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Cable stayed bridges : nonlinear elastic dimensional analysis 1988

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C A B L E S T A Y E D B R I D G E S : N O N L I N E A R E L A S T I C D I M E N S I O N A L A N A L Y S I S Walter F. Janzen B. A. Sc. The University of British Columbia, 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1988 © Walter F. Janzen In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C ^ ' L £A/6/#e£&V6 The University of British Columbia Vancouver, Canada Date ' A?&L 2T?tt1*f DE-6 (2/88) Abstract Cable stayed bridges are the state of the art in long span bridges. Developments in cable response analysis and computer hardware and software have allowed engineers to design and build many cable stayed bridges. Bridges of many different configurations with longer and longer spans are being built. With the long spans and high axial loads in the deck, the stability of cable stayed bridges becomes a growing concern. Current analysis procedures use a parabolic approximation to the true catenary response of cables, which is quite accurate for tight cables; however, for near instability conditions with the accompanying large deflections and consequently loose cables, a catenary model is called for. Herein, a study is conducted on the elastic stability of cable stayed bridges utilizing a true catenary model of cable response. A dimensional analysis of cable stayed bridge stability is undertaken as well as deck maximum deflection and moment under service load conditions. A computer program written by this author is the analysis tool used in this work. The program is given the acronym ULA which stands for Ultimate Load Analysis. ULA is a nonlinear plane frame program with catenary cable elements and an interactive graphics interface. The dimensional analysis is carried out by comparing the computer model of a cable stayed bridge to the theoretical model of a beam on an elastic founda- tion. Two preliminary simplified cable stayed bridge backspan models are studied first in order to develop the dimensionless ratios that are applicable to the bridge response and to gain an insight into the cable stayed bridge behaviour. The final model is that of an entire cable stayed bridge. Because of the multitude of parameters governing cable stayed bridge behaviour it is not possible to describe all cable stayed bridges. Instead, the dimensionless behaviour of a standard model is examined and the sensitivity of this standard to various parameter variations is given in the form of dimensionless charts. ii Finally, a preliminary design and analysis aid is developed from the dimensionless charts and is then applied to two existing bridge designs. iii T a b l e o f C o n t e n t s A b s t r a c t i i L i s t of T a b l e s v i i L i s t of F i g u r e s v i i i N o m e n c l a t u r e x i i S p e c i a l S u b s c r i p t N o t a t i o n x i v A c k n o w l e d g e m e n t x v 1 I n t r o d u c t i o n 1 1.1 Development of Cable Stayed Bridges 1 1.2 Advantages of M u l t i - C a b l e Stayed Bridges 6 1.3 Scope of Thesis , 7 2 A n a l y s i s S y s t e m 9 2.1 Current Analys is Procedures 9 2.1.1 Deck Stabi l i ty 10 2.1.2 Br idge Response 11 2.2 Proposed Analysis Procedure 13 2.3 Dimensional Analysis 15 2.4 Computer Models Analyzed 15 3 S i m p l i f i e d B a c k s p a n M o d e l 18 3.1 The M o d e l 18 iv 3.2 Stabi l i ty 20 3.2.1 Stabi l i ty Dimensionless Ratios 21 3.2.2 Parameters and Ratios from Exis t ing Bridge Designs 26 3.2.3 Results of Stabil i ty Analysis 28 3.3 Deflection and Moment 35 3.3.1 Deflection Dimensionless Ratios 35 3.3.2 Results of Deflection Analysis 39 3.3.3 Moment Dimensionless Ratios 44 3.3.4 Results of Moment Analysis 46 4 Backspan Model 53 4.1 T h e M o d e l 53 4.2 Stabi l i ty . . 55 4.2.1 Stabi l i ty Dimensionless Ratios 55 4.2.2 Parameters and Ratios from Exis t ing Bridge Designs 58 4.2.3 Results of Stabil i ty Analysis 59 4.3 Deflection and Moment 66 4.3.1 Results of Deflection Analysis 67 4.3.2 Results of Moment Analys is 71 5 Full Bridge Model 76 5.1 The M o d e l 76 5.2 Stabi l i ty 82 5.2.1 Stabi l i ty Dimensionless Ratios 82 5.2.2 Parameters and Ratios from Exis t ing Bridge Designs 85 5.2.3 Results of Stabil i ty Analysis 87 5.3 Deflection and Moment 98 5.3.1 Results of Deflection Analysis 101 v 5.3.2 Results of Moment Analysis 108 6 Application 116 6.1 The Model 116 6.2 Procedure 117 6.2.1 Stability 118 6.2.2 Deflection and Moment 118 6.3 Examples 119 6.3.1 Stability 119 6.3.2 Deflection and Moment 120 7 Conclusion 124 7.1 Further Research 125 Bibliography 128 A Catenary Cable 131 B TJLA Implementation of Catenary Cable 140 C Data From Existing Cable Stayed Bridge Designs 144 C. l Backspan Data from Bridge Designs for Simplified Backspan Model . . . 146 C.2 Backspan Ratios from Bridge Designs for Simplified Backspan Model . . 147 C.3 Backspan Data from Bridge Designs for Backspan Model 148 C.4 Backspan Ratios from Bridge Designs for Backspan Model 149 C.5 Bridge Data from Bridge Designs for Full Bridge Model 150 C.6 Bridge Ratios from Bridge Designs for Full Bridge Model 152 D Maximum Deflection and Moment 153 vi List of Tables 6.1 Check on applicability. 119 6.2 Summary of stability analysis 120 6.3 Summary of deflection analysis 121 6.4 Summary of moment analysis 122 vii List of Figures 1.1 First modern cable stayed bridges 3 1.2 Cable bearings in towers. 4 1.3 Modern cable stayed bridge 5 1.4 Truss model of cable stayed bridge 5 2.5 Progressive computer models of cable stayed bridges. (Models drawn to different scales.) •. . 17 3.6 Simplified backspan model 20 3.7 Derivation of elastic foundation modulus 23 3.8 Stability failure of simplified backspan model 29 3.9 Stability factor of safety with uplift pressure 30 3.10 Stability of simplified backspan model (F' jFJJ) 33 3.11 Stability of simplified backspan model 33 3.12 Stability of simplified backspan model—n varied 34 3.13 Examples of deflection decay in a beam on an elastic foundation. . . . 36 3.14 Dimensionless plot of Hetenyi's deflection 39 3.15 Deflection of standard simple backspan model 40 3.16 Magnification of maximum deflection. . . 41 3.17 Dimensionless deflection versus dimensionless cable tautness 42 3.18 Dimensionless deflection versus dimensionless decay length 43 3.19 Dimensionless deflection versus dimensionless load 44 3.20 Dimensionless plot of Hetenyi's moment 46 3.21 Moment of standard simple backspan model. 47 3.22 Bridge deck construction methods 48 viii 3.23 Magnification of maximum moment 49 3.24 Dimensionless moment versus dimensionless cable tautness 51 3.25 Dimensionless moment versus dimensionless decay length. . 51 3.26 Dimensionless moment versus dimensionless load . 52 4.27 Backspan model 54 4.28 Free-body diagram of a cable-deck connection 55 4.29 Stability failure of backspan model 60 4.30 Stability factor of safety of backspan model 61 4.31 Critical simplified backspan section 62 4.32 Critical backspan section 63 4.33 Stability of backspan model 65 4.34 Stability of backspan model—ID/IB varied 65 4.35 Deflection of standard backspan model 68 4.36 Magnification of maximum deflection. 69 4.37 Dimensionless deflection versus dimensionless decay length 70 4.38 Dimensionless deflection versus dimensionless cable tautness 70 4.39 Dimensionless deflection versus dimensionless load. . 71 4.40 Moment of standard backspan model 72 4.41 Magnification of maximum moment 73 4.42 Dimensionless moment versus dimensionless decay length 74 4.43 Dimensionless moment versus dimensionless cable tautness 74 4.44 Dimensionless moment versus dimensionless load 75 5.45 Bridge model (half shown—symmetric, except right bearing is a fixed pin) 77 5.46 Free body diagram of bridge model (half shown—symmetric) 80 5.47 Axial influence lines and load cases for instability. 88 5.48 Stability failure of standard bridge model—load cases 1 and 2 89 ix 5.49 Stability failure of standard bridge model—load cases 3 and 4 90 5.50 Stability factor of safety of bridge model 92 5.51 Critical deck section for each load case 94 5.52 Stability of full bridge model 95 5.53 Critical deck section for varying F/FJJ 96 5.54 Factor of safety vs F/FH 97 5.55 Stability of full bridge model—ID/IB varied 98 5.56 Stability of full bridge model—ACa/^2-Ac varied 99 5.57 Stability of full bridge model—ETIT/EI varied 99 5.58 Load cases for maximum backspan and mainspan deflection 102 5.59 Maximum backspan deflection (magnified 200 times) of standard full bridge model 102 5.60 Maximum mainspan deflection (magnified 200 times) of standard full bridge model 103 5.61 Magnification of maximum deflection 104 5.62 Dimensionless deflection versus dimensionless decay length 105 5.63 Dimensionless deflection versus dimensionless cable tautness 106 5.64 Dimensionless deflection versus dimensionless load. 106 5.65 Dimensionless deflection versus dimensionless cable area 107 5.66 Dimensionless deflection versus dimensionless tower stiffness 107 5.67 Load cases for maximum backspan and mainspan moment 108 5.68 Maximum backspan moment of standard full bridge model 109 5.69 Maximum mainspan moment of standard full bridge model 110 5.70 Magnification of maximum moment. . . I l l 5.71 Dimensionless moment versus dimensionless decay length 112 5.72 Dimensionless moment versus dimensionless cable tautness 113 5.73 Dimensionless moment versus dimensionless load. 114 x 5.74 Dimensionless moment versus dimensionless cable area 115 5.75 Dimensionless moment versus dimensionless tower stiffness 115 7.76 Plastic stability failure of bridge model 127 7.77 Plastic hinge failure of bridge model 127 A.78 Catenary cable element 131 A.79 Catenary cable 132 A.80 Pin ended member equivalent to catenary cable 135 A.81 Elongation of catenary cable 135 A.82 Bar extension under a change in tension 138 A. 83 Relative motion perpendicular to cable chord 138 B. 84 Behaviour of catenary cable under load changes 140 xi Nomenclature Ac cable cross sectional area Aca total anchor cable cross sectional area E deck elastic modulus Ec cable elastic modulus Ex tower elastic modulus F axial load in deck at tower due to dead load FH buckling load for a beam-column on an elastic foundation f dimensionless cable tautness H horizontal cable force h cable height hx tower height I deck moment of inertia IT tower moment of inertia k elastic foundation modulus L cable horizontal projected length IB backspan length ID backspan decay length IM mainspan length MB maximum backspan moment MM maximum mainspan moment MH Hetenyi's beam on an elastic foundation moment N number of deck intervals in the backspan Nc number of cables in the backspan or half the mainspan n number of half-waves in a buckled shape xi i P axial load in tower at base due to dead load PT Tang's critical axial load in the deck 5 cable chord length s cable spacing at deck or length of deck interval s' cable spacing at tower T dead load cable tension in the chord direction Tt service (dead plus live) load cable tension wc weight per unit length of cable tUjo uniformly distributed dead load U>L uniformly distributed live load a dummy variable f3 dummy variable pY Tang's elastic foundation constant q dummy variable 7c cable density A cable elongation in the chord direction 6 deck vertical deflection 6B maximum backspan deflection 8M maximum mainspan deflection SJI maximum deflection of a beam-column on an elastic foundation A load level o~c cable stress 6 cable chord angle of inclination to deck xi i i Special Subscript Notation avg average c critical cen centroid eff effective sec secant tan tangent ^ half-wave • standard model * value from computer analysis (superscript) xiv Acknowledgement I would like to thank Dr. Roy Hooley for his support and direction throughout this research. This work would not have been possible without the funding provided by the Na- tional Sciences and Engineering Research Council of Canada. I would like to thank the University of British Columbia for providing an excellent research environment, and I thank the Civil Engineering Department for their help and for the provision of excellent computer analysis tools by way of their Computer Graphics Laboratory. I would especially like to express my gratitude to the two ladies in my life. I thank my mother for her confidence in me and my wife for her patience and support. xv Chapter 1 Introduction 1.1 Development of Cable Stayed Bridges The concept of supporting a beam with stays is not a new one. Egyptians used the idea for their sailing ships and in some tropical regions vines attached to trees were used to support walkways. Several cable stayed bridges were built in Europe from 1784 to 1824 using timber, wire, chain and iron bar stays. However, the development of cable stayed bridges stopped there due to the collapse of several bridges. It was Navier who reported on these failures and made the comment that suspension bridges were preferable. This suggestion of Navier's was enough to cause bridge engineers to abandon the cable stayed bridge concept [1,2]. The use of stay cables was not entirely abandoned, as several suspension bridges had inclined ropes in addition to vertical hangers. However, the low strength of the stays precluded prestressing, so the stays were installed in a slack condition. These stays would simply elongate due to sag reduction when loaded, thus were ineffectual under service loads. It was not until 1938 that stay cables were again used in a bridge design as main load carrying members. Dischinger had the task of designing a railway bridge over the Elbe River and he found that the deflections of a suspension bridge could be significantly reduced if tight wires were used as supplemental stays. After World War II, the rebuilding of Germany's bridges brought the cable stayed bridge back into favour, largely due to the need for economy of steel. Earlier cable stayed bridges failed due to inferior materials and a misunderstanding of the structural behaviour of the cables. Now, with better materials and armed with Dischinger's 1 Chapter 1. Introduction 2 findings, which were published in 1949 [3], cable stayed bridges were proposed for the reconstruction of bridges over the Rhine River. The first modern cable stayed bridges were essentially beam bridges with only 2 to 6 stays in the main span as shown in Figure 1.1. The spans between cable supports were between 30 and 60 meters, with the cables acting as major supports, taking the place of piers. This reduced the bending stresses in the deck, allowing smaller cross sections; hence, a saving in steel. With the limited number of stays along the span, the cables had to carry major portions of the bridge dead load and live load, thus the cables were very large and the connections to the deck and anchor were quite complicated. To reduce tower moments, special roller and rocker bearings were built which allowed the cables to pass through the tower, as shown in Figure 1.2. Along with the heavy members and elaborate bearings, the construction of these bridges still required falsework. Several arrangements of cable layout and tower shape were used as can be seen in Figure 1.1. Radiating cable layouts with the cables converging to one area on the tower, and harp designs with the cables parallel to each other were used in conjunction with single plane cable, double plane cable and double inclined plane cable systems (A-frame towers). These configurations were used in single tower and multi-tower bridges. The deck was sometimes rigidly supported at the tower and in other designs the deck passed through the tower with only side bearings provided. Fixed-base and pinned-base towers were both used. As experience was gained in the design and construction of cable stayed bridges it was found that most difficulties disappeared if a larger number of stays was used such that the cable spacing at the deck was from 8 to 15 meters [4]. With this many cables, it was no longer possible to connect all the cables at one point for a radiating layout and complicated tower bearings for harp bridges had to be abandoned. Figure 1.3 shows a modern cable stayed bridge with a fan shape arrangement of cables. In this system, the deck behaves more like the bottom chord of a truss, rather Chapter 1. Introduction p i rS ^—r\—^ ^ \ 74 m 183 m 74 m Stromsund Bridge Stromsund, Sweden (1955) 108 m 260 m North Bridge Diisseldorf, West Germany (1958) 108 m 301 m 151 m Severin Bridge Cologne, West Germany (1960) 106 m 280 m Leverkusen Bridge Leverkusen, West Germany (1964) 106 m Figure 1.1: First modern cable stayed bridges. Chapter 1. Introduction 4 Rocker Figure 1.2: Cable bearings in towers. than a simply supported beam. Th i s analogy is outl ined in Figure 1.4. The basic element (a) is a p in connected truss made of four members, and the weight W is supported entirely by axial stress. The weight can be applied in different positions along the deck, and to accomodate this , extra members are bui l t into the truss (b). A complete bridge can be modelled as a p in connected truss (c) and al l loads on the deck panel points are supported by axial stress. W i t h the deck modelled as a continuous beam rather than p in connected beams, the structure s t i l l behaves as a truss wi th only secondary bending stresses in the deck. The final extension of the truss model is to distribute the cable connections along the towers (d). Th i s induces pr imary bending stress into the structure to produce a frame not a truss; however, the bridge can be made nearly moment free for dead load. The evolution toward modern cable stayed bridges was made possible by the de- velopment of orthotropic steel and concrete decks that could bear large axial forces, together w i t h the advent of the computer. W i t h the nonlinear cable behaviour mod- elled in a computer program and wi th the tremendous computing power available, the analysis of multi-cable stayed bridges became easy. Chapter 1. Introduction Alex Fraser Bridge Vancouver, British Columbia, Canada (1986) Figure 1.3: Modern cable stayed bridge. Chapter 1. Introduction 6 1.2 Advantages of Multi-Cable Stayed Bridges There are many advantages of multi-cable stayed bridges over other types of bridges. The original necessity that brought about cable stayed bridge designs was to have long span bridges that used a minimum of steel. With the cables acting as supports from above the deck, the cost of constructing deep water falsework was eliminated and the passage for navigation was kept open during construction. Cable stayed bridges share the advantage with suspension and arch bridges that the dead load stresses can be controlled by the erection procedure. With some skill during construction, the dead load of the bridge can be carried moment free. In addition, cable stayed bridges can have smaller live load moments. For suspension and three-hinged arch bridges the maximum live load moment is approximately WL12mI&Q, while for a cable stayed bridge it is about wLlM/440. Thus, the deck moments are greatly reduced and a smaller cross section is possible. This results in a lighter bridge that is easier to support and allows even longer spans. Both suspension and cable stayed bridges can have aerodynamically streamlined decks, but cable stayed bridges have inherent characteristics that make them superior to suspension bridges. In a cable stayed bridge, each of the cables—with the mass of the deck attached to it—has a different natural frequency, and if external forces try to set up resonant flexural oscillations of the deck, the vibrations are effectively dampened by the interference of cables with different frequencies. A more dramatic aerodynamic advantage is in the resistance to the antimetric tor- sional mode of oscillation—the mode which led to the collapse of the Tacoma-Narrows Bridge. In a suspension bridge the two cables can move against each other, one going up in half of the span and the other going down; thus, allowing torsional movement of the deck. In cable stayed bridges the cables themselves provide a large resistance against this torsional movement. The difference in deflection of the two cable planes is mainly due to the different deflection of the towers in each plane, and with the tower Chapter 1. Introduction 7 tops being tied with stiff back stays, the tower movements are much smaller than sus- pension bridge towers. Further torsional resistance is possible by connecting the towers as shown in Figure 1.1 (top bridge) or by constructing an A-frame as shown by the third bridge of Figure 1.1. Thus, cable stayed bridge decks need not have as large a torsional stiffness as suspension bridge decks. Further economic advantages are acquired in the ease of erection of cable stayed bridges. The need for heavy suspension cable anchors is gone, and with fixed base towers the cantilever method of construction can be employed, which eliminates the need for falsework. As the number of cables in a span increases, the cables become lighter, the connections are simpler and the construction cantilever length decreases. The construction misalignment at the center of the main span is easily adjusted by slackening and tightening the appropriate cables. Finally, and perhaps paramount to the public who use bridges every day, cable stayed bridges are very aesthetically pleasing. 