Journal of Engineering Mechanics

The paper considers the elastic buckling of skew plates subjected to in‐plane loadings. The buckling analysis is performed using the Rayleigh‐Ritz method with the newly proposed pb‐2 Ritz functions, which consist of the product of a two‐dimensional polynomial function and a basic function. The basic function is formed from taking the product of the equations of the boundaries, with each equation raised to the power of 0,1, or 2 corresponding to free, simply supported, or clamped edges; thus satisfying the kinematic boundary conditions at the outset. With pb‐2 Ritz functions, the analyst avoids the difficulty of searching for the appropriate function for any arbitrarily shaped plates with various combinations of supporting‐edge conditions. Using this efficient and accurate pb‐2 Rayleigh‐Ritz method, buckling solutions are obtained and presented in the form of design charts for skew plates with different edge conditions, angles, and aspect ratios. In addition, the differing viewpoints on the kinematic condition for a corner formed by two simply supported edges are discussed.

In structural dynamics, analytical models based on the finite element method are often inaccurate for complex systems. Corrections are then made by using experimental data as standard references. Due to a lack of reliable criteria to measure the effectiveness of corrections, most of the available model correction methods are empirical in nature. In this paper, we carry out a comprehensive evaluation of model corrections by incorporating both analytical properties and numerical criteria. It is shown that, in many correction procedures, neglecting the correct correspondence between the system's natural frequencies and the eigenvalues of the stiffness matrix often misleads the model corrections. A correspondence rule of complex damping ratio is then proposed to avoid this problem. Several improved model correction methods are discussed.

Analytic three‐dimensional elasticity solutions are presented for the thermal buckling problem of multilayered anisotropic plates. The plates are assumed to have rectangular geometry and an antisymmetric lamination with respect to the middle plane. The temperature is assumed to be independent of the surface coordinates, but it has an arbitrary symmetric variation through the thickness of the plate. No external loads are present, but the motion of the plate is partially restrained in its plane. A mixed formulation is used, with the fundamental unknowns consisting of the six stress components and the three displacement components of the plate. The prebuckling deformations are taken into account. Each of the plate variables is decomposed into symmetric and antisymmetric components in the thickness direction, and is expressed in terms of a double Fourier series in the Cartesian surface coordinates. Extensive numerical results are presented showing the effects of the prebuckling deformation on the critical temperature, as well as the effects of variation in the lamination and geometric parameters of composite plates on the importance of the transverse stress and strain components.

This paper suggests an optimization‐based methodology for the design of minimum weight structures with kinematic nonlinear behavior. Attention is focused on three‐dimensional reticulated structures idealized with beam elements under proportional static loadings. The algorithm used for optimization is based on a classical optimality criterion approach using an active‐set strategy for extreme limit constraints on the design variables. A first‐order necessary condition is derived and used as the basis of a fixed‐point iteration method to search for the optimal design. A fixed‐point iteration algorithm is used based on the criterion that at optimum design the nonlinear strain energy is equal in all members. A nonlinear analysis procedure for three‐dimensional structures is discussed and used in developing the optimization algorithm. Several examples are given to evaluate the validity of the underlying assumptions and to demonstrate some of the characteristics of the proposed procedures. The procedure is verified using two well‐known examples.

A numerical tool commonly used in digital signal processing, the exponential window method, is briefly reviewed in this paper and applied to problems in structural dynamics. This method allows one to carry out analyses of undamped structures in the frequency domain, and yields highly accurate results for both discrete and continuous systems. In essence, the solution involves: (1) Finding both the transfer function and the forward Fourier transform of the excitation for complex frequencies; (2) performing a standard inverse Fourier transformation into the time domain; and (3) removing the effect of the complex frequencies by means of an exponential factor (or window). Excellent results are obtained when this factor is chosen so that the power of the excitation and response signals at the end of the window are attenuated by some three orders of magnitude. In such case, it is found that a quiet zone (a tail of trailing zeroes) is not needed for accurate computations, and that temporal aliasing (folding) is negligible. This computational advantage is achieved at the expense of having to evaluate accurately the transfer functions at each frequency step, since interpolation schemes cannot be used in this method.

