Analysis of a dynamically loaded beam bridge in torsion

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Analysis of a dynamically loaded beam bridge in torsion

Pia Hannewald

TRITA-BKN. Master Thesis 236, Structural Design and Bridges 2006 ISSN 1103-4297

ISRN KTH/BKN/EX--236--SE

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Preface

This thesis was carried out at the Division of Structural Design and Bridges at the Department of Civil and Architectural Engineering at the Royal Institute of Technology (KTH). The supervisor has been Professor Håkan Sundquist, who also presented the idea for the work done in this thesis.

I would like to thank my supervisor Professor Håkan Sundquist and co-supervisor PhD student Johan Wiberg for making it possible for me to write this thesis and thereby take part in the research project at the Årsta Bridge. Thank You very much for Your help and guidance throughout the work and all the time You invested.

Stockholm, May 2006 Pia Hannewald

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Abstract

In this thesis, a simple model was developed to analyse the natural torsional frequencies of a concrete box-girder bridge with a simply symmetric, non-uniform cross-section.

The superstructure of the bridge is a slender pre-stressed construction with thin walls.

Torsional warping can have a significant influence on the behaviour of this kind of box- girder bridges. Therefore, both St.Venant’s torsional stiffness and warping stiffness were considered in the calculation and the influence of warping on the natural frequencies was examined. Vlasov’s theory for thin-walled beams was assumed applicable and used for the calculation of the frequencies. The influence of the non-coincidence of the shear centre and the centre of gravity, as well as the influence of cross-section deformations, were assumed negligible.

The calculation of the natural frequencies was done with the finite difference method of central differences. The differential equation for torsion was expressed by finite difference formulations and applied for a single bridge span with boundary conditions determined mainly by the character of the supports. Thereby, matrices representing the span’s torsional stiffness were derived. The eigenvalues and –vectors of these matrices, i.e. the bridge span’s natural frequencies and mode shapes, were calculated with the MATLAB^® software. Furthermore, an attempt was made to take the behaviour of the piers into account and thereby develop a model to calculate the frequencies of the whole bridge.

The calculated frequencies were compared with frequencies derived from the analysis of in-situ measurements as the bridge is equipped with accelerometers and other instruments for long term monitoring. Accelerometer signals from one main span during train passage were used to calculate the natural frequencies.

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Zusammenfassung

In dieser Arbeit wurde ein einfaches Model für die Berechnung der Torsionseigen- frequenzen einer Hohlkastenbrücke aus Beton mit einfachsymmetrischem, veränderlichem Querschnitt entwickelt.

Der Brückenoberbau ist eine schlanke, dünnwandige Spannbeton Konstruktion. Wölbkraft- torsion kann auf das Torsionsverhalten solcher Brücken einen großen Einfluss haben, weshalb bei der Berechnung der Eigenfrequenzen sowohl die St. Venant’sche Torsions- steifigkeit als auch die Wölbsteifigkeit berücksichtigt wurde. Der Einfluss der Wölbsteifig- keit auf die Torsionseigenfrequenzen dieser Brücke wurde dabei untersucht. Die Berechnungen der Torsionseigenfrequenzen basieren auf Wlassows Theorie für dünn- wandige elastische Stäbe, deren Gültigkeit für diesen Querschnitt vorausgesetzt wurde.

Des Weiteren wurde die ungekoppelte Differentialgleichung der Torsion verwendet unter der Vorraussetzung, dass es nur einen geringen Einfluss auf die Eigenfrequenzen hat, dass der Schubmittelpunkt nicht gleich dem Schwerpunkt ist. Eventuelle Querschnitts- verformungen wurden ebenfalls nicht berücksichtigt.

Für die Berechnung wurde die Finite Differenzen Methode mit zentralen Differenzen verwendet. Dazu wurde die Differentialgleichung der Torsion mit Hilfe von zentralen Differenzen formuliert und auf die Geometrie des Brückenträgers unter Berücksichtigung der durch die Lagerung vorgegebenen Randbedingungen angewendet. Die Eigenwerte und –vektoren der mit dieser Methode erhaltenen Torsionssteifigkeitsmatrix, d.h. die Eigen- frequenzen und –formen des Brückenträgers, wurden mit der Computersoftware MATLAB^® berechnet. Es wurde ausserdem versucht, nicht nur die Frequenzen eines einzelnen Feldes zu berechnen, sondern auch die Schwingungen der Brückenpfeiler zu erfassen und somit ein Model für die Berechnung der Frequenzen der gesamten Brücke zu entwickeln.

