material behave a model with springs and dashpots can be introduced. Where the spring represents the elastic component and the dashpot is represents the damping component of the model. The most commonly and simplest model used is the Kelvin model see Figure 2.9. The energy loss in the model is represented by a viscous dashpot. Meaning that the energy that is being lost from the system is dependent on the velocity or in other word the time rate of the deformation. Hence, the deformation is dependent on the frequency. This kind of damping is referred to as rate-dependent damping.
Figure 2.9 The rate-dependent Kelvin model represented with spring and dashpot, (Ishihara, 1996)
The Kelvin model can be described using the general form of expression correlating the stress and strain shown above. The total stress is given by
τ = Gγ + G′dγ
dt (2.47)
where G is the spring constant, G′ is the dashpot constant and γ is the strain equally imposed on
the two components. The part Gγ is the stress transmitted to the spring and the stress carried
by the dashpot is G′ dγdt. If the strain and stress are expressed according to equations 2.36 and
2.37 the total stress can be rewritten as
τaeiδ= (G + iωG′)γa (2.48)
and according to equation 2.41 resulting in
µ + iµ′ = G + iωG′ (2.49)
where the following relations are obtained
The relations shows that the elastic modulus µ is a shear constant and the loss modulus µ′ is a linear function of the frequency ω. The loss coefficient η is therefore given by
η = G
′ω
G =
µ′
µ = 2ξ (2.51)
where ξ is the damping ratio. The equation show that the loss coefficient increases linearly with the frequency ω the body is excited with. With reference to equation 2.41 and inserting the loss coefficient the following relationship is obtained
G = (1 + iωη)G (2.52)
or expressed with the damping ratio ξ as
G = (1 + iω2ξ)G. (2.53)
The rate-independent Kelvin Model
In the rate-dependent Kelvin model it was shown that the loss coefficient η is frequency dependent which comes from the use of the viscous dashpot. Meaning that the energy dissipation is dependent on the time rate of the deformation. A lot of materials exhibits frequency independent damping, therefore a more suitable model is presented in order to remove the rate dependency of the dashpot. The rate-independent or non-viscous Kelvin model see Figure 2.10 is the simplest model and fulfills the requirement. The dashpot in the model is a non-viscous and therefore removes the frequency dependency of the deformation.
Figure 2.10 The rate-independent Kelvin model represented with and a non-viscous dashpot, (Ishihara, 1996)
The following stress strain-relation may be used to express the model
τ = (G + iG′0)γ (2.54)
where G′0 is a dashpot constant. The equation contains the imaginary term iG′0γ which has
of soils. Using the same procedure as described for the rate-dependent Kelvin model the loss coefficient for the model is derived as
η = tan δ = G
′ 0
G (2.55)
where µ = G and µ′ = G′0. The loss coefficient is no longer frequency dependent and thus
the damping is frequency independent. In structural dynamics this is referred to as structural damping and gives good accuracy of the damping properties of the soil. The relationship can be described in the same way as equation 2.52 which gives
G = (1 + iη)G (2.56)
or in terms of damping ratio ξ
G = (1 + i2ξ)G (2.57)
the expression is also referred to as hysteretic material damping.
2.5.3 Impedance Functions of Foundations
A way of considering the soil-foundation interaction in dynamic analysis is using impedance functions. The approach is based on (Gazetas, 1991a) and (Gazetas, 1991b) and the simplest way of explaining the method is using a case of only vertical vibration. Figure 2.11 shows the dynamic equilibrium of the simple case from which the impedance function can be derived.
