Journal of Engineering Mechanics

In this work, schemes based on limited‐memory quasi‐Newton methods are investigated, as applied to solving large systems of nonlinear equations with sparse symmetric Jacobian matrices. Problems in mechanics typically give rise to such systems when the method of finite elements is employed to solve them. An attempt is made to develop algorithms that take advantage of sparsity and can effectively use a variable amount of storage according to the availability. The use of preconditioning matrices as initial approximations to the tangent stiffness matrix is suggested in order to accelerate convergence when the available high‐speed storage exceeds the needs of purely vectorial methods but is not sufficient to house a full factorization of the tangent stiffness. The limited‐memory quasi‐Newton methods are also combined with the concept of truncation, based on a preconditioned conjugate gradient iterative solver of the linearized equations, to produce quite efficient algorithms.

It is well known that certain structural systems that exhibit bifurcation buckling may be sensitive to the small structural imperfections. A general method for the reliability‐based design of such imperfection‐sensitive structures is developed. The method is based on a regular perturbation analysis in terms of the imperfection magnitude around the prebuckling equilibrium state of the corresponding perfect structure. The structural imperfections are represented as a weakly stationary random process with a prespecified autocorrelation function. The new approach results in simple analytical formulae for the design value of the imperfection magnitude at any desired reliability level. All the imperfection modes are taken into account, and no restrictions, besides the usual smoothness requirements, need be imposed on the imperfection patterns. The analysis holds equally well for both simple and compound buckling of conservative elastic systems, and allows the consideration of more than one different forms of structural imperfections at the same time. As an example, the reliability‐based design of a column on a linear elastic foundation is investigated. Numerical results on the effect of imperfections, which invalidate certain intuitively anticipated design improvements, are obtained.

A boundary element method (BEM) is developed for the analysis of plates of variable thickness. The plate may have arbitrary shape, and its boundary may be subjected to any type of boundary conditions. The nonuniform thickness of the plate is an arbitrary function of the coordinates x, y. Since it is practically not possible to establish the fundamental solution of the governing equation, which is a differential equation with variable coefficients, the proposed method uses the fundamental solution of the plate with constant thickness and treats the unknown terms as domain forces. The boundary value problem is formulated in terms of two differential and three integral coupled equations. The differential equations are solved using the finite difference method, while the integral equations using the BEM. The domain integrals are treated by employing an effective Gaussian integration over domains of arbitrary shape. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy.

The results of the stress analysis of a threaded tubular connection, obtained by using a hybrid analogue‐numerical method, are discussed in the paper. The model enabled both the tooth loads and the local tooth fillet stresses to be calculated. In particular, the local stresses resulting from cyclic axial loading and preload are discussed in detail. It is shown that the stress concentration factor at the critical tooth of a preloaded connection varies as axial load is applied from zero to its maximum value. However, the ratio of the local stress range at the root of the tooth to the nominal average axial stress range at this tooth remains constant. It is demonstrated that preloading a threaded connection reduces the local stress range and increases the local mean stress in comparison with connections without preload subjected to the same cyclic loading.

A nonlinear incremental structural analysis including thermal loads was performed on a lock monolith in order to assess the effects of creep, shrinkage, adiabatic temperature rise, and construction procedures with respect to tensile stresses and strains. Mass concrete structures are vulnerable to cracking as a result of high tensile stresses and strains due to thermal loads, material properties, and mechanical loads. The results of a parametric study considering upper and lower bounds on adiabatic temperature rise, creep, and shrinkage coupled with the incremental construction model are presented. Stress and strain history plots are shown and discussed with respect to typical locations within the monolith. The time history plots are essential in tracking the effects of the construction sequence. The results presented are used to give guidance for future analysis and designs of mass concrete structures.

The statistical linearization method is often inadequate for estimating spectral properties of random responses of nonlinear systems. This is due to the fact that the power spectra of responses of linear systems span only the frequency range of the excitation spectrum, whereas significant responses outside this range are possible for nonlinear systems. As an extension of the linearization method, a statistical “quadratization” method is presented. Specifically, the statistics of the nonlinear system are obtained from an “equivalent” quadratic system, which is constructed by replacing the nonlinearity by polynomials up to quadratic order. The nonlinear equivalent system has a form for which the solution can be approximated by using the Volterra series method. The method is formulated for a nonlinear single‐degree‐of‐freedom oscillator under stationary, Gaussian excitation. Stationary, non‐Gaussian response statistics are obtained. The non‐Gaussian response probability distribution is approximated by a third‐order Gram‐Charlier expansion. To demonstrate the applicability of the method, solutions are obtained for two sample systems. The corresponding results compare well with relevant Monte Carlo simulation data. Further, it is shown that the quadratization method is notably superior to the linearization method for the considered systems.

