Dynamic Analysis of the Skyway Bridge

STOCKHOLM SVERIGE 2017 ,

Dynamic Analysis of the Skyway Bridge

Assessment and Application of Design Guidelines

DANIEL ANDERSSON ERIC THUFVESSON

KTH

Department of Civil and Architectural Engineering

Division of Structural Engineering and Bridges

Stockholm, Sweden, 2017

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 me- chanically 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/s

. 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

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 lat-

erala 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/s

. 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.

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

Roman Letters

Notation Description Unit

A

Constant −

a Acceleration m/s

B

Constant −

c Damping N s/m

c Damping matrix −

d Displacement; Crowd density m; 1/m

E Young’s modulus P a

f Cyclic frequency of vibration; Force Hz; N

f

Step frequency Hz

F

(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 N m

m

Modal mass kg

n; N Number −

N

; n

Equivalent number −

p Force N

Q Static load P a

q

Harmonic function

S Surface area; Span length m

; m

S

Effective span m

t Time s

u Displacement matrix m

˙

u Velocity matrix m/s

¨

u Acceleration matrix m/s

u Displacement m

w Pressure load P a

α

Dynamic load factor, horizontal direction −

α

Dynamic load factor, vertical direction −

α

Numerical coefficient, horizontal direction − α

Numerical coefficient, vertical direction −

γ Reduction factor −

δ Displacement m

λ Reduction factor −

ξ Critical structural damping −

ρ Density kg/m

υ Poisson’s ratio −

φ

Mode shape; Eigenvector −

φ

Phase angle, horizontal direction rad

φ

Phase angle, vertical direction rad

φ

Phase angle, lateral direction rad

φ

Phase angle, vertical direction rad

ψ Reduction factor −

ω Natural circular frequency rad/s

ω

Circular frequency rad/s

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

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

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

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

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 com- plicated 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 verti- cal 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

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 us- ing 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 correspond- ing 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 re- quirements 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.

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.

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 contin- uous 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 longitu- dinal 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 lat- eral 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 VKR250x250x12.5

Columns VKR220x120x6.3

VKR250x250x12.5

Crossbeams HEB120

HEB300

The substructure consists of two rhombus-shaped support columns, oblique relative the Skyway bridge, with a cross section dimension of 900 × 2400 mm

with column heads with cross section dimension equal to 900 × 800 mm

. 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]

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 solv- ing 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.

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 nec- essary 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, f

, is equal to the sum of all forces acting on the system,

f

= f

+ f

+ p(t), (3.1)

where f

is the damping force, f

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,

f

= m¨ u, (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

f

= −c ˙u, (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

f

= −ku. (3.4)

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

m¨ u + c ˙u + 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 simpli- fications 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,

m¨ u + c ˙u + 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].

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 superpo- sition 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 fre- quencies and mode shapes are determined by solving the EOM,

m¨ u + ku = 0. (3.7)

The solution to the EOM is given by,

u = q

(t)φ

, (3.8)

where the mode shape, φ

, does not vary with time. The variation in time is given by the simple harmonic function,

q

(t) = A

cos(ω

t) + B

sin(ω

t). (3.9) By inserting Equation 3.8 into the EOM given by Equation 3.7 the following is obtained,

[−ω

mφ

+ kφ

]q

(t) = 0. (3.10) Equation 3.10 has two possible solutions, either q

(t) = 0 which implies no motion in the system or the non trivial solution given by,

[k − ω

m]φ

= 0. (3.11)

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,

det[k − ω

m] = 0. (3.12)

Equation 3.12 is the characteristic equation of the system. The solution to Equation 3.12 is a number of roots, eigenvalues, corresponding to ω

. The number of roots correspond to the DOF of the system. If the natural circular frequencies, ω

, are inserted in Equation 3.11 the eigenvector φ

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

k = km

c. (3.13)

The nodal displacement of the structure can be expressed by

u(t) =

N

X

n=1

Φ

q

(t), (3.14)

where q

is a generalized coordinate. Inserting the expression for the nodal dis- placement into Equation 3.6 gives the EOM for each natural mode, Φ

, according to

M

q ¨

+ C

q ˙

+ K

q

= P

(t). (3.15) Where M

, C

, K

and P

(t) are the generalized mass, damping, stiffness and force given by,

M

= Φ

mΦ

, C

= Φ

cΦ

, K

= Φ

kΦ

and P

(t) = Φ

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, Φ

. Thus we get N uncoupled

equations and the total response, u(t), is obtained by solving these equations for q

and then summarize there contribution according to Equation 3.14 [4].

3.2 Human Induced Vibrations

Human-induced loading on footbridges is frequently occurring and often the dom- inant 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 pedestri- ans 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 vibra- tions.

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.

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

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

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 pedes-

trian weight, through probability distribution functions. The focus of this report

will be on deterministic force models.

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]:

F

(t) = G +

∞

X

i=1

Gα

sin(2πif

t + φ

) (3.17)

F

(t) =

∞

X

i=1

Gα

sin(2πi f

2 t + φ

) (3.18)

Where:

G Pedestrian weight, usually 700N [N]

α

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

α

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

f

Vertical stepping frequency [Hz].

