1

Dynamic analysis of bridge girders subjected to moving loads Numerical and analytical beam models considering warping effects

João Serra

October 2014

Abstract

In the case of railway bridges traversed by eccentric moving loads, important torsion moments are introduced. Therefore, special consideration must be given to the torsional response of thin walled girders and warping of the cross sections cannot be neglected in structural analysis and design.

In the present work, a numerical and an analytical beam models to perform efficient dynamic analysis of continuous high-speed railway bridges are proposed.

By applying the Hamilton’s principle along with the Vlasov’s beam theory the coupled governing equations of thin walled beams are derived. These equations are solved by considering a modal analysis along with a direct integration scheme. The natural frequencies and vibration mode shapes of each problem are determined through the developed models and compared with the results given by a commercial shell finite element model.

In the developed numerical model, the behaviour of the bridge girder is described by a three-dimensional Euler-Bernoulli beam finite element, where an additional degree of freedom is considered to describe the warping displacements. Explicit expressions for the stiffness and the mass element matrices are presented for arbitrary cross sections.

The proposed analytical model allows to solve the governing equations in an exact sense. The mode shapes of vibration are calculated by analytical explicit expressions, yielding the exact solution for the problem of coupled bending-torsional vibrations.

The trains are modelled as series of moving masses acting eccentrically at constant velocity. Thus, the corresponding inertial effects of the moving loads are considered besides the gravitational force of mass.

Keywords: thin walled beams; dynamic analysis; high-speed; warping; moving masses; bridges.

1. Introduction

The number of works on coupled lateral bending-torsional vibrations of beams under moving loads is limited. Relevant studies are those of [Sophianopoulos and Michaltsos, 1999], [Stayridis and Michaltsos, 1999] and [Michaltsos et al., 2005]. In the last work, a numerical study on the subject is presented, being considered simply supported thin walled open cross section steel beams traversed by single eccentric moving loads. The analysed structures simulate highway bridges with eccentric lanes. Simplified single vehicles are taken into account and the structural damping is neglected. Through a modal analysis, the free and forced vibrations are investigated by utilizing symbolic algebra software.

Recently, [Lisi, 2011] developed a beam finite element model for the static and the dynamic analysis of thin walled beams under the action of single eccentric moving forces. His work constitutes a powerful contribution to the present work, which aims to extend Lisi’s study in order to include sets of moving masses.

2. The Governing Equations of Coupled Thin Walled Beams

The governing differential equations that describe the static and the dynamic behaviour of thin walled beam-like structures were derived by applying the Hamilton’s principle. For a discrete system with 𝑛𝑑𝑜𝑓 degrees of freedom,

consider both the potential energy 𝛱 and the kinetic energy 𝛬 described in terms of a discrete number of generalized coordinates 𝑞𝑖 and their derivatives, such that

𝛬 = 𝛬(𝑞𝑖 , 𝑞�̇�, �̇�𝑖′) and 𝛱 = 𝛱(𝑞𝑖 , 𝑞𝑖

′, 𝑞𝑖′′) ,

where 𝑖 is an integer between 1 and 𝑛𝑑𝑜𝑓. Thus, the Hamilton’s principle reduces to

[𝜕

𝜕𝑡(

𝜕𝛬

𝜕𝑞�̇�) −

𝜕𝛬

𝜕𝑞𝑖+

𝜕𝛱

𝜕𝑞𝑖] 𝛿𝑞𝑖 + [

𝜕

𝜕𝑡(

𝜕𝛬

𝜕�̇�𝑖′) +

𝜕𝛱

𝜕𝑞𝑖′] 𝛿𝑞𝑖

′ + [𝜕𝛱

𝜕𝑞𝑖′′] 𝛿𝑞𝑖

′′ = 0 .

The potential energy 𝛱 and the kinetic energy 𝛬 are respectively defined as functions of the element’s generalized displacements as follows:

𝛱 = ∫ [1

2𝐸(𝐴𝜉𝑥

′ 2+ 𝐼𝑦𝑦𝜉𝑧

′′2+ 𝐼𝑧𝑧𝜉𝑦

′′2+ 𝐼𝜔𝜔𝜙′′2

− 2𝑆𝑦𝜉𝑥′ 𝜉𝑧

′′ − 2𝑆𝑧𝜉𝑥′ 𝜉𝑦

′′ − 2𝑆𝜔𝜉𝑥′ 𝜙′′ + 2𝐼𝑦𝑧𝜉𝑧

′′𝜉𝑦′′ + 2𝐼𝑦𝜔𝜉𝑧

′′𝜙′′ + 2𝐼𝑧𝜔𝜉𝑦′′𝜙′′)

𝐿

+1

2𝐺𝐽𝜙′2

− (𝜉𝑥𝑓𝑥 + 𝜉𝑦𝑓𝑦 + 𝜉𝑧𝑓𝑧 + 𝜙𝑚𝑥 + 𝜉𝑦′ 𝑚𝑧 − 𝜉𝑧

′ 𝑚𝑦 + 𝜙′𝑚𝜔)] 𝑑𝑥 ,

𝛬 = ∫1

2𝜌 (𝐴𝜉�̇�

2+ 𝐼𝑦𝑦𝜉�̇�

′ 2+ 𝐼𝑧𝑧𝜉�̇�

′ 2+ 𝐼𝜔𝜔�̇�′2

− 2𝑆𝑦𝜉�̇�𝜉�̇�′ − 2𝑆𝑧𝜉�̇�𝜉�̇�

′ − 2𝑆𝜔�̇�𝑥𝜙′̇ + 2𝐼𝑦𝑧𝜉�̇�′ 𝜉�̇�

′ + 2𝐼𝑦𝜔𝜉�̇�′ 𝜙′̇ + 2𝐼𝑧𝜔𝜉�̇�

′ �̇�′)𝐿

𝑑𝑥

+ ∫1

2𝜌 [𝐴𝜉�̇�

2+ 𝐴𝜉�̇�

2+ 2𝑆𝑧

𝑃𝜉�̇��̇� − 2𝑆𝑦𝑃𝜉�̇��̇� + (𝐼𝑧𝑧

𝑃 + 𝐼𝑦𝑦𝑃 )�̇�2]

𝐿

𝑑𝑥 ,

where primes indicate differentiation with respect to 𝑥 and dots represent derivatives with respect to time 𝑡. 𝐸 represents the Young’s modulus, 𝐺 is the shear modulus, 𝐽 is the torsion constant, 𝐴 is the cross section area, 𝑆𝑖 and 𝐼𝑖𝑗

are respectively the first and the second moments of inertia (𝑖, 𝑗 = 𝑦, 𝑧), and 𝑆𝜔 and 𝐼𝜔𝜔 correspond to the first and the second sectorial moments, respectively. The variables 𝜉𝑖 represent the 𝑖-th cross-sectional translation component (𝑖 = 𝑥, 𝑦, 𝑧) and 𝜙 represents a rotation about the sectorial pole axis �̂�.

As shown in Figure 2.1, the applied torsion moment 𝑚𝑥 acts about the pole axis �̂� and the transverse loads 𝑓𝑧 and 𝑓𝑦 act through it. The bending moments 𝑚𝑦 and 𝑚𝑧, and the axial force 𝑓𝑥 are referred to in the (𝑥, 𝑦, 𝑧) coordinate system. The

warping moment 𝑚𝜔 is shown lying along the pole axis �̂� for consistency with the bimoment 𝑀𝜔 representation.

Figure 2.1 – element loading and internal forces.

By applying the Hamilton’s principle the governing differential equations of a thin walled beam of arbitrary cross section yield

𝜌(𝐴𝜉�̈� − 𝑆𝑦𝜉�̈�′ − 𝑆𝑧𝜉�̈�

′ − 𝑆𝜔�̈�′) − 𝐸(𝐴𝜉𝑥′′ − 𝑆𝑦𝜉𝑧

′′′ − 𝑆𝑧𝜉𝑦′′′ − 𝑆𝜔𝜙′′′) = 𝑓𝑥 ,

𝜌(𝐴𝜉�̈� + 𝑆𝑧𝑃�̈�) − 𝜌(−𝑆𝑦𝜉�̈�

′ + 𝐼𝑦𝑦𝜉�̈�′′ + 𝐼𝑦𝑧𝜉�̈�

′′ + 𝐼𝑦𝜔�̈�′′) + 𝐸(−𝑆𝑦𝜉𝑥′′′ + 𝐼𝑦𝑦𝜉𝑧

′′′′ + 𝐼y𝑧𝜉𝑦′′′′ + 𝐼𝑦𝜔𝜙′′′′) = 𝑓𝑧 + 𝑚𝑦

′ ,

𝜌(𝐴𝜉�̈� − 𝑆𝑦𝑃�̈�) − 𝜌(−𝑆𝑧𝜉�̈�

′ + 𝐼𝑧𝑦𝜉�̈�′′ + 𝐼𝑧𝑧�̈�𝑦

′′ + 𝐼𝑧𝜔�̈�′′) + 𝐸(−𝑆𝑧𝜉𝑥′′′ + 𝐼𝑧𝑦𝜉𝑧

′′′′ + 𝐼𝑧𝑧𝜉𝑦′′′′ + 𝐼𝑧𝜔𝜙′′′′) = 𝑓𝑦 − 𝑚𝑧

′ ,

𝜌[𝑆𝑧𝑃𝜉�̈� − 𝑆𝑦

𝑃𝜉�̈� + (𝐼𝑧𝑧𝑃 + 𝐼𝑦𝑦

𝑃 )�̈�] − 𝜌(−𝑆𝜔𝜉�̈�′ + 𝐼𝜔𝑦𝜉𝑧

′′̈ + 𝐼𝜔𝑧𝜉�̈�′′ + 𝐼𝜔𝜔�̈�′′) + 𝐸 (−𝑆𝜔𝜉𝑥

′′′ + 𝐼𝜔𝑦𝜉𝑧′′′′ + 𝐼𝜔𝑧𝜉𝑦

′′′′ + 𝐼𝜔𝜔𝜙′′′′) − 𝐺𝐽𝜙′′

= 𝑚𝑥 − 𝑚𝜔′ .