1.3 Scope of Thesis The work presented herein was undertaken in order to compare existing cable stayed bridges and to allow preliminary design and analysis without the use of a computer. A large number of cable stayed bridges exist, all of different spans, tower sizes, cable con- figurations and deck cross sections. In order to compare the wide range of cable stayed bridges, dimensional analysis is used to study the stability, deflections and moments of a cable stayed bridge computer model. Once the behaviour of cable stayed bridges is presented in the form of dimensionless charts, a preliminary design and analysis aid which uses these charts, is developed. This thesis is applicable to multi-cable stayed bridges with a fan type of cable arrangement (Fig 1.3). Double-tower bridges with fixed bases are modelled with an elastic support—equivalent to one cable—for the deck at each tower. The cables are Chapter 1. Introduction 8 attached to the towers rather than allowed to pass through. The response of the bridge computer model is presented as an analogy to a simple beam on an elastic foundation. Nonlinear elastic computer analysis is done for deck stability as well as for maximum deflection and moment. The response of the deck is determined for distributed live loads. A plane frame analysis is used, thus out of plane loads from wind or earthquake are not considered. Torsion is not considered and axial deformations of the deck and towers are neglected. C h a p t e r 2 A n a l y s i s S y s t e m 2.1 C u r r e n t A n a l y s i s P r o c e d u r e s Cable stayed bridges are complicated structures to analyze, just like any other bridge or structure. Analys is procedures for calculating the deflections, moments and buckling behaviour, must be derived based on a model of the actual structure and the materials it is made of. Some assumptions used in the model can be very true to the real s i tuat ion, such as Young's modulus being constant wi th increasing strain and plane sections remaining plane upon a change in curvature of a beam. B u t all models have their l imits of val idi ty such as Young's modulus changing when strains become plastic and plane sections not remaining plane for certain cross section shapes. Some of the modell ing assumptions in cable stayed bridges concern the deck stiffness, the axial load in the deck, the cable behaviour and the beam-column on an elastic foundation behaviour of the deck. The deck stiffness of cable stayed bridges can for the majority of the span be constant because of the nature of the moment distr ibution in the deck. However, because of the increasing axial load induced in the deck by the cables, the cross-sectional area of the deck near the towers might be increased, thus increasing the stiffness as a by-product. The cables are of varying cross-sectional area and tension and hang in the shape of a catenary which results in a varying nonlinear cable behaviour along the span. The combination of the deck stiffness and axial load variations and the cable variat ion w i t h the cable nonlinearity and deck beam-column nonlinear behaviour makes the beam-column on an elastic foundation model very complex. The following two sections are overviews of how some researchers have modelled the 9 Chapter 2. Analysis System 10 cable stayed bridge behaviour at buckling loads and at service loads. 2.1.1 D e c k S t a b i l i t y The most commonly used check for deck buckling is Tang's approximate formula [5] : PT = 2yjEI0T where PT — Tang's cr i t ical axial load in the deck E = deck elastic modulus I = deck moment of inertia PT Tang's elastic foundation constant PT = EcAc/sS Ec = cable elastic modulus Ac = cable cross-sectional area S = cable chord length s length of deck interval between cables Th i s is derived from an analogy wi th a beam-column on an elastic foundation where the axial load is constant throughout the beam-column and the beam stiffness does not vary. Th i s formula is for buckling wi th no variat ion of cable stiffness along the beam and wi th the cables themselves modelled as linear springs. Tang has compared the results from this approximation wi th that obtained from a computer model based on an energy method that includes the tower interaction wi th the deck and cables (modelled as linear springs). The computer model accounted for the varying deck axial load and stiffness as well as the varying cable stiffnesses. It was found that the two models give almost identical results for the cri t ical axial load. Tang's approximation is good for short span bridges only. The deflections of short span bridges are s t i l l fairly small near instability, so good analytical results can be Chapter 2. Analysis System 11 obtained from the linear cable model. W i t h cable stayed bridges being designed wi th ever longer spans and increasingly slender decks, closer investigation which includes cable nonlinearity, is required. 2.1.2 Bridge Response H a n d calculat ion procedures to determine several prel iminary cable stayed bridge pa- rameters are given in an excellent book by Gims ing [6]. Troi tsky [2] also gives approxi- mate hand calculat ion methods in a chapter of his book. However, because cable stayed bridges are highly statically indeterminate, computer solutions are needed to perform the final detailed analyses of these structures. Several authors have presented methods of cable stayed bridge analysis suitable for computer application. Smi th [7,8] developed a mixed force-displacement method for analysis of single and double plane cable stayed bridges. This was one of the first analyt ical tools developed for cable stayed bridges and the effects of cable sag and beam-column behaviour were not included. Troi tsky and Lazar [9] used a flexibility method to compare wi th their tests on a physical model . Thei r physical model included nonlinear effects but their analytical model was s t i l l linear. Lazar [10] also used the stiffness method and included nonlinear cables wi th movable tower-cable connections. Tang [11] employed the transfer mat r ix method and reproduced the nonlinear cable behaviour w i th fictitious loads. A three-dimensional linear finite element procedure was presented by K a j i t a and Cheung [12] for both static and dynamic analysis. Rajaraman, Loganathan and R a m a n [13] modified a plane frame stiffness program to include beam- column and cable nonlinearity. Como, Gr ima ld i and Macer i [14] developed a continuous model of cable stayed bridges based on their truss-like behaviour and presented the results in nondimensional form. A n energy method of analysis was developed by Hegab [15] in which a smal l number of Fourier terms for the assumed deflected shape of the deck was sufficient to allow quick convergence. Chapter 2. Analysis System 12 In all of the above references where cable nonlinearity was modelled, the Ernst [16] or Dischinger equivalent modulus of elasticity was used. For cable stayed bridges with small live to dead load ratios, as was the case for early bridges, the tangent modulus could be used: E c E t a n ~ + ^cLfAcEc 123? where Eton — cable tangent modulus of elasticity Tt — service (dead plus live) load cable tension L = horizontal projected length of cable wc = weight per unit length of cable and the other symbols are as defined previously. For multi-cable stayed bridges the deck is slender, thus lighter than early bridges which results in a higher live to dead load ratio. The initial cable tautness is much less than was the case for early cable stayed bridges, thus the change in cable tension due to the application of live loads is more important and the secant modulus should be used: E~ = (wcL)2AcEc(T~+JVj {2-l) 24T2T, where E t e c — cable secant modulus of elasticity T = dead load cable tension The derivation for these formulas assumed a parabolic rather than a true catenary cable shape and the force in the chord direction was taken as the cable tension. Ito and Maeda [17] presented a derivation based on a parabolic cable shape but used the Chapter 2. Analysis System 13 true cable tension. They noted however, that the difference between using the force in the chord direction and using the true cable tension or the force tangent to the cable was negligible. 2.2 Proposed Analysis Procedure The model used in developing the analysis procedure used herein incorporates cate- naries as the cable shapes with the secant equivalent modulus, P—A effects for beam- columns, and is limited to elastic behaviour. The analyses conducted for this thesis were done using a plane frame computer program written by this author. The program was designed for ultimate load analysis combining second order effects with plastic hinge formation and is given the acronym ULA. The nonlinear analysis is achieved by incorporating stability functions into mem- ber matrices to model the P-A behaviour and by using equivalent areas for cables based on the catenary shape to model the effect of cable sag. Further details of ULA con- cerning plastic hinge formation, moment axial interaction, strain hardening and the interactive graphics capabilities are given in a thesis by Mill [18]. Details of ULA that are important to this thesis are given in the following paragraphs. The stability functions used are standard and are presented by many authors in- cluding Weaver and Gere [19]. The stability functions depend on the axial forces, and the axial forces depend on the deflected shape. It is therefore necessary to use an iter- ative procedure to find the solution. This is handled by the interactive graphics format of ULA, where the user can decide upon convergence of the P—A effect by observing the changes in the determinant of the structure matrix. Most authors use an equivalent Young's modulus to model the stiffness of straight line cable members as actual sagging cables. Herein, an equivalent area is used instead. This choice is a matter of convenience only, in that it is the stiffness of the cable that changes with sag and not the area or elastic modulus. The data input requirements of Chapter 2. Analysis System 14 ULA are such that each cable can be assigned a unique area easier than an individual modulus. The equivalent cable area also depends on the axial forces and an iterative procedure is necessary just as for the stability functions. However, this does not mean that seperate iterations need to be done because the recalculations for cable sag can be performed during the same iterations that are required for P — A convergence. The convergence of cable area change is monitored by ULA in the same manner and at the same time as that for P— A . Most authors also use what they cite as a parabolic derivation of equivalent cable stiffness (Equation 2.1). The derivation is in fact a hybrid, in that a parabolic shape of the cable is assumed while the weight per unit cable length is used instead of the load per horizontal projection as for a true parabolic deflected shape. Authors were hesitant to use a full catenary derivation because of its complexity, but with series substitution it becomes manageable. The derivation by Hooley [20] is repeated in Appendix A. Note that the equivalent stiffness from this derivation is exactly the same as that from the hybrid derivation. This can be seen by manipulating Equation A . l appropriately until Equation 2.1 appears with A c in the place of EQ. As stated previously, the two derivations produce identical results because of the cable self weight manipulation. In a parabola, the self weight is a constant load per horizontal length; in a catenary, it is a constant load per cable length. The two self weights are related by cos 6 where 0 is the angle between the horizon and the tangent to the cable at any point along the cable. If the cable is tight, then 6 is approximately constant and the parabolic and catenary solutions are the same. For a slack cable, 6 is not constant and the parabolic solution will not be the same as the catenary. A comparison between the parabolic method and the catenary method has been made by this author and found that for tight cables ( / = 7) there is only a 0.1% to 1% difference in deflections. For slack cables ( / = 2), the parabolic solution is 10% to Chapter 2. Analysis System 15 15% in error from the catenary solution. A graphical representaion of cable tautness / is given in Appendix B along with an explanation of the implementation of cables into ULA. 2.3 Dimensional Analysis The response of a cable stayed bridge depends on a multitude of parameters. With the use of dimensional analysis, the parameters can be isolated and the behaviour of a structure can be studied in a more logical manner. The principles of dimensional analysis are well known, with a good explanation of the theory given by Goodier [21]. The procedure in dimensional analysis is to identify all the independent parameters affecting the structure and then construct independent dimensionless ratios using these parameters. A very useful feature of this technique is that the number of dimensionless ratios to be studied is two less than the number of parameters. Comparisons can be made between the complex structure at hand and a simpler similar structure. Thus, the complex structure can be thought of as the simple struc- ture with some adjustments. The use of a simple model is also an aid in choosing the parameters that would affect the real structure. The dimensionless ratios are indepen- dent, so one ratio can be altered and the response studied without affecting the other ratios. The results from dimensional analysis are applicable to all similar structures, for which the characteristic ratios are the same. 2.4 Computer Models Analyzed In order to gain an understanding of cable stayed bridge behaviour, the analogy to a beam-column on an elastic foundation will be made. Three computer models of pro- gressive complexity are studied. An extremely simple model is studied first to facilitate the derivations of the dimensionless ratios used to describe the model. The first model Chapter 2. Analysis System 16 is also designed to display very similar behaviour to that of a beam-column on an elas- tic foundation. This model is shown in Figure 2.5 and represents the backspan of a cable stayed bridge with the use of uniform cables of constant length. The first model has a constant foundation modulus and is not a good representation of a cable stayed bridge backspan; therefore, the second model is designed to be a truer representation of a cable stayed bridge backspan without adding too many complexities. The second model—also shown in Figure 2.5—has varying cable areas and lengths and the cables are attached to an immovable tower. This model will display the effects of a varying foundation stiffness without the added complexity of a flexible tower. The third model is of an entire cable stayed bridge. The boundary conditions are drawn on Figure 2.5 as well, with the deck vertical support at the tower being equivalent to one vertical cable stay. In all three models, dimensional analysis is applied to a standard bridge and then variations from the standard are studied. The standard bridge is defined by setting the dimensionless ratios that describe the bridge to representative values as determined from existing cable stayed bridge designs. With the understanding gained from the simple model and the intermediate model, the full bridge model behaviour can be explained, and the important parameters can be identified. Once the response of the full bridge model is understood, comparisons can be made with existing cable stayed bridges and an analysis procedure can be developed using beam-column on an elastic foundation theory and the results of the dimensional analysis. Chapter 2. Analysis System Backspan Model (Chapter 4) Full Bridge Model (Chapter 5) Figure 2.5: Progressive computer models of cable stayed bridges. (Models drawn different scales.) Chapter 3 Simplified Backspan Model 3.1 The Model A study of a complete cable stayed bridge would be complex, because the relative influence of the towers, the deck—backspan and mainspan—and the cables would be difficult to separate. Therefore, simple models are studied first, and the knowledge gained from these analyses is applied to the more complex model. Simple models are also useful to start with because there are fewer variables to study than in the complete model. Two models of the backspan—isolated from the tower and mainspan—are chosen as the simple models to study before looking at the full bridge model. The first backspan model—called "simplified backspan model"—is designed to have a uniform elastic foun- dation; the second model, which is discussed in Chapter 4, has a nonuniform elastic foundation. A sketch of the simplified backspan model, with all the possible parameters defined, is shown in Figure 3.6. In this model all the cables have the same modulus of elasticity, dead weight, area, and length and are attached to the deck at the same angle at equal intervals. The leftmost cable is included to simulate the backstay. The deck is modelled as having constant stiffness EI and the live load is applied as a distrubuted load. The definition of all the parameters shown in Figure 3.6 is given in the following: Ac = cable cross sectional area E = deck elastic modulus Ec = cable elastic modulus 18 Chapter 3. Simplified Backspan Model 19 F — maximum axial load in deck due to dead load h = cable height I = deck moment of inertia L = horizontal projected length of cable IB = backspan length N = number of deck intervals in the backspan S cable chord length s = length of deck interval between cables T = cable tension in the chord direction due to dead load wc = weight per unit length of cable WD uniformly distributed dead load UlJLc = critical uniformly distributed live load e cable chord angle of inclination to deck Initially the model is under dead load only and the cable tensions T are set up such that there are no moments or deflections in the deck under the dead load. All the cable tensions are the same, with the cable attached to the roller bearing given the same tension as well. The parameter F is the maximum axial load in the deck under dead load only. The axial load distribution induced into the deck by the cables is nonuniform and the maximum occurs at the rightmost end. The computer program ULA has the capability to have axial prestress input as part of a load case. This feature is used to model the dead load cable tensions and dead load deck axial load distribution. The structure dead load is in equilibrium with the prestress; thus, if the prestress is applied as part of a load case, the dead load is not. In other words, the cable tensions represent the dead load, and the distrubuted 20 H I M TTT1 I I I I I I I I I I » ! • I < I t t < I I T N = Number of deck spaces s Figure 3.6: Simplified backspan model. dead load is not actually applied as a part of any load case. This also means that the input geometry is the dead load geometry. It should be noted that the P—A effects are calculated from the stability functions using the the initial prestress plus the axial load from a specific load case. The following sections will deal with the buckling behaviour of the model and the deflections and moments of the deck under application of live load. 3.2 Stability Modern cable stayed bridges are being built with increasingly slender decks. With the longer spans and smaller deck cross sections of these bridges, the factor of safety against buckling becomes an ever more important consideration. Tang's formula for buckling load is acceptable for short span cable stayed bridges; however, as Tang points out, more detailed analysis is required for long span structures. Herein, a closer investigation of Chapter 3. Simplified Backspan Model 21 cable stayed bridge elastic stability is carried out. 3.2.1 Stability Dimensionless Ratios Not all of the 16 parameters shown in Figure 3.6 can be used as variables in dimensional analysis because they are not all independent of each other. For example, 5 is a function of L and h, so only two of these three parameters can be used in defining the response of the model, thus reducing the number of parameters that need be considered. A further reduction in the number of parameters is accomplished by combining E and I to get the bending stiffness of the deck EI, and combining Ac and Ec to get the axial stiffness of the cables AcEc- The 16 parameters reduce to nine independent ones and are given in the following function: Even though the distributed dead load is not actually applied in the computer analyis, the parameter " I W D " is retained because it is more convenient to use than T and F. The five parameters not used in Equation3.2 are related to the nine parameters of Equation3.2 by: wLt = ?[EI, ACEC, lB, / i , 6, N, wc, wD] (3.2) L = h tan 6 S = h sin0 T = VJDlB Ns'm6 and N Chapter 3. Simplified Backspan Model 22 The next step in dimensional analysis is to construct independent dimensionless ratios from the independent parameters. Buckingham's II theorem states that the number of dimensionless ratios required is the number of parameters less the number of fundamental units needed to quantify all of the parameters. Since the two units of length and force describe all the variables, seven ratios need to be constructed out of the nine parameters in Equation 3.2. The seven ratios chosen to represent this model are: s'md A h U>LC wD WDlB 2 tan 6 N AcEpEI sin* 0 lah IB 4 NACEC sin3 6 lBhEI WDIB Ns\n$, . ACEC (wch\ 2 \ 24 \txn0J All of these dimensionless ratio have a physical meaning—some more obvious than others—and the remainder of this section is devoted to deriving and explaining the significance of these dimensionless ratios. Since the stability behaviour of a cable stayed bridge is to be investigated, Hetenyi's [22] critical load for an infinitely long beam-column on an elastic foundation is used as part of one ratio. FH = 2VkEI (3.3) Chapter 3. SimpliBed Backspan Model 23 where Fff = buckl ing load for a beam-column on an elastic foundation k = elastic foundation modulus Note that F R , k, and EI in Equat ion 3.3 are al l constant along the beam-column. For the derivation of k, a single cable is isolated along wi th the t r ibutary length of deck associated w i th i t , as shown in Figure 3.7. Figure 3.7: Derivat ion of elastic foundation modulus. F r o m geometry: A = SsinB where A = cable elongation in the chord direction 6 = deck vertical deflection and from Hooke's law: T = A A C E C Chapter 3. Simplified Backspan Model 24 For a unit of deck deflection 6 — 1 and span s the foundation modulus k is V/s, where V is the vertical cable force Tsintf. This results in k = A c E c s [ n 2 e ( 3 4 ) SO Further substitutions using N = I B / S and 5 = h/s'md yield IB", Equation 3.4 can be compared with Tang's H T sS Tang omits the sin2 6 term, but the cable properties near the tower—where the deck axial load is greatest—are used in his formula. As long as the cable inclination at the critical deck section is greater than 72° the error in using Tang's approximation is less than 10%. Substituting Equation 3.5 into Equation 3.3 results in NAc ECEI sins0 F„ = 2̂  ' - - - - - - - - (3-6) To make a dimensionless ratio from this, the maximum axial load in the deck due to dead load alone F = WDIB / t&n 8 is used to compare with FH to give JL = (3 7) V There is no live load in the expression for F/FH', therefore, this represents the deck stability under dead load only. As stated previously, FH is the buckling load for an infinitely long bar. The true solution for the a finite length bar also involves the number of half-waves n in the buckled shape. Hetenyi gives the number of half-waves as Chapter 3. Simplified Backspan Model 25 IB J k n = 7 f e which in terms of the independent parameters is IB J n = —» 7T A NACEC sin* 0 lBhEI Thus, two dimensionless ratios—F/FJJ and n—are defined as a measure of deck stabil- ity. In order to explore the effect of cable nonlinearity on the response, the nondimen- sional cable tautness / as derived in Appendix A is used as one of the dimensionless ratios. where wDlB H = Tcosfl = 7Y tan 6 and P V 245s as is explained in Appendix A. The expression for / in terms of the independent parameters becomes / . , . (3.9) Ns\n6< ACEC (wch\' \\ 24 \tan^y There is no live load in the expression for / ; therefore, this represents the cable tautness under dead load only. A physically intuitive description of / is that at high values of / the cable is very taut and has nearly the same stiffness as a straight bar. In fact, at / = 2 the cable stiffness would still be 85% of the straight bar stiffness. The cable stiffness then decreases rapidly as / decreases to the limit of zero. The critical live load is an obviously important parameter and this will be repre- sented by the dimensionless ratio wie/u>D- Chapter 3. Simplified Backspan Model 26 TV, sin0 and h/lg, which describe the geometric characteristics of the model, are used as the last three ratios required. A complete set of dimensionless ratios has been found and can be proven to be independent of each other. The reader can convince himself of this by noting that all nine parameters are used in forming the seven ratios, and by going through a process of elimination to prove independence. The ratio / is obviously independent of the other ratios because it is the only one containing wp. Of the six remaining ratios, WLC appears only in U>LC/WD. With tvLe/wD eliminated, F/FJJ is the only ratio containing WD] therefore, is independent of the others. The three geometric ratios are obviously independent of each other, which leaves n as the only ratio left; thus, independent of all the others. The critical live load causing instability of the simple backspan model is described by the function: x r • A h r F N, sm6, —, /, n, — This function is based on beam-column on an elastic foundation stability theory with both the cable slackness and the P—A nonlinearities modelled. 