Journal of Engineering Mechanics

Spatial wave attenuation in elastic material is investigated in light of attenuating neighborhood theory of continuum mechanics. The attenuating part of the constitutive equation is expressed by an integral function of strain over each point of the material. It is shown that the spatial attenuation can be determined independent of the temporal attenuation in the viscoelastic material with fading memory, discussed by the writers in their previous paper, where the nonelastic part of the constitutive equation is determined by a functional of strain rate. However, the functional expressions for constitutive law are similar. The linear theory of attenuating neighborhood is briefly introduced and the solution to plane wave equation is derived. Then the characteristics of spatial wave attenuation are investigated in wavenumber domain. The critical conditions for wave propagation are also discussed. The fading memory is utilized in the previous paper by the writers to express all the temporal characteristics of wave attenuation. The present study shows that the attenuating neighborhood theory can be advantageously utilized to express the characteristics of spatial attenuation, such as the wavenumber dependence of $Q^{-1}$ factor.

The nonlinear behavior of rectangular thin plates with initial imperfections is studied in this paper by using an iterative boundary element and finite element method. The transverse and the in‐plane deformations of the plates are analyzed, respectively, by the boundary element method and the finite element method. The coupling between these two deformations and the nonlinearity of the problem considered are taken into account by introducing initial strains and through an iterative procedure. Spline functions are used in both boundary element and finite element analysis. Imperfections, which are expressed by double Fourier series, are considered in the numerical examples, although the method is also applicable to other imperfections. Numerical results, dealing with large deflection of imperfect rectangular plates with either simply supported or clamped boundaries, are presented, discussed and compared with the results obtained by using alternative approaches. Upon assuming the absence of imperfections, the corresponding large deflection analysis of a perfect plate occurs as a particular case.

We evaluate the least time to failure of a linear elastic system with uncertain time‐varying input. The damage is quadratic in the velocity. The uncertainty of the load history is represented by an ellipsoidal Fourier‐bound convex model. The maximum increment of damage during a duty cycle, accounting for the uncertain load history, is found as the solution of an eigenvalue equation. The mass, damping and stiffness matrices are updated after each duty cycle by assuming that they depend on the load‐history only through the amount of damage occurring during the cycle. The least number of duty cycles to failure is evaluated recursively by assuming that failure occurs at a specific value of damage. This lifetime function generates a convex‐model analog of the traditional S‐N curve, which accounts for uncertain time‐varying load histories. This leads to predicted least lifetimes. Lifetime and S‐N curves are evaluated numerically for a simple example.

This paper examines the importance of structural dynamics in the primary response of flexible pavements to fluctuating, moving wheel loads. The influence of vehicle speed and loading frequency are examined first, using a validated dynamic road‐response model with idealized loads. The dynamic calculation procedure is then simplified to provide a “quasi‐dynamic” calculation which accounts for vehicle speed but not the dependence of road response on loading frequency. The simplification decreases the necessary computer resources significantly and reduces the computation time sufficiently for use on a personal computer. The validity of the simplification is tested by comparing primary road responses and fatigue damage predicted by the two methods, for dynamic wheel loads generated by two representative mathematical vehicle models, on two different roads. It is found that differences in response are observable, but in practical situations they are likely to be small. It is concluded that the simplified quasi‐dynamic calculation approach is preferable.

To gain insight into the phenomenon of the interaction of stress waves with material defect, the transient problem of a half‐space containing a subsurface inclined semi‐infinite crack subjected to dynamic antiplane loading on the boundary of the half‐space is investigated in this study. The solutions are determined by superposition of the fundamental solution in the Laplace transform domain. The fundamental solution is the exponentially distributed traction on crack faces. The exact close form transient solutions of dynamic stress intensity factor are expressed explicitly in time domain. These solutions are valid for an infinite length of time and have accounted for the contributions coming from incident, reflected and diffracted waves. Numerical results of dynamic stress intensity factors are obtained and compared with the corresponding static values. The transient solution is shown to approach the static value after the first few waves interactive with the crack tip have passed.

This paper presents the first known solutions for free flexural vibration of Mindlin plates of arbitrary shape with internal point supports. A new approach for modeling the point supports in vibration analysis is proposed. The method is based on the use of a set of pb‐2 shape functions to account for prescribed boundary conditions of different shapes. Internal point supports are imposed as geometric constraints in the final formulation of the governing eigenvalue equation. The proposed method is employed to determine the vibration frequencies of square and circular plates having an arbitrary number of internal point supports. Comparisons with known thin‐plate solutions are carried out, where possible, to verify the accuracy of the proposed method.

