._ . 3238.81-29.35 ~20¢ 2d 2-! UO¢~7~ v3». "30¢; to: NO<~ In 12¢ Illililflifiil’iliil'l‘lfiilmil'flfiflilililil'fiilfliiliilll. ‘ “53's 3 1293 10641 8514 ‘ "' "if fliespm fi‘riafifi’sa wmfivamfiay This is to certify that the thesis entitled NONLINEAR ELASTIC FRAME ANALYSIS BY FINITE ELEMENT presented by Jalil Rahimzadeh-Hanachi has been accepted towards fulfillment of the requirements for Ph.D. degree in C.E. - Structures RIC/M, Major professor Dr. R. Wen Date Jug 30; 1981 0-7639 1+«rh 5 ‘ 1 n” §u ; # ’ " at. 7“ , '5 ". “.I . b k ‘ Q 0 ‘~. l.‘ -:,r r ‘ ’ "j r " Q ‘5 , I Itiungmu‘ :1 /: . . . N f" [Ta-WNW“: r‘JQTSFTrI s. IV1:)I_J Piece in LJJx :3: Op to LJBRARJES remcvc i”?$ h~rvsur from w: you text”, 1:? M” be characc if :Jck is :3 { rki.’ Cape firm .5; 1 ,. . i . _ .__. _ . r ' l i [A Ku 1?” fiJ?“ 1; “-9 K11 7 ¢1E9‘7<" NONLINEAR ELASTIC FRAME ANALYSIS BY FINITE ELEMENT by Jalil Rahimzadeh-Hanachi 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 1981 ,/ / “ya ABSTRACT NONLINEAR ELASTIC FRAME ANALYSIS BY FINITE ELEMENT BY Jalil Rahimzadeh-Hanachi Several methods of analysis of nonlinear elastic framed structures are discussed. A method of analysis is defined to consist of three components: (a) a finite element model, (b) local coordinates (Eulerian or Lagrangian) for the element, and (c) a solution process. The finite element models are based on a linear longitudinal displace— ment function and a cubic transverse displacement function. However, two versions of the contribution to the axial strain by the transverse displacement are considered: one quartic and one constant. Both the Eulerian and Lagrangian coordinates are considered for the specification of the element local displacements. In addition, two versions are employed for the Lagrangian formulation: one with a fixed coordinate system and the other a moving (updated) coordinate system. Solution processes considered include the Newton-Raphson, the one-step Newton-Raphson and a straight incremental procedure. Past contributions are pointed out in the framework as outlined above. They include the works of Martin, Jennings, Mallet and Marcal, Powell, Holzer and Somers, Ebner and Ucciferro, Oran, Bathe, Akkoush, et a1., and others. The finite element results are compared among themselves and with the numerical solutions corresponding to the "exact" beam— column formulation. Jalil Rahimzadeh-Hanachi In addition, the identification of bifurcation loads is discussed. The formulation of eigenproblems and the accuracy of their solutions as estimates of bifurcation loads are also considered. Recognizing that in practical applications the number of members in a structure system is likely to be large, emphasis is placed on the effectiveness of using a single finite element to represent a beam (- column) member in a framed structure. In this regard, the results seem to indicate that a most effective method would be using the finite element with a constant axial strain (which of course includes the effects of trans- verse displacements), Lagrangian, fixed coordinates, and the Newton- Raphson algorithm. ACKNOWLEDGMENTS The investigation reported in this dissertation has been made possible with the support of many people to whom I am greatly indebted. I would like to express special appreciation for the advice and en- couragement of my dissertation advisor, Dr. Robert K. Wen, throughout the long months of research and writing. Thanks are also due to mem- bers of the writer's Guidance Committee: Dr. William Bradley, Dr. Charles Cutts, Dr. James Lubkin and Dr. Nicholas Altiero. The support and encouragement of Dr. William Taylor, Chairman of the Department of Civil Engineering, are sincerely appreciated. I would also like to express my appreciation to the National Science Foundation for their support of the research and to Nancy Hunt and Vicki Brannan for their dedicated typing. ii TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii LIST OF TABLES vii LIST OF FIGURES viii CHAPTER I. INTRODUCTION 1 1.1 BACKGROUND 1 1.1.1 WORKS BASED ON BEAM-COLUMN MODEL 2 1.1.2 WORKS BASED ON FINITE ELEMENT MODEL 3 1.2 OBJECTIVE AND SCOPE 6 1.3 NOTATION 8 II. FINITE ELEMENT MODELS 12 2.1 FINITE ELEMENT MODELS FOR THREE AND TWO DIMENSIONAL 12 BEAM ELEMENTS 2.1.1 GENERAL 12 2.1.2 STRAIN ENERGY OF THREE DIMENSIONAL BEAM 12 ELEMENTS BASED ON QUARTIC AXIAL STRAIN FUNCTION 2.1.3 STIFFNESS MATRICES OF A THREE DIMENSIONAL 17 BEAM ELEMENT 2.1.4 STIFFNESS MATRICES OF A TWO DIMENSIONAL BEAM 19 ELEMENT 2.1.5 STIFFNESS MATRICES BASED ON "AVERAGE AXIAL 19 STRAIN" 2.1.6 GLOBAL EQUILIBRIUM EQUATIONS 20 2.2 INITIAL STRAIN STIFFNESS MATRIX 22 2.2.1 GENERAL 22 2.2.2 INITIAL STRAIN STIFFNESS MATRIX BASED ON 24 QUARTIC AXIAL STRAIN ASSUMPTION 2.2.3 INITIAL STRAIN STIFFNESS MATRIX BASED ON 25 THE AVERAGE STRAIN ASSUMPTION 2.2.4 INITIAL STRAIN STIFFNESS MATRIX BASED ON 26 LONGITUDINAL DISPLACEMENTS ONLY 2.3 EIGENVALUE PROBLEMS FOR BUCKLING LOAD ANALYSIS 27 iii CHAPTER III. IV. METHODS OF SOLUTION 3.1 GENERAL 3.2 NEWTON-RAPHSON METHODS CONCEPT 3.2.4.1 3.2.4.2 3.2.4.3 WWW WAD) NEWTON-RAPHSON METHODS FOR UPDATED COORDINATES 3 2 1 3.2.2 NEWTON-RAPHSON METHODS FOR FIXED COORDINATES 3 2 3 3 2 4 CONVERGENCE CRITERIA GENERAL CONVERGENCE CHECK BASED ON UNBALANCED FORCE VECTOR CONVERGENCE CHECK BASED ON INCREMENTAL DISPLACEMENT VECTOR "ONE-STEP" NEWTON-RAPHSON METHOD "STRAIGHT INCREMENTAL" METHOD SOLUTION OF EIGENVALUE PROBLEMS 3.5.1 LINEAR EIGENVALUE PROBLEM 3.5.2 QUADRATIC EIGENVALUE PROBLEM 3.6 COMPUTER PROGRAMS 3.6.1 GENERAL 3.6.2 PROGRAMS FOR PROBLEMS IN LAGRANGIAN FORMULATION 3.6.2.1 3.6.2.2 PROGRAM NFRAL3D PROGRAM NFRALZD 3.6.3 PROGRAM NFRAE2D FOR TWO DIMENSIONAL PROBLEMS IN EULERIAN COORDINATES NUMERICAL RESULTS 4.1 GENERAL 4.2 NONLINEAR LOAD-DISPLACEMENT BEHAVIOR 4.2.1 "LARGE DISPLACEMENT" PROBLEMS 4.2.1.1 4.2.1.2 4.2.1.3 4.2.1.4 4.2.1.5 4.2.1.6 CANTILEVER BEAM WITH TWO LATERAL LOADS 4.2.1.1.1 COMPARISON OF RESULTS FROM DIFFERENT SOLUTIONS 4.2.1.1.2 COMPARISON OF MARTIN'S METHOD AND FEA-UPDATED METHOD . .1.1.3 CONVERGENCE CRITERION . . .1.4 COMPARISON OF ONE-STEP NEWTON-RAPHSON METHOD WITH STRAIGHT INCREMENTAL METHOD OF SOLUTION CANTILEVER BEAM WITH A SINGLE TIP LOAD CANTILEVER BEAM WITH BOTH LATERAL AND AXIAL LOAD CANTILEVER BEAM SUBJECTED TO END MOMENT CURVED BEAM SUBJECTED TO A LATERAL LOAD DISCUSSION iv Page 29 29 29 29 30 32 34 34 34 34 35 36 36 36 37 38 38 38 38 4O 4O 41 41 41 41 41 42 43 44 45 45 46 47 47 48 CHAPTER IV. 4.2.2 "SMALL DISPLACEMENT" PROBLEMS 4.2.2.1 ONE SPAN PORTAL FRAME 4.2.2.2 TWO STORY FRAME 4.2.2.3 TWO BAY FRAME 4.2.2.4 PLANE ARCH FRAME WITH HINGED SUPPORTS 4.2.2.5 SPACE ARCH FRAME 4.2.3 "INTERMEDIATE DISPLACEMENT" PROBLEMS 4.2.3.1 HINGED HALF CIRCULAR ARCH WITH A CONCENTRATED LOAD AT CROWN 4.2.3.2 CANTILEVER BEAM WITH TWO LATERAL LOADS 4.3 BUCKLING LOAD STUDIES 4.3.1 GENERAL 4.3.2 PROBLEMS INVOLVING SYMMETRIC LOADING 4.3.2.1 ONE SPAN PORTAL FRAME 4.3.2.2 ARCH PROBLEM WITH A CONCENTRATED LOAD AT CROWN 4.3.2.3 SPACE ARCH FRAME 4.3.3 PROBLEMS INVOLVING ASYMMETRIC LOADING 4.3.3.1 GENERAL 4.3.3.2 HORIZONTAL AND VERTICAL LOADING 4.3.3.2.1 ONE SPAN PORTAL FRAME 4.3.3.2.2 TWO BAY FRAME 4.3.3.3 ASYMMETRIC VERTICAL LOADING 4.3.3.3.1 ONE SPAN PORTAL FRAME SUBJECTED TO ASYMMETRIC VERTICAL LOADS 4.3.3.3.2 A 90°-ARCH SUBJECTED TO TWO VERTICAL LOADS 4.3.3.3.3 A HALF CIRCULAR ARCH SUBJECTED TO AN ASYMMETRIC LOADING V. DISCUSSION AND CONCLUSIONS 5.1 ASSESSMENT OF METHODS 5.2 CONCLUDING REMARKS TABLES FIGURES LIST OF REFERENCES APPENDICES A. MATRICES [k], [n1], AND [n2] A.1 [k] MATRIX A.1.1 THREE DIMENSIONAL A.1.2 TWO DIMENSIONAL Page 49 49 50 50 50 51 51 52 52 53 53 56 56 56 57 58 58 58 58 59 59 59 6O 6O 62 62 63 66 7O 99 102 102 102 103 A.2 [n1] MATRIX A.2.1 THREE DIMENSIONAL A.2.2 TWO DIMENSIONAL A.3 [n2] MATRIX A.3.1 THREE DIMENSIONAL BASED ON QUARTIC AXIAL STRAIN FUNCTION A.3.2 TWO DIMENSIONAL BASED ON QUARTIC AXIAL STRAIN FUNCTION A.3.3 THREE DIMENSIONAL BASED ON AVERAGE AXIAL STRAIN A.3.4 TWO DIMENSIONAL BASED ON AVERAGE AXIAL STRAIN [kE ] INITIAL STRAIN STIFFNESS MATRIX FOR QUARTIC AXIAL O STRAIN FUNCTION B.1 THREE DIMENSIONAL B.2 TWO DIMENSIONAL ] GEOMETRIC STIFFNESS MATRIX THREE DIMENSIONAL TWO DIMENSIONAL [kG] MATRIX 000:3 wtora & COMPUTER PROGRAMS D.1 DESCRIPTION OF SUBROUTINES VARIABLES USED IN THE COMPUTER PROGRAMS PROGRAM NFRAL3D PROGRAM NFRALZD PROGRAM NFRAE2D UUUU Ulh.w m vi Page 103 103 105 106 106 109 110 113 114 114 116 118 118 118 118 119 119 120 128 166 193 TABLE LIST OF TABLES Comparison of Solutions for Cantilever Beam with Two Lateral Loads Numerical Results for Cantilever Beam Subjected to Two Lateral Loads Comparison of Eigensolutions and Load-Displacement Results for a Symmetric Arch Subjected to Concentrated Load at Crown vii Page 66 68 69 FIGURE LIST OF FIGURES End Displacements of Three Dimensional Beam Elements Cross Section of Beam Element Configurations of a Two Dimensional Beam Element at Successive Load Increments in Updated-Lagrange Formulation Nonlinear Load Deflection Relation Newton-Raphson Iteration Determinant Search Method Comparison of Martin's Method and PEA-Updated Method Effect of Convergence Criterion One-Step-NR versus Straight Incremental Method Comparison of Solutions for Cantilever Beam with a Single Tip Load Comparison of Solutions for Cantilever Beam with Both Axial and Lateral Load Comparison of Solutions for Cantilever Beam Subjected to End Moment (v-component) Comparison of Solutions for Cantilever Beam Subjected to End Moment (u—Component) Comparison of Solutions for Cantilever Beam Subjected to End Moment (6 - Component) Comparison of Solutions for a Three Dimensional Beam Curved in Space Subjected to a Lateral Load Comparison of Solutions for a One Span Portal Frame Comparison of Solutions for a Two Story Frame viii Page 70 7O 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 FIGURE 4-12 4-l3(a) 4-l3(b) 4-l9(a) 4-19(b) Comparison of Solutions for a Two Bay Frame Comparison of Solutions for an Arch Frame (No Horizontal Load) Comparison of Solutions for an Arch Frame (With Horizontal Load) Properties of and Loading on a Space Arch Frame Comparison of Solutions for a Space Arch Frame Solutions of a Half Circular Arch Load-Determinant Relation for Different Cases of Nonlinear Behavior Load-Displacement and Stability of a Symmetrically Loaded Portal Frame Behavior of a 1350-Arch Subjected to a Concentrated Load at the Crown Variation of Determinant of[KT]with Load One Story Frame Subjected to Asymmetric Vertical Loads A 9OO-Arch Subjected to Two Vertical Loads A Half Circular Arch Subjected to an Asymmetric Loading ix Page 86 87 88 89 90 91 92 93 94 95 95 97 98 CHAPTER I INTRODUCTION 1.1 BACKGROUND In recent years the concept of basing structural design on ultimate strength has gained increasing acceptance. The computation of the ultimate strength of a structure would generally involve load- displacement relationships that are nonlinear. In other words, non- linear structural analysis becomes necessary. Nonlinear behavior of structures may be the result of two sources: (a) "material nonlinearity" such as a nonlinear stress- Strain relation, and (b) "geometric nonlinearity" which represents the effect of the distortion of the structure on its response. In the present study we Shall exclude the effects of "material nonlinearity" and consider only "geometric nonlinearity", although the exclusion of material nonlinearity would place a limitation on the direct application of the results to the design of many types of engineering structures. The study, however, represents a fundamental step. For slender structures such as suspension bridges and perhaps arch bridges, elastic nonlinearity is of direct concern. Nonlinear elastic analysis of framed structures has been the subject of investigation by a number of researchers. In the following review, past works will be discussed as two groups. One group considers the basic beam element as a continuum. The "exact" method would use the correct expression for the curvature of beams, and is called the "theory of elastica" (Timoshenko [l]*). But most works have been based on the use of an approximate expression for the curvature (equal to the second derivative of the lateral deflection), and the resulting theory is referred to as "beam-column theory." (Timoshenko [l], Bleich [2]). The second group consists of those works that use finite elements to model the members of a frame. It is instructive to note that, in addition to the basic element model discussed above, a method of analysis has two more attributes. The first is the local coordinate system used which could be either Eulerian or Lagrangian. In the "Eulerian coordinate formu- lation" local displacements are measured with respect to the chord of the deformed member, and in the "Lagrangian coordinate formulation" local displacements are measured with respect to the axis of the un- deformed member. The second attribute is the method of solution. At least four methods have been used: the method of "Direct Substitution", Newton-Raphson, One-step Newton—Raphson, and "Straight Incremental" (Cook [3], Haisler [4]). The last three methods of solution have been used here and are described in Chapter III. 1.1.1 WORKS BASED ON THE BEAM-COLUMN MODEL The essence of the beam-column theory is the inclusion of the effect of the axial force on the bending moment of a deflected beam. Discussion of this can be found in various monographs (Timoshenko [l], Bleich [2]). Detailed expressions representing the exact solution of the * Number in brackets refer to entries in the list of references. 3 beam-column problem were given by_Saafan [5]. He also derived a tangent stiffness matrix [6] for use in a Newton-Raphson method of solution. However, the effect of bowing (flexural deformation) on the axial Shortening was neglected in the tangent stiffness matrix. Conner, Logcher, and Chan [7] I using the principle of virtual displacement,developed the stiffness and tangent stiffness matrices in two and three dimensions. The method was based on fixed Lagrangian coordinates (the local coordinates are fixed, i.e., are not updated after each incremental loading). It is good only for deforma- tions involving small rotations. Methods of solution discussed in- cluded that of successive iteration,Newton-Raphson and the straight incremental method. Oran [8,9] formulated for both two and three dimensional problems the "exact" tangent stiffness matrix in "Eulerian coordinates". Subsequently, he and Kassimali [10] applied these matrices to obtain solutions to a number of numerical problems. The Newton-Raphson and straight incremental methods were used. Nonlinear load-displacement behavior as well as stability were discussed. The accuracy of the solutions was shown to be generally excellent even for very large displacements. 1.1.2 WORKS BASED ON FINITE ELEMENT MODEL Martin KELlpresented one of the earliest finite element formu- lation to deal with geometrically nonlinear problems. The method is one of incremental loading, using the well-known geometric stiffness matrix [12] based on Lagrangian coordinates and updating the geometry of the structure at every load increment. This is referred to herein as the updated-Lagrange coordinates. Although in his method of 4 solution there is no Check on equilibrium, it is a very efficient approach. Jennings [13] highlighted his study by including the bowing effect on the axial strain in the finite element formulation. He derived stiffness and tangent stiffness matrices based on Eulerian coordinates for plane frames. Consequently the expressions may be used for very large displacements. Mallett and Marcal [14] presented relationships between the strain energy, the total equilibrium and incremental equilibrium equa- tion in terms of the usual stiffness matrix and two (nonlinear) incre- mental stiffness matrices. Lagrange coordinates were used in the formulation. Expressions for the stiffness matrices for two dimensional beam elements were derived based on the usual cubic Shape function for the lateral displacement. Since the contribution of that displace- ment to the axial strain is in terms of the square of its derivative, this model is referred to as the "quartic axial strain model." They presented no numerical results. Powell [15], in illustrating a general discussion of the theory of nonlinear structures, presented the stiffness and incremental stiff- ness matrix for two dimensional beams. He adopted the same shape func- tions as those by Mallett and Marcal [14] but the stiffness matrices were derived in Eulerian coordinates. Akkoush, Toridis, Khozeimeh and Huang [16] used the concept of geometric stiffness matrix for a three dimensional beam model in updated- Lagrange coordinates. It is essentially a generalized version of Martin's method for space frames. The method was used to generate a com- plete load-displacement path to study post-buckling and post—limit load behavior. Hozler and Somers [17] developed a method for the study of the nonlinear response of reinforced concrete and steel plane frames up to collapse. Material nonlinearity was also considered. Their formula- tion was based on a minimization of the energy function defined by generalized coordinates and forces in an Eulerian coordinate formula- tion. Bathe and Bolourchi [18] developed the stiffness matrices for three dimensional beam elements subjected to large displacements and rotations for the application to elastic, elastic-plastic, static or dynamic analysis. Their formulation was quite rigorously based on the theory of continuum mechanics. A number of numerical results were given which will also be considered later in this thesis. A theoretical and numerical comparison of methods including those of Martin [11], Jennings [13], Mallett and Marcal [l4], and Powell [15] was undertaken by Ebner and Ucciferro [19]. The study was limited to two dimensional problems. They presented derivations of the stiffness matrices related to these methods from a common starting point and thus made more clear the similarities and differences among them. Part of the numerical results they obtained for comparison studies have also been reproduced and discussed here. Most of the previous studies have dealt with two dimensional problems and structures with a small number of members. Since in practice, three dimensional and larger systems are frequently in- volved, this thesis is an effort to consider this class of problems. The specific objectives and scope are discussed intjuafollowing section. 1.2 OBJECTIVES AND SCOPE The primary objective of this work is to search for an effective method for the elastic nonlinear analysis of three dimensional framed SPEEEEEEeSI With a view to eventually applying it to structural sys- tems consisting of a relatively large number of members. It was anti- cipated that the finite element model would be more efficient than the more accurate beam-column model; that a formulation based on the Lagrangian coordinates would be more efficient than onewbasednondtheuv ..._ '-~w. \——-. _, _.——-..w . .2 -~-- I ”g... - L.. - .R __-. _,_ -.-—o-'r-—‘"‘ Eulerian coordinates; and that_the_fixed-Lagrange formulation of_the solution would bemore efficient than theupdated-Lagrange one. These considerations led, in the initial phase of the work, to a development of the M & M model (Mallett and Marcal) referred to previously [14] for three dimensional beam elements. However, pre— liminary results indicated certain basic problems for this model. That is, for very slender members, it produced grossly inaccurate results. This motivated a comperative study of the other finite element models discussed previously. They include Martin [11], Powell [15], Jennings [13] as well as a new model [20] developed in the course of the present research of which this thesis is a part. The new model is based on an "average axial strain assumption“, i.e., the axial strain due to the lateral de- flection is averaged over the element as discussed in Chapter 2. It is herein referred to as the FEA (Finite Element Average) model. For the comparative studies, the beam—column model was used as a basis. In addition to studies of load-displacement relations, this study also included the formulation of eigenvalue problems, using finite element models, from which estimates of "bifurcation load” or "limit load" can be obtained. The scope of this report is thus as follows: 1) To prepare computer programs for two dimensional problems based on the following methods: the beam-column method, the M & M method, Jennings' method and Powell's method. 2) To develop the stiffness matrices for three dimensional beams based on the "quartic axial strain assumption”. 3) To develop computer programs for three and two dimensional problems based on the FEA formulation. 4) To formulate and solve eigenvalue problems in order to obtain estimates of bifurcation or limit loads. Both linear and quadratic eigenvalue problems were considered. 5) To obtain and compare numerical results, using the devel- oped programs. 6) To assess the relative merits of the various methods. In the course of the study, it was found appropriate to divide the problems into three categories: problems of "Small Displacements", "IntemeClisEePisplaszsments"v and "Ls:29...P.iPP}P£=_¢T?—ntS"~ "a” .. - .— The comparison indicated that, for "Large Displacement" prob— ‘fiuwfiw-n-M~-~w .. .- .. ,. _. ., lems,the method would have to be based on either Eulerian coordinates or updated-Lagrange coordinates. For "Small Displacement" problems (although still involving load-displacement relationships that are .v’wflmq.“ .- _.,.,._ quite nonlinear), the fixed-Lagrange formulation considered here is ‘no' “hm-,m y-qwq..« satisfactory (that is, both the M & M method and the FEA method). How- ever, for "Intermediate Displacement" problems, the FEA method still produces reliable results while the M & M method seems to fail. The results obtained from eigenvalue problem studies indicated that, with little primary or no bending, both linear and quadratic eigenvalue solutions agreed with results obtained from com- plete load-displacement solutions. For problems with substantial primary bending,IJJKRHTeigensolutions still generally produced accep- table results if the structure-load system is symmetric. For asym- metric systems the significance of the eigensolutions deteriorated. However, in some cases certain linear eigensolutions were shown to represent reasonable estimates for "limit loads." 