Measurement evaluation and FEM simulation of bridge dynamics

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(2) Measurement Evaluation and FEM Simulation of Bridge Dynamics - A Case Study of a Langer Beam Bridge. by. Andreas Andersson and Richard Malm. January 2004 Technical Reports from Royal Institute of Technology Department of Mechanics SE-100 44 Stockholm, Sweden.

(3) Kungliga Tekniska Högskolan i Stockholm, c Andersson and Malm 2004. Valhallavägen 79,. Stockholm..

(4) Preface The research presented in this thesis was initiated by the structural engineering company Tyréns AB and the Department of Mechanics at the Royal Institute of Technology, KTH. It was carried out at the Department of Mechanics from June to December 2003 under the supervision of Adjunct Prof. Dr. Per-Olof Thomasson. The field measurements of the train induced strain were performed by Stefan Trillkott and Claes Kullberg of the Department of Civil and Architectural Engineering. The field measurements of the train induced acceleration were performed by Kent Lindgren of the Department of Aeronautical and Vehicle Engineering. We would like to thank them for letting us take part in the measurements and for their support in analysing the results. We give our sincere appreciation and gratitude to Adjunct Prof. Dr. Per-Olof Thomasson for introducing us to this project and foremost for his invaluable advice, guidance and support through this thesis. We would especially like to thank and express our deepest thankfulness and admiration to Professor Anders Eriksson for spreading his knowledge and founding our interest in structural mechanics through his courses at KTH. His encouragement and enthusiasm has been a great inspiration for us. A special thanks goes to Doctoral Student Mehdi Bahrekazemi at the Division of Soil- and Rock Mechanics for his help concerning signal analysis and time integration methods. We would also like to thank all the people at the Department of Mechanics that has shown interest in our work and helped us and we would especially like to thank Dr. Jean Marc Battini for always taking time to help us analyse the dynamic effects and Dr. Gunnar Tibert for his valuable support throughout the thesis.. Stockholm, January 2004 Andreas Andersson and Richard Malm. iii.

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(6) Abstract The aim of this thesis is to analyse the effects of train induced vibrations in a steel Langer beam bridge. A case study of a bridge over the river Ljungan in ˚ Ange has been made by analysing measurements and comparing the results with a finite element model in ABAQUS. The critical details of the bridge are the hangers that are connected to the arches and the main beams. A stabilising system has been made in order to reduce the vibrations which would lead to increased life length of the bridge. Initially, the background to this thesis and a description of the studied bridge are presented. An introduction of the theories that has been applied is given and a description of the modelling procedure in ABAQUS is presented. The performed measurements investigated the induced strain and accelerations in the hangers. The natural frequency, the corresponding damping coefficients and the displacement these vibrations leads to has been evaluated. The vibration-induced stresses, which could lead to fatigue, have been evaluated. The measurement was made after the existing stabilising system has been dismantled and this results in that the risk of fatigue is excessive. The results were separated into two parts: train passage and free vibrations. This shows that the free vibrations contribute more and longer life expectancy could be achieved by introducing dampers, to reduce the amplitude of the amplitude of free vibrations. The finite element modelling is divided into four categories: general static analysis, eigenvalue analysis, dynamic analysis and detailed analysis of the turn buckle in the hangers. The deflection of the bridge and the initial stresses due to gravity load were evaluated in the static analysis. The eigenfrequencies were extracted in an eigenvalue analysis, both concerning eigenfrequencies in the hangers as well as global modes of the bridge. The main part of the finite element modelling involves the dynamic simulation of the train passing the bridge. The model shows that the longer hangers vibrate excessively during the train passage because of resonance. An analysis of a model with a stabilising system shows that the vibrations are damped in the direction along the bridge but are instead increased in the perpendicular direction. The results from the model agree with the measured data when dealing with stresses. When comparing the results concerning the displacement of the hangers, accurate filtering must be applied to obtain similar results. Keywords: dynamic, railway, finite element analysis, vibration, measurement, frequency, fatigue. v.

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(8) Contents Preface. iii. Abstract. v. 1 Introduction. 1. 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2. Properties of the Bridge . . . . . . . . . . . . . . . . . . . . . . . . .. 2. 1.3. Aims of the Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 1.4. Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2 Evaluation Methods 2.1. 2.2. 2.3. 7. Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.1.1. Undamped Free Vibration . . . . . . . . . . . . . . . . . . . .. 7. 2.1.2. Viscously Damped Free Vibration . . . . . . . . . . . . . . . .. 9. 2.1.3. Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. 2.1.4. Half-Power (Band-Width) Method . . . . . . . . . . . . . . . 11. 2.1.5. 2D Continuous Beams . . . . . . . . . . . . . . . . . . . . . . 11. 2.1.6. Rayleigh Damping . . . . . . . . . . . . . . . . . . . . . . . . 14. Signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1. Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 16. 2.2.2. Discrete Fourier Transform (DFT). 2.2.3. Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17. 2.2.4. Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17. . . . . . . . . . . . . . . . 16. Time Integration Methods . . . . . . . . . . . . . . . . . . . . . . . . 18. vii.

(9) 2.4. 2.3.1. Numerical Approximation Procedures . . . . . . . . . . . . . . 18. 2.3.2. Newmark Beta Methods . . . . . . . . . . . . . . . . . . . . . 19. 2.3.3. Hilber-Hughes-Taylor Alpha Method . . . . . . . . . . . . . . 20. Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21. 3 Creating Finite Element Models 3.1. Modelling Procedures in ABAQUS/CAE . . . . . . . . . . . . . . . . 25 3.1.1. Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25. 3.1.2. Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26. 3.1.3. Analysis Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 27. 3.1.4. Contact Methods . . . . . . . . . . . . . . . . . . . . . . . . . 29. 3.1.5. Explicit versus Implicit Methods . . . . . . . . . . . . . . . . 30. 3.1.6. Contact Conditions for Train Simulations . . . . . . . . . . . . 30. 4 Measurements 4.1. 4.2. 4.3. 25. 31. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.1.1. Strain Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . 32. 4.1.2. Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . 33. Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2.1. Free Vibration Test . . . . . . . . . . . . . . . . . . . . . . . . 36. 4.2.2. Train Induced Vibration . . . . . . . . . . . . . . . . . . . . . 39. 4.2.3. Structural Damping . . . . . . . . . . . . . . . . . . . . . . . . 43. Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3.1. Errors due to Integration . . . . . . . . . . . . . . . . . . . . . 46. 4.3.2. Correction of Errors . . . . . . . . . . . . . . . . . . . . . . . 47. 4.3.3. Final Displacements . . . . . . . . . . . . . . . . . . . . . . . 48. 4.4. Plane Stress Variation . . . . . . . . . . . . . . . . . . . . . . . . . . 55. 4.5. Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.5.1. Assessing the Risk of Stress Concentrations . . . . . . . . . . 62. 4.5.2. Fatigue Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 65. viii.

(10) 5 Modelling Results 5.1. 67. Modelling Results in ABAQUS . . . . . . . . . . . . . . . . . . . . . 67 5.1.1. The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67. 5.1.2. Model Simplifications . . . . . . . . . . . . . . . . . . . . . . . 71. 5.1.3. Modelling of Rail Vehicle Components . . . . . . . . . . . . . 71. 5.1.4. The Train Model . . . . . . . . . . . . . . . . . . . . . . . . . 73. 5.2. General Static Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 74. 5.3. Eigenvalue Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78. 5.4. Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79. 5.5. 5.4.1. Convergence of the Results . . . . . . . . . . . . . . . . . . . . 80. 5.4.2. Rayleigh Damping . . . . . . . . . . . . . . . . . . . . . . . . 82. 5.4.3. Variation of the Train Parameters . . . . . . . . . . . . . . . . 84. 5.4.4. Deformation of the Hangers . . . . . . . . . . . . . . . . . . . 85. 5.4.5. Stabilisation of the Hangers . . . . . . . . . . . . . . . . . . . 92. 5.4.6. Stress Variation in the Hangers . . . . . . . . . . . . . . . . . 97. Study of the Turn Buckle. . . . . . . . . . . . . . . . . . . . . . . . . 99. 5.5.1. The Static Model . . . . . . . . . . . . . . . . . . . . . . . . . 99. 5.5.2. The Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . 101. 6 Conclusions. 103. 6.1. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103. 6.2. Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103. 6.3. Finite Element Modelling. 6.4. Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105. . . . . . . . . . . . . . . . . . . . . . . . . 104. Bibliography. 107. A Measurement Results. 109. A.1 Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 A.1.1 Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 A.1.2 Stress Variation . . . . . . . . . . . . . . . . . . . . . . . . . . 112. ix.

