We study volumetric deformation structures in stepover regions using numerical simulations and field observations, with a focus on small-scale features near the ends of rupture segments that have opposite-polarity from the larger-scale structures that characterize the overall stepover region. The reversed-polarity small-scale structures are interpreted to be generated by arrest phases that start at the barriers and propagate some distance back into the rupture segment. Dynamic rupture propagating as a symmetric bilateral crack produces similar (anti-symmetric) structures at both rupture ends. In contrast, rupture in the form of a predominantly unidirectional pulse produces pronounced reversed-polarity structures only at the fault end in the dominant propagation direction. Several observational examples at different scales from strike-slip faults of the San Andreas system in southern California illustrate the existence of reversed-polarity secondary deformation structures. In the examples shown, relatively-small pressure-ridges are seen only on one side of relatively-large extensional stepovers. This suggests frequent predominantly unidirectional ruptures in at least some of those cases, although multisignal observations are needed to distinguish between different possible mechanisms. The results contribute to the ability of inferring from field observations on persistent behavior of earthquake ruptures associated with individual fault sections.

The stability of steady slip and homogeneous shear is studied for rate-hardening materials undergoing chemical reactions that produce weaker materials (reaction-weakening process), in drained conditions. In a spring- slider configuration, a linear perturbation analysis provides analytical expressions of the critical stiffness below which unstable slip occurs. In the framework of a frictional constitutive law, numerical tests are performed to study the effects of a nonlinear reaction kinetics on the evolution of the instability. Slip instabilities can be stopped at relatively small slip rates (only a few orders of magnitude higher than the forcing velocity) when the reactant is fully depleted. The stability analysis of homogeneous shear provides an independent estimate of the thickness of the shear localization zone due to the reaction weakening, which can be as low as 0.1 m in the case of lizardite dehydration. The potential effect of thermo-chemical pore fluid pressurization during dehydration is discussed, and shown to be negligible compared to the reaction-weakening effect. We finally argue that the slip instabilities originating from the reaction-weakening process could be a plausible candidate for intermediate depth earthquakes in subduction zones.

Geophysical observations have shown that transient slow slip events, with average slip speeds v on the order of 10 −8 to 10 −7 m/s, occur in some subduction zones. These slip events occur on the same faults but at greater depth than large earthquakes (with slip speeds of order ∼ 1 m/s). We explore the hypothesis that whether slip is slow or fast depends on the competition between dilatancy, which decreases fault zone pore pressure p, and thermal pressurization, which increases p. Shear resistance to slip is assumed to follow an effective stress law τ = f ( σ - p ) ≡ f σ ¯ . We present two-dimensional quasi-dynamic simulations that include rate-state friction, dilatancy, and heat and pore fluid flow normal to the fault. We find that at lower background effective normal stress ( σ ¯ ), slow slip events occur spontaneously, whereas at higher σ ¯ , slip is inertially limited. At intermediate σ ¯ , dynamic events are followed by quiescent periods, and then long durations of repeating slow slip events. In these cases, accelerating slow events ultimately nucleate dynamic rupture. Zero-width shear zone approximations are adequate for slow slip events but substantially overestimate the pore pressure and temperature changes during fast slip when dilatancy is included.

Internal resonance in the vibration of a floating roof coupled with nonlinear sloshing in a circular cylindrical oil storage tank is investigated. The nonlinear system exhibits internal resonance when nonlinear terms of the governing equation have a dominant frequency close to a certain modal frequency of the system. Numerical results show that when internal resonance occurs, the responses of stresses in a floating roof exhibit a long-duration period of large amplitude despite a short duration of the earthquake excitation applied to the tank. Due to the presence of internal resonance, the underestimation of the stresses associated with the use of the linear theory becomes more marked, and thus the importance of nonlinearity of sloshing in the stress estimation is accentuated. It is illustrated that the magnitudes of the stresses increase with the increase in the liquid-filling level, and that the effect of internal resonance on the stresses noted in the case of sinusoidal excitation appears under real earthquake excitation. A method for reducing the stresses is proposed.

