Cumulative Damage of Structures Subjected to Response Spectrum Consistent Random Processes Citation Jeong, Garrett Duane (1985) Cumulative Damage of Structures Subjected to Response Spectrum Consistent Random Processes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/97b3-2427. https://resolver.caltech.edu/CaltechTHESIS:04052019-121036771 Abstract A theoretical analysis of the effect of duration on the damage of structures subjected to earthquakes is presented. Earthquake excitation is modeled as a nonstationary random process. Estimates of the first-passage probability of a simple oscillator are employed to choose modulated Gaussian random processes consistent with a prescribed response spectrum. The response spectrum is assumed to be specified independent of the duration. Expressions for the mean damage of a structure are derived using an approach similar to the Miner-Palmgren rule for failure caused by cyclic loads. The expected damage expressions are then evaluated for a structure subjected to modulated Gaussian random processes of varying duration. Two types of structures are examined: a steel structure and a reinforced concrete structure. Results are presented for systems with constant linear stiffness and a particular form of softening behavior. The nonlinearity of the softening system is accounted for by statistical linearization. The level of expected damage is found to be a strong function of both the duration of the excitation and the ductility of the response. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Applied Mechanics Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Applied Mechanics Thesis Availability: Public (worldwide access) Research Advisor(s): Iwan, Wilfred D. Group: Earthquake Engineering Research Laboratory Thesis Committee: Iwan, Wilfred D. (chair) Beck, James L. Hall, John F. Caughey, Thomas Kirk Knauss, Wolfgang Gustav Defense Date: 11 January 1985 Other Numbering System: Other Numbering System Name Other Numbering System ID EERL Report 85-03 Funders: Funding Agency Grant Number Caltech UNSPECIFIED NSF UNSPECIFIED S. D. Bechtel, Jr. Foundation UNSPECIFIED Record Number: CaltechTHESIS:04052019-121036771 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04052019-121036771 DOI: 10.7907/97b3-2427 Related URLs: URL URL Type Description https://resolver.caltech.edu/CaltechEERL:1985.EERL-85-03 Related Item Technical Report EERL 85-03 in CaltechAUTHORS Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 11441 Collection: CaltechTHESIS Deposited By: Mel Ray Deposited On: 10 Apr 2019 15:23 Last Modified: 20 Aug 2021 22:45 Thesis Files Preview PDF - Final Version See Usage Policy. 22MB Repository Staff Only: item control page

CUMULATIVE DAMAGE OF STRUCTURES SUBJECTED TO RESPONSE SPECTRUM CONSISTENT RANDOM PROCESSES Thesis by Garrett Duane Jeong In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1985 (Submitted January 11, 1985) -ii- ACKNOWLEDGMENTS I wish to express my sincere appreciation to Professor W. D. Iwan, my thesis advisor, for his guidance and encouragement throughout the course of this investigation. I would also like to thank the California Institute of Technology, the National Science Foundation, and the S. D. Bechtel, Jr. Foundation for their financial support during my years of graduate studies. The technical assistance of Sharon Beckenbach and Gloria Jackson in the typing and preparation of this manuscript is gratefully acknowledged. Lastly, I would like to thank my parents, to whom this thesis is dedicated, for making me realize the importance of learning and helping me whenever they could throughout my education. -iii- ABSTRACT A theoretical analysis of the effect of duration on the damage of structures subjected to earthquakes is presented. is modeled as a nonstationary random process. Earthquake excitation Estimates of the first- passage probability of a simple oscillator are employed to choose modulated Gaussian random processes consistent with a prescribed response spectrum. The response spectrum is assumed to be specified independent of the duration. Expressions for the mean damage of a structure are derived using an approach similar to the Miner-Palmgren rule for failure caused by cyclic loads. The expected damage expres- sions are then evaluated for a structure subjected to modulated Gaussian random processes of varying duration. Two types of structures are examined: a steel structure and a reinforced concrete structure. Results are presented for systems with constant linear stiffness and a particular form of softening behavior. The nonlinearity of the softening system is accounted for by statistical linearization. The level of expected damage is found to be a strong function of both the duration of the excitation and the ductility of the response. -ivTABLE OF CONTENTS PAGE ACKNOWLEDGMENTS ii ABSTRACT iii CHAPTER I: INTRODUCTION 1 CHAPTER II: RELEVANT CONCEPTS OF RANDOM VIBRATIONS, THE FIRST PASSAGE PROBLEM, AND STATISTICAL LINEARIZATION 5 2.1. 2.2. 2.3. Random Response of a Linear System to a Class of Nonstationary Excitation • • . . • • • . • • 5 The First-Passage Problem for a Lighly Damped Simple Oscillator • • . • • • • . • • • . • • . • • . . 12 Extension of the First-Passage Problem to a Nonlinear Simple Oscillator • • • • . • • • • • • • • • 21 RESPONSE SPECTRUM CONSISTENT RANDOM PROCESSES . 25 Probabilistic Determination of the Maximum Response of a Simple Oscillator Subjected to Modulated White Noise 25 Determination of Response Spectrum Consistent Random Processes 38 Maximum Response of a Nonlinear Simple Oscillator Subjected to Earthquake-like Excitation . . . . • 46 CHAPTER III: 3.1. 3.2. 3.3. CHAPTER IV: AN ANALYTICAL METHOD FOR COMPUTATION OF CUMULATIVE DAMAGE • • . • • • • • • • • • • • • • • • • • • • 59 4.1. Failure of Structural Members under Cyclic Loading • 59 4.2. Damage Accumulation for Narrow-banded Random Response 61 4.3. Damage Accumulation Applied to a Simple Oscillator 66 4.4. A Closed Form Expression for a Special Case of the . Damage Law 4.5. ... .... .... Normalization of the Expected Total Damage for a Linear System Subjected to Modulated White Noise ...... 69 73 -v- TABLE OF CONTENTS (CONCLUDED) PAGE CHAPTER V: ASSESSMENT OF THE EFFECTS OF DURATION ON THE DAMAGE OF A SIMPLE SYSTEM • • • • 77 5 .1. Introduction . 77 5.2. Expected Total Damage -Linear Model . 80 5.3. Expected Total Damage - Softening Nonlinear Model 84 5.4. Expected Damage Contours • • 90 CHAPTER VI: REFERENCES SUMMARY AND CONCLUSIONS 93 96 - 1 - CHAPTER I INTRODUCTION In many parts of the world, seismic considerations are the controlling factor in the design of structures. Building codes contain seismic elements which have been adopted for the safety and welfare of the public. For most structures, the building code recommends an equivalent lateral force analysis. Seismically induced loads are accounted for by equivalent static lateral loads. Although easy to implement and compu- tationally efficient. such an approach may be inadequate for structures such as schools, hospitals, nuclear power plants. and other important structures where the integrity of the structure is of great importance. Most important structures and special facilities are designed using a more detailed dynamic analysis in order to ensure their safety during a seismic event. Such a detailed analysis requires a specification of the anticipated nature of the ground motion as well as a complete description of the structure. If one or more characteristic earthquake accelerograms are specified as input, the equations of motion may be numerically integrated to give a detailed representation of the system response. However, numerical integration may be costly and the statis- tical nature of the seismic input makes specification of the input time histories difficult. Due to the many uncertainties in predicting the precise nature of the time history of earthquake ground motion, the design response spectrum has received wide acceptance as a measure of the design input. - 2 - The design spectrum provides a direct measure of the anticipated peak response of a structure or system under consideration as a function of its natural frequency and damping. In some cases, it may be argued that the peak response is the predominant factor affecting the safety and reliability of the system. However, in most cases the performance of the system will depend upon more than just the peak response of various components. In particular, for system components which fail due to repeated cyclic loading, performance will be a function of the entire history of cyclic oscillations. The failure of structures subjected to strong earthquake excitation is often caused by a low-cycle material failure. In this case, the structural components experience high strains and relatively few response cycles. For some materials, this type of failure is often referred to as low-cycle fatigue. Using numerical integration of the equations of motion of a simple hysteretic steel structure, Kasiraj and Yao [35] showed that low-cycle failure is not predictable by considering only the maximum response. Using random vibration theory, Miles [34] derived an analytical expression for the mean damage of a structure subjected to a stationary random process based upon the cumulative damage hypothesis of Miner [31] and Palmgren [32]. Roberts [33] and Lin [1] generalized the mean damage expression to the nonstationary case, but did not apply it to the lowcycle failure of structures subjected to earthquakes. In order to implement the expressions for expected damage to seismic structures, empirical constants are required for the assumed cumulative damage - 3 - hypothesis. · Bertero and Popov [27] and Yamada [28] have performed experimental tests on structural members to determine the necessary constants for structures experiencing a number of cycles of oscillation at large strains. Design spectra may be defined either deterministically or probabilistically. It is the objective of this thesis to examine the effects of the duration of excitation on the reliability of a system when the response spectrum is specified probabilistically. elements 'are combined to achieve this goal. Three basic First, the first-passage probability for a simple harmonic oscillator is used to define a nonstationary random process which corresponds statistically to a desired response spectrum. Although the first-passage problem for the simple oscillator has not been solved exactly, reasonably accurate approximations have been obtained by Vanmarcke [6] and Mason and Iwan [8]. Secondly, a model is postulated for the incremental damage of a system. Damage to the structure is based on the simple cumulative damage hypothesis proposed by Miner and Palmgren. Finally, random vibration theory is used to compute the accumulated damage of a simple structure and to determine the degree of total damage. In Chapter II, the relevant concepts from random vibration theory are reviewed. Analytical approaches to the first-passage problem and the method of statistical linearization are also discussed. In Chapter III, the accuracy of one approach to the first-passage problem is assessed through simulation and then applied to define a response spectrum consistent process. The probabilistic nonlinear - 4 response