1.3 NOTATI ON a1, 32, coo, 612 FEA Area of cross section; End nodes of an element; Parameters used for definition of shape functions; Young's modulus; Finite Element Average strain model; Shear modulus; Moment of inertia of cross section (Figure 2-2); Moment of inertia of cross section (Figure 2-2); Straight incremental; Torsional constant; Element and structural linear stiffness matrices; Element and structural initial strain stiffness matrices; 1 [n1*]: Axial load Length of element; M [n1]. [N1] [n1*], [N1*1 [n2]. [N2] & M NR {P AP C P B P N1+N2 {P {q {Q {Q } r C ref } } ref Mallett and Marcal's method; Element and structural first order nonlinear stiffness matrices; Element and structural first order geometric stiffness matrices; Element and structural second order nonlinear stiffness matrices; Newton-Raphson; External load vector; Load step (load increment); Axial load at the end of ith load increment in the element; Critical value of applied load; Critical load corresponding to Beam-Column solution; Critical load corresponding to eigenvalue solution (using N1); Critical load corresponding to eigenvalue solution (using N1*); Critical load corresponding to quadratic eigensolution; Reference external load vector; qur qZI ~o-1 q6: Q7: QB: ‘0'! quJTi (Element generalized displacement vector); Structural generalized displace- ment vector; Reference structural generalized displacement vector; Symbol for exact configuration of the structure; {AR}i R [S J 5 [ST] u, V, w ul, V1, w1 and 112! V2: W2 U TOTAL €+t U2! U3: U14» Vol. X,y,Z 10 Symbol for structural configura- tion at the ith iteration; Unbalanced force vector related to the ith iteration; Radius of circle; Structural secant stiffness matrix; Structural tangent stiffness matrix; Displacements along local x, y, z axes, respectively; Displacements for nodes 1 and 2 Of the beam element along x, y, z axes, respectively; Strain energy of the element; Initial strain energy of the element; Torsional strain energy of the element; Total strain energy of the element; U + U E t Quadratic, cubic and quartic parts of strain energy; Potential energy of external loads; Volume; Local coordinate axes; Global coordinate axes; Angle of opening of circular arch; Multiplier for asymmetric loading; Longitudinal strain; Initial strain at the beginning of ith load increment; Tolerance ratio for convergence check based on displacement variation; Tolerance for convergence check based on unbalanced force vector; Rotation about x, y, z axis, respectively; Rotation about x, y, z axis for nodes 1 and 2, respectively; Total potential energy; Chord rotations about 2 and y axis, respectively; Buckling load parameter; Incremental operator; Column vector; Row vector; Rectangular matrix; CHAPTER II FINITE ELEMENT MODELS 2.1 FINITE ELEMENT MODELS FOR THREE AND TWO DIMENSIONAL BEAM ELEMENTS 2.1.1 GENERAL In this chapter the strain-displacement relations for three and two dimensional beam elements are presented. Then the stiffness matrices (including the linear and nonlinear parts) are derived, and finally the equilibrium equations are written. 2.1.2 STRAIN ENERGY OF THREE DIMENSIONAL BEAM ELEMENTS BASED ON QUARTIC AXIAL STRAIN FUNCTION Consider a beam element in space as shown in Figure 2-1. The x-, y-, z-axes, a right-handed coordinate system, represent the local or member coordinates. The displacements and rotations corresponding to these axes are denoted by u, v, w and ¢, T, 9, respectively. The initial position of the element is AB. The length of AB is equal to 2. The displaced position is A181 and the projections of u u n u AlBl on the x-y plane and x—z plane are denoted by AlB1 and A1B1° It should be noted that the following assumptions have been used in our derivation. a) The material of the beam element is linearly elastic. b) Plane sections remain plane after deformation. c) The cross section of the beam is constant and has two axes of symmetry. d) The effect of torsional deformation on normal strain is negligible. 12 13 For a finite element analysis we assume linear shape functions for u and O and cubic Shape functions for v and w, i.e., u = a1+a2x _ 2 3 V — a3+aux+a5x +36X 3 w = a7+a8x+a9x2+a10x ¢ = a11+612X .-_ ..J [.rx %,n The boundary conditions are: at x=o. dv dw dx — 6" dx _ -W1, ¢ _ $1 at n=2 u — u2, v — v2, w = wz dv dw dX ‘ 62' dX — Ki121 ¢ — $2 (2-1) (2-2) Substituting Equation (2-1) into Equation (2-2), we obtain a system of linear equations for the unknowns a1, a2, ..., a12. Solving the equations and substituting the results back into Equation (2-1) we have V = V1+91X+%(-291-62+3e )X2+ '32- (61+62-28 )X3 0 R O 1 1 W = W1-W1X+E(2'y1+'¥2-3‘ijo)x2+ E72- (-'¥1-q’2+2wo)x3 ¢2-¢1 ¢ = $1 + g X (2-3) 14 in which _ V2“V1 eo — 2 and (2-4) _ -(W2-W1) lyo - 2 Following the usual beam theory assumption of plane sections remaining plane, the longitudinal strain at each point of the beam ele- ment may be written as: 2 2 dv dw C(XID:C) - €a(X) + D'a;§ + C 5;? (2-5) in which €a(x) is the axial strain at the centroid, and n and C are the coordinates of the point with respect to the principal axes of the cross section plane as shown in Figure 2-2. The axial strain at the centroid is du l dv 2 1 dw 2 =——+—— —— — €a(X) dx 2 (dx) + 2 (dx) (2 6) in which the last two terms represent the nonlinear effects of bending. Thus it is seen that when v and w are cubic functions of x, €a(x) is quartic. Using Equation (2-6), Equation (2-5) becomes: 2 2 2 2 du l dv l dw dv dw €(X.U,C) = a;'+ 5-(a;) +-§ (5;) + n 5;? + '5;§ (2-7) From equations (2-3) we obtain: 15 du _ Uz-ul dx 2 dv 2x 3x2 a; = 91 + E—2(—261—62+360) +‘Ef’ (91+92'290) dw _ 2x w 3x2 . d;'_ -T1 + E—'(2.1+W2-3WO) + E?- ('Vi-W2+2WO) (2‘8) 2:2 =.3 (-29 -e +36 ) +-§§ (6 +6 -28 ) dx2 2 1 2 o 22 1 2 o dzw _ 6X dx2 —'E (2Y1+W2-3w0) +IE? (-W1-W2+2WO) Using Equation (2-8), Equation (2-7) may be written as: 1 d 2 2 e(x.n,c) = a + 3'(b+%x+17x2) +-§ . .] = [—————J in which qi, qj represent the generalized coordinates such as u1, v1, ..., etc. It should be noted that [k] is the usual linear stiffness matrix, while [n1] and [n2] contain, respectively, linear and quadratic terms of the displacements. The calculations of [k], [n1], [n2] in Equation (2-16) are very lengthy, but straightforward. The intermediate computations are not presented here and expressions for each of the above matrices are given in Appendix A. It is of interest to note that if the terms containing rota- tional displacements are dropped from [n1], i.e., only terms involving the relative axial displacement (uz-ul) are kept, the resulting matrix is: _ AE +n—;7+dd V Cd {£1 del ( ) aZlUE _] (2-36) 1&t 1 = [(ike )m,n]= [qu: Sq The intermediate computations are not shown here. The major steps and final expressions for [R8 ] are given in Appendix 0 B for both the three and two dimensional cases. 25 2.2.3 INITIAL STRAIN STIFFNESS MATRIX BASED ON THE AVERAGE STRAIN ASSUMPTION In this case the contribution of each load increment to the initial strain is based on the previously-mentioned average strain assump- tion. Thus at the beginning of the ith increment, - 3 . i-l g U2 ‘U 9 (dV 2 dw 2 18o (X,C,U) = .2 [ 2 1 2210%_) dx+ 2210 ( &) dx 3=l dzv dzw + U dx2 + C dx2 (2-37) Since the right hand side of Equation (2-37) is independent of x by using Equations (2-37) and (2-7) in Equation (2-33) we have: 2 dw R d—") dang, Io (—) dx1+n——Y-+ P 2-u1 l U =EA'L[ “550(52- d2W R du l dv 2 l dw 2 d2v yo [dx+ 2 (3;) +3 (3;) + n -—dx2 + (1 (3:2) dx (2-38) . . l . . . in which UE finally can be shown as a function of generalized o coordinates. Similar to the previous case, by using Equation (2-36) we have: i i [kE ] = P [k6] (2-39) 0 in which - - 2 - p =EAZ [22f111+%(y_27vi)+%(32_2_w_1, —--2(261 -6192+2922 ) + 3% (2‘1’12-‘1’1‘112+2‘¥22 )] (2'40) 26 for three dimensional and 1-1 3 _ _ 2 p = EA_Z [3153* +~§ (VZQVS + 3%-<2elz-elez+2ezz)] <2-41) for two dimensional beam elements, and [kG] is shown in Appendix C. 2.2.4 INITIAL STRAIN STIFFNESS MATRIX BASED ON LONGITUDINAL DISPLACEMENTS ONLY The geometric stiffness matrix that appears in the literature cited previously (Martin [11] and Przemieniecki [12]) and given in Appendix C may be regarded as an initial strain stiffness matrix and derived as follows. In this case we take: - j . i-l le: (x.n.t;) = Z [—3———u gul] (2-42) 0 j=l Using Equation (2-42) and (2-7) in Equation (2-33) we have: i—l i _ twm flWlfllfii _ U60 — EA {jél [— ]}f: [dx+( 2 dx) +2(dx )] dx (2 43) By following a similar procedure: i _ i _ [REC] — P [kc] (2 44) in which i i-l j -u p = EA 2 [Big—L] <2-45) j=1 - 27 2.3 EIGENVALUE PROBLEMS FOR BUCKLING LOAD ANALYSIS In Section 2.1.6 we introduced the linear incremental equili- brium equation (Equation (2-29)). In the following section we are going to use these equations to formulate certain eigenvalue problems for the calculation of buckling loads. One usual way to evaluate the critical load of a structure is to set the incremental load vector {P} to null in Equation (2-29). This leads us to the following equation. ([K] + [N1] + [N2]) _{AQ} = {0} (2-46) {Q} For a buckling load analysis we look for a point ({5}, {5}) on the load displacement curve (Figure 2-4) which satisfies the above Equation (2-46). That {5} would be the buckling or critical load. The exact solution of (2-46) in general is complicated because of its nonlinear nature. But if we assume that the displacement of the structure is a linear function of applied loads just up to the point at which buckling occurs, then we have: {P H—I II [K] {Qref} (2-47) ref and } (2-48) {Qref} [KJ-l {Pr ef In (2-47) {Pref} is an arbitrary reference load vector. Since [N1] and [N2] are linear and quadratic functions of displacements, with {5} = A {pref}: 28 [N1({§})] [N1({Qref})]x (2-49) and [N2({§})] [N2({Qref})]A2 (2-50) in which A is a parameter. Since Equations (2-49) and (2-50) are supposed to be valid until buckling; we have [N1({§})] [N1({Qref})] Acr [N2({§})] 2 [N2({Qref})] Acr Thus Equation (2-46) can be written as: ([K] + Aer [N1] + Air [N2]){Q {A9} = {0} (2-51) ref Equation (2-51) is a quadratic eigenvalue equation. For sufficiently small displacements,matrix [N2] may be neglected and Equation (2—Sl) reduces to a linear eigenvalue equation: ([K] + A [N1]) {A9} = {0} (2-52) CI {Q ref Solution of Equation (2—51) or Equation (2-52) would yield }. A and, of course, the critical load vector is A {P cr cr ref CHAPTER III METHODS OF SOLUTION 3.1 GENERAL As mentioned previously, a method of analysis for the non- linear elastic behavior of framed structures may be regarded as con- sisting of three parts: (i) model, (ii) local coordinates, and (iii) method of solution. In the preceding chapter several finite element models have been formulated in Lagrange coordinates. In this chapter, the methods of solution that will be applied for the solution of these models are described. 3.2 NEWTON-RAPHSON METHOD 3 . 2 . 1 CONCEPT: Consider a structure subjected to a prefafined external load vector {P}. Let Q be symbolically the so called exact deformed con- figuration of the structure. If we assume an iterative process, and in the ith iteration the approximate configuration Qi is known, we are interested in improving Qi in such a way that it would get sufficiently close to Q. We write the load displacement relation as: {P}=={f(Q)} (3-1) using a first order Taylor series expansion about Qi we have: 3f {P} = {f(Qi)} + {3Q {Ag} j . c2i 1 29 30 in which, {f(Qi)} may be interpreted as representing the elastic re— 3f 35‘} sistance of the structure corresponding to Qi' and { as the tangent j Q. stiffness at Qi' Then the modification to Qi is: l '1 3f ‘1. {AQi} = {--% {P-f(Qi)} = f—-} iARi} 3Q. JQi in which,{ARi}is the "unbalanced force vector" at stage Qi. The modified displacement is: Q. 1+1 = Qi + A91 The process may be repeated until either AQi+k or ARi+k is sufficiently small. This process is graphically illustrated in Figure 3-1 for a one degree of freedom system. The preceding discussion was for the load applied as a single load increment. For many problems greater accuracy in the solution may be obtained by applying the load in increments (i.e., AP, ZAP, ..., etc.). For each increment the concept described previously applies, provided the stress state of the structure at the beginning of load increment is properly taken into account. At the beginning of the increment the geometry of structure may or may not be updated. Both cases are considered in the following sections. 3.2.2 NEWTON-RAPHSON METHODS FOR FIXED COORDINATES In this case the geometry of the structure is not updated. The steps of the calculation are as follows: 1) Set load increment (and check if the intended total load has been applied)( 2) 3) 4) 5) 6) 7) 8) 9) 10) 31 Form the structural tangent stiffness matrix as: Tangent stiffness matrix = [K] + [N1({Q})] + [N2({Q})] Solve for {AQ} from: {AQ} = [tangent stiffness matrix]-1{load increment vector} Add {AQ} to the latest {Q} to obtain a new {Q} If convergence check is based on displacement and {AQ} is sufficiently small, return to 1. Based on the new {Q} from step 4 evaluate N1({Q}) and N2({Q}). Form the tangent and secant stiffness matrices and resis- tance force vector as: Tangent stiffness matrix = [K] + [N1({Q})] + [N2({Q})] Secant stiffness matrix [K] +§tul1+§ mm] Resistance force vector - [Secant stiffness matrix] x {Q} Evaluate the unbalanced force vector as: Unbalanced force vector = Increment load vector - Resistance force vector. If convergence check is based on unbalanced force vector and it is sufficiently small, return to 1. Return to 2 but use the unbalanced force vector for the load increment vector. 32 3.2.3 NEWTON-RAPHSON METHOD FOR UPDATED COORDINATES This procedure is to be used to implement the theory as dis- cussed in Section 3.2.1. The loads are applied in increments. At the end of each increment the geometry of structure is updated. In addition to the usual stiffness matrices [k], [n1], [n2] there is the initial strain matrix (resulting from initial strain energy) as ex- plained previously. The steps of calculation are as follows: 1) 2) 3) 4) 5) Set load increment (and check if the intended total load has been applied). Determine the most up-to-date geometry of the structure by using the latest joint displacements, and update the linear stiffness matrix. Form the tangent stiffness matrix according to one of the following cases: a) b) For the first load increment: Tangent stiffness matrix = [K] + [N1({Q})] + [N2({Q})] For other load increments: Tangent stiffness matrix = [K] + D<€ ] + [N1({Q})] o + [N2({Q})] in which [KE ] is the initial strain stiffness matrix. 0 Solve for {AQ} from: {AQ} = [tangent stiffness matrix]'1{load increment vector} If convergence check is based on displacement and {AQ} is sufficiently small, return to l. 6) 7) 8) 9) 10) ll) 33 Add {AQ} to the latest {Q} to obtain a new {Q}. Based on the new {Q} evaluate [N1({Q})] and [N2({Q})]. Form tangent and secant stiffness matrices and resistance force vector as: a) For the first load increment: Tangent stiffness matrix = [K] + [N1({Q})] + [N2({Q})] Secant stiffness matrix = [K] + é-[N1({Q})] + é-x [N2({Q})] b) For other load increments: Tangent stiffness matrix = [K] + [K€ ] + [N1({Q})] o + [N2({Q})] Secant stiffness matrix = [K] + fi<€ ]+%5[N1({Q})] O L l + 3' [N2({Q})] Resistance force vector = [Secant stiffness matrix]*{Q} Evaluate unbalanced force vector from: Unbalanced force vector = Incremental load vector - Resistance force vector If convergence check is based on unbalanced force vector and it is sufficiently small, return to 1. Return to 4 but use the unbalanced force vector as the load increment vector. 34 3.2.4 CONVERGENCE CRITERIA 3.2.4.1 GENERAL In implementing the above Newton-Raphson method a convergence criterion is needed. In this report, two convergence criteria have been used. The first one is based on the unbalanced force vector and the second one is based on the incremental displacement vector. 3.2.4.2 CONVERGENCE CHECK BASED ON UNBALANCED FORCE VECTOR In this type of convergence check, a reasonable tolerance (which has the unit of force or moment) is prescribed first for each group of components (i.e., force or moment)of the unbalanced force vector. After the evaluation of the unbalanced force vector in each iteration the absolute value of each component of the vector is in— dependently compared with the prescribed tolerance. Convergence is considered achieved if, for each of the components, this absolute value is less than or equal to the tolerance. The feature of this convergence criterion is that it represents a real test of the equilibrium of the structure and it is an absolute check. The tolerance for this convergence criterion is denoted by E f times unit force or unit moment. 3.2.4.3 CONVERGENCE CHECK BASED ON INCREMENTAL DISPLACEMENT VECTOR In the displacement convergence check used herein, for each group of displacement components (i.e., translations or rotations) a reasonable tolerance ratio is defined. If we denote the incremental displacement vector by {Ax} and the total displacement vector by {x}, convergence is considered achieved if for both groups the following is 35 simultaneously satisfied. 1: 2 Zi (Axi) [.W] 4 Tolerance ratio = Ed i i in which i varies from 1 to the number of translation or rotation com- ponents of the displacement vector, and Ed is the tolerance ratio. It should be noted that this convergence criterion does not directly deal with equilibrium of the structure. Furthermore, its absolute tolerance would decrease as the total displacement increases. A comparison of the use of the two convergence criteria will be presented in Chapter IV on numerical results. 3.3 "ONE-STEP" NEWTON-RAPHSON METHOD This approach in general is the same as what was described in Section 3.2. The only difference is that we do not iterate more than once for each load increment. Thus there is no convergence check. Obviously the advantage of this approach, when compared to the Newton-Raphson method presented previously,is that it takes less computation. It should be noted that whenever this method or the straight incremental method (as described in next section) is used for beam- column models, the iteration process on the axial load of each element should be continued until convergence is satisfied. 3.4 "STRAIGHT INCREMENTAL" METHOD This approach is the same as the One-Step Newton-Raphson method except that not even one iteration would be used. Hence, there is no need to evaluate the secant stiffness matrix, resistance and unbalanced force vectors. Obviously the accuracy of this method would 36 depend on the size of the load increment more than the previously mentioned methods. 3.5 SOLUTION OF EIGENVALUE PROBLEMS 3.5.1 LINEAR EIGENVALUE PROBLEM In this report the inverse vector iteration technique as des- cribed by Bathe and Wilson [21] is used for solutions of the linear eigen- value problems.The technique may be regarded as a mathematical formula- tion of the Stodola method [22] in structural mechanics. The basic equation (2-52) could be written as: Aq = ABq (3—2) in which for simplicity symbols ([ ] and { }) have been dropped and A = [K], B = -[N1]. It is assumed that A is positive definite and B may be a diagonal matrix with or without zero diagonal terms. The technique used for computer implementation is as follows: (a) Start with a trial vector 31 for the first eigenvector T q, (XlBQI# 0.) (b) For i=1, 2, ..., etc. evaluate Axi+l = yi y1+1 = Bxi+l - _ i+1 i 0(xi+1) — _ T _ (3 3) x y. 37 in which 0 is the Rayleigh quotient (c) The iterative process is considered to have converged if: p(xi+l)-p(xi) _ ‘ Epsi (3-4) p(X;L+1) -AS epsi in Equation (3-4) should be less than or equal to 10 if the answer is required to be accurate up to ZS digits. If Equation (3-4) is satisfied for i=n the smallest eigenvalue will be taken to be: A = 0(X ) (3-5) and the corresponding eigenvector is: Xn+l n GP 37 )1. n+1 n+1 The computer implementation of this technique (Ref. [23]) is contained in the subroutine EIGENVL listed in Appendix D. 3.5.2 QUADRATIC EIGENVALUE PROBLEM Using Equation (2-51) the solution for a quadratic eigenvalue equation may be obtained by finding A for which: detl [K] +)([N1] + A2 [N2] | = o (3-7) {Qref} Since we are looking for the lowest buckling mode the smallest value of 38 A is required. The solution is carried out by evaluating the left-hand side of equation (3-7)1mfi1m;increasing values of A, starting from zero with small increments as shown in Figure 3-2. If for AA.det (AA)> 0, but for AB = AA + A),P1 In Table 4-3 are shown the numerical results for the above arch as well as arches with a 90° and 180° Opening angles. Again the values of P are very close to P . The difference between P and P * N1 BC N1 N1 seems to increase with the opening angle, i.e., the comparison improves in the case of the 90°-arch but deteriorates with the 180°-arch. The behavior of the load displacement and load-determinant curves for the 90° and 180° angles are similar to the ones presented for the 135° angle. The buckling modes corresponding to PN * and PN l l for all cases in this example are antisymmetric. 4.3.2.3 SPACE ARCH FRAME The finite element eigensolutions of the three dimensional space frame as described in Figure 4—14 have been obtained. For these calculations the horizontal load Q was set equal to zero. PN was found to be 87 kips which corresponds to a lateral 1 buckling with a mode shape which is antisymmetric and normal to the planes of either arch ribs. The same solution was obtained using the quadratic eigenproblem formulation. From the load-displacement curves plotted in Figure 4-15 it may be noted that the eigensolutions represent a good estimate of the limit load of the system. In fact, in the FEA-fixed solution the determinant of the tangent stiffness matrix did change sign (vanish) at P = 89.7 kips. 58 4.3.3 PROBLEMS INVOLVING ASYMMETRIC LOADING 4.3.3.1 GENERAL For this class of problems no bifurcation load exists in beam- column theory. However, eigenproblems may still be formulated using fi— nite element models. It is of interest to study the significance (or lack of it) of their solutions. Again judgment should be based on a comparison with the load-displacement, load-determinant curves of the beam-col-NR method. 4.3.3.2 HORIZONTAL AND VERTICAL LOADING 4.3.3.2.1 ONE SPAN PORTAL FRAME This problem involves a square portal frame subjected to two vertical loads and a small horizontal load. It has been considered previously (for nonlinear load-displacement behavior study) in Section 4.3.2.1. The load-displacement curves are shown in Figure 4-10. As expected,det [KT] did not vanish in the beam-col-NR solution (i.e., there is no bifurcation load). The load determinant plot belongs to Type (C) in Figure 4—17. However, solutions for all four types of eigenproblems have been obtained. They are as follows: = . I * = 4 . = 4 . PN1 4759 kips PN1 758 kips, PNl+N20 764 kips and = 47 9 k' . PNl+N2A 5 ips We can also note from the load-displacement curves shown in Figure 4-10 that any of these critical load values may be regarded as a 59 good estimate of the limit load of the system. This is not unexpected because the horizontal load is very small and the exact bifurcation load for this frame (in1flmaabsence of the horizontal load) is 4750 kips [7],which is extremely close to all the critical loads obtained from the eigensolutions. 4.3.3.2.2 TWO BAY FRAME This system is similar to the previous one, but larger. The load—displacement curve is shown in Figure 4-12. The following eigen- solutions have been obtained. :44 " *= ' = ° PN1 9 0 kips PNl 4940 kips and PN1+N2A 4939 kips From the load-displacement curve it is seen that these results may be regarded as a limit load. By checking the det [KT] we obtained: PBC = 4965 kips, and from the FEA-updated solution, the critical load is 5028 kips. Both of these are very close to the eigensolutions given above. 4.3.3.3 ASYMMETRIC VERTICAL LOADING 4.3.3.3.1 ONE SPAN PORTAL FRAME SUBJECTED TO ASYMMETRIC VERTICAL LOADS In Figure 4-20 the load displacement plots are shown for four cases of asymmetric vertical loading as illustrated therein. It should be noted that for clarity the curves begin at different points on the displacement axis, and for purposes of comparison the case of symmetric loading has been replotted from Figure 4-18. For asymmetric loading, no bifurcation load is expected. Indeed 60 the load-displacement plots obtained from the beam-col-NR solution (not shown) belong to Type (C) of Figure 4-17. As before, the quadratic eigensolutions did not converge, but the linear eigenvalue solutions have been obtained and marked on Figure 4-20. It is seen that at PN * the structure has gone substantially 1 into the nonlinear range, while at PN the displacements would appear to l begin their higher rate of increase. It would seem that one could use P or the average of P and P * as an index of a "limit load" N1 N1 N1 of the structure-load system. 