Deflections in Stiffening Trusses of Suspension Bridges Dorothy Goettler A Seminar submitted to the Faculty of Rensselaer at Hartford in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE Mechanical Engineering Approved by Ernesto Gutierrez-Miravete Rensselaer at Hartford Hartford, Connecticut December 1999 Table of Contents NOMENCLATURE ....................................................................................................................................................3 LIST OF FIGURES .....................................................................................................................................................4 ABSTRACT .................................................................................................................................................................5 INTRODUCTION .......................................................................................................................................................6 PROBLEM DESCRIPTION ......................................................................................................................................7 FORMULATION OF EQUATIONS .........................................................................................................................8 ANALYSIS OF CABLE ..................................................................................................................................................8 DEFLECTIONS OF UNSTIFFENED BRIDGES ................................................................................................................. 10 APPLICATION TO STIFFENED BRIDGES .................................................................................................................. 14 Dead load only .................................................................................................................................................... 14 Addition of live load ............................................................................................................................................ 14 DEFLECTIONS OF STIFFENING TRUSSES .................................................................................................................... 16 Development of equations ..................................................................................... Error! Bookmark not defined. Numerical Approaches........................................................................................................................................ 18 1. Successive approximations ........................................................................................................................................... 18 Solutions in one Variable ................................................................................................................................................. 22 2. Bisection Method ......................................................................................................................................................... 24 3. Fixed Point Iteration ..................................................................................................................................................... 26 4. Newton-Raphson Method............................................................................................................................................. 28 RESULTS, ERROR ANALYSIS AND DISCUSSION ........................................................................................... 29 CONCLUSIONS ........................................................................................................................................................ 29 REFERENCES .......................................................................................................................................................... 30 APPENDIX A............................................................................................................................................................. 31 APPENDIX B ............................................................................................................................................................. 32 APPENDIX C............................................................................................................................................................. 33 2 Nomenclature Ac Ec f fw H Hp Hw h k L l Mw Mp Mx P p s w Tensile area of cable Young’s modulus of cable Sag of cable Sag of cable under action of dead load Horizontal component of tensile force in cable Horizontal component of tensile force produced in the cable by the live load Horizontal component of tensile force produced in the cable by the dead load Difference in elevation of the ends of the cable Variable defined by Equation 16 Integral defined by Equation 12 Span of beam Bending moment due to dead load Bending moment due to live load Bending moment at given cross section Live load, acting at a point Live load, uniformly distributed over a given length Length of cable Horizontal Deflection of cable Vertical deflection of cable Dead load, uniformly