Journal of Engineering Mechanics

Under general conditions reliability analyses of elaborate structural models can be replaced by much simpler equivalent reliability analyses of corresponding idealized structural models such as those based on rigid‐plastic theory. The equivalence is in the sense of being with respect to probability of failure and is obtained by applying a random factor correction of the strength variables of the simple model. This model‐correction factor is called the effectivity factor, and it is approximated to the zeroth order by a constant or to the first order by an inhomogeneous linear function of all the basic variables in the realistic model. The coefficients of this function are calculated from the elaborate model (e.g., by finite element calculations) with values of the input variables chosen on the basis of a first‐order reliability analysis of the simple model (FORM). The model‐correction‐factor method is a special type of response‐surface method based on mechanical principles. It is illustrated quantitatively on a yield hinge frame model with strain softening mutual rotation‐moment behavior in the hinges upon the onset of yielding.

The long‐established equations for the motion of a free‐standing rigid block under periodic forcing have been studied numerically and analytically in detail by the writer in several papers since 1989, when chaos was first shown to be present in the simple system of piecewise linear equations. Other results include the presence of multiple solutions, symmetry‐breaking bifurcations, and period‐doubling cascades as well as exact analytic results for some stability boundaries. Experimental results have been explained, and excellent agreement found with the straightforward theory. Numerical calculations have been performed on the same problem by other authors. In this paper it is shown that their work can be explained and broadened by the theory of the writer. The opportunity is taken to present several new results, including forcing‐amplitude frequency diagrams for a range of coefficients of restitution. Results missed by these authors are also given.

Finite element analysis of dynamic foundation problems requires the use of transmitting boundaries to model the radiation of waves from the finite element mesh into the far field. Problems involving inelastic behavior of the soil in the near field are most readily solved in the time domain. The standard viscous boundary is widely used in such situations. However, in axisymmetric situations this boundary is inappropriate. This paper develops plane strain boundary equations for axisymmetric shear and dilation waves based on an approximation of the form of the outward traveling waves. These boundary equations are shown to be equivalent to mechanical systems with frequency independent components. The complex stiffnesses of the new boundaries are compared with the equivalent viscous and plane strain boundary stiffnesses, and the new boundaries are found to agree closely with the plane strain boundaries. The response of an extended axisymmetric finite element mesh subjected to a transient force of the type generated by pile hammers is computed and compared to the responses of the same mesh truncated with various transmitting boundaries.

Bubble functions are finite element shapes that are zero on the boundary of the element, but are nonzero at the other points. They have been hitherto used to augment finite element formulations to obtain rapid convergence. The paper augments bubble functions to the ordinary semi‐analytical complex finite strip treatment in order to calculate the elastic local buckling stress of plate assemblies. The results show that the use of bubble functions improves significantly the convergence of finite strip method in terms of strip subdivision, and leads to much smaller storage required for the structure stiffness and stability matrices. Numerical studies are included, including long plates in compression and shear, a channel section in pure compression and shear, a stiffened plate, and an I‐section with a longitudinal stiffener. These studies illustrate the power of the bubble function based method for studying the elastic stability of plates and plate assemblies.

Accurate analytical‐type solutions are obtained for the free vibration of thin rectangular orthotropic plates resting on point supports symmetrically distributed about the plate central axis. Solutions are obtained by the method of superposition. Convergence is found to be rapid. It is pointed out that the same method permits the obtaining of solutions regardless of the number of supports and even when the support points are not symmetrically distributed. Highly accurate eigenvalues are tabulated for the first four modes of the three families of free‐vibration modes characteristic of square plates with four symmetrically distributed point supports located on the plate diagonals. These are modes fully symmetric, and fully antisymmetric, about the plate central axis, as well as modes symmetrical about one axis and antisymmetric about the other. Dimensionless distances from the center of the plate to the support points are allowed to vary from 0.2 to 0.9. Parameters that characterize the orthotropic properties of the plate are permitted to vary within a limited range.

A new approximate asymptotic method is presented for the nonstationary response of classically damped MDOF (multidegree‐of‐freedom) linear systems to a general class of nonstationary stochastic excitations. The method is based on a simplification of the equations for the modal covariance response by neglecting secondary dynamical effects in the modal Liapunov equation. A comprehensive study is presented which provides general conditions involving the system and excitation characteristics for which the approximations are reliable. The proposed technique reliably predicts the essential features of the statistics of MDOF response for a broad range of system and excitation parameters, and provides valuable insight into the nonstationary response characteristics. Certain similarities and differences between the nonstationary and stationary response are revealed. The method is applicable to both lightly and heavily damped systems, and to various types of nonwhite excitations, including excitation processes with slowly varying intensity and frequency content encountered in earthquake engineering applications. The performance of the new method is demonstrated by some examples.

