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The problem of safety of steel structures subjected to seismic loading

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The problem of safety of steel structures subjected to seismic loading

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Content THE PROBLEM OF SAFETY OF STEEL STRUCTURES SUBJECTED TO SEISMIC LOADING by. Sun-ju Hung A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Engineering) June 1969 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90007 This dissertation, written by Sun=ju Hung under the direction of has... Dissertation Com- mittee, and approved by all its members, has been presented to and accepted by The Gradu- ate School, in partial fulfillment of require- ments of the degree of DOCTOR OF PHILOSOPHY ATH Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS The author wishes to express his gratitude for the guidance and encouragement provided by Professors Maciej P. Bieniek, Paul Seide, Victor I. Weingarten, and Walter T. Kyner. He, in particular, wishes to express his sincere appreciation for the helpful guidance and suggestions received from his research advisor, Dr. Bieniek. The work would never have been completed without his ideas and iriterest. Thanks are expressed to the University of Southern California Computer Sciences Laboratory and Systems Simulation Laboratory for the use of the Honeywell H-800 and the IBM 360/44 computers, and to the California In- stitute of Technology for the use of the IBM 7094 com- puter. The author is pleased to acknowledge the interest shown in his work by Drs. D. E. Hudson and G. W. Housner of the California Institute of Technology. He is also grateful to the American Iron and Steel Institute for the Engineering Fellowship award. Last, but not least, the author wishes to express his deepest gratitude to his parents and wife for their ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. constant encouragement and advice. Special acknowledg- ment goes to his wife Tina for her understanding and sacrifices which enabled the completion of his graduate work. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS PAGE ACKNOWLEDGMENTS 2. 2. 2 © « «© © © © © 8 © © © 2 «© «@ Li LIST OF FIGURES .« . « © «© «© © © © © © «© «© © «© © «@ vi LIST OF SYMBOLS . 2. © «© « © «© © © © © © © © © © «@ vil ABSTRACT . 2. « © e © © © © © © © © © © © © © © xiv CHAPTER I. INTRODUCTION . .« «© «© « © © © © © © ©» © © 2 1 II. SINGLE-DEGREE~OF~FREEDOM SYSTEM . .« « « « « 7 2.1 Shear Beam Structure . . . « « « « « 7 2.2 Concept of Complex Stiffness and Structural Damping . . . - « « «© « « 8 2.3 Determination of |z| .......-. 13 2.4 Response to Unit Impulse .....e. - 23 2.5 Response to Random Excitations ... 26 IIL. MULTI-DEGREE-OF-FREEDOM SYSTEM . . . «© « « 30 3.1 Matrix of Impulse Response Functions ... .« «© « « «© « «© © «© « 30 3.2 Mean Square Response Matrices ... 35 IV. SAFETY OF STRUCTURES OF ELASTIC-PLASTIC MATERIALS SUBJECTED TO SEISMIC LOADING. . 40 4.1 Probabilistic Characteristics of Ground Motion and Structural Response . « « « © © © © © © © « © « 46 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4.2 Computation of Mean Square Relative Displacements and Relative Velocities . . 2. « »« » « « » 4.3 Probability of Structural Failures .. 4.4 Fatigue Failure under Seismic Loading. 4.5 Safety of Structures .... V. NUMERICAL EXAMPLE . .« « « « « « « «@ 5.1 Scope of Investigation... 5.2 Procedure of Computations . VI. CONCLUSION . . «6 « «© «© © © © «© © «@ BIBLIOGRAPHY . 2 « « « «© © © «© © © © © « APPENDICES: COMPUTER LISTINGS AND FLOW CHARTS. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 44 47 52 55 57 57 63 80 83 88 LIST OF FIGURES FIGURE Ll. 2. Single-Degree-of-Freedom Shear Beam Structure . Vectorial Representation of Equation of Motion . Idealized Stress-Strain Relation . . ...-... Lateral Displacement of One-Story Shear Beam Structure . « « « «© © © «© © «© © © © © © © 2 Flexural Rigidity, Moment Diagram and Conjugate Beam e e. s e eo e e e * s e e e es e e e e e e e Shear Resistance-Displacement Relation ..... Hysteresis Loop e ° oe «a e e e °. s e e e eo e* * e N-Degree-of-Freedom Shear Beam Structure with Complex Stiffness ...... Example Structure . . 2. 2 «© « «© © «© © «© « e « « S-N Diagram of A36 Steel... Mean Square Relative Displacement (lst Story). . Mean Square Relative Displacement (2nd Story). . Mean Square Relative Displacement (3rd Story). . Mean Relative Displacement Story). . Square (4th Mean Square Relative Velocity (lst Story) ... Mean Square Relative Velocity (2nd Story) ... Mean Square Relative Velocity (3rd Story) ... Mean Square Relative Velocity (4th Story) ... Probability of Fatigue Failure . . 2. . « + 6 « « vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A A. = (or 3) [A(W) ] D = * D = D(s) = D(t; Ti,T2) D, (s) = Dis = Reproduced with permission of the copyright owner. LIST OF SYMBOLS cross-sectional area of the columns (at j-th story); adjoint matrix of [2(“)]; felt area (Eq. (4-6)); intensity process of the ground acceleration; mean value of the intensity; intensity of the earthquake; positive material constant (Eq. (5-2)); (M17 > (K" 1; coefficient of viscous damping; positive material constant (Eq. (5-2)); total damage; fatigue strength of the material; amount of damage caused by N(s) cycle of stress amplitude s; defined in Eq. (5-6); damage caused by one cycle of stress amplitude s; amount of fatigue damage produced at a critical section in the i-th story by one earthquake; amount of fatigue damage at a crit:ical section in the i-th story within a time interval T; amount of damage caused by the j-th cycle of stress amplitude s; Vil Further reproduction prohibited without permission. [h(t) ] depth of the columns (at j-th story); determinant of A; modulus in the elastic range; modulus in the plastic range; energy dissipation due to viscous damp- ing per cycle; flexural rigidity of the elastic column; flexural rigidity of the column within the yield length; expectation of A; error function; complementary error function; shear force; probability density function of B; probability density function of Mai probability density functions of Sii probability functions of S.i gravitational acceleration; normalized stationary Gaussian process; frequency response function; matrix of the frequency response functions; yield length factor; impulse response function; yield length of the column; matrix of the impulse response functions; vili Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [h(t)]= = transpose of [h(t)]; I = moment of inertia of the cross-section; [I] = identity matrix; T,, (si b,t) defined in Eq. (5-11); I,; (s,b) defined in Eq. (5-12); I,, (s,b) defined in Eq. (5-13); I,,(s) defined in Eq. (5-14); I,, (s) defined in Eq. (5-15); Ig, (s) defined in Eq. (5-16); [K"] = complex stiffness matrix; k = equivalent spring constant; kK” (or K's) = complex stiffness (at j-th story); Ka = elements of the complex stiffness matrix; L (or Ls) = story height (at j-th story); M = earthquake magnitude on the Richter scale; Fu] = diagonal mass matrix; Mn = number of earthquakes during T years period; Mn = expectation of Mj; Mp = end moment; My = yield moment; (M,] = ([{m,}{m,} 71; m (or m.) = mass (at j-th story); {m5} = mass column vector; N = number of cycles to fatigue failure; ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. n.(s; b,t) Prob [A] Q(t) {Q(t)} q(t) number of cycles of stress s; fatigue life (in cycles) of the material; average seismicity; number of intervals of t, b, s respec- tively; number of maxima per unit time at i-th story; expected number of maxima at stress amplitude s during one earthquake of intensity b; number of maxima at stress amplitude s during one earthquake of any intensity; number of intervals or number of stories; number of maxima at S. = s per unit time in an earthquake’ of intensity b; probability of failure; probability of failure of the building within T years; probability of survival of the building within T years; probability of failure of a critical section in the i-th story; probability of survival of a critical section in the i-th story; probability of survival of all the sections in the i~th story; probability of the event A; random excitation process; random excitation column vector; external excitation; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. gq = amplitude of the excitation; m R = seismic severity cof the earthquake; R, [A] = real part of A; Rag (ta7ty) = correlation function of g(t); Ry x, (tyr tg) = correlation function of Xo (t); Ryy (t,t) = correlation function of Y(t); Ryy (ty rt) = correlation function of Y(t); Ryy (t,t) = mean square value of Y(t); Ryy (t,t) = mean square value of Y(t); [Ryy (ty, t,)] = correlation matrix of {xX(t)}; [Ryy(ty,t,)] = correlation matrix of {X(t)}; [Ryy (t,t) ] = mean square matrix of X(t) ; [Ry (t,t) ] = mean square matrix of {X(t)}; S; = maxima of Yi; S, (b,t) = maxima of column stress during one earthquake of intensity b; s = stress level; sgn (A) = Sign of A; T = service period of the building, in years; Ty = duration of one earthquake; t (or t,t») = time; V(t: T1,T2) defined in Eq. (5~10) V[A] = variance of A; W (or Wa) = floor weight (at j-th story); Wo = work done by the external force per cycle; xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ww Xo (t) {x(t)}, {X(t)} v(t), Y(t), %(4) ¥, (t) ¥, (t) y(t), y(t), ¥(t) (or Ym? B(y) Bi 6 (t) At, Ab, As, At At, C (or S4) n wide flange; ground acceleration process; random response column vectors (displacement and velocity, respec- tively, relative to the ground); random displacement, velocity and acceleration process respectively; displacement of i-th story relative to i-lst story; velocity of i-th story relative to i-lst story; displacement, velocity and accelera- tion respectively; amplitude of the response (within the j-th cycle); yield displacement; mechanical impedance; impedance matrix; a parameter characterizing the duration of the ground motion; slope of the shear force-displacement curve; E,/Ei Dirac delta function; intervals of integrations; 27/2; non-dimensional constant; defined in Eq. (4-32); xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A = scalar constant; ha = eigenvalue of [C]; uw (or Hy) = ductility factor (at j-th story); T = product; ) = summation; oy = yield stress; t (OL T,,T2) = dummy time variable; Q = arbitrary constant; u) = forcing frequency; wo = undamped natural frequency; Wi,W2 (or ws) = complex frequencies; W (or Way) = real part of a complex frequency; Os (or Woy) = imaginary part of a complex frequency. xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Due to the fact that the characteristics of ground motions and the material properties of buildings are random in nature, the probabilistic theory is employed in the response and safety analysis of buildings bubjected to seismic loading. The shear building concept is used in the response analysis of a one-story building and then the results are extended analogously to the investigations of multi-story buildings. A bilinear stress-strain rela- tion is assumed for the material. Structural damping, caused by the plastic deformations, is accounted for by complex stiffnesses. The probable number of stress cycles within the service time of the structure is then estimated using the results of