distributed over length of bridge 3 List of Figures Figure 1 Components of a suspension bridge Page 6 Figure 2 Cable under uniform load, ends supported at different heights. Page 8 Figure 3 Unstiffened bridge under uniformly distributed live load. Page 10 Figure 4 Infinitely small element of cable from Figure 2 Page 11 Figure 5 Stiffened bridge under dead load only. Page 14 Figure 6 Stiffening truss under point load. Page 16 Figure 7 Total deflection of stiffening truss Page 21 Figure 8 Graph of equation to be solved, g(x) Page 23 Figure 9 Results of bisection method Page 25 Figure 10 Graphical representation of h(xk) Page 26 Figure 11 Graphical representation of h(xk) and g(x) Page 26 Figure 12 Divergence of fixed point iteration method Page 27 Figure 13 Results of Newton-Raphson Method Page 28 4 Abstract The stiffening truss of a typical single span suspension bridge will be analyzed in order to determine the maximum vertical deflections due to a single localized live load applied in addition to the dead load of the structure. This will be accomplished by developing the necessary equations for increasingly complex and accurate models of the bridge and solving them using the assumption of a specific geometry that will be described. The solution will be attempted using five different methods: successive iterations, trigonometric series, bisection, fixed point integration and the Newton-Raphson method. Finally, the results obtained from each of these will be compared and the results discussed. 5 Introduction The suspension bridge is a structure consisting of either a roadway or a truss suspended from two or more cables that pass over two towers and are anchored by backstays or towers to a firm foundation. In the simplest type of suspension bridge, applicable to short spans, the floor system attaches directly to the cables by hangers. However, the structure lacks rigidity, which allows wind loads and moving live loads to distort the cables and produce a wave motion on the roadway. For longer spans it becomes necessary to connect the floor systems to stiffening trusses or girders that distribute the moving loads more uniformly to the hangers. This method of distribution reduces the distortion of the cable by distributing the concentrated live loads over a considerable length. When the roadway is supported by a truss that is hung from the cable in this way, the structure is called a stiffened suspension bridge. Hangers Support Towers Roadway may be stiffened with a truss (see detail below) Roadway Stiffening Truss Figure 1 Components of a suspension bridge. Cables of larger sizes, such as those up to 36 inches in diameter used for the George Washington Bridge in New York City, are assembled (or spun) in the field by using pencil thick wires laid parallel and bound securely together. For smaller cables, strands of wire wound spirally in the factory are assembled into cables. Factory-made strands of parallel wires were used for the first time in 1968 for the 15 inch diameter cables of the Newport Bridge in Rhode Island. The hangers which connect the floor system to the cable are formed of twisted wire ropes. The floor system, stiffening truss and towers are constructed of rolled steel. The longest bridges in the world are all of the suspension type. The Akashi Straight Bridge in Japan is the world’s longest suspension bridge as of 1998 with a 6,633-foot center span. To provide a reference for the order of magnitude of these bridges, Appendix A contains a table of representative large suspension bridges in North America. 6 Problem Description Before the deflections in the stiffening truss of the suspension bridge may be determined, the equations to be solved must be developed. The first step in doing this is to analyze the most basic form of a suspension bridge, a simple cable supported from two rigid towers. Equations for the length and sag of this cable will be important for the calculations that follow and must be developed at this point. Also, the method used to determine the reaction forces, which will also be required throughout the analysis, will be used first for the case of a simple cable. Once the behavior of the cable is known, it is extended to the case of the unstiffened suspension bridge. This consists of a roadway or other structure suspended from the cable by a large number of wire rope hangers. The deflections are then calculated first for the straightforward case in which all of the reaction forces are known, then for the more challenging but realistic case in which the horizontal component of the tensile force produced in the cable by the live load in unknown. This situation will require a system of equations to be solved simultaneously for two unknowns. These equations can then be applied to the problem of a stiffened suspension bridge, from which the behavior of the stiffening trusses themselves can be determined. The deflections in the stiffening trusses will then be determined using a variety of methods, and the results compared. It should be noted that in most cases the analysis is performed first assuming only the weight of the structure itself, or dead weight. Once this form of the equations has been developed, they are extended to the more general case of a live load, which essentially represents a single pedestrian or vehicle crossing the bridge. It would also be possible to extend this to the more general case of a distributed live load over the entire span of the bridge. In order to reduce the tremendously complicated problem of a suspension bridge into one manageable enough to describe using mathematical equations, certain assumptions had to be made. These include but are certainly not limited to those listed below: 1. Both ends of bridge are assumed to be at the same height. 