4.3.3.3.2 A 90°-ARCH SUBJECTED TO TWO VERTICAL LOADS Similar to the case considered in the preceding example.the load-displacement curves for an arch with an opening angle equal to 90° (approximated by four elements), subjected to two symmetrically placed loads,are shown in Figure 4-21. Again, except for the case of two equal loads, there is no bifurcation load for the system considered. The load-displacement curves were terminated at points beyond which the number of cycles of iteration for convergence increased drastically and the converged re- sults did not appear physically reasonable. As before, the finite element linear eigenvalue solutions have been noted. It is of interest to note that while PN] values would * values noted along the 1 load axis were too high to be of any significance. provide a rough measure of the "limit load? PN 4.3.3.3.3 A HALF CIRCULAR ARCH SUBJECTED TO AN ASYMMETRIC LOADING The geometry of the structure and the load-displacement curves 61 are shown in Figure 4-22. It may be seen that the behavior is highly nonlinear. At P = 5.2 lbs., the arch "snaps" and it is so grossly distorted that part of it now lies below the chord. The values of PN and PN * were found to be equal to .94 lbs. 1 l and 68.70 lbs., reSpectively. In this case PN * is totally meaningless, and PN is too small 1 l to be of significance. CHAPTER V DISCUSSION AND CONCLUSIONS A comparative study involving a number of existing and some new methods was presented intimapreceding chapter. From the numerical re- sults obtained an assessment of the methods may be given as follows: 5.1 ASSESSMENT OF METHODS l) Martin's method [11], a "straight incremental" type, is very efficient and generally quite accurate for all types of problems. This method is very sensitive to the step size, but gives good results even with very few elements (even one element per beam or column). Un— fortunately, there is no equilibrium check (or convergence check of any kind) involved in his procedure. The only way to judge the results is by comparing them with known accurate solutions or by decreasing the step size and/or increasing the number of elements until a pattern of converging results emerges. 2) Jennings' formulation [13], when used with the Newton- Raphson procedure, produces very good results for all classes of problems with a small number of steps and a small number of elements. Because of the Eulerian formulation it requires coordinate transformation in every iteration, which tends to be time consuming. The straight incremental version is very sensitive to the step size and number of elements. It is ineffective for "large displacement" problems. 3) Mallett and Marcal's method [14], based on fixed-Lagrange coordinates, is effective for "small displacement" problems. It requires 62 63 small numbers of elements and number of load steps for an acceptable solu- tion. However, it is totally inaccurate for "large displacement";problem. The version based on fiuaupdated-Lagrange coordinates developed here did not result in any improvement. 4) Powell's method [15] is good only for "small displacement" problems. In general it.hsvery sensitive to the step size and number of elements both in the iterative and straight incremental versions. The method is not efficient because of the Eulerian formulation. 5) Based on continuum mechanics, Bathe's formulation [18] of three dimensional beam finite element should be very accurate. However, comparison indicated that simpler models used here with less computa- tion requirements can produce results that are of the same order of accuracy. 6) The FEA-updated and FEA-fixed methods which have been con- sidered throughout this study are quite efficient. For the "large dis- placement" problems, the FEA-updated method is competitive in accuracy with all approaches used above. They are not very sensitive to the step size and number of elements. In general,for the framed structures considered, one element per beam or column was enough. The FEA-fixed method is even more efficient as it involves no coordinate transformation. However, it should be used only for "small and intermediate displacement" problems. 5.2 CONCLUDING REMARKS The objective of this study as stated in Chapter I was to search for an effective method which could be used for nonlinear behavior study of relatively large space frames. From the preceding chapters, it seems 64 evident that there is no single method that is most useful for all structural load systems. The choice would depend on whether the displacemnet is "large", "small" or "intermediate". For "small displacement" (say of the order of 2% or less of the length of a typical member), it appears that the FEA-fixed is attractive. For "intermediate displacements" (approximately 2—15% of the length of a member), the FEA-fixed method is still good. For "large displacements" (over 15% of member length) a number of methods are effective. They include the Jennings' method [13] and Martin's method [11]. However, the FEAsupdated method is com- petitive with these two. It is appropriate to comment on the continuum beam-column method which has been used as a reference throughout this study. When using Oran's tangent stiffness matrix [10], this method is quite efficient as far as a continuum model goes. However, it is less efficient than the finite element models because it requires iteration for the calculation of the axial force and computations involving transcendental functions. Of course, it also wouldlxadifficult to extend the formulation to ele- ments that do not have a constant cross section. In lieu of a complete but time consuming load—displacement analysis, some index values useful for engineering purposes may be ob- tained by the solution of eigenvalue probelms. From the eigenproblems considered here, it appears that solutions of the quadratic eigen- ‘problems have little merit intflmzsense that whenever they are meaningful they are also very close to the 50111121011 of the simpler linear eigen- problems. 65 For the latter eigenproblems, the use of the usual geometrical stiffness matrix provides good results (PN1*) for classical problems of elastic stability, i.e., problems involving little or no primary bending. The use of the first order incremental stiffness matrix, N1, in the eigenproblems overemphasizes bending effects in the system. How- ever, it appears that in the same cases the critical load PN1 thus ob- tained could be taken as a rough estimate of the "limit load". This possibility seems to deserve further study. As mentioned previously, the present study is limited to geo- metric nonlinearity. For many practical problems, when geometric non- linearity becomes significant, effects of material nonlinearity would become important at the same time. Thus, future studies of finite element analysis of frame structures should include these effects. 66 mv.vv mv.HH am.oe ov.m OOH om Hoe; oceum.Hemzom a mH.m mo. mo.m Ho. om m Hm.mm om.a vo.oe mv.e om on mmeu mznm.eamzoa o Aam.eocve.em AHA.0mcHo.mH Amm.qmcoo.om Aoo.mcvm.e ooa om mafia ocHI.mmc«c:mo m Amm.oHme.o 14m.cmo. Amo.mvme.m 160.3Ho. om m ma.om mo.om mo.qm mo.m om m Hoe; azu.mmcaccmo v mm.mm no.0m om.mm mm.a om m HOHH oceueoonsmmm m ma.oo am.om mo.vm vo.m om m HOHH mzneoousmmm m mm.b© Ho.Hm vH.mm mm.w I I mvmu MUAummHm H A.cev 6 .ud i.eev o .ua A.cec m .um i.eev m .um .ocH .EmHm eosuwz 30m .Hemo .usm> .Hmmo .Nauom .Humo .usm> .Hmwo .Neuom mo .02 mo .02 mpmoq Hmnmumq 039 zuHB Emom um>waflucmu now mcoflusaom mo 20mflummEOO div mqmde 67 oa.me He.mm ma.am me.m cm m omumomsuamm NH mm.aee Hm.mn ma.mm mm.me om m 6mxemu .wamu .NHHO$ .HMOU .uuwxw .Hmwmu .NflHOE MO .02 MO .02 Aomsceucooc Hue mamas 68 cowumuom new 0 ‘0 #m COHUOOHMOQ HMHOHMQ > .0 am cowuowammo Hosapsufimaoq u 5 mOHO.I whom.w mvoo.l NmHN.I movm.mHI mmmm.Hl N0HN.I momm.le VMBN.HI mOHH. mmoo.l mmqm.l wmoo.l HNmH.I NHom.NHI mvmm.l thH.l vovm.NHl mmom.l mmmo. Nmoo.l vmwh.l mmoo.l omva.l ommm.oal ammo.l mvva.l bm©N.OHI Nwoo.l mmho. mmoo.l OHNB.I mmoo.l mmHH.! mmom.ml womm.l mMHH.I mono.ml HVBM.I mmmo. 0500.1 hmmo.l NNOO.I mmmo.l mvmm.ml thH.I NNmO.I Hth.ml mva.I mmvo. ¢®OO.I mmmm.l mHOO.I hmVO.I mmHm.ml OHho.l mmvo.l OOHm.MI 5050.! mmmo. mvoo.l wavm.l mooo.l mOHO.I thH.HI mhoo.l mmao.l thH.HI thO.I mmoo. m > s o > s o > s vase @mxam I 2 w 2 boxAMIflmm mZIHOOIEmmm Hue mucosa CH ozonm mm mpmoq Hmumumq 039 on pouommodm Emmm uw>mHHUCMU Mom muasmwm Hmoflumssz va mqmdfi 69 TABLE 4-3 Comparison of Eigensolutions and Load-Determinant Results for a Symmetric Arch Subjected to Concentrated Load at Crown E = 10000. psi A = .1875 in2 I = .008789 in“ R = 10. in 4 equal elements P * p a PBC PN * PN le PNl 1 1 BC BC 0 90 12.70 13.47 10.23 1.0606 .8055 O 135 8.56 8.56 4.33 1.00 .5058 O 180 4.71 5.62 2.06 1,1932 .3665 7O 111 z,w FIGURE 2-1 End Displacements of Three Dimensional Beam Element L FIGURE 2-2 Cross Section of Beam Element 71 x. 1+1 o ' X FIGURE 2-3 Configuration of a Two Dimensional Beam Element at Successive Load Increments in Updated-Lagrange Formulation 72 Load {) Unstable -\ Bifurcation Load Stable Fundamental Path Displacement FIGURE 2-4 Nonlinear Load Deflection Relation 73 Load [ Assumed Solution \\__ Converged Solution P—- [- o----——--- J —'1 --------—------. g V tO P. Q AQ i+l Deformation i J n— r I FIGURE 3-1 Newton-Raphson Iteration 74 12L 2 + TE 4+ T; _ A). FIGURE 3-2 Determinant Search Method 75 oosumz omumodousmm 6cm cosuwz m.:euumz mo comfiummeoo Hue mmoon AnUCHV oak one um newswomammflo HMOfluum> .om .om .on .00 .ov .0m .0m .0H 0.0 _ _ p _ _ . . 1 H. mamum 0H .m.ceuumz n mo 1 m. mmmum m .Ho. H mm .pmumeSIémm n so mmmum om .m.cfluumz n mu 1 m. mmmum m .Ho. H mm .UmuMUQSIfimm M NO Abomxwv Ho. u to .mzlaoolemmo n HO 1 v. 1 m. :CH nwaooo. u H 1 0. «CA N. n d ewe .OH x .oM u m u a. A ll 2 o = o. m we om m mm T m. m m m0 NU fl . :U MU m 1. m m . m . pa mm H DH mm Amoav m rum pmoq 76 coflwmufluu wocwmum>cou mo uommmm mlv mmeHm Arcane may 050 um pamEmomHmmHo Hmueuum> .00 .00 .oo .00 .0m .ov .0m .0m .0H 0.0 _ _ C — _ _ p e » .lH. IN. lm. mcoHuDHOm Ham Mow mmoum 0N H0. H Um ..Eme o .mZIHooIEmmn n 60 v. H0. H mm ..Emam v .Uwumpmslemm n mu Ho. u no ..Eme m .6mbmcdsuame n :o m. H00. H Um ..Emam v .mZIHooIEmmo n no . I U .. . I oonEmm u N H0 I m Edam v mz H n O :CH owaooo. n I0. Roomxmv H0. n Um..EmHm m .mZIHooIEmmo u HU . «CA N u ewe woe x cm u up. v 1 =mo.0m :m0.mm m. m we 5 m m H m N O O O O O O m :0. m . w . DH mm H DH mm Ambav Wm um Umoq 77 HmucwfimuocH ocmflmuum msmum> mZImmumlmco mlv mmDOHm Acucev one 0:0 pm unmewomommac Hmoeun0> .00 .00 .on .ov .om .om .0H 0.0 b P L L p . _ I H. 4N. I_m. rv. mcofluDHOw Ham Now poms mucmewaw m 0cm madam 0N m. OCHIHOUIEmwo n mu I 00 m .I 001 o n N u H H Em o O :cfl ..0. Abomxwv mZIHOUIEmmo n "O hoaooo. CA N. N I h. Hmm 80H x cm 1 . ] . 1 1m. :No 0m =m0 mm mo Ho No 0 m [w . . m l. m . . Amnev m 78 CMOH 0H9 0H0CHm m nqu Emmm nw>mHHucmo MOM mcoHusHom mo cOmHummEoo vlv mmDOHm AsosHv m u< coHuomHmoQ HmoHusm> .00 .00 .0» .00 .om .ov .om .0N .0H 0.0 h m _ _ — [ b m _ mcoHuoHom HHm I H. Mow pom: bcmeHw H can mmwum 0m Ho. u 00.60066091amm n :o m.CHuumE n mu Ho. n 00.6mumcmsnz a z n «0 N. Abomxmv MZIHOOIEmmo n HO 1 m . \ \ :CH .H H H \\ NQH .00H n d I v. N0 M0 .0 H0 Hmm .OOOH n m :.00H m m ws AmnHv rm HBOH 79 meH HmumUMH .oh UCM HMflXHN £00m SDHS Emmm H0>0HHUCMU MOM mCOflUDHOW MO COmHHMQEOU mlv EDUHW “SUCHV m on COHuomHmoo HMUHun> .00 .om .ov .0m .0m .0H 0.0 _ _ H _ r t a 1.0m ucwEme H n.0v mme .m u mNHm mmum Ho. n 60 .cmumcosusmm n No H0. u to .mZIHoousme n HO 52H .0H n H 1.06 NU NCH Ho ”AN no me 0000m u m ll? wees .Ha n 0 Ho 7) .111 1.0m =.00H m H Al WAN # mmHM 0 H. rm cmoq 8O HucmcomEOUI>v ucwEoz cam on pwuoanSm Emmm nm>mHHucmO Mom mCOHusHom mo comHummEou 01v mmoon Hmem u ummemumm ucweoz Hz 0H. 0. w. h. 0. m. w. m. m. H. 0.0 b p p p P p P P r _ .EmHm 0m .mmem 00 .flmHg H ////////vrwn I“ I ..H NcH m. u a r] Hmm 30H x N.H ll LL] 2 Ic>2 O O H u 02 >I°2 81 Hma .OH x N.H Hpcmcomeoo :0 ucwEoz mom on pwuoonosm Emwm um>mHHocoO Mow mcoHusHom mo cemHummEou hlv mmDUHm Hmem u e umumEmumm ucwEoz Hz 0H. 0. m. o. 0. m. e. m. m. H. 0.0 _ F _ H _ _ p h _ — IN. .Eme oN .mmmum om .HNHH azHos n mo flmHH COHquom HMUHu>Hmc« n .Eme m .Ho. n 00 .pmHMCQSIdmm n 1v. mmoum 0m .EmHm H .H0. n to .MZI.m0chcmh n .EmHo H .H0. u to .mZIHooIEmoo u IQ. .51 x-:-4 To. :cH mvoHo. n H z 5 . O NCHm H4 ”m [ [ FoH SIDE 82 Rococomeoulov ucweoz cam on pmuomflnsm Emmm Hm>wHHucmu H00 mCOHquom mo EOmHHmmEOU va mmDuHm Hmem I C HmomEmumm ocwEoz Hi I OH. 0. w. b. 0. m. v. m. m. H. 0.0 F H H H H H H H H p .Eme ON .mmoum om .HNHQ uso Emmm HmconcwEHo mouse fl H00 mCOHusHom mo COmHHmmEOU 0Iv mmDOHm III.“ x HmuoEmumm UMOH mmmum 00 \A com .EwHw HmconcmEHp mwucu 0H \\ \\ HO .Eme 6003 m .HmHH dZHod mu“““““““ .MW m> 1M. wwwum 0m .EwHw v m .H. n 0 .6mumomsuama .N .mm . _ m Iv. mmmum 0m EmeVIm.cHuHmz "H .flM \ Im. coHuowm mmOHO Snow #0. L ..I ...—H Hmm hOH I m 00 Ill. ,3, Tull. :H cH .00H u m uotqoatgea dII Ieuotsuemtpuou 84 mEmHm HmuHom comm woo a mo mcoHusHom mo COmHHmmEOO 0HIv mmDOHm ALUCHV m on coHuomHmwQ HmucouHuom OH. 0. w. h. m. m. w. m. N. b P — P b p b p _ .oEmE\.EmHm H .mme .omm n mNHm mmum Ho 0 , 0 02 .m0. u w .pwawlz w z n mu I Ho 0 a oz .mo. u 0 .6mxHHuamm u :0 Ho 0 0 02 .Ho. n w .pwumeSIz w z n no .HU U I mme come I a .mo. n 0 .mwumomsusmm u No .HU . . my a 0 02 m0 n m mZIHooIEmwo NO I mo .6 {U_ m o maHx .omnv u a 1 . CH 0H.0Hm n H . 3 =ONH CH 0 ..H AN N . as HH 0 m 0 H60. me .oooom u m . LIII 1 r .. .OOOH .000N .ooom .ooov .omhv .ooom mmHM oEmHm muoumloze d 00 mcoHuDHom mo comHHMQEOU HHIv mmDOHm HzooHv 0 pm COHuomHmmo HmucouHHom 85 0.N v.m N.N 0.m 0.H 0.H 0.H N.H 0.H m. 0. v. m. 0.0 H H r H e H H H H H H H H I.000N 11 LB :CH OH.OHM H H NEH FF.HH H AN 1. .000m .IIIN Hoo. me .OOOON u m .nEmE\.EmHm H q gnu H.o u 60 .60xHHIsmm n mo 0 Ho. n 60 .mmumomsuamm I No Till]. H0. H 00 .mZIHooIEmmn H HO I.0000 =.ONH1 1.0000 1.0000 mme m 86 manna saw 039 a Ho mcoHusHom Ho comHHmano NHIq HmouHN HSUCHV 0 pm QOHHUmHme HmucoNHHom .H a. m. a. 0. m. a. N. N. H. 0.0 r H H H H H H H H H A: In A» =.oNH . .6 .mI .. 8. - a a a .nsme\.EmHm H «I 1. . maHx .omN u mNHm amum .cH OH.on u =.oNH =.ONH Ho. m 60 .6mxHHIsmm u no I NCH hh.HH n H0. u 00 .meMCQSIflmm u NU me .oooom u Ho. u 0 .mZIHooIEmmn n Ho mUIIIIIIIIIIIlIIIIIIIIWIIIIIIIIIIIIIIIIIlIH} NU 4H/U .000H .000N .000m .ooov .0000 mme 87 HUMOH HmucouHHom ozv oEmHH coud cm How mcoHusHom mo comHHmmEoo HMVMHIV mmDUHH Hnoch m um COHuomHmwQ HMOHHH0> 00.m mn.m 00.0 mm.m 00.N 0n.H 0m.H mm.H 00.H mm. 0. mm. 0.0 H H H e H H H H H k H » I.oom .QEmE\.EmHm H mme .omm n mNHm mmum HO 0 oz .mo. u 60 .6waHIsma n mo I.OOOH .HO . mu . mnH NHNN n a mo. u 0 ompmoasIama I No .HO mnH omHN u a .mo. n 60 .szHooIEmwn n Ho H.oomH I.OOOH NU .HU .{ 0 .r {1 HI .OOmN mu 3 mm .- ONH .- mm ...—CH OHoOHm ” H ...mm 1 .000m NcH HH.HH u < o m mme me .oooom n 0 NH Ha a 88 HUMOA Hmucomfluom Sufl3v wEmum £0H< cm MOM mcoflpsaom mo comflummeou HQVMHIV mmDQHm ALUCHV m um cofluowamwo Hmucoufluom m. mo. v. mm. m. mm. m. ma. H. mo. 0.0 P _[ H b H H H H W m r .oom .QEwE\.E®Hm H mmfix .omm u mNHm mwum Mo . . o . o oz mo n o ooxflwu :H mH >I> H . >> gum mmH n H m H H H Ho xx mo mu om om HA Hum v. n H H Hum m. u a o¢| .w .:x _ w HmHHOH x ommq n m .o Hoo. n a V H H wk .mmm.mo Homv.Ho .mwm.mw HH+\\\\\\\L .Efixm mo mcme Om 9O wEmHh £0u< wommm < Mom mcoHusHom mo COmHHmmEoo mle mmDUHm Huwv COHuowHHw x :H U um :oHuonwwD HmucomHuom m. H. o. m. w. m. m. H. 0.0 H F F H H H m H 1.0H t.0m 1.0m r.ov .nEwE\.Eme H 1.0m z N +H mme .Hm n z zm H . mmflx .Hm n 2o z oo .. MU H , s M H moflx HH om u o moflx .m u o< H. u o ooxflmuamz H No MU .. . m oz .mon .m n o< HH. u mo .ooumomsuzmm u Ho 1 0H r.om «0 H0 It!!!» 1.0m mmflx r m 91 Loud HmHDUHHU HHmm M NO mcoHusHom lev mmDUHm H£UCHV CBOHU wsu um coHuuwHHmQ HMUHuHm> o.m v.m m.m o.m m.H o.H v.H N.H 0.H m. o. v. m. o.o H H H H H H H H H H H H H H mCOHHSHOm HHm How Ho. H Um H.8me Hmsvw v I.H .HU moH m. u m< .moH mm.H n o .ooomooouz w z u no HO moH m. n o< . o oz .ooxflmuz w z u .o .HU . moH m. u o< .moH Hm.m u o .ooumomsnmmm n mo 1 m .HU mQH m. H Q< H m oz chwaldmm n «0 H0 moH v. n oq .moH HH.m n o .manoousmon u Ho r.m mo 1.v cH OH H m 1.m .cH omnmoo. n H Nofl mHmH. u a mo Hmo .ooo0H u m :o r.o a H HmnHH 92 Det [Tangent Stiffness] B l A‘\ Load FIGURE 4-17 Load-Determinant Relation for Different Cases of Nonlinear Behavior 93 oEmum Hmuuom commoq >HHmoHuumEE>m a mo quHHQmum Ucm HcmEmoMHmmHolomoq wHIv mmeHm HSUCHV 0 pm ucwEoomHmmHo HmucowHuom mv. 0v. mm. Om. mm. om. mH. OH. mo. 0.0 H H H H - H H H H H TII+II4IIL 1.000H :o?ov=ov :cH OH.on H H 41 NCH no.HH u fl zomH l.ooom Hm o H . x OOOOM m D 0 Hz a a mmflx omom n o Hmouaw\.Eon m m m . I.ooom :EsHoo\.EoHo H moflx .omm n ma . I o . Ho I w mZIHOoIEme 1.ooov Hz mon .HNHH u H o um mm . H \\ m Hx ov0m .\\W\\v+\\ W.ooom 1.0000 mon 94 czouo moo om omoq ooomuucoocoo a 0» oooooHosm nouonom HmHoan mmoon Hnoch CBOHU mnu um COHuoonwQ .HH .0H .m .w .5 .w .m .v .m .N .H 0.0 H. H L H r H H H. r H H H . . n H cH 0H m moH mm.v u zo . :CH mmhmoo. u H NCH mHmH. n < Hmm .ooooH u m T Hz om mQH om.m u « m n # m > I s r .Eon .Hmowo v mQH m. n m< Ho. H Um HmzuHooIEmoQ I 00.0 OH. NH. moH Determinant 95 E = 10000. psi A = .1875 in2 I = .008789 in“ R = 10. in h ' Load FIGURE 4-19(b) Variation of Determinant of [K ] O T w1th Load 96 momoq HmoHuHm> OHHumEE>m¢ ou ompommnsm mamum wuoum mco omlv mmeHm HSUCHH U um coHuoonwo HmucomHHom .om .0H .oo .om .oo .0m .0m .oH o.o H H r H H H H H hooom z Noam memo oomH mvvo H o .. mHom mHmm momv mvmm Hz a .H n .uooov mH. om. mm. moflx oqom Hz mH. n o oHuoooo om. u o r.ooom H ow». 8.. on .H : H. I.OOOOH « m Hfi 4‘ H .O H \ xx wcoHusHom HHm How :cH OH.OHm n H Ho. n ow H.QEwE\.Eme H «CH hh.HH n < :.omH r.oooNH mzuHoouemoo me .oooom n m D U H 1.ooo¢H Ho o o mme lbs 30. 25. 20. 15. 10. 7 a ll 1. .75 .50 .25 0. 97 PN1 Jl7.96 7.83 6.73 5.69 4.87 PN * l9.67 11.05 12.89 15.46 19.29 1 beam-col-NR, Ed = .01, 4 equal elem. P = 9.35 lbs for a = 1. BC ”"PN * (0 = 0.) E = 10000. psi 1 A = .1875 in2 , I = .008789 in“ H- * -_— \ I - PN1 (01 ‘25) ‘3‘ R=lO. in "' * = .5 PNI (01 ) --PN * (a = .75) ~ 1 a - 1. a - .5 .25 .L__ 7‘ denotes PN1 I r’ 1 I u r— i 7 I 0.0 .1 .2 .3 .4 .5 .6 .7 .8 Horizontal Displacement at B (inch) FIGURE 4-21 A 90°-Arch Subjected to Two Vertical Loads 98 :H .oH :cH omHmoo. NcH mHmH. Hod .ooooH mCvaoq UHHuoEE>m¢ c¢ Ou Couoonnsm noud HmHDoHHU MHmm a mmlv mmDDHm HLUCHH m um mpcoEwomHmmHo HmucomHHom 6cm HmoHpHm> .h .w .m .v .m .N .H .o H H H H H H H F H2 H2 mQH Oh.w© H t m HmQH vm. H m II [11de .EmHo Hmswo v HHo. n ow mZIHooIEmwQ moH 10. ll. 12. 99 LIST OF REFERENCES Timoshenko, S. P., and Gere, J. M., "Theory of Elastic Stability", New York, McGraw—Hill, 1961. Bleich, F. "Buckling Strength of Metal Structures," McGraw-Hill Book Co., Inc., New York, N.Y., 1952. Cook. R. D., Concepts and Applications of Finite Element Analysis, John Wiley and Sons, Inc., New York, N.Y., 1974. Haisler, W. E., Stricklin, J. A., and Stebbins, F., "Development and Evaluation of Solution Procedures for Geometrically Nonlinear Structural Analysis by the Direct Stiffness Method," proceedings of AIAA/ASME 12th Structures, Structural Dynamics, and Materials Conference, Anaheim, California, March 1972. Vol. 10, No. 3, pp. 264—272. Saafan, S. A., "Nonlinear Behavior of Structural Plane Frames," Journal of the Structural Division, ASCE, Vol. 89, No. 8T4, Proc. Paper 3615, August, 1963, pp. 557-579. Saafan, S. A., "A Theoretical Analysis of Suspension Bridges," Journal of the Structural Division, ASCE, Vol. 92, No. ST4, Proc., Paper 4885, August, 1966, pp. l-11. Conner, J., Jr., Logcher, R. D., and Chan, S. C., "Nonlinear Analysis of Elastic Framed Structures," Journal of the Structural Division, ASCE, Vol. 94, No. 5T6, Proc. Paper 6011, June, 1968, pp. 1525-1547. Oran, C. "Tangent Stiffness in Plane Frames," Journal of the Structural Division, ASCE, Vol. 99, No. 8T6, Proc. Paper 9810, June, 1973, pp. 973-985. Oran, C. "Tangent Stiffness in Space Frames," Journal of the Structural Division, ASCE, Vol. 99, No. ST6, Proc. Paper 9813, June, 1973, pp. 987—1001. Kassimali, A. "Nonlinear Static and Dynamic Analysis of Frames". Ph.D. Thesis, Department of Civil Engineering, University of Missouri-Columbia, August, 1976. Martin, H. C., "A Survey of Finite Element Formulation of Geometri— cally Nonlinear Problems," Recent Advances in Matrix Methods of Structural Analysis and Design, Edited by Gallagher, R. H., Yameda, Y., and Oden, J. T., The University of Alabama Press, 1971, pp. 343-381. Przemieniecki, J. 3., Theory of Matrix Structural Analysis, McGraw— Hill Book Co., New York, N.Y., 1968. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 100 Jennings, A., "Frame Analysis Including Change of Geometry," Journal of the Structural Division, ASCE, Vol. 94, No. 8T3, Proc. Paper 5839, March, 1968, pp. 627-643. Mallet, R. H. and Marcal, P. V., "Finite Element Analysis of Nonlinear Structures," Journal of the Structural Division, ASCE, Vol. 94, No. ST9, Proc. Paper 6115, Sept., 1968, pp. 2081-2105. Powell, G. H., "Theory of Nonlinear Elastic Structures," Journal of the Structural Division, ASCE, Vol. 95, No. STlZ, Proc. Paper 6943, Dec., 1969, pp. 2687-2701. Akkoush, E. A., Toridis, T. G., Khozeimeh, K., and Huang, H. K. "Bifurcation, Pre- and Post-Buckling ANalysis of Frame Struc- tures," Computer & Structures, Vol. 8, June 1978, pp. 667-678. Holzer, S. M. and Somers, A. E., "Nonlinear Model, Solution Pro- cess, Energy Approach" Journal of the Engineering Mechanics Division, August 1977, pp. 629-647. Bathe, K. J. and Bolourchi, S. "Large Displacement Analysis of Three-Dimensional Beam Structures", International Journal for Numerical Methods in Engineering, Vol. 14, April 1979, pp. 961- 986. Ebner, A. M. and Ucciferro, J. J., "A Theoretical and Numerical Comparison of Elastic Nonlinear Finite Element Methods," Computers and Structures, Vol. 2 Nos. 5/6, 1972, pp. 1043-1061. Wen, R. K., Unpublished Research Notes on Work Done Under NSF Grant Eng—7822478, College of Engineering, Michigan State Univer- sity, 1980. Bathe, K. J., and Wilson, E. L., Numerical Methods in Finite Element Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976. Hurty, W. C., and Rubinstein, M. F., Dynamics of Structures, Prentice-Hall, Inc., 1964. Lange, J. S. "Elastic Buckling of Arches by Finite Element Method" Ph.D. Thesis, Department of Civil and Sanitary Engineering, Michigan State University, 1980. Frisch-Fay, R., "A New Approach to the Analysis of the Deflec- tion of Thin Cantilevers," Journal of Applied Mechanics, Trans- actions of the American Society of Mechanical Engineers, Vol. 28, Series E, March, 1961, pp. 87-90. 25. 26. 101 Oran, C. and Kassimali, A., "Large Deformations of Framed Struc— tures Under Static and Dynamic Loads", Computer and Structures, Vol. 6, 1976, pp. 539-547. Gere, J. M. and Weaver, W.J.,"Analysis of Framed Structures", D. Van Nostrand Company, 1965, page 291 — Figure 4—46. 102 APPENDIX A MATRICES [k], [n1], AND [n2] All matrices contained in this appendix and Appendices B and C are symmetric. Only non-zero entries are given here. A.l [k] MATRIX A.1.1 THREE DIMENSIONAL k (1,7) = -k(l,l) k(1,1) = k(7,7) = -k(1,7) =.__ 1231 k(2,2) = k(8,8) = ‘17—3' k(2,8) = -k(2.2) lZEI; k(3,3) = k(9,9) = “Ejr—‘ k(3,9) = -k(303) k(10,10) = k(4,4) = %§ k(4,lO) = -k(lO,lO) 4E1 k(6,6) = k(12,12) = 2 n 431 k(5,5) = k(11,ll) = x C 6EI k(618) = k(8112) = -k(2112) -6EI k(3,11) = k(3,5) = ——159 k(5,9).= k(9,ll) = -k(3,ll) .l. 2 k(6,12) k(5,11) 2E1 n l 2E1 2 TWO DIMENSIONAL k(l,1) k(l,4) k(2,2) k(2,5) k(2,6) k(3,5) k(6,6) k(3,6) k(4,4) = —k(1,l) k(5,5) = -k(2,2) k(2,3) k(5,6) k(3.3) 2E1 n 2 [n1] MATRIX THREE DIMENSIONAL n1(l.2) n1(ll8) n1(2I2) n1(2,8) n1(7.8) n1(2.7) n1(3l3) n1 (3'9) 103 12EIn £5 6EI n -k(2,3) 4E1 n = 2.13:. EA 101 = -n1(112) = n1(8,8) = n1(9,9) = 553—1513523333— EA -n1(2I2) n1(5,5) = n1(6,6) = n1(11,11) n (13) =n (79) =ELEA 1 ' 1 ' 102 n1(llg) = n1(317) = -n1(ll3) ..F,1 =———J EA n1(1:6) 3O n1(6.7) = -n1(l,6) n1(l 5) = ;G—5-l-EA ' 3O n1(517) = -nl(115) = ‘Fsz n1(l.12) 30 EA n1(7,12) = -n1(l,12) _ “G52 n1(llll) " '3'0— EA n1(7,ll) = -n1(l,11) n1(2,6) = n1(2,12) = n1(5,9) n1(3,5) = n1(3,ll) = n1(8.12) - = - 2&121 n1(5,1l) — n1(6,12) 30 in which: Pa = 91+62-1280 Gk = W1+W2-12W0 104 F51 = 481-92-390 = n1(9rll) — EA n1(12,12) = 2(u2-u1) 15 EA uz-ul 102 —n1(216) .2. 