Die berechneten Frequenzen wurden mit gemessenen Frequenzen verglichen. Diese Frequenzen wurden durch die Analyse von in-situ Messungen während Zugüberfahrten gewonnen. Die Brücke ist unter anderem mit Beschleunigungsmessern instrumentiert, um Langzeitmessungen durchzuführen.

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1 Introduction

1.1 Introduction

In this thesis, the torsional vibration of the New Årsta Railway Bridge was analysed.

Natural torsional frequencies were calculated with a finite difference method and compared with measurements. For both the finite difference calculation of the frequencies and the signal analysis the MATLAB^® Software was used.

The bridge, located south of Stockholm, was completed and inaugurated in the summer of 2005 after a construction time of five years. It is made of pre-stressed concrete and has a very slender superstructure. The superstructure is a non-uniform box-girder with varying height and diaphragms over the piers. In the calculation of the natural frequencies, the contribution of Saint Venant torsional stiffness and warping stiffness were considered.

Two bridge spans were instrumented with amongst others accelerometers to monitor the bridge during construction and in service. Accelerometer signals were analysed to determine the bridge’s natural torsional frequencies. Those were compared with the calculated ones to find out whether the finite difference model satisfactorily describes the actual behaviour.

An overview of the bridge location and shape is given in Figure 1.1. An existing bridge next to it is indicated, as well as the island Årsta Holmar and the lake Årstaviken crossed by the bridge.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( )

( ) ( )

( )

ÅRSTA HOLMAR

ÅRSTAVIKEN TANTO

ÅRSTA

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

NL P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 SL

( )

P5 P6 P7 P8 P9 P10 SL

P3 P4 P2 P1

NL EXISTING BRIDGE

Figure 1.1 Overview of the bridge location; from [21].

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1 1.2 Bridge Information

The new Årsta Bridge is a railway bridge with two tracks, located south of Stockholm between Södermalm and Årsta. It spans over a body of water called Årstaviken and a small island called Årstaholmen. Next to this new bridge is an older railway bridge constructed in the 1920s. The new bridge is part of a project to increase the train traffic capacity in the southern direction between Stockholm south and a new station called Årstaberg. The bridge has a pedestrian and cycle track on the west side and a service road on the east side.

Banverket, the Swedish National Railway Administration, initiated an architectural competition for the design of the bridge, which was won by a proposal made by Foster and Partners in collaboration with Ove Aarup A/S. The final design was made by COWI who was responsible for the structural design. The aim of the design was to make the new bridge fit the old bridge and the surrounding area. The superstructure follows a slight waveform in both the longitudinal and transversal direction and the piers are elliptical. In reminiscence of the typical falu-red Swedish wooden houses it is made of red concrete.

Construction of the bridge began in the summer of 2000 and was completed in the summer of 2005.

The bridge is a slender pre-stressed concrete construction, which, in addition to the pre- stressed tendons, contains a lot of reinforcement. The superstructure is a wave-formed box- girder with cross-sections as shown in Figure 1.2. The bridge has ten piers and nine spans with a length of 78 m each. The span between the southern abutment and the following pier is 65 m long, the one at the northern abutment 48.15 m, resulting in a total length of 833 m. The width of each cross section is 19.5 m and the height varies between 5.325 m over the piers and 3.425 m in the middle of the span. The navigational channel has a free height of 26 m.

Cross sections of the superstructure were instrumented by the Division of Structural Design and Bridges at the Royal Institute of Technology (KTH) to monitor the static and dynamic bridge behaviour during construction and the first ten years in service.

1.3 Aims of the Study

In this study, the natural torsional frequencies of a box-girder bridge were calculated with a finite difference method. Compared to a finite element model, the modelling procedure for the finite difference method is rather simple and fast. The aim of the study is to examine whether this model is able to produce satisfactory results. Therefore, the computed frequencies were compared with in-situ measurements. Furthermore, the influence of torsional warping on the natural frequencies was examined.

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Figure 1.2 Cross-sections of the bridge; from [21]. The upper picture shows the cross section at midspan, the lower picture the cross-section over the piers.

1.4 Thesis Layout

Below, each chapter is presented with a short description to give an overview of the thesis.

Chapter 2 contains a literature study and general information about the topics dealt with in this thesis. It includes information about the theory of torsion and the behaviour of concrete box-girder bridges subjected to torsion. Information about warping of concrete box-girders and the influence of warping on the natural torsional frequencies of beams is included as well as information on the application of the finite difference method for similar problems.

