Note for Structural Dynamics - SD By Tharach Janesuapasaeree

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Structural Analysis IV Structural Dynamics 4th Year Structural Engineering 2009/10 Dr. Colin Caprani 1 Dr. C. Caprani

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Structural Analysis IV Contents 1. 2. 3. 4. Introduction ......................................................................................................... 4 1.1 Outline of Structural Dynamics....................................................................... 4 1.2 An Initial Numerical Example......................................................................... 6 1.3 Case Study – Aberfeldy Footbridge, Scotland ................................................ 9 1.4 Structural Damping........................................................................................ 12 Single Degree-of-Freedom Systems ................................................................. 14 2.1 Fundamental Equation of Motion.................................................................. 14 2.2 Free Vibration of Undamped Structures........................................................ 19 2.3 Computer Implementation & Examples........................................................ 23 2.4 Examples........................................................................................................ 29 2.5 Free Vibration of Damped Structures............................................................ 32 2.6 Computer Implementation & Examples........................................................ 36 2.7 Estimating Damping in Structures................................................................. 39 2.8 Examples........................................................................................................ 41 2.9 Response of an SDOF System Subject to Harmonic Force .......................... 44 2.10 Computer Implementation & Examples .................................................... 50 2.11 Examples .................................................................................................... 55 2.12 Numerical Integration – Newmark’s Method ............................................ 57 2.13 Computer Implementation & Examples .................................................... 63 2.14 Problems..................................................................................................... 69 Multi-Degree-of-Freedom Systems.................................................................. 70 3.1 General Case (based on 2DOF) ..................................................................... 70 3.2 Free-Undamped Vibration of 2DOF Systems ............................................... 73 3.3 Example of a 2DOF System .......................................................................... 75 3.4 Case Study – Aberfeldy Footbridge, Scotland .............................................. 80 Continuous Structures ...................................................................................... 83 4.1 Exact Analysis for Beams.............................................................................. 83 2 Dr. C. Caprani

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Structural Analysis IV 5. 6. 4.2 Approximate Analysis – Bolton’s Method.................................................... 93 4.3 Problems ...................................................................................................... 102 Practical Design Considerations .................................................................... 104 5.1 Human Response to Dynamic Excitation.................................................... 104 5.2 Crowd/Pedestrian Dynamic Loading .......................................................... 106 5.3 Damping in Structures ................................................................................. 114 5.4 Design Rules of Thumb............................................................................... 116 Appendix .......................................................................................................... 121 6.1 Past Exam Questions ................................................................................... 121 6.2 References.................................................................................................... 126 6.3 Solutions to Differential Equations ............................................................. 128 6.4 Important Formulae ..................................................................................... 137 6.5 Glossary ....................................................................................................... 142 3 Dr. C. Caprani

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Structural Analysis IV 1. Introduction 1.1 Outline of Structural Dynamics Modern structures are increasingly slender and have reduced redundant strength due to improved analysis and design methods. Such structures are increasingly responsive to the manner in which loading is applied with respect to time and hence the dynamic behaviour of such structures must be allowed for in design; as well as the usual static considerations. In this context then, the word dynamic simply means “changes with time”; be it force, deflection or any other form of load effect. Examples of dynamics in structures are: • Soldiers breaking step as they cross a bridge to prevent harmonic excitation; • The Tacoma Narrows Bridge footage, failure caused by vortex shedding; • The London Millennium Footbridge: lateral synchronise excitation. k m (a) (b) Figure 1.1 4 Dr. C. Caprani