Dynamic Analysis of the Skyway Bridge Assessment and Application of Design Guidelines

EXAMENSARBETE INOM SAMHÄLLSBYGGNAD, AVANCERAD NIVÅ, 30 HP STOCKHOLM, SVERIGE 2017

DANIEL ANDERSSON

ERIC THUFVESSON

KTH SKOLAN FÖR ARKITEKTUR OCH SAMHÄLLSBYGGNAD c Daniel Andersson and Eric Thufvesson Royal Institute of Technology (KTH) Department of Civil and Architectural Engineering Division of Structural Engineering and Bridges Stockholm, Sweden, 2017 Abstract

In recent years the design of pedestrian bridges has become more slender. As a result the bridges has lower natural frequencies and are more prone to excessive vibrations when subjected to dynamic loads induced by pedestrians. Akademiska Hus are building such a bridge at Nya Karolinska Solna where the bridge will span over Sol- navägen connecting the hospital building, U2, and the research facility BioMedicum. Due to practical reasons, it is not possible to connect one of the bridge ends mechanically to the building which increases the risk for lateral modes in the sensitive frequency range of 0-2.5 Hz. The increased risk of lateral modes of vibrations within the sensitive frequency range as well uncertainties when determining the dynamic response led to this thesis. This thesis covers a frequency analysis of the previously mention bridge and an evaluation of the dynamic response under pedestrian loading by implementation of several design guidelines. A literature review was conducted with the aim of giving a deeper knowledge of human induced vibrations and the relevant guidelines for modelling of pedestrian loading. Furthermore, a parametric study was conducted for parameters which might be prone to uncertainties in data. The investigated parameters were the Young’s modulus for concrete and the surrounding fill material as well as the stiffness of the connection to BioMedicum. The parametric study yielded a frequency range of 2.20-2.93 Hz for the first lateral mode and 5.96-6.67 Hz for the first vertical mode of vibration. By including non- structural mass the lower limit for the frequencies were lowered to 2.05 and 5.59 Hz in the first lateral and vertical mode respectively. The parametric study also showed that the largest impact on the natural frequencies were obtained by manipulating the parameters for the supports, both for BioMedicum and the substructure. The implementation of the guidelines resulted in a lateral acceleration between 0.05 and 0.599 m/s2. No evaluation was conducted for the dynamic response in the vertical direction due to a natural frequency of 5.59 Hz, which is higher than the evaluation criteria stated in Eurocode 0. The results showed that the design of the Skyway bridge is dynamically sound with regard to pedestrian loading and no remedial actions are necessary.

Keywords: ﬁnite element analysis, ﬁnite element modelling, footbridge, human induced vibrations, natural frequencies, mode shapes, parametric study, pedestrian bridge

Sammanfattning

Under de senaste åren har utformningen av gång- och cykelbroar blivit allt mer slank och till följd av det har broarna längre egenfrekvenser och är mer benägna att utsät- tas för överdrivna vibrationer till följ av den dynamiska belastningen som fotgängare orsakar. Akademiska hus bygger en gångbro vid Nya Karolinska Solna som spänner över Solnavägen och förbinder sjukhusbyggnaden U2 med forskningsanläggningen BioMedicum. På grund av höga vibrationskrav är det inte möjligt att förankra ena änden mekaniskt till sjukhuset, vilket ökar risken för horisontella moder inom det kritiska frekvensområdet 0-2,5 Hz. Den ökade risken för horisontella moder inom det kritiska frekvensområdet samt osäkerheter vid bestämning av brons dynamiska egenskaper ledde till denna avhandling. Denna avhandling täcker en frekvensanalys av den tidigare nämnda bron och en utvärdering av det dynamiska beteendet vid fotgängarbelastning genom att imple- mentera olika guider för hantering av dessa dynamiska laster. En litteraturstudie utfördes med avsikten att ge en djupare kunskap om vibrationer och laster orsakade av fotgängare och om de relevanta guiderna för modellering av dessa fotgängarlaster. Vidare genomfördes en parametrisk studie för elasticitetsmodulen för betong och fyllnadsmaterial samt styvheten i anslutningen till BioMedicum för att undersöka deras eﬀekt på brons egenfrekvenser. Den parametriska studien gav ett frekvensintervall på 2,20-2,93 Hz för den första laterala moden och 5,96-6,67 Hz för det första vertikala moden. Den nedre gränsen för frekvenserna sänktes till 2,05 och 5,59 Hz för den första laterala och den första vertikala moden genom att inkludera massan från icke-bärande material. Den parametriska studien visade också att den största förändringen i frekvenserna er- hölls genom att ändra parametrarna för stöden, både för BioMedicum och grun- den bestående av en betongkulvert och omkringliggande fyllnadsmaterial. Imple- mentering av de beräkningsmetoder och lastmodeller, som erhölls ur guiderna för hantering av dynamiska fotgängarlaster, resulterade i en lateral acceleration mellan 0,05 och 0,599 m/s2. Ingen utvärdering utfördes för de vertikala accelerationerna då den första vertikala moden hade en frekvens på 5,59 Hz, vilket är högre än de utvärderingskriterier som anges i Eurocode 0. Resultaten visade att bron klarar de dynamiska krav som ges i Eurocode gällande fotgängarlaster och inga förbättrande åtgärder är nödvändiga.

iii

Preface

The topic of this master thesis was initiated by the bridge division at the consultant company Tyréns AB in Stockholm, Sweden together with the department of Civil and Architectural Engineering at the Royal Institute of Technology (KTH). The thesis was conducted during the spring semester of 2017 as the ﬁnal part of our degree in Master of Science in Engineering. We would like to thank our supervisor at Tyréns, Mahir Ülker-Kaustell, as well as our supervisor at KTH, Ph.D Emma Zäll, for the guidance and support during the writing of this master thesis. We would also like to thank Joakim Kylén at Tyréns for all the help that we have recieved during our time at Tyréns. A special thanks to Prof. Raid Karoumi for his inspirational lectures on dynamics and bridge design which lead us to the topic of this master thesis. Last but not least we would like to thank our family and friends for supporting us throughout our years at KTH. Thank you!

Stockholm, June 2017 Daniel Andersson and Eric Thufvesson

Notations

Roman Letters Notation Description Unit An Constant − a Acceleration m/s2 Bn Constant − c Damping Ns/m c Damping matrix − d Displacement; Crowd density m; 1/m2 E Young’s modulus P a f Cyclic frequency of vibration; Force Hz; N fpv Step frequency Hz Fv(t) Harmonic load N G Static weight of a pedestrian N i Order of the harmonic − k Conventional stiffness matrix of structure − k Stiffness; Number; Reduction coefficient N/m; −; − m Mass matrix of structure − m; M Mass kg M Moment Nm m∗ Modal mass kg n; N Number − 0 Neq; n Equivalent number − p Force N Q Static load P a qn Harmonic function S Surface area; Span length m2 ; m Seff Effective span m t Time s u Displacement matrix m ˙u Velocity matrix m/s ü Acceleration matrix m/s2 u Displacement m w Pressure load P a

vii Greek Letters Notation Description Unit αi,h Dynamic load factor, horizontal direction − αi,v Dynamic load factor, vertical direction − αn,h Numerical coeﬃcient, horizontal direction − αn,v Numerical coeﬃcient, vertical direction − γ Reduction factor − δ Displacement m λ Reduction factor − ξ Critical structural damping − ρ Density kg/m3 υ Poisson’s ratio − φn Mode shape; Eigenvector − φn,h Phase angle, horizontal direction rad φn,v Phase angle, vertical direction rad φi,l Phase angle, lateral direction rad φi,v Phase angle, vertical direction rad ψ Reduction factor − ω Natural circular frequency rad/s ωn Circular frequency rad/s

viii Abbreviations

Abbreviation Description

3D Three-dimensional DLF Dynamic load factor DOF Degrees of freedom EC Eurocode EOM Equation of motion FEA Finite element analysis FEM Finite element method MDOF Multi-degree-of-freedom ODB Output database SDOF Single-degree-of-freedom

Contents

Abstract i

Sammanfattning iii

Preface v

Notations vii

Abbreviations ix

1 Introduction 1 1.1 Background ...... 1 1.2 Aim of the master thesis ...... 2 1.3 Limitations ...... 3 1.4 Disposition ...... 3

2 Studied Bridge 4 2.1 Bridge Geometry and Material ...... 4

3 Theoretical Background 6 3.1 Basics of Structural Dynamics ...... 6 3.1.1 Single-degree-of-freedom System ...... 7 3.1.2 Multi-degree-of-freedom System ...... 8 3.1.3 Mode Superposition ...... 9 3.2 Human Induced Vibrations ...... 11 3.2.1 Characteristics of a Single Pedestrian ...... 11

xi 3.2.2 Force Modelling ...... 13 3.2.3 Pedestrian Crowds and Lock-In Eﬀect ...... 16 3.2.4 Human Perception of Vibrations ...... 17 3.3 Design Guidelines ...... 21 3.3.1 Eurocode 5 ...... 21 3.3.2 UK National Annex ...... 23 3.3.3 Acceleration Criteria ...... 26 3.3.4 SS-ISO 10137:2008 ...... 27 3.3.5 Sétra ...... 30 3.3.6 SYNPEX ...... 36 3.3.7 HIVOSS ...... 41 3.3.8 JRC ...... 42

4 Finite Element Analysis 43 4.1 General Description of the FE Modelling ...... 43 4.2 Beam Model ...... 44 4.2.1 Material Data ...... 46 4.2.2 Element Types ...... 46 4.2.3 Connections and Boundary Conditions ...... 47 4.3 Shell Model ...... 48 4.3.1 Material Data ...... 49 4.3.2 Element Types ...... 49 4.3.3 Connections and Boundary Conditions ...... 49 4.4 Sub Models of Supports ...... 50 4.4.1 Substructure ...... 50 4.4.2 Connection at BioMedicum ...... 51 4.4.3 Element Type ...... 53 4.5 Veriﬁcation of Models ...... 53

5 Parametric Study 54

xii 5.1 Selection of Design Parameters ...... 54 5.2 Methodology ...... 55

6 Results 58 6.1 Frequencies and Mode Shapes ...... 58 6.2 Parametric Study ...... 61 6.3 Veriﬁcation of Beam Model ...... 62 6.4 Acceleration Results ...... 67

7 Discussion 72 7.1 Frequencies and Mode Shapes ...... 72 7.2 Parametric Study ...... 73 7.3 Veriﬁcation of Beam Model ...... 73 7.4 Assesment and Application of Design Guidelines ...... 74

8 Conclusions 78 8.1 Frequencies and Mode Shapes ...... 78 8.2 Parametric Study ...... 78 8.3 Veriﬁcation of Beam Model ...... 79 8.4 Assesment and Application of Design Guidelines ...... 79 8.5 Suggested Direction for Further Research ...... 80

Bibliography 81

Appendix A Numerical Results 84

Appendix B Python Script 97

xiii xiv Chapter 1

Introduction

1.1 Background

Pedestrian bridges are often simple and intuitive structures such as continuous beams. In recent years, the design of footbridges has become more slender, re- sulting in lower natural frequencies of the structures and in turn a higher sensitivity to dynamic loads. Loads induced by pedestrians has a low intensity and is usually not a problem for big structures. However, with more slender bridges the dynamic effect of pedestrians often need to be considered. Design of footbridges with respect to human induced vibrations can become complicated due to uncertainties in theoretical models of the structure as well as in the loads induced by pedestrians. Practical and architectural aspects may also lead to problematic design situations where the designer cannot freely choose a structural system that is sound from both a static and a dynamic perspective. Vertical load frequencies from pedestrians ranges from 1.6 to 2.4 Hz for walking and from 2 to 3.5 Hz for jogging. In the transversal direction however, the load frequency is equal to half of the vertical load since each step exerts forces in the opposite direction [1]. Bridges with frequencies within these ranges are likely to be excited by pedestrians during normal use. The issue regarding vibrating bridges concerns the serviceability state and comfort of pedestrians. The requirements for pedestrian comfort is not clearly stated in the design codes and much is left to the designer. Verification of the comfort criteria, according to Eurocode, should be performed if the structure has lateral or torsional modes of vibration lower than 2.5 Hz and vertical modes of vibration lower than 5 Hz. The comfort criteria serves as recommended maximum levels of acceleration and it is up to the designer to determine whether it is necessary to make provisions in the design for damping after the structure is built [2]. There is often a high uncertainty in the data of the finite element modelling used to estimate the dynamic properties of structures and therefore also in the results. The bridge studied in this thesis will span over Solnavägen and connect the two

1 1.2. AIM OF THE MASTER THESIS buildings on each side. The structure will be supported by columns on each side of the road and the preliminary design indicates that the structure may be sensitive to lateral vibrations. Due to practical reasons, it is not possible to connect one of the bridge ends mechanically to the building, which increases the risk for lateral and torsional modes in the most sensitive frequency range of 0-2.5 Hz. The increased risk of modes of vibrations in the sensitive frequency range as well as uncertainties when determining the dynamic response led to the topic of this thesis.

1.2 Aim of the master thesis

The purpose of this thesis is to evaluate the dynamic properties of the bridge using the preliminary designs of the structure and design a structure which is sound from both a static and dynamic perspective. In order to achieve this a parametric study of different parameters, such as Young’s modulus and stiffness at the supports, will be carried out to determine their impact on the mode shapes and corresponding frequencies. A literature review regarding the dynamic properties of bridges, pedestrian loading and relevant design guidelines will also be carried out. The main objective is to find a structural system which fulfils the acceleration requirements of the Eurocode and, if necessary, propose remedial actions, which guar- antees that the built structure will do so. The bridge structure will be modelled using finite element software Brigade/Plus from Scanscot Technology and programming software Python. The aims of the thesis are as follows:

• Design a three-dimensional (3D) ﬁnite element model using ﬁnite element software Brigade/Plus from Scanscot Technology and programming language Python.

• Create 3D ﬁnite element models of the supports using Brigade/Plus from Scanscot Technology and programming language Python.

• Evaluate the dynamic properties using ﬁnite element analysis (FEA).

• Perform a parametric study of certain parameters and study their inﬂuence on the dynamic properties.

• Veriﬁcation of comfort criteria using design guidelines determined from the literature review.

2 1.3. LIMITATIONS

1.3 Limitations

The limitations of the thesis regards the finite element modelling of the bridge. As- sumptions and simplifications of the FE models will be further explained in Chapter 4. The verification of the comfort criteria stated in Eurocode will be performed using the design methods presented in Chapter 3.

1.4 Disposition

This thesis consists of 8 chapters including the introduction given above. Each chapter covers a diﬀerent part of the working process and a brief overview of the chapters is given below.

Chapter 2 General description of the Skyway Bridge, including geometry and the materials used.

Chapter 3 Theoretical background used in the thesis. The theoretical background is obtained through a literature review.

Chapter 4 Method used to construct the FEA. Presents the limitations and assumptions as well as diﬀerent modelling approaches and methods used to conduct the analysis.

Chapter 5 Parametric study. The parameters that was used, the combinations and how they are implemented in the model.

Chapter 6 Results from FEA including the parameters impact on the mode shapes and corresponding frequencies as well as acceleration results using design guidelines.

Chapter 7 Discussion regarding the FEA, the results from the parametric study and the acceleration results.

Chapter 8 Conclusions and proposed further research.

3 Chapter 2

Studied Bridge

2.1 Bridge Geometry and Material

In the preliminary designs the Skyway bridge is a 47.97 m long and 4 m wide continuous composite bridge divided in three spans of 11.07, 29.52 and 7.38 m supported on two columns on each side of Solnavägen. Due to different elevation heights of the two buildings, the bridge is constructed with a slight longitudinal inclination of 1:42 towards Nya Karolinska Solna (NKS). The superstructure will be constructed as a truss bridge with two parallel longitudinal chords at the bottom and at the top of the bridge. The bottom chords are connected with crossbeams spaced 1.23 m while the top chords are connected by lateral bracings. The top and bottom chords are connected by diagonals and verticals with a spacing of 3.69 m. All steel profiles are designed as hollow sections except the crossbeams which consists of HEB-profiles. The dimensions of the profiles in the superstructure, given in Table 2.1, are in accordance with preliminary design set by Tyréns A [3]. The bridge deck has a thickness of 120 mm and is placed on top of the crossbeams. The deck is a composite slab with concrete cast on top of a corrugated steel sheet with welded steel studs [3].

