ENELASTIC RESPONSE OF A BEAM SUBJECT£D TO A COASTING LOAD Thesis for the Degree of M. S. MECHEGAN STATE UNIVERSITY F. FARHOOMAND 1958 mm LIBRARY Midis!!! Stats ”W ' 7 H 800K HINDERYWENU, 'IIDD' ABSTRACT INELASTIC RESPONSE OF A BEAM SUBJECTED TO A COASTING LOAD by F. Farhoomand In this thesis a numerical method for analyzing the dynamic response of a beam subjected to a coasting load is presented. The method is based on a discrete model with lumped mass and stiffness. The moment-curvature relation is of a general elastic-plastic-strain-hardening type with hysteretic behavior. Numerical solutions are obtained using an iterative procedure for a simply-supported slender beam subjected to a coasting mass load. The distinguishing feature of the present analysis lies in the treatment of the kinematics of the beam defor- mations. The analysis corresponds to a large-deflection theory. In comparison with available experimental data, the present analytical results indicate a better agreement than those for the small-deflection analysis. Certain parametric studies are also included in the thesis. It is found that for a load lighter than the ul- timate load there exists a finite initial speed which is F. Farhoomand most damaging to the beam. It is also found that a mass load heavier than the ultimate load can cross the beam, resulting in only moderate permanent deflections, if its initial speed is sufficiently large. INELASTIC RESPONSE OF A BEAM SUBJECTED TO A COASTING LOAD BY F. Farhoomand A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil Engineering 1968 ACKNOWLEDGMENTS The writer wishes to express his gratitude to his academic advisor, Dr. Robert K. Wen, Professor of Civil En- gineering, whose advice proved to be invaluable. The writer is also indebted to Dr. Charles E. Cutts, Chairman of the Department of Civil Engineering, for his encouragement. Sincere appreciation is extended to the National Science Foundation and to the Division of Engineering Re- search at Michigan State University, for their support of this investigation. ii TABLE OF CONTENTS Page ACKNOWLEDGMENTS O O O O O O O O O O O O O O O O O O i i LIST OF FIGURES o o o o o o o o o o o o o o o o o 0 V I. INTRODUCTION . . . . . . . . . . . . . . . . l 1.1. General 1.2. Scope 1.3. Notation II. METHOD OF ANALYSIS . . . . . . . . . . . . . 6 2.1. General 2.2. Discretization of Beam 2.3. Equations of Motion 2.4. Jumps in Velocities and Accelerations III. METHOD OF NUMERICAL SOLUTION . . . . . . . . 17 3.1. Dimensionless Form of Equations of Motion 3 2. Description of Parameters 3.3. Numerical Integration of Accelerations and Velocities 3.4. Numerical Solution of Equations of Motion 3.5. Use of Computer IV. COMPARISON OF ANALYTICAL AND EXPERIMENTAL RESULTS I O C C O O O O O O O O O O O O O O O 26 General Midspan Deflection versus Time Interactive Force versus Time Permanent Sets Maximum Midspan Deflection versus Speed mbWNH O Dohtbubnb iii Table of Contents (Continued) Page V. INFLUENCE OF PARAMETERS . . . . . . . . . . . 30 5.1. General 5.2. Influence of Load Weight 5.3. Influence of Beam Moment-Capacity VI. SUMMARY AND CONCLUSIONS . . . . . . . . . . . 34 LIST OF REFERENCES 0 O O O O O O O O O O O O 0 O O 36 FIGURES O O O O O O O O O O O O O O O O O O O O O O 37 APPENDIX-—COMPUTER PROGRAM 0 o o o o o o o o o o o 52 iv Figure 1. 