Dynamic Soil-Structure Interactionof Soil-Steel Composite Bridges: A Frequency Domain Approach Using PML Elements and Model Updating

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IN

DEGREE PROJECT CIVIL ENGINEERING AND URBAN MANAGEMENT,

SECOND CYCLE, 30 CREDITS STOCKHOLM SWEDEN 2019,

Dynamic Soil-Structure Interaction of Soil-Steel Composite Bridges

A Frequency Domain Approach Using PML Elements and Model Updating

DIEGO FERNANDEZ BARRERO

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENT

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Abstract

This master thesis covers the dynamic soil structure interaction of soil-steel culverts applying a methodology based on the frequency domain response. At the first stage of this master thesis, field tests were performed on one bridge using controlled excitation. Then, the methodology followed uses previous research, the field tests, finite element models (FEM) and perfectly matched layer (PML) elements.

Firstly, a 2D model of the analysed bridge, Hårestorp, was made to compare the frequency response functions (FRF) with the ones obtained from the field tests. Simultaneously, a 3D model of the bridge is created for the following purposes: compare it against the 2D model and the field tests, and to implement a model updating procedure with the particle swarm algorithm to calibrate the model parameters. Both models use PML elements, which are verified against previous solution from the literature. The verification concludes that the PML behave correctly except for extreme parameter values.

In the course of this master thesis, relatively advanced computation techniques were required to ensure the computational feasibility of the problem with the resources available.

To do that, a literature review of theoretical aspects of parallel computing was performed, as well as the practical aspects in Comsol. Then, in collaboration with Comsol Support and the help given by PDC at KTH it was possible to reduce the computational time to a feasible point of around two weeks for the model updating of the 3D model.

The results are inconclusive, in terms of searching for a perfectly fitting model. Therefore, further research is required to adequately face the problem. Nevertheless, there are some accelerometers which show a considerable level of agreement. This thesis concludes to discard the 2D models due to their incapability of facing the reality correctly, and establishes a model optimisation methodology using Comsol in connection with Matlab.

Keywords: Dynamics, soil-steel bridges, soil-structure interaction, model updating, PML, particle swarm, corrugated steel plates, frequency domain assurance criteria, parallel computing

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Preface

This thesis has been done as the final part of my education at the Royal Institute of Technol- ogy, KTH. During the thesis Tyréns AB has given me the opportunity to carry it out in their central offices in Stockholm providing many resources that have been useful to successfully complete this master thesis.

First of all, I would like to thank my parents for the support given in all these years of education. Without them I would never have had all these extraordinary educational oppor- tunities, both in Spain and in Sweden. Then, I want to extend my sincerest gratitude to Johan Lind Östlund, supervisor of this master thesis and PhD. candidate at KTH, for the useful assistance, endless conversations and helping me to bear all the problems during the execution of this thesis.

After, I want to think Mahir Ülker-Kaustell who despite not being my official supervisor, has always find the time to advice and teach me about the thesis, and Andreas Andersson, PhD. at KTH, for the opportunity to help to perform the field tests. Also, to Jing Jong, from PDC support, for the interminable list of emails helping me on how to run the model faster.

Finally, I feel deeply grateful for all the amazing people I have met in Sweden who have not helped consciously to this thesis. However, they have done it just by being there whenever needed.

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Nomenclature

List of abbreviations

DAF Dynamic amplification factor

FAAC Frequency amplitude assurance criterion FEM Finite element model

FRAC Frequency response assurance criterion FRF Frequency response function

HPC High performance computer LAN Local area network

OH Overhead

PDC Parallelldatorcentrum PML Perfectly matched layer SCP Steel corrugated plate List of symbols

αs PML stretching function parameter α Parameter of the radiation damping β_s PML stretching function parameter

¨

u Acceleration [m/s²]

∆W Work of a cycle [J]

ij Strain in the i direction on the plane defined by j η Structural loss factor or structural damping γ shear strain

λ(x) PML coordinate stretching function

νij Poisson ratio in the i direction on the plane defined by j ω Angular frequency [rad/s]

ρ Density [kg/m³]

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Table of contents AF223X

σ_ij Stress in the i direction on the plane defined by j [Pa]

τ Shear stress [Pa]

ξ Viscous damping ratio Ai Wave amplitude for x = ri

B_ij Extensional properties of an orthotropic plate [N/m]

c_p Pressure wave speed [m/s]

c_r Rayleigh wave speed [m/s]

cs Shear wave speed [m/s]

Dij Bending properties of an orthotropic plate [N/m]

D Span of the culvert or diameter

E_e,i Equivalent young module in the i direction [Pa]

E_ij Young modulus in the i direction on the plane defined by j [Pa]

f Frequency [Hz]

f(x) Sub-function of the PML stretching function

Gij Shear modulus in the i direction on the plane defined by j [Pa]

H_k(ω) Frequency response function of the data set k

H_k^H(ω) Hermitian transpose of the frequency response function H_u(ω) Frequency response function

L_φ Dynamic equivalent length Lpml Length of the PML domain [m]

Ls Length of the soil in the FEM model [m]

m Mass or mass matrix [kg]

m Parameter of the PML stretching function

p_pml Curvature parameter of the PML domain in Comsol r_d Dynamic reduction factor for soil-steel culverts ri Distance from a wave propagating source [m]

s_pml Scaling parameter of the PML domain in Comsol S_up Speed-up coefficient

s number of cores

te Equivalent plate thickness [m]

t Plate thickness [m]

W Work of one load application [J]

x_j Cartesian coodinate

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1 | Introduction

In the recent years high speed lines been built or are being planned throughout the entire world, UIC (2015). Sweden will also be integrated in the European high-speed network and the project for the first section of the high-speed network, called Ostlänken, is currently in progress Trafikverket (2018b). The design standards for these lines are stricter, with train speeds exceeding 250 km/h, and the requirements in terms of passenger comfort and accelerations are more demanding. Consequently, the dynamic response takes a major importance in the evaluation and design of high speed projects. This constitutes the frame in which this project is developed.

That said, we must introduce the specific problem straightforward. In Sweden and the Nordic countries, Canada, Poland and other countries the use of soil-steel composite bridges holds an important percentage of all the bridges with small to medium span ranges, Beben (2014). They comprise a well-known, flexible and efficient solution for low speed ranges, therefore, the designers are aiming to implement a similar solution for the high-speed rail- tracks. As a result, several master thesis and research papers have been carried out in the last years. These documents are Mellat (2012) , Woll (2014), and Andersson et al. (2016).