In chapter 3 methods and formulas used for the calculations are explained. The finite difference method of centred differences is introduced and the formulations for the derivatives up to the fourth order are given. The differential equation for torsion of a beam and its notation using finite difference formulations are presented. The composition of the matrices, which are obtained by applying the finite difference equation for every calculation point and which are needed for the calculation with MATLAB^®, is presented as

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well. Furthermore, the convergence of the results of finite difference calculations towards the exact analytical result is examined.

Chapter 4 contains a short introduction to signal analysis. Fourier Series, which form the mathematical basis for the analysis, are introduced as well as concepts - windowing and filtering - that can be used in the signal analysis process. Moreover, the instrumentation of the bridge with accelerometers is described and the location of the instrumented cross sections as well as the location of the accelerometers in those sections is presented.

Chapter 5 contains the results of the finite difference calculations and the signal analysis.

The results of the finite difference calculations with a single span bridge model and with a multi span bridge model are presented. The graphs with the cross section properties used for the calculations are included in this chapter as well. The effect of the warping stiffness on the natural frequencies is examined with the single span model. The chapter also includes frequency spectrums from the signal analysis and the comparison of the frequencies in those signals with those from the finite difference calculations.

Chapter 6 contains conclusions from the work done in this thesis.

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2 Literature Study

2.1 Introduction

A literature study was conducted to study the theory of torsion and the finite difference method. Therefore, literature covering the following topics was searched:

• Torsion: Differential equation for torsion with regard to St. Venant torsion and warping, explanation of the theory and conditions for its application

• Concrete box-girders under torsion: studies on torsional behaviour of box-girders, application of torsional theory and significance of warping

• Influence of warping on vibrations: Significance of considering warping stiffness for the calculation of natural frequencies

• The finite difference method: Derivation of the finite difference formulations and statements about their accuracy, examples for the use of this method for similar problems

2.2 Torsion

The theory of torsion is mainly based on the works done by B. de Saint Venant and V. S.

Vlasov. St. Venant’s theory is usually applied when the cross-section is non-deformable out of its plain or those deformations are very small. In other cases the more general theory according to Vlasov [22] needs to be applied. Vlasov derives formulations for thin-walled elastic beams with regard to deformations out of the plain, i.e. warping, based on some assumptions regarding the beams geometry and material properties. To develop thin- walled behaviour, the beams length should be ten times greater than the width, and the width ten times greater than the wall thicknesses. Shear deformations are neglected in this theory and the twist is assumed to be small, so that the deformations can be linearised. The material behaviour is assumed linear-elastic and the shear stresses constant over the wall thickness.

Based on these assumptions a system of four differential equations, representing four equilibrium conditions, is derived by using geometrical proportions and equilibrium conditions. The system of equations is derived for a general case, where shear centre and centre of gravity need not be the same. If they are on the same axis, this system resolves into four independent equations, one for axial loading, two for bending and one for torsion.

In many cases, those four equations are independent of each other, or the interaction is assumed negligible. This was also assumed in this study and the uncoupled differential equation for torsion was applied.

2.3 Concrete Box-Girders under Torsion

Box-girders provide a very efficient way of building bridges with low weight and high torsional stiffness. The closed cell of a box-girder can take up and distribute the stresses

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resulting from eccentric loading. To spare weight, even concrete box-girder bridges have thin walls compared to the outer dimensions of the cross-section. Nevertheless, compared to a girder made of steel, concrete box-girders do not evidently appear to be thin-walled beams. This, and the fact that closed cells do not warp as much as open sections, often leads to the assumption that warping deformations are negligible and torsion can be treated according to St. Venant’s theory. The application of Vlasov’s theory requires not only thin walls but also linear-elastic material behaviour, which is only approximately fulfilled by concrete for rather low stresses. For these reasons, literature about the behaviour of concrete beams and bridges under torsion was searched.

Maisel et al. [9] conducted studies on concrete box-girders to examine amongst others torsional behaviour. Single cell rectangular box-girders with different geometrical proportions and loaded with an eccentric point-load at midspan were examined. The maximum longitudinal stress at the bottom line was increased through warping and distortion of the girder.

Waldron et al. [19] conducted studies specifically on the influence of warping on concrete box girders. Rectangular single-cell box-girders with different geometries, i.e. different ratios between wall-thickness and width, were examined in the study. The relation of longitudinal warping stresses to longitudinal bending stresses under an equally distributed eccentric load, instead of a point load as in Maisel’s study, was calculated, for a single span box-girder with diaphragms over the supports. The maximum warping stress at midspan of a single span bridge was 5 % for the girder geometry analysed in this example, instead of 29 % obtained through a point load. The maximum warping stress in a multi- span bridge was 6 % at midspan, but up to 22 % over the supports in the analysed example.

