Journal of Engineering Mechanics

A parabolic model for propagation of water waves, implemented with an efficient split‐step Fourier transform algorithm, is presented. The split‐step Fourier transform algorithm, widely used in acoustics to solve the parabolic form of the Helmholtz equation and modified to approximately account for wave number dependency in the direction of propagation, has been successfully applied to model refraction and diffraction of water waves. The solution algorithm is exact for a constant depth ocean and gives a good approximation for a wave spectrum on a uniform slope. The results of the model for an elliptical shoal on a uniform slope are presented for both linear and nonlinear wave dispersion relationships and show good agreement with experimental results. The wide‐angle capabilities of the model are also examined for a circular shoal of elliptical cross section on a flat bottom. The algorithm is very efficient and stable, permitting large computational step sizes in the direction of propagation.

A new stochastic fields simulation technique that can realize prescribed means and covariances with a substantially smaller sample size than that of other existing methods is developed. The method utilizes the modal decomposition of the covariance matrix of the correlated random vector and the spectral representation of random processes. The decreasing feature of the eigenvalues of the covariance matrix and the orthogonality of the trigonometric functions are taken advantage of for generating sets of independent random variables. The generated discretized stochastic field is Gaussian by virtue of the central limit theorem. The sample functions of the discretized stochastic field precisely reproduces, when ensemble‐averaged, the prescribed zero‐mean and covariance function. Hence, the proposed statistical preconditioning technique will, in general, dramatically reduce the large computational effort that Monte Carlo simulation involving stochastic fields would otherwise entail.

It has been observed that the application of cyclic shear loading to a dry sand and other cohesionless soils results in a progressive volume reduction and a relative density increase. Although this densification is larger and takes place faster in loose sands, it occurs in both loose and dense soils, and is caused by the tendency of any sand to contract under a small shear strain. A micromechanical statistical model was developed by the writers to interpret and compute the compaction of such a dry granular soil subjected to a strain‐controlled cyclic simple‐shear test. The method postulates that an important role is played in the compaction phenomenon by the spatial statistical distribution of porosity within the soil specimen. Numerical simulations of densification are presented using the proposed model, which are in excellent agreement with observed sand compaction in actual cyclic simple‐shear tests. These simulations provide new insight into the densification process, and especially on the significance of the evolution of the porosity distribution with the number of strain cycles.

This paper is concerned with the development of the integral equation method for the analysis of a spherical shell under axisymmetric loads. The governing equations of shell are traditionally described as a set of two ordinary differential equations with two unknown state variables. These equations are normalized by eliminating their first derivatives, then multiplied by a weighting function that is a selected Green's function. Finally they are repeatedly integrated by parts until their differential operator is shifted from acting on the state variables to the weighting function. Consequently, the differential equations are transformed into a set of integral equations. To complete the analysis procedures, efforts are made to insert various boundary conditions of a shell into the kernels of the integral equations, and to express the internal forces, moments, and displacements of a shell in terms of the state variables. Thus, the integral equations are readily available for the analysis as well as the optimum design of a spherical shell.

In a companion paper the integral equations for a spherical shell have been developed. This paper is concerned with the engineering applications of the integral equation method. A structure is selected to demonstrate how to apply the method in shell analysis. Two different types of construction are used for the shell: one has constant thickness, the other, piecewise linearly varying thickness. The solutions so obtained are compared with the corresponding ones found by the finite element method. They are also compared against the theoretical solutions whenever possible. A good agreement is attained. The aforementioned shell is also subjected to optimal design. An exterior penalty function method is selected to minimize the weight of the shell with stress and displacement constraints. The integral equation method is used in the analytical phase. Acceptable results are obtained.

A numerical procedure is presented for the solution of stationary two‐state Markov‐process problems with a single source. The proposed solution method is an efficient splitting technique that combines finite element and finite difference methods to take advantage of the specific form of the governing differential equation. A standard one‐dimensional finite element technique using linear interpolation and weighting functions is used to determine the solution for each row, while a variable‐weighted finite difference scheme is used to step in the other direction. This method is then illustrated by determining the stationary response of the Duffing oscillator and the statistical moments of the time to reach a critical crack size for the stochastic fatigue crack growth problem. Excellent comparisons are shown between this method, previous analytical studies, and experimental results with a significant reduction in computer processing time and storage.

The purpose of this study is to develop a general method for estimating the system reliability of offshore structures with the aid of the full distribution method and to introduce a probabilistic definition of structural redundancy. Structures are treated as systems of structural components. Failure of any number of these components results in a redistribution of the internal and/or external forces. The probability of structural failure is then evaluated by examining a limited number of significant sequences of member failures that produce collapse of the structure. The structure examined is an indeterminate deep off shore truss under fully developed sea conditions. Two different types of material behavior are considered to characterize the type of failure of the components: ductile and brittle behaviors. This reliability analysis and redundancy definition will form an important analytical basis for further investigation of offshore structural integrity.

A simple kinetic model for flows of nearly elastic granular materials in the grain‐inertia regime is presented, which includes the effects of frictional‐energy dissipation during collision. It is shown that frictional losses have the same effects as energy dissipation due to the inelasticity of granular particles. The special case of granular simple shear flow is treated in detail. Variations of stress components with solid volume fraction, coefficient of restitution, and friction coefficient are studied. The results are compared with the available experimental data for rapid shearing of glass and polystyrene beads and good agreement is observed. For small values of the coefficient of friction, it is shown that the shear and normal stresses decrease as frictional‐energy dissipation increases. The decrease in normal stress is more noticeable than that of the shear stress.

