Dynamic analysis of simply supported slab bridges

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Dynamic analysis of simply supported slab bridges

ANDREAS ANDERSSON

Report No. 1922

Stockholm, 2019

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TRITA-ABE-RPT-1922, 2019 ISSN 1103-4289

ISRN KTH/BKN/RPT–1922–SE

KTH Struct. Eng. and Bridges SE-100 44 Stockholm Sweden © Andreas Andersson, October 2019

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Summary

This report present results from simulations of simply supported concrete slab bridges for railway traffic. The geometry follows the Swedish standard deck models according to design drawing B2447-2 and B2447-8, with span lengths ranging from 2-8 m. For each bridge four different configurations are studied; straight or skewed bridge deck and short or long edge beams. In addition, a case of higher mass due to increased ballast depth is studied. In total 78 different bridge configurations are included.

According to the numerical models the first natural frequency range from about 15-80 Hz depending on span length and configuration. In all simulations the first three modes of vibration are included. The limit criteria is a peak deck acceleration of 3.5 m/s2 when loaded by the HSLM-A train model. Including a speed safety

factor 1.2 according to EN 1991-2 results in an allowable speed that range from 175-350 km/h depending on the bridge configuration.

The allowable speed is somewhat higher for the skewed bridges compared to straight bridges. Increased mass results in lower acceleration but also lower resonance speed. An increase in ballast depth from 0.6 to 1.2 m generally results in lower allowable speed, except for the shortest bridges in the study.

It should be noted that the above conclusions are based only on simulations. Before upgrading these bridges to speeds higher than 200 km/h experimental validation is recommended. On the other hand, most existing real trains are likely to result in significantly lower dynamic response compared to the HSLM-A trains.

Keywords: Slab bridge; Railway bridge; high-speed train; dynamic analysis; deck

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Introduction

1.1 Background

When upgrading existing railway lines to higher speed, dynamic assessment of the railway bridges may be required. Eurocode EN 1990 CEN (2002) stipulates a set of dynamic design limits to be fulfilled and EN 1991-2 CEN (2003) present different train load models. Eurocode does not provide much guidance in setting up accurate models for dynamic assessment and the choice of e.g. boundary conditions and load distribution may influence the results significantly, especially for short span bridges. According to the Swedish bridge management system BaTMan, there is a total of 488 concrete slab bridges with a span between 2-8 m, see Figure 1.1. Many of these were built using standard solutions with the same drawings.

2 4 6 8 0 50 100 150 200 18500 1900 1950 2000 50 100 150 200

Figure 1.1: Span length and construction year for slab bridges in Sweden.

1.2 Aims and scope

This report comprise dynamic assessment of simply supported concrete slab bridges following the standard deck solution according to drawings 2 and B2447-8. The aim is to estimate the allowable speed using the HSLM-A load model in EN 1991-2 and the design limits in EN 1990/A2. The results are based only on simulations and no experimental validation has been performed.

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CHAPTER 1. INTRODUCTION

1.3 Requirements in Eurocode

A set of dynamic limit criteria is stated in EN 1990/A2, as indirect criteria for traffic safety and passenger comfort. In most cases the main governing limit is the vertical deck acceleration, set to 3.5 m/s2 for ballasted tracks. The motive is to

avoid ballast instability. According to EN 1990 A2.4.4.2(4), this limit should be checked for a frequency range limited to fmaxin Equation 1.1, where f1and f3 are

the first and the third natural frequency. For the short span bridges in this study

fmaxwill most often be limited to the first three modes of vibration.

EN 1990 also gives additional limits for vertical deck displacement, deck twist and end rotations, but these are deemed unlikely to be exceeded for the short-span bridges in the present study. Hence the main criteria in this study is the vertical deck acceleration.

fmax= max{30, 1.5f1, f3} (Hz) (1.1)

Following the flowchart in EN 1991-2 Figure 6.9 results in the need for full dynamic analysis, either due to note 1 that the bridge is not a simple structure e.g. due to skewness, that nT<1.2n0 or that the modes are not simple. EN 1991-2 (6.4.3(3))

states that the results from the dynamic analysis shall be compared with the static analysis. However, since the studied bridges are designed for a significantly larger static load it is assumed that this will not exceed the dynamic response.