spectrum of a softening nonlinear elastic system is also examined. In Chapter IV, an incremental damage model is assumed based on the Miner-Palmgren failure rule for damage due to repeated cyclic loads. Through the further application of random vibration theory, a measure of the damage to a system subjected to a deterministically modulated Gaussian random process is derived. Damage in a simple structure is calculated in Chapter V for the response spectrum consistent random process defined in Chapter III. The effects of variations in duration of the excitation and ductility ratio of the response are discussed. Damage in a softening nonlinear elastic system is also computed through the use of statistical linearization. - 5 - CHAPTER II RELEVANT CONCEPTS OF RANDOM VIBRATIONS, THE FIRST PASSAGE PROBLEM, AND STATISTICAL LINEARIZATION 2.1 Random Response of a Linear System to a Class of Nonstationary Excitation The system to be considered is described by its equation of motion 9(t)w(t) (2.1) where ~ is the fraction of critical damping, w0 is the undamped natural frequency in radians per second, and 9(t) is a deterministic modulating function for w(t), a stationary Gaussian white-noise process with mean zero. The initial conditions for the system may be posed in one of two ways. First, they can be specified deterministically. A special case of this is the zero start where the system is assumed to be at rest when the excitation is applied. Alternatively, a probability distribution may be specified for the initial conditions. For the system described by equation (2.1) with 9(t) set to a constant, the stationary response may be described by a stationary probability distribution. If this sta- tionary probability distribution is used to specify initial conditions, this situation is known as a stationary start. In light of the fact that structural response to earthquake-like excitation is to be studied, a zero start will be assumed. - 6 - Equation (2.1) may be solved in 2n-space by first considering a system of n first-order ordinary differential equations written in matrix form as y(t) A(t)y(t) + Q(t)w(t) (2.2) Y(O) = y_o where A(t) is a time-varying n by n matrix, ~(t) is a time-varying vee- tor of order n, and w(t) is a stationary Gaussian white-noise process with mean zero. The fundamental matrix solution for this system is a time-varying n by n matrix denoted by I<t) which satisfies X(t) A(t)X(t) (2.3) I<O> = I where I is an n by n identity matrix. The solution to equation (2.2) may be expressed in terms of the fundamental matrix solution as y(t) (2.4) The function w(t) is a stationary white-noise process with mean zero which implies E[w(t)] =0 (2.5) where s 0 is a constant spectral density and 5(.) is the dirac delta - 7 - function. Furthermore, since w(t) is Gaussian and y(t) is related to w(t) through a linear operation. y(t) is a Gaussian random vector process. It is assumed that the process is specified at a given time by its mean vector and its covariance matrix; however, the autocorrelation matrix E[y(t+~)yT(t)] is also needed to completely define the process. By taking expected values of both sides of equation (2.4) and using equation (2.5), the mean vector~ y (t) may be written as ~y<t> = E[~<t>l = x<t>:0 (2.6) The covariance matrix Q(t) is defined as Q(t) - E{[y(t)-u <t>l[y(t)-u <t>lTJ y y (2.7) Substituting equations (2.4) and (2.6) into equation (2.7) and taking expected values yields Q(t) t t X<t>{J I- 1 <~')~(~')E[w(~')w(~)]~T(~)[I- 1 <~)]Td~'d~}IT(t). (2.8) J 0 0 Using equation (2.5) and performing the integration on~· gives the covariance matrix as Q(t) t 2rrS x<t>£f I-l(~)~(~)aT(~)[l-l(~)]Td~}IT(t) 0 (2.9) 0 An alternative to equation (2.9) for computing the covariance matrix can be obtained [1] by first taking the expected value of equation (2.2) - 8 - (2.10) Subtracting equation (2.10) from equation (2.2) and letting ~(t) = y(t)-E[y(t)] gives d dt~(t) = A(t)z(t) + ~(t)w(t) (2.11) Post-multiplying equation (2.11) by ~ T(t) leads to (2.12) Transposing equation (2.12) and adding the result to equation (2.12) yields (2.13) Taking expected values of both sides of equation (2.13) and noting that Q(t) = E[~(t)~T(t)], one finds d~Q(t) = A(t)Q(t) + [A(t)Q(t)JT (2.14) Using equations (2.4) and (2.5), the expected values in the third and fourth terms may be evaluated as E[w(t)z(t)J = ~s 0 ~<t> E[kT(t)w(t)J = ~s 0 ~T(t) (2.15) - 9 - Hence, equation (2.14) becomes (2.16) Since the initial conditions are specified deterministically, equation (2.16) has zero initial conditions. Note that using equation (2.9) requires solving n 2 first-order differential equations for the fundamental matrix solution and n(~+ 1 ) integrals for the covariance matrix. On the other hand, using equation (2.16), the covariance matrix is directly computed by solving n(n+ 1 ) 2 first-order differential equations. Except in the simplest cases where one can express the covariance matrix explicitly through equation (2.9), the use of equation (2.16) is probably numerically more efficient. Having defined the mean vector ~ (t) and the covariance matrix y Q(t), the joint probability density for the Gaussian vector process y(t) may now be written as p(y(t)) 1 1 T -1 "''( 7t)n exp{- 2[y(t)-J.b (t)] Q (t)[y(t)-~ (t)]}.(2.17) V' 2 det Q(t) Y Y The system described by equation (2.1) may be put in the form of equation (2.2) by letting 1 y -2~w 0 ] ' and The fundamental matrix solution is given by ~(t) (2.18) - 10 - X<t> (2.19) 0 where wd = w ~1-~ 2 • Since a zero start is assumed, equation (2.6) implies that the mean vector ~y(t) is the zero vector. Using equation (2.9) and the fundamental matrix solution, the covariance matrix Q(t) may be calculated. Choosing the alternative method, equation (2.16) and the definitions in equation (2.18) require the elements of the covariance matrix to satisfy qll = 2q12 q12 2 q22 - [wOqll + 2 ~w0q12 1 (2.20) 2 2 q22 = - 2 [w0q12 + 2 ~w0q22 1 + 2rrS 09 ( t) Equation (2.20) is a simple set of first-order differential equations where q .. is the element of Q(t) in the ith row and the jth column. l.J With the zero mean vector and the specified covariance matrix, the joint probability density for x and x at a given time t may be derived from equation (2.17) as p(x,x,t) where (2.21) - 11 -1 Q Another important statistic of this system is the average number of times per unit time the random process x(t) crosses a given positive threshold level from below. Let the threshold level be x = b where b is a positive constant and denote this statistic by ~(b,t). The expected frequency of up-crossing of a level b is related to the joint probability of x and x through [2] ~(b,t) J x p(b,x,t) dx (2.22) 0 Substituting equation (2.21) into equation (2.22) and performing the integration yields ~(b,t) • (2.23) Letting b = 0 in equation (2.23) one arrives at the expected frequency of zero crossings with positive slope as 'Vdet Q.(t) ~(O,t) 27tq11 ( t) (2.24) The steady-state solution for the situation where 9(t) is unity may be found by setting the left side of equation (2.20) to zero. specifies the covariance matrix as This - 12 - 0 (2.25) 0 This reduces equations (2.23) and (2.24) to ~(b) (2.26) ~ ( 0) 2.2 (2.27) The First-Passage Problem for a Lightly Damped Simple Oscillator The first-passage problem involves determining the probability that a random process exceeds a specified threshold level during a given time interval. Since first-passage probabilities are often associated with failure probabilities, it is appropriate to use the terms "safe" or "unsafe" to refer to the domain where the random process is respectively below or above the threshold. For a lightly damped simple oscillator subjected to stationary Gaussian white noise described by equation (2.1) with 9(t) set to unity, the displacement response is the random process of interest. Because of the relationship between the response spectrum and the response time history, a symmetric double barrier is considered. Figure 2.1 - 13 - . X UNSAFE REGION SAFE REGION UNSAFE REGION X Figure 2.1 Phase Plane Representation of the First Passage Problem with Symmetric Double Barrier. - 14 - illustrates this configuration in the phase plane. The safe domain is characterized by the region between two barriers at x = b and x = -b. The first-passage problem consists of determining the probability distribution of the time when the trajectory of the response first leaves the safe region and enters the unsafe region. Let W(T) be the probability that the magnitude of x(t) does not exceed a level b throughout the interval [O,T]. W( T) = Pr [ I x ( t ) Imax < b 0 Hence, (2.28) S. t S. T ] where Pr[A] denotes the probability that the expression A is true. W(T) is called the reliability function and is related to the first-passage probability density through p(T) _ dW dT (2.29) where p(T)dT is the probability that first passage occurs on the interval [T,T+dT]. It has been observed [4] that for small values ofT, the reliability function depends highly on its initial conditions. However, for large values ofT, W(T) tends to a decaying exponential regardless of the initial conditions. Mark [5] proposed a linear combination of n decaying exponentials of the form (2.30) as an approximation for W(T), but this requires a substantial amount of - 15 - numerical computation to use. When T is large, W(T) may be approximated by the dominant term of equation (2.30). Therefore, W(T) is assumed to be of the form W(T) (2.31) where a is the smallest ai in equation (2.30). The parameter a is known as the limiting decay rate of the first crossing density or the average crossing rate. A great deal of effort has been directed toward finding a good approximation for this decay rate. The assumption of independent level crossings leads to the simplest approximation for the limiting decay rate. The average rate of up- crossing of a level b is ~(b), and is equal to the average rate of downcrossing of the level -b. Hence, the average crossing rate is given by a = 2~(b) where ~(b) is given by equation (2.26). (2.32) The number of level crossings that occur constitute a Poisson process with average crossing rate a. The assumption of independent level crossings works well for high barrier levels. However, for low barrier levels, this approximation breaks down since the response is narrow-banded and the crossings are not independent. Other simple approximations for the limiting decay rate involve assuming independent peaks, independent envelope crossings, or independent envelope peaks. Each of these assumptions offers varied degrees of success in approximating the limiting decay rate. Those - 16 - assumptions involving the envelope statistics are better approximations since they take into account the narrow-handedness of the system. Reasonably good results have been obtained for the limiting decay rate by considering the response as a two-state Markov process. Because of the narrow-handedness of the response, the peaks above the threshold occur in clumps of duration T1 • The peaks are spaced at approximately 2 ~~ 0 ); therefore, the expected value of the duration of a clump is given by E[N c_ ] ____ 2~(0) where N is the number of peaks in a clump. c (2.33) Between each clump there is a period T where the peaks remain below the threshold that is