.V. V./...”.VV...T./?. Vinnf. yawn—5.. '51.“. J. fix... on :4?th J . V~ ”W. MWW. .unUuV.nv .V. Jar» flu! . f,- fin V . . .V. . 55”.. . 1». 1.: V... _. .7. . {an Ir.- r _1.32....V}:EV,.VVV.W».6:::I Vv I... . VV ,4.» V Vultfiaflrfidb .rVr u» o V6,. .. (VIVPZV 3.. .23 4r i .V ‘ 4...: 4h» .. 7.2.“... V .VV V....lV,..4V..V «7—1 :3me V. in. V, V 7:.1W4ruhzmtrr. .2.qu 5!er #LVV 5.} a, 2... V 2 I .flvryflrf. . V: . Vppvma; . V1.) V.;VV'..V..V V .V 1V 3.. 4.1.155... V. 1sz V. .3 V V wwixéizai... V.\...\..wl m). .V .. V lJ LIBRARY g 5“ IIIIIIWII 1, m This is to certifg that the thesis entitled A NONLINEAR DYNAMIC ANALYSIS OF LINE STRUCTURAL MEMBERS presented by John G. Janssen has been accepted towards fulfillment of the requirements for Ph. D. degree in Civil Engineering Major professor [x I x __ r“ Date W7? ' l .2 76 J 0-169 iwvm ABSTRACT A NONLINEAR DYNAMIC ANALYSIS OF LINE STRUCTURAL MEMBERS by John G. Janssen A general numerical procedure for the dynamic analysis of line structural members in which both geo- metric and material nonlinearities are considered is pre- sented. A lumped mass and lumped flexibility model is formulated by discretizing both the length and the cross- sections of the member. Thus, the method can take into account the effects of axial force-bending moment inter- actions, large displacements, inelasticity, as well as initial crookedness, residual stresses, initial static loadings, and movements of the supports, if any. The numerical procedure is based upon a step-by- step numerical integration of the equations of motion. A computer program was written in Fortran for use on Michigan State University's CDC 3600 computer. The reliability of the numerical approach is demonstrated by comparing solutions obtained from the present analysis with a variety of known solutions. From the comparison it was concluded that the method John G. Janssen of analysis presented does produce results sufficiently accurate for engineering purposes. Solutions to problems heretofore unsolved are then obtained to demonstrate the usefulness and versa- tility of the method. These problems include the inelas— tic dynamic snap—through of arches and the effects of strain hardening and residual stresses on the critical snap load. The failure of an inelastic axially loaded column subjected to transverse support motion was also studied. A NONLINEAR DYNAMIC ANALYSIS OF LINE STRUCTURAL MEMBERS by John G. Janssen A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil Engineering 1968 ACKNOWLEDGMENTS The author would like to express his deep gratitude and appreciation to his advisor, Dr. R. K. Wen, whose guid— ance and assistance were invaluable throughout the author's graduate program and especially during the course of this investigation. Thanks are also due the other members of the author's guidance committee, Dr. C. E. Cutts, Chairman of the Department of Civil Engineering, Dr. J. S. Frame, and Dr. L. E. Malvern, for their encouragement and inspira- tion. Special thanks are extended to the National Science Foundation for their support of this investigation through Grant #GK-590, and to the Division of Engineering Research. Also, the personnel of the computer laboratory deserve thanks for their advice and c00peration during the course of this study. The author will forever be indebted to his wife, Susan, whose help and encouragement have made his work immeasurably easier. ii TABLE OF CONTENTS Page ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . ii LIST OF TABLES . . . . . . . . . . . . . . . . . . V LIST OF FIGURES . . . . . . . . . . . . . . . . . Vi Chapter I. INTRODUCTION . . . . . . . . . . . . . . . l 1.1 General . . . . . . . . . . . . . l 1.2 Related Works . . . . . . . . . . 2 1.3 Organization of Report . . . . . . . 4 1.4 Notation . . . . . . . . . . . . . . 4 II. DISCRETE MODEL . . . . . . . . . . . . . . 7 2.1 Introduction . . . . . . . . . . . . 7 2.2 Subelement Arrangements Investigated . . . . . . . . . . 8 2.3 Formal Element Adopted . . . . . . 10 2.4 Spring System Representation of Flexibility . . . . . . . . . . 12 2.4.1 General . . . . . . . 12 2.4.2 Deformation of Springs . 13 2.4.3 Spring Force- Deformation Relation . . . . . . . 14 2.4.4 Internal Spring Force Resultants . . . . . . 15 III. METHOD OF ANALYSIS . . . . . . . . . . . . 16 3.1 Introduction . . . . . . . . . . . 16 3.2 Geometric Considerations . . . . . . 16 3.3 Equations of Motion . . . . . . . . 18 3.4 Support Conditions . . . . . . . . . 19 3.5 Initial Residual Stresses . . . . . 20 3.6 Arches and Initial Crookedness . . . 22 3.7 Initial Static Loads . . . . . . . . 23 IV. NUMERICAL PROCEDURE AND COMPUTER PROGRAM . 24 4.1 Introduction . . . . . . . . 24 4.2 Numerical Integration Procedure . . 24 4.3 Step— By- Step Numerical Solution . . 25 4.4 Stability of the Numerical Solution. 27 Chapter. 4. 4. 6. 4. 6. 2 4 6. 3 v. COMPARISON STUDIES .1 Introduction . . . . Fixed— Fixed Elastic Beam 5 Computer Program . . . 6 Energy and Impulse— —Momentum Check 1 General . . . . . . . . Energy Check . . Impulse-Momentum Check Elastic Circular Arch . Imperfect Elastic Column Simply Supported Inelasti C B 9-, o o n a VI. NUW RICAL EXAMPLES . . . . . . . . . . . 6.1 Introduction . . . . . 6. 2' Dynamic Snap Through of Arches . . 6.2.1 Inelastic Arches . . . 6.2.2 Effect of Strain Harden ing on the Critical Dynamic Snapping Load 6.2.3 Effect of Residual Stresses on the Criti— cal Dynamic Snapping Load . . . . . 6.2.4 Effect of Model Bias . 6.2.5 Comparison with Static Arch Buckling . . 6.3 Effects of Support Motion on a Column . . . . . . . . . . VII. CONCLUSIONS . . . . . . . . . . . . . . . BIBLIOGRAPHY . . . . . TABLES . . . . . . . . FIGURES . . . . . . . APPENDIX. COMPUTER PROGRAM . . . . . . . . . . iv 42 44 44 46 47 49 52 57 58 76 LIST OF TABLES Table Page 1. Comparison Study of Elastic Circular Arch . 57 2. Effect of Strain Hardening and Residual Stresses on the Critical Dynamic Snapping Load, Ps’ of a Pin—ended Arch . . . . . . . 57 LI ST OF FIGURES Figure Page 2.1 Spring System Representation of Lumped Flexibility . . . . . . . . . . . . . . . . 58 2.2 Model Element Assemblies . . . . . . . . . . 59 2.3 "Formal" and "Modified" Models . . . . . . . 60 2.4 Bilinear Stress—Strain Relation . . . . . . 61 3.1 Portion of Deformed Model . . . . . . . . . 61 3.2 Free-Body Diagrams of a Mass Point and a Massless Panel . . . . . . . . . . . . 61 3.3 Treatment of Boundary Conditions Involving Prescribed Support Rotation . . . 62 3.4 Residual Stress Representation . . . . . . . 63 4.1 Flow Diagram for One Step of Integration . . 64 5.1 Response History for Center Displacement . . 65 5.2 Response History for Quarter Point Displacement . . . . . . . . . . . . . 66 5.3 Elastic Circular Arch . . . . . . . . . . . 67 5.4 Large Displacements of a Column . . . . . . 68 5.5 Comparison of Large and Small Deflection Solutions for a Small Deflection Problem . . 69 5.6 Comparison of Large and Small Deflection Solutions for a Large Deflection Problem . . 70 6.1 Snap—Through of an Arch (N = 10) . . . . . . 71 6.2 Dynamic Snap—Through Loads of a Shallow Arch . . . . . . . . . . . . . 72 vi Figure Evaluation of Model Bias . . . . . . . Maximum End Displacement vs. Peak Support Motion . . . . . . . . . Permanent Lateral Set at E vs. Axial Load Page 73 74 75 CHAPTER I INTRODUCTION 1.1 General There has been much interest in the study of struc— tures subjected to earthquake or blast loadings. Under these loading conditions the ultimate strength and the associated deformations are of interest to the engineer. In order to determine these sufficiently accurately, an inelastic, large deflection (i.e., geometric nonlinearities considered) analysis is usually required. This thesis is concerned with the dynamic inelastic analysis of line structures (i.e., structures in which the length is much greater than either the depth or the breadth) which includes beams, columns, and arches. The study is limited to response contained in one of the principal planes of bending. In this investigation a lumped mass and lumped flexibility model is used to approximate a line member of arbitrary shape and cross—section. The cross—section of the member as well as its length is discretized so that the moment—thrust interaction is implicitly considered in the model. The equations of motion of the masses are integrated numerically. Large deflections are considered even though the deformation of each lumped flexible joint is assumed to be small. This analysis also takes into account the effects of axial inertia, residual stresses, inelasticity, and support motion. 1.2 Related Works The work of Newmark (32)* in obtaining a numerical solution of the buckling load for an elastic beam—column is well known. Salvadori (38) solved the same problem by finite difference techniques. Eppink and Veletsos (10) studied the dynamic response of elastic circular arches by using a lumped mass, lumped flexibility model instead of directly applying finite difference techniques to the differential equations for the continuum structure. Wen and Toridis (51) studied three discrete models for elasto-inelastic beams without considering geometrical nonlinearity and axial loads. The models studied were: model A, lumped flexibility and continu— ous mass; model B, lumped mass and lumped flexibility; and model C, lumped mass and continuous flexibility. Beylerian (6) also used a lumped parameter model to study the effects of shear deformations on the response *Numbers refer to references listed in the Bibliography. of inelastic beams. In the study the interaction of bending moment and shear on the elasto—plastic response was taken into account. Analytical studies of the response of an elastic column with a small sinusoidal imperfection under the dynamic loading condition that one end of the column is forced toward the other were made by Hoff (18) and Sevin (40, 41). Sevin considered the effect of axial inertia while Hoff did not. Bailey (2) improved upon the works of Hoff and Sevin by taking into account the effects of large deflections as well as axial inertia. In studying the inelastic static behavior of wide flange steel beam—columns Ketter, Kaminsky, and Beedle (22) developed a series of moment—thrust interaction curves. Galambos and Ketter (14) used these interaction curves to determine the inelastic critical load (i.e., the load at which Newmark's numerical procedure, when applied to the inelastic member, would just fail to con- verge). They also considered the effects of residual stresses on the moment thrust interaction curves and the aforementioned critical load. The static behavior in the inelastic range of line members with rectangular cross—sections was studied by Malvick and Lee (28) and Lee and Murphy (26). Both straight columns with small initial deflections and cir— cular arches were studied in the small deflection range. A feature of their analyses is the discretization of the cross—section into a finite number of subareas. 1.3 Organization of Report In Chapter II a basic element is formulated of which the model of the continuous member is constructed. Chapter III concerns the derivation of the equations gov- erning the behavior of the model. The treatment of the support conditions, residual stresses, initial crooked- ness and static loadings are also considered in this chapter. In Chapter IV the numerical solution of the governing equations is presented. In the same chapter a section on the computer program used in this study is also included along with the impulse—momentum and energy checks on the numerical solution. The credibility of the model is examined in Chapter V by comparing its numerical results for certain problems with solutions that have been presented in the literature. In Chapter VI the dynamic behavior of several arches, including the "snapping through" phenomenon, is depicted. The failure of a column subjected to a lateral motion of one of the supports is also considered. 