Journal of Engineering Mechanics

A finite element method, using versatile curved elements and some special techniques, is presented in this paper for nonlinear stability analysis of steel members. The method and its computer program can be applied to the analysis of elastoplastic local, overall, and local‐overall interactive instability problems of steel members with arbitrary cross sections and arbitrary boundary conditions. All initial imperfections, including initial deflections, initial eccentricities, and residual stresses, can be taken into account in analysis. Compared with the results of other numerical methods and experiments, the effectiveness and accuracy of the present theory are quite satisfactory.

In this paper, a new approach to the development of the transition plate‐bending elements with variable midside nodes is presented. These elements will facilitate abrupt changes in mesh refinement between domains consisting of quadrilateral elements. For compatible mesh gradation, special shape functions with slope discontinuity at midside nodes are used. The transition elements are formulated based on the Mindlin‐Reissner plate theory to take shear deformation into account. To evaluate the proper element stiffness pertinent to shear, the substitute shear strain polynomials in Cartesian coordinates are constructed. A modified Gaussian integration is adopted to eliminate the slope discontinuity problem in the element domain. Some numerical examples are presented that illustrate the validity and effectiveness of these transition elements for the efficient analysis of plate‐bending problems.

The formulation of the stability equations for continuously supported thin plates is reexamined. It is shown that the middle‐surface stress changes due to bending must be taken into account in a correct uncoupling of the equilibrium equations. The apparent absence of these stress changes in the final linear stability equations is shown to be a consequence of a little‐known orthogonality condition between the middle‐surface stresses in the plane configuration of the plate and the stress changes due to bending. With the knowledge of this orthogonality condition, previous inextensional arguments used in connection with plate stability are reassessed. A weighted‐integral approach is outlined for the stability of supported rectangular plates with generally distributed edge loadings. This is used to show practically how the stress changes associated with buckling can be determined from the linear eigenvalue solution without the need for further integration.

The formula for the covariances of the local averages of homogeneous random scalar fields over rectangular domains is first generalized to include the case of homogeneous random vector fields with quadrant symmetry. For nonhomogeneous random fields and/or nonrectangular domains, Gaussian quadrature is proposed to evaluate the means and covariances of the local averages of random vector fields. The stochastic finite‐element analysis based on the local averages of random vector fields is then formulated for static, eigenvalue, and stress‐intensity factor problems. The present stochastic finite‐element method (SFEM) is a generalization of the SFEM based on the local averages of random scalar fields and can be applied to any configuration of structure with several correlated random parameters and several correlated random loads. Numerical examples are examined to show the advantages of the present SFEM over the SFEM based on midpoint discretization, i.e., more rapid convergence and more efficiency.

Properties of discontinuous bifurcation solutions for elastic‐plastic solids are established for conventional test conditions that are pertinent to experiments on soil, powder, and rock. Typically, such conditions are characterized as plane strain and axisymmetry. A restricted form of nonassociativity is assumed, where only the volumetric portion of the flow rule is nonassociated. Apart from this restriction, the aim is to make as general predictions as possible without resorting to a specific elastic‐plastic model. It is shown that states of plane strain are far more susceptible to bifurcations than those encountered in the conventional triaxial apparatus (which represents cylindrical stress states). Moreover, the solutions for dilatant materials will rarely bifurcate under conventional triaxial compression (CTC) test conditions, while conventional triaxial extension (CTE) test conditions fall between the two extremes of plane strain and CTC. For Drucker‐Prager criterion it is shown that bifurcation is never possible in the hardening regime for CTC. These findings are in agreement with experimentally observed behavior when it can be ensured that the stress state is sufficiently homogeneous throughout the specimen.

Analytical solutions for steady seepage into long double‐walled cofferdams have been obtained by using complex variable techniques. Employing the method of successive conformal mapping, the correspondence between the complex physical and potential planes has been obtained in terms of elliptic integrals of the first and the third kind. The equations depicting the rate of seepage, the distribution of hydraulic heads on the various segments of the domain boundaries, and the exit gradients at the base of the excavation have been obtained. The graphs of solutions for the rate of steady seepage and the maximum exit gradient have also been presented in this paper in the form of dimensionless plots to facilitate computation.

The previously presented constitutive model of microplane type for nonlinear triaxial behavior and fracture of concrete is used in nonlocal finite element analysis of compression failure in plane strain rectangular specimens. For specimens with sliding rigid platens there is a bifurcation of the loading path at the beginning of postpeak softening; a symmetric (primary) path exists but the actual (stable) path is the nonsymmetric (secondary) path, involving an inclined shear‐expansion band that consists of axial splitting cracks and is characterized by transverse expansion. The secondary path is indicated by the first eigenvalue of the tangent stiffness matrix but can be more easily obtained if a slight nonsymmetry is introduced into the finite element model. In specimens with bonded rigid platens there is no bifurcation; they fail symmetrically, by two inclined shear‐expansion bands that consist of axial splitting cracks. The transverse expansion produces transverse tension in the adjacent material, which serves as the driving force of propagation of the axial splitting cracks. Numerical calculations indicate no significant size effect on the nominal stress at maximum load.

