OVERDUE FINES: 25¢ per day per item RETUMlfi LIBRARY MTERIALS: ‘\ . .. _ Place in book return to remove . ‘V’ charge from circulation records qfflmhjp .. ' ‘3 3"Pyfi #9 K117 ”40094 W. '4 7 K1 31 ’57ka 1-HT‘TW I ., 234 ELASTIC BUCKLING OF ARCHES BY FINITE ELEMENT METHOD By Jose G. Lange A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil and Sanitary Engineering 1980 ABSTRACT ELASTIC BUCKLING OF ARCHES BY FINITE ELEMENT METHOD BY Jose G. Lange A procedure for the computation of the elastic buckling load of arches is presented. The arch is represented by beam finite elements curved in one plane but deformable in three dimensional space. The curved axis of the element is represented by a fourth-order polynomial. The displacement functions are approximated by cubic polynomials. The expressions for the genera- lized strains include the linear and quadratic terms of the displacements. By using these functions the expres- sion for the strain energy of an element is derived. This expression consists of three parts: the quadratic, cubic, and quartic terms. Proper differentiation of these expressions yields the linear stiffness matrix (K) and the incremental stiffness matrices (N1 and N2) of the element. Assuming that the system is elastic and conserva- tive, the equilibrium equation is obtained from the first variation of the potential energy. This represents a set of nonlinear algebraic equations. The equation Jose G. Lange governing the linear incremental behavior is obtained from the second variation of the potential energy. A basis for obtaining the critical load of a structural system is the vanishing of the load increment vector corresponding to a change in the displacement vector. To avoid dealing with nonlinear equations, an estimate of the buckling load is obtained by assuming that the displacement increase linearly with the applied load until buckling occurs. This leads to a quadratic eigenvalue problem for the buckling loads and their associated buckling modes. Assuming that at buckling the displacements are sufficiently small the quadratic eigenproblem reduces to a linear one. The quadratic eigenproblem is solved by the deter- minant search method in conjunction with the modified regula falsi iteration technique. Inverse vector itera- tion is used for the solution of the linear problem. Eigenvalue problems are also formulated for the case of tilted loads (for example, due to the horizontal rigidity of the deck of an arch bridge). In addition, the buckling problem involving interactions between hori— zontal transverse loading and vertical in-plane loading is formulated. A computer prOgram was prepared for the implemen- tation of the linear equilibrium solution and the buckling load solutions. Numerical results were obtained involving arch ribs with in-plane and out-of-plane behavior. The Jose G. Lange influence of the number of elements on the accuracy of the results was investigated by considering both linear equilibrium problems and buckling problems. The types of buckling problems considered are: in-plane, out-of- plane, tilted loads, and the effect of out-of-plane horizontal loads on the in-plane buckling load. Good agreement was indicated by comparisons of the first three types of problems with existing analytical solutions based on the classical buckling theory. Results for the last type of problems indicated that while a small out- of-plane horizontal load may have little effect on the in-plane buckling load, the latter decreases rapidly with increases in the horizontal load. TO MARIA MARGARITA AND ANDRES JOSE AC KNOWLEDGMENTS The writer wishes to express his appreciation to his major professor, Dr. Robert K. Wen, Professor of Civil Engineering, for his guidance and numerous helpful suggestions during the conducting of the research and preparation of this dissertation. Thanks also to members of the writer's doctoral committee: Dr. William A. Bradley, Professor of Metallurgy, Mechanics and Materials Science, Dr. George E. Mase, Professor of Metallurgy, Mechanics and Materials Science, Dr. Harvey Davis, Professor of Mathematics. The writer also owes his appreciation to Ms. Ann Greenfield for her dedica- tion in the typing of this dissertation. Special appre- ciation is also due his wife Maria Margarita, and son Andres Jose, who make it all worthwhile. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS LIST OF TABLES LIST OF FIGURES CHAPTER I. II. III. INTRODUCTION 1.1 GENERAL 1.2 OBJECTIVE AND SCOPE 1.3 LITERATURE REVIEW 1.4 NOTATION FINITE ELEMENT MODEL FOR A CURVED BEAM NNNN O O O pwww 2.5 GENERAL DISPLACEMENT-STRAIN RELATION STRAIN ENERGY EXPRESSION FINITE ELEMENT FORMULATION 2.4.1 DEFINITION OF COORDINATE SYSTEMS 4 ELEMENT GEOMETRY .4 DISPLACEMENT FUNCTIONS .4 ELEMENT STRAIN ENERGY QUILIBRIUM EQUATIONS kwN MNNN BUCKLING LOAD ANALYSIS WWW 0 (AMP GENERAL FORMULATION OF EIGENVALUE PROBLEMS SOLUTION OF EIGENVALUE PROBLEMS 3.3.1 QUADRATIC EIGENVALUE PROBLEM 3.3.2 LINEAR EIGENVALUE PROBLEM BUCKLING OF ARCHES DUE TO TILTED LOADS EFFECT OF OUT-OF-PLANE LATERAL LOAD ON IN-PLANE BUCKLING LOAD OF ARCHES COMPUTER PROGRAM iv Page iii vi vfi OChFJF‘ ha 14 14 17 18 18 19 21 22 26 29 29 29 31 31 32 34 36 37 CHAPTER Page IV. NUMERICAL RESULTS 40 4.1 GENERAL 40 4.2 LINEAR EQUILIBRIUM PROBLEMS 41 4.2.1 CONCENTRATED IN-PLANE 41 LOAD (VERTICAL) AT CROWN 4.2.2 CONCENTRATED TRANSVERSE 43 LOAD (HORIZONTAL) AT CROWN 4.3 BUCKLING PROBLEMS 44 4.3.1 LINEAR VERSUS QUADRATIC 44 EIGENPROBLEM SOLUTIONS 4.3.2 IN-PLANE BUCKLING 45 4.3.3 OUT-OF-PLANE BUCKLING 45 4.4 BUCKLING OF PARABOLIC ARCHES 46 SUBJECTED TO TILTED LOADS 4.5 EFFECT OF HORIZONTAL TRANSVERSE 47 LOAD ON THE VERTICAL BUCKLING LOAD OF ARCHES V. CONCLUSION 49 5.1 DISCUSSION 49 5.2 SUMMARY 51 TABLES 54 FIGURES 62 LIST OF REFERENCES 77 APPENDICES A. METHOD FOR COMPUTING bi IN 82 EQUATION (2-17) B. MODIFIED REGULA FALSI ITERATION 85 TECHNIQUE C. COMPUTER PROGRAM 88 TABLE LIST OF TABLES LINEAR EQUILIBRIUM OF CIRCULAR ARCH SUBJECTED TO CONCENTRATED IN-PLANE LOAD AT CROWN LINEAR EQUILIBRIUM OF PARABOLIC ARCH SUBJECTED TO CONCENTRATED IN-PLANE LOAD AT CROWN LINEAR EQUILIBRIUM OF CIRCULAR ARCH SUBJECTED TO CONCENTRATED OUT-OF-PLANE LOAD AT CROWN LINEAR VERSUS QUADRATIC EIGENVALUE SOLUTIONS IN-PLANE BUCKLING OF CIRCULAR ARCH SUBJECTED TO UNIFORMLY DISTRIBUTED RADIAL LOAD OUT-OF-PLANE BUCKLING OF PARABOLIC ARCH SUBJECTED TO UNIFORMLY DISTRIBUTED VERTICAL LOAD BUCKLING OF PARABOLIC ARCHES SUBJECTED TO TILTED LOADS EFFECT OF HORIZONTAL TRANSVERSE LOAD ON VERTICAL BUCKLING LOAD OF A PARABOLIC ARCH EFFECT OF HORIZONTAL TRANSVERSE LOAD ON VERTICAL BUCKLING LOAD OF A CIRCULAR ARCH SOLUTION PROCEDURES FOR COEFFICIENTS bi vi Page 54 55 56 57 58 59 60 61 61 84 FIGURE 1-1 1-2 2-1 LIST OF FIGURES IN-PLANE BUCKLING UNDER SYMMETRICAL LOADING LOAD-DEFLECTION RELATION BEAM ELEMENT (CURVED IN x-z PLANE) CROSS-SECTION OF PRISMATIC MEMBER COORDINATE SYSTEMS TYPICAL ELEMENT TYPICAL ELEMENT AFTER TRANSFORMATION TO ELEMENT COORDINATE SYSTEM DETERMINANT SEARCH METHOD TYPICAL ARCH BRIDGES TILTED LOADS ON RIBS OF DECK AND THROUGH BRIDGES LINEAR EQUILIBRIUN OF CIRCULAR ARCH SUBJECTED TO CONCENTRATED IN-PLANE LOAD AT CROWN LINEAR EQUILIBRIUM OF PARABOLIC ARCH SUBJECTED TO CONCENTRATED IN-PLANE LOAD AT CROWN LINEAR EQUILIBRIUM OF CIRCULAR ARCH SUBJECTED TO CONCENTRATED OUT-OF-PLANE LOAD AT CROWN IN-PLANE BUCKLING OF CIRCULAR ARCH SUBJECTED TO UNIFORMLY DISTRIBUTED RADIAL LOAD vii Page 62 63 64 64 65 65 66 67 68 69 70 71 72 73 FIGURE viii OUT-OF-PLANE BUCKLING OF PARABOLIC ARCH SUBJECTED TO UNIFORMLY DISTRIBUTED VERTICAL LOAD EFFECT OF HORIZONTAL TRANSVERSE LOAD ON VERTICAL BUCKLING LOAD OF A PARABOLIC ARCH EFFECT OF HORIZONTAL TRANSVERSE LOAD ON VERTICAL BUCKLING LOAD OF A CIRCULAR ARCH MODIFIED REGULA FALSI ITERATION Page 74 75 76 87 CHAPTER I INTRODUCTION 1 . 1 GENERAE The purpose of this thesis is to develop a procedure for the computation of the elastic buckling load of arches in space. In order to achieve this a three-dimensional beam element curved in one plane that takes into account the nonlinear effects of geometry changes was formulated. A computer program has been prepared to implement the analy- sis of arches under different loading conditions, and the results of certain numerical problems are presented. This chapter describes the objective and scope of the present work, a literature review of related studies, and the general notation needed in the subsequent analysis. 