Table 2.1: Steel proﬁles in Skyway [3]

Member Proﬁle Chords VKR250x250x12.5 Diagonals VKR220x120x6.3 Lateral Bracings VKR80x80x5 Struts VKR250x250x12.51 Columns VKR220x120x6.3 VKR250x250x12.51 Crossbeams HEB120 HEB3001

1Above support column

4 2.1. BRIDGE GEOMETRY AND MATERIAL

The substructure consists of two rhombus-shaped support columns, oblique relative the Skyway bridge, with a cross section dimension of 900 × 2400 mm2 with column heads with cross section dimension equal to 900 × 800 mm2. The support columns are connected to the roof a concrete culvert located approximately 2.5 m beneath Solnavägen. The support columns has the length 11.415 m and 12.110 m. The diﬀerence is due to the inclination of the bridge [3]. The rhombus-shaped support columns and the rest of bridge can be seen in Figure 2.1.

Figure 2.1: Representation of the Skyway bridge including the culvert [3]

5 Chapter 3

Theoretical Background

This section aims to familiarize the reader with the necessary theoretical background used within this thesis. It contains theory regarding structural dynamics and human induced vibrations as well as the relevant design guidelines.

3.1 Basics of Structural Dynamics

Structural dynamics treats the behaviour of structures when subjected to dynamic loads. Dynamic loads are loads that vary in time, such as wind, pedestrians, earth- quakes or explosions. The structural analysis of the system aims to determine the displacement for all locations of the structure at all times, which is obtained by solving the systems equation of motion (EOM). Solving of the EOM consists of finding the equilibrium between all forces which are inertia forces, damping forces, stiffness forces and external forces [4]. Each structure has natural frequencies corresponding to natural modes of vibration. These natural modes of vibrations are deformed shapes in which the structure will vibrate. The corresponding frequency is the number of oscillations per second during free vibration. For simple structures, such as simply supported beams the natural modes, illustrated by Figure 3.1, and natural frequencies are easy to calculate. But for more complex structures it is often necessary to implement finite element software to determine the modes of vibration.

6 3.1. BASICS OF STRUCTURAL DYNAMICS

Figure 3.1: Illustration of the ﬁrst three mode shapes for a simply supported beam

3.1.1 Single-degree-of-freedom System

A single-degree-of-freedom system (SDOF) is the basic element of structural analysis and consists of a simple oscillator. This is the simplest way to describe a structure and it gives an understanding of more complex systems which will be described in the next subsection. The number of degrees of freedom is the number of independent displacements necessary to describe the displaced location of the mass of the system. A SDOF system is a spring-mass-damper system, Figure 3.2, in which the mass, m, is concentrated to one position and is only allowed to move in one direction [4]. The system has a linear stiﬀness, k, and a linear damping coeﬃcient, c.

Figure 3.2: Spring-mass-damper system

The inertia force of the system, fI , is equal to the sum of all forces acting on the system,

fI = fD + fS + p(t), (3.1) where fD is the damping force, fS is the stiﬀness force and p(t) is the external force applied to the system. The inertia force is proportional to the acceleration in accordance with Newton’s second law of motion,

7 3.1. BASICS OF STRUCTURAL DYNAMICS

fI = mu,¨ (3.2) where m and ü is the mass and acceleration of the structure respectively. The damping force is proportional to the velocity of the system by

fD = −cu,˙ (3.3) where c and u˙ are the damping coefficient and the velocity of the system respectively. The stiffness force is described by Hooke’s law where the stiffness, k, is related to the displacement, u, of the system by

fS = −ku. (3.4)

By substituting equation 3.2 - 3.4 into equation 3.1 and rearranging, the EOM for the system is obtained,

mu¨ + cu˙ + ku = p(t). (3.5)

3.1.2 Multi-degree-of-freedom System

A structure has an infinite number of degrees of freedom (DOF) but some simplifications are necessary to analyze the structures dynamic behavior. By discretizing the structure to a finite number of elements, with the motion of the nodes that sub- divides each element as the degrees of freedom, an approximation of the structure can be obtained. Such a system is called a multi-degree-of-freedom (MDOF) system where the simplest example consists of two degrees of freedom. The theory of MDOF systems is a generalization from one to N degrees of freedom, where the EOM of the system is given by,

mu¨ + cu˙ + ku = p(t), (3.6) where m, c, k are N x N matrices describing the mass, stiffness and damping of the system respectively. The displacement, velocity and acceleration for each node are given by the N x 1 vectors u, u˙ and u¨. The external force on each node is given by the N x 1 load vector p(t). In reality the mass of the structure is distributed over the entire structure, but as an idealization the mass of each element can be assumed to be concentrated to the nodes. As a result each structural member is replaced by lumped masses at the element ends. The stiffness matrix is obtained by assembling a local stiffness matrix for each element and the damping is generally specified by numerical values for damping ratios, based on experimental data [4].

8 3.1. BASICS OF STRUCTURAL DYNAMICS

The stiﬀness and damping matrix will have oﬀ-diagonal values, known as coupling terms, and the equations need to be solved at the same time. The EOM for such a system can be solved either by numerical integration or by the modal superposition method [4]. The modal superposition method will be explained in the next subsection and is also the method chosen for the FEA.

3.1.3 Mode Superposition

The response of a linear system with classic damping, which is a practical assumption of many structures, can be determined with the mode superposition method. With classical damping there are no coupling terms between the modes of the structure and mode superposition is a way to approximate the nodal displacement using linear combination of the natural modes. By using this method a coupled MDOF system with N degrees of freedom can be transformed to N uncoupled SDOF systems by introducing modal coordinates. The response of each natural mode of vibration can be computed separately and the modal responses can be combined to obtain the total response of the system [4]. To apply the mode superposition method the natural frequencies and natural modes of the structure during free vibration need to be determined [4]. The natural frequencies and mode shapes are determined by solving the EOM,

mu¨ + ku = 0. (3.7)

The solution to the EOM is given by,

u = qn(t)φn, (3.8) where the mode shape, φn, does not vary with time. The variation in time is given by the simple harmonic function,

qn(t) = Ancos(ωnt) + Bnsin(ωnt). (3.9)

By inserting Equation 3.8 into the EOM given by Equation 3.7 the following is obtained,

2 [−ωnmφn + kφn]qn(t) = 0. (3.10)

Equation 3.10 has two possible solutions, either qn(t) = 0 which implies no motion in the system or the non trivial solution given by,

2 [k − ωnm]φn = 0. (3.11)

9 3.1. BASICS OF STRUCTURAL DYNAMICS

This is called the matrix eigenvalue problem where the matrices k and m are known. The general non-trivial solution to the eigenvalue problem is given by,

2 det[k − ωnm] = 0. (3.12)

Equation 3.12 is the characteristic equation of the system. The solution to Equation 2 3.12 is a number of roots, eigenvalues, corresponding to ωn. The number of roots correspond to the DOF of the system. If the natural circular frequencies, ωn, are inserted in Equation 3.11 the eigenvector φn is obtained for every egeinvalue which deﬁnes the mode shape associated with each eigenfrequency [5]. When the natural frequencies and mode shapes are determined the response of a damped system can be determined by solving the EOM given by Equation 3.6. For classical damping, c is a square matrix with values along the diagonal. The damping matrix must satisfy the identity given by Equation 3.13 and the natural modes will then be real-valued and identical to those determined for the undamped system.

cm−1k = km−1c. (3.13)

The nodal displacement of the structure can be expressed by

N X u(t) = Φnqn(t), (3.14) n=1 where qn is a generalized coordinate. Inserting the expression for the nodal displacement into Equation 3.6 gives the EOM for each natural mode, Φn, according to

Mnq¨n + Cnq˙n + Knqn = Pn(t). (3.15)

Where Mn, Cn, Kn and Pn(t) are the generalized mass, damping, stiﬀness and force given by,

T T T T Mn = Φn mΦn,Cn = Φn cΦn,Kn = Φn kΦn and Pn(t) = Φn p(t). (3.16)

There are N equations like Equation 3.15 corresponding to the number of DOF and each equation depend only on one natural mode, Φn. Thus we get N uncoupled equations and the total response, u(t), is obtained by solving these equations for qn and then summarize there contribution according to Equation 3.14 [4].

10 3.2. HUMAN INDUCED VIBRATIONS

3.2 Human Induced Vibrations

Human-induced loading on footbridges is frequently occurring and often the dominant load case for a footbridge due to the nature of the structure – to allow for the passage of pedestrians. When crossing a bridge, pedestrians induce a dynamic force into the bridge deck and this force has components in the vertical, transversal and longitudinal directions. The time varying, dynamic, forces induced by pedestrians depends on many diﬀerent parameters, such as walking speed, pacing frequency and step length, and a lot of research has been conducted on the topic to better understand the load and create models that accurately describes the real behaviour [6]. This chapter aims to familiarize the reader with the subject of pedestrian induced vibrations by describing the characteristics of the induced load as well as give an understanding of the complexity regarding the modelling of human induced vibrations.

3.2.1 Characteristics of a Single Pedestrian

Due to the complexity of the pedestrian loading a lot of research has been conducted on the topic. One of the first measurements of pedestrian loads was conducted by Harper et al. [7] and Harper [8] in the 1960’s. The aim of that research was to determine the friction and slipperiness of different floor surfaces. With the use of a force plate, vertical and lateral force of a single step was measured and the research resulted in a force time history similar to the one presented in Figure 3.3. The time history of a single step was later confirmed by other researchers, including Andriacchi et al. [9], and are still used for modeling of the pedestrian load [6]. The pedestrian induced force depends on a variety of parameters, as mentioned above, which make the understanding and modelling of the loading complicated. These parameters vary between different persons but also between the steps of one single pedestrian. Due to the random variation of these parameters statistically based probabilistic methods are necessary to draw conclusions regarding them. Statistically based tests have shown that the frequency for normal walking vary between 1.6 and 2.4 Hz with a mean value of approximately 2 Hz, while the frequency for running usually vary between 2 and 3.5 Hz [6]. In the transversal direction however, the load frequency is equal to half of the vertical load since each step exerts forces in the opposite direction [1]. The ground reaction force in the lateral direction can be seen in Figure 3.4.

11 3.2. HUMAN INDUCED VIBRATIONS

Figure 3.3: Force time history of a single step [9]

Figure 3.4: Lateral movement and ground reaction force [10]

12 3.2. HUMAN INDUCED VIBRATIONS

3.2.2 Force Modelling

The forces induced by pedestrians need to be modelled analytically to be applied to a specific bridge in the design process. In the literature two types of analytical models exists: time- and frequency domain load models. The time dependent model is most common but in both cases the load is difficult to model accurate due to its complexity [6]. This thesis only covers the time-dependent load models. Živanović [11] categories the challenges of modelling the load into four groups, which are: • Randomness in the human loading: A single pedestrian cannot repeat two identical steps and the force also differs between individuals. • Human-structure interaction: Pedestrians interact with the moving bridge which will impact the dynamic force. • Human-human interaction: When walking in a crowd the pedestrians will be influenced by others and the dynamic force differs from when a single person walks unrestricted. • Human perception and response to vibration: The vibrations of the structure will impact the movement of the pedestrians. This factor is dependent on the perception level of a single pedestrian as well as the magnitude of the accelerations. Regardless of the complexity, models for the pedestrian induced force exist and are based on the assumption that both feet produce the same force and that the loading is periodic, see Figure 3.5 . However, all of the models are based on simplifications and the challenges described above are often neglected. However, it is important to notice that neglecting the variety of the load might result in an error up to 40 % of the dynamic response [12]. The time dependent load models describe the human loading as a function of time. These load models are categorized into two groups, deterministic and probabilistic models, where the former is most common. Deterministic models aims to establish a general force model for all types of human activities, while the probabilistic models incorporates the randomness of the parameters, e.g. walking frequency and pedestrian weight, through probability distribution functions. The focus of this report will be on deterministic force models.

13 3.2. HUMAN INDUCED VIBRATIONS

Figure 3.5: Periodic walking time histories [6]

3.2.2.1 Deterministic Force Models

If the force is assumed to be periodic, the vertical and lateral force component can be represented by a Fourier series as follows [13]:

∞ X Fv(t) = G + Gαi,v sin(2πifpvt + φi,v) (3.17) i=1

∞ X fpv Fl(t) = Gαi,v sin(2πi t + φi,l) (3.18) i=1 2

14 3.2. HUMAN INDUCED VIBRATIONS

Where:

G Pedestrian weight, usually 700N [N]

αi,v Dynamic load factor (DLF) vertical direction, ratio of force amplitude to the pedestrian weight [-].

αi,h Dynamic load factor (DLF) lateral direction, ratio of force amplitude to the pedestrian weight [-].

fpv Vertical stepping frequency [Hz].

φi,v The phase-angle of the ith harmonic in the vertical direction [rad].

φi,l The phase-angle of the ith harmonic in the horizontal direction [rad].

i The order of the harmonic [-]

In the literature, deterministic load models are all expressed with a Fourier series but researchers present diﬀerent values for the dynamic load factor (DLF). Most of the research on the subject of pedestrian induced vibrations has been focused on the vertical force and corresponding DLF, but recently the interest in the lateral induced vibrations has increased and therefore also the research of the DLF in the lateral direction. Rainer [14] derived values for the DLF in the vertical direction which were shown to be strongly frequency dependent, see Figure 3.6. Bachmann and Ammann [15] presented values for the dynamic load factor in the lateral direction with a dominance of the ﬁrst and third harmonic ( α1 = 0.039, α2 = 0.01, α3 = 0.043, α4 = 0.012, α5 = 0.015). More research has been conducted on the subject of DLF and for more detailed information the reader is referred to [6].

Figure 3.6: DLF for the ﬁrst four harmonics for (a) walking, (b) running and (c) jumping (after Rainer [14])

15 3.2. HUMAN INDUCED VIBRATIONS

3.2.2.2 Probabilistic Force Models

The probabilistic force models of the pedestrian load take the variability and uncertainty of the human induced load in to account. They are based on the fact that the the force-time history diﬀer during repeated experiments. The load is still assumed to be periodic but the randomness of the load is accounted for by introducing probability density functions for the model parameters, e.g. weight and pacing frequency. In order to construct such models it is necessary to have a lot of measurements of the loading. This thesis will focus on the deterministic force models and for a more detailed description of the probabilistic model the reader is referred to [6] and [16].

3.2.3 Pedestrian Crowds and Lock-In Eﬀect

Lock-in is a phenomenon where a pedestrian crowd, walking with random frequencies and phase shifts, gradually synchronizes to a common frequency and phase which coincides with the natural frequency and motion of the bridge [1]. The most famous occurrences of the lock-in phenomena refer to the Solférino Bridge in Paris and The Millennium Bridge in London. On both occasions the crowd traversing the bridge altered their walking to move in sync with the moving bridge. That excited the lateral movement of the bridge which in turn led to excessive accelerations and displacement in the lateral direction. Since these incidents the interest in the pedestrians eﬀect on the lateral movement increased and promoted research on the subject [1]. To better understand this phenomenon, further test where performed on the before mentioned bridges as well as on moving platforms in laboratories. Tests performed on the bridges showed that both the Solférino and the Millennium Bridge had lateral modes of vibration with frequencies around 1Hz which coincides with the frequency of the lateral force induced by the pedestrians (see Section 3.2.1). Bridges with frequencies around this value are prone to excessive lateral vibrations. In the model for determining the risk of lock-in (derived from tests on the Millennium Bridge) the pedestrian load is assumed to be proportional to the velocity and the load can be seen as negative damping. The lock-in eﬀect will cause an increase in the negative damping force, caused by an increase of the pedestrians altering their movement to coincide with the bridge motion. The model, called Arup’s criterion, gives a critical number of pedestrians for which the cumulative damping reaches the damping of the structure [17]. The critical number is expressed as,

8πξm∗f N = , (3.19) L k

16 3.2. HUMAN INDUCED VIBRATIONS where:

ξ Structural damping ratio [−]

m* Modal mass [kg]

f Natural frequency [Hz]

k Constant of 300 [Ns/m] over the range 0.5−1.0 Hz.

Sétra [1] presents another way of determining the risk of lock-in, which was derived from the tests on the Solférino footbridge. The Sétra working group measured the acceleration over time as well as the synchronization rate, which is seen as the ratio between the equivalent number of pedestrians (in synchronization with the bridge motion) and the pedestrians present on the bridge. The tests showed a clear threshold for the lock-in eﬀect in terms of an acceleration limit, beyond which the degree of synchronization rapidly increased. Sétra [1] set the acceleration limit in the lateral direction to 0.1 m/s2 for design purposes. Below this value the pedestrian’s behavior could be seen as random but beyond the limit the level of synchronization could reach a value of 60 % which in turn led to an increase in acceleration, from 0.1 to 0.6 m/s2.