(2.1)

(2.2)

(2.3a)

(2.3b)

(2.4a)

(2.4b)

(2.4c)

(2.4d)

3. Development of a Finite Element Solution

In this section, a beam finite element is derived for the static and the dynamic analysis of a thin walled girder considering the cross-sectional warping. The developed finite element considers the six nodal degrees of freedom of a three-dimensional Euler-Bernoulli beam and an additional warping degree of freedom, which represents the out-of-plane displacements of the cross sections when a beam is loaded in torsion, as depicted in Figure 3.1.

Figure 3.1 – thin walled beam element and the corresponding nodal generalized displacements referred to the centre of gravity.

The polynomial approximate solution 𝑢𝒩𝑒 for the element nodal generalized displacements 𝑢𝑒, say 𝜉𝑥

𝑒, 𝜉𝑧𝑒, 𝜉𝑦

𝑒 and 𝜙𝑒, and the corresponding derivatives when required, are written in the form

𝑢𝒩𝑒 (𝑥, 𝑡) = 𝓝𝑢

𝑒 (𝑥)𝒖𝑢𝑒 (𝑡) .

The element approximation functions corresponding to the axial extension problem are Lagrange interpolation functions, while the Hermite’s cubic interpolation functions are suitable to the bending and torsional problems. Thus,

𝓝𝜉𝑥

𝑒 (𝑥) = [𝒩𝜉𝑥1e (𝑥) 𝒩𝜉𝑥2

𝑒 (𝑥)] =1

2[1 − 𝜁 1 + 𝜁] ,

𝓝𝜉𝑧

𝑒 (𝑥) = 𝓝𝜉𝑦

𝑒 (𝑥) = 𝓝𝜙𝑒 (𝑥) = [𝒩𝜉𝑧1

𝑒 (𝑥) 𝒩𝜉𝑧′1𝑒 (𝑥) 𝒩𝜉𝑧2

𝑒 (𝑥) 𝒩𝜉𝑧′2𝑒 (𝑥)]

=1

8[2(2 + 𝜁)(1 − 𝜁)2 𝐿𝑒(1 + 𝜁)(1 − 𝜁)2 2(2 − 𝜁)(1 + 𝜁)2 𝐿𝑒(−1 + 𝜁)(1 + 𝜁)2] ,

being the transformation mapping between the element natural coordinate 𝜁 and the local axial coordinate system 𝑥𝑒

𝜁 =2𝑥𝑒

𝐿𝑒− 1 with 𝑥𝑒 ∈ [0, 𝐿𝑒] and thus, 𝜁 ∈ [−1,1] .

In order to obtain the discrete finite element equations the differential equations (2.4) are restated in an integral form called the weak form. After performing some mathematical treatments, where the Galerkin’s method is applied, the weak form leads to the element equations, written in the matrix form as follows:

𝑴𝑒�̈�𝑒 + 𝑪𝑒�̇�𝑒 + 𝑲𝑒𝒖𝑒 = 𝒇𝑒 ,

being 𝑴𝑒 the element mass matrix, 𝑪𝑒 the element damping matrix, 𝑲𝑒 the element stiffness matrix, and 𝒇𝑒 the element external force vector.

The contributions 𝑴𝑢𝑒 to the element mass matrix 𝑴𝑒 regarding the generalized displacements 𝑢 are given by

𝑴𝜉𝑥

𝑒 = [𝐿𝑒𝜌𝑒𝐴𝑒

6𝒎1 −

𝜌𝑒𝑆𝑦𝑒

12𝒎2 −

𝜌𝑒𝑆𝑧𝑒

12𝒎2 −

𝜌𝑒𝑆𝜔𝑒

12𝒎2] ,

𝑴𝜉𝑧

𝑒 = [𝜌𝑒𝑆𝑦

𝑒

12𝒎3

𝜌𝑒𝐴𝑒𝐿𝑒

420𝒎4 +

𝜌𝑒𝐼𝑦𝑦𝑒

30𝐿𝑒 𝒎5𝜌𝑒𝐼𝑦𝑧

𝑒

30𝐿𝑒𝒎5

𝜌𝑒𝑆𝑧𝑃𝑒𝐿𝑒

420𝒎4 +

𝜌𝑒𝐼𝑦𝜔𝑒

30𝐿𝑒𝒎5

] ,

𝑴𝜉𝑦

𝑒 = [𝜌𝑒𝑆𝑧

𝑒

12𝒎3

𝜌𝑒𝐼𝑧𝑦𝑒

30𝐿𝑒𝒎5

𝜌𝑒𝐴𝑒𝐿𝑒

420𝒎4 +

𝜌𝑒𝐼𝑧𝑧𝑒

30𝐿𝑒 𝒎5 −𝜌𝑒𝑆𝑦

𝑃𝑒𝐿𝑒

420𝒎4 +

𝜌𝑒𝐼𝑧𝜔𝑒

30𝐿𝑒 𝒎5] and

𝑴𝜙𝑒 = [

𝜌𝑒𝑆𝜔𝑒

12𝒎3

𝜌𝑒𝑆𝑧𝑃𝑒𝐿𝑒

420𝒎4 +

𝜌𝑒𝐼𝜔𝑦𝑒

30𝐿𝑒 𝒎5 −𝜌𝑒𝑆𝑦

𝑃𝑒𝐿𝑒

420𝒎4 +

𝜌𝑒𝐼𝜔𝑧𝑒

30𝐿𝑒 𝒎5

𝜌𝑒(𝐼𝑧𝑧𝑃𝑒 + 𝐼𝑦𝑦

𝑃𝑒)𝐿𝑒

420𝒎4 +

𝜌𝑒𝐼𝜔𝜔𝑒

30𝐿𝑒 𝒎5] ,

where 𝒎1 to 𝒎5 are the following constant matrices:

𝒎1 = [2 11 2

] , 𝒎2 = [−6 𝐿𝑒

−6 −𝐿𝑒6 −𝐿𝑒

6 𝐿𝑒 ] ,

(3.1)

(3.2a)

(3.2b)

(3.3)

(3.4)

(3.5a)

(3.5b)

(3.5c)

(3.5d)

(3.6)

𝒎3 = [

6 6

−𝐿𝑒 𝐿𝑒

−6 −6

𝐿𝑒 −𝐿𝑒

] , 𝒎4 = [

156 22𝐿𝑒

22𝐿𝑒 4𝐿𝑒2

54 −13𝐿𝑒

13𝐿𝑒 −3𝐿𝑒2

54 13𝐿𝑒

−13𝐿𝑒 −3𝐿𝑒2

156 −22𝐿𝑒

−22𝐿𝑒 4𝐿𝑒2

] and 𝒎5 = [

36 3𝐿𝑒

3𝐿𝑒 4𝐿𝑒2

−36 3𝐿𝑒

−3𝐿𝑒 −𝐿𝑒2

−36 −3𝐿𝑒

3𝐿𝑒 −𝐿𝑒2

36 −3𝐿𝑒

−3𝐿𝑒 4𝐿𝑒2

] .

In what concerns the contributions 𝑲𝑢𝑒 to the element stiffness matrix 𝑲𝑒 the following expressions are obtained:

𝑲𝜉𝑥

𝑒 = [𝐸𝑒𝐴𝑒

𝐿𝑒 𝒌1 −𝐸𝑒𝑆𝑦

𝑒

𝐿𝑒 𝒌2 −𝐸𝑒𝑆𝑧

𝑒

𝐿𝑒 𝒌2 −𝐸𝑒𝑆𝜔

𝑒

𝐿𝑒 𝒌2] , 𝑲𝜉𝑧

𝑒 = [𝐸𝑒𝑆𝑦

𝑒

𝐿𝑒 𝒌3

𝐸𝑒𝐼𝑦𝑦𝑒

𝐿𝑒3 𝒌4

𝐸𝑒𝐼𝑦𝑧𝑒

𝐿𝑒3 𝒌4

𝐸𝑒𝐼𝑦𝜔𝑒

𝐿𝑒3 𝒌4] ,

𝑲𝜉𝑦

𝑒 = [𝐸𝑒𝑆𝑧

𝑒

𝐿𝑒 𝒌3

𝐸𝑒𝐼𝑧𝑦𝑒

𝐿𝑒3 𝒌4

𝐸𝑒𝐼𝑧𝑧𝑒

𝐿𝑒3 𝒌4

𝐸𝑒𝐼𝑧𝜔𝑒

𝐿𝑒3 𝒌4] and 𝑲𝜙𝑒 = [

𝐸𝑒𝑆𝜔𝑒

𝐿𝑒 𝒌3

𝐸𝑒𝐼𝜔𝑦𝑒

𝐿𝑒3 𝒌4

𝐸𝑒𝐼𝜔𝑧𝑒

𝐿𝑒3 𝒌4

𝐸𝑒𝐼𝜔𝜔𝑒

𝐿𝑒3 𝒌4 +𝐺𝑒𝐽𝑒

30𝐿𝑒 𝒌5] ,

being 𝒌1 to 𝒌5 defined by

𝒌1 = [1 −1

−1 1] , 𝒌2 = [

0 10 −1

0 −10 1

] ,

𝒌3 = [

0

−10

1

0

10

−1

] , 𝒌4 = [

12 6𝐿𝑒

6𝐿𝑒 4𝐿𝑒2

−12 6𝐿𝑒

−6𝐿𝑒 2𝐿𝑒2

−12 −6𝐿𝑒

6𝐿𝑒 2𝐿𝑒2

12 −6𝐿𝑒

−6𝐿𝑒 4𝐿𝑒2

] and 𝒌5 = [

36 3𝐿𝑒

3𝐿𝑒 4𝐿𝑒2

−36 3𝐿𝑒

−3𝐿𝑒 −𝐿𝑒2

−36 −3𝐿𝑒

3𝐿𝑒 −𝐿𝑒2

36 −3𝐿𝑒

−3𝐿𝑒 4𝐿𝑒2

] .