The ratio of warping to bending stresses is influenced by girder geometry and loading and the rather simple geometrical criteria given by Vlasov were found to be insufficient to determine the warping influence. In general it can be stated that, depending on the geometry, torsional warping can have a significant influence on concrete box girder bridges.

Mehlhorn, Rützel examined warping torsion in thin-walled reinforced concrete beams with more accurate assumptions for the constitutive equations [12]. Because of the material properties of concrete, the deformation is underestimated when applying Hooke’s Law, especially after the rated strength value of concrete is reached. The stiffness depends on the loading and a low increase of the stress resultants causes a high decrease in stiffness. When a material has a non-linear behaviour, the resistance as well as the shear centre therefore do not only depend on geometry, but also on the stress resultants.

In another study by Waldron et al. [20], attention was given to deformation of thin-walled concrete box-girders and the stresses arising through that. Those stresses were significant and higher than the warping stresses in the analysed girders. Depending on the girder geometry, the influence of deformations is therefore considerable.

2.4 Influence of Warping on Vibrations

Studies on the effect of warping on the torsional vibrations of thin-walled elastic beams show that the natural frequencies increase when warping stiffness is considered. Li et al.

[8] examined the effect of warping on torsional vibration of members with open cross sections and coinciding centre of gravity and shear centre. The effect of considering

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warping in calculating the natural frequencies of beams with constant cross-sections and varying boundary conditions was investigated by setting the frequencies computed with and without regard to warping in relation to each other. The ratio of the jth angular frequencies of a simply supported beam calculated with considering warping stiffness, i.e.

ωj, and without considering warping stiffness, i.e. ϖ_j, is then

π ² 1

W 2 2

⎟ +

⎠

⎜ ⎞

⎝

= ⎛

l j C C

j j

ϖ ω

where CW is the warping stiffness, C the torsional stiffness and l the length of the beam.

The influence of warping decreases with increasing l and increases with increasing j, i.e.

becomes significant for short beams and high frequencies. The same result was derived in a study by Subrahmanyam, Kaza [18], where pretwisted cantilever beams were examined and the natural torsional frequencies with regard to warping stiffness were always higher than those calculated without it. Warping is especially important for beams with low aspect ratios, i.e. length/ width, and low thickness ratios, i.e. wall thickness/ width, as well as for higher modes.

The cross-section geometry and boundary conditions influence the significance of warping as well. In a study by Jun et al. [5] the natural torsional frequencies of axially loaded monosymmetrical Euler-Bernoulli beams with two different cross-section geometries were compared. Warping was significant for all frequencies of a beam with monosymmetrical channel cross-section. In contrast to that, the first three of the five computed frequencies of a beam with semi-circular cross-section were not significantly affected by warping but the next two. When this beam had clamped supports on both ends, all frequencies were influenced by warping as well.

2.5 Finite Difference Method

Finite difference formulations for ordinary differential equations can found in manuals for numerical methods with computer programmes [10][15]. Derivatives of a function may be expressed by finite difference formulations of different accuracy. To express the accuracy of a formulation, error terms obtained by Taylor series expansion can be used. Those terms depend on the grid spacing and therefore become smaller with a finer grid. The centred difference approach mainly used in this thesis depends on the squared spacing, but there are also more accurate formulations.

To calculate the eigenfrequencies of a beam, its differential equation is expressed by finite difference formulations and the eigenvalues are computed. Petersen [14] gives examples for the use of this method for eigenfrequency calculations of beams with constant cross section. Those calculations are rather simple, but accurate even with regard to elastic supports and only few points for the numerical calculation. A collection of examples for the expression of the differential equations of beams with variable cross-sections and varying boundary conditions is found in [23]. The Årsta Bridge is comparable with the continuous beam in this paper or, if only one span is considered, with a simple beam and the necessary boundary conditions.

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A finite difference analysis can be used to calculate the natural frequencies of non-uniform Euler-Bernoulli or cantilever beams with good results, as the following studies show. In a study where the natural frequencies of non-uniform Euler-Bernoulli beams were calculated, Richardson extrapolation was used to improve the results [13]. The relative error increased for higher modes but was in general very small. Another way of improving the results was chosen in another study, where finite difference formulations with better accuracy, i.e. error terms depending on λ⁴ instead of λ², with λ being the grid spacing, were used [17]. Nevertheless, the results obtained with lesser accuracy were also satisfactory. It can generally be stated, that the accuracy of the method increases with refined grid spacing, resulting in more points for the numerical calculation, and decreases for higher modes.