1.4 Notation The symbols listed below have been adopted in this investigation. A = area of the continuous cross-section; i' = jth lumped area of the ith discrete cross— ] section; Amp = rise of sinusoidal arch; dij = deformation of the jth spring in the ith spr1ng system; dhi = relative displacement of the ith spring system; E = Young's modulus; Ep = inelastic modulus; 1' force in the jth spring of the ith spring 3 system; Flexi = lumped flexibility at the ith joint; flex(s) = distributed flexibility; hi = deformed length of the ith element; h0 = initial element length; i = subscript referring to joints or panels; I = moment of inertia of the continuous cross— section; j : subscript referring to parameters dealing with the jth subarea; L = length of the member as measured along the centroidal axis; m = mass per unit length of the continuous member; i = ith lumped mass; i = moment at the ith joint; n = number of elements in the discrete model; pi = longitudinal load lumped at the ith joint; P = A0 ; Y Y magnitude of asymmetric loading; "(5 ll magnitude of uniformly distributed loading; transverse load lumped at the ith joint; radius of circular arch; strain hardening parameter, Ep/E; radius of gyration of the continuous cross— section about the axis of bending; distance measured along centroidal axis of the member; axial force in the ith panel; time, the subscript referring to a particular instant; time increment; displacement in the x—direction of the ith mass; shear in the ith panel; displacement in the y—direction of the ith mass; x—coordinate of the ith mass; initial x—coordinate of the ith mass; y—coordinate of the ith mass; initial y-coordinate of the ith mass; — distance of Aij from the neutral axis; rotational correction factor applied to the ith joint; relative rotation of the ith joint; density of member material; yield stress of the member material; angle between ith element and the x-axis. CHAPTER II DISCRETE MODEL 2.1 Introduction The model considered in this work has both the flexibility and the mass of the line structure lumped at discrete points along its length. This approach was hinted by the results of Wen and Toridis (51) which showed that a seemingly more accurate approach (either model A or C) would not necessarily lead to a more accurate solu— tion. The model for the entire member is composed of a series of elements joined together. Each element is made up of subelements. These subelements are rigid panels, lumped mass points (rotatory inertia not in— cluded) and lumped flexible systems. Each of the sub- elements represents the corresponding properties of length (geometry), mass, and flexibility for a portion of the continuous member. The flexible systems are composed of a finite number of parallel springs located between two rigid parallel faces that are perpendicular to the adjacent panels (see Fig. 2.1). In order to determine the deformation and the force in each spring of a flexible system, the rotation and compression distortion of the flexible system are required. Hereafter the flexible systems will be referred to as "spring systems." In this chapter three types of elements based on different arrangements of the subelements will be considered. The reason for adopting one in favor of the others will be discussed. In addition the manner of lumping the flexibility into spring systems will be described. 2.2 Subelement Arrangements Investigated In Fig. 2.2 are shown the elements and their corresponding assemblies considered in this chapter. The element, B-l, in Fig. 2.2a consists of a lumped mass at either end and two spring systems contiguous to the masses. A rigid massless panel connects the spring systems. In this configuration the lumped masses and flexibilities represent the mass and flex- ibility of the continuous member corresponding to one— half of the panel length. Due to its symmetric arrangement, this element has a strong intuitive appeal. However, it has one serious drawback. The model assembled from this ele- ment is kinematically indeterminate, i.e., it is not possible to determine the distortions of each spring system entirely from considerations of the kinematics of the displacements of the mass points alone. Con— sideration of any joint (see Fig. 2.2a for definition) reveals that, given the total relative rotation of that joint, it is necessary to proportion this rotation be- tween the two spring systems located at that joint. Since the mass lumped at the joint is assumed to possess no rotational inertia, it is required that the moments generated by the two spring systems be equal. An equi— librium condition must therefore be satisfied. Similarly by considering the relative displace— ments of the two masses at the ends of an element, it is seen that the total compression in this element must be appropriately proportioned between the two spring sys- tems at the two ends to satisfy the equilibrium condi— tion that the axial forces generated by these two spring systems be equal. It should be noted that these two equilibrium conditions are coupled among the successive joints and panels. It is possible, given the boundary conditions of the problem, to write a set of simultaneous equations which incorporates both the geometric and the equilib— rium conditions of the problem. A solution of this set of equations would yield the spring system distortions, provided that the solution exists and is unique, which would be true in the linearly elastic case. It is, 10 however, easily conceivable that in a nonlinear problem in which yielding takes place, the solution of these equations would be very difficult, if not impossible, to find. In Fig. 2.2b is shown element B—2. By lumping the flexibility at only one end of the element the difficulties associated with element B-l are eliminated. In this case the spring distortions can be uniquely de— termined directly from the mass point coordinates with— out the necessity of solving a set of simultaneous equa— tions. This greatly simplifies the analysis. This ele— ment, however, lacks the appealing symmetry of element B—l. It is possible to obtain a symmetric element different from B—l, by lumping the flexibility at the center as shown in Fig. 2.2c. It turns out that this element B-3, leads to a model which may or may not be kinematically determinate depending upon the boundary conditions. Even in the cases where the boundary con- ditions lead to a kinematically determinate model, a set of nonlinear simultaneous equations must be solved in order to determine the spring system distortions. 2.3 Formal Element Adopted Due to the complications which would arise in the utilization of element B—1 and B—3, they were not 11 used. Element B-2, in spite of its asymmetry, was adopted in this investigation because it makes the analysis man- ageable. The individual elements B-2 are then assembled to form a formal model of the continuous member as illus— trated in Fig. 2.3a. In this case the model represents a beam-column with one end fixed and the other end simply supported as shown at the top of the figure. The coor- dinate axes are shown as well as the numbering of the masses, the flexible systems, and the massless rigid panels. The flexibilities are lumped in the following manner: ho Flexl = 6 flex(s) ds Si+1 Flexi = f flex(s) ds .....(2—l) Si Flexn+1 = 0 where Flexi denotes the lumped flexibility at the ith joint, flex(s) the flexibility distributed along the length, and s the distance along the member axis. An inspection of the "formal" model in Fig. 2.3a reveals that it is asymmetric; there is a spring system adjacent to the left support but not the right. As will be shown later, the error introduced by approximating the continuum member with this model would be aggravated in this case if one or both of the ends of the member were 12 fixed. In order to alleviate this difficulty, a slight variation of the formal model is introduced. The modified model is shown in Fig. 2.3b. The difference between the formal model and the "modified" model lies in the manner of defining what portion of the continuous member each spring system represents. In contrast to Eqs. (2-l), the following equations define the lumped flexibilities for the modified model. ho/2 Flexl = f flex(s) ds 0 Si + ho/2 ’ Flex. = f flex(s) ds .....(2-2) 1 s - h /2 i o L Flex = f flex(s) ds n+1 L _ ho/2 It should be noted in passing that for a prismatic member the amount of flexibility lumped at an interior joint is the same for both the formal and modified models. 2.4 Spring System Representa- tion of Flexibility 2.4.1 General.-—The flexibility of the member is lumped in spring systems at discrete points along the mem- ber as described in the previous sections. In this section the discretization of the flexibility over the cross—section of the member is considered. This discretization is 13 consistent with the philosophy of the discretization of the mass and flexibility along the length of the member; As mentioned earlier the spring systems are each represented by a collection of parallel springs connected to two rigid faces. The cross—sectional area of the con- tinuous member is divided into an appropriate number of subareas of some appropriate shape. In Fig. 2.1 is shown a spring system which represents the cross—section of a wide flanged member. A“j represents the jth subarea of l the ith cross—section, and zij is the distance from the neutral axis of the member to the centroid of Aij' The positive direction of zij is opposite to that of y. Each spring in a spring system represents the axial flexibility of a prismatic bar whose length, ho’ is that of the rigid panel and whose cross-sectional area is that of the corresponding subarea. The spring is located at the same distance from the neutral axis as that of the centroid of the subarea to which it cor— responds. 2.4.2 Deformation of Springs.——In order to cal— culate the force in each of the springs in a spring system,. it is necessary to determine the deformation in each spring. There are two parameters which uniquely determine the de— formation of every spring in the spring system. These two parameters are the relative rotation, oi, and the relative displacement, dhi, of the faces of the spring system. 14 Referring to Fig. 2.1 and assuming ¢i is small, the de— formation of the jth spring in the ith spring system is given by: dij=dhi+¢izij ooooo(2"3) Spring deformations are considered positive if they are compressed, and positive rotations are those which cause compression in the springs with zij > 0. 2.4.3 Spring Force-Deformation Relation.——Once the deformation of a spring has been determined, the force in it can be found from the force deformation relation for that spring. The force deformation relation for any par— ticular spring is dependent upon the stress—strain rela- tion of the continuous member material, the length dis- cretization, and the area which that spring represents. For each spring the.force deformation diagram is obtained by respectively multiplying the scales of the stress and strain axes of the stress—strain diagram by the subarea, Aij and by the panel length, ho. Used in this work is the bilinear stress-strain relation in which hysteresis and strain hardening are taken into account as shown in Fig. 2.4. Of course, the method of analysis as presented is not restricted to this type of stress—strain relation. More general types such as trilinear or curvilinear relations may be employed. 