A mixed‐time integration method that had been developed for the finite element analysis of structural and thermal dynamics has been implemented for the study of both steady and unsteady fluid mechanics problems. The method to be discussed is capable of partitioning the domain into implicit and explicit regions in an attempt to capitalize on the desirable properties of each method, namely the stability and accuracy of the implicit method, and the manageable computational resource demands of an explicit method. In addition, the explicit region is further divided into subregions, each of which may have a different time step that is governed by the local stability criterion of an explicit method. To demonstrate the applicability of these methods to equation systems that govern fluid flow, several examples are presented. These include one‐ and two‐dimensional advection of a cosine hill, as well as two‐dimensional steady and unsteady inviscid, compressible flow problems. These examples will be used to show the favorable features of a multi‐time integration method, such as a reduction in CPU time, which can be directly attributed to the differing time steps used in the various subregions.

The buckling of columns with finite shear stiffness such as built‐up and laced columns is investigated. The methods of analysis associated with Engesser and Haringx are compared. The difference between the methods can be traced to the direction of the axial force and the shear force used in the analysis. It is concluded that when the usual shear stiffness of the column is used, the Engesser method is the correct one for columns modeled as continuous Timoshenko shear beams. The appropriate method to use for helical springs is not investigated. The effect of the shear stiffness on the column buckling load for the usual standard boundary conditions is also presented analytically, and in the form of graphs.

A finite element formulation combined with stochastic linearization and normal mode methods is developed for the study of geometrically nonlinear random vibration responses of beam and frame structures subjected to simultaneously spatial and temporal Gaussian stationary nonwhite and nonzero mean random excitations. Examples include dynamic random responses of simply supported beams, clamped‐clamped beams, a simply supported beam on an elastic foundation, and a fixed‐end portal frame under various random loadings. To compare and validate the present formulation and solutions, results are compared with alternative analytical solutions whenever available. The Monte Carlo simulation method is also used to generate solutions to compare with some of the present results when alternative solutions are not available.

An incremental elastic‐plastic constitutive model is proposed to relate force and displacement at the contact between two identical elastic rough spheres. The model is formulated in a three‐dimensional force space defined by the normal and two tangential contact force components. The normal and two tangential components of the relative displacement between the centers of the spheres are predicted for any arbitrary contact force history. The infinite yield surfaces in force space are cones parallel to each other, the apexes of which generally do not coincide with the origin of the force space; the axes of the cones translate without rotation during loading. It is a kinematic hardening model with a modified normality rule. The model identically reproduces all available closed solutions obtained by Mindlin, Deresiewicz and others for particular loading histories, leading to the conjecture that it may be a rigorous solution of the contact problem. The model has been coded into program CONTACT and used for numerical simulations of granular media using finite element and discrete element techniques.

Using Faber series expansion and the complex potential method in the plane theory of elasticity of an anisotropic body, stresses in an orthotropic laminate with two elastic pins having snug and interference fitting tolerances are accounted for. It is shown from the numerical results obtained that a little interference fitting tolerance in a mechanically fastened joint may probably be acceptable for a laminated composite structure, and that the case of two elastic pins is at an intermediate stage between two extreme cases: that of two absolutely rigid pins and that of free hole edges or of two absolutely soft pins.

A critical excitation is one that maximizes chosen response quantities subject to constraints related to duration and energy imposed on the excitation. A study of the response of single‐ and multi‐degree‐of‐freedom systems subjected to critical base excitations is presented. The effect of damping and duration of excitation on the peak response is studied. The critical peak relative displacement, relative velocity, and absolute acceleration are related to one another in the same manner as the spectral displacement, velocity, and acceleration are related in earthquake response spectra. Objective functionals for maximizing response of multi‐degree‐of‐freedom systems are developed. Examples illustrating system behavior are included, and a comparison is made of peak critical response with peak responses from a strong‐motion earthquake.