According to Beben (2014), corrugated steel plate (CSP) culverts are used typically for span ranges between 3 and 15 m, they fulfil of all the safety requirements with lower initial and long-term maintenance costs and a lessen construction time than other conventional solutions for short-span bridges. As a consequence, their usage is constantly increasing.

1.1 The future high speed rail-network in Sweden: Ostlänken

Accodring to Trafikverket (2018a), the future network will be composed of four sections:

Ostlänken between Stockholm and Linköping; the Central Section between Linköping and Borås which includes the intersection point of Jönköping that will be the most important crossing point; the section between Jönköping and Malmö that will connect through the Öresund Link with Copenhagen and the rest of Europe, and, finally, the section between Borås and Gothenburg. Thus, the three most important cities in Sweden will be connected by the high-speed rail-network. The travel time for every connection between Malmö, Stockholm and Gothenburg will be reduced to a bit more than two hours according to the plans of Trafikverket (2018b).

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1.1. The future high speed rail-network in Sweden: Ostlänken AF223X

STOCKHOLM UPPSALA VÄSTERÅS

ÖREBRO

HALMSTAD

SÖDERTÄLJE ESKILSTUNA

KARLSTAD

TROLLHÄTTAN

KARLSKRONA KALMAR

NORRKÖPING LINKÖPING

GÖTEBORG

BORÅS JÖNKÖPING

HELSINGBORG

MALMÖ LUND

VÄXJÖ

JÄRNA

KRISTIANSTAD

NYKÖPING

HÄSSLEHOLM

TRANÅS

LJUNGBY

ESLÖV LANDSKRONA

VÄRNAMO

Figure 1.1.1: The Swedish High-Speed Rail-Network. Trafikverket (2018a)

The starting section will be Ostlänken. The railway between Stockholm’s Central Station and Järna has been already built. Between the years 2033 and 2035 Ostlänken shall be completely finished. Figure 1.1.2 shows the planed alignment of the railway and the different subsections of the project, also a schematic view of the whole Swedish High-speed Network can be appreciated.

Figure 1.1.2: Sketch of the Ostlänken Project. Trafikverket (2018b)

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1.2. Dynamic soil-structure interaction AF223X

1.2 Dynamic soil-structure interaction

In this section a brief explanation of the SSI is introduced. Essentially, it is a discipline that binds together the geotechnical and structural aspects of a structure, which are tradi- tionally considered as a separated problem, in a fairly reasonable simplification, due to the complicated calculations required, the restrains imposed by the computation capacity and the absence of further interest on the topic. As stated before, in the recent years the interest in the dynamic aspects of the structural design have increased so has done the complexity of our structures, the speeds and the comfort criteria, NEHRP (2012).

The account of soil-structure interaction in our calculations may partly explain the higher damping ratios measured in several kind of structures when performing field tests. For this reason, considering it may reduce the necessity to implement further countermeasures to mitigate the dynamic effects involved such as tuned mass dampers (TDM), additional non- structural mass or other sort of solutions commercially available. Thus, the global costs of a construction project could be reduced.

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2 | Background

2.1 Soil steel culverts

According to Pettersson and Sundquist (2014), soil steel culverts are a type of bridges that can be used for span-ranges that vary from short to medium lengths. The term culvert, in this context refers to bridges which are generally composed by a pipe or an arch formed by several thin metal plates assembled together. Then the pipe or arch is covered by engineered quality soil, which is compacted, to ensure an adequate carrying capacity of the culvert thanks to the structural interaction between the soil and the culvert. Figure 2.1.1 depicts a typical configuration of a soil steel culvert.

Figure 2.1.1: Sketch of the basic steel soil culvert configuration. Source: Pettersson and Sundquist (2014)

The plate is generally corrugated, and the structural pipe can adopt several shapes according to the project restrictions and specifications. The gamut of commercial profiles gather circular, horizontal ellipse, vertical ellipse, three-radii pipe-arch, four-radii pipe-arch, single radius arch, several radii arch and box-arch culvert (Pettersson and Sundquist (2014),Viacon (2015)).

The static problem is considered in several international design codes and articles, for example Duncan (1979) and AASHTO (2017) in the United States, CSA (2014) in Canada, or Pettersson and Sundquist (2014) in Sweden. Consequently, there is no further need to inves- tigate the static behaviour of these structures. Furthermore, to have a deeper understanding on how well developed the static solutions are and how limited research has been done related with the dynamics, it is relevant to mention that in the United States and Canada there are two associations that gather all the producers of these corrugated steel pipes in each country.

They are, respectively, the National Steel Culvert Pipe Association (NSCPA) in the United

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2.1. Soil steel culverts AF223X

States and the Corrugated Steel Soil Pipe Institute (CSPI) in Canada. In the web pages of both associations further information, design guidelines, pertinent standards, research articles, brochures, installing procedures and indications on possible applications can be found.

The NSCPA has even developed its own standard for the structural design of this systems, NSCPA (2008). However, there is not a single section related to the structural dynamics.

2.1.1 Dynamics of Soil-steel Composite Bridges (SSCB)

In Pettersson and Sundquist (2014), de facto the Swedish standard for designing soil-steel culverts, the dynamic problems are briefly mentioned. Consequently, the calculation procedure to consider the dynamic effects is rather simple and conservative. It is based on the dynamic amplification factor (DAF) method gathered in CEN/TC-250 (2013) which is slightly altered.

The proposed modification is to set the value for the determinant length (Lφ) as 2 · D, where Dis the span length. In Section 3.5.1 of Pettersson and Sundquist (2014) it is also proposed a reduction factor (rd) of the DAF based on investigations done by Smagina (2001), this modification takes into account the reduction observed in the dynamic effects when the soil cover depth is increased. The standards AASHTO (2017) and AASHTO (2002) use a similar approach based on the DAF.

Other researches on the topic include the article by Beben (2013). In his investigation Beben remarks the absence of models which emulate properly the behaviour of a soil-steel bridge. The author affirms that the lack of agreement between the reality and the models might be caused by the extensive simplifications adopted in the models and the problems that emerge when modelling the interaction between the soil and the structure. The article explains the field tests performed on one of these bridges, and compares the dynamic and static displacements and strains. The first conclusion is that there are important differences between the static and dynamic response. Therefore, further research related to the dynamics should be done. Furthermore, the author compares the field test results with the dynamic impact factors (DAF) calculated according to the American Standards, AASHTO (2002) and AASHTO (2017), which underestimate the amplification factor according to the author’s investigations. Finally, the author concludes that the most important parameters are the type of culvert, the span length, the soil cover depth at crown and the speed of the vehicle. The measurements were realised on service train loads passes.