The contributions 𝒇𝑢𝑒 to the element external force vectors 𝒇𝑒 are written as

𝒇𝜉𝑥

𝑒 = [𝑓𝑥

𝑒𝐿𝑒

2− 𝑁1

𝑒𝑓𝑥

𝑒𝐿𝑒

2+ 𝑁2

𝑒]𝑇

, 𝒇𝜉𝑧

𝑒 = [𝑓𝑧

𝑒𝐿𝑒

2+ 𝑚𝑦

𝑒 − 𝑉𝑧1𝑒

𝑓𝑧𝑒𝐿𝑒

12+ 𝑀𝑦1

𝑒𝑓𝑧

𝑒𝐿𝑒

2− 𝑚𝑦

𝑒 + 𝑉𝑧2𝑒 −

𝑓𝑧𝑒𝐿𝑒

12− 𝑀𝑦2

𝑒 ]𝑇

,

𝒇𝜉𝑦

𝑒 = [𝑓𝑦

𝑒𝐿𝑒

2− 𝑚𝑧

𝑒 − 𝑉𝑦1𝑒

𝑓𝑦𝑒𝐿𝑒2

12− 𝑀𝑧1

𝑒𝑓𝑦

𝑒𝐿𝑒

2+ 𝑚𝑧

𝑒 + 𝑉𝑦2𝑒 −

𝑓𝑦𝑒𝐿𝑒2

12+ 𝑀𝑧2

𝑒 ]

𝑇

and

𝒇𝜙𝑒 = [

𝑚𝑥𝑒𝐿𝑒

2− 𝑚𝜔

𝑒 − 𝑀𝑥1𝑒

𝑚𝑥𝑒𝐿𝑒

12− 𝑀𝜔1

𝑒𝑚𝑥

𝑒𝐿𝑒

2+ 𝑚𝜔

𝑒 + 𝑀𝑥2𝑒 −

𝑚𝑥𝑒𝐿𝑒

12+ 𝑀𝜔2

𝑒 ]𝑇

.

The contributions to the element matrices have to be assembled in order to develop the element mass and the element stiffness matrices, 𝑴𝑒 and 𝑲𝑒, including all deformation effects.

The global finite element equations are written in the matrix form as follows:

𝑴�̈� + 𝑪�̇� + 𝑲𝒖 = 𝒇 ,

where the global mass matrix 𝑴, the global damping matrix 𝑪, the global stiffness matrix 𝑲, and the global external force vector 𝒇 are derived through a direct assembly procedure, [Fish and Belytschko, 2007].

The natural frequencies and the corresponding vibration mode shapes are obtained by solving the equation

𝑴�̈� + 𝑲𝒖 = 𝟎 .

Considering an harmonic solution

𝒖(𝑥, 𝑡) = 𝒗(𝑥) sin(𝑝𝑡)

the 𝑛-th mode shape 𝒗𝑛, in which the structure responds when vibrates with a frequency 𝑝𝑛, is obtained by

[𝑲 − 𝑝𝑛2𝑴]𝒗𝑛 = 𝟎 .

To quantify the damping properties a proportional relation regarding the modal mass properties is invoked. The modal mass properties are determined by considering the orthogonality of the vibration modes with respect to mass, i.e.,

𝒗𝑠𝑇𝑴𝒗𝑟 = 0 .

Considering the 𝑛-th normalized mode shape 𝝍𝑛, given by

𝝍𝑛 =𝒗𝑛

√𝒗𝑛𝑇𝑴𝒗𝑛

,

the mass and stiffness matrices, 𝑴 and 𝑲, may be rewritten for the modal coordinates, as follows:

𝕄 = 𝜳𝑇𝑴𝜳 and 𝕂 = 𝜳𝑇𝑲𝜳 , with 𝜳 = [𝝍1

𝝍2

⋯ 𝝍𝑛

⋯ 𝝍𝑛𝑑𝑜𝑓

] .

The elements of the diagonal matrices 𝕄 and 𝕂 are respectively expressed by

𝝍𝑠

𝑇𝑴𝝍𝑟

= 𝛿𝑠𝑟 and 𝝍𝑠

𝑇𝑲𝝍𝑟

= 𝛿𝑠𝑟𝑝𝑠2 ,

being 𝛿𝑠𝑟 the Kronecker delta.

Thus, by choosing the modal coordinates, the governing equations are rendered decoupled. The response calculations are symplified, being each of eqns (3.4) transformed into a single degree of freedom governing equation given by

𝕄�̈� + ℂ�̇� + 𝕂𝒖 = 𝕗 ,

in which 𝕄 and 𝕂 are the diagonal matrices of eqns (3.17). The modal loading is given by

𝕗 = 𝜳𝑇𝒇 ,

(3.7)

(3.8a)

(3.8b)

(3.9a)

(3.9b)

(3.10a)

(3.10b)

(3.10c)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

and ℂ is a diagonal matrix which elements are expressed by

2𝜍𝑠𝑝𝑟𝛿𝑠𝑟 ,

where 𝜍𝑠 represents the viscous damping ratio corresponding to the 𝑠-th vibration mode.

The structural response to a prescribed time-dependent loading 𝒇(𝑡) involves the solution of the ordinary differential equation (3.19) in time.

Using the modal superposition method, the nodal displacements vector 𝒖 is obtained by summing the 𝑛𝑑𝑜𝑓 products of

the normalized mode shape vectors 𝝍𝑛 and the modal amplitudes 𝛶𝑛 as follows:

𝒖 = ∑ 𝝍𝑛𝛶𝑛

𝑛𝑑𝑜𝑓

𝑛=1 .

The equation (3.19) is solved by adopting the Newmark’s β method, which consists in a direct integration method. By using the Newmark’s formulation, the displacement and velocity at time 𝑡 + Δ𝑡 are obtained through the equations

𝒖𝑡+Δ𝑡 = 𝒖𝑡 + Δ𝑡�̇�𝑡 + (12⁄ − 𝛽)(Δ𝑡)2�̈�𝑡 + 𝛽(Δ𝑡)2�̈�𝑡+Δ𝑡 and �̇�𝑡+Δ𝑡 = �̇�𝑡 + Δ𝑡�̇�𝑡 + (1 − 𝛾)Δ𝑡�̈�𝑡 + 𝛽Δ𝑡�̈�𝑡+Δ𝑡 .

The implicit recursive relation for 𝒖𝑡+Δ𝑡 can be converted to an explicit form by substituting eqns (3.23) into the governing equations (3.19) at time 𝑡 + Δ𝑡, [Wunderlich and Pilkey, 2003], which result is

𝓚𝒖𝑡+Δ𝑡 = 𝓕𝑡+Δ𝑡 ,

where the effective stiffness matrix 𝓚 and the effective loading vector 𝓕𝑡+Δ𝑡 are respectively defined as

𝓚 =1

𝛽(Δ𝑡)2 𝕄 +𝛾

𝛽Δ𝑡ℂ + 𝕂 , 𝓕𝑡+Δ𝑡 = [

1

𝛽(Δ𝑡)2 𝒖𝑡 +1

𝛽Δ𝑡�̇�𝑡 + (

1

2𝛽− 1) �̈�𝑡] 𝕄 + [

𝛾

𝛽Δ𝑡𝒖𝑡 + (

𝛾

𝛽− 1) �̇�𝑡 + (

𝛾

𝛽− 2)

Δ𝑡

2 �̈�𝑡] ℂ + 𝕗𝑡+Δ𝑡 .

The vector 𝕗𝑡+Δ𝑡 contains the magnitude of the nodal loads 𝒇 at time 𝑡 + Δ𝑡 transformed into modal coordinates. Whenever a point load is placed in the domain of an element 𝑒, the values of the considered loading at the nodes 𝑥1

𝑒 and 𝑥2

𝑒 are approximated by a linear interpolation.

Since 𝒖𝑡+Δ𝑡 is calculated using eqn (3.24), the velocity and the acceleration at the corresponding time step become

�̇�𝑡+Δ𝑡 = �̇�𝑡 + (1 − 𝛾)Δ𝑡�̈�𝑡 +𝛾

𝛽Δ𝑡[𝒖𝑡+Δ𝑡 − 𝒖𝑡 − Δ𝑡�̇�𝑡 − (

1

2− 𝛽) (Δ𝑡)2�̈�𝑡] , �̈�𝑡+Δ𝑡 =

1

𝛽(Δ𝑡)2 [𝒖𝑡+Δ𝑡 − 𝒖𝑡 + Δ𝑡�̇�𝑡 + (1

2− 𝛽) (Δ𝑡)2�̈�𝑡] .

4. Development of an Analytical Solution

In this section, it is proposed an analytical model to perform dynamic analysis of continuous thin walled beams with mono-symmetric cross sections. The exact natural frequencies and the corresponding mode shapes of vibration are obtained for the cases of decoupled vertical bending motion and coupled lateral bending-torsional motion. To this end, implicit expressions are derived. The dynamic response due to eccentric moving loads with constant velocity is obtained by considering a modal analysis along with the direct integration method.

The proposed formulation considers an 𝑁-span bridge of total length 𝐿𝑡𝑜𝑡, as depicted in Figure 4.1. Each span is considered to have uniform geometrical and material properties along its length. For clarity of notation, the properties of the generic span 𝑠 are denoted by the subscript 𝑠, being the local coordinate 𝑥𝑠 related to the global coordinate 𝑥 by

𝑥𝑠 = 𝑥 − ∑ 𝑙𝑠

𝑟

𝑠=1 .

