ELASTIC-PLASTIC RESPONSE OF BEAMS INCLUDING EFFECTS OF SHEAR AND ROTATORY INERTIA Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY B. NureI Beerryan 1965' «I;(«owgwmWlmmfiluwfl ABSTRACT . ELASTIC-PLASTIC RESPONSE OF BEAMS INCLUDING EFFECTS OF SHEAR AND ROTATORY INERTIA by B. Nurel Beyleryan In this investigation a physical model is constructed in which the shear deformation and rotatory inertia of a continuous beam, in addition to the bending deformation and lateral displacement inertia, are lumped at a discrete number of points. The model thus consists of rigid panels connected by shear and moment Springs. The inter- action between moment and shear on the material behavior at yield is taken into account. The mass, rotatory inertia, and external loading of the panels are lumped at the center of each panel. Solutions are obtained by numerical techniques, which have been programmed in the Fortran language for use on the CDC3600 system of Michigan State University. The convergence of the discrete model is indicated by the increasing degree of agreement of the numerical results as the beam is divided into larger numbers of panels. Numerical results are then obtained for simply supported and fixed-fixed beams subjected to a blast type loading. Taking the web thickness and the beam length of an I-beam as parameters, the B. Nur e1 Beyl eryan influence of the interaction between moment and shear is studied. It is shown that, as expected, as the web thickness or the span length is increased, the elastic-plastic solution including shear and rotatory inertia effects (the "Timoshenko" model) approaches that of the simple theory (the "Euler" model). For steel I-beams of usual preportions, the influence of shear and its interaction with moment was found to be quite significant for fixed-fixed beams and to a lesser extent for simply supported beams. The discrete model is also reduced, for the elastic case, to lesser forms such as one that excludes the effect of rotatory inertia. However, it is found that, rather unexpectedly, the computer time required when using the complete model is no more than any of the reduced models. The latter, therefore, do not seem to offer any practical advantage. ELASTIC-PLASTIC RESPONSE OF BEAMS INCLUDING EFFECTS OF SHEAR AND ROTATORY INERTIA by B. Nurel Beyle ryan A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOC TOR OF PHILOSOPHY Department of Civil and Sanitary Engineering 1965 ACKNOWLEDGMENTS The author wishes to acknowledge the valuable guidance of Dr. R. K. Wen, under whose direction this study was conducted. Thanks are also extended to the members of the author's Guidance Committee, Dr. C. E. Cutts, Dr. L. E. Malvern, Dr. G. E. Mase, and Dr. C. P. Wells, for their interests and help during the course of the author's studies at Michigan State University. The author wishes to express his special appreciation to the Chairman of the Civil Engineering Department, and the Head of the Engineering Research Division, for their support of the author's doctoral program. ii TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF FIGURES I. III. IV. INTR ODUCTION 1.1. General 1.2. Notations BASES OF ANALYSIS 2. l. Continuum Theory 2. 2. Discrete Theory 2. 3. Boundary Conditions METHOD OF NUMERICAL SOLUTION . General . The "Timoshenko" Model . The "Shear" Model . The "Rotary" and "Euler" Models . Time Increment . Use of the Computer wwwpuww l 2 3 . 4 . 5 . 6 RESULTS IN THE ELASTIC RANGE Introduction Convergence of the ”Euler” Model "Apparent" Convergence of the ”Rotary, " "Shear, " and "Timoshenko" Models 4. 4. Relative Importance of Shear and Rotatory Inertia “H“? wNo-v Page ii 19 21 Z6 Z7 29 30 31 31 31 33 34 V. ELASTIC-PLASTIC BEHAVIOR OF THE "TIMOSHENKO" MODEL 5. 1. Introduction 5. 2. Convergence of the "Timoshenko" Model in the Elastic-Plastic Range 5. 3. Response of Simply Supported I—Beams with Different Lengths and Web Thicknesses 5. 4. Response of Fixed-Fixed I-Beams with Different Lengths and Web Thicknesses VI. CONC LUSION BIB LIOGRAPHY FIGURES A PPENDIX 36 36 37 38 41 43 46 51 77 - w. . I - . ‘ A a- l I I ..II|LIIIIHI.IEIII1IIHIIIIIIII.III£. I Figure 2. I. Figure 2. 2. Figure 2. 3. Figure 2. 4. Figure 2. 5. Figure 3. 1. Figure 4. 1. Figure 4. 2. Figure 4. 3. Figure 4. 4. Figure 4. 5. Figure 4. 6. Figure 4. 7. Figure 4. 8. Figure 4. 9. LIST OF FIGURES Forces Acting on an Element of Continuum Plastic Potential Function Discrete Beam Model Deformed Configuration of the Discrete Beam Forces Acting on a Typical Panel of the Discrete Beam Finite Increment Treatment of Plastic Yielding Cros s -Sectional Preperties Convergence of the ”Euler" Model --Def1ection at Mid-span Convergence of the "Euler" Model --Moment at Mid-span Convergence of the ”Euler" Model --Shear at the Support ”Apparent" Convergence of the "Rotary" Model--Deflection at Mid-span "Apparent" Convergence of the "Rotary" Model--Moment at Mid-span ”Apparent" Convergence of the ”Rotary" Model—-Shear at the Support "Apparent" Convergence of the "Shear" Model--Def1ection at Mid-span "Apparent" Convergence of the ”Shear" Mode1--Moment at Mid-span Page 51 51 52 52 53 53 53 54 55 56 57 58 59 6O 61 I131 . Figure 4. 10. Figure 4. 11. Figure 4. 12. Figure 4. 13. Figure 4. 14. Figure 4. 15. Figure 5.1. Figure 5. 2. Figure 5. 3. Figure 5. 4. Figure 5. 5. Figure 5. 6. Figure 5. 7. Figure 5. 8. "Apparent” Convergence of the ”Shear" Model--Shear at the Support "Apparent" Convergence of the "Timoshenko" Mode1-—Deflection at Mid-span "Apparent" Convergence of the "Timoshenko" Model--Moment at Mid-span "Apparent” Convergence of the ”Timoshenko" Model--Shear at the Support Mid-span Moment Responses Influence of Shear, and Shear and Rotatory Inertia in the Elastic Range Shea r-Moment Inte raction Curves "Apparent" Convergence of "Timoshenko" Model in Elastic-Plastic Response --Moment at Fixed End ”Apparent" Convergence of ”Timoshenko" Model in Elastic-Plastic Response --Shear at Fixed End Locus of Stress State for Problem in Figures 5‘. 2 and 5. 3 Regions of Plastic Response for Simply Supported I—Beams with Different Lengths Deflections, Permanent Sets, and Permanent Slides for Simply Supported I-Beams with Different Lengths Deflections, Permanent Sets, and Permanent Slides for Simply Supported I-Beams with Diffe rent Thickness es Fixed-End Moment and Shear Responses of I-Beams with Different Lengths vi 62 63 64 65 66 67 68 69 70 71 71 72 73 74 Figure 5. 9. Figure 5. 10. Deflections, Permanent Sets, and Permanent Slides for Fixed-Fixed I-Beams with Different Lengths Deflections, Permanent Sets, and Permanent Slides for Fixed-Fixed I-Beams with Different Web Thicknesses 75 76 1. INT R ODUC TION 1. 1. General It is well known that the usual engineering theory of beam vibrations is based on the assumptions that deformations are caused by bending only, and only transverse inertia forces need be considered. In particular, the theory neglects the effects of shear deformations and rotatory inertia. The first modification of the theory by including the above mentioned effects was given as early as 1859, by Bresse (8); but, apparently it went unnoticed. Rotatory inertia effects were also discussed by Raleigh (36) in 1877. Today, for the more exact theory that includes shear deformation and rotatory inertia effects, the presentation of Timoshenko (41) is usually quoted. In fact, it is known as the "Timoshenko" beam theory. A derivation of this will be given in Section 2. 1. For several decades, after Timoshenko's contribution, work was generally directed towards obtaining estimations of the error introduced if effects of shear deformation and rotatory inertia were neglected. Solutions obtained with substantial rigor for various Special cases of the problem have appeared since 1948. Two approaches seem to dominate the literature covering the elastic vibrations of the Timoshenko beam: the wave method (7, 12, 21. 