Various research projects and field tests have been performed in the last years in Sweden and other countries. The aim and scope of these studies is normally to create a finite element model that fits well enough the actual data obtained from the field test. In the previous research several methods have been applied to match the results with the reality such as parametric analyses Woll (2014), as well as model updating procedures. The final objective of the studies is to give guidelines when creating a finite element model for soil-steel culvert bridges, to analyse the adequateness of the 2D models and to evaluate if these bridges fulfil the SLS requirements for high-speed railway loads. The field tests were also based on train passes.

In the study by Woll (2014) the conclusion states that the results are not reliable because the measured and the calculated results differ excessively. The reasons for these discrepan- cies are suggested to be caused by model uncertainties, mainly the estimation of the effective width of the 2D model, and the material parameters uncertainties. The author suggests further analysis based on 3D models calibrated with experimental data and special emphasis in a parametric analysis of the young modulus and the damping.

In Mellat et al. (2014) and Mellat (2012) the authors analyse a soil-steel bridge in Märsta creating a 3D model that is calibrated with the accelerations measured in the field tests.

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2.1. Soil steel culverts AF223X

Furthermore, he also proposes an innovative approach towards the 2D plain strain model defining the term effective width, which is calculated as the length needed for an evenly distributed stress with the same maximum value to have the same resultant load in the 3D model. This parameter is calculated for various layers and several train speeds. The results obtained with both models had good agreement with the measurements, except with some problems with the boundary conditions in the model. Finally, a parametric analysis was performed evaluating the response to different soil elastic moduli, damping ratios, Poisson’s moduli and soil densities. In this case, the author compares the model against train passages.

In the article by Andersson et al. (2012), the authors obtained a good correspondence between the measured data and the 2D and 3D finite element models. The plain strain width of the model, effective width was calculated assuming a 2:1 stress distribution. The recommendations for future research on soil steel culvert bridges (SSCB) are to use a combination of experimental testing jointly with model updating techniques. Additionally, they suggest that other bridge configurations such as longer spans and shallower fill depths should be studied. The field tests were based on train passages.

In conclusion, the available literature examination indicates that this problem has never been faced in the frequency domain with model updating techniques. In this paper, following the recommendations from previous research projects this approach is selected. Neither a controlled excitation of a SSCB has been made before, using a hydraulic exciter. These techniques for analysing soil structure interaction are well known and have been applied before for other kind of problems such as cap-pile foundations or portal-frame bridges.

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3 | Aim & scope

The final target of this master thesis is to create a finite element model that is able to represent accurately enough the dynamic behaviour of a soil-steel composite bridge subjected to a harmonic load.

Firstly, field tests were carried out in a soil-steel composite bridge located in Hårestorp.

During the field tests a dynamic analysis with a hydraulic exciter was performed. The results obtained are the frequency response functions (FRF) of the bridge accelerations at several points. These results will be used as a reference for the finite element models (FEM), and the base input for the model updating procedure.

Secondly, an additional intermediate objective was to determine if a 2D FEM of these structures would be sufficient to represent correctly, within a reasonable error margin, in their entirety the phenomena occurring when a steel-soil bridge is excited by a harmonic load. The results are compared against the field tests FRFs.

Additionally, a 3D model will be created in order to compare it with the latter model and the field tests results. If the 2D is not able to cope with the reality, the 3D model will gain a major importance. Otherwise, a comparative analysis considering the computational costs and the accuracy of the two models will be performed. As this model would require a vast computational effort, methods to parallelise the computations and optimise the performance would be required. Thus, a literature review on high performance computation, parallel computation and how to implement it in Comsol shall be carried out. As a result, this research project would provide a method as well to increase the computational efficiency of this particular problem in Comsol.

Finally, either the 2D model, the 3D model or both would be used for implementing a model updating procedure. The decision of which one would be the most appropriate for the task will be made according to the previous hypothesis and objective criteria. To implement the model updating algorithm, Matlab will be used, by reason of its powerful set of optimisation tools and the great connection that Comsol and Matlab have. A succinct analysis of the different optimisation algorithms available will be effectuated in order to select the one that is the most adequate for this topic. The goal of this algorithm will be to obtain the closest similarity in terms of the Frequency Response Functions (FRF) between the models and the field tests performed during this master thesis in the Hårestorp Bridge.

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4 | Theory

4.1 Frequency Response Function (FRF)

For the derivation of this section a single degree of freedom system has been used. The derivation for multi-degrees of freedom systems is analogous. The basic equation that rules the dynamic behaviour of an elastic body is Equation 4.1.1. For further information on how to derive this equation, Chopra (2007) could be reviewed.

m · ¨u + c · ˙u + k · u = p(t) (4.1.1) where m is the mass, ¨u is the second derivative of the displacement, c is the viscous damping coefficient, ˙u is the first derivative of the displacement, k is the stiffness, u is the displacement, and p(t) is a time dependent load.

For this master thesis, it is required to modify Equation 4.1.1 applying a harmonic load, p(t) = P · e^iωt. As well, the damping is set to be structural, η, and it is expressed as a ratio of the stiffness. The result of these operations is gathered in Equation 4.1.2.

m · ¨u + k · (1 + i · η) · u = 1 · e^iωt (4.1.2) According to Chopra (2007), Equation 4.1.2 admits the solution indicated in Equation 4.1.3 for the steady state.

u(t) = Hu(ω) · e^iωt (4.1.3)

where Hu(ω) is the frequency response function (FRF) of the displacement.

Therefore, the solution gathered in Equation 4.1.3 is substituted into Equation 4.1.2 obtaining Equation 4.1.4

Hu(ω) · eîωt·−ω²+ k · (1 + i · η) = eîωt (4.1.4) Finally, reordering and cancelling the term eîωt. The frequency response function (FRF) for the displacement, Hu(ω), can be computed through Equation 4.1.5.

H_u(ω) = 1

−ω²· m + k · (1 + i · η) (4.1.5) A similar solution could be found for the FRF of the accelerations, Ha(ω). This is realised just by obtaining the second derivative of the solution for the steady-state case, gathered in Equation 4.1.3 and substituting in Equation 4.1.5. The solution is reflected in Equation 4.1.6.