105 = - - 8 P52 462 81 3 0 G51 = 4w1—W2-3WO G52 = 4W2-El-3T o v -v e = 2 1 O 2 Wl‘wz W = O 2 TWO DIMENSIONAL n1(4,5) = n1(l,2) = [_ Siggg 6(VEEVI)J‘%§ n1(2,4) = n1(1,5) = -n1(l,2) mm = [92-481+3(VZ;VI)1§%_ mm) = Eel-49m (VT/1:31)] if: n1(4.6) = -n1(l,6) n1(3,4) = -n1(l,3) n1(5,5) = n1(2,2) = 6(u2-u1) ‘§%7 n1(2,5) = -n1(2,2) n1(2,3) = n1(2,6) = (uz-ul) 1%? n1(3,5) = n1(5,6) = -n1(2,3) n1(3,3) = n1(6,6) = 2(u2-u1) %%. n¢(3,6) = -(u2-u1) E5 30 106 [pg]¥MATRIx THREE DIMENSIONAL BASED ON QUARTIC STRAIN FUNCTION n2(212) n2(218) n2(3r3) = n2(319) = n2(273) = n2(2I9) = n2(216) = n2(6r8) = n2(3r6) = n2(619) = n2(6,6) = n2(515) n2(12,12) n2(ll,ll) n2(3I5) n2(519) n2(8r8) =(6B3-6Bg+2B5) 25' 2 -n2(212) n2(919) = (683‘6Bg+2B10) EE- R -n2(3r3) EA (6313'6B1u+2515) if -n2(213) (-3Bz+1083-7B.+285) %§- ”32(2r6) EA n2(5,8) = (-3B12+10313-7Blu+2315) if n2(215) = -n2(3l6) 22 EAQ (Bl-4B2+ 7f-B3f-4Bg+Bs) ‘3;- 22 2 (Be-487+ T;‘Bg-4Bg+Blo) E?— 4 EAR = (E'Ba-ZBu+Bs)'—§- 4 BAR — _ + — (3 Ba 2B9 BIO) 2 (3B7-IOBB+7B9-2B10) £25 -n2(315) 107 n2(5'6) = - (B11’4Blz+ 3321313-4311.+B1s) 221% “2(3'8) = ' (6313-6814+2815) {if n2(8,9) = -n2(3,8) n2(2,12) = (433-58u+2B5) %? n2(8,12) = -n2(2,12) n2(3,12) = n2(8,11) = (41313-5131u+2815) 1525 in n2(9,12) = n2(2,ll) = -n2(3,12) n2(3,ll) = (—438+539-2310)‘%? n2(9,ll) = -n2(3,ll) 11 EAR n2(6,12) = (-B2+ 783-3Bg+85) T 11 EAR n2(6'11) = I12(5,12) = (B12- TBI3+3Bll+-BIS) -——-2 ll EAR n2(5,ll) = (-B7+ 7;.83-389+Blo) —3-' -4 BAR n2(ll,12) = (7813+ZBIM'BIS) T which: 68 2 = % [2(812+622)+18802-380(81+82)-8182— Ty12+ 34222 + 6W02 +39wlwo-szo-17wlwz] _ l, 2 2 2 2 2 2 _ - 5 (91 +392 +1880 -68061-6162-104W1 +W2 +6Wo +58W1Wo 76W1W2) Bz 108 1 B3 = 3; (6612+27822+108602-458081+188082-98182-292W12+9W22 +36W02+489W0W1+6W0W2-213W1W2) 33 2 29 2 1107 2 9 9 33 =-——- + ——- - ————- - —2 —- - ——-e B“ 140 61 28 62 7 6° 5 6°61 + 5 6°62 70 162 _ 1949 2 13 2 9_ 2 117 339 _ 711 'IZE— W1 + EE'WQ + 7 We + —§—-V0W1 + —7—'W0W2 —76'W1W2 _ 2_. 2 31. 2 32. 2 _ 32. 31. _ 2.. BS 35 61 + 14 92 + 7 6° 14 9°91 + 14 9°92 14 6162 627 2 9 2 g_ 2 423 9 _ 183 35 VI + 14 12 + 7 1° + 14 W0?‘ + 14 9022 14 2122 B6 = é’[2(W12+w22)+18W02-3WO(W1+W2)-W1WZ- %§'612+ %-822 + 6802+396180’8280-178162] B7 = é-(912+3w22+18902-6wow1-Wlw2-1o4e12+e22+6eo2+588160-766162) Ba =-§§ (6912+27w22+108w02-4swowl+1BWOW2-9w122-292612+9e22 +36802+4898081+68082-2138182) _ 33 2 22_ 2 1107 2 2. 2. -.32 B9 - 140 VI + 28 WZ 7 We 5 Wowl + 5 WOWZ 7O WIWZ 1252. 2 l}, 2 2_ 2 117 339 _ 711 , 140 61 28 62 + 7 60 + '5'” 6081 + —7 9092 ""‘70 8182 .. 2.. 2 32. 2 22. 2 31. El _.2_ B10- 35 ‘1’1 + 14 W2 + .7 ‘Po 14 W0W1+ 14 Wowz 14 411912 .231 2 2_. 2 2_ 2 423 g__ _ 183 .3. 2 109 B11= i 911,1 + My— + 81912 + W180 + 629’} 15 5 15 5 5 l2 1 "' '5— 80‘90 +g' 60912 1 4 + -5- 92110 -1—§ 921112 B12: :2- 81111 + l 2 1 12 2 15 611,0 +1-5- 61‘1’2 + '5' W180 + .16 921/1 - '5" eowo - ‘5‘ GZWZ who -4 3 3 3 3 72 813:3—581W1 42781110 +3—5-81‘112 +7919) +-3—5-82W1 - 35-60410 6 6 18 35 80W2 - 35 ezwo - 35 ezwz 11 3 11 , 18 70- 81‘1’2 + g W180 + 77-6 82V1- 77- 80910 -11 3 v 811,: :76— 81111 + g 813/0 + 3 3 l3 - 2'7" 801% - “:7“ 829’0 -- It; 62% -6 9 3 9 , 3 18 815: 3'5“ 91W1+ ~171- 61% + 1*; 81112 + IZ ‘1’180 + 121- 82‘1’1 - 77— 80910 9 9 9 - 1—4- 6092 - I4 82%) - 7 821’2 TWO DIMENSIONAL BASED ON QUARTIC STRAIN FUNCTION 18 EA n2(3,3) = [122812+2822-328182+ E—'(V2*"1)2+3(V2-V1) (81-92)] 138 108 72 EA n2(2,3) = [-3912+3822+68182+ 2—2- (vz-V1)2-'E— 91(V2-V1H 2—80 n2(3,5) = -n2(2,3) 2 2 EA n2(3,8) = P3291 -3282 +428182-6(V2-V1)(61+92)1 350 18 BA n2(6,6) = [2812+122822-328182+ Ef-(V2-v1)2+3(V2-V1)(62-61)] 148 n2(2,6) = [3612-3622+68162+ 533 (V2-v1)2- 7—2 62(V2-vm 3‘- SL 2 280 .3. 110 n2(5,6) = -n2(2,6) n2(5,5) = n2(2.2) = [$33- 612 + i3 922 + 51—3—3 (VZ’V1)2 -'%%§ (Vz-V1)(91+92)] 1%5' n2(2,5) = -n2(2,2) THREE DIMENSIONAL BASED ON AVERAGE STRAIN = = FL. £2. 25 n2(2,2) n2(8,8) (100 + 25) R n2(2r8) = -n2(212) EA n2(2,3) = n2(8,9) = -F1,GL. m n2(2,9) = n2(3,8) = -n2(2,3) G} + £2. EA n2(3,3) = n2(9,9) ( 100 25 2 n2(319) = -n2(3l3) EA 2 + —— n2(2,6) (F31 G2) 300 n2(6,8) = -n2(2,6) EA n2(215) ‘ FHGSI 300 EA n2(6,9) = -n2(3,6) EA n2(3,5) = -(G31 + F2) ‘*" 300 n2(5I9) n2(616) n2(516) n2(5,5) n2(518) n2(2,12) Dz(8,12) n2(2,11) n2(8,11) n2(3,12) n2(9,12) n2(3,11) n2(9rll) n2(6I12) n2(6rll) n2(5,12) n2(5,11) H 111 -n2(315) (Eil.+ §2_§ EAR 300 225 EAR F51G51§36 (Efil.2.§L_) EAR 300 " 225 EA “651 3'55 EA + — (F32 62) 300 -n2(2,12) EA FuGsz #300 -n2(2,11) EA "(3413' 52 _300 -n2(3r12) EA — + _— -n2(3,11) EAR. F52Gs 1 _900 EAR F2 (G7 3 ) 300 n2(12,12) = (Eéé-+ 92—9 8A2 300 225 112 _ 8A2 n2(ll,12) — F52G52900 G52 F2 , = _—_.+ ___. in which: G21: F4 G32 F51 2 9812+9822-26182-368180-368280+216602 9W12+9W22-2W1W2-36W1W0-36W2W0+216W02 2812+2822—8162-36180-36260+18802 W12+Zsz-W1W2—3W1W0-3W2W0+18W02 61+62 -1260 WI+W2 ~12W0 -2812-2822+68182-26160-26280-3602 -2112-2922+ewlw2-2wlwo-2w2wo-3wo2 6812+822+28162-546180+66280+54802 6922+612+26162—548280+66180+54802 6212+w22+2w192-54wlwo+ew2wo+54wo2 8222+212+2w1W2-54W2wo+6wlwo+54wo2 461-92-380 113 F52 = 482-81—380 G51 = 421-92-390 G52 = 492-W1-3WO F61 = 8812+3822-48182-128180-28280+27802 F62 = 8922+3812-48182-128280-28180+27802 Gel = 8212+3w22—42122-122120-2w2wo+27202 G52 = 8922+3wl2-49192-129290-22lwo+27wo2 TWO DIMENSIONAL BASED ON AVERAGE STRAIN n2(2,2) = n2(5,5) = (9912+9822-28182-368180-368280+216802) 1532 n2<2,5> = -n2(2,2) n2(2,3) = (6812+822+26162-548190+66280+54802) 386 n2(3,5) = -n2(2,3) n2(2,6) = (6822+812+28182—548280+68180+54802) 386 n2(5,6) = ~n2(2,6) n2(3,3) = (8812+3612-46192-128160-28280+27802) §%% n2(3,6) = (-2812-2e22+66182—28leg-28280-3802) §%% n2(6,6) = (8622+3612-48162-126280-28180+27902 ggg in which 90 = XEZZL Q 114 APPENDIX B [kE ] INITIAL STRAIN STIFFNESS MATRIX FOR QUARTIC STRAIN FUNCTION O THREE DIMENSIONAL 36EA k 2,2 = , , = I = — E ( ) k8 (3 3) k6 (8 8) RE (9 9) (03 204+05) 2 o o o o kE (2,8) = Re (3,9) = —k8 (2,2) 0 o 0 RE (3.5) = Re (6.8) = 6 (Dz-503+7Du-305) EA 0 o kE (2,6) = k8 (5,9) = -k€ (3,5) 0 o 0 RE (6,6) = k8 (5,5) = (pl-802+2203-24ou+ 905) EAR o o kE (11,11) = Re (12,12) = (403—1201+905) EAR o 0 k8 (9,11) = kE (2,12) = 6 (203-502+305) EA 0 o k€ (3,11) = kE (8,12) = -kE (9,11) 0 o 0 k6 (5.11) = kE (6.12) = (-202+1103-1804+9Ds) EA2 0 o in which pl, Dz, ..., OS are evaluated from the following steps: 1) Initialize BTOi = 2) Save BTO. in BOL. 1 1 BLO. = ETC. 1 1 0.0 for i = 1,5 as: i II ..o W 3) 4) 5) 6) 115 Evaluate 01: 02: 03. 81, 82, Ba 35: 01 = 81 2 02 ='E (-3V1-291£+3V2-82£) 3 C13 = E (2V1+91£-2V2+62£) 81 = “W1 82 = %’(-3W1-2W1Q+3W2+W2£) Ba = -'(2W1-WIQ-2W2-V22) Evaluate b1,b2 , ..., b5 as: b1 = 22%21 +'% (812+812) b2 = O(10‘2‘1‘8182 b3 =‘% (022+822) + OLIC3‘3"'8183 b4 = 02G3+8283 b5 = %'(032+832) Updated BTOi as: BTO. = BOL. + b. i = 1,5 1 1 1 Evaluate pi as: 1 1 1 1 .=.—BTO +-—— +— +-.——BTO + pl 1 1 BT02 BT03 1+3 4 1+1 1+2 ~ 1+4 BT05 i=1,5 .2 & 116 TWO DIMENSIONAL k k E k 6 k E k E k E k E k k C in which pl, 2. (2,2) 0 (2,5) 0 (3.5) 0 (2,3) 0 (3.3) 0 (6,6) 0 (2,6) 0 (5,6) 0 (3,6) 0 36EA (Os-204+Ds) £ k€ (5,5) 0 -k (2,2) 8 o 6 (Dz-503+7Ou-305) EA -k€ (3,5) 0 k8 (2,2) = (pl-8p2+2203-24pg+905) EAR O (403—1294+905) EAR 6 (2p3-502+3p5) EA -kE (2,6) 0 ('202+1103-1804+9Os) EAR 02: ..., OS are evaluated as following steps: The same as the three dimensional case. Evaluate a1, G1 91 3 52, .31 £( 0.2, C13 as: (-3v1-281£+3v2-822) 2v1+812-2v2+82£) 5. &6. 117 Evaluate b1, b2, ..., b5 as: Uz-U1 1 2 b1 = -—ET—'+ E'dl b2 = @102 ha = % 0122811013 b4 = 8283 b5 = 9'3: The same as the three dimensional case. 118 APPENDIX C [n1*] GEOMETRIC STIFFNESS MATRIX C.1 THREE DIMENSIONAL n1*(2,2) = n1*(3,3) = n1*(8,8) = n1*(9,9) = %%%~(u2-u1) n1*(3,9) = n1*(2,8) = -n1*(2,2) n1*(5,5) = n1*(6,6) = n1*(11,11) = n1*(12,12) = %%§-(u2-u1) n1*(2,6) = n1*(5,9) = n1*(9,11) = n1*(2,12) = % (u2-u1) n1*(3,5) = n1*(6,8) = n1*(8,12) = n1*(3,ll) = -n1*(2,6) n1*(6.12> = n1*<5,11) = 3%}: (uz-ul) C.2 TWO DIMENSIONAL n1*(2,2) = n1*(5,5) = 2%9-(u2-u1) * = * ='—— - n; (2.3) n1 (2,6) 10% (U2 u1) n1*(2,5) = n1*(3.5) = n1*(5.6) = -n1*(2.2) 2EA n1*(3,3) = n1*(6.6) = 33-(u2-U1) -EA n1*(3,6) = 36—'(U2-u1) c.3 [kc] MATRIX fl = ~—-——————— * th t d three dimensional cases. [kG] EA(u2-u1) [n1 ] for bo wo an 119 APPENDIX D COMPUTER PROGRAMS D.1 DESCRIPTION OF SUBROUTINES A general description of the computer programs is given in Sec- tion 3.6. The listing of programs are presented at the end of this appendix with appropriate comment statements. In the following a brief description of the subroutines is given. The main programs (NFRAL3D, NFRALZD, NFRAE2D) direct the flow of computation by calling the appropriate subroutines for each step of the solution procedure. Subroutine NODDATA reads data regarding the overall geometry of the structure including coordinates and degrees of freedom. Coordinates for plane circular or parabolic arches may be generated. The equation numbers are generated by this subroutine. The subroutine ELEMENT reads data related to the element properties and node numbers. The subroutine BAND computes the semibandwidth, MBAND, that the stiffness matrix of the structure will have. Subroutines BEAM and TRUSS evaluate the linear stiffness matrices of the beam and truss elements, respectively. Subroutines TRANSFM and INVTRNS are used for geometric transformation from local coordi- nates to global coordinates and vice versa. Subroutines SBEAMEl, SBEAMEZ, and KEPSIOl, respectively, evaluate the non-zero entries of [n1], [n2], and [KS ]. The assembly of [k], [n1], [n2] and [K€(3 into the appropriate gloEal stiffness matrices is accomplished with sub- routine ASEMBLE. Subroutine LINSOLN solves the system of linear 120 equations by Gauss elimination. Subroutine STCONDN condenses the structural linear stiffness matrix and load vector into the degrees of freedom which have been established in subroutine NODDATA. Sub- routine RECOVER recovers the internal degrees of freedom of the struc- ture after using subroutine LINSOLN. Subroutine IDENT identifies the displacements obtained from LINSOLN with the nodal displacements similarly for those found in the recovery process. The solution of the linear eigenvalues problem and the quadratic eigenvalue problem is obtained with subroutines EIGENVL and NLEIGNP, respectively. The subroutine EIGENVL uses inverse vector iteration with Rayleigh quotient to obtain the lowest eigenvalue and correspond- ing eigenvector of the linear problem. For the solution of the quad- ratic problem, the subroutine NLEIGNP uses the modified regula falsi method of iteration by calling subroutine MRGFLS and the function sunprogram DET. Subroutine MULT is used for matrix multiplication and the function subprogram DETl evaluates the determinant of the structural tangent stiffness matrix. Finally, subroutines ENDFORC and STRESS evaluate the element end forces and stresses, respectively. D.2 VARIABLES USED IN THE COMPUTER PROGRAMS The variable names used in the programs are listed below in alpha- betical order: MAIN PROGRAMS NFRAL3D, NFRALZD, NFRAE2D A(M) = The cross-sectional area of ele- ment M; A7OLD(M), A7TOT(M) = parameters related to element M for evaluation of the initial strain stiffness matrix; BOL(M,J), BTO(M,J), BE(J) D(I) DTOT(I), DACTUAL(I) DETER, DETERMNT DN(I,1) E(N) ES(I,M) G(N) IA(N,I) IB(N,I) 121 Intermediate parameters for the evaluation of initial strain stiffness matrix; Displacement vector, found from the solution of the system S*D=R. I varies from 1 to NEQ; The same as D(I) but for total displacement measured with ref- erence to the beginning of each load increment or initial geome— try, respectively; Determinant of the structural se- cant or tangent stiffness matrices; End forces in global coordinates for each element. I varies from 1 to 6; Modulus of elasticity of element group N; End forces in local coordinates for element M. I varies from 1 to 3; Shear modulus of element group N; "Boundary condition code" of node N for its Ith degree of freedom. Initially it is defined as follows: IA(N,I) = 1 if constrained; = 0 if free After processing, IA(N,I) = 0 if initially = l; = equation number for the D.O.F. if ini— tially = 0; "Additional boundary condition codes." IB(N,I) 0 if free N if slave to node N; -1 if to be condensed. After processing, IB(N,I) is un- changed except, IB(N,I) = -(condensation number of the D.O.F. if initially IB(N,I) = -1); ICALI, ICALZ, ICAL3 ICHECK IDET IGOPTIN IPAR ISTRESS IXX(M) IZZ(M) KT(M) L(N,K) LE(M) NCOND NCOUNT 122 Variables controlling print-out (more details are indicated by "comment statement" in the listing of programs); Parameter used for Newton-Raphson approach in Lagrangian coordinates to control the type of computation needed in each load increment; Parameter used for evaluation of the determinant of the secant or tangent stiffness matrices either before or after Gauss elimination process; Parameter used to specify type of the geometry for plane frames (i.e., circular, parabolic arch or arbitrary geometry); Variable identifying appropriate "Tape" for storage of different structural stiffness matrices (i.e., [K]: [KE 1: [N1]! [N2])3 o If EQ. l, compute nodal forces and stresses in the structure. If EQ. 0, skip; Moment of inertia about the C-axis of the cross section of element M; Moment of inertia about the n-axis of the cross section of element M; Torsion constant of element M; Variable identifying the Kth ele- ment in the element group N; Length of element M; Semibandwidth of structure stiff— ness matrix; Total number of degrees of freedom to be condensed out; The order of load increment in incremental approaches; NE NEQ NODEI (M) NODEJ(M) NSIZE NUMEG NUMEL(I) NUMEL NUMITER NUMNP PI(N.I) PACTUAL(I) R(I) ROT(I,J), ROTRAN(I,J) S(I,J) SCALE SE(I,J), SEI(I,J), SE2(I,J) 123 Total number of elements in the structure; Total number of equations; Variable identifying the number of node I of element M; Variable identifying the number of node J of element M; Total number of degrees of free- dom, condensed and free, of the system. (NSIZE = NEQ + NCOND); Total number of element groups; Total number of elements in ele- ment group I (in NFRALBD); Total number of elements (in NFRALZD and NFRAE2D); Number of iterations at each stage of computation; Total number of nodal points; Load applied at node N, in the Ith direction; Applied load related to the Ith D.O.F. in the structural load vec- tor at each stage; Load vector of the system; Rotation and inverse rotation matrix for each element (I = 1, 6, J = l, 6), respectively; Tangent stiffness matrix of the system; Scale factor in the evaluation of the determinant of the structural stiffness matrix; Element stiffness matrices (i.e., [k], [n1], [n2], respectively); SXX(M) ULOC(M,I) USTAR(I,M) W(I,J), WCHK(I,J) WTOT(I,J) X(N) , Y(N) . Z(N) YPGM (M) , ZPGM (M) SUBROUTINE NODDATA ALFZERO RADIUS RISE SPAN SUBROUTINE TRANSFM Rcol(I) SUBROUTINE INVTRNS V(NP,I) 124 Section modulus about the C-axis of the cross section of element M; Identifies local displacement in the Ith direction of element M (I varies from 1 to 12 for three dimensional case and from 1 to 6 for two dimensional); Identifies the end displacement for the Ith direction of element M in Eulerian coordinates; Incremental recovered displacements (used in iterative process) related to node I in the Jth direction; The same as W(I,J) but for total displacements; Global X, Y, Z—coordinates of node N; Y and Z res ectivel ; PY PY ’ p y See Ref. [26]; Opening angle of circular arch; Radius of circular arch; Rise of parabolic arch; Span of parabolic arch; Identifies the entries of rota- tion matrix for three dimensional beam element. I varies from 1 to 9; Identifies the element local dis- placements for nodal point NP and Ith direction (I varies from 1 to 6); SUBROUTINE STCNDN Rc(I) SC(I,J) SUBROUTINE EIGENVL (EIGEN, IDATA) EIGEN EIGNVTR EPSI MAX SUBROUTINE ENDFORC DN(I) SUBROUTINE STRESS SIGMA(M) STRAIN(M) SUBROUTINE NLEIGNP A,B ERROR FL 125 Condensed structural load vector (1 = l, NEQ); Condensed structure linear tan- gent stiffness matrix; Eigenvalue; Eigenvector corresponding to EIGEN; Tolerance; Maximum number of iterations allowed; Rayleigh quotient; Vector that stores the approxima- tion to the eigenvector after each iteration; Stress resultants on the nodes of each element; Maximum stress of element M; Maximum strain of element M; Variables defining the interval in which the eigenvalue is enclosed; Upper bound on the computation of the eigenvalue after convergence; Value of the determinant of the matrix S = K + L*N1 + L*L*N2 at the converged value of the eigen- value; FTOL NTOL XTOL SUBROUTINE MRGFLS IFLAG FA FB FW FUNCTION DET DET K(I.J) 126 Convergence criterion for suffici- ently small value of the determi- nant of eigenvalue; Converged value of the eigen- value; L = (A+B)/2; Maximum number of iterations allowed; Tolerance; Variable defining the status of the iteration. If EQ. l, conver- gence was successful. If EQ. 2, no convergence after NTOL iterations. If EQ. 3, both endpoints, A,B, are on the same side of the root, hence method of iteration cannot be used; Value of the determinant of matrix S at interval endpoint A; Value of the determinant of matrix S at interval endpoint B; Weighted values of the root between interval endpoints A and B; Value of the determinant of matrix S at the weighted value W; Value of the determinant of the matrix S = K + L*N1 + L*L*N2 at a particular value of L; Part of element S(I,J) correspond- ing to linear stiffness K(I,J); Load parameter; 127 N1(I,J) = Part of element S(I,J) correspond- ing to matrix N1 (I,J); N2(I,J) = Part of element S(I,J) correspond- ing to matrix N2(I,J). 128 D.3 IWKXHUUCIEERAL3D 032‘..567R.901953Q.56750 01.23“: .57890123‘c. 678901234§JAQIR 90123NSE7°.° 0123.46.75.79 c.01 123N‘.6189111311311122222222223‘.B3333333QORRQQNNNRp—Jpfissssssss5666.b66666777777777788 ......OOQOOOOOOO00.0.00... TNG A SUBROUTINE THTS PROGRAM USES THE FINITE ELEMENT METHOD TO ANALYZE OOOOOOIOOOQQOOOOOO......00..........OQOOOOOOOOOOOOOOOQO0.0.00.0... STRUCTURE MADE UP OF STRAIGHT BEAN ELEMENTS IN THREE Y N R o cccccccccc LC33QIPARQICALIQTCALZQICAL3O 0E M’S’I’Iull” 2, ’I’EIIIR O [furl-LulchP D t a o o o o I N o o o 0 t o 3 . ... a o 0 a T o o O 6 a U E t o 3 C t L D o t O t 6 L o O D E T o X t 3 A e S E S N T a 0 a C U a N T T oUD E E N a 1. o T 3T 0 plans. TA 0 E c 5 A. t 9 I K 5C t GET. N ESS N N N a 2 o 0 CA .- TTR EE STU E A 7. TL a .... a F 3 EP 0 END. "P U G T H LA 3 O .0 6 Q N B O o .1 E0 R5 R R NU o a o 3 3 T 93 3 a Dis LL SA... E E E 00 c : «S C U P 30 O o NPN ES IT V H T E o N 0T X 6 nu.36 a: c .A 0 (EA N Y..L R a T .N x C 9 9C3 O a El. HF 1...... O 0 ET 0 D. QE 1.. c 1. 6T0 7 . HST CO NNR C F LE 0 O 0 ON 0 R N 30233 0 C A A T 0 F. 85 a! K U! 200 3 3 O 9 CT 05C 0 ACT EN NNA N I. 0 0K H 52 A0? 7 6 3 N 00912 a OTU o 0 COM. 0 T ERE c H C I“... Tan... 3 3 O T T969N .- R P HR... 3!. 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A RR T 0 E NLS o3P12PH:2cOoLD TCC3CC6CGNLRTDPSR007IFFZSOIO IOOIOOIZNOI ELROSIEOY .10F738N1:E0LE SA...CZICJTUJPA R a 90 O 96 .. 3E: : .. :EF : :1: = 53... :3 .. ....F O3 0 75 0...... G 9.... OR 00 AC NISEZD SI Oil/I P083 3331NNHKKNN NNONNITTHNENN N0091561XT9EK0 :N I/IIII/031267NNNN o 91 TIAHHTTIOTTPTT:U=TTNTAETR¢6NIONOP2LH01A 23‘50893011310000 055 SSSTTSC».DP::OP PPKRROPFPTHPE. 0081. DIONOCQAL RROOXXOOHOOT-LE oro T020 AJNCcDNCITTR ATR INSNNNNbNNN 'NNNNSSSSE.PPpPPPP00f3L—LIITIUUCGGSHLRTITC. T. O 6TTCTTNP 0 in. L8 ozoooosoOOUOOnUnuN‘NNNGO RRRRRRRIIHTT RRFFSS‘AAUTOOE.T F.H~Ro O CPR-LO R O o 'T 0 EM. LUTRMUHU H. OH.“ "6M.” ”Kf1Er.EEITTTTTTTNNTITPPTT‘H.HENNJOTFDSDTIPTODOMITOM: 96x1. DU ‘HuTHHUHVBU 0 MC”. 0 UN“. 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N . 0:9. .5 H D I PUGN G EUUU . E . 120 .NEV.GF.RT LNI IRTROO . H . AIO .IVLNXOS EAT. E—EXTHH . 0 . TSL .DIBITF UVDG DESSS . H 0 LP. .ASODF R LNAN RR . . EE1 OOERA 1A AEOI OOOESS . R . DHN .LCPON:E VGLD F THEE . o t 60 t C LOKN N1 A E TCC . F . TQP .LUE STI EELO OHD RR . . 1X0 .ASULHSL G AL RTENOOT. E . 300 .TCLAPU ICT E TOFFG. C . 91L .NHATAJT EINL ZEA N. N . 6/9 .ECVNR L TEA RLDODI. AR. 05U .HANE.DN RANT OAEENND. R0. IIL .EOEHNNO AREN T RIEEO. ET. 60A .RRGEOA EDRE 6 L C. LC. C1V .CPIRT G NACI. LN)D.TT . 0E. E25 .NPECUON TUNE AOSPNNT. TV. TEI .IA NE:I LOIRTUDNAEEN. . IQE . RINKT C5000 MINE. EE. R: O .RSO HA RRPNEEOTS::LI. LC. UIE .OSFRRCD OOOIT LTTLLC. BR. IPS.FE OORP FFF AT ANEEI. A0. TSTS. N FFEU. RNEOUE F. T NSSSOI OTQuWIOF. O .5 oEOR.IF.I TTOIIIFDI EQ....E. LD.10HRT. I 931 R:P HSSC. LE. thPS.IT2333 TIZSAOTUF SSD. AC.1 o 0 I. ..S: : : :HA : ._ : ..OC OREEE. :N.2KXT '.E EEEECRUUUUCII ORRT. 2A.EH016.PTPPPPAELLLL NNH TTA. AL.2C1 9N. YNYYTYOTAAAADOOESSSU. TA.CRCUO.TATTTTRIVVVVEPPTDIIL. LB.TET6P.OCOOOOP GGGGXDDSA A. EN.ATACD.RERRRRPOITIIIOOTOFFV. DU.NINDO.PSPPPPANEEEEFLLSLIIE. .RCRALQ .OFOE 0. .FIFRRO... UFOTIPTEOSFS 0.. 81 .3 99 66 TO 501 ....OQOQOIQQQiIiitii.it.IOQOOQQIOQOO........ittfiitttfififiitti.tittt. TFCPROTYPE.EO.2) CCCCCCCCCCCCCCCCCCCCCCC cccccccc 0 0.. .00 9: . TX“..TTX6= POTQPPUTG OITT OOIIT 00. 115 3 QNOSS 0N3 C6ITCC6IT N oDPNN oPP LE.ITI To. .IIn... . 0 FOR OTHER GEOMETRIES . . t t . . . . t . . . .RR . L .EE . A .TT. . N .11. . R .xx . .5. .AA . T .HM. C . X . O 9 I . E .2./LA A 66 L . .TTT T TT 0 . D .000 0 00 3 . E .TTT T TT A t N .PPP P PP R . I . 99' O O! A . F .224 A! 660 P . E 0 .CCC C6 CC6 . D R .NNN No NN. R . E .III I0 110 0 . M Z .PPP P1 PPI F . U . O 90 OF OQF . H 0 .22A A. T660 T 2 . 1 TS.TTT T.”TT.” : t X N.ITI I:6OII:6O . A L0.NNN N3 .6NN5 .6 . H. AI.ITI 1C0 oIICO. . UT.D.PP PNIOPPNIU . nu GA. 0!. IFIOQIFI . N ER.113 3P OFSSP 0F . A E.TTT T..OTT..O . . . . . . . . . . . . . . . . . . . . . . . 1 UP TO PINITGQPINCI UP TO PINC6 AND PTOTI UP TO PTOT6 L A T N E H E R U3.PDD 01.03.0100 C 0‘. 9'9 QFXXOOFXX N HU.113 390055000 I SN.TTT T.11TT.11 0. .III 1:0911300 LSIV..NNN N366NN566 ATTU.III IT a oIIT o o INTN.PPP PIOJPPIOO TENT. N11 N11 : INIX. TTTFF TIFF NORA . TT69P O OTTP O 9 IP N.T99 .9...93... v T: .9999? Q: :93 0: : EOSR.9T611X3A61X56 HCAE. 6 91—? OUT-C1 ROTC TT ET. '1 06110N OIIONE I DLI.069C6CTT06CTIU NEA XQ6C6TCTPP6CTPPN IROTA . CECAEA . . CEA . .I PALAH . DTDM. TN 9 OUT" 0 OT .AIARIRXXAIRXXN .r.R—LOROOO.LR°00° ..RURFUFIIRUF11C IGOPTTN=1 FOR CIRCULAR ARCHOIGOPTIN ..iiitfiifittfiifiiii......it...tttfifittfifiiiifiifiitfitfififitttit0.6.0.0.... AND IGOPTIN O O O O 9 9 7 9 9 3 1 6 T 8 U 1 1 1 sccccc O 5 N 0 D D 0 L 9 I A. ! N 2 0 I D. 9 D . nu : L .b 9 N 3 0 N P 0 D P Q 0 U 2 L 0 T. i L I O 0 . X 2 : 0 N 1 1 O N]. P 022 D PII 0 090 L 0.. 9 L: : T1 .35 IN 'NN 90 X00 2P OPP I3 10.0 ITNE Q 9 0 TTIP . XX PPTVI :0an OOPT E11, “.530 D. 9 9 Q. T .TSR Q 0 .I TXSEP .. 2 .- SOEROZRS T217T ONqu 05 ORSXOOE .1 O, QIOPPR . 91/1 OIDDT .06C56COOS .(OCTCNTLLI ‘. CEAEOA . . . .DTM.-TPN Q Q Q .AIRIDRXXX ..LROROAJUOO .RJFULFIII 0 O9. 7 5 8 6 2 T 5 8 EZgTITLEboNE.NUMNPQNUMEGoIOATAQICALlo READC6001010) . . . i . . . . .S .S ...L .N .F .F .1 OT .5 . .R D .A E ..L T .N N .T I .L R . P .L .A E .R B .U .TTO .CPT .UI .RKR .TSO .S T . 1C .0:E .NIV .AL . AT .RCN X IN LOCAL ETAILS OF EIGENVALUE it...AAAOAAAAAAAAQQAA‘AOAfittdttfifitifitfitfi..itiifiii ID ccccccccc 130 34567R.0.01234.36789012345679 o 01234—3678Q 012314..) FTOCO 0121.4... .9790. 012345578901234567R o 0123 66666667777777777889888888999QQO9c9O900000000301111111111222222222233333333324494 1111111111111111111111111111.1111111112222222222?222222222222222222222222222222222 'NODE‘QIOXQ'XCIT‘OIOXofiTCIT*.10X9*ZCIT0./T .............................................t........... E SEVIBANDUIDTH OF STRUCTURE STIFFNESS MATRIX A T A 0 _TT T 66 U TTT I T 5 D 0' o 123 C A E A 0 0 A .11 o 090 A. Z T P T o . 