15 2.4.4 Internal Spripg Force Resultants.--Once the spring forces are known, the internal force resultants can be calculated by the familiar equations of mechanics. The moment is given by: M. _ 2 fl 0200 g....(2_4) 1 . 13 13 J and the thrust is given by To = 2 fun 000.0(2-5) 1 j 13 where fij is the force in the jth spring of the ith spring system. CHAPTER III METHOD OF ANALYSIS 3.1 Introduction Up to now the main topic under consideration has been the formulation of the model used to approximate the actual continuum structure. Once the model has been established, the next step is to derive the equations which govern its behavior. 3.2 Geometric Considerations In Fig. 3.1 is shown a portion of the deformed structure. The deformed length of the ith element in- cluding the spring system, is given by _ 2 2' _ hi _.\/(xi+l — xi) + (yi+l - yi) .....(3 l) where xi and yi are the x and y coordinates of the ith mass. The compression in the ith spring system is dhi=h ‘h- 0.000(3_2) where hO was defined previously as the initial panel length. The angle that the ith panel makes with the 16 l7 x—axis is not explicitly needed in the computations. However the sine and cosine of this angle are. They are given by Y- ‘ Y- . _ 1+1 1 _ Slnei — T on... (3 3) and cosei = ——_-h____ .....(3—4) Denoting the angles of the adjacent panels by ei_l and Si, the relative rotation of the spring system is given by ¢. = 91—1 — 9. .....(3—5) Taking the sine of both sides of Eq. (3-5) and using the appropriate trigonometric identity and finally taking the arcsine of both sides, the required relative rota— tion is found to be ¢- = arcsine(cose.sine. 1 1 1—1 — Slneicosei_l) .....(3—6) Since the relative rotation of a spring system is assumed to be small, the multi—valuedness of Eq. (3—6) is inconr sequential. Note that the limitation of smallness of the 18 relative rotation of the spring systems does not pre— clude large rotations of the panels. 3.3 Equations of Motion In Fig. 3.2 are shown a free—body diagram of a mass point and of an adjacent massless panel. The forces acting on this typical mass and Ennel are shown in their positive sense. For simplicity the external loads are assumed to act at the mass points. The equations of motion for this mass point are obtained by applying Newton's second law of motion. miui = Ti-lcosei—l + Vi—lSlnei—l - Ticosei - Vis1n6i - pi .....(3—7) mivi = Ti—lSlnei-l - Vi-lcosei—l — T.sine. -+ V.cose. + q. .....(3-8) 1 1 1 1 1 where mi is the ith lumped mass, fii and vi are the accel- erations in the x and y directions respectively, Vi is the shear in the ith panel, and qi and pi are the external loads. Since the shears cannot be determined directly from the deformations which are being considered, it is necessary to determine them in some other man— ner. By writing an equation of moment equilibrium 19 about one end of a massless panel it is possible to find the shear resultant in terms of the end moments of the panel. This relationship is given by i+l—Mi) one-0(3_9) It should be pointed out that the element length hi in the above relation is not constant; it depends on the linear deformations of the spring system. 3.4 Spppprt Conditions For a continuum member, three boundary conditions may be imposed at each end of the member by prescribing either the displacement or the corresponding reaction for each of the coordinates: u, V, and ¢- The standard support conditions of "free," "roller," "hinged," and "fixed" are special cases of the preceding. In general the interpretation of the boundary conditions for the model is obvious and need not be dis- cussed except in the case where the rotation of the sup- port is prescribed in the continuum problem. If there is no spring system at the support point (as at the right end of the formal model), the rotation of the adjacent rigid bar should be taken to be the same as the pre- scribed support rotation since the bar is fixed to it. If there is a spring system at the support (as at the 20 left end of the formal model and both ends of the modified model) the appropriate face of the spring system is fixed to the support as shown in Fig. 3.3; and the adjacent rigid bar is allowed to rotate. The relative rotation of the spring system is then determined by finding the rotation of the rigid bar relative to the support. In the use of the modified model for prismatic members, the flexibilities of the spring systems lumped at the supports are half that of the interior systems. It is convenient from a computational point of view to consider all spring systems to possess the same flexi— bility and double the relative rotations of the spring systems at the supports. This has the same effect of halving the flexibility at the support as the model is formulated. 3.5 Initial Residual Stresses In certain cases of rolled steel sections there may be self-equilibrating residual stresses distributed over the cross—section due to the forming and subsequent cooling process. These stresses would influence the re- sponse of the member in the inelastic range. In this study it is assumed that the distribution of residual stresses remains constant along the length of the mem- ber (14). 21 The cross—section of the member is divided in a manner that would allow a reasonable approximation to the residual stress pattern. In dividing the flange areas, it may be necessary to lump at least two areas in order to consider the variation of the residual stresses there. For example, in a wide flanged member both tension and compression stresses exist in each flange (14). In this case it is necessary to consider at least two lumped areas to represent each flange, one area for tension and one for compression as shown in Fig. 3.4. The lumped area corresponding to the com- pression residual stresses in the flange represents the two exterior portions of the flange away from the web while the lumped area corresponding to the tension re— sidual stresses represents the remaining interior area of the flange. The problem is initialized by pre—setting the force intercepts of the force deformation diagrams ac- cordingly. They are pre—set in such a way that the force due to zero spring deformation is given by = f c dA ' .....(3-lO) where or is the residual stress acting over the subarea A... 13 22 3.6 Arches and Initial Crookedness Since the procedure is developed within the frame- work of§large displacement theory, it is capable of treat- ing structural members with initial shapes other than the perfectly straight. These would include arches which may be considered to have "alignments" which are far removed from perfectly straight or members with slight imperfec- tions or crookedness as occasionally imparted to rolled steel sections during manufacture. A method of handling these initial "misalignments" consists of describing the initial coordinates of the mass points such that they approximate the shape under consid— eration. The sum of the lengths of the massless panels which make up the model may be taken to be the length of the member. Each panel is assumed to be straight. The manner of computing the initial coordinates may consist of prescribing the value of the y—coordinate for each mass of the model, and then computing the x—coordinate in such a manner that the distance between two adjacent masses is equal to the length of the massless panel, ho’ i.e., _ 2_ _ 2' - xoi _W/ho (yoi yoi_l) + XOi—l .....(3 11) where xoi and yoi are respectively the initial x and y coordinates of the ith mass. 23 It should be noted that for an initially non- straight member there exists at each spring system some initial rotation which must be accounted for in the com— putation of the deformation of the spring systems. This may be done by applying to Eq. (3—5) a correction factor ooi, taken equal to the negative of the initial relative joint rotation. In this manner any bending rotations computed are relative to the initial configuration of the model. Thus Eq. (3—5) takes the form: ¢- = arcs1ne(coseis1n9i_l J. — s1neicosei_l) + ooi .....(3-12) 3.7 Initial Static Loads In order to make the treatment of one dimensional structural members more complete, the effects of the ini— tial static loadings should be considered. This can be accounted for by setting the initial values of the dis— placements and forces to those called for by static equi- librium requirements. CHAPTER IV NUMERICAL PROCEDURE AND COMPUTER PROGRAM 4.1 Introduction In the preceding chapters a nonlinear model of a line structural member and the equations governing its be- 3 havior have been developed. In this chapter the numerie A cal procedure used to calculate the model's response is presented. This numerical procedure includes the adapta— tion of a numerical integration technique to a step—by— step solution of the governing equations of motion. A computer program based on the method developed was written. This program is briefly discussed along with its capabili— ties and limitations. 4.2 Numerical Integration Procedure The particular numerical technique adopted for this study is from the class of self—starting single step marching integration methods studied by Tung and Newmark (49) and Newmark (33). In particular the method used in this work is the so—called B = 0 method. This method has the advantage of being non—iterative if the accelera- tions are not explicitly dependent on the velocities of 24 25 the mass points. This is the case in the problem at hand since damping is not considered. Let the acceleration, velocity, and displacement of a mass, say, the ith mass, in the x—direction be given by Hi, ii, and ui respectively. Then the B = 0 method pre- scribes the velocity and displacement at time tj + At in terms of the displacement, velocity, and acceleration of the mass point by the following relations: ui(tj+At) = ui(tj) + Atui(tj) l 2" 3 + E‘At ui(tj) + 0(At) 0.000(4—1) . _ . 1 " ui(tj+At) — ui(tj) + §At[ui(tj) + fii(tj+Ati] + 0(At2) .....(4-2) A similar set of equations is obtained for the y—direction by substituting v for u in the above relations. The man— ner in which these equations are employed in the solution process is explained in the following section. 4.3 Ste -B -Ste Numer1cal Solution The general procedure involved in extending the solution from time tj to tj + At is briefly explained be- low. A generalized flow diagram of a single step is pre— sented in Fig. 4.1, in which the boxes represent 26 intermediate stages in the completion of the step and the arrows represent advancements. It is necessary at the start of the problem to establish the initial values of the quantities relevant to the solution. These, for example, include displace— ments, velocities, and accelerations of the mass points. In addition, the force deformation relation for each spring must be pre—set to its initial state. Once the relevant quantities have been initialized the fundamental problem is to determine these quantities one increment or step into the time domain. Once this fun— damental problem has been solved it is only necessary to reapply the solution of the single step repeatedly to ad— vance the solution of the given problem the desired dis- tance into the time domain. At the beginning of each step, for example, the step starting at time tj, all the relevant quantities are known. The first advancement in the step is the determi- nation of the displacements ui(tj+At) and vi(tj+At). These are either prescribed or computed by numerical in— tegration of the equations of motion as the case may be. In general, the interior masses are free to move and are governed by the equations of motion while the end masses may be either case, depending upon the boundary conditions. Once the displacements of the mass points are known their coordinates are given by: 27 xi(tj+At) = xoi + ui(tj+At) .....