The flow field around a moving contact line, generated by an impulsive, horizontal motion of a vertical plate, is studied numerically. The two‐dimensional, unsteady Navier‐Stokes equations, coupled with the kinematic and dynamic boundary conditions on the free surface, are solved. The contact angle between the free surface and the vertical plate is found to be very close to that formed by the free surface of a fluid contained in a rectangular tank that moves with the same constant acceleration. The velocity of the moving contact line is thereby derived as a function of the acceleration of the plate based on the principle of mass conservation.

This paper is a finite element study of the nonuniform (necking) deformation of creep‐cavitating bars under constant load or constant load with superimposed hydrostatic pressure. The multiaxial constitutive model used in the analysis is a finite strain generalization of a physically based model that incorporates Dyson's micromechanical mechanism of constrained cavity growth. The mechanism governs the phenomenon of coupled grain boundary cavitation and creep in polycrystalline metals at elevated temperatures and within the low‐to‐moderate stress regime. The finite element study, which is based on the principle of virtual work and constitutive model in the convected Lagrangian view, includes calculations of elongation, area, stress, and damage histories. Comparisons of elongation history as well as necked profiles of various cases are carried out. The results predict (1) Transitional behavior of catastrophic necking and brittle fracture to the tendency for diffuse necking and ductile rupture under a constant load with increasing superimposed hydrostatic pressure; (2) the effects of imperfections on the fracture time; (3) the locations of failure initiation; and (4) the sensitivity of necking behavior to geometrical imperfection (varying radius) and material imperfection (varying Monkman‐Grant constant).

In this study, a set of rules is established that, when used in the modeling of dilatant soils, within the framework of associative plasticity, enables very successful shear and dilatancy predictions. The most important of the proposed principles are outlined as follows: (1) The plasticity model must have a loading surface that hardens kinematically and a failure surface that is perfectly plastic; and (2) experimental evidence shows that uniformly deformed sand samples dilate with a constant rate when they reach their ultimate strength value, while critical state is only achieved at very large strains (well in excess of 30%). There is a unique point A on the loading surface that corresponds to the experimentally observed dilatation rate. The hardening rule must, therefore, ensure that the stress point approaches A as it comes closer to the failure surface. The implementation of these rules to a plasticity model gives results that compare very well with experimental observations.

Recent developments related to cam‐clay models are summarized, and a different approach to interpret these developments in terms of physical parameters of clays is presented. A new state variable, i|), indicative of the fabric of clays, is introduced. Based on these concepts and a general failure criterion, a simple model to predict failure parameters of anisotropic clays for many commonly encountered stress paths is developed. The model capability to interpret in situ strength measured under a given stress path and transfer it to another stress path is illustrated. Finally, the ability to obtain failure parameters for any stress path using data from a single CIUC test is demonstrated.

Perfectly flexible cables with zero or low initial tension, subject to impulsive loads, have different dynamics than taut cables. The initial transient motion is that of an inextensible cable, and the resulting equations of impulsive, three‐dimensional response show that the developing velocities and tensions depend only on the curvature and not on the geometric torsion of the configuration of the cable. Furthermore, singularities may develop when either the curvature becomes zero, or its derivative is discontinuous. This singularity is removed by introducing the small bending stiffness of the cable. Boundary layers develop, however, having length proportional to the square root of the bending stiffness. Numerical simulations establish the validity of the methodology, and are applied, in conjunction with perturbation methods, to study the response of a freely hanging chain that provides a canonical example of transition from high‐ to low‐tension behavior.

Using a special class of basis functions, the Saint‐Venant flexure function suitable for the problem of bending of a rectangular bar with $\mathrm{4N}(N=1\text{,}2\text{,}3\text{,}••\text{.})$ circular‐cylindrical cavities is obtained. It is assumed that the load passes through the centroid of the symmetrical cross section of the beam. It is also assumed that all conditions needed for the Saint‐Venant solution of bending of prismatic bars are met. The linearly‐independent basis functions automatically satisfy a homogeneous outer‐boundary condition. They are generated by integrating the products of trigonometric functions and the Green's function for the solution of the Poisson equation with singularities symmetrically located in four quadrants. By shrinking the radius of the closed circular path of integration, and by certain limiting processes, the integrations are performed analytically, leading to the basis functions in the form of infinite series. The convergence of these series is accelerated by using an analytic function and its derivatives. The inner‐boundary conditions(s) is (are) satisfied with the point‐by‐point technique and method of least square error.