Journal of Engineering Mechanics

This paper describes a closed‐form solution based on thin‐walled member theory for the out‐of‐plane buckling of thin rings of an open cross section. The theory considers the practical features of restraint and loading conditions of rings situated at axisymmetric shell intersections, and it is applicable to any thin‐walled open‐section form. Numerical comparisons between the closed‐form solution and a finite‐element shell buckling analysis are then presented to demonstrate the accuracy of the closed‐form solution, and to study the effects on the buckling behavior of angle section rings of various parameters, including the ring width‐to‐thickness ratio, the load position, the restraint position, and the flange size. The effect of secondary warping on the buckling behavior of angle section rings is also discussed in some detail. Finally, a simple expression for use in design is established for the buckling strength of an angle section ring loaded and supported at its inner edge.

Elastic buckling of pressure‐loaded spherical space domes is reinterpreted using a recently postulated reduced‐stiffness model. By means of extensive nonlinear numerical studies into the effects of geometric imperfections, over a range of practical geometric parameters, this simplified analysis is shown to provide close approximations of the lower bound loads and their associated modes. As has been demonstrated for related shell‐buckling problems, the reduced stiffness method allows the load‐reduction effects of the complex, postbuckling nonlinearities to be conservatively reproduced. The analysis is sufficiently simple that it can be routinely adopted at the design stage.

In this paper the work on the thick‐plate equations and their solution for the problem concerning the bending of a rectangular plate with four free edges resting on the Pasternak foundation are described. The basic equations are those obtained from Reissner's 1945 theory, and are modified to include the Pasternak foundation. The solutions of the basic equations are arrived at by superposing the solutions of three elemental plates, one with four guided support edges; the other two with two guided support edges and two free edges acted upon by an unknown bending moment. Finally, numerical examples are also presented to examine the solution for convergence and validation. It shows that the presented solution can be used as a good mechanical model for the analysis of the plate structures supported by elastic foundation, where the transverse shear deformation and the local effects in the plates and the transverse connection in the foundation must be considered.

Linear potential theory is applied to the analysis of wave motion through a two‐layer porous structure. For special cases, the characteristics of waves in nondissipative, weakly, as well as strongly, dissipative media and their relations with the inertial and resistive properties of the media are explored. It is noted that in a nondissipative medium, the wave components are either nonpropagative or nondecaying. In a dissipative medium, however, they are always propagative and decaying as well. The reflection, transmission, and dissipation of monochromatic incident waves by a rectangular block, with typical dissipative characteristics and various thickness as well as submergence of its crest, are studied by the method of matched velocity potentials. It is found that there is an optimum thickness for a porous structure beyond which any further increase of the thickness may not lead to an appreciable improvement of its functional performance in reducing the transmission and reflection. It is also discovered that a medium of moderate permeability may be favorable in the design of a wide‐crested breakwater if the wave heights on both front and lee sides of the structure are required to be controlled.

The representation of directional random waves in deep water under locally homogeneous conditions is examined in terms of the effect of second‐order nonlinearities on the wave envelope and phase. Theoretical expressions describing the joint, marginal, and conditional distributions of wave envelope and phase are derived systematically, correct to the first order in the root‐mean‐square surface slope. Some immediate implications of these results are discussed in detail. In particular, it is found that the wave envelope is Rayleigh‐distributed, as in the case of linear waves. But, the wave phase and envelope are no longer statistically independent, and the phase distribution is nonuniform over the interval (0, 2π). As the root‐mean‐squre slope and thus the surface skewness increase, the phase distribution deviates from uniformity progressively, indicating an excess of values near the mean phase π and corresponding symmetrical deficiencies away from the mean toward 0 and 2π. Comparisons with four sets of wave data gathered in the Gulf of Mexico during a hurricane provide a favorable confirmation of these theoretical results, and thus reinforce the validity of the second‐order random wave model.

Stress analysis of anisotropic layers under various loading conditions is of interest in the mechanics of composites or geomechanics. In an attempt to develop an efficient numerical approach for such an analysis, a solution method is developed herein. Based on the state space approach and the Fourier transform, the fundamental solution for an orthotropic elastic layer under the action of arbitrary surface loads and volume forces is obtained in the form of infinite integrals, which are then evaluated numerically. In order to remedy the slow convergence of the numerical integrals associated with the fundamental solution, a procrustean technique is introduced. The fundamental solution is then implemented in the spline‐boundary‐element method and a computational strategy for the numerical implementation is discussed. As an illustrative example, a problem of anisotropic layer containing an elliptic cavity is considered for two different boundary conditions and numerical results are compared to the finite‐element solutions.

A revision to a previously developed method for solving the dynamic‐contact problem of a rigid, smooth striker impacting an elastically supported beam is presented. The revision is such that only a local‐static solution is superposed on an elementary‐beam‐theory solution that incorporates the dynamic effects. The local‐static solution is expressed in terms of the difference between an elastic‐finite‐layer solution and a static‐beam‐theory solution. The matching of boundary conditions leads to a Volterra‐integral equation of the second kind in terms of the pressure distribution and contact length as functions of time. These quantities are obtained numerically using a technique developed for the solution of non‐Hertzian‐contact problems. The revised method is shown to more accurately model the low velocity impact response of finite beams through a comparison of experimental results and theoretical predictions.