In the dynamic analysis the train load model HSLM-A is used, consisting of 10 train sets with axle loads from 17-21 tonnes, coach length of 18-27 m and total length of about 400 m. The response from the dynamic analysis is multiplied with the dynamic load factor 1+0.5φ00 _{to account for dynamic effects from track}

irregularities.

Damping is included according to EN 1991-2 Table 6.6, also shown in Equation 1.2 below. In the present study this corresponds to a damping of 2.3-2.8%. Addi-tional damping according to EN 1991-2 Equation 6.13 is not included since it may overestimate the effect of train-bridge interaction.

ζ= 1.5 + 0.07(20 − L) (%) (1.2)

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Chapter 2

Models for slab bridges

2.1 Geometry

The standard deck solution exist for simply supported slab bridges with a span from 2-8 m. The shortest spans of 2.0-4.5 m follow drawing B2447-2 and spans of 5.0-8.0 m follow drawing B2447-8. The cross-section is illustrated in Figure 2.1. The edge beams may be either short (0.6 m) or long (1.4 m) to include a pathway for inspection an maintenance. In addition the bridge deck may be either straight or skewed, as illustrated in Figure 2.2 and Figure 2.3. The maximum skewness is 60° for the B2447-2 bridges and 45° for the B2447-8 bridges. The resulting geometry is based on Table 2.1 and Table 2.2. According to both drawings concrete quality K400 shall be used. 0.1 3.3 0.6 0.1 a a h_b 0.25 0.2 0.3 0.25 1.4 d h₁ d h₁+0.1

Figure 2.1: Cross-section of standard deck B2447.

Table 2.1: Geometry for slab B2447-2, a = 0.25 m and h1= 0.32 m in all cases.

L(m) 2.0 2.5 3.0 3.5 4.0 4.5 angle

d(m) 0.23 0.23 0.23 0.25 0.30 0.32 90°

d(m) 0.25 0.25 0.28 0.30 0.32 - 60°

Table 2.2: Geometry for slab B2447-8, d = 0.32 m and h1= 0.40 m in all cases. L(m) 5.0 5.5 6.0 6.5 7.0 7.5 8.0

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CHAPTER 2. MODELS FOR SLAB BRIDGES 0.30 0.30 L 0.30 0.30 L

Figure 2.2: Plane view of standard deck B2447-2, straight and skewed versions.

0.30 0.30 L 0.30 0.30 L

Figure 2.3: Plane view of standard deck B2447-8, straight and skewed versions.

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2.2. MODELS

2.2 Models

The slab bridges are analysed using finite elements with general purpose 9-noded shell elements following Mindlin-Reissner plate theory. The element size is approx-imately 0.5 m. A view of the model is illustrated in Figure 2.4.

In all analysis the concrete is assigned to the following properties: Young’s modulus

Ec= 30 GPa, density ρc = 2500 kg/m3and Poisson’s ratio ν = 0.2. The ballast is included as an evenly distributed mass with density ρs = 1800 kg/m3 and height

hb = 0.6 m. In the study of increased mass a ballast depth of 1.2 m is used. The analysis is performed using modal dynamics including the first three eigen-modes and a time increment of 2.5 ms. This time increment was deemed sufficient based on a convergence study of the shortest bridges. However, a smaller time step is generally recommended to accurately describe higher frequencies and to reduce period elongation.

All of the HSLM A1-A10 trains are included, using a speed range 100-400 km/h in increments of 2 km/h. The load from each train axle is evenly distributed in the transverse direction on a width of 2.5 m and follows a triangular distribution along the track with a length of 3.0 m. For the case of skewed bridges the load distribution follows the direction of the track. More detailed description of the load distribution can be found in ERRI (1999) and Andersson (2018). The output from the analysis consists of the peak acceleration in the bridge deck from each train passage. The study includes a total of 78 different bridge configurations.

x y z w1 L h1 w2 t2 t3 t1 w1

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Chapter 3

Results

3.1 Eigenvalue analysis

The eigenvalue-analysis is based on the models presented in Chapter 2. The first natural frequency is presented in Table 3.1 and Table 3.2. The natural frequency is slightly higher for the skewed bridges compared to the straight. In all analysis the first three eigenmodes are included. This is illustrated in Figure 3.1 for the case of L = 8 m. The first mode resembles 2D bending but the second (torsion) and third (transverse bending) are 3D modes. For shorter spans also the first mode has a 3D behaviour. The three eigenmodes are presented for all studied cased in Appendix A. The first natural frequency is compared to the limits for n0,minand n0,maxin EN 1991-2. Figure 3.2 show that the slab bridges in this study generally

are close to or above the upper limit in EN 1991-2.