3.2.4 Human Perception of Vibrations

The perception of motion and vibration is highly subjective and hence diﬀerent for each pedestrian. HIVOSS [18] presents a variety of ‘soft’ attributes which are important in the assessment of vertical and horizontal vibration, such as:

• Number of people walking on the bridge

• Frequency of use

• Height above ground

• Position of human body (walking, sitting, standing)

• Vibration frequency

• Exposure time

• Transparency of the deck pavement and the railing

• Expectancy of vibration due to bridge appearance

17 3.2. HUMAN INDUCED VIBRATIONS

An example of these ‘soft’ attributes is given in the analysis of the Kochenhofsteg and Wachtelsteg footbridges in Germany, shown in Figure 3.7, where the percentage of pedestrians feeling disturbed by crossing the bridges was four times higher for the sturdier-looking Wachtelsteg Footbridge than for the Kochenhofsteg Footbridge despite having similar dynamic properties [18]. This was due to the expectancy of vibration.

Figure 3.7: Kochenhofsteg Footbridge, Stuttgart (left) and Wachtelsteg Footbridge, Pforzheim (right) [18]

3.2.4.1 Vertical Vibrations

Reiher and Meister [11] performed one of the first laboratory tests regarding the human perception of vibrations with the help of ten people in three different positions; laying, sitting and standing on a test rig. By exciting the test-rig with different amplitudes, directions and frequencies they could divide the perception of vibration into six categories spanning from imperceptible to intolerable (see Figure 3.8). Leonard [11] performed an experiment in a laboratory on a 10.7m long beam which was excited by a sinusoidal motion at different amplitudes and frequencies between 1-14 Hz. With the help of forty persons, while trying to establish the boundary between acceptable and unacceptable vibrations, and by the previous findings of Reiher and Meister he noted that walking pedestrians are less sensitive to vibrations than people standing still. Most research regarding the perception of vibrations in footbridges is done by inter- viewing the pedestrians as the bridge is subjected to different acceleration amplitudes and frequencies. Živanović et al. [6] however, took a different approach and defined the disturbing level as the acceleration of the footbridge at the moment when the pedestrian looses its step. This moment was denoted as when the simulated and measured responses started do differ. By performing this experiment on two test subjects and footbridges it was concluded that the pedestrians lost their step at 0.33m/s2 and 0.37 m/s2 depending on the bridge. However, the reliability of these results can be questioned due to the number of test subjects.

18 3.2. HUMAN INDUCED VIBRATIONS

Figure 3.8: Perception of vertical vibrations [6]

3.2.4.2 Horizontal Vibrations

Research regarding the perception of horizontal vibrations are not as comprehensive as for vertical vibrations. The motion of human walking has a dominant vertical frequency of approximately 2 Hz and a dominant lateral frequency of about half of the vertical, 1 Hz, as a result of the periodic shifting between one leg and the other [19]. Most data concerning lateral vibrations are done for high-rise buildings with a frequency in the lateral direction up to 0.2 Hz and therefore not likely to be applicable for bridges [20]. Research done by Nakamura [11] on the Nasu Shiobara Bridge in Japan provides valuable data from measurements of full-scale bridges about diﬀerent tolerance levels when exposed to crowd loadings. Nakamura found that a reasonable serviceability

19 3.2. HUMAN INDUCED VIBRATIONS limit of of horizontal accelerations is 1.35 m/s2 . He also concluded that most pedestrians tolerated an acceleration of 0.3 m/s2 while an acceleration of 2.1 m/s2 prevented them from walking due to discomfort.

20 3.3. DESIGN GUIDELINES

3.3 Design Guidelines

3.3.1 Eurocode 5

Eurocode 5 considers timber structures and the guidelines given in the code is produced for simply supported beams or truss systems [21]. The response model in Eurocode 5 is not material speciﬁc and can therefore be used for any kind of footbridge, as long as it is simply supported. Eurocode 5 does not specify load models to implement, only a way of calculating the acceleration response based on certain bridge properties. The vertical accelerations of the bridge caused by pedestrians excitation are calculated for one single pedestrian and a group of pedestrians crossing the bridge. For one single pedestrian walking or running over the bridge the acceleration in m/s2 is given by Equation 3.20 and 3.21 respectively.

 200  for f ≤ 2, 5Hz. Mξ vert avert,1 = 100 (3.20)  for 2, 5Hz ≤ f ≤ 5, 0Hz. Mξ vert

600 a = for 2, 5Hz ≤ f ≤ 3, 5Hz. (3.21) vert,1 Mξ vert

Where:

M Total mass of the bridge [kg]

ξ Damping ratio [-]

fvert Natural frequency of the bridge (vertical direction) [Hz]

For several people crossing the footbridge, either by walking or running, the acceleration in m/s2 is given by Equation 3.22.

avert,n = 0, 23 · avert,1nkvert (3.22)

21 3.3. DESIGN GUIDELINES

Where:

2 avert,1 Acceleration determined from either Equation 3.20 or 3.21 [m/s ]

kvert Reduction coeﬃcient [-] based on the vertical natural frequency fvert from Figure 3.9.

n Number of pedestrians [-]. Taken as 13 for a distinct number of pedestrians or 0,6A for a continuous stream of pedestrians where A is the area of the bridge deck.

Calculation of the horizontal acceleration in m/s2 of the footbridge is carried out similar to the vertical vibrations. The horizontal acceleration for one person crossing the bridge is calculated according to Equation 3.23. For several pedestrians crossing the bridge, the horizontal acceleration is calculated according to Equation 3.24.

50 a = for 0, 5Hz ≤ f ≤ 2, 5Hz. (3.23) hor,1 Mξ hor

ahor,n = 0, 18 · ahor,1nkhor (3.24)

Where:

fhor The fundamental horizontal natural frequency [Hz].

khor Reduction coeﬃcient based on the horizontal natural frequency fhor from Figure 3.9 [-]

Figure 3.9: Relationship between fundamental natural frequency and vertical (left) and horizontal (right) coeﬃcient [21]

22 3.3. DESIGN GUIDELINES

3.3.2 UK National Annex

BS EN 1991-2 [22] establishes models and representative values intended for design of new bridges, including bridges subjected to pedestrian traﬃc. Furthermore, two analyses are required for pedestrian bridges; a determination of maximum vertical acceleration and an analysis to determine the likelihood of excessive lateral responses. The guideline does not cover activities such as mass gatherings or delib- erate synchronization by pedestrians. The studied bridge should be categorized into a bridge class, ranging from A to D depending on group size and crowd density. Class A refers to seldom used bridges in rural locations and class D to bridges at primary access routes close to transportation facilities or sport arenas. The classes and corresponding density is given by Table 3.1.

Table 3.1: Recommended values for density (after[22])

Bridge Group size Crowd class (walking) density, ρ (walking) A N=2 0 B N=4 0.4 C N=8 0.8 D N=16 1.5

The maximum vertical acceleration of the bridge should be calculated for one single pedestrian or a group of pedestrians moving over the the deck of the bridge at a constant speed. By assuming that the pedestrians exerts a pulsating load on the deck, the load is calculated according to Equation 3.25 [22],

q F = F0 · k(fv) · 1 + γ(N − 1) · sin(2πfv · t). (3.25)

23 3.3. DESIGN GUIDELINES

Where:

F0 The reference load given as 280N and 910N for walking and jogging respectively [N]

fv Vertical walking frequency, chosen as the resonance frequancy [Hz]

γ The reduction factor for the eﬀective number of pedestrians, Figure 3.11 [-]

N The size of the group of pedestrians depending on bridge class [-]

k(fv) Factor considering the relative weighting of pedestrian sensitivity to response, harmonic responses and the eﬀects of a more realistic pedestrian population (Figure 3.10) [-]

t The time in seconds [s]

In crowded conditions the distributed pulsating force, w, exerted on the bridge deck is assumed to occur over enough time that steady state conditions is reached according to 3.26.

s F γ · N w = 1.8 · ( 0 ) · k(f ) · · sin(2πf · t) (3.26) A v λ v

Where:

N The total amount of pedestrians distributed over the span N = ρ · A [-]

λ A reduction factor which considers the load from an eﬀective number of pedestrians which contributes to the mode of interest [-]

24 3.3. DESIGN GUIDELINES

Figure 3.10: Factor k(fv) [22]

Figure 3.11: Reduction factor γ considering unsynchronized combinations of pedestrians [22]

25 3.3. DESIGN GUIDELINES

3.3.3 Acceleration Criteria

The vertical acceleration limit, given by Equation 3.27, is not dependent on the natural frequency of the bridge, but is instead determined by the parameters given in Table 3.2, 3.3 and 3.4.

alimit = 1.0k1k2k3k4 (3.27) k1, k2 and k3 (Table 3.2 to 3.4) are response modiﬁers depending on the site usage, route redundancy and structure height respectively and k4 is a project speciﬁc exposure factor taken as 1.0 unless determined otherwise.

Table 3.2: Recommended values for the site usage factor k1 (after[22])

Bridge function k1 Primary route for hospitals or other high sensitivty routes 0.6 Primary route for school 0.8 Primary route for sport stadium or other high usage routes 0.8 Major urban centers 1 Suburban crossings 1.3 Rural environments 1.6

Table 3.3: Recommended values for route redundancy factor k2 (after[22])

Route redundancy k2 Sole means of access 0.7 Primary route 1.0 Alternative routes readily available 1.3

Table 3.4: Recommended values for the structure height factor k3 (after[22])

Bridge height k3 Greater than 8 m 0.7 4-8 m 1.0 Less than 4 m 1.1

If the footbridge has lateral natural frequencies below 1.5 Hz, a check for stability needs to be performed. This is done by comparing the pedestrian mass damping parameter, D, to Figure 3.12. The pedestrian mass damping parameter is given by Equation 3.28.

m · ξ D = bridge (3.28) mpedestrian

26 3.3. DESIGN GUIDELINES

Where:

mbridge The mass of bridge per unit length [kg/m]

mpedestrian The mass of pedestrian crowd per unit length where one pedestrian weighs 70 kg [kg/m]

ξ The structural damping ratio [-]

Figure 3.12: Check for lateral stability [22]

3.3.4 SS-ISO 10137:2008

SS-ISO 10137:2008 [20] is an International Standard which gives recommendations about the evaluation of serviceability against vibrations of buildings and walkways. The load exerted on the bridge by one single pedestrian is expressed as a Fourier series for both the vertical direction (Equation 3.29) and the horizontal direction (Equation 3.30).

27 3.3. DESIGN GUIDELINES

k X Fv(t) = Q 1 + αn,v sin(2πnft + φn,v) (3.29) i=1

k X Fh(t) = Q 1 + αn,h sin(2πnft + φn,h) (3.30) i=1

Where:

Q The static load of a single pedestrian [N]

αn,v A numerical coeﬃcient corresponding to the nth harmonic in the vertical direction, see Table 3.5 [-]

αn,h A numerical coeﬃcient corresponding to the nth harmonic in the horizontal direction, see Table 3.5 [-]

f The vertical step frequency. For horizontal vibrations f is one-half of the activity rate of walking or running [Hz]

φn,v The phase-angle of the nth harmonic in the vertical direction. π Conservatively chosen as 2 [rad]

φn,h The phase-angle of the nth harmonic in the horizontal direction. π Conservatively chosen as 2 [rad]

n The number of harmonics of the fundamental considered [-]

k The number of harmonics that characterize the forcing function in the frequency range of interest [-]

The dynamic load produced by a group of people depends mainly on the weight of the partakers, the density of people per unit ﬂoor area as well as the degree of coordination of the partakers. To account for the non-perfect synchronization between the participants, the response of the structure will be reduced compared to an ideal scenario where the entire group is perfectly synchronized. The reduced response can therefore be approximated by multiplying with a factor C(N) according to Equation 3.31.

FN (t) = F (t) · C(N) (3.31)

28 3.3. DESIGN GUIDELINES

Where:

N is the number of people [-] √ N C(N) is taken as [-] N

Table 3.5: Design parameters for moving forces due to one person (after [20])

Activity Harmonic Common Numerical Numerical number, range of forc- coefficient coefficient n ing frequency, for vertical for hor- nf direction, αn.v izontal [Hz] direction, αn.h Walking 1 1.2 to 2.4 0.37(f-1.0) 0.1 2 2.4 to 4.8 0.1 3 3.6 to 7.2 0.06 4a 4.8 to 9.6 0.06 5a 6.0 to 12.0 0.06 Running 1 2 to 4 1.4 0.2 2 4 to 8 0.4 3 6 to 12 0.1 aThese higher harmonics are rarely significant where human perception is of concern, but may be important for more sensitive building occupancies such as vibration-sensitive instrumentation

3.3.4.1 Vibration criteria

SS-ISO 10137:2008 [20] does not provide any deﬁnitive data regarding criteria for accelerations for footbridges. However, it provides base curves for acceleration limits in buildings which should not be exceeded for walkways over roads or waterways, seen in Figure 3.13. The acceleration limits are determined by multiplying the value of the base curve, at the frequency of interest, by a factor of 60 or 30 for vertical and lateral vibrations respectively.

29 3.3. DESIGN GUIDELINES

Figure 3.13: Base curve for acceleration limit for lateral (left) and vertical direction (right)[20]

3.3.5 Sétra

The Technical Department of Transportation, Roads and Bridges Engineering and Road Safety, or Sétra in short, is a technical department within the French Min- istry of Transport and Infrastructure. The guideline presented in this section was prepared within the framework of the Sétra working group on “Dynamic behaviour of footbridges" [1]. Sétra’s technical guide for dynamic design of footbridges is based on a footbridge classiﬁcation concept as a function of traﬃc levels and on the required comfort level for the bridge. The methodology relies on the results from tests performed at the Solférino bridge as well as an experimental platform. The guideline concern normal use of footbridges and do not guarantee that comfort criteria is met during exceptional events such as marathons, demonstrations or vandal loading. However, vandal loading is mentioned in the guideline as a design check in the ultimate limit state [1].

30 3.3. DESIGN GUIDELINES

3.3.5.1 Design Methodology

As mentioned above, the guideline put forward by Sétra is based on a classification concept which in turn is based on traffic assessment and comfort levels. The owner of the bridge needs to define these levels before the design calculations begins. With the given levels of traffic the bridge can be categorized into a class and then the natural frequencies are to be determined. The footbridge class and the natural frequencies lead to a selection of one or several load cases, load cases that represents the possible effects of pedestrian traffic. Implementation of the load case results in a maximum acceleration which is compared with the chosen comfort level. Figure 3.14 shows a flow chart of the design methodology and each part will be presented further described in this section.

Figure 3.14: Flow chart of design methodology [1]

The class is related to the expected level of pedestrian traﬃc on the bridge. The classes range from I-IV where Class I refers to bridges in high populated areas frequently used be dense crowds, e.g. bridges at or close to railway stations. Class IV is used for bridges seldom used, e.g. bridges built to ensure continuous walking paths in motorway areas. It is up to the owner of the bridge to choose a class, however, Class IV does not require a dynamic analysis (as presented in Figure 3.14) and it is advised to at least use Class III to ensure a minimum risk control [1]. The concept of comfort is highly subjective and the experience of certain acceleration might diﬀer between individuals. The choice of comfort level is based on the population using the bridge. For instance, elderly or disabled people can be particu-

31 3.3. DESIGN GUIDELINES larly sensitive and hence a stricter comfort level is advised. The level of comfort can also be influenced be the length of the span, a shorter bridge has a shorter transit time and as a result a higher acceleration is acceptable. Sétra [1] categorizes these comfort levels as different ranges, for vertical and horizontal vibrations respectively. These ranges are presented in Table 3.6 and Table 3.7. Regardless of the acceleration ranges in the horizontal direction the accelerations limit is set to 0.1 m/s2 to avoid synchronized lateral excitation or the “lock-in” effect. The research group behind the Sétra guideline derived this value from measurements regarding acceleration and level of synchronization between the bridge and the pedestrians (see Section 3.2.3). When exceeding this threshold value of 0.1 m/s2 the synchronization rate changed rapidly from 10 to 60 percent which increased the accelerations to about 0.6 m/s2 [1].