H_a(ω) = −ω²· H_u(ω) = −ω²

−ω²· m + k · (1 + i · η) (4.1.6)

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4.2. Hysteretic damping AF223X

4.2 Hysteretic damping

In dynamic soil-structure interaction there are two types of damping involved, the radiation damping due to the wave propagation and the material damping inherent to the soil. In this master thesis the approach followed to model the material damping is based on the hysteretic damping which is explained in this section. In order to measure this damping, the material is subjected to a cyclic load and the work of a cycle is obtained. The expression of the material damping, also known as structural damping may be seen in Equation 4.2.1.

η = 2 · ξ = 1 2π·∆W

W (4.2.1)

being W the maximum potential energy of the cycle and ∆W the energy dissipated in the soil.

Equation 4.2.1 can be derived according to Ishihara (1996) assuming a linear-viscoelastic behaviour of the soil. Supposing that we have a cyclic acting shear stress that can be represented by Equation 4.2.2

¯

τ = τr+ τ · i (4.2.2)

And a complex shear deformation represented by Equation 4.2.3:

¯

γ = γ_r+ γ · i (4.2.3)

The stress-strain relation of a hysteretic cycle is defined by Equation 4.2.4:

τ = µ · γ + µ⁰·p

γ_a− γ² (4.2.4)

The viscous part of Equation 4.2.4 is showed in Figure 4.2.1, which is the part that behaves as a damper.

Figure 4.2.1: Viscous part of the stress-strain relation. Ishihara (1996)

If we integrate we can obtain the value of the work over a cycle, which is the energy dissipated by the damping behaviour. The result is showed in Equation 4.2.5:

∆W = Z

τ · dγ = µ⁰· π · γ_a² (4.2.5) And the maximum potential energy of the cycle is indicated in Equation 4.2.6:

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4.2. Hysteretic damping AF223X

W = 1

2 · µ · γ_a (4.2.6)

Thus, the structural loss factor can be obtain using Equations 4.2.5 and 4.2.6. The result is indicated in Equation 4.2.7

η = µ⁰ µ = 1

2π·∆W

W (4.2.7)

where µ is the elastic modulus and µ⁰ is the loss modulus which is a characteristic parameter of the energy dissipating properties of a viscoelastic material.

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4.3. Wave propagation problems AF223X

4.3 Wave propagation problems

A wave propagation problem, in the context of a three-dimensional elastic media, is dominated by the following equations. Equation 4.3.1 is the equilibrium equation, Equation 4.3.2 is the material constitutive equation and Equation 4.3.3 is the deformation tensor. The reference for this section is Basu (2008).

X

j

∂σ_ij

∂xj

= ρ · ¨u_i (4.3.1)

σij =X

k,l

Cijkl· ε_kl (4.3.2)

ε_ij = 1 2· ∂u_i

∂x_j +∂u_j

∂x_i

(4.3.3) Where σij is the stress in the in the direction of xi on the plane define by xj, xi is the cartesian coordinate i, ρ is the density, ¨ui is the acceleration in the xi direction, Cijkl is the constitutive relation tensor, ij is the strain in the direction of xi on the plane define by xj, and ui is the displacement in the direction xi.

This collection of equations admits four kind of solutions. Each solution represent a different kind of elastic wave. Thus, the solution is composed of four different waves. The P or pressure waves, the S or shear waves, the Rayleigh waves and the Love waves. The shape and propagation behaviour of the P-wave can be seen in Figure 4.3.1. Equation 4.3.4 is the expression to calculate the wave speed of the P-waves.

c_p = s

λ + 2 · G

ρ (4.3.4)

where λ is the 1^st Lamé parameter, G is the shear modulus and ρ is the density of the elastic media.

Figure 4.3.1: P-wave. Source: Mishra (2018)

The shape and propagation behaviour of the S-wave is depicted in Figure 4.3.2. The wave speed of the S-wave can be calculated with Equation 4.3.5

cs= s

G

ρ (4.3.5)

Figure 4.3.2: S-wave. Mishra (2018)

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4.3. Wave propagation problems AF223X

Figure 4.3.3 sketches the shape and propagation behaviour of the Rayleigh waves. Equa- tion 4.3.6 is the most normal relation between the shear wave’s and the Rayleigh wave’s speed. It can be used to calculate the wave speed of the latter.

c_r

c_s = 0.862 + 1.14 · ν

1 + ν (4.3.6)

where cs is the speed of the shear wave and ν is the Poisson’s modulus of the elastic media.

Figure 4.3.3: Rayleigh-wave. Mishra (2018)

Figure 4.3.4 shows the shape and propagation behaviour of the Love waves. There is no general expression to calculate the wave speed of the Love waves, however, many empirical formulas and wave speed ranges could be found in the literature.

Figure 4.3.4: P-wave. Mishra (2018)

4.3.1 Radiation damping

The radiation damping is the one due to the expansion of energy in the wave propagation process. It is based on the principle of conservation of energy. When the wave expands the wave front occupies a higher area. Hence the energy is distributed over a greater surface, and the amplitude of the displacements and accelerations is diminished. This damping is the most important contributor to the dissipation of energy in the area of interest. There are many models for the radiation damping, one of the most common ones is gathered in Equation 4.3.7 based on the technical report Björn Möller and Moritz (2000).

A₂

A₁ = r₂ r₁

−n

· e^−α·(r²^−r¹⁾ (4.3.7)

Where n = 0.5 for L-waves and R-waves, n = 1 for P-waves and S-waves, r1 is the inner radius and r2 is the outer one. The parameter α is defined by Equation 4.3.8 and depends on the damping ratio (ξ), frequency (f) and the wave speed of the shear-wave (cs).

α = 2 · ξπ

c_s (4.3.8)

One can see, analysing Equation 4.3.8, that the radiation damping is frequency dependent.

Therefore, it is a non-linear parameter.

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4.4. PML AF223X

4.4 PML

The PMLs are a kind of finite element that perform a transformation of coordinates between the real plane and the complex plane, they were originally developed in the article Berenger (1994) for electromagnetic waves and then applied to several physical problem by other authors. Such a manipulation of coordinates, if the correct stretching function is used, provokes that the solution of the wave equation fades away in the PML domain as explained in Basu (2008). This said, the PMLs are theoretically an almost perfect absorbing boundary conditions for both evanescent and propagating waves.

Despite obtaining good results for some standard configurations, an attempt to go further analysing a simple problem and comparing it to previous research studies has been carried out. Unluckily, the results are not conclusive, and it was impossible to obtain general recommendations on how to choose the parameters that rule the behaviour of these elements.

Until further research is done, the general recommendation is that a tailor-made sensitivity study for each PML’s parameter should be performed individually case by case and software by software. Figure 4.4.1 illustrates the basic behaviour of a PML region as an absorbing boundary condition for wave propagation.