Table 3.1: First natural frequency (Hz) for the deck, B2447-2.

2.0 2.5 3.0 3.5 4.0 4.5 L(m)

63.6 49.5 40.8 38.2 39.5 36.5 straight, short 78.1 61.7 55.7 49.8 45.1 - skew, short 38.7 36.2 33.3 32.9 35.7 35.5 straight, long 40.6 38.9 38.8 38.2 37.8 - skew, long

52.2 39.7 32.5 30.5 31.9 29.6 straight, short, high mass 70.6 50.2 45.1 40.3 36.6 - skew, short, high mass

Table 3.2: First natural frequency (Hz) for the deck, B2447-8.

5.0 5.5 6.0 6.5 7.0 7.5 8.0 L(m)

33.5 30.0 27.0 24.6 22.2 20.1 18.2 straight, short 34.9 31.2 27.9 25.4 22.8 20.7 18.7 skew, short 32.0 28.4 25.3 23.3 21.0 19.4 17.6 straight, long 33.3 29.3 25.9 23.8 21.4 19.7 17.8 skew, long

27.1 24.5 22.1 20.7 18.8 17.5 15.9 straight, long, high mass 28.1 25.2 22.7 21.2 19.2 17.8 16.2 skew, long, high mass

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CHAPTER 3. RESULTS -1 0 8 1 7 6 5 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 0 8 1 7 6 5 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 0 8 1 7 6 5 4 3 ₃ 2 ₂ 1 ₁ 0 0 Figure 3.1: The first three mode shapes for the case of L = 8.0 m.

2 3 4 5 6 7 8 10 15 20 30 40 50 60

Figure 3.2: First natural frequency of the studied bridges.

3.2 Response from passing trains

The dynamic response from passing trains are presented as an envelope of peak acceleration from train load HSLM A1-A10, including the dynamic factor 1+0.5φ00_.

The results for all studied bridge combinations are presented in Appendix B. The speed corresponding to the design limit amax = 3.5 m/s2 is presented in Table 3.3 and Table 3.4, including a speed safety margin of 1.2 according to EN 1991-2 (6.4.6.2). The results show an allowable speed that range from 175-350 km/h depending on the bridge configuration. The allowable speed is somewhat higher for the skewed bridges compared to straight bridges. Increased mass results in lower acceleration but also lower resonance speed. An increase in ballast depth from 0.6 to 1.2 m generally results in lower allowable speed, except for the shortest bridges in the study.

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3.2. RESPONSE FROM PASSING TRAINS

Table 3.3: Allowable train speed (km/h) for amax = 3.5 m/s2, deck B2447-2.

2.0 2.5 3.0 3.5 4.0 4.5 L (m)

190 175 213 228 262 260 straight, short

333 307 300 302 318 - skew, short

207 190 212 233 278 267 straight, long >350 305 297 297 305 - skew, long

230 222 202 215 253 250 straight, short, high mass

333 282 275 273 280 skew, short, high mass

Table 3.4: Allowable train speed (km/h) for amax = 3.5 m/s2, deck B2447-8.

5.0 5.5 6.0 6.5 7.0 7.5 8.0 L (m)

243 228 213 205 197 193 190 straight, short 270 243 223 213 207 203 197 skew, short 235 218 202 198 192 190 185 straight, long 260 232 212 208 197 197 190 skew, long

215 197 183 182 175 178 177 straight, long, high mass 232 207 200 195 190 185 182 skew, long, high mass

In Figure 3.3 the response using a time increment of 1/400 s and 1/1200 s is compared for the case of the 2 m straight slab. The estimated allowable speed change from 190 to 198 km/h. For longer bridges the difference is expected to be lower due to lower natural frequencies. A comparison is also done with a 2D model of similar total mass and natural frequency. The behaviour appears similar but with lower amplitude for the 2D model, possibly due to lower modal mass. This may result in an overestimation of the allowable speed.

100 200 300 400 0 1 2 3 4 5 6 100 200 300 400 0 1 2 3 4 5 6

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Chapter 4

Conclusions

4.1 Results from simulations

A total of 78 bridge configurations has been analysed, with spans from 2-8 m. According to the numerical models the first natural frequency range from about 15-80 Hz, which is often close to or above the frequency n0,max in EN 1991-2.

