• No results found

Dynamic behaviour of soil-steel composite bridges for high-speed railways

N/A
N/A
Protected

Academic year: 2021

Share "Dynamic behaviour of soil-steel composite bridges for high-speed railways"

Copied!
56
0
0

Loading.... (view fulltext now)

Full text

(1)

Dynamic behaviour of soil-steel composite bridges

for high-speed railways

(2)

TRITA-ABE-RPT-208, 2020 ISSN 1103-4289

ISRN KTH/BKN/RPT–208–SE

KTH Struct. Eng. and Bridges SE-100 44 Stockholm Sweden © Andreas Andersson, June 2020

(3)

Summary

This report deals with the dynamic behavior of corrugated steel culverts under railways. The main objective is to investigate the feasibility of these structures for high-speed trains. A combination of experimental testing and numerical simulations is presented.

Experimental testing is reported for three bridges, all built for conventional rail-ways. The main novelty of the work is a method for forced vibration tests. This is performed using a 50 kN load actuator from inside the culverts. The forced vibration tests enables experimental estimates of the frequency response functions at all sensor locations, both in the culvert and the adjacent track area. For the tested bridges, the first resonance frequency is in the range of 15-20 Hz and the damping is estimated to 4-8%. Tests with increased load amplitude generally show a decrease in natural frequency but increase in damping, possibly due to nonlinear effects of the soil-structure system.

Numerical models have been developed, mainly focusing on the largest of the stud-ied bridges. The parameters of the numerical model is fitted using a model updat-ing algorithm together with the experimental data. A full 3D model of the bridge is rather computationally demanding, especially when simulating passing trains. Therefore, a simplified 1D track model with modified parameters has been devel-oped. Relatively good agreement is found between the 1D and 3D model. Both models show that the studied bridge is likely to fulfill the dynamic requirements for high-speed trains, according to EN 1990.

The 3D model also shows relatively good agreement with experimental data of real train passages. At the centre crown point, the model somewhat underestimate the response.

Keywords: Railway bridge; high-speed train; dynamic analysis; deck acceleration; soil-structure interaction.

(4)
(5)

Preface

The work presented in this report has been funded by ViaCon Sp z o.o and carried out at KTH division of Structural Engineering and Bridges. The experimental work has been planned and performed by the author together with the technicians Mr. Stefan Trillkott and Mr. Claes Kullberg. The experimental work has also been assisted by PhD-student Johan Lind Östlund and MSc. Diego Fernandez Barrero. The numerical simulations have been performed by the author but also partly developed within two master projects by MSc. Diego Fernandez Barrero and MSc. Jonathan Ljung, all at KTH division of Structural Engineering and Bridges. Stockholm, June 2020

(6)
(7)

Contents

1 Introduction 1

1.1 Background . . . 1

1.2 Aims and scope . . . 4

1.3 Limitations . . . 5

1.4 Outline of the report . . . 5

2 Dynamic analysis of railway bridges 7 2.1 Requirements in EN 1990 and EN 1991-2 . . . 7

2.2 Governing parameters . . . 8

2.3 Some aspects of soil dynamics . . . 9

2.4 Literature review on soil-steel composite bridges . . . 12

3 Experimental results 15 3.1 Experimental equipment . . . 15

3.2 The Märsta bridge . . . 16

3.3 The Hårestorp bridge . . . 23

3.4 The Örsjö bridge . . . 29

3.5 Summary of experimental results . . . 33

4 Simulation results 35 4.1 3D-model of the Hårestorp bridge . . . 35

4.2 Analysis and model updating . . . 36

4.3 Track model . . . 37

4.4 Response from passing trains . . . 40

5 Conclusions 43 5.1 Experimental results . . . 43

5.2 Simulation results . . . 44

5.3 Suggestion for further work . . . 45

(8)
(9)

Chapter 1

Introduction

1.1

Background

Soil-steel composite bridges (SSCB) consists of a corrugated steel culvert supported by engineering soil. The culvert often consists curved corrugated plates, bolted to-gether to form either circular, elliptical or arch shaped geometries. SSCB are some-times considered a cost-efficient alternative to conventional bridges with application for both road, railway and fauna passages.

A typical cross-section of an SSCB for railway application is illustrated in Figure 1.1. These bridges are commonly built for conventional railways but its performance due to dynamic loading from high-speed trains is not well known.

L1

h hc

D

Figure 1.1: Typical cross-section of a soil-steel composite bridge.

Soil-steel composite bridges in Sweden

According to the bridge management system BaTMan, there are 75 soil-steel com-posite bridges for railway traffic in Sweden and at least another three in planning. The oldest bridge is from 1961 and the rate of construction has been relatively con-stant, Figure 1.3. Most bridges have a closed elliptic culvert profile and the average diameter D = 3.3 m. The height of cover hcis usually not less than 1.1 m. The data from the Swedish bridges is presented in Figure 1.4 and the location is shown in Figure 1.2. The corrugation profile is schematically illustrated in Figure 1.5 having a wave length c, height hcorr and plate thickness t. More detailed description of different culvert profiles and corrugation sections can be found in Pettersson (2007) and Pettersson and Sundquist (2014).

(10)

CHAPTER 1. INTRODUCTION 0 100 200 300 km Laxå Mora Ånge Eslöv Ystad Piteå Kalix Luleå Falun Mjölby Hoting Um eå Malung Värnamo Älm hult Alvesta Malm ö Älvsbyn Ludvika Gävle Bollnäs Borås Varberg Kalm ar Storum an Storlien Vetlanda Nässjö Em maboda Örebro Lycksele Boden Kiruna Mellerud Skövde Olofström Ängelholm Hultsfred Nynäsham n Västervik Uppsala Haparanda Gällivare Härnösand Ström stad Oskarshamn Arvidsjaur Hässleholm Sim risham n Trelleborg Nyköping Västerås Skellefteå Bastuträsk Hallstavik Fagersta Borlänge Herrljunga Svappavaara Riksgränsen Söderham n Sundsvall Uddevalla Halmstad Göteborg Karlstad Karlskrona Jönköping Linköping Eskilstuna Stockholm Östersund Vänersborg Falköping Helsingborg Norrköping Avesta Krylbo Katrineholm Hallsberg Charlottenberg Örnsköldsvik

Figure 1.2: Location of SSCB in Sweden. 2

(11)

1.1. BACKGROUND 19600 1970 1980 1990 2000 2010 2020 5 10 15 20 25

Figure 1.3: Construction year for SSCB in Sweden.

0 1 2 3 4 5 0 1 2 3 4 5 0 10 20 30 0 1 2 3 4 5

Figure 1.4: Geometrical data for SSCB in Sweden.

c

t

h

corr Figure 1.5: Basic profile of the corrugated plate.

(12)

CHAPTER 1. INTRODUCTION

Soil-steel composite bridges in Poland

SSCB are also rather common in Poland. Figure 1.6 shows an increased construc-tion rate from the mid 1990ies until today. The data is based on 101 SSCB under railways. Figure 1.7 presents geometrical data and show several bridges with rela-tively long spans but lower height to span ratio compared to the Swedish bridges. According to the data base, many of the bridges has a top length L1 longer than 30 m, possibly due to several tracks. Data for hc is not included in the database.

19950 2000 2005 2010 2015 2020 5 10 15 20 25

Figure 1.6: Construction year for SSCB in Poland.

0 5 10 15 0 2 4 6 8 0 20 40 60 0 2 4 6 8

Figure 1.7: Geometrical data for SSCB in Poland.

1.2

Aims and scope

The aim of this report is to study the dynamic performance of SSCB under railways, especially for high-speed applications. This is performed by a combination of

(13)

1.3. LIMITATIONS imental testing and numerical simulations. None of the tested bridges are built for high-speed application and the highest allowable speed is 180 km/h. Instead, the dynamic performance of the bridges is mainly estimated based on full-scale forced vibration tests. This experimental data is then used when updating numerical models that are used for predicting the response of future high-speed trains.

1.3

Limitations

The study is limited to short and medium span SSCB with elliptic cross-sections. Experimental testing has been performed on three bridges, the largest have a span of 4.9 m. The numerical simulations are mainly focusing on replicating the response from the largest bridge.