1.2 OBJECTIVE AND SCOPE Many engineering structures have components that may be considered as curved beams. Examples are the ribs of arch bridges and arch roofs, stiffening rings in aircraft and naval vessels, and horizontally curved highway bridges. When such elements are subjected to considerable compres- sion such as in the case of arch bridges, their stability becomes a major consideration. Although a considerable amount of work has been done (see literature review), there are a number of signifi- cant problems that have not been solved. Most of the previous works have dealt with the stability problem in the plane of the curved element. Past studies that had considered out of plane buckling have been either limited to circular or parabolic arches or made with the assumption that the curved element may be represented by a series of straight beam elements. The ouf-of-plane behavior of a truly curved element has not been studied. The objective of the present study is to develop a three-dimensional nonlinear curved beam finite element. The numerical model is applied to the study of the buckling of arch ribs. Figure l-l illustrates the general deformation behavior of a symmetrical arch under a symmetrical load- ing. It is seen that the buckled shape may be symmetric or antisymmetric. The curve "0C" represents the "exact" response which may be obtained by the solution of the (non- linear) equilibrium equations of the system. It is called the "fundamental path." Depending upon the properties of the arch and loading, a point (e.g., point A) of "bifurca- tion" may occur before the peak point G or after it (i.e., point A). Immediately beyond a bifurcation point on the fundamental path, the structure is unstable; so the behavior 3 would follow the bifurcated path AB or AP. If the bifur- cation point occurs before C, the buckling shape would be antisymmetrical (sometimes called "sidesway," generally occurring for "deep arches"). If the bifurcation point occurs at A, the arch would have buckled at C in a sym- metric buckling form ("snap-through," generally for "shallow arches"). The preceding represents the arch behavior described by an "exact" nonlinear analysis. The classical buckling theory would assume that up to the point when buckling takes place, the structure would maintain its original undeformed shape (point A'). At buckling, it goes into an adjacent equilibrium configuration (point B') which is unspecified in magnitude. The buckling load thus computed is called the "classical buckling load." The approach followed in this investigation is that outlined by Mallett and Marcal (27)*. The theory is essen- tially different from either that which follows from the nonlinear equilibrium behavior or the classical theory of stability. The system is assumed to be elastic. The strain energy is written in terms of displacement variables. Geo— metrically nonlinear effects are considered by including the quadratic terms of displacements in the expressions for * Number in parantheses refer to entries in the list of references. the generalized strains. Thus, the strain energy may be written as the sum of one part containing quadratic terms, one containing cubic terms, and one quartic terms. The first variation of the potential energy (assuming that loads are conservative, i.e., their direction do not change with structural displacements) produces the equilibrium equation. The latter equilibrium equation may be transformed into an eigenvalue problem by assuming that the displacements increase linearly with the load parameter that controls the magnitude of the applied loads, and at the buckling load the linear incremental stiffness (tangent stiffness) vanishes. This formulation yields a quadratic eigenvalue problem; the lowest eigenvalue corresponds to the lowest buckling load. For the finite element developed in this study, the curved shape of the element is represented by a fourth- order polynomial. This representation of the geometry can maintain continuity of position, slope, and curvature at two adjacent elements. The shape functions describing the displacements along each of the three coordinate axes and the twist along the longitudinal axis of the element are each expressed as cubic polynomials. This requires the introduction of eight degrees of freedom at each end. The linear and nonlinear stiffness matrices require numerical integration over the curved domain, for which the Gauss quadrature method was used. 5 The quadratic eigenvalue problem was established and solved by the determinant search method (5) in conjunction with the modified regula-falsi iteration technique (9). The effect of the quadratic term on the lowest eigenvalue was found to be'small and it seems reasonable to drop that term. Hence, the problem reduces to that of a linear eigenvalue. It was solved by using the inverse iteration method (5). A computer program was prepared to implement the linear equilibrium solution and the buckling load solution. Numerical results included systems with in-plane and out- of-plane behavior. In addition to some linear equilibrium problems solved to indicate the reliability of the model, the following types of buckling problems have been consid- ered: (a) in-plane, (b) out-of-plane, (c) "tilted load," and (d) the effect of out-of-plane loads on the in-plane buckling load. For the first three types, comparisons were made with existing analytical solutions based on the classical theory (except in two cases of tilted loading for which no analytical solutions are available). The agree- ments are in general good. Results of type (d) indicated while a small out-of-plane horizontal load may have little effect on the in-plane buckling load, the latter decreases rapidly with increases in the horizontal load. As far as is,known to the writer, this behavior has not been studied previously. 1 , 3 LITERATURE REVIEW Ashwell and Sabir (2) discussed the use and limita- tions of several types of shape functions for finite ele- ments for circular arches. Three types were considered: (a) polynomial expressions for the radial and circumfer- ential displacements, (b) polynomial expressions modified to include certain trigonometric functions so as to make the circumferential strain and the change in curvature equal to zero when the displacements correspond to a rigid body motion, (c) expressions corresponding to axial strain and linear curvature. The latter two admit true rigid body displacement representations. They showed that the type (c) functions are only slightly inferior to type (b) when used for shallow arches. They also indicated that the performance of the shape functions would depend on the geometry of the arch. Shape functions good for shallow arches may not be good for deep arches and vice versa. This work was limited to actions of circular arches in a plane. Dawe (13) pointed out the advantage of using higher- order polynomials for shape functions. His comparison included models using polynomials of orders quintic-quintic, cubic- quintic, quintic-cubic , cubic-cubic to represent the tangen- tial and normal components of the displacements, and also a constant strain, linear curvature model for the shape functions . 7 Mebane and Stricklin (29) pointed out that rigid body motion could be considered to be implicitly included in the poly- nomial form of the shape functions as the number of elements used to represent the structure increases. The preceding studies had been undertaken actually to investigate the optimal approach of using the finite element method for axi-symmetrical shells. The latter, like the arch deformed in its own plane, is a two-dimen- sional problem. For linear equilibrium problems of an element curved in one plane, the stiffness coefficients of the "in-plane" degrees of freedom (i.e., axial displacement, in-plane transverse displacement, and in-plane rotation) are uncoupled from the "out-of-plane" degrees of freedom. For circular elements the stiffness coefficients for in- plane degrees of freedom are well known (see e.g., Reference 37). For circular elements subjected to out-of-plane loading exact stiffness coefficients for sections with neg- ligible warping have been reported by Lee (26). Stiffness matrices including warping had been reported by El-Amin and Brotton (l6), and by Thornton and Master (40). The former employed the finite element method using cubic poly- nomials for the displacement functions and the stiffness coefficients were given explicitly. The latter presented expressions from which "exact" stiffness matrices may be computed from the inversion of certain component matrices. 