7-7). and the mode method (1, 4, 10, 20, 40, 43). The former usually employs Laplace transform techniques to yield solutions in closed form. The complicated superposition required, in order to accommodate various loading and boundary conditions, makes the method rather!unwieldy to apply. The mode method, presented in full in Referencel, has also proven to be inconvenient for applications to actual problems. Furthermore, convergence of the solution is not always guaranteed. While the preceding discussion applies to the linearly elastic case, the literature on the inelastic case is rather scarce. Two articles, by Salvadori and Weidlinger (38), and Karunes and Onat (22) dated 1957 and 1960, respectively, have considered the rigid-plastic response of beams. It is found in the former work that, in case of a simply supported beam, "plastic shear hinges" may deve10p at the supports in addition to a moment hinge at the mid-span of the beam. In the latter study, a free-free, rigid—plastic beam subjected to a concentrated load at the mid-span is investigated. In both references, the methods used can not deal with the interaction effects between the bending moment and shear when the material goes into the plastic range. Yet, this interaction is known to exist, and its effects on beam vibrations have not been ascertained. The previous paragraphs point out clearly what has been missing so far; namely: a method of analysis which can be used to calculate the elastic-plastic vibrations of beams, including shear deformation and rotatory inertia effects, with any usual boundary conditions, and subjected to any usual loading. This will be the general purpose of the present investigation. Thus, the first objective of the present work is to develop such a method. The second objective is to use the method to study the significance of the effects of shear deformation and rotatory inertia on the beam response in the inelastic range. Recognizing the intrinsic difficulties of the problem, particularly from a continuum point of view, the present work uses the discrete model approach. Briefly, the model used to represent the beam consists of rigid panels connected by moment and shear springs. The force-deformation characteristics of these springs interact when the deformations are in a plastic state. At the middle of the panels, there are lumped masses on which the external loads act. This model is amenable to analysis and numerical results are conveniently obtainable from a computer. Of course, it is not sufficient that the model produces results. It is also necessary to show that such results are trustworthy. To this end, the obvious way is to compare the model results with the III I." III In 4 exact analytical solutions (of the continuum). But exact solutions are not readily available. Hence, the credibility of the model is examined by comparing the numerical results yielded by models with different degrees of fineness (analogous to the mesh size in a formal finite difference approach). If the results seem to converge (referred to later as "apparent" convergence), then the model is regarded as reliable. There is an exception to the above approach. For one of the "reduced forms” of the model, exact solutions are avail- able-~and used for comparison. Furthermore, the reliability of the model is also judged from a physical point of view in that the behavior, as exhibited by the model results, must make good physical sense. In pursuit of the second major objective of this thesis, the model is used to obtain numerical results that would reflect the significance of shear deformation and rotatory inertia in elastic-plastic response. The data cover both simply supported, and fixed-fixed beams. In order to highlight the shear effects, I-beams alone are considered. The variables considered are the total permanent deflection, the permanent deflection due to shear effects alone, as well as the maximum deflection. Parameters considered are the web thickness and span length. The web thickness will be varied from 20% to 3% of the flange width and the length will be varied from 6 to 20 times the beam depth. In the remaining chapters of this thesis, the theoretical bases of this work are presented in Chapter II, and the numerical technique is presented in Chapter 111. Chapter IV and Chapter V contain, respectively, the elastic and elastic-plastic numerical results. Concluding remarks are made in Chapter VI. 1. Z. Notations The notation listed below has been adOpted in this investigation. a = Lumped change of curvature; Ge = Elastic part of a; up 2 Plastic part of a; 0. = Elastic change of curvature, correSponding to a Y change of moment A M=My; E = a a / Y [3 = Shear displacement (slide); De = Elastic part of [3 ; (3p 2 Plastic part of (3 ; (3 = Elastic change of shear slide, corresponding to a Y _ . change of shear AS-Sy, B - 13/5}, [5' = (3 divided by its tributary length; A = Prefix denoting "increment"; p = Density of material; 7 2 Time in plastic range; 4) == SIOpe of a panel of the model; CI) = Change of <1) with respect to time; Rate of change of (I) with reSpect to time; 6: ll kl 7 Cross-sectional area of the beam; Half of the flange width; A boundary constant; Arbitrary coefficient pertaining to applied load; Vector with components A; and A3; Elastic modulus of the material; Plastic potential function; Shear modulus of the material; Gravitational constant; Half of the beam depth; (Subscripted or not) Length of a panel; Moment of Inertia; Variable subscript to denote a point or a segment on the beam; The highest value of i; Constants; A cross-sectional constant; Length of beam; Moment; Maximum moment at yield; Non-dimensional moment M/My; Lumped mass; Number of panels into which a beam is divided; I . 4... . .. .o., . .. . I. . h n, u . .1 . . 9 ¢ . . .‘I u p |:. M 3.. - . I o I. .l , . ’13: I Put 1. . a... .- .5)? I311... .‘S . .e . 1 a 8 Lumped external loads acting on panel i; Shear; Maximum shear at yield; Non-dimensional shear S/Sy; Flange thickness of I-beam; Period of the fundamental mode of vibration; Period of the n-th mode of vibration; Time; Initial time; Web thickness; Non-dimensional time t/Tl; Non-dimensional web thickness tw/(ZB); Minimum stationary loading that will cause yield for a simply supported beam; Loading as a function of position and time; Length coordinate; Static deflection caused by a uniformaly distributed load of W; Vertical direction (unsubscripted); Deflection of the beam (subscripted); Velocity; Acceleration; SloPe; Non—dimensional deflection y/ Yy' II. BASES OF ANALYSIS 2. 1. Continuum Theory For the sake of completeness, a derivation of the more exact beam equations including the effects of shear deformation and rotatory inertia will be given below for a physical continuum (41). - EX I When a beam deforms, its slope d (or y ) may be x considered to consist of two parts: 4; , the lepe due to bending 29 . - - only, and dx (or [3 ), an additional slope due to shear ([5 denotes the shear deformation or "slide”). Therefore, y'=¢+B' 2.1 The kinetic equations can be obtained from D'Alembert's principle by summing the vertical forces and the moments. Referring to Figure 2. 1, one can write: 8 3%”. —w(x,t) 2.2 t 2 8M 6 S = -—8—){— + pI 2.3 at where t is time, A is the cross-sectional area, p is the density, and I is the moment of inertia; the other symbols are defined in the figure. 10 Equation 2. l is obtained from a consideration of geometry, and Equations 2. 2 and 2. 3 from equilibrium. Thus, they are general and valid for all materials. Further relations needed for an analysis must be obtained from the prOperties of the material. For an elastic and isotropic material one can write, M = — Eig—‘l’ 2.4 X s = Gk'Ap' 2.5 where E is the elastic modulus, G is the shear modulus, and k' is a sectional constant. For elastic beams, Equations 2. 1 through 2. 5 can be combined into a smaller number of equations, as various investigators have done (if combined into one fourth order partial differential equation, it becomes the "Timoshenko" beam equation (41)). However, since plastic as well as elastic cases will be treated herein, the equations will not be combined at this point. Equations 2. 4 and 2. 5 must be replaced by suitable relation- ships when the deformations go beyond the elastic range. When this happens, the material is assumed to be perfectly plastic. The term "perfectly plastic" is used here to mean that no work-hardening effects are considered. (As long as the material is not strained excessively, most mild steel can be assumed to be perfectly plastic.) The inelastic behavior will be assumed to be governed by the "plastic potential theory" (19), which is briefly explained below for the problem under consideration. Associated with a given cross-section there is a plastic potential function f(M, S). The curve in the M-S plane: f(M, S) = 0 2.6 is known as the "yield curve" or the "interaction curve. " This is illustrated in Figure 2. 