As a part of a doctoral thesis at KTH, the response of cable-stayed and suspension bridges to vehicles was studied using a finite difference method [6]. Even for this rather complex bridge model, taking into account cables and vehicles, the results were satisfying and considered accurate enough for preliminary studies. However, for a more detailed analysis, taking into account exact cable behaviour, for instance, this method was found to be unsuitable

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3 The Finite Difference Method for Torsional Problems

3.1 Introduction

The finite difference method is based on the numerical approximation for the derivative )

(x

f ′ of a function f(x):

λ λ

λ

) ( ) lim (

) (

' 0

x f x

x f

f = + −

→ (3.1)

An approximation of the derivative is therefore given by

λ λ) ( ) ) (

(

' x f x f x

f ≈ + − . (3.2)

Equation (3.2) is a so called forward difference approximation. Hereby, the derivative of the function is expressed through the derivative of a secant through the two points f(x) and f(x+λ). If instead of f(x) another value to the left is taken into account, the derivative can be approximated by a centred difference

) , 2 (

) ( ) ) (

(

' λ

λ

λ f x E f

x x f

f = + − − + (3.3)

where E(f ,λ) is the error. The centred difference approximates the first derivative more exact than a one sided difference. Using Taylor series expansion the error in this centred difference is dependent on λ², whereas it is dependent on λ in the one sided approach in (3.2). Error terms will be addressed later on.

3.2 Derivations with Finite Differences

As the differential equation for torsion is a differential equation of the fourth order, the derivatives up to the fourth order of a function f(x) are presented in this chapter using the finite difference method of central differences. The x-axis is therefore discretised by a uni- form grid with spacing λ. When the differential equation of a beam is to be expressed with this method, the x-axis can be placed in the centre of gravity of the beam, which is then divided into n sections with the length λ = l/n according to Figure 3.1.

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Figure 3.1Beam with n sections.

The derivatives of the function f(x) from the first to the fourth order at a point j on the grid are:

) 2 (

1 d d

1

1 +

− +

−

≈ f_j f_j

x f

λ

) 2

²( 1

² d

1

1 +

− − +

≈ f_j f_j f_j x

f λ

) 2

2

³( 2

1

³ d

2 1 1

2 − + +

− + − +

−

≈ f_j f_j f_j f_j

x f

λ

) 4

6 4

1 ( d

d

2 1 1

4 2 4 4

+ +

−

− − + − +

≈ f_j f_j f_j f_j f_j

x f

λ ^(3.4^a-d)

The error terms of the derivatives expressed with the formulas in equations (3.4 a-d) depend on λ². The accuracy of the finite difference expressions can be increased with other formulations. With an error term depending on λ⁴ the derivatives are:

(

2 8 1 8 1 2

)

12 1 d d

+ +

−

− − + −

≈ f_j f_j f_j f_j

x f

λ

(

2 1 1 2

)

2 2

2

16 30 12 16

1 d

d

+ +

−

− + − + −

−

≈ f_j f_j f_j f_j f_j

x f

λ

(

3 2 1 1 2 3

)

3 3 3

8 13 13

8 8 1 d

d

+ + +

−

− − + − + −

≈ f_j f_j f_j f_j f_j f_j

x f

λ

(

3 2 1 1 2 3

)

4 4 4

12 39

56 39

6 12 1 d

d

+ + +

−

− + − + − + −

−

≈ f_j f_j f_j f_j f_j f_j f_j

x f

λ ^(3.5^a-d)

0 1 2 3 …. j-1 j j+1 n-2 n-1 n

λ l

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In the following paragraphs, the derivation of the differential equation in finite difference notation and composition of the matrices is presented for equations (3.4 a-d) only. Matrices including the higher order difference equations (3.5 a-d) are obtained the same way.

3.3 Error Terms

The central difference formulas given in (3.4a-d) are approximations of the derivatives.

Taylor series expansion can be used to approximate the values for f(x+λ) and f(x−λ) and thereby describe the difference between the exact derivative and the numerical approach.

The functions f(x+λ) and f(x−λ) expressed by Taylor series expansion are

+K + ′′′

+ ′′

+ ′

=

+ 6

) ( 2

) ) (

( ) ( ) (

3

2 λ

λ λ

λ f x f x ^f ^x ^f ^x

x f

+K

− ′′′

+ ′′

− ′

=

− 6

) ( 2

) ) (

( ) ( ) (

3

2 λ

λ λ

λ f x f x ^f ^x ^f ^x

x

f (3.6)

These terms are subtracted and the Taylor series is truncated at the third derivative.