Table 3.6: Acceleration ranges, vertical direction (after [1])

Acceleration 0-0.5 0.5-1 1-2.5 2.5 < ranges [m/s2] [m/s2] [m/s2] [m/s2] Range 1 Max. comfort Range 2 Mean comfort Range 3 Min. comfort Range 4 Unacceptable

Table 3.7: Acceleration ranges, lateral direction (after [1])

Acceleration 0-0.15 0.15-0.3 0.3-0.8 0.8 < ranges [m/s2] [m/s2] [m/s2] [m/s2] Range 1 Max. comfort Range 2 Mean comfort Range 3 Min. comfort Range 4 Unacceptable

The calculation of natural frequencies and corresponding mode shapes of the footbridge should be performed with ﬁnite element software. However, simple analytical calculations can be used to determine the order of magnitude of the frequencies. The bridges natural frequencies are necessary to determine for Class I-III and it is necessary to simulate all possible mode shapes in the critical frequency range up to 5 Hz. Depending on the structure the mode shapes may vary and the most critical mode shape might also vary between vertical, transversal or longitudinal vibrations. Two mass assumptions are used in the design guideline, one calculation for an empty bridge and one with the bridge loaded through its bearing area, given the pedestrian load of 70 kg/m2. The frequencies for the empty bridge is used to determine the risk of resonance and the frequencies with increased mass is used when calculating the accelerations [1]. Sétra [1] presents four diﬀerent ranges of frequencies between 0-5 Hz, corresponding to a decreasing risk of resonance. Range 1 is for the maximum risk of resonance and Range 4 for negligible risk. These ranges are presented in Table 3.8 and Table

32 3.3. DESIGN GUIDELINES

3.9 for vertical and horizontal vibrations respectively. Depending on the class of the bridge and the frequency range one can determine which load cases to implement in the model. Table 3.8: Frequency ranges, vertical direction (after [1])

Frequency 0-1 1-1.7 1.7-2.1 2.1-2.6 2.6-5 5< ranges [Hz] [Hz] [Hz] [Hz] [Hz] [Hz] Range 1 Range 2 Range 3 Range 4

Table 3.9: Frequency ranges, lateral direction (after [1])

Frequency 0-0.3 0.3-0.5 0.5-1.1 1.1-1.3 1.3-2.5 2.5< ranges [Hz] [Hz] [Hz] [Hz] [Hz] [Hz] Range 1 Range 2 Range 3 Range 4

3.3.5.2 Load Application

Three different load cases are presented in the guideline which are simplified and practicable ways of implementing the effect of a sparse or dense crowds, by multiplying the effect of a single person with an equivalent number of pedestrians. The load cases are based on probability calculations and statistical processing to incorporate the effect of pedestrian crowds. The simulations assume random walking frequencies and phases of the pedestrians and each time assessing the equivalent number of pedestrians which, when distributed evenly, or in phase and at the natural frequency of the bridge will give rise to the same response as random pedestrians. The tests were repeated 500 times with a fixed number of pedestrians, fixed damping and fixed number of mode nodes. By varying these parameters two different laws for implementing the crowd effect were retained and are presented by Equation 3.32.

 q 10.8 · (N · ξ) for sparse or dense crowds Neq = √ (3.32) 1.85 · N for very dense crowds

The equivalent number of pedestrians, Neq, is the number of pedestrians who, at the same frequency and in phase, would give rise to the same eﬀect as random pedestrians. The combination of traﬃc class and frequency range will determine which load case to apply to the bridge model (Figure 3.15). The load should be applied to the

33 3.3. DESIGN GUIDELINES whole bridge structure with the direction of the load equal to the direction of the mode shape and inverted each time the mode shape changes direction to produce the maximum eﬀect of the applied load.

Figure 3.15: Load case veriﬁcation dependent on class and frequency ranges [1]

The three diﬀerent load cases corresponds to the following crowd conditions:

• Case 1 - Sparse or dense crowds

• Case 2 – Very dense crowds

• Case 3 – Crowd complement, 2nd harmonic

3.3.5.3 Load Cases

Load case 1 corresponds to sparse or dense crowds and are to be considered for bridges with a traffic class of III or II. The difference between the classes regard the density of the pedestrians (Table 3.10). The first load case given by Equation 3.33 is divided in to three loads, one for each direction of vibration. The vertical load should be applied to the vertical modes at risk and the transversal load to the transversal modes at risk and at each case the load frequency should be adjusted to the natural frequency of the structure. The static weight of the pedestrians is not included in the equations since it has no influence on the acceleration. However, the mass of pedestrians (calculated for the given density) should be incorporated in the mass of the bridge. This applies for load case 2 and 3 as well.

Table 3.10: Density of the crowd for Class III and II [1]

Class Density of the crowd III 0.5 ped/m2 II 0.8 ped/m2

34 3.3. DESIGN GUIDELINES

 s  ξ d · (280N) · cos (2πfv · t) · 10.8 · · ψ Vertical direction  n  s  ξ w(t) = d · (140N) · cos (2πfv · t) · 10.8 · · ψ Longitudinal direction (3.33)  n  s  ξ d · (35N) · cos (2πf · t) · 10.8 · · ψ Transversal direction  t n

Where:

w(t) Time-varying surface load N/m2.

d Density of the crowd, see Figure 3.10 ped/m2.

fv,l,t Natural frequency for the mode shape [Hz].

t Time, in seconds [s].

ξ Critical damping ratio, see Table 3.11 [-].

n Number of pedestrians on the bridge, n = S · d [-].

ψ Reduction factor, considers the risk of resonance (Figure 3.16 and 3.17) [-]

S Surface area of the deck [m/s2].

Load case 2 is used for very dense crowd situations and is considered for bridges in traﬃc class I. This load case is similar to Load case 1, with a diﬀerence in the density of the crowd and in the calculation for the equivalent number of pedestrians. The load per unit area for load case 2 is given by Equation 3.34.

 s  1 d · (280N) · cos (2πfv · t) · 1.85 · · ψ Vertical direction  n  s  1 w(t) = d · (140N) · cos (2πfv · t) · 1.85 · · ψ Longitudinal direction (3.34)  n  s  1 d · (35N) · cos (2πf · t) · 1.85 · · ψ Transversal direction  t n

Load Case 3 considers the second harmonic of the load and is only taken in to account for bridges with a traffic class of I or II. The load is similar to Case 1 and 2 but the force exerted by a single pedestrian is reduced to 70N in the vertical direction, 7N in the transverse direction and 35N longitudinally. The density of the crowd is set to 1 ped/m2 for traffic class I and 0.8 ped/m2 for traffic class II. As for the other load cases the load is uniformly distributed.

35 3.3. DESIGN GUIDELINES

Figure 3.16: Reduction factor for vertical and lateral vibrations, Case I and II. [1]

Figure 3.17: Reduction factor for vertical and lateral vibrations, Case III. [1]

Table 3.11: Damping ratio for diﬀerent types of bridges [1]

Type Critical damping ratio [%] Reinforced concrete 1.3 Pre-stressed concrete 1 Mixed 0.6 Steel 0.4 timber 1

3.3.6 SYNPEX

Research Fund for Coal and Steel has funded an extensive research and measurements to develop the report called Advanced load models for synchronous pedestrian excitation and optimized design guidelines for steel footbridges or in short, SYNPEX [23]. The load models were derived by measuring the pedestrian induced forces on a test rig that were able to vibrate both laterally and vertically. Verification of the load models were performed on 9 different footbridges to ensure its accuracy. The design methodology presented by SYNPEX is based on classification of the traffic and comfort levels of the studied bridge which will be further explained in the following subsection.

36 3.3. DESIGN GUIDELINES

3.3.6.1 Design Methodology

As mentioned above the guideline is based on assessment of traffic and comfort levels. These levels need to be determined before the structural analysis of the bridge is performed. The chosen traffic and comfort levels lead to different design situations that need to be assessed by the designer in the structural analysis. The design methodology starts with determining the natural frequencies of the structure and determining if the frequencies lies within the critical frequency range. The chosen traffic class and the calculated natural frequencies leads to a selection of different load models that need to be implemented in the analysis and the response of which are to be evaluated against the comfort criteria set by the bridge owner. A flowchart of the design methodology is given in Figure 3.18 and a more detailed description of the different sections are given below.

Figure 3.18: SYNPEX guideline ﬂowchart with section references to original report [23]

SYNPEX [23] presents two ranges of frequencies, 1.3 to 2.3 Hz and 0.5 to 1.2 Hz,

37 3.3. DESIGN GUIDELINES which corresponds vertical and lateral vibrations respectively. If the structure has natural frequencies within these ranges the bridge is prone to excessive vibrations during normal use. The traffic class, presented by Table 3.12, is chosen after the expected type of pedestrian traffic. It is important to consider the different traffic situations that might occur during the life cycle of the bridge, such as inauguration and the daily use, to check the serviceability for every situation in order to guarantee the comfort.

Table 3.12: Traﬃc classes [23]

Traffic Description Pedestrian Characteristics Class loading TC 1 Very weak Group of 15P; b=width of deck traffic d=15P/bl l=length of deck TC 2 Weak traffic d=0.2P/m2 Comfortable and free walking Overtaking is possible Can freely choose pace TC 3 Dense traffic d=0.5 P/m2 Significantly dense traffic Unrestricted walking Overtaking inhibited TC 4 Very dense d=1.0 P/m2 Freedom of movement restricted traffic Uncomfortable situation Overtaking is not possible TC 5 Exceptional d=1.5 P/m2 Crowding begins dense traffic One can no longer choose pace

SYNPEX presents diﬀerent comfort classes ranging from 1-4 with a decrease in degree of comfort. Depending on the chosen comfort class, acceleration limits are set for both the vertical and lateral response (Table 3.13). These acceleration criteria are to be compared to the calculated acceleration response from the structural analysis. SYNPEX [23] refers to the Sétra guideline regarding the acceleration limit in the lateral direction for lock-in eﬀect. The experiments that led to the SYNPEX guideline shows a risk for synchronisation with the bridge if a pedestrian has a walking frequency of +/- 0.2 Hz of the structures natural frequency.

Table 3.13: Comfort classes according to SYNPEX [23]

Comfort Degree of Acceleration Acceleration Class comfort level vertical level lateral [m/s2] [m/s2] 1 Maximum <0.50 <0.10 2 Medium 0.50-1.00 0.10-0.30 3 Minimum 1.00-2.50 0.30-0.80 4 Unacceptable >2.50 >0.80 discomfort

38 3.3. DESIGN GUIDELINES

SYNPEX also refers to the Sétra guideline (Section 3.3.5) for critical damping ra- tions.

3.3.6.2 Load Model

A general expression for the load on the bridge deck is given by,

n0 p(t) = G · cos(2πft) · · ψ. (3.35) S

Where:

G · cos(2πft) Harmonic load due to a single pedestrian [N] .

f Natural frequency under consideration [Hz].

n0 Equivalent number of pedestrians on loaded area S [−].

S Loaded area [m2].

ψ Reduction factor to take into account the probability that the footfall frequencies approach the natural frequency under consideration [-].

The amplitude of the dynamic force, G, and the reduction coeﬃcient ψ when considering walking are given in Table 3.14.

Table 3.14: Parameters of walking for load model (after [23])

G[N] Vertical Longitudinal Lateral 280 140 35 Reduction coeﬃcient ψ Vertical and longitudinal Lateral

39 3.3. DESIGN GUIDELINES

Footbridges are usually subjected to actions from several pedestrians simultaneously. Since the pedestrians are more or less synchronized, the actions from each pedestrian on the bridge cannot simply be added together to get the total force exerted on the bridge. The number of pedestrians that are assumed to be synchronized (equivalent number of pedestrians) depends on the loaded area and the density of pedestrians. With a density less than ped/m2 (TC1 to TC3), pedestrians are assumed to be able to move without restrictions. When the density increases above ped/m2 (TC4 and TC5) the degree of synchronization increases as the movements of the pedestrians become more homogeneous. The equivalent number of pedestrians can be determined through Equation 3.36.

 √ 10.8 ξ · n for TC1 and TC3 n0 = √ (3.36) 1.0 · 1.85 n for TC4 to TC5

Where:

n Total number of pedestrians. n = S · d [-].

S Loaded area of bridge [m2]

ξ Structural damping ratio [-]

The load model for jogging are considered in the vertical direction. The load model is based on that n jogger are perfectly synchronized (n = n0) with the parameters G and ψ according to Table 3.15.

Table 3.15: Parameters for jogging for load model (after [23])

G[N] Vertical Longitudinal Lateral 1250 − − Reduction coeﬃcient ψ Vertical and longitudinal Lateral

40 3.3. DESIGN GUIDELINES

3.3.7 HIVOSS

Research Fund for Coal and Steel has produced the report Human induced Vibration of Steel Structures or in short, HIVOSS, which is guidelines based on the previously mentioned SYNPEX research project (Section 3.3.6) [18] [10]. Therefore, HIVOSS implements the same guideline for footbridges as SYNPEX, which is visualized in the flowchart in Figure 3.19. The goal was to produce the first European guideline for vibration design of pedestrian bridges and floors. There are some additional information in the HIVOSS guideline which is presented in the following sections.

Figure 3.19: HIVOSS guideline ﬂowchart [10]

41 3.3. DESIGN GUIDELINES

3.3.7.1 Reduction Coeﬃcient

For footbridges with a vertical or longitudinal natural frequencies in the range of 2.5 to 4.6 Hz, the second harmonic of the pedestrian load should be considered. The reduction factor, ψ , for the 1st and 2nd harmonic can be seen in Figure 3.20.

Figure 3.20: Reduction factor ψ for the vertical and longitudinal (left) and the lateral (right) direction

3.3.7.2 Criteria for Lateral Lock-in

SYNPEX mentions that the lock-in phenomenon begins at a trigger acceleration amplitude of 0.1-0.15 m/s2. HIVOSS provides a method for calculating the triggering number of pedestrians at which a sudden ampliﬁed response is produced. This method is called Arup’s criterion and is given by the following equation,

8πξm∗f N = , (3.37) L k where:

ξ Structural damping ratio [-]

m∗ Modal mass [kg]

f Natural frequency [Hz]

k Constant of 300 [Ns/m] over the range 0.5-1.0 Hz.

3.3.8 JRC

JRC Scientiﬁc and Technical Reports [24] has based the report Design of Lightweight Footbridges for Human Induced Vibrations on the SYNPEX and HIVOSS guidelines, covered in Sections 3.3.6 and 3.3.7 respectively. JRC does not provide any additional information compared to SYNPEX and HIVOSS. Hence, it is not further explained. This guideline is however present in other literature and is therefore mentioned here as well.

42 Chapter 4

Finite Element Analysis

This section aims to describe the method used to construct the ﬁnite element models used to perform the dynamic analysis of the footbridge. The modelling is divided into several models and the method as well as how these models are combined will be further explained in the sections below.

4.1 General Description of the FE Modelling

The modelling part of this thesis consists of three different models; sub models of the foundation and the connection to BioMedicum as well as the truss system that makes up the bridge’s superstructure. Each model was built after preliminary drawings of the structure and Figure 4.2, 4.9 and 4.11 shows the different parts that together creates the entire structural system. The modelling work flow is visualized in the flowchart given in Figure 4.1. The modelling of the support at BioMedicum and the foundation was conducted using the graphical interface of Bridgade/Plus and the stiffness of each connection was extracted from the ODB-files using Python scripting. The stiffness of the supports was then included in the model of the superstructure, both for the beam and the shell model. The models were then submitted for frequency analysis and Python scripting was used to extract the frequencies. The frequencies were later imported to Matlab for post-processing of plots.

43 4.2. BEAM MODEL

Figure 4.1: Flowchart of the FE modelling

4.2 Beam Model

The geometry for the beam model was based on preliminary drawings of the structure and the cross section for each structural member of the truss system is given in Chapter 2. Python scripting was used to construct the truss structure and it was built as a single part. The deck was, however, modelled as a separate part using shell elements and connected to the crossbeams by tie constraints. The top part of the support columns (above ground level) was included in the model, as well as the column heads, and the stiﬀness extracted from the sub model of the foundation was introduced at the ends of the columns as connector elements. The connection to BioMedicum was also modelled using connector elements with a stiﬀness from corresponding sub model. The sub models are further explained in Section 4.4. A representation of the beam model is given in Figure 4.2. Early models of the superstructure incorporated the inclination of the bridge (see Figure 2.1 in Chapter 2) which was neglected in the later models. Instead of including the inclination, the support columns were given a length equal to the tallest of the columns.