3.2.2 Parameters and Ratios from Existing Bridge Designs In order to construct the standard1 simple backspan model to reflect actual bridges as closely as possible, data was aquired from four steel deck and five concrete deck cable stayed bridges. Appendix C.l shows the numerical values for the parameters, and the resulting dimensionless ratios are given in Appendix C.2. Typical values for concrete and steel E are chosen and the value for cable E is that for modern parallel strand cables. The density of the cables 7<? is taken to be 77 kN/m3. The moment of inertia of a deck varies in a real bridge but in this study I is kept constant. / is calculated from a representative section of the backspan which is usually a long central section of 1 Defined in the last paragraph of Chapter 2. Chapter 3. Simplified Backspan Model 27 constant cross section. For this model, constant values of cable area, height and angle are needed. To obtain values for these parameters the "centroid cable" is defined as the cable nearest the centroid of the triangle formed by the backspan, tower and backstay cable. The area, height and angle of this centroid cable are used as the representative values for the bridge and as the constant values for the model. The uniformly distributed live load wi for the standard model is assumed to be the AASHTO HS20 lane load for highway bridges. For railway bridges, the actual design specifications for live load are used. The dead loads are estimated from both the cable prestresses and the deck cross sectional area. N is the number of cables in the backspan with all the backstay cables counted as one cable. F is calculated from WDIBI tan The dimensionless ratios could be calculated for every cable in each bridge to get the average values for each bridge; however, to reduce the need for tedious calculations on the part of future users of this thesis, the centroid cable values of the dimensionless ratios are taken as the average values. The dimensionless ratio values actually used for the standard simple backspan model are given in Appendix C.2. These values are averages of the 9 cable stayed bridge values, and are repeated here: N 20 sin 6 0.70 0.40 / 7 n 3 F/FH 0.10 XVL/U>D 0.20 where, for the purpose of calculating / : u>c •A-CccnlC- Chapter 3. Simplified Backspan Model 28 3.2.3 Results of Stability Analysis The standard simplified backspan model of Figure 3.6 is constructed using the preceding values for the dimensionless ratios. In order to find the buckling load for this standard model, the structure is analyzed with the computer program ULA. The live load starts out at 20% of the dead load {VJL/WD = 0.20) and is increased until the critical live load that causes elastic instability is reached. (Structure instability is recognized when the determinant of the stiffness matrix becomes zero.) The standard live load is increased by multiplying it by the factor A, called the "load level." With this nomenclature the critical load ratio becomes U>Lc _ Awx, wD wD The structure buckles under the combination of 50 original live loads (A = 50) and one dead load—which is to say WLC/WD — 10. In other words, the buckling load is wD + Atuj, where A is taken as the factor of safety against failure. Thus, the factor of safety on the live load against elastic buckling of the standard model is 50. This is a very high factor of safety and cannot be related to cable stayed bridges because the model is too simplistic. The simple model is necessary, however, to be able to verify the computer program results against the theoretical beam-column on an elastic foundation before analyzing more complex models. To this end, the deflected shape of the deck just before stability failure is shown in Figure 3.8 along with the axial load (dead load plus live load) and moment distributions. Note that the deflection and moment shapes exhibit half-waves close to /jg/n or, in this case IB/3. The nonlinear distribution of deck axial load occurs because of the large deck deflections which cause cable tensions to increase beyond the linear live load tensions. As an aside, the possibility of a more severe loading condition for instability can be seen in Figure 3.8. If an uplift pressure (from wind and/or inertia) is applied in Chapter 3. Simplified Backspan Model Figure 3.8: Stability failure of simplified backspan model. Chapter 3. Simplified Backspan Model 30 the region of deck negative moment (the center section), a more severe deflected shape would result. This uplift pressure would also reduce the deck axial load which stabilizes the structure. To determine the overall effect of the uplift pressure, the factor of safety is calculated from computer analyses for a range of uplift pressure magnitudes over the region of negative moment. The result is shown in Figure 3.9 where the abscissa is the uplift pressure expressed as a percentage of the live load magnitude. It can be seen that for this backspan model the reduction in deck axial load overrides the more severe deflected shape and the factor of safety increases with increasing uplift pressure. 150 Uplift Pressure (% of Live Load) Figure 3.9: Stability factor of safety with uplift pressure. The backspan model buckled with F/FJJ = 0.10. However, the theoretical beam- column on an elastic foundation will only buckle when the constant axial load in the beam-column is exactly as calculated by FH = 2\/kEI. For this reason it might be expected that F / F H should equal 1.0 in order for the model to buckle. This discrepancy of a factor of ten exists because Chapter 3. Simplified Backspan Model 31 1. F does not include the live load, 2. the actual axial load distribution is nonuniform and nonlinear, 3. FH is for an infinitely long bar, not for the case at hand. Most of the discepancy is due to the live load not being in F. To analitically account for the live load, F is multiplied by (WD + XWL^/WD; which for this case (XWL/WD = 10) gives F wD + XwL _ i i Q FH wD instead of 0.10. Thus, the discrepancy is brought down from a factor of 10 to an error of 10%. It still remains to explain why there is still this 10% deviation. Since the axial load varies along the deck it is more reasonable to use an average axial load over a half-wave F a v i ^ as shown in Figure 3.8. An approximation to F a v g ^ is F ( l + \wL/wr))(2n — l)/(2n). This is approximate because it does not include the axial load induced by the backstay cable. For large values of N this approximation is fairly good. The exact formula is XU>L\ (2n — 1 1 \ Due to the P—A effect the real average is larger. Computer runs near critical show that the real average F*vg^ is l.lo6Favg~ for the particular case shown in Figure 3.8. The dimensionless ratio now becomes F H FH \ wD ) \ 2n 2N J A final refinement can be included by considering the boundary conditions of the model. Hetenyi's formula FH = 2\/k~EI is for an infinitely long bar, and needs to be changed to reflect the true case. The formula for the buckling load of a finite length pin-pin bar is given by Hetenyi as •*%EI l\k_ lB/ni + n2x2 Chapter 3. Simplified Backspan Model 32 where all the parameters are as defined previously. To make this applicable to the situation in Figure 3.8, /^/n is substituted by the measured effective length of the half-wave, 0.781B/n, to give Thus, the simplified backspan model is equivalent to a beam-column on an elastic foundation within 3%. In future stability plots the only adjustment made will be to multiply F/FH by (l + XWI/WD)- The deviation from Hetenyi's solution due to nonuniform axial load distribution and finite length will not be corrected, but will be left as a visual indication on the plots. The standard model is next varied by altering only F/FJJ and then determining the live load necessary to cause the deck to buckle. The result of this exercise is shown in Figure 3.10. The theoretical result for a beam-column on an elastic foundation is also plotted in Figure 3.10 and labelled as "Hetenyi." The discrepancy between the two results is due to the same three reasons as explained for the standard model. If the live load is included in F the resulting plot is shown in Figure 3.11. If the rest of the refinements are made, the ratio Flvg^jFJJ^ at the ordinate intercept, for example, is 1.05 as compared to F/FJJ = 1.32. Which differs from the the 0.97 obtained for wL/wD — 0.2 because of errors in measuring the half-wave effective lengths and obtaining F * V G ^ . The dimensionless ratio n is now varied to determine how n affects the model. The range of n in existing cable stayed bridge designs is from 2 to 4 and the stability curves for n — 2 and n = 4 are plotted in Figure 3.12 along with the curve for n = 3. There is a variation with n in that the three curves differ from each other. This variation is not expected with Hetenyi's theoretical result, and if all of the adjustments The ratio FH/FJJ^ is calculated to be 0.886 and the final ratio FH^ = 0.97 Chapter 3. Simplified Backspan Model Figure 3 . 1 1 : Stability of simplified backspan model. Chapter 3. Simplified Backspan Model 34 to F/FJJ are made to the curves as was done for the standard model, all three curves would plot as nearly the same. (They will not be exactly the same due to the errors mentioned previously.) The following table summarizes the result of these adjustments for the ordinate intercepts: F F" Iff FH FH~ 4 1.25 1.06 3 1.32 1.05 2 1.48 1.03 Thus, it can be seen that the stability of the simplified backspan model can be compared to a beam-column on an elastic foundation with reasonable accuracy. WD Figure 3.12: Stability of simplified backspan model—n varied. As mentioned previously, a nonlinear response is expected due to the changes in cable sag when the cables experience a load change. To investigate this, the standard Chapter 3. Simplified Backspan Model 35 model is varied by altering the initial cable tautness / . In the normal range of 5 to 9, the cable tautness ratio does not affect the model behaviour. Only when the cables are slack (/ < 2) is an effect seen, and then only for low live loads because higher loads tighten the cables. The curves for / = 5 and / = 9 are precisely the same as the curve for / = 7. The remaining dimensionless ratios, which are all based on the model geometry, are not investigated for this model. 3.3 Deflection and Moment It is important to check the deck for instability to assess the factor of safety against buckling failure, but this will usually not govern the design. In preliminary design analyses, the maximum deflection and maximum moment are the governing parameters for the deck. Herein, dimensional analysis is applied to the response of the deck and compared to the response of a simple beam on an elastic foundation. 3.3.1 Deflection Dimensionless Ratios For the dimensional analysis of cable stayed bridge deflection only one extra parameter is needed and that is the maximum backspan deflection 6B- All the other parameters are exactly those used in the stability analysis with the exception of using VJL instead of WLC. The definition of the backspan model deflection in terms of independent parameters is 6B = 7[EI, ACEC, IB, h, 6, N, wc, wD, wL) Buckingham's n theorem requires that one more dimensionless ratio be constructed. In order to compare the deflection of the backspan model to a simple beam-column on an elastic foundation, the dimensionless ratio 6B/6H is used. The symbol 6H refers to the maximum deflection of a simply supported beam on an elastic foundation under a uniformly distributed load, without an axial load. The deflection as a function of x is Chapter 3. Simplified Backspan Model 36 given by Hetenyi as cosh 72 cos 7(/B — x) + cos 7Z cosh 7(/B — x) 1 - cosh 7/5 + cos 7/^ (3.10) where 1 =  V ^ 4 £ 7 and is given by Eq. 3.5. The deflection for a beam on an elastic foundation rather than a beam-column on an elastic foundation is used so that the amplification of deflections due to the axial load will become apparent. Also, Hetenyi assumes a constant axial load in his derivation of beam-column on an elastic foundation deflections, while for cable stayed bridges the axial load varies along the deck. Complications due to the different axial distributions are thus avoided by comparing cable stayed bridge deflections to the beam on an elastic foundation deflections without axial load. In Eq. 3.10 the variable 7 is related to n, the number of half-waves in the buckled shape. For response analysis it is advantageous to change this equation to reflect the decay length of the beam rather than the buckled shape. The decay length /p is the length over which a local deflection or moment in a beam dies out. Two examples are shown in Figure 3.13. P Semi-infinite Beam Infinite Beam Figure 3.13: Examples of deflection decay in a beam on an elastic foundation. The equations for deflection are composed of two parts, an exponential decaying part and a sinusoidally varying part. For example, the deflection of the semi-infinite Chapter 3. Simplified Backspan Model 37 beam on an elastic foundation in Figure 3.13 is 6{x) = cos 71 and for the infinite beam on an elastic foundation the deflection is ^(z) = ~7Tre 1 1 (cos 7x-f sin 71) The decay length could be defined as the distance from the maximum deflection to the point at which the deflection first becomes zero. This is governed by the sinusoidal part, but the location of this point depends on the boundary conditions and whether it is deflection or moment that is of interest. Therefore, this method of defining decay length is not good. The best way to define decay length is to say it is the distance to the point where at least 95% of the maximum deflection or moment is dissipated. This occurs for all instances when c-""* < 0.05. For 7Z = 3, e~lx = .049, thus we get 7/75 = 3 or ID = 3/7. This definition of decay length is arbitrary and Hetenyi uses 7r instead of 3, which results in a longer decay length. For this historical reason and for convenience in relating ID to n, TT is used herein. The new dimensionless ratio to replace n will be ID/IB where A more meaningful expression for 6H[X) can now be derived using ID instead of 7. Specifically, using l_D _ IB l h Substituting for 7 and using Eq. 3.5 gives lD _ *y/2 J lBhEI (3.11) lB IB Sj NAcEcsm3e This equation can be simplified by use of Eq. 3.8 to give I D ^ V/2 IB n 7T ID Chapter 3. Simplified Backspan Model 38 then X 7T JC 7T cosh7r— cos — (IB — x) + cos 7r— cosh — (IB — x) '£> <£> *r> <D COSh 7T (- COS 7T — In the dimensional analysis of the backspan model, the maximum deflection of Hetenyi's model is compared to the maximum deflection of the computer model. Hetenyi's maximum deflection SR occurs when the derivative of SH(X) is equal to zero. This derivative (the slope) is equal to zero when x = f/p and because of the symmetry of the simply supported beam on an elastic foundation, the maximum value of x is lB/2 or IB 3, J = 4 D or IB 3 Thus, when ID/IB < 2/3 , x = \ID should be used and when ID/IB > 2/3, x = lB/2 should be used. This results in two equations for the maximum deflection 6H of a beam on an elastic foundation: OH k k 1 - . 3 / 3 . 3 i / T 3 . COSh -7TCOS(-— 7TJ + COS -7TC0Sh( -— 7r) 4 ID/IB 4 4 ID/IB 4 cosh 2 cosh 7T ID/IB 7T + COS IT ID/IB f ° r ^ 3 (3.12) COS 1 - Hp 11B 2lD/lB cosh + COS 7T IB ~ 3 ID/IB ' h/h For convenience and reference a dimensionless plot of Snk/wi versus ID/IB is given in Figure 3.14 as a graphical representation of Eq. 3.12. The dimensionless ratios used in describing the maximum deflection of the simple backspan model are listed by the function: AT . a h lD WL F N, sin0, —, / , —, — , — IB IB ™D FH Chapter 3. Simplified Backspan Model 39 IB Figure 3.14: Dimensionless plot of Hetenyi's deflection. With Su defined by Eq. 3.12 this function will show the magnification of the maximum deck deflection due to the deck axial load. 3.3.2 Results of Deflection Analysis The same standard simplified backspan model is used for maximum deflection analysis as that for the stability analysis {ID/IB = 0.47 for n = 3). The deflection of the standard model under a uniformly distributed live load (WL/WD = 0.20), is shown in Figure 3.15 along with the beam on an elastic foundation, deflection. This deflection is due to live load only, since the dead load deflection is zero. The live load deflection is magnified due to P—A effects from the combination of the dead and live load deck axial forces. The 6B/£>H ratio is equal to 1.024 which indicates that there is very little magnification due to the deck axial load in the standard model. The maximum live load deflection versus span ratio is 1/2000 (calculations are shown in Appendix D) Chapter 3. Simplified Backspan Model 40 Model 6 2000 Figure 3.15: Deflection of standard simple backspan model. which is well within the recommendation of 1/500.2 The standard model is now varied by altering the F / F R ratio while keeping all other ratios constant. The maximum deflection is then determined from a computer analysis and the 6B/Sff ratio is plotted against F/FR. The result is given as the solid curve in Figure 3.16 for a range of F/FH values. In the 6B/6ff ratio, the u6Bn is the magnified maximum deflection of the deck due to the presence of axial load in the deck. The a6#" is Hetenyi's maximum deflection of a beam on an elastic foundation without axial load. Thus, the SB/Sfj ratio is equal to 1.0 for F/FB: equal to zero, and increases as F/Fjj increases, showing the magnification behaviour. A comparison can be made with Hetenyi's maximum deflection for a beam-column on an elastic foundation with constant axial load Sjjaxiai where: cosh fix cos cc{lB — x) + cos ax cosh 0(lB — x) °Haxial — k cosh piB + cos alB , 2 2^ sinh fix sin a(lB — x) + sin ax sinh - x) -(/?>-a2): 2a/?(cosh piB + cos adB) 2Recommended by the Task Committee on Cable-Suspended Structures of the Committee on Special Structures of the Committee on Metals of the Structural Division. [23] Chapter 3. Simplified Backspan Model 41 Figure 3.16: Magnification of maximum deflection. where ct = 4EI 4EI and U L + J L 4EI 4EI F = wDlB tan 0 is constant and equal to the maximum value in the deck. A new ratio SHaxiai/6H is formed and plotted as the dashed curve on Figure 3.16. It can be seen that the computer model behaves very similarly to the Hetenyi model, with the small difference being due to the different axial distributions in the beam. It is interesting to note that the triangular axial load distribution in the computer model produces a more severe magnification of the maximum deflection than the constant axial load of Hetenyi's Chapter 3. Simplified Backspan Model 42 model. For interest sake, the standard amplification formula for a simply supported beam-column is applied to this model and is also plotted on Figure 3.16. It can be seen that the elastic foundation significantly suppresses the deflection of the beam-column. To assess the cable tautness effect on the maximum deflection, the / ratio is varied while all the other ratios are kept at their standard values and the maximum deflection is determined from the computer analysis. The result of this is shown in Figure 3.17. The cable tautness does not affect the deflection of the backspan for normal values of / (> 5). The importance of having tight cables is demonstrated however, by the magnification of the deflection for low values of / . 1.20 1.15- ^ 1.10 1.05- Standard Model ~ N = 20 sin 6 = 0.70 h/lB = 0.40 n = 3 F j FH = 0.10 wL/wD = 0.20 1^ 2 3 4 5 6 7 8 i 9 10 11 12 13 M 15 Figure 3.17: Dimensionless deflection versus dimensionless cable tautness. The deflection plot as a function of dimensionless decay length IDJIB is shown in Figure 3.18. 