BRIDGES’ DYNAMICS By GEORGE T. MICHALTSOS IOANNIS G. RAFTOYIANNIS PROFESSOR ASSISTANT PROFESSOR Civil Engineering Department Civil Engineering Department National Technical University of Athens National Technical University of Athens Cover photo: The Brooklyn Suspended Bridge in New York eBooks End User License Agreement Please read this license agreement carefully before using this eBook. Your use of this eBook/chapter constitutes your agreement to the terms and conditions set forth in this License Agreement. Bentham Science Publishers agrees to grant the user of this eBook/chapter, a non-exclusive, nontransferable license to download and use this eBook/chapter under the following terms and conditions: 1. This eBook/chapter may be downloaded and used by one user on one computer. The user may make one back-up copy of this publication to avoid losing it. The user may not give copies of this publication to others, or make it available for others to copy or download. 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Photocopying Information for Users in the USA: Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by Bentham Science Publishers for libraries and other users registered with the Copyright Clearance Center (CCC) Transactional Reporting Services, provided that the appropriate fee of US $25.00 per copy per chapter is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers MA 01923, USA. Refer also to www.copyright.com CONTENTS Preface i Keywords ii CHAPTERS 1. Introduction 3 2. Distressing - Loading - Modelling of Vehicles 15 3. Principles of Dynamic Analysis 48 4. The Problem of Moving Loads 81 5. Motion of Supports 119 6. Structural Analysis of Bridges 136 7. Aeroelasticity 195 8. Dynamic Instability Problems 215 9. Damping Systems 257 General Bibliography 279 Alphabetical Index 280 i PREFACE This book deals with the problems arising from the dynamic distress of a bridge. Up to now, there is not a book containing the main objects and factors related and affected the dynamic behavior of a bridge. The present book focuses on this goal. After an extensive introduction, in which it is exposed not only the evolution of bridges but, also, an historical review of the research and the influence of each bridge destruction on this research (first chapter), the dynamic loadings are studied (second chapter). The dynamic wind loads, the aeroelasticity principles, the earthquake loadings, the air blast ones, the moving loads and their modeling are the objects of second chapter. The third chapter deals with the dynamic analysis principles. Firstly, the needed mathematical concepts are exposed, as for example: elements of calculus of variations, the d’ Alembert principle, Lagrange’s equation, the Hamilton principle, the equations of Heilig, and the δ and H functions. Afterwards the general equations of dynamic equilibrium are concluded and the main methods for their solution are exposed, while the axial, bending and the coupled lateral-torsional free and forced vibrations are studied determining simultaneously the corresponding orthogonality conditions. It is also exposed the nature and the influence of the internal and external damping. The fourth chapter deals with the problem of moving loads. After a short historical review the strict theory of the moving load-mass is exposed and the general equation of motion is found through the use of Hamilton principle, while the influence of a lot of factors are studied, as for example the damping, the speed, the load’s eccentricity, the influence of a real (of two axles) vehicle, the influence of the mass of the load, and of the secondary mass-forces (Coriolis and centripetal). Finally, the real conditions of the support of a load are studied and also the influence of the deck’s irregularities and deck’s roughness. The fifth chapter deals with the movement or rotation of bridge supports and the determination of the influence functions, while are studied the one, two, and three span beams. Finally the influence of the pylons’ height and their different movements are studied. In the sixth chapter is studied the bridge as a static system under dynamic loadings. They are studied: the one, two, and three spans beam, the arch bridges (circular and parabolic), the cable-stayed bridges (of radial and harp system) through a new analysis aspect, the suspended bridges, and the curved in plan bridges. For all the above static systems are given formulae for their eigenfrequencies, shape functions and orthogonality conditions. The seventh chapter deals with the problem of aeroelasticity. Firstly the exact theory of Theodorsen is exposed and, after, the aerodynamic loads of bridges are studied and the main models are shown, while are also studied the