2. The live load is assumed to be small relative to the dead load. 3. The live load is assumed to be static. 4. The span of the bridge is assumed to be long relative to its width. 5. Environmental factors such as wind and temperature fluctuations are neglected. 6. The hangers are sufficiently close together that it may be assumed the dead load is uniformly transmitted to the cable. 7 Formulation of Equations Analysis of cable In this section, a simple cable suspended at both ends will be analyzed. The results from this analysis will be required for the development of the equations for the suspension bridge. Figure 2 Cable under uniform load, ends supported at different heights. For a uniform, perfectly flexible cable fixed at points A and B under a uniformly distributed load (see Figure 1), the equation of moments about point C is written as: Mx H h x Hy 0 l Equation 1 In this equation Mx denotes the bending moment at the cross section m-n of a simply supported beam of span l and carrying the load acting on the cable. In the particular case when the load of intensity w is uniformly distributed along the horizontal projection of the cable, as the dead weight of the actual suspension bridge would be, we find that: Mx wx l x 2 The result above is developed in more detail in Reference 11, chapter 7. Substituting this into Equation 1 gives: y wx l x h x 2H l This indicates that the curve is in this case a parabola with a vertical axis. If the ends of the cable are on the same level, as will be assumed in all future analysis, this may be reduced to: y wx l x 2H Equation 2 8 Applying this equation to the mid-point of the cable ( x = l / 2 ) where the y coordinate of the curve represents the sag f, we obtain: wl l l 4H 2 y f wl 2 8H Equation 3 This holds true in the more general case illustrated in Figure 2 as well if f is measured from the mid-point of the straight line between points A and B. To obtain the complete equations for the stiffening truss of a suspension bridge, the length of the cable will be required. It may be obtained from the following equation: l 1 2 s (1 y ' ) dx 2 0 Developing the expression under the integral sign into a series and substituting Equation 2 for y, we obtain from Reference 1: s l (1 8 f 2 32 f 4 256 f 6 ...) 3 l2 5 l4 7 l6 In the case of relatively flat parabolic curves, where f / l 1 / 10, the terms to this series rapidly become extremely small, and may be reasonably well approximated by taking only the first two terms. The simplified formula for the length of the cable is therefore: s l (1 8 f2 ) 3 l2 Equation 4 9 Deflections of unstiffened bridges This is the simplest form of a suspension bridge. Its analysis allows the equations to be derived for a simple case before they are applied to the more challenging application of stiffened bridges. We will therefore assume that the curve of the cable under the action of the dead load is again a parabola and will concentrate on the deflections in the cable produced by a live load. Figure 3 Unstiffened bridge under uniformly distributed live load. Figure 3 illustrates a symmetrical case in which in which a load of intensity p is uniformly distributed along the distance 2a of the span. The line drawn indicates the shape of the cable under the action of the dead load w only. We let fw and Hw denote the corresponding values of the sag of the cable and of the horizontal component of the tensile force in the cable. The length of the cable becomes: s l (1 8 f w2 1 w2l 2 ) l ( 1 ) 3 l2 24 H w2 We now add a vertical live load acting on the cable which still has both ends at the same level. Summing the moments, as was done previously in Equation 1, results in the following: y Mw Hw Equation 5 In this equation, Mw is the bending moment due to dead load calculated as for a simply supported beam, and Hw is the horizontal component of the tensile force produced in the cable by the dead load. If a live load is now applied, the bending moment calculated as for a simple beam becomes Mw + Mp , and the horizontal component of cable tension becomes Hw + Hp. Denoting by the vertical deflections of the cable we obtain from a moment equation: 10 y Mw M p Hw H p Subtracting Equation 5 from this equation, we obtain: Mp Hpy Equation 6 Hw H p The vertical deflection can be easily calculated, assuming the horizontal component Hp of cable tension produced by live load is known. This can typically be calculated from the geometry of the bridge. If Hp is not known however, it must be determined before a solution