(11) A.1.3 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . 116 A.1.4 Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 A.1.5 Plane Stress Results . . . . . . . . . . . . . . . . . . . . . . . 122 A.1.6 Fatigue Results . . . . . . . . . . . . . . . . . . . . . . . . . . 124. x.

(12) Chapter 1 Introduction 1.1. Introduction. This thesis deals with the dynamic effects on a Langer beam bridge during train passages. The bridge is located in ˚ Ange municipality in central Sweden. It was constructed in 1959 by request from the Swedish Railway Association (SJ). In 1967 fatigue fractures in the bridge were repaired. In 1984 further improvements had to be made, to stabilise the hangers of the bridge, which were vibrating during train passages. It was still not guaranteed to work as intended, and due to the complexity of the problem, no further confirmation could be made. Measurements of the bridge were performed in June 2003 using accelerometers and strain gauges. Here the new data has been analysed to gain further information about the stresses and motions of the bridge and to draw conclusions concerning the risk of fatigue in the hangers. A finite element approach has also been made, using the commercial software ABAQUS [11]. The intention of the finite element model is to get a more detailed understanding of the behaviour of the bridge and to compare the results with the measured data.. Figure 1.1: Langer beam bridge.. 1.

(13) CHAPTER 1. INTRODUCTION. 1.2. Properties of the Bridge. The bridge is a 45 m long Langer beam bridge, entirely made of steel, as seen in Figure 1.1. It is a one track rail bridge for both passenger and freight train traffic. The bridge is made of two rectangular hollow arches which carry the load of the carriageway through the hangers connected to the arch. The carriageway is made of I-beams along and across the bridge, as seen in Figure 1.3. During a train passage, the load is transferred from the train via the rail to the long beams, over to the cross beams, over to the main beams through the hangers and to the arches. The load is then transferred in the arches down to the supports of the bridge. The hangers are connected to the main beam with a triangular plate as seen in Figure 1.2. This makes the hangers rather clamped in direction parallel to the bridge and more or less pinned in the perpendicular direction. The connection with the hanger and the arch is shown in Figures 1.2 (a) and (c). The hangers are connected to the arch by a steel plate, so that the load can be distributed to the walls of the arch and the hangers themselves are connected to the plate by a screw. This connection can be seen as clamped in both directions. The hangers are pre-stressed using a turn buckle as seen in Figure 1.2. This is probably the most critical area of the bridge, because there will be local stresses in the threads that can lead to fatigue if the amplitudes of the stress cycles are too large. In 1984 a stabilising system were made, consisting of rectangular hollow section (RHS) beams that connect all hangers on each side of the bridge. The intention was to stabilise the hangers with themselves, because during train passages some of the hangers seemed to vibrate excessively. The result of the stabilising system have been analysed in Chapter 5 using finite element methods.. 2.

(14) 1.2. PROPERTIES OF THE BRIDGE. turn buckle. z. z. x. y. (a). (b). (c). Figure 1.2: (a) construction drawing of the hanger parallel to the bridge, (b) photo of the hanger and turn buckle, (c) construction drawing of the hanger perpendicular to the bridge.. cross beam long beam main beam z. y x. Figure 1.3: The beam structure.. 3.

(15) CHAPTER 1. INTRODUCTION. Table 1.1: Dimensions of the beams in the carriageway.. Beam: cross beam long beam main beam. tf (mm) 34 22 40. tw (mm) 18 12 16. tw. h (mm) 750 438 1550. b (mm) 300 297 450. h. tf b Figure 1.4: Cross section of the beam.. The main beams are reinforced at the supports where tw = 30 mm and b = 600 mm.. 1.3. Aims of the Study. The aim of this thesis is to analyse the dynamic effects of the bridge and the risk of fatigue in the hangers. The results are mainly based on the measurements performed in June 2003, considering accelerations and strains. From this information it is possible to determine the frequency spectra for each hanger, as well as the maximal stresses occurring during a train passage. These measurements are confirmed by detailed finite element models created in ABAQUS. From these models more detailed information about the dynamic effects of the bridge can be made, such as deflections and stress variations of the hangers. The result of the stabilising system will be analysed and new approaches will be suggested for damping of the hangers.. 1.4. Structure of the Thesis. A short description of each chapter is presented below to get an overview of the general structure of the thesis. In Chapter 2, selections of the theoretical aspects are presented, that constitute the basis of the calculations in this thesis. The theory chapter briefly describes dynamic. 4.

(16) 1.4. STRUCTURE OF THE THESIS. properties of a basic structural model in order to visualise the methods such as the Half-Power method and Rayleigh damping that are used to analyse the train induced vibrations. Other important theoretical aspects that are discussed are the signal analysis, time integration methods and fatigue. The signal analysis section includes subjects such as Fourier transform, filtering and windowing. The procedure of creating finite element models in ABAQUS/CAE is presented in Chapter 3. The algorithms within ABAQUS that are used in this thesis are briefly presented and the different dynamic analysis approaches are compared. The use of contact formulations in ABAQUS is also presented. In Chapter 4, the equipment used in the field measurements and the different train passages that were measured are presented. The main part of Chapter 4 describes the analysis of the measurements and the effect of the train induced vibrations. The analyses that have been made are frequency response of the hangers and their damping properties from both a free vibration test and excitation from train passages. Other important results are the displacements in the hangers and the train induced stress. Long term effects of the train induced vibrations are considered with fatigue analysis. The results of the finite element modelling in ABAQUS are presented in Chapter 5 and the analyses are divided into static analysis, eigenvalue analysis and dynamic analysis. The whole bridge is analysed during train passages and a detailed study of the turn buckle is performed. A comparison of the results is made with the measurements and conclusions of the modelling are drawn. Chapter 6 contains a discussion of the results and presents the main results. Some recommendations for further research are also suggested. Appendix A includes results from the measurements of some train passages that illustrate the behaviour of the bridge.. 5.

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(18) Chapter 2 Evaluation Methods 2.1. Structural Dynamics. There is an extensive amount of literature concerning the theory of structural dynamics and it is beyond the scope of this thesis to review this literature to any larger extent. However, some of the theoretical aspects that are used in this thesis are presented in this section. The intention of this section is to provide a background that can be useful when analysing dynamic properties of train induced vibrations and other problems in structural dynamics. All of the Figures in this Section are reproduced from Battini [2] except Figure 2.9 which is reproduced from Clough and Penzien [4].. 2.1.1. Undamped Free Vibration. The motion of a linear single degree of freedom (SDOF) system could be visualised by a rigid block with the mass m attached to a spring with stiffness k. The block is constrained with rollers so that it only can move in one direction and the single displacement coordinate u(t) defines the position of the block, as seen in Figure 2.1.. u. u (t ). k. m. uo. . uo. C. t. Tn (a). (b). Figure 2.1: (a) Undamped SDOF system, (b) displacement.. The structure is disturbed from its static equilibrium by either an initial displacement u(0) or velocity u(0) ˙ which results in free vibration. To define the equation. 7.

(19) CHAPTER 2. EVALUATION METHODS. of the motion for the structure D’Alembert’s principle is used by simply expressing the equilibrium of all the forces acting on the structure, see Figure 2.2.. .. u(t) fS =ku(t) .. f =mu(t) I. Figure 2.2: D’Alembert’s principle.. Equilibrium of forces acting on the structure: fI (t) + fS (t) = 0. (2.1). where fI (t) is the inertial force acting on the block, which is the product of the u(t). fS (t) is the spring force acting on the block mass and acceleration, fI (t) = m¨ and it is the product of the spring stiffness and the displacement, fS (t) = ku(t). The displacement of the block is a simple harmonic and oscillatory about its static equilibrium and has the solution: u(t) = u0 cos ωn t +. u˙ 0 sin ωn t ωn. (2.2a). or u(t) = C sin(ωn t + θ). (2.2b). u20 + ( ωu˙n )2 , cos θ = ωun˙ 0C and sin θ = uC0 . The natural circular frequency k , the natural period Tn = ω2πn and the natural frequency is defined as ωn = m where C =. fn =. 1 Tn. =. ωn . 2π. [4]. 8.