For mono-symmetric cross sections, being 𝑧 the axis of symmetry, and considering the principal global coordinate system (𝑥, 𝑦, 𝑧), the principal sectorial pole 𝑃 and origin 𝑂, the dynamic equilibrium of the bridge’s girder is given by

𝜌𝐴𝜉�̈� + 𝐸𝐼𝑦𝑦𝜉𝑧′′′′ = 𝑓𝑧 ,

𝜌𝐴�̈�𝑦

− 𝜌𝑆𝑦𝑃�̈� + 𝐸𝐼𝑧𝑧𝜉

𝑦′′′′ = 𝑓

𝑦 ,

−𝜌𝑆𝑦𝑃𝜉�̈� + 𝜌(𝐼𝑧𝑧

𝑃 + 𝐼𝑦𝑦𝑃 )�̈� + 𝐸𝐼𝜔𝜔𝜙′′′′ − 𝐺𝐽𝜙′′ = 𝑚𝑥 ,

where the warping effect is included and the rotary inertia of the cross sections is neglected. Considering the sectorial pole 𝑃 coincident with the centre of gravity 𝐶𝐺 of the cross section, the eqns (4.2) are rewritten as follows:

𝜌𝐴𝜉�̈� + 𝐸𝐼𝑦𝑦𝜉𝑧′′′′ = 𝑓𝑧 ,

𝜌𝐴𝜉�̈� + 𝐸𝐼𝑧𝑧𝜉𝑦′′′′ + 𝐸𝐼𝑧𝜔𝜙′′′′ = 𝑓𝑦 ,

𝜌(𝐼𝑧𝑧 + 𝐼𝑦𝑦)�̈� + 𝐸𝐼𝜔𝑧𝜉𝑦′′′′ + 𝐸𝐼𝜔𝜔𝜙′′′′ − 𝐺𝐽𝜙′′ = 𝑚𝑥 .

(3.21)

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

(4.1)

(4.2a)

(4.2b)

(4.2c)

(4.3a)

(4.3b)

(4.3c)

Figure 4.1 – model of the 𝑁-span bridge considered in the analytical formulation.

Since the present formulation deals with sets of moving point loads, 𝑓𝑧, 𝑓𝑦, and 𝑚𝑥 are of the form

𝑓𝑧(𝑥, 𝑡) = ∑ δ(𝑥 − 𝑥𝑘(𝑡))𝑛𝑙𝑜𝑎𝑑

𝑘𝑉𝑧𝑘(𝑡) , 𝑓𝑦(𝑥, 𝑡) = ∑ δ(𝑥 − 𝑥𝑘(𝑡))

𝑛𝑙𝑜𝑎𝑑

𝑘𝑉𝑦𝑘(𝑡) , 𝑚𝑥(𝑥, 𝑡) = ∑ δ(𝑥 − 𝑥𝑘(𝑡))

𝑛𝑙𝑜𝑎𝑑

𝑘𝑀𝑥𝑘(𝑡) ,

being 𝑥𝑘(𝑡) the position of the 𝑘-th load, δ(𝑡) the Dirac delta function, and 𝑛𝑙𝑜𝑎𝑑 the number of loads.

The homogeneous solution of the governing equations (4.2) is obtained by considering a harmonic function for the displacement components, say 𝜉𝑧, 𝜉𝑦 and 𝜙, which can be expressed as follows:

𝜉𝑧(𝑥, 𝑡) = 𝛯𝑧(𝑥) sin(𝑝𝑡) , 𝜉𝑦(𝑥, 𝑡) = 𝛯𝑦(𝑥) sin(𝑝𝑡) , 𝜙(𝑥, 𝑡) = 𝛷(𝑥) sin(𝑝𝑡) ,

being 𝛯𝑧, 𝛯𝑦 and 𝛷 the respective shape functions of the vertical displacement 𝜉𝑧, lateral displacement 𝜉𝑦 and torsional

rotation 𝜙, and 𝑝 is the corresponding circular frequency. Thus, by substituting eqns (4.5) into the homogeneous form of eqns (4.2), and noticing that sin(𝑝𝑡) can be non-null, leads to

−𝜌𝐴𝑝2𝛯𝑧(𝑥) + 𝐸𝐼𝑦𝑦𝛯𝑧′′′′(𝑥) = 0 , −𝜌𝐴𝑝2𝛯𝑦(𝑥) + 𝜌𝑆𝑦

𝑃𝑝2𝛷(𝑥) + 𝐸𝐼𝑧𝑧𝛯𝑦′′′′(𝑥) = 0 ,

𝜌𝑆𝑦𝑃𝑝2𝛯𝑦(𝑥) − 𝜌(𝐼𝑧𝑧

𝑃 + 𝐼𝑦𝑦𝑃 )𝑝2𝛷(𝑥) + 𝐸𝐼𝜔𝜔𝛷′′′′(𝑥) − 𝐺𝐽𝛷′′(𝑥) = 0 .

For the case of decoupled vertical bending motion the equation (4.6a) is simplified to the ordinary differential equation

−𝜆4𝛯𝑧(𝑥) + 𝛯𝑧′′′′(𝑥) = 0 ,

by introducing a frequency parameter 𝜆 given by

𝜆4 =𝜌𝐴𝑝2

𝐸𝐼𝑦𝑦 .

The solution 𝛯𝑧 of the eqn (4.7) may be expressed in trigonometric and hyperbolic terms as follows:

𝛯𝑧(𝑥) = 𝒜1 cosh(𝜆𝑥) + 𝒜2 sinh(𝜆𝑥) + 𝒜3 cos(𝜆𝑥) + 𝒜4 sin(𝜆𝑥) ,

being the 𝒜𝑗 constants such that the boundary conditions at the beam ends are satisfied. This allows to express three of the constants 𝒜𝑗 as functions of the fourth, which represents an arbitrary amplitude of the mode shape.

For the case of coupled lateral bending-torsional motion, the solutions 𝛯𝑦 and 𝛷 of the eqns (4.6b) and (4.6c) involve a

more cumbersome mathematical treatment. To start with, consider that these two equations are combined into one equation by eliminating either 𝛯𝑦 or 𝛷, [Banerjee et al, 1996]. This operation leads to the ordinary differential equation

[𝑑𝐷8 − 𝐷6 − (𝑎 + 𝑏𝑑)𝐷4 + 𝑏𝐷2 + 𝑎𝑏𝑐]𝜒(𝑥) = 0 ,

being 𝜒 either 𝛯𝑦 or 𝛷, 𝐷 is a differential operator and 𝑎, 𝑏, 𝑐 and 𝑑 are non-dimensional parameters defined as follows:

𝐷 =𝑑

𝑑𝑥 , 𝑎 = (𝐼𝑧𝑧

𝑃 + 𝐼𝑦𝑦𝑃 )

𝜌𝑝2

𝐺𝐽 , 𝑏 =

𝜌𝐴𝑝2

𝐸𝐼𝑧𝑧

, 𝑐 = 1 −𝑆𝑦

𝑃2

𝐴(𝐼𝑧𝑧𝑃 + 𝐼𝑦𝑦

𝑃 )and 𝑑 =

𝐸𝐼𝜔𝜔

𝐺𝐽 .

Considering an exponential solution for the differential equation (4.10) of the form

𝜒(𝑥) = 𝑒𝓈𝑥

leads to the characteristic equation

𝑑𝓈8 − 𝓈6 − (𝑎 + 𝑏𝑑)𝓈4 + 𝑏𝓈2 + 𝑎𝑏𝑐 = 0 .

The eqn (4.13) reduces to a quartic polynomial with real coefficients by substituting 𝓈2 by 𝜂, as follows:

𝑑𝜂4 − 𝜂3 − (𝑎 + 𝑏𝑑)𝜂2 + 𝑏𝜂 + 𝑎𝑏𝑐 = 0 .

Since, for a physical problem, 𝑎, 𝑏, 𝑐 and 𝑑 are always positive reals, being 𝑐 less than one, it is possible to show, [Bishop et al., 1989], that all four roots of the equation are reals and they are within four distinct intervals, namely

]−∞, −√𝑏[ , ]−√𝑏, 0[ , ]0, √𝑏[ and ]√𝑏, +∞[ .

Considering 𝜂1, 𝜂2, -𝜂3 and -𝜂4 to be the four real roots, the eight roots of the characteristic equation (4.13) are

𝛼 , −𝛼 , 𝛽 , −𝛽 , 𝑖𝛾 , −𝑖𝛾 , 𝑖𝛿 , and −𝑖𝛿 ,

46

The proposed formulation considers an ! -span bridge of total length ! ! " ! , as illustrated in Figure 4.1. Each

span is considered to have uniform geometrical and material properties along its length. For clarity of

notation, the properties of the generic span ! are denoted by the subscript ! , being the local coordinate ! !

related to the global coordinate ! by

! ! = ! − !!!

! ! !!.

Figure 4.1 – model of the ! -span bridge considered in the analytical formulation

The dynamic equilibrium differential equations of thin walled beams were obtained for a generic cross

section in Chapter 2, eqn (2.50). For mono-symmetric cross sections, being ! the axis of symmetry, and

considering the principal global coordinate system ! , ! , ! , the principal sectorial pole ! and the principal

sectorial origin ! , the dynamic equilibrium of the bridge’s girder is given by

! " ! ! + ! !! ! ! !!!!! = ! ! !,

! " ! ! − ! ! !! ! + ! !! ! ! !

!!!! = ! ! !,

−! ! !! ! ! + ! !! !

! + !! !! ! + ! !! ! !

!!!! − ! "! !! = ! ! !,

where the warping effect is included and the rotary inertia is neglected. Considering the sectorial pole !

coincident with the centre of gravity ! " of the cross section, the eqns (4.2) are rewritten as follows:

! " ! ! + ! !! ! ! !!!!! = ! ! !,

! " ! ! + ! !! ! ! !!!!! + ! !! " !

!!!! = ! ! !,

! !! !! + !! !

! ! + ! !! " ! !!!!! + ! !! ! !

!!!! − ! "! !! = ! ! !.

Both sets of equations, (4.2) and (4.3), will be required in the next discussion.

Since the present formulation deals with arrays of moving point loads, ! ! , ! ! , and ! ! are of the form

! ! ! , ! = δ ! − ! ! !! ! " #$

!! ! " ! !,

! ! ! , ! = δ ! − ! ! !! ! " #$

!! ! " ! !,

! ! ! , ! = δ ! − ! ! !! ! " #$

!! ! " ! !,

being ! ! ! the position of the ! -th load, δ ! the Dirac delta function, and ! !" #$ the number of loads.