for Equation 6 is possible. To do this, we now consider the infinitely small element ab of the cable, as illustrated in Figure 4. Figure 4 Infinitely small element of cable from Figure 3. The live load causes this element to elongate and take a new position a1b1. We denote by and the horizontal and vertical components of the small displacement of point a. The initial length of the element is: Equation 7 ds 2 dx 2 dy 2 The length of the same element after application of the live load is: (ds ds) 2 (dx d ) 2 (dy d ) 2 Equation 8 In the equation above, ds is the elongation of the element caused by the live load. Neglecting the small change in slope of the cable produced by the live load, we obtain from the standard equation for the deflection of a beam: ds ds H p ds Ac Ec dx Equation 9 11 Since Hp ds / dx is that part of the tensile force in the cable which is produced by live load and which is usually much smaller than the part produced by dead load, the unit elongation ds / ds is usually very small. If this is the case, (ds)2 in Equation 8 can be neglected. For the same reason, and from the observation that the curve of the cable is a flat curve, we also neglect (d)2. Combining Equations 7 and 8 we obtain: 1 ds ds dx d dy d (d ) 2 2 This may be written as: ds dy 1 d ds d d dx dx 2 dx d Substituting Equation 9 for ds in this equation and integrating, we obtain: Hp x ds ( dx ) Ac Ec 0 3 x dx y ' ' dx 0 1 x 2 ' dx 2 0 Equation 10 In the equation above, the primes indicate derivatives with respect to x. With the values of y’ and ’ which are typically encountered in long-span bridges, the value of usually does not exceed onethousandth of x. The maximum value of / x occurs near the supports where ’ and y’ usually have their largest numerical values. At the ends of the cable, vanishes and we obtain from Equation 10: Hp Ac Ec l ds ( dx ) 0 3 l dx y ' ' dx 0 1 l 2 ' dx 2 0 Equation 11 The integral on the left side of this equation for the assumed parabolic shape can readily be evaluated, and we obtain: 3 1 l ds 3 1 5 16 f 2 16 f 2 2 2 2 0 ( dx ) dx 0 (1 y' ) dx l{ 4 ( 2 l 2 )(1 l 2 ) 1 3l 4f 16 f 2 ln[ (1 2 ) 2 ]} 32 f l l l Equation 12 Integration by parts is used on the right hand side of Equation 11. Taking advantage of the fact that is zero at the ends of the cable and the previously developed Equation 2, we obtain: l y' ' dx y' 0 l y ' 'dx 0 l 0 w l dx H w 0 1 l 2 1 1 l 1 l l ' dx ' 0 ' 'dx ' 'dx 2 0 2 2 0 2 0 Equation 13 12 Substituting Equations 12 and 13 into Equation 11, and denoting the integral of Equation 12 by L, we obtain: Hp Ac E c L w l 1 l dx ' 'dx Hw 0 2 0 Equation 14 Equation 14, together with Equation 6, gives the system of equations necessary to calculate vertical deflections of the cable. 13 Application to stiffened bridges Figure 5 below illustrates the simplest type of stiffened suspension bridge: a single span cable stiffened by a simply supported truss of constant cross section. Figure 5 Stiffened bridge under dead load only. It is assumed that a for properly assembled suspension bridge the dead load of the structure is uniformly distributed along the span and entirely transmitted through the hangers to the cable which takes a parabolic form. The solid lines in the figure above represent this loading condition. When a live load is added, the deflections produced in both the cable and the truss are as seen by the dashed lines above. We assume that the elongation of the hangers and their small inclination to the vertical during deformation are negligible so that at any given position along the length of the bridge the deflection of the cable is equal to the deflection of the truss. This assumption is also made in Reference 1. It is also assumed that the spacing of the hangers is small compared to the length of the span so that the load may be considered uniformly distributed along the span Dead load only We first consider the case where the structure is carrying only dead load. The truss does not experience bending in this case and the equation of moments for the forces to the left of a cross section m-n may be written as: Mw Hw 0 Addition of live load When live load is applied and deflections are produced, there will be a bending moment M acting in a cross section m-n of the truss, and the equation of moments for the forces to the left of this cross section is: M w M p H w H p y M 0 Subtracting the equation for the dead load from the equation for the live load we obtain: 14 M M p H w H p H p y From this equation, the bending moment at any cross section of the truss can be calculated provided the horizontal component of the tensile force in the cable and the deflection are known. In the case of very rigid stiffening trusses, the deflections could be ignored, yielding the simplified equation: M Mp Hpy Under this assumption, the bending moment is independent of deflections and can be evaluated the same way as any other rigid statically indeterminate structure. Applications have demonstrated, however, that the stiffening trusses in large span bridges are usually very flexible. This means that the more complete equation including the deflection of the truss must be evaluated in order to calculate the bending moments. To do this, we begin with the differential equation of the deflection curve of a beam: EI d 2 M dx 2 This equation is applied to the bending moment equation that includes deflection of the truss, and yields: EI d 2 H w H p H p y M p dx 2 Equation 15 The quantity Mp in this equation can be calculated for any distribution of live load over the span. The quantities y and Hw are given by Equation 2 and Equation 3, and only the quantity Hp is unknown. It depends on the deflections , and Equation 14 is used to evaluate it. Equation 15 together with Equation 14 completely define the deflections of the stiffening truss. 