buffeting forces. Finally, the factors for the 2D and 3D problems are given. Afterwards the distress of a bridge caused by galloping, torsional divergence, and flutter is studied. For the completeness of the chapter, elements of the theory of modeling are given and the way of bridge’s study through the use of air-tunnels is briefly exposed. In the eighth chapter, the mathematical theory of dynamic instability is developed for problems in plane (2D) or in space (3D). The basic equation of Mathieu-Hill is studied and the Bolotin’s pioneer studies are exposed. Illustrative examples for the dynamic instability of the alone cable, or of a bridge pylon, or of a cable-stayed or a suspended bridge are given. In the last chapter, the absorb systems are studied. The passive, active and semi-active systems are presented and their properties and operation are analyzed in detail. The alone absorber and the system bridge-absorber are mathematically probed, while the equations governing the behaviour of such a system are given and solved. George T. Michaltsos Ioannis G. Raftoyiannis Civil Engineering Department National Technical University of Athens ii KEYWORDS Chapter 1 Ancient bridges, stone bridges, historical bridges, keystone, arch-type bridges, dome-type bridges, wooden bridges, concrete bridges, steel bridges, modern bridges Chapter 2 Dead loads, traffic loads, moving loads, vehicle modeling, snow loads, wind loads, aeroelastic loads, blast loads, earthquake loads, acceleration, deceleration, centrifugal forces, vehicle collision, support settlement, erection loads Chapter 3 Dynamic principles, energy axioms, dynamic equilibrium, equations of motion, damping, Ritz method, Galerkin method, free vibration, axial motion, flexural motion, torsional motion, critical damping, orhtogonality conditions, forced motion Chapter 4 Moving loads, moving mass, dynamic influence lines, velocity effects, damping effects, harmonic loads, vehicle modeling, vehicle mass, mass suspension, deck irregularities, surface irregularities Chapter 5 Support motion, support settlements, longitudinal motion, vertical motion, transverse motion, influence functions, support sliding, overturning, piers Chapter 6 Bridge modeling, bridge analysis, continuous systems, shape functions, frequency equations, bending vibrations, torsional vibrations, arched bridges, cable-stayed bridges, fan system, harp system, suspension bridges, curved-in- plane bridges Chapter 7 Aeroelasticity, wind flow, aerodynamic forces, buffeting forces, galloping instability, torsional divergence, fluttering instability, flutter coefficients, scaled models Chapter 8 Dynamic instability, Matheu-Hill equations, excitation parameter, Floquet solutions, critical eigenfrequencies, instability regions, instability boundaries Chapter 9 Damping systems, passive control, active control, internal damping, external damping, elastomeric dampers, slipping dampers, viscous dampers, gravity pendulum, spring pendulum, complex pendulum, rolling pendulum Bridges’ Dynamics, 2011, 3-14 3 CHAPTER 1 Introduction Abstract: This introductory chapter presents a brief historical review of bridge structures. From ancient times up to present days, bridge engineering is a continuously developing field of science although various construction materials have been used. From stone and wood in the past to concrete and structural steel at the present, various types of bridges have been designed and constructed. The most representative types of bridges are given schematically of in photographs. HISTORICAL EVOLUTION OF BRIDGES From Strabo 1.3.18*, one can conclude that the forerunners of bridges were dykewise passages like a sort of viaducts. Indeed, in the Greek language a bridge is called “γέφυρα” from the word “γαία”=earth, and the verbe “φυράω”= I make a dyke. In Latin, a bridge is called “ponte” and “pont” from the Greek word “πόντος”=the boundless sea, or abyss, or passage way. In Anglo-Saxon, the word bridge comes from the old Norse word brygga = landing stage, gangway or movable pier that has the same root with the old Greek word (Homeric word) βρύξ= the bottom of the sea, the sea’s abyss. In the ancient times, the quantity of mined iron and its difficult elaboration did not allow its use in structural works and, of course, in bridge building. Therefore, wood and stone were the main materials in use until the Middle Ages. Today, from the saved ruins we guess that the first bridges were constructed in Assyria, in Egypt and in Minoan Crete, although no historical reference regarding the above has been found anywhere. The introduction and use in buildings of plate-covered drains and domes in Assyria and Minoan Crete is very old. In excavations in Babylon’s Nineveh and Knossos, plate-covered