0 = = : III 9 Q A 2.“ 0 T 00 5 0 U L 1;. 1. T Cl}\ 9 I. D “(A nu T 5‘. 11. = .. 00 9. T AR7D Hun. A I .24 IRXU 8 3t. 1?. O .T T 3 L TT 0 C TTT 0.. C T 2* :R 2 33 22 T N I .L A A 1:10 ta T. T131 TQII 0 T N AUKDr 9 3!. C O A f: '1 000 0 NVJN TTT. T QT E 0:0 Try! 5 nXu 0 D u. .L A T NUéJ 0 cl 131T .ECC .T8 H .ITT.—LK .UO TT.—b .L C T D T I CC T D PPP XTZ 0 T. E NOHC 0 TT 0 L Q N N ITO : : : : : : : G C5 L OOCGCE T 00 7 O 2 T T T PPT 0 T “56 TTT X1 E GGL TH 00 565 A B L N T CC 5 T 300 III PGF 3 C o 65 ET A T P E O 90 C AAAP PCCC NT3 E RT T G v.0. E C O 0 R NNG T L SOON NXTZ T R Q R T UCTO T TU. M T T T P G O .3 0A LLLW M n. U 5 o STN CTN 3.T 3N0 A 1 2 T T T TT . EU PPPU U .T NTT T I .1 00C T .1 TQQETN . o L C S STOP E NTI23 N6 NTTT 059979 5 : EoTEr G E. IEIQNQ 050 A A 152N N . QCTTTTTT O 0 000.0 E9 X10 X P. NED oNT E NE 0N :TN1 E CE 3 T D 019% onuTATIIQ.fi.$11 1. o o o 090:90 ‘1 E ouv?T 0N u. .N 1 oNNC: .1. o o A N 202U EU:DNNN . o 0.. :0 :900 NRTTAT N T E OONEE U7 E . EENNLK N:N N D A 91 ON P0T:TTTEEEIJ o TEEE T 9 O 0 9 A T PKICPN N2 PK T. D .0. CE TTTT D 191. TO TPDPGGG OI oooEDT/ T/EEE H YCQLYEEoQE TCEU.NKY LN KP PEEE 0 0 D 606E T11T:::00.00::0XXXU06C66CUUUD U TEOETLUQSU TEUP.ATT5EUIOOIOUUU A N A C6CN 0:0C123GQCIITK2TTTNICTOCTNNNA N 0H6MOENE N 0HNN.BSO4NNQNU4UNNN E E ECE . REOEDDDEEEOOJCOFFFTREA9EATIIE : IRCCUR T oOT RCTO. URTU:3NS3STTT R R TDTN PL5VAAANNN33 9E3TITTPTM.4TH TTTR R OPTDNPLTNTT PTTC.LJP5NE5CM5MTTT AONCCACCLMIC Mu CCN ALCC CV: LC CNNN D. .. ..FF...FFAOFOO FF) .AFFOFAO ..FAJF 003 TNNTTRTTCCIGC TIC .CTTDTNDHTDTCCC TATC CA AOOOCCC TH CCCNCTD TRNNN RERF PCOSLLLFFFOJCCOFFFOFROOROOOO URUT TSDPPPPTTTDDUTDTTTCTUFDUFCCC ......OQIOAAQAQAAOCOQQO......O.AOOOOOOIOOIOOOOOOOOQOCOAQt......QOQ .....Cfittfifiiiitfitfifitiii......IOOQOOOOIIt.....fiififittfifitfitifiiittifili .....OOOOIOOOOQifiifitttfitfififiitfititit.........OOOOOIIQOOOOOOC0...... CALL 1 0 1 O 5 7641 7 2A 9 5 150 0 0 1 3 2 9 9992 2 2A. 3 2 3 441 O 0 0 0 0 9 9990 9 24 0 9 3 371 CCCC CCCC .3 C 5 3 1 5 A A QASCCC 5 2A. 1 5C SCC 552CC 131 4.567890123456799012345678901234567 9: 5123456789012345 b78a 91234557.! 901234 557 83 01234 64‘4445595555555666666666677777777778R899898839a99990:990000009000111111111122222 222222222222222222222222222222222222222222222222222222221.3333331.3333333333333333... VECTOR LOAD OQIOQOQQIIQ.if.OGOOI...fittitfittifitii......Qlfitiiittit...iiiitt... ASSENBLE INITIAL LOADS AND NODAL LOADS INTO SET ARRAYS -S- AND -R- EOUAL TO ZERO CCCCCC HBLE CH) EE T. LT Lrw A0 CC 3 3 3 3 .56 TTT TTT NNN TTT PPP .. : = 61 TTT 31 456 31 NNN 32 090 P9P 30 390 TT 123000 TTTLLL 00 TTTCCC GG NNNTTT TTTRRR PP PAAA T O = : :TTT ST E TTTSSS .1 N 123PPP E . OUNNN Nnu 1 0009TTT .E E COLT-.009 . o . PKI TOOOEEE YCT TLLLGGG RCOORRRREEE PTTSAAAANNN CC TTTTCCC FF ODSSSSFPP TTGOPPPPTTT .06 33 233 :33 K COO r.TT H COO T55. , TT1 11 o o .0 GOP. EE 0 0 OK KKC TTE CCCA 2112 2 ES PN TRRT TTSB 0V99 RN99 PT 9 0° KLTT CL EAOO HCGG C anJ TT T7,..N Qh..TPTA N .3 .58 CPL CONN" TNQF. 9 9 ul of. .11 TIUUPZK ON 3 2 0 332LREP11: I 2113 LCCC 1901 1801 GO TO 2101 GT Eh. 0 IT UQN NEC 9 OT TTT. 3N” NCA 88 89 99 “A 00 TT 00 66 ED Tatar. TCTA q: .(.8 .QNM. GE 0 O .... .11 .N : : KTTJ LEEECT IEBUUEPIT : o O 0 TA CLNMAC 7691 GO TO 7692 IT... :NH NCA CLODPONlT.E0.0.T GO TO 5010 .PSAV RED 07:11 CIA 258 ON». 0 9 9 ..—11 I: :0 KTJ . LPLEPEC o 2....3UUF.66 .- 9336 GO TO 9437 ass 0) U OT NON .EC 1 01 ..TU NNT Cs TLPPF.» ONISTTOTCIQJT GO TO 5010 S S R A O 0 A TTT n1. TRSCC N- SAISS ITN NSR R12TT NTTU E11 KTRJ . . ... :TRI: 000 KTJT:1TTEF. 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S(OENXHO EE.‘ .1 00’30H5N05N000)90 .C .... 0.REDDDEEEDDDEEEL: R((.NHTO.CRRF.T0 TE. 5PRTIAUC 0H. 0. 88J:1U IC 1299J..1 .NLT SaPLAAn—NNNOOONNNPTLPRR.ITAL.RPETIT IR. 1MPITHNN IT.LT.L 11 QRZNLTNLTNII 9R3 .(LN .(AOOOCCILLL(C((EL(((. .((T1.u w. . F.(HN:(( .L .L IA (LN(LN( IA .FAO .FCLLLFFFCCCFFFFDAFFF. .FFERU3 U . OTFUONFF .A .A OOCPOFAOFAOFOOCPO . ICC . ISPPPIIIRRRIIIIICIII . . IIOUNG N . ODINCNII . C . C DDSIDICCICCIDDSID . 5 8 1 9 1 7 5 85 7 0 0 2 0 3 9 0 7 5 71 7 8 1 0 9 C BCCCC 2 6 CCCCCC CTCCC 1. 77 1000?. CC CCC 1 2 3 I. 137 0123456789012345....79901234r) .317». a 0123.95.67 : 6.01.23.95.67 96 913345 6799912349679 3333333333444494449455555555556:666666o67777777777n9cQFggF880:ccocoon 777777777777777777777777777777777777777777777777777777777777777777777 ’ 0 A D. T ... A T. D 8 ... 1. U D 9 7 9 N .1 E 0 3 G T N 1. O E 0 P 1. G I D L 0 0 V 1 L N ..1 9 1| E R 5 D 0 G E 1. 9 0 1. L F .1 9 .L 0 9 9.. T = N 0 L o 0 RC 0 T L 3 E Er. P 0 A I I L TL 0 0 0 c 2 0.1 . IF 0 G 9 N [5 ..1 Hurt L R 2 V o .1 U0 1. O O N L0 1. NH 0 T 1 11. R A... N 97 9 1 ZNN . 956 HF 0 )9 1 1 4560 o NRRII CCFL 9 Dr 1.x 1 1 CCCG a R..NN NNN S: 0 N0 N o NNN E SSRR 1.1.1. 1.0 O 01. 0 0 1.1.1.1 . .1 o .9 9 P99. 1 N L Pl P E PPP’ 0 U IOOKK ... ... CT... 1. 05 D o 00.1. 1 2L9 IKKRR 21. 222 956 NND. D 0 o 0 N 955T 3 2A9 9RR= .. 10 I’.’ 000 I—LH I Lo L fl 0300 9 2V“ 2::1’ 7 ~56 AAA 9H5 1 (1 ( P AAAT o 9 no 1.1.1JJ 0 C36: 900 95 9 \1 «JF an o 0300 T 8 AVID (JTuQO Nnv LNNNW LITL 11K! 1 90) 9 T Llrft TET D 9 91.1. IT 1.111. 99.9. 0C5 N 1:5 1. 5 9.9.95 0 0 . AIII... .l PPP : : : AN... 0 DD... 0 D : : :8 G T 000 [(1.0 D. E0 : : 2123056 011/ 9 AA 9 A 01234:..5A GNG RSSSS PG 456CCCDDD L05 0 00.. 0 CCCCDDO . A nu A n2LCNNJTAAA ner o .LLQ::L NNV6:3AAT U G \r 9.1 0 1.1.1.1.! .11 NNM‘ITIOOO 00 L pHN—LP Alt-310006 0.1 1.22 N 101.01. 1.1. IIIPPPLLL L1. .1 1.1 Q 50L .0 PD,LLL 0 ON ..1 .0. 0A 1...... o. ...PPP. ..PPPQ .1HF 1 D... .1. 191A ....PPP) El. 0 000 EBKNQOQQQ 00 29.2 1.23 5.195 9 N .N N.1 9XTC.1 01.23 1 .2 ON Err—E NM.RRE:;LEE f3». ”/.1.1.1039.1.1\1M11afi 2: O 03 91 90AQ.9 0900.11.10 ‘17. an o o o 9 9 o o o o o o . 123955AAA955 9.1.9.. 90 D L0 1.1. 3119\(9 r.AAAQ=,.5A NI... EN ECU 11.1)NNNNN NR CCC . o .000 o . .1P1XA7D 00 :D 1. 9E13 L000 0 o .0 (A .0 PPLO : 31.111.11.111. IEZNNNEEELLLEEE:V. 950708U081. : 9’TT 9 oLLLpr—EEL9 LEE—LOCEYYADETJIITTTTTEEA6TTOIIIGGGPDPGGGIEIIL7L7‘L7 .1 .../1.51. DPPPGGGDQ E8UUNTUTTV9U 9 9PPPPPUU 9170 PD. 0 o o: .. .. o o . T61_.D.11\71\1.751. G(HD6RE: : = o o 01.2 HSNNOSNOOG N784600000NN0000. : : ..QQQIZSQGQSDCTH 0 SD 5(3‘T1:(ET123090$ U TIC IR9.107.00‘1.229.9.2111”.N2U0123EEEUDDF.FLF.3:EAobo:03:OSP:EAIRF.TEDDDE£EDSO NLTTNLTPPETTIIDDNNNNNTTIINNTCanNNNAAANNN31TV. 9TDTADT2HTTH 9F.TICAAANNNAT ‘LH‘N‘LN119‘1‘ N AA(‘(‘(~NUU1\1\ NM‘N“(OOO1\I\1\ IT-Rx L (\L rLEIQXTIMa‘OOO(l\l\l\ FAOOFAOFFFOOOOEEFFFFFOOEEF FOIIIFFF LLLFFF3‘R05030F000TOR005RUFLLL-...... FPO ICCCICCIIIGCDDRRIIIIICCRRIIGPPP11.1.0.9.91.1.1.0anIGUGIUGDDIUFIDUNIPPD1.11.1.6 O O O 1 9 9 5 9 7 09 9 2 87 9 5 9 1. 2 85 9 9 7 1.9 9 2 00 9 3 R. .U 0 3.9 3 .J 7 5‘ .5 9.. 1.1 9 3 1 7 7 72 1 7 1. 3 31123456790 0 900.000.60.103... 7: 988868983... MATRIX UDULD SS 0 IN SUBROUTINE NLEIGNP FOR NONLINEAR - a . (N1) STIFFN HOULD BE CONSIDERED ................................................... =1 IGEV GENV?LUE SOLUTION USING DETERQINANT SEARCH METHOD (SCALE) ESDUF; 0P AlLeu. .UN ECSAI. 06 v NVE. 01. A.LONI.10.L HHCE ...IL T GR.N:N ONEIO.EE TTBEF.GLL .IAL .ECA .ISC ......It.‘A...99.09..........tfiiififififiififififitIOtOQ'QOOQQO99.99.99... CONTINUE 99 C “rvcnbcccnv 138 1234.36789012349.67Q 90.12.34.357an 01.9.34.fi..b7§..°.01.0..34§.6759.11.2345. C7Q Q 01.928.033.079... 915.398.67.590... 11.11.111.1122222322323333333333 4 4 6 a.“ 4 44 44:..5555555: .566[=6656666777777777752.Q 9:... 3 9380.... 88888888888883.888888980.885869.89-89.9.R.BP.8R.830 8898888.888888Q.888R880 F.998985390.98838888 9 I, .. I, 2 NZ L OT A ....H 9C 72 I X1..L9 8 7H EX . ,6 RS 5 I 9 T. 9 I 1. .5X .UY 0 F I7. T 1. 9 9’ DH . . ../ ‘2 .D/ : Pt..U9 10.1 2 N1. LX F15 C H9 H8 9 01. N U: 9 9 95 9 ’1. N1 1.x :1: SP HL 9T )FQ. 0. 6A HRH 1. 9N 5 9 9C 72 N .0 1X 9.1. 2 9 0:1 F0 7H 9X P11. 91 [Au—L9 03A . 9 I 9 D 9 ONR :8 3x 02 LOT. 1. ..1 1.7 "H (PT C55 9’ 1 Dn:. N191 :/ 9 .0H IF 9 3 X 9L... D. 9. 1. H9 X11. .... 9 7.9 00 9 9..R E: Iv. 1... XZF. Nlnnlfl ’90 OTT HT 51 9X... 1.1.1. 6‘ 0 9 .0, 9~x 9o ‘x 51.5 81‘ x1109 1’1. 0PM .IH1759 F0 0 5. 0 ’61. x 91.5 I. 9 9.1 ’ 9 9LH : .2 FXXS OX...A1 05F. 9001. 1.731. 9 ‘1. 9 91.1.7 AlLTX’ 0F .. : 9 9 9 9,4135 L 9.1 1.880 03cm... 0 H 9",?! o 01.1 5..‘ST.55A1. 9.1NIN1.1. 9 0 .1le 9IFF00 50NM.6P 9 91.1. 110R .9 9 O‘F .1 . 9D 50 9.. .. 95 9..|./ 995 .3117LUT19112nu91.m91. 9 1.1.91.1.Hl.1..lo.11.1.1\l. TTTTDOLTTPPTTTT A‘AA 9 ...AA 9 t AAAA "Inn" 9 9"," 9 9HHN." RRRRxxRRXX—KRRR 00000009000000 FFFFIIFFIIFFFF 99 99 O 9 CONTINUE FORMAT! END Ofifizu 1.1.21. ...-.900 1.1.1.2 9 99 01.9 99 9C1113 67 2015 FORMAT1 00 2020 SUBROUTINE NODDATA(IGO9TIN) . . . 0 9 D. N. 9. . .39 R. p... B. H. U. NY. L. NE. SATIO ECTIV ..... -IA- AND ‘18- RESP . TO CALCULATE EQUATION NUMBERS AND CONDEN STORE THE" IN APRAYS .......................................... .................................................................. TO READ AND PRINT NOOAL POINT DATA c ccccccc ccccccc 1/NEQNU3NP0NUHE59LEC36).NUMEL(3).IPARQICALIOIC‘L2.ICAL39 G 1 . . . t . . . . . . i . . . . . . . . . . . . . 9 t . . . . . . . D . . F. . . G . . M. . . A . . H . . C . . . . E . . B . . o. . 0H.. . Lo... . UR. . 0A. . H . . Sr... . H. 9 1T. . N . o ..G. . YN. . 1.. . 3N. . NI. . ‘F. .A .... .T10. ..AN . .01.... . XV. .T 9.. . HQU. .IOC. .OF . .P O. . ST. .LN . 0‘06. .319. .05.... .NSD. . ER. .090. .APC. .EXC. .REA. 9 9 . . . . CCCCCCC SS OZUU 001.1. 1.200 AA CORR To! 9 9 00 CORR 65E...— 27.5 )IFFNP flundLL/N .1.1.1 9 9AAOM 00500 O U 011E: 1E“. “.30 o 0.132 9 222NN02F1. 9 9 9111.... 9L .. 6659.906: 9(I-119ALFINC) 2) 2) It IT 031 91 9 .. .AQUZOE SPAN*92) INE 2/1 .X(N)9Y(N).Z(N) II?! 1.1. 9 90 NNG ), .10? IOLN 03M. OZU NNG ” .10P 02”.. 00v. 02U 1. 9N11. 9N IIITT QIAIODREO 91.1.0955... 91. .O 91. o OSIUUO fiEloE—t 1.1.1.0061C10RSN 61.30.RNN61.N (DIN EEEGG‘ENO:...-101....0..::1.1.1..L 001...... . TTTIIDTIZII1TTh.72111TTDTNTDTN 11119.11... 1.1.1." A1. IIINNAII Anvil. RRRFFEQL31.1.1.DPJ..LRO1.((OO.LRFops-9...? uIU”IIRUADZXYCGRUDZXYCCRHIGRHI 201 232 000 229 o o 1 . . . . . . .D .N ..A I .S .R ...L .2. .w. .U .N . .N .0 .1. .T .A .U .Q .E . .D .N .1. ...r . .0 9T . .. .B .1. .. . ..U .N .I . .. .A .1. .— i .S .9. 9A .R .R .A . .S .S ..L .C .0 .R .D. . . . CCC 139 234557890123 95678O 013345;:qu 99.19.34.367; 0 01.2345678901234557Q 3 0123¢5678° 0123456746 01.2 goQoaa9Q0000000000111111111122223222223333333333444444444455555555556666666666777 888888886 9996 999 99999 9.9Q 0.0 999G 9C 990.6 990. 996.90 96.99999: 990.0 90 990 Q q a o 0.996 0.9: Go 6 999a 9Q. T t t c o D t t 0 H I t N a c X 2 3 ¢ A t .- 3 9 t c O 0 t 3 0 X o g 3 A O Q 0X 3 0 t L I a c 33 v a t A i t ...3 OY .— t C S O t D 9 XT t t I V. t t 0) 3H t g Q A t o C.1 O 39. t t 2 R 0 t OX 0 9 c t L R O 9 ”(NS \Ix t a A A t t O .1 0‘ 9 T20 O Q c t t 1. I ‘1 TAT Q 9 i 0 I N O i .. I) I TIT Nun ) Q t 9 I a i N 9/ I IHH (T 3 c t 1 t c I 0/ 072 AH .... o t L S t t \c 1. O N 9 O '12 O t t A a t o 6 .0 OXX H D : o t C D 0 t 0 51 A C13 7X 0 t t I N o o 1 1. o 2 O O OQ N a E. O 0 o t : F5 T V. OX XV. Q 0 o N. R C 0 t T. 91 «KQT 212 C QT... A N o t O :F A AEH 22H N QT. D. a t ‘1 S O .03.... 0 01. H c U. I 0 t .- 1 U: 0 NV 9’ Sc. 9 6 to. Q ‘0 N t t 0 TN UUX) REX 9 «R t ) Tr. A c Q N 0A 0N4N .1534 X .8. 3 ED. . Q T AP 8 0‘ IBM. ' 3 ‘1 .Ut ( PV. 5 t a 8 RS T HZZ)/MUY O E 051 L YT I t A. T HH TTHHS UNH 3 P e a E T0 0 O T: T 75 N V/ld. oAN 1. I v. «T. H 0R .5. o A. O Q. TSOOZT H0 9 T 0N0 U RD. N a 0. ) XX 1. OEXXIANTX : 0 «E. N P Q t t 6 00 )PTQR.F00/4 Q R 0 Mt v OK E a To 0 0 1.1 0/ A903 .150 F. D. *5. ‘1 KC R o N. 1. N 09 ILNY) QLTRX N 0 0L. 6 CE 0 t It : 0 00 P AIHNXAAEH H K .Et 3 EH. T t 0. .... C 11 A301‘40U.31 4 C t t 1! HC 5 t 5 Pt 9 N o o TOR OYIOQM. O 9 E OE. .L CI 0 0 t so 9 55 ANOXH ONEUX )X H 0T. L I 0 o 0 1. L. 1. 0 1.1 LDHOQQR. HNQ 53 C «A. 0 ON 3 t 0 A t 9 r. F.,... .60 O 0.106 ls .... O T. .1. G V‘ R 9 0 2 0 c N N 9 9) AL.) XX6EIN2 63 O o R t r. (S E o T 1 09 as Q ....5 ACTHS 9T '00 0.... N to... H MS 8 0 O No A 5 AF. 0 00XN.1 OXAXIX X 0 ( .0. U AU I. at 0 5 .- 1 2 F50 3.0.... !)1R5T1 1: 0;... N [R UV. 5 1. 0a ( I LII 0N20XNIEIAI 1.5 T 9 Pt 0 8T NL. 1. .... 0 S ARF OP 4‘ ON 03/ OZ N to t P E. P ‘1 0 T. N N HH3 NT... (X:.Er.~.’ 51 r. «A. M. LL NV. N 1. 0 0 A. C ’55 9’ UDL?HIGD.LI TS H. c t M E LL 0.... H o 1 0 N R. ‘1), ) 09933HP0A 046 009 6N F. 05: U P AA IT. U E ITN 0 Ft 000 0 IXXII3NN0XOOHNNN 9H L 9H. N Y CC Tc. N6N 00 C No 345 6 0002231 01XX2 0‘ X6 E 0T. 0 T A5. 0 9 0 )DC 9 N p... 000 0 5.11.1.1 9 “19 62 :3 6 0 a Q r. 0 SP. 11’ l’NN ..L 0 69 222 2 I'OOOQH H/OQ/ HI 0. E 9L1 N R ‘0’ NS. ....IOO.IO.01N 0 g 0 o o 9 FII5518H3))51H0H 5. N o L: I P 12 5E. NT 0:2: 90:2 0 :EEE E g 11.1 .1 2,111. 172TXT/727 .1 t I t A i 1 o 9 0R. 0 N’l’NN’lOTUUN T. 666 6 ((((1\((( 90((( )‘l‘ T 90.. ./SR 00 N t :50‘! IT:I EINN : I. ((()(N TTTTTTTT NITTT ,TT U i . NSF. EEN 08. 0022A '0 QBD QON 01.1... R. EEEPER AAAAAAAA ( QAAA XAA O .0. DEC . .R col. ..NIIINTNTNNT :NTTZ U. TTTNTU NF».H.MHU..M BZNMM. 0v." R 0 Ti MRE NNU i 00 (I. ((0.! G(NNI Q ITIMTT RRRRRRRR ....TRRR IRR 0 B t .- 0 TT ((T 0 o ECOOFAOAFCBOEAOOS . RRRURE 00000000 HH000 000 N U a i OSN FFE N t NNDDTIGIINIGNICCN t UUUNUR FFFFFFFF TZFFF ZFF E S t t C.1... 11R E O O o O O O O 9 01005 000 00 1 5 0 5 05 00911 239 56 0 0 .1 1 22 30390330 030 30 CCCCI 1 1 1 11 0000 012311222 222 22c CCCCCCC CCCCC C C CCCCCC 140 SUBROUTINE BAND c cccccccc NODES OF A CE BETUEEN THE ota.a9ooottc.oagootngottotototcooto0.9.0.......ototottttoctoc..tog ......OOOOOQOOOOQ0.969669... RE STIFFNESS NATRIX N E TO COHPJTE SE‘IBAN DONE 8T FINDING TH EQUATION NUMBERS A Ofifltfitfifitfiti..fit.tfiffi PARTICULAR ELEMENT QIXXTSGIQKTTSGI 'QNUHNPQNUHEGQLEC36IQNUHELCSIgIPARcICALloICALZQICALSO N T 6 S A ( Z 1/N I I I o E.O) GO TO 199 ’0 0 10: 0.1L E 0L 0 N 6?. a) 0,, 0 0’9 Ivnv ILI O 0..l\(:R OT. 31.9.1645 678901334567890123456780 0121.45678Q 0123456780 0123 3456783 0.123A567Q 6 0123456780 CUCCO.30000111111111.1222222222231.31.333333aa. 4 a 44 A a 4 Ar): 55 7777777888888898809Q9aoo9a9003000000OCUDDOOOOOCCOCUCCOUOUOUCUCDCCOOCOOCCOD0000300 u;79oév9ai99c;vfi6499649964$96:99a4?91:11121113111:111:111:1112111:1112111:111ill1t111t1112111§111€111§111:11 oLEoOD GO TO 809 "I II 0.. 1 = I. IJITINPFIID, .- OL 5E NN‘UNN9L : RUDDOOIANII R DTCOOOCAIIAAOT A0 2 .- (‘nJN‘ 2 0.. a. Q 9 0 9 Q 9 0 3 2 2 8 3 0 0 0 T T .I T o 0 0 Ad 5 G G G ’ ’ ) ) .0 00 00 00 0 o 0 .0 o 0 0 QE 0... or. CE .LL ELEL ..LL L o L 0L 0 L o a, o, o) o, ’6 06,6 ’5 3 O 3 93 Q 3 0 0J OI OJ OJ JN INJN JN .l‘l‘ N‘N‘ N‘ (A (A(A (A AT. AIAT. AT. I o I 0T. 0 T. o .0 .0 .0 03 DN ”NON 0N NA NANA NA A o A 0A 0 A o 00 00 cc .0 0 o 0 .0 o n... 0 0E 0E of... CE EL ELEL EL L o L 0L 0 L o o, o) o) o) )5 ISIS )5 22' 2920 29 0 OJ 0 OI OJ VJ OJN 0 INJN JN INC 7 NIN‘ BN1; (A (A(A U ‘A 3A.... 0 AIAI :AI TI 0 T T. 0T. 0 DI 0 09 03 00 N CD 03?». 0 DNDN ADN son‘A Po NARA «DNA A o A 0A 0 "A o 00 I .0 00 .0 In. 0 0 0 on. o 3 o I or. 0 2L OE I of. GEL E TEL—LL DEL 0L. L ILoLo NL. 65 o) o N a) 0’ A 0) 90,0 II.I4I4 at,“ 1L1. 0 JJ21 91. 0 9.1. O :R OJ 9!. 9 0 ON OI OJ 0 OJ JTXHNEQL. 9n€JTTfiTdN .IJN NN‘UQL UO‘AN R 2N~S~(N(IG~( ‘(BIA‘A O o‘A 6).LE.0) GO TO 900 3).LE.G. 9 1. 9 CI EINEEI UNCUU6 NIANNIN E L 02000) "BAND OCAIIOTIOAAAAIAIBSAIIAIIIER NNO NNBII oTIITNOTII oTTNNTIIII oI oPu,I oTI oTTTU (ION o: 1‘: :(jl‘fiu:(l\0 NTnJNNIT BCOIJOFFNOF IOCIOFFNOOCSjFZBF NFNBF FNOF NOORE HIDNNDIIACINGINDIIACGII.GINNIAIAHIIACIACCUR O 1001 199 99 .9 1332 O O O 9 9 9 9 a7 9 3 9 2 6 0 .UD 0 .JO 7 39 QISI FORMAT!010.2OHSENIBANDJIDTH HBAND END 300 SUBROUTINE BEANTNQICHECKQPROTYPEI ELEMENTS ARE CONSIDERED ....ifittittttttfifit.....ttilfittfititttittitlttttfitilttittfitiitifitifit BEAM ELEHENT SULROUTINE tttttfiiflfiitfiOOQQOQQQOOQQIOQOOttOfittttitttttttifittiiittcittttlfittlfi ONLY STRAIGHT BEAN czc ccccccc cccccc 7102(37) IINEoNUHNPoNUMESQLECSG)vNUMELTSTQIPAR.1CALIQICAL2gICAL3. QYTS EIGEN 141 456780 012345678901234567R 90123456790 5123456786 ...12345679o 0123455730 01234567590123.“ 935.555.... 641:5? 8.661667777777777 4..., Q .526 2.9 980 o c 6.96 3.0 G O,00000030001.111111111199L29.14222221....1.1:... 0000000000CUOOOOOOOOOOOOUCOCCC00.000200“.COOCOCH.OIIIIIIIII1.1.11.IIIIIIIIIIIIIIIIIIIIII 1111111111111..111111111111111111111111111111111111111111111111111111111111111.1111... t t t S. t t t p... 0 Q t T c t a S. A c a t T I N. a a N. It 9 t E. 01 I O t M: t 9‘ t 6 t t E. 0. 3 a a La 0. I t i O E. Ct T t t O I t 0 K t c I w. n.‘ Lt O t 9 M I At A t I i t I H. Et Ci 6 O a t M G 8.. 09 3 I t a G P c L. I 0 a a P 2 F.— t X 6 o a Z O 0. N. X I t t O I O I. I C t t I H X a t O P. a N. H I It 5. I O I t 0. I V., R o T. 6 I 2 t I. v- 5 T0 N9 3 o i a T. G P A. E. I 6 Q t A t 9 v. H. H a 2 A I I t M 9 v. O t E. 3 3 2 o O Q I o R. O I Sn L. t t a c I 6 z 0 a. I H. St E. i 0 II. I 6 O . a F o N I E. t I I HMI H 3 I J t No I A N. H. O u... H. II" I I U z 0 I. 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(INV‘IMI no11NM¢ND 9‘ TD 0. i M TV 0 \ FALCCCCOCCCCCOAAOKLOLLOOOO OFROOOOFAO:FRQIJ3C FAAQF:I\FROO(FD\OOOJO:.N U t 05000 IRRRRRGRRRRRDNVDJUDLUCCCC a1UFLCDINDHIUCNNDIINNDIJVIUCGVTUCCCFFRE S t CICCC O o 9 9 9 3210 Q 9 9 96579 9 9 9 0000 0 9 9 9 1 93010 6 7 Q 1111c: 1 2 3 9 9 11111 ccccccc ccc 146 9012345678901234567890123456789012345678901234 5678901234567 P 901234567 890123455789 5666666666677777777773839988938o9996999990000000000111111111122222222223333353333 4444444444 444444444444444“ 44 4 44 4 4444 444 4 455: R 5555555:.55555555r1.7:.5:.5r..55.2c6.5:.‘555535 111111111111111111111111111111111111111111111111111111111111111111111111111111111 ) 6 3 ‘ T K O ’ 5 9 3 \l 0| 0 x 6 x p! 1 C 9 R 1 0 1 6 1 2 3 o t 1 6 t A C 1 I. 4 1 1 G 2 6 O . 3 1 J l‘ a Z .d) 6 ( [6’ 1| 6 0377:.) 2 OIEELG t NH1Ru' I 96(1) ‘9 ,pbn'o 11. 1 52 9,6 00 V- 3 .‘lol‘ no - (1‘524 11. J 16 Q ’G a V- ..5370 9 003 ‘ 0136.! E 2T7. O OHC3AU L 2 “NGNPLU 9 OOnV g CPD 9‘ T TGG 0 ’V. C .1 K 1 3 0116 9 0 1 (150 0 Z G, x 65 OZ’SZpL 1v) . .37 VOU'IP 0122111111-0 "SOSA 09-12.. oN.”’)\a’x 3Z‘6TGYT2NQ QHHHHH nuts (21(31Y30051((((((( E1PS/01RQE.:111JJJT I:!I/flTIQPf2PKHF§ELFF:LR 5’89 10x ....QtfiiOi...fl.....iti......Otifififififittttitiit......tl... ATERIAL INFORHATION t 0H. OYC DDDDDDOCZ i QECN)OG(N) nYID 232 01.! 1191 IA7CI/932KRTEfiDOOKXUOnibU1fiutGnXb N3NNNOIEAOH5NNNNNNzN A.(6( 0'0000 CPRC5((((((’10EQE(E "3MWHELEIPI5XYZXYZHTTR9TDT ”(MqvnnlAT‘l‘l‘ 2 2 2 2 2 2‘ V . OLOOOCEMVFFFOIIIJJJ—Loa o COCCCSRIIIIDXYZXYZLCG c O O 5555 1A1 9759 ”IR”- 1001 READ ELE“:NT AND CROSS SECTION INFORMATION titOOQOOtiOttititOiittt00......ittitttttfifittttfititttttttltttttttfit URITE(6102821) CCC I‘V).NODEJ(H)9A(N,oYPGR(H)QZPGM(M)I C1(‘1'NODCJ‘M’QA‘"’OYPGH(‘)OZPGM(M’Q O 31ru) On NM» N1 9‘ if.” T "K K 0’ Q ’19..) OMDEH GUILJ‘ 0227. 12 02 0111 0 '6 O 61‘, GO TO 105 :EENJJSEU 1DD‘NN : NNN (HF;.1K0n3§l(I.:1R D(T(+ ONNCYZVKTU OAXTXK N 2 2 ..(LCCNT :5XRX:(1JA .. .35.: DE KRIUIKLNM‘ABCLICR O .9 105 CALCULATE AVG STORE LINEAR STIFFNESS HATRIX OF TRUSS ELEWENTS titttfititfitittttttfittttttltttfittttttttttii.tttttttfiitiiitittttttit 3 \IA NN 1| 9 L1 .L: E MK, U U K uv N0 9 1 :2” T 52‘ N u. L 0 A0: C N09 9 no CZCCCC CCC E. T o E(N)OA(M)/LE(H) 1 UOJNM‘11 00 911 9 9 451TT17 (Nanw‘l‘ . «JO—p.305: i DDSCCSS T’RHS OF STIFFNESS MATRIX OF TRUSS ELEMENTS IN LOCAL COORDINA Citlififlfififiifitfiififittiil.....tfifiitfifittit...tttifitfitififiiifitfil.... 00 0° 54 O.ti.......iti...4......tittitltttitiifitlfitfil......tfitii Y R T ... H. V. \l v. 2 S 1 9 V: 1 B : 1 x O 1 .l R 2 TI 1 A 1 9 9 01. 6.. F J 3 O Q T) F J L O O A GT. H 1| F. 9. ’5 E 1 0‘ P 1 01 P. 22 O 0 U 1:1J 9L) 9 .l‘ 00 NQ1T¢L S1. 1. : :S S I .1‘U..E—b.0£ Lt ,UR1U L t OOJNTI‘N 1 t 11 OISEI F . 221T1Tvl a (N‘IN a DUPLOFRO QDDSCIUC o 1 1 o noon: GLOBAL COORDINATE GLOBAL MATRIX IN FOR” 85(12912) FROM LOCAL TO STIFFNESS 1 IS THE thittttttititiittttiiit.itttttitttttiittttitttattttttitt RANS 2912 it. CALL TRANSFH(M) TO T SECI .ttctit 147 012345678Q 01234567901019.1445 673901234567R 901234567: 0 0123456780‘ 0123456790 0123456789 0 4444444 444555::.:55555 6‘. b; 56 666 5777 7777777 89 99R .9 9.? 9 8921.94 2. c o. G 90C0.40000001111111.1119. 555555555....55555555555::55:z..55555555555:.555555c.5555ch.55555c..555 5696666666‘. {666666666 1111111111111111111111111111111.11111111111111111111111111111111111111111111111111 . t . s . . . U i t . L . . . U . . . 0 . . . 0 Q 1 . . . M. X 1 . . . d 1 1 . . . 7 1 . . . Q Q K t . . X or. C . . . 6 L 0 . . . Q H L . . . R 2 B . . . E Q . . . B X H. . . . H 2 9 . . . U 1 I 0 . . . N Q I S T t . . H \r I S N . . . 6X H T E 1 4 I 0 Q1 T N N . . . X/ J E F T t t . 23 E) Mn 2. N . 1 . . /U 0) E 1 I. Q C . . IL 0”. L T H . Z . 1 o ) 2U NT E S 1 E . 1 . 1 1 2 . 1 1 2 10 HT, 2 L . S . 1 6 2 1 . 1 6 2 1 Q0 8K4S T T 1 E . N . 1 6 Q 1 Q . I: 6 Q 1 Q N Qu. as N ’N Q . Q . 2 Q 1 Q 7 . 2 Q 1 Q 7 RH X55U E 2E 1 S . 1 T . 1 1 = 7 2 T . 1 1 : 7 : 7 3 QIR M. 1n. 2 S . 2 N . Q 2 1 = 1 N . Q 2 1 2 1 E Q QXET E QE 1 U . 1 E . 1 1 Q 1 Q E . 1 1 Q 1 Q X ’03 L L Q R O Q I. t 2 Q 1 Q 1 Ir Q 2 Q 1 Q ‘I 82 ”1 QF. C RE .1 T . ... E . 1 .1 2 .1 2 E . 1 ) 2 1 2 1 TQ60 E 2 S. 0 L . Q 6 1 6 1 L . Q 6 1 6 1 H Q 1’ o H BH 1 HS. N E . 1 Q Q Q Q EL. 1 Q Q Q Q F) EH5X C1HC Q CE. A L. 2 1 7 1 7 A. 2 1 7 1 7 U07 0T11 A . UA 21 AN . 3 HA . 1 : : : : HB . 1 .. 2 : : HT OZFR ELNE 0: CF. 01" CC. Q J J J J C0. Q J J J J N21 NZST A 6J F. 11 Q A0. 1 Q Q Q Q AL. 1 Q Q Q Q \a QC H1 QA FBTF Q F1. 1 EL.1 2 1 1 1 1 [5.0 : I: 1 1 1 4 X1 BHXH \c 00N0 0) 0T. 00: .5 J J J J J .3 J J J J J oPlT Q56 1 LE TJ S. NNJ FN.2 Q Q Q Q Q FN.2 Q Q Q Q Q 0 15 X Q QS TJTSGHS Q S . 11 Q 01. 1 1 1 1 1 01. \u 1 1 1 1 1U QA 3X55 J JE EC 01 SR. UV) .0 J T T T T .0 J T T T T F FL Q91E31KJ1NL1 GT ptA. EEJ SX .T Q E .L E r. SX.T Q E r. r. ..L 301.... T Q QNLu C RIFZK 9.. Na... RR Q E1. 1 S S s 5 E1. 1 S S S S Q HH N’XF QKOKT HT, S F". 1 IRQO T T T T T 19.0 T T T T T 6R20TEW6FOCLCNX5NO 11‘ F1. \ITT RTOG “F. T T T T RT...“ NE T T T T o .169 T Q12080E11E 0T 1L.’ 11$ TA. E S TA. E S 06X Q oEX5TFL L.R Q.L O T t" 00‘ "H.,, "1‘111111’, ""61, "11111111,, 1’1H4X7LX1368HBQTXQA 0 SL.T GOT E .041A3T02426282 E .02)A3T02426282 52F3/41EIQHQ 9 XAOXC C, AQC ...-L s. QSNNS 1.3333333 S. OSNNS 333331C33 1 .32R Qt. HX3XH/H5Mu150 .0 ER. L o o) TS.E0T Q0100000000 TS.r_0T Q0100000000 .0 Q QA52H65319/9 Q Q QL $1 LU.B 550 NE. N2L12022222222 NE .N2L12022222222 115. ET Q8 Q Q. 1/.//S// SQ ST." 551 TN. .QE: Q1QQQQQQQQ 01N. oQE: Q1QQQQQQQQ 1 2E11HN6/X61/I1/IS/IN E1r.n.C .EEEE Q4PF. . 3110. H1 Q11111111EE1RF . 31M.N1 Q11111111E1E E23. SET/91. I]. [IE/I1 R1UCU05URR4 PFOL6U 6066666666UU PFQL6U 6166666666U U TTTTHMT‘QTTTTTTTNTT TTNSR.ANTTTD 1. ATN0T1TTTTTTTTNND 1.ATN2T1TTTTTTTTNDNN TTTTSETTTTTTTTTTFTTX SEIST. 1SSEN0T.CE:5ETEEEEEEEE11N0T.CE..4ETEEEEEEEE1N1R AAAA QLAAHAAAAAAAFAAI 1TTAS. LT11T1TS. 1TE2TDTTTTTTTTTTTTS.1TE2TDTTTTTTTTT1TU “HM.“ XEu.HTMH.Hu.M.N.H1u.M.R TIN .LNTTIU 0‘18. 1A11111111NNU .TIV. 1A11111111NUNT RRRR1HRRARRO‘2‘RRRTRRT D FRO .AOFFR—L IFRAORERRRRRD‘RROOE .FRAORERRRRRRRRO—LOE 00001900H0000030500A N 1UP» .lCCI-AUR OIUNDURUUUUUU‘UCCR OIUNDURUUUUUUUUCRCR FFFFQQFrflFFFFFFFO—VFM‘ E 40 O 000 01 2024632 34 2 0 01 2 0 120 22 9.333335 55 I. o. .05 4 .3 0000 39 0300000 00 -. . .. I. I. ..1112 22 92222222 22 C CCCCC 148 123456780,0123456789012345678901234:457890123456780 01234567.890123456789012345678901 222222229.33333333334444444 444E.555555.35565666£ 665677777777778399R‘9Q n.3Q Q O O O o, o 3.0. gnu/C0 566666666666666666666664.666666665666666666b66666666656666666 bFébiCéfifibée 06666677 111111111111111111111111111111111111111111111111111111111111111111111111111111111 . O . . . t . . t . . . . . Q . . . . 3 . . . . L . . . . A Q . . t O c 1 l i . . 1 6 . . . . Q 3 . . . . 2 C 1 . 1 . . E . O 0 L T 1 1 1 1 t T . .D . A K 1 1 o 1 1 o . A . .N . c Q Q 1 0 Q 1 0 . N . .A . 1 1 M Q 3 M Q 3 . 1 . . 0 Q 6 N C v: I C H. I . DL. .5 . 1 3 T C C 1 C C 1 . RA. .T . L C P 0 C H O C H . 03. .N . A X 0 L O C L 0 C . 00. .E . c x G U L A U L A . CL. 0" Q 1 1 1 . U C - U Q . G. OE O Q Q N 1 g 1 1 . 