The results from passing trains show an allowable speed in the range of 175-350 km/h depending on bridge configuration. This includes a safety margin 1.2 on the speed. The allowable speed is somewhat higher for the skewed bridges com-pared to straight bridges. Increased mass results in lower acceleration but also lower resonance speed. An increase in ballast depth from 0.6 to 1.2 m generally results in lower allowable speed, except for the shortest bridges in the study.

4.2 Suggestions for further study

The above presented report serve as a screening to investigate dynamic effects in short-span slab bridges. The generic model is simplified and partly based on design parameters in Eurocode. The conditions for in-situ bridges may differ significantly, mainly regarding natural frequency and damping. Especially for shorter spans with high frequencies soil-structure interaction may be significant and can potentially have beneficial effects on the inherent damping.

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References

Andersson, A., 2018. Simplified approach to dynamic analysis of railway bridges for high-speed trains. Report 1837, KTH, Struct. Eng. and Bridges.

CEN, 2002. Eurocode EN 1990: Basis of structural Design. European Committee for Standardization.

CEN, 2003. Eurocode EN 1991-2: Actions of structures - part 2: Traffic loads on bridges. European Committee for Standardization.

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Appendix A

Mode shapes

In this appendix the first three mode shapes are presented for all studied cases of the slab bridges. The x-direction is transverse to the track and the y-direction along the track. For the B2447-2 bridges with 2.0-4.5 m span length all modes are dominated by plate bending. For the longer spans of the B2447-8 bridges the first mode gradually resembles the first bending mode of a simply supported beam. The total linear mass of the models are presented in Table A.1.

Table A.1: Linear mass (kg/m) of the bridge models.

short long high mass

L(m) straight skew straight skew straight skew

2.0 7 050 7 230 7 560 7 740 10 620 10 890 2.5 7 050 7 230 7 560 7 740 10 620 10 800 3.0 7 050 7 500 7 560 8 010 10 620 11 070 3.5 7 230 7 680 7 740 8 190 10 800 11 250 4.0 7 680 7 860 8 190 8 370 11 250 11 430 4.5 7 860 - 8 370 - 11 430 -5.0 7 980 7 980 8 340 8 340 11 980 11 980 5.5 7 980 7 980 8 340 8 340 11 980 11 980 6.0 7 980 7 980 8 340 8 340 11 980 11 980 6.5 8 120 8 120 8 480 8 480 12 140 12 140 7.0 8 120 8 120 8 480 8 480 12 140 12 140 7.5 8 260 8 260 8 630 8 630 12 300 12 300 8.0 8 260 8 260 8 630 8 630 12 300 12 300

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CHAPTER A. MODE SHAPES

A.1 B2447-2

-1 3 0 2 1 2 1 ₁ 0 0 -1 3 0 2 1 2 1 ₁ 0 0 -1 3 0 2 1 2 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 4 1 3 ₃ 2 ₂ 1_{0 0} 1 -10 4 1 3 ₃ 2 ₂ 1_{0 0} 1 -10 4 1 3 ₃ 2 ₂ 1_{0 0} 1 -10 4 1 3 ₃ 2 ₂ 1_{0 0} 1 -10 4 1 3 ₃ 2 ₂ 1_{0 0} 1 -10 4 1 3 ₃ 2 ₂ 1_{0 0} 1 -1 0 5 1 4₃ 3 2₁ ₂ 1 0 0 -1 0 5 1 4₃ 3 2₁ ₂ 1 0 0 -1 0 5 1 4₃ 3 2₁ ₂ 1 0 0

Figure A.1: Mode shapes for B2447-2 slabs with length 2.0-4.5 m, short edge beams, straight deck.

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A.1. B2447-2 -1 4 0 1 3 3 2 ₂ 1 ₁ 0 0 -1 4 0 1 3 ₃ 2 ₂ 1 ₁ 0 0 -1 4 0 1 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 _{0 0} 1 -1 5 0 1 4 3 ₃ 2 ₂ 1 _{0 0} 1 -1 6 0 1 5 4 3₂ ₃ 2 1_{0 0} 1 -1 6 0 1 5 4 3 ₃ 2₁ ₂ 1 0 0 -1 6 0 1 5 4 3 ₃ 2₁ ₂ 1 0 0 -1 6 0 1 5 4 3 ₃ 2₁ ₂ 1 0 0 -1 6 0 1 5 4 3 ₃ 2 ₂ 1_{0 0} 1 -1 6 0 1 5 4 3 ₃ 2 ₂ 1_{0 0} 1

Figure A.2: Mode shapes for B2447-2 slabs with length 2.0-4.0 m, short edge beams, skewed deck.