Figure 2.11 Dynamic equilibrium of a vertically vibrating foundation block (Gazetas, 1991b)
The dynamic equilibrium in the vertical direction gives
where Pz(t) is the resultant of the reaction force of the soil, m is the mass, ¨uz(t) is the acceleration
and Fz(t) is the applied harmonic force as Figure 2.11 shows. The system experiences a harmonic
displacement uz(t) and it relates to the reaction force Pz(t) as
Pz(t) = Szuz(t) (2.59)
where Sz is called the dynamic vertical impedance and is usually express as a force displacement
ratio
Sz =
Pz(t)
uz(t)
(2.60) However, since the system experiences radiation and material damping for all modes of vibration
the reaction force Pz(t) is often out of phase with the displacement uz(t) it is useful to express
the impedance in complex form as
Sz = Kz+ iωCz (2.61)
where Kz the dynamic stiffness, Cz is the dashpot coefficient and are both dependent on the
frequency ω. The dynamic stiffness describes the stiffness and inertia properties of th soil. The dashpot coefficient reflects the radiation and material damping generated in the system. The
dynamic stiffness Kz = Kz(ω) is a product of the static stiffness Kz and the frequency dependent
dynamic stiffness coefficient kz(w) giving the expression
Kz(ω) = Kzkz(ω) (2.62)
indicating the frequency dependency of the dynamic stiffness. The dashpot coefficient Cz is
divided in two part and expressed as
Cz = Cradiation+ Chysteretic (2.63)
in which Cradiation is containing the radiation damping due to energy dissipation from wave
propagation away from the source and Chysteretic is containing the material damping due to
energy losses in the soil by hesteretic action. The hysteretic dashpot coefficient is given by
Chysteretic=
2Kz
ω ξs (2.64)
where ξs is the damping ratio of the soil. The dynamic stiffness and dashpot coefficient can
be calculated for different types of foundations and subsoil conditions in order fit the present soil-foundation system studying. The derivation of the impedance function can also been done in a similar way for each degree of freedom to study any mode of interest.
The impedance can be implemented to the equation of motion for the vertically vibrating block in shown in Figure 2.11. By combining equations 2.58 and 2.59 it leads to
and inserting equation 2.61 and assuming a harmonic response ueiωt and harmonic excitation
force Fzeiωt the equation of motion is given by
[(Kz− mω2) + iωCz]uz= Fz (2.66)
Once the spring stiffness and dashpot coefficient haven been obtain the equation of motion can be solved and the vertical displacements can be determined.
2.5.4 Transmitting Boundaries
The method used to analyze the soil-structure interaction is the finite element method. In finite element modeling the entire structure containing the bridge and surrounding soil is model. The soil is often modeled as finite continuum medium with transmitting boundaries at the border of the soil. This is done to make sure that reflecting waves do not disturbs the response of the system and properly capture the soil behavior. A way of implementing the unboundedness of the soil is using the viscous boundary method (Zangeneh, 2018). At each node dashpots are applied, one in the normal direction and two in the tangential direction, in order to absorb the dilatation
and shear waves, respectively. The normal Dn and tangential dampers Dt can be calculated as
Dn= ρCsAtrib (2.67)
Dt= ρCpAtrib (2.68)
where ρ is the density of the soil, Cs and Cp are the shear and dilatational wave velocities and
Atrib is the contributing area per node. The Figure 2.12 shows how the daspot are applied at the
border of a body.
Figure 2.12 Transmitting boundaries applied to finite body, (Zangeneh, 2018)
The viscous boundary method also reduces the computational time since the entire soil does not need to be modeled. If no proper transmitting boundaries are applied to the soil medium the following criterion
Lr≥
Cp
fmin
(2.69)
must be fulfill to receive a undisturbed steady-state response. Where Lr is the shortest path in
the finite domain from the region of interest to the boundaries and fmin is the lowest frequency
Method
In this thesis, the authors strove to create a model that accurately captured the behavior of the real case bridge. For this reason great emphasis was placed on modeling in three dimensions (3D) and verification in form of a simplified two dimensional (2D) model. Verification through simplified 2D models was useful for several reasons, e.g. to create increased understanding of behavioral patterns. If the 2D model verifies the 3D model, the simplified model can be used instead in order to derive desirable results in less computational time (CPU-time).
The methodology of this work was divided into cases, where each case yield results that was interpreted. The idea of dividing the methodology into different cases was to isolate behavior and thus reach conclusions about why certain phenomena occur. By doing this, the effect of controlled changes in parameters and input values was distinguished.
The bridge structure was modeled in both 2D and 3D according to drawings. The models followed the Method analysis described in Section 3.6 and was applied to three different cases.
• Case 1: Bridge models with ballast subjected to HSLM
• Case 2: Bridge models with surrounding soil subjected to HSLM
3.1
Used Software
In this thesis, different softwares have been used for different purposes. The following section will shortly describe used software throughout the time of the project.
Abaqus/CAE
Abaqus/CAE was used in the modeling approach. Abaqus is a FEM program for general purpose used to solve various problems. The software was used to create a model of the bridge, to extract data and to analyze the behavior of the structure when subjected to different kind of loading. The software provided understandable visualizations which were used to determine if the model was reasonable.
Python
Python is an programming language used for general purpose programming. Python was used to obtain FRF:s from the bridge in Abaqus. The Python script is designed to create a step in Abaqus that is consistent to the intended purpose.