(4.2a)

(4.2b)

(4.2c)

(4.3a)

(4.3b)

(4.3c)

(4.1)

(4.4a)

(4.4b)

(4.4c)

!! !! !! !! !! ! ! !!

! ! " !

! ! ! ! ! ! ! ! ! ! ! ! ! !

! ! " ! ! ! ! ! !

(4.4)

(4.5)

(4.6a), (4.6b)

(4.6c)

(4.7)

(4.8)

(4.9)

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

(4.15)

(4.16)

with

𝛼 = √𝜂1 , 𝛽 = √𝜂

2 , 𝛾 = √𝜂

3 , 𝛿 = √𝜂

4 , and 𝑖 = √−1 .

Substituting each of these roots into eqn (4.13) separately, one obtains the solutions for the lateral bending displacement 𝛯𝑦 and for the torsional rotation 𝛷 as follows:

𝛯𝑦(𝑥) = ℬ1 cosh(𝛼𝑥) + ℬ2 sinh(𝛼𝑥) + ℬ3 cosh(𝛽𝑥) + ℬ4 sinh(𝛽𝑥) + ℬ5 cos(𝛾𝑥) + ℬ6 sin(𝛾𝑥) + ℬ7 cos(𝛿𝑥) + ℬ8 sin(𝛿𝑥) ,

𝛷(𝑥) = 𝒞1 cosh(𝛼𝑥) + 𝒞2 sinh(𝛼𝑥) + 𝒞3 cosh(𝛽𝑥) + 𝒞4 sinh(𝛽𝑥) + 𝒞5 cos(𝛾𝑥) + 𝒞6 sin(𝛾𝑥) + 𝒞7 cos(𝛿𝑥) + 𝒞8 sin(𝛿𝑥) .

The relations between the two sets of constants ℬ𝑗 and 𝒞𝑗 are given by

𝒞1 = 𝜅𝛼ℬ1 , 𝒞2 = 𝜅𝛼ℬ2 , 𝒞3 = 𝜅𝛽ℬ3 , 𝒞4 = 𝜅𝛽ℬ4 , 𝒞5 = 𝜅𝛾ℬ5 , 𝒞6 = 𝜅𝛾ℬ6 , 𝒞7 = 𝜅𝛿ℬ7 and 𝒞8 = 𝜅𝛿ℬ8 .

where the coefficients 𝜅𝛼, 𝜅𝛽, 𝜅𝛾 and 𝜅𝛿 are defined as follows:

𝜅𝛼 =𝜌𝐴𝑝2 − 𝐸𝐼𝑧𝑧𝛼4

𝜌𝑆𝑦𝑃𝑝2

, 𝜅𝛽 =𝜌𝐴𝑝2 − 𝐸𝐼𝑧𝑧𝛽4

𝜌𝑆𝑦𝑃𝑝2

, 𝜅𝛾 =𝜌𝐴𝑝2 − 𝐸𝐼𝑧𝑧𝛾4

𝜌𝑆𝑦𝑃𝑝2

and 𝜅𝛿 =𝜌𝐴𝑝2 − 𝐸𝐼𝑧𝑧𝛿4

𝜌𝑆𝑦𝑃𝑝2

.

In order to calculate the bending natural frequencies and the respective mode shapes of a continuous beam, consider that the 𝑛-th vibration mode shape of the 𝑠-th span is given by

𝛯𝑧𝑛𝑠(𝑥𝑠) = 𝒜1𝑛𝑠 cosh(𝜆𝑛𝑠𝑥𝑠) + 𝒜2𝑛𝑠 sinh(𝜆𝑛𝑠𝑥𝑠) + 𝒜3𝑛𝑠 cos(𝜆𝑛𝑠𝑥𝑠) + 𝒜4𝑛𝑠 sin(𝜆𝑛𝑠𝑥𝑠) , 𝛯𝑦𝑛𝑠(𝑥𝑠) = ℬ1𝑛𝑠 cosh(𝛼𝑛𝑠𝑥𝑠) + ℬ2𝑛𝑠 sinh(𝛼𝑛𝑠𝑥𝑠) + ℬ3𝑛𝑠 cosh(𝛽𝑛𝑠𝑥𝑠) + ℬ4𝑛𝑠 sinh(𝛽𝑛𝑠𝑥𝑠) + ℬ5𝑛𝑠 cos(𝛾𝑛𝑠𝑥𝑠) + ℬ6𝑛𝑠 sin(𝛾𝑛𝑠𝑥𝑠)

+ ℬ7𝑛𝑠 cos(𝛿𝑛𝑠𝑥𝑠) + ℬ8𝑛𝑠 sin(𝛿𝑛𝑠𝑥𝑠) , 𝛷𝑛𝑠(𝑥𝑠) = 𝒞1𝑛𝑠 cosh(𝛼𝑛𝑠𝑥𝑠) + 𝒞2𝑛𝑠 sinh(𝛼𝑛𝑠𝑥𝑠) + 𝒞3𝑛𝑠 cosh(𝛽𝑛𝑠𝑥𝑠) + 𝒞4𝑛𝑠 sinh(𝛽𝑛𝑠𝑥𝑠) + 𝒞5𝑛𝑠 cos(𝛾𝑛𝑠𝑥𝑠) + 𝒞6𝑛𝑠 sin(𝛾𝑛𝑠𝑥𝑠)

+ 𝒞7𝑛𝑠 cos(𝛿𝑛𝑠𝑥𝑠) + 𝒞8𝑛𝑠 sin(𝛿𝑛𝑠𝑥𝑠) .

The constants 𝒜𝑗𝑛𝑠 and ℬ𝑗𝑛𝑠, since 𝒞𝑗𝑛𝑠 depends upon ℬ𝑗𝑛𝑠, are calculated by imposing the boundary conditions at the ends of the 𝑠-th span. For a continuous beam, the kinematic boundary conditions of the 𝑠-th span require that

𝛯𝑧𝑛𝑠(0) = 0 , 𝛯𝑧𝑛𝑠(𝑙𝑠) = 0 , 𝛯𝑦𝑛𝑠(0) = 0 , 𝛯𝑦𝑛𝑠(𝑙𝑠) = 0 , 𝛷𝑛𝑠(0) = 0 , 𝛷𝑛𝑠(𝑙𝑠) = 0 ,

𝑑𝛯𝑧𝑛𝑠(𝑥𝑠)

𝑑𝑥𝑠

|𝑥𝑠=0

=𝑑𝛯𝑧𝑛𝑟(𝑥𝑟)

𝑑𝑥𝑟

|𝑥𝑟=𝑙𝑟

,𝑑𝛯𝑦𝑛𝑠(𝑥𝑠)

𝑑𝑥𝑠

|𝑥𝑠=0

=𝑑𝛯𝑦𝑛𝑟(𝑥𝑟)

𝑑𝑥𝑟

|𝑥𝑟=𝑙𝑟

,𝑑𝛷𝑛𝑠(𝑥𝑠)

𝑑𝑥𝑠

|𝑥𝑠=0

=𝑑𝛷𝑛𝑟(𝑥𝑟)

𝑑𝑥𝑟

|𝑥𝑟=𝑙𝑟

.

From continuity over the inner supports, the static boundary conditions imply the following relations:

𝐸𝑠𝐼𝑦𝑦𝑠

𝑑2𝛯𝑧𝑛𝑠(𝑥𝑠)

𝑑𝑥𝑠2

|𝑥𝑠=0

= 𝐸𝑟𝐼𝑦𝑦𝑟

𝑑2𝛯𝑧𝑛𝑟(𝑥𝑟)

𝑑𝑥𝑟2

|𝑥𝑟=𝑙𝑟

, 𝐸𝑠𝐼𝑧𝑧𝑠

𝑑2𝛯𝑦𝑛𝑠(𝑥𝑠)

𝑑𝑥𝑠2

|𝑥𝑠=0

= 𝐸𝑟𝐼𝑧𝑧𝑟

𝑑2𝛯𝑦𝑛𝑟(𝑥𝑟)

𝑑𝑥𝑟2

|𝑥𝑟=𝑙𝑟

, and

𝐸𝑠𝐼𝜔𝜔𝑠

𝑑2𝛷𝑛𝑠(𝑥𝑠)

𝑑𝑥𝑠2 |

𝑥𝑠=0

= 𝐸𝑟𝐼𝜔𝜔𝑟

𝑑2𝛷𝑛𝑟(𝑥𝑟)

𝑑𝑥𝑟2 |

𝑥𝑟=𝑙𝑟

.

For an easier understanding of the assembly procedure presented by [Chan and Ashebo, 2006], which is being extended to coupled vibrations, the free bending decoupled motion about the 𝑦 axis is firstly considered.