(20) 2.1. STRUCTURAL DYNAMICS. 2.1.2. Viscously Damped Free Vibration. Introducing damping in the SDOF system can be symbolised as in Figure 2.3 (a) and result in a decay of the motion which is seen in Figure 2.3 (b).. u −ξωnt. Ce. un. u ( t) c k. u n+ p. tn + p. tn. m. TD =. (a). t. 2π ωD. (b). Figure 2.3: (a) Damped SDOF system, (b) displacement.. By using D’Alembert’s principle, the equilibrium of the forces could be written as: fI (t) + fS (t) + fD = 0. (2.3). where fD is the force acting on the block from the damper and is the product of the ˙ [3] The damping constant c and the velocity, fD = cu(t). √ equation for the motion has the solution for an underdamped system c < cr = 2 km: u˙ 0 + ξωn u0 sin ωD t u(t) = e−ξωn t u0 cos ωD t + ωD. (2.4a). u(t) = Ce−ξωn t sin(ωD t + θ). (2.4b). or. ˙0 2 ˙0 nu nu where C = u20 + ( u˙ 0 +ξω ) , cos θ = u˙ 0 +ξω and sin θ = ωD ωD C c c √ ξ is defined as ξ = cr = 2 km and the damped pulsation is ωD = ωn 1 − ξ 2 . [2]. 9. u0 . C. The damping ratio.

(21) CHAPTER 2. EVALUATION METHODS. 2.1.3. Resonance. When a structure is subjected to a time varying force, it will after a while vibrate with the same frequency as the applied force. This is called steady state response. The amplitude of the vibration is equal to the product of the static deformation multiplied with a dimensionless dynamic factor Rd : 1 Rd = 2 1 − ( ωωn )2 + (2ξ ωωn )2. (2.5). where ωn is the natural circular frequency of the structure and ω is the circular frequency of the load. For an undamped structure, ξ = 0, the dynamic factor tends to infinity as the frequency ratio approaches unity, i.e. ωωn → 1. As the damping coefficient ξ increases, the value of the dynamic factor reduces, as seen in Figure 2.4. Steel bridges have normally a very low material damping coefficient, ξ ≤ 0.02. According to Johnson [12] the damping of large steel structures is commonly assumed to be 0.5%, independent of the mode and the amplitude of the vibration. If ξ = 0.02, the dynamic deformation at the resonance frequency is 25 times larger than the static one. 6. ξ=0 ξ =0.1. 5. Rd. 4. 3. ξ =0.2. 2. ξ=0.5. 1. 0. 0. 0.5. 1. 1.5. ω/ωn. 2. 2.5. 3. Figure 2.4: Variation of dynamic factor with damping and frequency.. It is seen in Figure 2.4 that the maximum steady-state response amplitude occurs ata frequency ratio slightly less than unity. Resonance is reached when ω = ωn 1 − 2ξ 2 [4].. 10.

(22) 2.1. STRUCTURAL DYNAMICS. 2.1.4. Half-Power (Band-Width) Method. The most commonly used experimental method to determine the damping in structures is the Half-Power (Band-Width) method. This method is used in the frequencydomain. It is seen in Figure 2.4 that the shape of the frequency response is controlled by the amount of damping in the system, therefore it is possible to derive the damping ratio from many different properties of the curve. The Half-Power method calculates the damping by using the relationship between the frequencies Rd , with Equation (2.6). It is seen in Figure 2.4 that the response corresponding to √ 2 peaks are more narrow for a lower value of the damping coefficient. [12] ξ=. f2 − f1 f2 + f1. (2.6). Rd Rd √ 2. Rd √ 2. f1. f2. Figure 2.5: Half-Power method to estimate damping.. 2.1.5. 2D Continuous Beams. To obtain an equation for the transverse vibration in a two-dimensional beam the following structure is studied. The beam is subjected to an external force and has a distributed mass m and flexural rigidity EI which can vary with position and time, which is shown in Figure 2.6.. u. p dx. p ( x, t ). x x. dx. M. EI m (kg /m) (a). ∂u. V. 2. ∂ t2. ∂V V ∂ x dx. ∂M M ∂x d x. dx. (b). Figure 2.6: (a) Beam and applied force, (b) forces acting on an element.. 11.

(23) CHAPTER 2. EVALUATION METHODS. The differential equation describing the transverse vibration of the beam is expressed in Equation (2.7). d2 u d4 u EI 4 + m 2 = p(t) dx dt. (2.7). To obtain a unique solution of this equation the boundary conditions and the initial displacement u(x, 0) and velocity u(x, ˙ 0) must be defined. [2] Eigenfrequencies For a beam with constant bending stiffness and mass distribution over the length, the following expression for the eigenfrequencies can be developed: µ2 fn = n 2π. . EI ml4. (2.8). Where µn depends on the boundary conditions and is given in Table 2.1. Table 2.1: Eigenvalues. Boundary conditions pinned-pinned clamped-pinned clamped-clamped. Eigenvalue µn n=1 n=2 n=3 π 2π 3π 5π 4. 9π 4. 13π 4. 7.730. 7.853. 10.996. If a simply supported beam is subjected to a constant axial load N , the eigenfrequency will increase and Equation (2.8) can be expressed as: 2 EI µ N + 2 2 (2.9) fn = n 4 2π ml ml µn The hangers can be assumed to be clamped at the connection with the arch. In the connection with the main beam is it likely to believe that the hanger is pinned in the y-direction (perpendicular to the rail) and that it is nearly clamped in the x-direction (parallel to the rail). The connections with the arch and the main beam are shown in Figure 1.2. The first three eigenmodes for a beam which is clamped at one end and pinned at the other is visualised in Figure 2.7. [8]. 12.

(24) 2.1. STRUCTURAL DYNAMICS. Figure 2.7: The three lowest eigenmodes.. Beam Element Let a uniform beam lie on the x-axis. This 2D beam element has a node at each end and each node has three degrees of freedom (D.O.F); axial translation, lateral translation and rotation, as seen in Figure 2.8. Transverse shear deformations are taken into account by the Timoshenko beam theory, which is usually applied when beam vibration is studied.. Y. v1 1. v2 θ1 E, I, A, m u1 2 L. θ2 u2. X. Figure 2.8: 2D beam element.. The stiffness matrix for this Timoshenko beam element  X 0 0 −X 0  0 Y Y 0 −Y 1 2 1   0 Y Y 0 −Y 2 3 2 k=  −X 0 0 X 0   0 −Y1 −Y2 0 Y1 0 Y2 Y4 0 −Y2. 13. is defined as:  0 Y2   Y4   0   −Y2  Y3. (2.10).

(25) CHAPTER 2. EVALUATION METHODS. where X=. AE L. (4 + φy )EIz Y3 = (1 + φy )L. Y1 =. 12EIz (1 + φy )L3. (2 − φy )EIz Y3 = (1 + φy )L. Y2 =. 6EIz (1 + φy )L2 (2.11). 12EIz ky φy = AGL2. Note that as an element becomes more and more slender, φy approaches zero. A/ky is the effective shear area for transverse shear deformation in the transverse direction. [5] The consistent mass matrix for the beam element is:   140 0 0 70 0 0  0 156 22L 0 54 −13L    2  mL  0 0 13L −3L2  22L 4L  (2.12) m=  70 0 0 140 0 0 420     0 54 13L 0 156 −22L  0 −13L −3L2 0 −22L 4L2 The corresponding HRZ lumped mass matrix is [2]:  35 0 0 0 0 0  0 39 0 0 0 0  2 mL   0 0 L 0 0 0 m= 148   0 0 0 35 0 0  0 0 0 0 39 0 0 0 0 0 0 L2. 2.1.6.        . (2.13). Rayleigh Damping. There are several different ways to introduce damping in a finite element model. A very common and easy way is to introduce material damping. One way to introduce this material damping is to use Rayleigh damping. The Rayleigh method assumes that the element damping matrix can be expressed as a linear combination of the mass and the stiffness matrices as: c = a0 m + a1 k. (2.14). where c is the damping matrix, m is the mass matrix and k is the stiffness matrix. a0 and a1 are proportional constants which can be chosen to control the material damping and have the units of s−1 and s.. 14.