6B/6~H depends very little on ID/IB with there being only a slight increase in magnification due to the axial load as the decay length reaches the backspan length and beyond. As the decay length becomes very long, the effect of the initial axial load Chapter 3. Simplified Backspan Model 43 diminishes and the SB/SH ratio reduces asymptotically to 1. A t the other end of the scale, the increase in SB/SJJ is not due to the short decay length per se; rather, due to a deviat ion of s tructural behaviour from a beam on an elastic foundation. To maintain accuracy in modell ing a beam wi th a continuous elastic foundation by a beam with discrete springs, there should be at least 4 or 5 springs over the decay length. In other words, ID/IB should be greater than 4/N. Thus , for the standard backspan of 7Y = 20 and ID/IB less than 0.20, the response is no longer s imilar to a beam on an elastic foundation response. To demonstrate this graphically, the curve for 7Y = 80 is plotted as the dotted line. 1.20 i 1.15 ^ 1.10H 1.05- Standard Model I—N = 80 N = 20 sin 6 = 0.70 h/lB = 0.40 f = 7 F/FH = 0.10 wL/wD = 0.20 I I I I r 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 l_D IB 2 2.2 2.4 2.6 Figure 3.18: Dimensionless deflection versus dimensionless decay length. Figure 3.19 shows the relationship between 6B/6H and the live load to dead load rat io WL/WD- It can be seen that the max imum deflection of the standard simple backspan model is closely represented by Hetenyi's equation (Eq. 3.12) for any value of live load. The small slope shown in Figure 3.19 is due to the nonlinear P— A effects Chapter 3. Simplified Backspan Model 44 1.20 1.15- ^ 1.10H 1.05- iStandard Model N •= 20 sin 6 = 0.70 h/lB = 0.40 f = 7 n = 3 F/FH = 0.10 1 T ™ " " " ™ ^ I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 WD Figure 3.19: Dimensionless deflection versus dimensionless load, induced by the extra live load axial force in the deck. 3.3.3 Moment Dimensionless Ratios The dimensional analysis of cable stayed bridge moment is carried out exactly the same as that for the deflection analysis, with the substitution of the maximum backspan moment MB for the maximum backspan deflection 6B. The definition of the backspan model moment in terms of independent parameters is MB = 7\EI, ACEC, lB, h, 6, N, wc, wD, wL) In order to compare the moment of the backspan model to a simple beam-column on an elastic foundation, the dimensionless ratio MB/Mn is used. The symbol MH refers to the maximum moment of a simply supported beam on an elastic foundation under a uniformly distributed load, without an axial load. The moment as a function Chapter 3. Simplified Backspan Model 45 of x is given by Hetenyi as u>i sinh^isin^^B — x) + sin ^x sinh 7 (/B — x) MH(x) = 2 72 cosh 7/fl + cos 7/13 where 7 = AEI and A; is given by Eq. 3.5. Using this formula, the amplification of moments due to the axial load will become apparent, just as in the deflection analysis. A more meaningful expression for MH(X) using ID instead of 7, is X . 7T ,. . . X . . 7T MH{x) = _ wLlD 2?r2 sinh7r— sin — (IB — x) + sin7r— sinh — (IB — x) lp lp lp lp 1 IB . ^B COSh7Tt—h cos7r— lp lp The maximum moment M# occurs when the derivative of MH(X) is equal to zero. This derivative (the shear) is equal to zero when x = lD/4 and because of the symmetry of the simply supported beam on an elastic foundation, the maximum value of x is lB/2 or IB 2 lp 4 or r = 2 « B Thus, when ID/IB < 2 , x = ID/4 should be used and when ID/IB > 2, x — lB/2 should be used. This results in two equations for the maximum moment MR of a beam on an elastic foundation: 7T 7! \ 1 27T 2 UP sinh — sin(-—— 4 Klp/lB 7T . 7T . - - +sm-sinh-4 4 /r>//B cosh IT } B . sinh cosh 7T W ' B 7T + COS lp/h for  lf < 2 *B sin 2/P//B 2lD/lB h/h + COS ID/IB \ (3.13) for > 2 ' B Chapter 3. Simplified Backspan Model 46 For convenience and reference a dimensionless plot of MH f[wilB) versus ID/IB *1S given in Figure 3 .20 as a graphical representation of Eq. 3 .13 . 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 IB Figure 3 .20: Dimensionless plot of Hetenyi's moment. The dimensionless ratios used in describing the maximum moment of the simple backspan model are listed by the function: . . h lD wL F , sint/, — , / , — , — , — - IB IB U>D FH . With M H denned by Eq. 3 .13 this function will show the magnification of the maximum deck moment due to the deck axial load. 3 . 3 . 4 Results of Moment Analysis The same standard simplified backspan model is used for maximum moment analysis as that for the deflection analysis. The moment of the standard model with wL/wD = 0.20, is shown in Figure 3.21 along with the beam on an elastic foundation moment. The Chapter 3. Simplified Backspan Model 47 MB/MH ratio is equal to 1.057 which indicates that there is very little magnification due to the deck axial load in the standard model. The maximum moment is wil\/262 (calculations shown in Appendix D) which is a tremendous reduction from the simple span maximum moment of wLlB/S. lB Figure 3.21: Moment of standard simple backspan model. In Figure 3.21 it can be seen that local fixed ended moments of the deck between stays are superposed on the overall structure moment diagram. These local bending moments are caused by the application of the live load as a continuous distributed load. The effect of the local moments is to add small negative moments at the cable-deck connections and to produce a nonlinear moment distribution along the deck between any two connections. These effects are small—and decrease with increasing N—and will not be explicitly regarded in the analysis. The maximum backspan moment MB is taken as the maximum moment at a cable connection, with the nonlinear effect included. Note, however, that the dead load moments are not included in the model, which is to say the dead load is applied as point loads at the cable—deck connnections. This method of applying the dead load is used to achieve a moment and deflection free model as a starting point. The live load being uniformly distributed over the deck implies that the construction method of having floor beams attached directly to the main longitudinal girders is used Chapter 3. Simplified Backspan Model 48 between cable-deck connections, as shown in Figure 3.22. However, the local dead load deflections and moments are not accounted for in this study but can be superimposed on the results of this study if required by the reader. The other construction method shown in Figure 3.22 makes use of stringers that span between floor beams placed at cable-deck connections. This layout transfers most of the live and dead load to the floor beams, thus the distributed loads act as concentrated loads on the main longitudinal girders at the cable-deck connections. Realizing that the dead loads are applied at the panel points and that the local effects of live load are relatively small, the results of this study can be applied to both types of deck construction. Figure 3.22: Bridge deck construction methods. The standard model is now varied by altering the F/FH ratio while keeping all other ratios in the function constant. The maximum moment is then determined from a computer analysis and the MB/MJJ ratio is plotted against F/FJJ. The result is given as the thick solid curve in Figure 3.23 for a range of F/Fn values. Chapter 3. Simplified Backspan Model 49 F_ Figure 3.23: Magnification of maximum moment. In the MB/MH ratio, the "A/jg" is the magnified maximum moment of the deck due to the presence of axial load in the deck. The U M H " is Hetenyi's maximum moment of a beam on an elastic foundation without axial load. Thus, the MB/MH ratio should be equal to 1.0 for F/Fg equal to zero, and increase as F/Fg increases, showing the magnification behaviour. Note that the ordinate intercept is actually less than 1.0. This is due to the fact that the deck moments are dependent on the locations of the cable connections to the deck. If a different layout of cable spacing is used such that a connection occurs at the point on the deck corresponding to the maximum moment location for a continuous elastic foundation, then the deck would be subjected to the maximum possible moment. Thus, it can be seen that the curve obtained for N = 20 is really a happenchance result (within a narrow range) and for design purposes should be replaced with the curve that represents the maximum possible moment. The cable layout in the simplified Chapter 3. Simplified Backspan Model 50 backspan model could be adjusted to achieve this, but a more convenient way to get the maximum possible moment is to increase the number of cables to 80. Increasing the number of cables also has the effect of reducing the local negative bending moments, which increases the maximum positive moment as well. The result for N = 80 is plotted on Figure 3.23 as the dotted curve. This curve behaves as expected and should now be the one to refer to in future discussions. A comparison can be made with Hetenyi's maximum moment for a beam-column on an elastic foundation with constant axial load M#oz,0j where: where a and 0 are the same as defined for deflections. The ratio M#-ai,0j/M# is formed and plotted as the dashed curve on Figure 3.23. The standard amplification formula for a simply supported beam-column is applied to this model and is also plotted on the figure. The model moment amplification is much higher than the deflection ampli- fication, which may be because the supressed deflection causes severe curvatures. It is again noted that the triangular axial load distribution in the computer model produces a more severe magnification of the maximum moment than the constant axial load of Hetenyi's model. Figures 3.24 to 3.26 show the model moment magnification for varying cable taut- ness, deck decay length and live load. Looking at the curves for N = 80 it can be seen that the moment magnification is slightly more severe than the deflection magni- fication, but the general behaviour with varying dimensionless ratios is similar to the deflection behaviour. MH axial = EIwL (/?2-fa2)2 k 2a/3(cosh 01B + cos a/̂ ) [sinh 0xsin a(/s — x) + sin ax sinh 0(IB - i)] Chapter 3. Simplified Backspan Model 1.20 1.15- 1.10- H 1.05- I Standard Model sin 6 = 0.70 h/lB = 0.40 n = 3 F/F„ = 0.10 U!L/wD = 0.20 N = 80 N = 20 i i i i i i i i i i i i i i I 1 2 3 4 5 6 7 8 9 10 11 12 13 U 15 Figure 3.24: Dimensionless moment versus dimensionless cable tautness. 1.20 Mi M 1.15- 1.10- H 1.05- Standard Model sin 6 = 0.70 h/lB = 0.40 f = 7 F/F„ = 0.10 wLfwD = 0.20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 [D IB Figure 3.25: Dimensionless moment versus dimensionless decay length. Chapter 3. Simplified Backspan Model Chapter 4 Backspan Model 4.1 The Model The second model is a more faithful representation of a cable stayed bridge backspan as shown in Figure 4.27. This model has varying cable lengths, inclinations and areas, and all the cables are attached to an imaginary immovable tower. It may seem that the model has changed drastically from the first model, but in fact the result of all the individual changes is only to produce a nonuniform elastic foundation and a slightly different distribution of deck axial load. The deck has a constant stiffness and the live load is applied as a distrubuted load, just as in the first model. All of the model properties shown in Figure 4.27 have been identified in Chapter 3 except for s', which is the cable spacing at the tower. However, most of the properties now have a subscript. The general subscript is " i " where i = 1 signifies the cable or deck section closest to the tower and i = N is the backstay and corresponding deck section. The subscipts are necessary to describe the model, but in the dimensional analysis of this model, representative values will be used when referring to the backspan parameters and dimensionless ratios. This representative value shall be indicated by the subscript uN/2n where N/2 is as calculated for even values of N, and for odd values of T V , N/2 signifies the average value of two cables. For example, if N = 11 then AcN/2 would be the average of Ac and Act • The major geometric change from the first model is the varying cable height / i , . The parameters (h^) and (hi) will define the top and bottom cable locations. The 53 Chapter 4. Backspan Model 54 m n i u w i i i i i i i r n i i i i i u i w»i i i i i i i i i n Figure 4.27: Backspan model. remainder are at a constant spacing of s\ where , hN — hi 8 =IT^T This nomenclature allows the entire range of cable layouts to be defined, from the radiating shape to the harp arrangement. The relationship of the first model cable height h to this model is h = hjt/i. Initially the model is under dead load only and the cable tensions Ti are set up such that there are no moments in the deck at input geometry. The cable tensions represent the dead load, and the distributed dead load is not actually applied, just as in the first model. A common desire in cable stayed bridge design is to have all the cable dead load stresses the same. Since the cable inclinations vary, it is also necessary to vary the cable areas in order to maintain constant cable stresses. The free-body diagram of a cable-deck connection in Figure 4.28 helps to explain this requirement. The symbol "<7c" represents the cable stress which is to be the same for all cables. By equilibrium Chapter 4. Backspan Model 55 of forces: wDs = oCACI sin Oi or , U>DIB Noc sin 0, The constant o"c can be determined from the representative cable as VJDIB Oc — NACN/3 sin 0N/2 so that A C N / 7 sin g^ /2 •Ac.- = sin Bi which clearly shows that the cable areas vary. Ti = ocACi Figure 4.28: Free-body diagram of a cable-deck connection. The parameter F is the maximum axial load in the deck induced by the cables in the initial state, i.e., for dead load only. The Fi represent the axial load in the j'th deck section due to Tit i.e., F< = T{ cos 0t and F = Ff. 4.2 Stability 4.2.1 Stability Dimensionless Ratios The critical live load of the system is defined by the function wLc = J\EI, A C N / 3 E c , h, hlt hN, N, wCfi/3, wD) (4.14) Chapter 4. Backspan Model 56 All nine of the parameters in Equation 4.14 are independent of each other. The parameters that do not appear in the equation are functions of the ones given in Equation 4.14, i.e., tan Otfji 2 where and i>N/2 = hi sin eN/2 IB S = N , hs — hi s = N - 1 rr WDIB TNI* = TFZT. TV sin ON/2 F = J2Ticos8i=X^f:™^ 1=1 ^ «=i f cote - f i l ( N ~ * ) i a hi + hN{l - ^ hi+ hN(l - ~) hN/2 = 2 ~ N for both even and odd values of N. Note that because hx and hN are used as inde- pendent parameters, 0^/2 and h^/2—which were used in the previous model as 6 and h—cannot be used for this model. Chapter 4. Backspan Model 57 Seven dimensionless ratios need to be constructed out of the nine parameters in Equation 4.14. The seven ratios chosen to represent this model are: N hi h]y IB WDIB A . a 1=1 NAcN/7ECEI sin3 6N/2 ID IB 7Ty/2 N A C N / 3 E C sin3 BN/2 f = wDlB N sin ON/2 \ AcN/3Ec ( wcN/7hx/2 24 y tan ON/2 J The last three dimensionless ratios above are the same as the ones used for the first model, the only difference being in the use of the subscript "iV/2" and F = T^?=1Fi. The ratios N and VJLC/U>D are exactly the same as those used in Chapter 3. The only new ratios are the ones that define the cable layout, h i / l B and h^/lB- In summary, the stability of the backspan model is described by the function: = J hi hN j lD F IB IB IB FH This function is based on beam-column on an elastic foundation stability theory with both the cable slackness and the P—A nonlinearites modelled. Chapter 4. Backspan Model 58 4.2.2 Parameters and Ratios from Existing Bridge Designs The same nine cable stayed bridges that were used for the first model are used to acquire data for the construction of this model. However, the subsripting system introduced for this model suggests a choice of representative values for cable area, height and angle different from the "centroid cable" of Chapter 3. The new representative cable is the one (or average of two, as explained previously) corresponding to t = N/2. The area, height and angle of this cable are used as the representative values for the bridge and as the "t = N/2" values for the model. Appendix C.3 shows the numerical values obtained for this backspan model. All the other parameters are determined from the design drawings in exactly the same manner as described in Chapter 3, except for F. In the dimensional analysis of this model F is calculated as F = ^^f* 22£Li cot 0, but this is too unwieldly to use in acquiring data from design drawings. A simple method is to use F = NF^/2 or F = WQIB/tan0^/2- This is based on a linear distribution of F,; however, since the actual distribution for this model is concave in shape, Fs/2 is larger than that for a linear distribution and NF^/i is a conservative value for F. The representative dimensionless ratios for each of the nine bridges are calculated using the data in Appendix C.3. The averages of these nine ratios are used to construct the new standard backspan model. These values are given in Appendix C.4 and are repeated here: N 20 hills 0.30 hx/h 0.50 / 7 ID/IB 0.50 F/FH 0.10 Chapter 4. Backspan Model 59 wL/wD = 0.20 where, for the purpose of calculating / : VJCn/2 — AcN/27c- 4.2.3 Results of Stability Analysis The standard backspan model of Figure 4.27 is constructed using the preceding values for the dimensionless ratios. Using the computer program ULA, the critical load ratio for this standard model is found to be WU/WD = XWL/WD = 18.2, which is to say, since WI/WD — 0.20, that the factor of safety on the live load against elastic buckling of the standard model is 91. Just as in Chapter 3, the factor of safety for this model does not apply to cable stayed bridges because only the backspan is modelled. However, the very much higher factor of safety over the 50 obtained for the simplified backspan model should be explained. The reason for the increase is the nonuniform elastic foundation. Due to the increasing cable angle, a much stiffer foundation exists at the high axial load area of the deck than in the previous model; which means that higher axial loads are needed to cause instability. The deflected shape of the deck just before stability failure is shown in Figure 4.29 along with the axial load (dead load plus live load) and moment distributions. The nonlinear distribution of deck axial load is mostly due to the varying cable inclinations and to a smaller degree due to the deck deflections. The possibility of an uplift pressure causing a more critical loading condition is investigated for this model in the same manner as for the previous model. The resulting factor of safety plot is given is Figure 4.30. In this case, the forced deflected shape is more influential than the reduction in deck axial load and the factor of safety decreases with increasing uplift pressure. This decrease, however, is not severe—there is only a 8.2% drop in the factor of safety when the uplift pressure is 50% of the live load magnitude. There is a further drop of 20.1% for an uplift pressure of 100% of the live load magnitude. High uplift pressures are possible, [24] but the occurrence of a full live Figure 4.29: Stability failure of backspan model. Chapter 4. Backspan Model 61 load at very high wind velocities is improbable, therefore, the high uplift pressure range of Figure 4.30 is not truly applicable. Wind and dynamic loads are beyond the scope of this thesis and the uplift pressure load case will not be used in further analyses for this model, but it should be kept in mind that a slightly more severe result is possible. 150-1 . _ oS 100- Uplift Pressure (% of Live Load) Figure 4.30: Stability factor of safety of backspan model. The fact that the backspan buckled at F/FH = 0.10 and not at F/FJJ = 1.0 is accounted for by the same three reasons given for the simplified backspan model. However, the critical deck section is not as easy to determine as for the simplified backspan model. For the previous model it was obvious that the deck section with the highest axial load was critical because the elastic foundation was uniform. In this case, •with the varying foundation stiffness and the fairly uniform axial distribution over two half-waves, a closer investigation is needed to identify the critical section. A plot of F*VG^/'FH~ V S X/IB is shown in Figure 4.31 for the simplified backspan model of the previous chapter. The curve is calculated by taking the average of /.g/n Chapter 4. Backspan Model 62 (in this case lB/n = 20/3) deck axial forces nearest a cable and dividing by FHr^ as given in Chapter 3. Figure 4.31 clearly shows that the critical deck section is defined by the half-wave nearest to the tower. 1.1-r 1- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x TB Figure 4.31: Critical simplified backspan section. The corresponding plot for the current backspan model is shown in Figure 4.32. Three curves are plotted, each based on the assumption that only one of the three half- waves is critical. The curves are calculated using average deck axial loads as was done for the simplified backspan, with the variation of foundation stiffness of this model averaged out in the same manner. From this figure it can be seen that the middle half-wave is the critical section. It is academically interesting to identify the critical deck section, but this knowledge is of little practical use. Therefore, for simplicity in future calculations and with only a small loss in accuracy, the deck section with the highest average axial load—whether it spans over one or more half-waves—shall be assumed to be the critical section. With Chapter 4. Backspan Model 63 I I 0.8 0.9 • • • I I I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x Figure 4.32: Critical backspan section. this in mind, the three adjustments to F/FR will be made for the standard backspan model. The first adjustment gives ^ ^ + ^ = 0.1(18.2+ 1.0) = 1.92 FH W D The second adjustment is to use the average axial load over the critical deck section. In this case, that will be over the two half-waves nearest to the tower. The uFavg^Jl nomenclature is retained to represent this, with the subscript symbolizing one or more half-waves. The linear formula for this average is A t " z A /2n - 2 1 \ Due to the nonlinear axial distribution this is not the real average. Computer runs near critical show that the real average F*VG^ is 1.006.Fau^ for the particular case shown in Figure 4.29. Chapter 4. Backspan Model 64 F0' F / \wL\ (2n-2 1 \ - ^ p - = 1.006— 1 + — - ( — + — J = 1.336 FH FH V WD j V 2n 2N The dimensionless ratio now becomes The final refinement is to replace FH with which reflects the boundary condi- tions and average foundation stiffness of the critical region. The effective length of the critical half-wave is 0.51/^/2 as is shown in Figure 4.29, and the properties of cable 5 are used to calculate the average foundation stiffness, i.e., F - ( 2 V V*FT i fO-^B\2NACiEcsin3es F»~ ~ \*MTb) * E I + {—) *HBh5 The ratio FH/FH<~. is calculated to be 0.680 and the final ratio is F* F* Fu = 1.336 x 0.680 = 0.91 FH~ FH FH~ Thus, the backspan model stability mechanism can be understood in terms of a beam- column on an elastic foundation if the effect of the nonuniform elastic foundation is taken into consideration. The standard model is next varied by altering only F j FH and then determining the live load necessary to cause the deck to buckle. The result of this exercise is shown in Figure 4.33, where the F/FH ratio is adjusted to include the live load. The theoretical result for a beam-column on an elastic foundation is also plotted in Figure 4.33 and labelled as "Hetenyi." The discrepancy between the two results can be eliminated if all the adjustments as explained for the standard model, are made. For example, if the calculation for the ordinate intercept were carried out, the ratio FLa~lFH~ would be 1.01 as compared to F/FH = 1.64. The dimensionless ratio ID/IB is now varied in the same manner that n was varied for the simplified backspan model. The range of lD/lB in existing cable stayed bridge designs is from 0.40 to 0.65 and the stability curves for ID/IB = 0.40 and ID/IB — 0-65 are plotted in Figure 4.34 along with the curve for ID/IB = 0.50. There is a variation with ID/IB in that the three curves differ from each other. Just as in the simplified backspan model, this variation is not expected with Hetenyi's Chapter 4. Backspan Model 65 Figure 4.33: Stabil i ty of backspan model . 