1.4

Outline of the report

Chapter 2 gives a short background to the requirements for dynamic analysis ac-cording to Eurocode EN 1990 and EN 1991-2. Further, the main governing pa-rameters in dynamic analysis is described, both is structural dynamics and soil dynamics. A brief literature review is given, limited to previous studies on the dynamic performance of SSCB for railway application.

Chapter 3 presented the experimental testing on three bridges. The instrumentation and test procedure is described and the analysis of the experimental results is presented.

Chapter 4 describes the numerical models, focusing on the largest bridge in the study. The parameters of a 3D solid model is fitted to the experimental results by using a model updating algorithm. A 1D track model is also presented, show-ing relatively good agreement with the 3D model but much more computationally efficient.

(14)
(15)

Chapter 2

Dynamic analysis of railway

bridges

2.1

Requirements in EN 1990 and EN 1991-2

When following the Eurocode, limit criteria for dynamic analyses are found in EN 1990/A1 CEN (2002). The main aim is to avoid excessive vibrations that may result in changes of the track geometry, reduction in wheel-rail contact forces or passenger discomfort. This is controlled by limits for deck acceleration, deck displacements, bridge end rotations and uplift, deck twist and lateral vibrations. For short and medium span bridges the limit for vertical deck acceleration is usually the governing parameter, for ballasted tracks limited to γbt= 3.5 m/s2. This limit origins from the first high-speed line between Paris to Lyons in the early 1980ies, where track misalignment was found on certain bridges due to excessive vibrations. Experimental tests first carried out by SNCF and later confirmed by BAM within ERRI D214 ERRI (1999) has since been included in EN 1990. The experimental tests by BAM, performed up to 20 Hz, showed an increased risk of ballast instability at about 0.8g and by simply using a safety factor 2 ended up as the current limit of 3.5 m/s2. It should also be noted that this refers to the acceleration of the bridge deck and not the track itself. The frequency limit in EN1990 is currently max(30; 1.5f1; f3) Hz.

Figure 6.9 in EN 1991-2 presents a flowchart to determine when a dynamic anal-ysis is required. For train speeds over 200 km/h and non-simple structures a full dynamic analysis is required. The analysis is for those cases performed with train load model HSLM-A, consisting of 10 trains with axle loads ranging from 17-21 ton/axle and total length of about 400 m. This load model is intended to cover a large range of real trains according to EN 1991-2 Annex E. In the dynamic analysis speed up to 1.2vmaxshould be checked, where vmaxis the maximum train commis-sioning speed at location of the bridge. EN 1991-2 also gives values of damping to be used in analysis, but are intended for beam like structures of steel or concrete. For structures embedded in the embankment, e.g. portal frame bridges or soil-steel composite bridges, the damping may be significantly higher. Additional damping denoted ∆ζ is given to account for beneficial effects from train-bridge interaction. This factor may however be non-conservative and should therefore not be used. The results from the dynamic analysis should be multiplied with a factor 0.5ϕ00to account for dynamic effects due to track defects and vehicle imperfections.

(16)

CHAPTER 2. DYNAMIC ANALYSIS OF RAILWAY BRIDGES

It should be noted that the current regulations for dynamic analysis are mainly in-tended for resonance phenomena of beam like structures. For SSCB the response is likely more governed by the geotechnical conditions and the dynamic characteristics of the track.

2.2

Governing parameters

The main governing parameters in dynamic analysis is the mass m, flexural rigidity EI and damping ζ. The peak deck acceleration is independent of EI but inversely proportional to m and ζ, as illustrated in Figure 2.1. The resonance speed vi is a linear function of the natural frequency n0 and the wavelength λ according to Equation 2.1 where d is any axle spacing of the train. For a simply supported beam with constant linear mass and flexural rigidity the first natural frequency is given by Equation 2.2, the proportionality generally holds for other boundary conditions. In the present example with 1.0m and 1.0EI results in n0 = 3.1 Hz which for HSLM-A1 with D = 18 m gives a resonance speed of 200 km/h.

150 200 250 0 1 2 3 4 5 150 200 250 0 1 2 3 4 5 150 200 250 0 1 2 3 4 5

Figure 2.1: Dynamic response of a simply supported beam, train load HSLM-A1.

vi= n0λi, λi = d/i i = 1, 2, 3, ... (2.1) n0= π 2L2 r EI m (2.2)

The upper and lower limits for n0is given by EN 1991-2 and presented in Figure 2.2. The dynamic factor ϕ00depends on n

0and the determinant length LΦ, which may result in a factor ranging from 1.25 to 1.40 for short spans. Again it should be

(17)

2.3. SOME ASPECTS OF SOIL DYNAMICS noted that this is intended for beam-like structures and its application to soil-steel composite bridges is not obvious.

4 6 8 10 15 20 5 10 15 20 30 5 10 15 20 1 1.1 1.2 1.3 1.4 1.5

Figure 2.2: Natural frequency n0 and dynamic amplification factor ϕ00 as function of the determinant length LΦ.

2.3

Some aspects of soil dynamics

Soil dynamics with railway applications is a vast research field and is not covered in detail in this report. Instead, the main governing parameters are briefly presented that will be used in further analysis. For SSCB it is assumed that the steel culvert is surrounded by well compacted granular material of well defined quality. It is further assumed that the passing trains induce small additional strains in the soil so that an ideal linear elastic condition can be assumed.

Wave propagation

There are three main types of wave propagation; dilatational (P-waves), shear (S-waves) and surface waves. The P-wave moves parallel to the wave direction and has the fastest wave speed. The S-wave moves perpendicular to the wave direction and is slower. The most common surface wave is the Rayleigh wave, that is caused by interaction of S- and P-waves at the surface and is often compared to ripples on water. The wave speeds are given by Equation 2.3.

vs= s G ρ, vp= vs r 2 − 2ν 1 − 2ν, vr≈ vs0.862 + 1.14ν 1 + ν (2.3)

One of the most well-known cases of high-speed train induced vibrations on em-bankments is the case of Ledsgård, Sweden. The site is part of the West-coast line in Sweden with a design speed of 200 km/h. Shortly after opening in 1997,

(18)

CHAPTER 2. DYNAMIC ANALYSIS OF RAILWAY BRIDGES

large vibrations of the embankment was observed and the allowable train speed was reduced to 130 km/h. Due to layers of very soft cohesive soil, cross-hole tests showed a shear wave speed as low as 40 m/s, near critical speed related to the first Rayleigh surface wave was reached during train passages. Extensive experimental work was conducted and by using a track loading vehicle the first resonance was found at 2-4 Hz. The embankment was improved by lime-cement columns in 2000 and the allowable train speed was later increased to 180 km/h, Smekal and Berggren (2002). Experimental results of vertical track displacement during train passage is presented in Madshus and Kaynia (2000). At 70 km/h the response was quasi-static with a downward displacement of about 5 mm. Corresponding values for stiffer em-bankments are typically 1-2 mm. At 200 km/h the downward displacement was more than 10 mm and the upward displacement more than 5 mm. A numerical model based on a layered viscoelastic half-space was developed in Kaynia et al. (2000). Simulations of passing trains was solved in frequency domain. The simu-lation results were generally in good agreement with the experimental results and was also used to demonstrate the effectiveness of strengthening the embankment for mitigating ground vibrations.

Foundation vibrations

Dynamic loading of structures on a flexible foundation may experience unwanted dynamics effects. The fundamental frequency f of a homogenous stratum of depth H can be described by Equation 2.4. The equation of motion is given by Equa-tion 2.5 where the impedance Z(ω) = F (ω)/x(ω). For load frequencies lower than the fundamental frequency of the layer the dynamic stiffness is characterised by Re(Z) = K − ω2M starting at the static stiffness for ω = 0 and descending to 0 at the fundamental frequency. The damping C(ω) = Im(Z)/ω = C is mainly governed by material damping and is relatively low.

fp =

vp

4H (2.4)

F(ω) = x(ω)(−ω2+ iωC + K) (2.5)

Beyond the fundamental frequency the response is instead governed by wave propa-gation that even for fundamental cases shows a relatively complex behaviour, often with a significant increase in damping and large variation in dynamic stiffness. Con-sider a rigid circular slab on a homogenous shallow stratum according to Figure 2.3. A numerical analysis is performed using a 2D axi-symmetric model with H = 2 m, R= 1 m, E = 400 MPa, ρ = 2000 kg/m3, ζ = 5%. A dimensionless frequency a

0 is calculated according to Equation 2.6. Normalized results are presented in

(19)

2.3. SOME ASPECTS OF SOIL DYNAMICS ure 2.4 for different values of ν. The fundamental frequency range from 60-70 Hz which corresponds to a0= 1.56. a0= ωR vs (2.6) z E, ν, ρ, ζ R H F(ω)

Figure 2.3: A circular rigid slab on a homogenous shallow stratum.