(26) 2.1. STRUCTURAL DYNAMICS. Combined ξn ξm. Stiffness proportional a0=0 Mass proportional a1=0 ωm. ωn. Figure 2.9: Relationship between damping ratio and frequency for Rayleigh damping. It is important to note that the dynamic response generally will include contributions from all the eigenmodes even though only a limited number of modes are included in the uncoupled equations of motion. Thus none of these types of damping matrix is suitable for use with a multi degree of freedom (MDOF) system in which the frequencies of the significant modes span a wide range because the relative amplitudes of the different modes will be seriously distorted by inappropriate damping ratios. The Rayleigh damping method is convenient for direct time integration analysis, but suffers from the disadvantage that the material damping becomes frequency dependent. The relationship between the damping ratio and the frequency is: ξ=. a1 ωn a0 + 2ωn 2. (2.15). It is obvious that the proportional constants a0 and a1 can be evaluated by the solution of a pair of simultaneous equation if the damping ratios ξm and ξn are known. [4] The two modes with the specified ξm and ξn should be chosen to ensure reasonable values for the other modal damping ratios. In practice, the lowest mode and the third or fourth lowest should be used to determine a0 and a1 . [2] If Equation (2.15) is written for both eigenmodes ωm and ωn , the constants can be obtained from: ωm ωn −ωm ωn a0 =2 2 (2.16) 2 a1 −1/ωn 1/ωm ωn − ωm Because detailed variation of damping ratio with frequency seldom is available, it is usually assumed that ξm = ξn ≡ ξ which leads to [4]: . a0 a1. . 2ξ = ωn + ωm. 15. . ωn ωm 1. (2.17).

(27) CHAPTER 2. EVALUATION METHODS. 2.2. Signal Analysis. 2.2.1. Fourier Analysis. The response of a system acted upon by an arbitrary force can be determined by the time domain analysis procedure for any single degree of freedom system. It is sometimes more convenient to transform the signal to frequency domain. This is especially suitable when the equation of motion contains parameters which are frequency dependent. Such parameters can be the stiffness k or the damping c. A simple periodic function can be separated into harmonic components by using Fourier series: ∞ (2.18) an cos ω n t + bn sin ω n t p(t) = a0 + n=1. In which the natural circular frequency ω n = nω1 = n T2πn and Tn represent the period. [4] The Fourier coefficients are: a0 an bn. Tn 1 = p(t)dt Tn 0 Tn 2 = p(t) cos(ω n t)dt Tn 0 Tn 2 = p(t) sin(ω n t)dt Tn 0. (2.19) n = 1, 2, 3 . . .. (2.20). n = 1, 2, 3 . . .. (2.21). The periodic function can also be written as:. p(t) = c0 +. ∞ . cn sin(ωn t + φn ). (2.22). n=1. The coefficients are: c0 = a0 , cn =. . a2n + b2n and φn = arctan abnn. The Fourier coefficient cn represents the magnitude and φn is the phase angle. A plot of magnitude versus frequency is known as the Fourier amplitude spectrum. [10]. 2.2.2. Discrete Fourier Transform (DFT). Note that the time function is denoted by lowercase letter and the Fourier transform of the function by the same letter in uppercase. The frequency domain analysis of a dynamic response requires that both the Fourier transform of p(t) and the inverse. 16.

(28) 2.2. SIGNAL ANALYSIS. Fourier transform of the complex response amplitude U (t) are determined. Analytical evaluation of these direct and inverse Fourier transforms is not possible except for excitations described by simple functions applied to structural systems. The integrals have to be evaluated numerically for excitations varying arbitrary with time, complex vibratory systems, or situations where complex frequency response (or unit impulse response) is described numerically. Numerical evaluation requires truncating these integrals over infinite range to a finite range, and becomes equivalent to approximating the random time-varying excitation p(t) by a periodic function. The discrete Fourier transform is defined as [3]: N −1 2πnm 1 p(tm )e−i N Pn = N m=0. where t = tm = m∆t. 2.2.3. (2.23). m = 1, 2, 3, . . . , N. Filtering. Filtering is often used to minimize high frequency signals (noise) in order to make the primary pulse more readable. Filtering can attenuate the unimportant parts of the signal, but it can also be misapplied if the signal is over-filtered. This will lead to a distortion of the data, which normally reduces the signal peak amplitude. To prevent over-filtering, the filter frequency should be at least five times greater than the highest frequency of interest. The most common type of filter is a low-pass filter, which attenuates the high frequency signals while the low frequency signals are unmodified. Another common type of filter is a band-pass filter which attenuates signals with frequencies that not are within a specified interval. Filters can either be mechanical or applied digitally. Mechanical (analogue) filters are used when measuring the signal, while digital filtering only is possible when the signal has been digitalised, such as with PC-based instrumentation systems. Digital filtering is accomplished in three steps. First the signal has to be Fourier transformed and then the signals amplitude in frequency domain should be multiplied by the desired frequency response. Finally the transferred signal must be inversely Fourier transformed back into time domain. The advantage with a digital filter is that it does not introduce any phase errors and that the original unfiltered signal still can be stored. [10]. 2.2.4. Windowing. The Fourier transform has been described for a periodic signal, but a measured signal is obviously not periodic. Fourier transform of a non-periodic signal can result in several spurious amplitudes. [12] Windowing is the multiplication of the input signal by a weighted function to reduce spurious oscillations in the frequency domain, which forces the signal to be periodic. Signals obtained from measurements are made over finite time intervals, while Fourier transforms are defined of infinite. 17.

(29) CHAPTER 2. EVALUATION METHODS. time intervals. This means that Fourier transforms of a measured signal is an approximation. Consequently, the resolution of the Fourier transform is limited to 1/T Hz, where T is the finite time interval of the measured signal. The resolution of the Fourier transform can only be improved by sampling at longer intervals. The measured signal has several errors and the reason for these spurious oscillations is that the signal is being instantly turned on in the beginning of the measurements and suddenly turned off at the end. These spurious oscillations are called leakage, which means that energy is distributed into adjacent frequency bands. [10] There are many different windowing functions, three of them are presented in Figure 2.10. The mathematical expression of the Hanning windowing function is given in Equation (2.24). 1. Hanning Hamming Bartlett. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -x0/2. x0/2. Figure 2.10: Different windowing functions. whanning (x) =. 2πx 1 x0 1 + cos |x| < 2 x0 2. (2.24). Because of the windowing procedure, energy is lost from the spectra, and the peaks to get broader. This is a problem when the damping of the structure is calculated with the Half-Power method which is based on the difference between the frequencies Rd . [12] at √ 2. 2.3 2.3.1. Time Integration Methods Numerical Approximation Procedures. The time step methods employ numerical procedures to approximately satisfy the equations of motion at each time step, using either numerical differentiation or integration. A brief summary is presented here to give the general ideas of how these numerical step methods works and may be applied for solution of structural dynamic problems:. 18.

(30) 2.3. TIME INTEGRATION METHODS. 1. The method may be classified as either explicit or implicit. In an explicit method, the result in each step depends only on the quantities obtained in the preceding step, so the analysis proceeds directly from one step to the next. The implicit method on the other hand is based on that the expression for a certain step includes one or more values pertaining to the same step. This means that trial values of the necessary quantities must be guessed and these are refined by successive iterations. Even though the equations required for each step are very simple, the cost of iteration within a step may be ruling. 2. The primary factor to be considered in selecting a step method is efficiency, which concerns the computational effort required to achieve the desired level of accuracy over the range of time for which the response is needed. Accuracy alone cannot be a criterion since any level of accuracy can be obtained with any method if the time step is small enough, but with obvious increase of costs. In any case the time steps must be made short enough to provide an adequate definition of the loading and the response history. A high frequency input or response cannot be defined with large time steps. 3. Factors that may contribute to errors in the results obtained from well defined loadings include: (a) Round off: resulting from calculations being done using number expressed by too few digits. This is normally not any problem with the computer programs used today. (b) Instability: caused by amplification of the errors from one step during the calculations of subsequent steps. Stability of any method is improved by choosing a smaller time step. (c) Truncation: using too few terms in series expressions of quantities. 4. Errors resulting from any causes may be manifested by either or both of the following effects: (a) Phase shift or apparent change of frequency in cyclic results. (b) Artificial damping, in which the numerical procedure removes or adds energy to the dynamically response system.. 2.3.2. Newmark Beta Methods. A general step method was proposed by Newmark and the equations for the velocity and displacement in step i is defined as: ui−1 + γ∆t¨ ui u˙ i = u˙ i−1 + (1 − γ)∆t¨ ui = ui−1 + ∆tu˙ i−1 +. 1 2. 19. − β ∆t2 uï−1 + β∆t2 uï. (2.25a) (2.25b).