2 for the positive quadrant. When the values of M and S acting at the section are such that f(M, S) < 0, the laws of elasticity apply. When f(M, S) = 0, the section is in a plastic state, and yielding or plastic deformation takes place. The value of f(M, S) can never be positive. When plastic deformation occurs, it is governed by the following rule: the rate of the plastic deformation corresponding to M (or S) is proportional to the M-component (or S-component) of the gradient Vf, or the normal to the interaction curve; see Figure 2. 2. Expressed mathematically, yielding is to follow the relation: 8f(M, S) 3f(M, S) 3M - S K 2 7 5 (1‘35 ) _ a «55) 1 ' 3T axp .57 ‘a—xp in which the subscript p denotes the plastic component of the deformations, and K1 is a scalar. A simultaneous solution of the preceding equations, even for the elastic case, has proven to be very difficult. For the 12 elastic-plastic case, an analytical solution seems almost impossible. One could try the usual finite difference approach. However, he would soon encounter difficulties in the treatment of boundary conditions and the changing material properties. 2. 2. Discrete Theory In order to obtain an approximate solution of the problem, the present author suggests the use of a discrete model for the beam. It is understood that changing a continuous beam into a discrete model inevitably involves a loss in "resolution'' and even some distortion. However, it is believed that the essential features of the beam have been kept in the model. Figure 2. 3 illustrates the discrete beam in an undeformed state, and Figure 2. 4 shows the deformed configuration of the model. The various prOperties of continuum are retained in a manner as given below. a) The continuous beam is divided into a discrete number of panels, N. These panels may, in general, be of different lengths, hi‘ b) The panels are assumed rigid at all times. However, in order to account for the deformations and rotations that a continuous beam undergoes, moment and shear springs are inserted between the rigid panels. The sections where these Springs are 13 placed are called the ”force points. " c) The rotation of a moment spring corresponds to the sum of the curvature within a length of h/2 on either side of the panel point; that is, the flexibility of a panel is lumped apally at the Springs to the left and right of a panel. At a typical spring i, the lumped elastic flexibility is ZEI/ (hi-1+hi)' d) Similarly, a shear Spring lumps the shear flexibility from a tributary length of h/2 on either side of the panel point. For example, the flexibility of the i-th shear spring is 2Gk'A Bi/ (hi-1+hi)' e) In inelastic action, when interaction is considered between shear and moment, the moment and shear Springs are made to obey laws that are direct generalizations of the plastic potential theory described earlier for a cross-section of the continuous beam. Detailed description of the procedure is given in Section 3. 2. f) The mass and rotatory inertia within each panel are lumped at the center of the panel, which is referred to as a "mass point .. " The external Loading will be similarly lumped. In Figure 2. 4 the symbols mi,(P Ih)i’ Pi' denote, respectively, the lumped mass, rotatory inertia, and load. Some of the implications of this model may be noted here as follows. The characteristic of rigid panels gives the beam a 14 discontinuous look. In particular, the deflections are discontinuous at the shear springs. Consequently, the apparent slope of a segment is given by ¢i , which is the slope due to flexure alone. Since equations of motion will be written for the mass points, deflections are defined only at mass points; stress resultants are defined only at the force points. It is also apparent that concen- trated external loads must be applied at mass points. As a consequence of the lumping, one might expect that the displacements, moments, and shears essentially represent the values of the corresponding quantities in the continuum, averaged over appropriate lengths. Relations regarding boundary conditions between the model and continuum will be dealt with in Section 2. 4. The equations governing the motion of a discrete beam system are written in the same way as for a continuous beam. However, the infinitesimal increment along the beam is replaced by a segment of finite length, hi' Writing the equation of motion in the vertical direction for a typical panel 1, (Figure 2. 5) one obtains, / 15 Similarly, the equation for rotatory motion becomes 824). h. 1 - - _1_ (pIh)i at2 " M1 M1+1+ 2 (51+Si+ 11 2.9 If elasticity prevails, the constitutive equations are, in discrete form *2EIai Mi = ET'TT— 2°10 1 1-1 and s = Gk'As ———Z——— 211 1 i hi+hi_1 ' where 0’1 and Bi denote, reSpectively, the deformations of the moment and shear springs. They are related to yi and (bi by geometry and the relations are given in Section 3. 2. When plastic yielding is considered, Equations 2. 6 and 2. 7 are applied to the springs of the discrete beam. At a force point, the yield curve is given by f(M,,S.) = O 2.12 1 1 This is similar to Equation 2. 6 which pertains to a cross-section of the continuum. Since the curvature and the shear slide are lumped at the force points, Equation 2. 7 can be generalized for a typical force point as 3f(Mi, Si) 3f(Mi, Si) 3Mi BSi magi ’ 8113p)i " (K1). 2'13 T T in which (up)i and (13p)i are the plastic components of o.i and [ii , respectively. It must be pointed out that the governing equations of the discrete model are spacewise discrete, but the linear and angular accelerations, yi and (Iii , are continuous in the time dimension. Therefore, temporally continuous solutions of these equations could be sought. However, these equations, as discussed later, will be integrated herein numerically. 2. 3. Boundafl Conditions In the use of the model, supports will be made to coincide with force points. However, in the prototype (continuous) problem, boundary conditions are not always given in terms of stress resultants. Thus, whenever a boundary condition is given in terms of displacement in the prototype problem, it is necessary to interpret this condition as a stress resultant condition for the model. Since the stress resultants are directly related to the deformations of the moment and shear Springs, the prototype .4 I....I.'. 153.1. I: 17 displacement conditions must be used to compute these deforma- tions. As will be shown later, this can be done through geometry. But first, the following relations at a boundary point should be noted. singly. 2 3y _ Z shear [Y1 model ’ [Y + f0 5; ] continuum 2' 14 11. 53¢) = 2 [CID] model [CID + f0 5'; dx] continuum 2. 15 If y = O for the continuous case, Equation 2. 14 reduces to h 8yshear = E 6 [Y] model [ f0 5x dx] continuum 2' 1' If (I) = O for the continuum, Equation 2. 15 will yield [<1] = [ IE ggdx] 2.17 continuum The usual types of boundary conditions can now be taken Simple Supports: The condition that moment vanishes is straightforward and needs no further remark. The other condition of vanishing (total) displacement for the prototype implies, by Equation 2. 16, that the model will have some shear deformation at the support representing the shear deformation of the continuum l8 lumped from a tributary length of h/2 adjacent to the support. For the model, the value of this shear deformation can be determined from considerations of geometry. If j (j=N+l) denotes a support at the right end of a beam, then the lumped shear deformations at the two ends of a Simply supported beam are given by Knowing the deformation (or its increment), the shear force (or its increment) at the support can be computed. M Supports: In this case, Equation 2. 17 gives the fixed end slope of a model as the lumped curvature due to bending, contributed by a length of h/2 adjacent to the support. From geometry, the rotations of the moment springs at the two fixed supports are given by a. :—¢ 2. 19 Knowing the rotation (or its increment ), the moment (or its increment) at the support can be computed. Shears at fixed ends are found in precisely the same way as for simple supports. mm: Since both boundary conditions correspond to specifying stress resultants--moment and shear vanish--no special interpretations are necessary. III. METHOD OF NUMERICAL SOLUTION 3. 