A condensation procedure for the coupled analysis of multidegree‐of‐freedom primary‐secondary (PS) structural systems is developed based on energy principles. It provides a simple and consistent approach to the response analysis of secondary systems in a vibrational environment for both tuned and detuned cases without explicit knowledge of the frequency characteristics associated with the combined PS system. In this method, a multidegree‐of‐freedom (larger than three degrees) P‐system is always substituted by a three‐degree‐of‐freedom system before condensing a 1S‐3P model to a 1S‐2P model based on the equivalence of the S‐system energy. All the energy parameters in the 1S‐2P and 1S‐3P models are analytically calculated from closed‐form expressions so that an increase in the number of degrees of freedom does not substantially increase the computational effort. Some of the solution properties such as its convergence as the mass ratio approaches a small value and the accuracy and efficiency of the solution procedure are discussed in some detail. Two numerical examples are presented to demonstrate these properties.

The relative merit of the different control configurations cannot be foreseen by considering only the controllability and observability conditions. A measure of the degree of controllability has to be defined to help choose the best control configuration. The optimal control configuration that maximizes the control effectiveness and minimizes the control cost is considered in this paper. The vibrational mode shapes, the structure‐controller interaction, the control strategy, the control objective, and the control spillover are among the factors influencing the optimal placement of the control actions. The contribution of these factors in the control distribution problem is assessed through numerical examples. It is found that the structure‐controller interaction, a factor usually neglected in previous studies, greatly affects the optimal control distribution. Quantitative measures of the degree of controllability are proposed. Finally the paper presents a methodology for dealing with the optimal control configuration problem, a problem that does not have a unique solution.

Circulatory change and alteration of salinity in Galveston Bay is investigated numerically by using a three‐dimensional hydrodynamic and transport model. Galveston Bay is an extremely complex and dynamic estuarine system. Tides, freshwater inflows, wind, and bathymetry all affect the circulation patterns and salinity distribution. A thorough understanding of the physical hydrodynamic and environmental impact on the estuary, due to the influences of stream inflows, wind, tides, bathymetry, and pollutant transport, is essential to develop a rich and healthy estuarine ecosystem. A three‐dimensional hydrodynamic and salinity transport model is applied to simulate the whole Galveston Bay. This model solves coupled full Navier‐Stokes equations and salinity transport equations in a curvilinear coordinate system. By inputting freshwater inflows, tide, and wind data into the model, the time variation of the three‐dimensional circulation patterns, free‐surface elevations and salinity profiles are obtained to describe this dynamic system. A curvilinear grid of the Galveston Bay is generated for computation. A monthly simulation has been conducted to study the tide and freshwater induced circulation. The free‐surface elevations and salinity distribution are also presented. The predicted free‐surface elevations in the bay are in good agreement with the field measurements. The results also indicate that the bottom salinity in the bay increases during a monthly tidal‐forcing. The impact of velocity and the salinity field caused by the freshwater inflows are discussed.

A preliminary experimental investigation is carried out to assess the feasibility of the adaptive feedforward control approach for the response reduction of structures excited by seismic ground motions. Although adaptive control has been used in many areas of engineering, its applications to control of civil engineering structures remains unexplored. The proposed methodology is presented together with a brief description of the theory. The adaptive feedforward methodology overcomes many of the shortcomings of previously proposed methods for structural control through the use of adaptive filters in conjunction with feedforward control. An experimental demonstration of the proposed method was carried out using a small‐scale model of a three‐story structure. The experimental study yielded very encouraging results thus showing the potential benefits of applying the adaptive feedforward method in structural control.

