A theoretical analysis of the effect of duration on damage to structures exposed to earthquakes is presented. Estimates of the first passage probability for a simple oscillator are used to select modulated Gaussian random processes according to a prescribed response spectrum. In some cases, it can be argued that the peak response is the dominant factor affecting system safety and reliability.
The solution to equation (2.2) can be expressed in terms of the fundamental matrix solution as Choosing the alternative method, equation (2.16) and the definitions in equation (2.18) require that the elements of the covariance matrix are satisfied. Due to the relationship between the response spectrum and the response time history, a symmetric double barrier is considered.
UNSAFE REGION
REGION
The first-passage problem consists of determining the probability distribution of the time when the trajectory of the response first leaves the safe area and enters the unsafe area. The parameter a is known as the limiting decay rate of the first crossing density or the average crossing rate. The average rate of crossing up of a level b is ~(b), and is equal to the average rate of crossing down of the level -b.
The parameter y is a measure of the bandwidth of the response and is determined by the spectral moments of the response as . Approximations of the limiting decay rate found for the stationary excitation together with the instantaneous response statistics are used to calculate a limiting instantaneous decay rate, a(t). A criterion for minimizing ~ is to require that the mean of the scalar product ~T~ be minimal, i.e.
CHAPTER III
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). To determine the accuracy of the analytical approach used to determine the reliability function, a Monte Carlo simulation study of equation (3.1) was performed. Since the applied excitation is assumed to be a straight line segment at each time interval, the solution to equation (3.1) can be solved by a digital computer in a purely arithmetical way [19].
The total response for a time interval was then obtained by analytically matching the initial conditions of the current interval with the final conditions of the previous interval. In this way, no numerical approximations other than the white noise approximation and rounding due to the digital representation of the response were introduced. The threshold levels are normalized by the stationary standard deviation of the system described by equation (3.1) with 9(t) = 1.
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The stationary standard deviation of the response is inversely proportional to the square root of the damping ratio. In Figure 3.5, the probability that the peak response is less than a constant threshold level is plotted as a function of excitation duration for several values of damping. Figures 3.6–3.8 show the probability that the maximum response of a simple oscillator is below a specified level for several threshold values and several damping values using the envelope of Eq.
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 different values of damping, of the maximum displacement plotted versus the natural. Smooth curves are then chosen to describe the shape and normalized level of the response spectrum for a given confidence level.
This can be achieved by performing a first-pass probability analysis for a simple oscillator to select power spectral density ordinates for the process from the design response spectrum. For a fixed response spectrum and a fixed confidence level, the level of the power spectrum increases with duration. A shift in the distribution of the power towards the lower frequencies as the duration decreases is evident from.
If a structure behaves nonlinearly, the usual linear response spectrum may not characterize the response of the structure. The response of the compliant system is often described by the ductility ratio; that is, the ratio of the maximum response to the maximum elastic response. The nonlinearity parameter defined herein is comparable to the ductility ratio of the compliant system.
In Figures 3.17 and 3.18 the statistical maximum response of the softening non-linear elastic system is plotted against the small displacement frequency. Second, 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 to the instantaneous effective. This variation depends on the shape of the power spectral density of the excitation process.
CHAPTER IV
If we normalize the displacement in equation (4.2) by the yield displacement, the damage law can be written in terms of the ductility factor as Using the least squares method to fit the data, the relationship between the constant deflection amplitude of the test specimen and the number of cycles to failure can be written as The ductility value for N = 1 provides the upper limit of the allowable ductility factor of the structure.
Large stresses can develop within the components of the structure, so that the second term of equation (4.6) determines the failure. In such a situation, the first term of the right-hand side of equation (4.6) can be neglected and the failure relationship can be written as. The simple rule proposed independently by Miner [31] and Palmgren [32] assumes that damage accumulation is a linear function of the number of cycles of constant stress amplitude, cyclic.