2 - 1.9- 1.8- 1.7 1.6 1.5 tc 1.4 H £ 1.3 -^1.2 1.1 | 0 . 9 H 0.8- 0.7- ^ 0 . 6 0.5- 0.4- 0.3- 0.2- 0.1- 0 - = 0.40 Hetenyi —j— 0.1 N = 20 hi/lB = 0.30 hN/lB = 0.50 f = 7 0.2 0.3 0.4 0.5 0.6 0.7 0.8 —r— 0.9 Figure 4.34: Stabil i ty of backspan model—Ip/ls varied. Chapter 4. Backspan Model 66 theoretical result, and if all of the adjustments to Fj'FJJ are made to the curves as was done for the standard model, all three curves would plot nearly the same. The following table summarizes the result of these adjustments for the ordinate intercepts: l_D F_ FgVg^ IB FH FH^. 0.40 1.62 1.01 0.50 1.64 1.04 0.65 1.71 0.99 Thus, it can be seen that the stability of the backspan model can be compared to a beam-column on an elastic foundation. Just as in the previous model, altering the initial cable tautness / in the normal range of 5 to 10, did not affect the model behaviour. Only when the cables are slack (/ < 2) is an effect seen, and then only for low live loads because higher loads tighten the cables. The remaining dimensionless ratios, which are all based on the model geometry, are not investigated for this model. 4.3 Deflection and Moment The choice of deflection and moment functions to represent this backspan model is based on the same rationale presented in Chapter 3. The definition of the backspan model deflection in terms of independent parameters is 6B = 7[EI, AcN/2Ec, lB, hu hN, N, wc„/7, W D , WL] and the maximum nondimensional deflection of the backspan model is described by the function: 6B f = 7 <>H N hy h N lp wL F '£ lB *B Wp f}i Chapter 4. Backspan Model 67 With 8H defined by Eq. 3.12 and k = _ NAcN/2Ecsin36N/2 this function will show the magnification of the maximum deck deflection due to the deck axial load. All other ratios in this function are the same as the ones defined for the stability analysis of this model. The definition of the backspan model moment in terms of independent parameters is MB = T[EJY A C N / 3 E C , IB, hu hN, N, u>cN/i, U>D, U>L] and the maximum nondimensional moment of the backspan model is described by the function: MB MR = J N  ^1 hN lD wL F »B «B *B U>D * R With MR- defined by Eq. 3.13 this function will show the magnification of the maximum deck moment due to the deck axial load. 4.3.1 Results of Deflection Analysis The same standard backspan model is used for maximum deflection analysis as that for the stability analysis. The deflection of the standard model under a uniformly distributed live load {WI/WD = 0.20), is shown in Figure 4.35 along with the beam on an elastic foundation deflection. The SB/SR ratio is equal to 1.420 which is a significant deviation from the Hetenyi deflection. This is mostly due to the fact that this backspan model has a varying foundation stiffness, which makes it a significantly different structure from the Hetenyi model and the backspan of Chapter 3. The effect of the varying foundation in this model is readily seen in that the right side or "stiff foundation" deck deflection is much less than the left side deck deflection. Also, the slight deflection magnification in the previous model was at the high axial load deck section, as expected; whereas, in this model the large deflection occurs in the low axial Chapter 4. Backspan Model 68 load deck section, which indicates that the flexible foundation in this region is the cause of the large deflections. The maximum live load deflection versus span ratio is 1/1500 which is well within the recommended value of 1/500. IB Figure 4.35: Deflection of standard backspan model. The standard model is now varied by altering the F/FJJ ratio while keeping all other ratios constant. The maximum deflection is then determined from a computer analysis and the 6B/&H ratio is plotted against F/FJJ. The result is given in Figure 4.36 for a range of F/Fg values. The deflection magnification is apparent in that the &BIO"H ratio increases from 1.4 to 2.05 as F/Fg increases from 0.0 to 1.2. This is a less rapid rate of increase in 6B/b~H than the simplified backspan model amplification (Figure 3.16) but the actual deviation from the Hetenyi model is greater in this model for F/Fg > 0.8. The high initial deflection is due to the low foundation stiffness at the left deck section and the low rate of magnification is due to 6B being at the low axial load deck section. The deflection plot as a function of dimensionless decay length ID/IB is shown i n Figure 4.37. It can be seen that 6B/6g is highly dependent on ID/IB hi the normal range [ID/^B from 0.40 to 0.65) for cable stayed bridges. This dependancy is not present i n the simplified backspan model, in fact at the standard value of ID/IB — 0.50, the 6B/6H ratio is almost 1.0 for the previous model. The varying foundation stiffness of the Chapter 4. Backspan Model 69 2.5 2.4- 2.3- 2.2 2.1- 2 - 1.9 <*B 1.8 SH 1-7 1.6 1 . 5 H 1.4 1.3- 1.2- 1.1- 1 Standard M o d e l 0.1 0.2 0.3 0.4 0.5 N = 20 = 0.30 h N / l B = 0.50 f = 7 U)L/U>D = 0.20 = 0.50 0.6 F_ 0.7 0.8 0.9 1 1.1 1.2 Figure 4.36: Magnif icat ion of max imum deflection. current model produces a value of 1.42 for 6B/6B at lz>/lB — 0.50 (and F/Fg = 0.10) instead of 6B/6B « 1.0, and 6B/6H = 1.40 for F/FH = 0.0. If Figure 4.36 were to be used for design purposes i t would be necessary to plot a series of curves, each wi th a different value of ID/IB. F r o m Figure 4.37 it is apparent that the curves for ID/IB > 2.0 would a l l be the same and 6b/SH would be equal to 1.0 for F/Fg = 0.0. Note, that for ID/^B > 2.0 the foundation stiffness would be essentially constant over the backspan length and the behaviour would closely follow that of the simplified backspan model. The cable tautness graph is shown in Figure 4.38. The cable tautness effect is the same as for the simplified model except for the asymptote being at 1.42 instead of 1.02. The importance of having tight cables is again demonstrated by the magnification of the deflection for low values of / . Figure 4.39 shows the relationship between 6B/SH and the live load to dead load rat io WL/WD- It can be seen that WL/WD has very litt le influence on SB/6n, even less Chapter 4. Backspan Model 70 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 Standard Model N = 20 = 0.30 hN/lB = 0.50 wL/wD = 0.20 F/FH = 0.10 ID/IB = 0.50 I I I I I 1 I I 1 I I I I I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 / Figure 4.38: Dimensionless deflection versus dimensionless cable tautness. Chapter 4. Backspan Model 71 than was exhibited for the simplified model. Again, this is because 6g is at the low axial load deck section where the nonlinear P— A effect is not as pronounced as at the high deck axial load section. N 20 1.9- hi/la 0.30 1.8- • Standard Model hN/lB 0.50 1.7- f = 7 1.6- F/F„ = 0.10 1.5- ID/IB — 0.50 1.4- " 1.3- 1.2- 1.1- 1 T i i i i i i i i i 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 WD Figure 4.39: Dimensionless deflection versus dimensionless load. 4.3.2 Results of Moment Analysis The same standard backspan model is used for maximum moment analysis as that for the deflection analysis. The moment distribution of the standard model with WL/VJD = 0.20, is shown in Figure 4.40 along with the beam on an elastic foundation moment. The MB/MJ} ratio is equal to 1.685. This large deviation from the Hetenyi moment is due to the relatively flexible foundation of the model in the leftmost region, which allows more severe curvatures than the Hetenyi and Chapter 3 models. The maximum moment is WLIB/140 which is an 85% increase over the simplified backspan model moment. Chapter 4. Backspan Model 72 IB Figure 4.40: Moment of standard backspan model. The standard model is now varied by altering the F/FH ratio and the MB/MH ratio is plotted against F/FH in Figure 4.41. It can be seen that this plot is quite different from the corresponding plot (Figure 3.23) for the simplified backspan model. The moment magnification follows a different path, the ordinate intercept is not zero, and the curve for N — 80 is not as significant as it is for the previous model. The analysis for 7Y = 80 is done in order to increase the chance of having a cable connected to the deck at the theoretical maximum moment location. In this case, the flatter curve of the moment diagram (compare Figure 4.40 with Figure 3.21) improves the chances of the twenty cable layout to produce the maximum possible moment. In fact, the two curves of Figure 4.40 cross each other which indicates that the local bending moment influence is greater than the cable location dependency at high F/FH ratios. It is interesting to note that there is very little moment magnification due to F/FH until F/FH is greater than 0.7. The magnification increases when F/FH increases beyond 0.7, but this is well beyond the existing cable stayed bridge maximum F/FH ratio of 0.3. Thus, there is very little moment magnification of the standard model and the models within normal ranges, but there is a high initial deviation from the Hetenyi moment. This is due to the same reasons given for the deflection magnification plot, i.e., the low foundation stiffness at the left deck section and MB is at the low deck axial Chapter 4. Backspan Model 73 2.5-r 2.4- 2.3- 2.2- 2.1- • Standard M o d e l 2- 1.9- MB 1.8- N = 80 MH 1.7- N = 20 1.6- 1.5- 1.4- 1.3- 1.2- 1.1- 1- hi/lB hN/lB f 0.50 0.20 0.50 0.30 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 _F_ FH Figure 4.41: Magnif icat ion of m a x i m u m moment. 1.1 1.2 load region. The moment plot as a function of dimensionless decay length ID/IB is shown in F igure 4.42. It can be seen that MB/MJJ is highly dependent on ID/IB in the normal range (ID/IB from 0.40 to 0.65) for cable stayed bridges. For ID/1B < 0.5 the model wi th twenty cables looses accuracy and the model w i t h iv" = 80 should be used for analysis purposes. Just as for deflection, if Figure 4.41 were to be used for design purposes it would be necessary to plot a series of curves, each wi th a different value of ID/IB, wi th the curves for ID/IB > 2.0 al l being the same and going through MB/MJJ = 1.0 at Figures 4.43 and 4.44 show the model moment deviation for varying cable tautness and live load, respectively. It can be seen that the moment deviation is more severe than the deflection deviation, but the general behaviour w i th varying dimensionless ratios is s imilar to the deflection behaviour. F/FH = 0.0. Chapter 4. Backspan Model 74 MH 1.9- 1.8- 1.7- 1.6- MB_ 1.5- 1.4- 1.3- 1.2- 1.1- 1 0.9-r I Standard Model .N = 80 M B = 0.30 hN/lB = 0.50 f = 7 u>i/u>D = 0.20 F/FH = 0.10 i i i r— 0 0.2 0.4 0.6 0.8 I I I I 1.2 1.4 1.6 1.8 h IB - r 2 2.2 2.4 2.6 Figure 4.42: Dimensionless moment versus dimensionless decay length. 2 1.9 1.8 MB MH 1.5 H 1.4 1.3 1.2 1.1 1 \ N - 20 hi/lB = 0.30 • Standard Model hs/la = 0.50 wL/wD = 0.20 F/FH = 0.10 h / h = 0.50 7 8 9 10 11 12 13 14 15 Figure 4.43: Dimensionless moment versus dimensionless cable tautness. Chapter 4. Backspan Model MB MH 2- 1.9- • Standard Model 1.8- 1.7- 1.6- N = 20 1.5- 1.4- hi/lB = 0.30 1.3- hN/lB = 0.50 1.2- / = 7 1.1- F/FH = 0.10 ID/IB = 0.50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 1 1 0.8 0.9 1 Figure 4.44: Dimensionless moment versus dimensionless load. Chapter 5 Full Bridge M o d e l 5.1 The M o d e l The final model is a representation of an entire cable stayed bridge, half of which is shown in Figure 5.45. The cables are attached to the back and main spans at constant spacing 5 and to the towers at constant spacing s'. The towers are fixed to the ground but not directly connected to the deck. A spring, equivalent to a single vertical cable, is connected to the deck at the deck-tower intersections to keep the deck spans between cable stays constant at the towers. The cables are arranged in a fan shape with the anchor stay cables attached to a single point on the deck. The mainspan stays are arranged such that a deck section of length s is at the center of the bridge. The deck stiffness is constant along the entire bridge and the live load is applied as a distributed load. The definition of all the parameters used in this model is given in the following: Ac, — t'th cable cross-sectional area total anchor cable cross-sectional area E — deck elastic modulus Ec = cable elastic modulus Ex = tower elastic modulus F = maximum axial load in deck due to dead load Fi t'th axial load in deck due to t'th cable K t'th cable height 76 Figure 5.-15: Bridge model (half shown—symmetric, except right hearing is a fixed pin). Chapter 5. Full Bridge Model 78 hi tower height I = deck moment of inertia h = tower moment of inertia U horizontal projected length of t'th cable IB = backspan length IM mainspan length N number of deck intervals in the backspan NC = number of cables in the backspan or half the mainspan P maximum axial load in tower due to dead load Pi = t'th axial load in tower due to t'th pair of cables Si = t'th cable chord length s = length of deck interval between cables s' length of tower interval between cables Ti = t'th cable tension in the chord direction due to dead load = weight per unit length of t'th cable WD = uniformly distributed dead load critical uniformly distributed live load Oi t'th cable chord angle of inclination to deck There are four new parameters introduced in Figure 5.45: ETIT, hT, lM and ACa- The tower stiffness EJIT is modelled as being constant along the entire length hT. (The tower height is measured from the tower foundation; whereas, the cable heights are measured from the deck.) The anchor cable in the previous models was a single cable that was modelled in the same manner as the rest of the backspan cables. That single cable is present in this model with the addition of several more cables acting as anchors. This group of cables is referred to as the anchor cable and the total area Aca Chapter 5. Full Bridge Model 7 9 of this anchor cable is independent of the rest of the backspan cables. The number of cables that comprise the anchor cable is determined by the number of extra cables required to keep the cable spacing constant over the mainspan length IM. Thus, each tower has the same number of cables attached to the mainspan side as to the backspan side. In this model the maximum value of the subscript " t " is Nc where Nc is the number of cables attached on one side of a tower. This is different from the previous model which has a maximum value of N—the number of deck spaces in the backspan. The representative values for cable properties is still taken as the "TV/2" value. The maximum cable height, however, is now kNc with the bottom cable location at hi. The remainder are at a constant spacing of s', where , _ hNc - hi Nc-1 This nomenclature allows the entire range of cable layouts to be defined, from the radiating shape to the harp arrangement. The relationship of the previous model cable height hN to this model is hN = h N c . Initially the the input geometry is the dead load geometry and the cable tensions Ti are calculated such that there are no moments or deflections in the deck or towers. This is shown in Figure 5.46 which is an exploded view of Figure 5.45. The horizontal and vertical components of the cable tensions are drawn at the cable ends and the reactions of these forces are drawn at the corresponding tower and deck locations. The dead loads are also shown, but they are not actually applied in the computer model. That is, the cable, tower and deck axial forces at input geometry are in equilibrium with the deck dead load. For a moment free tower at input geometry, the horizontal components of the anchor cable tensions and the corresponding mainspan cables must be equal. This means that the anchor cable vertical forces are not equal to the corresponding mainspan cable vertical forces, as is shown in Figure 5.46. The effect of this is a slight eccentricity of FNa tan 0Nc WpS^WpS wDs WDSE, WDS \~'~\ w D s w D s l - H WpS WpS I — H TJ/V° Ft ' a n 0* WpS WpS WpS WpS WpS WpS WpS I L 1 L U I L I Yj£ c Fi tan^ w p s i i D5 J D s J D s ti ca wDs J D s I I I I I I I I I I tUoS 1 0 D - S WpS WpS WpS WpS WpS WpS WpS WpS I I I I I I I I I WpS WpS WpS WpS WpS WpS WpS WpS WpS WpS p 8 to o a. Figure 5.46: Free body diagram of bridge model (half shown i—symmetric). oo o Chapter 5. Full Bridge Model 81 the tower axial load if the cable to tower connections are not on the tower centerline. The effect of this horizontal eccentricity of the cable connections is neglected in this analysis. The parameter P is the maximum axial load in the tower induced by the cables under dead load only. The P,- represent the axial load in the t'th tower section due to the pair of cables attached above that section. For non-anchor cables, P,- = 2wps. For the anchor cables, P,- = Tisin0,- + wps where 71 and 0, are the anchor cable tensions and angles. The axial load in the tower at any location is the sum of the P, above that point, and P = £ ^ ^he prestress due to the tower self-weight is small compared to the cable induced prestress and is not considered in this model. The parameter F is the maximum axial load in the deck induced by the cables under dead load only. The F, represent the axial load in the t'th deck section due to Ti, i.e., Fi = Ti cos 0, and F = J^Ji for the mainspan. The backspan deck i only goes to TV, but the sum of the mainspan Fi and backspan F,- are equal which means that the backspan F^ is equal to the mainspan "E^Jtf Ft. To fulfill the desire to have all the cable dead load stresses the same, the stay cable areas are again determined by A C N / , sin 6N/2 Ad = T~2 sin 0, The anchor cable area Aca is independent of the stay cable areas, but the proportion- ing of the individual anchor cables is determined by the constraints of constant dead load cable stress and the sum of the individual anchor cable areas being equal to the aggregate anchor cable area. This involves solving a system of linear equations; i.e., the requirement of constant cable stress AcN _ ACfl+l _ A c N c produces Nc — N equations while there are Nc — N + 1 unknowns (the individual cable areas of the anchor cable, i.e., Ac{,i = N,NC—the cable tensions are known). The Chapter 5. Full Bridge Model 82 final equation comes from the anchor cable area relationship: No where Aca is acquired from existing cable stayed bridge designs. An approximate analytical method of determining the anchor cable area is derived from the assumption of dead load stress in the anchor cable being equal to the other cables. That is, (hi. _ u\ ,„„ ... ^ (If - lB) wD _ _ WDi = oc = ACa sin 6Nc NAcN/i sin 0N/2 Rearranging terms and using IB/IM = 0.4 (from the standard model) gives _ NAcN/a sineN/2 Aca — " : 2— 4 sin 6Nc where for the standard model again, sin 6Na which results in N A C N / I This formula could be used in preliminary design values for the anchor cables, but in this study the actual bridge anchor cable areas are used. 5.2 Stability 5.2.1 Stability Dimensionless Ratios The critical live load of the system is defined by the function wLc — T[EI, ETIT, A C N / 2 E C , A C A , IB, IM, hi, h N c , hT, N, VJCN/3, VJD] (5.15) All thirteen of the parameters in Equation 5 .15 are independent of each other. The parameters that do not appear in the equation are functions of the ones used in Equation 5 .15 , i.e., hN/2 IB Chapter 5. Full Bridge Model 83 ^N/2 = sin 6 N/2 IB S = N , _ h N c - hi Nc-1 rp WDIB •1 N/2 = N sin P « WDlM Nc .„ i Nc F = £ r , c o s * , = ^ V > t 0 , where t is the counter along the mainspan from the tower to midspan and h co 1 hNMN-i)+hN{i-i)\ h! + hN{l^jj) N/2 = 1 £ tan 6 hNli = : s — and l^/2 = 7y—r Z N hN = h N c - (Nc —n)s' Chapter 5. Full Bridge Model 84 for both even and odd values of N. Eleven dimensionless ratios need to be constructed out of the thirteen parameters in Equation 5.15. The eleven ratios chosen to represent this model are: N h IB IB hx EX IT EI U>Lc F_ FH N i-l N ACN/2ECEI sin3 6N/2 hhN/2 IJD *V2 IB iBhN/jEI IB \ NAcN/3Ecsm36N/2 y Chapter 5. Full Bridge Model 85 , _ w D / B The dimensionless ratios N, hi/lg, WLC/WD, ID/IB, and / are the exactly the same as the ones used for the previous model. The ratios hxc/lB and F/FH are respectively equivalent to hs/lB and F/FH of the previous model. That is to say F is the same in both models but the individual Fi are different in the two models, and as stated previously, hftc is the same as of the backspan model. The new ratios are the ones that define the tower, mainspan and anchor cable properties. The ratios HT/IB, IB/IM and ETIT/EI have obvious physical meanings and AcJ 1 2 Ac is simply the ratio of anchor cable area to backspan cable area. In summary, the stability of the full bridge model is described by the function: U>D hi hNc hr IB EJIT ACa , _F_ ' / a ' IB ' / « ' / M ' EI ' £ A c ' U lB' FH This function is based on beam-column on an elastic foundation stability theory with both the cable slackness and the P— A nonlinearites modelled. 