Taylor’s theorem states that there is a value c with |c – x| < λ so that

6 ) ( ) 2

( 2 ) ( ) (

λ3

λ λ

λ f x f x ^f ^c

x

f ′′′

′ +

=

−

+ (3.7)

This equation is solved for f ′

( )

x , which yields 6

) ( 2

) ( ) ) (

(

λ2

λ

λ f x f c

x x f

f′ = + − − − ′′′ (3.8)

The first part of the formula on the right side is the finite difference formula for the first derivative as in equation (3.4a) and the second part is the truncation error. As the truncation error depends on λ² this finite difference formula is called a formula of order O(λ²). The error terms for the difference formulas (3.5 a-d) are derived in a similar way.

In numerical calculations, the round-off error, which is depending on the number of decimal places, might also become considerable [10]. However, with the precision of a computer program this should not be a problem, as the number of decimal places is high.

3.4 Differential Equation for Torsion

The partial differential equation for torsion for a beam with both variable warping stiffness and St. Venant torsional stiffness along its axis and coinciding centre of gravity and shear centre is:

) ,

² ( ) ² ( )

² ( ) ²

² (

²

R

W m x t

x t x J

x x C x x

x C =

∂ + ∂

⎟⎠

⎜ ⎞

⎝

⎛

∂

− ∂

⎟⎠

⎜ ⎞

⎝

⎛

∂

∂ φ φ φ

(3.9)

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With:

CW = ECSC = Warping stiffness kNm⁴

E = Modulus of elasticity MPa

CSC = Warping constant m⁶

C = GIR = Torsional stiffness kNm²

G = Shear modulus MPa

IR = Torsional moment of inertia m⁴

J = Mass moment of inertia per unit length kgm²/m

= ρ⋅

(

I_y +I_z

)

=ρ⋅IP

ρ = Mass density kg/m³

Iy, Iz = Moments of inertia relating to axis y, z m⁴

φ = Torsional angle rad

mR = External twisting moment per unit length kNm/m

Equation (3.9) is a partial differential equation including derivatives with respect to space, i.e. x, and time, i.e. t. In the function φ( tx, ) the variables can be separated by making the following harmonic approach

x t

t T x t

x Φ Φ ^ω

φ( , )= ( )⋅ ()= ( )eⁱ (3.10)

where ω is the angular frequency and i= −1 . This changes the partial differential equation into an ordinary differential equation and a harmonic time function:

) , ( e

) d (

)d d (

d d

)d d (

d

R i

2 2

2 2 W

2

t x m x

x J x x C x x

x C

t =

⎥⎦

⎢ ⎤

⎣

⎡ ⎟−

⎠

⎜ ⎞

⎝

− ⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ Φ Φ ω Φ _ω _(3.11)

Expressed by the finite difference notations of equations (3.4 a-d) and applied for a grid point j the partial differential equation (3.11) becomes:

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{ [ ( )

( )

( ) ]

[ ( )

( )

( ) ]

}

e ()

25 1 . 0 25

. 0 2

25 . 0 25

. 0

5 1 . 0 5

. 0

6 2

2 10

2

2 6

5 . 0 5

. 0

, R i

2

1 2 1

1

1 1

1

1 4 , W ,

W 1 , W 2

, W 1 , W 1

1 , W ,

W 1

, W

1 , W ,

W 1

1 , W ,

W 1 , W 2

t m J

C C

C

C C

C C C

C C

C

C C

C

j t j j

j j

j

j j

j

j j

=

−

+ +

− +

−

+ +

− +

+

− +

+

− +

+

−

+

− +

+

−

+

−

+

−

ω ω

Φ Φ λ Φ

Φ λ

Φ Φ Φ Φ Φ

(3.12)

This equation is applied for all j = 0, 1…n, which results in a system of equations that can be written in matrix form

[ ] [ ] { }

(

A − B − J ω²

) { }

Φ eⁱ^ω^t =

{

m_R

( )

t

}

(3.13)

The matrices [A] and [B] are sparse band matrices containing warping and St.Venant torsional stiffness, respectively. {J} is a vector containing the mass moment of inertia for each point j; {Φ} is the vector with the torsional angles, and the external twisting moments at each point are contained in the vector

{

m_R

( )

t

}

.