9a. 9b. 10a. 10b. LIST OF FIGURES Initial State of Beam-Load System . . . Moment-Curvature Diagram . . . . . . . Free Body Diagrams . . . . . . . . . . Discrete Model . . . . . . . . . . . . Midspan Deflections versus Time . . . . Interactive Forces versus Time . . . . Permanent Sets . . . . . . . . . . . . Maximum Midspan Deflection versus Initial Speed (Analytical and Experimental) . . . . . . . . . . . . Maximum Midspan Deflection versus Initial Speed (Effect of Load Weight) Permanent Midspan Deflection versus Initial Speed (Effect of Load Weight) Maximum Midspan Deflection versus Initial Speed (Effect of Beam Moment Capacity) . . . . . . . . . . Permanent Midspan Deflection versus Initial Speed (Effect of Beam Moment Capacity) . . . . . . . . . . Page 37 38 39 40 41 43 46 47 48 49 50 51 I. INTRODUCTION 1.1. General In recent years the problem of moving loads on structures has been the subject of many theoretical and experimental investigations. Most of the works reported in the literature have been limited to the linearly elastic range of structural behavior. However, more recently there have been some studies that considered the inelastic range. The history of past work in the latter category was reported in Ref. 5. In that reference was also presented a method of analyzing the dynamic inelastic response of beams subjected to moving loads. The method was based on a bilin- ear type of moment-curvature relation and a small-deflection approximation, i.e., the angle of the slope at any point of the beam was approximated by its sine. The same problem was further considered in Ref. 3 in which both experimental and analytical results were pre- sented. The analysis utilized the same approach as in Ref. 5 except that a more general type of moment-curvature rela- tion was employed. Comparisons between the theoretical and experimental responses generally seemed satisfactory. HOW- ever, serious discrepancies emerged near the end of crossings 1 for those cases in which the beam suffered appreciable per- manent set. In such cases the analysis predicted a complete collapse of the beam while the experiments showed only a finite permanent set. These discrepancies were reasoned to have resulted from the assumptions of the small deflection theory which led to two consequences: (1) the moving load maintained a constant horizontal speed and (2) the load effectively stayed on the beam over a longer period of time. Accordingly, it was felt that by using more exact geometrical relations in place of the small-deflection assumption it would be possible to improve on the theoretical analysis. This consideration, in fact, motivated the present work. In passing it may be added that in recent years, the ultimate strength theory has been gaining increasing accep— tance in structural engineering. However, comparative ex- perimental and analytical works have been scarce in inelastic dynamics of structures. In this connection, the present study may have some value beyond its apparent scope of moving loads, as it also reflects the validity of the same general approach for other loading conditions. 1.2. Scope The physical system considered is first defined in Chapter 2. There the method of analysis, including the der- ivation of the equations of motion, is also presented. In Chapter 3, the numerical method of solution is described. Chapters 4 and 5 contain the numerical results of the study. In Chapter 4, analytical results are compared with experi- mental ones. ‘In Chapter 5, the influence of three important physical parameters on the response is studied. The last chapter comprises a summary of the present study. 