Journal of Engineering Mechanics

General differential equilibrium and stability equations are derived for thin, prismatic quasi‐shallow arches with an arbitrary centroidal‐axis shape under dead loading. Sanders‐type strain‐displacement relationships are used in the derivation, subject to Donnell‐type quasi‐shallow approximations and Kirchhoff assumptions. No simplifications about arch shallowness are made other than those connected with the Donnell approximations. A general criterion is obtained that determines whether the arch equilibrium becomes critical with respect to snap‐through or bifurcation buckling. Bifurcation occurs when a particular coefficient vanishes, which takes place when the arch membrane‐stress resultant reaches the value necessary to buckle an equivalent straight bar. The equations derived here reduce to those for circular arches as a special case. Discretized versions of the differential equations for pin‐ended arches are obtained by using series expansions for the various variables. The veracity of the resulting nonlinear algebraic equilibrium equations is confirmed by the fact that when the appropriate initial‐curvature approximations are introduced, they agree with corresponding equations obtained by previous authors using different, but less general, methods.

The motion of oscillators governed by the standard equations for the aerodynamic galloping of square prisms is studied for two cases: a harmonically forced, single elastically mounted bar immersed in a uniform flow, and an autonomous, elastically coupled pair of such bars. It is shown that the behavior of the forced oscillator has similarities to the behavior of the standard circle map. Thus it is possible to describe how locked‐in oscillatory forms are organized within the forcing amplitude/frequency parameter space and to identify transitions from quasiperiodicity to chaos and turbulent intermittencies. For the coupled pair of oscillators, two stable attractors were identified on which the orbits are topologically similar, respectively, to the two normal modes of the associated linear system. Depending upon the system parameters, one of the attractors contains orbits that may be periodic, quasiperiodic, or chaotic. Beyond a critical flow velocity this attractor vanishes. For the other attractor, only periodic orbits were identified. This work is the first stage of a numerical and experimental investigation aimed at assessing the potential role of chaotic dynamics in bluff body fluid elasticity, with a view to application in ocean engineering.

In composite structures characterized by lightweight, thin‐walled members, the linear buckling load is one of the most important design considerations. Plate structures (bridge decks and ship hulls) are often subjected to differential compression due to nonuniform bending during their service life. This paper presents a general formulation for the buckling of rectangular, anisotropic, symmetric, angle‐ply composite laminates under linearly varying, uniaxial compressive force using the energy method in conjunction with orthogonal polynomial sequences, generated by a Gram‐Schmidt process. Orthogonal polynomials provide a simpler and efficient tool for handling complex combinations of simple and clamped boundaries. The present study highlights the unusual insensitivity of the buckling load of anisotropic laminates to fiber orientation under in‐plane tension‐compression‐type loading. Such behavior is not known to exist under idealized loading of constant, uniaxial compression, and orthotropic material behavior. The paper introduces an effective approximation technique of differential quadrature as an alternative to energy methods and discusses its credibility for solving complex plate stability problems against benchmark solutions provided by the Ritz method.

A method is presented for the nonlinear analysis of an element of a laminated composite using the Ramberg‐Osgood stress‐strain relation and considering transverse shear, degradation of elastic moduli due to fracture, and deformation theory of plasticity generalized for anisotropic inelastic material. An efficient solution is obtained employing the Newton‐Raphson method for proportional or nearly proportional loading in incremental steps. At the end of each step failure criteria are applied to each layer as well as to the laminate to study failure due to yielding, fracture, delamination, and degradation of laminate stiffness. The method gives a better comparison with experimental results than previous inelastic analysis. Because of its generality, the method can be used as an effective tool for the selection of allowable stresses in the design of laminated composites.

The mean and the variance are investigated for the maximum absolute value of a nonstationary process that represents the response buildup of a linear oscillator. The mean and variance are computed both by simulation and approximate analytical techniques, with emphasis on a comparative evaluation of available techniques and proposed procedures. The cumulative distribution function of the maximum value is represented by a form that is a function of a conditional rate of barrier crossings. This rate is computed by using a nonstationary approximation of Poisson crossings, and also by using available empirical expressions. In addition, existing expressions are used to estimate the mean and the variance of the maximum value of a nonstationary process by defining a shortened duration for an “equivalent” stationary process. Finally, the first passage problem is also posed as one governed by classical state‐space moment equations, and a modified Gaussian closure technique is used to obtain an approximate solution.