1 . L . .C . R 1 Q 7 1 1 7 1 1 . AN. .A . A 6 N Q 7 C Q 7 C Y. 81. .L O P 3 T N Q 1: up Q 1..... R . 0 . .P . 1 C P C H 6. 1 C n. 6. T . LX. .S . Q A O C C .1 H C 1 C .1 E. G1. .1". 1 Q U 0 C 21 C 0 H C 21 H. R. .DE. 3 1 S L .0 C1 E L .C 0 C1 H. OT. . T. C N 6 I” U 5L E Q L U 5: L ..L Q Y. 1 TA. .55. L F. 3 Q C :1U SH I C :1L U S»- S. 2 u... ONT . E G C 1 . 1/C .C 1 O 1” C .C . 1 L I .15. H 1 J 9 o 21: :C 0 0H . 211 = ..C Y. Q AS. .5 . U E E1 C 6 1M1 10 0101C 6 1MH 1 10 B. 1 CS. .UE. N 1 DGTL : QC1 8L TSTSA : QCC 1 8L . : OE. . T. Q Q 0350 1 2A1 0U 1 0. . . . . 1 2AA 1 .U X. 1 LN. .1A. 1 D NCDC 9 1.9 6. 1H01oooo1 9 1.. Q 6. 1. Q F. .NN. 6 N .HIR 9 C19 C1 HCTS33331 9 C11 9 C1 R. 1 HF. . 1. 3 A 1GQQ 9 E1C C7 CE......C 9 E11 C [7 T. 2 01. .F0. C B 6P11 C SCE S Q9EL 0 01111.... C SCC . o E S 9 A . 11 RT. .OR. E H 3202 9.... :ES :H9L/222165. ... :EF. . . 0 0 S :H H. 0 Q F5. . 0. L 9 C Q21 95 1. : 1C1/11111 Q Q1 5 1 . . 0 0 3 3 : 1C . 61 . .50. Q 0 11 QC :11111 2C 11 c . Q QM“. .1 : 1111113 3 I I 1 2c F . 2 11. .EC. 5 N E60U 0122119 1001911v WCC0212211 .3, / 1 1 9 13 0. OJ 21. .1 . E 0 036 Q T8. Q .1 O OLT2 Q21CCCC1 98. Q 010 .1 1 I u. 0 QL . T 9 1H. 1 .RL. I” C 0CC1 Q. 21 0518U 9H11CC00/1 0.21 Q11" N. C C 518U F . 1 . C. N .TA. U N NHPZ 081C1HC1CCOV C 9 DOCLL4C81C1V../CC C A A C1CC L. OJ 2 . C .NC. N O 1.31 GCv-r.CCr.u.E.GCCNU.LLUUFr.CM.r.CC4r.A A . . r.» E. A. G Q 15. 1 .E9. O 011.? .0 ECSr.CSCS: COCCUU. . .SECSCCGS. . 1111SCS: H. 1 CH. E . L. P E23716 SE.SO:€.:1 OLCC. .11C.Sr..SOC.1 1 1211..E..1 . C ST. N .F . N N1C Q03 1 :L: : L1L121LU001121: : :L: : L: ..1111C1C11L12 R . 1.: S . A .ON. I. O OG12C E 01. 11U9.. 111U . LL6511111.11U11C6CSE 2L 02.11 E. 105 3.1 E . 1. U E2Q6 QC 0. 22 .3.96.1.19..1UU900Q873 096.97ECE0.1.1101Q ul.2 1.C R1." 8 .N . N 7.11300 1A 11 005Q91000601360vHUH QOQ5091OQ.1.12C2C .0Q6 0.11.1C E .C S .05. Q 1C3C6L NN Q 0 ESC367213CES .11CCCC725C367731C1C5E5F.213C L. Q QJr. F1. H .1“. F. SEC7CJ C Q 11... .C’CC QCCCF. . IWSRJCCCCCCC/CC QCC5_._:JE: SGSCCCPL .11C .1 52. F E .TO. N NSEZS/ L1 = : .NE1.LEHE/ESNU.C O QOOOOECE1Cr_u.—:EFSGS. . . .E/ES N . z : E50 N1. 5 N .A1. I I’l1’2 E: 1J0TSHSSCS1S :TCCHHLLLLSSSHSSCSS- . u . = : : :515 : 1.1Jss1 A . . N 1 .UT. 1 245 .91 HK1 :P:C::C:H:1PC0CCUUUU::..C::C::........1111..v.:1£ . :51... R2.A T . LA. [SI/’1/IEU K 101A1101C1100LCC. . . .111A11011111122111C11.U L. 19.1U T1. R U .AT. NSNNNGNNLNO .55JGZ.85L6A516LU00 o a . .282.85L39 675711116A51N L. 66JTCN C.T 0 .VO. OE000300 :2N00 O1 Q1QOU Q. 9Q1UCLL4444QQQ1OIUQOOOQQOOQQQ. Q1 1.00 0351 DE. R ...LR . HRHHH OH v.LEZC111N2125C2135NC : UU .. : : :112125C11161517172135T F .1111TT TS. L B . . ”THHHSNHA" L CCCCCC.C1CCC :0: ..1122CCCCCC.CCCCCCCCCCC1CCN . CCIN .L U . . OSOOOCOOF.AO=OOEFF.EP.E .r.CE—LF01445555?.E.Lr:tr. .rsr.r.EEEEEV.Er.CE—Lo .OOEFRO .A s . . CQCCCIOLCCRNDNDDSIS.SSZSESS1TSFGFGFGSSS.SSZSSSSSSSSSSSESSC .DDS1HC .C O . 6 . . O 5 9 6 1 0 9 9 0 0 CCC CCCCCC 1 9 1CCC 1 6CCCCC 149 23456789012345.6789012345678901234567R.901234.§E739012345679901234216780.0123456789012 000000001111111111222222222233333333354444444444555555555566606669667777777777958 77777777777777777777777777777777 7777777777 77777777 7777 77777 777 77 77777777777777777 111111111.111111111111111111111111111111111111111111111111111111111111111111111111 . . . . . . . . O . . . . L . A . R . U . T . c . U . R . T . S . t 0 . 1 T . E N . Z I. . I 1 i S 2 T . N 1 N . Q 0 E . 1 1 N . = = E . 1 1 L . Q Q E . 1 1 . D 2 H . N 1 C . A 2Q A . B 01 E . N 6: . 12Q J F . 1 0Q 0 . 33: T1 . NNJ J s . 110 O Q S 0 ”U1 51 E . EEJ C N . RRQ E F . 1 1S F I 11C 0C 1 .1 113 .C T .N o .C 0 S .C 00C E R.E EE .1 ES.L ..1 $0 LE.8 550 51 EN." 551 EQE NF.EEEEQ6 RZU EFOSURR6 TCN $1. ANTTCD SET. ST. 1SSEN ITT As. L711T1 CTN .LNCC1U FRO .AOFFRE 1UC .CC11UR 2 0 FJ 9. (£21.15) NT SUBROUTINE KEPSIOICN) C3610NUHCLC31.IPARQICALIQICALZQICAL3. THE INITIAL STRAIN STIFFNESS MATRIX QNUUNPQNUHEGQL TO HAV- .0........OOQOOQQOIQOQ.00..........QOQOOOOO...0.0.0.0.00000000009. - b - ...tiitifififittfiititfitt...........OIQOIQOOOOC0.00QOIO.CIOQOOQOOOQQOQ VII/N COHMO cc 1 ccccccc Pun-argue 1 Q N . . C I o . . . r. . 6789 L . III, . 1 . 0 .1111 1 2 O 1 55555 1 W . . 6 ’0QQQ. H C . . 3 1WWHH1 C A 1 . C SCCCCW E . W . T 1 QJOUOC 1 L 1 C . . K 5 HTTTTA H . 1 E 5 . Q C C5888. C 11 C L 1 . 1 E OOOOO. A . Mu E . I . 6 N8 T....6 . 1 CC . . 1 . 3 TO 856783 1 5 AA 1 5 1 . C an .I)S(/. 1. 0 ... :4 C .I1 . x 05 011111 C R 11 o ’ E1 O X 50 444443 E . 11 Q 1 .M . 1 16 ’ Q Q Q Q0 . 0 CC . 1 1C . 0 N3 1MHHMR . 9 EE . C NE . 1 QC 4CCCCO 6 O .. 9 E CL . 6 N0 000004 . 4 1. O . A. 7. 3 TT HTTTTO 1 0 56 4 1 .1 R. C P3 CBBBBR 5 R 0. O H .0 T. A 0Q OOOOO. o O R1 R C 21 a... Q U1 Too... R . .5 . A .C W. 1 59 345672 . 4 .0 . . 7’ H. N 6 ”Q O/l”. . 2 9R 8 7 A1 1 Y. E 3 .6 111113 3 . O. 1 A :1 . S. G C 1.3 333330 c 3 4 o . . 1C 0 . 1 J 9C ’0000R 4 0 03 3 . 2E 3 Y. E E1 CL 1HHHHC 0 R RO 0 6 1.11 C B. 1 DETLO 3CCCC: R . .4 R : .721 i . . CJEOB 000001 . . .0 . 1 2A0" 1 X. D NCDCQ HTTTT9 . 2 2R . 9 1.6C 1 1. N QW1R1 1 CBBBBQ 7 2 1. 1 Q C1CE C R. A 10006 H 0OO909 O O .0 1 9 EMEL E T. 8 69113 C TooooC 3 2 35109C SCS. . A. H 3202C T 933456E 0 0 0.929E :A.. 1 H. 0 C0210 0 90/Il/S R R R3001S 1::0 N . 0 11QCL T .1111: 1.1. .31R :11111 C F. N ESOUD 7 0222921 2.3. .R1.012199C A 0. 0 03607 A T/00Q0R QSQB 4.C.TSQQQQ/ . . C 0CC1A : 1MHHHQ 2.5- CoEz Q21121 7 F. N NUDZQ 7 02CCCC3 C2C1 :25.08C1111 A L. Q Q6511 A G QanJaJOC For nu 1C - CGCF.CCCC . A. 0110. Q06 HTTTTE SRSR 2.... .. 7.8...EE—t1: H. E237163 CBSQBS .C.C 1111 3.555.21 . N1C003C 1 13.OOO: :::: Q2361:::::105 R. QQG12CT E E 1 1T....1 111112QQQ211111M2Q E. 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Elk A12 .3 .L. ..O V73 9 11 T. 9 9 1 000030000 1 55.0 PL. 12 ’1 .UA. E 0361 ‘6 5.91 J ”222220;..52 1 L.5 1 . 1D L/H. .T . H C110 (1 . 5 9 AOQC90999 55.1 R .. EH A14 .5 0. U NNP .0 EC 0.01 021 0 811111111 0 RN. CT. Q.» 110 .H.5. N ODS... TT E.T.L 5K1 6 ”.55686666 T UA.O 6 6 SA. 03 T13 N .31. 9 9991 11 S. T 95 911111111 T1.T O O R“. . 1H 19 1 D .T . P 5115 D Q N . DY. 011 0 1r.E.Lr:LEEE 0. CF. ITJJ UR. J N 9 N .go. V V539 UH {.Cq TK1 T KTTTTTTTT GD UT .0 : = : : SO . J1 1R( 0 . M. . u 0 0215 a... U 11111111 N. R .GEEEE .F. OD HOQ C .EA. U ET90U11 N.) 011 O 1RRRRRQRR 10 TR. 11.-yd J . 1N 3TH T .5 . N 2306A0.C 0.01 550 G EUUUUUUUU 0C SF..1 JO. 1A “C9. 5 .H. A. 9 11.315 0 . C. .2 12 1 ON U .11111 c p... 18 9... O . 7.1. ... 311.....133 Nu. ... o 1100 11 011111111 nu 9 "0.11212 D.1$H .V. ..L .D . N NPS/ OFF. U.NO K722 K7 201234567 EIK 1L...... EN..:. 1 0 N .NS. I ///01.. ..E .099 .. 9.0.0.... .... .TOGQD GA .11TEEEE . H . 1 .DY. 1 38° 1011 5.1 o 0.111 DEUlOOQOQOOQE DKE LT..LF=LF2L ”8.1JGUUUU 121 T .CA. ISI/I/ZLL T....R uvL6EBBNLSSEEEEEEEEU N 21 LL...... I ..JoNNNN N T1T U . Q. NSVNNNOAA 1.AA A.11OOA. (cocooooouv 021. 1N.J1122 HF.J 9.01.111 R A A 0 .OR . OEOOOn OCC R. C? 835519.350E333333331 ClSL FO.JKKKK C°.J1JTTTT U M. M R .TA. URHHHM 611 U.1118HKTTKKHKTTKKKKKKKKT ~1~L .11111 .211NNNN T R R D E . . ”TMMPR111 .112221112221 111111111~ 1 22 .FFFFF .J1F0000 E D 0 ..N U . . DSOODDCFF .FF123FRRlO..3FORFFFFFFFFO FOLK .1111... .JSICCCC R F F E S . . CICCCCSII .IIKKKIHHKKKIGHIIIIIIIIC IDLK . O . 0 0 05.15 0 1 5566 0 0 0 0 0 CCC CCCC 1111C CZ 2C CCCCCCC CCCCCC CCC 5 6 9C 8°.012345.57890121..456780 01234567 890121.“...67nbo 012545570 0.0123 45.979.9OIOLS45670.901.236.5578 88°Q99969o99000000000011111111113222222222333335333344444444¢45559555555566954£66 11111111111122?22222295222222.4222.4634222232222222222222222229:5222nL229LA,._A...222222221..9.22 2222222222222222222222222222....2222222222222222222222222222222222222222222223222222 . . . . . . . . . . . . . . . . . . O O . 1 i . . 1 . . . 1 S O O t 1 I! t . 11111111 0 a) . . QOQQGQQG . l\ .L . . EEEEEEEE . R . . NNNNNNNN . F. U . . 0 Q O O Q 0 O O . Ox T . . 11111111 . U C . . : = .. : .. : .. .. ._ T U . . 11111111 . C R . . O O O 9 O O O C E . U T i 0 11111111 R i R s . . 1 00030000 U. T . . 0 NNNNNNNN T. S F1 . . r. AAAAAAAA C. 0’ . . N 35833358 U. F I 1 . . O NV."M.VMVH R. O S1 / . . 1 1 9 O 9 9 Q 0 O 9 T . 51 I . . : 0 11111111 5.1 S E‘ 1 . . 1 E KKKKKKKK .0 5 NR 1 . . O N ........:::: F.E E ..r 1 . . 1 9 JJJJJJJJ 0.N N1 ...-R R . . 1 1 O O 9 O O 9 9 9 . 0 F4 10 . . l. = 11111111 <..1 F1 TT R 5 . . R 1 JJJJJJJJ 8.: 19 SC 0 5 . . O 9 0009909! [.1 TH E T 1 . . 1 1 11111111 N. 9 56 RV C . . l. G. 1.1.1111“ F.1 U A E . . K SSSSSSSS F.D R0 ED1V1 O .11 C ‘1‘“(‘1‘1‘ .1.”. AR NAG 6 H . .00 1 1 ((1‘1“( T.A [H 10.0. U . .78 0 K 8.3 NT LL6A6 o D A. .00 o 3 11111111 .H T.” 101 1 . T. .22 0 J 71234560 3.. L7 DDFLF 1 1 A.0 N. 0! 2 9 22222222 A.1 I. EEO Q 1 1 0.50 0.115 9 1 00000000 .... : DA. 55:0: 00 1 .15 1.6681 J 022222222 N.J C1 NN1E1 TN 0 00 N. 1 T.((16 0 O 0 NQOOOOOQQ 1.. SO EEHSH 0 A 0 UK 0.0 A .5... 1 821 8 A11111111 L.1 ”.5 ”BEN-2 103 1 1K 1.T0 R.TTOE 1K( 1 855666566 .J [N NNOEQ 16” l! T. T 7.111TT OS "((‘(l‘l‘Tl‘ D. 9 nun-1111111 00202 O 0 OS A.O N.QR I 01( 0 OEEEEEEEE C.T VU55555651CCTN1 01K T T g 5.600 D .....UUAI.R TK‘ T 1TTTTTTTT S .1 0L . . . o o . .5NN 00 O TOK 1 N. GNIN G. GU K173111111 N.S C06666616 .UU(C( o( ODOJ [.1 0.A1 .111 01) O 1RRRRRRRR E.‘ )HCIIIIIZIGHHRHR L OOSLDGNGJ D.C1CDBJ A.0001 6:0 5 SUUUUUUUH D.‘ SKHEEEEEECISSHéfl LOG-L, ON A O N. .2NNM. 0 T . . o .2 12 1 N. 1.478234567542922 ON .11A181L 0.3.03 OK A.QQ.;. 1190 11 011111111 0.1 .O’QOOQQOOOQOQOO 1A11J :BOHO‘ C.EF.101(1 D.EENQ K722 K7 201231567 C .0 1. . . . . . . . . . . . . . . :BOJ OH" 0. .5 . .N:N:SK . ...E ..O. ..0!........ .1 21.0000000011 1 LflovKHOEKszEEE E.OoT.J:(E K.111. 0E11 0561030090905 5.04 E............... OEKK LLKL1UUUU R . ND E 1R U c . LLLR ~L6688NL16E?.EEF.EEEU R . “ (11(((((l\“l\(“( OKLKTO. .. .JNNNN D.OAOZOJ:N E.AAAA A .(l‘. .A o 1 o o o o o o o .N 0.10 N TTTTTTTTTTTTTTTT 1K.(SOHJH1J1111 T.CP§13011 H.CCCP 835E12830E333333331 T.EN R AAAAAAAAAAAAAAAA 1.JS:1HJN,1'TTTT S.N11S111T C.111118NKTTKKNKTTKKKKKKKKT 5.7.1 U "V MMMHU-MWHPM.HV-UH L11" :(:(LNNNN .(( N (‘N .((((:::(11:::( 111((((((N .19 T PRQRQQDQRRRRRRRR D 0:.PFU0JF1FI‘OOOO .FF 0 ..OCCO .FFFF123F RQ123FOQF c FFFFFFO . RE F. 3300333050300000 N DJ11DDJ1115CCCC .TIDKDSRC .1111KKK1UUKKKTGU1111111IC .UR R FFFFFFFFFFFFFFFF E 05011.5456700900 0025 000 0 0 5 012222222235578 0111 34:3 6 8 3 0010030303300300 1111CCC 1110C...» 1 1 100: C C1222222229.222222C C nnnnnnnnnnnn nnnnnn 000000000 (TOUGH nnnn (1 0CD UT 00 O nnnnuq N nnnnmm CO A OPRIATE SUBROUTINE S: STUCTURE'S LINEAR STIFFNESS D: VECTOR 0F D.O.Fo.s R: LOAD VECTOR GAUSS ELIMINATION EQUATION SOLVERO BANDED FORPAT FROM BOOK BY ROBERT D. K. FIG. 2.8.1.9 PAGE 45 CONCEPTS AND APPLICATIONS OF FINITE ELEMENT ANALYSIS 0.0.00.0...ttiitttttoo.ottottttttotacoooctccfitatttototatoQOQQQQtI. §g¥gggglINEQNU'NPQNUHEGOLE(36)QNUHELCBTQIPARQICALIoICALZQICAL3. O COMMON/ZINSIZEONEOo NCONDgfiBANDOIEIGEN COVNDN/QIPI(3706).PN(3796IQR(107) CONWON/9/S(60g20)oSP(60.20)QIDET C0“HON/10/3(50)oGI¢bDIgGZI60I953¢ GOIoGQISOTQRCIGOTO SC(GOQPOTQIGAUS COMMON/ISIPRIOPTN INTEGER PRIOPTN IFCIGAUSoEO-I) GO TO 9° FILL-IN ARRAY D(I) HITH VALUES OF LOAD VECTOR R( (I) AFTER SJLUTION D(I) HILL CONTAIN THE DISPLACEMENT VALUES totttttctfiootttotttttitOttttno.a.t...tttt¢to.httottctttttittact... DO 110 I=10NEO D(I)=R(II 156 SUBROUTINE LINSOLN ......i....IOQOQO......OQOQ‘......O...‘.....ittfifitilttfifififittit...‘ TgPSOLVE SYSTE" OF LINEAR EQUATIONS SOD=R BY CALLING THE CHECK DATA GENERATION FDR SOLUTION 0F EQUATIONS 0.00.0.0...09.00.0001.....fitiittifififitti....Ottfifiifitfiififititfififititflt IFCPRIOPTNoEOoII ICAL2=0 IF (ICAL2.EQ.0) URITE(51 2020) IF (ICALZoEQoOT URITE¢619 2010’ (IoD(I)oI=1vNEO) SOLVE SYSTEM OF ~NEO- LINEAR EQUATIONS FORHARD REDUCTION OF MATRIX (GAUSS ELININATION) ....IQOOCOOQQOOQQOO'OOQQO......i.tiifififiifiiifiififiiittitit......Ofi... DO 790 N=IONEQ U L: ZQHPAND NDL)OEOOOO, Go To 780 LIISINQII ( 9 O K=LvH8AND ) I N H (IQJI'C¢S(N9K’ zzHc om: -44¢oloq a~oa~a HHl-LHUI ZF'UKD \ rufimuudacrumno 000'- H 0 II H II ‘fio FORUARD REDUCTION OF CONSTANTS (GAUSS ELIMINATION) ..‘iififfi.......ififififfi‘.......Q.....‘.......Q..*‘........0....‘.t.. EI'ngg?) 50 TO 320 )-S(N.L)tD(N) ~)/5(N.1) v: FJR uwxwouws av BACK SUBSTITUTION .OQQOOOIOOOifitiififififitfififitfifiiflfifitfiifitfifififiifitififititlfiItiififitii DO 860 N: ZQNEG =NEG -M DO 850 L=2vHBAND IWH of‘ NNNAJNNNNMNNNNNNNNNNNNNNNMMNNNMMNMNNNNNNNMNNNNNNNNNNNMNNNNNMNNNNNNMMNNMNNNNNNNNNNN 'II’A'JHIA'JA'JHA'JHJUJI'J'Jl'flUtuU‘U'fl'fiUU‘J‘U"J‘UUDIUU'(”M'IHIIUOUAIN'JI'J‘U‘U'JOHJUU'JHJOMUDMMNNNNB)MMMNNMNNNTJNNPJMNNNTONMNNNN : a I: a #b a 43> c: '1")!(JUA'JI(NU'fi'N’HV‘JV‘JI‘JT‘JTQFJT‘JNNTQHHHHHHHHHHOOOOOQOOQO'O\D O on o a O-n'n'nx'nm'fimm om'nslsaflflwlslflflfldm \noostcr'fibmrowo-nndomavx~Ha\n-n~lmmammwoomqownbumuosebummbunnowmqubuwwonmummaumwommuombumwo.o 157 0I23456799012345678901234567 Rd 01234567990123455739 01234.: 6790.0I23a 5 67-90 012.: 4567090 55.559.555.554 .5554.5666.27777777777R : 990‘ 99389: a. c Q Q o a Q: G C~.~00000000IIIIIIIIII...a:222322223 33.0.333133?.333333331.‘~332¢‘.13332.13.31.13.5133331.1:31..31..ICIVSRLXEKVXHILa a 4 4 “4 4 A. 44 4 H a 4 .4 4 4 4 a .44 4 4 4 a.“ 44 4 4 A 22222229.2222222222222229.222222222222222222222222222222222222222222222222....22222222 . . . . . . . . . . . . . . . . . . Q . . . . . 3 . . . . . L . S. . . . A . S. . . . C . E. . .R . I . C. . .E . O . 0. . .T . 2 .FR. . .F . L .30.. . .A . A . . . . . C .NY. . .E . I .09... . I .R . Q 9 .IE. . I .U . I I .TV. . I .T . L 0 .U0. . N I .C . A 6 .LC. . o I .U . c I .OE. . I I .R . I C .SR. . T N .T . O D‘ . . . U 0 .S . R 0 .EE. . L I . . A I .HH. . 0 T .E . P 0 .TT. . S U .H . I 6 . . . L .T . 9 I .NN. . R 0 . . I 4 .II. . A S .F . 3 G . . . E .0 . I N9 .00. . N R . . L EI .NN. . I A .S . E GO .UU. . l. E .0 . I. I6 .00. . N . .S. U E... I .FF. . H I .FN. N I3 E . . . I OIL . .I. O .G 7. I .55. . 0 R5 .UT. I 90 I I5 .TE. . E FIR . .A. 6 NI I S .I .NN. . N .0 .UU. 3 A0 0 N F. .EO. O 0 $5.? . a. I 36 N 9 .5 .N . 0 . I T2 .LE. E "I A N 02 .EE. 5 . 0 .. NER .A . L .2 8 : .E .CH. 8 0 N. 04 I {.90 .NF. 0 DG 00 H 0 I 0. .AT. A. 0. ..I 9 N2T .Rnd. 5 N9 :2 9 2 O : .L t 0 I I. 2 I EIC .E . E DI 21 J I I LI .90. T I T. LO I CHE .TM. H CO L O) ( I AH .SN. K 0 A. AT I A2V .NE. U N6 A0 OI C 0 ( N2 .IA. 0 I T R. C D L. .IT. N 1‘ CT 1!. S T 0 R. .0 . G D E. IO 9 P30 R . S. 9 01 I D I 0 E3 . R. I . 0 N. G I SIA E .EY. P EG N. O O. I 0 I TI .E... . I G E. I I I90 V .HS. N no. T IGDJ TI M G I N9 T .HD. 0 L 6! II OIL 0 OT 0 Hv OISDN I NO I U I... N .Ttt . O I . .nJII HDH C . E. U EOUOT .IOD OT. 0 II I H0 E . S. 0 N I A. 0.00 A.HI E .RH. N ZbAIP QOCN 050 . 00 0 5H 0 .Y . E I o T. E501 .323 a .917... Q IIAORO EoNO KN J I .0 I 1.2 I .Foo. . S 0 A. .N00 900 .V . E SDIDI .0 CC .AI‘ J E0 0 90 .TN. I . E D. N.22 ... E .00. N N] O/R NEIN 180C ( N2 2 .. E .NO. L I . . T2 0’ N .CN. I /0I6P T.... :H.S C .9 9 I N .EI. 9 NEES K. PLII ... I .EI. I 2I0] PDJE IOEOER 2I I .. I .DT. NI‘UUU C. 0A661 (In. T .RV. IS/I2/R ON 20 JLMUI L6161 (1‘ T .IA. (.DNNA E. ICC‘:N TTT U . L. NSNN ONE I00I.10J .UN: A(QC:N TT U . U. SL..IIG H. RIEE2R AAA 0 .00. DEDOUOG RC150.0.I0II CEOE2R AA 0 .00. (.ITTI C. P‘TTLU HUM R .TS. HvaéflE PNIN:JIII:TJ ITETLU HH R .TE. NNNN( . ( IIAT RRR D B . . MTUMIHT (I :MJ :(NNJ (INIAT RR 0 B . . F:(00F . FFRRCE 030 N U . . DSODCDN FFQJU:3IFU3( FR:O\CE 00 N U . . IKDCCI . IIUUIR FFF E S . . CICCSCI IIOJDKDIIDCD IUNUIR FF E S . . O O 900 00 00 0 012 00 0 01 56 4 1.33 01. 9. 30 88 cccc 1c222c CCCC#9CC abccccc nu IIC IC22C CCCACCCC ccccc 158 1234567R 90123456796 0123a...2.730.012345578901234»:67990121..“567a 0 0123456795 01234:. b75° 01 33333333344444 “RNA a:.5555555555... :4 .5 .. 5666777777 77772.3Q 95.51.80. H.043 Q 993 a a a a o 020302000311 R‘R.:3fiuc.u‘g.c_2_:. 222222222222229.2222222226.2229.22222229.2222220524,.2n¢29.0.22nc9.220.22229.«Lfic29.2n¢2222222.202é2022 “444“““NQQQNNNNQNQQQNNNQQ“NHQNQQNNQ“QQQQQQQQNQONNONQQQNNQQ““04““40“14“....“ IINEoNUHNPoNUHEGQLEI36I9NUHEL¢3I.IPAR.ICALI.ICAL2.ICAL3. .IXXI36IOKTISGI. 3 3 60I.GQ(60I.RC(60I. N‘DHI AXIGO B 'L.P6 "I3Z‘ '6‘ 02 OQIIG N7_._6 9 0303I CIOIO NBNH.6 OT. 95... 0 OIPI EI‘.YG N6... 0 O 0(3706IoVI3796I . N IT 0 'GIISZPN E7 CGUUIOT Z-JISEA‘IP Ids-D(IGNRO. SAIZDIDPI NIEZ’ 'IIR ’I’IUIIbP 235'ICII OSOOOIOCOON CICCCLCSCCI 6. IDENTIFICATION OF DISPLACEMENTS .................................................................. cccc , I ) ) I I I I I I I I D D D D D C D. D. P P P D N N N N. N N ( ( ( ( ‘ ( U U U U ”I u. 9 O 9 O O 0 I I I I I I D D U D 0 Q P P D. P P P N N N N N N 9 9 O O O 9 N M. I“ Mn N H M I I I I I I I I o o flu 0 0 o o o o 0 0 4 I 2 2 7 2 2 2 2 o 2 O o o 1 o o o 0 2 2 A... 2 2 0 2 2 2 0‘ 0 O 0 O I o O I V C Q :1 T I 5 1 1 T 1 2 1 1 I 26 5 I 5 6 6 O 6 5 5 LI. 0 l. O l. l. o l. 0 l\ ... l. Ar... 6 E 5 E E G E 0 E E .L CT T 5 T T T 2 T T T II I I I I I I I 9 I I I RGO R Ind 9 R R 0 9n 0 0‘ R R IUE. U NN 0 U n. . 0 U 99 U U H I "0 .L 2: ’h T p: 1?. .IU...IMI I DO I I I L N I N I I I GUN DNA 0 AJNN I0 0 . OID IOIO 0 I0 EO'INN 0 0 6I L. I I IL. IIL. O L. .31...‘ O QIIIII OIINQ 09 II IN). IIH‘Q 73 INO NE : NLI EM M :12IIIIE .E OI .(E 9 ’12.. .E ICE T .N‘E...’ .I‘I . ... 9 90 .00 .00 030.9 .0v'v 0 .00 .0 OD . P2NLM. KK2IJN00ID D :22 : 22ND Q N .. 22N.r. :22 :22u :2EEE 0L .LU Glut-r. FEE NNIL2IL2INIIIL2I‘IL2IL2NILUUUI IAOHNONADDO..O((IA IA BI BIA ABIA IA (IANNN:N RCQU ..3NCOD3II2AA 9C0 OCOIBOI iCOII 9C0 ..COA ICIII2R PI2NE2IINN2KK2IID ITPITIIT . PIT... . PITPITIPITTTLU (I I! LI:: II (:NI NI : :N( :N‘ NI :N‘NNNAT FFDFAD:FIJDFFAJFLIFD(FOFNOLIFDFLIFOIFadLIFaJOOCE IIDINDHINNOIIDINJIGUIGINGNUIGINUIGUIGNUICCCIR 0 5 0 0 00 0 0 030 :1. 5 O 7 3.4. I I 230. 1 I I I 11 2 2 29.2 9E25.15I H..12.1H0011.2HI L$LACEHENTS ON EACH ELEMENT) S 3 1 II. T(*1*.3_ T(c-..7 AAA 000 330 END SUBROUTINE EIGENVL(EIGEN.IDATAI A U SfiX=-(LAHBDAI*SI*X VECTOR ITERATION HITH THE ESSEST EIGENVALUE AND CORRESPONDIN V .L M Y TH S I T ISENVALUE PROBLEM ' s . v .OQQOOQO‘AQQOO......l...‘fittfifitiittiti....C...A...iifititi C’ O 0 C222C CCCCCCC CCCCCC 159 2345673901.234567890123455? 8Q.0121.4.36789012345a.78o. 0123456799 01.23.4567 8901234557 c.9012 1111111122222222223333333333444444444455555555555é¢6545655777777777792Q9;RFQCQQQQ 55555555: .55; 555555c .5pw.c..:.55555: ......cT55.:555555555555:.n..:.r355555555rd: 5555; ,.35:..:1.55c .:.R..c.55.355n.. 22222264222229.22222222222222222222222222222229.222222222222222222222222222232222221.. . . . . . . . . . . . . . . . . . . . . . . . . . 9 . . . . . . . . 3 . . . . . . . . L . . . . . . . . A . . . . . . . . C . . . . . . . . T. . . . . . . . . O . . D. . . . . . 2 . . r... . . . . . L . . D. . . . . . A . . U. . . . .1.: . . C . . L. . .N. . CO . . I . . C. . .1n.. . [E . . 9 . . N. . II. . NN . . 1 . . I. . (T. . 9 O . . L . . . . BU. . 1.1. . . A . . T. . XL. . : .. . . c .1 . . a. . 0. . II . . 1.. O . . N. . RS. . 0 O . . Q 6 1 . . . . 0 . . T) . . R 1. N . .1 . L. . FS. . DD . . A R A . I . A. . S. . .NN . . P T Q . c. . N. . 3U. . AA . . T V R . X . .1 0. . NA. . Q3 . . O N O o. . . 0 G. . IG. . Ha- L. . \u 6 TD. .1 . .1 .L A. . V . . 9. 0. . 3 .1 CE. J . O N .1. . LR. . 11 S. . l. N E EU. 0 . E O nu. . 00. . : : S. . L E 0 V0. ... . N 1. Q. . SF. . JJ U. . E G .1 L. 1. . Q : .1. . . . 9 O A. . H I 0 GL. 1 . 1 I I. . D). . T.. G. . U E 6 NA. 51. z 9 1.. .1. N1. . JJ . . N IT... T. . :3. I ..1 X. T.. A... V.,. 9' .... . 0 ...LT T5. .1 . O D .. ... 8. L1. .1.... N. . .1 00. RN. IX. .1 .... .1. X. .1X. NH. (1. I. . 6 N...) A0. (I. D A J. .. I . 0.. 55 T. O 3 AQO TI. TR. N B 9. .1. O (.0. K. loll U. . a. 3’6 ST. 00 T. A In I. 0 J. 5 TT. K1... 1... 0. . E H01. 0A. HH HA. 8 9 .... 1 ... 1 :N. . ox. . L 92X R. RR 0" . I. 1 1. 1 1.. 0 .1.... RR. .1.! 3. . Q 9.. TE. 0. R . .. : S. C. O) 6 T. . 00. 00 6 U. . 6 “0.13 FT. IT. FF. 1 J . 0 O 1.. TI 1 (.1. FF. 11 7 S. . E 060U 1...... SS 0. : O F. 3 0.1 3 S. .1 Bl. . 9.. 1 . . H C(SA H . PP .1 . J .1 0. 1. TI 1 . 01. 00 X... A9. A9 0. . U NP(G SF. 55 IN. 0 J . I F. GX bT .Y. . (1. 0 T. . N 058.1 0. 99 (G. .1 9 P. 0 01. 0 9. . 1 .1 . 5E. 30 T . . 0 COT! G . XX Y1. J I 5. T GX T . .1.! 0 JF. PP. AA T. . P r.) 9.1 NR. AA 5. 9 l. r... . P. .J 06 '0. AA. ...—L O N. . N No.10 .15. MM. R . I 3 S .U. 09.1.1 0 5. OJ T Y. . TT. PR 5 E. . H 0202 T3. GE. 1.. P. l. 5. SN .J G .L. . 9 .1 (S. . S. . U E060 Ru ..1 .10 T6. 5 0A) 1. . 0 A00 U. Q 0!. 00 SE. X0 00. ).1 .1 . . N 7.31.0 AU.0)0.L CN. ... 55'... L. E )3 .I .1 S. E 5.... GM. U. Ar. NM. 01 C A. . O .1636 TN. IOON ....A . 1. NM. 0 .1 A. N QWQI. Q . N .1. A 5L. MN AA. . . o T . . ... S(Xl. S . 003 O .VH. . 0 9.1 0 T. O ..OES ._ L. 9 ’5 .13 NA. 0 O . 0.0 p: A . . N NS/C ”.2021 C. .1 11.. 1. N. 1 N2 .. N A. 1 J. 0” 0V. 1.1.101... ..LE N D. . I [’05 EU. '1. O: N . 0 = :S 03 0. : 0...!) . N. : J.- . .0 I. . = :1... : . o o . . . 1 ?91. uH.1.11 IT. 16TJ.EC$1 2. IO TJJT TEE 0. I 91 OTQE TE. KI(AA. AAQOS T. . [SI/I) U1... 606 . AS. 0 :UUI I. o G .CIGUU G. 1(11E51U AD . TTT. TT L N. . NSNNNO SX.(6(01 TR. SDTS)NN(D R. 001 .OIY. .NN A. 0 IT. 6 . . N U0. 052AA. AADDA I. . 0.5.0006 SA.E(E0: 8.1. (NOOJIIEN 0. 3:.12‘:IIII I. 6 (..IJIJOI 0T. 06.100. DONNC R. . "RHMH‘I. AH.TDT1.1 OF. 0.11.1 OTTTT. H. 1.111.113.1IITT D. IIISTIJIJTT ...-5. 31.1.11. 11.11.... P. . "TM HM-..» .TAT. I . AU INNIU . .1.... 1.1:..NM. . : ..(I: :1... N . C . (CH-It. . . 050009. .REQ‘OT . EEOC‘OORE . 0(1FOF‘IF00 . OIJFCIJFFOO . 003 . FFEEF . . CvflCCCN . URUDX . RRDDSCCUR . DYIIDITIICC . DIJIYIJIIGC . DDX . IIRRI . a 57 o 00 0 U 9 5 0 00 1 23 4 5 .3 6 cc CCCC Irvcccc 11. Cn.cc 1 IICCFLC 1 1 ICCCCC lccabc ccc 160 345678a 0123456739012345678a 01234567890123456790 019.348.6789 912345678901354562.190 0123 96 Q Q o 9O.000000000.0111.11.11.111.222222292423333333333a 444.944“ .4“5555555552..g .24 ....b& 66.557777 savage—.55.“. DEL}? 366666566556655 Ltbzbrbéérbéba 6‘L..h56.~.6666.b6 5655666565.. 5.56.5. begitéééééfié b65566 2222222222222222222222222222222222222222222222229.222222222222234.22222232922922222 . )\I)\l”\l\l 30003303 ....EEEPLVZLE NNNNVNNN . ' . . Q . ‘ C 11111111.. 2 2 2 2 2 2 2 2 1.0111111! . Q . O C . . ' ”’,)\I\I\I 00000000 AAAAINA‘A BQVBBQBQVDU H.Hu-M.w.U.UI. . . . ‘ . . ‘ . 11111111 KKKKKKKK .. 2 : : = 2 = 2 JJJJJJJJ . . C C . 9 C . ODIAGONAL NOT IVCLUDED 0............fifitictttttfitt... IQNEG) ‘1. IONEO) ’,’,,”) JJJJJJJJ ' . . C . ' Q . IIIIIIII ‘(((((‘( SSSSSSSS ““((‘( ((((““ K10K2)01 C . ,,’,’,,, 01234565 22222221. nfll1fll1:11 "22222222 AnOOO-IIOOO 811.11.11.11 N 6566666 .0 IQNEO) 1. 8’ 1018AN0)91 o . OJ,'J EOURTIONS S(I,J)*X8(I)=Y(I) 17¢ 10K? (SCI X 60 TO 230 .‘(l‘o‘l‘ll‘ l\ IEpLFZtEpLE—L KTTTTTTTT 11111111 I . 0 P OF Sl(XvJ)¢X9(I) .Qtfittttttitit.tit...Oitltfittttttfiifittittitttt.ttttittttitttttittt , ‘11 .l.‘ Q‘B EYY FA Nit Mv‘ 9” 0’ 111 1, :(‘221 (XS(I)vI Q BS(02)*'.5) ’RRRRD‘D‘RDH” OUUUHUUHnInUfi. Go 1.1 1. 33 1 1,22 1, 1”),,’Ip’11 o x c so TO 17~ 3 so to 230 9.0) un11£(61.2030) 1.~£o : a) MBAND) OBTRIN VECTOR YE(I, FROM YB‘I’351‘IOJ,.XB(I) Ottittittfitttttictttttttittfittiifititttttituttttittttttttltttttiot. LN 0) ((StleDQJ ’9. K7Lv0 K7.a50195:336793c 10 .011 ..290000000009 .1. Dr7b6 Dr31100330302u01t1E 9L6 NI7|(RXUNItlérFiLEr=LEr7b6nu N‘ A0 90A. (coo-0000((N 0E B3EE12830E333333335EI K718"KTTKKHKTTKKKKKKKKTTT (1......(1122 :l‘ 1((l‘l‘l‘l‘l‘l‘IIN FR123FRD‘123FORFFFFFFFFRRO 'fliKKK?3!UKKR¥2VUI§3§III§3IUHF. GE 0 a ) O oNEG) GO TO 200 20 02" .... o TJJB TEE 0 G OYIGUU 0:1. .01.: O oNN 0,0I9(’IIII ZIIIISIIITT (:l‘ ((:(U.N OBIFOFSIFOO DYIIDIYIICC I: vJ 8( a 0T1: IYIIEGZU Oznu¢¢1tUY H241205: R0200:2’ .... ....0 I 01:912HX9( RODGGRDY SOLVE SYSTE" 0F ......tttitiitotfiiifittiOttttttitttttttfiiQtttttittttfitt..OQQQOQQIQO HORIZONTAL SHEEP OF S COVPUTE RAYLEIG” GUJTIENT Qtttttltiittfiiifitfitiitfi..ttttttttttitIifititttlttifitttttattttitttt. ......ittafifittittittttifi N G q 0 O DIAGONAL SUE 8 6 o 00 o o 7 7 8 B19 1.. 2 3 1 1Cflvcc cccc cccc 1 IZCCCC 2 2 172 174 240 250 161 45678991234567o.o.0125456780 91.23456739012345673Q 01234 56780 O125¢56739O12545978O 01234 7777779. a Q EFFREESG q Q 99: o. o a Q COOO35.33.001111111111222222223253333333334 44 4 4 4 “44455555 666666666 btééé béébéfia. ‘oéb :67777 7777777777777777777 77? 