0 1 2 3 4 5 6 -1 0 1 2 0 1 2 3 4 5 6 0 1 2 3 4 5

Figure 2.4: Vertical impedance for a circular rigid slab on a homogenous shallow stratum.

Soil modulus for friction material

The accuracy in estimating the dynamic response in soil- and soil-structure inter-action problems highly depends on how well the material properties can be esti-mated. The properties of the soil highly depends on the in-situ state of stress and strain. Site specific data is often sparsely available and estimates are then based on geotechnical surveys or data from similar sites.

The maximum shear modulus of frictional soil can be described according to Equa-tion 2.7. K is a funcEqua-tion of the void ratio, compacEqua-tion rate, over-consolidaEqua-tion

(20)

CHAPTER 2. DYNAMIC ANALYSIS OF RAILWAY BRIDGES

ratio and plasticity index. According to TR Geo 13, K may range from 15 000 for sand to 30 000 for crushed stones. The mean effective stress σ0

m is given by Equation 2.8. The shear modulus decrease with increased shear strain as suggested by Equation 2.9. The damping D increase as suggested by Equation 2.10. The re-lation between shear modulus G and Young’s modulus E is given by Equation 2.11 for reference. Despite the stress- and strain dependent properties of the soil, it is sometimes possible to find an equivalent modulus that fit numerical or experimental data sufficiently well.

Gmax= Kpσ0m (2.7) σ0m= σx0 + σ0y+ σ0z /3 (2.8) G Gmax =1 + 1600γ (1 + 101 −2000γ), 0 6 γ 6 10−2 (2.9) D= 0.8 + 181 + 0.15 (100γ)−0.9 −0.75 (2.10) G= E 2(1 + ν) (2.11)

2.4

Literature review on soil-steel composite bridges

There is a relatively vast amount of literature on the topic of SSCB, mostly devoted to different aspects of static design and construction methods. Several experimental campaigns are also published, comprising static load tests to failure, fatigue tests and dynamic tests. In this report however, only work related to dynamic analysis and tests on SSCB for railway traffic is included.

Skivarpsån, Sweden

One of the earlier work on that topic related to the bridge over Skivarpsån in South of Sweden. It consists of a semi-circular corrugated arch with D = 11 m, h = 4.3 m and hc = 1.8 m that was installed outside of an existing concrete arch bridge in 2003. The new bridge is founded on separate supports and does not transfer any load to the old bridge that is still in place. A SuperCor S37 corrugation with c = 380 mm, hcorr= 140 mm and t = 7.0 mm was used.

(21)

2.4. LITERATURE REVIEW ON SOIL-STEEL COMPOSITE BRIDGES The first experimental campaign is reported in Flener (2003), comprising the con-struction stage followed by static and dynamic load tests with trains. Strain gauges were installed on the intrados of the corrugation, at quarter point and the crown. An LVDT was installed at the crown, measuring the relative displacement between the new and the old bridge. The result from a passing freight train showed about 1 mm vertical crown displacement and about 13 MPa stress in the corrugation. The same tests were later published in Flener et al. (2005).

More comprehensive dynamic tests were performed in two following campaigns, reported in Flener (2004) and Flener (2005). A 78 tonne RC4 locomotive was used, running at different speeds from 10 to 125 km/h. The measured response showed a crown displacement from 0.7 to 0.9 mm and about 55 to 70 µ in the culvert. Both tests were performed with similar test schedule, in May and October 2004.

The results was later published in Flener and Karoumi (2009) and Flener and Karoumi (2010). It was concluded that there was no significant difference between the two tests and that the ballast acceleration was less than 0.5 m/s2. The work on the bridge over Skivarpsån is also presented in Flener (2009).

Märsta, Sweden

In 2009 and 2010, experimental testing was performed on a double track closed arch culvert in Märsta outside of Stockholm, Sweden, first reported in Andersson et al. (2012). The geometry is D = 3.7 m, h = 4.1 m and hc = 1.9 m and a VE-profile with c = 150 mm, hcorr= 50 mm and t = 5.5 mm.

The culvert was instrumented with strain gauges, accelerometers and LVDTs and the response from passing trains was recorded with a top speed of 175 km/h. The results showed a peak displacement of about 0.5 mm, acceleration of 0.8-1.4 m/s2 and 30 µ in the culvert. Accelerometers were also installed in the track area before, on and after the bridge. The peak ballast acceleration was in the range of 0.5-3.5 m/s2.

2D and 3D models were developed with relatively good agreement compared to experimental results, reported in Mellat (2012) and Mellat et al. (2014). The load distribution from the track to the culvert was found to have a significant impor-tance and the 2D-models were partly based on a best-fit 3D-model in that aspect. Simulations of high-speed trains indicated potential exceedance of the accelerations for train speeds above 250 km/h. It was pointed out however that the models were afflicted with great uncertainties.

Additional simulations using 3D models was done by Aagah and Aryannejad (2014), studying the influence of soil modulus and culvert-soil frictional contact. Extensive

(22)

CHAPTER 2. DYNAMIC ANALYSIS OF RAILWAY BRIDGES

parametric studies was done by Woll (2014) based on 2D-models calibrated against the aforementioned experimental data. In Andersson and Karoumi (2017) a best fit 3D model was found for a soil E-modulus of 120 MPa and simulations indicated that the acceleration limit was reached at about 300 km/h.

A hydraulic system for forced vibration tests on railway bridges was later devel-oped and first tested in Andersson et al. (2015). This equipment was used when performing new tests on the culvert in 2017 and on two additional bridges in 2018. One of the last bridges have recently been studied by Barrero (2019) and Ljung (2019) and is also the studied in more detailed in this report.

Culverts in Poland

Experimental testing of a double track twin culvert is reported in Beben (2011). The closed arch culvert has the geometry D = 4.4 m, h = 2.8 m and hc = 2.4 m and the centre distance between the two culverts is 5.9 m. The corrugation has a geometry with c = 150 mm, hcorr = 50 mm and t = 3.0 mm.

The vertical crown displacement was measured using an interferometric radar sen-sor and a total of 40 trains with speed from 20 to 120 km/h was recorded. The results showed a vertical peak displacement of about 0.6 mm. Based on the same data, further analysis reported in Beben (2013) showed a dynamic amplification factor ranging from 1.1 to 1.4. It should be noted however that the resolution and accuracy of the measurement system was rather limited. The same bridge was later instrumented with strain gauges, accelerometers and LVDTs, reported in Beben (2014). The results confirmed previous displacements of about 0.6 mm. In addition, a peak strain of 54 µ was recorded. From a passing train at 120 km/h the peak acceleration was 0.7 m/s2 in the culvert and 1.2 m/s2in the ballast. Experimental testing of a similar bridge but with a single span is reported in Beben (2018). The geometry was D = 4.7 m, h = 2.9 m and hc = 2.4 m and with the same corrugation as the previous bridge. Trains passing at 40 to 130 km/h was recoded during 24 h and relatively similar results were obtained as the previous tests; 0.65 mm vertical crown displacement, a peak strain of 60 µ and 0.7 and 1.3 m/s2 in the culvert and the track respectively.

(23)

Chapter 3

Experimental results

3.1

Experimental equipment

The main part of the experimental equipment consists of the actuator system for forced vibration tests. The schematics is illustrated in Figure 3.1. It consists of a 50 kN MTS load actuator that is powered by an oil pump with a peak pressure of 210 bar and flow rate of 120 litre/min, powered by an integrated 40 hp diesel engine. The total mass of the oil pump is about 1000 kg. The actuator is placed inside the bridge and the load is transmitted to the crown by an aluminium truss system that can be rebuilt to adjustable heights. A load cell is placed on top of the truss and both the lower and upper points of the system is hinged. Load controlled harmonic sweeps are performed by using an MTS FlexTest SE controller. The data is collected by an MGCPlus data acquisition system, consisting of the input force, actuator displacement and the sensors on the bridge.