15 Deflections of stiffening trusses The first case considered is that of a single concentrated load P acting on the truss. Taking the second derivative of Equation 15, we find that the deflections of the truss in this case are the same as those occurring in a simply supported beam subjected to a combination of loads: 1. an axial tensile force Hw+Hp 2. a uniformly distributed upward lateral load of intensity Hpw/Hw 3. a concentrated load P This state of combined loading is shown in Figure 6. Figure 6 Stiffening truss under point load. Hw H p EI k2 Equation 16 Under the loading conditions above, and using the notation defined in Equation 16, the deflections in the stiffening truss produced by the load P in the beam to the left of this load ( x < l – c ) may be written as: 1 P sinh kc Pcx sinh kx H w H p k sinh kl ( H w H p )l Equation 17 For the portion of the beam to the right of the load ( x > l – c ), the deflections are: 1 sinh k (l c) P(l c)(l x) P sinh k (l x) H w H p k sinh kl ( H w H p )l Equation 18 The deflections produced by the upward pull are: 2 Hp Hw wl 2 Hw H p cosh( kl / 2 kx) x(l x) 1 2 2 2 2 2l 2 k l cosh( kl / 2) k l Equation 19 The total deflections of the truss may be obtained by superimposing deflections 1 on deflections 2. 16 1 2 To determine the magnitude of tension Hp, which is required to solve Equations 17, 18 and 19, Equation 14 is simplified by omitting the second term on the right hand side, as seen below: Hp Ac E c L w l dx H w 0 It can be shown that in most cases this has only a small effect on the magnitude of Hp. Substituting Equations 17, 18 and 19 into Equation 17 and performing the integration we obtain the following equation: H w H p L 1 8 f 2 12 24 kl Hp 1 2 2 3 3 tanh 2 k l Ac E c l 12 l k l 8 f 1 c c 1 sinh kl sinh kc sinh k l c P 1 2 2 l 2 l l k l sinh kl Equation 20 Once the value of Hp has been determined, it can be used to determine the deflection in the stiffening truss. 17 Numerical Approaches Four different methods will be used to solve Equation 20. The results will then be compared. In order to effectively compare the results, it is important to begin with the same geometry and loading conditions for each case analyzed. The assumptions made are therefore summarized below: l = 125 ft Hw = 270,000,000 lbf I = 210,000 in4 f = 12.5 ft Ec = 30,000,000 psi A = 4500 in2 c = 0.75l P = 2,000 lbf These values will be used in all subsequent calculations. 1. Successive approximations This is perhaps the most obvious method of solving this problem. The emphasis will be on developing an acceptable first approximation to the solution of Hp through manipulation of Equation 20, and refining it through iteration until it is within the desired tolerance. This accepted value of Hp will then be substituted into the appropriate equations to develop the actual deflection in the stiffening truss. H w H p L 1 8 f 2 12 24 kl Hp 1 2 2 3 3 tanh 2 k l Ac E c l 12 l k l 8 f 1 c c 1 sinh kl sinh kc sinh k l c P 1 2 2 l 2 l l k l sinh kl Equation 20 In the case of long-span bridges the quantity kl is typically a relatively large number. For example: Ambassador Bridge (Detroit) George Washington Bridge kl = 9.52 kl = 35 This indicates that the terms in Equation 20 that contain k are small and could be neglected to obtain a first approximation of the result. The term (Hw + Hp) / AcEc may also be considered very small and therefore omitted. This simplifies Equation 20 considerably, and allows it to be rewritten as: Hp 3 l c c P 1 4 f l l For c =0.75 l, this gives: Hp 9 l P 64 f 18 Using the given assumptions for bridge geometry and loading, this allows us to obtain a first approximation for Hp. 9 (2,000lbf )(10) 64 Hp Hp = 2812.5 lbf The accuracy of this first approximation will depend largely on the magnitude of kl. For values of kl greater than ten, it has been found that the approximation is typically quite good. To increase its accuracy, the approximate value of Hp is used to calculate k from Equation 16. Hw H p EI k2 Equation 16 270,000,000lbf 2812.5lbf k2 4 (30,000,000 psi )( 210,000in ) k = 0.0065465708 This value of k is then substituted back into Equation 20, which gives a second approximation of Hp. In order to do this, the value of L must first be defined: L 1 5 l. . 4 2 2 16. f . 1 2 l 1 1 2 2 2 2 16. f 2 l 3. l . 4. f ln 32. f l 1 16. f 2 l The right hand side of the equation must also be entered into Mathcad so that it may be used to obtain the solution: RHS 8. f 1 c P. . . . 