5.2.2 Parameters and Ratios from Existing Bridge Designs The same nine cable stayed bridges that were used for the first two models are used to acquire data for the construction of the full bridge model. The representative cable is the one corresponding to i = N/2 as explained previously. The other parameters that are applicable to both the backspan model and the full bridge model are determined from the design drawings in exactly the same manner as described in Chapter 4. Indeed, the parameters and dimensionless ratios given in Appendices C.3 and C.4 are used for the full bridge model. The fact that F/FH is made the same as the previous model is of no consequence in the dimensional analysis, but it does require some explanation in regards to the actual Chapter 5. Full Bridge Model 86 values for the existing designs. The full bridge model F is the sum of Nc deck forces, whereas the previous model F is the sum of N deck forces. In the cases where Nc is greater than N, F is obviously greater for the full bridge model than the backspan model, which is also true for F/FH—FH being the same for both models. The new values for F and F / F H along with Nc are given in Appendices C.5 and C.6 along with all the other paramaters and ratios. F is calculated using the approximation of F = N C F N C / 2 , or p = W d 1 m 2 tan 9Nc/2 which is—as explained in Chapter 4—a conservative calculation of F. The average value of FI FH for the full bridge model is now greater than 0.1, but 0.1 is chosen as the representative value for the sake of continuity with the previous models. The additional parameters necessary to model the towers, mainspan and anchor cables are acquired from the bridge designs and listed in Appendix C.5. (Unfortunately, the tower and anchor cable information was not available to this author for three of the nine bridges.) As stated previously, the tower height hx is measured from the tower foundation and the anchor cable area Aca is the sum of the cross sectional areas of the cables acting as anchors. The tower moment of inertia J is modelled as being constant throughout its length. In the real cable stayed bridge designs this is rarely the case, and for the varying J towers a substitute tower with a constant / is used. The substitute tower / is calculated to make the free standing tower deflections of both the real and substitute towers the same for a unit lateral point load at the top. The tower moment of inertia for the Alex Fraser concrete design is so much larger than the other bridges—most probably because of architectural considerations—that it is not included in finding the representative ratios. The representative dimensionless ratios of the standard full bridge model are N = 20 hjla = 0.30 Chapter 5. Full Bridge Model 87 0.50 hT/lB 0.80 0.40 Ex IT I EI 50 0.50 7 ID/IB 0.50 F/F„ 0.10 wL/wD 0.20 where, for the purpose of calculating / : wcN/3 = AcN/7ic- 5.2.3 Results of Stability Analysis The standard bridge model of Figure 5.45 is constructed using the preceeding values for the dimensionless ratios. In order to find the minimum live load at which the model buckles, the most severe load distribution must be determined. To this end, influence lines for maximum axial load in the deck are calculated using the computer program ULA. Four load cases, which result from the influence lines for maximum axial load at the towers and at the backspan anchor connections, are indicated from this study as shown in Figure 5.47. The load cases are numbered one to four, indicating maximum deck axial load from left to right. The bridge deflected shape, deck axial load (dead load plus live load) and deck moment just before stability failure are shown in Figures 5.48 and 5.49 for all four load cases on the standard model. The four axial distributions are drawn to the same scale and the four moment diagrams are drawn to the same scale as well, but different from the axial scale. Using the computer program ULA, the critical load ratio for the standard model Chapter 5. Full Bridge Model 88 Load Case 1 I J M M M M M A = 1 Load Case 3 A = 1 Load Case 4 u J J w H t < H I 1 Figure 5.47: Axial influence lines and load cases for instability. Chapter 5. Full Bridge Model 89 Deflection A x i a l Moment 1 ^TTTTfTfrrrrti. Deflection Load Case 2 1 1 1 1 ( i n i m m i Axial M o m e n t V i ln ius Figure 5.48: Stability failure of standard bridge model l—load cases 1 and 2. Chapter 5. Full Bridge Model Deflection Axial Moment Deflection - Load Case 4 l| 4 4 4 4 4 i i i i u i i _ Axial Moment i l k , Figure 5.49: Stability failure of standard bridge model—load cases 3 and Chapter 5. Full Bridge Model 91 under each load case is found, and the factor of safety against elastic stability failure is calculated. For example, the critical load ratio for load case 4 is found to be VJLC/VJD = \WL/U>D = 2.37, which is to say, since WL/WD = 0.2, that the factor of safety on the live load against elastic buckling of the standard model is 11.9. This is far less than the factor of 91 obtained for the backspan model. This is mostly due to the extra backspan cable tensions from the live load in the mainspan and—as will be explained shortly— due to a less stiff deck section causing instability. The critical load ratios for the four load cases and the corresponding factors of safety against elastic stability failure are: Load Case 1 2 3 4 The influence of an uplift pressure on the factor of safety is investigated by placing uplift pressures on the deck regions that are not loaded (as indicated by the axial influence lines). The resulting factor of safety plot for a range of uplift pressures on all four load cases is given is Figure 5.50. The uplift pressure has a pronounced effect on the factor of safety, with a near linear decline of 37% to 30% uplift pressure. The rate of decline decreases for higher uplift pressure with a further drop of 52% for uplift pressures of 100% of the live load. Wind and dynamic loads, and the probabilities of load combinations are beyond the scope of this thesis and the uplift pressure load case will not be used in further analyses, but it should be kept in mind that a more severe result is possible. The previous discussions were concerned about the effect of various loading patterns on the elastic stability of the standard model with F/FJJ = 0.10. The fact that the bridge buckled at all (one might expect that instability would occur only if F/Fg = 1.0) — - Factor of Safety 2.63 13.2 2.50 12.5 5.24 26.2 2.37 11.9 Chapter 5. Full Bridge Model 92 0 I i i i i I I I I I ! 0 10 20 30 40 50 60 70 80 90 100 Uplift Pressure {% of Live Load) Figure 5.50: Stability factor of safety of bridge model. is accounted for by the same three reasons given for the previous models; that is, the live load axial force must be accounted for, along with the nonuniformity of the axial distribution, and Fu must be calculated for the critical deck section. Analytical adjustments cannot be made for the axial load in this model as was done for the previous two models. Instead, numerical values of average deck axial loads over a half-wave are taken as the values for F*VI^. The corresponding values of Fg~. also cannot be calculated in the same manner as previously because the critical deck section is not a pin-pin beam-column on an elastic foundation. The deflected shapes and moment diagrams in Figures 5.48 and 5.49 suggest that semi-infinite boundary conditions are appropriate. The deck section in the mainspan near the right tower—for load cases 1, 2 and 4—with the near zero moment and smooth deflection, behaves like the end of a semi-infinite beam-column on an elastic foundation. In load case 3, the deck section just to the left of the left tower can be Chapter 5. Full Bridge Model 93 thought of as the end of a semi-infinite beam-column on an elastic foundation. Hetenyi gives the critical axial load for a semi-infinite beam-column on an elastic foundation as yfkEI and F#_s is calulated as the average value of this formula over a half-wave. That is, _ /AcEcEI sin3 B F h ~ = \ 7h where Ac is the final cable area after the changes are made for the catenary calculations. Plots of F*vg^/Fjj^ along the deck for each of the four load cases are given in Fig- ure 5.51. It can be seen that the rightmost deck section is critical {F*vg^/FH^ = 1) ex- cept for load case 3 where the critical deck section is between the center of the mainspan and the left tower. However, load case 3 does not govern in stability considerations and in general the worst case bridge stability failure is caused by the combination of low cable stiffness near the end of the bridge and the high deck axial loads induced by the backspan cables which are acting as tiebacks for the mainspan cables which in turn, support the mainspan loads. Thus, the full bridge model stability mechanism can be understood in terms of a beam-column on an elastic foundation if the effects of the nonuniform elastic foundation, varying axial loads, and applicable boundary conditions are taken into consideration. The standard model is next varied by altering only F / F H and then determining the live load necessary to cause the deck to buckle. The results of this exercise for each load case are shown in Figure 5.52. The F / F H ratio is not adjusted to include the live load as was done in the previous chapters because of the noncontinuous load distributions of the four load cases. The standard model is represented by the solid square in Figure 5.52 and load case 4 produces the lowest factor of safety for this model. Load case 4 is also the governing load case for all values of F / F H , and this load case will be used for the rest of the stability analyses. It is interesting to note that load cases 1, 2 and 4 exhibit nearly identical behaviour in Figure 5.52 yet load case 3 is unique. This is because load case Chapter 5. Full Bridge Model 94 4 Load Case 1 1 1 H i 1 i 4 J J J _ j F' FM~ * Load Case 2 • i n n F' Load Case 3 * ^ ^ ED F' Figure 5.51: Critical deck section for each load case. Chapter 5. Full Bridge Model 95 1.5 0 0.5 1 1.5 2 2.5 3 3.5 A Wp Figure 5.52: Stability of full bridge model. 3 causes the bridge instability to occur at a different deck section, as demonstrated in Figure 5.51. The nine existing bridge designs from Appendix C are plotted on Figure 5.52 as the numerals 1 to 9. The scatter of the numerals qualitatively shows how the existing bridge designs compare with each other. The theoretical result for a beam-column on an elastic foundation would plot on Figure 5.52 as a horizontal line at F/Fg = 1.0. This apparent discrepancy between the Hetenyi and computer results can be eliminated if F*vg^/Fg^ is used instead of F/Fg. This is illustrated by Figure 5.53 where plots of F*vg^/Fjj^ along the deck for load case 4 are shown for different values of F/Fg. It can be seen that as F/Fg increases, the boundary conditions of the critical deck section change from a semi-infinite to a pin-pin beam-column on an elastic foundation. In order to get the expected Hetenyi result the apparent boundary conditions must be used to calculate Fg^ along with the appropriate deck section F* . Chapter 5. Full Bridge Model 96 Load Case 4 \ i Figure 5.53: Critical deck section for varying F / F H . Looking again at Figure 5.52 it can be seen that the factor of safety against buckling decreases with increasing F/FJJ. The rate of this decrease is shown in Figure 5.54 for the standard model with wi/wp = 0.20 and load case 4. The possibility of F jFjj being 0.20 for a bridge with WL/WD = 0.20 is quite likely—as indicated by the distribution of actual bridges on Figure 5.52—and the factor of safety for this bridge is 6.5, which is a drop of 45% from the standard model factor of safety of 11.9. This is comparable to a 40% uplift pressure on the standard model. Figure 5.54 can be used to determine the factor of safety on the live load if a load factor is applied to the dead load. This is calculated by using the fact that F is directly proportional to wp. Thus, if tup is multiplied by 1.5, for example, then F/FH is multiplied by 1.5 and the corresponding factor of safety for the new F/FH ratio is read off of Figure 5.54. This procedure yields 8.7 as the factor of safety against elastic stability failure. In other words, instead of wp + 11.9u>£ causing instability, Chapter 5. Full Bridge Model 9 7 25-| 20- 15- o s 10 5- 0- Standard Model 7 wL/tvD = 0.20 ID/IB = 0.50 0.50 ETIT/EI = 50 N = 20 h i / l B = 0.30 hNc/lB = 0.50 = 0.80 IB/IM = 0.40 I I I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 FH Figure 5.54: Factor of safety vs F/Fg. 1.5WD + 8.7WL causes instability. As another example, the factor of safety on the live load is 6.7 when the dead load factor is 2; i.e., 2.0ttf£> + 6.7u>£ also causes instability. The decay length to backspan length dimensionless ratio ID/IB is now varied in the same manner that ID/IB was varied for the backspan model. The range of ID/IB in existing cable stayed bridge designs is from 0.40 to 0.65 and the stability curves for ID/^B = 0.40 and ID/IB — 0.65 are plotted in Figure 5.55. There is only a very small variation with ID/IB, a nd the stability factor of safety for the standard model varies from 11.2 to 12.4 which is ± 5 % from the standard value of 11.9. Just as in the previous model, altering the initial cable tautness / in the normal range of 5 to 10, does not affect the model behaviour. The two new ratios of Aca/Y. Ac and ETIT/EI are also varied in the same manner as ID/IB- The normal ranges of AcjUAc and ETIT/EI are 0.4 to 0.6 and 25 to 100 respectively, and the model be- haviour is only slightly affected within these ranges, as shown in Figures 5.56 and 5.57. Chapter 5. Full Bridge Model 98 1.5 Figure 5 .55 : Stability of full bridge model—ID/IB varied. The stability factor of safety varies from 11.2 to 12.4 for the variation of Aca/HAc which is ± 5 % from the standard value of 11.9. For the variation of ETIT/EI the fac- tor of safety ranges from 11.2 to 12.4 as well, and is again a ± 5 % variation from the standard. The remaining dimensionless ratios, which are all based on the model geometry, are not investigated. 5.3 Deflection and Moment The choice of deflection and moment functions to represent the full bridge model is based on the same rationale used in Chapters 3 and 4. The mainspan deflection and moment is investigated as well as the backspan deflection and moment. This is nec- essary because the maximum deflection will be in the mainspan and the maximum Chapter 5. Full Bridge Model 1.5 U>£> Figure 5.56: Stability of full bridge model—Ac./23 Ac varied. Chapter 5. Full Bridge Model 100 moment will be in the backspan. For the sake of continuity and consistency the deflec- tions and moments for both the backspan and mainspan will be compared to that of a simply supported beam on an elastic foundation without axial load. The definition of the full bridge model backspan deflection in terms of independent parameters is 8B = JB[EI, E T I T , A C N / 2 E C , A C A , IB, IM, h u h N c , h T , N, W C N / 2 , W D , VJL] and the maximum nondimensional backspan deflection of the bridge model is described by the function: S * - T hi h N c h T lB E T h ACA , h F • * > ! > ! > i > , > o r I r-v i I / l i ! ' r - t IB* IB ' IB* IM* EI ' £ A c ' ' ' B ' W C ' Fg The maximum nondimensional mainspan deflection of the model is described by a function that uses exactly the same ratios as the backspan function, but the functions themselves will be different. The mainspan deflection function is Sg hi hsc h T IB ETIT AC, lp wL F IB' h ' * B ' / M ' E I ' £ Ac' h ' wD' Fg With Sg defined by Eq. 3.12 and k = N A C N / 2 E C sin3 0N/2 these functions will show the magnification of the maximum deck deflection due to the deck axial load. All other ratios in these functions are the same as the ones defined for the stability analysis of this model. The definition of the bridge model backspan moment in terms of independent pa- rameters is M B = ?B[EI, E T I T , A C N / 2 E C , A C A , IB, IM, h u h N c , h T , N, wCfl/2, wD, wL\ and the maximum nondimensional backspan moment of the bridge model is described by the function: M B Mg N, hi h^n h T lB ETIT A C a , h wL F , J, IB' h ' h ' V EI ' £ A C ' " lB' ™D Fg Chapter 5. Full Bridge Model 101 Just as for deflection, the maximum nondimensional mainspan moment is described by the same ratios but a different function. The mainspan moment function is N hi hNc hT /JB ETIT ACA lD wL F M„ With MJI defined by Eq. 3.13 these functions will show the magnification of the maximum deck moment due to the deck axial load. 5.3.1 Results of Deflection Analysis The same standard full bridge model is used for maximum deflection analysis as that for the stability analysis. However, different load cases are necessary in order to get the maximum deflection in the backspan and mainspan. The envelope of maximum deflection influence lines is determined using ULA, and the maximum backspan and mainspan deflection influence lines are shown in Figure 5.58 along with the resulting live load distributions and the locations of the maximum deflections. It should be noted that influence lines are only valid for linear structures. The cable stayed bridge being studied herein is nonlinear in that P—A effects and catenary cables are included in the analysis; therefore, the influence line analysis is only approximate. It turns out, however, that the deflection influence lines work very well for this structure because the load patterns for maximum deflection cover long sections of the deck and would not change, no matter how nonlinear the cable stayed bridge is. The deflection of the standard model (magnified 200 times) under the maximum backspan deflection load case with WL/WD = 0.20, is shown in Figure 5.59 with the Hetenyi deflection superposed on the left backspan. The 6B/£>H ratio is equal to 3.155 which is a significant deviation from the Hetenyi deflection and cannot be entirely attributed to magnification due to the deck axial load. In fact the majority of the backspan deflection is due to the mainspan deck flexibility which allows the tower to move; thus, allowing greater backspan deflections than in the previous models. The model deflected shape under the maximum mainspan.deflection load case is Chapter 5. Full Bridge Model 102 Figure 5.59: Maximum backspan deflection (magnified 200 times) of standard full bridge model. Chapter 5. Full Bridge Model 103 shown in Figure 5.60. The Hetenyi deflection is again superposed on the left backspan and the deflections are magnified 200 times. The 6M/6n ratio is equal to 7.177. This is more than twice the value for 6B/6JJ which is mostly attributable to the mainspan being 2.5 times as long as the backspan. The bridge deflection does not resemble a beam on an elastic foundation deflection because of the tower movements which are made possible by the flexible backspans. Figure 5.60: Maximum mainspan deflection (magnified 200 times) of standard full bridge model. The maximum live load deflection versus span ratios, 6B/IB and 6M/IM are 1/1000 and 1/1100 respectively, which are well within the recommended value of 1/500. The standard model is now varied by altering the F/Fg ratio while keeping all other ratios constant. The maximum deflection is then determined from a computer analysis of the maximum backspan deflection load case and the maximum mainspan deflection load case. The 6g /6g and 6M/6B ratios from their respective load cases are plotted against F / F H in Figure 5.61 for a range of F/FH values. The mainspan curve governs and the backspan curve is shown for interest sake. In the 6M/6H ratio, the u6M" is the magnified maximum deflection of the deck due to the presence of axial load in the deck. The u6jjn is Hetenyi's maximum deflection of a beam on an elastic foundation without axial load. Thus, the 6M/6H ratio is expected to Chapter 5. Full Bridge Model 104 15 14 13 12 11 10 9 6 8 7H 6 5 4- 3 2 1 0 / = 7 JV = 20 0.20 • Standard Model hi/lB = 0.30 h/lB = 0.50 0.50 0.50 0.80 ETIT/EI = 50 IB/IM = 0.40 Mainspan (f̂ ) ' BackTpan (ij) 0.1 0.2 0.3 0.4 0.5 0.6 _F_ 0.7 0.8 0.9 1.1 1.2 Figure 5.61: Magnification of maximum deflection. be equal to 1.0 for F/FH equal to zero, and as F/FH increases the &M/6H ratio should deviate from 1.0, showing the magnification behaviour. The ordinate intercept is 7.1, not 1.0. This high initial deflection can be attributed to the relatively soft foundation at the mainspan midpoint and the flexible towers. It must also be noted that the model mainspan deflection is still being compared to a Hetenyi beam of span IB not IM- The deflection magnification with increasing F / F H is visible in Figure 5.61 but the increase is not very rapid because of the low deck axial load at the midspan. The deflection increases rapidly after F / F H reaches 0.75 due to the large deflections of the backspan at this point which allows the mainspan to sag. As expected from Figure 5.52, the bridge becomes unstable at F / F H = 0.87. The deflection plot as a function of dimensionless decay length ID/IB is shown in Figure 5.62. It can be seen that 6M/6H is highly dependent on ID/IB and this is especially true in the normal range of Ip/ls from 0.40 to 0.65 for cable stayed bridges. Chapter 5. Full Bridge Model 105 If Figure 5.61 were to be used for design purposes it would be necessary to plot a series of mainspan deflection curves, each w i th a different value of Ip/ls- 15 Figure 5.62: Dimensionless deflection versus dimensionless decay length. The cable tautness graph is shown in Figure 5.63. The cable tautness has very litt le effect on the mainspan deflection in the normal range of / . However, the importance of having tight cables is demonstrated by the rapid increase of the deflection for low values of / . Figure 5.64 shows the influence of the live load to dead load ratio wifwp. It can be seen that the m a x i m u m deflection of the standard full bridge model is not affected by this ratio. The deflection plot as a function of the ratio of anchor cable area to backspan cable area Ac,/T,Ac is shown in Figure 5.65. It can be seen that 6MI&H is also quite dependent on ACtf H Ac in the normal range of ACo/T, Ac from 0.40 to 0.60. If Figure 5.61 were to be used for design purposes it would be necessary to have a series of Chapter 5. Full Bridge Model 106 15-I 14- 13- 12- 11- 10- 9- 6 8- 7- 6- 5- 4- 3- 2- 1- 0- Standard Model 0.20 F/FH = 0.10 ID/IB = 0.50 0.50 ETIT1 EI = 50 N = 20 hi/lB = 0.30 hsc/lB = 0.50 hT/lB = 0.80 IB/IJ* = 0.40 Mainspan ( f ^ ) Backspan ( f t ) 2 3 4 5 6 7 8 9 10 11 12 13 U 15 Figure 5.63: Dimensionless deflection versus dimensionless cable tautness. 6_ 15-i- 14- 13- 12- 11- 10- 9- 8- 7- 6- 5- 4- 3- 2- 1- 0-- Standard Model / = 7 F/FH = 0.10 ID/IB = 0.50 ACJY,*C = 0.50 ETIT/EI = 50 N = 20 — 0.30 w * 0.50 — 0.80 0.40 Mainspan (f̂ ) Backspan ( f t ) i 0.1 0.2 0.3 0.4 0.5 0.6 Wp 0.7 0.8 0.9 1.1 1.2 Figure 5.64: Dimensionless deflection versus dimensionless load. Chapter 5. Full Bridge Model 107 Act Figure 5.65: Dimensionless deflection versus dimensionless cable area. 6_ 1 5 T 14- 13- 12- 11- 10- 9- 8- 7- 6 - 5- 4- 3- 2- 1- 0-- Standard Model / = 7 Wi/tVp = 0.20 F/FH = 0.10 0.50 0.50 Mainspan (jjj) 50 I 100 El N — 20 hi/lB 0.30 hf/c /lB — 0.50 hrlh 0.80 IB/IM — 0.40 Backspan ( ^ ) 150 200 Figure 5.66: Dimensionless deflection versus dimensionless tower stiffness. Chapter 5. Full Bridge Model 108 figures. Each figure would be valid for a specific value of Ac./"52 Ac and on the figure would be a series of mainspan deflection curves, each with a different value of ID/IB- The final graph of the tower stiffness effect is shown in Figure 5.66. The tower stiffness has very little effect on the mainspan deflection, which means that the cable stayed bridge stiffness is dependent on the anchor cables and hot the towers. 