MatLab
A MatLab script was used to manage the FRF:s obtained from Abaqus. The FRF:s was multiplied by created train loads(HSLM). The MatLab script also controlled other parameters such as time-steps, frequency-steps etc. MatLab was also used for presentation purpose. Data from excel was used in order to create more desirable presentations of result.
MathCad
Mathcad was used to when dealing with more advanced calculations such as impedance functions and spring coefficients.
AutoCad
3.2
Modeling
Following section describes the modeling procedure throughout this thesis. A simplified 2D-model and a detailed 3D-2D-model was created in Abaqus/CAE. The 2D-models was based on drawing according to Appendix A. The idea with this section is to clarify the modeling procedure as a whole but also simplifications and details in modeling. The choice of material parameters was based on the Eurocode and previous research.
The concrete used in both models was assumed to be cracked as the bridge is not newly produced. Following conditions apply for cracked concrete:
• Density, ρ = 2400 [kg/m3]
• Modulus of Elasticity, E = 28 [GPa] • Poisson’s ratio, ν = 0
The E-modulus for concrete is assumed to be 0.8E according to (Tahershamsi, 2011). This reduced stiffness accurately captures the effect of cracked concrete. Poisson’s ratio was taken from Eurocode 1991-1.1:2005, 3.1.3 (4) where ν = 0.2 for uncracked concrete and ν = 0 for cracked concrete.
The structural damping ζ of the concrete for bridges was chosen according to Table 3.1. In this thesis, ζ = 1.5 was chosen.
Table 3.1 Minimum values of structural damping according to SS-EN 1991-2, 2003
Bridge Type ζ Lower limit of percentage of critical damping [%]
Span L < 20 m Span L ≥ 20 m
Steel and composite ζ = 0.5 + 0.125 (20 - L) ζ = 0.5
Prestressed concrete ζ = 1.0 + 0.07 (20 - L) ζ = 1.0
Filler beam and reinforced
concrete ζ = 1.5 + 0.07 (20 - L) ζ = 1.5
Boundary Conditions
The boundary conditions were set to be the same in both the 2D- and 3D-model. Two different boundary conditions have been tested according to Section 3.8.1 where the influence of subsoil was analyzed. In the first case, the boundaries were set to be clamped to the bedrock. In the second case, the influence of the underlying soil was taken into consideration and impedance functions were modeled used springs and dash-pots. Support 1-3 has an underlying layer of 0.5 m soil while Support 4 has an underlying layer of 5.5 m soil, see Figure 3.1. The boundary conditions in this thesis was assumed to be fixed.
(a) Support 1 (b) Support 2
(d) Support 3 (e) Support 4
3.2.1 2D Bridge Model
A simplified model of the real case is a 2D-model. A 2D-model requires less CPU-time than a 3D-model but can never the less be used as a verification of the 3D-model. Differences between the actual case and the 2D model have arisen due to the inability to model certain details in Abaqus. Modified sections was created in order to capture the properties present in the 3D-model such as mass, stiffness and damping. Instead of modeling all details, increase of mass trough lumped mass modifications was used, see Figure 3.2. Lumped mass modifications contributes to a simplified model that requires less CPU-time but sufficiently captures the dynamic properties of the model. Beams were used as elements in the model. See Figure for 2D-model.
Ballast
The influence of ballast was accounted for through a non structural mass applied over the length of the bridge. The weight over the area in 3D was recalculated as weight over length in 2D. By using non structural mass as ballast, the 2D-model remains simplified and the structure does not face changes in stiffness.
Figure 3.3 Ballast as non structural mass
Support Conditions
The support conditions at the supports was set to movable in x-direction at the supports and fixed at the pillars according to Figure 3.4 below.
Figure 3.4 Support conditions 2D-model
(a) No movement (b) x-axis
Figure 3.5 Explanation of movable directions of supports
(a) Coupling constraint (b) Tie constraint
3.2.2 Modeling of Soil in 2D
The effect of backfill soil was accounted for in two different approaches. Figure 3.7 shows the foundation-soil system where the soil depth to bedrock was set to 3 m. The bedrock was assumed to have infinite stiffness. The first approach was to set clamped boundary conditions to the end-shields. This was done due to evidence provided by the 3D-model where clamped end-shields was tested. Figure 3.8a below illustrates the clamped end-shields. In the second approach, the end-shields were connected to impedance functions involving springs and dashpots. Frequency independent static springs and dashpots in the vertical direction were used as a simplification. Furthermore, in order to take into account the wing-walls which were not modeled in the 2D-model the impedance for these parts were calculated as well. Figure 3.8b illustrates the end-shields connected to the impedance functions.
