Journal of Engineering Mechanics

A bending or torsional moment may be generated internally as the stress resultant of a cross section or externally by mechanical devices. Depending on how the moments are generated, they do not exhibit identical behaviors when subjected to three‐dimensional rotations. In the buckling analysis of space frames, all types of moments, whether internal or external, must be specified in the buckling position. This is particularly important in writing the equilibrium conditions for internal joints with members meeting at angles and in setting up the natural boundary conditions for external joints subjected to loadings of the moment type. Emphasis of the paper is placed on the effects of torsional loadings generated by external mechanisms, which may be of the semitangential type or of the quasitangential type (first kind or second kind), on the buckling of structural frames and members. The solutions presented herein are exact and are useful for theoretical as well as for practical purposes.

Vlasov and Yoo's curved‐beam theories have been criticized for not applying the principles of mechanics directly to the curved beam but deriving their equations as the analogies of straight‐beam equations. Although later researchers have worked directly on the curved beam, they have inevitably introduced ambiguities in maintaining consistent orders of magnitude for the terms involved in their formulations. The reality is that not all the theories are consistent with each other. This paper intends to demonstrate that by considering a curved beam in the limit as an infinite number of infinitesimal straight beams—the only kinematic assumption—the classic straight‐beam equations can be manipulated to derive a new set of curved‐beam equations. Because the ambiguities involved in previous formulations are totally circumvented, the presented method is believed to be most convincing. For the two cases of uniform bending and uniform compression, the presented results are compared with those obtained by couples of existing theories.

This paper examines the response of hemispherical, air‐supported structures to turbulent wind, both theoretically and experimentally. Wind‐tunnel tests were conducted on an aeroelastic model of a hemispherical, air‐supported structure to investigate the wind‐induced response and the internal pressure fluctuations for different gradient wind speeds, exposures, enclosure volumes, and mean internal pressures. Also, a semi‐analytical approach, which depends on external pressure measurements and static deflections, is established for predicting the wind‐induced response of air‐supported structures. To verify this methodology, a rigid hemispherical model was wind‐tunnel tested to measure the external pressures, and static deflections were calculated using the finite element method. The response predicted using the semi‐analytical approach agrees well with the experimental data. The experiments resulted in a number of observations: The mean membrane deflections in strong wind are very large, compared with conventional structures, and mostly outward except on the windward side. The dynamic response is broad band in character, does not feature any resonance amplification, and is much smaller than the mean response. The roof response is very sensitive to the variations in the mean internal pressure. A marked increase of the response occurs when the internal pressure is reduced.

A hybrid method is presented for analyzing scattering of time‐harmonic plane strain waves by cracks in an infinite laminated composite plate. The modeling is achieved by dividing the domain into two regions: an interior region that consists of the cracks and a finite region of the plate around the cracks; and an unbounded exterior region. The hybrid method combines a finite element formulation in the interior region with a wave‐function expansion representation in the exterior region. The method is illustrated through solving the problem of scattering by a symmetric normal‐edge crack. Numerical results for the reflection coefficients are presented for an isotropic plate, a homogeneous fiber‐reinforced plate, and a laminated fiber‐reinforced plate. The validity and accuracy of the results are checked by satisfaction of the reciprocity relations and the principle of energy conservation. The technique presented can be used to characterize cracks in laminated composite plates.

A mixed‐variational approach is presented to analyze the axisymmetric flexural interaction between a thin circular plate with a restrained edge and an isotropic elastic half‐space. In this approach, both the plate deflection and the flexural moments are treated as independent variables and are approximated by power series in terms of the radial coordinate and a set of unknown constants. The effect of the restrained boundary is accounted for by including an appropriate term in the mixed‐variational functional (irs), which is applicable for a thin semi‐infinite cylindrical shell connected monolithically with the circular plate at its edge. The unknown constants associated with the approximation functions are evaluated using the stationary property of the functional (irs). Numerical results are presented to demonstrate the effects of edge restrainments on the plate deflection and the flexural moments. It is found that the edge restrain can significantly influence these quantities. The mixed‐variational approach presented here is computationally more efficient than the potential energy‐based methods and can be used effectively to study the flexural response of restrained and unrestrained plates of practical interest.