This paper presents a new analytical method for determining the state of stress in a homogeneous, general anisotropic, and elastic half‐space limited by an irregular and smooth outer boundary. The half‐space represents a rock mass with an irregular and continuous topography. The rock mass is subject to gravity and surface tractions. The stresses are determined assuming a condition of generalized plane strain, and are expressed in terms of three analytical functions following Lekhnitskii's complex function method. These analytical functions are determined using a numerical conformal mapping method and an integral equation method. As an illustrative example, it is shown how the proposed method can be used to determine the state of stress in long isolated and symmetric ridges and valleys in orthotropic or transversely isotropic rock masses. It is found that the magnitude of the stresses is of the order of the characteristic stress $\rho g\left|b\right|$, where $\rho$ is the rock density, $g$ is the gravitational acceleration, and $\left|b\right|$ is the height of the ridge or depth of the valley.

This paper presents a simple semi‐Loof plate bending element. Unlike Irons' semi‐Loof element (reported in 1976), it is not necessary to construct any Kirchhoff‐constraint equations for the proposed element. The complementary energy functional in which only the boundary deflection needs to be defined is employed. Flexibility and leverage matrices of the present element can be derived explicitly. Computational cost incurred by the numerical integration is thus eliminated. The element is deemed to be the most efficient hybrid semi‐Loof plate element, because of its minimal assumed moment space. It is also frame invariant, insensitive to nodal numbering, of the correct rank, and able to pass the constant‐moment patch test. Popular benchmark problems are examined and the accuracy of the proposed element is found to be comparable to that of Irons.

This paper discusses frequency domain optimal control strategies for active control of civil engineering structures under seismic loading. The basic problem formulation is presented, and a description of two specific techniques, the $H_2$ and $H_{\infty }$ control methods, are given. Numerical examples showing $H_2$ control of a seismically excited eight‐story building are given to illustrate the application and delineate the advantages of the frequency domain approaches. The main thrusts of the examples are to show that frequency‐domain design methods can be effectively used to: (1) Optimally control a structure using output feedback of acceleration measurements; and (2) selectively shape the output structural response in a desirable manner. In contrast to previously reported time‐domain‐based controllers, the numerical studies show that these control techniques are capable of reducing the building's response in both the first and second mode response using an active mass damper. The frequency domain optimal control design methods are shown to be flexible and to offer a good match between control concepts and engineering practice.

A finite element‐based time domain system identification procedure is proposed to evaluate existing large structural systems at the element level. The procedure does not need any information on the input excitation forces. Since the input exciting forces are not required, there is no restriction on the type of exciting force, only a small number of observation time points are required and no information is required on the modal properties of the structure. The unknown exciting forces can be applied at the ground level representing the seismic excitation. The procedure is particularly applicable to identifying an existing structure. The method is verified using three examples. For verification purposes, both the noise‐free and noise‐included output responses are considered. In all cases, the proposed method identified the structural parameters very well. The errors in the estimation of the parameters are considerably smaller than those in the other methods currently available in the literature. The proposed method is very economical, simple, and robust.

The response and reliability of a linear secondary system attached to a multi‐degree‐of‐freedom yielding primary structure under white‐noise or filtered white‐noise excitations are studied by using equivalent linearization and digital simulation. It is shown that the effect of the primary‐structure yielding on response and reliability of the secondary system mainly depends on the ratio of the secondary‐system natural frequency to the natural frequencies of the primary structure and on the excitation spectral density shape in the range of natural frequencies of the primary structure. Under white‐noise excitation, amplification of the secondary system response results due both to tuning of the secondary system with one of the modes of the postyield primary structure and to an increase in excitation spectral density values at natural frequencies of the postyield primary structure. High‐frequency amplification is possible in this case. It is also shown that amplification of the secondary system response can be alleviated by increasing the damping ratio of the secondary system and by positioning the secondary system at higher levels of the primary structure.

The paper examines reliability‐based optimization problems formulated as the minimization of an objective function with reliability constraints that model the requirements for the reliability index of component failure modes. It is briefly described how these problems can be solved. The main contribution of the paper is given in the sensitivity analysis of the optimal solution at the optimal stationarity point, i.e., the sensitivity of the objective function and the optimal solution point due to changes in the various model parameters (nonoptimization variables). For the case of sensitivity analysis of the objective function with respect to a model parameter, it is shown by use of perturbation techniques that the total derivative of the objective function (taken at the optimal point) equals the partial derivative of the Lagrange function of the problem. Hereby, the sensitivities of the objective function can be achieved quickly and efficiently. In the case of sensitivity analysis of the optimal point due to a change in a model parameter, an equation system is presented. It is shown that the second‐order derivatives of the element reliability indices are necessary. Expressions for these second‐order derivatives are then derived by perturbation of the optimality conditions of the underlying reliability‐index‐optimization problem.