taken to 0 be an independent exponentially distributed random variable with parameter a. Therefore, 1 a (2.34) Since there are Nc level crossings during the period T0 +T 1 , the expected value of that period may be taken as (2.35) Vanmarcke [6] combines equations (2.33), (2.34), and (2.35) and arrives at - 17 - 1ihl] - .ru.hl [ -1 a - E[N ] 1 - ~(0) c ( 2 .36) He then estimates the expected number of peaks in the clump as where a is the standard deviation of the response. The parameter y is a measure of the bandwidth of the response and is defined by the spectral moments of the response as y (2.38) where CD and G(w) is the one-sided spectral density of the response. It has been shown [7] that y varies between 0 and 1 and is small for narrow-banded processes and relatively large for broad-banded processes. Substituting equation (2.37) into equation (2.36) yields a = An alternate approach is chosen by Mason and Iwan [8]. (2.39) Using equa- tions (2.33), (2.34), and (2.35), the average crossing rate is written - 18 - as 1 1ihl_ [ 1ihl_] -1 a- E[T ] ~(0) 1 - ~(0) 1 (2.40) Assuming a probability density for T as 1 PT (t) 1 =C t exp(-~t) (2.41) 1 where C is a normalizing constant and ~ is a parameter of the density, the expected value of T 1 is given by (2.42) The limiting value as n~~ of the conditional probability that a clump which already contains n crossings will continue for at least one more crossing is found from the probability density of T 1 P* = exp(- ~) where P* denotes that limiting value. to be (2.43) By considering the response of the system for one-half cycle of oscillation after a peak greater than the threshold b, an integral equation for the stationary probability density for successive peaks greater than the threshold is obtained for which p~ is the eigenvalue. The eigenvalue is approximated and the limiting decay rate is given by a where = -~(b) [1 - ~~~~]- 1 ln(P*) (2.44) - 19 - P* !=IT~] [erf(y1) 1 b {t[1 erfc<vra> ~ rr(1-c 2 ) + rr Q[exp(-y~) - exp(-yi)] + erfc(y1 )) 1-c for which c = exp(- y y 2 rre ~1-~2 ) 2 1 = 1c ( .~ y2a + 12 ~1T ( 1-c ) ) = max [ _lL ~a It has been observed that the Mason and !wan approach is somewhat less conservative than Vanmarcke's approach and corresponds well with numerical simulations . In the case of nonstationary excitations, the modulating function 9(t) is allowed to vary in time . The approximations of the limiting decay rate found for stationary excitation along with the instantaneous response statistics are used to compute an instantaneous limiting decay rate, a(t). Equation (2.31) is then replaced by td W(td) = exp( - J a(t) dt) 0 where td is the duration of the excitation. process may be extended in this fashion. The two-state Markov (2.45) - 20 - Using Vanmarcke's approximation [9], the spectral density is allowed to vary in time. The moments may then be calculated based on this evolving spectral density and used to calculate an instantaneous y(t). The limiting decay rate is then given by 2\) ( b, t) [ 1 a(t) 1 \)(b.t) (2.46) \)(O,t) For modulated white noise, y(t) is a constant and the nonstationary behavior is accounted for by the time dependence of a(t), \)(b,t), and \)(O,t). Mason and !wan [8] propose the use of an alternate probability density for T when using their method for nonstationary excitation. 1 That density is given by exp(-~t) (2.47) 2 where a 2 (t) is the instantaneous variance of the response and a (t) is s the stationary response variance associated with the instantaneous value of the excitation. The ~ a term reflects the nonstationarity by a 4 ( t) s accounting for the greater independence of barrier level crossings when the response is broad-banded. Note that for stationary response this term is equal to unity, thus reducing the density to that for the stationary case. ta tion yields Using a derivation analogous to that for stationary exci- - 21 -2~(b,t) a(t) 1 + [ ln[P*(t)J (2.48) ~] [1 - ~ ( b, t)] cr4(t) ~(O,t) s where ~(O,t), ~(b,t), and P*(t) are implicitly time-varying, based on the instantaneous covariance values. 2.3 Extension of the First-Passage Problem to a Nonlinear Simple Oscillator The foregoing analysis may be extended to a nonlinear simple oscil- lator by using the method of statistical linearization to compute approximate response statistics from which estimates may be made of the first-passage probability. Independently introduced by Booton [11] and Caughey [12] at about the same time, the method of statistical linearization is an extension of the equivalent linearization technique of Kryloff and Bogoliuboff [13]. Since the simple oscillator is a special case of an n-degree of freedom system, the n-degree of freedom system will be discussed. Caughey used modal decomposition to uncouple the linear part of a nonlinear system of equations, and applied the linearization technique to each of the resulting single degree of freedom equations. Foster [14] generalized the method by developing an approximate closed form solution for the equivalent linear damping and stiffness matrices, but this required inversion of a 2n by 2n matrix for an n-degree of freedom system. Using a simple physical interpretation for the effective linear - 22 parameters, Iwan and Yang [15] determined the terms of the equivalent linear damping and stiffness matrices as simple scalar equations. Using 2n-space notation a nonlinear n-degree of freedom system may be written as y (2.49) h(y) + f(t) where h(y) is a nonlinear vector function of y and f(t) is a random excitation vector. Consider an auxiliary system of linear differential equations of the form y = A[S(t)]y + t<t> (2.50) where A[S(t)] is an arbitrary matrix dependent on the time-varying response statistic S(t) chosen such that the solution to equation (2.50) approximates the solution to equation (2.49). The error in estimating the nonlinear system by a linear system may be defined by the difference between the equations involved. ~ Hence from equations (2.49) and (2.50), h(y) - A[S(t)]y (2.51) where ~ is the vector of the equation differences. The appropriate choices for the elements of A[S(t)] will be those that minimize ~ in some sense. A criterion for minimizing ~ is to require the mean of the scalar product ~T~ to be a minimum, i.e., E[~T~] = minimum (2.52) It is noted that minimization of the equation difference does not imply - 23 - minimization of the solution difference. However, widespread usage of the method in conjunction with experiment or numerical simulation suggest that the technique approximates the nonlinear response well. The necessary condition for equation (2.52) is a T -,-E[~ ~] uaij -2E[y.e.] J l. 0 i,j=1,2, ... ,n (2.53) It has been shown [16] that for £(t) Gaussian this will be a true minimum (as opposed to a maximum). Applying the definition of~ leads to 0 (2.54) Atalik and Utku [17] showed that if y is a jointly Gaussian random vector process with mean zero and h(y) is sufficiently smooth so that first partials with respect toy. for i=1,2, ... ,2n exist, the condition l. in equation (2.54) reduces to the elements of A[S(t)] being given by ai. = J E[~h . (y)l uyj (2.55) i, j=1, 2, . •• , 2n l. - A[S(t)] is an implicitly time-varying matrix. If we assume A[S(t)] is actually continuous in time, a unique fundamental matrix solution exists and equation (2.9) defines the approximate covariance matrix. For Gaussian excitation the response statistic S(t) may be taken as the covariance matrix Q(t) and equation (2.16) becomes a system of nonlinear differential equations easily implemented numerically. - 24 - For a nonlinear simple oscillator, obtaining an approximation for the covariance matrix allows one to arrive at approximations for the response statistics necessary to compute a limiting decay rate and estimate the first-passage probability. - 25 - CHAPTER III RESPONSE SPECTRUM CONSISTENT RANDOM PROCESSES In the following applications of the first-passage problem for a simple oscillator, the Mason and Iwan two-state Markov process approach has been used to compute first-passage probabilities. Vanmarcke's two- state Markov process approach or any other method of estimating firstpassage probabilities may have been chosen. The accuracy of any appli- cation of the first-passage problem will depend upon the accuracy of the first-passage probability estimate. 3.1 Probabilistic Determination of the Maximum Response of a Simple Oscillator Subjected to Modulated White Noise The solution for the first-passage problem for a simple oscillator may be used to give the probabilistic specification of the maximum response of the oscillator subjected to modulated white-noise excitation. Recall from equation (2.1) that the equation of motion of the oscillator is given by 9(t)w(t) (3.1) where ~ is the fraction of critical damping, w0 is the undamped natural frequency, and 9(t)w(t) is a modulated white noise. tions will be assumed. Zero initial condi- - 26 - The probability, W(td), that the maximum response of the oscillator is less than or equal to a level b after a time td is the reliability function defined in equation (2.28). Hence, W(td) = Pr[ lx<t>l max < b (3.2) where td is the duration of the modulating envelope, 9(t). The relia- bility function is evaluated using the analytical approach of Mason and Iwan. To check the accuracy of the analytical approach used to determine the reliability function, a Monte Carlo simulation study of equation (3.1) was performed. An ensemble of sample functions was generated to represent stationary Gaussian white noise. Each sample function was constructed from a sequence of independent normally distributed numbers with zero mean and unit variance. The numbers were used as ordinates of the function at equally spaced time intervals, ~t. assumed to vary linearly over each interval. The function was The time scale was chosen such that the initial point was uniformly distributed on the interval [-~t,O]. The numerically generated unit variance sample functions were multiplied by [2::oJ~ to give a process with a power spectral density of [18] S(w) S 6 0 S(w) approaches a constant s 8cos(w~t) + 2cos(2w~t) (3.3) (w~t)4 0 as w~t approaches zero. for S(w) remains within 5% of s 0 The expression for w~t < 0.57, and within 10% for - 27 w~t < 0.76. Thus , the time interval may be chosen sufficiently small to approximate a white-noise process to within a given tolerance out to any desired frequency. In this study, ~t was set equal to T where T is the 20 undamped natural period of the oscillator. Since the applied excitation is assumed to be a straight line segment on each time interval, the solution to equation (3.1) may be solved by digital computer in a purely arithmetical way [19]. The exact analytical expression for the response of a damped single-degree-offreedom system with arbitrary initial conditions was written for a linearly varying excitation. The total response for a time interval was then obtained by analytically matching the initial conditions of the current interval to the final conditions of the previous interval. In this way, no numerical approximations were introduced other than the white-noise approximation and the round off due to the digital representation of the response. The simplest modulating envelope is a rectangular pulse of unit amplitude and duration, td (Figure 3.1a). In Figures 3.2-3 . 