To calculate the natural frequencies, the external twisting moment in equation (3.13) must be set to zero. As eⁱ^ω^t ≠0 for all ω and t, the equation that needs to be solved then is

[ ] [ ]

(

A′ − B′ −ω²

) { }

Φ =0 (3.14)

where

[ ]

A′ and

[ ]

B′ are the matrices [A] and [B] from (3.13) where each row j is divided by Jj. Equation (3.14) is an eigenvalue problem with the eigenvalues ωj² and corresponding eigenvectors {Φ}. The square-roots of the eigenvalues are the natural circular frequencies.

3.5 Equation Matrices

The two band matrices

[ ]

A′ and

[ ]

B′ are composed using equation (3.12) and (3.14). For a beam divided into n sections, matrix

[ ]

A′ is a (n – 1)×(n – 1) band matrix, consisting of the part the differential equation that contains the warping stiffness. Each row contains five elements, which are the five expressions in brackets with the warping constants in equation (3.12) divided by Jj. The indices of the matrix elements show which angle they are multiplied by. The first index specifies the row and the second the column or angle. In row j element aj,-2 is then multiplied by angle Φj,−2, element aj,-1 by Φj,−1 and element aj, which is placed on the main diagonal, is multiplied by angle Φj.

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1 4 , W ,

W 1 , W 2

,

1 2

1

jλ

j j

j

j C C C J

a ⎟

⎠

⎜ ⎞

⎝

⎛ + −

= ₋ ₊

−

(

W, W, 1

)

₄

1 ,

2 1

6 ^j ^j _jλ

j C C J

a ₋ = − + ₊

(

2 _W_, ₁ 10 _W_, 2 _W_, ₁

)

¹ ₄

jλ

j j

j

j C C C J

a = − ₋ + − ₊ (3.15 a-e)

(

W, 1 W,

)

₄

1 ,

6 1

2 ^j ^j _jλ

j C C J

a = ₋ −

1 4 , W ,

W 1 , W 2

,

1 2

1

jλ

j j

j

j C C C J

a ⎟

⎠

⎜ ⎞

⎝

⎛− + +

= ₋ ₊

[ ]

B′ is a (n – 1)×(n – 1) band matrix with three elements in each row containing the St.

Venant torsional stiffness. Those elements are also taken from (3.12):

1 2 1

1 ,

1 4

1

jλ

j j j

j C C C J

b ⎟

⎠

⎜ ⎞

⎝

⎛ + −

= ₋ ₊

−

(

²

)

¹_λ₂

j j

j C J

b = − (3.16 a-c)

1 2 1

1 ,

1 4

1

jλ

j j j

j C C C J

b ⎟

⎠

⎜ ⎞

⎝

⎛− + +

= ₋ ₊

The matrices have the size (n – 1)×(n – 1) because of the first boundary condition. If equation (3.12) is applied for all points 0, 1, 2…n it yields (n + 1)×(n + 1) matrices. As Φ0

and Φn are the angles at the supports of the beam and therefore known to be zero, the first and the last column would be multiplied by zero and are not taken into account. The first and last row of the (n + 1)×(n + 1) matrices are crossed out as well, to obtain square matrices of size (n – 1)×(n – 1).

3.5.1 Matrices for Single Span Model

Two different boundary conditions were considered in the one span model to test which one describes the actual support condition best. The support of the piers itself was assumed to resemble a fork bearing, where the torsional angle and its second derivative are zero, i.e.

warping is not constrained. The boundary conditions are then:

• Φ(x=0)=Φ(x=l)=0

• 0

² d

) (

² d

) 0 (

²

d = = = =

x l x x

x Φ

Φ _(3.17_a-b)

Written in the finite difference notation these conditions become:

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• Φ₀ =Φ_n =0

• ( ) 0

² ) 1 2

²( 1

1 1 1

0

1− + = ₋ + =

− Φ Φ

Φ λ Φ

λ Φ ^→Φ₋¹ =−Φ¹

0 )

²( ) 1 2

²( 1

1 1 1

1− + ₊ = ₋ + ₊ =

− n n n n

n Φ Φ

Φ λ Φ

λ Φ ^→Φⁿ⁺¹ ⁼⁻Φⁿ⁻¹ (3.18 a-c)

It was assumed to be another possibility that the support is rigid and warping constrained because of the stiffness contribution of the adjacent spans. In this case, the torsional angle and the first derivative of the angle are zero.