1.3. Notation The symbols and letters used in this report are listed in alphabetical order, with English letters preceding Greek letters. They are also defined where they are first introduced. A letter with an upper bar represents a dimensionless variable. By the same token, a letter with one or two upper dots indicates a first or second derivative with respect to time. a1, a2, a3 = auxiliary variables Ai = auxiliary variable B1’ B2, B3 = auxiliary variables C = auxiliary variable di = auxiliary variable dt = finite increment in time B = modulus of elasticity g = acceleration of gravity h = length of panel X-component of internal force between joint (i) and panel (1) X-component of internal force between joint (i) and panel (i-l) auxiliary variable dummy subscripts moment of inertia subscript identifying the panel being traversed by load length of beam lumped mass at any interior joint of model bending moment at joint (i) yield moment; see Fig. 2 number of panels interactive force between load and model 4My/L; yield load total mass of beam mass of load time time when load is passing joint (i) time infinitesimally after ti time infinitesimally before ti fundamental period of elastic vibration of beam smallest period of elastic vibration of beam initial speed of load Y-component of internal force between joint (i) and panel (i) Y-component of internal force between joint (i) and panel (i-l) X-coordinate of load X-coordinate of joint (i) Cartesian coordinate Y—coordinate of load Y-coordinate of joint (i) Cartesian coordinate PyL3/(48EI); maximum elastic deflection of beam when Py is applied at midspan angle of relative rotation of two ad- jacent panels connected to joint (1) angle of deviation of panel (i) from X-axis II. METHOD OF ANALYSIS 2.1. General Consider a system of beam and load as shown in Fig. The beam is straight, slender and simply-supported. It has a uniform distribution of mass and stiffness. The left—hand support is hinged at the origin of the coordinate axes OX and OY. The right-hand support is allowed to slide along the X-axis. The load consists of a single unsprung mass. It enters the beam at time t = O with an initial speed v, and is to coast on the beam from the left to the right. The relation between the bending moment and curva- ture of the beam is of the general elasto-inelastic type described in Ref. 3. Assuming a rectangular cross-section and referring to Fig. 2 the relation for loading is given by M = EIk for k 5 ky (la) 2 M = 1.5M - 0.5M k k f r k < k < 10k lb y y(y/) o y“ _ y () s k (1c) M = 1.5M + 0.03EI k - 10k for 10k Y ( Y) Y where M denotes the bending moment, k the curvature, My the yield moment, ky the yield curvature, E the modulus of 6 elasticity, and I the moment of inertia. The relations for unloading and reloading, after the initial elastic region is exceeded, depend on the history of deformation. They follow a hysteretic pattern as fully explained in Ref. 4. The assumptions made in this study are outlined below. 1) Axial and shearing deformations are negligible. 2) Bending deformations are not affected by axial and shearing forces. 3) The beam has no rotary inertia. 