1 l 2 l c l 1 . ( sinh( k. l) sinh( k. c ) sinh( k. ( l c) ) ) 2. 2. k l sinh( k. l) Finally, the entire equation is entered and a solution obtained: H p. Hw Hp L . A c. E c l 2 1 . 8. f . 1 12 l 12 2. 2 k l 24 . k. l tanh 3 3 2 k .l RHS H p = 2685.4814691737 lbf 19 For practical applications, this second approximation is typically accurate enough, though this procedure may be repeated until the desired accuracy is achieved. 2 k Hw Hp E c. I k = 0.0065465693 in RHS H p. 1 8. f 1 c P. . . . 1 l 2 l Hw Hp L . A c. E c l c l 1 . ( sinh( k. l) sinh( k. c ) sinh( k. ( l c) ) ) 2. 2. k l sinh( k. l) 2 1 . 8. f . 1 12 l 12 2 2 k .l 24 . k. l tanh 3 3 2 k .l RHS H p = 2685.4814599255 lbf 2 k Hw Hp E c. I k = 0.0065465693 in 1 The results of this iteration are summarized in the table below: Hp (lbf) 2812.5 2685.4814691737 2685.4814599255 k (in-1) 0.0065465708 0.0065465693 0.0065465693 As seen above, it takes only three iterations for the value of Hp to converge to a result accurate to within 10-4. The second approximation would certainly have been adequate for the construction of an actual bridge. Because the value of k has converged even more quickly to an accuracy of 10-10 it is impossible to further refine the approximation of Hp. 20 Once Hp has been calculated in this way, the deflection curve may be found using Equations 17, 18 and 19 with the method of superposition. 1 P sinh kc Pcx sinh kx H w H p k sinh kl ( H w H p )l For x < l - c 1 sinh k (l c) P(l c)(l x) P sinh k (l x) H w H p k sinh kl ( H w H p )l For x > l - c 2 Hp Hw cosh( kl / 2 kx) x(l x) wl 2 1 2 2 2 2 H w H p k l cosh( kl / 2) k l 2l 2 1 2 Microsoft Excel was used to tabulate and graph these values for 0 < x < l. Some of these results are summarized in the Appendix B1 so that they may be compared to the values that will be obtained using various other methods. Total Deflection - Method of Successive Iterations 0 Deflection -0.0005 0 500 1000 1500 -0.001 -0.0015 -0.002 -0.0025 -0.003 Distance along Bridge Figure 7 Total deflection of stiffening truss. 21 Solutions in one Variable For the following three methods, the deflections of the stiffening trusses will be based on the assumed value of Hp developed through the method of successive iteration. The emphasis will instead be on developing a more accurate value for k and using this to determine the actual deflections. The first step in doing in this was to plot the function of k. This permitted the applicable methods of approximating the solution to be determined. To begin, Mathcad was used to plot Equation 20, as seen below. H w H p L 1 8 f 2 12 24 kl Hp 1 2 2 3 3 tanh 2 k l Ac E c l 12 l k l 8 f 1 c c 1 sinh kl sinh kc sinh k l c P 1 2 2 l 2 l l k l sinh kl Equation 20 This will be rewritten in the following form so that the root of the equation may be found: H w H p L 1 8 f 2 12 24 kl g (k ) H p 1 2 2 3 3 tanh 2 k l Ac Ec l 12 l k l 8 f 1 c c 1 sinh kl sinh kc sinh k l c 0 P 1 2 2 l 2 l l k l sinh kl Initially, k is defined as a range variable taking on values from 0.0065 in-1 to 0.0066 in-1 in increments of 0.00001. This range is based on the approximate solution obtained using the method of successive approximation and is wide enough that we may be confident that the exact value is included. k 1 1 0.0065. in , 0.00651. in .. 0.0066. in 1 Equation 20 is now defined as g(k) for the program. For formatting reasons in Mathcad, it was necessary to separate it into left and right hand sides of the equation. This does not, however, affect the final result. 22 Below is the equation as defined in Mathcad: Hw Hp L . A c. E c l Equation20 RHS( k) H p. Equation20 LHS( k) 8. f 1 c P. . . . 1 l 2 l g( k) Equation20 RHS( k) 2 1 . 8. f . 1 12 l c l 1 12 2 2 k .l 24 . k. l tanh 3 3 2 k .l . ( sinh( k. l) sinh( k. c ) sinh( k. ( l c ) ) ) 2. 2. k l sinh( k. l) Equation20 LHS( k) The graph below shows the values of the function defined above over the given range of k. Because the default units for length in Mathcad are feet, it was necessary to divide to values of k in the graph by 12 in order to be consistent with the values used in other calculations. 0.5 0 g( k ) 0.5 1 0.0065 0.00655 0.0066 0.00665 k 12 Figure 8 Graph of the equation to be solved, g(x) From this graph, the viable methods for approximating k may be determined: 23 2. Bisection Method It may be seen on Figure 8 that the beginning and end points of the function have opposite signs. This indicates that the function crosses the line g(k)=0, and that the bisection method may be used to determine the location of this zero. Below is the Mathcad program used to implement the Bisection Method. root( N ) a 0.0065. in 1 b 0.0066. in 1 for i 1 .. N p a b a 2 p FP g( p ) FA g( a ) a p if FA . FP > 0 (b p ) otherwise Line Function 1 The function root requires an input of N, which is the desired number of iterations. The long vertical line represents the length of the program. The variable a is defined to be the value of the lower end of the range for k as previously determined. 