(31) CHAPTER 2. EVALUATION METHODS. The factor γ provides a linearly weighting between the influence of the initial and the final accelerations on the change of velocity and β provides the same weighting between the initial and final accelerations for the displacements. According to Clough and Penzien, [4], studies of this formulation has shown that the factor γ controls the amount of artificial damping induced by the step procedure and if γ = 1/2 there is no artificial damping. Direct time integration methods based on the Newmark formula are summarised in Table 2.2. [4] Table 2.2: Different Newmark methods.. Method Trapezoidal rule Linear acceleration Central difference. Type Implicit Implicit Explicit. β 1/4 1/6 0. γ 1/2 1/2 1/2. The stability criterion for Newmark’s method is: 1 ∆t 1 ≤ √ √ Tn π 2 γ − 2β For the Trapezoidal rule,γ = tion (2.27). [3]. 1 2. and β =. 1 , 4. this condition is presented in Equa-. ∆t <∞ Tn This means that the Trapezoidal rule is stable for any time increment.. 2.3.3. (2.26). (2.27). Hilber-Hughes-Taylor Alpha Method. The Hilber-Hughes-Taylor Alpha method is used when damping is introduced in the Newmark method, without degrading the order of accuracy. The method is based on the Newmark equations, whereas the time discrete equations are modified by averaging elastic, inertial and external forces between both time instants. The parameters γ and β are defined: 1 − 2α 2 (1 − α)2 β= 4 γ=. where the parameter α is chosen so that: 1 α ∈ − ,0 3. (2.28a) (2.28b). (2.29). The result of this is an unconditional stable second-order scheme and it is a logical replacement of the Newmark algorithm for non-linear problems in which it is necessary to control the damping during the integration. [8]. 20.

(32) 2.4. FATIGUE. 2.4. Fatigue. Loads suddenly applied to structures are termed shock or impact loads and result in dynamic loading . This also includes rapidly moving forces such as those caused by a railroad train passing over a bridge. Structural members subjected to repeated fluctuating, or alternating stresses, which are smaller than the ultimate tensile strength σu or even the yield strength σyp , may nevertheless manifest diminished strength and ductility. This is termed fatigue and is mainly influenced by minor structural discontinuities, the quality of the surface finish and the chemical nature of the environment. Generally the fatigue fracture has its origin at points with high stress concentrations. This type of failure, through the involvement of slip planes and spreading cracks, is progressive in nature. Tensile stress, and to lesser degree shearing stress, lead to fatigue crack propagation, while compressive stress probably does not. [15] The dynamic loads are normally loads from moving vehicles and/or wind loads. A load that is time-dependent will induce stress variations in the structure and if they are large and many they could lead to fatigue fractures. When designing for fatigue the stress range σrd is used. Stress range is the difference between maximum and minimum stress in the studied point during the stress variation. The stress range σrd is compared with the characteristic fatigue strength frk . The characteristic fatigue strength is dependent on the detail category, the number of stress cycles nt and the stress collective κ. For a stress collective with constant stress range, κ = 1, the following expression can be used for the characteristic fatigue strength:. frk.  2 · 106 1/3    C if 103 < nt < 5 · 106 ,   n t     2 · 106 1/5 = 0.885C if 5 · 106 < nt < 108 ,    nt       0.405C if nt > 108 .. These equations are visualised in Figure 2.11.. 21. (2.30).

(33) CHAPTER 2. EVALUATION METHODS. Characteristic fatigue strength frk. [MPa]. 400 300 200. C C = C = 4 C==3 40 5 30 5. 100 80 60 40. 20. 10 3 10. 4 10. 5 10. 6 10. Number of stress cycles nt. 7 10. 8 10. Figure 2.11: Characteristic fatigue strength, with varying number of stress cycles.. Loads with stress cycles less than 103 can be neglected. For stress collectives with varying stress range, the Palmgren-Miners linear damage rule is used according to the fatigue strength in Equation (2.30). In practice, the value of the stress range should be multiplied with the factor 1.1γn , where γn is the safety factor. When the stress collective is created, the 100 largest stress cycles and the stress cycles that are smaller than the fatigue strength at nt = 108 can be neglected. The stress cycles should then be grouped into appropriate intervals. The following design criterion is valid for varying stress range: ni nti. ≤ 1.0. (2.31). where ni is the number of stress cycles at a stress range σri , nti is the number of stress cycles for a specific fatigue strength with the stress range σri . [9] This rule is based on that the measure of damage is simply the cycle ratio with assumptions of constant work absorption per cycle. Failure is reached when a characteristic amount of work has been absorbed. The energy accumulation leads to a linear summation of cycle ratio or damage. Failure will occur when the summation of cycle ratio is greater or equal to 1. [6] This method is based on empirical studies and is therefore a quite coarse tool for calculating the risk of fatigue failure in structures. However, this is the most commonly used design criterion and if the sum of cycle ratios is much larger than 1 there is great risk of failure. According to BSK94, [9], the value of the detail category for a threaded construction. 22.

(34) 2.4. FATIGUE. element should be less than 45. A rolled thread and thread turned in a lathe should, respectively, have 90% and 70% of the dimensioning value frd at C = 45.. 23.

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(36) Chapter 3 Creating Finite Element Models This chapter deals with the modelling techniques that are used to analyse the bridge in the commercial software ABAQUS. A brief summary of the different routines that have been used are presented, how they were used and the advantages of different analysis approaches. The finite element models were created in ABAQUS/CAE which includes the Graphical User Interface (GUI). This method of creating models is easier than coding an input file, especially when the models are large, as in the case with 3D models and dealing with solid elements.. 3.1. Modelling Procedures in ABAQUS/CAE. The ABAQUS/CAE environment is divided into different modules, where each module defines a logical aspect of the modelling process; for example, defining the geometry, defining material properties, and generating a mesh. The GUI interface generates an input file with all information of the model, to be submitted to the solver, using ABAQUS/Standard or ABAQUS/Explicit routines. The solver performs the analysis and sends the information back to ABAQUS/CAE for evaluation of the results.. 3.1.1. Modules. Most models created in ABAQUS/CAE are assembled from different parts. Parts are created separately in the part module. Different parts may need different material properties, which are defined in the property module. ABAQUS provides a full range of material properties, such as elastic and plastic behaviour, as well as thermal and acoustic behaviour. The model is then assembled in the assembly module, by combining the different parts. In the step module the analysis is divided in different analysis step, such as static and dynamic analyses. These can be combined in a way to resemble the physical problem that is to be analysed. At this stage, the parts in the model do not interact with each other. In the interaction module the parts are connected to each other using constraints, by defining the degrees of freedom. 25.

(37) CHAPTER 3. CREATING FINITE ELEMENT MODELS. to be connected with another part. Interactions between parts are also defined, such as contact methods and friction behaviour. Connector elements can also be defined, to simulate for example spring or dashpot behaviour. The loads acting on the model are defined in the load module, as well as boundary conditions. The loads and the boundary conditions can be defined to vary over time as well as over different steps. Physical kinematical behaviour, such as velocity, can be applied either as boundary condition or field procedure. Velocity is generally more stable when defined as boundary condition than field variables. The whole model is then meshed in the mesh module. The meshing techniques vary with the element type and the geometry of the model. Different meshing techniques can be applied to the same part if partitioning is used. This is useful when regions of the same part requires different accuracy, or if the geometry is complicated.. 3.1.2. Elements. All elements used in ABAQUS are divided into different categories, depending on the modelling space. The element shapes available are beam elements, shell elements and solid elements and the modelling space is divided into 3D space, 2D planar space and axisymmetric space. Beam Elements A beam element is an element in which assumptions are made so that the problem is reduced to one dimension mathematically. The primary solution variable is then functions of the length direction of the beam. For this solution to be valid, the length of the element must be large compared to its cross-section. There are two main types of beam element formulations, the Euler-Bernoulli theory and the Timoshenko theory. The Euler-Bernoulli theory assumes that plane cross-sections, initially normal to the beams axis, remain plane, normal to the beam axis, and undistorted. All beam elements in ABAQUS that use linear or quadratic interpolation are based on this theory. The Timoshenko beam theory allows the elements to have transverse shear strain, so that the cross-sections do not have to remain normal to the beam axis. This is generally more useful for thicker beams. [11] Shell Elements The shell elements defined in ABAQUS are divided into three categories: thin, thick and general-purpose elements. The thin shell elements are based on the Kirchoff shell theory and the thick shell elements are based on the Mindlin shell theory, which includes shear deformation and therefore are better suited for thicker elements. The general-purpose shell elements can provide solutions for both thin and thick shell elements. In ABAQUS/Standard all three types are available, while ABAQUS/Explicit only provides general-purpose elements. [11]. 26.