(a) Plan view of foundation soil structure (b) Profile view of foundation soil structure
Figure 3.7 Soil structure system of 2D-model
(a) Fixed end-shield (b) Spring-dash pot
Spring stiffness coefficient
The static spring stiffness was calculated for three different parts since the end-shield and the wing walls have different geometrical properties as Figure 3.9 shows.
Figure 3.9 Spring coefficients for individual parts
The total stiffness was then calculated as a rough approximation by adding each part together according to
Kz =
X
Kz,i (3.1)
The static spring stiffnesses Kz,i for each individual part was determined as approximation by
Kz,i = Kz,sur 1 + 1 21 D B(1 + 1.3χ) 1 + B H 0.5 + BL (3.2)
where Kz,sur is the stiffness for an arbitrarily shaped foundation on the surface of homogeneous
halfspace and can was calculated as
Kz,sur=
2GL
1 − ν(0.73 + 1.54χ
0.75) (3.3)
in which G is the shear modulus and ν is poisson’s ratio. The parameters D, B, H and L are the embedded depth, half of the foundation width, the soil depth to bedrock from the foundation surface and half of the foundation length, respectively. The factor χ can be calculated as
χ = Ab
2L2 (3.4)
where Ab is the surface area of the foundation. The part containing the variable χ in equation
3.2 is a factor considering the embedded part of the foundation and the last part of the equation takes the underlaying bedrock into account.
Dashpot coefficient
The material dashpot coefficient Chysteretic was roughly estimated as
Chysteretic=
2Kz
ω β0 (3.5)
where β0 hyteretic damping ratio. The natural frequency ω was set according to the frequency of
According to (Gazetas, 1991a) operation frequencies below the first resonant frequency of the soil stratum of each mode of vibration the damping is zero or negligible.
3.2.3 Complete 2D-model
When adding the effect of ballast and the effect of backfill to the 2D-model, the complete 2D-model was created. This model is later on compared to the 3D-model.
3.2.4 3D Bridge Model
Following section presents how the 3D-model was modeled. The 3D model was set to the same as the 2D model regarding material properties, constraints, boundary- and support conditions. This was done in order to achieve consistency. Moreover, a detailed and a simplified 3D model was created. The simplified model was used in the verification with the 2D model. The detailed model was used to display the differences in dynamic response when adding more details in order to achieve a better representation of the true dynamic behavior of the bridge.
Detailed Bridge Model
The bridge located at Sidensjövägen is a slab bridge designed with end-frames and pillar supports to support the span of 49 m. The modeling in Abaqus is based on drawing. The total length of the bridge is 49 m with edge beams extending 1.85 m over of bridge length. The width of the bridge deck is 6.2 m and a total width with edge beams included is modeled to 7.1 m. The bridge deck has varying thickness around the pillar supports. The partitions seen in Figure 3.11 are made in order to apply different thickness. Shell elements were used to model the bridge.
X Y
Z
Ballast & sleepers
Ballast and sleepers were modeled according to drawings. Simplifications regarding the geometry was done in such way that the inclination of top surface height was simplified to an average height. Figure 3.11 below shows the modeled ballast and sleepers. The ballast is 4 m longer on each side of the bridge deck in order to capture the real case accurately. Solid elements were used to model the ballast and sleepers.
X Y
Z
Figure 3.12 Abaqus ballast & sleepers model
Support Conditions
A total of eight supports conditions were set in respect of movement in the principal axis. The bridge deck is allowed movement in x-direction at the supports. The support conditions for the pillar and bridge deck connection is modeled to be fixed conditions. See Figure 3.13 & Figure 3.14 below for explanation of how the support conditions were defined.
Figure 3.13 Support conditions for 3D-model
(a) No movement (b) x-axis
The constraints in the 3D-models was set to coupling and tie constraints in similarity to the 2D model. However, the connection between the elements differ. In the 3D models, the constraints was set to specific areas that represent the area of supports and pillars. Figure 3.15 illustrates the different constraints.
(a) Coupling constraint (b) Tie constraint
3.2.5 Modeling of Soil in 3D
The soil backfill was modeled behind the end shields according to Figure 3.16. The soil width of the backfill is the same as the width of the bridge deck, i.e 6.2 m, 3.5 m high(over the bedrock) and continuous 6 m behind the end shields.