Figure 4.4.1: Behaviour of a PML layer as an absorbing boundary. Source: Kucukcoban and Kallivokas (2013)

4.4.1 Literature research

The variety of stretching functions, the number of relevant parameters, the diversity of im- plementations and the difficulty to analyse the problem further away than a conglomerate of mathematical equations makes the tuning a rather complicated problem. In the articles Kucukcoban and Kallivokas (2013) and Fontara et al. (2018) an analysis of the absorbing efficiency of the PMLs against the stretching function parameters was effectuated. Both articles start with the following general stretching function given in Equation 4.4.1.

λ (x) = α_s− i · β_s

ω (4.4.1)

Where αs, the real part of the function, is called the scaling function. It scales the coordinates and imposes an attenuation on evanescent waves. βs is the attenuation function.

It accounts for the amplitude decay of the propagating wave once it enters the PML as

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4.4. PML AF223X

explained in Fontara et al. (2018). According to the same article some formulations for soil-structure interaction problems ignore completely the attenuation of evanescent waves by setting αs equal to 1. On the other hand, high values of the scaling can cause the numerical pollution of the results. Consequently, the authors propose a stretching function which is different for the scaling and the attenuation parts. The mapping function is gathered in Equation 4.4.2.

λ (x) = 1 + f (x) − i · f (x) α0

with α0= b · ω

c and f (x) = f0· x Lp

m

(4.4.2)

Kucukcoban and Kallivokas (2013) propose the following formulation for the stretching function which can be seen in Equations 4.4.3 and 4.4.4.

αs(s) =

(1 , 0 ≤ s ≤ s₀,

1 + α0

h(s−s0)·ns

LP M L

im

, s0 < s < st, (4.4.3)

βs(s) =

(0 , 0 ≤ s ≤ s₀,

1 + β0

h(s−s0)·ns

LP M L

im

, s0 < s < st, (4.4.4)

The conclusions in terms of absorption performance can be explained by Figures 4.4.2 and 4.4.3.

(a) ADF for several β0 with m = 2 (b) ADF for several m with β₀= 4 Figure 4.4.2: Amplitude Decay Factor (ADF) across the PML domain for an m and β₀ parametric analysis. Source: Kucukcoban and Kallivokas (2013)

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4.4. PML AF223X

(a) |R| for several f0 with m = 1 (b) |R| for several m with f₀= 10

Figure 4.4.3: Amplitude of the reflected wave (|R|) for an m and f₀ parametric analysis.

Source: Fontara et al. (2018)

As stated in both articles, high values of the scaling parameter lead to numerical inconsis- tencies. Moreover, it is recommended to have a decaying function that varies as much linear as possible within the boundaries of the PML domain. As a conclusion values of β0≈ f₀ ∈ (1, 8) and m ∈ (1, 4) will lead to fairly good results. Apart from these, other properties such as the wave speed, the frequencies analysed and the relative length of the PML in comparison with the maximum wavelengths are also decisive factors.

4.4.2 PML in Comsol

The parameters that affect more the behaviour of the PML are the scaling parameter (spml) and the celerity of the reference wave (cref). The Comsol Manual defines the stretching function of the PML according to Equation 4.4.5, 4.4.6 and 4.4.7:

∆x = λ · f_i(ξ) − ∆_w· ξ (4.4.5)

where

f_p(ξ) = s_pml· ξ^p^pml· (1 − i) (4.4.6) and

ξ = xpml

L_pml (4.4.7)

Not being completely sure about the correlation of these equations and the PML functions implemented in Comsol reflected in Section 4.4.2 based on Comsol (2018), it seems reasonable to think that the parameters α0 and β0 as well as the parameter f0 from Fontara et al.

(2018) stretching function are related somehow with the parameter spml of the Comsol PML implementation. Likewise, the parameter ppml from Comsol is connected with the parameter m of the articles by Fontara et al. (2018), Kucukcoban and Kallivokas (2013) and Basu (2008).

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4.5. Corrugated steel plates (CSP) AF223X

4.5 Corrugated steel plates (CSP)

The corrugated plates can be theoretically approximated to orthotropic equivalent plates with different flexural and extensional properties, as investigated in Wennberg et al. (2011). This reduces dramatically the degrees of freedom of the finite element model. In order to solve this problem the plate should be modelled with an orthotropic material and a finite element that has an orthotropic behaviour in which the axial stiffness is defined by an elastic modulus that is different to the one used for the bending stiffness. The equivalent parameters, which are based on the article by Wennberg et al. (2011), could be obtained with Equations 4.5.1 and 4.5.2:

Exx = 12 · (1 − ν_xy· ν_yx) t³ · D_xx E_yy = 12 · (1 − ν_xy· ν_yx)

t³ · D_yy Gxy = 6 · Dxy

t³

(4.5.1)

Where Eii is the Young’s modulus in the xi direction, νij is the Poisson’s ratio in the direction of xion the plane defined by xj, t is the thickness of the corrugated plate, Diiis the flexural stiffness in the xi direction, Gxy the shear flexural modulus and Dxy the torsional stiffness.

E_xx^e = B_xx t E_yy^e = Byy

t G^e_xy = B_xy

t

(4.5.2)

Where E_ii^e is the extensional Young’s modulus, Bii is the axial stiffness, Bxy is the shear stiffness and G^exy is the extensional shear modulus.

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4.6. Model updating AF223X

4.6 Model updating

4.6.1 Optimisation problem

An optimisation problem could be defined as the search of the best feasible solution according to a certain objective and subjecting the variable space to certain constraining conditions. In advanced problems, in order to be able to find a solution, the use of computational methods is required. All the methods are based on a certain solver which modifies the relevant parameters or variables taking into account all the restraints until the global minimum of the objective function chosen is reached. According to Matlab (2018) an optimisation technique or optimisation computational method is a set of procedures that "are used to find a set of design parameters or decisions that give the best possible result". The most relevant concepts are more extensively elaborated:

• Solver: a solver is a mathematical software that takes as input a description or char- acterisation of a problem and provides as output the solution of the problem previously defined as an input. In this particular case, the result is a set of variables that optimise the objective function. There are several kinds of optimisation solvers that can be gathered in two basic groups, local and global optimisation solvers. The local optimisation solvers are based on mathematical techniques which are able to solve linear, quadratic, least squared and nonlinear optimisation problems. Each of the problems has one or more associated solvers that will optimise satisfactorily the solution.