A nonlocal continuum model for strain‐softening damage is derived by micromechanics analysis of a macroscopically nonhomogeneous (nonuniform) system of interacting and growing microcracks, using Kachanov's simplified version of the superposition method. The homogenization is obtained by seeking a continuum field equation whose possible discrete approximation coincides with the matrix equation governing a system of interacting microcracks. The result is a Fredholm integral equation for the unknown nonlocal inelastic stress increments, which involves two spatial integrals. One integral, which ensues from the fact that crack interactions are governed by the average stress over the crack length rather than the crack center stress, represents short‐range averaging of inelastic macro‐stresses. The kernel of the second integral is the long‐range crack influence function which is a second‐rank tensor and varies with directional angle (i.e., is anisotropic), exhibiting sectors of shielding and amplification. For long distances $r$, the weight function decays as $r^{-2}$ in two dimensions and as $r^{-3}$ in three dimensions. Application of the Gauss‐Seidel iteration method, which can conveniently be combined with iterations in each loading step of a nonlinear finite element code, simplifies the handling of the nonlocality by allowing the nonlocal inelastic stress increments to be calculated from the local ones explicitly. This involves evaluation of an integral involving the crack influence function, for which closed‐form expressions are derived. Because the constitutive law is strictly local, no difficulties arise with the unloading criterion or continuity condition of plasticity.

A study of the dynamic characteristics of rigidly supported upright circular cylindrical tanks containing two different liquids is presented. The governing differential equations for the tank—two‐liquid system are obtained by application of the Rayleigh‐Ritz procedure in combination with Lagrange's equation. The response functions examined include the fundamental natural frequency, the associated mode of vibration and hydrodynamic pressure exerted against the tank wall. Unlike the cases of tanks containing one liquid in which the dynamic response is controlled by four parameters, the dynamic response of a tank that contains two liquids is controlled by six parameters. The numerical results are presented in tabular and graphic forms, and are compared with those of the identical tank filled with one liquid. Also, a simple approximate equation for evaluating the fundamental natural frequency for preliminary design is proposed.

An analytical procedure is proposed to determine the three‐dimensional response of a multiple‐cylindrical‐cavity system deeply buried in a viscoelastic medium subjected to moving or seismic loading. The tunnels are treated in the local coordinate systems. The interaction between the tunnels is taken into account by a coordinate system transform. The accuracy of the coordinate transform has been investigated. A numerical study describes the dynamic through‐soil interaction between a main tunnel and a service tunnel when the main tunnel is subjected to vehicles moving with constant velocity and applying pressure over rectangular areas. The effects of the Mach number of the traveling load, the space between the tunnels, and the vibration frequency of the traveling load are examined. The numerical results show that the through‐soil interaction is important for the service tunnel when it is close to the main tunnel where vehicles travel at high speed.

Structural series‐system reliability analyses formulated solely by use of a finite number of random variables can often be done with sufficient accuracy in terms of first‐ or second‐order upper and lower bounds on the system failure probability. The bounds are of the first order if they are defined solely by the single‐element‐failure probabilities, and of the second order if they also include the probabilities of the intersections of the failure events of any two single elements. A reexamination of the proof of the most well‐known pairs of second‐order bounds shows that almost the same reasoning can be used to obtain similar formulas for upper and lower bounds on the outcrossing rate of a general vector process out of the safe set of a series system. For polyhedral safe sets and smooth Gaussian vector processes, the needed outcrossing formulas are given explicitly in a form that is well suited to be put on computer.

This note deals with the buckling mode change of antisymmetric angle‐ply laminates on the basis of YNS plate theory. The cusps phenomenon due to the change of buckling mode is presented in the relations of critical buckling loads versus some parameters, such as lamination angle, moduli ratio, aspect ratio, length‐to‐thickness ratio, and boundary condition. An analytical approach is applicable in conjunction with the concept of state‐space to investigate the Levy‐type solution of the title problem. The rank of final matrix, which is used to determine the critical buckling load, has been reduced to half of the relative one given in previous publications due to shifting the location of boundary conditions. Because the rank of matrix is smaller, the computing time is shorter and the calculation is of higher precision.

The estimation of structural‐system reliability often requires the computation of the intersection probability of multiple‐failure events. Bounds on this joint probability may be less time‐consuming to compute and have adequate accuracy for some practical problems compared with approximate numerical solutions. This technical note compares the accuracy among different second‐order upper bounds on the probability of intersection of multiple‐failure events. An upper bound has been derived in the literature from the Ditlevsen lower bound on the probability of union of multiple‐failure events; the equivalence of its different expressions commonly used in literature is explained, and its accuracy is numerically investigated and compared with other possible second‐order upper bounds. The accuracy of this upper bound reduces to the joint failure probability of the two least probable events for practical problems. The different bounds are ranked in terms of computational effort and accuracy.