The sensors consists of uniaxial accelerometers (SiFlex SF1500S), strain gauges (120Ω from HBM) and displacement transducers (HBM WA10 LVDT). The strain gauges were glued to the intrados of the corrugated plate, the LVDTs were mounted on a frame under the bridge and connected to the crown, the accelerometers were fastened both inside the culvert and in the adjacent track.

hydraulic oil pump MGC Plus controller PC load cylinder (MTS) piston load cell bridge load frame oil F(t), d(t) sensors signal F(t) d(t) d(t+∆t)

(24)

CHAPTER 3. EXPERIMENTAL RESULTS

3.2

The Märsta bridge

The bridge at Märsta is a double track closed arch bridge located about 50 km North of Stockholm. The geometry is D = 3.7 m, h = 4.1 m and hc = 1.9 m and a VE-profile with c = 150 mm, hcorr = 50 mm and t = 5.5 mm. A photo of the bridge is shown in Figure 3.2. The first tests were performed in 2009 and 2010, comprising train passages ranging from 100 to the allowable speed of 175 km/h. The results are reported in Andersson et al. (2012). New field tests were made in September 2017, involving both passing trains and forced vibration tests with the load actuator system.

Figure 3.2: View of the Märsta bridge, during field tests in 2009.

Instrumentation

The instrumentation from 2017 is presented in Figure 3.3 and Figure 3.4. The uniaxial SiFlex accelelerometers are denoted a1 to a16; a1 to a5 inside the culvert intrados, a6 on the ground and a7 to a16 in the track. The vertical displacement is measured in d1 to d4, whereof d1 is from the actuator system and d2 to d4 via LVDT’s mounted on a frame inside the culvert. Strain gauges denoted e1 to e12 are installed on the culvert intrados, odd numbers installed at the through point of the corrugation and even numbers at the crest, see Figure 3.4. During all tests the actuator was located near the West track, denoted USP.

Train passages

A total of 30 train passages were recorded, all consisting of different types of pas-senger trains with speed ranging from 90 to 175 km/h. An example of the vertical crown displacement is shown in Figure 3.5 with a peak displacement of about 0.4 mm. The displacement can also be estimated by integrating the acceleration. This is done in Figure 3.6 that illustrate the peak vertical displacement in the crown

(25)

3.2. THE MÄRSTA BRIDGE 2.25 2.25 a3 a4 a5 a6 USP NSP a10 a9 a8 2.25 2.25 a1 a11 a7 1.5 1.3 19.8 1.3 a13 a12 a10 a9 a8 a11 a7 a14 a15 a16 N a2 let si Le 1.75 11.1 F d2 d3 d4 d1 e1, e2 e3, e4

e5, e6 e7, e8 e9, e10 e11, e12

1.95 1.80 1.0 y x z 1.0y x z cx1 0.2 cx2 1.95 1.95 3.90 0.225 cx1y cx2 z zy 3.35 Arlanda Knivsta

Figure 3.3: Instrumentation of the Märsta bridge in 2017, plan and elevation.

e1 e2 a14 a13 a12 a8 (a10) a6 1.6 1.5

e1, e2e5, e6e3, e4 Arlanda

Figure 3.4: Instrumentation of the Märsta bridge in 2017, section and detail of corrugation.

(26)

CHAPTER 3. EXPERIMENTAL RESULTS 0 0.5 1 1.5 2 2.5 3 3.5 -0.6 -0.4 -0.2 0

Figure 3.5: Vertical displacement at d3, X55 train passage on NSP.

-6 -4 -2 0 2 4 6

-0.6 -0.4 -0.2 0

Figure 3.6: Vertical peak displacement of the culvert estimated from accelerometer a1-a5, X55 train passage on NSP and USP.

for a train crossing on the NSP and USP track respectively. The results indicate that the displacement is distributed on a width of about 6 m along the culvert. The stresses in the culvert is calculated using Equation 3.1 where Es = 210 GPa is assumed. The peak stress from bending and axial components are presented in Figure 3.7 when the train runs on USP. The axial stress is mainly concentrated to the section closest to the loaded track whereas the bending moment is also partly distributed to sensor e7 and e8 about 2.2 m to the side.

σM = Es(εbot− εtop) /2

σN = Es(εbot+ εtop) /2 (3.1)

(27)

3.2. THE MÄRSTA BRIDGE e 1-2 e3-4 e5-6 e7-8 e9-10e11-12 -5 -3 -1 1 3 5 e 1-2 e3-4 e5-6 e7-8 e9-10e11-12 -5 -3 -1 1 3 5

Figure 3.7: Peak stress, X55 train passage on USP.

EN 1990 stipulates limits for peak acceleration of the bridge deck, for ballasted tracks set to 3.5 m/s2 for frequencies up to 30 Hz. This can be seen as an indirect limit for ballast instability that may occur in the track area. For SSCB it is however not obvious what limit to use and at what section it should be evaluated. Train induced vibrations usually include relatively high-frequency components and the peak acceleration may therefore be sensitive to the filter limit. This is illustrated in Figure 3.8 where the peak acceleration at different low-pass filter limits fLP is plotted. A somewhat unexpected results is that the peak acceleration inside the culvert is higher than in the ballast and for NSP even higher than on the sleeper. In Figure 3.9 the peak acceleration in the ballast vs. the culvert is plotted for all train passages. The results with 100 Hz cutoff limit is one order of magnitude higher than with 30 Hz cutoff.

0 100 200 300 0 1 2 3 4 0 100 200 300 0 1 2 3 4

(28)

CHAPTER 3. EXPERIMENTAL RESULTS 0 0.1 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 0 1 2 3 4 0 2 4 6 8

Figure 3.9: Peak acceleration with 30 Hz and 100 Hz filter, all trains.

Forced vibration tests

Forced vibration tests were performed with the actuator located at sensor a2 close to the USP track. Harmonic frequency sweeps were performed from 1-80 Hz with a rate of ∆f = 0.05 Hz/s. Three tests were performed with a load amplitude of 1 kN and three tests with 10 kN. The complex-values transfer function H(ω), hereafter abbreviated F RF for Frequency Response Function, is calculated based on the Fourier transform of the output acceleration a(t) and input force F (t) according to Equation 3.2. Further in this report, the FRF will have the unit kN/m/s2 unless stated otherwise.

H(ω) = A(ω)/F(ω) (3.2)

The imaginary part of the FRF is presented for the culvert in Figure 3.10 and the track in Figure 3.11, both for the case of Famp= 1 kN. The gap at 27 Hz and 37 Hz is due to disturbance from passing trains. In Figure 3.12 the response in the culvert (a2) and the track (a8) is compared. Both curves show similar response up until about 25 Hz, indicating that both sensors and hence the bridge-track system moves in-phase. For higher frequencies however the track and bridges moves more out-of-phase. Finally in Figure 3.13 the response with 1 kN and 10 kN load amplitude is compared. The difference in the curves indicate that the system is amplitude dependent. At 1 kN load amplitude the first natural frequency is about 20 Hz, which decrease to about 18 Hz for 10 kN load amplitude. Still the peak acceleration is only 0.6 m/s2 in the latter case, far from any design limits.

The damping is estimated using the Half-Power Bandwidth method according to Equation 3.3, where ∆fi is the bandwidth for a 3dB (1/

2) decrease of the peak 20

(29)

3.2. THE MÄRSTA BRIDGE 10 15 20 25 30 35 40 -0.05 0 0.05 0.1 0.15

Figure 3.10: FRF, response in the culvert, Famp = 1 kN.

10 15 20 25 30 35 40 -0.05 0 0.05 0.1 0.15

Figure 3.11: FRF, response in the track, Famp = 1 kN.

response. For the 1 kN response the damping is estimated to 6% at sensor a1 and 7% at sensor a8. For the 10 kN response the corresponding values are 8% and 10% respectively. This indicate that firstly the damping is higher in the track than the culvert and secondly that the damping increase with increased load amplitude. Despite increased damping the accelerance amplitude is higher for the 10 kN load case. The reason is not clear but may be due to change in boundary conditions or the equivalent modal mass.