2.2. Discretization of Beam In order to accomplish a numerical analysis of the problem described in the preceding section, the continuous properties of the beam are lumped or discretized. The man- ner of discretization corresponds to that for "model B" discussed in Ref. 4. Accordingly, the beam is replaced by a finite number of massless rigid panels connected by flex- ible joints with lumped masses. The panels are further assumed to be of equal length h. The lumped mass at any interior joint of the model is m = Qb/n, where Qb is the total mass of the beam and n the number of panels. Furthermore, at a boundary joint, the lumped mass is m/2 = Q /2n. b The moment-rotation relation for each interior joint (i) is obtained from Eqs. 1 by replacing M by Mi and k by ¢i/h, where Mi is the bending moment at joint (i) and ¢i the angle of relative rotation between panels (i-l) and (i). 2.3. Equations of Motion Let k identify the panel being traversed by the coasting load at some typical instant t. Referring now to Fig. 3a, application of the linear momentum theorem to the coasting load yields P Sln 6k = QZX (2) and ng " P COS 6k = QZY (3) in which P denotes the interactive force between the model and the moving load, 6k the angle of deviation of panel (k) from the X-axis, QZ the mass of the load, (i,§) the accel- eration components of the load, and g the acceleration of gravity. Similarly, considering Fig. 3b, application of the linear momentum theorem to a generic panel (i) as well as to the specific panel (k) gives - H? = 0 where i # k (4a) 1 H' - H+ P s'n e — 0 (4b) k" 1 k— and = 0 where i # k (5a) V - V + P cos 6 = 0 (5b) k in which (HI, VI) and (HE, Vi) are the internal forces act- ing on joint (i) transmitted by panels (i) and (i-l), respectively. Another application of the same theorem to a generic joint (i) in Fig. 3c provides Hf - HT = mx. where i # n + 1 (6a) 1 1 1 + - _ m Hn+1 - Hn+1 7 xn+1 (6b) and + - _ 00 where (xi, yi) denotes the acceleration components of joint (1). Furthermore, application of the angular momentum theorem to the left end of a generic panel (i) as well as to the left end of the specific panel (k) gives M. - M. - HT h sin ei + VT 1 1+1 1+1 1+1 h cos 6i = 0 where i # k (8a) Mk - Mk+1 - Hk+l h Sin 6k + Vk+l h cos 6k X-X +P m=o (8b) 10 in which x and xk are the X-coordinates of the load and joint (k), respectively. To carry out an analysis of the system, Equations 1 through 8 must be supplemented by certain kinematical equations. These equations, with reference to Fig. 4, are readily given by . yi+l - yi 6i = Arcs1n h (9) i x. = h cos 6. (11) y = yk + (x - xk) tan 6k (12) where (x,y) and (xi,yi) are the coordinates of the load and joint (1), respectively. Taking the first partial derivative of Equations 11 and 12 with respect to time they are transformed into x 0 II I MP“ .=2 (yj - yj-l) tan ej-l (13) 3 . . . . (x-xm'r -§r> y = yk + (x - xk) tan 6k + k 3k+l k h cos 6k (14) in which (x,y) and (xi,yi) are the velocity components of the load and joint (i), respectively. 11 Taking once more the first partial derivative of Equations 13 and 14 with respect to time they are further transformed into . - - 2 1 (Y. - y~_l) fi- = - Z (9. - §._ ) tan e._ + 3 3 1 j=2( 3 31 31 hcos36. j-l (15) y = yk + (x - xk) tan 6k + (X ’ xk)(yk+l ’ Yk) h cos3 a k + 2‘* ' xk)(yk+l ' Yk) h cos3 9k + 2 4’ tan 8k (16) h cos 6 k Elimination of i and y between Equations 2, 3 and 16 leads to the following equation for P. x - x k n P = Q - cos 6 y Z h cosZ_ek k k x - x k u - QZ;____§__—yk+l + 019 cos 6k cos 6k (i-a’cuiz -9) + Q i sin 8 - 20 k k+1 R Z k k Z h cos2 e k . . 