4, the prob- ability that the maximum response will be less than a given threshold level is plotted versus duration of excitation for several threshold levels and several values of damping based on analytical first-passage estimates. The results are displayed in a dimensionless form. durations are expressed in multiples of the natural period. The The thres- hold levels are normalized by the stationary standard deviation of the system described by equation (3.1) with 9(t) = 1. The stationary standard deviation, a , may be written from equation (2.25) as s - 28 - 8 ( t) 1 t 0 0 (a) Rectangular Envelope 8 ( t) 1 t 0 0 (b) Earthquake-like Envelope Figure 3.1 Modulating Envelopes for the Stationary Gaussian Random Process - 29 - 0 0 ---- --- ----------· -- .......... \ ------ \ --- --- -- . . . A \ \ --------- b = 3a s \ \ (!)\ ----- b = 2a \ b \ [!] C) A \ ..---!0 -IJ) \ s 1 mu lat I ·O(t_ ~ '\ ...o· '\ do s = 1 as = 21. '\ ...0 0 "tQ "' L ()_IJ) N 0 . '' ' ' 'Ql 0 0 10.00 20.00 30.00 40.00 Durat Jon/PerJod Figure 3.2 Probability of Not Exceeding the Threshold Level versus Normalized Duration for Rectangular Modulating Envelope, ~ = 2%. - 30 - 0 0 0 . \ \ \ .---.0 \ -lf) ..o· \ do --------- b = 3 \ - - - - - 'G( ..0 0 \ L \ ' (Llf) 0 . l2J (!)A sJmulatJon = 5% 0 0 -~-- 10.00 Our at Figure 3.3 b = 2 b = 1 30.00 on/Per od 20.00 40.00 Probability of Not Exceeding the Threshold Level versus Normalized Duration for Rectangular Modulating Envelope, ~ = 5%. - 31 - 0 0 b b = 3a = 2ass b = 1a s Ln r-c s 1 mu 1 at 1 on ~ = 1 0 I. ...--<0 -Ln .D· do .D \ 0 L \ \ \ Q_Ln \ ("\J \ 0 \ (!)" 0 0 0 "' '' ---- .,_----------~----------~--------~~~~------~ 0.00 10 . 00 20. 00 30. 00 40. 00 Duro.t on/Per1od Figure 3.4 Probability of Not Exceeding the Threshold Level versus Normalized Duration for Rectangular Hodulating Envelope, ~ = 10%. - 32 - a (3.4) s where s 0 is the constant power spectral density of the Gaussian white noise, w(t). Also shown in Figures 3.2-3.4 are the results of the simu- lation study for the same threshold levels and damping values using an ensemble of 1000 sample functions. It is observed that the probability that the maximum response is less than a given threshold level decreases with increasing duration. Therefore, the probability distribution of the maximum response varies with duration. This suggests that duration is an important parameter when probabilistically determining the maximum response. The analytical method agrees reasonably well with simulation. From Figures 3.2-3.4, it may appear that the response of a system with more damping is more likely to surpass a given absolute level. However, this is not the case. The stationary standard deviation of the response is inversely proportional to the square root of the damping ratio. Therefore, the normalized threshold levels which are constant multiples of the stationary standard deviation of the response become smaller as the damping ratio increases. In Figure 3.5 the probability that the maximum response is less than a constant threshold level is plotted versus the duration of the excitation for several damping values. From this figure it is evident that the probability that the response of a system will not exceed a given threshold increases as the damping in the system increases. - 33 - 0 0 b a (t;=S%) s l.[') [' 0 ~a -l.[') .o· do .0 0 L o._l.[') N 0 0 0 93.00 10.00 20. 00 30.00 40.00 Durat Jon/Per od Figure 3.5 Probability of Not Exceeding the Threshold Level versus Normalized Duration for Rectangular Modulating Envelope. Threshold Level held constant, independent of damping ratio. - 34 - To represent earthquake-like excitation, it is appropriate to choose a modulating envelope that models the nonstationary character of actual accelerograms. Specific forms for such envelopes have been sug- gested by a number of authors (e.g., [20], [21]). Real accelerograms often consist of three phases: a phase where the excitation builds, a strong motion phase where the shaking remains fairly constant, and a phase where the motion dies out. Earthquakes of larger magnitude tend to have envelopes of longer duration. One form of modulating envelope which has been suggested [20] is shown in Figure 3.1b. This envelope may be expressed as 2 56.25(tt > d 9(t) = 2 0 ~ t ~ 15td 1.0 2 lstd < t ~ 1 2td exp[-2.976(tt- 1)] 2 d 1 2td < t ~ td where td is defined as the duration of the excitation. (3.5) When td = 30 seconds, the modulating envelope, 9(t), is similar to the Caltech B-type earthquake envelope given in Reference [20] which was designed to represent shaking close to the fault in a Richter Magnitude 7 or greater earthquake. Figures 3.6-3.8 show the probability that the maximum response of the simple oscillator is below a specified level for several threshold levels and several values of damping using the envelope of equation (3.5). Results from Monte Carlo simulation are also shown. cal approach agrees well with the simulation results. The analyti- As in the case of - 35 - 0 0 ------ 0 ----.It------ . ...-10 -t..n ..o· 00 ..0 0 = 3o s = 2os b = lo s L --------- b Q_Ln ----- b N 0 . [!] C) e. s1mulat I On I; = 2% 0 0 9J. 00 Figure 3.6 10.00 Durat 20.00 on/Per 30.00 od 40. 00 Probability of Not Exceeding the Threshold Level versus Normalized Duration for Earthquake-like Modulating Envelope, I; = 2%. - 36 - 0 0 --- --- ---;zh- ______ _ ---<!1-----. \ \ \ \ 0 b = ~--- ---- - \ \ b = '\ '\ --<o -Ul 3a s ----- b = 20 s (9\ lo s [!] C) .1:. sJrflulat 1on '\. I; = 5/. '\. ..o· do '\. '\. '& ..D 0 (__ '' Q_u; C\1 ' ' .... .... 0 0 0 10. 00 Durot Figure 3.7 20. 00 30. 00 40 . 00 on/Per od Probability of Not Exceeding the Threshold Level versus Normalized Duration for Earthquake-like Modulating Envelope, I; = 5%. - 37 - c c 0 \ \ \ ----·----- \ --'o -lf) _Q· do \ \ (')\ \ ...0 b = ----- b \ 3os = 20 s = 10 s b [!] C) .,1,. slmulat 1on = 10% E, 0 (_ Q_lf) N c ... (') 0 0 10 . 00 20. 00 30. 00 40 . 00 Duro.t on/Per od Figure 3.8 Probability of Not Exceeding the Threshold Level versus Normalized Duration for Earthquake-like Nodulating Envelope, E, = 10%. - 38 - the rectangular modulating envelope, the apparent variation in the probability distribution suggest that duration of excitation is an important parameter when probabilistically specifying the maximum response of a simple oscillator. In Figure 3.9, the probability that the peak response does not exceed a constant threshold level is plotted versus the duration of excitation for several values of damping using the earthquake-like envelope. Like the results for the rectangular envelope, the likelihood that the threshold level will not be exceeded increases as the damping in the system increases. 3.2 Determination of Response Spectrum Consistent Random Processes The response spectrum has been widely accepted as a way of describ- ing some of the aspects of earthquake ground motion that are of interest to the engineer. The response spectrum is defined from the behavior of a single-degree-of-freedom system. The given ground acceleration is applied to the base of a simple oscillator and the maximum displacement is measured or calculated. The maximum displacement depends on the applied excitation, the value of damping in the system, and the natural frequency of the system. The family of curves, for various values of damping, of the maximum displacement plotted versus the natural frequency make up the response spectrum for the ground motion. - 39 - 0 0 b a (?;=5%) s 10% 0 0 10. 00 0 urot Figure 3.9 20. 00 30. 00 40.00 on/Per od Probability of Not Exceeding the Threshold Level versus Normalized Duration for Earthquake-like Modulating Envelope. Threshold Level held constant, independent of damping ratio. - 40 - Studies of response spectra for actual accelerograms [22] have shown that important features of the response spectra can be represented by simplified smoothed curves for design purposes. These curves are derived by computing the response spectra for a number of earthquakes and then normalizing the spectra in such a way that the spectra may be compared. The mean and the standard deviation of the normalized response spectrum values are calculated over the entire frequency range and used to compute parameters of a probability distribution. Smooth curves are then chosen to describe the shape and normalized level of the response spectrum for a given confidence level. design response spectra. The curves make up the When scaled to reflect the maximum ground motion, they are used to specify seismic input for structural design. Design response spectra of the type discussed above have been adopted by the U.S. Nuclear Regulatory Commission (NRC) for the seismic design of nuclear power plants [23]. The NRC design response spectrum is made up of straight line segments when plotted on logarithmic tripartite graph paper. The vertices or control points of the straight line segments for each value of damping are specified by the frequency at which the points occur and the ratio of the response pseudo acceleration to the maximum ground acceleration. Based on the work reported in Reference [24], these design response spectra have a confidence level of 84.1%. Analytically, it is convenient to model earthquake excitation as a Gaussian random process defined by its power spectral density and a deterministic modulating envelope. If the seismic input is specified in - 41 - this way, it should correspond statistically to the design response spectrum. This may be achieved by performing a first-passage prob- ability analysis for a simple oscillator in order to select power spectral density ordinates of the process from the design response spectrum. A linear oscillator subjected to the artificial earthquake process may be described by 9(t)g(t) (3.6) where 9(t) is the deterministic modulating envelope and g(t) is a stationary Gaussian random process with spectral density S(w). Response spectra are usually constructed only for lightly damped systems where the response is narrow-banded about the natural frequency. Hence, for ~ sufficiently small, a good approximation for the response of the system described by equation (3.6) may be obtained by replacing g(t) with a stationary Gaussian white-noise process, w(t), with constant spectral density s 0 = S(w ) for each w . 0 0 Since the analytical approach of Mason and !wan for calculating first-passage probabilities of a simple oscillator was directed toward modulated Gaussian white-noise processes, it may be used here to compute the power spectral density ordinates. The design response spectrum may be described by = (3.7) where SD(w 0 ,~> is the target spectrum value for natural frequency w0 and - 42 fraction of critical damping, spectrum. ~~ and Ps is the confidence level of the The confidence level, Ps , corresponds to the reliability function defined in equation (2.28), with the threshold level b replaced by the value of the target response spectrum, SD(w 0 ,~). Hence, equation (2.45) may be written as (3.8) where td is the duration of the excitation, a is the limiting decay rate, and s 0 is the constant power spectral density of the excitation. An increase in s 0 causes a decrease in W(td) and vice versa. simple iterative process, Using a s0 may be varied in equation (3.8) until W(td) is within some predetermined tolerance, for example, 0.5%, of P • s The value of s0 is then assigned to S(w ). 