Journal of Engineering Mechanics

A method to obtain the response of a rigid circular foundation resting on a uniform elastic half‐space and subjected to a spatially random ground motion is presented. The solution of the resulting mixed boundary‐value problem is based on the use of an integral representation of the response of the foundation in terms of the free‐field ground motion and of the contact tractions between the foundation and the soil obtained in the course of calculating the dynamic compliance functions for the foundation. The results obtained indicate that the effects of the spatial randomness of the ground motion on the response of the foundation are qualitatively similar to deterministic wave passage effects. Both effects involve a reduction of the translational components of the response at high frequencies and the creation of rotational response components.

A mode synthesis method is developed for obtaining the modal properties of a combined single‐degree‐of‐freedom (SDF) equipment and structure system in terms of the modal properties of the individual systems. The eigenvalues of the combined system are obtained as the roots of a simple non‐linear characteristic equation. A simple Newton‐Raphson solution technique can be used to solve this equation. Also, the closed form expressions are provided to calculate the eigenvectors. These combined eigenproperties can be utilized to calculate the structure or equipment response, such as floor response spectra, which will incorporate the effect of the dynamic interaction between the two systems. As no assumption is made about the size of the equipment in the development presented here, this approach can be used equally effectively with light as well as heavy equipment.

A finite difference approach to the Reissner's theory of plate bending is presented. The theory for the numerical analysis of rectangular plates is presented. The results of an example of a plate with simply supported edges are compared with other forms of solution, i.e., finite element, etc., for the uniformly loaded and concentrated load situations. The numerical analysis procedure for simply supported, fixed, free, and combinations of the three type rectangular plates is presented along with methods of solution.

An efficient computational procedure is presented for the analysis of laminated anisotropic shells of revolution and assessing the sensitivity of their response to anisotropic (nonorthotropic) material coefficients. The analytical formulation is based on a form of the Sanders‐Budiansky shell theory, including the effects of both the transverse shear deformation and the laminated anisotropic material response. Each of the shell variables is expanded in a Fourier series in the circumferential coordinate, and a two‐field mixed finite element model is used for the discretization in the meridional direction. The three key elements of the procedure are: (1) use of mixed finite element models in the meridional direction with discontinuous stress resultants at the element interfaces; (2) operator splitting, or decomposition of the material compliance matrix of the shell into the sum of an orthotropic and nonorthotropic (anisotropic) part; and (3) application of a reduction method through the successive use of the finite element method and the classical Rayleigh‐Ritz technique. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Rayleigh‐Ritz technique. The potential of the proposed procedure is discussed and numerical results are presented to demonstrate its effectiveness.

Ludwig Prandtl's 1904 paper “Ueber Flüssigkeitsbewegung bei sehr kleiner Reibung” is generally considered to have marked the beginning of fluid mechanics. By chance, it appeared at the time that human flight was attracting considerable attention. Although empirical hydraulics and theoretical hydrodynamics had long been in existence, they were of little evident use to aeronautics, the one being too limited in scope and the other too impractical. Prandtl's boundary‐layer theory, on the contrary, gradually found use in the analysis of airfoils, propellers, and the behavior of immersed bodies in general. However, convenient as it is to consider the dated establishment of a new science, the fact remains that—as is true of all knowledge—no principle is formulated independently of prior developments, and these in turn are based on still earlier experience. The true origins of fluid mechanics lie in the hydraulics and hydrodynamics of the previous century, and the beginnings of these can likewise be traced back to the continued growth of civilization itself, as outlined in this paper.

The paper first classifies various kinds of pressure loads, then linear and nonlinear finite element buckling analyses of cylindrical shells under uniform and nonuniform loads are presented. In particular the influence of displacement dependent loads on the buckling load of cylindrical shells under water and wind loading is discussed. Finally, the paper summarizes the design rules for wind loaded cylindrical tanks recently developed in Austria and Germany. The study has three goals: to clarify the problem and the effect of follower loads, to demonstrate the capability of nonlinear finite element analyses, and to discuss some results for design practice.

A two‐dimensional finite element formulation for imbricate non‐local strain‐softening continuum is presented and numerically demonstrated. The only difference from the usual, local finite element codes is that certain finite elements are imbricated, i.e., they regularly overlap while skipping the intermediate mesh nodes. The element imbrication is characterized by generating proper integer matrices that give the numbers of the nodes for each finite element and the numbers of the imbricate elements overlapping each local element. The number of unknown displacements remains the same as for a local finite element code, while the number of finite elements approximately doubles. Numerical results show that stable two‐dimensional strain‐softening zones of multiple‐element width can be obtained, and that the solution exhibits proper convergence as the mesh is refined. The convergence is demonstrated for the load‐displacement diagrams, for the strain profiles across the strain‐softening band, and for the total energy dissipated by cracking. It is also shown that the local formulations exhibit incorrect convergence; they converge to solutions for which the energy dissipation due to failure is zero, which is physically unacceptable. Stability prob?ems due to strain‐softening are avoided by making the loading steps so small that no two mutually nonoverlapping elements may enter the strain‐softening regime within the same load step.

An “equivalent homogeneous, orthotropic” model that includes edge effects and an accompanying finite element analysis is presented for elastomeric bearings. The model is developed for three‐dimensional configurations with horizontal layers, and for linear, elastic, small deformation conditions. The equivalent homogeneous theory, in addition to capturing. the overall response characteristics of the layered elastomeric bearing system, approximately models the important edge effects, which occur at and near boundaries that cut the layers, and the stress concentrations at layer interfaces. The primary dependent variables for the theory have been selected such that the highest derivatives appearing in the strain energy function are first‐order, thus requiring only C° continuity of the finite element approximations. As a result, the finite element analysis is simple and computationally efficient. Numerical examples are presented to verify the theory and to illustrate potential applications of the analysis.

Previous authors have reported difficulty in modeling the free warping torsion of slit cylindrical shells using flat triangular shell elements. During the current study it has been determined that the theoretical basis for traction boundary conditions, rigid body motion characteristics of the finite element model, and kinematic mode singularities are all contributing factors to the finite shell element difficulties with the slit cylinder problem. The scope of this paper is concerned with the formal derivation of the thin cylindrical shell traction boundary conditions and their relation to the finite element analysis of the slit cylinder subjected to warping torsion. It is shown that with the foregoing cumbrances resolved, the flat and shallow shell finite elements respond very accurately for cylindrical shell warping problems.

The fundamental frequency of transverse vibration of a beam with intermediate supports and partially embedded in a Winkler-type foundation is determined using the Rayleigh-Schmidt approach. The displacement function is approximated using polynomials that contain an exponential optimization parameter.