8 Chauduri and Shore (8) developed a thin-walled curved beam element that also included warping effects and established a consistent mass matrix for the element. The area of classical buckling analysis of curved structures has been investigated by several researchers. Austin (3) summarized the state of the knowledge of the in-plane bending and buckling of arches. His presentation was concerned with the available experimental and analytical data and their relations to design applications. Austin and Ross (4) compared the solution of the in-plane elastic buckling of arches between the classical buckling theory and the exact, nonlinear, buckling load analysis. They found that, except for buckling in the symmetric mode (snap-through), the buckling load obtained with the class- ical theory is very close to that obtained with the exact theory. I. Ojalvo and Newman (31) reported a basic theoreti- cal work on the elastic stability of a curved beam in space. The governing differential equations of a curved element in space were derived and a solution procedure akin to the "shooting method" was outlined. The shooting method is one that solves a boundary value problem as an initial value problem (30). If a curved beam is analyzed, the solu- tion would be carried out from one end of the beam with the known boundary conditions plus certain assumed boundary con- ditions there as the "initial conditions." The solution proceeds toward the other end where, in general, it would not agree with the prescribed boundary conditions. MOdi- fications would then be made of the boundary conditions assumed for the "initial end" such that the conditions at the "final end" would be met. M. Ojalvo, Demuts, and Tokarz (32) followed the preceding works to study the out-of-plane buckling of a member curved in one plane. The theory was also applied by Shukla and M. Ojalvo (38) to calculate the buckling loads when they may be tilted such as those which would result from a horizontally rigid deck of an arch bridge. Extensive numerical data were presented covering ranges of parameters involving the ratio of the torsional stiffness to the out-of-plane bending stiffness and the ratio of the rise to the span length. Laboratory results that corrob- orated the theoretical data were presented by Tokarz (41). As a particular case of the equations presented in Refer- ence (32), Tokarz and Sandhu (42) developed linear differ- ential equations that define the lateral-torsional buckling of a parabolic arch subjected to a uniformly distributed load. They also made a comparative study with those results obtained experimentally by Tokarz (41). Godden (20) studied the buckling load of a tied arch by use of the Rayleigh-Ritz method. The tilted 10 hangers (on account of the assumed horizontal rigidity of the deck) were replaced by a continuous membrane along the arch. He verified the solution with experimental results. In a subsequent work Donald and Godden (14) reported a numerical procedure of the shooting method type for the analysis of curved beams (replaced by straight chords between panel points) subjected to loads normal to the plane of the structure. 1.4 NOTATION The notation shown below has been used in this report: A = area of cross-sections; A, B = end nodes of an element; b2,b3,bu = element geometry coefficients (Eq. 2-12); B = Young's modulus of elasticity; G = shear modulus; H = rise of the arch; HD = difference in elevation between crown of arch and deck; Ixx, Iyy = moment of inertia of cross-section (Fig. 2-2); 15' I = moment of inertia of cross-section ” (Fig. 2-2); [K] = structural linear stiffness matrix; [Km] = modified structural linear stiffness matrix; [k] = element linear stiffness matrix; K = torsion constant of cross-section; rr Kx,Ky,Kz kx,ky,kz [N1] [N2] [n1] [n2] {P} {545} ll changes in curvature about x, y, z axes; current curvatures about x,y,z axes; initial curvatures about x,y,z axes; curved length of element; span or arch; first order structural incremental stiffness matrix; second order structural incremental stiffness matrix; first order element incremental stiffness matrix; second order element incremental stiffness matrix; vector of applied loads; vector of applied horizontal loads; a given load, a given load vector (Eq. 3'2); critical value of applied loads; reference load, reference load vector (Eq. 3-2); generalized coordinates, displace- ment vector; displacement vector due to applied horizontal (out-of-plane) load; displacement vector due to applied vertical (in-p1ane)load; reference displacement vector (Eq. 3-1) 0 I X,Y,Z 8X! BY! 82 12 reference displacement vector (EQ- 3‘2); radius of curvature radii of curvature at ends of an element; longitudinal axis of curved beam member; diagonal transformation matrix (Eq. 3-14); diagonal transformation matrix (Eq. 3-16); displacements along x,y,z axes, respectively; strain energy of an element; strain energy due to longitudinal strain; strain energy due to torsion; quadratic, cubic, and quartic parts of strain energy; structure global coordinate system; relative position of end nodes of an element (Eq. 2—10); element coordinate system; coordinates of node B in element coordinate system; angle of opening of circular arch; twist of cross-section about z-axis; rotations about x,y,z axes, respec- tively; normalized variable (Eq. 2-19); longitudinal strain; {} L] E] 13 angle of tangent at node B (Fig. 2-5); rotation about x-axis; rotation about y-axis; buckling load parameter; incremental operator; column vector; row vector; rectangular matrix. CHAPTER II FINITE ELEMENT MODEL FOR A CURVED BEAM 2.1 GENERAL In this chapter the displacement—strain relation and the expression of the strain energy of a curved beam are first presented. Next, the geometric representation and the displacement functions of the element are described. The strain energy expression of a typical curved element is developed. Finally, the general equations that govern the equilibrium and the linear incremental behavior of a struc- ture are derived. 2 . 2 DISPLACEMENT-STRAIN RELATION Consider a beam element curved in one plane as shown in Figure 2-1. The centroidal axis curves in the x-z plane with radius of curvature R (which may vary). The x, y, and z axes form a right-handed coordinate system with corresponding displacements u, v, and w as indicated in Figure 2-2. The cross-section of the element is taken to be constant. Assuming that plane sections remain plane after bending deformation, the expression for the longi- tudinal strain at a section 5, measured along the curved 14 15 centroidal axis, may be written as e) = e ) + n K - E K (2-1) in which Ez)o is the longitudinal strain and Kx and Ky are the changes in curvature of the centroidal axis. For the general case of a beam curved in space, the changes in curvature have been derived by I. Ojalvo and Newman (31) as follows: - — - as K=k-k=k -k +323 x x x y 8z 2 By s - _ _ d8 K=k-k=-kB+kB+——1 (2-2) Y y y x z z x ds _ - de Kz = k2 - k2 = kx By - 1"y 8x + —d; in which kX and kX are the current and initial curvatures about the x-axis, respectively, similarly for ky, ky and k k and 8X, By, 82 are the rotations about the x, y, z z' 27 axes, respectively. The latter rotations are given by: " _ ‘ _ 91’. du - ‘ _ g3 w _ By - a; - V k2 + ky W d5 fi (2 3) B = 8 in which 8 is the twist of the cross-section about the z-axis (note that‘kX = k2 = 0). Substituting Equations 16 (2-3) into Equations (2-2), a: II + 2 du dwl 1.1. "‘—‘ “g d(R) (2 4) The longitudinal strain at the centroidal axis may be written as: V72 1 dy'Z +§) + 5(a) (2-5) 11 l du _. 91". _ - (ds - R) + 2 (ds 5 ) 2o in which the terms in the first parenthesis are the usual linear hoop strain for a curved element and the next two terms (which are nonlinear) represent the contribution to the strain by the rotations of the centroidal axis about the y- and x-axis, respectively. Substituting Equations (2-4) and (2-5) into Equation (2-1) the expression for the longitudinal strain for any point in the cross-section is obtained, i.e., dw u l du w 2 1 dv 8 = —— - — + — _— + _ + — __) flag,” (ds R) 2(ds R) 2(ds 8 dzv + ”(R - dsz) (2-6) dzu 1 dw d 1 ' 5 [rs-,2 + 'R as: was (RU in which, it may be noted again that u, v, w, and B are displacements of the centroidal axis of the beam. 17 2.3 STRAIN ENERGY EXPRESSION The expression for the total strain energy of the system may be written as: U = UE + Ut (2‘7) where U8 is the strain energy due to the longitudinal strain and Ut is that due to torsion of the cross-section along the axis of the beam. They are given by the expres- sions (2-8) G K 1 - t _ 2 Ut._.L 2 (BS + R vs) ds in which a is the longitudinal strain as expressed by Equa- tion (2-6), A is the cross-sectional area, B and G are the Young's modulus and shear modulus, and Kt is the torsion constant of the cross-section. In the preceding equation, the common notation of using a subscript to represent a differentiation has been used, e.g., 85 E dB/ds. This notation will also be used subsequently. The total strain energy becomes (as + % vs)2ds (2-9) r E 52 U=JJ TdAds+J s A 8 Equation (2-9) will be used in Section 2.4.4 to obtain the strain energy of the curved element, and by proper differ- entiation to obtain its stiffness matrices. 