ζi= ∆fi

2fi

(30)

CHAPTER 3. EXPERIMENTAL RESULTS 10 15 20 25 30 35 40 -0.05 0 0.05 0.1 0.15

Figure 3.12: FRF, compare response in culvert and track, Famp = 1 kN.

10 15 20 25 0 0.02 0.04 0.06 0.08 0.1 10 15 20 25 0 0.02 0.04 0.06 0.08 0.1

Figure 3.13: FRF from a1 and a8 for load amplitudes Famp = 1 kN and 10 kN.

(31)

3.3. THE HÅRESTORP BRIDGE

3.3

The Hårestorp bridge

The bridge at Hårestorp is a single track closed arch bridge located about 400 km South of Stockholm. The geometry is D = 4.9 m, h = 3.9 m and hc= 1.2 m and an MP 150 profile with c = 150 mm, hcorr = 50 mm and t = 5.0 mm. A photo of the bridge is shown in Figure 3.14. Experimental tests were performed in September 2018 consisting of the same equipment as for the Märsta bridge.

Figure 3.14: View of the Hårestorp bridge, during field tests in 2018.

Instrumentation

A similar instrumentation as for the Märsta bridge was performed, as illustrated in Figure 3.15 and Figure 3.16. An unusual feature for this bridge is the shallow foundation to bedrock, as illustrated in Figure 3.16. For the accelerometers in the track area only a12 and a15 are installed on the sleepers and the remaining sensors in the ballast.

Train passages

A total of 8 train passages were recorded, running at a speed ranging from 120 to 150 km/h. The allowable speed at the location is 160 km/h. The peak vertical dis-placement during train passage is about 0.8 mm. By integrating the accelerometer a1 to a5, the displacement along the culvert is presented in Figure 3.17.

The peak acceleration from the culvert and the track during train passages is pre-sented in Figure 3.18. Similar to the Märsta bridge, the peak acceleration using the 100 Hz LP-filter is significantly higher than for the 30 Hz LP-filter. The relatively large scatter in results become less when comparing individual sensors for the same type of trains at similar speed.

(32)

CHAPTER 3. EXPERIMENTAL RESULTS N Alvesta Rydaholm 1.2 a5 a4 a3 a2 a1 d1 d3 d2 e1, e2 a6 1.35 1.35 1.5 1.5 a7 a8 a9 a10 a11 a7 a8 a9 a10 a11 a13 a14 a15 cx1 cx2 2.1 2.1 2.1 2.1 a16 1.5 32 sleep ers (appro x 20 m) a12 1.3 x y x y

Figure 3.15: Instrumentation of the bridge at Hårestorp in 2018, plan and elevation.

(33)

3.3. THE HÅRESTORP BRIDGE 3.5 e1 e2 4.2 0.4 Rydaholm

Figure 3.16: Instrumentation of the bridge at Hårestorp in 2018, section and detail of corrugation.

-5 -4 -3 -2 -1 0 1 2 3 4 5

-1 -0.5 0

Figure 3.17: Vertical peak displacement of the culvert estimated from accelerometer a1-a5, X14 train passage.

An example of the stress in the crown during train passage is shown in Figure 3.19. The largest bending stress is about 7 MPa. The combination of axial and bending stress results in a peak stress of about 4 MPa in tension at e1 and 12 MPa in compression at e2.

Forced vibration tests

Forced vibration tests were performed with the actuator in the centre of the bridge near sensor a3. Tests were performed with load amplitudes from 1 to 20 kN. For the 1 kN case FRFs in the culvert and the track are presented in Figure 3.20 and Figure 3.21 respectively. Similar to the Märsta bridge, the vibration of the culvert and the track seems to be in phase up to about 30 Hz. When comparing the FRFs for different load amplitudes, Figure 3.23, increasing the load amplitude results in a clear decrease in natural frequency, ranging from 16.4 Hz to 14.4 Hz. At the same time the estimated damping increase from 3.8% to 6.5%.

(34)

CHAPTER 3. EXPERIMENTAL RESULTS 0 2 4 6 8 0 2 4 6 8

Figure 3.18: Peak acceleration with 30 Hz and 100 Hz filter, all trains.

0 0.5 1 1.5 2 2.5 3 -6 -4 -2 0 2 4 6 8

Figure 3.19: Stress in the crown of the culvert, X14 train passage.

(35)

3.3. THE HÅRESTORP BRIDGE 10 15 20 25 30 35 40 -0.2 -0.1 0 0.1 0.2

Figure 3.20: FRF, response in the culvert, Famp = 1 kN.

10 15 20 25 30 35 40 -0.05 0 0.05 0.1 0.15 0.2

Figure 3.21: FRF, response in the track, Famp = 1 kN.

10 15 20 25 30 35 40 -0.05 0 0.05 0.1 0.15 0.2

(36)

CHAPTER 3. EXPERIMENTAL RESULTS 10 12 14 16 18 20 0 0.05 0.1 0.15 0.2 10 12 14 16 18 20 0 0.05 0.1 0.15 0.2

Figure 3.23: FRF, compare response in culvert and track for different load ampli-tudes.

(37)

3.4. THE ÖRSJÖ BRIDGE

3.4

The Örsjö bridge

The Örsjö bridge is located about 90 km East of the Hårestorp bridge, and is also a single track closed arch bridge. The geometry is D = 2.9 m, h = 2.7 m and hc = 1.2 m and an MP 150 profile with c = 150 mm, hcorr = 50 mm and t = 6.0 mm. A photo of the bridge is shown in Figure 3.24. Experimental tests were performed in September 2018 consisting of the same equipment as for the Märsta and the Hårestorp bridge.

The bridge is founded on moraine and according to geotechnical surveys the depth to bedrock is estimated to range from 3 to 5 m.

Figure 3.24: View of the Örsjö bridge, during field tests in 2018.

Instrumentation

The instrumentation is shown in Figure 3.25 and Figure 3.26 and have a similar layout as the Hårestorp bridge. The actuator was located 0.6 m from the track centre line due to an obstacle in form of a lamp in the centre point.

Train passages

A total of 5 passenger trains were recorded, crossing at speeds from 115 to 125 km/h. Based on integrated accelerations the displacement along the culvert is estimated in Figure 3.27. The peak displacement is about 0.4 mm. The peak acceleration in the culvert and the track is presented in Figure 3.28, showing similar results as the previous bridges.

(38)

CHAPTER 3. EXPERIMENTAL RESULTS 1.3 N Emmaboda Nybro a5 a4 a3 a2 a1 d3 d2 e1, e2 a6 a7 a8 a9 a10 a11 2.1 2.1 d1 2.1 2.1 a7 a8 a9 a10 a11 a13 a14 a15 cx1 cx2 a12 a16 1.35 1.35 1.5 1.5 1.5 1.2 x y x y 0.6 32 sleep ers (appro x 20 m) Mn

Figure 3.25: Instrumentation of the bridge at Örsjö in 2018, plan and elevation.

(39)

3.4. THE ÖRSJÖ BRIDGE 2.7 2.9 2.4 e1 e2 0.4 Emmaboda Mn

Figure 3.26: Instrumentation of the bridge at Örsjö in 2018, section and detail of corrugation. -5 -4 -3 -2 -1 0 1 2 3 4 5 -0.5 -0.4 -0.3 -0.2 -0.1 0

Figure 3.27: Vertical peak displacement of the culvert estimated from accelerometer a1-a5, commuter train passage.

0 1 2 3 4 5 6 7 8 0 1 2 3 4

(40)

CHAPTER 3. EXPERIMENTAL RESULTS

Forced vibration tests

Due to a malfunction of the hydraulic actuator, only one forced vibration test was performed, using 1 kN load amplitude. The FRFs for the culvert and the track is presented in Figure 3.29. The first natural frequency is found at 18.4 Hz and the damping is estimated to 4.5%. Comparing the FRF in the culvert and the track shows similar results in Figure 3.30.

10 15 20 25 30 -0.05 0 0.05 0.1 0.15 0.2 10 15 20 25 30 -0.05 0 0.05 0.1 0.15 0.2

Figure 3.29: FRF, response in the culvert and the track, Famp = 1 kN.