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 finite element model. Therefore, the solvers considered in this section are based on global optimisation techniques. In the frame of global optimisation, there are several solvers. Whereas some of them are based on biological or natural phenomena, such as Particle Swarm, Genetic Algorithm, or Simulated Annealing, others, such as Pattern Search, are based on direct searching procedures dynamically meshing the parametric space. Surrogate Optimisation is another kind of solver that builds up an estimation of the objective function after a prefixed number of iterations and uses it to calculate the global minimum faster. This solver is good for time-consuming objective functions. The last solvers consider for global optimisation in this master thesis apply gradient-based techniques starting at several points. They look for local minima that lead the whole optimisation process to a global minimum. These solvers are Global Search and Multiple Starting Point. The main disadvantage they have is that the functions to optimise must be smooth. In this report the multiple objective solvers have been discarded, because a well defined objective function could combine the assurance criteria. Moreover, they limit the range of solvers that can be used.

• Objective function: the objective function is the mathematical expression that gathers all the goals that are the object of the optimisation. The goals can be multiple depending on the field, however the most used ones are reducing the costs, or the use of a certain material or, as in this report, improving a model by increasing the similarity of the results when compared to the reality. When evaluated, the objective function must give a scalar value that reflects simultaneously all the objectives of the problem in order to lead the solution to the desired optimum.

• Constrains: constraint are set limits for the variables object of model updating when solving an optimisation problem. They could represent many conditions such as reasonable values, or minimum and maximum values gathered in the standards among others.

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These restraints can be linear, non-linear, a lower boundary or an upper boundary. In this mater thesis the constrains used are lower and upper boundaries.

4.6.2 Objective function

The theory behind the objective function is based on the article Zang et al. (2001) in which the authors reviewed the main frequency-domain criteria for correlating and updating dynamic finite element models. The authors propose a new objective function that gathers frequency and amplitude criteria. Equation 4.6.1 reflects the frequency response assurance criterion, which has as output a scalar value between 0 and 1. It evaluates the degree of correlation of the frequency response between two FRFs, being 1 a perfect correlation and 0 an absolute lack of correlation.

F RAC_j =

{H_X(ω_i)}^H_j · {H_A(ω_i)}

2

({H_X(ω_i)}^H_j · {H_X(ω_i)}_j) · ({H_A(ω_i)}^H_j · {H_A(ω_i)}_j) (4.6.1) Where HX(ω_i) is the FRF of the field tests in vector format that corresponds to the frequency ωi, HA(ω_i)is the FRF calculated in a model in vector format for the frequency ωi, and H^H indicates the hermitian transpose of the respective FRF.

Equation 4.6.2 calculates the degree of correlation between the field measured FRF and the model computed FRF in terms of amplitude response. Likewise the FRAC, it takes scalar values between 0 and 1, where values closer to 1 indicate a better degree of correlation.

F AAC_j =

2 ·

{H_X(ω_i)}^H_j · {H_A(ω_i)}

({H_X(ω_i)}^H_j · {H_X(ω_i)}_j) + ({H_A(ω_i)}^H_j · {H_A(ω_i)}_j) (4.6.2)

As explained in Section 4.6.1, it has been rejected to consider the problem as a multiple objective one given that there is no particular advantage in formulating this specific problem following that approach. Consequently, it is needed to gather the two previous theoretical equations with a unitary mathematical expression that combines both, the evaluation of the frequency correlation, measured in terms of FRAC, and the amplitude correlation, measured in terms of FAAC. In order to do that, the expression proposed in Zangeneh et al. (2017) for a model updating procedure based on FRAC and FAAC indicators is used. The expression is showed in Equation 4.6.3. Performing a simple analysis of the formula indicates that, contrary to the previous indicators, lower values reflect a better fitting being 0 a perfect correlation.

Given that the expression is a summation for every evaluation point of a function that can take values up to 2, the expression could take any value in the range between 0 and 2 · N being N the number of points in which it is being evaluated, in this case accelerometers.

Obj =X

j

(F RAC_j− 1)²+ (F AAC_j− 1)² (4.6.3)

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4.6.3 Procedure

Figure 4.6.1 clarifies how the model updating technique will work. The updating of the variables is performed with the solver optimisation algorithm. Some convergence criteria could be also set in advanced to stop the procedure when some parameters reach a certain value.

Figure 4.6.1: Flowchar of the model updating procedure. Source: Svedholm (2017)

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4.7. Parallel computing AF223X

4.7 Parallel computing

4.7.1 Introduction to parallel computing

According to Wikipedia (2019a), parallel computing is "a type of computation in which many calculations or the execution of processes are carried out simultaneously". In the last decades the processors have increased the computational capacity increasing the number of cores, getting rid of the previous paradigm based on increasing the processor’s base frequency. In order to take full advantage of the capabilities of multi-core processors and multi-processors computers, parallelisation software techniques must be used. The particular implementation of these techniques are dependent on the hardware, software, and type of problem. The most relevant factors are the total number of cores, the number of cores assigned to each job, the communication interfaces between the cores and the memory inside the cluster and inside the node. Furthermore, the degree of parallelisation of the problem and how the RAM is installed across the memory have also a major impact. To clarify the most important concepts of parallel computing the following references have been used: Comsol (2019a), Comsol (2019b), Barney (2019) and Wikipedia (2019b). The basic concepts required for the problem are the following:

• Multi-core: when a single processor or central processing unit (CPU) is made up out of 2 or more cores or processing units which can run instructions independently. The cores are connected to each other through a certain network interface, normally all the cores have the same set of instructions and the same characteristics. Although, some of them could be heterogeneous or differ significantly from the standard configurations.

• Multiprocessor: if a computer system is built up with more than one CPU then it constitutes a multiprocessor computing system. All the processors are linked by means of a certain network interface like a computer bus, and share the memory and peripherals in order to carry out tasks in a parallel way.

• Node: in high performance computers (HPC) the computer system is, indeed, a network of computers. Put it in another way, they are distributed systems. Normally these computers have several computation nodes that act themselves as a proper independent computer, and all of them communicate through the local area network (LAN) to act as a single computer.

• Cluster: a set of 2 or more computation nodes in a LAN constitute a computer cluster.

The main advantages of using this computer system is the flexibility, adaptability and expandability compare to single computers. They are a cheaper alternative than single computers. For HPC, a parallel computer would be less costly than a single computer with equivalent characteristics.