2 The variable b is defined to be the value of the upper end of the range for k. 3 This is the beginning of a for loop. The short vertical line represents the length of the loop. The contents of the loop will be repeated until i has stepped through each integer between 1 and N. It is not necessary in Mathcad to manually increase the value of i with each step, the default action if no other is specified is to increase it by one. 4 The variable p is defined to be the midpoint of the distance between a and b. 5 The variable FP is defined to be the value of the function g(k) at point p. 6 The variable FA is defined to be the value of the function g(k) at point a. 7 If FA*FP > 0, indicating that point p was to the left of the zero in the function, the value of p is reassigned to the variable a. 8 Otherwise, the value of p is reassigned to the variable b. The range of k is now one half of what it was originally. If the desired number of iterations has not been satisfied, the loop will begin once more using an updated value for either a or b. 9 This indicates that the final value of p should be displayed when the function is called out. Bisection Method Program 24 0.00656 0.00655 root( i ) 12 0.00654 0.00653 0.00652 0 10 20 i Figure 9 Results of bisection method Above is a plot of the values obtained for N = 20, or 20 iterations as output by Mathcad. It appears that by the end the values for k have converged. It is interesting to note the abrupt dip that occurs between the first and second iterations. This emphasizes the importance of performing sufficient iterations to ensure that the results have actually converged. The table summarizing the output of the program for N = 20 may be found in Appendix C. The values in the table confirm that the results have converged. 25 3. Fixed Point Iteration The next method of approximation used was Fixed Point Iteration. In order to ensure that this was indeed a viable option, the function h(xk) = xk was plotted to ensure that such a point did in fact exist. xk 0.0065, 0.00651.. 0.0066 h( xk) 0.00665 xk 0.0066 h( xk ) This is a plot of the function h(xk) = xk. As expected, it is a straight line extending from the origin and bisecting the first quadrant. This curve will be superimposed on the plot of the function to be analyzed using fixed point integration. 0.00655 0.0065 0.0065 0.00655 0.0066 0.00665 xk Figure 10 Graphical representation of h(xk)=xk k 1 1 0.0065. in , 0.00651. in .. 0.0066. in g( k) Equation20 RHS( k) 1 0.5 Equation20 LHS( k) g( k ) 0 h( xk ) 0.5 Here the plot of the function to be analyzed is shown superimposed with the plot of h(xk) = xk. Because the two curves intersect as seen on the plot, it is known that there is in fact a fixed point and this method of approximation may be pursued. 1 0.0065 0.00655 k 0.0066 0.00665 , xk 12 Figure 11 Graphical representation of h(xk)=xk and g(x) 26 Because the function g has a fixed point at some point p, it is known that the function defined by m(k) = k - g(k) has a zero at p. fixedpoint( N ) 0.0065455. in p0 1 for i 1 .. N p To the right is the program that was written to determine the solution using this method. g( p0 ) p0 p 0.01 0.01 Because of the extremity of the differences in scales, however, this method does not converge to a solution. The plot to the right is the same as above, simply viewed at a more realistic scale. It is obvious from this graph that with the two curves very nearly horizontal and vertical, no projection can be made from one to the other. g( k ) 0.005 h( xk ) 0 0 0.0065 0.00655 0.0065 0.0066 k 0.00665 0.0066 12 Figure 12 Divergence of fixed point iteration method 27 4. Newton-Raphson Method The final method of solving Equation 20 for k is the Newton-Raphson Method. One requirement of this method is that the function be differentiable. It appears from the graph that this condition is met. Below is the Mathcad program used to approximate the solution to the equation using the NewtonRaphson Method: newton( N ) 0.0065. in p0 1 for i 1 .. N p p0 p p0 p g p0 d g p0 dp 0 Line Function 1 The function newton requires an input of N, which is the desired number of iterations. The long vertical line represents the length of the program. The variable p0 is defined to be the initial approximation for k as previously determined. 2 This is the beginning of a for loop. The short vertical line represents the length of the loop. The contents of the loop will be repeated until i has stepped through each integer between 1 and N. It is not necessary in Mathcad to manually increase the value of i with each step, the default action if no other is specified is to increase it by one. 3 The variable p is defined to be the difference of p0 and the ratio of g(po) over g’(po). 4 The variable po is reassigned the value of p which was obtained from the calculation in line 3. This is theend of the for loop. 5 This indicates that the final value of p should be displayed when the function is called out. Newton-Raphson Method Program i 1.. 4 newton( i ) 12. ft 0.00654657 0.0065466 1 0.0065462312 0.0065465692 0.0065465693 0.0065465693 newton( i ) 1 0.0065464 12. ft 0.00654623 0.0065462 1 1 2 3 i 4 4 Figure 13 Results of Newton-Raphson method. As seen above, the Newton-Raphson method converges very rapidly to an approximation. 