A probabilistic approach for model updating and damage detection of structural systems is presented using noisy incomplete input and incomplete response measurements. The situation of incomplete input measurements may be encountered, for example, during low-level ambient vibrations when a structure is instrumented with accelerometers that measure the input ground motion and the structural response at a few instrumented locations but where other excitations, e.g., due to wind, are not measured. The method is an extension of a Bayesian system identification approach developed by the authors. A substructuring approach is used for the parameterization of the mass, damping and stiffness distributions. Damage in a substructure is defined as stiffness reduction established through the observation of a reduction in the values of the various substructure stiffness parameters compared with their initial values corresponding to the undamaged structure. By using the proposed probabilistic methodology, the probability of various damage levels in each substructure can be calculated based on the available dynamic data. Examples using a single-degree-of-freedom oscillator and a 15-story building are considered to demonstrate the proposed approach.

A relatively simple and straightforward procedure is presented for representing non-stationary random process data in a compact probabilistic format which can be used as excitation input in multi-degree-of-freedom analytical random vibration studies. The method involves two main stages of compaction. The first stage is based on the spectral decomposition of the covariance matrix by the orthogonal Karhunen-Loeve expansion. The dominant eigenvectors are subsequently least-squares fitted with orthogonal polynomials to yield an analytical approximation. This compact analytical representation of the random process is then used to derive an exact closed-form solution for the nonstationary response of general linear multi-degree-of-freedom dynamic systems. The approach is illustrated by the use of an ensemble of free-field acceleration records from the 1994 Northridge earthquake to analytically determine the covariance kernels of the response of a two-degree-of-freedom system resembling a commonly encountered problem in the structural control field. Spectral plots of the extreme values of the rms response of representative multi-degree-of-freedom systems under the action of the subject earthquake are also presented. It is shown that the proposed random data-processing method is not only a useful data-archiving and earthquake feature-extraction tool, but also provides a probabilistic measure of the average statistical characteristics of earthquake ground motion corresponding to a spatially distributed region. Such a representation could be a valuable tool in risk management studies to quantify the average seismic risk over a spatially extended area.

Three optimal control algorithms are proposed for reducing oscillations of flexible nonlinear structures subjected to general stochastic dynamic loads, such as earthquakes, waves, winds, etc. The optimal control forces are determined analytically by minimizing a time-dependent quadratic performance index, and nonlinear equations of motion are solved using the Wilson-θ numerical procedures. The optimal control algorithms developed for applications to nonlinear structures are referred to as the instantaneous optimal control algorithms, including the instantaneous optimal open-loop control algorithm, instantaneous optimal closed-loop control algorithm, and instantaneous optimal closed-open-loop control algorithm. These optimal algorithms are computationally efficient and suitable for on-line implementation of active control systems to realistic nonlinear structures. Numerical examples are worked out to demonstrate the applications of these optimal control algorithms to nonlinear structures. In particular, control of structures undergoing inelastic deformations under strong earthquake excitations are illustrated. The advantage of using combined passive/active control systems is also demonstrated.

An approximate method for nonstationary random vibration analysis is presented. This method utilizes properties of the stationary solution for simplifying the analysis. This approach has previously been applied by the author to linear and nonlinearly damped SDOF systems. In the present paper the concept is extended to linear MDOF systems and applied to nonstationary earthquake-type loading. Comparisons with available exact solutions show very good agreement in numerical results with the additional benefit of reducing computer time by more than one order of magnitude.

An experimental method has been developed for generating oblique forces with known orientations and time histories. Recorded signals from several forces were analyzed by an iterative deconvolution method to determine their orientations and time histories. The recovered values agree closely with the exact ones for these controlled sources. These experiments are a valuable test of source characterization methods that may be applied to seismic data from earthquake sources or to signals recorded from the acoustic emission of cracks.

A simple yet efficient method is presented for the on-line control of nonlinear, multidegree-of-freedom systems responding to arbitrary dynamic environments. The control procedure uses pulse generators located at selected positions throughout a given system. The degree of system oscillation near each controller determines the controller’s activation time and pulse amplitude. The direct method of Liapunov is used to establish that the response of the controlled nonlinear system is Lagrange stable. Simulation studies of three example systems (conducted with digital and analog computers) demonstrate the feasibility, reliability, and robustness of the proposed active-control method. These systems, which include one with a hysteretic nonlinearity, are structures representative of modern tall buildings; they are subjected to nonstationary random excitation representative of earthquake ground motions.