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CHAPTER A. MODE SHAPES -1 3 0 2 1 2 1 ₁ 0 0 -1 3 0 2 1 2 1 ₁ 0 0 -1 3 0 2 1 2 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 4 1 3 ₃ 2 ₂ 1_{0 0} 1 -10 4 1 3 ₃ 2 ₂ 1 _{0 0} 1 -10 4 1 3 ₃ 2 ₂ 1_{0 0} 1 -10 4 1 3 ₃ 2 ₂ 1 ₁ 0 0 -10 4 1 3 ₃ 2 ₂ 1 ₁ 0 0 -10 4 1 3 ₃ 2 ₂ 1 ₁ 0 0 -1 0 5 1 4₃ 3 2₁ ₂ 1 0 0 -1 0 5 1 4₃ 3 2₁ ₂ 1 0 0 -1 0 5 1 4₃ 3 2₁ ₂ 1 0 0

Figure A.3: Mode shapes for B2447-2 slabs with length 2.0-4.5 m, long edge beams, straight deck.

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A.1. B2447-2 -1 4 0 1 3 3 2 ₂ 1 ₁ 0 0 -1 4 0 1 3 ₃ 2 ₂ 1 ₁ 0 0 -1 4 0 1 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 _{0 0} 1 -1 5 0 1 4 3 ₃ 2 ₂ 1_{0 0} 1 -1 6 0 1 5 4 3₂ ₃ 2 1_{0 0} 1 -1 6 0 1 5 4 3 ₃ 2₁ ₂ 1 0 0 -1 6 0 1 5 4 3 ₃ 2₁ ₂ 1 0 0 -1 6 0 1 5 4 3 ₃ 2₁ ₂ 1 0 0 -1 6 0 1 5 4 3 ₃ 2 ₂ 1_{0 0} 1 -1 6 0 1 5 4 3 ₃ 2 ₂ 1_{0 0} 1

Figure A.4: Mode shapes for B2447-2 slabs with length 2.0-4.0 m, long edge beams, skewed deck.

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CHAPTER A. MODE SHAPES -1 3 0 2 1 2 1 ₁ 0 0 -1 3 0 2 1 2 1 ₁ 0 0 -1 3 0 2 1 2 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 3 1 3 2 ₂ 1 ₁ 0 0 -10 4 1 3 ₃ 2 ₂ 1 _{0 0} 1 -10 4 1 3 ₃ 2 ₂ 1 _{0 0} 1 -10 4 1 3 ₃ 2 ₂ 1 _{0 0} 1 -10 4 1 3 ₃ 2 ₂ 1 ₁ 0 0 -10 4 1 3 ₃ 2 ₂ 1 ₁ 0 0 -10 4 1 3 ₃ 2 ₂ 1 ₁ 0 0 -1 0 5 1 4₃ 3 2₁ ₂ 1 0 0 -1 0 5 1 4₃ 3 2₁ ₂ 1 0 0 -1 0 5 1 4₃ 3 2₁ ₂ 1 0 0

Figure A.5: Mode shapes for B2447-2 slabs with length 2.0-4.5 m, short edge beams, straight deck, high mass.

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A.1. B2447-2 -1 4 0 1 3 3 2 ₂ 1 ₁ 0 0 -1 4 0 1 3 ₃ 2 ₂ 1 ₁ 0 0 -1 4 0 1 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1 ₁ 0 0 -1 5 0 1 4 3 ₃ 2 ₂ 1_{0 0} 1 -1 5 0 1 4 3 ₃ 2 ₂ 1 _{0 0} 1 -1 6 0 1 5 4 3₂ ₃ 2 1_{0 0} 1 -1 6 0 1 5 4 3 ₃ 2₁ ₂ 1 0 0 -1 6 0 1 5 4 3 ₃ 2₁ ₂ 1 0 0 -1 6 0 1 5 4 3 ₃ 2₁ ₂ 1 0 0 -1 6 0 1 5 4 3 ₃ 2 ₂ 1_{0 0} 1 -1 6 0 1 5 4 3 ₃ 2 ₂ 1_{0 0} 1

Figure A.6: Mode shapes for B2447-2 slabs with length 2.0-4.0 m, short edge beams, skewed deck, high mass.

