Journal of Engineering Mechanics

Analytical solutions are presented for deflections beyond the elastic limit for prismatic and nonprismatic beams (beams with linearly varying depth) restrained and loaded in any way. The analytical solutions provide by differentiation the rotation, curvature, moment, and the shearing force. Simply supported and built‐in beams with uniform cross section and with linear variations in depth symmetrically with respect to midspan, loaded at midspan, are analyzed; and the results are given in a dimensionless form. The displacement occurring just prior to the collapse mechanism is 2.243 of that occurring at the limit elastic behavior for simple and built‐in uniform beams. For ratios between the minimum and the maximum depths of the beam less than 0.41, the collapse mechanism occurs with plastic hinges located at midspan and along the beam. Five elastoplastic behaviors for built‐in beams with linearly varying depth are established, and the limit ratios between the maximum and the minimum depths of the beam separating these behaviors are determined.

A new method for efficient discretization of random fields (i.e., their representation in terms of random variables) is introduced. The efficiency of the discretization is measured by the number of random variables required to represent the field with a specified level of accuracy. The method is based on principles of optimal linear estimation theory. It represents the field as a linear function of nodal random variables and a set of shape functions, which are determined by minimizing an error variance. Further efficiency is achieved by spectral decomposition of the nodal covariance matrix. The new method is found to be more efficient than other existing discretization methods, and more practical than a series expansion method employing the Karhunen‐Loève theorem. The method is particularly useful for stochastic finite element studies involving random media, where there is a need to reduce the number of random variables so that the amount of required computations can be reduced.

The radiation condition for the second‐order diffraction problem was studied. A transient elementary flow was considered, representing the interaction of a progressive plane wave train having a submerged pulsating source. A transient second‐order solution is obtained using the Fourier transform in space and the Laplace transform in time. The radiation condition is obtained by carrying out the asymptotic analysis for large time and far field. To the leading order, the waves at far field consist of a free and forced component. The free component, satisfying the homogeneous form of the free‐surface condition, is shown to represent outgoing waves, decay like $\sqrt[R]{1}$, and satisfy the Sommerfeld radiation condition. The forced component satisfies the far‐field form of the nonhomogeneous free‐surface condition. This forced wave component, modulated by the incident wave $e^{-{\mathrm{iv}}_{0}x}$, represents outgoing or incoming waves, depending on the relative magnitudes of the incident wave and source frequencies of oscillation. The azimuthal variation of the far‐field wave disturbance was also examined.

In system reliability analysis, it is frequently necessary to resort to Monte Carlo simulation (MCS). The importance sampling method (ISM) is an advanced MCS method that may greatly improve the efficiency and accuracy of simulation approaches. Its application to structural system reliability analysis is focused here. Importance of proper determination of the sampling density is emphasized based on a critical review of other suggested sampling densities. A new sampling density, the weighted general normal sampling density (WGNSD), is proposed. This density is proportional to the ideally optimal one at the most important points of the variable space, and it is general for structural system reliability analysis. A variety of applications of this method are presented for illustration, as well as for demonstration of success with the proposed sampling density, including nondifferentiable failure surfaces and higher‐dimensional problems.

A mathematical model is developed for simply supported plain concrete columns, accounting for their nonlinear material characteristic. The axial and transverse motions of concrete columns are found to be second‐ and fourth‐order nonlinear partial differential equations, respectively. These equations are then reduced to a set of coupled nonlinear differential equations in terms of time‐dependent modal coefficients. Approximate solution of the axial vibration and its substitution in the equation governing the transverse vibration of the column results in a single‐degree‐of‐freedom Hill's equation. This equation together with a method based on the Floquet theory are employed to investigate parametric stability of a simply supported concrete column. Results, obtained from Strutt diagrams, indicate the existence of instability regions corresponding to principal and secondary resonances, and potential for failure under sinusoidal amplitude of magnitudes far below that of static buckling. Moreover, the vibration parametric stability signature of columns are found to be relatively invariant with respect to their material and geometric characteristics.

The Mohr‐Coulomb yield condition with a nonassociated flow rule is often used to model the behavior of dry or drained sand. A previous paper demonstrated, using a single shear band with a rotating orientation, that a genuine but limited instability would occur in such a model at each state of triaxial stress in the inelastic range. This paper develops the equations, explores the quantitative behavior of the model, and examines possible alternative patterns of deformation. The extent of the unstable jumps in plastic strains (0.01) and displacements, and the largely hidden transient stress jumps (15%), are calculated for such a model of sand with properties similar to those given by Lade in his papers reporting triaxial test stability in the wedge region. When the incremental elastic response is too small to stabilize the plastic, any similar model with a nonassociated flow rule likewise is unstable. The mathematical requirements for stability and the modeling of stable granular media are commented upon briefly.