1. General The method essentially consists of a step-by-step numerical integration of the system of equations presented in the foregoing chapter. The problem here may be formulated as follows: at some given time t=to, the system is known to be in an elastic state, and the values of all the displacements yi:yi(to) (and (bi = (bi (to)) and their first derivatives vine) (and ((51 (to)) are known. Furthermore, the external loading Pi(t) is completely prescribed. It is required to determine the displacements and stress resultants at time tl=to+ At, where At is a small time increment. For easy reference, the model presented earlier which contains the mechanism of shear deformation and rotatory inertia will be referred to as the "Timoshenko" model. Obviously, the model can be reduced to lesser forms. Thus, the ”Shear" model refers to the case in which shear deformation is considered but rotatory inertia is neglected. The "Rotary” model refers to the case in which rotatory inertia is considered but shear deformation is neglected. Finally, when both effects are neglected, the model reduces to the ”Euler" model. The symbols T(N) will be used to denote: "Timoshenko model; beam divided into N equal panels. " The symbols S(N), l9 20 R(N), and E(N) are defined in a similar fashion. The analytical solution of the continuum case will be referred to as the ”Exact" solution. The term "continuous beam” will be used to mean the continuum case; it does not refer to a multi-span beam structure, which is not treated here. Theoretically, it is possible to obtain solutions of the "Shear, " Rotary, ” and "Euler" models directly from the "Timoshenko" model. A solution of the ”Shear" model could be obtained if the shear stiffness is taken as infinity. However, since this is not practical for numerical work, C may be assumed to be very large, thus letting the ”Timoshenko" model approach the "Shear" model. Similarly, if the rotatory inertia term is taken close to zero, the "Rotary" model could be approached. By simultaneously using a large value for G and a small value for (pIh) the ”Euler" model can be approached. This approach was not used here mainly due to technical difficulties. First, it is not clear how big or how small G and (p Ih) have to be in order to lead to satisfactory results; and secondly, the time increment needed for stability in the numerical integration has proven to be very sensitive to changes of these quantities (G and p Ih I. In one instance, approaching the "ShearH model from the "Timoshenko" model required a time increment almost 10 times smaller than the ”Shear” model requires. 21 In order to avoid these difficulties, it was found more convenient to deve10p the ”Shear, ” ”Rotary, " and ”Euler" models individually even though the numerical procedure of solution differs slightly from one model to another. The procedure of solution for the different models will thus be described separately in the following. It should be noted that while the procedure is given for the "Timoshenko” model for both the elastic and the plastic range, for the other models, only the elastic case is considered. 3. 2. The ”Timoshenko” Model Elastic RanLe 1. From geometry (see Figure 2. 4), the initial (t=to) deformations of the shear and moment springs can be computed from known initial displacements (assuming 4) to be sufficiently small): ciao) = w 1 - «1,4101 3.1 l 0 two) = yi> 3.2 2. So long as elasticity prevails, one can compute the moments Mi(to) and shears Si(to) from Equations 2. 10 and 2. 11. 3. Knowing Mi(t0), Si(to)' and the loading Pi(to), the accelerations y'i(to), and 5p.i(to) can be computed from Equations 2. 8 and 2. 9, respectively. Thus, all quantities that enter into the problem are known for t=to. 4. The changes in the displacements at t=t1 can now be determined by a forward numerical integration procedure. The 22 formula used in this thesis is: o l 2 vial) = yiuo) + (tl-to)yi(to) + EItrto) Vino) 3. 3 and similarly for ¢i(tl) (see Reference 32). 5. Knowing the displacements, one goes through the same procedure as outlined in steps (1) through (3) and obtains yi(t1) and ¢1“1L 6. The velocities at time t1 can be computed by a numerical integration. The formula used here is '(t)—'(.)+l(t t)<"(t)+”(t)) 34 Yil_yio 21'oyio yi1 ‘ and similarly for ‘I’iItlI’ 7. Thus, one is ready to repeat the process to solve for the re5ponse at t2=t1+ At, etc. Elastic-Plastic Range Successive applications of the cycle of integration in the elastic range will, at some time, yield values of M and S that violate the plastic potential theory; that is to say, f(M, S) will become positive. Smaller time increments will then be tried until a time To is found when f(M, S) = 0 (within a prescribed degree of accuracy). The numerical method, representing a finite incremental form of the plastic potential theory, is given below for one step of integration in the plastic region. 23 Let plasticity start at time ti: To. Since Equations 3. 1 through 3. 3 apply, the increments A0. and A13 are known for the next time increment A71 ; however, the material property Equations 2. 10 and 2. 11 can not be applied to the whole of Au. and M3. There- fore, the problem becomes one of splitting A0- and 136 into elastic and plastic portions, of which Age and Afie (the elastic portions of Au and A13 ) will produce changes in the stress resultants, but the plastic portions, Aap and App do not affect the stress resultants. A graphical representation of the necessary technique is given in Figure 3. 1 for a typical force point. The yield curve is not specified, indicating that the method is general, and it applies to any plastic potential function. In Figure 3. 1 the ordinate is assigned the dual scales of m (the non-dimensional moment M/My) and AE(the non-dimensional angle change AQ/Gy , where ay is the change of rotation corresponding to a totally elastic change of moment AM=My at the particular force point). Similarly, the abscissa is assigned 5 (s=S/Sy) and AE(A3=AB/(3y 1 where fly is the shear slide corresponding to a totally elastic change of shear AS=S at the force point). It is Y important to note that, this scaling has made possible a direct graphical correspondence betwen AM and Ade , and similarly for shear. 24 Let A (Figure 3. 1) be the position of the stress state at time To . The vector d (with components A; , AE- ) is laid on the graph with its tail at A (AD). Next, a circle is constructed with AD as a diameter to intersect the yield curve at B (and of course, A). Then, Bis the stress state at time To + A'rl . Moments and shears are read directly from the graph at point B. Furthermore, the components of AB on the Ad and A3 scales are the elastic parts of these quantities. The components of BD (the perpendicular to AB) are the plastic parts of A; and A3 . The validity of the preceding statements is explained in the following. There are three conditions to be satisfied by the division of AG- and AB into the elastic and plastic parts. First, A0. = Ade + A0. A = + :5 Ase Asp secondly, the new stress state at time 7°+A71 computed from Ace and Ape (using Equations 2. 10 and 2. 11) must satisfy the yield condition, i. e. . Equation 2. 12; and thirdly, the plastic flow rate vector must be normal to the yield curve (Equation 2. 13). The first condition is satisfied here from simple geometry considerations. The second one is obviously satisfied since point B is on the yield curve. As for the third ., since Figure 3. 1 is constructed for a very small time interval, the vector quantities Adp and AEP essentially represent the rates of the plastic 25 deformations. Therefore, the flow rule of Equation 2. 13 is satisfied on a finite increment basis. When the vector AD points inward, elasticity is resumed and Equations 2. 10 and 2. 11 are validated for the whole of A0, and AB . It must be noted that, while one or more force points may go plastic and require the treatment described above, other points that remain elastic will, or course, be handled according to the elastic rules. Conceptually, the graphical procedure outlined above is simple and straightforward. However, the programming of it on a computer, though feasible, is not convenient. Therefore, to facilitate programming, the further assumptions are made that the arc AB can be approximated by a circle whose curvature, and center of curvature are those of the actual yield curve at point A, as shown in Figure 3. 1. Since the length AD can be controlled by the size of AT, the error introduced due to the above approximation can be kept as small as needed by using a sufficiently small A7. The method is, in general, consistent with the forward integration method used throughout this study. Note that, so far as the flow rule of Equation 2. 13 is concerned, the procedure incurs no error except that of approximating an arc by a chord, which is inherent in the numerical integration method. 26 3. 3. TE "Shear" M The "Shear" model differs from the "Timoshenko" model in that the rotatory inertia is neglected. Thus Equation 2. 9 assumes the form h. 0=M.