Let m be the total number of spikes per unit time and p(e;tlm) be the conditional probability density for the stress response amplitude given the number of spikes per unit time. If the total number of peaks per unit time is assumed to be independent of the peak amplitude, the following relationship exists. Therefore, the expected fraction of peaks above level b is approx. b,t)/~(O,t), and the probability distribution of peak sizes at time t can be approximated as.
The average probability density of the highest magnitudes at time t can be obtained by differentiating equation (4.20) with respect to b. A characteristic of the steady-state response of a simple oscillator is that there is a displacement response and a velocity response. Using this form of the covariance matrix, the derivative of the frequency of upward transitions with respect to the transition level can be written as
The expected total damage of the nonlinear system depends on the shape of the power spectral density.
CHAPTER V
System response statistics, in particular the covariance matrix, are needed to evaluate the expected total damage. The methods discussed in section 2.1 can be used to calculate the response statistics of the system. The response statistics can then be applied to equations (4.39) and (4.43) to calculate the expected total damage.
For a fixed ductility level, the total expected damage is primarily a function which initially increases rapidly with increasing duration. l.[). As the ductility level increases, the material is cycled further into its plastic range and the total expected damage increases. Damage is therefore a stronger function of ductility ratio for reinforced concrete structure than for steel structure.
Equation (4.35) is used directly to calculate the expected total damage due to the variation in power spectral density caused by the time-varying frequency shift. The expected total damage of a softening nonlinear elastic system is qualitatively the same as that of a linear system. Based on a linearly behaving system, the contours for expected total damage equal to unity can be represented as the ratio of the response ductility factor to the excitation duration.
In this way, it is easy to identify combinations of duration and ductility factor for which the expected total damage is greater or less than unity. Therefore, the allowable ductility factor for a given duration shown in Figure 5.8 can be considered an upper limit of the design ductility factor of the structure. However, it is assumed that the functional dependence of expected damage on ductility factor and duration is representative.
Because the expected total damage from the mitigating nonlinear system is less than that for the linear system, the contours for the linear system can be used conservatively in place of the contours for the nonlinear system.
CHAPTER VI
The role of the first-pass problem was reversed to find the maximum response of a statistically linearized nonlinear system. The maximum response of the softening nonlinear system as a function of frequency was found to be comparable to a linear response spectrum translated along an axis of constant displacement in a log-log pseudo-velocity plot. For a special case of the law on damages, a closed-form expression has been formulated for the expected damage percentage.
Damage to a reinforced concrete structure is much more dependent on the level of ductility of the response than damage to a steel structure. Based on the results presented here, it is concluded that the duration of the excitation and the design level of response ductility can have a strong influence on the expected damage. Using the response spectrum alone to determine the structural input ground motion takes into account the dependence of damage on the response ductility factor, but ignores the effects of excitation duration.
Crandall, S.H., "First-Crossing Probabilities of the Linear Oscillator", Journal of Sound and Vibration, vol. Vanmarcke, E.H., "On the distribution of the first passage time for normal stationary random processes", Journal of Applied Mechanics, ASME, vol. Vanmarcke, E.H., "Properties of spectral moments with application to arbitrary vibrations", Journal of the Engineering Mechanics Division, ASCE, vol.
Corotis, R.B., Vanmarcke, E.H., and Cornell, C.A., "First Passage of Nonstationary Random Processes," Journal of the Engineering Mechanics Division, ASCE, vol. Booton, R.C., "The Analysis of Nonlinear Control Systems with Ran- dom Input," Proceedings of the Symposium on Nonlinear Circuit Analysis, Brooklyn, New York, 1953, vol. Atalik, T.S., and Utku, S., "Stochastic Linearization of Nonlinear Multidegree-of-Freedom Systems," Earthquake Engineering and Structural Dynamics, vol.
Nigam, N.C., en Jennings, P.C., "Berekening van responsspectra van Strong-Motion Earthquake Records", Bulletin of the Seismological Society of America, vol.