(38) 3.1. MODELLING PROCEDURES IN ABAQUS/CAE. Solid Elements ABAQUS provide solid elements in two and three dimensions. The two-dimensional solid elements allow modelling of plane and axisymmetric problems. In three dimensions the isoparametric hexahedra element are most common, but in some cases complex geometry may acquire tetrahedron elements. Those elements are generally only recommended to fill in awkward parts of the mesh. ABAQUS provide both first-order linear and second-order quadratic interpolation of the solid elements. The first-order elements are essentially constant strain elements, while the secondorder elements are capable of representing all possible linear strain fields and are more accurate when dealing with more complicated problems. [11]. 3.1.3. Analysis Type. ABAQUS provides several different analysis types which are divided in two main groups: general and linear perturbation. General analysis defines a sequence of events and the state of the model at the end of one step provides the initial state for the next step. Linear perturbation analyses provide the linear response of the model about the state reached at the end of the last general nonlinear analysis. In ABAQUS/CAE those different analysis types are managed under the step module. General Static Analysis The general static analysis can involve both linear and nonlinear effects and is performed to analyse static behaviour such as deflection due to a static load. A criterion for the analysis to be possible is that it is stable. A static step uses time increments, not in a manner of dynamic steps but rather as a fraction of the applied load. The default time period is 1.0 units of time, representing 100% of the applied load. If nonlinear effects are expected, such as large displacements, material nonlinearities, boundary nonlinearities, contact or friction, the NLGEOM command should be used. When dealing with an unstable problem, such as in buckling or collapse, the modified Riks method can be used. It uses the load magnitude as an additional unknown, and solves simultaneous for loads and displacements. This method provides a solution even if the problem is nonlinear. [11] Linear Eigenvalue Analysis Linear eigenvalue analysis is used to perform an eigenvalue extraction to calculate the natural frequencies and the corresponding mode shapes of the model. The analysis can be performed using two different eigensolver algorithms, Lanczos or subspace. The Lanczos eigensolver is faster when a large number of eigenmodes are required while the subspace eigensolver can be faster for smaller systems. When using the Lanczos eigensolver, one can choose the range of the eigenvalues of interest while the subspace eigensolver is limited to the maximum eigenvalue of interest. [11]. 27.

(39) CHAPTER 3. CREATING FINITE ELEMENT MODELS. Dynamic Implicit Analysis The dynamic implicit analysis method is used to calculate the transient dynamic response of a system, e.g. a moving body interacting with other parts of the model. When nonlinear dynamic responses are studied, a direct time integration of the system must be used. In linear analysis, modal methods can be used to predict the response of the system, using eigenmode extraction. This is generally less expensive than direct integration, because in direct integration the global equations of motion of the system must be integrated through time. Implicit schemes solve the dynamic quantities at time t + ∆t and the nonlinear equations must be solved. This method uses the Hilber-Hughes-Taylor operator, which is an extension of the trapezoidal rule. If the equations are highly nonlinear, it may be difficult to obtain a solution. Nonlinearities are easier to handle in dynamic procedures than in static ones, which make the implicit scheme applicable in most cases that do not deal with extreme nonlinearities. The time step used in the implicit scheme can be controlled by the “half step residual”, introduced by Hibbit and Karlsson (1979). The half-step residual is the equilibrium residual error halfway through a time increment, t+∆t/2 and once the solution at t+∆t has been obtained, the accuracy of the solution can be assessed and the time step adjusted appropriately. The choice of the time increment depends on the type of analysis performed. In dynamic problems, a smaller time increment than the stable one might be used, to get an accurate result depending on the variations in the structure. In a static model on the other hand, the time increment usually does not have the same physical meaning and corresponds to a fraction of the applied load rather than physical time. The time increment can be defined using either the automatic or the fixed incrementation. The automatic incrementation is based on the half step residual and is recommended for most analysis except in cases when the problem is well understood, or when convergence is not achieved with the automatic incrementation. Even if convergence is achieved, the results are not guaranteed to be correct. The automatic time increments are chosen by defining initial, minimum and maximum increment sizes. If no convergence can be found with the initial increment, a smaller one is used until convergence is achieved, down to the minimum increment defined. If the solution converges with the initial increment size, an attempt with a larger one will be used. No increments will be attempted that are larger than the maximum stated. The routine for these procedures are based on empirical studies. [11] Dynamic Explicit Analysis The dynamic explicit analysis available in ABAQUS/Explicit is to be used with short dynamic response time and extremely discontinuous processes. It also allows general contact conditions. Due to its large deformation theory, models are allowed to be heavily deformed, such as in explosions or collisions. The dynamic explicit routine performs a large number of small time increments efficiently using an explicit central difference time integration rule. In this method, each increment is relatively inexpensive compared to the direct-integration method because there is no solution. 28.

(40) 3.1. MODELLING PROCEDURES IN ABAQUS/CAE. for a set of simultaneous equations. The explicit central difference operator satisfies the dynamic equilibrium equations at the beginning of the increment, t and the accelerations calculated at time t are used to advance the velocity solution to time t + ∆t/2 and the displacement solution to time t+∆t. The stability increment limit is given in terms of the highest frequency of the system as ∆t ≤ 2/ωmax . When introducing damping to the system, the time increment is given by ∆t ≤. 2 ωmax. 2 ( 1 + ξmax − ξmax ),. (3.1). where ξmax is the fraction of critical damping in the mode with the highest frequency. An approximation to the stability limit can be written as the smallest transit time of a dilatational wave across any of the elements in the mesh, ∆t ≈ Lmin /cd , where Lmin is the length of the smallest element in the mesh and cd is the dilatational wave speed. [11] The wave speed for steel is approximately 6000 m/s. If the smallest element size is 0.1 m, the stable time increment becomes 17 µs.. 3.1.4. Contact Methods. Contact methods are used in ABAQUS to define contact either between bodies or with the body itself. Contact between bodies can be made with both rigid and deformable bodies. There are two types of contact algorithms in ABAQUS/Explicit: kinematic and penalty method. The kinematic contact method does not allow any penetration of the contact bodies while the penalty method does. When defining a contact between two bodies, one serves as a slave body and the other one as master. When contact occurs between these two bodies, it is determined which slave nodes penetrates the master surface, the depth of each penetration and the mass associated with it. The force required to move the slave nodes to the master surface is then calculated. This force is distributed to the master surface without deforming it and is used to adjust the acceleration of the nodes. A second adjustment is then performed to ensure that no other parts overlap each other. When using hard kinematic contact method it is still possible for the master surface to penetrate the slave surface after the correction. Such penetrations can be minimised by refining the mesh on the slave surface. If softened kinematic contact is used, it will allow penetrations since its corrections are made to satisfy the pressure-overclosure relationship at the slave nodes, not the condition of zero penetration. Sliding Formulation There are three approaches to account for the relative motion concerning contact formulations: finite sliding, small sliding and infinitesimal sliding. Finite sliding is the most general and allows any arbitrary motion of the surfaces, small sliding. 29.

(41) CHAPTER 3. CREATING FINITE ELEMENT MODELS. assumes relatively small relative motions between the contact surfaces, although the bodies themselves have large motions. The infinitesimal sliding and rotation assumes that both relative and absolute motions are small. The last two types cannot be used with the penalty contact algorithm. [11]. 3.1.5. Explicit versus Implicit Methods. Using the dynamic implicit scheme instead of the explicit can reduce the solution time radically. The main difference lies in the definition of the stable time increment. As discussed in Section 3.1.3, the stable time increment for the explicit analysis is Lmin /cd , where Lmin is the smallest element in the mesh and cd is the dilatational wave speed. This is generally most useful when dealing with models that suffer from large deformation during a short time interval, e.g. explosions, collisions or buckling analysis. The train simulation in this thesis does not have any large deformation and have rather long time duration, more than 10 seconds. In the explicit scheme it is recommended to use automatic incrementation to achieve convergence, because changing the size of the increment easily makes the solution diverge. When using the implicit scheme it is also recommended to use the automatic incrementation, but convergence can be accomplished with a fixed time increment if the problem is well understood. Even if the solution converges it is not guaranteed that the results are accurate, so it must be verified with different sizes of the increment.. 3.1.6. Contact Conditions for Train Simulations. The dynamic simulations of the train in this thesis were mostly performed with dynamic implicit scheme because the calculation time is just a fraction of the time for the explicit routine. Both the train and the rail were made of solid elements with contact formulation. Contact was set with a penalty friction formulation with the friction constant µ = 0.1 in the tangential direction. In the normal direction the contact was set with hard contact and no allowance to separate after contact. The contact was performed with a surface to surface contact method, with each wheel as a separate master surface and the rail as slave surface. The rail was constrained with the sleepers using a TIE connection to constrain every degree of freedom. The bridge was then made in the same model with either 3D beam elements or 3D shell elements. The simulation was made with both automatic and fixed time increments with different time steps to achieve accurate results, which are further presented in Chapter 5. The loads, in this case gravity load from the bridge and the train load, were made in the static step and then propagated into the dynamic step. If the load was created in the dynamic step it was applied in a dynamic manner, causing the structure to oscillate severely before equilibrium was reached.. 30.