28 Results, Error Analysis and Discussion Of the four solution methods attempted, only three were successful. The results obtained using these three different methods were however identical if sufficient iterations were performed. The table below summarizes the number of iterations required for each method to converge to a solution accurate to 10-10. Successive Iteration Bisection Method Newton-Raphson Method k = 0.0065465693 k = 0.0065465693 k = 0.0065465693 2 iterations 19 iterations 3 iterations Error analysis in the case of this problem is trivial. As there is no way to calculate the exact theoretical value for the deflection of the stiffening truss of a suspension bridge, the numbers above can only be compared against one another. Given sufficient iterations, the results converged to the same value, accurate to 10-10, regardless of the method. In doing this, it is obvious that the level of accuracy far exceeds that which would be required for an application of this problem in the actual physical environment. Conclusions Each of the methods that proved successful in calculating the deflection in the stiffening truss yielded the same result, indicating that all were valid. As the values obtained for k using each of the methods were identical, it was not necessary to recompute the actual deflection numbers. Perhaps the most interesting case was the one that was not successful. All of the stated required conditions were met, yet the fixed point iteration method did not converge to a result, even when the given initial value was extremely accurate. The reason for this became clear with a more detailed examination of the plotted functions and the discussion in the appropriate section of this text, but serves as a reminder that the ideal solution method for each problem must be determined individually. 29 References 1. Theory of Structures, Timoshenko and Young, McGraw-Hill, Second Edition, 1965. 2. Standard Mathematical Tables and Formulae, CRC Press, 29th Edition, 1991. 3. Numerical Analysis, Burden and Faires, Brooks/Cole, Sixth Edition, 1997. 4. Standard Handbook for Structural Engineers, Merritt, McGraw-Hill, 1968. 5. Design of Bridge Superstructures, O’Connor, Wiley, 1971. 6. Scientific Encyclopedia, Considine, VNR, 1976. 7. Mechanics of Materials, Gere and Timoshenko, PWS, Fourth Edition, 1990. 8. Design of Welded Structures, Blodgett, Lincoln Arc Welding Foundation, 15th Printing, 1996. 9. Analysis of Numerical Methods, Isaacson and Keller, Dover, 1966. 10. Numerical Mathematics and Computing, Cheney and Kincaid, Brooks/Cole, Third Edition, 1994. 11. Roark’s Formulas for Stress and Strain, Young, McGraw-Hill, Sixth Edition, 1989. 30 Appendix A Representative Large Suspension Bridges in North America [Reference 6] Verrazano-Narrows New York, NY 4,260 ft span Golden Gate San Francisco Bay, California 4,200 ft span Mackinac Straits of Mackinac, Michigan 3,800 ft span George Washington Hudson River, New York 3,500 ft span Transbay San Fransisco Bay, California 2,310 ft each span (total two) Bronx-Whitestone East River, New York 2,300 ft span Quebec Road Montreal, Quebec 2,190 ft span Delaware Memorial Wilmington, Delaware 2,150 ft span Walt Whitman Philadelphia, Pennsylvania 2,000 ft span Vincent Thomas Los Angeles Harbor, California 1,500 ft span Appendix B Deflection of Stiffening Truss – Method of successive iterations 1 x 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 2 3 4 5 deflection1 (x < l - c) deflection1 (x > l - c) deflection1 deflection2 0 -0.009317804 0 0 -0.000623796 -0.005990911 -0.000624 -2.53E-06 -0.001277899 -0.004173232 -0.001278 -4.86E-06 -0.002006079 -0.003139771 -0.002006 -6.87E-06 -0.002884999 -0.002513805 -0.002514 -8.53E-06 -0.004058271 -0.002099582 -0.0021 -9.78E-06 -0.005800225 -0.001795385 -0.001795 -1.06E-05 -0.008637747 -0.001548358 -0.001548 -1.1E-05 -0.013584265 -0.001331039 -0.001331 -1.1E-05 -0.022589819 -0.001129155 -0.001129 -1.06E-05 -0.039407077 -0.000935288 -0.000935 -9.78E-06 -0.071257905 -0.000745584 -0.000746 -8.53E-06 -0.132040611 -0.000558034 -0.000558 -6.87E-06 -0.248502226 -0.000371587 -0.000372 -4.86E-06 -0.472116917 -0.000185683 -0.000186 -2.53E-06 -0.901945802 0 0 0 6 total deflection 0 -0.000626322 -0.001282754 -0.002012953 -0.002522331 -0.002109363 -0.001806008 -0.001559405 -0.001342085 -0.001139778 -0.000945069 -0.00075411 -0.000564908 -0.000376443 -0.00018821 0 Column 1 lists the position along the bridge. The total length is 1500 in. Column 2 uses Equation 17 to determine 1 for the portion of the beam to the left of the applied load. Column 3 uses Equation 18 to determine 1 for the portion of the beam to the right of the applied load. Column 4 contains an ‘if’ statement to determine which of columns 3 or 4 should be used, based on x. Column 5 uses Equation 19 to determine 2. Column 6 sums columns 4 and 5 to determine the total deflection. 32 Appendix C Deflection of Stiffening Truss – Bisection Method root( i ) 12. ft 1 0.00655 0.006525 0.0065375 0.00654375 0.006546875 0.0065453125 0.0065460938 0.0065464844 0.0065466797 0.006546582 0.0065465332 0.0065465576 0.0065465698 0.0065465637 0.0065465668 0.0065465683 0.0065465691 0.0065465694 0.0065465693 0.0065465693 33