2 (x - x )(y - y ) k k+l k — 30Z 2 3 tan 6k (17) h cos 6k 12 Elimination of H: between Equations 4 and 6 leads to H; = -mxi + Hi+l where i # k, n + 1 (18a) H; = -mxk + Hk+l - P sin 6k (18b) Hn+1. = Jilin+l (18c) It should be recognized that in the above transition the boundary condition HH+1 = 0 has been used. Equations 18 are now solved for Hi' n Hl - §xn+l — mjéix - P s1n 9k for 1 s k (10a) Hi = -§xn+l - mgixj for k < 1 s n (19b) - _ _IE.. Hn+1 _ 2xn+l' (19C) For convenience in subsequent computations an auxiliary variable Hi (which has no apparent physical meaning) is introduced. 1.. n .. H. = -§Xn+l - E X. (20) Equations 19 are thus written as H. = mHi - P sin e for i s k (21a) k H. = mH. for k < 1 (21b) 1 1 13 Solution of Equations 8 for v; with the latter sub- stitution for H; yields Vi+l = mdi+l - P Sin 6k tan 6i for 1 < k (22a) _ x - xk vk+1 = mdk+1 ’ P 2 (22b) h cos 6 k Vi+1 = mdi+l for k < 1 (22c) where Mi ' Mi-l d1 = Hi tan 61-1 + mh cos 61-1 Elimination of v: between Equations 5 and 7 yields myi = Vi+l - Vi where 1 # k (23a) myk = Vk+l - Vk + P cos 6k (23b) Finally, substitution of Equations 22 into 23 gives the following differential equations of motion. " _ _ _ §_ . _ yi — di+l di m Sln 6k(tan 8i tan ei-l) for i < k (24a) x— X k .. P . yk — dk+1 - dk + 5(51n 8k tan ek-l + cos 6k - (24b) h cos2 6 k ) (24c) 91 = d. - d. for k+l < 1 (24d) 2.4. Jumps in Velocities and Accelerations It is evident that the velocity and acceleration of a mass load coasting on a smooth beam with continuous dis- tribution of stiffness are continuous functions of time, so long as the load is in contact with the beam. However, if the discrete model used in the present analysis replaces the beam, these functions are no longer continuous. In fact, when the load is passing a joint, the velocity and acceleration of the load will experience sudden changes or "jumps." At the same time, similar jumps will also take place in the velocity and acceleration of the very joint being passed by the load. 1) Jumps in Velocities Let the time when the load is passing a joint (1) be denoted by ti’ and the instants immediately before and after that by t; and ti. Then, Equation 13 at times t; and t: becomes xi = -(yi - yi-l) tan 91-1 - 2 (y - yj-l) tan 6j_1 15 and i-l 0+_-0+-0 - 0-0 xi - (yi yi-l) tan ei_l jEZ (yj yj-l) tan ej-l in which the superscripts "-" and "+" refer, respectively, to t; and t1. Subtracting one of these two equations from the other, there results 0+ 0— X. = X. l 1 0+ 0- - (yi — yi) tan ei_l (25) Letting x = xi at time t1, Equation 14 becomes -+ _ + _ °- -+ y — (x Xi) tan 6i + yi (26) Furthermore, conservation of linear momentum in the X and Y directions provides 0+ 0- 0+ .— Q(x - x ) + m(xi - xi) = O (27) and on? - 9‘) + mu}: - 9;) = o (28) -+ Elimination of 4*, y , and k: between Equations 25 through 28 produces ._ y- - i; - (x- - ii) tan 91 = y- + (l + m—)(1 + tan 6 tan 6 ) (29) Q1 1 i-l Next, i: is determined by back substitution of 9: into Equa- tion 25. Using i: and §:, Equations 27 and 28 yield x+ and .+ y , respectively. 