44 4.2. BEAM MODEL

The bridge deck is a composite deck comprised of a concrete deck cast on top of a corrugated steel sheet (TRP) with welded steel studs, see Figure 4.3. Two different versions of the deck were used in the modelling, one model with the 120 mm of concrete with a density calculated from the cross section (Figure 4.3) to incorporate the steel sheet and one model where the steel was neglected to check the effect of low interaction in the composite deck. That model had a concrete deck with a thickness of 75 mm and the mass from the remaining concrete and the steel sheet were incorporated by increasing the density. Mass from non-structural members was also included in the beam model of the superstructure. Calculations for the non-structural mass is given in Appendix A.3. The calculated mass was incorporated in the model by manipulating the density of the four chords. Non-structural elements has the potential to influence the dynamic behavior of the footbridge significantly but the amount of contributing mass and stiffness is uncertain. By adding the extra mass of non-structural members but neglecting their contributing stiffness, a lower limit of the natural frequencies can be obtained. The result is presented in Chapter 6.

Figure 4.2: Beam model of the superstructure

45 4.2. BEAM MODEL

Figure 4.3: Drawing of the composite deck

4.2.1 Material Data

The concrete and steel is set as elastic isotropic materials with values chosen for Young’s modulus, Poisson’s ratio and density. Poisson’s ratio and density is given by Table ??. The Young’s modulus vary between diﬀerent models due to the parametric study and will be further presented in Chapter 5 covering the parametric study.

Table 4.1: Values for density and Poisson’s ratio

Material Poisson’s ratio Density [−] [kg/m3] Concrete 0.3 2400 Steel 0.3 7800

4.2.2 Element Types

The beam model consists of beam elements except for the deck plate which was modelled with shell elements. The truss structure is modelled with type B31 elements, which is a shear flexible (Timeshenko) 3D beam element with reduced integration. The shear flexible beam elements take the transverse shear deformations into account and these elements are also used for the columns and the column heads. Shell elements are used to model structures where one dimension, usually the thickness, is much smaller than the other directions. Hence, shell elements are useful for modelling of the bridge deck. S4R is the element type used for the shell comprising the deck plate. It is a 4-node general-purpose quadrilateral element with reduced integration. General-purpose means that it gives a good solution for both thin and thick shells because it utilizes both the Kirchhoff shell theory and the shear flexible Mindlin theory. With reduced integration, the number of Gaussian points is reduced

46 4.2. BEAM MODEL to one which is positioned in the center of the shell element. Shear calculations are performed for the Gaussian point and assumed constant over the entire element. Therefore, values of shear strain, force and stress will remain constant over the element. Reduced integration requires more elements to generate the same results as fully integrated elements, but the computational time is less for the elements with reduced integration [25].

4.2.3 Connections and Boundary Conditions

The supports are modelled separately, as previously mentioned, to ease the modelling and to reduce the computation time. The stiffness extracted from the sub models was introduced as connector elements at the bottom of the cut off columns and at the top of the truss where the bridge is connected to BioMedicum. Reference points were placed close to these connections and the connector elements were placed in between the connection points and the reference points. Fixed boundary conditions was then introduced at the reference points, making the displacements and rotations depend on the stiffness of the connector element. The connection category for the connector elements were set to basic with Cartesian and Rotation set as the translational and rotational type respectively. Coupled linear elastic behavior was used with six-DOF representing the upper triangular of the stiffness matrix extracted from the sub models. A detailed view of the connector elements at BioMedicum is presented by Figure 4.4. The truss was created as a single part, no extra conditions to ensure the connection between the members was necessary. However, connections needed to be introduced for the bridge deck and the columns. The deck was partitioned along each crossbeam and coupled together with the underlying crossbeam by tie constraints. The bearings, which connect the truss structure to the column heads, was modelled using couplings. Two reference points was introduced between the column heads and the corresponding crossbeam at the position of the bearings. Each reference point was connected to either the crossbeam or the column head and between these reference points the displacement conditions were introduced. The displacement conditions are in accordance with Figure 4.5.

47 4.3. SHELL MODEL

Figure 4.4: Connector elements used to model the stiﬀness at BioMedicum

Figure 4.5: Bearings for the footbridge

4.3 Shell Model

A shell model, Figure 4.6, of the superstructure was created as a verification method of the previously mention beam model. The shell model was created using the graphical interface of Brigade/Plus with sketches created in AutoCad. The sketches are based on the drawings of the centerlines of the structural members. The cross section was created by offsetting the centerlines in each direction. The truss structure was created from four different parts, one part for each side and separate parts for the crossbeams and the roof of the truss. The dimensions for each stuctural member is given in Chapter 2.

48 4.3. SHELL MODEL

Figure 4.6: Shell model of the superstructure

4.3.1 Material Data

Only one shell model of the bridge was created and the chosen material parameters were in accordance with the beam model with the lowest natural frequency. Values for Poisson’s ratio and the density was chosen according to Table ??. The material parameters will be further presented in the Chapter 5 covering the parametric study.

4.3.2 Element Types

In the shell model, shell elements make up the bridge deck as well as the truss structure. S4R is the element type used for the shells, which is a general-purpose quadrilateral element [25]. The columns and column heads remain as beam elements. Description of the element types is given in Section 4.2.2.

4.3.3 Connections and Boundary Conditions

The diﬀerent parts of the truss were connected using tie constraints in the interaction module. The tie constraints were introduced between the crossbeams and the bottom cords and between the roof and the top chords. The deck plate was partitioned over the crossbeams and the interaction was ensured by tie constraints as well. Reference points were introduced at the ends of the chords and the points are tied together with the entire cross section instead of a single node, see Figure 4.7. Con- nector elements were placed between these points and the reference points with the ﬁxed boundary conditions, as in the beam model.

49 4.4. SUB MODELS OF SUPPORTS

Figure 4.7: Detailed view of the reference point representing the end of the chord

4.4 Sub Models of Supports

Special consideration was paid to the supports for the bridge since they play a big part in the dynamic behavior of the structure and each support has been carefully modelled in an isolated model, so called sub model. The model was divided into different sub parts to ease the modelling and to reduce the computational cost. These sub models were later combined in the model for the superstructure by introducing their individual stiffness as connector elements. To extract the stiffness of the supports, unit loads and unit moments were introduced at the connections in three directions (U1-U3 and UR1-UR3) and the stiffness was calculated as the force divided by the simulated displacement. The stiffness from each model was extracted using Python scripting as seen in the flow chart (Figure 4.1).

4.4.1 Substructure

The bridge is supported by two concrete columns which in turn are founded on the roof of a concrete culvert, approximately 2.5 meters’ underneath Solnavägen, running along the length of the bridge. The sub model of this support was created using the graphical interface of Brigade/Plus with the use of both shell (culvert) and solid elements (columns and surrounding fill). The amount of contributing fill material as well as the dimensions for the culvert were calculated from preliminary designs of the culvert, see Figure 4.8. As seen in Figure 4.8, the culvert rests on a thin layer of fill material which in turn rests on the bed rock. The surrounding fill was assumed to be fixed to the rock. The concrete columns where cut at the ground level and the unit loads were introduced at the top of the cut off columns. Cut outs were created in the fill at the position of the columns, as seen in Figure 4.9, to remove any overlapping material.

50 4.4. SUB MODELS OF SUPPORTS

Figure 4.8: Drawing of culvert used to model the substructure, after [3]

Figure 4.9: Model of the substructure with cut outs for columns

The columns, the culvert and the surrounding fill was put together in the assembly module and the interaction between them was ensured by introducing tie constraints. Partitions were created at the top of the culvert were the columns are founded. The columns were tied to the partitioned surface via tie constraints. Each side of the culvert, except the before mentioned partitions, were connected to the surrounding fill. Fixed boundary conditions were in turn introduced at the outside of the fill, representing the surrounding rock.

4.4.2 Connection at BioMedicum

A sub model of the connection to BioMedicum was modelled with shell elements to evaluate the stiﬀness of the connection and to implement the stiﬀness in the model of the superstructure. The model of the connection was, as for the other models,

51 4.4. SUB MODELS OF SUPPORTS based on preliminary drawings of the connection, see Figure 4.10. The model of the connection to BioMedicum was limited to this column-beam connection which was considered to have the most influence on the stiffness. Fixed boundary conditions were introduced at the ends of each one-story column and the interaction between the columns and the beam was ensured by introducing tie constraints between the beams cross section and a corresponding partitioned area at the columns. The beam was modelled as a VKR 250x150x10 and the columns as VKR 250x250x12,5. The shell model is represented by Figure 4.11. The stiffness was derived by introducing a unit load at a reference point tied to an area of the beam representing the connection. The connection areas are closely spaced and are somewhat correlated, meaning that a displacement at one connection will cause a displacement at the other connection point. This was not taken into account and the unit loads were introduced separately. The stiffness of the plates and bolts, that make up the connection between the beam and the lateral bracing of the truss, was ignored. The preliminary design of the connection shows no fastening in the lower part of the truss structure and a lower connection was therefore not modelled in the sub model. However, the influence of steel plates welded between the lower part of the bridge and the building is tested in the parametric study.

Figure 4.10: Drawing of the connection to BioMedicum, after [3]

52 4.5. VERIFICATION OF MODELS

Figure 4.11: Shell model for the support at BioMedicum

4.4.3 Element Type

In the sub models, shell elements make up the beams for the connection at BioMedicum and the concrete culvert in the substructure. S4R is the element type used for the shells and the element type is previously described in Section 4.2.2. The ﬁll material surrounding the culvert is modelled with solid elements, type C3D10M, which is a robust modiﬁed 10-node second order tetrahedral element suitable for large deformation problems [25].

4.5 Veriﬁcation of Models

Veriﬁcation of the FE models are done by comparing the simulated mass of the superstructure to hand calculations. A convergence analysis of the the bridge deck is also conducted to verify the mesh size. The shell model serves as a veriﬁcation of the beam model with a comparison of mass, frequencies and mode shapes.

53 Chapter 5

Parametric Study

A parametric study of the bridge was performed for the properties which may be prone to uncertainties in data. The objective was to identify parameters which had a major impact on the natural frequencies as well as the mode shapes. A 2n-test was therefore performed for two parameters, Young’s modulus and stiﬀness, presented further in the following sections.

5.1 Selection of Design Parameters

Two major parameters were identiﬁed based on the preliminary designs from Tyréns[3]. The chosen parameters may be prone to uncertainties and the studied parameters are:

• Young’s modulus for concrete (columns, culvert and deck) as well as the surrounding ﬁll • Stiﬀness of the connection between the bridge and BioMedicum

Fill consisting of crushed stone has a recommended Young’s modulus of 50 MPa which was set as the lower limit value [26]. This was considered low with regard to the surrounding rock. A higher limit value was chosen as 250 MPa to see the effect on the frequencies and mode shapes. Concrete of strength class C20/25 was chosen with Young’s modulus equal to 30 GPa [27]. However, for cracked concrete the modulus was set to 0.6 · Ec in accordance with Bro 2004 [28]. This way of introducing a cracked concrete section was applied to all structural parts modelled as concrete. Aside from Young’s modulus, two values were tested for the stiffness of the lower part of the connection to BioMedicum. The lower limit value was chosen as zero assuming no connection. The second value was derived from the axial stiffness of four steel plates welded to a beam at BioMedicum. All of the investigated parameters are shown in Table 5.1.

54 5.2. METHODOLOGY

Table 5.1: Investigated parameters and values for parametric study

Fill Culvert Columns Deck Stiﬀness Ef [MP a] Ec[GP a] Ec[GP a] Ec[GP a] k [GN/m] 50 250 18 30 18 30 18 30 0 2.1

5.2 Methodology

The parametric study was divided into two separate studies, one for the substructure and one for the superstructure. The chosen values and combinations are visualized in Table 5.2 and 5.3. The first study was performed for the sub model of the substructure described in Section 4.4.1. A unit load test was performed in the sub model of the substructure where a force of 1 N was applied on a reference point tied to the top of the columns. The loads were applied in separate load cases, one for each principal direction. The long distance between the columns resulted in a very low interaction between the two columns and the load was therefore applied to both columns at the same time. The variation of the parameters resulted in eight different combinations, see Table 5.2. Each combination (A-H) refer to a specific stiffness, later used in the second part of the parametric study. The displacements were extracted from ODB-files using Python scripting and the stiffness was calculated through Equation 5.1. The rotational stiffness’s were evaluated by inserting a moment of 1 Nm in every principal direction and extracting the rotation. The rotational stiffness’s was calculated through Equation 5.2.

F k = (5.1) δ

M k = (5.2) θ Where:

F is the applied force [N]

δ is the displacement produced by the force along the same degree of freedom [m]

M is the applied moment [Nm]

θ is the rotation [rad]

By combining the eight combinations from the substructure (Table 5.2) with the variations of the parameters for the superstructure, a total of 32 combinations were

55 5.2. METHODOLOGY obtained, presented in Table 5.3. Python scripts were created for each combination and submitted to Brigade/Plus for frequency analysis. The ﬁrst ten natural modes of vibration were studied in the visualization mode in Brigade/Plus and the frequencies corresponding to each mode were extracted to Excel using Python scripting and later imported to Matlab for post-processing of scatter plots. The results from the FEA and the parametric study is presented in Chapter 6.

The values for Ec for the columns, in the parametric study of the superstructure, are chosen in accordance with the combinations of the substructure shown in Table 5.2. The columns are continuous throughout the foundation and must therefore be assigned the same values. If the columns are cracked in the substructure they must be cracked in the model for the superstructure as well, hence Combination A, C, E and G will only be combined with Ec equal to 18 MPa and the remaining combinations with 30 MPa (Table 5.3).

Table 5.2: Combinations for parametric study of the substructure

Fill Culvert Columns Ef [MPa] Ec[GPa] Ec[GPa] Combination 50 250 18 30 18 30 A x x x B x x x C x x x D x x x E x x x F x x x G x x x H x x x

56 Table 5.3: Combinations for parametric study of the superstructure

Columns Deck BioMedicum Substructure Ec[GPa] Ec[GPa] k [GN/m] Combination Combination 18 30 18 30 0 2.1 1 A x x x 2 A x x x 3 B x x x 4 B x x x 5 C x x x 6 C x x x 7 D x x x 8 D x x x 9 E x x x 10 E x x x 11 F x x x 12 F x x x 13 G x x x 14 G x x x 15 H x x x 16 H x x x 17 A x x x 18 A x x x 19 B x x x 20 B x x x 21 C x x x 22 C x x x 23 D x x x 24 D x x x 25 E x x x 26 E x x x 27 F x x x 28 F x x x 29 G x x x 30 G x x x 31 H x x x 32 H x x x Chapter 6

Results

This section presents the results from the FEA, including frequencies and mode shapes for the models and the results from the parametric study. The application of the design guidelines and the veriﬁcation of the model are also presented in this section.

6.1 Frequencies and Mode Shapes

The frequency analysis performed in Brigade/Plus generated the results presented in this section. The ﬁrst four frequencies and corresponding mode shapes for the beam model of the footbridge is illustrated by Figure 6.1, 6.2, 6.3 and 6.4 corresponding to mode 1, 2, 3 and 4 respectively. The ﬁrst two natural modes of vibrations are transversal modes and the third and fourth are vertical modes of vibration. These modes and frequencies corresponds to the worst case determined from the parametric study. The complete parametric study is presented in Section 6.2.

58 6.1. FREQUENCIES AND MODE SHAPES

Figure 6.1: Transversal mode 1, Figure 6.2: Transversal mode 2, ft = 2.1967 Hz ft = 3.5056 Hz

Figure 6.3: Vertical mode 1, Figure 6.4: Vertical mode 2, ft = 5.9556 Hz fv = 6.7064 Hz

The eﬀect of including non-structural mass in the model is visualized by Figure 6.5 to 6.8 . The mass was added to the chords of the superstructure and the calculations are presented in Appendix A.3. The mass was added to the worst case determined from the parametric study. Including non-structural mass resulted in reduced frequencies where the maximum decrease, 7.6 %, occurred for the second vertical mode (Mode 4). No change occurred in the mode shapes.

Figure 6.5: With non-structural mass. Figure 6.6: With non-structural mass. Transversal mode 1, Transversal mode 2, ft = 2.0534 Hz ft = 3.3392 Hz

59 6.1. FREQUENCIES AND MODE SHAPES

Figure 6.7: With non-structural mass. Figure 6.8: With non-structural mass. Vertical mode 2, Vertical mode 2, ft = 5.5911 Hz fv = 6.2312 Hz

The eﬀect of no interaction in the composite deck is visualized by Figure 6.9 to 6.12. The thickness of the bridge deck was reduced but the density included the entire deck structure as described in Section 4.2. As with the extra mass due to non-structural members, the reduced thickness of the bridge deck was evaluated for the worst combination. No interaction in the bridge deck resulted in a decrease of the frequencies with less than 1 % for all modes. The mode shapes remained unchanged.