• Φ₀ =Φ_n =0

•

( ) ( )

₀

d d d

0

d = = = =

x l x x

x Φ

Φ _(3.19_a-b)

Written in the finite difference notation these two conditions become:

• Φ₀ =Φ_n =0

•

( )

0

2 1

1

1+ =

−Φ₋ Φ

λ ^→Φ₋¹ =Φ¹

( )

0

2 1

1

1+ =

−Φ_n₋ Φ_n₊

λ ^→Φⁿ⁺¹ ⁼Φⁿ⁻¹ (3.20 a-c)

The matrices

[ ]

A′ and

[ ]

B′ are composed according to (3.14) with equations (3.15 a-e) and (3.16 a-c). When the equations are applied for the points 1 and n – 1 the boundary condi- tions have to be considered. Considering the boundary conditions (3.17) then yields the equations for j = 1

(

1

)

1 1 1,1 2 1,2 3

(

1 1 1,1 2

)

2 ,

1 Φ aΦ a Φ a Φ bΦ b Φ

a ₋ − + + + − + (3.21)

and for j = n – 1:

(

1

) (

1, 1 2 1 1

)

2 , 1 1 1 2 1 , 1 3 2 ,

1− − − − − − − − − − − − − −

− _n + _n _n + _n _n + _n − _n − _n _n + _n _n

n a a a b b

a Φ Φ Φ Φ Φ Φ (3.22)

Considering (3.21) and (3.22), matrices

[ ]

A′ and

[ ]

B′ are the following:

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⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

′ =

−

2 , 1 1 1 , 1 2 , 1

1 , 2 2

1 , 2 2 , 2

2 , 3 1 , 3 3 1 , 3 2 , 3

2 , 2 1 , 2 2 1

, 2

2 , 1 1 , 1 2 , 1 1

n n n

n

n n

a a a

a

a a

a a a

a a a a

A O O (3.23)

[ ]

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

′ =

−

1 1 , 1

1 , 2 2 1 , 2 1 , 3 3 1 , 3

1 , 2 2 1 , 2

1 , 1 1

n n

n n n

b b

b b b

b b

B O O (3.24)

When instead the boundary conditions in equations (3.19) are applied, matrix

[ ]

A′ is changed in the first and last row. By contrast, matrix

[ ]

B′ is the same as in (3.24) as it is only affected by the condition that the angle is zero. Matrix

[ ]

A′ becomes:

[ ]

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

+ +

′ =

−

2 , 1 1 1 , 1 2 , 1

1 , 2 2

1 , 2 2 , 2

2 , 3 1 , 3 3 1 , 3 2 , 3

2 , 2 1 , 2 2 1

, 2

2 , 1 1 , 1 2 , 1 1

n n n

n

n n

a a a

a

a a

a a a

a a a a

A O O (3.25)

3.5.2 Matrices for Multi Span Model

The multi span bridge model consists of several equal spans with the piers modelled as torsional springs. In Figure 3.2 this is illustrated for four spans.

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l l l l

k

^p

k

^p

k

^p

Figure 3.2 Model with four bridge spans.

The piers were considered to be rigidly supported at the foundations and free at the top, i.e.

the superstructure was not assumed stiff enough to prevent horizontal movement. Based on these assumptions the torsional spring constants of the piers are calculated with the following equation:

( )

pier pier

p l

I

k E⋅

= (3.26)

where (E⋅I)pier and lpier are the bending rigidity and length of the pier, respectively. The cross-section properties of the pier and the modulus of elasticity are [21]:

• Moment of inertia Ipier = 43.8186 m⁴

• Pier length lpier = 25 m

• Modulus of elasticity E = 36 000 MN/m²

This resulted in a spring stiffness kp = 63 099 MNm.

Considering springs changes the matrices for the calculation. To regard a spring the beam is divided at the point with the spring and virtual points are inserted as shown in Figure 3.3 where the virtual points are marked with an index v. For the cross section properties this index is not used as the cross section at the virtual points are the same as at the other points.

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h h+1, v h+2, v h-1

h-2

h-2, v h-1, v h, v h+1 h+2

k^p

Figure 3.3 Intersected beam with virtual points.

At the point of intersection the following boundary conditions were applied:

1) constant torsional angle φ_h_,_v =φ_h

2) no break in the beam, i.e. constant first derivative of the torsional angle

φ_h′_,_v =φ_h′

3) constant warping moment C_Wφ_h′′_,_v =C_Wφ_h′′

4) changing twisting moment M_R_,_h −k_pφ_h =M_R_,_h_,_v 5) constant twisting moment per unit length m_R =m_R_,_h_,_v

For condition 5) equation (3.12) was used and the equation for condition 4) was derived out of (3.9):

R 2

3 3 2 W

2

W J dx M

C x C x

C x

x − =

∂

− ∂

∂ + ∂

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

∂

∂ ^φ ^φ ^φ

∫

^ω ^φ ^(3.27)

Analysis of a dynamically loaded beam bridge in torsion