18 2.4 FINITE ELEMENT FORMULATION In the previous section the strain energy expres- sion for a general three-dimensional beam curved in a plane has been presented. In this section, the geometry and the strain energy of a finite element model of a curved beam will be developed. 2.4.1 DEFINITION OF COORDINATE SYSTEMS Figure 2-3 illustrates an arch with a typical con- stituent curved finite element AB. Additional coordinates for the element are described in Figure 2-4. Two coordi- nate systems are used in the analysis: 1) Structure Global Coordinate System. This system consists of a single set of cartesian axes with the origin located at the crown of the arch (Figure 2-3). The system is oriented with the X-axis hori- zontal, the Y-axis vertical, and the Z-axis perpendicular to the plane of curvature. The positions of the nodes of the total structure are expressed by means of this system. 2) Element Coordinate System. This system is illustrated in Figure 2-4. It con- sists of one set of cartesian axes with its origin located at node A, with the x-axis in the radial direction, the y- axis normal to the plane of curvature, and the z-axis tan- gent to the curved centroidal axis forming an angle ¢A with the global X-axis. Node B denotes the end node of the element. 19 2.4.2 ELEMENT GEOMETRY Referring again to Figure 2-4, the coordinates of the nodes A and B with respect to the structure global system are (XA' YA) and (X8, YB) respectively. Further- more, their relative position is defined by XL = XB - XA and YL = YB - YA. To represent the element geometry in the x and z coordinates the element is redrawn in Figure 2-5. The coordinates (xB, 2B) of node B in the element coordinate system are given by: xB =-XL sind>A + YL cosqu (2-10) zB = XL cosoA + YL sinoA in which ¢A' XL and YL are defined in Figure 2-4. The angle ¢ in Figure 2-5 varies from zero at node A to G at node B, s varies along the longitudinal axis, and the radii of curvature R at nodes A and B are respectively R1 and R2. At any point along the curve the following rela- tions hold dz ds cos ¢ (2-11) dx = ds sin ¢ The curve will be approximated by a fourth-order polynomial s =.bo + b1 ¢ + b2 ¢2+ b3 ¢3 + b“ ¢“ (2-12) 20 the boundary conditions needed to solve for the coefficients bi are (see Figure 2-5): (1) at ¢ = O, S = 0 (2) at ¢ = 0, R = R1 (2-l3a) (3) at ¢ ll 0 (I) II t“ where L is the curved length of the element (4) at ¢ = G, R = R2 1 6 G I dx J sin o R do (5) x = = B 0 o (2-13b) 6’55 e (6) 2B = f dz = I cos ¢ R do 0 0 From (1) it is found that bO = O. The curvature is obtain- ed by differentiating Equation (2-12): R = 53 = b1 + 2 b2¢ + 3 b3¢2 + 4 bu¢3 (2-14) From condition (2), b1 = R1. When condition (3) is appli- ed to Equation 2-12, the expression defining the length of the element is obtained as L = R19 + bZOZ + b393 + buO“ (2-15) Condition (4) yields: R2 = R1 + 2b,@ + 31:392 + 413.93 (2-16) Assuming that condition (3) is automatically satis- fied if conditions (5) and (6) are, and that the equation defining the curve is known (which implies that R1 and R2 are prescribed) a system of three linear equations for 21 three unknowns (b2, b3, b,) can be established, i. e., R1 + 2b2 e + 3b362 + 4b,.e3 = R2 R1 (1 - cos a) + 2b; (sin()- 9 cos 8) + 3b3 (-62 cos a + 26 sin e + 2 cos a - 2) + 4b..(-G3 cos 9 + 3stin e + 6(3cos e - 6 sin 9) KB (2-17) Rl sin 6 + 2b2 (0 sin 6 + cos 0 -l) + 31:3 (stin e + 2(3cos e - 2 sin e) + 4b“ (Basin 0+ 392cose- 6Gsine - 6 cosG + 6) = ZB Once the coefficients b2, b3, and b“ are obtained from the solution of Equations (2-17) the geometry of the finite element is completely defined by Equation (2-12). It may be pointed out that the values of b2, b3, and b,+ seem to be somewhat sensitive to the solution pro- cedure used for Equations (2-17). A comparison of the solu- tion obtained by several procedures is given in Appendix A. It is also shown that indeed condition (3), i.e., the length of the element, is satisfied by the procedure used. 2.4.3 DISPLACEMENT FUNCTIONS As indicated in Equations (2-6) and (2-9), the strain energy of the curved element considered here depends on four independent displacement functions, i. e., u, v, w, and B, the displacements along the x, y, z axes and the rotation about the z-axis, respectively. For the finite 22 element, these functions will be approximated by cubic polynomials in the variable o (Figure 2-5) u = a1 + azo + aaoz + au$3 v = a. + a6¢ + a7¢2 + ae¢3 (2-18) 2 3 w + “10¢ + “11¢ + a12¢ I] Q m B = 0‘12. + alu¢ + a1s¢2 + 0‘15‘1’3 Note that ¢ is related to the arc length, 5, by ds/do = R = radius of curvature. For simplicity, the independent variable ¢ in the preceding equations may be normalized by defining Y = o/O, and the displacement functions become: - 2 3 u — Al + A27 + A37 -+l“y v = A5 + lay + A7y2 + 1873 (2-19) - 2 3 w - 19 + Aloy + Ally + Alzy 3 B = A + A1uY + A y 2 13 Y +A1 15 6 2.4.4 ELEMENT STRAIN ENERGY In terms of the new variable Y the longitudinal strain may be rewritten from Equation (2-6) as: = w _ u l u up: 1 Zle 211. _ (Re)2 (2-20) + n [g - v y Yss] 23 in which = 91 - d¢ 1-— l ' (2-21) = - l , ,‘t L 2" 2 Yss fiTgt RS 7 , . L ‘I _.:L a ( 2 ) 1 YSY" (RH - Re yss (2-23) *0 Using the same change of variables for Equations (2-8), 2 U = I I E 5 dA ROdY 0 A 2 1 (‘ K - J t U _ [a 2 (BY Y (2-24) 1 2 + — v Red R Y Ys) Y s The expression for the strain energy of an element is now obtained by substituting Equations (2-20) and (2-24) into Equation (2-7) 1 , - _§_ 2 22 1 EL. 3:. I L; U - 2 [{[(WYYS) + (R) 4-; (uyvs + R) + 4 (VYYS) JA 0 + I (E - V {'2 - v Y )2 E R YY S Y 55 + I (u y2 + u y + n s Y 55 R 1 , 2 - w Y + WY Y ) Y 5 SY s / + "- 2w 3 + w + 3 2 L YYS R Ys (uYYs R) (2-25) ‘ W + w v 2 - E u + ~ 2 YYS ( YY5) R ( YYS R) _ u ' 2 1 ‘E2 2 , R (VYYS) + 2 (UYYS + R) (VYYS):]A }R®d\ 1 + GKt (3 y + l v ys)2 ROd) 24 Equation (2-25) may be divided into three parts in the form: U = U + U + U (2-26) 2 3 u . in which U2, U3, and U“ contain respectively the quadratic, cubic, and quartic terms of the total strain energy. In explicit form, after some simplifications, they are l U = E { [ l: (w 6 2 J .O - 2 _ :11__ ’ 22 Y a“) + IE R7 (RB 02 vYYssR ) I KY 2 2 1 2 + ‘TnT u + u R G + w G + w 1%) dY R e ( YY YYss Y YSY 3 1 GK t l 2 + 2 L R39 (REY + VY) dY (2 27a) EA 1 U3- if I E767 (wY-Gu) [(uY + 8w)2 + vY§]dY (2-27b) O 1 1 EA 2 2 2 U = -— -———- + e + 2-27 h 8 J 4R3G3 [(uY w) VY] dY ( c) 0 Upon substituting the displacement functions given by Equations (2-19) into Equations (2-27), U2, U3, and U. become functions of the coefficients Xi in the displacement functions. These coefficients will be replaced by certain degrees of freedom related to the displacement variables at the ends of the element. These degrees of freedom are 25 chosen to be: uA, uB = the radial displacement; VA, VB = the transverse displacement; wA, wB = the longitudinal displacement; BA, BB = the twist about the longitudinal axis; = du W = ' .. ' . GYA'GYB (a§ + fi) the rotation about y aXis, A,orB . ._. _dv = : _ - . GXA'GXB ( d§)A, or B the rotation about x aXlS, dw dw . . (——) —— = part of aXial strain; ds A'(ds)B d8 d8 . —- —— = rate of twist; may (ds)B For subsequent analysis the above degrees of freedom will be represented by the generalized coordinates {q}: . _ dw [-ql q2"‘qe’ q9"'q16-l— LuA VA WA 8A eyA GXA(H§)A d8 dw d8 _ ( u v w B G G (—) (ES-)8] (2 28) ——) ; dsA BBBByBdesB The relation between U;}and {A} may be obtained by use of Equations (2-19) and (2-28). Thus, U2, U3, and U, can be expressed in terms of the element nodal degrees of freedom {q}: 1 " [ f2({q}) dY o 1 (2-29) 3 — [ f3({q}) dY 0 C.‘ I G I C'.‘ II 1 I f“({q}) dY' O 26 in which f2, f3, and f“ are, respectively, quadratic, cubic, and quartic functions of the q's. The linear element stiffness matrix [k] may be obtained as [k] = [k..] = [—2:21_ ] = [f azfz dY ] (2_30) The "incremental stiffness matrices" [nlJand [n2]are defined to be: 2U r 82f = I. = [g]: —3 ‘ - [n1] [mun] aqiaqj [Jo aqie jdf] (2 31) 1 2 [n2] - [(n2)lj] -[__qiaqj] - [Jo quaqj dY] (2 32) The expressions for the integrands in terms of the q's in Equation (2-30), (2-31), and (2-32) are too lengthy to be presented here. They are, however, explicitly given in the subroutine NUMINT of the computer program in Appen- dix C. It may be noted in passing that the integrals them- selves were evaluated numerically by Gauss quadrature. It should also be noted that the elements of the matrices [n1] and [n2] are respectively linear and quadratic functions of the displacements. 