(4-3) yi(tj+At) = yoi + vi(tj+At) .....(4—4) From these coordinates the spring deformations are computed from the geometric relations derived in Chapters II and III. The second advancement in the step is the deter— mination of the spring forces from the spring deformations. From these forces the internal force resultants are obtained from Eqs.(2—4) and (2-5). These forces along with any ex- ternally applied forces and considerations of the geometry changes of the structure obtained in the first advancement are then used to compute the accelerations of the mass points in the third advancement.' Once the accelerations are deter— mined, the velocities of each mass point can be found by ap— plying Eq. (4—2). At this intermediate stage in the step, all the relevant quantities have been established for the next time, tj+At. The fourth and final advancement consists of in- crementing the time such that tj+At becomes tj+l and the situation is similar to that at the beginning of the pre- vious step. In this manner the equations of motion may be integrated along the time axis as far as desired. 4.4 Stabilit of the Numer1cal Solut1on Tung and Newmark (49) have shown that stability of the numerical integration method, 8 = 0, requires that the 28 maximum time increment be less than % times the smallest period of elastic vibration of the system under considera— tion. In general it is not convenient nor necessary to exactly determine this period in order to choose a time increment. It is only necessary to estimate the value of the smallest period of vibration and then take a fraction of it to insure that the time increment is less than the critical value. For the model under consideration there are two basic types of motion, transverse or bending Vibrations and.longitudinal.or axial Vibrations. The time increment is selected by first finding the nth (where n is the num— ber of rigid massless panels in the model) elastic bend- ing period of vibration of a simply supported span which possesses the same length L, modulus of elasticity E, moment of inertia I, area A, and distributed mass m as the continuum structural member under consideration. The nth elastic axial period of vibration of a column with both ends fixed is similarly computed. The time incre— ment is then taken as l/lO of the smaller of these two values as shown below. 2 _ . 1 2L - m . 1 2L m _ At_m1n'[l—0' E2 ETITUTVE—A] -----‘4 5) This approach of choosing a time increment was used by Wen and Toridis (51) with considerable success, although 29 in their case they only needed to consider bending vibra— tions. In order to test the stability of the numerical solution based upon the above method of time increment selection, the response of a particular member was calcu— lated by using several values for the increments. The time increments used in the solution were multiples of the "standard increment" as given by Eq. (4—7). The par— y;”‘ 4 ' ticular system studied is the fixed—fixed uniformly loaded elastic beam (with n=10) presented in section 5.2. The following multipliers of the standard time increment were considered: l/2, l, 3, 4, and 10. The re- sponse of the beam for multipliers l/2, l, and 3 varied by less than 1%. However, the response of the beam for the multiplier 4 was different from the above three by 150%. The response for multiplier 10 was erratic and obviously unstable. The stability limit for the problem under con— sideration thus seems to lie between 3 and 4 times the standard time increment, as expected. 4.5 Computer Program The numerical method presented in the foregoing was programmed for solution on the Michigan State CDC 3600 computer system using 3600 Fortran. The program can con— sider a model with up to 40 panels and 10 discrete cross— sectional areas. In order to simplify the program, however, 30 it has been.limited to prismatic members with bilinear stress-strain relations. Two versions of the program have been developed; one considers circular arches, the other straight members with sinusoidal imperfections of the form yoi+l = Amp°sin %; ; i = o, l, ..., n .....(4—6) Both versions of the program solve the problem in exactly the same manner. Only the-schemes in which the geometric properties and the loadings are input and initialized are different. The program for the circular arch is presented in the Appendix. The input for the program consists of the geometric and material properties of the member, the support condi- tions, the external loadings, prescribed motions, if any, of the supports, the strains due to existing static load- ings, the strain intercepts for residual stresses, if any, the number of panels used, interval to be covered by the solutions, and the number of print requests per unit time. At periodic intervals of time, as specified in the input, the computer prints out a graphic display of the displaced member, the coordinates of the mass points, the Spring de- formations and the corresponding spring forces, the joint rotations and the element compressions, the moments and thrusts, and the energy check parameters (to be discussed in section 4-6)° At the end of each problem, the computer 31 also presents a graphic display of the response history of the center mass point. In problems in which support motion is a factor, the following restriction exists. Support motion may not be prescribed for either rotational degree~ of freedom.- In addition, of the four remaining translational degrees of freedom only one may have a prescribed motion. This restriction could be removed, if necessary, without a great deal of reprogramming. The compiling time on the CDC 3600 computer for the program is about 1 min. and 15 sec. The run time will de- pend mainly upon the number of panels and the length of time for which the solutionis sought. For a typical problem with 10 panels, 4 discrete areas in the cross-section, and a total problem time of 3 fundamental elastic bending periods of vibration, a run time of 4 minutes is about average. 4.6 Energy and Impulse- Momentum Check 4.6.1 General.--In the analysis and the computer program presented there are at least four sources of error. The first source arises from the discrete approximation of a continuum member (the effectiveness of this approximation is studied in Chapter V by comparing results with known con— tinuum solutions). The second source arises from the trun- cation of the numericalintegration formulas, Eqs. (4-1) and (4—2). The third source, round off error, occurs 32 in the numerical computations associated with the problem. The fourth source of error is the mistakes which may occur in programming the method of solution on the computer. The energy and impulse—momentum checks incorporated in the program are intended to indicate programming errors] and unacceptable round off errors. Since these checks must necessarily be based upon the numerical solution to the equa- tions governing the behavior of the model, they cannot sug— gest the error of approximating the continuum member by a discrete model nor measure the truncation error of the numer— ical integration formulas. These checks may be easily re- moved from the program to decrease run time. 4.6.2. Energy Check.-—For a system initially at rest the input energy (the energy due to external loads and support motions) is balanced by the strain energy ab- sorbed by the springs and the kinetic energy. The concept of input and absorbed energies is that of force moving through a distance, while kinetic energy is the typical mv2/2. Since the velocity for each mass point is known at the end of every step, it is easy to compute the kinetic energy at each instant of time. On the other hand, the in— put and absorbed energies must be accumulated throughout the duration of the problem. Since the force varies dur— ing the time interval, the average value is used in the computations of either the input or absorbed energies. 33 Conceptually the input and absorbed energies can be com- puted as follows: t _ n as Eg — of F(T)fidT l n = f i:l{[F(ti) + F(ti_l)] '[S(ti) - s(ti_l)]} .....(4—7) where Eg represents energy, F is force, which may repre— sent load, reaction, or spring force, and S is displace- ment, which may represent deflections, support motion, or spring deformations, and t denotes time. By observing inconsistencies in the energy check several programming errors were detected. These errors would have been overlooked without this check as their effects on the numerical results were quite small. For example, an error existed in the initialization of the numerical integration procedure which, for a step loading, induced a small initial velocity to the mass points. Re— sults obtained with the erroneous programs were however, not all discarded since the difference in the maximum re— sponSe was only 1/2% (less than that obtained by varying the time increment) from the corrected solutions. Once the errors were corrected, the difference between the input energy and the absorbed and kinetic energies was consistently less than 0.001% of the total input energy. 34 4.6.3 Impulse-Momentum Check.--This is in reality two independent checks, one in the x—direction, and the other in the y—direction. The vector components in the x and y directions of the external loads and the reactions are numerically integrated in the time domain, in a similar fashion to that for energy calculations, to form the im— pulses applied to the mass system. These impulses are com— pared to their respective momentums, which like the kinetic energy, are computed at the end of every time interval. Any differences which may exist are noted. In general,the differences obtained from the check were of the order of 0.000001% depending on the problem and the length of time the solution had progressed into the time domain. CHAPTER V COMPARISON STUDIES 5.1 Introduction In the preceding chapters the model representa- tion of the continuous member and the analysis of the model have been presented. In order to obtain numerical results, this analysis has been programmed on the Michigan State University CDC 3600 computer. The purpose of this chapter is to compare the results obtained from this anal— ysis with known solutions in order to judge the accuracy and dependability of the present numerical procedure (as well as the correctness of the computer program prepared). Four problems for which the solutions are known are examined. The scope of these extends from an elastic small displacement problem to an inelastic large displace— ment one. Taken together these problems contain: four types of support conditions including support motion, in— itial crookedness, elasticity and inelasticity, and small and large displacements. The comparisons made between known solutions and the numerical results obtained show that the procedure presented here does in facu,produce reliable results. 