Substituting the boundary conditions (4.22) and (4.23) in the equation of motion (4.21a), the constants 𝒜𝑗𝑛𝑠 can be obtained from explicit expressions. Considering that one of the four 𝒜𝑗𝑛1 constants relative to the first span represents

an arbitrary amplitude of the shape function 𝛯𝑧𝑛1, one can obtain for the first span

𝒜1𝑛1 = 0 , 𝒜2𝑛1 = 1 , 𝒜3𝑛1 = 0 , 𝒜4𝑛1 = −sinh(𝜆𝑛1𝑙1)

sin(𝜆𝑛1𝑙1) ,

and for the 𝑠-th inner span

𝒜1𝑛𝑠 =1

2

𝐸𝑟𝐼𝑦𝑦𝑟

𝐸𝑠𝐼𝑦𝑦𝑠

𝜆𝑛𝑟2

𝜆𝑛𝑠2

[𝒜1𝑛𝑟 cosh(𝜆𝑛𝑟𝑙𝑟) + 𝒜2𝑛𝑟 sinh(𝜆𝑛𝑟𝑙𝑟) − 𝒜3𝑛𝑟 cos(𝜆𝑛𝑟𝑙𝑟) − 𝒜4𝑛𝑟 sin(𝜆𝑛𝑟𝑙𝑟)] ,

𝒜2𝑛𝑠 =𝜆𝑛𝑟

𝜆𝑛𝑠

[𝒜1𝑛𝑟 sinh(𝜆𝑛𝑟𝑙𝑟) + 𝒜2𝑛𝑟 cosh(𝜆𝑛𝑟𝑙𝑟) − 𝒜3𝑛𝑟 sin(𝜆𝑛𝑟𝑙𝑟) + 𝒜4𝑛𝑟 cos(𝜆𝑛𝑟𝑙𝑟)]sin(𝜆𝑛𝑠𝑙𝑠)

sin(𝜆𝑛𝑠𝑙𝑠) − sinh(𝜆𝑛𝑠𝑙s)

− 𝒜1𝑛𝑠

cos(𝜆𝑛𝑠𝑙𝑠) − cosh(𝜆𝑛𝑠𝑙𝑠)

sin(𝜆𝑛𝑠𝑙𝑠) − sinh(𝜆𝑛𝑠𝑙𝑠) ,

𝒜3𝑛𝑠 = −𝒜1𝑛𝑠 and

𝒜4𝑛𝑠 =𝜆𝑛𝑟

𝜆𝑛𝑠

[𝒜1𝑛𝑟 sinh(𝜆𝑛𝑟𝑙𝑟) + 𝒜2𝑛𝑟 cosh(𝜆𝑛𝑟𝑙𝑟) − 𝒜3𝑛𝑟 sin(𝜆𝑛𝑟𝑙𝑟) + 𝒜4𝑛𝑟 cos(𝜆𝑛𝑟𝑙𝑟)] − 𝒜2𝑛𝑠 .

Then, by imposing the boundary condition on the bending moment at the end of the last span 𝑁,

𝑑2𝛯𝑧𝑛𝑁(𝑥𝑁)

𝑑𝑥𝑁2 |

𝑥𝑁=𝑙𝑁

= 0 ,

(4.17)

(4.18a) (4.18b)

(4.19)

(4.20)

(4.21a)

(4.21b)

(4.21c)

(4.22)

(4.23)

(4.24a)

(4.24b)

(4.24c)

(4.24d)

(4.24e)

(4.25)

the natural frequencies for the decoupled bending motion about the 𝑦 axis are given by the roots of the equation

𝒜1𝑛𝑁 cosh(𝜆𝑛𝑁𝑙𝑁) + 𝒜2𝑛𝑁 sinh(𝜆𝑛𝑁𝑙𝑁) − 𝒜3𝑛𝑁 cos(𝜆𝑛𝑁𝑙𝑁) − 𝒜4𝑛𝑁 sin(𝜆𝑛N𝑙𝑁) = 0 .

Notice that the equation (4.25) has not been presented in [Chan and Ashebo, 2006]. In fact, similar defining expressions for the constants 𝒜𝑗𝑛𝑠 have been derived, but the procedure of finding the natural frequencies is unclear.

The formulation that has been presented is now extended to coupled lateral bending-torsional vibrations of the continuous beam illustrated in Figure 4.1.

The eight constants ℬ𝑗𝑛1 of the first span are obtained by substituting the remaining boundary conditions (4.22) and

(4.23) in the equations (4.21b) and (4.21c), yielding

ℬ1𝑛1 = 0 , ℬ3𝑛1 = 0 , ℬ5𝑛1 = 0 , ℬ7𝑛1 = 0 ,

ℬ6𝑛1 =ℬ2𝑛1 sinh(𝛼𝑛1𝑙1) (𝜅𝛿1 − 𝜅𝛼1) + ℬ4𝑛1 sinh(𝛽𝑛1𝑙1) (𝜅𝛿1 − κ𝛽1)

sin(𝛾𝑛1𝑙1) (𝜅𝛾1 − 𝜅𝛿1) ,

ℬ8𝑛1 =ℬ2𝑛1 sinh(𝛼𝑛1𝑙1) (𝜅𝛼1 − 𝜅𝛾1) + ℬ4𝑛1 sinh(𝛽𝑛1𝑙1) (𝜅𝛽1 − 𝜅𝛾1)

sin(𝛿𝑛1) (𝜅𝛾1 − 𝜅𝛿1) .

For the generic inner span 𝑠 the expressions of the eight constants ℬ𝑗𝑛𝑠 can be easily obtained using symbolic algebra software. By computing the explicit expressions for the constants ℬ𝑗𝑛𝑠, it is straightforward to apply the presented

analytical procedure to girders with any number of spans.

The natural frequencies 𝑝𝑛 for the coupled lateral bending-torsional vibrations are obtained by imposing the static boundary conditions at the end of the 𝑁-th span,

𝑑2𝛯𝑦𝑛𝑁(𝑥𝑁)

𝑑𝑥𝑁2

|𝑥𝑁=𝑙𝑁

= 0 and𝑑2𝛷𝑛𝑁(𝑥𝑁)

𝑑𝑥𝑁2

|𝑥𝑁=𝑙𝑁

= 0 ,

and finding the roots that satisfy both equations.

Consider that the boundary conditions (4.28) are written only as functions of ℬ2𝑛1 and ℬ4𝑛1 as follows:

[𝑆22 𝑆24

𝑆42 𝑆44] {

ℬ2𝑛1

ℬ4𝑛1} = 𝟎 ,

where the numerical value of each component 𝑆𝑘𝑙 can be readily obtained by making ℬ𝑘𝑛1 and ℬ𝑙𝑛1 equal to zero and one, respectively, when the first span is evaluated, eqns (4.27). By solving the matrix equation (4.29) it is possible to express seven of the eight constants ℬ𝑗𝑛1 in terms of the eighth, which represents an arbitrary amplitude of the mode shape components 𝛯𝑦𝑛𝑠 and 𝛷𝑛𝑠.

By giving ℬ2𝑛1 a numerical value, say unity, and therefore the constants ℬ𝑗𝑛1 (4.27) become

ℬ1𝑛1 = 0 , ℬ3𝑛1 = 0 , ℬ5𝑛1 = 0 , ℬ7𝑛1 = 0 , ℬ2𝑛1 = 1 , ℬ4𝑛1 = −𝑆22

𝑆24

,

ℬ6𝑛1 =𝑆24 sinh(𝛼𝑛1𝑙1) (𝜅𝛿1 − 𝜅𝛼1) + 𝑆22 sinh(𝛽𝑛1𝑙1) (𝜅𝛽1 − 𝜅𝛿1)

𝑆24 sin(𝛾𝑛1𝑙1) (𝜅𝛾1 − 𝜅𝛿1) ,

ℬ8𝑛1 =𝑆24 sinh(𝛼𝑛1𝑙1) (𝜅𝛼1 − 𝜅𝛾1) + 𝑆22 sinh(𝛽𝑛1𝑙1) (𝜅𝛾1 − 𝜅𝛽1)

𝑆24 sin(𝛿𝑛1𝑙1) (𝜅𝛾1 − 𝜅𝛿1) .

Finally, the natural frequencies 𝑝𝑛 for the coupled bending-torsional motion are calculated by seeking for the roots of the following equation:

det ([𝑆22 𝑆24

𝑆42 𝑆44]) = 0 .

Once the frequencies of vibration have been calculated, the 𝑚 or the 𝑛-th mode shape, in which the entire continuous beam is said to respond when vibrates with a frequency 𝑝𝑚 or 𝑝𝑛, is established through the assembly of the 𝑚 or 𝑛-th mode shape functions calculated to each span, as follows:

𝛯𝑧𝑚(𝑥) = ∑ [Η (𝑥 − ∑ 𝑙𝑟

𝑠−1

𝑟=1) − Η (𝑥 − ∑ 𝑙𝑟

𝑠

𝑟=1)] 𝛯𝑧𝑚𝑠(𝑥𝑠)

𝑁

𝑠=1 ,

𝛯𝑦𝑛(𝑥) = ∑ [Η (𝑥 − ∑ 𝑙𝑟

𝑠−1

𝑟=1) − Η (𝑥 − ∑ 𝑙𝑟

𝑠

𝑟=1)] 𝛯𝑦𝑛𝑠(𝑥𝑠)

𝑁

𝑠=1 ,

𝛷𝑛(𝑥) = ∑ [Η (𝑥 − ∑ 𝑙𝑟

𝑠−1

𝑟=1) − Η (𝑥 − ∑ 𝑙𝑟

𝑠

𝑟=1)] 𝛷𝑛𝑠(𝑥𝑠)

𝑁

𝑠=1 ,

being Η the Heaviside step function.

(4.26)

(4.27)

(4.28)

(4.29)

(4.30)

(4.31)

(4.32a)

(4.32b)

(4.32c)

To quantify the damping properties a proportional relation to the modal mass properties must be established. A procedure for determining the modal mass properties is by considering the orthogonality of the vibration mode shapes with respect to mass, which can be demonstrated for a continuum system by application of the Bettis’s law.

Considering the modal superposition method, the beam’s generalized displacements 𝜉𝑧, 𝜉𝑦 and 𝜙 are defined as

𝜉𝑧(𝑥, 𝑡) = ∑ 𝛯𝑧𝑘(𝑥)𝛹𝑘(𝑡)∞

𝑘 , 𝜉𝑦(𝑥, 𝑡) = ∑ 𝛯𝑦𝑙(𝑥)𝛶𝑙(𝑡)

∞

𝑙 , 𝜙 (𝑥, 𝑡) = ∑ 𝛷𝑙(𝑥)𝛶𝑙(𝑡)

∞

l ,

where 𝛹𝑘 and 𝛶𝑙 are the time-varying amplitudes of the 𝑘-th and the 𝑙-th vibration modes.