0 In Figure 3.10, the NRC design response spectra outlined in Regulatory Guide 1.60 are shown for a maximum ground acceleration of 50% g. Power spectra were derived for several durations and are shown in Figures 3.11 and 3.12 for damping values of 2% and 10% of critical damping. A confidence level of 84.1% was assumed. The overall shape of the power spectrum is somewhat similar as duration is varied. For a fixed response spectrum and fixed confidence level, the level of the power spectrum increases as the duration decreases. and vice versa. A shift in the distribution of the power toward the lower frequencies as duration decreases is evident from the - 43 - u Q) 1j) 'c Q) > 0 ""0 :::J Q) lJl Q_ 1 0 0 1 0 -1 10 ° Frequenc!d Figure 3.10 10 1 (Hz) NRC Horizontal Design Response Spectra, scaled to O.Sg Maximum Ground Motion. - 44 - 0 0 !.[) .---.N (Y) ----·- ·---- **oc me -........c td = 10 sees ----- td = 20 se e s td = 30 .sees ; = 2/. (DC 1-... I 1 I 1 I 1 I (\!- **c I I ~c I I I '.... '.... '', '.... ....... I -0 /1 ' } J ./" ' ... ...., ____ ~/ "' I I !.[) I I ~[' I -I-' I I I I I lJ) / I / Co (l;O 1- - oo ----- / !.[) --1 u L -1-'C (JC; (\) Qt.n . I I "J en (_ (\) 3c oo o._ 0 1 0-1 10 ° Frequenc!J Figure 3.11 10 I (Hz J Comparison of Derived Power Spectra for three Durations, ; = 2%. - 45 - 0 0 Lf) ,--..(\J (Y) - --- --- ·· - **uc td = 10 se e s --- - - td = 20 se e s = 30 se e s ~ = 10% tl;C td IJ)c:) ,c C"\J~ **c -0 '-'c Lr ::J;l'- I +--' I I I I t.n i Cc I (\)0 I oc lD ...-< d (_ +--'c oc (\) I I . Lf) 0..(\J c.n (_ (\) 30 oo 0....... 0 10-1 10 ° Frequenc\:j Figure 3.12 10 I (Hz l Comparison of Derived Power Spectra for three Durations, = 10%. ~ - 46 - peak that occurs at 0.25 Hertz. Since the response spectra for different values of damping are derived from a single earthquake time history, one might expect that a single process may be derived from the design response spectrum for different values of damping. This is not the case for the power spectra derived above. The design response spectra for different values of damping produce slightly different power spectra. This may be expected since the design response spectra are constructed from a number of real accelerograms each with its own duration, frequency content, etc. In Figures 3.13-3.15, the power spectra are replotted for each duration to compare their agreement for different values of damping. Better agree- ment is found at the high frequencies for shorter durations and at the low frequencies for longer durations. Figures 3.13-3.15 suggest a dura- tion slightly longer than 20 seconds will give good agreement between the power spectrum derived using 2% damping and the power spectrum derived using 10% damping for the design spectra considered here. This may be expected since the earthquakes upon which the design response spectrum is based have durations primarily between 20 and 30 seconds. 3.3 Maximum Response of a Nonlinear Simple Oscillator Subjected to Earthquake-like Excitation If a structure behaves nonlinearly, the customary linear response spectrum may not characterize the response of the structure. In Section 3.2, a Gaussian random process was constructed to agree statistically - 47 - 0 0 L() ,.........0.1 (Yl * *uo ----- E, = 21. E, = 10% td = ~0 10 sees U1a '-.....0 (\J ....... * *c- 0 .___0 L() r, ::J)r" I I / +-' U1 co / ' ,; "--- ~0 oa L() ..-; d (__ +-'0 uo ~ I I I I . Q~ en (__ ~ 30 oo o_ 0 10-1 10° Frequency Figure 3.13 10 1 10 2 (Hz J Comparison of Derived Power Spectra for two values of Damping, td = 10 seconds. - 48 - 0 0 Ln ,._.N (Y) ....... = 21. t;, = 10/. t d = 20 sees t;, **uo - - -- - il)O U1c::) -.........o (\J ....... **c -0 ~0 Ln ::J)f'+-' U1 Co il)O oa Ln I ....-! I d I L I +->o uo il) I I • I Q_~ en I I \ I L il) 30 oo CL o 10 -t 10 ° Frequenc!J Figure 3.14 10 1 (Hz l Comparison of Derived Power Spectra for two values of Damping, td = 20 seconds. - 49 - 0 0 t..n ~N (I) = 21. = 10/. td = 3 0 s e e s [, *uo * - -- - - Q)O [, U1c) -. . . . o (\] ....... * *c - 0 ~0 t..n :J)I' 4-J lJ) Co / Q)O / oa / t..n .-; d L r r +-'o u o Q) \ \ I • \ I Q_~ I en \ \ I \ I I L Q) 30 oo o_ a 1o -I 10 ° Fr e qu e nc~ Fi gur e 3 . 15 10 I ( H z) Compariso n of Der i ved Power Spectra for t wo values of Damping, t d = 20 s econds . - 50 - with a specified design response spectrum. Using that process, statis- tical linearization, and the analytical approach to the first-passage problem for a simple oscillator, the maximum response of a nonlinear simple oscillator may be obtained statistically for excitation defined by a design response spectrum. As an example of a nonlinear simple oscillator, consider a softening elastic restoring force that has a force-deflection relationship described by (3.9) f where fu is the ultimate force and k ness. 0 is the small displacement stiff- The nature of this restoring force characteristic is shown in Figure 3.16. f Let the quantity _Q be denoted by x ko response, xmax , to x y be denoted by ~· Y and the ratio of the maximum Then, the value of ~ is a measure of the nonlinearity of the system and may vary from 0 to oo with ~ = 0 corresponding to a linear system. corresponding to a restoring force f x y represents the displacement for a linear system with stiffness u k , and is similar to the elastic limit displacement of a yielding 0 system (see Figure 3.16). Equation (3.9) may be expressed in terms of~ and xmax as 2 koxmax f = - 1T ~ -1 (1T tan x 2 max - ~-x--- ) (3.10) - 51 - f I f u -----~------------------ I I X X Figure 3.16 y Softening Elastic Restoring Force Characteristic - 52 - The nonlinear force-deflection relationship described above is similar to the backbone curve of a yielding system in that the stiffness softens as the displacement increases. However, when a yielding system is loaded beyond some point, the unloading path differs from the loading path. This inelastic behavior is not modeled by the nonlinear relation- ship described by equation (3.10). However, the nonlinear relationship is a better approximation to a yielding system than a linear relationship and is useful to assess the effects of softening on system response. The response of the yielding system is often described by the ductility ratio; i.e., the ratio of the maximum response to the maximum elastic response. The nonlinearity parameter ~· defined herein, is similar to the ductility ratio of the yielding system. The equation of motion for a simple oscillator with the particular softening behavior discussed above may be written as (3.11) where ~ and w0 are the fraction of critical damping and undamped natural frequency, respectively, associated with small displacements. Writing the equation of motion in 2n-space notation yields y where 4(y) + a<t)w(t) (3.12) - 53 - ytl l -2~w h(y) y 0 2 tan -1(1T yl )) 211xmax and 2< t) = J 0 le<t> } Assuming that Y is a jointly Gaussian random vector process with - mean zero, equation (2.55) leads to the equivalent linear system given by Y A[Q(t)]y + ~(t)w(t) (3.13) where 0 A[Q<t>J exp(y 2 ) erfc(y) and y V2 xmax 1T!l )'qll(t) Hence, the instantaneous equivalent linear natural frequency w and e damping ratio ~ e are - 54 - (1) e ~r= w [,ny exp(y 2 ) erfc(y)] l/z. 0 (3.14) ~e = ~ [vny exp(y ) erfc(y)]-o/z 2 Substituting A[Q(t)] into equation (2.16), it is seen that the covariance matrix approximation for the nonlinear system satisfies (3 . 15) with ·zero initial conditions. Using the equivalent linear system and the analytical approach to the first-passage problem outlined in Section 2.2, a statistical design response spectrum may be calculated for the nonlinear system excited by a Gaussian random process defined by its power spectral density S(w) and modulating envelope G(t). An equation analogous to equation (3.8) may be written for the equivalent linear system as .J l td -Ja<x ,S(w ),w .~ ,t) dt max e e e 0 l I (3.16) J where td is the duration of the excitation, a is the limiting decay rate, xmax is the unknown maximum response, and S(we) is the spectral density of the excitation evaluated at the equivalent natural frequency. An increase in x causes an increase in W(td). max It is a simple matter - 55 - to vary xmax in equation (3.16) until W(td) is within some tolerance of the confidence level, P • s Statistical design response spectra have been calculated for softening nonlinear elastic systems with ~ equal to 0, 2, 5, and 10 for the Gaussian random process generated with the linear system from the NRC design response spectrum of Section 3.2 with a duration of 30 seconds. In Figures 3.17 and 3.18, the statistical maximimum response of the softening nonlinear elastic system is plotted against the small displacement frequency. The curve for ~ = 0 is the NRC design response spectrum from which the excitation process was derived since ~ = 0 corresponds to a linear system. The instantaneous equivalent linear natural frequency, w , of the e softening nonlinear elastic system described in equation (3.14) is always less than or equal to the small displacement natural frequency, w • 0 A frequency shift occurs in the statistical maximum response spectrum for the softening nonlinear elastic oscillator due to the downward shift in the natural frequency. This frequency shift is most noticeable in Figures 3.17 and 3.18 at the control points in the response spectrum. The maximum response of the nonlinear simple oscillator is affected by two factors. First, an overall increase in the maximum response occurs due to the assumed level of softening, ~. in the system. Secondly, the frequency shift can cause a variation in the maximum response by increasing or decreasing the value of the power spectral density of the excitation corresponding the instantaneous effective - 56 - 1 0' 0 il.) Ln .......... ~ 10 2 .::J) +-> 0 0 ]J 10 ]J 5 ..--; il.) > 10 1 0 "'0 :J il.) Ln (L 10 0 1 0 - 1 10 ° Frequenc!d Figure 3.17 10 1 10 2 (Hz J Statistical Maximum Response of the Softening Elastic System, ~ = 2%, td = 30 seconds. - 57 - 1 0' 10 % E, td u ID lJ) ......... ~ 102 :J) +-' u 0 --< ID > \1 10 -u \1 5 :J ID \1 2 \1 0 10 I 0 lJ) Q_ 10 0 10 - I 10 ° F r e qu e nc\:;j Figu r e 3. 