response analysis and the statistics of the earthquake occurrence and intensity. Safety analvsis of the steel structure is performed on the basis of a statistical cumulative fatigue damage theory. As a numerical example, the safety of a four-story steel struc- ture is investigated with the aid of electronic digital computers. XLV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission CHAPTER I INTRODUCTION It has been observed, on the basis of the results of the elastic response analyses of structures subjected to seismic loadings, that the typical building codes are not safe enough to protect buildings against earthquake failure. In contrast, past experiences reveal that many buildings designed by following the building codes have withstood much stronger earthquakes with little or no damage to the main frames. The above two contradictory observations can be explained by the fact that a structure will not behave elastically after the magnitude of the seismic loading exceeds a certain level. An appreciable amount of energy is absorbed by the plastic action of the structure. More- over, non-structural elements like walls, partitions etc. in the conventional buildings also play an important role in consuming the input energy of the loading. During a moderate earthquake, the input energy caused by the ground motions is first absorbed by the non~structural elements, while the main building frames are protected against noticeable damage. Not until the intensity of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. earthquake becomes stronger will the non-structural ele- ments pass the duty of resisting the earthquake motion over to the main frames. It is this energy absorption property of the plastic deformations of structures that prevents many buildings from disastrous failure. The advantages of the plastic design of steel frames under static loading conditions are generally recognized. The safety of the plastic design for static loadings has been well established within certain material and design requirements. In the case of building frames subjected to seismic loading, the significance of the plastic deformations of steel elements appears to be quite unique and not fully understood. In addition to the usual effect of the plastic design procedure on the dimensions of the beams and columns, the plasticity of steel seems to have a very strong influence on the response of the frames to earthquake excitations and, consequently, controls the magnitude of the equivalent lateral forces. In general, this effect of the plastic deformations is highly beneficial, although its magnitude is not known with sufficient confidence, and it is, as a rule, disregarded in standard design practice. A serious argument against a very extensive reliance on the effects of plasticity Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. >) is the possibility of fatigue type fracture of a steel element subjected to several cycles of plastic strain. The importance of this phenomenon also is not clear, since it depends on a very complicated cycling process of stresses and strains during an earthquake and it involves the relatively less known fatigue behavior of steel elements and joints in the plastic range. Many investigations in the area of inelastic response of structures to earthquake motion have been done in the past decade. Several investigators (1, 2, 3) presented analyses of nonlinear single-degree-of-freedom systems, with and/or without damping, subjected to earth- quake motions. Jennings (4) elaborated Jacobson's (5) idea of replacing a nonlinear system by an "equivalent" linear system whose behavior will be an approximation to that of the nonlinear system and extended the well-known results of linear analysis to simple yielding structures. Berg (6) presented an analysis of elastic-plastic multi-story steel frames taking into account the flexural resistance of the frame, shear resistance of the walls and viscous damping resistance. Penzien (7) studied two idealized six-story elastic-plastic shear buildings of equal story heights. The equations of motion are solved by step-by-step integration procedure to obtain the com- plete time history of response for each floor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 Clough et al. (8) analyzed the nonlinear response of multi-story buildings to arbitrary time-varying lateral forces by a step-by-step procedure. Within each short time increment, the structure is assumed to behave in a linear elastic manner. The elastic properties may be changed from one interval to the next, thus the non- linear response is obtained as a sequence of linear response of successively differing systems. Saul et al. (9) and Lee at al. (10) also presented analyses of a typical multi-story bilinear inelastic shear building subjected to idealized earthquake motions. The equations of motion are solved by using a numerical integration procedure in which the variation in the momentum-force and deflection-velocity relationship is assumed to be linear over a small time interval. Due to the fact that the ground motion in an earthquake is a random process and that the occurrence of an earthquake is not a deterministic event, the safety analyses of structures are best oriented in the direction of applying the probability and statistic theory. Eringen (11) introduced a method «f difference equations in finding the mean square displacements and correlation functions for a multi-story building. The building is approximated by a series of masses and dashpots which take into account both interfloor and external damping. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a Various correlation functions are calculated in a numer- ical example for a ten-story building subjected to a purely random ground displacement process. Several papers, considering the earthquake motion as non-stationary random input, have been published. Barstein (12) suggested an approach for finding the "design seismic load" based on the mean square structure response to seismic loading. Bolotin (13) and Shinozuka et al. (14) investigated the safety of linear systems using the first-excursion failure (see Section 4.3) as failure criterion. Unlike the static strength, the fatigue strength of steel has large scatter. In the safety analysis of steel structures, the fatigue strength of steel will be described in terms of its statistical properties. In estimating the cumulative damage during a series of earth- quakes, a statistical theory of fatigue will be used. In the investigation to be performed, the effect of the plastic deformation in the elements of the struc- ture will be taken into account by assuming certain equivalent damping, whose magnitude corresponds to the energy dissipation capacity of the actual structure (15). The necessary statistical descriptions of the structure will be provided by the response analysis. This, and the statistics of the earthquake occurrence and intensity, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. will be used to estimate the probable number of stress (or strain) cycles within the service time of the struc- ture. A combining of the expected stress (or strain) cycle history with the cumulative fatigue damage theory will lead to the determination of the degree of safety related to the existence of plastic deformations in the structure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER II SINGLE-DEGREE-OF-FREEDOM SYSTEM 2.1 Shear Beam Structure An actual structure, even as simple as a column or a simple beam, is far from being a single-degree~of- freedom structure. It is just too complicated to in- vestigate its true structural properties without making some simplifications. For practical purposes, an actual building can be simplified by assigning to each structural element an equivalent mathematical model in the response analysis of the structure under random excitations. For simplicity, the shear building concept will be used in this investigation with the following assumptions (7, 16): (1) The distributed mass of the structure is considered to be lumped at the floor levels. (2) The floors are infinitely rigid subjected to no rotation during the horizontal deformation of the structure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3) The horizontal deformation of the structure is of shear type. (4) Damping between floors is similar to viscous damping. A single-story building as shown in Fig. 1 can, under the foregoing assumptions, be considered as a single-degree-of-freedom structure. The governing differential equation of motion of the structure sub- jected to the external excitation q(t) is in the form: m¥(t) + cy(t) + ky(t) = q(t) (2-1) where m, c, k are the mass, the coefficient of viscous damping and the equivalent spring constant, respectively, of the structure. 2.2 Concept of Complex Stiffness and Structural Damping In the course of mechanical vibrations, the steady-state response of an undamped mechanical system as a result of an external harmonic excitation is also harmonic. That is, for every harmonic excitation iwt . . . Ine there is a corresponding harmonic response y(t) = yer. The quantities Qin and Ym are the real il q(t) amplitudes of the exciting force and response respec- tively and are related as: a = BY (2-2) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fie, 1 Single-Derree-of~Freedom Shear Beam Structure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 where the stiffness k is real for the undamped system. The frequency w is the same, in the steady-state, for both the excitation and the response. For a system with structural damping, it has been common practice to add an imaginary part to the svtiff- ness to account for the damping, i.e., k*¥ = k{1 + ilé]sgn(w)] ([e] < 2) (2-3) where i = /-IT. The quantity k* is the complex stiffness of the system which represents both the elastic spring force and the structural damping force at the same time. And Ic| is a non-dimensional constant which depends on the property of the material and is usually less than unity. The function "sgn" takes the value +1 or ~1, according to whether the argument is positive or nega- tive. It assigns a proper sign for the imaginary part of the complex stiffness k*. This may be justified by considering the vectorial representation of all the "forces" acting on the system. Consider a single-degree-of-freedom system sub- jected to a harmonic excitation, as represented by Eq. (2~1) with the right-hand side replaced by qe. All the terms contained in the differential equation of motion are sinusoidal functions of the same frequency W., As the vectors which represent the various terms Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 11 rotate with the same circular frequency, they maintain fixed positions relative to one another. The vectorial sum of the vectors my, cy and ky should be equal to the ° representing the extermal force. The vector qe" vector cy which represents the damping force is equiv- alent to the vector induced by the imaginary part |z|« of the complex stiffness k*, Thus, it is clear from Fig. 2 that |¢] has a positive sign if the frequency w is greater than zero and a negative sign if w is less than zero. Now, suppose that the external excitation is in the form: q(t) = Relg,e*°*y (2-4) then, accordingly, the response of the system to this excitation will be: wt Y(t) = Rely e* ] = Y¥,COs wt (2-5) where the symbol Re stands for "the real part of." The work Wo done by the external force per cycle, then, 1s 27 = i! Wd q(t)¥(t)dt = m/c{kye (2-6) 0 : Use has been made of Eqs. (2-2) and (2-3) in obtaining Eq. (2-6). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 Fig. 2 Vectorial Representation of Equation of Motion 13 In the meantime, the energy dissipation Eq due to the visccus damping per cycle is 2a E, = ® ey (E) H(t) dt 0 mowy (2-7) Upon comparing Eqs. (2-6) and (2-7), the following rela- tion is obtained: c= lelk (2-8) W Therefore, knowing that ¥(t) = iwy(t) if y(t) = yae Eq. (2-1) can be written, by virtue of Eq. (2-8) as: my(t) + k*y(t) = q(t) (2-9) where k* is as defined in Eq. (2-3). 2.3 Determination of |¢| The idealized stress strain relation of the steel structure under investigation is assumed to be of bi- linear type as shown in Fig. 3. This assumption is justified for structures with wide-franged I-section columns whose cross-sectional areas are assumed to be concentrated at the two extreme edges away from the neutral axis of the cross-section. The yield stress Oy and the moduli E in the elastic range as well as EL in the plastic range are well defined from the material property of the structure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Wh Fig. 3 Idealized Stress-Strain Relation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 The shear force-displacement relation of the structure can be determined once the dimension and the Material property of the structure are known. Due to the lateral displacement Yin of the floor, the columns of the structure are subjected to end moment M, as shown in Figs. 4 and 5b. This end moment M, would exceed yield moment My as Y., increases beyond the yield displacement L? Ww (o,, = on) (2-10) where W is the weight of the floor, L, d, A are, respec- tively, the height, the depth, and the cross~sectional area of the columns. Here, both columns are assumed to be identical in shape, size and are of the same material. In the meantime, the ends of the columns start to yield, and the yielding gradually extends toward the central portion of the columns to form "yield length" at both ends of the columns. The flexural rigidity within the yield length of the columns is modified to E,1 due to the yielding while that of the central portion remains un- altered (Fig. 5c). The end moment and the yield moment maintain a linear relation (Fig. 5d): My M, = ToRGyY (2-11) where Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o~"e ia NL Me SN 7" £ Fen Fig. 4 Lateral Displecement of One-Story Shear Beam Structure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 (a) Original Glumn (b) Deflected Column Fig. 5 Flexural EL EI (c) Flexural Rigidity cd) Momen} Diagram bs ¥-€) (y2z-E)7 (2-4 Q 5 9 4 (e) Conjugate. Bearn Rigidity, Moment Diagram and Coniurate Beam 18 and EI is the flexural rigidity of the elastic column, h, 0 <h < er is the yield length factor. It is pointed out that the length of the yield length, hL, at each end of the columns is dependent on the magnitude of lateral floor displacement Yin* By using the conjugate beam method (Fig. 5e), and making use of Eqs. (2-11) and (2-12), the following relation is obtained: Yy where Ei Bi =e (2-14) Eq. (2-13) can be rewritten as n?(y) ~ Sh*(y) + 303 + PE yntyy + Stee = 0 (2-15) where be (2-16) is the ductility factor. Eq. (2-15), a cubic equation of h, always contains a single real root and two complex roots in conjugate pair. The real roots are readily obtained for every given up and every given $8; by numerical method if preferred. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 The shear force fn is, thus, found to be: _ 7, Yy £ ere ew (2-17) L 1=2h m ly ey, where k = 12EI ~ “Ts (2-18) with values of h(y) obtained from Eq. (2-15) for Yn? Yyr For y,, S Yyr Eq (2~17) degenerates simply to fa Kk Y (2-19) The slope &(y) of the shear force-displacement curve is, then, Vy 4 1 & bearer] Ym 7 ¥y? Bly) = dy (2-20) 1 (yA < Yy? Note that for the elastic-plastic material (Bi = 0), 8 has a constant value of zero which is independent of the magnitude of Yin Ym > Yy) due to the fact that h(y) = 0 for 8: = O. The shear force-displacement relation is shown in Fig. 6. Although hysteresis loops for shear force- displacement can be constructed using this variable 8 (y) for various values of $,, it is suggested for simplicity that constant values of B(y) at y = Ym be used for each Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 Fig. 6 Shear Resistance-Displacement Re).ation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 8, for the portion of the loop such that yy <y <y, (Fig. 7). The energy loss per cycle Ea is the area of the parallelogram loop abdina (17): Eg = 4kyy: (1-8 (y,)] + (yycyy) (2-21) This energy loss represents the mechanical energy dissi- pated by the material during one complete cycle of vibration. Since the damping force is the only force that dissipates mechanical energy, the amount of energy lost in one cycle of vibration must be equal to the work done by the external force in that cycle. Upon equating Eqs. (2-6) and (2-21), |t| is, therefore, obtained to be: AUB Ym} pen (u > 1) Oe - Iz] " " ~ (2-22) 0 (uw <1) The area of the parallelogram loop abdlna obtained by using constant B(y_) and that of the curved hysteresis loop constructed by using variable B(y) are reasonably close for all practical ranges of Vine Consequently, using a fixed value of B(y) at y = Ym in the computation of |t| seems justifiable. It is clear from Eq. (2-22) that, for each fixed value of u(t > 1), [t| attains its maximum and minimum at Bly) = 0 and Bly) = 1 respectively: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 22 _ > N _ eS ee be) Fig. 7 Hysteresis Loop Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 4-1 , , 151 mae = oT Fr if 8 = 0 (elastic-plastic case) (2-23) lol min = OF if 8 = 1 (elastic case). The largest value of ra occurs at yp = 2 in the elastic- Plastic case: lla = 7 = 0-318 (2-24) 2.4 Response to Unit Impulse Fourier transforms are used in finding the impulse response function--response of the system to unit impulse excitation. The Fourier transform of the equation of motion with q(t) = 6(t), where 6(t) is the Dirac delta function, has the form: -w?mH(W) + k*H(w) = 1 (2-25) where © H(w) = y(t) eat (2-26) and k* is the complex modulus introduced by Eq. (2-3). The function H(w) is called the frequency response func- tion of the system and can be expressed explicitly in terms of the complex frequency and the complex stiffness of the system, by virtue of Eq. (2-25), as W(0) = Tae Re (2-27) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 The denominator Z(W) = —mw* + ke, a complex function of the exciting frequency w, is known as the mechanical impedance of the system. The impulse response function h(t) is obtained by taking the Fourier inverse transform of Eq. (2-27): h(t) = oe (2~28) —_ 2 — —— oo m(w n ) In performing the integration, the method of residues may be used, as zeros of the mechanical impedance can easily be found. The poles of the integrand in Eq. (2-28) are obtained from w? - EE = 9 (2-29) the solution of which is Wy t = twovl + if (2-30) Wg = [E (2-31) is the natural frequency of the system in the absence of hysteretical damping. Use has been made of the relation (2-3) in obtaining Eq. (2-30). Expanding Eq. (2-30) in binomial expansion, the following is obtained: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 Wi lL. 1 i. w, = two(legilel+ glol*ergilel’- zal |**z5g41 61 Fees) = twol (1+ 3] e]?- olelteeeyshife] ae 4 alel?+ pagel" oe | or Wi Wo twp tiw i (2-32) in which wy = wo (Lt Zlel*- gogloltters 1 1 Ws = Zwo[e| (1- glcl*+ rogltltnee ) (2-33) Thus, the impulse response function of the system is ob- tained as n(t) = = (238) 1im(2—~). lim (=) -m W2W1 'W-We 4 WWe i . _ Ljexpli(urtiwi)t] , exp [i (rudd) b]. m 2 Wy a, _ tL _ i, ivrt_ -iwrt “it mu, | x (e e Je - 2 hit fog sin wyt (2-34) with the understanding that the impulse response function vanishes for negative arguments. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 For small [ze] or |z] << 1 WW, F Wo Jr ° a Ws i 2 the impulse response function simply degenerates to Lj h(t) = maT e- glslwotcin wt (t > 0) (2-35) 2.5 Response to Random Excitations Replacing the forcing function in Eq. (2-9) by a random process Q(t) and denoting the random response of the system by Y(t), the following stochastic differential equation of motion is obtained: mY (t) + k*Y(t) = Q(t) (2-36) The solution to Eq. (2-36), assuming that the random excitation begins at t = 0, is in the form of Duhamel integral: Y(t) = Q(t)h(t-t)dt (2-37) in which h(t-t) is the response of the system at time t due to the unit impulse excited at time t. Since h(t-t) = 0 for t >t, as was pointed out earlier, Eq. (2-37) may be written as: t Y(t) =m Xo (T)h (tT) at (2-38) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 in which Q(t) has been replaced by mass m times the ground acceleration process X,(T). The first and second-order statistical properties of the response process are of major importance in the investigations to be performed. The mean of the response process is simply ob- tained by taking the expectation of both sides of Eq. (2-38), i.e., t E[Y(t)] =m | E[Xo (t) Jh(t-t) dt (2-39) 0 The functions E[Y(t)] and E[X)(t)] are the first-order moment functions of Y(t) and Xp (t) respectively, and are called mean functions. The second-order moment function is defined to be the expectation of the product of two random variables, ¥(ti) and Y(t2), at two different instances in general, i.e., ti te E(Y¥(ti)¥(t2)] = | |, E(Xo (11) Xu (12) Jh (t1-T2)h(t2-T2) dt,dt2 (2-40) The second-order moment function of Y(t) and Xo (t) are denoted by Ryy (ty, ty) and Ry x, (tL! ty) and are called Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 the correlation function of Y(t) and Xp (t) respectively. Consequently, Eq. (2-40) can be rewritten as: ti pte Ryy (tirte) = m* Ry yo (THe t2)h(ti-T1)h(t2-Tz) dtidtz 0 jo (2-41) In obtaining the results in Eqs. (2-39) to (2-41), the liberty of interchanging the order of integration and expectation has been taken by assuming that the follow- ing conditions are met: t | | et (T)]h(t-T)dT| < © (2-42) 0 for all t, and ti -te | | Ryyyg (Tie T2)h (ei-Ti)h (ta-T2)dtidt2) < (2-43) u for all ty and to. The mean-square response of Y(t) is obtained by letting ty = to = t in Eq. (2-41): ELY* (t) ] tet ll 3 Ry y yo (TL T2)A(t-Ti)h (t-T2) dT i dltz (2-44) 0 ] The correlation function of the velocity Y(t) follows from Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 2*Ryy (ti, te) yy (t1,tz) = dt 10te (2-45) i.e., ti ty oh (ti-T1) oh (te—T2) oe = 2 os on ———— re | D y(tirte) m Ry i (T,,T2) ti 55 T10T2 (2-46) And the mean-square velocity E[Y?