Figure 6.9: Without interaction. Figure 6.10: Without interaction. Transversal mode 1, Transversal mode 2, ft = 2.1898 Hz ft = 3.4941 Hz

Figure 6.11: Without interaction. Ver- Figure 6.12: Without interaction. Ver- tical mode 1, tical mode 2, ft = 5.9170 Hz fv = 6.6658 Hz

60 6.2. PARAMETRIC STUDY

6.2 Parametric Study

The result from the parametric study is given in Figure 6.13, where the the natural frequency for the first lateral and the first vertical mode of vibration is plotted for each combination. The parametric study yielded a natural frequency range between 2.20-2.93 Hz for the lateral mode and 5.96-6.67 Hz for the vertical mode of vibration. The lower limit values were obtained from Combination 1 and the higher limit values from Combi- nation 32. Each subcombination, A-H, is given a different color in the scatter plot to visualize the impact of the parameters regarding the substructure. An increased stiffness from the substructure results in an increase in the frequencies, mainly for the first lateral mode of vibration. An increase in the Young’s modulus for the fill increases the natural frequencies for the lateral mode while having a small impact on the vertical frequency. This is visualized by the difference between Combination 1 and 9. The same goes for the modulus of the culvert, where the increase is given by the difference between Combination 1 and 5. A larger increase in the frequencies, both lateral and vertical, is obtained by changing the Young’s modulus for the concrete columns from 18 to 30 GPa. The increase is given by the difference between Combination 1 and 3. Introducing a stiffness in the lower part of the truss at the connection to BioMedicum results in an increase of the natural frequency of approximately 0.6 Hz for the vertical mode (Combination 2 and 18) and 0.1 Hz for the lateral mode (Combination 1 and 17). This effect is seen by comparing all combinations with the increased stiffness (Combination 17-32) to the combinations without the extra stiffness in the lower part of the connection (Combination 1-16). The effect of an uncracked concrete deck (higher Young’s modulus) results in a small increase of the frequencies for both the vertical and lateral mode. However, this effect is greater when combined with the extra stiffness at the connection to BioMedicum. The effect of an uncracked deck can be seen by comparing the frequencies for any consecutive combination in Figure 6.13.

61 6.3. VERIFICATION OF BEAM MODEL

All Combinations 6.7 28 32 Comb. 20 A 24 B 6.6 C 26 30 D 18 E 22 27 31 F 19 G 6.5 23 H 25 29 17 21 6.4

6.3

6.2 8 16 12 15

Vertical Natural Frequency [Hz] 4 6.1 7 14 3 11 6 10 2 13 6 5 1 9

5.9 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 Lateral Natural Frequency [Hz]

Figure 6.13: Scatter plot from parametric study of all combinations

6.3 Veriﬁcation of Beam Model

As a quality assurance of the beam model a convergence analysis of the mesh size was conducted for the bridge deck. As seen in Figure 6.14, a mesh size of 0.1 captures the shape of the finer mesh, size 0.05, but has lower peaks over each cross beam. With smaller mesh size these peak values will tend to infinity due to the singularity that occurs at the connection between the deck and the cross beams. The chosen size should give a good solution at the edge of the beams instead of at the center line. As seen Figure 6.14 the mesh size of 0.1 generates similar results as the finer mesh at the edges of the cross beams and was therefore chosen as the mesh size for the bridge deck.

62 6.3. VERIFICATION OF BEAM MODEL

Convergence Plot 2000 Mesh 0.05 Mesh 0.1 Mesh 0.2

1500

1000

500 SM1 [Nm]

-500

-1000 0 5 10 15 20 25 30 35 40 45 True Distance [m]

Figure 6.14: SM1 over the true length of the bridge deck, measured at the middle

As further quality assurance of the model, the simulated mass of the structure was compared to hand calculations. Table 6.1 shows the simulated and the hand calculated mass of the structure as well as the difference in percent. The difference in mass between the hand calculated value and the simulated value for the beam model is 0.1 % whereas the difference of mass between the beam and the shell model is 2.3 %.

Table 6.1: Total mass of the model, hand calculation and simulated values

Hand calculation [kg] 75098 Beam model [kg] 75199 Shell model [kg] 73479 Diﬀerence (beam/hand calculation) [%] 0.1 Diﬀerence (beam/shell) [%] 2.3

To evaluate the beam model the natural frequencies and corresponding mode shapes were compared with the shell model. Figure 6.15, 6.16, 6.17 and 6.18 shows the comparison of the two first transversal and the first two vertical natural frequencies and corresponding natural modes of vibration. The frequencies and mode shapes for the two models are similar and the largest difference in frequencies occur in the second vertical mode with an increase of 2 % for the shell model. The second transversal mode for the shell model is out of phase with the mode for the beam model, as seen in Figure 6.16.

63 6.3. VERIFICATION OF BEAM MODEL

Figure 6.15: Comparison between shell and beam model, Transversal mode 1, fbeam = 2.1967 Hz fshell = 2.1866 Hz

Figure 6.16: Comparison between shell and beam model, Transversal mode 2, fbeam = 3.5056Hz fshell = 3.2973 Hz

64 6.3. VERIFICATION OF BEAM MODEL

Figure 6.17: Comparison between shell and beam model, Vertical mode 1, fbeam = 5.9556 Hz fshell = 6.0815 Hz

Figure 6.18: Comparison between shell and beam model, Vertical mode 2, fbeam = 6.7064 Hz fshell = 7.0406 Hz

The ﬁrst and second vertical mode for the shell model occur as the fourth and sixth mode. In between these modes the shell model exhibits transversal modes for the chords which are not captured by the beam model, see Figure 6.19.

65 6.3. VERIFICATION OF BEAM MODEL

Figure 6.19: Shell model mode 3. Transversal mode for the top chords

Local deformations at the joints occur and a detailed view of these deformations is given by Figure 6.20 with a deformation scale factor equal to 5.

Figure 6.20: Local deformations, shell model mode 3

66 6.4. ACCELERATION RESULTS

6.4 Acceleration Results

The natural frequencies were simulated to 2.05 Hz and 5.59 Hz for the ﬁrst lateral and the ﬁrst vertical mode of vibration respectively (Section 6.1). According to Eurocode 0 [2], accelerations in the vertical direction need to be evaluated for frequencies below 5.0 Hz, hence, no evaluation was necessary. For the lateral mode, however, an evaluation of the acceleration was necessary due to a frequency below 2.5 Hz [2]. Load models are presented for each guideline to enable a comparison between the load models. The models are presented for both vertical and lateral directions, given the frequencies and properties regarding the studied bridge. Only lateral load models are implemented in the FE model for evaluation of the accelerations. Eurocode 5 does not give any load models but considers acceleration limits in each direction.

6.4.1 Eurocode 5

Three cases are evaluated for the accelerations in the lateral direction. The ﬁrst case regard a single pedestrian crossing the bridge and the other two regard two cases for pedestrian groups, one for 13 pedestrians and one for 83 pedestrians. The accelerations are presented in Table 6.2.

Table 6.2: Acceleration results from Eurocode 5

Case Group Acceleration size [m/s2] 1 N=1 0.11 2 N=13 0.093 3 N=83 0.599

Calculations for accelerations are given in Appendix A.1.1

6.4.2 UK National Annex

According to the UK National Annex two cases was considered regarding pedestrian loading on footbridges. One case regarding the maximum acceleration in the vertical direction and one case regarding the likelihood of large lateral responses. The guideline also states a serviceability limit for vertical accelerations. According to the guideline no unstable lateral responses occurs for frequencies over 1.5 Hz (Section 3.3.2). Hence, no evaluation of lateral accelerations was performed. No calculations of the vertical acceleration were necessary due to the criteria from Eurocode 0. However, the limit for the vertical acceleration was calculated to 0.91 m/s2. The load model for the vertical direction is given by Equation 6.1.

67 6.4. ACCELERATION RESULTS

w(t) = 1.288 sin(2πfvt)[P a] (6.1)

Comprehensive calculations for the acceleration limit and the load models are given in Appendix A.1.2

6.4.3 SS-ISO 10137:2008

No evaluation was necessary for the vertical accelerations but no restrictions regarding the lateral frequency occurs in the guideline. The ISO 10137 guideline [20] presents two load models for the pedestrian load. One for a single pedestrian and one for a group of pedestrians where the dynamic response is reduced compared to a group with perfect synchronization by multiplication with a factor C(N). These load models are presented in Appendix A.1.3 but are not directly applicable in the FE model. By assuming a uniformly distributed load and a density of 1 pedestrian per m2 the load models could be modiﬁed and applied to the FE model. These models are given by Equation 6.2 and Equation 6.3 which represents the vertical and lateral load model respectively. Assumptions and comprehensive calculations of the load model are presented in Appendix A.1.3.

π w (t) = 3.56 · sin(2πf t + )[P a] (6.2) v v 2

π w (t) = 0.594 · sin(2πf t + )[P a] (6.3) h h 2

Implementation of Equation 6.3 generated the lateral acceleration response given by Figure 6.21. The load was simulated over 50 seconds with a critical damping ratio of 0.6 %. The acceleration converged to a value of 0.07 m/s2.

68 6.4. ACCELERATION RESULTS

Acceleration Plot 0.08 Acceleration A2

0.06

0.04 ] 2 0.02 m/s

-0.02 Acceleration [

-0.04

-0.06

-0.08 0 5 10 15 20 25 30 35 40 45 50 Time [t]

Figure 6.21: Lateral acceleration results for SS-ISO 10137:2008

Calculations for the acceleration limits, according to the ISO guideline [20], results in 0.108 m/s2 for lateral vibrations and 0.3 m/s2 for vertical vibrations.

6.4.4 Sétra

Given the natural frequencies of the ﬁrst modes, the bridge falls into frequency range 3 for the lateral direction and range 4 for the vertical direction (see Figure 3.15 in Section 3.3.5.2). No evaluation was necessary for the vertical accelerations due to the criteria in Eurocode 0 and according to the Sétra guideline. However, the third load case regarding the second harmonic of the lateral load was evaluated. Both Class I and Class II was implemented in Brigade/Plus to consider every possible scenario during the bridge’s life span. The load models are given by Equation 6.4.

 1.0984 · cos(2πf t)[P a] for Class I w(t) = t (6.4) 0.445 · cos(2πftt)[P a] for Class II

The loads were simulated over 50 seconds with the recommended critical damping ratio of 0.6 %. The lateral acceleration for Class I is given by Figure 6.22 and the lateral acceleration for Class II is given by Figure 6.22. The acceleration converged towards 0.12 m/s2 and 0.05 m/s2 for Class I and II respectively.

69 6.4. ACCELERATION RESULTS

Comprehensive calculations of the load models can be seen in Appendix A.1.4.

Acceleration Plot Class I 0.15 Acceleration A2

0.1 ]

2 0.05 m/s

-0.05 Acceleration [

-0.1

-0.15 0 5 10 15 20 25 30 35 40 45 50 Time [t]

Figure 6.22: Lateral acceleration results for Class 1

Acceleration Plot Class II 0.06 Acceleration A2

0.04 ]

2 0.02 m/s

-0.02 Acceleration [

-0.04

-0.06 0 5 10 15 20 25 30 35 40 45 50 Time [t] Figure 6.23: Lateral acceleration results for Class 2

70 6.4. ACCELERATION RESULTS

6.4.5 SYNPEX

According to the SYNPEX guideline [23] pedestrian bridges are prone to excessive vibrations during normal use if the frequencies are within 1.3 to 2.3 Hz and 0.5 to 1.2 Hz for vertical and lateral vibrations respectively. The load model include a reduction factor which considers the risk of resonance. For frequencies outside of the ranges stated by SYNPEX, which is the case for the studied bridge, this factor is equal to zero which reduces the load to zero as well. Hence, no load models were implemented in Brigade/Plus. For calculations and general load models from the SYNPEX guideline the reader is referred to Appendix A.1.5.

6.4.6 HIVOSS

As with the SYNPEX guideline, the load model was reduced to zero since the natural frequencies of the bridge fall outside of the critical ranges. HIVOSS is based on the SYNPEX report and implement the same load model and critical frequency ranges. However, the triggering number of pedestrians required for lateral lock-in was calculated to 21 pedestrians. Calculations of the critical number of pedestrians are presented in Appendix A.1.6.

71 Chapter 7

Discussion

7.1 Frequencies and Mode Shapes

The footbridge studied in this thesis exhibits natural modes of vibrations primarily in the transversal direction which was expected due to the shape of the structure and the support conditions. The values for the natural frequencies was, however, higher than expected and the risk of resonance and excessive vibrations due to normal use is low. The load frequencies from pedestrians ranges from 1.6 to 2.4 Hz in the vertical direction and the bridge has its first vertical mode of vibration at 5.96 Hz. The load frequency in the transversal direction can be assumed to be half of the vertical frequency, approximately 1 Hz, which is much lower than the bridge’s first transversal mode at 2.20 Hz. Introducing non-structural mass to the model resulted in lower natural frequencies for all modes of vibrations, but no difference in the mode shapes. This was the anticipated effect of including the mass without regard to the stiffness of non-structural elements. The frequency of the first transversal and vertical mode, 2.05 and 5.59 Hz, are used when implementing the design guidelines. The model without interaction resulted in a slight difference in the frequencies with a reduction of less than 1 % for the studied modes. No interaction was modelled by reducing the thickness of the deck which in turn reduced the stiffness. The deck was still modelled with the same mass by increasing the density. The effect was less than anticipated but it is still reasonable due to the results from the parametric study, which showed an increase of the frequencies with 0.2 and 0.6 % for the lateral and vertical mode respectively, when increasing the Young’s modulus for the deck. Thus, only manipulating the values for the bridge deck has a low impact on the dynamic properties. The high values of the frequencies can be explained by the chosen cross sections of the structural members and the design of the truss system. The beam model might, however, overestimate the stiffness of the truss structure, which is why the beam model is verified by a shell model.

72 7.2. PARAMETRIC STUDY

7.2 Parametric Study

The parametric study generated a frequency range for the first vertical and lateral mode of vibration where an increase of stiffness at the supports, both at BioMedicum and the substructure, had the greatest impact on the frequencies. Improving the conditions in the substructure led primarily to an increase of the frequencies for the lateral mode while an increase in stiffness at BioMedicum led, mainly, to an increase of the frequency for the vertical mode. This effect was expected due to the importance and impact of boundary conditions on the bridge’s dynamic behaviour. The exact stiffness of the supports is uncertain but by gradually improving the conditions, as in the parametric study, the range of possible frequencies can be captured. To model the support conditions with more certainty, tests need to be performed on the built structure and then compared to the FE model. This is of course not possible before the structure is built and therefore is a parametric test, as the one performed, of great interest for the designer. The effect of manipulating the parameters for the superstructure is visualized by comparing the combinations within each subcombination. Each improvement led to a slight increase of the frequencies and the greatest increase for the vertical mode occurred when improving the bridge deck after increasing stiffness at BioMedicum. This effect is larger than when only improving the deck which could indicate a correlation between the two parameters. Each combination with uncracked columns returned higher values of the natural frequencies. They had the highest individual impact and the effect was larger for the lateral mode than for the vertical which is a reasonable result due to the bridges free end at NKS. The columns are a sensitive part of the structure and increasing the Young’s modulus of the columns improves the bridge’s dynamic behavior.

7.3 Veriﬁcation of Beam Model

The beam model was verified by comparing the mass to hand calculated values as well as the shell model. Both returned a difference below 5 % (0.1 and 1.2 %) which assures the quality of the model. No significant difference occurred in the comparison of frequencies and mode shapes between the beam and the shell model. The shell model exhibits two transversal modes for the chords as the third and fifth mode which is due to a lower stiffness in the connection between the structural members compared to the beam model. Local deformations of the structure at the connections is also a result of the reduced stiffness. The shell model was created as a verification method for the beam model and the effects previously described was expected. However, the shell model has slightly higher natural frequencies for the vertical modes of vibration which suggest a higher bending stiffness than for the beam model. This effect is assumed to be caused by the chosen cross sections of the members comprising the truss structure.

73 7.4. ASSESMENT AND APPLICATION OF DESIGN GUIDELINES

The dimensions of the members, that intersect in the vertical plane, are similar which minimizes the local deformations at the connections and increases the stiffness. The consistency between the models assures the quality of the beam model and the verifications show that it is possible to capture the over all behaviour of the bridge by modelling with beam elements. However, the shell model exhibits modes of vibrations for the top chords which are not captured by the beam model. These modes occur due to the reduced stiffness in the connections between the members compared to the beam model. If these modes and the corresponding local effects are of interest, a shell model is necessary.