2.5 EQUILIBRIUM EQUATIONS The element linear stiffness matrix, [k], and the incremental stiffness matrices [n1] and [n2] have been de- rived in the preceding section. The structural linear 27 stiffness matrix, [K], and the incremental stiffness matrices [N1] and [N2] may be obtained by "assembling" or adding up the corresponding element matrices, provided that they are all considered to be of the "structure size" and refer to the same global degrees of freedom. As mentioned previously, the formulation followed in this section is that described by Mallett and Marcal (27). Assuming that the system is elastic and conservative, the potential energy of the system is: op = U + v (2-33) in which U is the strain energy and V is the potential of the external loads. The total potential energy may be expressed as: T = I Egg dvol + V vol LqJ [é— [K] + g. [N1] + $13 [312]] {q} (2-34) - [q] {P} in which [N1] and [N2] appear on account of the nonlinear terms in the expression for the longitudinal strain (Equa- tion (2-6)). The first variation of the potential energy produces the equilibrium equation: [[K] + % [N1] + % [N2]] {q} = {P} (2-35) This represents a set of nonlinear algebraic equations. 28 The equations governing the linear incremental behavior follow from the second variation of the potential energy and are given by: ( [K] + [N1] + [N2] ){a} {Ag} = {AP} (2-36) where {q} is the reference equilibrium position. This equation will be used in the following chapter to develop the eigenproblem model for the buckling analysis. CHAPTER III BUCKLING LOAD ANALYSIS 3.1 GENERAL The governing matrix equation of the linear incre- mental equilibrium for a nonlinear elastic structure (Equation (2-36)) was introduced in Section 2.5. In this chapter, the calculation of the buckling load as a quad- ratic or linear eigenvalue problem will be discussed. In addition, the formulation required for the computation of the buckling loads of arch ribs subjected to "tilted loads," and the effect of horizontal loads on the vertical buckling load will be presented. The implementation of the solu- tion procedures used will also be given. 3.2 FORMULATION OF EIGENVALUE PROBLEMS A basis for obtaining the critical load of a struc- tural system is the vanishing of {AP} in Equation (2-36). This leads to: (ER) + [N1] + [312]){51} {Aq}={0} (3-1) Equation (3-1) may be written for each point on the "fun- damental path" (Figure 1-2). That path, of course, can be determined only by a solution of the nonlinear equilibrium equation, i.e., Equation (2-35). For a given load vector {P} and displacement vector {3} on the fundamental path, 29 30 if a nontrivial solution of {Ag} is obtainable from Equation (3-1), the load {P} would be the "exact" buckling load (the bifurcation or snap-through load). Of course, for a given {P}, {8} can be found only by the solution of the nonlinear equilibrium equations. In order to avoid dealing with the nonlinear equations, however, an estimate of the buckling load may be obtained from Equation (3-1) by assuming that the displacement of the structure increases linearly with applied load until buckling occurs. Thus, letting {qo} be some reference displacement, (e.g., {qo} = [K]'1{PO}, {R} being some reference load = pO {PO}), the incremental stiffness matrices may be computed as: [Nl({§})] [N1({qo})] (g1) (3-2a) O and [N2({§})] - 2 [N2({q })] (ll) (3-2b) 0 p o where {a} = [K].1 5 {HQ , and the squared term in Equa- tion (3-2b) follows from the fact that the elements of [N2] are quadratic in the displacement variables. At buckling, [N1({q})] pC r [N1]{qo}(5;) (3-3a) and p [N2({q})] = [132%q }(3§5)2 (3-3b) O O 31 Thus, Equation (3-1) may be written as ([K] + A [N1] + A2 [N2]){q } O {Aq} = {0} (3-4) where A= per/p0. This equation represents a quadratic eigenvalue problem for estimating the buckling loads and their associated buckling modes. If it can be assumed that at buckling the displace- ments are sufficiently small, then the matrix [N2] may be neglected. Thus, Equation (3-4) reduces to a linear eigenvalue problem, i.e., K +x N1 {a}= {o} 3-5 ([1 [ DMD} q ( ) In the following sections the solution procedures for the quadratic and the linear eigenvalue problems will be described. 3.3 SOLUTION OF EIGENVALUE PROBLEMS The determinant search method in conjunction with a modified regula falsi iteration technique is used to obtain a solution of the quadratic problem. For the linear problem the inverse vector iteration is used. 3.3.1 QUADRATIC EIGENVALUE PROBLEM A solution of the quadratic eigenvalue problem (Equation (3-4)) may be obtained by finding a value of A for which, det | [K] + A [N1] + A2[N2] |{q }= 0 (3-6) 0 For the purpose of this study only the smallest value of A is of interest as it corresponds to the lowest 32 buckling load and higher buckling loads have no practi- cal significance. Therefore, the solution is carried out by evaluating the left-hand side of Equation (3-6) with increasing values of A, starting from zero with small in- crements as shown in Figure 3-1. It may be noted that det (A==O)>0. Let AA be such that det (A = AA)>O but det (A = AB = AA + AA)<0. Then the solution A = A must lie in the interval [AA, AB]. A modified regula falsi iteration technique (9) was used to obtain a closer estimate of the root A. This technique is described in Appendix B. The computer imple- mentation of the solution of the quadratic eigenvalue ,problem is given in subroutine NLEIGNP’ of the computer program contained in Appendix C. 3.3.2 LINEAR EIGENVALUE PROBLEM As mentioned in Section 3.2 that the linear eigen- value problem is obtained from the quadratic problem by neglecting the matrix [N2] from Equation (3-1). In this thesis the linear eigenvalue problem is solved by using the inverse vector iteration technique as described by Bathe and Wilson (5). The technique may be regarded as a mathematical formulation of the Stodola method (23) in structural mechanics. From Equation (3-5) the basic relation for the inverse vector iteration is A q = A B q (3—7) in which for simplicity the usual symbols [ ] and { } 33 used for matrices and vectors have been dropped, and A = [K] and B = -[Nl]. The method assumes that A is posi- tive definite and B may be a diagonal matrix with or with- out zero diagonal terms or may be a banded matrix, as is the case in this report. A technique suitable for computer implemen- tation is as follows: (i) Assume a trial vector x; for the first eigen- vector q1 and that x? B q1 M 0. (ii) lfor i = l, 2,..., evaluate Axi+l = yi yi+1 = B Xi+1 _T _ x. y. D(x.+l) = $+1 : (3-8) 1 i. y i+l i+l i7. = i+l y1+1 -T _ (Xi+l Yi+l)% in which 9 is the Rayleigh quotient. (iii)The preceding iterative process is consid- ered to have converged if o(§. > - o(§ ) - ”(1),; )1 :EPSI (3-9) i+l 2 . S or smaller when the answer is where EPSI should be 10- required to zsedigit accuracy. If "n" is the last itera- tion, i.e., Equation (3-9) is satisfied for i = n, then 34 the smallest eigenvalue will be taken to be: ) (3-10) and the corresponding eigenvector is 3? = n+1 _ qi (i T’ Y' )5 (3 ll) n+1 n+1 The computer implementation of the techniques presented above is described in the subroutine EIGENVL listed in Appendix C. In the next two sections two special formulations of the buckling problem will be described, namely, buck- ling due to tilted loads, and the effect of horizontal loads on the vertical buckling load. 3.4 BUCKLING OF ARCHES DUE TO TILTED LOADS In considering the out-of-plane buckling of the ribs of arch bridges, researchers (38, 41) were concerned with the effect of the rigidity of the bridge deck. If the deck is assumed to be perfectly rigid in the horizon- tal direction, the vertical load transmitted through the columns (or hangers) to the rib would be tilted as the arch undergoes a buckling displacement. Typical deck, through, and half-through arches are illustrated in Figure 3-2. Figure 3-3a illustrates a section A-A of a deck arch bridge. The horizontal compo- nent of the tilted load is PH = P v/H. This horizontal 35 load will enter in the equilibrium equation (Equation (2-35)) corresponding to the horizontal degrees of freedom in the out-of-plane direction. In this case, the load vector should be modified to include these horizontal loads PH. The incremental form of the equilibrium equation is given by Equation (2-36) as ([K] + [N1(q)J + [N2(q)]) {Aq} = {AP} (3-12) in which {AP} represents the change in load during buckling. The change of the vertical loads are of a higher order of magnitude and may be neglected. Therefore, the right-hand side of Equation (3-12) contains only the change in hori- zontal loads: {AP}= (3-13) I’U 000:]:[>Oo 0. <1 Now Av is a component of the vector {Aq}; or, considering such loadings from all the columns, {Av} is a subset of {Aq}. Then Equation (3-13) can be written as {AP} = [T] {q} (3-14) in