777777771 77777777 777777777777 2222222222229.22222222222222222222222222222222222222222.42222229.2222222222222222222 35) KQEIGEN EN )9EIGNVTR(I)QI=10NEG) .051 CORRESPONDING EIGENVECTOR G(”’, £53000 VX0045 0.. 0.19.0 EoEPSI) GO TO 310 A 7 T q I I R 0 0 3 EIGENV‘LUE IVO OttttttfittfittitOOQOOtt.......Otiifittcfitttttitititttttt.tittttttttt (RHO-RGDIRHO) (L 1’5N22 5 00:1 0 o 99 BKHI‘3311E Inca QLLGSU K HNBVCCVILI cerizuwvllvlvlol ......IAu GI. (11" HFIOIFFRRO CIP§UEI;£UUr. CHECK CONVERGENCE TO DESIRED EIGENVALUE itotottifititfittittit..0......Ittttttttitttttfitttfittfitfittttitttifiat i 5 T h A ( c N OBTAIN o 260 o cccc scccc \I a E MW . , 1 5 2 O I i O t 1 ’ I 2 ‘ a 9‘ 1| vl S v 8 TN...‘ ‘ 556 1‘ A61 I LIE ‘1 1ft. I : OT : (((N NZVTEF=LR E3N577TU G rchlIIol IOILRRRE EDEIJUHR O O 12 33 Q X 9 ) . ’ ’Vx I \l [H 9 I 3 ’1?! 3 I N 0E 1 0 OXG 1 : ,IZI : T 67/: ‘ s oA/H H \I ‘ OR/S 2 I L 1EG O 9 I 1 r;INX X 1 H” 0115 1 1 6 : 71 1’ 1| 0 DEC: 9 9/ 8 X H517 0’ X 5 PRHH 51 0 £51 1‘ S R V 9 F3 O 0 HNHX \I 9x T 6175 9X l. C [(11 .131 0 .L I H 0 52M“ 5 V In. K 1051 S N SEVC [9 9 U E .§LOF..J!.)HH 1 no 081" 9x.3lv.u G I OOTC 011 0U E «(RAH 52510 O H FPR5 1.Q(R 7 1 9 E0 COIXSH 1 :ETX .09 T T O 1U14 :53... N X SL 1 N1CO E 9 PAR 9 E595 S O EVOO G 9 .5H E NTHH.1XRHXU O U HECR ES1A1 ’ L SEEN. H 055 9 1 A ’1V3 590 O ‘ v IF. 0 0 0X91 V. N I EX1X530 9 )E 1 995 9 3936/51 OFS IAR1’ C.,..C ..4 \U 01 O \I ‘72.. '/3 O. OAMI, ’1”’\IYIR..E .3 .5:NVBRIXSTUQ,1899Q.BO C1H)1 1X1nIYT 00 1A.... .9 o o o o o 06:09.... 0‘ L1HV..3_€.305 .666665V 01 o O 10» HHZNK 0 9H C16111111HX 05X RJZHSB Q5H3X4HE1F.[EF.EE7OX1O 1F“43X19.16373E234557315€2 .29005?.000'9'99‘Q0Q9999' 1;.tnnqfinoata.I-otgat.T-cato 25 . 11 97.. 1 . OOOOOOOO . O1 . f.1att50tt.tcttt009t0ttott (((l‘l‘XX“(((I\I\‘(‘(I\I\I\(((( 77777”.9‘777777777777777777 AAAAA21A‘AAAAAAAAAAAAAAAA HHHFM VHHHMMMMVNMHMMYVMM RPRQR RRPQQRQQQRRQDRQRRR 3 33030 930300009090033000 N FFFFFOOFFFFFFFFFFFFFFFFFF E 0000 500305012345635090 0013 3‘5011222222233789 OOOOO 030111111111111033 C11222 222222222222222222C CCCCCCC SURROUTINE ENDFORC TO CO”°UTE ELEMENT ENO FORCES ttfifitttfitttQttifittiitfitttitttttfiitOttfittfii..Oitfititttttttttfittitfit iiiitiitfifififififihfifittltit...Otiifiiiffitiiiii.ifittififitfiii....itfltilfififi aucccc JC3619£€36).1XX(36)9KT(36)9 (9).“SU0PTN0NISOPTN 6.1237) '6 N511! 96,3 0 O 95) 92‘ F.72 Q6719» 2311.53‘O 1((31‘VL Qua—717.100 .NUMNP.NUMEGQLE(36)QNUVELCS)oIPARgICALIQICAL2QICAL3. N 7 NISEZPII III/1’12 2345 0811 ’S/’//’/’/ NSNNNM‘.bNu!rd OFOOOOSOOO HR» ”Nu- OHM”. M TN,unu-V30.v..vu OSOOOOI‘OOO CICCCCWCCC O IIN 162 567890123“567390123456789012345678O 0.12345679901234567800123456739012345.2783 012345 5555566666.06666777777777788q 99R Dvn.833.q Q. c Q o q a. Q 903335505301111111111?27.27.329.33333333 777777777777777777777777777777777777777777777R H.999 9.3g 88% R .09. ... : R RBOR “dc R L o c, 3 .7. Sq ..e.£.89. 2229.2222222222222222222222222222222?2222222222?2222229.229.29.2222222222222222222222 PICLN.I)=O. 00 1cccc PROCESS EVERY ELEMENT OF EACH ELEMENT GROUP 0.00.0...9.0......ttfifitfififittfit.ltfifitfitiittttttttittitttItittttttt. 1012101=1012) 1012101=10121 O o 2 2 2 O J J T O O 1 \- O J J G O O Q 51 1 O 1!. l‘ O E E G o r. 05 S '10 U. TlI It I E A 1‘ 0| U o1~ 0 N K O 00 O’lll O) O 1KL2 ”"11 O :(EK)((2011 KLMJKKIJ .15 OE EU KEEO O1OU OHNO CDDE7 1N OU29KOO o‘O‘I OFAO 21JF...0E0 DINDUNNIRGRC O1 .35 11ccccc RESULTANT LOADS OBTAIN ........OOIOOQO0.00000I..OQOOOOOOQAOOO......O.......ttfittttfitttifit 1 In J J O O H H I! l‘ P. c O1 O L1 L U‘ U .N 0 ’0 O JO Iv O) O 7.1 '1 l. O 0| r.~ r. ”ESL 2 ZS 6 191‘ 1 10 O O11 O O) 1 11p 7 11 .. :(z . ...:s 1 th.) 1 oJVE O 01 O DU 520:. 0:52" 141O1NJ61511 N1111LN1111T : S (O: O ‘N MIOVOVINOMIOUO LDDDDPLDODDC O5 SO 4 A. .35 11 11cccc cc12c SULTAVT LOADS OF THE NODES OF EACH ELEMENT acocottctthooattgonaa.......ontctttootottgttcntotaton.a..gacocfita URITECSIQZOIO) Ho‘DNlI)01 URITE R- 7912) 106,9(ON(I)QI= 71 DON "NR 11U UUT Epic. RRR 5.9/l CH ELE’ENT/li) ‘1'6E1 A E E0 RRR Q OOOX FFFQ O OO 01 00° c1052 t 0 O i t t t t a a t a t t a t ... a t i t o t t Q a t t t t t O t .E: o aHX 9 QT. . t 1 a QFZ 0 .ON a t t t QELDQ cUtnu. tLLHQ tAOT. OV1EQ QNNNt .EI O OGLIQ RR .105. CC tEKL. NN i (A. 11 «c; F. .UO tHun t '9 t7—LAQ Ll. t LL. OO tEBUt TT ITOG. NN aURE: 90 1 QPPRQ LL 0 E 0v: 1 OO 1 L ..O—LDC TT 1 A tCUEt FF. C t L1. 99 O S .LAFt LL AT l. tLVII 00 F D. tINDt TT '0 N QHEOa XX AG 5 9 run.- I Cf...‘ O O E ONCE. ‘7) Eli L .11 Ht 100 L0 N .TCTa T 013 A21 .UI t E OOO CO. E .UT.). 0 022 SZO N GRAEQ 199 QOoRO 1 o R50 L 911 A1TCOE T QSDUa A O66 (6LN1U U 01A Q NL 6“ T101 N O tHUTt R (EE EEA001 R tTOIt EL OTT .OTFOTT 0 R. t I TA A11"...1(A‘ N N. U Q t was». r.D\D\,2AR—r:03 E S t . ER RUUAFUIAGC o 1 1 100 c cccccc ccccccc c B=A 2836 Azl-DTNCR 2857 CALL HRGFLSKOETvoBoXTOLoFTOLoNTOLoIFLAGQSCALE) 23‘s IF(IFLAG.GT.2) GO TO 500 2839 L=(A¢B)l2. 2940 ERROR:A3$(3-A)I2. 2941 FL=OET(L'SCALET 2842 URITE(61.2000) LoERRORgFL 2H43 500 COHTINUE 2944 RETURN 2945 C 2946 1000 FORMAT(2F10.791109F10.7) 2847 2000 FOR'ATCIIIIIQH THE ROOT TS oE25.15o10X912H PLUS/MINUS .E25.15/I 284? 0 15H DETERMINANT =.E25.15) 2849 2010 FORMATTOIOoZBHOUADRATIC EIGENVALUE PROBLEM/[16H XTOL=9F10.7/l/ 2350 9 6H FTOL=oF10.7///5H NTOL=.ISIII7H DINCR=.F10.7) 2951 2020 FOP"AT(t--.E25.15o5x.625.1511) 2932 £030 FORPAT¢Ill/13Xq6HLAHBDAo17xo11HOETERHINANT/I) 32:2 END 29:5 C 2936 C 2857 C 2858 C 285° C 29453 C 23:1 C 2962 P SURROUTINE HRGFLS(FcAoSoXTOLoFTOLoNTOLoIFLAGoSCALE) 325; C octoocacactaatcoco.toottattotoa.tgc.tcottcootttooaaoottongttttt*092°55 C ITERATES TO A SUFFICIENTLY SVALL VALUE OF THE DETERMINAVT 2956 C OR TO A SUFFICIENTLY SHALL INTERVAL HHERE THE ROOTS MAY 2E5? C BE FOUND 2853 C OfitttttltttatttlititfiitttctiititOtittfittiltifiti0itfittit00.010.909.2960 C 2‘70 IFLAG=0 2971 FA=F¢A.SCALE) 2272 SIGNFA:FA/A%S(FA) 2973 FB:F¢B.SCALE) 2874 C 2975 C CHECK FOR SIGN CHAVSE 2°76 C ......lfifitttttotttnt...I...tittittttttattttfittctttttttfitttttttttttz877 C 2878 IF(SIGNFA¢FB.LE.0.) GO TO 100 2879 IFLAG= 2E?0 URITE¢61¢2010) A.e 28E1 RETURN 2592 2833 100 Uzn 2994 FH=F‘ 2525 00 ~00 N=quTOL 2886 c 29c? c CHECK FOP SUFFICIERTLY SMALL INTERVAL 2999 C ......tittifitfitttfito...tittttfittttfifitttttitfiittitttitttittttittttt2889 C 2930 C IF(ABS(B-A)I2..LE.XTOL) RETURN 5:35 C CHECK FOR SUFFICIENTLY S"ALL DETERHINANT VALUE 2823 E PROTYPE=3 FOR INCREMENTA; LOADING IN “DVING COORDINATES 32:; IFTABS(FH).GT.FTOL) GO TO 200 2896 A=U 28:7 B=H 2858 IFLAG=1 289° RETURN 2200 200 U=(FA-8—FBtA)/(FA-FB) 2°31 PREVFH=FHIABS(FU) 2302 FuthuoSCALE) 2°03 C 2=c~ C TEMPORARY PRIHT OUT 2°35 C .ttfiitiitfififiifitittitfiit....titfifiifittiIOtttttIttfittttittiifictitttti2gae C 2°07 NH1=N-1 2908 C HRITE(6192020) NNIQA9U980FAQFH0FB 52%: C CHANGE TO NEH INTERVAL 2911 C tttfifiitttit.it.tiitttifiiilittitttitttt...oatititttifi0tttttfitittfittPQIZ C 2913 IF(SI GNFAtFU.LT.0.) GO TO 300 2914 A:H 3q15 FA=FH 2916 163 164 7890123456789012345678901234567R Q 51234567890.12345678901254567Rc..012545579.901234567 11122222222223333353333444 4 94a #4 455.35u.5....5556 5666666567«(77777777899899.912288Ga .... 3 7.9 ed. 999999990 9Q999O Q 996,6 90.9990 96 Q 39.99O 90 : COO/G $99999 .4 QC,G.,90.0_.90 c Q o 0.0 a o. 39 r. c.3 c c c a Q 990 0.0.9Q 0,9 222222222222222222222229.22222222222222222222222222222222229.2222222222222.222222222 a c n a t a t t t a a o a t o o t w t 2 t t t t t o t t g a \I t 0 t t t I t i t a t 9 I a o t a a 5 «F t o t a S 1 .0 t o 9 t T o t t t c n N 5 .T c t Pt 9 I 2 .N t . Et 2 0 E... 0‘ t t T. t D. 33 t N t t S. a D ON 0T. 0 a t t N SO 9" 0 so. H o t ..L ET. 0R 0 N. C. t UT if. 0 Do A. t 0 LA QT t T.. all t H AR or. g T u a t T VE QD . A. R 9 a .T o t N. .C. c H o H a W. F. 9 T HH «T t E. A a t 91. Q t L. t a T 01 0F c N F... E. a A X 9 ..O a E c U. o N 45 t c G S. Le a .... N IT. or. o T. S. A a 2 G G I O .U i E U. SVt i I T. 5" 0L a IT A. t t E S .11. 9‘ c CE 5. X. a o IT 0 0V t 03 (9 15¢ a OF. ..L SE 0 9 NT. X1 RLt 9 DD 0 o In 2c OE t A 9 It TA. a N1 2 2 I [N 9H 0 P.) R. 0 AC.- t A 9 I I 5 35 «T 0 H0 To 3 "St a 8) B A 95 t t 02 A. 3 a t 0 v.0 F F F. SR .3 o D 9 H. F. t a 02 : : 0 [E or. t NO .- 3 \a 0A.. 9 D 9 3 AC UV 0.120 35 30 F. T. K a O USU F. F S LN «UN.- C‘ 55 09 K TY. t 06 0 1 A0 0 P O 0 ND. 2“ t 0 9 NB. \: 9 CC 2 VC .1 t M 19 95 N t G N A 2 .E .. NP 2 \I \a L ) . pt 03M... 0 9 00 0. L Nu t L a OS 0 o o x LO .1. .C.. 7:! TT It \I \a D l\ T51. E A O 09 0 U 0 T C N A .- Kt “C T.. 3 o 1. n1 3 “T. L C 0 r.) T o o N F. H C 0V... 0?. 00 D C. v.0. 0 A a P»... A. .3 0 N0 T T 9 S .35. E9 66 H: UQOAO N a; C 59. C O 9 Q2 0 G G .1 H !H U. .T. t 7.0 OA)D¢EEQ L n . Ta 0 05 L o E 925 o o 0 3 59 1 tTXI .16 )TrfiJJEQNM .L 1‘ O .1 EN. 5’ ( 0 7.0M. U H 3 R .11 T .C.]. SC 12”" 99.9 0'. S L J DH. N) T a .16! F F. 0 I OI E oNRn NS 0 o OCT. .12) I L 9 3. 01. .L 0 3‘1) V V 2 I 0’ D t UT 0 0011(00 .. : L .1 : 1C [0* 1 9 D a ”SN 0 E0 E O I .I 99.5». II EE::PRONLL1L K as: T t ...l C II 00 9‘0 RE 1 I) 9’ N o M t 29 o oIJSA ..LL 0. L K S’CEE Us... T....r. N. 0 29L o P.‘ PU26 (5“ 0 OS 9 II TT :Ut NLO :LUUU DLt SUT 0 0 II 00 t oN:(N TlTT .1 9.1:... NN EEOO)R..UOLLN 0 )LNNN MA. OtNDN I g NNKE U0 HUI—DER A 0“ T «NH. 00 0099JOQQBCOL 51d 0.1.11 0C0 .OTI:D\ T Q 00 0 FT FFTATU ".5!" C .TT. 3.». IT.“ OFtluXuSNl‘ 3O ONTTT CS. IQDTIU C Q 0 “LL l. ...-:(NLIT P2RR D N t 9 Hm. (I. T. 9 (L50 JILNVN c : :NTT D N t Hutu—2‘ F0 ..BFOFO‘.L 0.2.00 N U a t 00 FFOD‘ o 00:: 2 : :0 ..(‘000 a TOTO—L... N U o OOEF IGBFICIUR Fan—rt. E F t 9 CC IIDDS QDDIICJDJSSCCC t UDDCDR r. F a CCRT. 0 00. O 0 1. 23 0 0 0 000 0 9 0 0 03 9 5 5 895 0 3 Q. :2 22c CCCCCCP. CCCCCC C “Crud. 3 332CCCC~L 4 C CCCCCCC CCC 165 8901.2345678901234567. 89912...“ 547 a. c, 012 q, 9000000000511111111112 229.229.29.223:.12. 9900000“ 0.0000000.096000020erdri .[Cnugc 2233333333333333333333331.3331.333335 ) 0 .L M1 Q 1 z I 11 0 NM ..- O 0 N 8 "I A 3 00 8 1.1 n. 0 C. 9 T :0 69 1 K .I:\ : 0 K 00 J G 9 AI 2 9 N E: V \I D L RR . J 3) \a .n..\ ( E D L O .No 1. A S L N! t T. All...» 0 B t A 0A ,IvOTL 1‘ p18. 1" M C QC EB 12NO S NNO L Q . r...) ) NnKoo 1 1| .0.». a! L in NI 5 '0 03 N l. 1.20 S L J 9) 1 11.. $31,. ....) I 2 9 1.1 o ....‘I 0 .0L ) ZNLL \u K 1C : 9 1 IJON N1 4 OOQGQILLLIL K at : T.T. 2 If... 0Kr.r.31 0.1.. 5’55 (... r. OGGZ:UU20 NLO :LUU 5U ( 00§111)NN QDDDDOULLN nu )LNN 0.. NTNT 10(EE(JIIOCNNNN98( O L 51.0 0T2] .OTIDRA 2207.10 OTTTDIITI‘.BSN( 39 ONTTIRDT .. UM A((RQLNN .fiuUUU .fiLSO .¥}LVN: .INTTPnu 00EFFE(OOOEEEEEODF : : ..O : ((90TOT3EEON DDRIIRSCCGRRRRRDDIICJDJSSCCDDDCDRFE 00 00 0 09 0 01 23 5 9.9 ...U 0 22 22 3 33 a. 1 3099 3033$LINES PRINT. IS 00“ SP .00 “q 166 D.4 PROGRAM NFRALZD 1.23 01230:.6789012305678901.21.45.978... 012305678901234567890123456789012345679 a. 31 A.56789111111111122222222223....33333333QAAQ0AQAA 45555555555666666666677777777.1788 O 9 a Q o o ‘ o o 0 .- t 0 o .0 Q c O t O t O O 2 t \l t a o T t n o c T c G a o i 0 o t o o O o N o O O s 1 O t O N t O t I H t 0 0 S T O t o a 0 F. a : a 0 C 9 t t .L «A 1 \a o a 0 I o 0 N t O A a 0 t R X N 5 o t 9 HT 0 T t C .... a t t T s o 3 o o 0 EU T O P t D t O O s 9 9 0| 0 o 0 LL N t 0 a T .L T o 0 o T )3 D T T P o . 0 80 A t T ..R M. T TA 9 9 .0 ' 60 0 5 5 H o a t 05 N t r. .0 EON N o 0 .... o 3 91 L C 3 r. 9 fi 0 R I o 0 .1.? TE... Cr. 0 t V 0 L 60 O r. 0‘ T o o 0 PE... u I 9 5. K CTR HP .- 0 I O A (1 3 5.10 D a t a UH R o O 1.0 ...C INP .LO 0 t s t C 2‘ T 95A 0 0 o I ELT F. n X 0.: RF. F1. LL 0 o S o .1 Fr. 1 IiXu T o t t UA 7... 0 ‘ccRsriu FRQ. .LS o 0 .I. O O 5' N SAL 2 a o 0 LV F. t 1 ...INHC EPN . .0 c O 2 O) .1 OLD. 1 Q o o ANF 0 0 OT TTaUOH C 0 Ht. t 0.... c t L ’0 P 0A 0 O o a o VEO c [P PPoaer. DE... C0 0 0" “U a A .61 O 13D) :4 c o . “VG .r c 90 n209rtlfir. .LBTfiu A 0 0A 5 t C 9‘ 3 (T233 . o o ETT T o 2F FFORAEN T A0 EN 0 cR o a .1 6A 0 0.1.12... 0 t a 5...“... E. TL L.,... RV... DAOTH 0 t 0F530 0 O (X A TAOI? . o 0N NT A 0V0 0A AAcFEEG EUTUT 91 o o EON 0 1 1:... O 9P5 ON 0 c .0 OETN ET. .H HHcTT R TL DE 0T 0 tLTEA o L .L O L O 0353‘ o o .TNT .1.... TT 0 .. Q 9. . ICC AASHH FC 0 9AA! . A S) T P TT(35 0 9 .TOTRAH AA. NLC COcE ZV UVTO D N o ONNFL t C 00 3 O 550‘ O t a N OPTPAUR N50 TAN NX. 2F...” 0 LELCLOTU 9 0011A o I )1. O 2 9311’ o c T 007.0... E TE. PVA AOQIOSO E A U AHEF o OTDCT 0 0 6‘ 1 T 0(5N3 o a P o P ASTT MN. 06R 931.5 C D VESETTG o OSREN Q R 0A INT TTPR O o c 0 9505!. OF. R o TIL L 90 RP DU [BETNEAC Q ONOPE o A 6 0 (TN (DESI . c T c5 GLND ES. EEO OlsPr=LREL RACHRI c QEOSH . P (3 K91 LTK 91 o c ... cEENNO T.....- DTT T 9.58TODC ET 1H ET a OHC F. o T N0 T HOP CD 9" 0 o D QNGAODE a Q. OOTMSFUNNB OED—LNVA I 01; ER" 0 C .71 E rhuo a~03fi5Luo o O .FLJKN Hup.Xo .LO) \:1.$n. L13; NHF:H0Aau a .DNBC 0 L RA D U52 9’51HT. 0 L ...-A5 .LT 81.. AXZ 3 on NEEC TDDTMCSCD 0 a A N . .... ,TJ T ORD ’53 OTP. 09 A oIRARU Ra V00 OSQF HCNEPLL RNHNA . .0.-.TT . H 10.... O .1 0A OSCSPOO .1. V .TGLO N DT. 510 OloLDANIEOUUTETPIU a .UGA 0 U 1RD ) 3)O 1(L3FT. 3. G nSA FEOXLA. TQQ QFnAESA 2OONT AAO n OTNNT t N (90 2 01L (TACEEQ P. T . L0 5 IUH. €20 OQgHN RETNHHANTRR o o A H a O TTN 1 loP ORUKRDQ Co E «T E ALRO c 110 199 15589 SSUIH TA 0 oSRSG t \p 060 9 1.6 O LATR '0. K. T tNDT CDTUSQ 096 6:05F4L NR 5N5 a ...-.50... t 0 T 0.1,.) ‘(1 0TC5‘IT. o 9 0A...A RA 5* r... as (....VETODDOSSTTTD 5 a tSAOA o 1 16053 TUT TSAOSPa 12R. C .CIDTETflflic P:D ECcID TLLFSSSNATTI a t TLHR a C 1(13‘ 0 CI APD) 0.0 c KNNO . P o ETP .. HN ON c YEA TM. 0 SEE UU EEUARUN . 0L TT 0 E (TTTP TTN QOOZIF. F. Y oSFUETOSTF. TPE TAQUPSEOOERRJUTCAN a 9A5ES t L XOTRS H61 TilllLoQQ: .— T a P CSTF. DYR RR. PULHHVTT SNTIT: .NNI. O t D .0 9R... 0 O 0 OP 05.3 OCAt 000 o D a QR RYR EUTQ TT Ur. o R 9550555.... SAN 0 CA! N O 19”., 00” To 0 133.3TflcFFF-Jo T QOOOTOONCTQ Ha I: LtOMqOA 9 HURRNREO 0 560 o SNAS‘IGSZ 31.1. ((0.39 '0 1.. N oFFFDFNFf.5t T’STOoFTTUllA ODOTU . OHUNS 0 PHB OSN 01’ 0(037... E(TN. O. 0 T Fun 0 w 111T a CNN IOFFFFCSE. 0A TH t EUPIQOT C51CAN0U.VDST.Q6910 o M TET. To .0... F. L 5.. ...-.1 Lo ORSUP 00 NN Olé’lpfisloanuTEALC-PO O O 91. \a 0 9 T N. 1. Canto '0 c L ASS r... 0 GEOAN . O OGTCICSCCLLD 7T50£0o 325 O. 0 0 12319051.... 9 OXr.1r_X g INOAIO 55910212 0 COR-Lana. pttpLA-tl‘oll‘DUUD ....APD 5.110 0000.. 0,. .. .. : .. .. .. :5. a.) 0 o On... .....2 .... ....1............ ......tr. QRUL.1. LNNTSEPSIIIIO/ RoTlSla OSQEEELLNTNNc AlESElaTGTCVNMRPNNKKNKO. 0PTONT. ’IIIIIIUI23L7NNNNP. c AToPPPAATNTA. o QPTP OtPRDNTTATTTTHHTT ,- o CFOA. X123459.01111016000 .5595. QSoYYYVVFrSY t IIYFY/aDc OATTSSSPDCCDDTo .5” TR 0 XII/I’ll/l/IZIIIIIO t ...r 5:... OTOTTTGGOGO t GTTTT‘ uFVFc PP 11.0099 00». t oTREH—Lo TNNNNMIVNNNNNVNSSSSE.PFPDVo GTODDDIITVTEQ (TDTOTgLuILLOOE IIEEUUT‘ .HTHETO 0090300000003NNNM.GOAAAAQ (A.TTTE.LEAV.H0 EATATAQAOAOITHFFRRTTSSOO .TSTNIO Lu.“u.u.v-0.u.w MHMDFF.EEFEOTTTTO DHQPPMIIDTDTQ TM v\n.v M tHCHTNNTTIPPITV-NPo 0 a AYMVUHuuvaMDMflHQYTo 9 AR. 0 TRCRCR. o 0 o 503000000000ODTIITN o a ...0 a o ROFOF? a t c t RCCCCCCCCCCCMLCDDDDTo 0 RF. 9 UFTFTF n t 0 0 ccccccccc o nbcccccc Ncccccccccccc 3 o o Nccccccccccccccccccc 1)PRTOPTN9N2OPTINQNIOPTTN91T£RCHK¢R§UOPTNoISTRESS HKvESUOPTNQISTRESS RC 2/ 167 5 . . S . 5 . F. . T . R . N . T . .L t 5 .5 NS. 1 .C 0T. . .N .wV. 9 9.... Hr... x .6 ON. 0 AR C0. 1 ...... D. 9 CV EN. 2 .N C0. 1 .0 RC. Q .9» 0 . .. . FT. N .R N. T .0 DE. D. .F EN... 0 . CE. U .5 NC. 5 0: AA. N .C LL. H .N AP. 8 .A 35. Q .R N1. X .... U0. 0 0L . 1. .nvhNN. I .T 00. 2 t . .1 .r._KK. D .I..LC. .. .8 EC. K .A NN. H .U CC. c to t R QtL CE. E 20!. CC. T 0.A NN. 1 Q. EE. H ...... 55. 3 nu.RH.KR. .. T.A EC. X . “VV. 0 0.2 NN. 1 GOA 00. I .T CC. 2 .L . .1 )tr; 9?. O 1.nu.00. .. .. FF. N 0.0 . 1 EON 19.. T ..A 3:. P E. TT. .0 P.112PP. 1 Y.AK00. N TOTCKKQ H)0.L.LHH. 993:.IEQLC. 1w..DC11. X‘(t 0 .U.F. . 1:10 t +0 ccccccc DELTAIQDELTA291CHKOPT URITE(61.6931) DELTAI.DELTA2.ICHKOPT READ(60.69Q8) 9123456706 01234:..éTBO 017.345 OTBC 0123A5€7390123A5670 90123056739012 23456789012345678...0000000000111111111129.222a 22293333333335“4a445449455... c 55555:. bbS BAEOQREXUBQA-9Oé£wPuév9131111:111;:1113:111Tf111:111;f111:111?:111?§111?f1113§1112111:111:111:111:§11:11 1 12=9E21015o9X.8HICHKOPT=.15./l TERNAL CONCENTRATED 0 N ODPONAQLODPON50LODPON6 .............................. ER OF D.O.F.(IN LINEAR HEPS NSQL .... 0RD O E E S POM T O P D H U "BER OF [TERATIONS 6 O I E T 0 A . THOD PUT HAX1TER ...................................... .P.uxu LP X0.TEN UL 00 . TOT A 1L.F.A F.1T lcO.RIRESCN SZOAEE NF. NU PE UABLE ENTAL ..... 1N. RTNIH .0.5 L.,..Pr. 1P.N)PL OR 0!... 20.05P86C LE. EO.PNAOTN LR. OL.DO R11 AC. .....015PN9. C. 11.LTT 1L5HN. 1N. ANP.PATU1t SOQOUEU 1V" . DP.T0V.L0Tr.1T. ED. EOATTNXH. H0.P HV NOAG. 6L.UF VP1PH1. O . 09....U u-..A. X1.1 05 .LORR. 01.NH 11HCET. E1OOQESETT TS. U(O.PTD 1 01 . NTGQDSARNEAXRQ 1A(.n.Y001R0A0. TMD.L5LFPALHF. NRA. 00:. CFR. .00 41 23 09 :06 cccccccccccc RR [E T7. 11 XX AA NH 90 4 66 92“ T, TT TTT O. 00 000 T8 TT TTT P. PP POP 95 '0’ '90 A1 660 22“ CF CCB CCC NO NN. NNN 1. 115 111 P: PP1 PPP 93 00F .99 AC/1669 1 224 TNO’TTOOI TTT 118'11zg. 111 NP08NN508 NNN 1.5.11CS. 111 lesples PPP 'XF1001F1 9'9 309F55P9F 113 T1.9TT..0 TTT 09:.000:. 000 TX4:TTX6: TTT POTQPPOT6 PPP O11TvO11T 909 30N055QNO 113 CSITCCB1T CCC NopPNNoPP NNN 150.115.. 111 P10'DP199 PPP OFXX'OFXX 0!! 300055000 113 T.11TT.11 TTT 1:0911:09 111 NSRRNNSBB NNN 1T..11T.. 111 915599155 PPP N11 N11 )11FF 11FF 1163P0019Pv0 199.9...99... 99937 9.. :98 9.. : 97611X3461X56 691F90TC1'0TC 01 .6110” 9110N 060(611105‘T1 G(CTCTPPS‘TPP (En-ASA . . (EA . . CTDfiTWOQDTNQO A1AR1RXXA1RXX ...Rr.0R000r5\000 RURFUF11RUF11 O O O O 9 9 T 1 1593 1899 no F5R PARABOLIC ..rnn..mnm..nu..un.ruu..nnw.nu..un..nn..nu..mnm..un... FOR CIRCULAR ARCH IGOPTIN:2 ” FOR OTHER GEOMETRIES AT LSASTIPINITI SHOULDNoT ea EQUAL TO 2 awn 160511~ .................................................................. TGOPTIN ccccccc .OQAMHMOOO ".ICALS Q’N 0N4 10X 3P0 GDP 12. *L0‘ NUVNPQICA 3 L 2 9 O 9 o N N 1 E C 0 . N PRINTED AND STRUCTURAL STIFFNESS MATRIX 85 p ...mnm..rnum..rnn...un1..nnm..mu....mnm... HRITETSIQZOIO)TITLEl.TITLEZ.TITLE3.NE.NUHNP.ICALIQICAL2.ICAL3 ER K! T 61L. .RKRTRK. 15H..1.050N05. 6890.F F1... . 170N1. 1 R 1. 08100.0..0P0... 190P1...1.. ...... 9110.01L253Lt 061.00. LALBLA . GTTLéiACA AC. (EAO(.C1COC1. OTMQDQTCITITQ A19XA. a :RDOE. . RUF1R. O O 5 6 T 8 READ NODAL POINT DATA .................................................................. CALL NODDATATIGOPTIN) READ AN) STORE 1N1T1AL LOAD DATA .................................................................. ccccccccc cccc cccc cccc 168 34567890123‘56789012345679.a .1 1.2345678... 01234....97n c 01236: ... 7Q 901.. 3456756 91234557 0 o 0123 66666667777777777388969998809aHagan99DDDODDODCD1111111111?92222222235335335334“44 11111111111111111111111111111111111112222229.22222222229..22222222292222222222222222 . . t I . . . . . . . . . . T..“ . N.1 . E.C . No.4. . 0. . P.D . M.T . o. . C.D . .G . T. . N.) 5. E.C E. H.~ S. E.A S. C.R .L. A.L R. L.D T. D..T S. 5.. . 1..». D. . 00L ”4. 3 . o A. I T.) . 1 S.) S. J E.1 E. 11 ! G.N C. 166 1 6 D‘.D R. 6.9 ( nv A.P D. '11 2 A560 L.D c... 1:: E DDDA 9 6 .D . :11 S A AAA 0 5 63 E.L 8L. 1.! O D 0000 D 3 11 H.‘ 3A. 9)) o D $.56 LLLT Q 4 DD T.D 2”. ’66 2 Q TTT 0.9.9. A“ .I QR. 609 I 111 : : :D D D R . 1 E. 011 1 D NNN 1115 T o T 03 0.) 0T. 1..: J T 111 A56 3 TT F.1 TN. ..JJ 0 PPP NNN D I D .N 1. JO! 1 o = = .. On..0\l G 2 G 00 K t o o 0.. 0’) I. G 45.0 PPol N 1 GD Coo. GS. 1JJ 1 000 DDDT D D 92 E E.D T. J90 E D AAA 1230001 11 D u. N .N L ".0 N.L011 S 1 N 000 DDDLLLN 0 .N u. GA GA IR )A 1) C.L )E.E1(( . 1 1 DA LLL AAA(((1 E RN EA E8 E8 12N. DC 711 .(1 1F.H(12 1 o H EB PPP DDDRRRP N E. NB NH NMKNVRI .5 0.1 E.Dl oE.UE_tE J1 E l. NI... 123 LLL . 01TD ON '0 Q'RDR.H« QT NDEN C..N EL.NSSS&.6 Q". N ET 9. .TTT1111P9P’)10 111E S1. 111 211 SJR ED: LENO “.00 NE. $111,011. . LN 11 3111Q56...::455E :(uv . N:1 E:: 5:: o. .N 0...? ....LD. . .1(((11(S E BE :20 DNNV...R)’1.... 1DURTR1:D HIJD H1J111OK LG SKRDB 5.008 SE.=)’,::ES P "MD 1J0H0111EEEE123EEE100 :NENT J. A o A 111KR A10 NHEDD R.UDO $V.H1111JSE Y EEN DADPPPGGGTVVNGGGD:= 1:TEV 04 E D E 1110‘..EQ69VEDE1CTLDE—L.(LOPEEA. 111 :R T SLA 55..r.1....: .ooIDODoooARRToIRIDNDD: B11: 822009..)U GL9ULR1T4UV.((QURH.7009881T D1AE822218:12300°XPPPDDDDTEDTEHI11113511’4511A691JNDDD1N N END N~.SD.NT .DT13DDJS T.. :DOJ EDDDEEEADDDEEELNTOPTU DDJ: DDJ: DD(((J'1NNNE 01 TU:DID.B:DISD.1(((119 HRLLLRAQ 'LLAAANNNHODDNNNPU1QH1NLLQ. QRLQQ QRLAQDDD O1T1111LTTL1NDTTC.ADTT1T. ODDDQQIL (ALLLA 1LADDDTC‘:LLL((((DI EMCLL 1AL 1AL AAA1CNHUUI‘L NL((L MI .(L N1 . AAA (L FPAAAD.DD(ACLLLFFFN.|((FFFFCUDTUFAADDCPADD‘PAODEEETPDEEEFADDAFFDDD .FDDDF .DEEEDDEA IICCCIDDSCSPPP111NRRR11111NDDNICCDDSICDDSICDDRRRSSCRRRICGCCIIUGC .IUGCI .DRRRDDSC 5 5 73 D .1 2 6 3 Q 8 2 D DD 1 1 1 5 1 1 D D D 30 D D D 3 D D 1 A A. AA. A. 4 A. A. QCC 4 CC A. 169 1.567893123A56780 D123A5L.7a.0 012345678 c.0123A567R a 319.345.3786 D10.1..A:.E.7a 9019.311.“ 67590123... AAAAAA55555555556666656666777 7777777175899989.88:o c u 0 90 c a Q DDDDCDDDDD11111111112220.2 22222222222222222222222222222222222222222222222222229.222333333533333.033333353314519 . t t . 1 A D i. 9 . Q . 9 6 . 6 9‘ 9 D . D r. . A . A T S o A 0 I 1 L . L X F P t p A 9 9 . I H 1 . .0 t 1 I 2 5 . 5 D O 3 ON . AN N t C ,0 C )0 A s N 2P S . 2P E 1 ND E. ND C N P DD 5. 00 N 1 . PL 5 . D L 1 1 T 0 DC E. D‘ I P X DD R. ODD 9 R 3 L I T . DL O 2 S U 0 (5 S. 915 1 1 D 9 DD . DD 0 H F 0 0A D. D .A . T 5 20 N . T20 D ._ D 1 DL A. DL D R H E F AD. 1 . DAD. 9 E T V O D 9 I 5. GD 9 T 1 L . L, 9 E. L1 D 1 H A : PA 5 c. pA T In H 2 ON 1 R. 1 AN U E C 10 o D. 6 ’10 A560 N1 C N 3N 19. I 1 cu . A 119. CCCG .D N E C1 ND 9 1 2 . 1 .ND NNN R 93 E E ND. 00 5 I E L. z 000 111) EX1 G B 1. PL 1 A 9 33 A. 1 EPL PPP) T.DA R1 P A DC . 5 . 22 N. O .01 ...1 IOQDEZ S ....X DD 1 1 ..D 11 R. ) NOD A56T NZIDVD A 22223 L O 2 0 TD AA E. 6 TL .. DDDD U169NA H IIIC ! (A EI I I I I I. N9 T. O PCA AAAT N91 D A56N9 DD 0 9’ CI 9’ AI 0’ 9: A DD V. 1 ODD DDDP 9.EDCD E CCCT. . 9A . 5 95 05 05 05 053 ND TT 1. .. T QA LLL‘ R ..TT T z NNNPS 1D :15151515151N 1T . J E10 PPPS TD‘1 R 1 111 1 1 DL 1 .1 .1 .1 .1 .1 .0 RH 00 0. O RDDL : = :3 NTRDDD 5 PD 91F 1 AP D1. .1 .1 .1 .1 .11 ERD 55 S. 1 E oAD. 123A56A Uku-GFG : : :C 912 l. D ORA21212121212T TED T . J TDD ORCCCDDD . DU P A56N.CC D L1EDE2E2E2E2E2EA1ET N.LQ 1ENL1ENNNAAAT 1CD Y E CCC1:NN PSTL CE 9?. 9E 9r. 0?. .REDE ,1 F . r11 EDAPA.T111DDDG 41C11A’ T NNNP111 D D “.19.. 9. 0. v. .. 9.EL D1 11 H .UT.... L . NIPPPLLL . 9‘ TDDUA S 111 CPP E 1E, )Duu..........:.:.:TA). . .. E.Ur.) A) .10”...PPP1 T’. o. .1. ...PPP’N.. N15N7 7PU.1:2:3:A:5=61C899 EE LoN§M C807PU123 1 N9OODDE20 222 1112 11 01 1DNKC2C3CAC5C6C 51X . NNN SE. .(( $1 .1DNDDD))\:D UPXEENLZX III1113PDD 11111 DD ODEDEDEDEJEDE: (CDT L . . N .113 (DEDD QAAAASSA D10 . 0. .DD 123A561.AA :DA:A AL’1LALALALALALD1A1L9 DSRTR.L. :15 1ALAL1000 . . .05CA1TT .RAl CCC . . .A 900 1.. 01 O '(6 OFDFDFDFDFDF T 9 O .1 1SSENTV. H1E T O . 9(6LLLEEELD1 A OPPXT O QDNNNEEE OXLL E713E ’1 1 1DNIF.Lt.L-LLELELV.D.L1IRD :Nf.TEVA. 1RE7F.E1R1DNPP PGGGPD:1/009N1IDI1165619PP U U51696 6 'D‘DPDPDPDPDPDND6‘EA RIR1DNH.9 .TU UD6E6 OD: : : o . .(AR6CFF‘U619PPP . o .61.... NDDDN1((1(3(3PT............:(TT 1ELTH11 .D7SNDN:(T(391QSDDDS TCTLLTD‘T :::ODQ(T12 1NNNIDPE1E : EDDA 9 O 0 O O O Q Q 9 0 9 OR EAED : T SU D. 1‘ 1N1REEEDDDDDEEEBDNEAAAACEA0123EEEEADD T111TANTATTTADNXXXXXXXXXXXXETHDTTILINLLT. ADLT1TETDTADAAANNNATUTNHHHITM TCCCNNNTM AA NUUUN E1 1:.10LRDDDDDDDDDDDDT19 1 EV LAT-LL O ALNUNTI‘IIOLOfiaol‘Al“ 01R ((R‘IR NNN‘E‘IROO OEEEDDTRDRDRL‘D111111111111.LRDFDDUAFFAA .DEADEDERFRL‘LLLFFFFDCRDFEDFRDDI11FFFRDLL CRRRCDDUDU1UPDF 9 . O 9 . 9 9 0.. o . .DUFIGINCITCC . DRCCRCDUIHPDPPPIIIIGIUFTIFIUFGPPPIIIUFPP O .. 0...... ... .. 7 8 5 96 7 5 6 9 3 9 9 9 D 2 1 1 D 3 1 11 1 1 D . D 2 1 D 2 3 2 2 3 1 2 D 10 D D D 1 1 D D 1 1 D D 1 A A A AA A A A CC A A A A A A A A A 170 567890123456780 Dl23A567R6 012545; 786 912345.57: c.01234.367990123458790 0123456790 01234:. 