A method for analyzing the earthquake response of deformable, cylindrical liquid storage tanks is presented. The method is based on superposition of the free lateral vibrational modes obtained by a finite-element approach and a boundary solution technique. The accuracy of such modes has been confirmed by vibration tests of full-scale tanks. Special attention is given to the cos θ-type modes for which there is a single cosine wave of deflection in the circumferential direction. The response of deformable tanks to known ground motions is then compared with that of similar rigid tanks to assess the influence of wall flexibility on their seismic behavior. In addition, detailed numerical examples are presented to illustrate the variation of the seismic response of two different classes of tanks, namely, “tall” and “broad” tanks. Finally, the significance of the cos nθ-type modes in the earthquake response analysis of irregular tanks is briefly discussed.

This paper presents an analytical study of the covariance kernels of a damped linear two-degree-of-freedom system that is subjected to spatially correlated nonstationary stochastic excitation consisting of modulated white noise. A unit-step intensity function and an exponential function, resembling the envelope of a typical earthquake, are considered in conjunction with a propagating disturbance. Results of the analysis are used to determine the dependence of the peak transient mean-square response of the system on the uncoupled frequency ratios, mass ratios, wave propagation speed, shape of the intensity function, and system damping.

A rigid mass driven by Coulomb friction from a randomly moving foundation is used as a simple model of a possible earthquake isolation system for a stiff compact structure. The structure is protected from accelerations greater than that produced by the limiting friction at the expense of a residual relative displacement with respect to the foundation. The RMS residual displacement is predicted approximately, both analytically and by computer simulation. The analytical approximation makes use of equivalent linearization applied to a nonstationary response problem. The computer simulation marches out exact response histories for an ensemble of independent earthquake realizations. There is good agreement between the two predictions. The trade-off between peak accelerations and residual displacements for typical earthquake intensities is displayed.

A rigid rectangular foundation, embedded at an arbitrary depth below the surface of an elastic half space is subjected to a plane, transient SH-wave. The Laplace and Kantorovich-Lebedev transforms are applied to derive the equation of motion for the foundation during the initial time period required for an SH-wave to traverse the base width. The peak impulse response is found to occur during this time and the response there-after appears to be valid based on a comparison with the known, long-time limit. Consequently, the results presented here can be convolved with an earthquake accelerogram to yield an accurate foundation earthquake response.

This paper establishes a new upper bound on the failure probability of linear structures excited by an earthquake. From Drenick’s inequality max|y(t)| ≤ MN, where N 2 = ∫ h 2 , M 2 , = ∫ x 2 , x(t) is a nonstationary Gaussian stochastic process representing the excitation of the earthquake, and y(t) is the stochastic response of the structure with impulse response function h(τ), we obtain an exponential bound computable in terms of the mean and variance of the energy M 2 . A numerical example is given.

Discussed are the mean-square response exceedance characteristics of a single-tuned system to amplitude modulated noise. The results bear on the accuracy of spectral estimates of nonstationary data, and subsequently, relate directly to the design, analysis, and testing of structural systems in environments as gusts, earthquakes, and ignition transients. For noise correlated as an exponentially damped cosine, the nonstationary response may exceed its stationary value by a factor in excess of two. A time-varying shaping filter explanation is offered for this behavior. For white noise, such exceedances do not occur.

The interaction of lateral structural inertia forces with horizontal seismic motion is formulated in terms of an integral equation of the Volterra type. By means of normal mode theory the inertia force at the base of the structure is expressed as a function of the foundation motion. After the motion of the two-dimensional elastic half space resulting from a uniform horizontal foundation force varying arbitrarily with time over a specified interval on the boundary of the half space has been determined, the interaction equation is derived. Numerical studies for two free-field acceleration inputs are made for different ground stiffnesses and structural characteristics. The first of these free-field inputs is a ramp sine function and the second is the east-west ground acceleration recorded at Golden Gate Park during the 1957 San Francisco earthquake. The interaction effects for structures similar to nuclear power plants prove to be significant.

The response of a linear system to the excitation of a nonstationary random process is studied. The random excitation considered belongs to a general class which may be generated by passing a nonstationary shot noise throuh a linear filter. The behaviors of different filters are discussed. In principle the covariance and the variance functions of an arbitrary excitation process can be simulated by properly choosing a filter and a non-stationary strength function in the shot noise. It is shown that, even with a first-order filter, the variance function of a complicated process such as a typical earthquake can be well reproduced, and the determination of the structural response to such excitations becomes relatively simple.

This paper analyzes the transient response of a simple harmonic oscillator to a stationary random input having an arbitrary power spectrum. The application of the results of this analysis to the response of structures to strong-motion earthquakes is discussed.