• Shared memory: when an application or a process is run in a computer it creates several derived processes, normally called threads. If the memory that the original process requires is shared among all the threads, and all the threads have access to all the variable and data, the memory is shared. The main advantage is that, as the threads have direct access to all the variables, there is no need for further communication. However, this is not completely true because there is always some need for synchronisation in order to avoid threads to perform the same task or read the same information simultaneously. When a problem is parallelised, it is quite an important requirement to use shared memory. The main problem is the difficulties one could face trying to increase the amount of memory, put it in another way, it does not scale well.

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Figure 4.7.1: Scheme of a computer node with shared memory. Mattsson (2014)

• Distributed memory: as opposite to the previous case, in this one each individual thread is assigned a certain amount of memory. Therefore, there is a need of explicit communication between the different threads. However, there are certain computational techniques to minimise the communication and synchronisation needs. The main advantage is that more memory could be easily added to the computer system, so called scalability.

Figure 4.7.2: Scheme of a computer node with distributed memory. Mattsson (2014)

• Hybrid parallel computing: modern computer clusters or HPC use both shared memory, at the node level, and distributed, at the cluster or network level. The use of both types of memory improves the scalability, minimises the message passing overhead and reduces the hardware initial investment. At the same time, it increases the programming complexity.

Figure 4.7.3: Scheme of a computer node with distributed memory. Mattsson (2014)

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• Overhead: the overhead is the time expend applying some of the computational resources to operate with the data, packaging and transmitting it. It is needed when a program requires communication between different tasks. Other times there is also a need for synchronisation, and the program just needs to wait until the synchronisation is finished to proceed further. The worst case is when the network bandwidth is saturated. The performance, to an important extend, depends on the bandwidth, the maximum data rate that can be transmitted by the network interface, and the latency, time spend to send a minimal communication between two points. One could effortlessly deduce that if the program sends many small messages the overhead will mainly be provoked by the latency. On the other hand, if the software requires to send a few messages containing vast amounts of information, the overhead will more likely be caused by the bandwidth. Other factors to consider are the topology of the network, the scope of the communications and if they are synchronous or asynchronous.

4.7.2 Parallelisation techniques

According to Barney (2019), there are two approaches when it comes to design a parallel program. One is data parallelisation, also called domain decomposition. The other is func- tional decomposition, frequently referred to as task parallelisation. They are both based on partitioning and decomposition, which is breaking the problem in discrete subtasks of work that can run simultaneously. Without parallelisation these would run one after the other.

The following text explains briefly the differences between both techniques:

• Data parallelisation: the data that defines the problems is split up in as many portions as we decide. Normally these portions are equal, and every parallel task performs basically, the same operations but working on different data.

• Task parallelisation: in this case the focus is put on the tasks that are required rather than on the data. Therefore, the parallel tasks are not equal and in many occasions they depend, up to a certain extent, on each other forcing the communication of the parallel tasks. An example of this could be to model the climate, in which we can split the task in the atmosphere, hydrology, ocean and land but it is clear that all the elements are interconnected.

• Mixed parallelisation: certain problems require a combination of both approaches.

The mixed parallelisation procedures will use both, data and task parallelisation, simultaneously.

The batch sweep functionality of Comsol is a parallelisation technique based on data parallelisation procedures.

4.7.3 Theoretical Speed-up. Amdahl’s and Gustafsson’s Laws

The Amdahl’s Law derivation is gathered in Equations 4.7.1, 4.7.2 and 4.7.3. The main assumptions are that the workload is constant, independently from the number of cores available, and the jobs we run are not completely parallelisable.

T = (1 − p) · T + p · T (4.7.1)

T = (1 − p) · T + p

s· T (4.7.2)

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T (s) = (1 − p) · T + p

s · T ⇒ S_up(s) = T · W

T (s) · W = T

T (s) = 1

1 − p · ^s+1_s (4.7.3) Where T is the total time, p is the parallelisable percentage, s is the number of cores, W is the workload and Sup is the speed-up coefficient.

Figure 4.7.4 depicts the speedups coefficients for several parallelisation percentages as a function of the number of cores. It can be clearly seen that for each parallelisation degree there is a maximum speedup value and this value is reached for a relatively low number of cores, specifically for low parallelisable problems. Consequently, this implies that for each problem there is a trade-off between the number of cores and the increment in performance.

In order to clarify this further, lets take 2 examples from Figure 4.7.4. On the one hand, for a parallelisation degree of 50% installing processors with more than 16 cores seems totally unreasonable because it is close to the limit of 4 times in performance increment. On the other hand, for high parallelisable problems, like the case with 96% of parallelisable processes, the efficient number of cores will be around 512 and the maximum speedup factor 25. This law reflects better an immediate behaviour when a model is run on one relatively low-performance computer and, alternatively, in an HPC. For example if we would have a relatively advanced 3D model, and we try to run it on a normal computer, then we would probably realise that the amount of time it takes to run is not practical for some purposes, e.g. parametric analysis.

Then, we look for a more powerful system, in order to speed-up the model. This law does not take into account the observed psychological effect of tending to run more advanced models when a better hardware is available.

Figure 4.7.4: Amdahl’s law for several parallelisation percentages

the derivation of the Gustafson-Barsis’s Law is given Equation 4.7.4 and Equation 4.7.5.

In contrast with the Amdahs’s Law, the work load in this case is a function of the number of cores.

W = (1 − p) · W + p · W ⇒ W (s) = (1 − p) · W + s · p · W (4.7.4)

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Sup(s) = T · W (s)

T · W = W (s)

W = 1 − p − s · p (4.7.5)

Where T is the total time, p is the parallelisable percentage, s is the number of cores, W is the workload and Sup is the speed-up coefficient.

Figure 4.7.5 depicts the Gustafson-Barsis’s Law for several parallelisation percentages.

One can see that, in this case, as the workload (W) depends on the number of cores (s), with the same computing time, the parallelisation speedup varies linearly with the number of cores. This law reflects a more general tendency towards the future than an actual case.

Therefore, it is relevant when there is a need to install a more powerful computer. It is caused by a psychological tendency observed that affirms that the people tend to run more advanced and sophisticated models when they have access to a more powerful computer. The principle behind is that for a given initial computer if with a simplified model it was assumed a time (T) as reasonable, with a more powerful system, a typical person will tend to increase the work load until the model runs in the same reasonable assumed time. Thus, the workload is risen by increasing the complexity of the models run.

Figure 4.7.5: Amdahls law for several parallelisation percentages

4.7.4 Communication costs

Equation 4.7.6 reflects the communication costs in a simplified theoretical way. It is based on the concept of overhead discussed in the Section 4.7.1. The overhead (OH) is measured by units of time, which is added to the total time of the Amdahl’s Law. Therefore, the speedup (Su) is diminished with respect to the original Amdahl’s Law.