which the transformation matrix [T] is a diagonal matrix with diagonal terms equal to zero except for those terms corresponding to the v-components of vector {Aq}, in which case the diagonal term is equal to Pi/Hi' where Pi is load in and Hi the height of the column concerned. Similar 36 analyses for the through bridge (Figure 3-3b) and half- through bridge would yield the same equation as Equation (3-14) provided Hi is computed from Hi = Yi - HD in which HD is the difference in elevation between the crown and the deck and Yi is the Yécoordinate of the node on the rib to which the ith column or hanger is connected. Substituting Equation (3-14) into Equation (3-12) one obtains: ([K] + [N1(q)] + [N2(q)] + [T]) {Aq} = {0} (3-15) Introducing Equations (3-3) into Equation (3-15) a modified version of the quadratic eigenvalue problem is obtained: ([K] + A ([N1] + [To]) + A2[N2]){q {Aq} = {O}(3-16) } o in which [To] corresponds to the loading {P} = {P0}. Equation (3-16) defines the quadratic eigenvalue problem when tilting loads are considered. If matrix [N2] is neglected, the linear eigenvalue problem is: ([K] + A ([N1] + [TO])) {Aq} = {0} (3-17) {qo} 3.5 EFFECT OF OUT-OF-PLANE LATERAL LOAD ON IN-PLANE BUCKLING LOAD OF ARCHES A curved beam may be subjected to a combination of out-of-plane lateral load and in-plane load. For example, a rib of an arch bridge may be subjected to a horizontal wind load normal to the plane of the rib in addition to the in-plane gravity load. 37 The approach used herein to an analysis of this problem is as follows. The horizontal load {PH}is applied first. The corresponding displacement {qH}is obtained from: [K] {qH} = {PH} (3-18) Next the vertical load is applied and additional displace- ment {qv} would result. The magnitude of the vertical load is gradually increased until buckling takes place. Noting that [N1] is linear in {q}, Equation (3-1) may be written as (dropping the [N2] term): ([K] + [N1({q})]){Aq} = ([K] +[Nl({qH})]+ [N1({qv}] ){Aq} = {0} (3-19) Writing as previously Nl({qv}) = A Nl({q0}) (see Equation (3-3a)), the above equation may be written as ([Km] + A [N1({qo})]) {Aq} = {0} (3-20) in which [Km] = [K] + [Nl({qH})] (3-21) Thus the vertical buckling load can be calculated from Equation (3-20), the influence of the horizontal load being accounted for in [Km] as indicated by Equation (3-21). 3.6 COMPUTER PROGRAM An outline of the program developed for this study is presented in this section; the program itself is given in Appendix C. The major steps in the program are described in the same order in which they are executed: l) 2) 3) 38 The basic information concerning the physical description of the arch is input. This infor- mation includes the number of elements, the type of arch, the type of load, and the type of eigenvalue problem to be solved. The global coordinates of the nodes are also input with the parameters defining the boundary condi- tions of the arch. The element data is input. The input of ele- ment properties is general for prismatic members of any cross-section. These properties include the modulus of elasticity, shear modulus, den- sity of the material, area of the cross-section, moments of inertia about the two principal axes and the torsion constant of the cross-section. The program has been prepared such that separate properties for each element may be used if this is desired (other types of elements may be included in the structural system by providing additional subroutines for their stiffnesses). Next, the geometry of each curved element is defined, i.e., radius of curvature at each node, coordinates of the nodes in the element system. With these coordinates (x8, 23) the coefficients bi are computed. 4) 5) 6) 7) 8) 39 The applied loads are next input. Their orien- tation and type of loading (i.e., concentrated, uniformly distributed) was defined with the information input in 1. From the information input in l, the semiband- width of the structure stiffness matrix is com- puted. The element linear stiffness matrices are computed and assembled into the linear stiff- ness matrix of the structure. This matrix is assembled in banded format and due to symmetry only the upper semibandwidth is constructed. A linear analysis of the arch is performed to obtain displacements and forces due to the applied loads. The displacements so determined are used to compute, for each element, the matrices [nl] and [n2] which are assembled also in banded format into the structure incremental stiffness matrices [N1] and [N2]. From the information input in l the type of eigenvalue problem that is to be solved is defined. The lowest eigenvalue and its corresponding eigenvector' for the specified arch and loading conditions are computed and output. CHAPTER IV NUMERICAL RESULTS 4.1 GENERAL In this chapter a number of numerical examples of arch behavior were considered using the computer program which embodies the solution methods developed in the previous chapters. Initially a comparison of the finite element solutions of linear equilibrium problems was made with analytical solutions to test the reliability of the element. The program is then used to solve both in-plane and out-of-plane buckling problems for circular and parabolic arches under several loading conditions using formula- tions of both linear and quadratic eigenproblems. Next, the buckling loads of the deck type and through type para- bolic arches subjected to tilted loads are calculated. Comparisons of the numerical solutions obtained with avail- able analytical data are presented. The effect of an out- of-plane transverse loading on the in-plane buckling load is then considered. It should be noted that in all the numerical solu- tions presented in this chapter the degrees of freedom dw/ds and dB/ds have been included since the formulation 40 41 of the buckling problem does not allow condensation of any degrees of freedom. 4.2 LINEAR EQUILIBRIUM PROBLEMS Two types of problems were solved. They are linear equilibrium problems for a concentrated load at the crown applied in the plane of the arch or normal to that plane. The examples were used to verify the reliability of the element and to consider the effect of the number of ele- ments on the accuracy of the solution. 4.2.1 CONCENTRATED IN-PLANE LOAD (VERTICAL) AT CROWN The solution was obtained for two types of arches, circular and parabolic. In both cases the symmetry of the load and of the structure were used to reduce the number of equations. Figure 4-1 shows, for different~numbers of elements, for a semi-circular arch the difference between the computed displacement at the crown and the analytical solution. Two sets of data are shown in the figure. They differ in the treatment of the degree of freedom dw/ds at the support. In one case this d.o.f. is restrained, i.e., set equal to zero. For the other it is free, i.e., a cor- responding "equilibrium equation" is formally assembled and it would generally take on a value different from zero. It is seen that the two sets of data are the same except for cases including small numbers of elements. 42 Since dw/ds enters in the expression for axial strain (Equation (2-6)) it seems logical to let it be a free degree of freedom. The subsequent results presented here- in all correspond to this specification of the boundary condition. Table 4-1 shows the numerical data for this case. The data indicated that the differences with the analytical solution decrease rapidly with increase in the number of elements. However, at larger number of ele- ments (say, greater than 10) the difference increases (for example, to 2.5% at 18 elements). It was first thought that the reason might be the sensitivity of the geometry coefficients b2, b and b“ (Section 2.4.2). For a circu- 3. lar arch these coefficients should be zero but numerical calculations would sometimes produce non-zero values. However, when these coefficients were set equal to zero in the program, no difference in the results was observed. Thus, it appears that the sensitivity of these coefficients was not the cause. Such behavior would seem to be the result of the round-off errors accumulated from the increas- ing amount of computation as the number of elements was increased. It may be noted that the use of the semi-circular arch in this case may be considered as a stringent test on the convergence behavior because it is a "deep arch" (2). 43 For shallower arches, the convergence should be better, and this is illustrated in the following. Figure 4-2 (see also Table 4-2) shows similar pattern of results for a parabolic arch. It is observed that the solutions converge very rapidly from 0.61% for 2 elements to 0.0041% for 10 elements. However, as men- tioned before, the results for larger number of elements, say greater than 12, are not as good as those for lesser number of elements. 