Explicit and implicit integration of elastoplastic constitutive relations subjected to the constraint of undrained behavior is investigated. The effect on numerical efficiency of various modes in terms of total stress and strain components is considered. The suggested explicit integration algorithm employs the appropriate tangent relation that incorporates the incompressibility condition and requires no iterations. The fully implicit algorithm used in this paper, on the other hand, is based on complete strain control and satisfies, in an iterative manner, the incompressibility constraint as well as equilibrium pertinent to any prescribed total stress component. The constitutive equation that is considered is a three‐invariant generalized cam‐clay model with volumetric hardening/softening as well as deviatoric hardening. Both the explicit and implicit methods seem reliable and efficient for reasonable load steps, even for nonconventional control paths.

Multilayered elastomeric bearings are a basic component in base isolation systems. Several design formulas that have been developed and used in the past consider the elastomer as incompressible. In this paper, the effect of bulk compressibility is included, and the governing equations for the hydrostatic pressure in the elastomer are developed and solved for a pad with a long, rectangular cross section. The resulting expressions are simplified and recommended for the design of elastomeric bearings. The formulas that include the effect of bulk compressibility predict the real behavior of such bearings more accurately than the conventionally used ones. The compressive and flexural stiffnesses of a rubber pad and the pressure distribution on its loaded surface depend on the bulk compressibility of the elastomer, and this dependence is more significant for larger shape factors.

A general framework for finite element reliability analysis based on the first‐ and second‐order reliability methods, FORM and SORM, is presented. New expressions for the required gradients of the response of geometrically nonlinear structures are derived and implemented in an existing finite element code, which is then merged with a FORM/SORM reliability code. The gradient computation does not require repeated solutions of the nonlinear response and is free of the errors inherent in the perturbation method. The proposed reliability method offers significant advantages over the conventional Monte Carlo simulation approach. The method is illustrated for a plate problem with random field properties, random geometry, and subjected to random static loads. The example represents the first application of the finite element method in conjunction with SORM, for a system reliability problem, and involving non‐Gaussian random fields. Extensive analyses of reliability sensitivities with respect to parameters defining the random fields are carried out.

A finite element technique is developed for two‐dimensional dynamic analysis of systems composed of a gravity dam, the water contained in the reservoir, the sediment accumulating on the reservoir bottom, and the underlying foundation. Of particular significance in this development is the modeling of sediment as a two‐phase medium, specifically, a fluid‐filled poroelastic solid. It is shown that this model of the sediment can be accommodated rigorously in the derivation of a hyperelement, which recognizes the coupling between the water, sediment, and foundation and permits convenient discretization of the upstream region. The hyperelement is obtained by synthesizing the motion in the water‐sediment‐foundation stratum from the modes of wave propagation and a particular solution representing the applied ground motion. In the companion paper, results of an application of the present technique to the study of sediment effects on the seismic response of dams are reported and discussed.

The finite element technique developed in the companion paper is applied to the analysis of an idealized concrete gravity dam interacting with the water, sediment, and foundation. Of interest in the study is the evaluation of the effects of fluid‐filled poroelastic sediment on the response of the dam to harmonic ground motion. Since specific information on the sediment behind dams is not currently available, the sediment properties are developed on the basis of measurements near river deltas. The results indicate that the effects of fully saturated sediment are limited to a slight decrease in the maximum acceleration of the dam and the maximum hydrodynamic force. However, if the sediment is only partially saturated, there is a significant decrease in the fundamental frequency along with a substantial reduction of the peak acceleration under horizontal ground motion. These findings hinge on the assumed sediment properties and suggest that the latter be measured in actual dam sites before any definitive conclusion is reached regarding the importance of sediment effects.