16 2) Jumps in Accelerations It is recalled that the interactive force between the load and the model is always normal to the panel being traversed by the load. Thus, in passing from t; to ti, the interactive force suddenly changes direction. This causes jumps in accelerations. However, all equations of kinetics must be satisfied. In order to account for these jumps, the transition from t; to t: is treated as a step in the general numerical integration procedure, to be dis— cussed in the next chapter, with the following modifications: (1) dt = t? - t7 = 0 l l (2) k is increased by one. III. METHOD OF NUMERICAL SOLUTION 3.1. Dimensionless Form of Equations of Motion In computing numerical results it is convenient to deal with dimensionless equations. To this end, the fol- lowing dimensionless variables and parameters are introduced. _ h _ M H-=_H. M,=—l— 1 2 1 1 M V Y -_P -_v P-P—— t—Ht Y §_§ i=3}- - h v __h.. -_yi x-T‘ Yl-h— v ' _ i 3 _ yl-V— Yi‘—Yi v 2 V QZ QQZ °“'h‘1'>_‘ B=p Y Y Y=31 gig. —hM Qb y Where Py = 4My/L denotes the "yield load." 17 18 With the use of the dimensionless variables and parameters, Equations 9, 10, 1, 15, 20, 17, 24, and 2 are, respectively cast in the following dimensionless form. 3| 3| 3| Xu Xv Arcs1n (yi+l - yi) (30) ei-l - 6i (31) C¢i for §¢i s 1 (32a) 1.5 - 0.5(C<1>i)-2 for 1 5 C¢i s 10 (32b) 1.5 + O.O3(C¢i - 10) for 10 S C¢i (32C) 1 _ (§j - §j_l)2 -.§ (yJ - yj-l) tan 8 l + 3 j—2 cos 6 j-l (33) 12 n 2 - §xn+l - .E. Xj (34) 3—1 B17k + BZyk+1 + B3 (35) Ai + a3P for 1 < k (36a) Ak + aZP (36b) Ak+1 + alP (35C) Ai for k + 1 < 1 (36d) P . a s1n 6k (37) 19 in which i-S’ck B1 = a -——7——- - cos 6k cos 6 k 82 = — Bl - a cos 6k - 2a 2 cos 6k - 3a tan 6 cos3 6 k k A1 = Hi+l tan 6i - Hi tan ei_l yn i+l M1 M1 - M1-1 + 4a cos 6. cos 6. 1 1-1 a1 = 13}. X ' xk d cos 9k a = 12(sin 6 tan 6 + cos 9 ) - a 2 a k k-l k l — _Y_n - _ a3 - a Sln 6k(tan 6i tan ei_l) For completeness, Equations 29, 25, 28, and 27 are also put in the following dimensionless form. III. METHOD OF NUMERICAL SOLUTION 3.1. Dimensionless Form of Equations of Motion deal with dimensionless equations. In computing numerical results it is convenient to To this end, the fol- lowing dimensionless variables and parameters are introduced. where P Y F 1 N: XI WI w. P- =—H. 174'. V2 1 1 _P - _ 5. t Y _§ 5;- ‘h =h_;; g, V2 1 =_i = V yl v2QZ ‘53— B y .32. C Qb 4My/L denotes the "yield load." 17 M .1 M Y 2 k yk+1 k - 3a tan 6 cos3 9 k k H +1 tan 6i - Hi tan ei_l + In i+l Mi _ Mi _ M1-1 4a cos 6. cos 6. 1 1-1 is x"xk a cos 6k Yn - _ —E(S1n 6k tan ek-l + cos 6k) al yn Sin 6k(tan 8i — tan ei-l) For completeness, Equations 29, 25, 28, and 27 are also put in the following dimensionless form. 20 1+ = 1- + n 7 ‘ 91 ’ (R ’ 21) tan 61 (38) Y1 Y1 Y (1 + yn)(1 + tan 6i tan 91—1) &+ 1- 1+ 1- i — xi - (yi - yi) tan 61-1 (39) z+ _ :- _ i i y - y Yn (40) :+ _ :- §+ = fi— _ _£____£ (41) yn 3.2. Description of Parameters The problem under consideration has eight dimen- sional physical parameters: n, L, Qb, EI, My’ v, 01’ and 9. According to the theory of dimensional analysis these can be grouped into five independent dimensionless para- meters which may be chosen to be n, a, B, y, and Q, as listed in the preceding section. Alternatively, any other independent combination of these parameters can be adopted. It is of some interest to interpret the physical character of the parameters a, B, Y, and C. The speed parameter a = VZQZ/hpy is directly proportional to the ini- tial kinetic energy of the load. The weight parameter 8 = gQZ/Py is a measure of the load weight in terms of the yield load. The mass parameter y = QZ/Qb stands for the ratio of the load mass to the beam mass. The stiffness parameter Q = EI/hMy is such that its reciprocal represents the angle of rotation ¢i when Mi = My' 21 3.3.Numerica1 Integration of Accelerations and Velocities In order to integrate the accelerations and veloci— ties numerically, the so-called 8 method of integration as outlined