4.2.2 CONCENTRATED TRANSVERSE LOAD (HORIZONTAL) AT CROWN For this case, the solution was obtained only for the semi-circular arch as described in Figure 4-1. Figure 4-3 shows, for different numbers of elements, the differ- ence between the computed transverse displacement at the crown and the analytical solution. For 2 elements the difference is 2.5% but decreases rapidly as the number of elements is increased (0.37% for 10 elements). Table 4-3 contains the numerical values for this case. It may be mentioned in passing that the linear stiffness matrix, for this case of a circular beam sub- jected to out-of-plane loads, had been compared with that given in Reference (16) and found to be essentially the same . 44 4.3 BUCKLING PROBLEMS The computation of the buckling load was discussed in Chapter III. It was pointed out that the load may be computed from the solution of a quadratic eigenvalue prob- lem, Equation (3-4). If the quadratic term that involves the [N2] matrix is neglected, the buckling load may be computed from a linear eigenvalue problem, Equation (3-5). 4.3.1 LINEAR VERSUS QUADRATIC EIGENPROBLEM SOLUTIONS To study the-importance of the matrix [N2] in the solution of the eigenvalue problem, three types of problems were considered: first, the in-plane buckling of a 900 circular arch under a uniformly distributed radial load, second, the out-of-plane buckling of the same type of arch under the same loading, and third, the out-of-plane buckl- ing of a parabolic arch subjected to a uniformly distribut- ed vertical load. All three arches were simply supported. For the out-of-plane cases, the rotation degrees of free- dom about the x-axis (Figure 2-3) at the supports were restrained. Table 4-4 shows the values and the ratios of the critical load as computed from the linear eigenproblem to that from the quadratic eigenproblem. The number of ele- ments was kept constant at 12. It is seen that in each case the ratio is very close to unity. This would indicate that the inclusion of the matrix [N2] in the eigenvalue 45 problem would not give significantly different results from the case where it is neglected. Since much computer time can be saved by ignoring [N2], this has been done in obtaining the following results. 4 . 3 . 2 IN-PLANE BUCKLING To illustrate the effect of the number of elements on the computed values of the in-plane buckling loads, the critical load of a circular arch under a uniformly distri- buted radial load was calculated for different numbers of elements. Figure 4-4 shows the results in terms of percen- tage differences with respect to the analytical solution given in Reference (39). Table 4-5 lists the numerical values. The buckling mode of all solutions was that of antisymmetry or sidesway (Figure l-lb). It is seen from Figure 4-4 that the results "converge" rapidly. However, instead of converging to the analytical buckling load they converge to a value approximately 6% higher than the ana- lytical value. This discrepancy is thought to be due to the inherent difference of the methods that produced the results. This point will be discussed again in Chapter V. 4 . 3 . 3 OUT-OF-PLANE BUCKLING The effect of the number of elements on the out-of- plane buckling of a simply supported parabolic arch (with the rotation d.o.f. about x-axis at supports restrained) under a uniformly distributed vertical load was considered. 46 The critical loads in terms of percentage differences with respect to the analytical values given in Reference (24) are shown in Figure 4-5 (see also Table 4-6). In this case, the symmetry of both the geometry of the arch and loading were utilized to halve the number of degrees of freedom for the problem. This can be done because the buckling mode is symmetric. Again, as the number of elements is increased, a fast convergence is seen from 9.04% for 2 elements to 2.42% for 10 elements, beyond which some oscillation of the results was encountered. 4 . 4 BUCKLING OF PARABOLIC ARCHES SUBJECTED TO TILTED LOADS The analysis for the buckling load of arches under tilted loads was given in Section 3.4. The numerical data obtained for this study involved all three cases of arch ribs: deck, through, and half-through as illustrated in Figure 3-2. It may be seen that basically, in the case of buckling the columns and/or hangers rotate about the deck as the rib undergoes out-of-plane buckling. Table 4-7 shows for deck (HD = 0) and through arches, the values and the ratios of the buckling loads obtained in this research to the analytical values published in Ref- erence (38). It is seen that the agreement between the two sets of results are quite good. It should be noted that the formulation developed in Section 3.4 has an advantage 47 in that, for the deck arch, the deck is not required to be tangent at the crown of a symmetric arch (HD = 0, Figure 3-3). For the case of HD # 0 a lower critical load and a symmetric buckling mode (contrary to an antisymmetric mode for HD = 0) would be expected. This was verified by using one of the arches of Table 4-7 (type 1, 20 elements), but with HD = 3 in., for which the critical load was found to be only 54% of that when HD = 0. In addition, the analysis presented in Section 3.4 is also applicable to half-through arch ribs for which no analytical data is available. The buckling load for such a rib (type 1, 20 elements, HD = 2.4 in.) has been calcu- lated to be 325.81 lb/in. which, as expected, is in between the buckling loads of 151 lb/in. for the deck type (HD = 0) and 566.39 lb/in. for the through type. 4.5 EFFECT OF HORIZONTAL TRANSVERSE LOAD ON THE VERTICAL BUCKLING LOAD OF ARCHES The analysis for this loading case has been given in Section 3.5. Numerical data are presented here for a parabolic and a circular arch. Several levels of uniformly distributed transverse loads were considered. In each case a given level of transverse load was applied first, and then a uniformly distributed vertical load was added and the corresponding critical value of that vertical load was determined. 48 Figure 4.6 shows these results for a parabolic arch. The same data is also shown in Table 4-8. Since the dimensions of the arch considered correspond to those of a realistic arch bridge, it is of some interest to express the transverse load also in terms of wind velocities in mph. For low wind velocity the vertical buckling load is little affected. As the velocity increases the rate of re- duction in the vertical buckling load rapidly increases. A point is reached where the wind load would be enough to make the arch buckle in tension (at about 150 lb/ft). Sub- sequent increases in the wind load will result in negative vertical buckling load (upward) to keep the out-of-plane symmetrical buckled configuration. It must be noted, how- ever, in real arch bridges, the two ribs would be braced together and the response could be quite different from that of a single arch. Figure 4-7 shows similar kind of behavior for a cir- cular arch. Note again that increasing the transverse load rapidly decreases the vertical buckling load of the arch. Once again a point is reached (at about 0.1034 lb/in) at which buckling would be achieved with tension in the rib. As in the case of the parabolic arch, the buckling mode was out-of-plane and symmetric. CHAPTER V CONCLUSION 5.1 DISCUSSION In the preceding chapter results obtained using the finite element developed for this study were compared with those of analytical solutions. The differences were of the order of 6% which should be acceptable, at least for engin- eering design purposes. However, the fact that the numeri- cal results did not in all cases converge to the analytical solution should be considered. It should be noted that for all comparisons the ana- lytical solutions correspond to the classical theory of elastic stability (39), i.e., at the buckling load an equi- librium configuration exists adjacent to the original unde- formed configuration of the structure. In this sense, the elastic deformation prior to buckling is neglected. The numerical method used herein may be regarded as an offshoot from a nonlinear equilibrium analysis (27). That is, if one formulates the tangent stiffness of the non- linear structure (which is a function of the elastic defor- mation) and makes the assumption that the displacements increase linearly with the load, an eigenproblem for the 49 50 buckling load is obtained as described in Chapter III. This approach does not appear to be identical to the class- cal theory, and hence the results should not be expected to be exactly the same. Although the differences of the buckling