φ

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

φ

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 ( α

= 0.039, α

= 0.01, α

= 0.043, α

= 0.012, α

= 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])

3.2.2.2 Probabilistic Force Models

The probabilistic force models of the pedestrian load take the variability and uncer- tainty 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 proba- bility 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,

N

= 8πξm

f

k , (3.19)

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/s

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/s

.

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

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 hu- man 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/s

and 0.37 m/s

depending on the bridge. However, the reliability of these

results can be questioned due to the number of test subjects.

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

limit of of horizontal accelerations is 1.35 m/s

. He also concluded that most

pedestrians tolerated an acceleration of 0.3 m/s

while an acceleration of 2.1 m/s

prevented them from walking due to discomfort.

3.3 Design Guidelines

3.3.1 Eurocode 5

Eurocode 5 considers timber structures and the guidelines given in the code is pro- duced 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 foot- bridge, 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 ex- citation 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/s

is given by Equation 3.20 and 3.21 respectively.

a

=



 



 



200

M ξ for f

≤ 2, 5Hz.

100

M ξ for 2, 5Hz ≤ f

≤ 5, 0Hz.

(3.20)

a

= 600

M ξ for 2, 5Hz ≤ f

≤ 3, 5Hz. (3.21) Where:

M Total mass of the bridge [kg]

ξ Damping ratio [-]

f

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

For several people crossing the footbridge, either by walking or running, the accel- eration in m/s

is given by Equation 3.22.

a

= 0, 23 · a

nk

(3.22)

Where:

a

Acceleration determined from either Equation 3.20 or 3.21 [m/s

] k

Reduction coefficient [-] based on the vertical natural frequency f

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/s

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.

a

= 50

M ξ for 0, 5Hz ≤ f

≤ 2, 5Hz. (3.23)

a

= 0, 18 · a

nk

(3.24)

Where:

f

The fundamental horizontal natural frequency [Hz].

k

Reduction coefficient based on the horizontal natural frequency f

from Figure 3.9 [-]

Figure 3.9: Relationship between fundamental natural frequency and vertical (left)

and horizontal (right) coefficient [21]

3.3.2 UK National Annex

BS EN 1991-2 [22] establishes models and representative values intended for de- sign of new bridges, including bridges subjected to pedestrian traffic. Furthermore, two analyses are required for pedestrian bridges; a determination of maximum ver- tical 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

class

Group size (walking)

Crowd 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],

F = F

· k(f

) · ^q 1 + γ(N − 1) · sin(2πf

· t). (3.25)

Where:

F

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

f

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(f

) 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.

w = 1.8 · ( F

A ) · k(f

) ·

s γ · N

λ · sin(2πf

· t) (3.26) 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 [-]

Figure 3.10: Factor k(f

) [22]

Figure 3.11: Reduction factor γ considering unsynchronized combinations of pedes-

trians [22]

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.

a

= 1.0k

k

k

k

(3.27)

k

, k

and k

(Table 3.2 to 3.4) are response modifiers depending on the site us- age, route redundancy and structure height respectively and k

is a project specific exposure factor taken as 1.0 unless determined otherwise.

Table 3.2: Recommended values for the site usage factor k

(after[22])

Bridge function k

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 k

(after[22])

Route redundancy k

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 k

(after[22]) Bridge height k

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.

D = m

· ξ

m

(3.28)

Where:

m

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

m

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).

F

(t) = Q ^ 1 +

k

X

i=1

α

sin(2πnf t + φ

) ^ (3.29)

F

(t) = Q ^ 1 +

k

X

i=1

α

sin(2πnf t + φ

) ^ (3.30)

Where:

Q The static load of a single pedestrian [N]

α

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

α

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]

φ

The phase-angle of the nth harmonic in the vertical direction.

Conservatively chosen as

[rad]

φ

The phase-angle of the nth harmonic in the horizontal direction.

Conservatively chosen as

[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.

F

(t) = F (t) · C(N ) (3.31)

Where:

N is the number of people [-]

C(N ) is taken as

√ N N [-]

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

number, n

Common range of forc- ing frequency, nf

[Hz]

Numerical coefficient for vertical direction, α

Numerical coefficient for hor- izontal direction, α

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

4

4.8 to 9.6 0.06

5

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

a

These 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.

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].

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 accelera-

tion 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-

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/s

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/s

the synchronization rate changed rapidly from 10 to 60 percent which increased the accelerations to about 0.6 m/s

[1].

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

ranges

0-0.5 [m/s

]

0.5-1 [m/s

]

1-2.5 [m/s

]

2.5 <

[m/s

] 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

ranges

0-0.15 [m/s

]

0.15-0.3 [m/s

]

0.3-0.8 [m/s

]

0.8 <

[m/s

] 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 foot- bridge 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 nec- essary 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/m

. 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

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

ranges

0-1 [Hz]

1-1.7 [Hz]

1.7-2.1 [Hz]

2.1-2.6 [Hz]

2.6-5 [Hz]

5<

[Hz]

Range 1 Range 2 Range 3 Range 4

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

ranges

0-0.3 [Hz]

0.3-0.5 [Hz]

0.5-1.1 [Hz]

1.1-1.3 [Hz]

1.3-2.5 [Hz]

2.5<

[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 multi- plying 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.

N

=







10.8 · ^q (N · ξ) for sparse or dense crowds 1.85 · √

N for very dense crowds (3.32)

The equivalent number of pedestrians, N

, 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