Substituting the eqns (4.33) in the governing equations (4.3), the Betti’s law stands for

∫ 𝛯𝑧𝑚(𝑥)𝛹𝑚(𝑡)𝜌𝐴𝛯𝑧𝑛(𝑥)𝑝𝑛2𝛹𝑛(𝑡)

𝐿

0

𝑑𝑥 = ∫ 𝛯𝑧𝑛(𝑥)𝛹𝑛(𝑡)𝜌𝐴𝛯𝑧𝑚(𝑥)𝑝𝑚2𝛹𝑚(𝑡)

𝐿

0

𝑑𝑥 ,

∫ [𝛯𝑦𝑚(𝑥)𝛶𝑚(𝑡)𝜌𝐴𝛯𝑦𝑛(𝑥)𝑝𝑛2𝛶𝑛(𝑡) + 𝛷𝑚(𝑥)𝛶𝑚(𝑡)ρ(𝐼𝑧𝑧 + 𝐼𝑦𝑦)𝛷𝑛(𝑥)𝑝𝑛

2𝛶𝑛(𝑡)]𝐿

0

𝑑𝑥

= ∫ [𝛯𝑦𝑛(𝑥)𝛶𝑛(𝑡)𝜌𝐴𝛯𝑦𝑚(𝑥)𝑝𝑚2𝛶𝑚(𝑡) + 𝛷𝑛(𝑥)𝛶𝑛(𝑡)𝜌(𝐼𝑧𝑧 + 𝐼𝑦𝑦)𝛷𝑚(𝑥)𝑝𝑚

2𝛶𝑚(𝑡)]𝐿

0

𝑑𝑥 .

Subtracting both sides of eqns (4.34) from one another the following orthogonality conditions are obtained:

∫ 𝛯𝑧𝑚(𝑥)𝛯𝑧𝑛(𝑥)𝐿

0

𝑑𝑥 = 0 and ∫ [𝜌𝐴𝛯𝑦𝑚(𝑥)𝛯𝑦𝑛(𝑥) + 𝜌(𝐼𝑧𝑧 + 𝐼𝑦𝑦)𝛷𝑚(𝑥)𝛷𝑛(𝑥)]𝐿

0

𝑑𝑥 = 0 with 𝑝𝑛 ≠ 𝑝𝑚

The decoupling of the governing equations (4.3) is accomplished by considering the modal coordinate expressions (4.33), which leads to

∑ [𝜌𝐴𝛯𝑧𝑘(𝑥)�̈�𝑘(𝑡) + 𝐸𝐼𝑦𝑦𝛯𝑧𝑘′′′′(𝑥)𝛹𝑘(𝑡)]

∞

𝑘= 𝑓𝑧(𝑥, 𝑡) ,

∑ [𝜌𝐴𝛯𝑦𝑙(𝑥)�̈�𝑙(𝑡) + 𝐸𝐼𝑧𝑧𝛯𝑦𝑙′′′′(𝑥)𝛶𝑙(𝑡) + 𝐸𝐼𝑧𝜔𝛷𝑙

′′′′(𝑥)𝛶𝑙(𝑡)]∞

𝑙= 𝑓𝑦(𝑥, 𝑡) ,

∑ [𝜌(𝐼𝑧𝑧 + 𝐼𝑦𝑦)𝛷𝑙(𝑥)�̈�𝑙(𝑡) + 𝐸𝐼𝜔𝑧𝛯𝑦𝑙′′′′(𝑥)𝛶𝑙(𝑡) + 𝐸𝐼𝜔𝜔𝛷𝑙

′′′′(𝑥)𝛶𝑙(𝑡) − 𝐺𝐽𝛷𝑙′′(𝑥)𝛶𝑙(𝑡)]

∞

𝑙= 𝑚𝑥(𝑥, 𝑡) .

Multiplying eqn (4.36a) by 𝛯𝑧𝑚(𝑥), integrating with respect to 𝑥, and applying the orthogonality condition (4.34a) yields

𝑀𝑧𝑚�̈�𝑚(𝑡) + 𝑝𝑚2𝑀𝑧𝑚𝛹𝑚(𝑡) = 𝑃𝑧𝑚(𝑡) ,

being the generalized mass 𝑀𝑧𝑚 and the generalized loading 𝑃𝑧𝑚 associated with the 𝑚-th mode of vibration given by

𝑀𝑧𝑚 = 𝜌𝐴 ∫ 𝛯𝑧𝑚2(𝑥)

𝐿

0

𝑑𝑥 and 𝑃𝑧𝑚(𝑡) = ∫ 𝛯𝑧𝑚(𝑥)𝑓𝑧(𝑥, 𝑡)

𝐿

0

𝑑𝑥 .

Multiplying eqns (4.36b) and (4.36c) by 𝛯𝑦𝑛 and 𝛷𝑛, respectively, integrating with respect to 𝑥, adding them to one another and applying the orthogonality condition (4.34b) gives

𝑀𝑦𝜙𝑛�̈�𝑛(𝑡) + 𝑝𝑛2𝑀𝑦𝜙𝑛𝛶𝑛(𝑡) = 𝑃𝑦𝜙𝑛(𝑡) ,

where the generalized mass 𝑀𝑦𝜙𝑛 and the genralized loading 𝑃𝑦𝜙𝑛 associated with the 𝑛-th mode are defined as follows:

𝑀𝑦𝜙𝑛 = ∫ [𝜌𝐴𝛯𝑦𝑛2(𝑥) + 𝜌(𝐼𝑧𝑧 + 𝐼𝑦𝑦)𝛷𝑛

2(𝑥)]𝐿

0

𝑑𝑥 and 𝑃𝑦𝜙𝑛(𝑡) = ∫ [𝛯𝑦𝑛(𝑥)𝑓𝑦(𝑥, 𝑡) + 𝛷𝑛(𝑥)𝑚𝑥(𝑥, 𝑡)]𝐿

0

𝑑𝑥 .

Thus, the decoupled modal equations are written as follows:

𝑀𝑧𝑚�̈�𝑚(𝑡) + 𝐶𝑧𝑚�̇�𝑚(𝑡) + 𝑝𝑚2𝑀𝑧𝑚𝛹𝑚(𝑡) = 𝑃𝑧𝑚(𝑡) and 𝑀𝑦𝜙𝑛�̈�𝑛(𝑡) + 𝐶𝑦𝜙𝑛�̇�𝑛(t) + 𝑝𝑛

2𝑀𝑦𝜙𝑛𝛶𝑛(𝑡) = 𝑃𝑦𝜙𝑛(𝑡) ,

where the generalized damping coefficients 𝐶𝑧𝑚 and 𝐶𝑦𝜙𝑛 are given by

𝐶𝑧𝑚 = 2𝜍𝑧𝑚

𝑝𝑚

𝑀𝑧𝑚 and 𝐶𝑦𝜙𝑛 = 2𝜍𝑦𝜙𝑛

𝑝𝑛

𝑀𝑦𝜙𝑛 .

being 𝜍𝑧𝑚 and 𝜍𝑦𝜙𝑛 the modal viscous damping ratios of each vibration mode.

As in the discrete case, the Newmark’s 𝛽 method is used for solving the governing equations (4.41). The corresponding explicit form of the method is written as follows:

𝒦𝑢𝑡+Δ𝑡 = ℱ𝑡+Δ𝑡 ,

being the effective modal stiffness 𝒦 and the effective modal loading ℱ𝑡+Δ𝑡 respectively defined as

𝒦 =1

𝛽(Δ𝑡)2 𝑀 +𝛾

𝛽Δ𝑡𝐶 + 𝐾 ,

ℱ𝑡+Δ𝑡 = [1

𝛽(Δ𝑡)2 𝑢𝑡 +1

𝛽Δ𝑡�̇�𝑡 + (

1

2𝛽− 1) �̈�𝑡] 𝑀 + [

𝛾

𝛽Δ𝑡𝑢𝑡 + (

𝛾

𝛽− 1) �̇�𝑡 + (

𝛾

𝛽− 2)

Δ𝑡

2 �̈�𝑡] 𝐶 + 𝑃𝑡+Δ𝑡 ,

(4.33)

(4.34a)

(4.34b)

(4.35)

(4.36a)

(4.36b)

(4.36c)

(4.37)

(4.38)

(4.39)

(4.40)

(4.41)

(4.42)

(4.43)

(4.44a)

(4.44b)

in which 𝑀, 𝐶 and 𝐾 represent the modal mass 𝑀𝑧𝑚 or 𝑀𝑦𝜙𝑛, the modal damping 𝐶𝑧𝑚 or 𝐶𝑦𝜙𝑛, and the modal stiffness

𝑝𝑚2𝑀𝑧𝑚 or 𝑝𝑛

2𝑀𝑦𝜙𝑛, respectively. The generalized loading 𝑃 represents either 𝑃𝑧𝑚 or 𝑃𝑦𝜙𝑛, where the magnitudes of the

applied loads at time 𝑡 + Δ𝑡 are transformed into modal coordinates.

5. Dynamic Response of Continuous Beams under Moving Loads

In the sequel, the proposed numerical and the analytical beam models are applied in order to perform the dynamic analysis of a multi-span bridge eccentrically traversed by moving vehicles. The longitudinal model is shown in Figure 5.1 and the prescribed boundary conditions at the supports are listed in Table 5.2. The dynamic responses are obtained by taking into account the high-speed universal trains HSLM-A10 proposed by the [EN 1991-2] travelling at 80 and 120 ms-1. The mass effect of the moving loads is also considered.

An open double-T cross section is considered, being the material and the geometrical properties presented in Table 5.1.

Table 5.1 – material and geometrical properties of the cross sections considered in the practical examples.

Cross Section

𝐸 (kN/m2)

𝐺 (kN/m2)

𝜌 (ton/m3)

𝐴 (m2)

𝐽 (m4)

𝐼𝑦𝑦 (m4)

𝐼𝑧𝑧 (m4)

𝐼𝑧𝜔 (m5)

𝐼𝜔𝜔 (m6)

Double-T 33 000 13 750 2.5484 9.5065 1.2245 9.1618 122.3711 175.7444 362.5405

The effectiveness of the developed thin walled beam models is verified by comparison with a shell finite element model implemented in the commercial software SAP2000. The selected thin-shell elements use a four-node formulation that combines separately the membrane and the bending behaviour. To simulate the cross-sectional in-plane indeformability of the bridge’s girder, body constraints are assigned to the element nodes.

Figure 5.1 – longitudinal model of the three-span bridge considered in the practical examples (m).