loads calculated certainly are not excessive,in- asmuch as this method is an approximate one and is applied here for the first time, as far as is known to the writer, it should be used only with caution. Of course, like in most cases of finite elements, the method developed here may be used for problems intrac- table by analytical methods. The present method has the advantage of being able to represent practically any curved shape with prescribed curvatures as well as slopes at the ends. In addition the [N1] and [N2] matrices can be used for the more exact nonlinear equilibrium analysis to deter- mine the fundamental path (Figure 1-2). Also worthy of note is that the method can be applied to a structural system (including more than one type of element) in as straightforward a manner as in the linear structural analy- sis so far the assembling of the different matrices are concerned. A significant contribution of this work is the formulation of the tilted load problem as presented. It would produce buckling loads not only for the deck and through types arch ribs but the half-through type also. 51 Another feature of this study is the formulation of the problem of the interaction of horizontal and vertical loads on the stability of arches. Extension of this study should include an attempt to gain an in-depth understanding of the role of the matrices [N1] and [N2] as used herein. Further applications may include the stability of curved structures under different types of loading, the stability of ribs in half-through arch bridges, the stability of bridge systems (in which the ribs are braced together). More in-depth study should be made of the interaction of out-of-plane lateral load and in-plane load on the stability of arches. Finally, the more exact nonlinear equilibrium behavior may be studied using the [N1] and [N2] matrices developed herein. 5.2 SUMMARY In the preceding chapters, the development of a three dimensional beam element curved in a plane has been presented. The solution method has been embodied in a gen- eral computer program written in FORTRAN. The analysis uses the finite element method. A nonlinear formulation of the displacement-strain relations has been employed. Numerical integration was utilized to obtain the stiffness matrices of the curved element. The geometrically nonlinear effects are accounted for by the matrices [N1] and [N2] determined from the cubic and quartic parts of the strain energy expression, respectively. The linear equilibrium solution of circular and para- bolic arches under a concentrated vertical or horizontal transverse load at the crown was carried out by using the Gauss elimination procedure. Convergence, with respect to the number of elements used, was indicated when the results obtained with the computer program were compared with the analytical results. The applications of the computer program have also included the computation of the buckling load of arches sub- jected to different loading conditions. Increasing number of elements were used to obtain a better measure of the reliability of the element developed in this report and also to provide guidance for selecting the number of ele- ments to be used for later applications. It appears that effective solutions may be obtained with approximately 10 elements. Comparative studies between the quadratic and the linear eigenvalue problems were carried out. The solutions did not differ by more than 0.5% which indicated that the inclusion of the matrix [N2] in the eigenvalue problem would not give significantly different results from those obtained from the linear eigenvalue problem for which the matrix [N2] was neglected. The out-of-plane buckling of parabolic arches under tilted loads was also considered. Comparisons with data published for deck and through arch types in Reference (38) 53 indicated reasonably good agreement. While earlier results were limited to cases in which the deck was tangent to the crown, it was shown that the formulation developed in this 'investigation could be used to compute the buckling load for the more realistic case when the deck is not tangent to the crown (HD # 0, Figure 3-3), and also for the half- through type arch ribs. Results on the influence of a uniformly distributed transverse load (normal to plane of arch) on the vertical buckling load of a parabolic and a circular arch were pre- sented. It was found that while a small transverse load has little effect on the vertical buckling load, the latter decreases rapidly as the transverse load increases. 54 TABLE 4-1 LINEAR EQUILIBRIUM OF CIRCULAR ARCH SUBJECTED TO CONCENTRATED IN-PLANE LOAD AT CROWN NUMBER OF DISPLACEMENT DIFFERENCE** ELEMENTS AT CROWN* (%) (in.) 2 0.0137649 6.20 3 0.0135398 7.73 4 0.0143005 2.55 5 0.0143899 1.94 6 0.0146852 0.03 8 0.0146179 0.39 10 0.0146035 0.48 12 0.0149135 -l.63 15 0.0145392 0.91 18 0.0142818 2.67 20 0.0144516 1.52 * Analytical Solution ** % Difference = Analytical-Numerical 0.0146740 in. Analytical 55 TABLE 4-2 LINEAR EQUILIBRIUM OF PARABOLIC ARCH SUBJECTED TO CONCENTRATED IN-PLANE LOAD AT CROWN NUMBER OF DISPLACEMENT DIFFERENCE** ELEMENTS AT CROWN* (%) (in. x 10'“) 2 0.464243 '0.61 3 0.465878 0.26 4 0.466563 0.11 5 0.466852 0.049 6 0.466870 0.046 8 0.466928 0.034 10 0.467066 0.0041 12 0.467152 -0.014 16 0.469314 -0.48 20 0.464965 0.45 * Analytical Solution = 0.467085 x 10'“ in. ** % Difference = Analytical-Numerical Analytiéal 56 TABLE 4-3 LINEAR EQUILIBRIUM OF CIRCULAR ARCH SUBJECTED TO CONCENTRATED OUT-OF-PLANE LOAD AT CROWN NUMBER OF DISPLACEMENT DIFFERENCE** ELEMENTS AT CROWN* (%) (in.) 2 2.755249 2.50 3 2.783664 1.50 4 2.795598 1.07 5 2.801654 0.86 6 2.806454 0.69 8 2.811643 0.51 10 2.815347 0.37 12 2.812205 0.49 15 2.816225 0.34 16 2.824705 0.04 18 2.822199 0.13 20 2.820483 0.19 * Analytical Solution = 2.825930 in. ** % Difference = Analytical-Numerical Analytical Hmm 60H x mm "mm.sw we u A .CH m.m n m "nou< Omaonmumm awn koa u m .oca u a .GA on n m "monoud HmasouflOn mm mm.o hm.¢ma Hw.mma oo.m «(ca x mnmn.a om.H quHBmm> mzmAEIROIBDO UHAOm~ mmmm.a «lea x momm.o oo.H quamm mz¢qmlm0|eso mdqoomHU mm.o vo.am H¢.Hm «(ca x mmhm.o mica x monm.o mnma.o quadm mzmqmle mmqDUmHU 035 033833 5&qu in: fat Etc: 984 £92 1.5}: 98m .2355 EH cos c mo mm? mo mm? mZOHBDAOm qumommszHm UHBEQABO mammms ~35qu vlv mum»? TABLE 4-5 IN-PLANE BUCKLING OF CIRCULAR ARCH SUBJECTED TO UNIFORMLY DISTRIBUTED RADIAL LOAD NUMBER OF CRITICAL LOAD* ' DIFFERENCE** ELEMENTS (lb/in) (%) 3 41.55 -14.91 4 48.43 - 0.82 5 50.95 4.34 6 51.69 5.86 8 51.95 6.39 9 51.85 6.18 12 51.41 5.28 15 52.05 6.59 18 51.25 4.96 20 51.42 5.30 * Analytical Solution = 48.83 lb/in ** % Difference = Numerical-Analytical Analytical 59 TABLE 4-6 OUT-OF-PLANE BUCKLING OF PARABOLIC ARCH SUBJECTED TO UNIFORMLY DISTRIBUTED VERTICAL LOAD NUMBER OF CRITICAL LOAD* DIFFERENCE** ELEMENTS (lb/in) (%) 2 206.07 9.04 3 197.82 4.67 4 195.59 3.49 5 194.67 3.01 6 194.43 2.88 8 193.79 2.54 10 193.57 2.42 12 189.61 0.33 16 192.79 2.01 20 193.56 2.42 * Analytical Solution 188.99 lb/in ** % Difference = Numerical-Analytical Analytical 60 Hmnw 90H xGN " m .cH mmkm.o.uwx ..:H om.o..>sH . .EH om.o.uxxH .NcH enm~.m."« “m «axe .573613354 x 3.5 8.2"»? .1373 x EEHuxxH . .5 om.Hn< "H 6&8... em.o GMHG moon av m.m~ m cm mm.o GMHG mmnm mv m.m~ N 6H mo.H mo.emm mm.mom we m.m H om Go.H mm.vmm 6H.m6m m4 o.m H 6H mom< monomme mo.H eGmH «mom we m.m~ m cm mo.H AmmH meow we m.mm N 6H mo.H mm.mmH oo.HmH we G.m H om ~H.H m~.mmH ~m.mmH av m.m H 6H Ho n one moma xomo oHecm HHX ——lr- HALF-THROUGH ARCH BRIDGE Figure 3-2. 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N.00.200000 00 00<0 000<00<30 0>400 . 0.00.200000 00 00<0 0<0204 0>400 . 0.00.200000 00 2040000 034<>20000 0>400.ooo..oo UUUUUU 0 020300 0 020300 0002.0n0.002<02.0u3.03.00000 000.0000003 0~.00.2000000 00 0002.0n0.002<02.0n3.00.0000. «00.0000003 00.00.2000000 00 0002.0u0.002+02u0202 000 000 00 00 00.000 0000N.00000003 0202x0+00.000u00.000 00.000 00000.00000003 0202.2 00¢0N.00000003 020>I02u0202 000 000 00 00 00.00.200000 00 00 mm 00 me 0¢ mm 174 00.000.” 02 00.000." \\\\0.000. 020 .m0omm0.xmm.«+«00<2000 000.000.xmm.«+«00<2000 00.00." 020.N0.02 20.«0«00<2000 0\\0200204 223400 200 00 223400 20<0 20 0<04 200\\\\00<2000 20\\\N0.n 20000 2m\\\000000 2030020 200\\\\00<2000 02 20\\\N0.u 20000 2m\\\000000 2000 200\\\\00<2000 0\\ 00<0 0<04 000400 200.«0«00<2000 00.000.00.042000 00.0m0v0<2000 000m 0000 0¢0N 000m 0N0~ 000m 000m 0000 00 00 00 00 TILTED 100 175 TILTED LOAD SOLUTION FOR HALF-THROUGH ARCHES In this case statements 32 to 44 in the subroutine should be replaced by the following: H(N) = Y(N) - HD WRITE (61, 2040) N, H(N) IF (H(N). EQ. 0.) GO TO 100 S(I, l) = S(I, l) + P/H(N) CONTINUE IE5 "‘limitfijujllflll'flmfl”wfl7fl@”