In recent years, several analytical studies have been conducted on the behavior of cross‐bracing members subjected to out‐of‐plane buckling. Most of these studies do not consider boundary conditions such as the relative stiffness of the end connections and adjoining members and eccentricity in loading. In this paper closed‐form relationships are obtained for the direct evaluation of the critical loads of cross‐bracings with semirigid ends for any value of relative stiffness of the end connections and any ratio of the dimensionless parameters of tension and compression members. The criteria are formulated for the general case, i.e., tension and compression braces have different section properties, lengths, and axial loading. Parametric solutions are graphically displayed to clarify distinct behavior including the boundary separating symmetrical and antisymmetrical modes of buckling. Analytical expressions for eccentrically loaded cross bracings with simple end connections are further derived and studied to yield a result that the tensile force does not contribute substantially to the load‐carrying capacity of the bracing system.

A mechanical test is described that can be used to measure the edge strength of annealed and heat‐strengthened glass. The mechanical test apparatus stresses only one of the two edge‐lines on the edge of a glass specimen. The testing concept employs the theory of unsymmetrical bending. The test device loads glass specimens in a manner that places tensile stresses on the tested edge‐line while keeping the other edge‐lines at zero stress or in compression. In annealed glass tests, the scored edge‐line was found to be 20% weaker than the edge‐line opposite the scored edge‐line. Comparisons between the edge strengths of annealed and heat strengthened glass revealed that the mean edge strength of heat strengthened glass is larger than that of annealed glass by about 50%.

A numerical method for solving stochastic mechanics problems by representing the solution using a small number of random parameters is presented. In essence, the method is a Galerkin approximation in the sample space. The associated projection of the solution into the space of simple random variables reduces the stochastic problem to a set of deterministic problems. Alternatively, this method can be viewed as a modified—for computational efficiency—stratified sampling method. Several examples are considered involving the use of the Loeve‐Karhunen expansion for a stochastic field approximation. The examples deal with the determination of the natural frequencies and of the seismic response of a beam with random rigidity.

A random‐vibration methodology is formulated for the seismic‐response analysis of linear multisupport‐structural systems. It reduces the response evaluation to that of a series of linear one‐degree systems in a way that fully accounts for the multiple‐support input and the space‐time correlation structure of the ground motion. Three spectral parameters are introduced and interpreted in terms of cross‐correlation coefficients and spectral moments associated with the response of one‐degree systems. The influence of local‐spatial variation of ground motion on the spectral parameters is also studied.

A finite‐element based, numerical analysis methodology has been developed for the nonlinear analysis of building‐soil systems. The methodology utilizes a reduced‐order, nonlinear continuum model to represent the building, and the soil is represented with a simple nonlinear two‐dimensional plane strain finite element. The foundation of the building is idealized as a rigid block and the interface between the soil and the foundation is modeled with an interface contact element. The objectives of the current paper are to provide the theoretical development of the system model, with particular emphasis on the modeling of the foundation‐soil contact, and to demonstrate the special‐purpose finite‐element program that has been developed for nonlinear analysis of the building‐soil system. Examples are included that compare the results obtained with the special‐purpose program with the results of a general‐purpose nonlinear finite‐element program.

Consider a reliability problem with a limit‐state function involving one unknown parameter. We seek to determine the unknown parameter such that a prescribed first‐order reliability index is attained. In this paper, an efficient algorithm is developed for the solution of this inverse reliability problem. The algorithm is an extension of the well known Hasofer‐Lind‐Rackwitz‐Fiessler algorithm used in reliability analysis, where the search space of the random variables is extended here to include the unknown parameter. A search direction and a merit function are introduced to guide the sequence of iteration points to the design point and the parameter solution in a balanced manner. The proposed method can be used in reliability‐based design optimization, or in determining the admissible response threshold for a specified first‐order probability, without performing repeated reliability analyses. Two examples are presented: one demonstrating the characteristics of the convergence sequence, and another describing application of the algorithm to finite‐element reliability analysis of an elastoplastic plate.

Based on generalized laminate plate theory (GLPT), the formulation of a one‐dimensional laminated beam finite element with layerwise constant shear (BLCS) is presented. BLCS formulation is equivalent to a first‐order shear deformation beam theory (Timoshenko beam theory) on each layer, and a cross section of the beam therefore does not necessarily remain plane through the laminate but only through each layer. Plane stress is assumed through both the thickness and width of the beam in the constitutive equation for a lamina. Details are presented for transforming the layerwise constant shear stresses obtained from constitutive relations into parabolic shear stress distributions. The layerwise representation of in‐plane displacement through the thickness results in the formulation of a relatively simple beam element. Numerical analyses are presented for a three‐node BLCS element integrated with two Gauss points. The accuracy of the element is evaluated by comparing the predictions to elasticity and experimental results.