35 36 5.2 Fixed-Fixed Elastic Beam The results of an elastic continuum fixed-fixed beam subjected to a uniform step loading (i.e., suddenly applied constant load) are compared with results using the model under consideration. The parameters were se— lected to insure small deflections in order that the two solutions could be compared. Solutions were obtained for both the formal and modified model (see section 2.3). In Fig. 5.1 the response of the center of the beam is shown for all three solutions corresponding to the formal and modified models with 20 panels and the continuum with 20 terms taken in the series solution. With respect to the first peak response of the continuum solution, the response of the modified model was in error by 2% while that of the formal model was in error by 1%. With respect to the time corresponding to that response the modified model was in error by 1.1% while the formal model by .25%. In Fig. 5.2 the response of the mass point at the quarter point of the beam is shown for all three cases. For the formal model the responses of both the right and left sides of the beam are shown since the model is asymmetric. It should be noted that in the elastic case under symmetric conditions the modified model yields symmetric response. Therefore, as in the continuum case, the response of only one quarter point 37 is needed. It can be seen that in this case the modi— fied model gives the better results. The formal model was in error by 12% and by 15% for the left and right sides, respectively, while the modified model was in error by only 3%. It is of interest to note that convergence of the modified model to the continuum solution's response is slower for the fixed—fixed support conditions than for the simply supported conditions. For 10 panels, the response of the model differs from the continuum solution in the simply supported case by 0.7% while in the fixed— fixed case the difference is 7%. A probable reason for the difference in the two cases is that, for a simply sup— ported beam, all boundary conditions are exactly satisfied, while for a fixed—fixed beam the rotational displacement boundary conditions are only approximately satisfied. 5.3 Elastic Circular Arch To compare the responses of a structure with a large "initial crookedness," an elastic arch problem studied by Eppink and Veletsos (10) is solved. The prob- lem consists of a two hinged circular arch of dimensions and properties as shown in Fig. 5.3. A uniform step loading normal to the structure is applied. The magni— tude of the loading is such that the displacements would be small. Eppink and Veletsos used a discrete model and 38 obtained results using both a direct numerical integra— tion of the equations of motion and a modal analysis ap— plied to their model. The comparison of the numerical results is shown in Table 1. All the results are based upon a discrete model with 12 panels. It is seen that the results of the present method agree closely with those obtained by Eppink and Veletsos. The maximum dif— ference with respect to the results of the modal analysis is 3%, while the average difference is 1.2%. 5.4 Imperfect Elastic Column Bailey (2) has studied the large elastic displace— ment behavior of a slender pin-ended column with a sinusoi— dal initial crookedness as shown in Fig. 5.4. The right end of the column is forced toward the left end at a con- stant velocity of 0.001 times the velocity of sound in the material. Bailey's solution is a finite difference solution of the continuum equations with a mesh size of L/16. In the corresponding solution obtained with the present model 16 panels were also used. The results of both solutions are shown in Fig. 5.4. Both solutions predict a bimodal (one complete wave) re- sponse of the column, although quantitatively there is a slight discrepancy. One point of difference is that Bailey's results did not indicate any horizontal displacement of the right support when in reality there should be a displacement 39 to the left. There is, otherwise, good general agree— ment of the results. 5.5 Sim 1 Su orted Inelastlc Beam Wen and Toridis (51) investigated the dynamic re- sponse of a simply supported beam using a small displace- ment, lumped mass, lumped flexibility model. Their beam was subjected to a uniformly distributed blast loading which decreased exponentially with time. In the small displacement range, the model considered in the present work should yield similar results. In this comparison study, the present analysis con- sidered so—called "ideal" or "sandwich" beams in order that the bilinear stress strain relation used in the present work will produce a bilinear moment rotation relation that was used in (51). Two sets of data are presented in Fig. 5.5 and Fig. 5.6 corresponding to, respectively, a small displacement problem and a large displacement one. In each figure the solution obtained from Ref. (51) is com— pared with that obtained by the present model. It is seen that for the small displacement problem in Fig. 5.5 both solutions agree well. However, for the large displacement problem in Fig. 5.6 the two solutions are appreciably different; the maximum mid-span deflection yielded by the small displacement analysis of Ref. (51) is 13% larger than that by the present large displacement 40 analysis. This is thought to be due to the "cable action" in the beam in the large displacement range. This mecha- nism is included in the large displacement analysis but not in the small displacement one. CHAPTER VI NUMERICAL EXAMPLES 6.1 Introduction In order to demonstrate the usefulness and versa— tility of the method of analysis presented in this thesis, g two categories of problems are considered, namely arch and column problems. Each individual problem is concerned with demonstrating how the procedure developed could be applied to study a particular type of problem that heretofore had not been analyzed. The arch problem indicates how the method could be applied to problems which consider strain hardening and residual stresses. The second category, column problems, complements the first by considering ground motion and ini— tial static axial loadings. Both categories,of course,in- volve material as well as geometric nonlinearities. 6.2 Dynamic Snap Through of Arches 6.2.1 Inelastic Arches.——The problem consists of a circular pin—ended arch that subtends an arc of 90 degrees with a radius of 136 inches. The material has a modulus of elasticity of 5,000,000 psi, a yield stress of 6000 psi, 41 42 and a strain hardening parameter, of 0.02. The cross— Rsh’ section has the shape of a wide flange 8WF31 section with the web in the plane of the deformation. The cross-section is divided into four areas, symmetrically located about the centroidal axis, one area representing each flange and two areas representing the web. A uniform step loading normal to the centroidal axis is applied to the member. A length discretization of ten panels is used. By trying various magnitudes of the loading, the VI critical (lowest) dynamic snapping load was found to lie be- tween 300 and 400 ppi. For the 400 ppi. loading the arch readily snapped and this "snap" is shown in stop action in Fig. 6.1. The time at which the arch configuration was captured is shown in the upper left of each picture. For this particular arch and loading, the failure shape corresponds to one and a half waves with two interior nodes, or briefly "tri-modal." Also in this particular case the arch snapped in a symmetric fashion about the center line of the arch. It should be mentioned here that on account of the asymmetry of the model, the response of the arch was not perfectly symmetric, and the vertical deflections of symmetrically located masses were not exactly the same. However, the difference is small--of the order of 3% with respect to the rise of the arch. 6.2.2 Effect of Strain Hardening on the Critical Dynamic Snapping Load.——Problems concerning initial 43 crookedness need not be limited to circular arches as above, but may consider other shapes as well. Consider for example the pin-ended arch subjected to a uniform step vertical load as shown in Fig. 6.2. The shape of the structure approximates a half sine wave as computed by Eq. (4—6) and its cross-sectional shape is the same as the previous arch problem. The material properties are: E = 30,000,000 psi, Oy = 30,000 psi, and Rsh = 0.0. The magnitude of the load is increased with each trial until the arch snaps. In this way the upper and lower bounds of the critical dynamid snapping load may be deter— mined. Also in Fig. 6.2 the maximum vertical crown dis— placement (within the range of time covered by the solu— tion, i.e., 1.5 Tf) is plotted against the load magnitude for each trial. The upper bound on the snapping load for the RS = 0.0 case is 962 ppi and the lower bound is 935 h ppi. Working the same problem for a strain hardening material with RS = 0.03, the critical snapping load in- h creased to an upper bound of 1229 ppi, and a lower bound of 1202 ppi. This is an increase of about 25% over that obtained from the no strain hardening case. It is of interest to note that at dynamic loads less than 800 ppi. there is no appreciable difference in the response of the two cases. 44 6.2.3 Effect of Residual Stresses on the Critical Dynamic SnappingpLoad.——The above two cases of section 6.2.2 (with RSh = 0.0 and 0.03) were reworked considering the existence of residual stresses. The residual stress pattern used here was shown in Fig. 3.4. The magnitudes of the residual stresses varied from 0.30Y (compressive) in the flanges to 0.3120y (tensile) in the central portion of the flanges and the web. This residual stress pattern was assumed to be constant along the length of the member. The results of the analysis are shown in Table 2. It can be seen that for the problems considered,the effect of residual stresses on the critical dynamic snapping load is much less than that of strain hardening. This is of course due to the fact that residual stresses affect only initial yielding. Once that has taken place the material behaves just as though there were no residual stresses. Hence residual stresses would have a significant influence on the response if the inelastic strains are small. On the other hand, the effect of strain hardening persists even for large strains. In the problems considered the strains are large. Hence, the previously noted observa- tions obtain. 6.2.4 Effect of Model Bias.-—As pointed out in Chapter II, the model used in this analysis is asymmetric. Therefore, it would produce "biased" results when used to 45 approximate a continuum member. In order to investigate the influence of the bias, the following method was em- ployed. The problem considered in section 6.2.2 with Rsh = 0.0 was modified in such a way that superimposed on the uniform vertical load, pb, was a full sine wave distributed load the magnitude of which pa could be varied. This additional loading was increased until the asymmetry of the response had been reflected about the centerline of the arch. This is shown in Fig. 6.3. A measure of the bias can be obtained by comparing the amount of asymmetric loading needed to cause the re— sponse to reflect about the centerline. In Fig. 6.3d is shown the deformed arch with pb = 990 ppi and pa = 0 at t = 1.5T . In Fig. 6.3e is shown the deformed arch with f pa = 27 ppi at the same values of pb and t as in Fig. 6.3d. It is seen that with this addition of pa the deformation pattern has been reversed. In this particular case the ratio of pa to pb is 0.03 and this may be taken as a meas— ure of the "bias." These results are for a length discretization of ten panels. It seems clear that as the discretization of the length becomes finer the effect of this bias would de— crease. The preceding discussions and the fact that the bias did not prevent the model from predicting an overall symmetric response as can be seen in Fig. 6.1 would indi- cate that the effect of the bias is small. 