C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C X Y Z
Figure 3.16 Applied dashpot illustration
The soil is resting on bedrock which was assumed to be rigid. Therefore, all three translation degrees of freedom was restrained at the bottom surface of the soil. As previously mentioned in Section 2.5.4 the soil is often modeled as finite continuum medium with transmitting boundaries at the border of the soil. The transmitting boundaries was modeled using the standard viscous boundary method as one normal and two tangential viscous dashpots at each node as shown in Figure 3.16.
Dn= ρCsAtrib (3.6)
Dt= ρCpAtrib (3.7)
The soil parts were tied to the end-shields and wing-walls assuming full contact between the soil and the bridge structure. The mesh size for the soil medium was chosen according to a mesh convergence with the following criterion in mind
le,max≤ 1 5 − 1 8 λmin, λmin = Cs fmax (3.8)
in which le,max is the maximum element size and fmax is the highest frequency of interest. The
equation is governed by the minimum shear wave length λmin of concern in order to accurately
3.2.6 Complete 3D-model
The parts were composed into one model which represents the case. Constraints were used to connect the surfaces on the different instances together. The complete model is presented in Figure 3.17. The used material properties is presented in table 3.2.
X Y
Z
3.3
Method Procedure
Following section describes the method procedure throughout this thesis. The aim of this section is to inform how the authors have come up to presented results and conclusions. The following flowchart, illustrated in Figure 3.18 below, is the outline of the method procedure.
Figure 3.18 Flow chart of method procedure
The 2D-model is compared to the simplified 3D-model. The authors strive to create a 3D-model as similar as the 2D-model as possible in order to verify the detailed 3D-model that is believed to capture the real life situation. The same quality assurance is carried out for both the 2D-model and the simplified 3D-2D-model. The quality assurance include mesh convergence analysis, comparison of static deformations and comparison of the first three vertical mode shapes with corresponding frequencies. The result of the quality assurance is evaluated and used in order to apply model refinements such as mass-, stiffness- and model modifications.
3.4
Quality Assurance
A vital part of the design work is to ensure the quality of model. A poorly constructed model will not represent the reality and give inaccurate results. A quality assurance of the model and modeling approach is therefore essential during the design work. The following subsections describes the quality assurance system used.
3.4.1 Convergence Analysis
A convergence analysis was performed to determine a appropriate mesh size and suitable element types for the components of the bridge.The analysis was done through studying the resulting
section forces for the self-weight. The first and third eigenvalues were used to study the
convergence of the different mesh sizes. Once convergence was reached refining the mesh more was not needed and the final mesh density and element type could be chosen with consideration to both CPU-time and the convergence analysis performed. For the 2D-model, three different densities of the mesh and element types were tested in different combinations according to Table 3.3 below to investigate which combination was most suitable for evaluating the bridge. The results of the mesh convergence analysis is presented in Table 3.4.
Table 3.3 Element types and mesh sizes tested for 2D-Model
Case 1 Element type Element size
Beams B21 0.75 0.5 0.25
Case 2 Element type Element size
Beams B22 0.75 0.5 0.25
Case 3 Element type Element size
Beams B23 0.75 0.5 0.25
Table 3.4 Assigned element types and mesh sizes in 2D-model
Shape Element type Mesh size
Bridge deck Wire B21 0.5
End-walls & head beams Wire B21 0.5
The 3D-model was also tested in three different cases of mesh size and element type. The combinations presented in Table 3.5 were all tested with both S4R and S8R elements since the bridge deck was sensitive to element selection. The results of the mesh convergence analysis is presented in Table 3.6 and further results can be found in Appendix B. The mesh size for the soil medium was chosen according to the criterion 3.8 in Section 3.2.5.
Table 3.5 Element types and mesh sizes tested for 3D-model
Case 1 Element type Element size
Beams B31 0.75 0.5 0.25
Deck S4R/S8R 0.75 0.5 0.25
Case 2 Element type Element size
Beams B32 0.75 0.5 0.25
Deck S4R/S8R 0.75 0.5 0.25
Case 3 Element type Element size
Beams B33 0.75 0.5 0.25
Deck S4R/S8R 0.75 0.5 0.25
Table 3.6 Assigned element types and mesh sizes in 3D-model
Shape Element type Mesh size
Bridge deck Shell S8R 0.5
End-walls & head beams Shell S8R 0.5
Abutments & supports Shell S8R 0.5
Pillars Wire B31 0.5
Ballast & sleepers Solid C3D10 0.4