Dimensional analysis is applied to a simple model of stiffened laminated cylindrical shells to determine the dimensionless parameters that characterize the buckling of these structures. Donnell shell theory is used to describe the kinematic deformation of the shells. The axial and ring stiffeners are modeled using the smeared technique. A nondimensional load is defined as a function of nine nondimensional parameters, which are a combination of the material and geometric properties of the shell. Additional assumptions regarding the construction of the shell walls can be used to reduce the number of nondimensional parameters. Some simply supported shells subjected to hydrostatic pressure are examined to demonstrate the use of the dimensionless parameters. Estimates of the imperfection sensitivity of these shells are made using Koiter's asymptotic theory. The dimensional analysis provides a framework for the systematic investigation of the relationship between material, geometry and the buckling loads of stiffened composite shells. The results of this analysis are intended to provide information that can be used in the preliminary stages of a shell design.

Closed‐form solutions are presented for random vibration response integrals arising in the analysis of multi‐degree‐of‐freedom (MDOF) systems to stationary nodal and/or support excitations Any pair of excitations must either be fully coherent (i.e., have identical frequency distribution) or totally incoherent. Fully coherent excitations may propagate with constant velocity, and have local amplitude variation. Solutions are presented for the response spectral moments under commonly used excitation spectra, including white noise, band‐limited white noise, rational spectra, and spectra that are piecewise linear in log‐log scale. These solutions provide complete generalizations of existing solutions, can save a great deal of computational effort in the random vibration analysis of large systems, and avoid difficulties that may be encountered in numerical integration when the integrands are highly oscillatory due to slow propagation velocities. It should be noted, however, that the solutions presented cannot be applied when the excitations are partially coherent.

A mathematical formulation is proposed to describe equilibrium and stability behavior of the cohesive‐crack model (mode 1 cracking). Under the condition that no unloading occurs, total potential energy of the system can be defined, including energy dissipation in the process zone. Displacement and crack propagation can be formally solved from the potential energy. The properties of stable crack propagation are discussed using second‐order potential‐energy variation. The slope of load‐deflection curve is determined as a function of the second‐order variations of potential energy. Particularly, the second‐order variation of the potential energy with respect to displacement is found to be critical at incipient crack instability, a feature very useful in calculating critical quantities such as peak load. Based on the proposed stability theory, an analytic model is derived for three‐point beams to demonstrate the validity and usefulness of the formulation. The predictions of the analytical model compared excellently with numerical‐simulation results. Comparison with experimental data is not attempted, since the cohesive crack model was already validated experimentally by Hillerborg (1976) and Hillerborg et al. (1983) and many other authors.

Rotary screens are an important unit operation in materials processing. Much of their design has been empirical. Herein, a mathematical model is developed with three major components: particle rise on the screen, particle trajectory through the air, and screening of the particle while in contact with the screen. All were implemented on a computer using numerical methods, allowing use of the original theoretical expressions with simplifications for solvability. Particle rise incorporates friction; particle trajectory incorporates drag. The screening element makes use of an entirely new probabilistic theory that differs from previous work by immediately considering the depth of the bed. Results show very good predictive capabilities. Analysis shows the delicate nature of rotational velocity, the significance of drag on the particles in terms of particle density and airflow in the trommel, and the importance of the coefficient of friction.

This paper presents a modification to the classical cap model by including the third stress invariant and the kinematic hardening rule in the formulation. An isotropic‐kinematic hardening rule for cohesionless soils is proposed that is applicable to both monotonic and cyclic loading conditions. For characterizing strength variations along compression and extension paths, the shape of the failure surface on the octahedral plane is assumed to be triangular with rounded corners. A linear kinematic hardening rule is considered in the formulation, in conjunction with the isotropic rule to account appropriately for the Bauschinger effect during cyclic loading. The model incorporates 12 parameters that can be determined from simple experiments, e.g., isotropic compression, cyclic triaxial, triaxial compression, and triaxial extension testing. The model is shown to realistically predict the soil response for triaxial experiments in both compression and extension. Thereafter, the model is used in a finite element formulation to solve a boundary value problem of a rigid model footing resting on sand. A comparison between the model prediction and experimental observations is presented.

The stability of columns with continuous flexural rigidity monotonically changing along the length of the beam is considered. A new analytical solution for the buckling load is developed in terms of Airy functions for pin-ended columns. Buckling loads for different modes and upper bounds are given when the flexural rigidity at any cross section multiplied by a linear function of position stays constant along the beam. The buckling load for any mode is less than the critical load corresponding to a column of the same length and of constant flexural rigidity whose value is the minimum along the beam of variable flexural rigidity studied. The solution has applications to tapered pin-ended columns with variable modulus of elasticity.