(t)] is then obtained by letting ty =t, = t in Eq. (2-46): 2 Rog (t,t) = BLY? (t)] trt , dh (ti-T1) dh (t2-T2) =m Ry i, (ti pT2)—3 EO HE att Ooju €,=t t2=t (2-47) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER IIT MULTI-DEGREE-OF-FREEDOM SYSTEM 3.1 Matrix of Impulse Response Functions All the basic results developed in the foregoing enable the investigation of the response of multi-degree- of-freedom structures to random excitations. The matrix notation is found most handy in dealing with this type of problems. The results for a multi-degree-of-—freedom structure, expressed in matrix notation, have perfect analogy to those of a single-degree-of-freedom structure. An n-degree-of-freedom shear beam structure with complex stiffness as shown in Fig. 8 will be investigated. The stochastic differential equation of motion for the whole structure is [uJ { xe} + [K+] ec) = {ace (3-1) where fu] = diag (Mi, Mo, eee, m,) (3-2) X, (t) X2 (t) {ce} = (3-3) X(t) 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig, 8 N-Degree-of-Freedom Shear Beam Structure with Comnlex 5tiffMmess Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 Qi (t) Mm; 7 Qe (t) M2 fore} =4. 0 Pays ) Korey (3-4) Q(t) mh, and * o* * k,+ke -ke k ok * kotks -k3 0 x oo * k3tky, —ky * [K ] = eovere (3-5) Symmetric * * _+* Ky-atky ka) * Ky Note that in the above expressions, X,(t) is defined to be the random displacement response of the i-th story relative to the moving hase, and Xo (t) is the random ground acceleration. The matrix of the frequency response functions, ([H(w)J, can be obtained by letting {a(t)} and {x(t)} in Eq. (3-1) be tb fore} = aiut (3-6) Ri eee Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 1 1 : fxcey} = tH(ay1 4: ) etvt (3-7) 1 Thus, «77H [H(w)] = Es [mu] + [kK i (3-8) The n by n matrix [Z(w)] = | -w [mu] + ral (3-9) is called the impedance matrix of the system. The matrix of the impulse response functions, [h(t)], is obtained from the relation: co In(t)] = 5 | [2(w)17*e" Faw (3-10) The matrices [H(w)] and [h(t)] are also n by n, and are perfect analogies to Eqs. (2-27) and (2-28). Again the method of residues may be used to evaluate the integral in Eq. (3-10). ‘The poles of [H(w)] are located by setting det |Z(w)| = O which is equivalent to det |[cC] - A[rI]| =o (3-11) in which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 * k Ki1 Kiz2 eee ki, m ‘Mm, Mm, *& *& * Kar Ka2 ,., Kan M2 Mg M2 - * : tcl = [uJ 7K) = | eee eee. (3-12) x * x Kn. kn2 Knn M, My Mn haw (3-13) and [I] is the identity matrix. This precisely is an eigenvalue problem. The poles of [H(w)] are determined first by finding the eigenvalues Ar, Any *°%, A, OF the matrix [C], then by a relation analogous to Eq. (2-32): Wn - 23-1 _ ay, tin... (j= 1, 2, ¢**, n) (3-14) jx jt where 1 (A5)2 =w5 tw, (j =1, 2, °**, n) (3-15) As the inverse of matrix [Z(w)] is well known to be [z(w)J7t = — fA (3-16) det]Z(w) | where [A(w) J adjoint matrix of [2(w)] (3-17) ll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and det|zZ(w) | {h(t)] then {h(t} J Application [h(t)] 35 | det| -w? fm] + [K']| i} det| fm] | - det] -w?[Z] + [c]| n 2 2n = (~1)" I m.+ 1 (w-w.) (3-18) jel 2 js. J becomes a ee ta (wet? ta (3-19) 27 n 2 2n (-1)" Im TI (w-w. ) of the method of residues yields 2n . i 5 (w-w,) [A(w) Jen?® = lim (3-20) n 7 k=1] ww 2n (-1)° TT m, k TI (w-w.) j=1 ja? assuming that all W'S are distinct. It is pointed out that proper sign for |¢| as suggested in Eq. (2-3) should be used in the matrix [Z(w)] in order to find each correspondent adjoint matrix [A(w)]. 3.2 Mean Square Response Matrices After the matrix of impulse response functions, {h(t)], is determined, the random response {x(t)| is simply given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. t 36 {x (4)} = | fh(t-t)] face at (3-21) 0 . The structure is assumed to be at rest prior to the excitation. Thus, the correlation matrix of {x (| af {xcers ffx cay | tipte - races] {acral ° otra} | [h(t,-12)] at, dt, is (18) tt [Ryy (1 rte )] Ul 0 0 ti -t2 = Re 3, (ta, T2) (h( tit) J Me] fh(te-td 1” o Jo dt;at2 (3-22) where T mi mi mM 3M, mm, seems Mz m2 Mam, M2M2 °* *MzM, [Mo] = . . = ee er (3-23) : : M,M1 MyM °° °mM My m m and Ry oxo (Tie t2 ) is the correlation function of the ground acceleration Xo(t). Use has been made of the relation Eq. (3-4) in obtaining Eq. (3-22) and the symbol "tT" stands for transposing the matrix to which T attaches. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 The mean square matrix is obtained by letting ti = te = t in Eq. (3-22): 2 [face] {co} J tt " [Rx (» €)] Ry x, (Tre t2) [h(t-11) ] [Mo] [h(t-s) ] Taniate ojo (3-24) The correlation and mean square matrices, respec- tively, of the relative story velocities, analogous to Eqs. (2-46) and (2-47), are t; toe [Reg (ti rte) ] = Ry x, (Tarte) goth ti-t.) ] [Me] o Jo Fey th (te-m2)] "at idee (3-25) and e e T [Ryy (t,t) ] = 2] { ke] { kee) t-t = Ry X_ (THe Te) epee) [M2] oju t1=t Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 in which the partial derivative eth (t) } of the impulse response function may be expressed explicitly as n+1 2n vs tmeun.) VT ekwt Pthity= y tim Story) Tato) le (3-27) T m. OMe oT (wew.s) je. J kel i Now, for the safety analyses of multi~degree-of- freedom structures to be performed in the following chapter, the mean square values of the relative displace- ments and velocities between adjacent stories are of major importance rather than those of the story displace- ments and velocities relative to the moving base. Denote the displacement and velocity of i-th story relative to i-lst story as ¥,; (t) and ¥, (t) respectively. It is clear that Y,(t) and ¥, (t), i= 1, 2, ***, n, have mean square values BIN, (t)] = BIX, (t)] + BX; y(t) ]- 2E(X, (t)X,_, (€)] (3-28) 2 2 .? 5 ° BIZ, (t)1 = BIR, (t)] + BLK, _)(t)] - 260%, (eR, y(t) (3-29) since Y; (4) = X; (t) - XxX 1) (3-30) f(t) = k(t) - &_, (8) (3-31) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 39 All the necessary informations on the right hand sides of Eqs. (3-28) and (329) are readily available from Eqs. (3-24) and (3-26). It is pointed out that the expressions thus far obtained in this chapter and previous chapter are per- fectly general in nature. No assumptions have been made as to whether the loading is stationary or nonstationary. The characteristics of the input process will be dis- cussed in the following chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER IV SAFETY OF STRUCTURES OF ELASTIC-PLASTIC MATERIALS SUBJECTED TO SEISMIC LOADING 4,1 Probabilistic Characteristics of Ground Motion and Structural Response The ground acceleration process X encountered at the base of the structure is, clearly, random and non- stationary in time. Unfortunately, there have not been enough actual strong ground motion records available for use in the structural response analyses. Attempts have been made by several authors in simulating artificial earthquake motions (19, 20, 21, 22). An expression sug- gested by Bolotin (13) and Shinozuka (23) is to be adopted as input function in this analysis. The ground acceleration Xp is assumed in the form: X (b,t) = Be toby (4) (t > 0) (4-1) where B is the intensity of the ground acceleration, g(t) is a normalized stationary Gaussian process with zero ensemble average; and a, a parameter characterizing the duration of the ground motion, #9, the undamped fundamental frequency of the structure. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4] The intensity B is a random variable with probability density function £, (b) assumed to be in the form: f(b) = 7 b/B (4-2) B i.e., the intensity B has exponential distribution. The parameter B is the mean value of the intensity. The number of earthquakes, Mavs occurring during T years is a random variable with the mean EM] and the variance ViM,J- Since the earthquake records in most of the regions are far from being complete and far from being sufficiently long enough such that the values of EM] and VIM] could be determined statistically. Thus ETM] and VIM] are to be computed based on an empirical equation suggested by Housner. Suppose that M,, is distributed exponentially with T probability density function f,, (m) defined as: Mp £ (m) = —~ e/ Mep (4-3) Mn Mi, Then EIMp] and Vim are easily found to be: “1/Mn an = if (4-4) E[Mp] H 3 e| a oO 2 E(M2] - (EIM,])" = My (4-5) VIMp] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 The mean value M , which is defined as the expectation T of earthquakes having magnitudes greater than M during T years, is described as (24) M, = A T No e W/R (4-6) where A is the felt area in square miles, No is the average seismicity of the region and the parameter R characterizes the seismic severity of the earthquake process. The magnitude M is a measure of earthquake magnitudes on the Richter scale. As to the characteristics of the response process, the maxima of the functions Yar i = 1,2,...,n, Will be denoted by Sj. The following approximate expression for the probability density functions of Sir due to S. O. Rice (25), will be used: s s? , f, (s) oO EX inode (i = 1,2,°*° ,n) i E[Y; (b,t)] 2E [3 (b,t) ] (4-7) with the corresponding probability functions gs? Fo (s) = 1 - exp —_——_-———- (i = 1,2,°°°,n) (4-8) i 2E[¥3 (b,t) ] The numbers of maxima per unit time are approximately given by (25): Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 02 1 1 ELY; (b,t)] 2 Nj (Prt) = oh er es (i = 1,2,°**,n) (4-9) ELY, (b,t)] Now, once the input function is defined, the structure response may be expressed more specifically. Since the correlation function of %) is in the form a te (titte) = 2 - - Ry | Xo (t1,tz) = B Rog te ti) (4 10) where Rog (tet) is the correlation function of g(t), the mean square matrices of {X(b,t)} and {X(b,t)} follow from Eqs. (3-24) and (3-26) respectively: tt 2| {xm.e0}{ xo,e7} | = B? NTE Rg g(t) [h(t-1,)] [Me] [h(t-t2)] atidte tet (4-11) T 5 | { Roee }{km,e] = B¢ eT two (tr ttal a (ce-ti) oju Z) ) T aa [h(t -<.1] [M2] ax [h(te-t2) J dti,dtz (4 i tet 2 ( dty ae? Vee , (4-12) In particular, the mean square displacement and velocity for a single-~degree-of-freedom system are, respectively, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 B E(y?(b,t)] =— ' we fag ite) exp [~2u,t+(w;-aw, ) (Ti+T2)]- , . sin w (t-1)) sinw (t-g)dtidt2 (4-13) and tpt o 2 E[y? (b,t)] = = Ryg(t2-T1) exp [-2w, t+(w;-owo) (titt2) ]- wW © jojo 2 * . : eiW. S <7 Ss t- “—W. Ss . rf ; Sin w At 2) Sin wf To) 0; W in{w -[2t-(titt2) Io, cos w (ton) cos u(tere)f at iars (4-14) Eqs. (4-13) and (4-14) are obtained directly from Eqs. (2-44) and (2-47), respectively, using Eq. (2-34). 4.2 Computation of Mean Square Relative Displacements and Relative Velocities Upon observing Eqs. (4-11) through (4-14) it is noticed that the mean square values of both relative dis- placement and relative velocity are expressed implicitly in terms of functions which are indirectly dependent on the mean square values themselves. It will be desirable that a practical procedure of computation be proposed in obtaining solutions for the mean square values. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 The following are described, in particular, for single-degree-of-freedom systems. For multi-degree-of- freedom systems, a procedure analogous to that for single- degree-of-freedom systems is readily visualized. It is obvious that the amplitude Ym of the random response Y is not deterministic. However, it shall be assumed that during each "cycle" of oscillation Y¥ is approximately sinusoidal in nature, i.e., ¥5(¢) = Yn Sin ot ((j-L)Ati<t<jAtiz; j = 1,2,3,...-) (4-15) where Q is an arbitrary constant, At, = 2n/Q, Y(t) and Yin. are, respectively, the displacement and the amplitude of the response within the j-th cycle. The mean square values of Ys within the time interval (j-1)At,<t<jAt,, then, are: pata 2 _ 1 | - 2m 4,24, . Le _ E(Y. (t)] Re, | Yn, sin t) dt = 5 Ym3 (4-16) 0 Thus, re Yn. = (2E[Y.