22222333333333344¢AAAAAAA55555555556nt退66667777777777a9q0998Q8860¢c:9OGCO3p9000 333333333333333333333335333333333333333333333333333333353333333333333333333AAAAAA 1000 N PLLLDA .PPPEB 3 ""1 0111.01 AAA-3611‘ 0...:2P5 LEEE1JHD P055 50 .. o 0 ..Qg.\v9I.A1 3000220 DEEEDD:0 INNNAA1T all“ I LFFFDD‘D P1110005 3 2 gnu A CONTINUE 0A nu CCCACCCCC GO TO A026 .EL01DN ETBD:.E YHED A56 TTT 111 NVN 111 PPP .. .. .. 1” A55 "NM 000 DP? 000 123000 TTTLLL .I..I..I..l..l‘_ll NNNDDD 111‘AA PLOAD‘11=0. ‘4 oppoooo.~ .. 3 ._ LLLE1O 123 o~P 111PPPW ODD N~N1111p0 33395530L PPP...10. P1H010HDDDDEEE LE =ENOOOGGG o” E TUS171LALLL...BE.U DNA:2153(1‘0033N1N222181LEH9002..211" T: [0‘ 4027 .loclo 2:0 1J.‘ DA 88:5 DDDEEED..1:DDJ 1V~A A 4035 1LEL(( , FP‘COTA‘LLLCFFJFA0903109AFF01J0€FJFLCDCOJOFFFF 1ICSDDCCPP9111D1.CIDDSCICI1DVND11DIVJGHCCC1111 A028 A029 1d N“ 2: L 115 01! 29p ..U ONNSA CON 1 0 NEE! : 11 L..111111=11: : OKT:HqK12111181 SHPM“ .. Q9121 VCD 1d 00 DPQBAEEE 9281482999830 D(NL) o O 0 “MN” ...nn1.nu..un..nu..nn..mn...un... S E S s E 9 T s D N A S E c R 0 F L 99 A 66 N 22 SR. AA NE. TRT. OONT‘. TTEVIO ON . LLT‘ DEE LLuiLIRR 11IAE.E( c1ccqtfls EOUS SECI.J1¢SE1(19J112.¢SE2(19J1/3. SE(1.J1 E:11L.N(11661 VN2ZE.O(22'92 CE..A. ETAA .0 .IL ..11. KSDGE.:1EE::E HSEEV.M1NN1JN .ERET..E 19K7EE7CEQNNDAPUUUQRRRH.DODQ11DUTU RDUD1 .1.. ESS¢H1 .170‘11ANSLDAAN31 TC300311SAA(0(111T3PP0.1(DP1191 IVDDID‘ HRLLATLLAAANNVA1DTRAAOLTL11ANNAKKA11KTKTTT1111T.AC11AA1TLT111TAT ‘DLA 0LL000111 :: (1 (:H HNNN 9379 2228 8882 A999 4830 ‘(“ ‘ A(( (NLHJ‘1N O .111 4119 A269 .DEFt13FDADEFFDDT .DR11DD1CCCR11CDD 171 6780 012‘.AG.67 R.Q 012.3Azurb—Iao 010¢\.AR.67PQTD123¢ 63:57:11.0 012349.67R q,012~\.A.fi.,OTQ..Q 0123A56789012349.6 00001111111111222222222231.‘.~s~.§2.33~33AAA AAA.“ AAAG.¢.= a... R.,—.KJCICIE6656‘..06667777777777880109888 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA ......9...‘ltfititilfifiitififitttfifi... O D 9 . 3 0 I T 2 N 10 P ’5 . 2 2 O 1 . . 5 N O t 1 R 1 1 1 .1 t 1. M 2 2 2 11 5 1. 1 1 O O R 2 . . 1. V. 11 1 1 0 ED . D. 81 AD DD A 1 1 6R 2 OF. K 7.. 31 D. 99 3 K T 2R 3 . . R 51. 00 A1 99 22 8 H 0 BE 1 :N : DJ. A. H 33 AA A C T A O A TT 1 A O. 6 1 6 DJ 00 9 U U 1 "P J 1. 3 OJ 3 T A AA 00 3 D . . DR 0 AD 9 01. D T 9 D H. TT 3 . 222211T0 T NT 1 TP. A 1 A 01 00 8 0 ....3A R 1r:— 1 St 01 G1 TT 00 A : ....HHOR 0 RND P 0:. 0 GD. 0 D 56 9 1 A 1111UUGF. G [R . 5 61. T S T 18 30 L 1J 2 L "9131353 TED 1. . .. 60 E N" 3 N UN. 0 9 0 'II 1ETN D D D 2 1. 0011 0 D1 11" : : 8 1 591111121 1EDFA OH H... N N1 19 . G...» .1 G .1 01U 29p A T ..U1111HH0 0L 0 . [6 DA 0‘ 19.90 DH.0 An... 0 Q1 1 . .N ANNB 0 D SNKKTTUU .1 .11. . NB :3 EEKNN. . .C.—L 8 .P E EP 0 00 Q 1 Q T . I. OHHDDSSDB E01 AD 9! NM NV RRR1E 0.1.. N 1» DH 1 N .M N 17.51 = 11 D D U1CCTT11E3 NS1.X . S 1 O 1 9 Q 2 O v NN EP. '0 0 .EE 0 0 1E Au 1 . ...11111:11: : 5:;Inau-UTT .0 .1007. N :1 r. 11 511 R . . .1 . E2 .T ... 1 WT 13 1KTHHHK12111121 :1....RRN9 K1A1L TR 1..-UH. : ..OH: :111.K KE. :0 N..1P N : JP 0H THO. 11 . . O 0131 2 123AQDT 9 HT Q 0 . NT :0 .0 1J ..A1J111KH HV.1: ..U-u: . 1 Q: . . OCD 1d 00 PP '3 OEEH HV‘MHSSP1FECF1IREEV DE DE 111RCEA69CA. 1 T 91 TEE «.1 0T TRK1EE1EE2NNPAPUUUSUUUU: ..06UURD61EUDN 22 :8 33:84.4 0 9 O:RU RH . 61 57111GUU9 111156 DEN—”3003 . .311" NNNSOsSSSS1211NNpL..1TTN113331§A331533A691ENDDDE .311 .311. .NNS 11. . . . ..TCBDDB118‘A1O111:8: : : :RRRE11TREAE1 ..DDJ ..DDJ DD111JT1NNNT0. DP. MDPD u.u..11nu PPMFHM 111ANNAKKA11TTTTT1A123A00PTTT1£THDTLLRAA OLPAA OLAADDD 01T1111T . AM11ASM11TTA11SM1J1J 111 :: 11 1:0 CNN" HAUMRR11NN1T1R1NLLA ILA 1L AAAI‘NHUH1 . 3:1 15:19N ::(E::11 1FF01J0FFOFLT0TOOUDUUUURRFROOFERDFOAAPDDI.‘POD1ADO—L—Lrb1FO—Lr21—P . Drum-FOP.— TMnC-000MII.FTuJMu—PF DIIDNNDIIDINHGUCCSDSSSSEEIUCCID"FICCCIDDSCIDOSCDHAKRRSICRRD‘T. t DPIIDIPIICCDIJIPIJII o A 321 5 6° 1 2 2 3 A 8 T.D o 1 3 3 333 2 29 3 3 3 3 3 3 33 A A 0 8 888 8 32 0 1 o o 0 D 00 0 0 A A AAA A AA A A A A A cc A A.A A A 172 789D123A56789D123A5679QD1234: 67 AG 01234.1 679: 01234:; 70 a 0.123.451. 7.8931234... .370 a 012345 E? 8880 9999999.. 9DD00000.00011111111114.122a.2222223333333333A AAAAAA AAA:;3; cE c.5555 r.» 1. (.5? 56.: AAAAAAAAAAAAA5555555555555:.555555:-&..55555555555.....5555559 5555555: 5555555555555555555 . t t . Q t t . . . t . . . t . 1 . . 3 . 1 . N . 2 . 0 O . A . D. 1 Q T . D 3 . L . 0 N . 3 t. . L 0 . 9 D . 1 P O . 2 . . P D X . A T S. M. 0R 0 t G E. E LE 1 Q 0 o 9.. T 1T 0 t T 3 S. 9 T1 5 . T E. O10M 1 . 0 R R . C 16TU . O 5 A T. a ...NjN 1 . P S. 9 N0 0 9 2 t . . DD 11 r. . 1 R 0. 0 P036 9 t 2 0 N. . T CONN . . A 111 . A. 3 DLDDI I : . T 1111 . I 0 L1PP O 0 2 . L 123A 5. 1 6 1D 00? 0 T t E O 9 AT p... 11 J PU 00 . 1 R . 0 PPPL C. 166 . WELLS . A Q 0 NNNE R O 6 A A 1 ET111 5 P . T 1110 3 0. C11 1 1 TPTTF 1 3. . 6 OPP. T F. 1..: 2 1 P.000 P T. . . M HHT P . :11 E . 017...». 1 9 Run . 2 TTTG A L. 1 9 Q S D 1500:IIIIII .IIII 1 AD .2R PPP. P 3A. 911 O E 1NOA1OOOQOO:OOAOI, P1 . 90 PFF2 0 SN . 166 . . N01118888881DDDD A I. O 0 .2R 1 EEET 2 2R . 6 0 O 2 N 0915”. . . . . . .1111135 25 .AR 5 RRRR T A.... .11 I T o DNN0555555N . . . . .1 T1 9 E D . a o A R T . 1: z 1 P “.0000 111-1110.35.5935 9 R . E.0 . A 11 111P A 0N. :JJ J 0 DLPDDFFFFFFD11111. A1 1C.TR 0 LL 123. P T1. J00 Q T 1L1DDDQOOOQODFFFFP: P2 TN. 0 0 NH 009R 9 . 911 1 E 11P00L......0.QQOOS 9E 1E.0. 5 11 P990 1 0.. 1JJ 1 RD 1PHLL1::::::L.....N 19 PG.G1 0 DP NNN. T 55. JO! 1 E. TME11P1111111:::::0 T. IR. A A AH P1111 R T . .11 E TD 0r.TTTv.9..3A566T111111 R:1 Er... T L 1J 0 DE 0n NTTTA an N. L111 8 15:1. TTPGOENNNNNV023A35T 01/1 TV.1L E NN 9 LT . . HQRRT P 1..-...(12 . EDA. DP .TTTOOOOODTK.\N NNA PT 0. 0PN.1E I. : = A PP 00 UTTTL 1U. . ”FL—LP, 1 L . G 1DDPD PD D D PDODODDR 95R [.0. .D U 2P0 0 .... .... NSSSE 1 .....USSS J1 A1 .1E11A1 9.009330.PPPPPE 1‘15 N1c.0 . N ONNBA 11 11 QFFFD D r.L.N11156 0H C105N79N51 0000000 GODDDDT DP .T .1 .ET 9 1 91 11 11 1EEE . 6 NE. 0111 O .11 $3 .1 Q1535AXLLLLLLX000001 5.11 11K . .3 111:11111 0 O O O ..QRD‘T n4 . . 1 1115 10511109...v‘nJ111111DLLLLL 0 92¢. D :DC.T .1 ..HM112..11PP9PP P11103A SE. :111: ..ES 1ALA:AADA01D PPPPP111111FAAXEU A 1AE.P16 H11K . .1 O ONNANN NSSS .5 O SV . H1111JSE T O . O1 9 00 .P ONWHHHM OTTTTTDS 90 .N DE 0H . 0R9E TJ 30 D P11011EEE 59910155,...» . 111 : 9E713EE10 1 11L10Ir.r.Er;L—LI03000 D11 . : AUSLC.K0AU7EEBEE9NNTDATPUUUZAAATAbUURH.3...AA1TU UD6563661601TTTTTT1TTTTTOA61:R NAP . HR NADDA . .A11RH RU. NNNrJ : : :R (NNT . 371333JSN000N :1T1111P1LTPPPPPPTDDDDDN 1T3E 010:0.CROIDDOD110AATT0TT1110123A0E1150.1111110 1NNN1REEE1EEHE1A......A......OEATT TTA1T.1ETTANNAKKA11SPTSPTTTATTTPTTTT1T.ADDDAATLT111TETDTATTETTN'QOOQQN'Q'QOOTTNR1 N .11 N :2 11 1:FF FFNNN 99R1 1NN1 . AAA 1LNEHUNT111 11T100¥XXXXX9XVXXXY TRhM 0001 .FFOODTJOFFDFLEEOEEOOOOAAAFOROOF .05r3:00...A.JCEEO.;RFADARD ATODODDODDOODDOOJAOPU GCDR .11GCDNND11D1NRRGRRCCCDPPP1SUCCI .DRRRDDSCCRRRCDH1UDUUOUDP111111F1111110HF.N I . . . ...... ...... . 9 3 2 1 D 997. 21 A 3 3 8 5 6 30 3 3 A 9 5 5 AAA 56 3 3 3 1 5 5 56 9 D D 2 D D 000 09 1 1 2 1 D 0 DD 2 A AC?» A A A AAA AA cc A A A A A A AA A 173 8a. 0123A567c 90123A5670 Q 012345678O 01234557 93.9123455790 312340679 0.011239% 57990123A557n. 66777 7777777Q.R‘88q188888°.° Q C 0 C G 0.3 D.oooooorvooc111111111Aficndnucn‘ZGLA/bfldzz‘»1...‘ ts‘133333“ “ A “A“qqq 55555555555555555555555555555555b6655666666 065666666666 56665.666556566666666 b66666 . 0 : ) 1 1 1 3 R N O 0 1 P 3 0 N 1 0 0 6 L P N 1 0 9 30 0 0R5 NP A LEO A.3,b 00 0 11A CCC P0 L T1 NNN 0L P10N0 111 0. O6TUT PPP LE 1ND” #06 .N 20000 111 E. ND11G A36 N1 0026 A56 VVN .. PONN CCC 000 10 0L00 NNN oDP .N 01PP1 111 000 0A L0001 DDP 000 ”‘0 1A001 A5 DOLLY. O O O 19..DLLL 111CCC111 GO TO A061 9 .N AL110 ASSNNNDDD 2 20 0PTTT NNN111AAA 0 NP L '00P 000P90000 A 00 P1111 PPPOOOLLL A56 PO 05005 000111PPP 0 ccc 0L 1N O 08 000123 .. : = T "N" 0. 1011A 123LLLNNN111 .11... LE "9.15 O ccc111000A5;b 0 PPP .N ODNNT NNNRRRPPPNNN G =:: E. POOOL 111:::000000 111 N1 0LDD. PPP11100009P 1 A56 .. 01001 ..OASSLLLDDU R R NNN 10 L0001 111NNN111000 E E 000 .N‘ 1ALL1 123000000LLL T T PPP 0A flunk-.1.“ NVNPPPAAA111 1 E 000 Mi . ALTTO 0000054000030 X10 000 AA 0PO0P PPPOOOLLLAAA A: 123LLL .M‘ L OTTO 1000LLLPD Pa. O 0 I L CCC111010 P1000 0000111: .. ..LLL .A1 NNNRRRENP 1A19L .LLLRRR111PPP ...-CB 111 N00 =4~611 0111 1933 LS... 9.99 111 APO .305AD ERRR111~V~111 .10 : = : ASSIDL CPDVA .......A.35330A.15 R1A 111...:0. AOA00 K111...PPP... 9 ET OO123F=LE1LE 90 OD LOH125EEE000EEE12 TE10NNNGGG .NE1L10PDCNNNGGGOOOGGG:0 1069000...HE.U616019R000...LLL...RA Az1 PPP0005N1N101LS EPPP000111000E UREODDDEEEO..1EAE180TDDDEEED0DEEET0 NETTOOONNNAIIDTTOTTAT1000NNNAAANM‘N1T. 1T1 LLL11‘ 1N~1L101 1LLL111000111V FE90111FFF0FA0RPRTF0F111FFFLLLFPFU0 1DUGRRR11101.CUQUDIG1RRR111PPP111NG O O O 7 8 A9 1 5 5 55 6 0 0 00 0 A A A A A [*QIOXQOX111‘010X9‘Y111*9/1 ............................................................ 0 o A 0 T : b A 1 5 1 D 1 A56 9 1 1 TTT A 1 .l v! 111 V- V. o ”NV 0 .) :L T .511 T 1R“ H 0 PPP 1 . E = : 2 .. 0 15 L 1 a 3 5 G 1.1 E 1 0009. OF 1 AAAN P 09.12 E L «.00H ...-.11 1 ALMV O 3‘ 0A LLLU U 10. 0 ”v 5 0 EU PPPNS va .1.U11 T NT123 9 9 N11 07 OM‘Q Q S . QCTTT11111 Q11 5.3.x 0.31. A.1A11..AG.L.: : 18‘ 090190 ) OzoMuNN . . .1lu0 =06 ~A1:A1 N 01:111EEE . 111 1991'. A 50 1P0 D 556 01 11EP1/ 1/EE U111:::...00::011UO61Qb1UU0 N=01121..:00111K2XTN11T51TNNA. 17.0EDDDC.F_E..J°JCO .. ..1REAO.EA11E. TLSVAAANVN-Dtu .F_311TPT0-AT.M.TTR . ,NA A000111 1H 11Am11q 1.K\V . OCOSLLLFFF301CD110FADORDOO . CSDPPPP11100UIDXYCIUFDUFCC . 6 1 0 1 0 7 996 2 0 1 0 2 5 55.3 0 0 0 0 0 9 999 A 5 3 1 3 A AAACCC 'YI1**291YJ-YI1**21 [111110 : N1111x 0.25;! Anqq"... ...McN. 111111 T. .EA:11JIUT 1K9..90.Ef:r:LD\ ”CEO 00303 UPLLARJOOODQL Nun—L . .DNNNN: ..c 05111.11 9‘1L15XYYYM. l .11 GO TO 4093 A1L : : : :1 PPAOOIIJJE 11CGDXTXTL 3 5 9 5 0 5 A 5 3333 0TH 0F STRUCTURE STIFFNESS MATRIX 174 90123456730 01234567: 0 012345670 c.0123456733 ...123456731 0123456790 01234567 9.0 C12345.t7e.9 ASSSSSSCJSBes666666666677777777770.90 9‘0 2.583.590 o G 0.0 a 90 90000000.! 001111111q111a._2n1.172.225.2422 666666666666666666666666666666666666666666666.66666677777777777777777777777. 7777777 6 . . . . . . . . . . . . . . . . . . . . . . . . R. . . 0. . . T . . . C. . . E. . . v. . . . . . 0. 0 . . A. 1 . t o. 0 S. Q L. q. E. 1 . . S . J . 0. 0 S. O . T . T E. 1 . N. R . 1 . 1. 0 T. 1 . . G S. E Q s. O Q. . 0. 0. O t A . 1 N. 1 . 0. . A . J . L. 0 . Q . . . S . 1 1 . L. 456 0 E. 1 1 . A. TTT E C. 11 E 0 . 0. 111 . R . 6 O S 1 . 0. NNN 1 0. 00 : . N. 111 1 F. 11 1 0 1 . 0 0 . PPP N . :1 J N J . . . 0. .. : : S 0 L. 10 9 1 O . 0 0 N . 111 6 N P 9A. .A 1 u- 1 . z = A. A56 3 R21 22 0 SN. 1E 1 E 1 . 1 1 . 5 NNN 3 T010 99 0 2R. 6R 5 R 0 . N 1 S. 9 000 3 V61 66 L AE. O S 1 . 1 0 0. 0 0 PP N72 77 1 T. 111 1 1 S . 0 N A . A 00 3 0 1 E 0N . : .6 . . P . L 1 0. 123000 T 00 00 V T1. J0 9 0 0 E . 0 L L. 0 TTTLLL LTT TT A . 9 .1 . . K . 7 0 . T 111111 0 L S 0 A. 1:... E E O . A B L. NNNTTT G A00 QC D 55. Ju.1 N N R RK . A. 0 111RRR 0 060 GG . T. 9 . 9 . . A AR . L 1. G PPPAAA V 0 0 R 0 M..L111 1 1 0 TRT: . E1 1 T. 0 .... ..TTT 0A 1 1 v.. 1 N 0 N 1......5116 1 1 H. 51.31 . I.1 2 1. 1 1 E 111555 1 EB 0101A 109A . 0A 1a.. . U ray 9 N N CA 1T1J . U . . N. H 1 N 123PD P 1 NN.. .3 ..LB 3 .53 2 EB .E. U301 0 1 0 EB ”SN 0 . N050 1. 1 . CONNN . O O E .QNH .ONM . NM EL. N1P :660. n. P Nu K 91R... . IE OE . 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Q Q QC H C Q Q 1 IQ2.5QCRR6Q0QQ2QQH6IQ Q0 C660 C 00 5 15C2C25QQCQE56C66CQQ1 66JE1 PQQE1 E 1FE 1 1Cr..JECL Q Q52 QCCF.CCt.11C Q QC :0 F65 Q0 L 0 Q Q Q QESRSEQ ‘25 . ~EESEP.L: ..E 11:51 SCC51 B 155 1 HS.C.S1CC.CTSS.SS.1JS ....55 Q NTTSQ H QSS2 2 C = : : = :H .. : = :P z 2 : : z : :EQIJ..EOFHILLE1EEE2.LP.1 E 011111C11110111111111U 1R1URUUR1USU1RR C T2232366253U222333§QJN66JTCNTHHCCNANC110NY BQQQQQQ QQQQS QQQQQQ00 Q100 QS—LI SE1 IESSNRA : 2553316666H23553611IT1111TTLLL1TTLTT111UH 5CCCCC1CCCCCCCCCCC C" CCINLLLCINLNICCUTRfi» OEEthECEEEEFE—LEV.EEOOEO00EFR06A6F ROAORFFE.LON RSSSSSESSSSQISSSSSSDDSCDDSIUCCCCIUCCCU11R RF...— Q 1 Q9 6 1 2 0 9 09 U U 0 2 o 9 11 1 6 6 2 QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ SUBROUTINE SBEAHEZ 1 ccccccc cc SS QSXXClO) LEC101QNUHELQIPARQICALIQICALZQICALSQISTR S ) DIS‘J D QDRE Q Q Q 9 N Q0106 MIA1O~J1C "86NQ1CT U” Q Q QCO N Q6150R QGC13LpL EFL—LCCUS NNSES/ [III/2N 12459.10 ’I’II’IE B I T Q 2 Q B Q A B T Q .l T Q 1 Q Q A A 1 6 T VQ u. T Q .0 C Q 1 88 E 1 I. V2 L H C C, Q C E Q1 Q E L QM. Q L Q 8C Q Q Q 0‘ 2 Q 3 1Q Q 3 o 191 Q1. 1 2 ”81 D1fl.2 In Q CTC TCQ C Q AQE QQAQ A B QAQ 1QB Q vl 1..-.1 H17! 1 Q 1Q1 C1. 1 1 CQvn EC1 C In E6C LEM E C QQE QQC Q E 12L Q1: 1 9 L QQ/ 3QL Q 9 Q 0Q1 .0Q 0 .3 2 “BA 28Q A. Q 1TV Q22 1 0 Q (Q. QC1 C .l A [QB A/Q1/ T113V T12H1 0 QH1QC Q1QC1 G 1Cq2Q 1BQEA HEleA uuTQuLol L 1 CL.QT1CQ.I/- E 1 E/AAQSEAQZB M. Q L2TTIQQLQI1QT. U66 0 QQCQ22QCH.QC ~00 E1111 QQQ Q7C QQC1Q Q1110 336252115 ....31501 1:: QNQQQQ1QA.1S.ALVA ..1JUTHI.HHCVVC2 - CVC . v H :PCCCC....=Q:=-:B. 10CCCC1BB1 Q 11B1VB NNNNNNSLUQQIUUOOOOSVV31S6V6CV OOOOOON 20° QSLLLL QCC Q04 Q QC QQC ‘1an pM-EL211IUIUUUU3Q Q2C33Q 6 QQ ”MMUHHMA CC: : .. ..C Q QCpLCC QCe Q OOOOOOIEODOEF‘8ABC.83€.L.L.£6F.13 CCCCCCDRDDDSITTVVS1 Q SCSS - S Q Q Q... 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((032; 00?. on... 0)?) a) Q 9‘ o) :66 o) L ) o o 3vsr. . 5 CE). S6)56)SB.J . 38611::50 («“50 B 055 (0.83.9053 0.03.06.00‘ 0‘ 9....351 NTTS) H 1339 : a : : 0 :9L: ’6 : 2 0.: : 9: :u: ..u :v )J..f. CrilLLE OFFLE Or}. )8)) O )1U)”1))"))I.))l‘)l\)l\ 6765).: (2(253‘56‘63C6C6CbGJTl‘NTI.HT‘NANCTTNNT 0* 9 OH 00: 9C. 0 0C Q QC 0 00 00 9000 OSEI 2 055(2T020022032053L3L6L1111TTLLLITTLTTI)HUM H‘LH00L00LI1‘U‘U‘U ((IN.LLL(IVLMI((ETQHJ nLVIEEI-EOREUREP.UrzlurzL0.... p... 93.LFROA‘AFRO‘ORFFRFNOV s - 58,56 s. .SSOSSOSS OS 05 ODDSIUCCCCIUCCCUII RFF. (97)!:( O O O. O O O O O 399 199 106 SE) IESSIR Au 602 220 )R3URUUP..DUSU9RRD (E21015) 0......it...OOQOOQIOOOQIOOOOOOO0.0000........Otttitttttittifittiitfi SU9ROUTINE AQEWBLE(M) 1 ccccccc cc OSXX(10) LECIO)oNUHELvIPARoICALloICAL2QICAL3gISTRESS 1 I VOID SO 90 ORE 0 CT. pni) 90)) IJA“~)AUI.20 n B OSN '15 UK.) V '1 O M1 916)): ) .0“)(51° r.r_Ar.l:ll\ or. NNISED SEN. 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(l.NNKw(‘TCD?\flU OJFFOFJOFFFF—rdl‘FOOOOC.” U 0 COAJJOD‘FFDOF 2 .. :0..(1\«JOOOF ..(0‘0:OF:(ODFFEDOO..1 CDIIDIJCIIIIIJSICCCCRE So CCCCCDDIIDDIICJDJSSCCDDIIDCDONDIKDCCIIR..rFFE 000 5 05 0505 0 0 00 00 03 01.2 1. 20 .3556 1 .3 8:. 23 .35 000 1 11 11.1.1 CCCCCCC CC 1 7 71. 38 98 222 CCCCCC 230567890123456789012345670.9012345678901234567 80 0121.456740 0123456799 0123456789012 005000331111111111222222222233333333334944444444555555555566666566?b7777777777388 777777777777777777777777777777777777777777777777777777777777777777777777777777777 111111111111111111111111111111111111111111111111111111111111111111111111111111111 ...Itfitiifitfiifiifittififiifit...ttittfifiitfii ) J(10)oA(10)oIXX(10)oL(1010)QSXX(10) LE(10)ONUHELQ1PAR91CAL101CAL2Q1CAL39151RESS .tttgtctctattittn09......919.....ttttttotatttttct ......OOQOQOQOOOOOOOCOO.........‘Q.OQOQOOOOOOOQOICOOOOOOO......‘t. 0 3 7 h 8 1 1 ( K H T C E U ) D O N ) 1 1E ) )O O J 0 0 1D 3 C. I” O ) 8 (O 9 0t ( M 2 3 V." 1 5. 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OE T t 1 1230 O 91 LAOO D 1 T 1 ((1(1(11 a 99.9.? 99.0L a 8 090. 1 L 12 DNII (S E 1 1111111. 5 R020 1:9‘L o t 3 DPDR T L Tr. ‘19. $3 0 1 111111 0nd ELDxt ORIOA Gr: A $9.13 T A Y 1 940.3 EA o 0 PPDPPPDN TTE: TOR 0C Ct 9 ) D((( o R C 9‘: ED11173QRIDA o D 7 .JA ) 1TT) 901 VOL 1d 0 L NCCCS A AI. TE 0 .CCCCCC 0T N 1 A o 1 v 011 099.1 E¢.L MN 9 H .0 VKWMI 9. OT )...Tfl.nqunivoH1mvn\TrJ A O )))))) 0“ Y1UTH. Mo GERS) )G o v. 2 2 t. l‘ o Uh AA < ) 2. cap-3.214.111.1118. o o D. 1 15.5 .4 c.2111“. D ) A 9cm“ UT E91 chU 290 Q 0‘ 0 VLLLP 1. A L 1h: FOODDPD o) 11 V 0 VM».\..\.V\“\C M‘ 1 v.2 onD )).E o .EtN CNN.) 2 2 93.3 «2». T o )D. A). QOOOOO¢O)3 00 H 7 1.3000330. n... O 0E))QO 19 O .0 0V. 0 1 O ) ) 1on U o 5 0 EH C1 O. 915344.361: 0T U31 PPPD 90.90 U M. TT93 93 .TXTE F_u.\t1 2 1) 1 1. ..UUUT 8 7. 8L. 53X: 20099909A [N M. 0 9.3993900 "31‘ GI3NXL 3303 o 00. 2 ))1))..)) 9 O a ((1.3 3 o 3 O (“JawaAAAAAAnJ oJ 91) DUOOODOL 9 01 09935.4( £91 .5 TCtK MU K12111POP N533 044 SBQX 111DDOOOOOOOL N0 12))LLLLLLL o 1111RUD P1T o O 015 P Q (at o o I QNBN 98.519 9 $7 90 0 T Q 9AALLLLLLLPOTCC 2111 o o o o o 0 0F. 2 :POEM- v0 '0 1.1/93:...0Kt )1J 99 O 9:33:11... AAAT31r.r.311)2:. [1,0090 PD DPDCGPV:N 0 vQQOQGQEm.F::VI:CT:10/T L6AOD.UKCO9KE.L9C_EQNNC9CUU32 2 2T “(DURQ/D‘RJzuU DS‘LL: 2 2 2 2 ..(590 29 1F.uvr:Lr._LEEN. OUU )1196Ll‘0 A(TRTNHEQT 0907 o 03“” NNN9123R (NT (T1 N121TPP12345553 TTElOC‘ o o o o o o .TNNGZDT. H5117. . CEARSICHa31003113AAA0A113TTTAOEISOEA0012PEAt1DDUDQUBACENTCTAA111111111177COUTETAO TTHETTICQQ‘NNAKKQITLTLTTQTTTPTTTITTHITTTETM. O QAAAAAAA oTDUIRlTITTTTTTT oTT33NTN1T00-X CTR‘CNC . L2 2 (l. 12.x PM». D 0:!" INC 1.9.0. NETTRXXOOOOH 019‘ (On (..((((I\((n,..\.o\ ( (u. 1.....ng PROF FOFOQO..TJDFFAJFLVOVOOJAAAFOROFOROC.OJD.~.RJOOLLLLLLF30FCUOJC-1FFFFFFF9039310FU9331 IUFIICITtDHNNDIIDINUGUCCDPPPTGUCIGUFQGCIDUFIIPPPPPPToGINNDDIIIIIITIIACCDDPGINUvFO o O O O o O 6 o o 87 T 2 109 3 9.3 97 1 Q 1 9 33 7 8 387 8 89 72 0 0 01 5 nu 3 00 3 3 333 3 33 37 4 3 70 7 2 0 99 4 CC A. 44.9 A. 94 43 1 1 19 3 5 9 202 6534554780 1.19.345.b7€n°.fl¢12~3“.C. b7; 69 011_.1.4§.67C\ C 01051~4spb70 G .Jlnds‘. fitlnctuaéfi. 37:,3 ClficzdnwcTOwIQAu «01‘s.; “a. 572;: °.q7.q C 3 O O G.,OAUF.3AJOOOOn,.1111111111.29.fie«éfiinl.ndficfiaOL-«uty191335931.KzfaQ :4 4 Q “a q h «:.:.C.:...R.r7..3.k..: “VORTE 43.? D.O66.5777 30.: Q..0.0. 8888888896 O O Q._9,Q.9Q..Q.D QQQ O ... O a.D 0.9.0 0 0 0.0.0 C 9q 0 90 D 0.0 a c 99.3.0 0 Q Q C = .- )) ‘13 NN.! 00p ‘1 PD”. 1.. “40v.” OOU N LLN3‘ (It '01.. 7.71111 00:..P0 TTNI..OP{E DD 0 l 3391 VT Q Q77 QOIA XXSSNTTM. «L0 11.00.1000“. Q QUDP GCFE Q. .....ifitttttifififififitt.................IDQOD...iiiitifittfiiti.Cfifittfit SUQROUTINE NADJATAHGW’TIN) 573 ccccccc cc I. L A C I Q 2 L A C I ) ) Q C D. 1 M \I ‘1’ Ms L I N u. u. A F (I (ll U C l. v: YY N T. A Q Q Q Q Q Q \I )’ 1 3‘ ) \I M: .1». : A 1 AL ll (IQ N D. . Q x XX Q I I t 0 O. 9 \I Q 1| hi it "a 3 L O A )3 1 3 Q E Q: ‘l P 3 Q Q Q 1. H 1. SS 0) 5 Q1. 1.1 : U 2 OZUU 22 ...l. 1.: : .. I N C 00.11 I. NI :1. I... 5 Q Q V- 1293 0. ’2 .1 Q Q Q 0 \r ‘l 0 AA R) Hi. 0,0 )\10 1 I 0 \I 009:1 5!. u!” A o ’19 T310 Q 2 1 TT Q Q 2‘ AA P) I Q Q Q1 0 N. 1‘ 2 00 FX PP ST. QNO N” .l 1| F. ( 009R L. 95 0‘ N‘T ((0 A L Y 50...... A3. Q Q \rvx (A AAT ..U T. Q3 Q 7.2 . Q ...r. 1.. A10 .1.... G (G D M.) FFr.P(Q SSD. . E III...» ((3 QE ”A3 "LLNK‘S IINIQ. 0! Q Q Q5 D \1 NN.. “a Q n.9AA/HVU 9.9.0.1:1 QM. NV N 1. 1‘ UN...” 0 O OUII U09 N), ‘1’ M: O \I)’, N ,OLOQRJQ onJanu. N 0. )CP ’00 U E «H.000 Q9‘11—L.L\J)E Q. A ‘1 Q0... 0 1.9.N 3.2M. N3”. 1 ASL. ....LAWAJ Q 00.0.7.1.ba1 \Ifi.1/“ 0; V 30V- 9 Q o 0 fl 0520 N..»T.22N.412F:U( 03:». 0?.U 02U 1.1) 1 .L 2222 [I] Q QII Q QLIIT 41 QIAE 1 QNvll QM. : ..100.1N Q Q Q Q 1...311TT01A DD 50 Q1 D7 SEE Q1. on... Q1. . NV] Q:2 : Q : E51111 IllSéDP 56: AOUZOS SIUU0.br..10 3L n\)l)0)UU5666 NNN((00((C1RSV 6‘3.QNN6(N 5‘“ 50‘! IEINN((((N 000EEGGOENO:..10‘50: ...-51‘: 0011.». 009.2A Q0 QN QIIErL—LER ”v.0.TTIIATIZ’)TTH;I?_’)TT—UTNTDTMV..IIIVTN.=NTTTTTTU UHVII(( IF IIN AI YINVAI( AI(Q (‘ (9(MVIIIIT ODOR—(\FF 7.RL~J((OO.LRO((OopLRFO—ZD‘C,EOJFAOArwAODD‘quq.L CCCUHIID‘ .UADXYCGR quYCCRHIGRHINDDIIGIVICCHHHUR 1 2 3“ a 1 5 05 o o 300 a 0 o 22 2 2 220/ 1 1 1 11 OD QQGO 3Q 9.0.00 3 c QQQQQIQQAVQ. Q x ...? Q S n; D O C \l N. I 0 I) I, 9’ T’ 0’ II 1.Q «.JQ 0: NM. .31 0‘ 10 CA FR... 1, Q1 YH) Q .C. R7N X : Q A Q‘ 2 Oct DXVI 2 U: ...le Q IN U15.“ )0 DA OQQ as A? Bax AP st r8 Tr. it )TBQ AD.» QQ ’NxM, DP. XX QIUN U, 00 .ON‘ LNQ 11. AD. x A X QQ 7- HH. 0N3 00 AL74)003 11.. DA, QSNI Q o o 05X 0 T9 55 LOEOZDA) 1.1.. ANTZIEUI FF 34A QFTQ.Q) QQ ODVZZAENX i i) ”SIT QD.‘*(T : :5 QDHXE QAH AC. 0 TXQODQNXI?‘ an» U30 erhqufi ’ LII D. Qox QG Q QX’ ARC: NEC4r\.tr.X45 ’1 .2 .10 Q1. in Q1 0 Q Q Q) .OTYSXOIYS 1XX1.3 QMwNH QCN QH Q 000711X ....le 11.x 511533 «J Q5 Q 9 Q6 1QQQQ1 PXQ/ EXQ D.,/55 QH “Ra/H845 nL/III/7L QvL/7U» Q1 ((‘(l‘l‘l‘AXI‘l‘l‘UX‘ TTTTTTTDHTTTNHT AAAAAAAOIAAA 1A ”VVVMMVNQWVU. QM“ DRR?.RC.FHXRRR X9 300000035003t50 FFFFFFFZIFFF/IF O O Q 0 0105 000 0 01.11 234 .3 0000000 000 0 CCIZSIIZZ 222 2 203 FORMAT(Q-Q10HNEO=113) END 2060 ......OOICIDODO‘DIDQIDD....DO‘I"......DDODQOIADOQOQO..OOQOOOOOOODD SURROUTINE ELEHENT ccccccc cc (20)1A(20)OIXX(20)1L(1120)1SXX(20) LolPARoICALIQICIL211CAL3 E ( ( NUHELQECI) NU“EL1€(1) U1 Q ‘1, N2, I110 Q1|1 022 —..A1\—L100 NIV.L022 Ill Q1 Q Q 13.?! Q1111 Il’x0U/b6 "NNIDQA 000 (CE MM.» LOTT MuVAAIIOIDXAIK1(TRRRRRD 03059.9.3‘:a\31r3\ :(FEJJOUOV CCCRRUUKUF 1 R HKLIRFFFFFE 31234:.679D 0123A5578c 012124.?570.6.0121..An..h.73o 012305575c 01.23 345.0711“. 01651.A:.,07a.o.01nc34: .57: 3 300.00? a.n..u?.1111111111ngahncfi..222:?nc1.5331.31.31.31» A L A 44.44.44: 1:25 777777788R.8Q.80.39..83 = c :6 a c a : a 000....nanCPJCDCOrufitocq.OCOabfikocooococoauccccc0000.0:anfiLCCflRlfliUCOC 9999999Q9096990999999999999111111111111111111111111111111111111111111111111111111 Q x 0 11 Q Q ) In ( A 1 Q x «a 1 1', Q Mun. 0 ‘1| \1 xx I. XX 1‘ 53 J Q Q r. ’) 0 U 4 H. (II N xx ‘1 1 xx I Q 11. Q X Q Q 1 3 \I) ‘I Q “9. 11 0 1|( 1| ‘1 AA. .1. M Q Q 1 0 ’, ' I u.“ v», r. (1| 06 0 JJ 11 o Er. Q5 N 00 11 i 00 Sty Q NN. T3 x Q Q ~ Q 3 1', 5 pix Q’uxu. 0 H5 Q/‘II 1 E Q T QII L5 N.p1hr 0 ’El r.|1«.JPu T (C Q H.900 \1 1:.X E(Mvuw 0 6705 L! Q Q G \1 111 Q Exp" GuanJti ’15 ‘1 \1 11.2”]. 5 Q1)? L 0.... Q1 Q 2X Q0? .... 1.3x QX 03K30 v F. QBXS 2 Q302 U Q5 Q3 Q Q/ Q1. Q N 51.5 Q6 1’. Q1 v 1 1131/1. 01:1an ...... (((‘C (TM.6| ’NMcTTTTT EA|(E1K 19‘AAAAA TMXUTO QKUMMHM u M. O 0001.). 12222 00000 111.222 2025 105 ...ODODODOOQOQDODODIIDODQDDDQIOQ.....DDOODIDOQQI......OOCOAODO‘... SUBROUTINE BAND CCCCADCC cc 53 T3 199 (20)13(20).IXX(20)QL(1120)1?KX(20) 2’.LEOOOAND.IA(".IQS’OLi-og’ LECZO)QNUMELOIPA“CICAqulCAL211CAL3 )J IPL 20 11 Q 0 (0 OT. 0 TN 0”. 8 Q Q 11‘ 1!) A o 10 01 T 2a... T 1 (1| 0 0 X1. ON 5 93 Qr. GA 94,3 1 \1 ”Axes ,0 9 v. 3 QN 1. 1 1 UH... Q of E N Qfir.’ f. OIL L Q0111 N 3... 1 1 r..._AlI Q’) Q 1‘1 \1, NN.-1F. lfluv.1L1 II [III 0..1|1|=RQ QQ 11 1.235 ..HIJITIEIIQ... III/0L Er. NNUNVgL MINNN..RUDDCO(N1\( R ooaonTCOOOCAIAAOT 9 99 ..J 0 9 99 C 0 A. 2?. 8 9 0 30 0 0 T TT T T «U fi~o 0 «J G 55 G G 0 ,) 0 0 0 01. 0 o 1 1 1 1 1 .L [E E E L LL L L 1 1 1 1 1 , ,, ) , 3 33 o: 3 Q Q Q Q Q Iv 1..-u J I n.» ”X N u. ( (0 ( 0 A AA A A I II 1.1 I 1 1 1 1 1 0 DD 0 D o\ ”N N. N. A AA A A 1 1 1 1 1 0 OC 0 0 1 1 1 1 1 r. [E E ..L L LL L L 1 1 1 1 1 0 ’0 0 0 2 22 2 2 2 Q 0 Q Q Q Q OJ 0 1.1-J J '1 0M: 7 PIN N N 1‘ (1| 8‘ 1| A 0 AA MIA A. 01 T 1.1 ...I. I T 1 1 1 D 1 1 0 0 00 V9 0 ON 6 N...» A11 N GA AA BA A 1 .1 1 1 M. 1 1 ’0 n. 30.. 1.1 0 1. 1 1 1 1 ‘1 1 1 15 F. )EE DE E 0L .L QILL MwL L 35 1 1 N 1 1 A 1 1 Q 0’ ’1’- 1!, 3’ I1 1L1 Jnuzll H1 1 ..R Q 91. 9 Q Q" Q Q 1 Q Q IUTJF.9= qluav‘vld TJEQA MINUQL 2NN§N~IGNUN 00(N R 1|1|31|1|Q 11|Hul\ OCAIOTIOAAAAABBAIA Mn... v. u...MyN9MNBIITITTNAJTIITTNNTIIIIIV v ....TvI vvaAaJ .... ((Ma(_. 0: 1|(MI1 0.. (: ..‘(..l\l|~1| «JOQOBCAJIJDFFOFlauCl)FF03€30F).3FFR.FF0F. .LCPZLWIAUNMH¥IIr¥1Na31Nn111rXYlHc¥1NM1:1WYQALI 1001 199 99 1002 399 ~w9 299 700 Q567Q.90121.455789011913455.7.150 01.234.51.71 9114123456789 310.34.39780 011(1..4§.67590121.Q.35701901234 R 2:15:25; PL 2,...5,» 65.657777117777714 Q; G .1 2 2 a. 9.3 O a 0,9: 090 3 q CauabnuonunVAanU31111111111111.a<2fignsfi¢ficn1?2.fl~!.§133 0000010000000000000C00n1000n.n.1vn.0110007.0000v00(00001.1.1..11111.1.1111111111111111111111111111.1111. 11.111.111.11111111111111111111111....1111.11.1111.11111111111111111111111111111111111.11111111111 1 3 3 1 1 1 1 1 1 1 II \1 \1 0 1 0 Mn 2 u‘ 1 I 2 1| 1 l1 3 1 1| 11?. C r. 2 1 Q x )L 1 L ‘1 1 .1 X HI .1 I M 1 0 S 11111|2 3 2 Q 1 2 Q 11112.1 1 1 2 1 Q Q “IIL1 1 1 Q n. 1 3 0 ((IZ \1 C 3 1 1 3 1 2 )EE)1 n 1 .| O 1 L 3 Q )LLE) 1‘ 11 M : 1 A l. 1. u II(v. E v R 11 1 c L 1| 1|)|1v|1| L 1| |I|1 A V. 1 11 A l. EBBRX I X "M. 1| Q 1 Q N Q L((QX \1 X (11 S 2 1 9. 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