-M. +7‘(si+s. 3.6 1 1+1 1+1) In case of the "Timoshenko" model both yi and (pi are independent quantities. For the "Shear" model they are not independent and the relationship is to be obtained as follows: a) By substituting (3 i's from Equations 3. 2, Equations 2. 11 are written out for Si in terms of Y1 and ¢i . b) Equations 2. 10 are written out for Mi in terms of 4’1 . c) The expressions for shear from (a) and the expressions for moment from (b) are then related by Equation 3. 6. Thus the following relationship between (pi and Vi is obtained (taking hi=h for all i): th (b)<1>1 + (-1 +—2) <1», = K2Y2+K2Y1 th th th ('1 "I" _2 )(IZ‘1+(2 + —2 I‘I’Z +('1+ —2 N53 = sz3-K2Y1 th ('1 +‘ Z N’n-l +(b) ¢n = - KZYn-l—szn 3. 7 _ k'AGh _ . . where KZ -—TEI— , b depends on the boundary cond1t1ons, for 27 Simple supports b=l+l. 5K2h, for fixed supports b=3+,.l.5K2h. The second equation from the top is typical. Except for the step that dai's are obtained from Equation 3. 7 (instead of by numerical integration) the numerical procedure for the "Shear" model is the same as the one outlined in Section 3. 2 for the "Timoshenko” model. 3. 4. The ”Rotary" and "Euler" Models The "Rotary" model differs from the "Timoshenko" model in that the shear deformation is neglected. Hence, the displacement of the "Rotary" model does not exhibit any jumps due to shear. Consequently, for a beam with N panels, there can be only N-l independent displacements yi. Similarly, all ¢i's are fixed by geometry once the yi's are determined. The supplementary geometry equations may be taken as, yn=yn_1-yn_z+yn_3-... 3.8 and, by setting (3i: 0 in Equation 3. 2 and solving, one obtains, _ l ¢i — E(Zyi’_ 4yi_l+4yi_2....) 3. 9 ¢n = — T The numerical procedure for this case differs from that for the ”Timoshenko" model in another aspect. Since (31 = 0 and G =0 Equation 2. 11 can no longer be applied to calculate the shears which are now governed only by the kinetic equations. 28 Using Equations 3. 8 and 3. 9, Equations 2. 8 and 2. 9 can be combined to eliminate all the second time derivatives so that the shears S 1, . . . Sn are related only to the bending moments and the external loads. (Note that the moments are computed from Equation 2. 10 as previously.) The resulting system of equations appears as, l+K3 l-K'3 0 0 S1 Cl—I -2K3 1+3K3 l-K3 0 0 S2 C2 2K3 -4K3 l+3K3 l-K3 0 0 -2K3 +4K3 -4K3 1+3K3 l-K3 0 . = _:I:(1+K3) -2-2K3 2+2K3 -2—2K3 3+K§J Sn Cn ‘ I. .1 _ L where K3 = 4:121) by taking equal hi's. A typical Ci is given 3' 10 by 2 Ci — K3P.1 - 2K3Pi-l + 2K3Pi-2 - . . . - E(Mi - Mi+l) but 2 Cn — Pn-(1+K3)Pn_1+(l+K3)Pn_z-... -E(Mn-Mj) The set of simultaneous Equations 3. 10 are solved to obtain the shears. Equation 2. 7. The shear at the last force point, Sj’ is obtained from The shears can now be used in Equation 2. 8 to calculate Vi which are, in turn, used to compute the displacements I III. IrrvII. ,IIIII0.IrI—I.Iq I)... . ... Pal 51':in lullIIvIl 11,1 . 29 Vi for the next time interval. The cycle is then completed. The ”Euler" model differs from the "Rotary" model in that the rotatory inertia is neglected. The numerical procedure is the same as that for the "Rotary" model except that, in arriving at Equation 3. 10, Equation 3. 6 is us ed instead of Equation 2. 9. 3. 5. Time Increment Mathematically, the method used here is analogous to a numerical integration of a system of partial differential equations which are, in general, nonlinear. The size of the time increment for each step of integration obviously plays a dominant role. Unfortunately, there seems to be no rigorous method in existence of estimating the apprOpriate values of At to use. A trial method, therefore, was used to determine a satisfactory time increment. The following is a conservative listing of the time increments in terms of T1, which is numerically equal to the fundamental period of vibration of a simply supported, elastic continuous Euler beam. a) "Timoshenko" and ”Shear" models: T At = g- —12 for Issections and all values of N; for N > 20, N At can be based on N = 20. r-l At = 1— 1 for rectangular cross-sections and all values 10 N2 of N. b) ”Rotary'l and ”Euler" models: 3O _1— T T _ l _ l . . At - 10 .1? or At — 1:1—3 whichever 13 smaller. It must be emphasized that the preceding is based on the numerical experience of the present study which has dealt with a rather simple kind of loading. The above listing may not apply if the loading is substantially different. 3. 6. Use _cg the Computer The computer work for this study was conducted on the digital computer CDC3600 at Michigan State University. The Fortran language was used. It is found that the time of comPilation, loading, etc. , is about one minute. For N = 10 and for a length of time t = 3T1 the execution of the program takes somewhat less than a minute. For the problems solved, the maxima are usually reached at t < 0. 5T1 (but it may be necessary to carry the solution as far as t = 3T1 in order to estimate the permanent set). For larger values of N, the time needed is slightly more than that calculated according to the square of the ratios of N. The time corre3ponding to the "Shear, ” ”Rotary, " and ”Euler" models can be estimated using the information given in Section 3. 4. I‘II I I 1 I I I I I I I I IV. RESULTS IN THE ELASTIC RANGE 4. 1. Introduction The results presented in this chapter deal exclusively with response in the elastic range. The convergence of mid-span moments, end shears, and maximum deflections are shown for the four models considered here. Also investigated is the relative importance of shear and rotatory inertia in elastic vibrations. According to the notations in Figure 4. l the example beam has the following cross-sectional dimensions: B = 5", H = 6", ?w = o. 0346, T = o. 577. The beam is simply supported at both ends, and divided into panels of equal length. Except for the data presented in Section 4. 4, the length of the beam is 10 ft. A blast type loading, applied uniformly on the beam, is given by the expression -2t/Tl w(x, t) = cW e 4.1 where c is a parameter representing the load intensity, and W = 8My/ L2 is the load necessary to cause yield in the mid-span of the beam. 4.2. Convergence ELIE "Euler" M1 The "Euler" model is the simplest of all the models studied. 31 32 In fact, it can be considered to be a special case of each of the other models. For this model, an exact analytical solution is obtainable, and can rightly be regarded as the true limit to which the discrete model should converge. Strictly speaking, the convergence to the true limit of a special case does not necessarily imply similar convergence of more general cases. However, it should certainly strengthen the case for the other models (for which there are no exact analytical solutions available for comparison). It is essentially in this regard that the study of the convergence of the ”Euler" model is significant for the purpose of the present work. The "Exact” solutions (see Appendix), as well as the ”Euler" model solutions for various values of N are plotted in Figures 4. 2, 4. 3, 4. 4 for the center deflection, center moment, and end shear, respectively.(The deflections are scaled by Yy, the maximum static deflection. ). For more precise comparisons, the values of the maximum responses and their times of occurrence are also noted in the figures. As expected, the deflection converges to the "Exact" solution considerably faster than the moment and the Shear. For N = 21, the deflection is so close to the exact solution that they are represented by a single curve in Figure 4. 2. (That odd number of panels is necessary for mid-span deflection is a consequence of the , I‘Mfl...“ ,nnN‘flzv Ivil'vlbn‘...‘ 1’ ‘ 0'1” 0 Ip 33 c0nstruction of the model: the deflections are defined only at mass points, and the supports are made to coincide with force points; see Chapter II. ) For N = 20 the moment values still have noticeable differences from the ”Exact" curve at some intervals. Although the exact maximum moment is approximated well even by using the small value of N = 4, it is important to note that the comparison should be viewed for the entire range of response. Thus, it is seen that overall agreement increases with larger values of N. The preceding is also generally true for end shear. In order not to clutter the illustration, shear responses are shown in Figure 4. 4.only for N = 10 and the exact solution. The agreement is seen to be good, and it improves with larger values of N (not shown). 4. 