An approximate analytical solution of the boundary conditions of Pasquarell and Ackermann is presented for the simple shear Couette flow of identical, smooth, slightly inelastic disks in the limit where a parametric measure of the wall form roughness is small. Excellent agreement is shown between the approximate solution and a numerical solution of the original integral equations. The solution predicts that simple shear Couette flow only occurs for unique combinations of wall and particle properties. As the wall form roughness decreases, so must the wall elasticity in order to sustain simple shear. The theory is applied to previous experimental work in which shear cells were used to study granular flows. Experimental observations comparing soft to hard walls and bumpy to smooth walls are explained in the context of the theory. It is shown that the accuracy of such experimental results is greatly increased when the walls in the shear cells are nearly elastic and roughened with closely spaced elements of dimensions on the same order or larger than the diameter of the sheared particles. These boundary conditions may be employed in problems involving more dissipative disks than those considered herein.

In this paper, nonlinear plane oscillations of a pendulum, whose support is in steady, circular motion in a vertical plane, are investigated. Regular and chaotic motions are shown to be possible in both damped and undamped cases. Numerical methods are used to obtain time histories, Poincaré maps and their fractal dimensions, Lyapunov spectra, and chaos diagrams. At high speeds of the support, the pendulum oscillates quasi‐periodically about a radial line. It is shown herein that, at low speeds, both damped and undamped motion of the pendulum about a radial line become chaotic. The effect of damping decreases the critical speed at which chaos is initiated and produces Poincaré plots with strange attractors that become more intense as the damping is increased.

A simple, convenient isotropic failure surface is developed that includes the classical failure criteria of maximum‐octahedral‐shear stress and Mohr‐Coulomb as special cases. In addition, the failure surface is general enough to generate a smooth failure curve in the octahedral plane that passes through experimental data for triaxial compression (TXC), a deviatoric state of torsion (TOR), and triaxial extension (TXE) as functions of pressure. To demonstrate the versatility of the failure surface, examples using data for limestone and Monterey sand are considered. Detailed analysis of the normal‐to‐the‐failure surface in stress‐space indicates that this normal is uniquely defined everywhere on the surface but that it may be discontinuous at TXC and TXE. The relationship of this result with the common notion that this normal is undefined at the “corners” of the Tresca failure surface is explained.

This paper explores the potential of a time‐domain identification procedure to detect structural changes on the basis of noise‐polluted measurements. The method of approach requires the use of excitation and acceleration response records, to develop an equivalent multi‐degree‐of‐freedom (MDOF) mathematical model whose order is compatible with the number of sensors used. Application of the identification procedure under discussion yields the optimum value of the elements of equivalent linear system matrices. By performing the identification task before and after potential structural changes (damage) in the physical system have occurred, quantifiable changes in the identified mathematical model can be detected. The usefulness of the identification procedure under discussion for damage detection is demonstrated by means of an example three‐degree‐of‐freedom (DOF) linear system. This system is used to conduct synthetic experiments to generate noise‐polluted “data” sets that are subsequently analyzed to determine the mean, variance, and probability density function corresponding to each element of the identified system matrices. Different versions of the model are investigated in which the location as well as the magnitude of the “damage” is varied. On the basis of this exploratory study, it appears that determining the probability density functions of the identified system matrices may furnish useful indices that can be conveniently extracted during an experimental test, to quantify changes in the characteristics of physical systems.

Solution is presented for the fluid‐pressure response surrounding insertion of a continuous, moving point dislocation within an initially saturated porous‐elastic medium. Included in the presentation is a unified framework for the analysis of penetration‐generated pore pressures in soils for partially drained states and an exposition of the overt penetration rate dependence of pore‐pressure generation and dissipation. Scaling parameters for data reduction are returned as a natural consequence. The analysis allows rational determination of permeability and consolidation coefficients in situ. Postarrest pore‐pressure dissipation at the tip enables $C_v$ to be determined independently. Nonuniqueness in pore‐pressure dissipation along the shaft precludes independent determination of consolidation behavior.

A new method is proposed to deal with a class of probabilistic mechanics problems. It is based on the application of the boundary element concept to problems involving random excitations. The method circumvents the issue of adequacy of the mesh size associated with the application of the finite element concept. Only the boundary of the domain of interest is discretized. The method is exemplified by analyzing a beam that is continuous over multiple supports and is excited by a force that exhibits spatially correlated random fluctuations in time. For this problem, the boundary of the domain consists of a discrete set of points and the exact solution is obtained. In addition, the Karhunen‐Loeve expansion is incorporated to improve the numerical efficiency of the method by uncoupling the double integrals involved and permitting their calculation as products of single integrals. The new method is applicable to a broad class of probabilistic mechanics problems.