7.4 Assesment and Application of Design Guide- lines

The guidelines presented in the Theoretical Background (Section 3.3) presents different ways of determining the acceleration response. These guidelines are based on different assumptions but there are two general methods for evaluating the dynamic performance: direct calculation of the acceleration response given certain bridge properties or defining a load model based on the characteristics of a single pedestrian. These load models are based on a Fourier series which are often reduced to the first harmonic. Eurocode 5 [21] considers timber structures and provides a guideline for determining the maximum acceleration for simply supported, beam-like structures. However, the guideline is not material specific and can be used for any type of footbridge under the assumption that the structure behaves as a simply supported beam. The calculations for determining the amplitude of the acceleration in each direction is based on single pedestrian, modelled as a harmonic force with a forcing frequency equal to the resonance frequency. The acceleration response for n pedestrians is obtained by multiplying the response of a single pedestrian according to Equation 7.1 and Equation 7.2.

avert,n = 0.23 · avert,1 · n · kvert(fvert) (7.1)

ahor,n = 0.18 · ahor,1 · n · khor(fhor) (7.2)

The response of a single person, avert,1 and ahor,1, depend on the total mass and damping of the bridge as well as the fundamental frequency. The response is reduced for frequencies above the normal forcing frequency of pedestrians. The method proposed by Eurocode 5 is a simple method for calculating the maximum acceleration response but the method is not applicable to more complex structures. The method does not consider the change in forcing amplitude when the

74 7.4. ASSESMENT AND APPLICATION OF DESIGN GUIDELINES walking frequency is altered and the guide does not provide any load models to implement in the FE model, which is also seen as disadvantage. The UK National Annex to BS EN 1991-2:2003 provides a guideline for determining the maximum vertical acceleration and a method for determining the likelihood of large lateral responses under pedestrian loading. The guide presents a load model for crowd conditions, Equation 7.3, which is deﬁned as unit area load dependent on the area of the bridge deck and the number of pedestrians.

s F N w = 1.8 · 0 · k(f ) · γ · · sin(2πf t) (7.3) A v λ v

F0 is the forcing amplitude of a single pedestrian, A is the deck area and N is the total number of pedestrian depending on the area and the crowd density. The factor k(fv) account for the probability of walking at the resonance frequency, γ considers the correlation of pedestrians and λ considers the eﬀective number of pedestrians. No load model exists for the lateral direction but the likelihood of excessive accelerations is evaluated by determining the factor D, which is the pedestrian mass damping parameter. If the mass damping parameter falls below a certain limit, excessive accelerations are expected. This is a simple method for checking the lateral response but not intuitive and the method is only applicable for lateral frequencies below 1.5 Hz and frequencies regarding higher harmonics is not considered. This is a disadvantage of the UK National Annex guide which otherwise is easy to follow and to apply to the model. SS-ISO 10137 provides a load model, similar to the other guidelines, where the load of a single pedestrian is expressed with a Fourier series. However, the guide only presents models for single pedestrians or pedestrian groups but no model for crowded conditions, which is considered to be disadvantage for the ISO 10137 guideline. The model for pedestrian groups is obtained by multiplying the eﬀect of a single pedestrian with a coordination factor C(N), which is given by Equation 7.4. N is total number of pedestrians on the bridge.

√ N C(N) = (7.4) N

The load models are not directly applicable to 3D FE models and the group model was therefore modiﬁed to generate a unit area load applicable to the FE model (see Appendix A.1.3). The modiﬁed load models for the vertical and horizontal direction is given by Equation 7.5 and Equation 7.6.

√ π N w (t) = Q + Q · α · sin(2πf t + ) · (7.5) v 4,v v 2 N √ π N w (t) = Q + Q · α · sin(2πf t + ) · (7.6) h 2,h v 2 N

75 7.4. ASSESMENT AND APPLICATION OF DESIGN GUIDELINES

The static part of the pedestrian load is included in the mass of the bridge, as in the Sétra and SYNPEX guidelines. ISO 10137 presents DLF’s for the first five harmonics of the pedestrian load in the vertical direction but only one for the lateral direction. A frequency of 5.59 Hz coincides with the common forcing frequency of the 4th harmonic and has a DLF, α4,v, equal to 0.06. A frequency of 2.05 Hz, as for the lateral mode of vibration, is higher than the first harmonic of the lateral pedestrian force and to determine the load model for the lateral direction a DLF that represents a higher mode is necessary. For the load model in the lateral direction, given by Equation 7.6, the DLF, α2,h, was chosen to 0.01. Comprehensive calculations and assumptions are given in Appendix A.1.3. The SYNPEX and the Sétra guidelines are the most comprehensive of the ones presented in Section 3.3. Both methods provide a load model, defined as a unit area load, for the lateral and vertical direction which are applicable to 3D FE models. These load models are based on the characteristics of a single pedestrian and the models for pedestrian crowds use an equivalent number of pedestrians, n0 as a multiplication factor, which considers the synchronization of pedestrians. The calculation for the equivalent number of pedestrians is the same for both guidelines (Appendix A.1.4 and A.1.5). The amplitude of the load is also multiplied by a reduction factor, ψ, which considers the risk of resonance between the natural frequency of the bridge and the average pacing rate. The reduction factor differs between the guidelines which will generate different load amplitudes for certain frequencies. The load model presented by Sétra is rewritten to incorporate the total number of pedestrian, n, instead of the equivalent number, n0. The models for crowded conditions presented by Sétra and SYNPEX are given by Equation 7.7 and Equation 7.8 respectively, corresponding to the load case with a density equal to 1 ped/m2.

s 1 w (t) = d · G · cos (2πf · t) · 1.85 · · ψ (7.7) vert vert n

n0 p (t) = G · cos(2πf t) · · ψ (7.8) vert vert S

G is the static weight of a pedestrian which changes between different directions and harmonics. Sétra presents a value for the static weight regarding the second harmonic in the lateral direction which is not stated in SYNPEX. S is the loaded area of the bridge deck which depends on the studied mode shape and d is the crowd density. The HIVOSS guideline is based on the SYNPEX report and uses the same load models. However, the HIVOSS guideline is rewritten and easier to follow and implement. HIVOSS also includes a reduction factor regarding the second harmonic for the vertical load which is not covered by SYNPEX. Implementation of the guidelines and the corresponding load models resulted in different acceleration results. Eurocode 5 offers a direct calculation of acceleration response which generated a lateral acceleration of 0.11 m/s2 for a single pedestrian,

76 7.4. ASSESMENT AND APPLICATION OF DESIGN GUIDELINES

0.093 m/s2 for a group of 13 pedestrians and 0.599 m/s2 for 83 pedestrians. This method should be used for simply supported beam-like structures which is not the case for the bridge studied in this thesis and the response greatly exceeds the response from the other guidelines. SS-ISO 10137:2008 presents load models for single pedestrian and groups which are not directly applicable to the FE model. The load models were modified to unit area loads and implementation in the FE model led to a lateral acceleration of 0.07 m/s2. This result is for a steady stream of pedestrians with a density of 1 ped/m2 and the response is way below the value from Eurocode 5. The modified load considers the second harmonic of the lateral force. Implementation of Sétras load model, considering the second harmonic of the lateral load, generated an acceleration of 0.05 m/s2 and 0.12 m/s2 for a density of 0.8 and 1 ped/m2 respectively. Given the same density the Sétra load model results in higher accelerations than the ISO method due to a higher multiplication factor for the synchronization of pedestrians. Due to the dynamic evaluation criteria from Eurocode 0, no vertical load models were implemented. However, given the amplitude of the load model from the UK National Annex and the modified ISO model (1.288 compared to 3.56) it is safe to say that the modified ISO model will generate a higher acceleration. The frequencies of the studied bridge is mostly above the risk levels for resonance which results in a reduction factor equal to zero for the Sétra and SYNPEX load model. With a reduction factor equal to zero the amplitude equals zero as well. The accelerations obtained from the load models are based on a steady stream of pedestrians with a density of 0.8 or 1 ped/m2. This load case is highly unlikely to occur during the life span of the bridge because of its location and intended usage and the loading can be classified as an exceptional crowd condition. According to Eurocode 0 [2] the acceleration limit for such conditions is 0.1 m/s2 which is exceeded by the load model given by Sétra. However, this acceleration limit is a recommended maximum level and the simulated acceleration of 0.12 m/s2 is considered acceptable. Sétra [1] recommend 0.1 m/s2 as a design limit for lateral acceleration regarding lateral lock-in. Hence, the risk for lock-in exists for exceptional crowd conditions but are highly unlikely to occur during normal usage.

77 Chapter 8

Conclusions

8.1 Frequencies and Mode Shapes

The design of the bridge is dynamically sound, based on the models and the analyses performed to complete this MSc thesis. The bridge exhibits a transversal mode of vibration as the ﬁrst mode, which was expected due to the design of the structure and the supports. The frequencies are not within the sensitive frequency range with regard to pedestrian traﬃc.

8.2 Parametric Study

A parametric study was performed for two different parameters, Young’s modulus (culvert, columns, deck and fill) and the stiffness in the lower part of the connection between the bridge and BioMedicum. The test resulted in 32 different combinations with a natural frequency range between 2.20-2.93 Hz in the first lateral mode and 5.96-6.67 Hz in the first vertical mode. By including the non-structural mass the lower limit frequencies were lowered to 2.05 and 5.59 Hz in the first lateral and vertical mode respectively. The largest impact on the frequencies was obtained by manipulating the values for the supports, both at BioMedicum and the substructure. An extra stiffness at BioMedicum resulted in the largest increase in the frequency for the vertical mode of vibration and an increased stiffness of the substructure resulted in the largest increase of the lateral frequency. The concrete columns had the highest individual impact, beside the supports, on the dynamic behaviour and the effect was larger for the lateral mode than for the vertical mode.

78 8.3. VERIFICATION OF BEAM MODEL

8.3 Veriﬁcation of Beam Model

The verification of the beam model was conducted by comparing the mass of the beam model with hand calculations as well as comparing the mass between the beam and the shell model. Comparing the mass resulted in a difference of less than 5 % for both cases which assures the quality of the model. A shell model of the same structure was created to verify the frequencies and mode shapes from the FEA of the beam model. The frequencies and mode shapes were similar but the shell model had two transversal modes for the top chords which were not captured by the beam model. These modes were a result of the reduced stiffness of the connections of the truss structure when modelling with shell elements.

8.4 Assesment and Application of Design Guide- lines

The dynamic evaluation was carried out by implementing several design guidelines, including: Eurocode 5, UK National Annex, ISO 10137:2008, Sétra, SYNPEX and HIVOSS. An evaluation of the lateral accelerations was necessary due to a natural frequency below 2.5 Hz and implementation of the guidelines resulted in a maximum lateral acceleration of 0.599 m/s2 (Eurocode 5). Eurocode 5 uses a direct calculation of the acceleration response and considers simply supported structures. The studied bridge is more complex and a more comprehensive method for determining the accelerations were deemed necessary. By implementing load models from the guidelines the maximum acceleration was simulated to 0.12 m/s2 which was considered more realistic. The maximum acceleration relates to a load case considering a steady stream of pedestrians with a density of 1 ped/m2. The load case corresponds to an exceptional crowd condition which is unlikely to occur during the life span of the studied bridge. The recommended design guidelines are Sétra, SYNPEX and HIVOSS. Sétra and SYNPEX provide thorough background information regarding the research used to construct the guideline as well as information necessary to understand the characteristics of pedestrian loading. The Sétra and SYNPEX methods are similar and the dynamic evaluation involves determining traffic classes and comfort criteria which leads to certain load cases. Of the two, Sétra is the easiest to implement. The HIVOSS guideline is based on the SYNPEX report and presents the same load models, traffic classes and comfort criteria as SYNPEX. The HIVOSS guideline is easier to follow and therefore easier to implement in the FE model. This assessment is based on the implementation of the load models and not on how well the load models correspond to the reality, since this needs to be verified by measurements.

79 8.5. SUGGESTED DIRECTION FOR FURTHER RESEARCH

8.5 Suggested Direction for Further Research

A suggestion for further research is a validation the FE model by comparing the result with measured data. Given correct measured data, the FE model can be updated which would help further modelling of bridges with similar design and support conditions. This was originally intended to be included in this thesis, but was omitted due to delays in the building process. The connections of the structural members in the truss system for shell model would be interesting to study in order to capture the real dynamic behaviour of the bridge. However, this would include modelling of stiﬀeners in each connection which would be a time-consuming process.

80 Bibliography

[1] Sétra - The Technical Department for Transport, Roads and Bridges Engi- neering and Road Safety, Footbridges: assessment of vibrational behaviour of footbridges under pedestrian loading, Paris, 2006.

[2] SS-EN 1990, Eurocode 0: Basic of structural design - Appendix 2: Application for bridges. EN 1990 Annex A2 (2005).

[3] Tyréns AB, Ramhandling FO Karolinska - SKYWAY mellan Biomedicum och U2, Nybyggnad, Stockholm, 2016.

[4] A. Chopra, Dynamics of Structures - Theory and Applications to Earthquake, 4th Edition, Prentice Hall, 2014, iSBN-10 0-273-77424-7.

[5] Staﬀan Sunnersjö, FEM i praktiken, 2nd Edition, Industrilitteratur AB, Udde- valla, 1999.

[6] S. Živanović, A. Pavic, P. Reynolds, Human−structure dynamic interaction in footbridges, Bridge Engineering 158 (BE4) (2005) 165–177.

[7] F. Harper, W. Warlow, B. Clarke, The forces applied to the ﬂoor by the foot in walking, National Building Studies (Research Paper 32).

[8] F. Harper, The mechanics of walking, Research Applied in Industry 15(1) (1962) 23–28.

[9] T. Andriacchi, J. Ogle, J. Galante, Walking speed as a basis for normal and abnormal gait measurements, Journal of Biomechanics 10 (1977) 261–268.

[10] Human induced Vibrations of Steel Structures, Design of Footbridges - Guide- line (2007).

[11] S. Živanović, A. Pavic, P. Reynolds, Vibration serviceability of footbridges under human-induced excitation: a literature review, Journal of Sound and Vi- bration 279(1) (2005) 1–74.

[12] F. Venuti, V. Racic, A. Corbetta, Pedestrian-structure interaction in the vertical direction: coupled oscillator-force model for vibration serviceability assessment, in: Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014, Porto, Portugal, 2014.

81 BIBLIOGRAPHY

[13] F. Venuti, L. Bruno, Crowd-structure interaction in lively footbridges under synchronous later excitation: A literature review, Physics of Life Reviews 6 (2009) 176–206. [14] J. Rainer, G. Pernica, D. Allen, Dynamic loading and response of footbridges, Canadian Journal of Civil Engineering 15(1) (66-71). [15] H. Bachmann, W. Ammann, Vibration in structures induced by man and ma- chines, in: Structural Engineering documents, Vol. 3a, IABSE, Zurich, 1987. [16] V. Racic, A. Pavic, J. Brownjohn, Expermimental identification and analytical modelling of of human walking forces: Literature review, Journal of Sound and Vibration 326. [17] P. Dallard, A. Fitzpatrick, A. Flint, S. L. Burva, A. Low, R. R. Smith, M. Will- ford, The london millennium footbridge, The Structural Engineer 70 (22). [18] Human induced Vibrations of Steel Structures, Design of Footbridges - Back- ground Document (2007). [19] Y. Fujino, D. Siringoringo, A conceptual review of pedestrian-induced lateral vibration and crowd synchronization problem on footbridges, Journal of Bridge Engineering 21(8). [20] Swedish Standards Institute, Bases for design of structures – Serviceability of buildings and walkways against vibration (2008,). [21] SS-EN 1995-2:2004, Eurocode 5: Design of timber structures – Part 2: Bridges (2004). [22] British Standards Institution, UK National Annex to Eurocode 1: Action on structures - Part 2: Traffic loads on bridges, NA to BS EN 1991-2:2003, London, 2008. [23] C. Butz, M. Feldmann, C. Heinemeyer, G. Sedlacek, B. Chabrolin, A. Lemaire, M. Lukic, P. Martin, E. Caetano, A. Cunha, A. Goldack, A. Keil, M. Schlaich, Advanced load models for synchronous pedestrian and optimised design guidelines for steel footbridges. [24] C. Heinemeyer, C. Butz, A. Keil, M. Schlaich, A. Goldack, S. Trometer, M. Lukic, B. Chabrolin, A. Lemaire, P. Martin, A. Cunha, E. Caetano, Design of Lightweight Footbridges for Human Induced Vibrations (2009) 1– 82doi:10.2788/33846. [25] Hibbitt, Karlsson, Sorensen, ABAQUS/Standard User’s Manual, Pawtucket, 6th Edition (2006). [26] Trafikverket, Trafikverkets tekniska krav för geokonstruktioner-TK Geo 13, tDOK 2013:0667 (2013). [27] SS-EN 1992-1-1, Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings, ref. No. EN 1992-1-1:2005 (2005).