5.3.2 Results of Moment Analysis The same standard full bridge model is used for maximum moment analysis as that for the deflection analysis. However, different load cases are again necessary in order to get the maximum moment in the backspan and mainspan. The envelope of maximum moment influence lines is determined using ULA, and the maximum backspan and mainspan moment influence lines are shown in Figure 5.67 along with the resulting live load distributions and locations of the maximum moments. Backspan Moment Load Case Mainspan Moment Load Case ^— MM 1 Figure 5.67: Load cases for maximum backspan and mainspan moment. Chapter 5. Full Bridge Model 109 The fact that the influence line analysis is only approximate for this structure is more of a concern here than for deflections. This is particularly true for the maximum mainspan moment load case. The length of the short segments of loaded deck in the mainspan are not expected to remain constant to produce the maximum moment as the nonlinear analysis of the bridge model progresses. As the standard model is altered by varying the dimensionless ratios, large nonlinearities are expected and unusual phe- nomenon may appear in the mainspan moment plots. However, the mainspan moment does not govern, and the structure is not too nonlinear at service load levels, so the influence lines are still applicable for the standard model. The moment of the standard model under the maximum backspan moment load case with WL/WD = 0.20, is shown in Figure 5.68 with the Hetenyi moment superposed on the left backspan. The MB/MJJ ratio is equal to 3.513. The reason for this large deviation from the Hetenyi moment is twofold: 1) the mainspan deck flexibility allows the towers to move which allows large backspan deflections and 2) the large backspan deflections occur in a relatively soft foundation region which allows a more severe curvature than the Hetenyi model. Figure 5.68: Maximum backspan moment of standard full bridge model. The model moment diagram under the maximum mainspan moment load case is shown in Figure 5.69. The Hetenyi moment is again superposed on the left backspan. Chapter 5. Full Bridge Model 110 The Sm/6JJ ratio is equal to 2.718. This is less than the value for MB/MJJ which is perhaps attributable to the anchor cable pulling back on the left tower which limits the curvature of the deck. That is, the backspan curvature is limited by the mainspan cables pulling on the towers but the mainspan curvature is restrained by the anchor cables which is a much stiffer system. Figure 5.69: Maximum mainspan moment of standard full bridge model. The maximum backspan live load moment is wilB/70 and the maximum mainspan live load moment is tuilB/90. These results can be compared to the moment of WLIB/140 obtained for the backspan model. It is more common, however, to express the maximum live load moment in terms of the bridge mainspan length. For suspension and three-hinged arch bridges the linear live load moment is approximately WLIM/60. The maximum nonlinear cable stayed bridge backspan and mainspan moments for the ratio of IB/IM = 0.4 are WLIM/440 and u>£,/j^/560, respectively. This is a sevenfold reduction in live load moment from other long span bridges and is a major testament to the economy of cable stayed bridges. The standard model is now varied by altering the F/Fg ratio while keeping all other ratios constant. The maximum moment is then determined from a computer analysis of the maximum backspan moment load case and the maximum mainspan moment load case. The MB/MH and 6M/6JJ ratios from their respective load cases are plotted Chapter 5. Full Bridge Model 111 against F / F H in Figure 5.70 for a range of F / F H values. The backspan curve governs and the mainspan curve is shown for reference. M MH 10- 9 8 7 6 5 4' 3 2< 1 0 N = 20 hi/la = 0.30 hsjla = 0.50 hT/la = 0.80 IB/IM = 0.40 Standard Model Mainspan 0.1 0I2 o!3 o!4 0*5 0I6 0I7 o!s O!9 _F_ FH Figure 5.70: Magnification of maximum moment. 1.1 1.2 The moment magnification is evident in that the Ma /MR ratio increases with in- creasing F / F H . The mainspan moment behaves similarly to the backspan moment except that the load pattern used for the mainspan moment causes instability at a lower F / F H ratio than for the backspan moment load case. Even though the mainspan moment increases rapidly just before buckling, it never exceeds the backspan moment. The slightly different behaviour of the mainspan moment curve from the backspan curve suggests that the load case used to produce the maximum mainspan moment should be altered as F / F H varies. The moment plot as a function of dimensionless decay length ID/IB is shown in Figure 5.71. It can be seen that MB/MH is highly dependent on ID/IB\ however, in the normal range of ID/IB from 0.40 to 0.65 the curve is much less steep than for Chapter 5. Full Bridge Model 112 M_ MH 10-1 9 8- 7- 6- 5- 4- 3 2- 1- 0 - Standard Model / = 7 wL/rvD = 0.20 F/FH = 0.10 0.50 EJIT/EI — 50 N = 20 = 0.30 hNC/LB = 0.50 = 0.80 IBIIM = 0.40 Backspan ( & ) \ M H ) Mainspan ( ^ f ) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 h IB Figure 5.71: Dimensionless moment versus dimensionless decay length. surrounding values of ID/IB- The curve for mainspan moment also exhibits a plateau for the normal range of ID/IB with steep sections on either side of this section, but it does not mimic the backspan curve as is the case for all other plots. This suggests that the load pattern must change for changing ID/IB in order to truly obtain the maximum moment. To this end, influence lines are obtained for various values of ID/IB and the appropriate load patterns are applied to get new MM/MR ratios. The corrected curve is plotted in Figure 5.71 as the dotted line. The corresponding curve for the maximum backspan moment is exactly that already plotted because the load pattern does not change with ID/IB in this case. The fact that the new curve is different from the original MM/MU curve shows that using the load pattern obtained for one ID/IB ratio is not the load pattern that will produce the maximum mainspan moment for other values of ID/IB- The new curve actually governs for ID/IB greater than 1.2. A last point that should be brought up Chapter 5. Full Bridge Model 113 is that for low values of Ip/la the moment distribution follows a jagged path rather than a smooth one as in Figure 5.69, and the cable layout used in this model may not produce the maximum moment possible. To get the maximum moment, more cables could be used in the model or the existing cables could be attached to different points on the deck. In any case, the discussion on M ^ / M H VS ID/IB is academic because the backspan moment governs for all practical cases. The cable tautness graph is shown in Figure 5.72. The cable tautness has very little effect on the mainspan moment in the normal range of / . However, the importance of having tight cables is again demonstrated by the rapid increase of the moment for low values of / . 10 wL/wD = 0.20 F/FH = 0.10 ID/IB = 0.50 N = 20 9- hJlB = 0.30 hsc/lB = 0.50 hr/lB = 0.80 IB/IM = 0.40 8- • Standard Model 7- AcJ^Ac = 0.50 6- ETIT/EI = 50 1- Figure 5.73 shows the influence of the live load to dead load ratio WL/VJD- The influence of wi/wp on the maximum backspan moment is negligible in the normal range of w^/wp. There is a marked increase in MB/Mm however, for wi/wp greater Chapter 5. Full Bridge Model 114 than 0.8. The moment plot as a function of the ratio of anchor cable area to backspan cable area Ac./ £ Ac is shown in Figure 5.73. It can be seen that dependence of MB /M# to Ac.IX) A c is not as great as for deflection . However, the importance of heavy anchors is demonstrated by the large increase in backspan moment for Ac./^2 Ac values less than 0.3. The effect of varying the tower stiffness is shown in Figure 5.75. The tower stiffness has very little effect on the maximum backspan moment, which means that the towers act in their most efficient capacity; i.e., as axial members vertically supporting the cables, not as bending members tying the cables back. M 10- 9- 8- 7 6 5- 4- 3- 2- 1- 0 Standard Model / = 7 F/FH = 0.10 ID/IB = 0.50 AcJEAc = 0.50 ETIT/EI = 50 Backspan T — i 1 r 0.2 0.3 0.4 0.5 0.6 0.7 0.8 XJJD Figure 5.73: Dimensionless moment versus dimensionless load. Chapter 5. Full Bridge Model 115 M 10' 9 8 7- 6- 5- 4- 3- 2 1 0-f Standard Model / = 7 u>i/wp = 0.20 FIFH = 0.10 h/h = 0.50 0.50 Backspan 200 Figure 5.75: Dimensionless moment versus dimensionless tower stiffness. Chapter 6 Application The dimensionless stability, moment and deflection plots of Chapter 5 can be used for a preliminary design and analysis aid of cable stayed bridges. A procedure for finding the factor of safety against elastic stability failure and maximum service deck deflection and moment is outlined in this chapter. A conceptual design could determine the materials to use, the tower and deck sizes, and the number and sizes of cables. However, the mainspan length and the live load are usually determined before the bridge design proceeds (bridges are designed to carry specified loads over fixed spans), so a suggested starting point for the conceptual design is to use MB = WLIM/440 and 6M = /M/500 to obtain preliminary values for the deck E and 7. Once the bridge design is more detailed and ready for computer analysis, a great saving of effort can be made by utilizing the results of this thesis to analyze and rean- alyze progressive bridge designs, instead of modelling and analyzing several different designs by computer. When the analysis of the bridge designs utilizing the dimen- sionless ratio models has served its usefulness, a more detailed analysis could then be performed by computer. 6.1 The Model The dimensional analysis model has constant deck and tower stiffnesses and a uniform distribution of cable areas such that each cable has the same dead load stress. The cables are at a constant spacing along the deck and tower. A representative geometric model is used and it is assumed that the model behaviour is independent of the number 116 Chapter 6. Application 117 of cables as long as at least four or five are within the decay length of the model deck. The model representative cable is taken as the one (or average of two) attached to the center deck section of the bridge backspan. The area, height and angle of this cable are used in the model as the representative values for the bridge. The deck moment of inertia of the bridge backspan center section is taken as the constant moment of inertia of the model deck, and the model dead load is determined from the bridge deck cross section at this point. The tower moment of inertia of the model is an equivalent one, such that the free standing tower deflections of the bridge and model are the same for a unit lateral load at the top. Once the representative parameters are determined, the dimensionless ratios are calculated as given in Chapter 5, Section 5.2.1. Note that the cable area ratio is not calculated using the representative cable area; rather, the actual bridge anchor and backspan cable areas are used. Also, the maximum dead load axial force F in the model deck is calculated using representative values, i.e., 6.2 Procedure Once the eleven ratios have been given a preliminary estimate, it is necessary to de- termine if the parameters of the bridge to be analyzed falls into the domain of the procedure described in the next sections. That is, the geometric dimensionless ratios hi/lB, hNa/lB, hxllB and IB/IM should be reasonably close to the values of 0.30, 0.50, 0.80 and 0.40, respectively. The number of cables over the decay length must also be greater than four. In other words, ID/IB should be greater than 4/N. The decay length F = WDIM 2 tan 6Ng/2 is given by where k = Chapter 6. Application 118 6.2.1 Stability The stability factor of safety is found by first determining the critical live load to dead load ratio \ W L / V J D from Figure 5.52 for the value of F/FH calculated for the bridge to be analyzed. The factor of safety on the live load is calculated as the ratio of the final and original live to dead load ratios, i.e., M ( - ) 6.2.2 Deflection and Moment The procedure for determining the maximum backspan and mainspan deflection is slightly more complicated than that for stability because of the influence of ID/IB, Ac* / YlAc and to a lesser degree F / F H . The first step is to obtain the S/6H ratio from Figure 5.62 corresponding to the bridge ID/IB ratio. This 6/6H ratio is valid if F/FH = 0.10 and ACa/T.Ac = 0.50. To adjust 6/6H for a different value of ACa/^2Ac, for example, simply multiply by the 6/6H ratio from Figure 5.65 corresponding to the bridge value of AcjYLAc, then divide by the B/6H value for the standard model. This assumes a linear relationship, and in symbolic form is This adjustment is repeated in a similiar manner for nonstandard values of F / F H , ETIJ/EI, f and WL/WD although they may not be necessary because they are of relatively smaller influence. Once the bridge 6/6H ratio is determined, the maximum deflection can be calculated by multiplying the ratio by SH as given by Equation 3.12 or Figure 3.14. Chapter 6. Application 119 The procedure for determining the maximum backspan and mainspan moment is exactly the same as that for deflection, using the appropriate corresponding moment graphs. The beam on an elastic foundation moment is given by Equation 3.13 or Figure 3.20. 6.3 Examples Two cable stayed bridge designs are taken as examples of the analysis procedure and then compared to computer runs of the actual bridge models. The A L R T steel and concrete designs are used for this purpose. The representative parameters and ratios are given for these two bridges in Ap- pendices C.5 and C.6. Before proceeding with the analysis, the geometric ratios are checked to see if they are close to the standard model ratios. Table 6.1 shows the per- cent deviations of the two designs from the standard model values, along with the check on the number of cables within the decay length. There is a sufficient number of cables for both bridges and the geometric ratio deviations are small except for the ratio that defines the shape of the cable layout, hi/la- The A L R T steel design is close to being a radiating shape and the A L R T concrete design is more of a fan shape. The effects of the two different cable layouts is assumed to be small, and the analysis procedure is deemed to be applicable to the two bridge designs. hT/lB IB/IM hlh > 4/iV Standard Model .3 .5 .8 A A L R T Steel .478 59% .548 10% .841 5% .419 5% .418 > .364 A L R T Concrete .234 22% .516 3% .819 2% .406 2% .423 > .333 Table 6.1: Check on applicability. 6.3.1 Stability The stability analysis using the suggested procedure and the computer analysis of the actual bridges are summarized in Table 6.2. The simple analysis gives conservative Chapter 6. Application 120 factors of safety. The live load factors are 16% and 38% lower than the computer analysis for the concrete and steel bridges, respectively. The radiating cable layout of the steel bridge may be the reason the simple analysis is further away from the computer analysis than the concrete bridge. Sim] pie Analysis Computer Analysis wL/wD F/FH \wL/wD Factor of Safety Factor of Safety Standard Model .20 .10 2.4 11.9 11.9 ALRT Steel .36 .12 2.1 5.83 9.33 ALRT Concrete .17 .24 1.1 6.47 7.69 Table 6.2: Summary of stability analysis. An attempt to improve the correlation between the simple analysis and the com- puter analysis can be made by adjusting the simple analysis factors according to the actual values of ID/IB, Aca/^2Ac and ETIT I EI. However, for these two bridges the adjustments counter one another and the net effect would be minimal. 6.3.2 Deflection and Moment The deflection analysis using the simple procedure and the computer analysis of the actual bridges is summarized in Table 6.3. The 6/SH ratios are calculated for the steel bridge as 6B SH I'M (3.8)^ .418 (3.0) <Zfl__ .595 (3.155)D (6.7) - (7.177)c 3.61 7.28 and for the concrete bridge as Chapter 6. Application 121 Simple Analysis Computer Analysis Standard ALRT ALRT ALRT ALRT Model Steel Concrete Steel Concrete IB — 138 138 138 138 IM — 329 340 329 340 F/FH .1 .12 .24 .12 .24 Ac/E Ac .5 .595 .502 .595 .502 ID/IB .5 .418 .423 .418 .423 — 16 16 16 16 k — 211.5 279.3 — — 6k/wL — 1.07 1.07 — — SH — .0809 .0613 — — 6B/6H 3.155 3.61 4.10 — — 6M/ 6H 7.177 7.28 7.93 — — 6B — .292 .251 .254 .261 6M — .589 .486 .661 .603 6BHB 1/1000 1/473 1/550 1/544 1/530 6M I^M 1/1100 1/559 1/700 1/498 1/564 Table 6.3: Summary of deflection analysis. The simple analysis gives unconservative maximum mainspan deflections. The mainspan deflection ratios are 11% and 19% lower than the computer analysis for the steel and concrete bridges, respectively. The corresponding backspan ratios are 15% higher and 4% lower, which indicates that the simple analysis gives conservative results for the maximum backspan deflections. The moment analysis using the simple procedure and the computer analysis of the actual bridges is summarized in Table 6.4. The M/MH ratios are calculated for the steel bridge as M B (2-7)Jft= 418 (2-6) x 0 . _ = .595 MI M M (3.6) P = .418 •fl (2.718)D (3.3) = 2.58 c„ _ .595 H (3.513): = 3.38 Chapter 6. Application 122 and for the concrete bridge as M, B (2.7K = =.423 (2.8) .24 Mi H (2.718): = 2.78 Mi Mi M H ( 3- 6)^=.42sJ [ ( 3 - 7 ) ^ = - 2 4 (3.513)D = 3.79 Simple Analysis Computer Analysis Standard ALRT ALRT ALRT ALRT Model Steel Concrete Steel Concrete IB — 138 138 138 138 IM — 329 340 329 340 F/FH .1 .12 .24 .12 .24 Ac .IT. Ac .5 .595 .502 .595 .502 ID/IB .5 .418 .423 .418 .423 — 16 16 16 16 MH/wLl'B — .0035 .0035 — — M„ — 1066 1066 — — MMjMH 2.718 2.58 2.78 — — MB/MH 3.513 3.38 3.79 — — MM — 2750 2963 2807.8 2751.1 MB — 3603 4040 3731.0 3480.1 MM/wLl2M 1/560 1/630 1/624 1/617 1/672 MB/wLlM 1/440 1/481 1/458 1/464 1/531 Table 6.4: Summary of moment analysis. There is fairly close agreement between the simple analysis moments and the com- puter analysis maximum moments, and any significant deviations are on the conserva- tive side. The backspan moment ratios are 4% lower and 16% higher than the computer analysis for the steel and concrete bridges, respectively. The corresponding mainspan ratios are 2% lower and 8% higher than the computer analysis. The power in the simple procedure is in being able to change the design and get the effect of the change without computer analysis. Thus, many concepts and alterations can be tested quickly and cheaply. Once the cable stayed bridge design has been sized Chapter 6. Application 123 via the iterative use of the simple procedure, the bridge can then be modelled for computer analysis. At this stage in the design process, the designer would also have a good understanding of the bridge behaviour and be better prepared to interpret the computer analysis results. Chapter 7 Conclusion The elastic stability of cable stayed bridge decks, as well as the deflection and moment of the deck are successfully modelled and analyzed using the computer program ULA (Ultimate Load Analysis). The computer analysis includes P— A effects and true cate- nary cables. The dimensional analysis used to study the the general behaviour of cable stayed bridges shows that the comparison to a beam on an elastic foundation is a valid one to make. Eleven dimensionless ratios are used to describe the model. Five are geometric ratios and are set to representative values and kept constant for this study. The remaining six ratios describe the material properties, structural elements and the loading of the model. These six ratios are varied from the standard and the effect on the cable stayed bridge model is observed. The elastic stability of cable stayed bridges is governed by the laws that govern a beam-column on an elastic foundation. The direct correlation is seen when the critical section of deck is isolated and examined. The critical load case for stability is identified and the factor of safety on the live load against elastic failure for the standard model is 11.9. This factor of safety is dependent on the maximum axial load in the deck under dead load only (F/Fu) and the magnitude of the servive live load (VJL/WD)- The remaining ratios have little (±5%) or no effect over the normal range of values. Deck stability could be the governing criterion if load factors are applied to the dead load, for example, 2.0IVD + 6.7U;L is a load case more severe than the standard WD + 11.9tU£,. Thus, the elastic stability of a cable stayed bridge could sometimes be the limiting design criterion. The maximum deflection and moment of the cable stayed bridge model is compared 124 Chapter 7. Conclusion 125 to that of a beam on an elastic foundation. The maximum live load deflection versus span ratios, 6B/IB and 6M/IM are 1/1000 and 1/1100 respectively, which are well within the recommended value of 1/500. The maximum nonlinear cable stayed bridge model backspan and mainspan moments are WL12m/440 and wLlM/560, respectively. This is a sevenfold reduction in live load moment from other long span bridges and is a major testament to the economy of cable stayed bridges. Both the deflection and moment are dependent on the decay length of the deck and on the anchor cable size. Magnification of deflection and moment is observed for increasing deck dead load axial force. It is also important to note that the cable tautness has no effect on the model response as long as the cables do remain tight. The cable stayed bridge response is also independent of the tower stiffness which is also a cost saving feature because the towers do not have to be designed for heavy bending and the tower foundations do not have to transfer large moments, except perhaps during erection. A simple analysis procedure is developed based on using the charts in this study. Two example analyses are carried out and compared to actual computer runs. For the two examples, the simple procedure is up to 40% conservative compared to the computer runs for elastic stability. The maximum mainspan deflection is up to 20% unconservative and the maximum backspan moment is up to 20% conservative for the two examples used. The large deviations of the simple analysis results from the computer results indicate that caution must be used in interpreting the simple analysis results, especially when the cable layout differs significantly from the standard model layout. However, the simple procedure can still be used to compare geometrically similar cable stayed bridge designs. 