10 15 20 25 30 -0.05 0 0.05 0.1 0.15 0.2

Figure 3.30: FRF, compare response in culvert and track, Famp = 1 kN.

(41)

3.5. SUMMARY OF EXPERIMENTAL RESULTS

3.5

Summary of experimental results

The experimental results from the three bridges is summarized in the tables below. Based on the forced vibration tests, the natural frequency and damping for the first mode is presented in Figure 3.1. Increased load amplitude shows a clear decrease in natural frequency and an increase in damping. The largest vibrations were obtained using 20 kN load amplitude at the Hårestorp bridge, corresponding to about 3.5 m/s2 in the culvert and 3.0 m/s2 in the track, both at the resonance frequency. Higher vibrations may however occur at higher frequencies, e.g. using the test with 10 kN load amplitude at 80 Hz results in about 8 m/s2 in the track for the Hårestorp bridge.

The peak acceleration during train passages is presented in Table 3.2. For the Märsta bridge several of the accelerometers in the track area were mounted on the sleeper and experienced over-load during train passages. The results are therefore based on sensor a9 in the ballast between the two tracks. The acceleration closer to the track is likely higher. For the Hårestorp and Örsjö bridges sensor a9 was instead mounted in the ballast between the sleepers. The general trend for these bridges is that the acceleration is larger in the track than the culvert. It should however be pointed out that a relatively small number of train passages was recorded.

The peak displacement during train passages is presented in Table 3.3. For the Märsta bridge the results are based on sensor d3 under the NSP-track. For the Hårestorp and Örsjö bridges the load actuator was located at the track centre line. The displacement during train passages recorded by the actuator was deemed unreliable and the results were instead extimated by integration of accelerometer a3. The accuracy of the integration was studied by comparing the response from d2 and d3 with integrated results from a4 and a5.

The peak stress during train passages is presented in Table 3.4. The results are based on the pair of strain gauges at the crown but not separated in bending and axial stresses. Negative values correspond to compression.

(42)

CHAPTER 3. EXPERIMENTAL RESULTS

Table 3.1: Estimated natural frequency and damping, based on forced vibration tests.

Famp = 1 kN Famp = 10 kN Famp = 20 kN bridge f1 (Hz) ζ1(%) f1(Hz) ζ1 (%) f1 (Hz) ζ1 (%)

Märsta 20.0 6.0 18.0 8.0 -

-Hårestorp 16.4 3.8 14.8 5.8 14.4 6.5

Örsjö 18.4 4.5 - - -

-Table 3.2: Peak acceleration during train passages, 30 Hz LP-filter.

aculvert aballast

max mean std max mean std

Märsta 0.78 0.56 0.11 0.56 0.24 0.09

Hårestorp 1.74 1.47 0.23 2.28 1.85 0.32

Örsjö 0.84 0.58 0.15 1.50 1.05 0.29

Table 3.3: Peak displacement during train passages. dculvert

max mean std

Märsta 0.51 0.45 0.04

Hårestorp 1.24 0.92 0.31

Örsjö 0.56 0.50 0.07

Table 3.4: Peak stress in the crown during train passages.

σmax σmin

max mean std min mean std

Märsta 2.90 2.22 0.49 -7.06 -6.41 0.48

Hårestorp 6.71 4.33 1.26 -13.98 -11.99 1.34

Örsjö 7.28 6.34 0.55 -8.65 -7.05 0.90

(43)

Chapter 4

Simulation results

A 3D finite element model of the Hårestorp bridge has been developed using Abaqus, based on the geometry from the construction drawings. An optimization algorithm has been used to estimate the optimal parameters of the model to fit the experi-mental results from the forced vibration tests. The fitted model is then used both for estimating the influence of the culvert as well as estimating the variation of track stiffness and ballast acceleration, information that is then used for fitting a more time efficient 1D model.

4.1

3D-model of the Hårestorp bridge

The 3D model of the Hårestorp bridge is illustrated in Figure 4.1. Due to the sym-metric loading from the forced vibration tests, only a quarter-section of the model is included in the analysis. The ballast, sub-ballast, subgrade, sleepers and rail pads are modelled with 20-noded quadratic brick elements (C3D20R). To mitigate reflecting waves along the track, the far end of the embankment is modelled with 12-noded infinite elements (CIN3D12R). The rail is modelled with beam elements and each rail seat is connected by node to surface translational constraints. The corrugated culvert is modelled with 4-noded shell elements (S4R) with orthotropic properties. The backfill and the culvert is rigidly connected and the lower horizontal surface of the subgrade is rigid, motivated by the shallow foundation to bedrock.

subgrade 3.55 2.35 1.65 h 1:2 sub-ballast ballast hc x y z Figure 4.1: 3D model of the Hårestorp bridge.

(44)

CHAPTER 4. SIMULATION RESULTS

The orthotropic properties of the corrugated shell is based on plane stress assump-tion according to Equaassump-tion 4.1. Following the notaassump-tions i Figure 1.5, the equivalent thickness teqis calculated according to Equation 4.2, where the principal E-modulus

E1is calculated according to Equation 4.3. The E-modulus in the transverse direc-tion, E2is based on the flexural rigidity for transverse bending. The in-plane shear modulus G12is given by Equation 4.4. The transverse shear moduli is given by G13 and G23. The equivalent density is given by ρeq. The parameters are summarized in Table 4.1.    ε1 ε2 γ12    =   1/E1 −ν12/E1 0 −ν12/E1 1/E2 0 0 0 1/G12      σ11 σ22 τ12    (4.1) teq= EsAc E1c (4.2) E1= EsAc1.5 12c r 12 I , E2= Es  t teq 3 (4.3) G12= √ E1E2 2(1 + ν), G13= E1Ac 2c(1 + ν), G23= E1tc 2(1 + ν) (4.4) ρeq= ρsAc teqc (4.5) Table 4.1: Properties of the corrugated plate and the orthotropic shell.

Geometry Material Orthotropic

c 150 mm Es 210 GPa teq 62 mm

t 5.0 mm ρs 7850 kg/m3 E1 21.4 GPa

hcorr 55 mm νs 0 E2 0.1 GPa

Ac 6.4 mm2/mm G12 0.76 GPa

Ic 2060 mm4/mm ρeq 800 kg/m3

4.2

Analysis and model updating

The forced vibration tests are simulated in the FE-model by a steady-state analysis in modal domain. The first step is to calculate the eigenvalues within the studied range, which is used as input for calculating the steady-state response. The FE-analysis is performed in Abaqus and the model updating is performed using the Pattern-Search function in Matlab. The variables consist of the E-modulus, density and Poisson’s ratio for the backfill as well as the modal damping. Despite the stress-dependent properties of the soil a sufficiently good fit is obtained by constant parameters for all layers of the ballast, sub-ballast and the subgrade.

(45)

4.3. TRACK MODEL In the optimization process the objective function g(x) is defined by Equation 4.6, based on the Frequency Response Assurance Criterion F RAC and Frequency Am-plitude Assurance Criterion F AAC according to Equation 4.7 and Equation 4.8, which operates on the shape and amplitude of the FRFs, Grafe (1998). It essen-tially works as a least square fit of experimental versus simulated response, where HXi (ω) denotes the complex-valued measured frequency response vector at sensor location i and HA

i (ω) denotes the corresponding model prediction.

g(x) = n X i=1 (1 − F RACi)2+ (1 − F AACi)2 (4.6) F RACi= (HXi (ω))HHAi (ω) 2 (HX i (ω))HH X i (ω) (H A i(ω))HH A i (ω) (4.7) F AACi= 2 (HXi (ω))HHAi (ω) (HX i (ω))HHXi (ω) + (HAi (ω))HHAi (ω) (4.8) The experimental data with 10 kN load amplitude is used for reference. The results from the model updating is presented in Figure 4.2 for selected sensors in the track (a8-a10) and the culvert (a2-a4). Given the complexity of the model and the level of uncertainties, a reasonable fit has been obtained. Least good fit is obtained at a3 where the load is applied. The difference may be due to either disturbance in the measurements or due to the model stiffness at that location. A simple sensitivity analysis is performed by varying all variables by 10%, resulting in a permutation of 16 simulations illustrated as the gray shaded area. The final parameters from the optimization is presented in Table 4.2.