1 8 10 I ( Hz ) S t a t is ti c al Haximum Respo ns e of t he Sof t en ing El as ti c Sy s t em, E, = 10 %, td = 30 second s . - 58 - natural frequency. This variation depends on the shape of the power spectral density of the excitation process. In Figures 3.17 and 3.18 , variation in the maximum response are most evident for frequencies less than 0.25 Hertz where the frequency shift causes a sharp decrease in the power spectral density and for frequencies greater than 9 Hertz where the frequency shift causes a sharp increase in the power spectral density. An increase in the structural damping decreases the response for the nonlinear system in the same manner as for the linear system. The net effect of the softening nonlinear elastic restoring force is that the statistical maximum response spectrum resembles a linear response spectrum which has been translated along an axis of constant spectral displacement (a line with a slope of one on the log-log pseudovelocity diagram). These results are similar to the inelastic response spectra for the hysteretic system described by Iwan and Gates [25] and Iwan [26]. - 59 - CHAPTER IV AN ANALYTICAL METHOD FOR COMPUTATION OF CUMULATIVE DAMAGE 4.1 Failure of Structural Members under Cyclic Loading When a structure is subjected to strong earthquake ground motion, large displacements can occur. The displacements may be associated with large strains in the structural members. The repeated application of large strains may cause failure in the structural members. Experimental tests to determine the behavior of structural members under large cyclic strains have been performed by Bertero and Popov [27] and Yamada [28,29]. The structural members were subjected to cyclic bending deflections of constant amplitude until fracture occurred. The relationship between the number of cycles to failure, N, and the deflection amplitude may be expressed as (4.1) where ~ is the ductility factor of the deformation, defined as the ratio of the maximum deflection to the maximum elastic deflection, and s and C~ are the empirical constants determined from the experimental tests. Yamada [29] defines the failure constants for steel structural members from tests on wide-flange steel columns. axially loaded in compression to The columns were t of the ultimate strength of the centrally loaded column and cycled in bending to failure . Failure was described as local flange buckling followed by local buckling of the web leading to torsion about the member axis and loss of axial resistance. - 60 - The damage law may be expressed approximately as Nb 1 • 85 = 81.93 (4.2) where b is the amplitude of the cycling in terms of the displacement of the actual test specimen. simple beam theory. The yield displacement may be found using Normalizing the displacement in equation (4.2) by the yield displacement, the damage law may be written in terms of the ductility factor as (4.3) 167.1 The exponent in equations (4.2) and (4.3) is approximately equal to 2, a widely accepted value for steel. It has been observed that failure of reinforced concrete members may also be described by equation (4.1). Data from bending tests performed on rectangular reinforced concrete columns axially loaded in compression tot of the yield load are given in Reference [28]. Using the method of least squares to fit the data, the relationship between the constant deflection amplitude of the test specimen and the number of cycles to failure may be written as 5 88 Nb • = 239.1 (4.4) Here, failure was described as when the concrete fell down due to the buckling of the longitudinal reinforcement. By considering a reinforced concrete column as a beam of two materials, namely concrete and steel, the yield displacement in bending may be computed. Hence, in terms of - 61 - the ductility ratio, the damage law becomes N~ 5 • 88 = 416.3 (4.5) The damage laws discussed for steel and reinforced concrete are shown in Figure 4.1. Note that the steel requires more cycles to fail than the reinforced concrete for the same level of ductility. The value of the ductility for N = 1 provides an upper limit to the allowable ductility factor of the structure. The damage laws are similar to some of those that are used to describe low-cycle fatigue failure of materials. 4.2 Damage Accumulation for Narrow-banded Random Response Cyclic failure in a material may be of two basic types. High-cycle failure occurs from repeated application of stresses below the yield stress of the material. The number of cycles to failure for high-cycle failure is of the order 10 4 . Low-cycle failure occurs when the material is strained repeatedly beyond the yield point. typically required for a low-cycle failure. Less than 10 3 cycles are When low-cycle failure is of concern, it is more meaningful to relate the number of cycles to failure to the strain amplitude rather than the stress amplitude. One approach, used to predict fatigue failure, is to model the relationship between the number of cycles to failure and the cyclic strain amplitude for a material under constant strain amplitude loading. - 62 - Steel Reinforced Concrete ~ :J) -J-...1 10 -J-...1 0 :J 0 1 1 10 Number Figure 4.1 100 of C8cles. 1000 N Damage Laws for Steel and Reinforced Concrete. - 63 - A widely accepted formula, proposed by Travernelli, Coffin and Manson [30] is given by (4.6) where N is the number of cycles to failure at the constant principal strain amplitude e, E is the modulus of elasticity, and the other variables are empirical constants determined from experimental test data. The first term on the right-hand side of equation (4.6) dominates when e is in the elastic range and reflects the classical S-N curve known as the Basquin equation for high-cycle fatigue. The second term is equal to the amount of strain beyond the elastic strain range at which the material must be cycled to fail in N cycles. The second term dominates for high-strain low-cycle fatigue. During an earthquake, a lightly damped structure may be subjected to severe random forces causing cyclic nonstationary response. High strains may develop within members of the structure such that the second term of equation (4.6) governs the failure. In such a situation, the first term of the right-hand side of equation (4.6) may be neglected, and the failure relationship may be written as ce (4.7) where N is the number of cycles to failure at the constant strain amplitude e, and s and C8 are positive empirical constants. By normalizing the strain by the elastic limit strain, equation (4.7) is - 64 - equivalent to equation (4.1). A cumulative damage hypothesis may be used to relate the failure of a system with varying response amplitude to a system with constant response amplitude. The simple rule proposed independently by Miner [31] and Palmgren [32] assumes damage accumulation to be a linear func- tion of the number of cycles of constant strain amplitude cyclic loading. Hence, the incremental damage due to the application of n. l cycles at a strain level ei is D.l ni (4.8) N(e.) l Furthermore, it is assumed that the order of application of different strain levels has no effect on the total damage.* Thus, the total damage for varying strain levels is (4.9) Failure occurs when D reaches unity. Substituting equation (4.7) in equation (4.9) yields D c-1 e [ n.e.s l l l (4.10) The concept of total damage may easily be extended to random strain response by assuming the number of peaks occurring in the strain response is synonymous with the number of cycles in the strain response. * There is evidence that this assumption may not be valid for all systems. However, it is employed herein because it allows the Palmgren-Miner theory to be used in the case of random strains. - 65 - This assumption is reasonable for lightly damped structures where the response is a narrow-banded nonstationary cyclic process. Let m be the total number of peaks per unit time and p(e;tlm) be the conditional probability density for the strain amplitude of the response given the number of peaks per unit time. m p(e;tlm) The quantity represents the number of peaks at a level e given m peaks per unit time. Hence, the expected number of peaks per unit time at a level e is given by E[n 8 (t)] = J m p(e;tlm) p(m;t) dm (4.11) 0 where p(m;t) is the time-varying probability density for the total number of peaks per unit time. Based on equation (4.8) with equation (4.7) substituted and using equation (4.11) the expected rate of damage accumulation due to peaks of strain amplitude e may be written as c:o dD ] E [ dt(e;t) 3 =~ J m p(e;tlm) p(m;t) dm (4.12) e 0 Integrating equation (4.12) over all levels of strain yields the expected rate of damage as c:o 1 c 8 c:o f es f m p(e;tlm) p(m;t) dm de 0 0 (4.13) - 66 - If the total number of peaks per unit time is assumed independent of the amplitude of the peaks, the following relationship holds CX> CX> J m p(e;tlm) p(m;t) dm p(e;t) 0 J m p(m;t) dm p ( e ; t) E [M ( t ) ] ( 4 .14) 0 where E[M(t)] is the expected number of peaks per unit time. Substi- tuting equation (4.14) into equation (4.13) yields CX> E [~(t) l e f es p(e;t) de (4.15) 0 which may be integrated over the duration of the excitation to give the expected value of the total damage as E[!d~~(t)dt] c f td E[D(td)] = ..1... e 4.3 0 CX> E [M ( t)] f es p(e;t) de dt (4.16) 0 Damage Accumulation Applied to a Simple Oscillator A lightly damped system is classified as a narrow-banded system. The predominant frequency components of the response are contained in a narrow band near the natural frequency of the system. The response appears to be a slightly distorted sine function with slowly varying amplitude and phase. The cumulative damage of a lightly damped system - 67 - may be computed by considering the cyclic nonstationary response of a simple oscillator. Assuming strain is proportional to displacement, equation (4.15) may be written in terms of displacement as s Q) = E[M(t)] cb bs p(b;t) db (4.17) 0 The damage model may be expressed in terms of displacement as (4.18) For the cyclic nonstationary response of a lightly damped simple oscillator, the number of peaks per unit time is approximately equal to the number of zero crossings per unit time. E[M(t)] Hence, \)(O,t) (4.19) where \)(O,t) is the frequency of zero crossings defined by equation (2.24). The probability density p(b;t) may be found by considering the frequency of up-crossing of a simple oscillator at a level b at time t, \)(b,t), which is given by equation (2.23). Since the number of up- crossings at a level b is approximately equal to the number of peaks above the level b, \)(b,t) is approximately equal to the expected number of peaks above b per unit time. The expected total number of peaks per unit time is equal to the frequency of zero crossings, \)(O,t). Hence, the expected fraction of peaks above the level b is approximately - 68 ~(b,t)/~(O,t), and the probability distribution of the peak magnitudes at time t may be approximated as 1 Pr[peaks < b; t] ~(b,t) (4.20) - ~(O,t) The average probability density of the peak magnitudes at time t may be obtained by differentiating equation (4.20) with respect to b. This yields p(b;t) 1 a~(b,t) ~co,t> ab (4.21) Substituting equation (4.19) and (4.21) into equation (4.17), the expected rate of damage accumulation is given by = - JL cb f bs a~(b,t) db ab (4.22) 0 a~(b,t) is found by differentiating equation (2.23) to be ab a~(b,t) ab Vdet Q.