7.1 Further Research Research on the effect of the geometric dimensionless ratios, in particular the ratio that defines the cable layout (hi/Is) would complement this study. Research on single tower Chapter 7. Conclusion 126 cable stayed bridges would be interesting as would a space frame analysis on bridges with A-frame towers. Further research on the elastic stability of cable stayed bridges is still called for, however, the real mechanism of cable stayed bridge collapse would involve the formation of plastic hinges in the deck. Therefore, plastic analysis of cable stayed bridges would be a more urgent topic of research. This involves the addition of two more parame- ters, namely the plastic moment capacity Mp of the deck and—for the moment-axial interaction—the deck cross sectional area A. This then requires two more dimensionless ratios say, MplM EI and A EAC It turns out that Mpl\f/EI is approximately equal to 0.6 and A/J^Ac is approxi- mately equal to 10 for steel deck bridges. Using these values of MplM/EI and A/ Y,Ac and the model values of IM, EI and YJ Ac values for Mp and A are input into ULA. The plastic stability analysis under load case 4 is carried out and a plastic hinge forms at 6.76wL/xvD which causes the bridge to become unstable. The factor of safety on the live load for this case is 6.76 which is a 43% drop from the elastic stability factor of safety. The deflected shape of the failed bridge is shown in Figure 7.76. This is a catastrophic mode of failure because the bridge collapses as soon as one hinge forms—there is no warning of imminent failure. The single hinge mode of failure occurs because the axial load in the deck is high enough to cause a P—A stability failure of the bridge with a hinge in it. A more likely mode of failure is plastic hinge formation with the post hinge structure remaining stable. Additional loading would then cause more hinges to form until enough hinges are in the structure to form a mechanism. This occurs for the maximum backspan moment load case, for example. Hinges form at both ends of the bridge at 5.76wL/wp Chapter 7. Conclusion 127 and the bridge is still able to carry even heavier loads until at 9.25wL/wD another hinge forms at the center of the mainspan. The bridge is still stable at this point but cannot carry heavier loads. The deflected shape of the deck with three plastic hinges is shown in Figure 7.77. If the formation of the first hinge is defined as failure, the factor of safety against failure is 5.76, which is lower for this mode than the plastic stability mode. However, this mode of failure may be preferred because there is a warning of collapse. Figure 7.76: Plastic stability failure of bridge model. H \ L >, • Hinge locations Figure 7.77: Plastic hinge failure of bridge model. Bibliography [l] Leonhardt , F . and Zellner, W . , "Cable-Stayed Bridges: Report on Latest De- velopments," Canadian Structural Engineering Conference, 1970, Canadian Steel Industries Construct ion Counc i l , Toronto, Canada. [2] Troitsky, M . S., Cable-Stayed Bridges, Theory and Design, Crosby Lockwood Sta- ples, London , England, 1977. [3] Dischinger, F . , "Hangebrucken fur schwerste Verkehrslasten," Der Bauingenier, M a r c h 1949, pp. 65, 107. [4] Leonhardt , F . and Zellner, W . , "Cable-Stayed Bridges," IABSE Surveys S-13/80, I A B S E Per iodica 2/1980, M a y 1980, pp. 21-48. [5] Tang, M . C , "Buck l ing of Cable-Stayed Girder Bridges," Journal of the Structural Division, A S C E , V o l . 102, No . S T 9 , September 1976, pp. 1675-1684. [6] G ims ing , N . J . , Cable Supported Bridges, Concept and Design, John Wiley & Sons L t d . , Chichester, England , 1983. [7] Smi th , B . Stafford, "The Single Plane Cable-Stayed Girder Bridge: A Method of Analysis Suitable for Computer Use," Proceedings of the Institution of Civil Engineers, V o l . 37, M a y 1967, pp.183-194. [8] Smi th , B . S., " A Linear Method of Analysis for Double-Plane Cable-Stayed Girder Bridges," Proceedings of the Institution of Civil Engineers, V o l . 39, January 1968, pp. 85-94. [9] Troitsky, M . S., and Lazar , B . E . , "Mode l Analysis and Design of Cable-Stayed 128 Bibliography 129 Bridges," Proceedings of the Institution of Civil Engineers, March 1971, p p . 439- 464. [10] Lazar, B. E., "Stiffness Analysis of Cable-Stayed Bridges," Journal of the Struc- tural Division, ASCE, Vol. 98, No. ST7, July 1972, pp. 1605-1612. [11] Tang, M. C , "Analysis of Cable-Stayed Girder Bridges," Journal of the Structural Division, ASCE, Vol. 97, No. ST5, May 1971, pp. 1481-1496. [12] Kajita, T. and Cheung, Y. K., "Finite Element Analysis of Cable-Stayed Bridges," IABSE Publication SS-II, 1973. [13] Rajaraman.A., Loganathan, K. and Raman, N. V., "Nonlinear Analysis of Cable- Stayed Bridges," IABSE Proceedings P-37/80, IABSE Periodica 4/1980, November 1980, pp. 205-216. [14] Como, M., Grimaldi, A. and Maceri, F., "Statical Behaviour of Long-Span Cable- Stayed Bridges," International Journal of Solids and Structures Vol. 21, No. 8, pp. 831-850, 1980, pp. 831-850. [15] Hegab, H. I. A., "Energy Analysis of Cable-Stayed Bridges," Journal of Structural Engineering, ASCE, Vol. 112, No. 5, May 1986, pp. 1182-1195. [16] Ernst, H. J., "Der E-Modul von Seilen unter Berikksichtigung des Durchhanges," Der Bauingenieur, Vol. 40, No. 2, February 1965, pp. 52-55. [17] Ito, M. and Maeda, Y., Discussion, "Commentary on the Tentative Recommen- dations for Cable-Stayed Bridge Structures," Journal of the Structural Division, ASCE, Vol. 104, No. ST2, February 1978, pp. 372-374. [18] Mill, A. J., "Ultimate Load Analysis of Fixed Arches," M.A.Sc. Thesis, The Uni- versity of British Columbia, 1985. Bibliography 130 [19] Weaver, W. and Gere, J. M., Matrix Analysis of Framed Structures, D. Van Nos- trand Company, New York, 1980. [20] Hooley, R. F., "Cable Equations for Cable Stayed Bridge," Unpublished notes, The University of British Columbia, 1986. [21] Goodier, J. N., "Dimensional Analysis," in Handbook of Experimental Stress Anal- ysis, Ed. M. Hetenyi, John Wiley &: Sons, New York, 1950. [22] Hetenyi, M. , Beams on Elastic Foundations, Ann Arbor: The University of Michi- gan Press, 1944. [23] By Committee, "Tentative Recommendations for Cable-Stayed Bridge Struc- tures," Journal of the Structural Division, ASCE, Vol. 103, No. ST5, May 1977, pp. 929-939. [24] By Committee, "Recommended Design Loads for Bridges," Journal of the Struc- tural Division, ASCE, Vol. 107, No. ST7, July 1981, pp. 1161-1213. A p p e n d i x A Catenary Cable For the free-body diagram of Figure A.78 which is an element from the catenary cable shown in Figure A.79, the sum of the forces in the vertical direction produces the governing differential equation. Hy' + u>dc = H{y' + dy') where w is the load per unit of cable length. H dx -4 Figure A.78: Catenary cable element. Simplifying and using dc = y/dx2 + dy2 yields Integration yields y' = sinh + A 131 Appendix A. Catenary Cable 132 Figure A.79: Catenary cable. Appendix A. Catenary Cable 133 and integrating again yields H , {wx \ y = — cosh — + A + B w \ H / where A and B are constants of integration. With the boundary conditions at x = 0 ; y = 0 and at z = L ; y = h the constants of integration are evaluated as wh \ wL and H ^ A B = cosh A w The cable arc length c is obtained by integrating the elemental cable length through the length of the cable. c = f dc= fL ^-dx= fL Jl + y'2 dx Jo Jo dx Jo v Appropriate substitutions and simplifications yield c = \lh2 + -wTsmh — Using the fact that tan 6 — y' the vertical components of cable tension v0 and vx can be determined and from equilibrium of moments, the point of application ah of the total cable weight wc can be evaluated. The results are shown on Figure A.79. The elastic elongation A e of the cable is T . fL H sec 6 fe ,  s fl / —— dc = —— sec 6 dx Jo AE Jo AE which reduces to HL (wh2 y wL 1 H , wL\ *' = AE\2HL C ° t h 2~H + 2 + 2 ^ 1 8 m h If) The above expressions are exact but inconvenient to work with. For tight cables, where r = wL/2H < 1/2 it is convenient to expand the hyperbolic functions as power Appendix A. Catenary Cable 134 series. The series below are used and terms up to and including r4 are retained. r 3 r5 sinh r = r + 1 + 1- . . . ~6 120 cosh r = r2 r4 r6 1 + + — + =— 2 24 720 coth r = 1 r r3 2r5 - + — — + 7T-r 3 45 945 From these expressions and using y/l + r = l + r/2 + r2/8 + -- - the final results are c = 5 and r ! / L ' \ r<l In most computer structural analysis the catenary cable is modelled with a straight line member that lies along the chord of the cable and is pin-connected at both ends. The cable behaviour is then duplicated by making use of the equations for c and A e . Free body diagrams that show how the catenary cable and the bar are made equiv- alent are given in Figure A.80. In the cable, the tensions T0 and Ti include the force necessary to support the cable self-weight; consequently, the cable tension is not con- stant along the cable. The cable tension force is, of course, in-line with the catenary shape of the cable. For the bar, the self-weight is made the same as that for the cable, but the application of this weight is at the center of the bar, and the support for this weight is provided by equal external reactions at the member ends. Thus, if the weight of the cable is applied at the two member ends as a dead load force equal to tuc/2, the cable tension becomes constant and acts in-line with the chord. The only approxima- tion made is in placing half the dead load of the cable at each end instead of in true proportion according to a, where for tight cables, a = .5 -f hr/6S. It only remains to duplicate relative motion of B with respect to A in both the cable and the bar. The relative motion of B with respect to A is split into the motions parallel and perpendicular to the chord line. The parallel motion is shown in Figure A.81. When the , cos20 , 1 + — 2 cos20 cos4 6 45 72 HL sec2 6 A< = —AE— Appendix A. Catenary Cable wc Catenary Cable Pin Ended Member Figure A.80: Pin ended member equivalent to catenary cable. Figure A.81: Elongation of catenary cable. Appendix A. Catenary Cable 136 horizontal end thrust changes from H0 to H, point B moves to B' for an elongation A. This chord line elongation is accounted for by assigning an equivalent area to the bar, so that the calculation for the elastic elongation of the bar gives the actual elongation of the cable due to a change in sag as well as the elastic elongation of the cable. In order to calculate A , the unstressed length of a cable USL is introduced. The USL is the length of an unloaded cable layed horizontally and straight on a uniform continuous surface. USL = c- A e rat = s ( i + £ « ' » ) - f § ~ ' j ( i - t . £ ) _T_, _ H L * -r S r2 «— HLT2 « — USL = S - —— sec2 0 + cos2 0 — sec2 0 AE 6 ZAE Neglecting the HLr2 sec2 0/3AE term because it is second order compared to the Sr2 cos2 0/6 term, and substituting for r and the trigonometric terms results in U S L = S + ^ - ^ 24SH2 AEL Ignoring the elastic elongation of the cable between B and B' the elongation of the cable is given as the relative displacement of points A and B in direction S, as shown in Figure A.81. A = USLo-USL w2L* HS2 A = S + tv'L4 H0S2 24SHI AEL S + 24SH2 AEL S 2 / w2LlAE\ S2 f w2LsAE' [ 24S*H2 ) AEL [ 0 24S3tf2 t AEL If two new parameters are introduced o i w2L5AE 0 = r— H 24S3 and Appendix A. Catenary Cable 137 the elongation is expressed as Gathering all the known initial condition terms along with a known cable elongation and equating them to the unknown final horizontal thrust terms gives A /?3 fi3 ^  + Ho~m = H~H~i To express this in dimensionless terms let H = //? where / is a dimensionless variable which is a measure of the cable tautness. After simplification this yields A_ 1 _ 1 Let G equal the left hand side and the final result is Thus, given G, the parameter / can be found and then the final horizontal thrust H of an elongated catenary cable can be calculated. The reduced area A of a pin ended member to give the same elongation A with this final thrust H is now derived. Figure A.82 shows the bar extension with a change in tension. The elastic elongation is A ^ ( r - r 0 ) g AE or (H -Hg)S* LAE which gives LEA v 1 This equivalent area is used as the bar area in the next cycle of calculations to produce the elongation of the cable. Appendix A. Catenary Cable 138 Figure A.83: Relative motion perpendicular to cable chord. Appendix A. Catenary Cable 139 The relative motion perpendicular to the cable chord is shown in Figure A.83. The effect of the lateral motion 6 is handled by augmenting the member stiffness matrix with the stability functions for a pin ended member. The 6 by 6 member matrix is augmented by the following matrix: 0 0 0 0 0 0 0 T/S 0 0 -T/S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -T/S 0 0 T/S 0 0 0 0 0 0 0 The implementation of catenary cables in the computer program ULA is described in Appendix B. I Appendix B U L A Implementation of Catenary Cable The equation relating cable deflection to cable thrust using dimensionless parameters as derived in Appendix A is A graphical representation of this equation is given in Figure B.84. (B.17) Figure B.84: Behaviour of catenary cable under load changes. It can be seen that for large values of / the curve is linear with G ta f and for small values of / the curve equation essentially becomes G = — l / / 2 . These observations are important because it will be necessary to calculate / from a known value of G. Note: 140 Appendix B. ULA Implementation of Catenary Cable 141 as point B goes to B' in Figure A.81, the cable action goes from B to B' on the graph of Figure B.84 to show how the cable tautness changes with the cable elongation. A table look up procedure is necessary due to the nature of the equation being solved. The table should extend from G = —oo to + 0 0 , however, the previous obser- vations suggest that the table can be cut off. The cut off points used are G = — 4 and 3 which limit the error in calculating / from the approximate equations (after four iterations) to 0.0001%. The values used in the table look up procedure are calculated from Equation B.17 and are as follows: G / Error -4 .473 .035% -3 .532 .017% -2 .618 .005% -1 .755 .016% 0 1.000 0% 1 1.466 .029% 2 2.206 .019% 3 3.104 .006% The calculation of / is carried out as follows: • If G < -4 then / = If-^G then iterate on / = l/y/-G + f four times. • If — 4 < G < 3 then use the table look up and interpolation procedure. • If G > 3 then f = G then iterate on / = G + l / / 2 four times. The implementation of catenary cables in ULA is necessarily an iterative one. The first iteration calculates forces and displacements for all members based on their original member stiffness matrices. If P—A effects are neglected, this single iteration would produce good results for most members. For a cable however, the highly nonlinear effect of the change in sag must be accounted for, and this is done by substituting Appendix B. ULA Implementation of Catenary Cable 142 an effective stiffness into the cable stiffness matrix. Either E or A can be altered to produce an effective stiffness EA. A will be used as a matter of convenience only, due to the data requirements of ULA. The procedure for catenary cable calculations in ULA is as follows: 1. Cable constants 0 and 7 are calculated. (u> is calculated using the input cable area and density. Absolute values of distances are used.) 2. Set A = A. 3. Build the stiffness matrix using A. 4. Solve the system of equations. 5. Calculate A = TS/AE where T is the change in cable tension. 6. Calculate G where G = A , PQ = input cable prestress load (+ tension). 7. Calculate / . 8. Calculate reduced area AA to look after the loss of stiffness due to sag. . AA = (4lfP ~ Po^ S ^L\ J EA For a very small change in cable tension this equation becomes indeterminant in the form 0/0. To avoid this, a test is made such that S L,FP-Po if where A is the original input area. < .03P0 then AA = A Appendix B. ULA Implementation of Catenary Cable 143 9. Set A = AA. 10. Calculate the error (AA — A)/A and print it on the terminal screen. 11. If the error is too large, continue the calculations at step 3. A p p e n d i x C D a t a F r o m E x i s t i n g C a b l e S t a y e d B r i d g e D e s i g n s This appendix contains the data acquired from four steel deck and five concrete deck cable stayed bridges. The data was obtained from design blueprints kindly provided by Bush, Bohlman and Partners. The bridges and the design dates are as follows: • Steel Deck Bridges — ALRT - Advanced Light Rail Transit Bridge; Vancouver, British Columbia; February 1986. — AF - Alex Fraser Bridge; Vancouver, British Columbia; September 1983. — Q - Quincy Bridge; Quincy, Illinois; May 1983. — S - Sunshine Skyway Bridge; Tampa Bay, Florida; May 1982. 144 Appendix C. Data From Existing Cable Stayed Bridge Designs 145 • Concrete Deck Bridges — ALRT - Advanced Light Rail Transit Bridge; Vancouver, British Columbia; January 1986. — AF - Alex Fraser Bridge; Vancouver, British Columbia; September 1983. — Q - Quincy Bridge; Quincy, Illinois; May 1983. — WS - Weirton-Steubenville Bridge; Weirton, West Virginia - Steubenville, Ohio; February 1983. — EH - East Huntington Bridge; East Huntington, West Virginia; October 1980. Appendix C. Data From Existing Cable Stayed Bridge Designs 146 C . l Backspan Data from Bridge Designs for Simplified Backspan Model Steel Deck 1,000,000 1,000,000 kN/m3 m m m m kN/m kN/m kN deg ALRT AF Q S E 200. 200. 200. 200. Ec 195. 195. 195. 195. 7c 77. 77. 77. 77. I .030 .281 .171 .180 Acccnt .00235 .00466 .00656 .00478 IB 138.00 182.75 134.00 146.80 70.6 68.7 44.7 75.5 WD 44. 135. 76. 127. wL 16. 28. 9.33 18.66 F 4847. 29929. 14069. 16494. Occnt 51.4 39.5 35.9 48.5 N 11 20 7 10 Concrete Deck 1,000,000 1,000,000 kN/m3 m m m m kN/m kN/m kN deg ALRT AF Q WS EH E 28. 28. 28. 28. 28. Ec 195. 195. 195. 195. 195. 1c 77. 77. 77. 77. 77. I .297 1.470 1.730 5.866 .636 .00350 .00420 .00656 .00668 .00355 IB 138.00 195.25 134.00 209.70 185.3 hccnt 58.3 62.4 44.0 83.8 72.1 U)D 96. 196. 112. 270. 87. 16. 28. 9.33 18.66 9.33 F 14009. 52097. 19916. 59873. 16988. Bccnt 43.4 36.3 37.0 43.4 43.5 N 12 30 12 21 14 kN/m2 kN/m2 kN/m2 kN/m2 Appendix C. Data From Existing Cable Stayed Bridge Designs 147 C.2 Backspan Ratios from Bridge Designs for Simplified Backspan Model Steel Deck Concrete Deck Value used 20 .7 .4 3 7 .1 .2 ALRT AF Q S ALRT AF Q ws EH TV- 11 20 7 10 12 30 12 21 14 sin ectnt .782 .636 .586 .749 .687 .592 .602 .687 .688 hcent/lB .512 .376 .334 .514 .422 .320 .328 .400 .389 n 3.52 2.95 2.32 2.62 3.49 3.51 2.50 2.79 3.56 f 5.62 5.99 6.65 8.69 8.08 7.31 5.78 8.12 7.22 F/FJJ .063 .103 .069 .073 .134 .199 .060 .104 .131 wL/wD .36 .21 .12 .15 .17 .14 .083 .069 .11 (.47) ID/IB .402 .480 .610 .540 .405 .403 .566 .507 .397 Appendix C. Data. From Existing Cable Stayed Bridge Designs 148 C.3 Backspan Data from Bridge Designs for Backspan Model Steel Deck ALRT AF Q S E 200. 200. 200. 200. 1,000,000 I .030 .281 .171 .180 m .00235 .00466 .00546 .00478 „ 2 m IB 138.00 182.75 134.00 146.80 m hx 65.9 44.5 35.7 64.1 m hN 75.6 95.0 53.6 81.6 m h-N/2 70.0 66.7 43.2 75.5 m u>r> 44. 135. 76. 127. kN/m wL 16. 28. 9.33 18.66 kN/m F 5520. 33800. 14760. 18100. kN sin 0N/2 .74 .59 .57 .72 N 11 20 7 10 kN/m2 Concrete Deck ALRT AF Q WS EH E 28. 28. 28. 28. 28. 1,000,000 kN/m2 I .297 1.470 1.730 5.866 .636 m4 A-CN/2 .00350 .00350 .00546 .00627 .00365 m2 IB 138.00 195.25 134.00 209.70 185.3 m 32.3 28.5 35.0 45.7 58.0 m hN 71.25 89.9 53.0 103.6 83.9 m h~N/2 55.7 56.0 43.0 83.1 71.4 m WD 96. 196. 112. 270. 87. kN/m WL 16. 28. 9.33 18.66 9.33 kN/m F 15960. 59800. 23450. 71670. 20400. kN sin 0N/2 .64 .54 .54 .62 .62 N 12 30 12 21 14 Ec = 1c = 195,000,000fcJV/m2 llkN/mz Appendix C. Data From Existing Cable Stayed Bridge Designs C.4 Backspan Ratios from Bridge Designs for Backspan Model 149 Steel Deck Concrete Deck Value ALRT AF Q S ALRT AF Q WS EH used N 11 20 7 10 12 30 12 21 14 20 M B .478 .244 .266 .437 .234 .146 .261 .218 .313 .3 hN/lB .548 .520 .400 .556 .516 .460 .396 .494 .453 .5 f 5.47 6.08 8.15 8.50 8.20 9.41 7.05 8.55 6.94 7 F/FH .077 .13 .082 .085 .17 .27 .09 .15 .18 .1 WL/WD .36 .21 .12 .15 .17 .14 .083 .069 .11 .2 .418 .503 .645 .557 .423 .440 .640 .554 .426 .5 Appendix C. Data. From Existing Cable Stayed Bridge Designs 150 C.5 B r i d g e D a t a f r o m B r i d g e D e s i g n s fo r F u l l B r i d g e M o d e l Steel Deck ALRT AF Q S E 200. 200. 200. 200. 1,000,000 Ex 28. 28. 28. 28. 1,000,000 I .030 .281 .171 .180 m 4 IT 9.25 49.3 14.0 58.5 m 4 A c N / , .00235 .00466 .00546 .00478 m AcN .01466 .04153 .00896 .02652 m T.AC .02463 .08292 .03392 .04534 m IB 138.00 182.75 134.00 146.80 m IM 329. 462. 274. 366. m hi 65.9 44.5 35.7 64.1 m hN 75.6 95.0 53.6 81.6 m hi 116.0 143.5 69.5 135.0 m hn/2 70.0 66.7 43.2 75.5 m Wp 44. 135. 76. 127. kN/m WL 16. 28. 9.33 18.66 kN/m F 8630. 46780. 15000. 29050. kN sin 0N/2 .74 .59 .57 .72 tan 0Nc/2 .838 .667 .694 .800 N 11 20 7 10 Nc 13 24 7 12 kN/m2 kN/m2 Appendix C. Data. From Existing Cable Stayed Bridge Designs 151 Concrete Deck A L R T A F Q WS E H E 28. 28. 28. 28. 28. 1,000,000 kN/m2 ET 28. 28. — — — 1,000,000 kN/m2 I .297 1.470 1.730 5.866 .636 m 4 IT 28.7 202.4 — — — m 4 A-CH/7 .00350 .00350 .00546 .00627 .00365 m Aca .01863 .04456 — — — m £ A C .03714 .10572 — — — m IB 138.00 195.25 134.00 209.70 185.3 m IM 340. 463. — — — m fci 32.3 28.5 35.0 45.7 58.0 m hx 71.25 89.9 53.0 103.6 83.9 m hT 113.0 150.0 — — — m 55.7 56.0 43.0 83.1 71.4 m WD 96. 196. 112. 270. 87. kN/m 16. 28. 9.33 18.66 9.33 kN/m F 22720. 74360. 23900. 87160. 38800. kN sin eN/2 .64 .54 .54 .62 .62 tan BNc/2 .718 .610 .642 .694 .706 N 12 30 12 21 14 Nc 15 35 12 24 16 Appendix C. Data From Existing Cable Stayed Bridge Designs 152 C . 6 B r i d g e R a t i o s f r o m B r i d g e D e s i g n s f o r F u l l B r i d g e M o d e l Steel Deck Concrete Deck Value ALRT AF Q S ALRT AF Q WS EH used N 11 20 7 10 12 30 12 21 14 20 KllB .478 .244 .266 .437 .234 .146 .261 .218 .313 .3 .548 .520 .400 .556 .516 .460 .396 .494 .453 .5 .841 .785 .519 .920 .819 .768 — — — .8 f 5.47 6.08 8.15 8.50 8.20 9.41 7.05 8.55 6.94 7 F/FH .12 .18 .083 .136 .24 .34 .09 .18 .34 .1 wL/wD .36 .21 .12 .15 .17 .14 .083 .069 .11 .2 ID/IB .418 .503 .645 .557 .423 .440 .640 .554 .426 .5 IB/IM .419 .396 .489 .401 .406 .400 — — — .4 Ex IT 1FI 43.2 24.6 11.5 45.5 96.6 137.7 — — — 50 Aca/T,AC .595 .501 .264 .585 .502 .422 — — — .5 Ac. AcJEAc NC i=N N-l EM 1=1 EJIAT Ac{ E & 1 ACi A p p e n d i x D M a x i m u m D e f l e c t i o n a n d M o m e n t The maximum deflection and moment is expressed in terms of the span length for deflection and wjj} for moment. The actual data from the computer analyses are used. (The actual values are not chosen to reflect real bridges—they are dictated by the dimensionless ratios.) The following table lays out the data and calculations for the maximum deflection and moment for the standard models: Simplified Backspan Backspan Full Bridge Backspan Mainspan / 180. 180. 180. 441. VJL 32.7 27.2 19.0 19.0 6* 0.0873 0.114 0.177 0.403 M* 4039. 6000. 8734. 6759. 6'/l 1/2062 1/1579 1/1017 1/1094 M'/WLI2 1/262 1/147 1/70 1/547 153

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