A method of analysis based on analytical techniques is presented to determine the axisymmetric response of a cylindrical borehole in a poroelastic medium. The medium is assumed to be fully saturated and governed by the classical Biot's theory. A set of general solutions relevant to the present class of problems are derived through the application of Laplace and Fourier integral transforms with respect to the time and the vertical coordinate, respectively. Boundary‐value problems corresponding to axisymmetric radial tractions or fluid sources applied over a segment of the borehole surface are solved and analytical solutions in the Laplace transform space are presented. Solutions corresponding to ring‐type loads and a fluid source are also presented. These solutions can be used to derive the solutions corresponding to nonuniform distribution of tractions and fluid sources applied over a segment of the borehole surface and a variety of other problems. Time‐domain solutions are obtained by using a numerical technique for the inversion of Laplace transform. Selected numerical results for displacements, stresses, pore pressure, and radial discharge are presented to portray the effects of poroelasticity on the response of the borehole.

A new p‐version finite element based on higher order theory is developed for the two‐dimensional modeling of thin‐to‐thick plates undergoing three‐dimensional (3‐D) deformations. The special case of cylindrical bending is also considered. In each case, the displacement is expressed as the product of two functions—one in terms of the out‐of‐plane coordinate and the other in terms of in‐plane coordinates. The displacement fields are based on integrals of Legendre polynomials. The computer implementation allows arbitrary variations of p‐level in the plane of the plate and in the thickness direction up to a maximum value of 8. A number of test problems on thin‐to‐thick isotropic and orthotropic plates are considered to evaluate the performance of the proposed scheme. Convergence characteristics of pointwise values of displacement and stress, as well as that of total potential energy, are studied. The superior performance of the scheme is established by comparing the results with those in the published literature.

An innovative approach to characterizing the resistance of asphalt concrete mixtures to fatigue‐crack propagation is presented. A fundamental equation based on the thermodynamics of irreversible processes is employed to extract parameters characteristic of the pavement's resistance to fracture. These parameters are the specific energy of damage and a dissipative characteristic of the paving mixture. The applicability of the proposed methodology has been demonstrated in view of experimental data generated for an AC‐20 paving mixture. In addition, the applicability of the present methodology has been extended to examine fatigue‐crack propagation (FCP) data reported in the literature by other researchers. It is found that the proposed model describes the FCP behavior of two asphalt mixtures over the entire range of crack driving force. The proposed methodology provides a basic rationale upon which the relationships between microstructure and/or processing conditions and resistance of pavement to fatigue‐crack propagation can be constructed. Such relationships can guide the development of paving mixtures with superior resistance to fracture and aid the lifetime assessment of load‐bearing structures manufactured from these mixtures.

The kinematics and kinetics of a planar double‐slip system are used to model the response of a single crystal in plane large plastic deformations. Based on the kinematics of the double‐slip element, the orientation distribution function (ODF) of an infinite number of such elements, modeling a polycrystalline material, is obtained analytically for plane deformations by solving its differential equation of evolution. The ODF models the texture development of the polycrystalline aggregate. Based on the corresponding kinetics of the double‐slip element, the ODF is used to obtain by integration in closed‐form analytical expressions for the deviatoric stress components and associated yield surface of the polycrystal during general plane large plastic deformations, accounting for the ongoing texture development. The general theoretical development is illustrated by examples for simple shear and plane strain rolling deformation processes, and in the former case is compared successfully with experiments.

The present work deals with the problem of flexural buckling of a thin column supported at its ends. The column has a constant cross section, but its principal axes of inertia rotate as a function of its axial coordinate. The problem is solved with a direct variational method using Fourier series and shows that this particular geometry improves the critical load of the column with respect to the case in which the orientation of the section of the column remains constant. The flexural buckling load increases on increasing the rotation between the two end cross sections of the column (pretwisting). The algorithm converges rather rapidly. However, on increasing the pretwisting of the column a greater number of Fourier series terms must be considered to reach the necessary precision. The calculations have been Carried out hypothesizing that the rotation angle of the principal axes of inertia varies linearly with the axial coordinate. The problem can be easily extended to other modes of varying this angle.