3. "Apparent" Convergence _cgth_e ”Rotary, " "Shear, " fl ”Timoshenko" Models Since exact analytical solutions for the limiting cases (N = cl3) of these models are not available, the "convergence'l is considered by comparing numerical results using different values of N. Results for the "Rotary" model are shown in Figures 4. 5, 4. 6, and 4. 7 for deflections, moments, and shears. Results for the "Shear'l model are shown in Figures 4. 8, 4. 9, and 4. 10 for the same quantities. Data illustrating the apparent convergence of the "Timoshenko" model are presented in Figures 4. ll, 4. 12, and 4. l3, . - ' I, 1‘ . ‘ . ‘ -’ "t ' $- s ‘e '7’ 34 respectively, for deflection, moment. and shear. From an examination of all these data it may be reaSOnably concluded that the results converge to some limit, and are trustworthy. From a comparison of the shear reSpOnses by the ”Rotary" and "Shear" models (see Figures 4. 7 and 4. 10) it is of interest to note that by considering shear deformations the shear response becomes appreciably smoother. 4. 4. Relative Importance ng_he_a_r_a£d Rotatory Inertia A comparison of the moment and shear response curves presented in the preceding sections will show that neither the maximum moment nor the maximum shear varies appreciably from one model to the other. Particularly, the maximum stresses shown by the "Rotary" model and the ”Euler" model are well within 1% of each other, whereas the "Shear" and "Timoshenko" models exhibit almost identical behavior in all cases. This is illustrated, for the moment, in Figure 4. 14. The differences between the "Rotary” and "Euler" models or between the ”Timoshenko" and ”Shear" models represent the influence of the rotatory inertia. It is seen that this influence is small indeed. The differences between the "Shear" and ”Euler" models or between the ”Timoshenko" and ”Rotary" models represent the influence of shear deformations. This influence is seen to be appreciable. 35 The preceding data pertain to a beam with a 10 ft. span. Additional data for Shorter Spans are presented in Figure 4. 15 for considerations of the effects of rotatory inertia and shear deformations. All curves plotted in the figures represent the differences between the results given by the discrete models (N = 10) and those correSponding to an exact solution of the continuous Euler beam. As expected, the influence of shear deformations on deflections is to increase the deflections with decreasing Span length. The effect of the shear deformations is to decrease both the midspan moment and end shear. For moment, the reduction stays essentially constant for span length greater than 6 ft. For shorter spans, the reduction increases. The shear reduction exhibits an oscillatory pattern. The differences between the pairs of graphs are obviously due to rotatory inertia. It is seen that the qualitative effects of rotatory inertia are to decrease the deflection and the moment, but to increase the end shear. However, the magnitudes of these effects are very small. In addition to the preceding, a number of beams with different cross-sectional prOperties and lengths were also solved to see whether the contributions due to rotatory inertia could be significant. It was found that neither the stresses nor deflections were affected appreciably by rotatory inertia. Hence, the data are not presented herein. V. ELASTIC-PLASTIC BEHAVIOR OF THE "TIMOSHENKO'l MODEL 5. 1. Introduction In this chapter, the apparent convergence of the elastic-plastic response of the "Timoshenko" model is considered first. Then, the inelastic behavior of simply supported and fixed-fixed I-beams is studied. The parameters considered are the web thickness and the span length. The variables are'the maximum deflection, permanent set (permanent total deflection), and permanent shear slide. Except for varying length or web thickness, the beams analyzed in this chapter are generally the same as described in Section 4. l. The ranges of the parameters are, in the notation of Figure 4.1, 0.2 > t_w >0. 025, and 6' < L < 20'. The loading used is the same as given by Equation 4. 1. How- ever, the parameter c is set equal to unity in order to carry the problem into the plastic range. Furthermore, after all the maximum reaponses have been reached, and elasticity resumed, the external loading is removed in order to obtain the permanent set from the subsequent free vibration. As described in Chapter II, for an analysis in the plastic range, it is necessary to define a yield or interaction curve. (This cross-sectional prOperty will be generalized for the discrete model 36 37 as a discrete joint prOperty. ) Though a considerable amount of theoretical work has been reported on this subject (13, 19, 25, 29, 30, 31), experimental data is very scarce (9, 15). For the numerical work here, two yield curves will be used: mlz + e2 = 1 5.1 m2 + s‘2 = l 5. 2 where m and s are, respectively, the non-dimensional moment M/My, and shear S/Sy. The moment capacity, M , is computed Y for the cross-section by assuming that the whole section has yielded at 433000 psi. The shear capacity, Sy, is obtained as k'A (the "active shear area") times the shear yield stress, taken to be 18000 psi. The relationship of Equation 5. 1 may be regarded as a good approximation to the actual behavior of I-sections, and agrees well with certain relationships which were formulated with some experimental basis (9, 15). On the other hand, Equation 5. 2 under- estimates the strength of the section; thus, it should be considered to be a lower bound. Figure 5. 1 illustrates the interaction relation- ships of Equations 5. 1, 5. 2, and the one given in Reference 9. 5. 2. Convergence _ci the ”Timoshenko" Model 3291—9 Elastic-Plastic Range In Chapter IV the convergence of the "Timoshenko" model in the elastic range was shown. Here, an elastic-plastic example is 38 treated. The beam is a 12WF53 (I-beam), 10 ft. IOng, and fixed at the ends. The yield criterion of Equation 5. 2 is assumed. As before, the deflection converges very rapidly, therefore, it is not presented. The moment and the shear at the support are plotted, respectively, in Figures 5. 2 and 5. 3 for several values of N. In the case of the moment, N = 20 and N = 40 give results that are very close to each other. As for the shear, one may note that the convergence is even more satisfactory than for the elastic case! (Compare Figures 4. 10 and 5. 3. ) (Seemingly, plastic yielding serves to attenuate the higher modes' effects. ) It is of interest to trace the locus of the stress state of the elastic-plastic response. This is shown in Figure 5. 4. Numerals on the locus correspond to those times similarly noted in Figures 5. 2 and 5. 3. Of course, the ranges between points (3), (4), and (5) are plastic, while the rest of the locus is elastic. 5. 3. Response o_f_ Simply Supported I-Beams fl Different Length! E m Thicknesses It is reasonable to expect that, for shorter beams, yielding would start first at the supports, then spread toward the mid-span. For longer beams, yielding would start at the mid-span, then spread outward. This behavior is illustrated in Figure 5. 5 for beams with 39 constant non-dimensional web thickness t—W = 0. 0346 but with spans varied from 6 ft upwards. The yield curve of Equation 5. 2 is used. Further dividing lines are placed in this figure to show the degree of interaction. (For instance, the region between the 80% and 95% lines, has yielded at values of maximum shear that are only 80% to 95% of the shear capacity. This is, of course, caused by the small amount of moment present in this region.) In Figure 5. 6 are plotted, for the same parameters as above, the maximum deflection, the permanent set, and the permanent slide. Full lines correSpond to the data obtained by use of Equation 5. 2 as the yield curve; dotted lines correspond to Equation 5. 1. It is noted that the displacements are scaled by the maximum (elastic) static deflection. For the present set of parameters, the elastic-plastic "Euler" model gives the maximum non-dimensional deflection as a constant equal to l. 56 for all span lengths. (Indeed, the choice of the external loading and the scaling of the deflections are reSponsible for this constancy.) Therefore, it is apparent that the difference between the maximum deflection curves and the constant ”Euler" solution is entirely due to the effects being considered here: rotatory inertia, and shear. (The latter, of course, is mostly responsible for the difference.) For longer beams, the maximum deflection curves are seen to approach the ”Euler" case, indicating that shear and rotatory inertia effects become less important. 