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CHAPTER A. MODE SHAPES

A.2 B2447-8

-10 1 4 3 2 ₁₂ 0 0 -10 1 4 3 2 ₁₂ 0 0 -10 1 4 3 2 ₁₂ 0 0 -10 6 1 4 3 2 ₁₂ 0 0 -10 6 1 4 3 2 ₁₂ 0 0 -10 6 1 4 3 2 ₁₂ 0 0 -10 6 1 4 3 2 ₁₂ 0 0 -10 6 1 4 3 2 ₁₂ 0 0 -10 6 1 4 3 2 ₁₂ 0 0 -10 1 6 4₂ ₃ 2 1 0 0 -10 1 6 4₂ ₃ 2 1 0 0 -10 1 6 4₂ ₃ 2 1 0 0 -10 1 6 4₂ ₃ 2 1 0 0 -10 1 6 4₂ ₃ 2 1 0 0 -10 1 6 4₂ ₃ 2 1 0 0 -10 8 1 6₄ 3 2_{0 0}₁₂ -10 8 1 6₄ 3 2_{0 0}₁₂ -10 8 1 6₄ 3 2_{0 0}₁₂ -10 8 1 6₄ 3 20 012 -10 8 1 6₄ 3 20 012 -10 8 1 6₄ 3 20 012

Figure A.7: Mode shapes for B2447-8 slabs with length 5.0-8.0 m, short edge beams, straight deck.

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A.2. B2447-8 -10 1 6 4 ₃ 2_{0 0}₁2 -10 1 6 4 ₃ 2_{0 0}₁2 -10 1 6 4 ₃ 2_{0 0}₁2 -10 8 1 6₄ 3 2 ₁₂ 0 0 -10 8 1 6₄ 3 2 ₁₂ 0 0 -10 8 1 6₄ 3 2 ₁₂ 0 0 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012

Figure A.8: Mode shapes for B2447-8 slabs with length 5.0-8.0 m, short edge beams, skewed deck.

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CHAPTER A. MODE SHAPES -10 1 4 3 2 ₁2 0 0 -10 1 4 3 2 ₁2 0 0 -10 1 4 3 2 ₁2 0 0 -10 6 1 4 3 2 ₁2 0 0 -10 6 1 4 3 2 ₁2 0 0 -10 6 1 4 3 2 ₁2 0 0 -10 6 1 4 3 2 ₁2 0 0 -10 6 1 4 3 2 ₁2 0 0 -10 6 1 4 3 2 ₁2 0 0 -10 1 6 4 3 2_{0 0}₁2 -10 1 6 4 3 2_{0 0}₁2 -10 1 6 4 3 2_{0 0}₁2 -10 1 6 4 ₃ 2_{0 0}₁2 -10 1 6 4 ₃ 2_{0 0}₁2 -10 1 6 4 ₃ 2_{0 0}₁2 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12

Figure A.9: Mode shapes for B2447-8 slabs with length 5.0-8.0 m, long edge beams, straight deck.

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A.2. B2447-8 -10 1 6 4 ₃ 2_{0 0}₁2 -10 1 6 4 ₃ 2_{0 0}₁2 -10 1 6 4 ₃ 2_{0 0}₁2 -10 8 1 6₄ 3 2 ₁₂ 0 0 -10 8 1 6₄ 3 2 ₁₂ 0 0 -10 8 1 6₄ 3 2 ₁₂ 0 0 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012

Figure A.10: Mode shapes for B2447-8 slabs with length 5.0-8.0 m, long edge beams, skewed deck.

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CHAPTER A. MODE SHAPES -10 1 4 3 2 ₁2 0 0 -10 1 4 3 2 ₁2 0 0 -10 1 4 3 2 ₁2 0 0 -10 6 1 4 3 2 ₁2 0 0 -10 6 1 4 3 2 ₁2 0 0 -10 6 1 4 3 2 ₁2 0 0 -10 6 1 4 3 2 ₁2 0 0 -10 6 1 4 3 2 ₁2 0 0 -10 6 1 4 3 2 ₁2 0 0 -10 1 6 4 3 2_{0 0}₁2 -10 1 6 4 3 2_{0 0}₁2 -10 1 6 4 3 2_{0 0}₁2 -10 1 6 4 ₃ 2_{0 0}₁2 -10 1 6 4 ₃ 2_{0 0}₁2 -10 1 6 4 ₃ 2_{0 0}₁2 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12

Figure A.11: Mode shapes for B2447-8 slabs with length 5.0-8.0 m, long edge beams, straight deck, high mass.