sewers or sewers covered by domes of bricks made in 2,500 B.C. were found. In the reconstruction of Photo 1 the historical bridge of Nabuchadnezzar the second, in Babylon (600 B.C.), is shown. The oldest known existing bridges in the form of bridge-drain of small span (from 1 to 2.5m) are dated in the Minoan period in which we have the first significant morphological and static evolution. Photo 1: Ancient bridge in Babylon *“On the road to Syracuse there is a bridge that connects this town to the continental country, as the Roman poet Ibykos says, made from selected stone named hand-picked” George T. Michaltsos and Ioannis G. Raftoyiannis All rights reserved - © 2011 Bentham Science Publishers 4 Bridges’ Dynamics Michaltsos and Raftoyiannis Photo 2: Ancient bridge over water flow From the primitive plate-covered drain in Draconera of Argolis, Photo 2 (about 1,400 B.C.), until the most known “bridge of Kazarma”, Photo 3, on the side of the highway of Nafplion-Epidavros (about in 1,300 B.C.) one can see the progress in the span increase through the use of the corbel construction system, and finally the use of the functional keystone. According to the above corbel system, spans from 2.5 to 6 m were achieved, as one can see in the Eleftherna bridge of the later classic period, that is in use even today, with a span of 4.3 m (Photo 4). Photo 3: Ancient bridge in Epidavros Photo 4: Ancient bridge in Eleftherna Introduction Bridges’ Dynamics 5 The invention of keystone (or crown) led to the primitive evolution of the simple arch and later of the three-hinged one. There are two reasons that ancient Greeks did not make arch bridges from stone with spans bigger than 10 to 15m. The first reason is the non-existence of big rivers in Greece and Asia Minor, while the second and most important reason was due to military purposes. Indeed, the Greeks did not wish to have “permanent” bridges but bridges with a wooden deck that could be removed fast or destroyed or burned when the enemy came. This type of bridge with the above characteristics is the one called “Assos type”, with a wooden instead of a stone deck and was dominant until the Roman conquest of Greece. Some remains of this type of bridge (of the 4th century B.C.) are found in the town of Assos in Asia Minor with a total length of 52m and dense rhomboid hydrodynamic stone bases [3]. All saved bridges, mainly near or inside ancient cities, show the technical development and the perfection in the construction of stone bridges in the old years. In Photo 5, the bridge in Rodino in Rhodes with a span of 8 m, erected in 227 B.C. still in use is shown. Photo 5: Ancient bridge in Rhodes In Photo 6 one can see the bridge on Kifissos, in Elefsis, with a total length of 30m, a width of 5.30m and four arched spans of 7m in diameter, for the middle arch, 4.30m for the end spans with the use of keystones. It was erected in 125 A.D. after the order of emperor Adrian for his initiation in the mysteries of Elefsis. The Romans adopted the Greek arch and boldly enough built bridges with spans up to 30 m, while for bigger spans (35 to 40 m) they used wooden bridges. The more frequent type of stone bridges was the one with a semi-circular dome of one center for the construction of bridges on roads. These bridges showed great strength and many of them have been saved until today. Photo 6: Ancient bridge in Elefsis 6 Bridges’ DDynamics Micchaltsos and Raftooyiannis It is notewworthy that the above bridges were so finelyy constructed aand the dome-sstones without voids betweenn them so well fittted with each other that one cannot see thee joints. Eight bridges were erected on thee Tiber only. TTwo of them, the FFabricius and tthe Aelius briddges are still in use. In Photo 7 the Fabriciuus bridge is shoown while in Phhoto 8 one of the best saved briddges, the Pont ddu Gard erecteed by Hyrippa, a friend of Auugustus is showwn. Photo 7: Faabricius bridge Photo 8: Poont du Gard bridge During thee Middle Ages the span limitss changed onlyy a little. In the 18th century thhe Frenchman Perronet consttructed a stone briddge with a spann of 54m, whille the Swiss Grrubenmann connstructed a wooden bridge off 119m. At the end of the 18th cenntury starts, caautiously at firsst and then systematically, thhe use of iron inn the constructtion of bridges andd hence a new era in bridge bbuilding opens.. The first irron bridge in tthe world was made from caast iron by thee factories “Gooalbrookdale Irron Works” ovver the Severn river in Brosely -- England (Figg. 1) during 17776-79. This brridge has a spaan of 30.48m, arch height 122.10m, width of 7..30m and a totaal weight of 3778 tons. This brridge is still in use. Figure 1: TThe first iron briddge in Brosely - England