OH(s) = c · f (s) ⇒ Sup(s) = 1

p/s + (1 − p) + c · f (s) (4.7.6) Where c is a constant, and f(s) is a function of the number of cores that depends on the network and the problem to solve.

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Figure 4.7.6 depicts the communication cost as a function of the number of cores available.

The different plots indicate what happens when different theoretical communication functions are used. The function that better evaluates a particular problem depends on the topology of the network, how the software manages the information, if the programmers have applied good practices and on the nature of the particular problem to be solved.

Figure 4.7.6: Speedup for different overhead functions

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5 | Method

For modelling purposes, the finite element method (FEM) was used. The choice of the finite element software was based firstly on the self-imposed requisite to compare Infinite against Perfectly Matched Layer (PML) finite elements through, both, a literature review and a parametric analysis. The intention was to verify the initial hypothesis of the PML being a superior quality boundary condition in terms of absorbing efficiency than the infinite elements. Therefore, a software with a commercial implementation of the PMLs needed to be used, restraining the list of finite element model (FEM) software suitable for the purpose.

As a result, among other interests related to future research projects in which the Bridge Department of KTH will take part in, it was decided to use Comsol as the main FEM software of this master thesis.

5.1 Field tests

As a part of this research field tests have been carried out, in order to obtain the measurements that will be used to compare the model against. The field tests were performed in collaboration with the PhD. Andreas Andersson, KTH and ViaCon. The campaign consisted of performing a dynamic response analysis in three steel-soil composite bridges located in the south of Sweden. The equipment required to execute the test was based on a hydraulic load cell, a pump, a signal controller and analyser, uni-axial accelerometers, tri-axial accelerometers, displacement transducers, strain gauges and two computers. Only one bridge was analysed in this master thesis, the one situated in Hårestorp. Table 5.1.1 indicates the GPS coordinates of the examined bridge:

Table 5.1.1: Coordinates indicating the location of the bridge Bridge GPS Coordinates

Hårestorp 56°55’25.1"N 14°26’29.9"E

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5.1. Field tests AF223X

Figure 5.1.1 displays the location of the Hårestorp bridge in which the field tests were planned to be carried out. The point in light blue indicates the actual location.

Figure 5.1.1: Location of the bridge Hårestorp. WeGo (2019)

Figure 5.1.2 shows the loading equipment including the hydraulic exciter or load cell and the hydraulic pump.

(a) Hydraulic exciter (b) Hydraulic pump

Figure 5.1.2: Loading equipment

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Figure 5.1.3 depicts partially the measuring equipment. It could be seen the cables that connect the accelerometers with the signal controller and the computers that are used to store the data of the measurements.

(a) Computer and signal controllers (b) Computer for the triaxial accelerometers

Figure 5.1.3: Computers for data collection and signal processing units

The measuring procedure starts with the installation of all the equipment and measuring devices connecting the accelerometers to the computer and signal processing unit using cables.

Figure 5.1.4 shows where the triaxial and uniaxial accelerometers are positioned, as well as the installation point of the strain gauges and the displacement sensors. The devices in red called ai are the accelerometers. The points in green with names di indicate where the displacement sensors are located. In blue, one can see the tri-axial accelerometers, cxi and the strain gauges with the symbol ei. The structure located in the mid-span section of the bridge is the exciter.

20.7

3.9

4.9

N Alvesta

Värnamo

0.150

0.050 0.005

1.2

4.2 4.2

A A

A-A B

B

B-B

0.9 1.1

a5 a3 a4

a2 a1

3.5 d1

d2 d3 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

e1 e2

2.1 2.1 2.1 2.1

a16 1.5

32 sleeper s (appr

ox 20 m)

a12

scale 1:10

10 m 5

0 scale 1:100

01 2018-09-18 1:100 (A3) A. Andersson

Instrumenta�on drawing Järnvägsbro Hårestorp

km 212+382, Värmano - Alvesta Instrumenta�on and tes�ng performed 2018-09-10 Data collected with MGCPlus

Sample rate 1200 Hz, 500 Hz LP-ﬁlter a1-a16: SiFlex DF1500S uni-axial accelerometers cx1,cx2: Sensr CX1 tri-axial accelerometer e1,e2: HBM 1-LY11-6/120 strain gauges d1: displacement measured by the MTS hydraulic actuator d2,d3: HBM WA10 LVDT

1.3

4.2 0.4

Värnamo

x y

Figure 5.1.4: Instrumentation planned for the analysed bridge in Hårestorp.

Afterwards, once the installation had been finished, it was required to ensure that every-

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thing was properly connected. Among other checks, it had to be verified that all the sensor were emitting their correspondent signal to the computer. The hydraulic load cell applied harmonic loading to the structure. The frequency was gradually increased with a frequency sweep rate. By this means, if the sweep rate is slow enough, the steady-state response is gained for all frequencies within the frequency interval.

Several frequency sweeps, with different load amplitudes and frequency ranges were exe- cuted. Also, the acceleration from commercial train passages were measured simultaneously with the accelerations provoked by the exciter. Table 5.1.2 gathers the set of values used in the field tests for each sweep and the number of train passes measured. The reason to perform tests with other load amplitudes was to detect a possible non-linear behaviour. The sweep rate is for all the sweeps equal to 0.05 Hz/s and the sampling rate was set to 1200 Hz.

Table 5.1.2: Description of the field test succesful sweeps Number f_min [Hz] fmax [Hz] Famp [kN] Trains

1 1 80 1 1

2 1 80 1 0

3 1 80 1 2

4 1 80 10 0

5 10 40 20 1

6 10 25 5 0

The data was processed afterwards with a Matlab script using the Fast Fourier Transform (FFT) to obtain the Frequency Response Function (FRF) from the accelerations. Further- more, a Tukey Window was applied in order to remove the FRF component of the frequencies excited by the trains that passed by during the tests to have a clear response of the dynamic load applied. Finally, a smoothing was carried out, using the smooth filter of Matlab, on the FRF as an attempt to reduce small peaks caused by the measurement error or negligible ab- normalities in the material properties or the geometry that would be impossible to represent in a Finite Element Model (FEM).

Unfortunately, the data from the uniaxial and triaxial accelerometers was collected in two different computers and it was impossible to activate the data collection simultaneously.

Therefore, the synchronisation of both data series was needed. Lamentably, it was not possible in this Master Thesis to carry out a successful synchronisation. Consequently, only the data from the uniaxial accelerometers has been used.