40 A similar observation can be made for the permanent set curve. Although the numerical data from the "Euler” solution have not been obtained, it is seen that for longer beams increasing the length did not change the essentially constant behavior of the sets. Also, as expected, the permanent slide converges to zero as the beam length is increased. On the other hand, as the beam becomes shorter, the permanent slide constitutes a higher portion of the permanent set. It is seen that the two yield criteria used give results that show little difference from each other. This can be explained by the fact that, for simply supported beams, yielding is dominated by either moment (mid-span region) or shear (support region). For these stress conditions the two yield curves used are quite close (see Figure 5. 1). Figure 5. 7 shows a set of curves similar to those just discussed. In this case, the beam length is held constant at 10 ft, and the web thickness is varied. It is seen that these curves have shapes similar to those in the previous figure. Recognizing that, so far as shear effects are concerned, a decrease in span length has the same qualitative effect as a decrease in web thickness, one can make observations about these data analogous to those made in connection with the preceding figure. It may be noted that, in this case, most structural I-beams, except for those with very thin webs, 41 are not greatly affected by shear, if they are simply supported. In the following section, it is shown that such is not the case for fixed-fixed beams. 5. 4. ResEonse 2f Fixed-Fixed I-Beams with Different Lengths and Web Thicknes 3 es A feature of fixed-fixed beams is that, at the supports both moment and shear can be large and thus strong interaction would take place in the plastic response. The moment and shear forces at the fixed end are shown in Figure 5. 8 for an I-beam with t—W = 0. 0346 and a varying length (as marked on the curves). The yield curve of Equation 5. 2 is used. It is seen that for longer beams yielding takes place due to relatively higher values of the moment; for shorter beams yielding is mostly due to shear. This is further illustrated in Figure 5. 9 which shows the maximum deflections, permanent sets, and permanent slides. Unlike simply supported beams (see Figure 5. 6), the results corresponding to the two interaction curves are quite distinct. This can be explained by referring to Figure 5. 1 and noting that there is an appreciable difference between the two yield curves in the region where both moment and shear play a substantial role. In Figure 5. 10 the maximum deflection, permanent set, and permanent slides are presented for a constant length of 10 ft and 42 varying web thickness. Together with the results shown in Figure 5. 9, the graphs in Figure 5. 10 indicate that, similar to the case of the simply supported beams, as the Span length or web thickness is increased, the reaponse curves seem to level off, indicating a decrease in the shear (and some rotatory inertia) effects and an approach to the "Euler" case. On the other hand, Figure 5. 10 indicates that at a length of 10 ft, most I-beams are very sensitive to shear effects. VI. C ONC LUSION In this thesis a method of analysis of the elastic-plastic vibrations of beams has been presented. The analysis has included the effects of shear deformations, rotatory inertia, as well as the interaction of moment and shear forces on the yield behavior. The method employs a discrete physical model. This, together with the use of a numerical procedure, makes it possible to handle beams with different loading and boundary conditions which have, in general, limited the practicality of the continuum approach to this type of problems. In the absence of exact analytical solutions, the reliability of the model is established essentially empirically by the ”apparent convergence" of the deflections, moments, and shears, as the beam is divided into larger numbers of panels. (An exception is the ”Euler" model in the elastic range, which has yielded solutions that converge to an exact analytical solution. ) Extensive numerical results have been obtained to study the influence of the web thickness and span length of I-beams on the relative importance of shear deformation. It is shown that as the length (or web thickness) is increased, results given by the ”Timoshenko'l model converge to that of the (elastic-plastic) ”Euler" 43 44 model. As expected, this analysis indicates that shear and rotatory inertia effects become negligible for longer (or thicker webbed) beams. However, it is also shown that the shear effects are substantial, and hence should be considered, even for beams of quite usual proportions. For example, for a 10 ft. long simple span (typical I-section), the permanent shear slide accounts for 35% of the total permanent displacement. The percentage could be much higher, depending on the form of the interaction curve, if the beam is fixed at the ends. It is further indicated that the interactions between shear and moment on the yield behavior play a significant role in the inelastic response. This is particularly true for beams with fixed ends. From experience gained in the numerical work of this investigation, the following observation regarding the relative merits of the different models considered herein is noteworthy: The efficiency of a model seems to improve as its number of degrees of freedom (for equal values of N) is increased. Thus, the "Timoshenko" model is seen to converge the fastest, that is, it yields sufficiently accurate results for lower values of N and for larger values of the time increment! 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"On the Dynamic Strength of Rigid-Plastic Beams Under Blast Loads, " Journal of the Engineering Mechanics Division, Proc. of the ASCE, Vol. 83, No. EM4, October 1957. Snowden, J. C. "Response of a Simply Clamped Beam to Vibratory Forces and Moments, " The Journal of the Acoustical Society of America, Vol. 36, No. 3, March 1964. . Szidarovszky, J. "Natural Vibration of a Bar under Axial Force, Taking into Consideration the Effect of Shearing Force and Rotatory Inertia, " Acta Tech. Acad. Sci. Hungaricae, Budapest, 39 1/2, 1962. Timoshenko, S. "Vibration Problems in Engineering, " D. Van Nostrand Co. , N. Y. , 1928. Toridis, T. G. "Dynamic Behavior of Elasto-Inelastic Beams Subjected to Moving Loads, ” Doctoral Dissertation, Civil Engineering Department, Michigan State University, September 1964. Tseitlin, A. I. "On the Effect of Shear Deformation and Rotatory Inertia in Vibrations of Beams on Elastic Foundations, " Applied Mathematics and Mathematics, 25, 2, 1961. 44. 45. 50 Tung, T. P. and Newmark, N. M. "A Review of Numerical Integration Methods for Dynamic Re3ponse of Structures, " Civil Engineering Studies, Structural Research Series, No. 69, University of Illinois, Urbana, Illinois, March 1954. Wen, R. K. and Toridis, T. "Discrete Dynamic Models for Elasto-Inelastic Beams, " Proc. of the ASCE, Journal of the Engineering Mechanics Division, Vol. 90, No. EMS, Part 1, October 1964. IY Figure 2.1. Figure 2. 2. (S. M) I l I +X I I . I L_ de——~' pI— 32.—.1 Mcs 111—... I . 1 Forces Acting on an Element of Continuum. f(S, M) = f(S, M) < 0 Plastic Potential Function. 52 Lumpe d Rotato ry Ine rtia Linear Spring (for shear) Lum ed Mass 5 + —|— Moment Spring (for bending) Mas sless Rigid Bars Figure 2. 3. Discrete Beam Model I hi/2 I hi/2 I h1+1 I I I 1... +x i+1’%+l +y1+1 “1+1 \ \ J + 6- HI + D—l S i h Segment i+l t Segment —-———---_—-_ I Segment +Y Figure 2. 4. Deformed Configuration of the Discrete Beam. 53 Figure 2. 5. Forces Acting on a Typical Panel of the Discrete Beam. G ene ral Inte raction Cu rve Radius of Curvature at A k?» Center of Curvature for ‘ III D ‘CD Figure 3. 1. Figure 4. 1. [> Drl‘ an an; Finite Increment Treatment of Plastic Yielding. I111." L__ t w =2—B 't' w Cross -Sectional PrOperties. Non-Dimensional Deflection ; (y/Yy) 54 N Max. Deflection At Time . 6 *‘ Exact . 668 . 4325 5 .659 . 4280 -—---‘. 11 . 667 . 4289 /’ “q ,/ ‘\.\ I \\ _ \ 5 / 13(5) / \\ , \ \ _LLLLL/\ Exact. E(Zl) . 4 " 1 l l I I .1 .2 .3 .4 .5 Non-Dimensional Time F (t/Tl) Figure 4. 2. Convergence of the "Euler" Model--Def1ection at Mid-span. . T b ‘.I . I. I . . // . . ,‘f ' '5 . - ‘_ .'_~_ . . ., ~ . _ a ' .. .. . 1: _ 7‘ __ _._ 4‘ 55 N Max. Moment At Time E(ZO) - 7 - Exact . 6957 .4485 1, 4 . 6981 . 4125 13(4) ;I ‘~.\\ 10 .7023 .4330 \‘ / “\‘x v" \ \. /<’/ \‘x e 6 F i E 10) \\\ / \ 1’7 'I/ Exact 5 _ I // A>‘ l, 2 " E .0, .4 L I E / E, 'I Q) .I E l o .’ 