46 It may be noted in passing that the shape of the load distribution has an effect upon the snapping shape of the arch. In the case of a half sine wave vertical step loading, the arch failed symmetrically while, under a uniform step loading, its failure shape is asymmetric. 6.2.5 Comparison with Static Arch Buckling.—— Recently Lee and Murphy (26) have published some data on the static buckling load of shallow arches. It was thought that it would be of some interest to compare their data to the dynamic solution. The particular problem considered corresponds to their Arch No. 3 which has a radius of 50 inches and sub- tends an arc of 40 degrees. The depth of its rectangular cross-section is 1/2 inch and the width is 1 inch. The arch material is aluminum alloy 3003-0 which possesses a curvi— linear stress strain relation. To approximate this with a bilinear relation the following material properties were used: E = 5,000,000 psi, OY = 6000 psi. and a strain hard- ening factor Rsh = 0.075. For the present analysis the model was divided into 10 panels and the cross—section into 4 equal discrete areas. The critical dynamic snap—through loading was found to be between 36.25 psi. and 37.5 psi. Lee and Murphy found the static critical load to be slightly above 41.2 ppi. It seems reasonable to expect that the dynamic critical load be smaller than the static critical load. In addition, 47 they observed that an asymmetric buckling mode developed from a symmetric displacement pattern as the loading on the arch increased. From the dynamic solutions obtained for the present study, an asymmetric snapping mode shape \was also seen to develop from a symmetric displacement pat— tern as the magnitude of the loading is increased. 6.3 Effects of Sppport Motion on a Column Consider a vertical column fixed at each end under a constant vertical load initially acting through the cen— troidal axis as shown in Fig. 6.4. For the particular column considered, the axial yield load of the column is far less than the Euler load (P Euler = 3.65Py). Th1s col- umn is then subjected to a horizontal support motion at the base which may idealize an earthquake. Two cases are considered. For case I, the vertical load is held constant at one half the yield load (based on the cross-sectional area). The problem is solved for support motions with various amplitudes but a fixed duration. The maximum longitudinal end displacement of the column is plotted against the amplitude of the support motion. It is seen that the displacement increases rap- idly with the amplitude of motion. At an amplitude equal to 0.075L the column failed. For case II, consider the support motion to have a constant magnitude of 0.015L. The vertical load is then 48 increased. In Fig. 6.5 is shown a plot of the permanent transverse displacement at the column mid—height versus the axial load on the structure. It is seen that the per— manent displacement increases rapidly with an increase in the axial load. Failure occurs at P = 0'9Py' An interesting feature of the response is the re- versal of the direction of the permanent set as the collapse load is approached. As the column fails the mid-point of the column actually bows to the left whereas the support motion was to the right. CHAPTER VII CONCLUSIONS In this thesis a method of analyzing the dynamic response of line structural members has been presented. A discrete model was used for the analysis in which both the length and the cross—section were discretized. In constructing the model, an asymmetric element was chosen, as the symmetric ones considered would lead to complica- tions not easily resolved. In the formulation of the analysis, both material and geometric nonlinearities have been included. Thus the method can take into account the effects of axial— force bending-moment interactions, large displacements, inelasticity, as well as initial crookedness, residual stresses, initial static loadings, and movement of the supports, if any. A computer program has been prepared incorporating most of the capabilities mentioned. Impulse— momentum and energy checks were found to be useful in de- tecting programming errors. They should also indicate unacceptable round—off errors. Using the program developed, a number of problems have been solved. These problems fall into two categories: problems which have been solved by other researchers and 49 50 problems which have not. The first category is used to check the reliability of the method and the correctness of the computer program. The second is used to exhibit the usefulness and versatility of the method. The first category problems included a fixed—fixed elastic beam, an elastic circular arch, an imperfect elastic column and a simply supported inelastic beam. By comparing the model's response with known solutions of these problems, it was demonstrated that the method presented here is sufficiently accurate for most engineering purposes. In the second category, two types of problems were considered: the snap-through behavior of elasto-inelastic arches, and the failure of an axially loaded column sub— jected to transverse support motions. In studying the snapping phenomenon of a shallow arch, it was found that strain hardening significantly increased the critical dy- namic snapping load while residual stresses did not affect it appreciably. Because of the asymmetric nature of the model, the numerical solutions indicated a certain amount of "bias." However, it is shown that the corresponding error is rela- tively small (of the order of 3% for 10 panels). In conclusion it may be said that a discrete model approach is a feasible and powerful method for studying the dynamic response of nonlinear (both geometric and material) line structural members. In the present study this model 51 is limited to motion in one plane. As a possible exten— sion of this study, the general case of motion in space may be considered. In this case, however, torsional deformation appears to be a significant factor. Conse— quently, the model would have to have a mechanism to in— corporate shear deformations. 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"Guide to Development and Use of Elec- tronic Computer Programs: Introduction," Journal of the Structural Division, ASCE, Vol. 90, No. ST6, Proc. Paper 4151, December 1964, pp. 1—5. . "Guide to Development and Use of Electronic Computer Programs: Program Planning," Journal of the Structural Division, ASCE, Vol. 90, No. ST6, Proc. Paper 4153, December 1964, pp. 19-24. Galambos, T. V., and Ketter, R. L. "Columns Under Com— bined Bending and Thrust," Transactions, ASCE, Vol. 126, Part I, 1961, p. 1. Galambos, T. V. "Strength of Round Steel Columns," Journal of the Structural Division, ASCE, Vol. 91, No. STl, Proc. Paper 4219, February 1965, pp. 121- 140. Goldberg, J. E., Bogdanoff, J. L., and Glauz, W. D. "General Computer Analysis of Beams," Journal of the Engineering Mechanics Division, ASCE, Vol. 90, No. EM3, Proc. Paper 3943, June 1964, pp. 135—146. Hauck, G. F., and Lee, S. L. "Stability of Elasto— Plastic Wide Flanged Columns," Journal of the Structural Division, ASCE, Vol. 89, No. ST6, Proc. Paper 3738, December 1963, pp. 297—324. Hoff, N. J. "The Dynamics of the Buckling of Elastic Columns," Journal of Applied Mechanics, Trans. 18, l (1951), PP. 68—74. Horne, M. R. "The Stability of Elastic—Plastic Struc— tures," Progress in Solid Mechanics, Vol. II, Edited by I. N. Sneddon and R. Hill. Amsterdam: North Holland Publishing Co., 1961. Huddleston, J. V. "Analysis of an Inelastic Column," Journal of the Engineering Mechanics Division, ASCE, Vol. 90, No. EMD4, Proc. Paper 3992, August 1964, pp. 1—21. Jennings, P. C. "Earthquake Response of a Yielding Structure," Journal of the Engineering Mechanics Division, ASCE, Vol. 91, No. EM4, Proc. Paper 4435, August 1965, pp. 41—68. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 54 Ketter, R. L., Kaminsky, E. L., and Beedle, L. S. "Plastic Deformation of Wide-Flange Beam Colums," Transactions, ASCE, Vol. 120, 1955, p. 1028. Lay,M.(L "The Yielding of Uniformly Loaded Steel Mem— bers," Journal of the Structural Division, ASCE, Vol. 91, No. ST2, Proc. Paper 4580, December 1965, pp. 49—66. Lay,M.(L,and Galambos, T. V. "Inelastic Steel Beams Under Uniform Moment," Journal of the Structural Division, ASCE, Vol. 91, No. ST6, Proc. Paper 4566, December 1965, pp. 67—93. Lay,lm G."Flange Local Buckling in Wide Flange Shapes," Journal of the Structural Division, ASCE, Vol. 91, No. ST6, Proc. Paper 4554, December 1965, pp. 95— 116. Lee,L.H.N” and Murphy, L. M. "Inelastic Buckling of Shallow Arches," Journal of the Engineering Mechanics Division, ASCE, Vol. 94, No. ET/Il, Proc. Paper 5804, February 1968, pp. 225-239. Lu, L. W. "Inelastic Buckling of Steel Frames," Journal of the Structural Division, ASCE, Vol. 91, No. ST6, Proc. Paper 4577, December 1965, pp. 185—214. Malvick, A. J., and Lee, L. H. N. "Buckling Behavior of an Inelastic Column," Journal of the Engineering Mechanics Division, ASCE, Vol. 91, No. EM3, Proc. Paper 4372, June 1965, pp. 113—127. Massey, C., and Pitman, F. S. "Inelastic Lateral Stability Under a Moment Gradient," Journal of the Engineering Mechanics Division, ASCE, Vol. 92, No. EM2, Proc. Paper 4779, April 1966, pp. 101—111. Miranda, C., and Ojalvo, M. "Inelastic Lateral— Torsional Buckling of Beam Columns," Journal of the Engineering Mechanics Division, ASCE, Vol. 91, No. EM6, Proc. Paper 4563, December 1965, pp. 21- 37. Neal, B. G. The Plastic Methods of Structural Analysis. London: Chapman & HalI, Ltd., 1956. Newmark, N. M. "Numerical Procedures for Computing Deflections, Moments, and Buckling Loads," Trans- actions, ASCE, Vol. 108, 1943, p. 1161. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 55 Newmark, N. M. "A Method of Computation for Structural Dynamics," Transactions, ASCE, Vol. 127, 1962, p. 1406. Onat, E. T., and Prager, W. "The Influence of Axial Forces on the Collapse Load of Frames," Proceed- ings of the First Midwest Conference on SoI1d Mechanics, University of Illinois, Urbana, Illi- n01s, 1953, pp. 40-42. Osgood,VL R. "The Double Modulus Theory of Column Action," Civil Engineering, Vol. 5, No. 3, March 1935, pp. 173—175. Pfrang, E. O., and Siess, C. P. "Predicting Struc- tural Behavior Analytically," Journal of the Structural Division, ASCE, Vol. 90, No. STS, Proc. Paper 4110, October 1964, pp. 99-111. Rosenbluth, E., and Herrera, I. "On a Kind of Hyster— ical Damping," Journal of the Engineering Mechanics Division, ASCE, Vol. 90, No. EMD4, Proc. Paper 3999, August 1964, p. 37. Salvadori, M. G. "Numerical Computation of Buckling Loads by Finite Difference," Proceedings, ASCE, Vol. 75, December 1949. Schreyer, H. L., and Masur, E. F. "Buckling of Shallow Arches," Journal of the Engineering Mechanics Division, ASCE, Vol. 92, No. 4, Proc. Paper 4875, August 1966, pp. 1—19. Sevin, E. "Digital Computer Solutions of the Dynamic Column Buckling Equations" Proceeding of the First Conference on Electronic Computation, ASCE, 1958, pp. 237-256. . "On the Elastic Bending of Columns Due to Ax1al Inertia," Journal of Applied Mechanics, ASME, Vol. 27, No. 1, 1960, pp. 125—131. Shanley, F. R. "The Column Paradox," Journal of the Aeronautical Sciences, Vol. 13, No. 12, December 1946, p. 678. . "Inelastic Column Theory," Journal of the Aeronautical Sciences, Vol. 14, No. 5, May 1947, pp. 261-267. 