In the conventional design of a spillway invert and flow bucket, circular, elliptic, and parabolic curves are generally used to connect straight boundaries. Because of the abrupt change in curvature at tangency points, considerable drop of pressure may occur, which sometimes causes cavitation damage. Two kinds of curves with gradually varied curvature, CGVC 1 and CGVC 2, are suggested in spillway design to connect two given points. The slope and the curvature at both ends are specified for a CGVC 1 curve. For a CGVC 2 curve, with other conditions the same as CGVC 1, the curvature at the downstream end is not prescribed. A variable domain finite element scheme, in which the location of the free surfaces of the jet is determined simultaneously with the stream function, is then briefly described. The proposed curves with gradually varied curvature are used in four sample designs. Computational results with the finite element scheme indicate that the new designs give better pressure distribution than the conventional one.

The elastostatic problem of a nonhomogeneous incompressible elastic half‐space containing a partially or fully embedded axially loaded, axisymmetric rigid inclusion is analyzed. The shear modulus of the half‐space is assumed to vary linearly with depth. The rigid inclusion is perfectly bonded to the half‐space. The associated boundary‐value problem is solved by using an indirect boundary integral equation method based on exact displacement and traction Green's functions of the nonhomogeneous elastic half‐space. The general solutions of the nonhomogeneous half‐space can be derived by employing Hankel integral transforms. Green's functions appearing in the boundary integral equation correspond to axisymmetric ring loads in radial and vertical directions acting in the interior of the half‐space. Explicit solutions for Green's functions are presented. Numerical techniques are adopted to solve the integral equation and to evaluate Green's functions. Selected numerical results for rigid cylindrical inclusions are presented to illustrate the influence of degree of nonhomogeneity of the half‐space on the axial stiffness and load transfer curves.

This paper formulates a new theorem of virtual work for finite deformation with the following qualities: (1) Any actual large displacement in a body may be adopted as the virtual displacement; and (2) the internal virtual work is composed in terms of the physical components of stresses and their conjugate finite virtual strains in the Lagrangian description plus the additional terms; these terms become the rigid‐body rotations for axial strains and the corrections due to elongations for shear strains. Fundamental matrix equations are derived for the finite deformation analyses of pin‐jointed trusses and rigid‐jointed frames on the basis of the present theorem. These formulations are simpler than previous approaches and lead to the exact solutions of nonlinear analyses, particularly for plane and space pin‐jointed trusses. Further, when using these equations, the larger increments of load and displacement can be accommodated in the numerical computations. This theorem unifies the principle of virtual work and the principle of complementary virtual work for finite deformation.

A nonlinear model for the fracture process zone (FPZ) that exists at the tip of a crack in concrete is developed through the interactive use of experimental data and finite element analysis of crack‐line‐wedge‐loaded double‐cantilever‐beam (CLWL‐DCB) specimens. That model relates the clamping stress to the crack opening displacement (COD) and is characterized by three straight‐line stress variations segmented by three critical CODs. In the first segment the clamping stress equals the concrete's tensile strength. In the second segment the crack closure decreases sharply with increasing CODs from the tensile strength at about $0.4\times {10}^{-3}$ in. (0.01 mm) to 0.3 times that stress at about $2.0\times {10}^{-3}$ in. (0.05 mm). The validity of that model is demonstrated by showing that it can predict the results of tests on five different groups of CLWL‐DCB specimens subjected to mode I loadings.

A comparison of the performances of three different base‐isolation systems, the laminated rubber bearing system, the New Zealand system, and the resilient‐friction base isolator system, is made under the same set of criteria. Conclusions pertaining to the ranges of the applicability of the systems for three earthquake sizes, M 6.0, 6.7, and 7.3, are obtained for a site located in the vicinity of the earthquake source (nominal source‐to‐site $\mathrm{distance}=10\mathrm{km}$). Two different seismological models of the earthquake source were used in modeling the spectral content of the earthquake input. The stochastic response of one‐story and five‐story base isolated structures is computed. The equivalent linearization technique is used to linearize the equations that govern those systems that are nonlinear. This study confirms the conclusions reached in previous studies by demonstrating again that friction plays an important role in energy absorption. This is the reason for the apparent superior performance of the resilient‐friction base isolator; system. However, more studies are needed to assess the feasibility of this system and to better understand its behavior by testing.

A sensitivity study is conducted to investigate the effect of uncertainties in system identification on the performance of the active control system for seismic‐excited buildings. Four control algorithms feasible for applications to seismic‐excited buildings have been examined, including the linear quadratic optimal control algorithm and three instantaneous optimal control algorithms. A simple simulation approach is used for the sensitivity analysis. Both the active tendon control system and the active mass damper system have been considered. Numerical examples are worked out to illustrate the effect of uncertainties in identifying stiffnesses and dampings on the active control system. The effect of parametric uncertainties on the performance of active control systems for seismic‐excited buildings depends on the particular control algorithm used. It is demonstrated that, while dampings of structures are difficult to identify, the accuracy of damping estimation has a negligible effect on the performance of the control system. For some control algorithms, however, care should be taken to estimate the structural stiffness.