This work, consisting of two papers, introduces the “weighted integral method” for calculating both the response variability and the reliability of stochastic frame structures. In the first paper, an exact expression of the stochastic stiffness matrix of the structure is calculated in terms of integrals of the stochastic field describing the random material property multiplied by a deterministic function. These integrals are random variables called weighted integrals. As a consequence, the finite‐element mesh that would be used in a deterministic analysis can be used for any value of the correlation distance of the stochastic field involved in the problem. Two approaches are used to derive the stochastic element stiffness matrix. The first approach is based on the principle of stationary potential energy and the second on the principle of virtual work. The potential energy approach produces a stochastic stiffness matrix that is an approximation of the corresponding exact one obtained using the virtual work approach. Finally, stochastic shape functions are introduced describing the stochastic displacement field of the beam element with random material properties.

After obtaining in a companion paper an exact expression of the stochastic stiffness matrix in terms of random variables called weighted integrals, the response variability and the safety index of stochastic frame structures are calculated in this paper. The response variability is calculated using a first‐order Taylor's expansion and the safety index using the advanced first‐order second‐moment approach. It is concluded that the potential energy and virtual work approaches produce identical results for the mean value and the variance of nodal displacements and internal forces. On the contrary, the two approaches produce different values for the safety index of both nodal displacements and internal forces. These values for the safety index obtained using the two approaches compare very well. It is noted that the stochastic stiffness matrix obtained using the potential energy approach is an approximation of the corresponding one obtained using the virtual work approach. Finally, the effect of statistical dependence or independence among the stochastic fields of different elements is examined.

Separation of crack growth signals is of fundamental importance for detecting, locating, and determining the significance of an internal flaw using acoustic emission techniques. The methodology used to determine the acoustic emission signal in this paper incorporates a source model that is an actual crack propagation and arrest event. This source model includes the effects of the following physical parameters: location of the crack, crack orientation, crack tip velocity, duration of crack propagation, and mode of fracture. The integral equation method is used to examine a semi‐infinite crack in an infinite, linear elastic body loaded under the conditions of plane strain. The dynamic mode I stress caused by a crack propagating with a prescribed velocity is determined, and then the displacements at any point are calculated. These time‐dependent displacements are the analytical form of the acoustic emission waveforms. Numerical results examine the effect of crack tip velocity, duration of propagation, and distance from the crack source to the point of observation on the calculated acoustic emission signals.

The method of stochastic averaging, as applied previously to analyze the energy envelope of a nonlinear system under random excitations of independent ideal Gaussian white noises, is generalized to include the case of correlated physical white noises. The excitations can be additive or multiplicative, or both. The key to the generalization is to separate the well‐known Wong‐Zakai correction terms into two parts and identify them as additional contributions to damping and restoring force, respectively. It is shown that this generalized procedure yields the same stationary joint probability density for the displacement and velocity as that obtained from another procedure, called dissipation energy balancing. Furthermore, the second‐moment stability conditions obtained for a reduced linear system are exact. Two examples are given for illustration.

Statistically equivalent solutions $\mathbf{Û}=\mathbf{h}\left(\mathbf{X}\text{,}\mathbf{\theta }\right)$ are developed for two types of stochastic problems: Nonlinear mappings $\mathbf{U}=\mathbf{g}\left(\mathbf{X}\right)\text{,}$ and equilibrium conditions $\mathbf{K}\left(\mathbf{X}\right)\mathbf{U}=\mathbf{S}\left(\mathbf{X}\right)$ involving a random vector X. The approximate response $\mathbf{Û}$ depends on a specified function h with arguments the vector of random parameters X and a set of unspecified parameters θ. These parameters can be determined from the condition that the difference between U and $\mathbf{Û}$ is minimized in some sense. Several objective functions are considered for finding θ and corresponding approximation of U. It is shown by examples that the accuracy of the approximating response $\mathbf{Û}$ significantly depends on the functional form of h, particularly when solving equilibrium problems.