82 BIBLIOGRAPHY

[28] Vägverket, Vägverkets allmänna tekniska beskrivning för nybyggande och för- bättring av broar (BRO 2004), publikation 2004:56 (2004).

83 Appendix A

Numerical Results

A.1 Guideline Calculations

This appendix presents the numerical calculations for the implementation of the guidelines, the veriﬁcation of the model through mass comparison and the weight of the extra mass.

84 A.1. GUIDELINE CALCULATIONS

A.1.1 Eurocode 5

Indata Vertical natural frequency fvert = 5.59 Hz Lateral natural frequency fhor = 2.05 Hz Total mass of superstructure M = 75199 kg Structural damping ratio ξ = 0.6 % Width of deck b = 2.9m Length of deck L = 47.97 m

A.1.1.1 Vertical Acceleration

No check for vertical acceleration is required since fvert > 5.0 Hz.

A.1.1.2 Lateral Acceleration

The horizontal acceleration for one person crossing the bridge is calculated through Equation A.1.

50 50 m a = = = 0.111 (A.1) hor,1 M · ξ 75199 · 0.006 s2

The area of the bridge deck is calculated by Equation A.2

A = b · L = 139.11m2 (A.2)

The horizontal acceleration for several persons crossing the bridge is calculated for two cases. The ﬁrst case considers a predetermined number of 13 people while the second case is determined from a density of 0.6 people per square meter in accordance with Section 3.3.1. The two cases can be seen in Equation A.3.

 13 n = (A.3) 0.6 · A = 0.6 · 139.11 = 83.468

The horizontal acceleration for several persons crossing the bridge is given by Equa- tion A.4 which, for the two cases, results in Equation A.5. The coeﬃcient, khor(fhor) is given as 0.36 in accordance with Figure 3.9 in Section 3.3.1.

m a = 0.18 · a · n · k (f ) (A.4) hor,n hor,1 hor hor s2

85 A.1. GUIDELINE CALCULATIONS

 m 0.18 · 0.111 · 13 · 0.36 = 0.093 for n=13  s2 ahor,n = m (A.5) 0.18 · 0.111 · 83.468 · 0.36 = 0.599 for n=83.468  s2

86 A.1. GUIDELINE CALCULATIONS

A.1.2 UK National Annex

A.1.2.1 Vertical Load Model

Indata Vertical natural frequency fv = 5.59 Hz Lateral natural frequency fh = 2.05 Hz Crowd density d = 0.8 per m2 Width of deck b = 2.9m Length of deck L = 47.97 m

The general load model for pedestrians in crowded conditions is given by Equation A.6. s F N w = 1.8 · 0 · k(f ) · γ · · sin(2πf t)[P a] (A.6) A v λ v

In Section 3.3.2 the reference load, F0, is deﬁned for walking and running. The reference load is given in Equation A.7.  280N for walking F0 = (A.7) 910N for running

The factor k(fv) considers the relative weighting of pedestrian sensitivity and is given in Figure 3.10. For the natural vertical frequency ,fv, Equation A.8 is obtained.  0.12 for walking k(fv) = (A.8) 0.16 for running

The number of pedestrians on the bridge, simultaneously, is given by Equation A.9.

N = d · b · L = 0.8 · 2.9 · 47.97 = 111.29 (A.9)

From Figure 3.11 in Section 3.3.2 the reduction factor, γ, is obtained as 0.05 for a structural damping ratio of 0.6 %. The reduction factor for the eﬀective number of pedestrians is given by Equation A.10.

S λ = 0.634( eff ) = 0.634for S = S (A.10) S eff

If Equation A.7 to A.9 is inserted in Equation A.6, the load model represented by Equation A.11 is obtained.  1.288 sin(2πf t)[P a] for walking w = v (A.11) 5.581 sin(2πfvt)[P a] for running

87 A.1. GUIDELINE CALCULATIONS

A.1.2.2 Vertical Acceleration Limit

The vertical acceleration limit is calculated through Equation A.12 in accordance with Section 3.3.3. The response modiﬁers are given in Table A.1.

Table A.1: Response modiﬁers

Response modiﬁer Explanation Value Site usage(k1) Major urban centres 1.3 Route redundancy(k2) Primary Route 1.0 Bridge height (k3) Greater than 8m 0.7 Other factors (k4) - 1.0

m a = 1.0 · k · k · k · k = 1.0 · 1.3 · 1.0 · 0.7 · 1.0 = 0.91 (A.12) limit 1 2 3 4 s2

A.1.2.3 Lateral Acceleration

According to NA.2.44.7 [22] no unstable lateral responses will occur for frequencies over 1.5 Hz. Here, fh = 2.05 Hz.

88 A.1. GUIDELINE CALCULATIONS

A.1.3 SS-ISO 10137:2008

Indata Vertical natural frequency fv = 5.59 Hz Lateral natural frequency fh = 2.05 Hz Static load of one person Q = 700N Bridge deck area A = 139 m2

A.1.3.1 Vertical Load Model

The vertical load model presented in ISO 10137 is given by Equation A.13.

k X Fv(t) = Q 1 + αn,v sin(2πnft + φn,v) · C(N) (A.13) i=1

The design parameter αn,v is the DLF for the vertical direction, describing the ratio between the forcing amplitude and the weight of a pedestrian, and is taken as 0.06 according to Table 3.5 in Section 3.3.4 for the fourth harmonic (n = 4, 4.8Hz ≤ f ≤ 9.6Hz). The degree of coordination between the pedestrians is considered by the coordination factor C(N), given by Equation A.15. ISO 10137 does not state a crowd density for calculating the number of pedestrians, N, on the bridge. Assuming a crowd density, d, equal to 1 ped/m2, the total amount of pedestrians on the bridge can be calculated through Equation A.14.

N = d · A = 1 · 139 = 139 (A.14)

√ √ N 139 C(N) = = = 0.0848 (A.15) N 139 π The phase angle, φ , is chosen conservatively to 90o = in accordance with Section n,h 2 3.3.4. The ﬁrst term in Equation A.13 represents the static load of the pedestrians and is not dependent on the synchronization between the pedestrians, therefore the static part is not reduced by the factor C(N). This gives the load model in Equation A.16. To be able to implement it in Brigade/- Plus, the load model is multiplied with the crowd density (1 ped/m2), see Equation A.17.

π F (t) = 700 + 3.56 sin(2πf t + )[N] (A.16) v v 2

π w (t) = d · F (t) = 700 + 3.56 sin(2πf t + )[P a] (A.17) v v v 2

89 A.1. GUIDELINE CALCULATIONS

A.1.3.2 Lateral Load Model

The lateral part of the pedestrian load is described by Equation A.18.

k X Fh(t) = Q 1 + αn,h sin(2πnft + φn,h) C(N) (A.18) i=1

Values for C(N) and φn,h are the same as in the vertical load model. The value for the DLF, αn,h, is chosen to 0.01. This is the DLF for the 2nd harmonic determined by Bachmann and Ammann [15] (see Section 3.2.2.1). No values for the DLF regarding higher harmonics in the lateral direction is given in the ISO 10137 guideline. No reduction occurs for the static part, as for the vertical load model, and in order to implement the load model in Brigade/Plus the load model is multiplied with the crowd density (ped/m2). The lateral load models are given by Equation A.19 and Equation A.20.

π F (t) = 700 + 0.594 sin(2πf t + ) (A.19) h h 2

π w (t) = d · F (t) = 700 + 0.594 sin(2πf t + )[P a] (A.20) h h h 2

A.1.3.3 Acceleration Limits

The acceleration limit in the vertical direction, given fv = 5.59Hz, is given by Equation A.21. The values are given in Figure 3.13 in Section 3.3.4.1.

2 av,lim = 0.005 · 60 = 0.3 [m/s ] (A.21)

The acceleration limit in the lateral direction, given fh = 2.05Hz, is given by Equa- tion A.22. The values are given in Figure 3.13 in Section 3.3.4.1.

2 ah,lim = 0.0036 · 30 = 0.108 [m/s ] (A.22)

90 A.1. GUIDELINE CALCULATIONS

A.1.4 Sétra

Indata Critical damping ratio ξ = 0.6 % Vertical natural frequency fv = 5.59 Hz Transversal natural frequency ft = 2.05 Hz Area of deck A = 139m2

Figure 3.15 in Section 3.3.5.2 is used to determine which load case to implement depending on frequency range and traﬃc class. The studied bridge falls into frequency range 3 and 4 for the lateral and vertical direction respectively. Therefore, no calculation are necessary in the vertical direction but load case 3, considering the second harmonic of the lateral load, needs to be evaluated. The load models in the Sétra guideline includes a reduction factor that considers the risk of resonance between the stepping frequency and the natural frequency of the bridge. With a natural frequency of 5.59 Hz in the vertical direction, this reduction factor is equal to zero which reduces the load to zero as well. Both Class I and Class II was implemented in Brigade/Plus to consider every possible scenario during the bridge’s life span. The load models for each class is given by Equation A.23.

 s  1 d · (7N) · cos (2πf · t) · 1.85 · · ψ[P a] for Class I  t n w(t) = s (A.23)  ξ d · (7N) · cos (2πf · t) · 10.8 · · ψ[P a] for Class II  t n

The density of the crowd is dependent on the traﬃc class, see Equation A.24. The number of pedestrians on the bridge (Equation A.25) is determined from the area of the bridge as well as the crowd density.

 1 ped/m2 for Class I d = (A.24) 0.8 ped/m2 for Class II

 d · A = 1 · 139 = 139 for Class I n = (A.25) d · A = 0.8 · 139 = 111 for Class II

From the indata and Equation A.24 to A.25, Equation A.23 can be rewritten. This is done in Equation A.26. The coeﬃcient, ψ, is set to 1 in accordance with Figure 3.17 in Section 3.3.5.3.

 1.0984 · cos(2πf t)[P a] for Class I w(t) = t (A.26) 0.445 · cos(2πftt)[P a] for Class II

91 A.1. GUIDELINE CALCULATIONS

A.1.5 SYNPEX

Indata Vertical natural frequency fvert = 5.59 Hz Lateral natural frequency flat = 2.05 Hz Weight of pedestrian, vertical G = 280 N Weight of pedestrian, lateral G = 35 N Structural damping ratio ξ = 0.6% Width of deck d=2.9m Length of deck L=47.97m Loaded area of deck S=139 m2

The natural frequency for the ﬁrst vertical mode, fvert, is outside of SYNPEX’s frequency ranges where the vertical vibrations need to be investigated.

A.1.5.1 Vertical Load Model

The general load model, in the vertical direction, is given by Equation A.27. The pedestrian weight for the ﬁrst harmonic equals 280 N in accordance with to Table 3.14 in Section 3.3.6.2.

n0 p = G · cos(2πf t) · · ψ (A.27) vert vert S

Depending on the crowd density given by the traﬃc class the number of pedestrians on the bridge is calculated by Equation A.28.

 √ 10.8 ξ · n for TC1 and TC3 n0 = √ (A.28) 1.0 · 1.85 n for TC4 to TC5

For traﬃc class 3 and 4 the density equals 0.5 and 1 ped/m2 respectively (Table 3.12 in Section 3.3.6.1), the equivalent number of pedestrians on the bridge is calculated through Equation A.29 and Equation A.30.

 d · S = 69.5 for TC3 n = (A.29) n = d · S = 139 for TC4

 √ 10.8 ξ · n = 6.97 for TC3 n0 = √ (A.30) 1.0 · 1.85 n = 21.82 for TC4

For the given vertical frequency, the reduction factor, ψ, equals zero (Table 3.14 in Section 3.3.6.2).

92 A.1. GUIDELINE CALCULATIONS

A.1.5.2 Lateral Load Model

The lateral load model is given by Equation A.31, and is identical to the vertical load model. The weight of the pedestrian is, however, changed to 35 N for the ﬁrst harmonic of the load.

n0 p = G · cos(2πf t) · · ψ (A.31) lat lat S

For the given lateral frequency, the reduction factor, ψ, equals zero (Table 3.14 in Section 3.3.6.2).

93 A.1. GUIDELINE CALCULATIONS

A.1.6 HIVOSS

Indata Lateral natural frequency flat = 2.05 Hz Structural damping ratio ξ = 0.6% Modal mass m∗=20000kg Constant k = 300 Ns/m

The triggering number of pedestrian required for lateral lock-in is calculated through Equation A.32.

8πξm∗f 8πξ · 20000 · 2.05 N = lat = = 20.61 (A.32) L k 300

94 A.2. MODEL VALIDATION - MASS CALCULATION

A.2 Model Validation - Mass Calculation

Hand calculation of the weight of the superstructure is given in Table A.2 to A.4.

Table A.2: Mass of the trusses of the superstructure

Part Proﬁle Weight Number Length Total [kg/m] [-] [m] weight [kg] Chords VKR250x250x12.5 91.9 4 47.97 17633.8 Diagonals VKR220x120x6.3 32 26 4.92 4093.4 Lateral bracing VKR80x80x5 11.6 13 3.5 527.8 Crossbeams HEB120 26.7 36 3.82 3671.8 HEB300 117 4 3.82 1787.8 Columns VKR220x120x6.3 32 20 3.155 2019.2 VKR250x250x12.5 91.9 8 3.155 2319.6 Struts VKR250x250x12.5 91.9 4 3.82 1404.2 VKR80x80x5 11.6 10 3.82 443.1 Total 33900.7

Table A.3: Mass of the deck of the superstructure

Part Depth Width Length Density Total [m] [m] [m] [kg/m3] weight [kg] Deck 0.12 2.98 47.97 2400 41197.4

Table A.4: Comparison between the mass of the hand calculation and the simulated mass from Brigade/Plus

Hand calculation [kg] 75098.1 Brigade/Plus [kg] 75199.0 Diﬀerence[%] 0.1

95 A.3 Extra mass

Calculation of non-structural mass is given in Table A.5.

Table A.5: Non-structural mass from preliminary drawings

Part Length Width Heigth Density Total [m] [m] [m] [kg/m3] weight [kg] Glass facad 2 · 47.97 3.255 0.02 1200 7494.8 Mineral wool 47.97 3.15 0.5 40 3022.1 Total 10516.9 Appendix B

Python Script

B.1 Python Script

B.1.1 ODB-ﬁle reader import odbAccess from abaqus import * from abaqusConstants import * from visualization import * import numpy as np import openpyxl def gatherFrequency(odbPath,odbfolder,odbFileName,steps):

odbFile=odbPath+odbfolder+odbFileName odb = odbAccess.openOdb(path=odbFile)

region = odb.steps[steps].historyRegions[’Assembly ASSEMBLY’] freqData = region.historyOutputs[’EIGFREQ’].data

#Creates 2 empty 1x2-matrices

frequency1 = np.zeros((1, 2)) frequency3 = np.zeros((1, 2))

#Saves the mode number and corresponding frequency in the previously empty matrix

#Mode 1: Transveral 1 frequency1[0][0] = freqData[0][0] frequency1[0][1] = freqData[0][1]

#Mode 3: Vertical 1 frequency3[0][0] = freqData[2][0] frequency3[0][1] = freqData[2][1]

97 B.1. PYTHON SCRIPT

odb.close()

return frequency1, frequency3 numberOfParameterSets = 32 frequency1 = [] frequency3 = [] odbpath = r’C:/BRIGADE Plus Work Directory/Combinations/ Combinations 170511/’ steps = ’FreqStep’ for i in range(numberOfParameterSets):

odbfolder = ’Combination-’ + str(i+1) +’/’ odbFileName = ’Combination-’ + str(i+1)+’.odb’

f1,f3 = gatherFrequency(odbpath,odbfolder,odbFileName,steps)

frequency1.append(f1) frequency3.append(f3)

#Saves the mode numbers and frequencies to an excel sheet. filename=’Frequencies_170511.xlsx’ wb=openpyxl.Workbook() ws1=wb.active ws1.title=’Frequencies_Paramstudy’ for i in range(numberOfParameterSets): ws1[’A’+str(i+1)] = i+1 ws1[’B’ + str(i + 1)] = frequency1[i][0][1] ws1[’C’ + str(i + 1)] = frequency3[i][0][1] wb.save(filename)

98 B.1. PYTHON SCRIPT

99 TRITA Examensarbete 507, Brobyggnad 2017 ISSN 1103-4297