44. 45. 46. 47. 48. 49. 50. 51. 52. 56 Shinozuka, M., and Henry, L. "Random Vibration of a Beam Column," Journal of the Engineering Mechanics Division, ASCE, Vol. 91, No. EM5, Proc. Paper 4510, October 1965, pp. 123—143. Thurlimann, B. "New Aspects Concerning the Inelastic Instability of Steel Structures," Journal of the Structural Division, ASCE, Vol. 86, No. STI, Pro . Paper 2485, January 1960, p. 99. Timoshenko, S. P. History of Strength of Materials. New York: McGraw—Hill, 1953. Timoshenko, S. P., and Gere, J. M. Theory of Elastic Stability, Engineering Societies Monographs, McGraw— Hill, New York, 1961, 2nd ed. Tong, A. L. "Elasto-Plastic Analysis by Numerical Procedures," Journal of the Engineering Mechanigs Division, ASCE, Vol. 86, No. EM6, Proc. Paper 2690, December 1960, pp. 73—88. Tung, T. P., and Newmark, N. M. "A Review of Numer— ical Integration Methods for Dynamic Response of Structures," Technical Report to Office of Naval Research, Contract N6 ori-071 (06), Task Order VI Project NR—064-183, Department of Civil Engineering, University of Illinois, Urbana, Illinois March 1954. Van Kuren, R. C., and Galambos, T. V. "Beam Column Experiments," Journal of the Structural Division, ASCE, Vol. 90, No. ST2, Proc. Paper 3876, April 1964, pp. 223—256. Wen, R., and Toridis, T. "Discrete Dynamic Models for Elasto-Inelastic Beams," Journal of the Engineer— ing Mechanics Division, ASCE, Vol. 90, No. EM5, Proc. Paper 4081, October 1964, pp. 71-102. Wood, R. H. "The Stability of Tall Buildings," Pro— ceedin s, Institution of Civil Engineers, Vol. 11, 1958 (Sept.—Dec.), p. 69. 57 Table l.—-Comparison Study of Elastic Circular Arch Time Vertical Crown Displacement/(pRz/AE) Eppink and Veletsos t/ZWRw/g Model Response _ Numer1ca1 Modal 0.5 2.130 2.128 2.128 1.0 2.021 2.024 2.029 1.5 2.073 2.061 2.057 2.0 0.916 0.902 0.890 2.5 1.064 1.075 1.088 3.0 1.704 1.698 1.697 Table 2.--Effect of Strain Hardening and Residual Stresses on the Critical Dynamic Snapping Load, PS, of a Pin-Ended Arch Number Residual . . of R h = E /E Stresses gounds onLthg Cg1t1ca1 Panels S p Present? napping oa ’ s' 5 0.0 No 962ppi < PS < 989ppi No 935ppi < P < 962ppi 0.0 5 Yes 908ppi < PS < 935ppi 10 No 1202ppi < P < 1229ppi 0.03 S Yes 1202ppi < PS < 1229ppi 58 ’centroidal axis 3 Fig. 2.1 Spring System Representation of Lumped Flexibility Element B—l \\\\~— rigid panel lumped flexibility lumped mass (a) i-2 i-l 1 1+1 1+2 WWWW—Mm— L4 joint 1 h 0 Element B—2 mp ‘ (b) \\\\———r1gid panel i—2 1+1 i+ WOWOWMOW 79" ...___—°___.-l Element B-3 (c) rigid panels Fig. 2.2 Model Element Assemblies 60 rat I Formal Model (a) Modified Model (b) pn+1 Fig. 2.3 "Formal" and "Modified" Models ._ 61 Y Fig. 2.4 Bilinear Stress-Strain Fig. 3.1 Portion of Deformed Relation Model qi i—l \ m°fii ——.- /. ‘—_pl V \ 1-1 T V. 1 1 m V Vi i i T. 1 Mi 91 v1 Mi+1 h. T 1 1 Fig. 3.2 Free-Body Diagrams of a Mass Point and a Massless Panel ' 62 Formal Model ¢1 [—7 cbn+1 = \ O Flexl = g flex(s) ds Flexn+1 = 0 Modified Model ¢ n+1 hO/2 L Flex = f flex(s) ds Flex = f flex(s) ds 1 0 n+1 L- ho/Z Fig. 3.3 Treatment of Boundary Conditions Involving Prescribed Support Rotation 63 Residual Stress Pattern of Reference 14 Discrete Cross- section Adaptation ' 0.0415T 0.0776C 3 Aij .0361T 1 a 6 1.18 in2 2 & 5 2.36 in2 3 & 4 1.02 in2 C = Compression T = Tension 3.78" Residual Spring Forces Fig. 3.4 Residual Stress Representation 64 HALL RELEVANT QUANTITIES i START i TIME t: INCREMENT t. + At 3 KNOWN AT t = tj (numerical integration) L I (geometric relations) PRESCRIBED DISPLACEMENTS SPRING DEFORMATIONS (force deformation relations) I INTERNAL FORCE RESULTANTS ”may .. (equations of motion) I , EXTERNAL LOADINGS ACCELERATIONS & VELOCITIES I Fig. 4.1 Flow Diagram for One Step of Integration 65 “coEoomHmmHo umpcoo Mom knoemflm Uncommom H.m .mfim Hm F mW\\/.mm \ofifle N m.o v.0 m.o N.o H.o o _ _ _ _ o z/H -o.m // \ /#/ .Q / z \ .\x 1 a, wnrw\ EH m.mm u H IIT- a man OH H a ewe ooo.ooo.om n m AmEHmu omv qoflpdaom Esscfipcoo -ilk-llv =ooa n A _ to.m Romucv Hmtoz HmEHom ..I.«.u|. “P 1“ smug ago: 663302 i m .fi\.mna oma _ T7_01 x q36uaq/quaweoexdsrq Jequeo 66 ucoEoomHmmflo pcwom Hmeuwsa Mow whopmflm Uncommom N.m .mfim Hm : AW ll \oEHB NqN moo V-O MuO NoO HoO O _ HI _ _ _ _ \\ //#V/ / \x // ‘\\ I Moo // «a \\ x / x X // ..x \ ca . n S.m$ H // \\ 104 \ Nqfl OH H d % // .um v\H Dflmflh Illllii \\ / x and . . u . \ . ooo ooo om m / a/ , \\\ .\ // // \X \ \.\ lm H w z x .. // 4 \ //x 4/ \\ . \\ NV . . // /¢/I..<\ \ \ IO.N or / \ \4 w. I I :11; \ K x \ /f /JTI\¥ \4 // I \\ lMoN smug Hmeoz Hmfiom It: Alix. Aomnav HOUOS UOHMHUOS .Ik-|| I.o.m Amfinmu omv cofluSHOm msoscflpcoo II+II 01 x quuaq/auemeoeIdqu qqud Jeqxenb P— 67 p = 12.62 ppi 87.21° E = 30,000,000 psi R = 72.5 in 2 . 4 p = 0.000732 lb-sec /1n n = 12 Fig. 5.3 Elastic Circular Arch 68 L/r = 144 vS =1/7‘E p un+1 = —o.001vS <— 0.25r / a :9: ./;/I) b§0 ./o/ \. t _ 20L Modelin/O / \ 1 _ vs /o-—Ref (2) Q /% %/° / \\O F /%,/ ee/ \ 5 /o-'0~o l o 0—0\ 0 / / o \ v 3 d§© ./3/ t2 _ 2L Y/r o o v 5 ‘\0 06¢/ S O§. {/ O~Q 9. 10 03.0 F F x/L Fig. 5.4 Large Displacements of a Column Centerline Displacement in Inches 69 1663 #/in-e_2t/Tf [INTI IIIIIIIIIII —js,,.. Afi> ,QE7 X-section L = 160" A = 12 in2 2.007- E = 30,000,000 p31 I = 100 in4 1.75__ 0y = 30,000 psi O/;o-£=QQ\ ' Cb _ / . RSh — Ool ./ \ , n = 10 ./ (’\ $ 1.50—- o f ‘0 o \o / \o ‘*——-———' Ref 51 1.25.. A;y/ ( ) \. o .I 1.00—- // . o 0"——————- Large Deflection Analysis 0.75- o o 0.50— // O I /0 0.25.. I. / 0 ’00 I I I I I 0 0.2 0.4 0.6 0.8 1.0 . 2L m Fig. Comparison of Large and Small Deflection Solutions for a Small Deflection Problem Centerline Displacement in Inches q = 4.167 #/in-e-2t/Tf QMILMIH : 12" l44"— ,4; I L L = 480" _I ’ , _ 11.1 X-sectlon A = 2 in2 E = 30,000,000 psi _7 . 4 120”-— I — 2 1n Oj = 30,000 psi 0"°‘~o Rsh — 0.1 0” $ 1’ o o o ”"’ I‘I“O \\ 96"—- / \o -\ Ref (51)—>0 0 \ / \. o 72u___ /. /0 8-‘—————— Large deflection analysis 48". / o o 24"— // o //o O " I I I J 0 0.2 0.4 0.6 0.8 . 2L2 m Tlme/ 7? E? Fig. 5.6 Comparison of Large and Small Deflection Solutions for a Large Deflection Problem A |__¢‘. i. ”a; t=0.OTf ) t=0.206Tf ) I t=0.412Tf > E t=0.567Tf > =0 618Tf > < fr 0 670T o t=0.824T f Fig. 6.1 71 t=0.876T t=0.927T t=0.979Tf t=l.030Tf t=l.082Tf t=l.133Tf t=1.185Tf =1.236Tf < t=1.288Tf < t=l.340Tf < Snap—Through of an Arch (N = 10) Maximum Crown Displacement/Length 72 WI 1: Eh I 0.015_‘ I 16" ' 156,, 8WF31 ' . L———————-J . I—ll-r-l a «.-I O. O. a. n. QI'O. 0.012— E = 30,000,000 9* 9* 0N m 3 S 8'3 - O = 30,000 m Ch -—I .—I y I "III II n s: c s: x: 3.3 3's 0.009— .0 n 12.1: _o___. R = 0.0 HIM H :4 sh o U ml o - 5's 5 E —o— RSh=0.O3 Pals H': I . ' ° I 0.006—— I I ' | _ I | 0' O I o 0.003— /0'°I ' 0’. | l — | .. / l o I I 05—”1””T’ I I I I I I L . I I I. l 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Load pb in Hundreds of Pounds per Inch Fig. 6.2 Dynamic Snap Through Loads of a Shallow Arch 73 . Zns a Sln (—E—) P pb (b) /\ 8WF3l 16" (c) __ EE:___ A L_ Span = 156" [V o / \ o 0“ 5ll_. . 0 lo"_ 5"— j (e) 0 ./ \. j P .— o a — 27 ppi ll \ ./ - 5 — ol\~.//’ Pb = 990 ppl lO"—' t/Tf = l 5 Fig. 6.3 Evaluation of Model Bias Longitudinal End Displacement, un+l,/Length 74 .03 I r— 8WF3l I ' P 0 SP 180 . ' = . L = II Y I _ l L/r = 52 I , I ”0 ”HT E = 30,000,000 [35' l ‘ a _ , o 0y = 30,000 p51 I '43 _ _un+l . I z I fl I 8. .02—— m I 5% _ I H m I O o '3 _ I-H H ///z///// I 0 —__’— G _ I 0 Y = h(t) g I "g _ I o o m / ' 8 .01“ I o g / 'D . l / ' ///o A -'__ h(t) l t I o _ ./// Tf 2Tf I o/ l 0 I I I iJ I I I I I 0 .01 .02 .03 .04 .05 .06 .07 A/L Fig. 6.4 Maximum End Displacement vs. Peak Support Motion 75 P = aP Y i 8WF3l /// 111/, L = 180" L/r = 52 , VS E = 30,000,000 psi 3 O = 30,000 psi 4.! .0047 Y 2‘ A/L = 0.015 g ////l llll' \\ ~ 0 > a} .003_ $2: ~H r—I u w 4.) a m U I; .002— 4.) m U) r—I S a) O 13’ q .001 '— 2 o a) O a g I collapse w m 0 I I . I I I I I I I I 0 0.5 1.0 a = P P / Y Fig 6.5 Permanent Lateral Set at E vs. Axial Load APPENDIX COMPUTER PROGRAM A.l Introduction For completeness the computer program written for this study is given here. The program is divided into four basic subroutines governed by the main program, "UNIAX." These subroutines represent the four advance- ments mentioned in section 4.3 and shown in Fig. 4.1. The first subroutine, "BTA," is concerned with the geometry and kinematics of the member, i.e., the de- termination of the mass point coordinates, and the bend- ing and compression deformations. The effect of support motion is also considered in this subroutine. Subroutine "FRC" is used to determine the internal force resultants from the deformations that are transmitted from "BTA." "FRC" also considers residual stresses, if any. In sub— routine "EQM" the equations of motion, (3-7) and (3—8), are used to compute the mass point accelerations from the internal force resultants and the prescribed external loadings. The velocities of the mass points are then com— puted using Eq. (4—2). The last of the four basic sub— routines "THK," increments the time and performs other 76 77 tasks associated with the completion of a step of inte— gration, such as finalizing the accelerations and con— trolling the print-out sequencing. The remaining subroutines and function subprograms perform certain tasks associated with the main program and the previously mentioned subroutines. Subroutine "XSEC" discretizes the cross—section of the member and computes the zij's and the Aij's and other quantities associated with the cross—section. Subroutine "EGK" checks the energy and impulse-momentum of the system as discussed in section 4.6. Subroutine "PLT" is used to graphically display the deformed configuration of the member at periodic intervals of time and, after the problem is completed, to display a response history of the motion. Function "DPNL" com— putes the distance between two adjacent masses given their coordinates. "DX" computes the change in the x—coordinate between two adjacent masses given the initial y-coordinates and the initial panel length, ho. Finally "DEF" computes the support displacement by linear interpolation between adjacent data points for problems in which support motion is prescribed. The program is presented in the following in three parts. The first part is a list of variable names and their meanings. Detailed flow charts of the main programs and of the subprograms are included. Finally a listing of the entire program is given. The particular program being 78 presented contains procedures expressly for inputting and initializing the geometry and load of circular arches sub— jected to normal uniform step loadings. By changing these procedures structural members other than circular arches could be considered. A.2 List of Computer Program Variables All variables which occur in common blocks are listed under the blocks in which they occur. Then vari- ables which occur in the main program and in subsequent subprograms are listed where they first appear. All lists are in alphabetical order except the common blocks. All variables are defined only once. However, there may be variables with the same name but with different meanings in different subprograms. For example, T means thickness in "XSEC" while it means time in all other routines. These doubly defined variables are listed in each routine where their meaning is different from