Table 4.2: Fitted parameters from model updating of the 3D-model.

ballast subgrade Eb 250 MPa Es 250 MPa νb 0.30 νs 0.30 ρb 1800 kg/m3 ρs 1800 kg/m3 ζmod 6%

4.3

Track model

The updated model is used when comparing the response with and without the culvert. In addition, the load is applied to the rail instead of the culvert which enables estimate of the track stiffness. A rail pad stiffness kpad = 130 MN/m is assumed. The dynamic track stiffness (inverse of the rail receptance) and the accelerance at sensor a9 is presented in Figure 4.3 for the case of an embankment

(46)

CHAPTER 4. SIMULATION RESULTS 10 15 20 0 0.05 0.1 0.15 0.2 10 15 20 0 0.05 0.1 0.15 0.2 10 15 20 0 0.05 0.1 0.15 0.2 10 15 20 0 0.05 0.1 0.15 0.2 10 15 20 0 0.05 0.1 0.15 0.2 10 15 20 0 0.05 0.1 0.15 0.2 FEM, ±10%

Figure 4.2: FRF for the updated 3D-model.

without the culvert and the reference model with the culvert. For the model with the culvert, the static track stiffness at the crown is 92 MN/m. The corresponding value for the embankment is 131 MN/m. The peak accelerance at a9 is 0.11 m/s2 at 14.5 Hz with the culvert and 0.04 m/s2 at 19.5 Hz for the embankment model.

0 10 20 30 -50 0 50 100 150 200 250 0 10 20 30 0 0.05 0.1 0.15

Figure 4.3: FRF for track stiffness and ballast acceleration, 3D model with and without the culvert.

Based on Wanming Zhai (2004), a 1D track model according to Figure 4.4 is de-38

(47)

4.3. TRACK MODEL veloped. The parameters for the rail and rail pad are set constant, the remaining parameters are varied in another optimization scheme with the objective to fit the dynamic track stiffness of the 3D-model without the culvert. The results are pre-sented in Figure 4.5 with parameters according to Table 4.3. The main difference compared to the parameters presented in Wanming Zhai (2004) is that the present model gives higher values for the equivalent ballast mass mb as well as the shear stiffness of the ballast and the stiffness of the subgrade. Since no direct dynamic tests of the track has been performed, the presented results should be interpreted as model fitting parameters rather than strictly physical values. Still, the 1D model seems to reproduce the track stiffness and accelerance relatively well.

cksw cb cp ks mb kb s kp ms cw EIr, mr

Figure 4.4: 1D track model.

0 10 20 30 -50 0 50 100 150 200 250 0 10 20 30 0 0.05 0.1 0.15

Figure 4.5: FRF for track stiffness and ballast acceleration, embankment model. The presence of the culvert will result in a change in track stiffness that is assumed to vary smoothly and have a minimum at the crown. In an attempt to account for this in the 1D track model, the subgrade parameters ks and cs and modulated by the factor κ according to Equation 4.9. By simple fitting the parameters α = 0.75 and β = 7 is obtained. The total length of the model is L = 20 m. The results

(48)

CHAPTER 4. SIMULATION RESULTS

Table 4.3: Parameters for the 1D track model, half-track model.

rail pad ballast subgrade

kp 130 MN/m kb 153 MN/m kw 764 MN/m ks 161 MN/m

cp 50 kNs/m cb 0 kNs/m cw 1000 kNs/m cs 198 kNs/m

mr 60 kg/m ms 125 kg mb 10300 kg

are presented in Figure 4.6 which show a decreased accuracy in static stiffness but very good agreement for the ballast accelerance.

κ(α, β, L) = 1 − α sinπx L β (4.9) 0 10 20 30 -50 0 50 100 150 200 250 0 10 20 30 0 0.05 0.1 0.15

Figure 4.6: FRF for track stiffness and ballast acceleration, model with the culvert.

4.4

Response from passing trains

The updated 1D track model is further used for analysing the dynamic response from passing trains. The train is represented by a series of moving loads traversing the rail. The response is solved using direct time integration and the output consists of peak acceleration in the ballast and peak displacement at the rail. All results are low-pass filtered at 30 Hz. The high-speed load model HSLM A1-A10 according to EN 1991-2 is analysed for the speed range of 100-400 km/h. The envelope of the response is presented in Figure 4.7. According to the analysis HSLM-A2 results in the highest dynamic response. The vertical displacement of the rail is about 0.8 mm for the embankment model and 1.0 mm for the culvert model, relatively independent of the speed. The peak ballast acceleration range from 0.5-1.5 m/s2 for the embankment model and maximum about 1.8 m/s2 for the culvert model. No clear resonance peaks are obtained in either of the analyses.

(49)

4.4. RESPONSE FROM PASSING TRAINS 100 200 300 400 0 1 2 3 4 100 200 300 400 0 0.5 1 1.5 2

Figure 4.7: Envelope of results for train HSLM A1-A10, 1D track model. The corresponding results from the 3D-model is presented in Figure 4.8. When simulating passing trains, the model in Figure 4.1 has been extended to a half model in the direction along the track. Due to computational time only HSLM-A2 is analysed, which was found to produce the largest response in the 1D track model. The results show similar trend as the 1D model but with somewhat higher acceleration response for the culvert model. Still, the response does not exceed the design limit in EN 1990. 100 200 300 400 0 1 2 3 4 100 200 300 400 0 0.5 1 1.5 2

Figure 4.8: Envelope of results for train HSLM A2, 3D model.

Finally, Figure 4.9 presents the time response from a real train passage and a comparison between the experiment and the 3D-model. It appears that the FE-model underestimated the response at the centre line, a9 and a3, but shows better agreement for the adjacent points.

(50)

CHAPTER 4. SIMULATION RESULTS 0 1 2 -1 0 1 0 1 2 -1 0 1 0 1 2 -1 0 1 0 1 2 -1 0 1 0 1 2 -1 0 1 0 1 2 -1 0 1

Figure 4.9: Time response from an X14 train at 130 km/h, experiment vs. 3D-model.

(51)

Chapter 5

Conclusions

The results presented in this report is based on a combination of experimental testing and numerical simulations. The aim is to give better understanding of the dynamic behaviour of corrugated steel culvert for high-speed railway applications. Due to the inherently large interaction with the surrounding soil, the system is a combination of structural dynamics and soil dynamics. Despite comprehensive experimental testing and extensive updating of the numerical models, relatively large uncertainties still exist. Therefore, the conclusions below should be considered indicative rather the definite. The conclusions are also limited to the range of studied bridges.

5.1

Experimental results

Experimental testing has been performed on three culverts, all with closed elliptic sections and built for conventional railways. Two of the bridges are single track and one bridge double track. Testing has been performed using both forced vibration response and passing trains.

• The forced vibration tests show a first natural frequency in the range of 15-20 Hz. The peak is more pronounced for the single track bridges.

• From the forced vibration tests, the damping of the first mode is estimated to 4-8%.

• Increased load amplitude generally results in a lower natural frequency and higher damping.

• The vibration of the track and the culvert seems to be in-phase up until about 30 Hz.

• At higher frequencies, dynamic soil-structure interaction appears to be more pronounced and the culvert may be out-of-phase with the track.

• The peak acceleration from passing trains range from 1-2 m/s2 for conven-tional trains.

• The peak vertical displacement of the culvert during train passages range from 0.5-1.2 mm.

(52)

CHAPTER 5. CONCLUSIONS

• The peak stress during train passage range from 3-7 MPa in tension to 7-14 MPa in compression.

5.2

Simulation results

The main part of the simulations has been devoted to the Hårestorp bridge, because it has the longest span and most comprehensive experimental data. Based on the experimental data it also has the lowest natural frequency and highest amplitude of vibrations.

The following procedure has been adopted for the numerical model of the Hårestorp bridge.

1. Develop a quarter-model of the bridge using 3D solid elements and calculate the FRFs at the sensor locations from the experiments.

2. Use a model updating algorithm to estimate the material properties of the model, with an objective function based on the experimental and simulated FRFs.