(t)1)2 (4-17) J (j-L) At <t<jAti Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 It is clear from Eq. (4-17) that the amplitudes, Yn? j and thus the ductility factors of each cycle of oscilla- tion are related to the mean square displacement within that particular cycle. Thus, Eq. (2-22) is modified as: 4{1- B (y,. 14. ‘ i (us > 1) as [c5[ = (4-18) O Mu. < 1) where Yn, .=e a - Us Vy (4-19) Due to the dependence of [ol on time, the complex stiffness of the system then will also be time dependent. Furthermore, the complex frequencies, being functions of Ie.1, are also time dependent. The mean square value, EIY,), for each particular time interval (j-1)At,< t < Jat, is then computed by using the modified time dependent impulse response function in which the time dependent complex frequencies are inserted. Note that the mean square value BI] is dependent on impulse response function which is further dependent on the complex frequencies. But the magnitudes of the complex frequencies are affected by the magnitude of Ic5| 2 which is a function of Ely]. Thus, a numerical method Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 rather than an analytical method will he employed. The procedure goes as following: (1) Letting loi] = O, the complex stiffness and the complex frequencies for the first time interval are obtained. Then the impulse response function and thus the mean square values E[Yi] and E(Y1] are obtained. These quantities are to be considered as the mean square values at t = At. 2 (2) For the second time interval, E[Y1] is used to compute Ying and |t2|!. The same steps as in the first time interval are followed to obtain E[Yo] and E(Y21, the mean square values at t = 2At. (3) The same steps are used to obtain E(Y31, EY], E(Ya], ETY],°°° up to a reasonable length of time when the mean square values from then on are small. The procedure described above is considered as adequate and justifiable if the time interval At is picked small enough. 4.3 Probability of Structural Failures In the course of safety analysis of structures, several structural failure mechanisms can be encountered. Of these the most important ones are the following: 1. First-excursion failures: Structural failure occurs the very first time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 the response process reaches a certain fixed level. 2. Fatigue failures: Structural failure occurs due to an accumulation of damage. Damage to the structure accumulates as each excursion of the response process inflicts an increment of damage which is not large enough to cause first-excursion failure. Failure occurs when the accumulated damage reaches 100 per cent of fatigue life of the structure. For structures of elastic-plastic materials the probability of failure based on the first-excursion beyond the yield point will be overestimated, since an elastic-plastic structure usually does not fail under a single occurrence of stress in the plastic range. If, on the other hand, certain critical plastic deformation is used as the criterion of failure, the probability of failure may be underestimated. This is due to the fact that several cycles of plastic strain may develop before the critical state is reached; consequently, fatigue damage of the material may accumulate to the extent causing failure of the structure. In this section, the probability of structural failure is described based on the analysis of fatigue Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 failure proposed by E. Parzen (26). According to Parzen's analysis, fatigue accumulation under cyclic loading is treated as a renewal process (27, 28). It is an approach which avoids making assumptions about the probability law of N(s), the number of cycles to failure, and makes use of the mean and variance of N(s). The experimental data required in the course of analysis are easily obtained from constant~amplitude fatigue tests in the laboratory. The basic relations are the following. Let the amount of damage caused by the j-th cycle of stress amplitude s be D.(s). Then failure is said to occur if n * Y D.(s) > D (4~20) jal 7 where D™ is a number characterizing the fatigue strength of the material. The damage D. (s) is a random variable. It is assumed that the distribution of D, (s) is the same for every cycle j of stress s. The mean value of damage caused by one cycle of stress s is E[D, (s)] and the variance is VID, (s)]. The smallest integer n for which Eq. (4-20) is satisfied is called the fatigue life (in cycles) and denoted by N’(s). Note that N’ (s) is a random variable k * with the mean E[N (s)] and the variance V[N (s)]. Using Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 certain results of the renewal theory, the following relations are observed: * E([D; (s)J = —_>___ (4-21) EIN (s)] VIN“ (s)] .*2 VID, (s)] = D (4-22) E7 (N (s)] For a given material, the values of EIN’ (s)] and VIN’ (s)] presumably can be determined in constant-amplitude fatigue tests. The amount of damage caused by N(s) cycles of * stress s(N(s) < N (s)j)“ is assumed to be N D(s) =) D. (s) (4-23) j J 1 If N(s) is a random variable and the variables D.(s) are assumed to be independent, the mean and variance of D(s) are (29): E[D(s)] = E[N(s)].EI[D, (s)] (4~24) VID(s)] = E[N(s)].V(Di(s)] + VIN(s)].B7 [Di (s)] (4-25) Suppose now that there are several stress levels Sj with the number of cycles at each level being N(s;). Assume that the total damage D is the sum of partial damages D(s;), i.e., Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 = | D(s,) (4-26) L It may be shown that D has mean and variance E[D] = 2 EIN(s,)].B(Di (s;)] (4-27) 1 V[D] = % BIN(s;) 1.V[Di (83)] + ve v(s,) 1D. (63)) (4-28) where Di(s;) is the damage done to a given structure by one cycle of load application at stress level Si Substituting Eqs. (4-21) and (4-22) into Eqs. (4-27) and (4-28), the following expressions are obtained: els | = een ssaeen (4-29) D i EIN (Ss, )] VIN" (s,)] N(s.) _ v| Be = 2, B(N(s;)] ar) + V >—+_— (4-30) D i tw’ (s,)] [FEIN (s,)] The probability of failure is defined as the * probability of the event D/D > 1. Assuming asymptotic * normality of the relative damage D/D , the probability of failure of the structure is found as: ao 1 1, P, = prob Po> 1 = e” 2% dy (4=31) D v¥2T 7 where 0) 1-E [— ;] D 1 + pT (4-32) Misaii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 4.4 Fatigue Failure under Seismic Loading The theory presented in the previous section will be applied to the study of fatigue failure under seismic loading. The analysis is to be performed by consider-~ ing cycles of stress. The symbol Ss; will stand a maximum of stress at a critical section in the i-th story of the building. In an earthquake of intensity b, the number of maxima at S, = s, per unit time, follows from Eqs. (4-7) and (4-9): n, (s;b,t) ds £ _(s)dsn, (b,t) S i i Ss s? = Ny (b,t) — 7 ~—eXp] - —_—_— ds i E[S, (b,t) ] 2E[S; (b,t)] (4-33) where E[S; (b,t)] are the mean square values of the Maximum column stress. Since the relative story dis- placement processes Y, and the maximum column stress processes S;(b,t) have a relation analogous to Eq. (2-10), 2 L. W. = - - ¥,; (b,t) = 3Ed, S,; (b,t) on (4-34) the mean square stresses then can be expressed in terms of mean square relative displacements as: Reproduced with permission of the Copyright owner. Further reproduction prohibited without permission 53 2 ; 2 W. E[S, (b,t)] = (2221) -E[Y, (b,t) ] (zm (4-35) . 1 L due to the fact that the expectations of Y; vanish. The expected number of maxima at the level s during one earthquake of intensity b is Ty 2 Ni (s,b)ds = | N, (b,t) ——~—— exp|- ——=———_| dtds 7 Oe E(S, (b,t)] 2E[S; (b,t)] (4-36) where T) is the duration of one earthquake. The number of maxima at the level s during one earth- quake of any intensity is Ny (s). Its mean value is i E(N, (s)]ds = N, (s,b) £, tb) db ds (4-37) 1 U and the variance ° 2 vin, (s) | ds = 1s, (2,0 - E(N, (s)1| f, (b) db ds (4-38) i i 0 Let D be the amount of fatigue damage produced l. 1 at a critical section in the i-th story of the building, by one earthquake. The mean value of the relative damage * D4/> follows from Eq. (4-29), i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 [oo] D.. E(N,.(s)] E Ea = | —tiii as (4-39) ou LN (s)] k while the variance of D,,/D is obtained from Eq. (4-30) with the order of variance and summation interchanged, D,: * VIN, . (s)] v Ea = | pin,,(s)] UNIS) as + | = 43 — as (4-40) 0 E'[N (s)] oE [N (s)] The amount of fatigue damage Dit at a critical section in the i-th story, accumulated within a given time interval T is the sum of damages caused by con- secutive earthquakes. Thus, + Di. + ... + D,, ; (4-41) T As discussed in the first section of this chapter, Mn where M. is the number of earthquakes within T years. is a random variable with its mean and variance given by Eqs. (4-4) and (4-5) respectively. The partial damages Day have identical distributions and are in- dependent. Their mean values and variances are given by Eqs. (4-39) and (4-40), respectively. The mean value and the variance of the sum D,/D" of the relative damages are (29): D. Di; =| 3 = EIM,].E| —$ (4-42) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 D. D.. Q.. i _ li z li v| =| = 2 «v2 + VIM] Pas) | (4-43) D D D The probability of fatigue failure of a critical section in the i-th story within the time period T is is) Pp. = Prob ~ > 4 (4-44) Py p* ~ Again assuming asymptotic normality of the variables D;/D™, the probability of failure P,, can be evaluated i by using the relations (4-31) and (4-32). 4.5 Safety of Structures In this section, after the probability of fatigue failure of each critical section of the columns in each story is found, the safety of the entire building as a whole will be investigated. Denote the probability of survival of a critical section in the i-th story, the probability of the event b,/D* <1, as Poo i.e., i Po = Prob [D,./D* <1] (4-45) L, 1 Then, by definition P = 1-P (4-46) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 In each story of the building, there are four critical sections, the sections adjacent to the floor levels, with the same probability of survival. Since the events, that is, the survival of each section, are independent, the probability of survival P, of all the sections in the i i-th story will be = 4 . 