in Ref. 2, with B = O, is used. Introducing dE to denote a small increment in t, the formulas for integrating § and i are §(E) + 0.5 §(E) + §aE + 0.532(E>dE2 (42b) Similarly for §i and §i, they are yi(t) + 0.5 yi(t) + yi(t + dt) dt §i(E + dE) (43a) §i .cmmam _mwslca mmmz mcstamcz m—mcwm m to mcwpmwmcou use; mcwummou a. A W 38 Emtmmwo mtzpm>tzuuazmsoz .N mtzmwl Axv mtzpm>gsu z x xo_ x nv z x > N11\ xv 2m.o - zm._ u z a: mcwumopcz sZm.1 a s 4//||| A 1-xvammo.o + 2m._ u z (w) iuawow 5u1puaa Figure 3a. Free Body Diagram of Moving Mass k <:}\Sk+l E HEH VE+1 Figure 3b. Free Body Diagram of Panel (k) V? 1 . Mi H; was ———(> H; Mi + Vi Figure 3c. Free Body Diagram of joint (i) 40 C: Pmcma >3 F+x _mvoz mpmgomwo _.+_.6 V .a mgzmwm >. OF I: Midspan Deflection (in.) 41 Dimensionless Time (vt/L) 0.2 0.4 0.6 .8 \\ \\ ,’ .’ \ / j / I / . \\ / // ~\\ / / \ \\ /./ O/- \\ / \ / -1,,/ / (a) v = 6.56fps. \\\\.\‘ \Q>\\‘ \ \. \\ \‘ \. \‘\ \\\\ \.\\ ‘\.\\ \‘\ \\ \\\ \\\ / \‘\\"."\ I \\ j M . I \‘U . \ Experiments ~\ ------ Large-Deflection Analysis \‘ —-—-—- Small-Deflection Analysis (b) v = ll.30fps. Figures 5a, b. Midspan Deflection vs. Time Midspan Deflection (in.) 0. 0. l l. 2 4 8 .2 .0 .2 0.4 L 42 Dimensionless Time (vt/L) 0.6 0. ...L— 8 -4 (c) v T I? l Experiments Large-Deflection Analysis Small-Deflection Analysis 13.50fp5. Figure 5c. Midspan Deflection vs. Time ) Y Interactive Force/Yield Load (P/P l. 43 Dimensionless Time (vt/L) 0.2 0.4 0.6 0. 8 l l I l l r —I)— Experiments -- Large-Deflection Analysis —- Small-Deflection Analysis v = 6.56fps. Figure 6a. Interactive Force vs. Time Interactive Force/Yield Load (P/Py) N —l O .—l O —l O 44 Dimensionless Time (vt/L) 0.2 0.4 0.6 0 .8 l I I Experiments --- Large-Deflection Analysis -—- Small-Deflection Analysis v = ll.30fps. ~1— H Figure 6b. Interactive Force vs. Time U"! y) Interactive Force/Yield Load (P/P .h 60 N —l 0 45 Dimensionless Time (vt/L) 0.2 0.4 0.6 0.8 .L i i i l . I l I l _ l i I . Experiments / ------ Large-Deflection Analysis / __ — ----- Small-Deflection Analysis (c) v = l3.50fps. ‘ CA/ \\*'*" ’//, \\ //” Figure 6c. Interactive Force vs. Time Permanent Deflection (in.) 46 Distance From Left End/L 0.2 0.4 0.5 l 1 J \ -\_\l-\ I 1 \\\ ‘\ \,\ \ “'— // \\ \-\ 5 ‘// \\ ’7’ / ‘7 4 \\ / \\ / \\ / (a) v = ll.30fps. “\.\, / 7-8 \\ 11 \\\\\\\\ Z \\ ‘‘‘‘‘ \ l \\ '\‘ . F 4 \\ \‘ /I \\ I, \ . \\\ \‘\_/'//I L 8 \\\ / \ / __ \\ / 12 \\\\~ / \\\ /// ’lFlO Experiments ------ Large-Deflection Analysis: Scale on Left — — — Small-Deflection Analysis: Scale on Right (b) v l3.50fps. Figure 7. Permanent Sets 47 m.mp AFmpcmetmaxm new FMUwp>ch cowpompemo :mamuwz Ezewxmz A.matv emmam _aaaw=1 m.N_ o.o_ _ _ :lE! _ fl A m 0 ,— II A Ch 0 II .m mczmwm m.o o.o so me _aocaspm=< Fmpcmswcmaxm Faucmswcmaxu mmwi.o.m o.¢ V/uogioallaa uedspiw wnwixew 48 mp Asgawmz one; to sumttmv camam _mpr:H .m> cowpompwmo :mamuwz E:E_xmz .mm mtzmwm mmx\umaam Palaces e_ N, o_ m a e A h _ _ _ _ _ _ _ A _ v/uogsoallaa uedspLN wnmgxew 49 my finesse: anon to pumttmv vmmam mepwcH .m> cowuumpemo :mamuwz pcmcmstma .nm acumen mm\\ummam _m131=1 a— NF o_ m m a _ b — b _ _ _ u m\> N.P .lIllnl'll'ld‘i-‘lltin'lil .1!)I§‘Il.|'!ll'tlvil l(.l )3).-. l.)()). —(.— -( KO v/uoiiaaljaa uedspr 1uauewaad Axbwomamo ucmsoz scam to pomeeuv ummam FowpmcH .m> cowpumpmmo cmamuwz Esswxmz .moF 813mm; mm\\uaaam _mcp1=1 S 3 S S m o s N b —r— — _ _ A _ _ _ 50 Np @— v/uoizoallaa uedspgw unmixew 51 mp ap uaaam _w1p_=H NF .Axpwowaou pcmsoz seam to uomemmv .m> :owuumFCmo commuwz “cognacma mm\\ummam _a_pwca o_ w a .nop casual _ _ q d.) N F to P v/uogsoallaa uedspgw suauewuad APPENDIX COMPUTER PROGRAM In this Appendix, the computer program as well as certain pertinent information are presented. 