2 / c .3” O '8 C Q) .E 1' e C . 2 I— '0’, O '1 z .' x/ .1 — ,I 0’” l/’ A ' I I I l l “v .1 .2 .3 .4 .5 Non-Dimensional Time t- (t/TII Figure 4. 3. Convergence of the ”Euler” Model--Moment at Mid-span. 56 “1-- ‘‘‘‘‘‘ \\\ \nl‘.‘ ’ -‘J \| ‘ 1 III\\ IIIII ’ I II t III ’ C ‘l‘l'-‘v a 7...: x 1 . I IV E ..... .1 fllltl "‘ a IIIII l lllllllll J A n" llllllll I . “"-"‘ fl IUII I ’ \l \ A I I, ’1’” ’I’ I \ ‘’ I 0 11:: "l’ 1 ‘1 \ E 111 fl II I |" 4”, II, I’, " ,\’ / "'l" ' u ”I ‘ \ E ‘v "“4 fl \'\ \ \ “II ’Ill"’ 7 6 5 4 3 Z 1 a» A m\mv m Assam dmzofimsofififlucoz Non-Dimensional Time t (t/Tl) Convergence of the ”Euler" Model--Shear at the Support Figure 4. 4. U'I Non-Dimensional Deflection V (y/Yy) L1 26 A N 57 N Max. Deflection At Time 11 .659 .433 21 .665 .437 31 .666 .437 //-—~\ / / / M31). RIZII \ / R 11 / / / / / / / / / 1 1 1 1 1 .1 .2 .3 .4 .5 Non-Dimensional Time 1:— It/Tl) Figure 4. 5. "Apparent" Convergence of the ”Rotary" Model-- Deflection at Mid - span. y) Non-Dimensional Moment m (M/M 58 N Max. Moment At Time R120) 7 10 . 6828 , 4000 A .. 20 . 6911 .4535 /.: .\ 30 .6983 .4471 \\ ,\ \ ._ \ // R(30) '/ 5 / / / — / / — / R 10 / / / / / 1 1 1 L 1 \J .1 .2 . 3 . 4 .5 Non-Dimensional Time t_ (t/Tl) Figure 4. 6. Moment at Mid - s pan. ”Apparent” Convergence of the "Rotary" Model-- Non-Dimensional Shear s (S/Sy) 59 N Max. Shear At Time .71— 10 .663 .453 1\ 20 .689 .420 V’u 30 .710 .401 , I, I ’\ I , \‘ .64— I, \ I, I V, 1‘ 1 , \ ’ I v / k I , R 20) I 12(30) ,’ .5” I I I I I .4- I I 'I / \ I \I III I .31— ’ I I I I I I I 02— /\ / ‘\ I .1” I L l l I .1 .2 .3 .4 .5 Figure 4. 7. Non-Dimensional Time F (t/Tl) ”Apparent" Convergence of the ”Rotary" Model-- Shear at the Support. Non-Dimensional Deflection V (y/Yy) U1 4:. 0 U) N 60 N Max. Deflection At Time Figure 4. 8. 11 . 7546 .4636 21 . 7555 .4723 — 31 . 7556 .4699 S(ll), S(Zl), S(31) l l I J I .1 .2 . 3 .4 .5 Non-Dimensional Time t— (t/Tl) "Apparent" Convergence of the "Shear Model-- Deflection at Mid-span. A-,H\4\w): J b . ~ ' ’1?" ' . g. 3:. . I _ . ‘. 3‘ -‘ 'T .. . a. I, ~ . ‘ __.....-_ 49.5"? .-'. .i;.,-=_~ _' . 1 I. a _y- 61 .7 .. N Max. Moment At Time 10 .6521 .5120 20 .6586 .5110 __.-. 30 .6613 .5082 ’ / 6_ / 520 / / s 10 ’3‘ 5 5 E h—- E ‘8 E -4” 530 O 2 7‘8 G O '6; c .3- Q) E 9 I: O z .2»- 01,—. 1 J I l I .1 .2 .3 .4 .5 Non-Dimensional Time t_ (t/Tl) Figure 4.9. ”Apparent" Convergence of the "Shear" Model-- Moment at Mid-span. Non-Dimensional Shear s (S/Sy) U1 4; W N 62 Figure 4.10. Non-Dimensional Time t_ (t/Tl) Shear at the Support. N Max. Shear At Time _ 10 . 7034 . 438 ,\ 20 .6955 .440 .’\.. 30 .6960 . 429 I \ / L ' x \ \ 5(20) 47/ L— s 10 .- . ~/ / ,/ _ I s 30 / l I / I / '- f. \/ x, 1 1 1 .1 . 3 .5 ”Apparent" Convergence of the ”Shear" Model-- (y/Yy) Non-Dimensional Deflection y 63 N Max. Deflection At Time 11 . 75254 . 4645 21 . 75272 . 4730 31 . 75272 . 4706 T(ll).T(21).T(31) l l l l . 1 . 2 . 3 . 4 . 5 Non-Dimensional Time t (t/Tl) H Figure 4. 11. ”Apparent" Convergence of the "Timoshenko Model--Def1ection at Mid-span. 64 N Max. Moment At Time -7 ” 10 .6501 .5190 20 .6590 .5175 30 .6615 .5156 x / ’ T(10)/ .6 — ’ A>~ E .5 4 E E 4.: T20 . T(30) C Q) .4 r— E o 2 '8 C.‘ .9. 2 .3 — a) E Q i C‘. 2’ 02 ‘h‘ .1 - _ I 1 1 I l .1 .2 .3 .4 .5 Non-Dimensional Time t— (t/Tl) Figure 4. 12. "Apparent" Convergence of the "Timoshenko Model--Moment at Mid-span. Non-Dimensional Shear s (S/Sy) 65 Figure 4. 13. N Max. Shear At Time 7 _. 10 .6935 .4420 20 . 6976 . 4428 ‘R‘\ 30 .6874 .4324 , \\ I '1 I "\ I) I \ .6 — / ’1. I \\ .’\ , \“ . 5 - 4,3. ‘ / // . ,- T 20 II '4 7 I T(30) I \ T510) . 3 - l / / 02 I'- / [’7’ 5 // 1 _ I l’ l L L l .1 . 2 . 3 . 4 . 5 Non-Dimensional Time t— (t/Tl) ”Apparent" Convergence of the "Timoshenko" Model --Shear at the Support. y) Non-Dimensional Moment m (M/M U'I uh 0 L» N ”Exact" R(30) S(30) T(30) Figure 4. 14. 66 Max. Moment At Time . 696 . 4485 / . 698 . 4471 /' \ .661 .5082 ’ 4.9% .662 . 5156 ”Are" 4 ‘\ ’7' ‘ \ 5 s 30 \ 1 1 1 1 .2 .3 .4 Non-Dimensional Time t— (t/Tl) Mid-Span Moment Respons es. x 100 Maximum Shear or Shear and Rotatory Inertia Contribution Maximum Euler Beam Response 100 80 60 40 20 -20 -16 -12 Figure 4. 15. 67 \§\ Mid-span Deflection Contribution of Shea r \ Contribution of Shear and Rotatory Inertia \ Mid-span Moment Contribution of Shear and Rotatory ner 1a End Shear Length in Feet Inertia in the Elastic Range. Contribution of Shear and Rotatory Inertia Influence of Shear, and Shear and Rotatory ) Y Non—Dimensional Moment m (M/M 68 Reference (9) l l l I l l I l .1 .2‘ .3 .4 .5 .6 .7 .8 .9 1.0 Non—Dimensional Shear s (S/Sy) Figure 5. 1. Shear-Moment Interaction Curves. y) (M/M Non -— Dimensional Moment m 69 / . / x I 13 //| FLA \STIC I i \ “40) / | RAN GE \ l/ \ I / \ \ f\\_ / l T 10 ‘ \ I ' ‘ / / / / . 2 2 / Yield Curve: 8 + m = l l l l J l .1 . 2 . 3 4 . 5 Non-Dimensional Time t— (t/Tl) Figure 5. 2. ”Apparent" Convergence of the "Timoshenko" Model in Elastic-Plastic Response -- Moment at Fixed End. Non-Dimensional Shear s (S/Sy) 70 Yield Curve: 32 + m l .4 .5 L Ci 0 Non -Dimens ional Time HIM)“- u/Tl) Figure 5. 3. "Apparent" Convergence of "Timoshenko" Model in Elastic-Plastic ReSponse -- Shear at Fixed End. Length in Feet 71 E Yield Curve: \\ E m2+SZ:1 \‘ E o .6- 2 3 8 o .4— E a) 6 5 g .2“ 2 7 X 9 . g . z 1 l a l 1 i .2 I’ .4 .6 .8 1.0 Non -Dirnensiona1 Shear s Figure 5. 4. Locus of Stress State for Problem in Figures 5. 2 and 5. 3. Yield Curve: 32 + m = 1 p—a U‘l ELASTIC ELASTIC PLASTIC \ 20% / 4 l l 307 50% \ I “""""' WWW mm<1mmm mmmmmm ELASTIC 95% Figure 5. 5. Regions of Plastic Response for Simply Supported I-Beams with Different Lengths. 2.5 '3. >~ \ 3; 2.0 |>~ G .9 ‘6 Q) t531.5 o 78’ c: O '5; C‘. $1.0 a I C.‘ O z .5 72 _. Yield Curve: 5 —— Yield Curve: 32 + m12 = 1 Maximum Deflection Pe rmanent Set b I— Length in Feet Figure 5. 6. Deflections, Permanent Sets, and Permanent Slides for Simply Supported I-Beams with Different Lengths. Non-Dimensional Deflection V (y/Yy) p—u o N U1 H O O ‘1 U1 0 U1 0 .25 73 Maximum Deflection Range of most _ FM? _ Yield Curve: s2 + m2 = 1 structura I-Beams --- Yield Curve: s2 + m12 = 1 Permanent Set (both yield curves) .2. fl". {\A Permanent Slide \ A L— A. 1 ‘4 . 05 . l . 15 . 2 Non-Dimensional Web Thickness Fw (tw/ZB) Figure 5. 7. Deflections, Permanent Sets, and Permanent Slides for Simply Supported I-Beams with Different Thicknesses. 74 A ( S/S) 3 1'9qu Teuoisuetun-uoN .msumnofl anonflflfl £33 madam-H mo momaoamom udoam pad «Goaoz paw-voxwh Zak: P055 32320529282 .m .m 6.33m ZHKS P685. HmcowmnoEHHIaoZ m. N. H. o v. . N. H. _ _ _ q u g N m \lsm.’ \/ . IV 44 / I \I / A e / ’ 7 s , 9 :0/47 uo>u90 Ugo“? w ( W/W) U1 mauiow reuoisuemtqhuoN A ) Y Non-Dimensional Deflection v (y/Y 75 10* __ Yield Curve: 3 +m - --- Yield Curve: 8 + m Maximum Deflection Permanent Set Permanent Slide 1 1 I 6 7 8 9 10 Length in Feet _ _ Figure 5. 9. Deflections, Permanent Sets, and Permanent Slides for Fixed-Fixed I-Beams with Different Lengths. (y/Yy) V Non-Dimensional Deflection 76 — Yield Curve: s2 + m = 1 --- Yield Curve: 5 +m Maximum Deflection Range of\\ most common Strictural I-Beams Permanent Set 45‘ A 0.5 .1 .15 .2 NOn-Dimensional Web Thickness t—w (tw/ZB) Figure 5. 10. Deflections, Permanent Sets, and Permanent Slides for Fixed-Fixed I-Beams with Different Web Thicknesses. APPENDIX "EXACT" ELASTIC SOLUTION OF EULER BEAM For a uniform, simply supported beam subjected to a uniformly . -2t T applied load cW e / l, the deflection is readily found to be -2t/T1 2 y(x t) = 1: 4W6 5111325 pie -picos(pit)+T1 sm(pit) ’ 1=1,3, . , "pA L ( ) . 2 4 l The bending moment M, and the shear S can be computed, respectively, from the second and third derivatives of the deflection function. If the loading W is written in terms of the yield moment (W = 8 My/ L2), the responses can be made non-dimensional as follows, WC t—) _ g 1536 sin in? (e-Zt— coszniZF+ sin21TiZt—) ' ‘ -_ —_2’4— ' "T—. 1—1, 3 5103(0 1 +1) 1n M8 -— Sizn3c — m(§nt): T Y(étt) p—a , 3 i: 3 3 m . M _ s16?) = 1223 %J M») where t—= t/Tl, 1’, = x/L. 77 78 It is of interest that through this non-dimensionalization process, no material and physical prOperties appear in the expressions for deflection and moment. (Indeed, the choice of W = 8 My/ L2 made this possible in the case of the moment.) It may be noted that all of these expressions are convergent. The deflection converges at 1/15, the moment converges as l/i3. and shear converges as l/iz. it'll 1| 1ICHIGAN STATE UNIV. LIBRARIES 1|HHWIWIIWIW1||||lllililllHIWIWWWWI 31293200871618