4 - Pe = (P,, } 7 (2 Ps} (4-47) i i P. is the probability of survival of i-th story with i failure occurring in none of the sections in that story. Furthermore, since the events that each story will survive are again independent, thus, the probability of survival of the entire building is n n 4 Py (T) =, 1 TI 1-P (4-48) vs, *41 ! F,) 1 1 and the probability of failure of the building within a given period of T years is 4 n P, (7) =l1- P,(T) = 1 - il (1 - Pa) (4-49) isl iL Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER V NUMERICAL EXAMPLE 5.1 Scope of Investigation A typical four-story shear beam structure as shown in Fig. 9 is investigated. The structure has story height of 12 ft, bay width of 20 ft, with floor area of 20 ft by 20 ft, and a lumped weight of 40 kips per floor. The floor girders are assumed to be infinitely rigid. Suppose that ASTM A36 steel (oy = 36 ksi) is used for the structure. The modulus E of steel in the elastic range is known to be 29,000 ksi while Eye in the plastic range, is assumed to be one tenth of E, i.e., By = 0.1. The shapes of the columns are: 8 W 40, 8 W 35, 8 W 28 and 8 W 24 respectively in each story in the sequence starting from the ground floor. Their corre- sponding moments of inertia are 146.3 in’, 126.5 in’, 97.8 in* and 82.5 in’, The story shear stiffnesses of the structure are calculated to be: 34.1 kips/in, 29.5 kips/in, 22.8 kips/in and 19.2 kips/in respectively. The correlation function Rag (t) of the normalized input acceleration process g(t) assumes the form: 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5A 450. in* = 18.2 Kips /in 200 int 24.1 Kips Zin 250 in* 20.1 Kips /in 300 in* 36.2 kips fin Fig. 9 Example Structure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 T2 Rog(t) = (1 - sar) e 0.02 (5-1) which, as investigated by Shinozuka (30), shows close agreement with those from real earthquake records. The parameter a which designates the durations of the ground motions is set equal to 0.2. This value is picked such that the illustrative results could be ob- tained within reasonable computation time. It is important to note that the magnitude of |z5| determined for each time interval (Eq. (4-18)) does not take into account the damping due to the non- structural elements, in both the elastic and the plastic ranges. In this example, a constant value of le5l= 0.1 is added to the results obtained from Eq. (4-18) through- out the entire period of time. In evaluating M, let the felt area A = 50,000 square miles, the seismicity for the regions of California, Ny = 1.2 per square mile, and the severity R = 0.48. The felt area of 50,000 square miles is about one third of the area of the State of California. This factor of 1/3 is introduced due to the thought that some less intensive earthquakes occur in the areas remote from the particular site of interest would cause negligible or no damage to the structures under investigation. The value of M in Eq. (4-6) is set, from the engineering point of view, equal to 4.75 (24). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 Due to lack of reliable low cycle fatigue data, the values for E(N (s)] and vin (s)] will be assumed for the purpose of illustrating the method of computation in this analysis. The computation will be based on the well known formula ns? =c (5-2) where s is the stress amplitude, N is the number of cycles to fatigue failure, and b and c are positive material constants. For ASTM A36 steel, the S-N diagram obtained by Payvar and Vasarhelyi (31) in a series of constant ampli- tude fatigue tests will be used as reference. The values of E[N’(s)] at various stress levels within the range 21.3 ksi, the endurance limit, and 59.31 ksi, 90 per cent of the ultimate static strength of A36 steel, are assumed to coincide with the N values of the S-N diagram. The corresponding number of cycles are 10° and 107 re- spectively. For the range where the stress levels exceed 59.31 ksi, two different assumptions are made. (1) straight line extension of S-N diagram on the double logarithmic scale up to 65.9 ksi, the ultimate static strength; (2) connecting the points qio3, 59.3) and (1, 65.9) with a straight line, also on the double logarithmic scale, which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 designates that the expected number of cycles to failure at the ultimate static strength is l. The following relations are derived from Eq. (5-2) with the values of b and c properly determined (Fig. 10): for curve il, 21.3 s )) (21.3 CORRDY(N, NT) CORRVY(NSNT) ce Oe eee aes RN DAE a ED Sy DD SAD NS ND SOD RS ED SO RS SYS SD SEER nn NE AED SD SD SP SRD A SAD GEG CD OED NS ED GRE SY ED OD A A NY OO A ET SN ES OE A ED SED NAD SED GAD CD Ne DS Ge COMPLEX DETM COMPLEX OMEGA( 8,40) sDENOM(Bs40) 2AZ1 4542 320) sHKLU4, 4941) ,HV1(4, 4941 l ) ; 7 . i co3 06 ‘UOISssIWJad jNOYyWM paliqiyold UoHONpoJdeu JayUN4 “JauMO YHUAdoo ayy Jo uoIssiwad YM paonpoiday 1 1 COMPLEX CKM (4,4) yEIGV( 594) yOMEGAN(4) 921494) » ZMINOR(393) 2ADSZ (494) B(4,1),0MEG(8) DIMENSION CL(4),CEPTH(4),AREA(4) ,ALOAD(4) ,H( 422) DIMENSION TAU1L(41) 2TAU2(41),RG(41),AMM(4,4)3S(4y4) DIMENSION AK(5),W1T(4),STOR(100) ,YYIELD(4) ,DUCT(4),BETA(4),ZETA(5) DIMENSION HXR1(424) 2HXR2(494) sHVR1(494) sHVR2 1454) pRX(1494) 0 RX1(494) sRV(424) sRV1(494) DIMENSION CCRRX(454) »CORRV(424) sCORRDY(4,40) »CORRVY{4,40) COMMON/CO1/DELT,DELTAU, ALPHA, BMEAN,BETAL,ZETAL COMMON/CO2/CM, THETA COMMON/CO3/0MEGA,CENOM,AZ,HX1,HV1L COMMON/CO4/CKMsEIGV,OMEGAN,ZeZMINOR,ADJZ 2B, OMEG COMMON/CO5/ TAU], TAU2ZsRGyAMMyS COMMON/CO6/AK,WTeSTGRs YYITELD,DUCT, BETA,ZETA COMMON/COT/HXR1y,HXR2 FHVR1I,HVR2gRXyRX1RVIRVI COMMON/CO8/CORRX,CORRV, CORRDY 2» CORRVY NCHECK=0 N=4 NT=40 ALPHA=0.2 DELT=0.1 DELTAU=G.1 ZETA1=0.1 NML=N-1 Ni=N4+1 N2=Ne#2 NST=N#*(S#N+5) NZ=NT#N2 C04 COS | C06 CO7 C08 16 ‘UOISSIWad jNOYYM payigiyosd uoNONpoidas JeyNJ JauMo JYyHuUAdOD ay} Jo UOISsiWWad YIM peonpoidey 703 704 TIME=DELT#®FLOATINT) M=NT+1 READ(5-54) SIGMAY,BETA1,ELAST READ(5,54) (AK(1)sT=13N) READ(5,54) (WTET):T=1,N) READ(5,54) (AREA(T),1=1,N) READ(5,54) (CL(I),1=1,2N) READ(5,54) (DEPTH(1),I=1:N) DO 301 I=1,sM TAUL( I) =DELTAU#FLOAT(I-1) TAU2(T)=TAULIT) WRITE(6,7001) FORMAT (1H1,25X,1OHeeeeenaeee ) WRITE(6, 7002) FORMAT{26X,10HINPUT DATA) WRITE(6, 7003) FORMAT { 26X,lOHHHHeeHHHEE/ //) ' WRITE(6,701) OELT . FORMAT(7X,6HDELT =,F6.2) WRITE(6,702) DELTAU,TIME . FORMAT(5X_y8HDELTAU =;F6e2,9X,6HTIME =,F6.2/) WRITE(6,710) ALPHA,ZETAI FORMAT(//6X_ THALPHA =,F70e327X_yTHZETAL =,F602) WRITE(6,703) SIGMAY,BETAL,ELAST FORMAT (5X, 8HSIGMAY =7F4.0,10X,7HKBETAL =, F602,10X,7HELAST =,F720//) WRITE(6,704) (CLU1I),T=1,N) FORMAT (9X,4HCL =,4F10.3) _WRITE(6,705) (DEPTH(T),T=19N) ~ 26 “‘UO!SsILWad JNOYUM payigiyosd uoHONposdad JOYyUN4 “JOUMO 1YBUAdOO ayy Jo UOISsiIMUed YM psonpoidey = 705 FORMAT(6X_,7HDEPTH =,;4F10.3) WRITE(6,706) (AK(I).I=1,N) 706 FORMAT(9Xs4HAK =54F10.3) WRITE(6,707) (WT(1T),1=1l9N) 707 FORMAT(9X,4HWT =34F10.3) Cc WRITE(6,711) (AREA(I),I=1,N) 711 FORMAT(7Xs6HAREA =,4F10.3) DO 102 NW=1,N ALOAD(NW)=0. I=NW GO TO 1022 1021 I=I+l 1022 ALOAD(NW)=ALOAD(NW)+4+WT( I) IF(IeLT»N) GO TO 1021 102 CONTINUE DO 103 I=1lyN 103 YYILELD(I)=(SIGMAY-ALOAD(1)/(2.#AREA(I))) #CL(1)##2/(03.#ELAST#DEPTH 1(1)) WRITE(6,709) (ALOAD(I),I=1,N) 709 FORMAT(6X,7HALOAD =,4F10.3) © WRITE(6,708) (YYIELD(1),I=19N) 708 FORMAT(///5X,8HYYIELO =,4F10.3) DO 105 NG=1sM TAU=FLOAT(NG-1) *#DELTAU IF (TAU-025) 1051,1051,1052 1052 RGING)=0. GO TO 105 1051 RGING)=( Le-TAU##2/0201) #EXP (-TAU##2/0.02) 105 CONTINUE | —— _. , a. Lek. €6 ‘UOISSIWJad jNOYWM payqiyold UoHONpoJdeu JayNN4 “JauMO YBAdoo au} Jo uoIssiuad YM paonpoiday 106 153 152 501 104 101 DO 106 I = 1,N DO 106 J = I4N . AMM(I,J) = (WTCL)#WT(J))/0032.22126 ) #*2) IFC (CI-J).EQ.0) GO TO 106 AMM(J,1)=AMM(I J) CONTINUE I=] OM=WT(1)/(32.2#12.) I=I+1 IF (1.GT.N) GO TO 152 DM=DM#WT(1)/(32.2%12.) GO TO 153 CONTINUE KG=0 READ(5,51) GAMMA BMEAN=GAMMA#32.2#12.6 KG=KG+tl1 WRITE(6,60) GAMMA DO 104 I=1lyN DUCT(1)=0. OO 101 [=lyN ZETACI)=ZETAL ZETA(N1)=06 DO 1000 L=1,NT T=FLOAT(L) ®CELT LL=L+1 DO 107 l=1,sN DO 107 J=1,N CORRX(1,J}=0- 46 _C “UOISSILUAd JNOYWM payqiyoud uoNONpode JayLN4 “JauMO JYBLAdoo 84} JO UOISSILUJad YUM psonpoJday 107 CORRV(1,J)=0. CALL SQUARE(NaN1l¢N2_NST sMeh sy TyOMEGAF 2 DETM, ILL 2H) ITF(NCHECK.EQ.1) GO TO 11ll WRITE(6e71) L 71 FORMAT(// 3X_,3HL =,13) . WRITE(6,72) (DUCT(I),T=1)N) 72 FORMAT(/7X,6HDUCT =,4F10.4) WRITE(6273) (ZETAC1),I=1:N) 73 FORMAT(7X;,6HZETA =24F10-4) WRITE(6;91) 91 FORMAT(//6X,7HOMEGA =/) WRITE(6792) (OMEGA(K,;L),K=1,N2) 92 FORMAT(20X,E12.4,5X,E12-4) IF(L.EQ-1) GO TO 1002 c CALL CROSS(N»N2sMeloLloT ) 1002 CONTINUE DO 150 I=1,N BDO 150 J=1,N CORRX(1,J)=CORRX(1,J) *DELTAU## 22 BMEAN##2/4. 150 CORRV(I,J)=CORRV(1I,J) *DELTAU##2 #BMEAN##2/4. CORRDY(1,L)=CORRX(1,1) DO 117 I=2,N . 1L7 CORRDY(I,L)=CORRX(1,1)—-2.#CORRX (1, I-1)+CORRX(I-1L2I~-1) CORRVY(1,L)=CORRV{1,1) 00 217 I=2,N : . 217 CORRVY(1I,L)=CORRV(I,1I)~2-#CORRVII,I~1L)+CORRVCI-1l,I-1) WRITE(6,69) LyT,(CORRDY(1I,L).IT=1,N) WRITE(6269) LoTy(CORRVY( 19h) 2T=15N) S6 “UOISSILUAad JNOYWM payqiyoud uoHONpode Jayyiny “JaUMO yUBLIAdoO 94} JO UOISSILUJEd YUM paonpoJday 1000 CONTINUE WRITE(6,62) WRITE(6,61) DO 601 L=1,NT T=FLOAT(L)#DELT 601 WRITE(6:69) LeT,(CORRDY(IsL),1=12N) WRITE(6,65) GAMMA WRITE( 6,63) WRITE( 6,61) DO 602 L=1,NT T=FLOAT(L)#*#CELT 602 WRITE(6269) LyT,(CORRVY(1,L),1=1,N) WRITE{6,;65) GAMMA PUNCH 7000, ((CORRDY(I eh), T=1laN) ei =1eNT) PUNCH 7OOO, ((CORRVY(I,L),I=lyN) yL=1ly,N7) IF(KG.LT.«2) GO TO 501 STOP 1111 WRITE(6,801) 51 FORMAT(F10.3) 54 FORMAT (4F10.3) 60 FORMAT{1H1,11HFOR GAMMA =,F4.1///) 61 FORMAT( 5SXs4HTIMEs3Xe1LLHFIRST STORY,1X,12HSECOND STORY,2Xs1L1HTHIRD 1 STORY,1X,12HFOURTH STORY//) 62 FORMAT(1H1L,60HTHE MEAN SQUARE VALUES OF THE RELATIVE STORY DISPLAC LEMENTS #* ///) 63 FORMAT(1LH1,57HTHE MEAN SQUARE VALUES OF THE RELATIVE STORY VELOCIT 1IES # ///) 65 FORMAT(///1X,13H*® FOR GAMMA =,F4.1) 96 “UOISSIWJd JNOYWM paliqiyold UoHONpoJdeu JayN4 “JauMO YBAdoo ay} Jo uoIssiWad YM paonpoiday 69 FORMAT( LXsl2eFGe2y1XeEl2e4slXeEl2o4e lL Xs El 204,1X2E12.4) 7000 FORMAT(4E15.4) B01 FORMAT(//92H THE MEAN SQUARE VALUE OF THE RELATIVE STORY DISPLACEM LENT IS LESS THAN ZERO, JOB TERMINATED. ) STOP END L6 Subroubne : 98 SQUARE Generate. OMEGA ,DENOM K Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AXR1, HXR2Z AVRIL, HVR2 L2=LZ+1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 “UOISSIWEd }NOYUM peyiqiyold uoHonpoidar Joyuny “J8UMO JYBUAdOO ay} Jo UoISssiWuad YM peonpoiday | SUBROUTINE -SQUARE(NyNLsN2aNSTsMoly Ty OMEGAF »DETMy ILL eH) COMPLEX DETM,CMPLX COMPLEX OMEGA(8,40) »CDENOM( 8,40) AZ (4 _4p320) 9HX1 (494941) pHV1(424241 COB 1 ) COMPLEX CKM( 454) pEIGV(5 94) yOMEGAN( 4) 9Z64,4) y2ZMINOR (393) 2ADJZ(494) 5 COG Bl4,1) ,OMEG(8) cos l * SIMENS TON TAULI41) + TAUZ(41)3RG(41),AMM(4; ode Skas 4) DIMENSION AK(5).WT(4),STOR(100) sYYIELD(4) »,DUCT(4),BETA(4),ZETA(5) C06 DIMENSION HXR1(494) @HXR2 (494) pHVRI (44) pHVR2 (494) pRK6 494) 5 cov RX14494) »RV( 494) »RV1L(474) : DIMENSIGN CORRX(4,4) ,CORRV(4,4) sCORRDY (4240) »,CORRVY({ 4,40) cos COMMON/COL/DELT,DELTAU, ALPHA, BMEAN, BETA, ZETAL COMMON/CO2/0DM, THETA ~ SOMMON/CO3/0MEGA,0ENOM,AZsHX1yHV1 COMMON/CO4/CKM,EIGV,OMEGAN, Z,y ZMINOR,ADJZ 9B, OMEG COMMON/COS/TAU1,TAU2sRGsAMM;S COMMON/COG/AK,y WT ,STOR,YYIELD,DUCT,BETAsZETA COMMON/CO7/HXR1,HXR2 sHVRIsHVR2yRXPRXIL SRV ORV COMMON/CO8/CORRX,CORRV» CORRDY » CORRVY NM1=N-1 DO 1001 LZ=loLl LZ1=LZ+1 IF(L~EQ.1) GO TO 101 IF(LZ.LT.«L) GO TO 1500 DO 119 I=1,eN O0T “UOISSILUAd JNOY"M paygiyoud uoHONpodad JayyNy ‘JaUMO JYyBUAdOO aU} Jo UOISsIWUad YM peonpoJday IF(CORRDY(IT,L-1)-LT.-O2) GO TO Llil DUCT(1)=SQRT(2.*#CORRDY(I,L-1)) /YYIELO(T) IF(DUCT(I)-LE-~1-) GO TO 1191 CALL DAMP(N,N1,DUCT,HsBETA,ZETA,1) GO TO 119 ZETAC(T)=ZETA1 CONTINUE CONTINUE CALL FRONCY (NagN1leN2yNSTsAK,WTs ZETA, CKM,EIGV, STOR »OQMEGAN, LGMEGAF,GMEG ;L} 104 THETA=AL PHA#OMEGAF DO 104 K=1;K2 OMEGA(K,L)=OMEG(K) DO 105 K = 1,N2 I=l IF (K GT. 1) GO TO 1051 I= 2 DENOM(K+L)=DM#(OMEGA(KyL)-OMEGA(iyL)) I = I+l IF (1 .EQ. K) GO TO 1053 DENOM(K,LI=DENOM(KyL)#(OMEGA(K,L)—~OMEGA(I;L)) IF (I eLTe N2) GO TO 1052 CONTINUE K=0 K=K+1 KL=(LZ~-1) #*N24+K T=i CONTINUE TOT “UOISSILUAd jnoYyYM payqiyoud uoHonNposdes JayyN4 “JauMO JYyHUAdOO ay} Jo UOISsIWUed UyM psonpojdoy | J=1 9791 CONTINUE IF(I-J) 1031,1030;1032 1030 ZCI, J)=CMPLX((AK(I) +AK(141)),(-1. yee (KEL) #(ZETALT) #AKLT) #ZETACT +1) 1#AK(1+1)))-OMEGA(K,L) ##2*#WT(1)/(32.2*12.) GG TG 163 1031 IF((J-1)-NE.1) GO TO 1033 Z(1,J)=-CMPLX(AK(J) 2, (-1. ee (KFLIZETALIISAK(I}) GO TO 103 1033 Z(1,J)=(0.,0.) GO TO 103 1032 IF((I-J).NE.1) GO TO 1033 ZilsJ)=Z(J5e1) 103 CONTINUE J=J+1 IF(J LE. N) GO TO 9791 T=I+1 IF(I .LE.