that of the main program. Mention should be made of subscripted variables denoted by "(I,J)*." In these arrays the term "jth direc- tion" denotes the x—direction for J=l and y-direction for J=20 COMMON BLOCK BC BCXl = boundary condition at left end in the x—direction given as either FREE, FIXED, or DEFINE; 79 BCYl = boundary condition at left end in the y-direction given as either FREE, FIXED or DEFINE; BCMl = rotation boundary condition at the left end given as either FREE or FIXED. BCXN = boundary condition at right end in the x-direction given as either FREE, FIXED, or DEFINE; BCYN = boundary condition at right end in the y-direction given as either FREE, FIXED or DEFINE; BCMN = rotation boundary condition at the right end given as either FREE or FIXED; FREE = BCD code representation of "FREE" used to compare with boundary con- ditions; FIXED = BCD code representation of "FIXED" used to compare with boundary con- ditions; DEFINE = BCD code representation of "DEFINE" used to compare with boundary con— ditions. COMMON-BLOCK BATEGK X(I,J)* = coordinates of mass point I; J = 1 gives x—coordinate, J = 2 gives y-coordinate. COMMON BLOCK EQMEGK P6(I) = transverse load at the ith mass; P(I) = longitudinal load at the ith mass. COMMON BLOCK L G = number of print lines in a graphi- cal display (in PLT G is RNOFL). COMMON BLOCK N IN INP INP2 INP3 COMMON BLOCK P P2 P3 P4 P7 P8 P10 PIE PIESQR COMMON BLOCK SEC A(I) Z(I) 80 number of elements in model; IN + 1; IN + 2; IN + 3. depth, D, divided by length, L; 54.. !I - length, L, divided by radius of gyration, RX; area, A, divided by radius of gyration squared, RXZ; strain hardening ratio; plastic modulus E divided by the elastic modulus, E; yield stress, SY, divided by the elastic modulus, E; uniformly distributed load which will just cause yield in a simi— lar simply supported span; TT; 2 W 7 ith discrete area of the cross— section divided by the total area; distance of the ith area from the neutral axis divided by the depth of the member. COMMON BLOCK X PROGRAM DISP(I,J) ID UNIAX ALPHA AT BM(I) BS(I) C(I) DA BS(I) HI(I) IJKLMN ISEC IX LC 81 response history of member; J = 1 gives time, J = 2 gives displace- ment; counter to keep track of subscript I in DISP(I,J). angle subtended by circular arch; total area of cross-section; bending moment at joint i at t+At; bending strain at joint i at t+At; cosine of the angle the ith panel makes with the x—axis at t+At; depth of the member; angle subtended by an element; elastic modulus; axial deformation in the ith element at t+At; initial element length, ho; deformed length of ith element; input parameter used to determine if more data exists; number of discrete areas in cross- section; moment of inertia of discrete cross-section; length of member; logical variable used to specify convergence or nonconvergence; LP LSTART MASS PCR PPI Pll RATIO RX S(I) SY TF THS THT TI TRST(I) 82 logical variable used to enter print routines; logical variable used to enter initialization routines; mass per unit length of member; critical elastic buckling load of a pin-ended column; uniformly distributed normal loading on arch; auxiliary variable; ratio of inelastic modulus to the elastic modulus; radius of circular arch; coefficient of standard time increment used in varying the time increment; ratio of the nth period of axial vibration to the nth period of bending vibration; radius of gyration of the discrete section; sine of the angle the ith panel makes with the x-axis at t+At; yield stress; time; n EI ' auxiliary variable; auxiliary variable; time increment; axial force in the ith panel; UDB(I,J)* UDDA(I,J)* UDDB(I,J)* V(I) ZN SUBROUTINE XSEC B DENSITY DP K NF NW TF TW SUBROUTINE BTA BSO(I) BSO(I) ICL 83 velocity of the ith mass at t+At in the jth direction; acceleration of the ith mass in the jth direction at time t; acceleration of the ith mass in the jth direction at time t+At; shear in the ith panel at time t+At; number of elements in model in decimal form. width of flange or rectangular section; density of member material; auxiliary variable; auxiliary variable; number of discrete areas representing the flange; number of discrete areas representing the web; thickness dimension of a discrete area; thickness of the flange; thickness of the web. initial rotation of the ith joint; initial compression of the ith element; number of the mass point chosen for response history plot; JJ MM STB STE UA(I,J)* UB(I,J)* UDA(I,J)* XO(I,J)* SUBROUTINE FRC CST CST2 FA(I,J) FB(I,J) FUNl(A,B) FUN2(A,B,C,) FUN3(A,B) JA(I,J) JB(I,J) ST STA(I,J) 84 dummy index; dummy counter; temporary storage of BSO(I); temporary storage of BSO(I); displacement of the ith mass in the jth direction at time t; displacement of the ith mass in the jth direction at time t+At; velocity of the ith mass in the jth direction at time t; initial coordinates of the masses. program constant; program constant; force in the jth spring at the ith section at time t; force in the jth spring at the ith section at time t+At; upper bound of the yield envelope of the force deformation relation; elastic portion of the force de— formation relation; lower bound of the yield envelope of the force deformation relation; zone factor for the jth spring at the ith section at time t zone factor for the jth spring at the ith section at time t+At; spring deformation increment; deformation of the jth spring at the ith section of time t; STB(I,J) STOA(I,J) STOB(I,J) STR(I,J) STRAIN SUBROUTINE EQM SUBROUTINE EGK APK BMA(I) BSA(I) CA(I) C1 C2 C3 C4 C5 C6 DT BA 85 deformation of the jth spring at the ith section at time t+At; elastic strain axis intercept of the jth spring of the ith section at time t; elastic strain axis intercept of the jth spring of the ith section at time t+At; stress at the jth area of the ith section; temporary storage of the residual strains. All variables have been previously defined. auxiliary variable; bending moment at the ith joint at time t; bending strain at the ith joint at time t; cosine of the angle the ith panel makes with the x—axis at time t; program constant; program constant; program constant; program constant; program constant; program constant; time increment; energy absorbed by straining; EAP EI EIP EISASK EK EKP BSA(I) FDH FDV FH FV SA(I) TRSTA(I) VA(I) VH VV XA(I,J)* XAO(I,J) SUBROUTINE THK IP IRATIO IRUT NPR 86 scaled energy absorbed by straining; input energy; scaled input energy; EIP-EAP-EKP; kinetic energy; scaled kinetic energy; axial deformation in the ith element at time t; horizontal impulse; vertical impulse; FDH-VH; FDV-VV; sine of the angle the ith panel makes with the x—axis; axial force in the ith panel at time t; shear in the ith panel at time t; horizontal momentum; vertical momentum; coordinates of the ith masspoint in the jth direction at time t; unused variable. print request counter; auxiliary variable; auxiliary variable; number of print requests per funda— mental period of bending vibration; 43‘; a _, _._ U « 4J3 NPRT TMAX SUBROUTINE PLT BLANK ICOUNT IX(I,J)* K LINE(I) MAX MIN POINT RANGE RMAX FUNCTION DX DH DX DY Y1 Y2 87 number of steps of integration between print request; length of time the solution is to progress into the time domain. BCD code representation of a blank; counter; mass point coordinates converted to scaled integer numbers; auxiliary variable; represents 120 space print line to be assembled; maximum value of the scaled integer y—coordinate; minimum value of the scaled integer y-coordinate; BCD code representation of a "."; RMAX—RMIN; maximum value of the y-coordinate; minimum value of the y—coordinate; scale factor; n BCD code representation of x" h and h 2; o o increment in x—coordinate between adjacent masses; 2 (yi_l - yi) and (Yi—l - yi) , y coordinate of the i-l mass; y—coordinate of the ith mass. FUNCTION DPNL DPNL DZ X1 X2 FUNCTION DEF DEF SLP TY(I,J) 88 deformed element length (x - Xi) and (xi+l - X-I ; i+l x-coordinate of the ith mass; x—coordinate of i+l mass. support motion at time t+At; number of data points to read in on input; counter; slope between two adjacent input points; ith point in prescribed support motion; J=2, ground displacement; J=l, time. 89 A.3 Flow Diagrams FLOW DIAGRAM FOR UNIAX (ENTER UNIAX BRANCH TO gSUBROUTINE XSEC READ IN & PRINT OUT INITIALIZE CONSTANTS, GEOMETRY, LOADING COMPUTE & BRANCH'TO SUBROUTINE BTA SET BENDING DEFORMATION I BRANCH TO SUBROUTINE FRC I BRANCH TO;:> SUBROUTINE EQM I SET END MASS VELOCITIES ADJUST TIME INCREMENT BRANCH TO FINALIZE SUBROUTINE RESPONSE pLT HISTORY THK BRANCH TO SUBROUTINE EGK BRANCH TO SUBROUTINE PROBLEM OVER? 90 FLOW DIAGRAM FOR SUBROUTINE XSEC READ IN & PRINT—OUT X-SECTION DATA ENTER XSEC FLANGE THICKNESS X—SECTION IS RECTANGULAR X-SECTION IS WF COMPUTE Aij & zij FOR THE FLANGE THICKNESS IDEAL X-SECTION COMPUTE Aij THE WEB & 2.. FOR ‘ l COMPUTE ' Ir AI r! RETURN MASS, ISEC 91 FLOW DIAGRAM. FOR SUBROUTINE BTA ENTER BTA FALSE RECORD DISP FOR TIME HISTORY OF RESPONSE INITIALIZE SUBROUTINE BTA BRANCH TO SUBROUTINE I PLT COMPUTE INITIAL X_COORD‘ PRINT OUT I AT COORD- INATES L—v INTEGRATE TO FIND DISPLACE— MENT CONSIDER EFFECTS OF FINALIZE BOUNDARY INTEGRATION CONDITIONS STEP 92 FLOW DIAGRAM FOR SUBROUTINE BTA (cont.) COMPUTE PANEL LENGTH BY DNPL COMPUTE: SIN Bi, COS 6i,‘ & ¢i. TEST LSTART FALSE REINITIALIZE ES, BS, BSO, ESO, BASED ON INIT. DEF &AXIAL DEF. INITIALIZE ESO & BSO 93 FLOW DIAGRAM FOR SUBROUTINE FRC INITIALIZE SUBROUTINE (:ENTER FRC FRC READ IN & PRINT OUT FALSE RESIDUAL STRAIN INITIALIZE RESIDUAL SPRING DEFORMATION FINALIZE INTEGRATION STEP COMPUTE RESET STORE SPRING p— CERTAIN RESIDUAL DEFORMATIONS VARIABLES STRAINS COMPUTE No A, CHANGE IN SPRING DEFORMATION @ NO CHANGE IN SPRING FORCES FB = FUN2 ‘___l 94 FLOW DIAGRAM FOR SUBROUTINE FRC (cont.) Ez>——> FB =.FUN2 ( FB < FUN3 FB > FUND FUN3 POSITIVE NEGATIVE C < FB < Y ELD SET _ _ YIELD SET IS = -1 FUNl JB ' =+1 REVERSAL FIND STRAIN INTERCEPT SET JB = 0 FB FUNl 95 FLOW DIAGRAM FOR SUBROUTINE FRC (cont.) ~ COMPUTE 10 BM, TRST, ‘ & STR FALSE PRINT OUT BM, TRST STR & Cm ) FLOW DIAGRAM FOR SUBROUTINE EQM ENTER EQQ FALSE INITIALIZE SUBROUTINE COMPUTE ACCELERATIONS ‘ . FROM EQUATIONS OF MOTION SET LC>= TRUE* COMPUTE F #’ VELOCITIES RETURN *B = 0 method converges automatically 97 FLOW DIAGRAM FOR SUBROUTINE EGK CENTER EG9 INITIALIZE . STEP I COMPUTE EI, EA, EK FOR INTERIOR \ JOINTS INITIALIZE SUBROUTINE (NOT DEFINE DEFINE:> ADJUST EI SET FOR SUPPORT RETAINER ADJUST EK- * -MOTION VARIABLES (:NOT DEFINE ‘ DEFINE :> RETURN ) { ‘ ADJUST EI ADJUST EK FOR SUPPORT MOTION GOT DEFINE DEFINE ) ADJUST EI ADJUST EK - ' FOR SUPPORT MOTION 98 FLOW DIAGRAM FOR SUBROUTINE EGK (cont.) (NOT DEFINE DEFINE\ I ADJUST'EI ADJUST EK FOR SUPPORT ; MOTION COMPUTE MOMENTUM FOR INTERIOR I,MASS POINTS (:NOT FREE' FREE :> I . I COMPUTE ADJUST IMPULSE OF HORIZONTAL HORIZONTAL MOMENTUMs , REACTION . FOR END MASS (NOT-FREE FREE ) COMPUTE» ADJUST IMPULSE OF VERTICAL. VERTICAL ' MOMENTUM REACTION FOR END MASS (:NOT FREE FREE :> COMPUTE ADJUST IMPULSE OF , HORIZONTAL HORIZONTAL MOMENTUM REACTION 9 FOR END MASS 99 FLOW DIAGRAM FOR SUBROUTINE EGK FREE :> I (NOT FREE COMPUTE- ADJUST IMPULSE OF. VERTICAL VERTICAL l MOMENTUM' REACTIONX, FOR END MASS ‘V COMPUTE IMPULSE'DUE TO EXTERNAL LOADINGS I ‘RESET RETAINER VARIABLES I RETURN I SCALE ITEMS APPROPRIATELY & COMPUTE DIFFERENCES I PRINT OUT CHECK PARA- METERS I (m) 100 FLOW DIAGRAM FOR SUBROUTINE THK SET LSTART FALSE TO FALSE SET LP TO FALSE INITIALIZE SUBROUTINE SET LSTART TO TRUE FINALIZE THE W‘———- @ . 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