(42) Chapter 4 Measurements 4.1. Introduction. ˚nge and The field measurements were performed at the Langer beam bridge in A took place in 24–25 June 2003. This project was initiated because it had been detected that the hangers on the bridge vibrated excessively during train passages and evaluation of the effect of these vibrations had to be made. To analyse the measurements the software MATLAB was used. The five different train passages were measured are presented in Table 4.1. Table 4.1: Measured train types.. Type of train. ID number. locomotive. 009 011 012 018 019. Rc4 Rc4 Rc6 Rc6 Rc4. freight freight passenger passenger freight (copper train). number of carriages 24 10 6 6 19. Velocity [km/h] 85 75 75 70 80. All of the results presented are taken from the last train passage with the copper train, ID019, since this train induced the largest vibrations and strains in the bridge. The results from a selection of the other train passages are displayed in Appendix A. The two types of measurements that were performed on the hangers measured the induced strain and acceleration. The instruments used to measure the train induced acceleration were: • DAT-recorder -Sony PC216AX • Amplifier-UNO-MWL 006 • Low-pass filter-PCP-848 LP-filter 20Hz • Accelerometers-Terra Technology 31.

(43) CHAPTER 4. MEASUREMENTS. The instrumentation used to measure the train induced strain was: • Amplifier-Hottinger MGCplus • Strain gauges-N11-FA-5-120-11 The signals from the accelerometers and strain gauges had, respectively, the sample frequency of 6000 Hz (samples/s) and 2000 Hz (samples/s). The accelerometer and the strain gauge used for these measurements are shown in Figure 4.1.. (a). (b). Figure 4.1: (a) Strain gauge attached to a hanger, (b) three accelerometers.. 4.1.1. Strain Gauge. The strain was measured with four foil strain gauges assembled 90◦ apart and 100 mm above the threaded section on the perimeter of each hanger. There are 11 hangers on each side of the bridge and they are numbered from 1 to 11, where hanger 1 and 11 are shortest and hanger 6 is the longest. The strain gauges were attached to hanger 2 to 5. The gauges were numbered from 1 to 16, where the odd numbers measured bending parallel to the rail, x-direction, and even numbers perpendicular to the rail, y-direction. The assembly of the gauges is shown in Figure 4.2. 15. 11. 7. 3. 14 hanger 5 16. 10 hanger 4 12. 6 hanger 3 8. 2 hanger 2 4. 13. 9. 5. 1. Figure 4.2: Placement of the strain gauges.. 32.

(44) 4.1. INTRODUCTION. Because of the placement of the strain gauges it is possible to calculate the plane stress in the perimeter of each hanger, which is done in Section 4.4. Several mechanical, electrical and optical systems have been developed for measuring the average strain at a point on a free surface. The most common method employs the bonded electric wire or foil resistance strain gauges. The bonded wire gauge consists of a grid of fine wire filament cemented between two sheets of treated paper or plastic backing, as seen in Figure 4.3. The backing insulates the grid from the metal surface in which it is to be bonded and functions also as a carrier so that the filament may be handled conveniently. Generally the diameter of the wire in the gauge is 0.025 mm. The grid in the bonded foil gauges, is constructed of a very thin metal foil, approximately 0.0025 mm, instead of a wire. Since the filament cross-section on a foil gauge is rectangular, the ratio between the surface area to the cross-sectional area is higher than of a round wire. This leads to increased heat dissipation and improved connection between the grid and the backing material. The ratio of the unit change in resistance of the gauge to the unit change of length (strain) of the gauge is called gauge factor and is 2.14 ± 1% for the strain gauges used. The metal of which the filament element is made is the principal factor determining the magnitude of the gauge factor. The strain gauge is made of Constantan, which is an alloy composed of 60% Copper and 40% Nickel. The operation of the bonded strain gauge is based on the change in electrical resistance of the filament that leads to a change in strain. Deformation of the hangers where the gauge is bonded results in a deformation of the backing and the grid. A variation in the resistance of the grid will manifest itself as a change in voltage across the grid and finally the voltage variation can be converted to strain with the relationship 1v = 100µ. [15]. lead wires. filament. paper backing. Figure 4.3: Strain gauge.. 4.1.2. Accelerometers. The accelerometers were assembled 0.23lhanger from the centre of the threaded section, where lhanger is the length of the hanger, defined from the centre of the threaded section to the connection with the arch. The position of the accelerometers are presented in Figure 4.4 and Table 4.2.. 33.

(45) CHAPTER 4. MEASUREMENTS. Table 4.2: Length of the hangers and the position of the accelerometers.. ltot (m) 4.613 6.244 7.356 8.003. lhanger (m) 4.043 5.674 6.786 7.433. 0.23lhanger (m) 0.93 1.30 1.55 1.70. lacc (m) 1.545 1.918 2.172 2.320. accelerometers. lhanger. Hanger 2 3 4 5. lacc. ltot. turn buckle. z x. Figure 4.4: Position of the accelerometers.. Each cylinder has three accelerometers, one for each direction in space, as seen in Figure 4.1. The accelerometers are piezoelectric. This type is primary made of piezoelectric materials, i.e. natural or man-made quartz, which produce electric charges in response to the strain in the material. By applying a seismic mass of known value to the piezoelectric material, a known force due to the acceleration of the seismic mass is created. The piezoelectric material has a force-voltage strain response which produces an electric charge proportional to the acceleration. Since the force-voltage strain response is nearly quadric it requires either a static load, or as in this case, a DC bias voltage to centre the response in a linear range. Therefore the frequency may be constant all the way down to 0 Hz. All accelerometers have a particular sensitivity, which refers to the ratio of electrical output to the mechanical input. Thus it is possible to calculate the acceleration by measuring the output charge or voltage from the accelerometer. [10] A low-pass (LP) filter was used on train passages ID018 and ID019 to reduce the frequencies above 20 Hz. This will show in the acceleration graphs were the mea-. 34.

(46) 4.1. INTRODUCTION. surements made on passages ID009, ID011 and ID012 has much greater amplitudes, almost by a factor 10. The effect of the LP-filter is shown in Figure 4.5 where almost all frequencies above 20 Hz is reduced in ID019, compared with ID009 where the frequency spectra spans the whole interval. ID019. 180. 180. 160. 160. 140. 140. 120 100. 120 100. 80. 80. 60. 60. 40. 40. 20. 20. 0. 0. 10. 20 30 Time (s). ID009. 200. Frequency (Hz). Frequency (Hz). 200. 0. 40. 0. 5. 10 15 Time (s). 20. Figure 4.5: Spectrogram for train passages ID019 and ID009.. 35.