3. Calculate the FRFs with a 3D model of only the embankment, with previously estimated material parameters.

4. Use a similar model updating algorithm to fit a simplified 1D track model to the 3D track model.

5. Fit a function that reduce the subgrade modulus of the 1D model to account for the track stiffness variation due to the culvert.

6. Use the 1D model for calculating the dynamic response from passing trains. 7. Extend the fitted 3D solid model to a half-track model and simulate the

response from passing trains.

Due to the relatively large size of the 3D models, the simulations are rather com-putationally demanding. To reduce the simulation time, the FRFs are calculated based on a modal approach. The stiffness modulus of granular material is usually stress-dependent. For dynamic loading it is usually expressed as a shear wave speed rather than a static modulus. The simulations does however show that it is possi-ble to fit a model using a constant average modulus. For the Hårestorp bridge, the fitted average E-modulus is 250 MPa for both the ballast layer and the subgrade. The fitted FRFs generally show an acceptable level of agreement compared with the experimental data. The main difference is that the response at centre of the crown is underestimated with the model.

(53)

5.3. SUGGESTION FOR FURTHER WORK The 1D track model shows surprisingly good agreement compared to the 3D model. The reduction factor for the subgrade modulus fits well with the estimated track stiffness variation of the 3D-model. Simulations of passing trains using the 1D track model shows a peak ballast acceleration in the range of 0.5-1.5 m/s2 for speeds in the range of 100-400 km/h.

Simulations of the critical high-speed train on the 3D model show a peak accel-eration in the culvert ranging from 1-3.5 m/s2 for the same speed interval. The corresponding peak acceleration in the ballast far away from the culvert range from 0.5-2 m/s2. When comparing the time response of a real train, the experimental re-sults show somewhat higher peak acceleration at the crown compared to the model. However, in adjacent positions better agreement is found.

In conclusion, the numerical models generally show acceptable agreement with the experimental results and also indicate then even the largest bridge in the study is likely to comply with the dynamic requirements according to EN 1990.

5.3

Suggestion for further work

The results presented in this report is based on a relatively small sample of bridges, all with relatively short spans and similar sections. The presented numerical mod-els generally show acceptable agreement with the experimental results but may be improved, both in accuracy and computational efficiency. Further research is proposed below.

• Experimental testing of bridges with larger spans and other sections. • Combine the work with railway track dynamics, especially on the dynamic

characteristics of ballasted tracks.

• Further improvement of the developed numerical models. • Extended parametric study of bridges with larger spans.

(54)
(55)

References

Aagah, O., Aryannejad, S., 2014. Dynamic analysis of soil-steel composite railway bridges. Master thesis 436, KTH.

Andersson, A., Karoumi, R., 2017. A soil-steel bridge under high-speed railways. In: Archives of the Institute of Civil Engineering. Poznan University of Technology, pp. 45–52.

Andersson, A., Sundquist, H., Karoumi, R., 2012. Full scale tests and structural evaluation of soil-steel flexible culverts for high-speed railways. In: Archives of the Institute of Civil Engineering. Poznan University of Technology, pp. 43–53. Andersson, A., Ülker-Kaustell, M., Borg, R., Dymén, O., Carolin, A., Karoumi,

R., 2015. Pilot testing of a hydraulic bridge exciter. In: EVACES 2015. MATEC Web of Conferences, pp. 1–6.

Barrero, D. F., 2019. Dynamic soil-structure interaction of soil-steel composite bridges. Master thesis 1921, KTH.

Beben, D., 2011. Application of the interferometric radar for dynamic tests of cor-rugates steel plate (CSP) culvert. NDT & E International 44 (5), 405–412. Beben, D., 2013. Experimental study on the dynamic impacts of service train loads

on a corrugated steel plate culvert. Journal of Bridge Engineering 18 (4), 339–346. Beben, D., 2014. Corrugated steel plate culvert response to service train loads.

Journal of Performance of Constructed Facilities 28 (2), 376–390.

Beben, D., 2018. Experimental testing of soil-steel railway bridge under normal train loads. In: EVACES 2017. Springer, pp. 805–815.

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

ERRI, 1999. Rail bridges for speeds >200 km/h. European Rail Research Institute. Flener, E. B., 2003. Field tesing of a long-span arch steel culvert railway bridge

over Skivarpsån, Sweden (Part I). Report 72, KTH, Struct. Eng. and Bridges. Flener, E. B., 2004. Field tesing of a long-span arch steel culvert railway bridge

over Skivarpsån, Sweden (Part II). Report 84, KTH, Struct. Eng. and Bridges. Flener, E. B., 2005. Field tesing of a long-span arch steel culvert railway bridge

(56)

CHAPTER REFERENCES

Flener, E. B., 2009. Static and dynamic behaviour of soil-steel composite bridges obtained by field testing. Doctoral thesis.

Flener, E. B., Karoumi, R., 2009. Dynamic testing of a soil-steel composite bridge. Engineering Structures 31 (12), 2803–2811.

Flener, E. B., Karoumi, R., 2010. Testing of a soil-steel bridge under static and dynamic loads. Bridge Engineering 163 (1), 19–29.

Flener, E. B., Karoumi, R., Sundquist, H., 2005. Field testing of a long-span arch steel culvert during backfilling and in service. Structure and Infrastructure En-gineering 1 (3), 181–188.

Grafe, H., 1998. Model updatin of large structural dynamics models using measured response functions. Doctoral thesis.

Kaynia, A., Madshus, C., Zackrisson, P., 2000. Ground vibration from high-speed trains: prediction and countermeasure. Journal of Geotechnical and Geoenviron-mental Engineering 126 (6), 531–537.

Ljung, J., 2019. Parametric studies of soil-steel composite bridges for dynamic loads. Master thesis 19441, KTH.

Madshus, C., Kaynia, A., 2000. High-speed railway lines on soft ground: dynamic behaviour at critical train speed. Journal of Sound and Vibration 231, 689–701. Mellat, P., 2012. Dynamic analysis of soil-steel composite bridges for high speed

railway traffic. Master thesis 373, KTH.

Mellat, P., Andersson, A., Pettersson, L., Karoumi, R., 2014. Dynamic behaviour of a short span soil-steel composite bridge for high-speed railways - field mea-surements and FE-analysis. Engineering Structures 69, 49–61.

Pettersson, L., 2007. Full scale tests and structural evaluation of soil steel flexible culverts with low height of cover. Doctoral thesis.

Pettersson, L., Sundquist, H., 2014. Design of soil steel composite bridges. Report 112, KTH, Struct. Eng. and Bridges.

Smekal, A., Berggren, E., 2002. Mitigation of track vibration at ledsgård sweden, field measurements before and after soil improvement. In: Eurodyn 2002. A.A. Balkema Publishers LISSE, pp. 491–496.

Wanming Zhai, Kaiyan Wang, J. L., 2004. Modelling and experiment of railway ballast vibrations. Journal of Sound and Vibration 270, 673–683.

Woll, J., 2014. Soil steel composite bridges for high-speed railways. Master thesis 421, KTH.

References

Related documents

Using FEC the inertial behaviour of the dynamic simulation model can be adjusted with the car mass, wheelbase, track width, centre of gravity, moment of inertia, mass effects

The scope of this master thesis is to find global optimal points that maximise the correlation between the data from the field tests performed and the results obtained in a

The Fourier series may be used to represent arbitrary periodic functions as series of trigonometric functions. The Fourier transform is used to decompose signals in the time domain

Two strengthening methods for a steel railway bridge have been investigated. The impact on fatigue life was studied together with vertical bridge deck acceleration from high speed

FDU ZKHUH SHRSOH ZLWK D XQLYHUVLW\ HGXFDWLRQ KDYH D VLJQLILFDQWO\  KLJKHU FRVW IRU WKH FDU MRXUQH\ WKDQ UHVLGHQWV ZLWK RQO\

Assessment and verification of strengthening solutions and decision schemes to selected bridges (case studies) Other case studies proposed to be deepen >>> steel mobile

The highest modal damping ratio and frequency change is seen when the impedance functions from Arnäsvall and Ångermanälven pile groups were applied.. The Arnäsvall impedance

The dynamic increment considered the maximum dynamic response y dyn , and the corresponding maximum static response y stat , at any particular point in the structural element, due