1. Identification of Parameters and Variables The parameters and some important variables are identified in the following: N = n ALPHA = a BETA = B GAMMA = y ZETA = c DELTA = A/L NS = number of steps between two con- secutive print-outs DTIME = of TOL = tolerance of iterations K = k z = i - ik s = k + (i - xk)/cos ek T = E 52 H(I) TH(I) AN1(I) BMA(I) AXl VXl DXl AX(I) vx(I) AYl(I) VY1(I) DY1(I) DYD(I) DYMAX(I) SDYMA(I) TDYMA (I) PMAX SPMA TPMA PA(I) PD(I) 53 ME MI >u- xu 3| XI. yi/A maximum yi/A S where yi/A attains its maximum T when yi/A attains its maximum maximum P S where P attains its maximum T when P attains its maximum permanent oi permanent yi/A 54 2. Print-outs At the beginning, all the parameters as well as DELTA, NS, DTIME, and TOL are printed out. During the process of step-by-step solution, S, DXl, VX1, AXl, P, T, and DYD(I) are also printed out. At the end are printed out DYMAX(I), SDYMA(I), TDYMA(I), PA(I), PD(I), PMAX, SPMA, and TPMA. C *** *iv'l’ 602 DIMENSION AXI25)cVX(25)oAYl(25)9AY2(25)9AY3(25)9VY1(25)9VY2(25)o IDYl(25)vDYZIZS)vDYD(25)oSTGA(25)qSTGB(25)oBMA(25)98MB(25)oBMC(25)o ZBMUIESIOBMLIZSIOANI(25)0AN2(25)9ANG(25)OPATH(25)0THPLIZSIQCMB(25)O 3CMT(25)9CAB(25)oCAG(25)oPREM(25)oPREA(25)qPA(25)9PD(25)oDYMAX(25)o 4SDYMA(25)OTDYMA(25)9TH(25)QDA(25)QH(25)9A(25)08(25) PARAMETERS **x N: ALPHA: BETA: GAMMA: 7FTA= AUXILIARY VARIABLES *** N1=N+l DELTA=N/(12oO*ZETA) D=DELTA*N E=GAMMA*N F=E/ALPHA G=F*N/4 0O TOL=IoOE-3 DTIME=O.16/SQRT(G*ZETA) I=1.0E+3*DTIME+O.5 DTIME=1oOE-3*I NS=SoOE-l/DTIME COMPUTER PROGRAM PRINT 6029N9ALPHAQBETAOGAMMAQZETAQDELTAONSQDTIMEQTOL FORMAT(1H192(/)Q4H N =I 1F5029 5X07HGAMMA :F6029 2F7040/95H NS =13! 8X¢7HDTIME =F6o39 3c 9X97HALPHA 5X96HZETA =F6029 C *** INITIALIZATION OF VARIABLES *** ASSIGN 426 TO NAP K=L=O=VX1=I.O T=DX1=AX1=AX2=PMAX=O DO 600 I=19N1 AYl(I)=AY2(I)=AY3(I)=VY1(I)=VY2(I)=DY1(I)=DY2(I)=DYMAX(I) BMA(I)=BMB(I)=BMC(I)=AN1(I)=AN2(I)=ANG(I)=PREM(I)=PREA(I) CAB(I)=CAG(I)=CMB(I)=CMT(I)=PD(I)=O STGA(I)=STGB(I)=loO BMUII)=+1.0 55 =F6029 6X05HTOL 6X96HBETA = 5X97HDELTA = =E80192(/)) O O OK) 600 *** 604 608 *** 300 298 297 296 295 294 293 292 *** 18 *** *** 201 503 501 500 231 502 235 271 56 BML‘I)=-100 RETURN *** M=NS DT=DTIME LOCATION OF COASTING LOAD *** DO 298 I=19N DY2+0.5*DT**2*AYl(I+I) TH(!)=ASINF(DY2(I+1)-DY2(I)) IFtK.E0.0) GO TO 292 DX2=DX1+DT*vx1+0.5*DT**2*Ax1 x=o DO 297 I=19K X=X+COSF(TH(I)) IF(L.E0.0) GO TO 295 IF(DX2-X) 29492959296 DT=(-VX1+SORT(VX1**2-2*AX1*(DXl—X)))/AX1 L=O GO To 300 Z=COSF(TH(K)) ASSIGN 422 To NAP L=l GO TO 293 Z=DX2-(X-COSF(TH(K))) S=K+(DX2~X)/COSF(TH(K)) CONTINUE ANGLE OF ROTATION *** DO 18 1:29N AN2(I)=TH(Idl)-TH(I) DA(I)=AN2(I)“AN1(I) BENDING MOMENT *** DO 240 I=Z9N IF(195-STGA(I)) 200,200,201 STAGE ONE *** BMB(I)=BMA(I)+DA(I)*ZETA IF(BML(I)-BMU(I)) 50295029503 IF(DA(I)) 50095009501 IF(BMB(I)-BML(I)) 24092409271 IFIBMB(I)-BMU(I)) 23192409240 CMB(I)=PREM(I) CAB(1)=RREA(I) STGBII)= 291 GO TO 222 IF (DA(I)) 23592369236 IFIBMB(I)-BML(I)) 27192409240 STGB(I)=2.3 (III. II. .I. III 57 PREM(I)=CMT(I) PREA(I)=CAG(I) GO TO 222 236 IF(BMB(I)-BMU(I)) 24092409231 C *** STAGE TWO *** 200 IFIDAII)*PATH(I)) 22192229222 221 BMU(I)=BMA(I) STGB(I)=1.0 BML(IJ=BMU(I)+SIGNF(2.O9DA(II) CMB(I)=0.5*(BMU(I)+BML(I)) CAB