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A.2. B2447-8 -10 1 6 4 ₃ 2_{0 0}₁2 -10 1 6 4 ₃ 2_{0 0}₁2 -10 1 6 4 ₃ 2_{0 0}₁2 -10 8 1 6₄ 3 2 ₁₂ 0 0 -10 8 1 6₄ 3 2 ₁₂ 0 0 -10 8 1 6₄ 3 2 ₁₂ 0 0 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 8 1 6₄ 3 2_{0 0}12 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 1 8₆ 4₂ ₃ 2 1 0 0 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012 -10 10 1 8 6₄ 3 20 012

Figure A.12: Mode shapes for B2447-8 slabs with length 5.0-8.0 m, long edge beams, skewed deck, high mass.

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Appendix B

Response from passing trains

In this appendix the envelope of peak deck acceleration during train passages is presented. Each envelope is the result from the HSLM A1-A10 trains, including a dynamic amplification factor 1+0.5φ00_.

B.1 B2447-2

100 200 300 400 0 1 2 3 4 5 6 100 200 300 400 0 1 2 3 4 5 6

Figure B.1: Deck acceleration for B2447-2 slabs with length 2.0-4.5 m, short edge beams, straight deck.

100 200 300 400 0 1 2 3 4 5 6 100 200 300 400 0 1 2 3 4 5 6

Figure B.2: Deck acceleration for B2447-2 slabs with length 2.0-4.0 m, short edge beams, skewed deck.

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CHAPTER B. RESPONSE FROM PASSING TRAINS 100 200 300 400 0 1 2 3 4 5 6 100 200 300 400 0 1 2 3 4 5 6

Figure B.3: Deck acceleration for B2447-2 slabs with length 2.0-4.5 m, long edge beams, straight deck.

100 200 300 400 0 1 2 3 4 5 6 100 200 300 400 0 1 2 3 4 5 6

Figure B.4: Deck acceleration for B2447-2 slabs with length 2.0-4.0 m, long edge beams, skewed deck.

100 200 300 400 0 1 2 3 4 5 6 100 200 300 400 0 1 2 3 4 5 6

Figure B.5: Deck acceleration for B2447-2 slabs with length 2.0-4.5 m, short edge beams, straight deck, high mass.

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B.1. B2447-2 100 200 300 400 0 1 2 3 4 5 6 100 200 300 400 0 1 2 3 4 5 6

Figure B.6: Deck acceleration for B2447-2 slabs with length 2.0-4.0 m, short edge beams, skewed deck, high mass.

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CHAPTER B. RESPONSE FROM PASSING TRAINS

B.2 B2447-8

100 200 300 400 0 1 2 3 4 5 6 100 200 300 400 0 1 2 3 4 5 6

Figure B.7: Deck acceleration for B2447-8 slabs with length 5.0-8.0 m, short edge beams, straight deck.

100 200 300 400 0 1 2 3 4 5 6 100 200 300 400 0 1 2 3 4 5 6

Figure B.8: Deck acceleration for B2447-8 slabs with length 5.0-8.0 m, short edge beams, skewed deck.

100 200 300 400 0 1 2 3 4 5 6 100 200 300 400 0 1 2 3 4 5 6

Figure B.9: Deck acceleration for B2447-8 slabs with length 5.0-8.0 m, long edge beams, straight deck.

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B.2. B2447-8 100 200 300 400 0 1 2 3 4 5 6 100 200 300 400 0 1 2 3 4 5 6

Figure B.10: Deck acceleration for B2447-8 slabs with length 5.0-8.0 m, long edge beams, skewed deck.

100 200 300 400 0 1 2 3 4 5 6 100 200 300 400 0 1 2 3 4 5 6

Figure B.11: Deck acceleration for B2447-8 slabs with length 5.0-8.0 m, long edge beams, straight deck, high mass.

100 200 300 400 0 1 2 3 4 5 6 100 200 300 400 0 1 2 3 4 5 6

Figure B.12: Deck acceleration for B2447-8 slabs with length 5.0-8.0 m, long edge beams, skewed deck, high mass.

Dynamic analysis of simply supported slab bridges