(47) CHAPTER 4. MEASUREMENTS. 4.2. Frequency Analysis. The frequency analysis was made according to the theory in Section 2.2 where Equation (2.23) was used to transform the measured signal from time domain to frequency domain. The result from this procedure is a complex number and when taking the absolute value of it, the Fourier magnitude is created. The frequency spectra should have the interval ∆f and is obtained by the number of samples N and the sampling rate, by the relation ∆f = N · rate. During the field measurements it was detected that the entire arch vibrated perpendicular to the rail. This is shown in the frequency spectrum as several frequency responses in the y-direction during and after a train passage. To estimate the eigenfrequencies of the hangers, Equation (2.8) was used which produced the results shown in Table 4.3. Table 4.3: Estimated eigenfrequencies.. x-direction Hanger Mode 1 [Hz] 2 17.32 3 9.45 4 6.81 5 5.75. y-direction Hanger Mode 1 [Hz] 2 11.94 3 6.51 4 4.70 5 3.97. Mode 2 [Hz] 47.74 26.06 19.23 13.53. Mode 2 [Hz] 38.68 21.11 15.21 12.85. These eigenfrequencies were calculated with the assumption that the hangers are clamped in the connection with the arch in both x- and y-directions. In the connection with the main beam it were assumed that the hangers are clamped in the x-direction and pinned in the y-direction. A connection is almost impossible to design so that it is clamped. Hence, the eigenfrequencies in Table 4.3 are overestimated. The eigenfrequencies are calculated using the entire lengths of the hangers, defined as ltot in Table 4.2.. 4.2.1. Free Vibration Test. To determine the eigenfrequencies of each hanger, free vibration tests were performed. The free vibration was initiated by a swift knock on each hanger. This made it possible to easily view the eigenfrequencies from the hangers without disturbance from global eigenmodes of the bridge. The result from the free vibration test is presented in Figures 4.6– 4.9.. 36.

(48) 4.2. FREQUENCY ANALYSIS. a/amax. 1.5. acc2x 15.96. 1. 0.5. 0 0. 5. 10. f (Hz). 15. 20. a/amax. 1.5. 25. acc2y 10.94. 1. 0.5. 0 0. 5. 10. f (Hz). 15. 20. 25. Figure 4.6: Frequency analysis of the accelerations in hanger 2.. a/amax. 1.5. acc3x 7.86. 1. 23.09. 0.5. 0 0. 5. 10. f (Hz). 15. 20. a/amax. 1.5. acc3y 18.84. 1. 0.5. 0 0. 25. 6.15. 5. 10. f (Hz). 15. 20. Figure 4.7: Frequency analysis of the accelerations in hanger 3.. 37. 25.

(49) CHAPTER 4. MEASUREMENTS. a/amax. 1.5. acc4x 19.23. 1. 0.5. 7.21. 0 0. 5. 10. f (Hz). 15. 20. a/amax. 1.5. 25. acc4y 6.04. 1. 16.50. 0.5. 0 0. 5. 10. f (Hz). 15. 20. 25. Figure 4.8: Frequency analysis of the accelerations in hanger 4.. a/amax. 1.5. acc5x 13.53. 1. 0.5. 0 0. 4.31 5. 10. f (Hz). 15. 20. a/amax. 1.5. 25. acc5y 11.40. 1. 0.5 3.61 0 0. 5. 10. f (Hz). 15. 20. Figure 4.9: Frequency analysis of the accelerations in hanger 5.. 38. 25.

(50) 4.2. FREQUENCY ANALYSIS. 4.2.2. Train Induced Vibration. During excitation from the train the hangers are forced to vibrate with the frequency induced by the train. After the train has passed, the hangers are in free vibration and vibrate with their natural frequencies. The frequency response obtained from the accelerometers after the train passage are presented in Figures 4.10– 4.13. The frequency response from the strain gauges assembled on hanger 5 is shown in the Figure 4.14, while the frequency responses of the other hangers are shown in Appendix A. In the frequency response graphs of hangers 4 and 5, obtained from the accelerometers, the second eigenmode is generally dominating, while the graphs from the strain gauges show the first eigenmode as the largest peak. The acceleration is higher when the hangers are vibrating with the second eigenmode than with the first and this is seen in the frequency response graphs as a higher peak. The reason for this is that a higher eigenfrequency means that the hanger vibrates with more cycles per second and because of this it has a higher acceleration. The frequency response obtained from the strain gauge shows which eigenmode that causes the largest strain in the hangers. The motion of the hangers will be further discussed in Section 4.3. 1.5. acc2x 16.07. a/amax. 1. 0.5. 0. 0. 5. 10. f (Hz). 15. 20. a/amax. 1.5. 25. acc2y 2.17. 1. 11.10 8.45 9.48. 0.5. 0. 0. 5. 10. f (Hz). 15. 20. Figure 4.10: Frequency analysis of the accelerations in hanger 2.. 39. 25.

(51) CHAPTER 4. MEASUREMENTS. a/amax. 1.5. acc3x 7.83. 1. 0.5. 0. 0. 5. 10. f (Hz). 15. 20. a/amax. 1.5. acc3y 6.14. 1. 0.5. 0. 25. 2.17. 0. 6.34. 5. 10. f (Hz). 15. 20. 25. Figure 4.11: Frequency analysis of the accelerations in hanger 3.. a/amax. 1.5. acc4x 7.21. 1. 0.5 19.20 0. 0. 5. 10. f (Hz). 15. 20. a/amax. 1.5. acc4y 2.17. 1. 16.41. 6.03. 0.5. 0. 25. 0. 5. 10. f (Hz). 15. 20. Figure 4.12: Frequency analysis of the accelerations in hanger 4.. 40. 25.

(52) 4.2. FREQUENCY ANALYSIS. 1.5. acc5x 13.55. a/amax. 1 4.31. 0.5. 0. 0. 5. 10. f (Hz). 15. 20. a/amax. 1.5. 25. acc5y 3.62. 1. 11.41 9.48. 0.5. 0. 2.17 0. 5. 10. f (Hz). 15. 20. 25. Figure 4.13: Frequency analysis of the accelerations in hanger 5.. ε /εmax. 1.5 1. 2.16. 0.5. gauge 13 3.62 4.32. ε /εmax. 0 0 1.5. 5. ε /εmax. f (Hz). 15. 20. 24.00 25 gauge 14. 4.32. 1 0.5. 13.57. 0 0 1.5 1 0.5 0 0 1.5 ε /εmax. 11.41 10. 1. 5. 10. f (Hz). 15. 20. 25 gauge 15. 2.16 3.62 4.32 5. 9.46 11.41 10 f (Hz). 15. 20. 24.00 25 gauge 16. 4.32. 0.5 0 0. 13.57 5. 10. f (Hz). 15. 20. 25. Figure 4.14: Frequency analysis from strain gauges in hanger 5 (gauges 13 and 15 correspond to the x-axis and 14 and 16 to the y-axis).. All of the signals from the train induced vibration have been subjected to windowing, which forces them to be periodic and makes the frequency response more legible.. 41.

(53) CHAPTER 4. MEASUREMENTS. This makes it easier to study the frequencies, because it removes noise, but the frequency response curve is not as smooth as earlier. A number of eigenfrequencies that appear in the frequency response graphs are global eigenmodes of the bridge. One of these eigenfrequencies is the peak around 2.2 Hz which is the first eigenmode of the arch. This eigenmode is shown in Figure 5.11. The measured eigenfrequencies for the hangers are presented in Tables 4.4 and 4.5.. Table 4.4: Frequencies from accelerometers.. x-direction Hanger Mode 1 (Hz) 2 15.97 3 7.86 4 7.21 5 4.31. y-direction Hanger Mode 1 (Hz) 2 10.94 3 6.15 4 6.04 5 3.61. Mode 2 (Hz). 19.23 13.53. Mode 2 (Hz). 16.50 11.40. Table 4.5: Frequencies from strain gauges.. x-direction Hanger Mode 1 (Hz) 2 16.05 3 7.84 4 7.19 5 4.32. y-direction Hanger Mode 1 (Hz) 2 11.03 3 6.11 4 6.11 5 3.62. Mode 2 (Hz) 22.97 19.19 13.57. Mode 2 (Hz) 18.86 16.49 11.41. Since the hangers have different lengths, their eigenfrequencies should differ if they all have the same boundary conditions and the same axial pre-stress. The measured eigenfrequencies for hanger 3 and 4 are almost identical, even though their lengths differ with about 1.1 m. This indicates that hanger 4 is harder constrained or more pre-stressed than the other hangers. Table 4.6 shows the corresponding eigenfrequencies from ABAQUS and Table 4.7 and 4.8 shows the ratio between the measured and calculated eigenfrequencies, fmeasured /fABAQUS .. Table 4.6: Frequencies from ABAQUS.. x-direction Hanger Mode 1 [Hz] 2 16.29 3 8.60 4 6.17 5 5.23. y-direction Hanger Mode 1 [Hz] 2 11.08 3 6.06 4 4.34 5 3.67. Mode 2 [Hz]. 17.46 14.62. 42. Mode 2 [Hz]. 13.82 11.62.

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