~ N) GO TQ 93790 4 CALL ADJNTI(N, NMLeZsZMINORs DETM,ADJZyBs ILL) DO 111 I=1l)N DO 1li J=l1,N oo a L111 AZ(T,JrKULI=ADIZII, J) IF(KeLT~NZ2) GO TO 2000 1500 CONTINUE . CALL HMAT(N)N2oM,LZeLZ1,TsTAUL) DO 12 Li=LZyLZ1 DO 12 L2=LZsLZi 00 120 I=1,N DO 120 J=1,yN OT "UOISSIW9d yNOYM poyiqiyosd uoHOnNposdas JayN J “JaUMO yYyHUAdOD ay} Jo UOISSIWWed YM pesonpoiday 120 1321 1322 132 HXR1(1,J)=REAL(HX1( 1 9J9L1)) HXR2( 1 ¢J)=REALCHX1( 1 eJyb2))- HVR1(1,J)=REAL(HVI( 1 2J2L1)) HVR2(1¢J)=REAL(HV1IC TI eJ9b2)) CALL SCMPRD(AMM,HXR2,S;N) CALL SQMPRD(HXR1L»,S?RX 1N) CALL SQMPRD(AMMyHVR22SN) CALL SQMPRD(HVR1,SsRV 2N) TAU=ABS(TAUZ(L2)-TAUL(L1)) DO 132 I=1lyN DO 132 J=17N TF(TAU-0.5) 1322%1322,1321 RX(1,J)=0- RV{I,J)=0. GO TO 132 EXPOQ=EXP(—THETA#(TAULI{L1)+#TAU2Z(L2))) NG=TAU/DELTAUt+1.5 RXCIL,J)=RXC1,5) #RGING) #EXPO RV(1I,J)=RV01,3) #RGI NG) #EXPO CONTINUE , DO 133 I=lyN DO 133 J=1,N CORRX(1,J)=CORRX( 1, J)+RX( 19d) CORRV(1,J)=CORRVIT 2s J)4+RVI 12d) CONTINUE CONTINUE RETURN NCHECK=1 RETURN END cot 104 S ubroubine : CROSS Generale Call Compute UXR1,HXR2 SQMPRD HVR1,HVR2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ‘UOISSIWEd yNOYUM payqiyod UoHONpoJdad JayN4 “JaUMO JYHUAdOD ou} Jo UOISSIWed YM paonpoidey SUBROUTINE CROSS(NsN2sMyLyiboT) COMPLEX OMEGA( 8,40) »DENOM( 8240) 2AZ (4242320) pHX1 (6424241) 2HV16-454241 1 ) DIMENSION TAUL(41)_TAUZ(41) »RG(41) 2AMM( 454) 951494) DIMENSION HXR1(494) 2HXR2(494) sHVR1 (454) 2HVR21 494) RX 404) 1 RX1(494) 9RV(494) 29RV1(454) DIMENSION CORRK(4,4) »CORRV( 494) sCORRDY( 4340) ,CORRVY( 4140) COMMON/CO1/CELT yDELTAU, ALPHA, BMEAN,BETAL sZETAL COMMON/CO2/CM_ THETA COMMON/CO3/O0MEGA,DENOM; AZsHX1,HVL COMMON/COS/TAUL,TAU2s;RG,AMM,S COMMON/COT/HXR1gHXR2 pHVR1 yg HVR2¢RXe2XLeRVIRVI _ COMMON/CO8/CORRX,CORRV,CORRDY,CORRVY | DO 1400 KS=2,L LZ=KS-1 CALL HMAT(NsN2pMgLZeLL »T,TAUL) - DO 141 LS=KSyL a . LS1=LS+1 DO 14 L1=LZ.KS DO 14 L2=LS,LS1 DO 140 I=1,N DO 140 J=lyN HXR1( Ie J)=REALCHX1( I ed eh Ll) ) HXR2{1,J)=REAL(HX1(19JdyL2)) HVR1(IeJ)=REALCHVL(IeJeL1)) . 140 HVR2(1,J)=REAL (HVLC I sJyLh2) ) cQ3 co5 CO7 cos SOT “‘UOISSIWWAd JNOYYM paygiyosd UoHONpodal JeyUN4 “JoUMO JYyBUAdOD aU} Jo UOISSIWed YM paonpoidey 142 1321 1322 132 133 14 141 1400 CALL SGMPRD(AMM,HXR2,S,N) CALL SGMPRDCHXR12Se¢RX oN) CALL SGMPRD(AMM,HVR2;5;N) CALL SQMPRO(CHVR1sSeRV 2N) CALL SGMPRD(AMM,HXR1,S,N) CALL SEMPRD(HXR2,SyRX1_N) CALL SQMPRD(AMM,HVR1,S53N) CALL SCMPRDO(HVR2,SsRV1,N) DO 142 J=1,N DO 142 [=l,)N RX(IpJJ=ERX CIyJI+RX1LGI 9d) RV(ITeJI=ERV (1,J)4+RVICI,J) TAU=ABS (TAU2Z(L2)-TAUL(L1)) DO 132 [=1,N DO 132 J=1,N IF(TAU-0.5) 1322,1322,1321 RX(I3;J)=0. RV(1,J)=0-4 GO TO 132 EXPO=EXP(-THETA#(TAUL(L1)+TAU2Z(L2))) NG=TAU/DELTAUt1.5 RX(1sJ)=RX( 1 eS) #RGING) EXPO RVCIeJ)=RV( 1.5) e#RGING) EXPO CONTINUE 0Q 133 [=1,N DO 133 J=l,N CORRX(1,J)=CORRX(1I,J)+RXC 19d) CORRV( I,J) =CORRV( I, JItRV(II2 JS) CONTINUE CONTINUE CONTINUE RETURN ; Soca rte ce END “UOISSIWJEd JNOYYM payqiyoud uoHonpojdes JayN4 “J9UMO JYHUAdOO ay} Jo UOIssiUad YUM psonpolday AAAMAAAN 100 40 42 43 45 SUBROUTINE DAMP(NsN1,DUCT»H,sBETAsZETA,1) eT CTT TT TTT eT eT eee eee eee Te Tee Perec ieee ree res COMPUTE THE IMAGINARY PART OF THE COMPLEX STIFFNESS ROOTS OF THE CUBIC EQUATION ARE FOUND BY BAIRSTOW,S METHOD FUNCTIONS CONVERGE FOR BETA1=0.0001--0.99, P1=Q1=8. Tee eS SEEPS RSE EEE ERPS SE ERE SEES EP ERE RE SAL RAR RRS ES RARER EO DIMENSION A(3) 9803) »C(3) sDUCTIN) sHIN:2) »BETAIN) sZETA(NL) COMMON/CO1L/DELT»DELTAUs ALPHAs BMEAN, BETALsZETAL P1=8. Ql=8. | EPSON=.0001 DELDUC=.1 KK=0 KK=KK+41 NP=3 Al(1)=~1.5 A(2)=(3.+DUCT(1I)*#BETAL/(1.-BETA1))/4. A(3)=BETAL#(1.-DUCT(1))/(8.#(1.-BETA1)) IF(NP~1) 80;42,43 D={-1.)#A(1) GO TO 80 TFINP=2) 45,745,546 P=A(1) Q=A(2) GO TO 8 LOT “UOISSIWUJEd JNOYYM payqiyoud uoNONposdes JayN4 “J9UMO JYHUAdOO ay} Jo UOIssiUed UM psonpolday 46 51 83 P=P1 Q=Q1 M=1 B(1)=A(1)-P B(2)=A(2)-P*B(1)~-Q DO 6 K=3_NP B(K)=A(K)—P#B(K-1)-Q*B(K-2) L=NP-1 C(1)=B(1)-P C(2)=B(2)-P#C(1)-Q DO 7 K=3¢L C(K)=B(K)-P#C(K-1)-Q#C(K-2) CBARL=C(L)-BIL) IF(NP-3) 70371;70 DEN=C(NP~2) #C (NP-2)-CBARL GO TO 72 DEN=C(NP=2) #C (NP~2)-CBARL#C (NP-3) IF (DEN) 75,48,75 IF(NP=3) 47973547 DELTP=(B(NP—1) #C(NP—2)-B(NP))/DEN GO TO 74 DELTP=(B(NP=—1) #C(NP—2)-B(NP)#C(NP-3) )/DEN DELTQ=(B(NP)#C(NP=2)—B(NP=1) #CBARL)/DEN P=P+DELTP Q=Q+DELTQ ABSDP= ABS(DELTP) ABSDQ= ABS(DELTQ) SUM=ABSODPtABSDQ IF (M-1) 80;,83,84 SUM1L=SUM GO TO ll got "UOISSIWW9d jNOUYWM peyqiyosd uoHONposdad JayyN4 “JauMoO 1YyHUAdOD ay} Jo UOISSILUAd UM pesonpoiday 84 IF (M-5) 11354,11 54 IF (SUM-SUM1) 11,55,55 LL IF (SUM-EPSON) 8,8,53 53 M=M+1 GO TO 51 §5 WRITE (6,57) 57 FORMAT(9X,s,69HFUNCTIONS DIVERGING IN SUBROUTINE DAMP =-=- TRY NEW VAL LUES OF Pi AND Ql ) GO TO 80 8 D=-P/2. F=Q-P#P/4. IF (F) 30,30;31 30 AF=ABS(F) E=SQRT(AF) T=0 S=E E=0. D=T+S NP=NP=1 S=(-1.)#S D=T+S 32 CONTINUE NP=NP-1 IF(NP) 80,80;81 31 AF=ABS(F) E=SQRT (AF) E=(-1.)+#E NP=NP—~1 GO TO 32 81 00 82 K=1,NP 82 A(K}=B(K) 60T “‘UOISsILWAd JNOYUM payigiyosd UoHONposdad J9YyUNy “JOUMO JYyBUAdOS au} Jo UOISsIMUed YUM psonpoidoy GO TO 40 48 WRITE (6:49) 49 FORMAT(65HDIVIDED BY ZERO IN SUBROUTINE DAMP —- TRY NEW VALUES OF LPL AND Q1 ) 80 CONTINUE H(IT,yKK)=0 DUCT(I)=DUCT(1I)+CELDUC IF(KKeLE~1) GO TO 100 DUCT(1)=DUCT(1)-2.#DELDUC BETACT) =(LlLe/ (Lem 2e#H( 192) Pm Le/(1le-2e.4#H(1,1)))/DELDUC ZETACI) =(4e8#(1-—-BETA(I) )/3214159) #(DUCT(1)-1L.)/(DUCT(1)##2)+ZETAL RETURN END OTT “UOISSILUad jNOY"WM payqiyold uoNoNpode JayyN4 “JaUMO JUBUAdOO au} Jo UOISSIWUad YM peonpojday red AAMAANMAAN o SUBROUTINE FRQNCY (NygNlLeN2_eNST¢ AK, WT, ZETAyCKM,EIGV,STOR,OMEGAN, LOMEGAF,QOMEG ;L) 101 1027 1028 J= 1021 1022 1023 1024 ME RE EE EE COMPUTE THE UNDAMPED FOUNDAMENTAL FREQUENCY AND COMPLEX FREQUENCIES OMEGAF--UNDAMPED FOUNDAMENTAL FREQUENCY OMEG----COMPLEX FREQUENCIES SRR MRM MERE RE REET HEHSHE TELS HKESHEESEGHRERT EH EHH UNITS ARE IN INCH-KIP-SECOND COMPLEX CMPLXsCSQRT + CKM(NsN) yEIGVIN12N) sOMEGANIN) ,QMEG (N2) DIMENSION WT(N) sAK(N1)» STOR(NST) sZETA(N1) , LN=2# (N41) DO 101 [=l1,N1 DO 101 J=1,N EIGV(I,J)=(0.,0.) I=0 I=I+1 J=0 J+1 IF (I-J) 1021,1024,1025 IF ((J-1I)-NE.1) GO TO 1023 CKM( 1, J)=-CMPLX(AK(J) #3222412. JWTUL) pZETAC J) #AK( J) #32. 2*12./WT(1)) 60 TO 102 CKM(1I,J)=(0.,04) GO TO 102 CKM( 1, J)=CMPLX(CAKCI) FAK (I 41) )#32.24120/WT( IT) ,(ZETAC I) #AK( TI) +ZETA TIT “UOISSILUIad jNOYyM payiqiyosd uoONposdas JayUNy “saUMO jYyBUAdOO BY] JO UOISSILUad UM paonpasdey LCI+1)#AK(14+1))#32.24#12./WT(1)) 1025 1026. 102 104 GO TO 102 IF ((I-J)-NE.1) GO TO 1023 CKM( 1,5) =-CMPLX(AK(1)#32.2"12./WTU1) sZETACT) #AK( 1) #32.28#12./WT(1)) CONTINUE ; IF(J.LT.N) GO TO 1028 IFCI.LT.N) GO TO 1027 CALL RWGE3I (CKMy1lsNaNsOsN2,EIGVyLNeNEIG,STOR) DO 103 I = 1,N OMEGAN(I) = CSQRT(EIGV(1,1)) IF(L.NE.1}) GO TO 1033 T=1 OMEGAF=REAL (OMEGAN(1)) T=I+1 IFC I.GT.N) GO TO 1033 IF COMEGAF.GTe-REAL(OMEGAN(T))) GO TO 1031 CONTINUE GENERATE COMPLEX FREQUENCIES DO 104 I = IsN K = 28] OMEG (K-1)=OMEGAN(T) OMEG (K)=CMPLX(-REAL (COMEGAN(1)) sATMAG(OMEGAN(1))) CONTINUE RETURN END zqt Subroutine ; 113 ADINT My Miz Mat, M2 Call Subroutne CSLECD Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. “‘UOISSILWAd JNOYYM payigiyod UoONpoidal JeyN 4 “JaUMO ]YBUAdOD Oy} JO UOISsIWWAd YM peonpoidey oO anNnaga 2011 2021 1003 2012 1005 2022 SUBROUTINE ADJNT(N,NM12ZsZMINOR;DETM,ADJZ,8, ILL) SESE ME TE He ME HE HE SE TE ME TE TE HE SE SE HE Se FE HE SE SE AE SE TE SE SE AE SE HE ESE SE EE EE SS SE EE HE HE SE Se Et EE COMPUTE THE ADJOINT MATRIX ADJZ OF A COMPLEX MATRIX Z Pee CPT TTP e Te eee eee ee eee tee etree ere Tie eee TS | COMPLEX Z(NsN)sZMINOR(NM1,NM1) sADJZ(NeN) »BIN,1) »DETM DO 200 I=1,N , ; Lome DO 200 J=1,N IM1=I-1 JM1=J-1 IFCI.EQ-1) GO TO 1001 - IFCI.EQ.N) GO TO 1002 IF(J-EQe-1) GO TO 1003 , ee IF(J-EQ.N) GO TO 1004 DO 2011 MI=1,i1M1 BDO 2011 MJ=1,JM1 ZMINOR (MI eMJ)=Z(UMI MJ) DO 2021 MI=IyNM1 DO 2021 MJ=l,JML oo oo. ZMINOR(MIsMJ)=Z(MI+19MJ) 00 2012 Mi=1l;IM1 DO 2012 MJ=J,NM1 ZMINOR(MI eMJ)=Z0M1I yMJt+1) DO 2022 MI=IyNM1 00 2022 MJ=J,NM1_ ae _ ZMINOR (MI sMJI=Z(MI4+1 2MJ+1) GO TO 2040 ae "UOISSILUJ8d yNOUWM payiqiyojd UoHONpoJdeu JayN4 “JauMo }YHuUAdOo au} Jo UOISsIWad UM paonpoiday 1001 2122 1006 2121 1002 2111 1007 . 2112 1004 2221 1008 2211 2040 200 IF(JsEQ.1) GO TO 1005 IF(JsEQ.N) GO TO 1006 DO 2122 MI=I,NM1 DO 2122 MJ=J,NML ZMINOR(MI»MJ)=Z(MI+1,MJ41) DO 2121 MI=I,NM1 DO 2121 MJ=1,JM1 ZMINOR(MIyMJ)=Z(MI+1sMJ) GO TO 2040 IF(JsEQ.1) GO TO 1007 IF(J.EQ.N) GO TO 1008 DO 2111 MI=1;1M1 pO 2111 MJ=1,JM1 ZMINOR(MI,MJ)=Z(MI gM) DO 2112 MI=1,IM1 DO 2112 MJ=J,NM1 ZMINOR(MIgMJ)=Z(MI pMU+1) GO TO 2040 DO 2221 MI=I,NM1 00 2221 MJ=1,JM1 ZMINOR(MIgMJ)=Z(MI +1 MJ) DO 2211 MI=1;IM1 DO 2211 MJ=1;JM1 ZMINOR(MIyMJ)=Z(MI9MJ) CONTINUE CALL CSLECD (ZMINOR,¢NM1,8,0,DETMy ILL) ADJZ (Je T)=C-1.) #140) #DETM CONTINUE RETURN ENO Stt “UOISSILUAd JNOY"M payqiyoud uoONpode JayyiN4 “JauUMO JUBLIAdOO 84} JO UOISSILUJEd YUM paonposday 1 1091 1092 109 SUBROUTINE HMAT(N sg N29MeLZettlyT,TAUL) COMPLEX E,AC,CEXP COMPLEX OMEGA( 8240) ,DENOM( 8,40) 5AZ (4942320) 2HX1 (1424941) sHV16494%241 CO3 ) DIMENSION TAUL(M) COMMON/CO3/CMEGA; DENOM, AZ sHX1eHV1 DO 109 LI=LZ,LL1 DO 109 I = 1,N 00 109 J = IyN HXLCIyJ,L1)=(0.20.) HV1( 1, J3aLbL1)=(002C0.e) . IF(TeLTeTAUL(L1I)} GO TO 1092 DO 1091 K=1,N2 KL=(LZ—-1) #*N2+K E =CEXP((0e,1.)*OMEGA(K,LZ)#(T—TAUL(L1L))) AD =AZ(I_JyKL) #E/((—-1e) ##N*#*DENOM(K,LZ)) HX1( I pJeLL)=HXL(II,J3,L1)4+(00,1.) #AD HVL(I_¢JeL1)=HV1L(IeJeL1L)-OMEGA(K,LZ)#AD . CONTINUE HX1(J,T,LLI=HX1(01T,JSeh1) ; \ HVL(J2TeLLJ=HV1L(I,J,L1) , CONTINUE RETURN END git “UOISSILUAd jNoY'M pa}iqiyold uoHONpoJded JayyN4 “JOUMO JYBUAdOD ay} Jo UOISsIWUed YM paonpoidey 101 102 SUBROUTINE SQMPRD(A,BsS:N) DIMENSION A(NeN) sB(NeN) ,S(NQN) DO 101 .I[=l+N DO 101 J=1,N S(I,J)=0. DO 102 L[=1,N DO 102 J=l,N DO 102 K=1,N S(T, JIHSCT eS) +ACIK) BK) RETURN END LIT | “‘UOISSILWAd JNOYUM payigiyosd UoHONposdad JaYyUNy “JOUMO JYyBUAdOS ayy Jo UOISsiIMUed YM psonpoidoy AOAMAN 18 20 21 SUBROUTINE CSLECD{ Ay Ms Bs Ny DETy ILL) $e 6 SE BE SE AG gE SE AE Se SE SESE SE HE GE HE JE HE HEE aE HE HE HE Ea aE HEE aE SEH HE AE EE THE EE SOLUTION QF COMPLEX SYSTEM OF LIN.EQUAT.WITH N RIGHT HAND VECTORSKWJCSLED ANO/OR COMPUTATION OF COMPLEX DETERMINANT KWJCSLED SEKGLAHA RARER GE RRAEEEHREREREEEEEHEGRAEERRHEERARR AREER AE EE HE KWJCSLED DIMENSION A(MgM )}e BUMei ) COMPLEX Ay Bs ATs FAC, DET KWJCSLED ILL= 0 KWJCSLED CALL OVERFL(1I0O) KWJCSLED SIGN= +1 KWJCSLED IMA= M-1 KWJCSLED KWJCSLED DO 35 [=l,IMA KWJCSLED AMAX= REAL(A(IsI)) * REALCACT21)) + AIMAG(A(T,1I)) * AIMAG(A(T21)) KWJCSLED JMAX= I KWJCSLED Ii= I+1 KWJCSLED DO 20) J=112M KWJCSLED ARE= REALCA(J,1)) KWJCSLED AIM= AIMAG(ALJs1)) KWJCSLED AJI= ARE*ARE + AIM#AIM KWJCSLED IF(AMAX-AJI) = 18,201220 KWJCSLED AMAX= AJI MWJCSLED JMAX= J KWICSLED CONTINUE KWJCSLED IFCAMAX) 21,90,21 KWJCSLED TFCI~JMAX) =23425,23 KWJCSLED KWJCSLED 23 SIGN= =SIGN ett “UOISSILUAd jnoYM paygiyoud uoHoNpode JayyN4 “JauMO JYyHUAdOD ay} Jo UOISSIWUad YIM peonpojday A Aa” 24 241 25 30 32 35 40 DO 24 K=I_9M KWJCSLED AT= AC(I,K) KWJCSLED ACI,K)= ACJUMAX,K) OS oo - a KWJCSLED AUC JMAX,K)= AT KWJCSLED IF(IN.LE.O) GO TO 25 © KWJCSLED DO 241 K=1,N KWJCSLED AT= B(I,K) . KWJCSLED B(I,K)= B(JMAX,K) KWJCSLED B(JMAXsK) = AT KWJCSLED DO 35) J=IlyM KWJCSLED FAC= A(J,I)/AC11-1,I11-1) KWJCSLED DO 30 K=I1,_M . KWJCSLED AlJdeK)= AlJSyK) — FAC#A(I2K) 7 KWJCSLED IFIN.LE.0O) GO TO 35 KWJCSLED DO 32 K=1,N . KWJCSLED BlJ,K) = BlJ,K) —-FAC#B( 12K) KWJCSLED CONTINUE KWJCSLED KWJCSLED TRIANGULAR MATRIX READY KWJCSLED KWJCSLED IF(N.LE.O) GO TO 70 7 KWJCSLED KWJCSLED IF( (CABS(A(M,M)))-EQe- O.)2 6&0 TO 90 DO 40 K=1,N KWJCSLED B(MsK) = Bi M,K)/: AUM,M) gKWJCSLED DO 60 I=1,IMA “KWJCSLED J= M-1 . . . . KWJCSLED Kl= Jel KWJCSLED DO 50 K=K1,M KWJCSLED DO SO L=1yN _ -—«KWJCSLED 6TT “UOISSILUAd JNOY"M payqiyoud uoNONpode JayyiN4 “JauUMO JUBLIAdOO 24} JO UOISSILUJed YUM psonpoJday nd 50 60 70 74 90 92 B(JyL)= Bldel) -AlJsK) *#BIKsL) DO 60 L=l1,N BlJeL) = Bidet) / AlJeJ) DET= All+1) DO 74 1=29M DET= DET#A(I,1) DET= DET# SIGN CALL OVERFL(1i0) IF(IO.EQ.