Table 5.2 – prescribed boundary conditions at the supports.

Support Nodal Coordinate Kinematic Boundary Conditions Static Boundary Conditions

𝑆1 𝑥 = 0 m 𝜉𝑧(𝑥) = 𝜉𝑦(𝑥) = 𝜙(𝑥) = 0 𝜉𝑧′′(𝑥) = 𝜉𝑦

′′(𝑥) = 𝜙′′(𝑥) = 0

𝑆2 𝑥 = 30 m 𝜉𝑧(𝑥) = 𝜉𝑦(𝑥) = 𝜙(𝑥) = 0

𝜉𝑧′(𝑥−) = 𝜉𝑧

′(𝑥+) 𝜉𝑦

′ (𝑥−) = 𝜉𝑦′ (𝑥+)

𝜙′(𝑥−) = 𝜙′(𝑥+)

𝜉𝑧′′(𝑥−) = 𝜉𝑧

′′(𝑥+) 𝜉𝑦

′′(𝑥−) = 𝜉𝑦′′(𝑥+)

𝜙′′(𝑥−) = 𝜙′′(𝑥+) 𝑆3 𝑥 = 70 m

𝑆4 𝑥 = 100 m 𝜉𝑧(𝑥) = 𝜉𝑦(𝑥) = 𝜙(𝑥) = 0 𝜉𝑧′′(𝑥) = 𝜉𝑦

′′(𝑥) = 𝜙′′(𝑥) = 0

The natural frequencies calculated by the two beam models are compared in Table 5.3 with the values given by the shell finite element model.

The dynamic influence lines of the vertical displacements and accelerations at the track’s midpoint of the central span are presented in Figure 5.2 and Figure 5.3.

In the next the mass effect of the moving vehicles is taken into account in the dynamic response of the three-span bridge. Therefore, the considered loading is written as follows:

𝑓𝑧(𝑥, 𝑡) = ∑ δ(𝑥 − 𝑥𝑘(𝑡))𝑛𝑙𝑜𝑎𝑑

𝑘[𝑉𝑧𝑘(𝑡) − 𝓂𝑘

𝑑2𝜈𝑘

𝑑𝑡2 ] and 𝑚𝑥(𝑥, 𝑡) = 𝑓𝑧(𝑥, 𝑡) 𝑒𝑐𝑐 ,

being 𝑥𝑘(𝑡) the position and 𝓂𝑘 the mass of the 𝑘-th load, 𝜈𝑘 is the vertical displacement of the moving mass 𝓂𝑘, δ(𝑡) the Dirac delta function, 𝑛𝑙𝑜𝑎𝑑 the number of loads and 𝑒𝑐𝑐 is the eccentricity of the track’s position.

30 40 30

𝑆1 𝑆2 𝑆3 𝑆4

𝑥

𝑒𝑐𝑐=2,5 m 𝑃𝑧

𝑧

𝑦 𝐶𝐺

Since the load is acting on a moving coordinate and therefore, [Ouyang, 2011]:

𝜈𝑘(𝑡) = 𝜉𝑧(𝑐𝑡, 𝑡) ,𝑑𝜈𝑘

𝑑𝑡=

𝜕𝜉𝑧

𝜕𝑡+ 𝑐

𝜕𝜉𝑧

𝜕𝑥and

𝑑2𝜈𝑘

𝑑𝑡2 =𝜕2𝜉𝑧

𝜕𝑡2 + 2𝑐𝜕𝜉𝑧

𝜕𝑡

𝜕𝜉𝑧

𝜕𝑥+ 𝑐2

𝜕2𝜉𝑧

𝜕𝑥2

where 𝑐 represents the velocity of the moving masses. Notice that it is assumed that the masses do not separate from the beam during their travel and vertical vibration.

Table 5.3 – natural frequencies of vibration of the double-T cross section girder.

Natural Frequencies of Vibration in the plane 𝑥𝑧 (Hz)

Frequency Number

Model of Analysis

Numerical Analytical Shell FEM

Inc. Rotary Inertia

Negl. Rotary Inertia

Inc. Rotary Inertia

1 4.5777 4.5942 4.4265

2 7.4770 7.5202 7.0536

3 8.9727 9.0216 8.1927

4 17.0665 17.2890 15.0470

5 25.8383 26.4031 22.1980

6 28.1767 28.7827 23.3090

Natural Frequencies of Vibration in the plane 𝑥𝑦 (Hz)

Frequency Number

Model of Analysis

Numerical Analytical Shell FEM

Inc. Rotary Inertia

Negl. Rotary Inertia

Inc. Rotary Inertia

1 4.8724 4.8853 4.7617

2 7.4149 7.4469 7.0966

3 8.5349 8.5700 7.9137

4 15.7794 15.9336 14.1110

5 17.1743 18.1063 12.7420

6 23.5509 23.9380 20.4600

In Figure 5.4 the influence of constrained warping in the structural dynamic response is also evaluated by using the numerical model. It is observed that both the displacements and accelerations in the 𝑧 direction are lower than when the warping effect is considered.

The results obtained by the numerical and the analytical models are in good agreement. Both deflections and accelerations given by the analytical model are larger, which is related with the fact of having been neglected the cross-sectional rotary inertia.

(a) – Velocity: 80 ms-1

𝑧-d

isp

lace

men

t (m

)

(b) – Velocity: 120 ms-1

𝑧-d

isp

lace

men

t (m

)

𝑥 (m)

∎ Numerical Model ∎ Analytical Model ∎ Shell FEM Model

Figure 5.2 – dynamic influence lines of the 𝑧-displacement at the track’s midpoint of the central span.

-1.0E-03

0.0E+00

1.0E-03

2.0E-03

3.0E-03

0 50 100 150 200 250 300 350 400 450 500

-3.0E-03

0.0E+00

3.0E-03

6.0E-03

0 50 100 150 200 250 300 350 400 450 500

(a) – Velocity: 80 ms-1 𝑧-

acce

lera

tio

n (

ms-

2)

(b) – Velocity: 120 ms-1

𝑧-ac

cele

rati

on

(m

s-2)

𝑥 (m)

∎ Numerical Model ∎ Analytical Model ∎ Shell FEM Model

Figure 5.3 – dynamic influence lines of the 𝑧-acceleration at the track’s midpoint of the central span.

Velocity: 120 ms-1

𝑧-d

isp

lace

men

t (m

)

𝑧-ac

cele

rati

on

(m

s-2)

𝑥 (m)

Including the Warping Effect & Neglecting the Mass Effect: ∎ Numerical Model ∎ Analytical Model Including the Warping Effect & Including the Mass Effect: ∎ Numerical Model ∎ Analytical Model

Neglecting the Warping Effect & Including the Mass Effect: ∎ Numerical Model

Figure 5.4 – influence of the mass effect on the dynamic influence lines of the 𝑧-displacement at the track’s midpoint of the central span.

-8.0E-01

-4.0E-01

0.0E+00

4.0E-01

8.0E-01

0 50 100 150 200 250 300 350 400 450 500

-3.0E+00

-1.5E+00

0.0E+00

1.5E+00

3.0E+00

0 50 100 150 200 250 300 350 400 450 500

-3.8E-03

-1.9E-03

0.0E+00

1.9E-03

3.8E-03

5.7E-03

0 50 100 150 200 250 300 350 400 450 500

-3.0E+00

-1.5E+00

0.0E+00

1.5E+00

3.0E+00

0 50 100 150 200 250 300 350 400 450 500

6. Conclusion

The main objective of the present work was to provide two beams models capable of performing dynamic analyses of railway bridges, being considered the warping displacements and the mass effects of the moving loads.

In the previous sections, a numerical and an analytical beam models were developed and applied to the dynamic analysis of an open cross section girder. The torsional response was shown to be significantly influenced by the warping deformations.

The developed beam models gave good results, being in good agreement with the shell finite element model implemented in a commercial software. Thus, the developed models perform efficient dynamic analyses that consider the warping effect, which may be very useful at early design stages.

The presented models could serve as a starting point for further in-depth study of the subject and may be considered as a first step tending to more complete formulations accounting for other aspects that have not been considered.

References

1. Banerjee, J.R., Guo, S. and Howson, W.P., Exact dynamic stiffness matrix of a bending-torsion coupled beam including warping, Computers & Structures, 59(4), 613-621, 1996.

2. Bishop, R.E.D., Cannon, S.M. and Miao, S., On coupled bending and torsional vibration of uniform beams, Journal of Sound and Vibration, 131(3), 457-464, 1989.

3. Chan, T.H.T. and Ashebo, D.B., Theoretical study of moving force identification on continuous bridges, Journal of Sound and Vibration, 295, 870-883, 2006.

4. EN 1991-2: Actions on structures – Part 2: Traffic loads on bridges, CEN – European Commitee for Standardization.

5. Fish, J. and Belytschko, T., A First Course in Finite Elements, John Wiley & Sons, England, 2007. 6. Lisi, Diego, A Beam Finite Element Model Including Warping – Application to the Dynamic and Static Analysis of Bridge

Decks, IST, Universidade Técnica de Lisboa, 2011. 7. Michaltsos, G.T., Sarantithou, E. and Sophianopoulos, D.S., Flexural-torsional vibration of simply supported open

cross-section steel beams under moving loads, Journal of Sound and Vibration, 280, 479-494, 2005. 8. Serra, J., Dynamic analysis of bridge girders subjected to moving loads: numerical and analytical beam models, IST, Universidade

de Lisboa, 2014. 9. Sophianopoulos, D.S. and Michaltsos, G.T., Combined torsional-lateral vibrations of beams under vehicular

loading – I: formulation and solution techniques, Automatic Control & Robotics, 2, 877-886, 1999. 10. Stayridis, L.T. and Michaltsos, G.T., Eigenfrequency analysis of thin-walled girders curved in plan, Journal of Sound

and Vibration, 227, 383-396, 1999. 11. Wunderlich, W. and Pilkey, W.D., Mechanics of Structures – Variational and Computational Methods, Second Edition,

CRC Press, 2003.