(t) 2nq 11 (t) J -b ( cJll (t)b2) \ qll ( t) exp 2 exp (- b2 ) erfc (cJ12(t)b)) 2q11<t> V2cJ <t> 22 . (4.23) - 69 - 4.4 A Closed Form Expression for a Special Case of the Damage Law The expected rate of damage accumulation given by equation (4.22) may be easily evaluated numerically for any appropriate value of s. For integer values of s, equation (4.22) may be evaluated in closed form. This was shown by Roberts [33] for odd integers and can be extended to even integers by the evaluation of a single integral. It is convenient to introduce the correlation coefficient p(t) defined as p(t) (4.24) where qij(t) are the elements of the covariance matrix Q(t). It is also helpful to normalize the displacement by the root mean square value of the response and denote this new dimensionless variable by ~ where (4.25) Substituting equation (4.23) into equation (4.22) then yields ( - 12 2 n 2 ) 0 (4.26) where the functional dependence of p and q .. on t has been omitted for l.J - 70 - brevity. Integrating equation (4.26) by parts gives E [~~ ( t)] (4.27) where (X) J(p,s) J ~s exp (- nf-) erf(v2(1 pn- p2 ) ) d~ (4.28) 0 and the gamma function f(a) is defined as (X) f(a) f xa-1 exp(-x) dx 0 The function J(p,s) cannot easily be expressed in a simple manner as a function of p and s. However, it does satisfy a reduction formula obtained by integrating equation (4.28) by parts to give s-1 2 J(p,s) = (s-1)J(p,s-2) + ~[2(1 - p )] 2 r(i) (4.29) Hence, to define the expected damage rate for all integral values of s, one needs only to evaluate equation (4.28) for s = 0 and s = 1. Performing this operation yields - 71 - 00 J(p,O) J exp (- ~) erff 0 2 pn ) dTt \-\J2(1 - p2) 2 In -~;tan !2":. -1("1 lpl - p )] sgn(p) [ ~2 (4.30) and 00 J(p,1) = J Tl exp (- n:_) err( pn 2 ) dTt 2 ~(1 p ) 0 (4.31) p The expected rate of damage accumulation may now be expressed as lj, q 22 (4.32) ( t) K ( p ( t) , s) where K(p(t) ,s) + sp'Yri r(s;2) K(p(t),s) is shown in Figure 4.2. of p. s+l 1 • (4.33) [r(s;l) + (t) 2 J(p,s) J K(p(t),s) is positive for all values Hence, the expected damage rate is positive. Note that K(O,s) is equal to unity for all values of s. A characteristic of the stationary response of a simple oscillator is that the displacement response and the velocity response are uncorrelated (i.e., p = 0). ary response is given by Thus the expected damage rate for station- - 72 - 0 0 ro ~ Lo il)O s = Q -i-1 ..... o :JO :L..; il) -i-1 d CCo 0 il) • Q)0J d E. d Do 0~~~~~~------~------~--------~ 0 Figure 4.2 -1.00 -0.50 Correlot 1 on 0. 00 0. 50 coeff 1c1ent. p 1. 00 Damage Rate Multiplier versus Correlation Coefficient. - 73 - s 22 r(T) (4.34) 2rrCb which is consistent with the results given by Hiles [34]. 4.5 Normalization of the Expected Total Damage for a Linear System Subjected to Hodulated White Noise Integrating equation (4.22) with respect to time, the expected total damage may be written as td CD 1 E[D(td)J =- cbf f bs ~(b,t) db dt (4.35) 0 0 Consider the special case where a simple linear oscillator with natural period T is subjected to modulated white-noise excitation. If the modulating envelope is a function of only the normalized time, ~ = ~· given the duration td of the excitation, a convenient normalization of the expected total damage problem exists. In the situation described above, the covariance matrix Q(~) may be expressed as (4.36) where the f .. (~) 's are dimensionless functions and s 0 is the power ~J - 74 spectral density of the white-noise excitation. Using this form of the covariance matrix, the derivative of the frequency of up-crossings with respect to the crossing level may be written as (4.37) where b Performing a change of variables on equation (4.35) yields (4.38) It may be shown that the first-passage problem defining the power spectral density from the response spectrum given in equation (3.8) may be written as td T exp - J a(r, ~ ,·t') d't' (4.39) 0 where Ps is the confidence level of the response spectrum, a is the limiting decay rate, and r is the response spectrum normalized by the excitation level given by - 75 r = SD \fs 0 T3 The expected total damage may be expressed in terms of r as td T CD f f (4.40) Recall from equation (4.7) that for low-cycle failure the damage law is given by where e is a strain which is greater than the elastic limit strain e y of the material. If the strain is normalized by the elastic limit strain, the damage law may be written as c!1 (4.41) where 11 is known as the ductility factor given by ....f... e y and c !1 Since strains are assumed proportional to displacements, it follows that - 76 - SDs /Ss SDs cb s Y. JL c /ss b y ll c (4.42) where Sy is the displacement corresponding to the elastic limit strain, ey. Using this relationship in equation (4.40) yields td s T oo 1 cll r s f f .JL- (4.43) 0 0 where r is implicitly defined by equation (4.39). The expected total damage may be computed given the duration of the excitation in natural periods of the system, the ductility factor of the system, and the confidence level associated with the ductility and the damage model. Note that the overall shape of the response spectrum is not important. Only the value of the response spectrum at the natural frequency of the system matters in its relationship to the ductility factor, j.l, of the response. For the softening nonlinear elastic system of section 3.3, the power spectral density of the excitation is not a constant with respect to the time because of the time-varying frequency shift. The expected total damage of the nonlinear system depends on the shape of the power spectral density. In this case, it is better to use equation (4.35) directly to compute the expected total damage. - 77 - CHAPTER V ASSESSMENT OF THE EFFECTS OF DURATION ON THE DAMAGE OF A SIMPLE SYSTEM 5.1 Introduction In order to assess the effects of various system parameters on the expected total damage of a structure, consider the simple frame shown in Figure 5.1. When the frame is subjected to strong earthquake ground motion, large displacements can occur. Large bending moments develop at the top and the bottom of the columns of the frame. The large bending moments cause large strains which may be associated with low-cycle failure. A simple analytical model for earthquake excitation is a modulated stationary Gaussian random process. The equation of motion for the simple frame subjected to such a process is that of a simple oscillator and given by (5.1) where ~ is the fraction of critical damping, w0 is the natural frequency of the system, 9(t) is the deterministic modulating envelope, and g(t) is a stationary Gaussian random process with spectral density S(w). Zero initial conditions are assumed. The modulating envelope will be taken to be the earthquake-like modulating envelope used in Section 3.2. terms of the dimensionless time, ~. as The envelope may be written in - 78 - X M 8(t)w(t) Figure 5.1 Simple Frame Structure. - 79 - 56.25 2td 0 ~ 't' ~ 15T 't'2 (td/T)2 9('t') 2td td < 't' ~ 2T 15T 1.0 exp[-2.976(td't'/T - t>] (5.2) td td << 't' - T 2T where 't' = 1 T and T is the natural period of the system. Response statistics of the system, in particular the covariance matrix, are needed to evaluate the expected total damage. If the system is lightly damped, the response will be narrow-banded about the natural frequency. Hence, the same approximation used in Section 3.2 for ~ sufficiently small may be used here. The process g(t) may be replaced with a stationary Gaussian white-noise process, w(t), with constant spectral density s 0 = S(w ). 0 The methods discussed in Section 2.1 may be used to compute the response statistics of the system. The response statistics may then be applied to equations (4.39) and (4.43) to compute the expected total damage. - 80 - 5.2 Expected Total Damage- Linear Model Equation (4.22) may be evaluated numerically for any value of the damage law exponent, s. However, to study the effects of duration on the expected total damage, it is computationally more efficient to use integer values of s. In that case the closed form solution described in Section 4.4 may be applied to evaluate the expected rate of damage accumulation. Numerical integration in time may then be used to compute the total expected damage. For steel, the damage law exponent will be taken to be 2 and equation (4.3) becomes N~ 2 = 167.1 (5.3) For reinforced concrete, s is approximately 6 in equation (4.5) so the damage law will be taken as N~ 6 = 416.3 (5.4) The normalization discussed in Section 4.5 for the linear elastic system will be applied here . Time variables will be normalized by the natural period of the system. The earthquake-like modulating envelope described by equation (5.2) will be used. The duration of excitation td will be expressed a 1r natural periods; hence, varying the natural period is equivalent to varying the duration. The expected total damage is given in equation (4.43) as - 81 - td T s .J,L.__!_ c fl. r s (X) Clj3 (j3 , 't) dj3 d't f f 13s ~ 0 (5.5) 0 where r satisfies a(r,~,'t) d't (5.6) The integration over j3 is performed by applying equation (4.32) and the integration over 't is performed numerically. Since large amounts of yielding are assumed, the system will behave in a nonlinear manner. However, the effects of a system nonlinearity can often be accounted for approximately by a simple shift in the damping factor and natural frequency of a linear structure. Therefore, by studying a linear system, a fundamental understanding of the effects of duration and ductility ratio on damage may be obtained even for a nonlinear system. In Figures 5.2 and 5.3, the expected total damage is shown for a linearly elastic system. Curves are shown for several levels of ductility and two values of damping. Figure 5.2 represents a damage behavior close to that of a steel structure and Figure 5.3 approximates the behavior of a reinforced concrete structure. For a fixed ductility level, the expected total damage is primarily a function which initially increases rapidly with increasing duration, - 82 - l.[) (\J I E, = 2/. E, = 10/. I I]J I 10 0 0 I 11)..--< I CJ) I ]J d I E 5 d Ln Dr-I .---;0 I 0 -j...J 0 I- 0 u LJ: rl)o -j...J u (1) Q XLn w": 0 0 0 / 50. 00 100. 00 0 uro.t Figure 5.2 1 1 so . 00 on/Per 1 200. 00 250 . 00 od Expected Total Damage versus Normalized Duration for a Linear Steel Structure. - 83 - 1..0 N --- -- ~ = 21. ~ - 10/. / / / / / 2. 5 j.l / / / / / / / / / (j / / of( / / / / / 0 / / +-' / / 0 f- / / / ::::::; -,, ._, t.r. / / / / / / (J / il.J / Q_ X l..J: j.l w~ c -- -- ----- --- --- --- c 0 2.0 = j.l = --- --- 1.5 - - ------------------------- -------- ---0. 00 50 . 00 100. 00 0 uro.t Figure 5.3 1 150. 00 on/Per 1 200. 00 25 0. 00 od Expected Total Damage versus Normalized Duration for a Linear Reinforced Concrete Structure. - 84 - then approaches a nearly constant positive slope corresponding to the stationary damage rate. For a steel structure the expected damage is a somewhat uniformly increasing function of duration. However, for a reinforced concrete structure, there is a region below 50 periods where the expected damage is rather insensitive to duration. As the ductility level increases, the material is cycled further into its plastic range, and the expected total damage increases. From equation (5.5), it is apparent that E[D] is proportional to ~s for a linear system. Therefore, the damage is a stronger function of the ductility ratio for the reinforced concrete structure than for the steel structure. The steel structure is capable of sustaining much greater levels of ductility than the reinforced concrete structure without failure. In Figures 5.2 and 5.3, structural damping is treated independently of the ductility factor. Realistically, however, an increase in damping reduces the system response, thereby also reducing the ductility factor. Hence, the net reduction in the damage due to an increase in damping is greater than the reduction implied in the figures. 5.3 Expected Total Damage- Softening Nonlinear Model The effect of system softening may be included to better approxi- mate a yielding system by considering a simple oscillator with a softening nonlinear elastic spring. Using the same system as in Section 3.3, - 85 - the equation of motion for such a simple oscillator is given by (5.7) where ~ and w0 are the fraction of critical damping and the undamped natural frequency, respectively, associated with small displacements, xmax is the maximum response of the system, and ~ is the nonlinearity parameter. Statistical linearization techniques may be used to define an equivalent linear system for which response statistics and expected total damage are computed. Equation (4.35) is used directly to compute the expected total damage due to the variation in the power spectral density caused by the time-varying frequency shift. In Figures 5.4-5.7, the expected total damage for the softening nonlinear elastic system subjected to the processes derived in Section 3.2 from the NRC design response spectrum is shown for several values of the nonlinearity parameter ~· Recall that ~ may vary from 0 to any finite value with ~ = 0 corresponding to a linear system, and that ~ is similar to the ductility ratio of a yielding system for values greater than unity. Figures 5.4 and 5.5 represent damage behavior based on the damage law for steel [equation (5.3)], and Figures 5.6 and 5.7 represent damage behavior based on the damage law for reinforced concrete [equation (5.4)]. The results are shown for various combinations of duration, ductility ratio, and damping ratio and plotted against the undamped - 86 - lD (\! ....... = 3 0 se e s --- - - t d = 10 s e e s ~ = 2 1. td 0 I 10 \.1 ~ ~ I I I 11}- I I C) 0 I E I 0 I I 0~ I I I I ....--<0 I 0 I 5 \.1 I I I -1-' n....... 1- I I I I I I I 0 / I D~ / I / ~0 / / -1-' / {' u / ~ / "LL. , XLD w '~ 0 _... _... _... _... ,, / / / \.1 = 2 _... _... 0 0 0 10 -1 Frequ e n c hJ Figure 5. 4 10 I 10 c (Hz l Expected Total Damage versus Natura l Frequency for a Softening Steel Structure, ~ = 2%. (NRC Design Response Spectrum) - 87 - Ln N 0 0 td - - -- - td ; = 10% = 30 sees = 10 se e s I I I I 10 ]J I I I I I I I I I I I ~o I d +' I I 0 f-- I 5 ]J I I I I I 0 I I Dlii ~0 I I I I I / +' I / () / / ~ / / / / Q / / X Ln / w~ / / 0 0 0 o 1o-1 Frequenc~ Figure 5.5 10 I 10 ~ (Hz) Expected Total Damage versus Natural Frequency for a Softening Steel Structure, ; = 10 %. (NRC Design Response Spectrum) - 88 - L[) N ·- - - - - s = 2% 0 td td = 30 sees = 10 sees ~ O-J ~0) ]J d = 2.5 / / / / E / / 0 0~ ........ o ,._,--/ . d +-' 0 1- 0 . ~l.D v ~0 +-' (_; ~ Q_ XL[) w '"": 0 --- ----- ------- 0 0 o 1o-1 F requenc~ Figure 5.6 10 I 1 0° (Hz) Expected Total Damage versus Natural Frequency for a Softening Reinforced Concrete Structure, = 2%. (NRC Design Response Spectrum) s - 89 - / ).l = 2.5 // / / ~~-~-~-~~ - ~-~-~ - ~------- - --- 0 oL_~~~~~-~--~--~~~Iill O 10 - I 10 O Frequenc~ Figure 5.7 10 I (Hzl Expected Total Damage versus Nitural Frequency for a Softening Reinforced Concrete Structure, ~ = 10 %. (NRC Design Response Spectrum) - 90 - natural frequency associated with small displacements. The expected total damage of the softening nonlinear elastic system is qualitatively the same as that of the linear system. The nonlinearity affects the expected total damage in three ways. First, an increase in the response amplitude occurs because of the assumed level of softening in the system resulting in an increase in the amount of damage per cycle of response. Secondly, the downward frequency shift reduces the number of response cycles executed for a fixed duration of excitation. Finally, the frequency shift also changes the effective power spectral density of the input excitation. The power input to the system may increase or decrease depending on the slope of the power spectrum near the small displacement natural frequency of the system. This, in turn, modifies the response amplitude accordingly. 5.4 Expected Damage Contours A useful way to display expected damage information is shown in Figure 5.8. Based on the system which behaves linearly, contours for expected total damage equal to unity may be presented as a relationship between the ductility factor of the response and the duration of excitation. In this way, combinations of duration and ductility factor for which the expected total damage is greater than or less than unity may readily be identified. From Figure 5.8 it is evident that the range of the allowable - 91 - 0 0 0,-r---------,------------.-----------.-----------.----------· \ I I E, E, I = 2% = 10% E[D] > 1 \ \ \ \ \ \ \ \ " " ',E[D]=l -+-' 0 ::Jo . . . .... oo ...0 d 30 -- --- --- - .._-- . Oo ..---' for steel ......... _ ~tructure ---------- ---- - - - - E[D] = 1 for reinforced concrete structure ..-JN CI E[D] < 1 0 0 50.00 100.00 0 urat Figure 5.8 1 150.00 200.00 250.00 on/Per 1 od Contours of Expected Total Damage equal to unit y versus Allowable Ductility Factor and Normalized Duration for Linear Steel and Reinforced Concrete Structures. - 92 - ductility factor is much smaller for the reinforced concrete structure than for the steel structure. The rapid increase in damage associated with short durations, along with the strong dependence upon the ductility ratio, leads to the existence of a sharp threshold for failure in the reinforced concrete structure. The expected total damage reflects a mean value of the damage of a structure. Hence, the allowable ductility factor for a given duration shown in Figure 5.8 may be taken to be an upper bound of the design ductility factor of the structure. The value of the damage is, of course, a function of the failure model employed in the analysis and may not be applicable to all structures. However, the functional dependence of the expected damage on the ductility factor and duration is thought to be representative. A considerable amount of computation is required to compute expected damage contours for a nonlinear system. Since the expected total damage of the softening nonlinear system is less than that for the linear system, the contours for the linear system may be conservatively used in place of the contours for the nonlinear system. - 93 - CHAPTER VI SUMMARY AND CONCLUSIONS In Chapter I, it was proposed that three basic elements be combined in order to study the effects of duration on the damage of structures. The first element was to define a nonstationary random process which models earthquake excitation and corresponds statistically to a desired response spectrum. The second element was to postulate a model for the incremental damage of a system. The last element was to compute a measure of the damage in a simple structure. In Chapter II the necessary tools to accomplish these tasks were reviewed. In the first section, random vibration theory was discussed including methods to compute the response statistics needed to implement the first-passage probability estimates and cumulative damage expressions. The second section dealt with analytical approaches to the first- passage probability of a simple oscillator and defined the equations which would later be used to synthesize the response spectrum consistent processes. The last section recalled the method of statistical lineari- zation in order that a nonlinear system could be considered by the firstpassage probability and accumulated damage estimates. In Chapter III, modulated Gaussian random processes were assumed to model earthquake-like excitation. fully defined by its duration. The modulating envelope was assumed Analytical estimates of the first- passage probability of a simple oscillator were calculated for several threshold levels. The result were found to be reasonably accurate when - 94 - compared to Monte Carlo simulation. Using the analytical first-passage probability estimates, power spectrum ordinates were chosen such that the maximum response exceeded the design spectrum with a predetermined probability. Actual random processes were computed consistent to the NRC design response spectrum. The magnitude of the power spectrum found in this manner varied significantly with duration. The maximum response of a nonlinear softening system to excitation specified by a response spectrum was discussed. The role of the first- passage problem was reversed to find the maximum response of a statistically linearized nonlinear system. The maximum response of the soften- ing nonlinear system as function of frequency was found to be similar to a linear response spectrum which had been translated along an axis of constant displacement on a log-log pseudo-velocity diagram. In Chapter IV, the behavior of structural members under cyclic loads was discussed. A rule analogous to the Miner-Palmgren rule approach to fatigue was postulated. The rule was used to compute the mean damage of a system with narrow-banded random response. In particu- lar, it was applied to a simple harmonic oscillator subjected to a modulated Gaussian random process. For a special case of the damage law, a closed form expression was formulated for the expected damage rate. In Chapter V, the degree of damage in a simple structure subjected to excitation specified by a response spectrum was determined. Two types of structure were considered: a steel structure and a reinforced concrete structure. The damage law was defined from cyclic bending tests - 95 - performed on steel and reinforced concrete structural members. The response spectrum consistent random processes computed earlier were used as input to the system. It was observed that the damage of a linear structure initially increases rapidly with increasing duration, and then approaches a constant positive slope corresponding to the steady-state damage rate. A steel structure is capable of sustaining much greater levels of ductility without failure than a reinforced concrete structure. Damage for a reinforced concrete structure displays a much stronger dependence on the ductility level of the response than the damage for a steel structure. When system softening is accounted for through statistical linearization, results are found similar to the linear system, but with lesser degrees of damage. A relatively straightforward analytical procedure has been presented to estimate the effect of earthquake duration on the damage of a structure. Based on the results presented herein, it is found that the duration of excitation and the design ductility level of the response can have a strong effect on the expected damage of the structure. Use of the response spectrum alone to specify a design input ground motion accounts for the dependence of the damage on the ductility factor of the response, but ignores the effects of the duration of the excitation. Therefore, it is recommended that some measure of duration be provided more often in earthquake design specifications and utilized in the analysis of structural reliability. - 96 - REFERENCES 1. Lin, Y.K., Probabilistic Theory of Structural Dynamics, New York: McGraw Hill, 1967. 2. Rice, s.o., "Mathematical Analysis of Random Noise," in Selected Papers on Noise and Stochastic Processes, Nelson Wax, ed., New York: Dover Publications, Inc., 1954. 3. Mason, A.B., Some Observations on the Random Response of Linear and Nonlinear Dynamical Systems, Ph.D. Thesis, California Institute of Technology, 1979. 4. Crandall, S.H., "First-Crossing Probabilities of the Linear Oscillator," Journal of Sound and Vibration, vol. 12, no. 3, 1970, pp. 285-299. 5. 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