-1) GO TO Yi RETURN DET= (0.70.2) WRITE(6;92) FORMAT(46HODET A = 0 OR OVERFLOW ILL= -1 RETURN END IN SUBROUTINE CSLECD ) KWJCSLED KWJCSLED KWJCSLED KWJCSLED KWJCSLED KWJCSLED KWJCSLED KWJCSLED KWJCSLED KWJCSLED KWJCSLED KWJCSLED KWJCSLED KWJCSLED KWJCSLED KWJCSLED —KWJCSLED ozt 121 Main program 2 Compute, PFSECT, PFSTRY EDAM,VDAM, ETA Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. "UOISS|WJad jNOYWM poeyqiyosd uoHonNposdas JayyNng “sauMO WYyHuAdOD au} Jo uolssiuued uM psonpojdey a FORTRAN IV 0001 0002 . 0003 0004 0005 0006 0007 AAAANMNAMAAAMAANANHA -MODEL 44 PS VERSION 3, LEVEL 2 DATE WED MAY 7, 1969 SE EN Be BE SHC OS 2K FIC OK ONE fe ae OK Oe IE ae ic ae Ok FHS oe aE NE IR IS Be Oe NE Be BE NE AE he OIE EE He Oe OE OE ale ONE NE OC OIE OE RE OE OE OE NE EK OE EE OE OE OIC OKC OE HS COMPUTE THE PROBABILITY OF FAILURE OF N-STORY SHEAR BEAM BUILDING PFSECT(I) = PROB OF FAILURE OF A SECTION IN THE I-TH STORY PLSECT(I) = PROB OF SURVIVAL OF A SECTION IN THE I-TH STORY PFSTRY(I) = PROB OF FAILURE OF THE I-TH STORY _ PLSTRY(I) = PROB OF SURVIVAL OF THE I-TH STORY - PFBLDG = PROB OF FAILURE OF THE BUILDING PLBLDG = PROB OF SURVIVAL OF THE BUILDING ; Be afe Be KE ae ae ae Ae ae oe ahs RC AR ae a aie oe ae i he ae Be ae Be he he a i i oe RE OK ie SEE fe OE OE OE ONE Oe IE AE IC ONC COC SE OE OE NE OK AS OE OE OK EO IS UNITS ARE IN INCH-KIP=SECOND DIMENSION WT(4) sAREA(4) 4CL (4) 9 DEPTH(4) sCORRDY (494052) LCORRVY (494092) yALOAD(4) ¢B(10) sDYSQ(4940710) 2VYSQ(4240,10), 28SQ(4440¢10) sANO(4¢405 10), PDFB(10) »S(47),ENS( 50),VNS( 50)¢ 3F1(40),ANSB(47 910)9F2(10)-¢EN1S (47) 9VN1S(47) 9 ENL (47) »F3(010)¢ 4F4(47) 9F5(47) 9F6(47) 2ED1 (4) 9VD1(4) ,ETA(4) » EDAM(4) »VDAM(4) » SPFSECT(4) sPLSECT(4) » PLSTRY(4) gPFSTRY (4) GAMMA (2,5) ,PFBLDG( 295) EQUIVALENCE (ANO,VYSQ),_ (SSQsDYSQ) - N=4 NT=40 NB=10 NS=47 A=500006 eat ‘Uolssiuuad jnoyM payqiyoid uoONposdas JO4YN “J9UMO IYBLAdoo ay} Jo UOIssIWad YUM psonpoiday YEAR=100. SEISM=1.2 RICHT=4.75 SEVER=0.48 EM=A*YEAR*SEISM*EXP(-R ICHT/SEVER) VM=EM¥*2 DEL T=0.1 BETA=0.1 DELB=BETA*32 -2%1240 DELS=1.0 READ(5,51) ELAST READ( 5,54) (WT(1I),IT=1,N) READ(5,54) (AREA(1),I=1,N)_ READ(5,54) (CLI) ¢IT=1—9N),(DEPTH(I),IT=1,N) READ(5 +541) (C(CORRDY(T obo IC) oe T=l9N) oh =1940) ,1C=1,2) READ(5,541) (C(CORRVY(Iy le IC) s1=leN) pL =1940) yIC=1y92) WRITE(&,690) WRITE(6, 701) WRITE 6,702) WRITE(6, 703) WRITE( 6571) ELAST WRITE(69 72) (WTC I), IT=19N) WRITE( 6975) (AREA( I)» I=1y9N) WRITE(6973) (CLI) ¢1=19N) WRITE( 6574) (DEPTH( I), I=19N) DO 102 NW=L1,N ALOAD(NW)=0. I=NW ‘ “GO TO 1022 1021 I=I+1 | 1022 ALGAD(NW)=ALOADINW)+WT(T) “UOISSIWEd }NOYWM payqiyosd UoNONposdau JoyUNy “JOUMO JYHUAdOO au} jo UoIssiuued YM peonpoiday 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054 0055 0656 0057 0058 0059 0060 0061 0062 0063 0064 0065 0066 IF(I.LTeN) GO TO 1021 102 CONTINUE 2011 2012 2013 201 DO 201 IB=1,NB B(1B)=FLOAT(1IB)/10.*32.2*12. DO 201 [=1,N ; 00 201 L=1l,NT IFC FLOAT(IBI-O43/BETA) 2011%9201272013 DYSQCIgL sy IB) =CORRDY( Teh o1)/(063/BETA)*FLOAT (IB) VYSQ( I yb _I1B)=CORRVY(T9Lb91)/(063/BETA) *FLOAT(IB) GO TO 201 DYSO(I+L,sIB)=CORRDY(1,L 51) VYSQ( IL» IB)=CORRVY( Ielo1) GO TO 201 DYSQ(IyL,1B)=(CORRDY( 1yLy2)-CORRDY(IyL91))/(0.3/BETA) *(FLOAT(IBI- 10-3/BETA)+CORRDY(Tebel1) VYSQ(IyLy IB) S(CORRVY(Iphe2)-CORRVY( 19h 91))/(063/BETA) *(FLOAT(IB) = 10-3/BETA)+CORRVY(I,L 21) IF(DYSO(I,L,IB) LTO.) DYSO( IL, 1B) =0. TFCVYSQ(TeLy IB) eL T0060) VYSQ(1sL,1B)=0. CONTINUE DO 101 IB=1,NB DO 101 I=1y%N DO 101 L=1,NT ANO(I9L,IB)=SQRT(VYSQ(IyLsIBI/DYSQ( I 9b y1B))/(22*3214159) 101 SSQ (Iyl+IB)=(3.- *ELASTSDEPTH(LY/(CLAT)##2) )¥R2eDYSO(T ob 9 1B) + L(ALOAD(T)/(2-0*¥AREA(1)) )¥¥2 S€ 1)=21.3 DO 202 IS= 2939 202 S(IS)=20e+FLOAT(IS)*DELS 00 204 18=41446 $(40)=59.31 Het “UOISSIWJad JNOYWM psyiqiyoud uoNonpoJdes JayN4 “JaUMO }YHUAdOD au} Jo UOISSILUJad UM peonpoiday 0067 204 S(IS)=19e+FLOAT(IS)*DELS 0068 $(47)=6509 0069 NCASE=2 0070 DO 5000 NCASE=1¢2 0071 WRITE (67690) 0072 “WRITE(6760) NCASE 0073 DO 203 IS=1;NS 0074 IF(NCASEeEQ.2) GO TO 2033 0075 IF(IS.GT.40) GO TO 2032 0076 2033 ENS( 1S )=1060**(6.+66748*(ALOG10(21-3/S(1IS)))) 0077 GO TO 2031 0078 2032 ENS(IS)=10.*%*(32+65256*( ALOGG10(59.31/S(1S)))) 0079 2031 VNS(IS)=ENSCIS) *¥2 0080 203 CONTINUE 0081 DO 555 I1G=175 0082 READ (5 551) GAMMA (NCASE, IG) 0083 BMEAN=GAMMA (NCASE,1G) ¥3262*12.6 0084 WRITE(69408) GAMMA(NCASE »IG) 0085 DO 302 IB=1,NB 0086 , 302 PDFBC(IB)=EXP(- B(TB)/BMEAN) /BMEAN 0087 FO=0-6 0088 DO 1001 L=1,N — 0089 DO 1032 IS= isNS 0090 DO 1031 IB=1+NB 0091 DO 1030 L=1;NT 0092 IF(SSQ( 9b 1B) oGTe65e9%*2) SSO I yb e1B)=65.9%*2 0093 AA=S (18 )**2/(2e0¥*SSQ(I rb 21B)) 0094 IF(AAeGTe100-0) GO TO 1033 0095 FSBT= ANOLIsL 1B )#(S(1S)/SSQ( Ty be 1B) #EXP (= ~AA) 0096 GO TO 1034 . 0097 1033 FSBT=0-0 S2T ‘UOISSILUAd jNOY"M payiqiyold UoHONposdas JayN4 “JaUMO JYBUAdOD ayy Jo UOISsIWUad UM Paonposday * 0098 - 0099 0100 0101 0102 0103 0104 0105 0106 0107 0108 0109 0110 0111 0112 0113 0114 0115 0116 0117 0118 0119 0120 0121 0122 0123 0124 0125 105 CONTINUE FL(L)=FSBT CALL INTEG(NTyDELT,FOsF1sAl) ANSBCTS?IB)=Al F2C1B)=ANSB(1S,1I8)*PDFB( 1B) CALL INTEG(NB,DELB,FOsF22A2) EN1S( 1S) =A2 DO 104 IS= 1,NS DO 1041 IB=1,NB BB=ANSB(IS,IB)-ENIS(IS) IF (ABS(BB).LTe1-E-35) GO TO 1042 F3(18)=BB**2*PDFB(IB) GO TO 1041 F3(1B)=0.0 CONTINUE CALL INTEG(NB,DELB,FOsF3,A3) VNIS(IS)=A3 DO 105 IS= 1yNS F4(1S)=EN1S(IS)/ENS(IS) F5( 1S )=EN1S(IS)*VNS( 1S) /ENS( 1S) **3 F6(IS)=VN1LS(1S)/ENS(IS)#*#2 F401=F4(2) FSOL=F5(2) F601L=F6(2) F402=F4(41) . F502=F5(41) F602=F6(41) AGOR(F4(1)4F4(2))/2 0% 0 TH(F4(39) 74140) )/20%031 g2t ‘UdISsiWad jNOYM payqiyosd UoHONposdas JayLny “J8UMO YBIJAdoo ay} Jo UoIssIUUad YIM peonpojday 0126 0127 0128 0129 0130 0131 0132 0133 0134 0135 0136 0137 0138 0139 0140 0141 0142 - 0143 0144 0145 0146 0147 0148 0149 0150 0151 0152 0153 0154 C C ASO=(FS(LI+F5(2) 1/2 0% 0 TH+ (F5(39)4F5(40))/2.¥*.31 A60=(FE(1I+F 612) 5/2 0% oe TH (F6(39)4+F6(40) )/20% 031 A40=A404(F4(40)4F 4141) 1/20 % e694 (F4( 465) FF 4147) 1/2 0% 69 A50=A504(F5(40)4F5(41))/2 ee b69t(FS(4O)4F5 (47) / 20% 09 A60=A60F(F6L40) FF 6141) 120% ce b941FE( 46) +F6(47) 20% 09 DO 106 1S=1,37 K=1S+2 F4(1S)=F4(K) -FSCIS)=F5(K) 106 F6(IS)=F6(K) CALL INTEG(37 sDELSsF401,F4,A41) CALL INTEG(37 ,DELS,F501;,F5,A51) CALL INTEG(37 sDELSsFOOirF6xA61) OO ~ DO 107 IS=175 K=[S+41 F4( IS )=F4(K) FOCIS)=F5(K) 107 F6éCIS)=F6(K) CALL INTEG(5,DELS9F4023F42A42) CALL INTEG(5,DELS F502 »F5sA52) CALL INTEG(5,DELS»F602¢F62A62) | A4=A40+A41 4A42 A5=A50+A51+A52 A6=A60+A61+A62 EDL (1 )=A4 VD1 (1) =A5+A6 1001 CONTINUE DO 108 f=1,N ETA(T)=0- Let { “UOISSILUad jnoYWM payqiyoid uoNONpodea JayyN4 “JauMO JYyHUAdOD au} Jo UOISsIWUed UM peonposdey t | 0155 0156 O157 0158 0159 0160 0161 0162 0163 0164 0165 0166 0167 0168 0169 0170 O171 0172 0173 0174 0175 0176 0177 0178 0179 0180 * 0181 0182 0183 0184 555 5060 666 EDAM(1)=EM*EDI(1) VDAM( 1) =EM*VD1(1)+VM*ED1(1)**2 IF(VDAM( IT) «LTeO0LE-70) GO TO 1081 ETA(I)=(1--EDAM(1))/SQRT(VDAM(T)) PFSECT(1)=0.5*ERFC(ETAC(I)/SORT(2.)) GO TO 1082 PFSECT(1)=0. PLSECT(I)=1.-PFSECT(1) PLSTRYCI)=PLSECTCT ) x4 PFSTRY(1T)=1e—-PLSTRY( 1) I=1 PLBLDG=PLSTRYC(T) I=I+1 PLBLDG=PLBLDG¥PLSTRY(1) IF(IeLTeN) GO TO 1091 PFBLDG(NCASE»? IG)=1--PLBLDG WRITE(6,80) WRITE(6982) (ETACT)sT=1:N) WRITE(6,83) (PFSECT( I). I=1,N) WRITE( 6984) (PFSTRY(T) sT=1yN) WRITE(6,61) PFBLDG(NCASE, IG) CONTINUE CONT INUE WRITE(63690) DO 666 NCASE=1,2 WRETE(6,60)NCASE WRITE(6,62) (GAMMA(NCASEs IG), IG=1 25) WRITE(6963) (PFBLDG(NCASE,? IG), 1G=1,5) WRITE (6,694) CONTINUE Bel ‘UOISS|WJad }NOYWM payqiyosd uoHONpoidas JayLny “JEUMO jYBIJAdoo ay} Jo UoIssiuuad YIM psonpoiday 0185 0186 0187 0188 0189 0190 0191 0192 0193 0194 0195 0196 0197 0198 0199 0200 0201 0202 0203 0204 0205 0206 0207 0208 408 FORMAT(//1X_7HGAMMA =,F6.3) 51 FORMAT(F10.3) 54 FORMAT (4F10.3) 541 FORMAT(4E15.4) 60 FORMAT(1Xs11HS-N DIAGRAM,12//) 61 FORMAT(/ 3X»46HTHE PROBABILITY GF FAILURE OF THE STRUCTURE TS+ LE2004//) 62 FORMAT (2X_5HGAMMA 9 10X_ 510 F72395X)/) 63 FORMAT(2X,15HPROB OF FAILURE,5E12.4) 690 FORMAT (1H1) 694 FORMAT(////) TOL FORMAT (31X 9 LOHR Re Ee ) 702 FORMAT(31Xs1OHINPUT DATA) 703 FORMAT (31X49 LOHR RE / / ) 71 FORMAT(8Xs7HELAST =,F12.3/) 72 FORMAT (LLXs4HWT =94F12.3) 73 FORMATCLIXs4HCL =94F12.3) 74 FORMAT (8X, 7HDEPTH =74F 12.3) 75 FORMAT(9X,6HAREA =94F1223) 80 FORMAT(/1LX,9HSTORY NOes 9X_1H1914X_1H2914X-1H3914X%91H4) 82 FORMAT( 3Xy,6HETA 24E15 04) 83 FORMAT( 3Xs6HPFSECTs4E15e4)} 84 FORMAT( 3X_y6HPFSTRY?4E15.4) STOP END 6eT “UOISSILWAd jNoYM peyqiyoid uoHONposdas JeyN4 ’euMo }YHUAdOo oy} Jo UOISSILUOd UM peonpoiday FORTRAN IV 0001 0002 0003 0004 0005 9006 0007 0008 0009 MODEL 44 PS 101 VERSION 3, LEVEL 2 SUBROUTINE INTEG(N,DEL »FO-F eA) DIMENSION F(N) SUMF=0 ° NL=N-1 DO 101 T=1l-sNi1 SUMF=SUMF+F{(1) A=DEL* (( FO+F(N) 1/2 -0+SUMF) RETURN . END DATE WED MAY 7, 1969 OT ‘UOISSIWWEd }NOYYM payiqiyold uoHONpodal Jeyyng “saumo yYyBUAdoo au} Jo UOIssiUad YM peonpoidey ae kesh eae a ake QCM ENPUT DATA ))UOUOUOUOUOO™O™~—ST eek fe eek ak tk ake ee a a a a a a en ee ee ae ee a me Me ce le a FN A hk A eh A te eh Hw weeeeee---- DELTAU = 0.10 0 TEME = 400 - = Oe = SIGMAY = 36.6 BETAI = 0.19 ELAST = 29000. 40 G00 40.000 4. COD BCG .CCO «769 10.306 8.230 7.06C 160.060 120.00C 80.090 40.000 Pd wv m > upon roy ~ SR a a a ke ee se ae a ee ee oe ae ose ne ane ee ore ee oe Oy a re ae ee me ee ee es ae I A en a en a ec a Re eS Ee ae ne ~YYTELO = 0.844" —CS SB BS]SC—~“C~S DC “TEL “UOISSILUEd JNOYYM payigiyold uolONpoJdas JayN4 “JaUMO YBAdOO ay} JO UOISSIWUAad YM peonpoiday S“N DIAGRAM 1 GAMMA PROB OF FAILURE S-N DIAGRAM 2 GAMMA PROB OF FAILURE 0-075 0-1301E 00 0.200 0. 23584E-06 0-100 0O-7975E 00 0-300 0.71 76E-04 0.2125 "0.9863E 00 0-400 0-3288E-03 0-150 0-9992E 00 0-500 0 65433E-03 0.175 0-9999E 00 0.600 0.5843E-03 cf This dissertation has been microfilmed exactly as received 70-5217 HUNG, Sun-ju, 1933- THE PROBLEM OF SAFETY OF STEEL STRUCTURES SUBJECTED TO SEISMIC LOADING. University of Southern California, Ph.D., 1969 Engineering Mechanics University Microfilms, Inc., Ann Arbor, Michigan Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Asset Metadata

Creator Hung, Sun-ju, 1933- (author)

Core Title The problem of safety of steel structures subjected to seismic loading

School Graduate School

Degree Doctor of Philosophy

Degree Program Engineering

Degree Conferral Date 1969-06

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag applied mechanics,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c17-193945

Unique identifier UC11351205

Identifier 7005217.pdf (filename),usctheses-c17-193945 (legacy record id)

Legacy Identifier 7005217.pdf

Dmrecord 193945

Document Type Dissertation

Rights HUNG, SUN-JU

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA