Journal of Engineering Mechanics

A Monte Carlo procedure for reliability estimation of structural systems, subject to multiparameter time‐varying loading and formulated in the space of the load processes, is described. For most realistic reliability problems, this space is of low order, significantly simplifying the required multidimensional integration. The structural resistance is considered to be time‐ and structural‐behavior‐invariant, and is described only in terms of time‐independent, random‐variable strength parameters. The integration to obtain both the outcrossing rate and the zero‐time failure probability is performed using directional simulation of a general type. Example applications are given for loadings modeled as continuous Gaussian processes and for a two‐bay, rigid‐plastic, portal frame structure. Both completely dependent member resistances and member resistances having a given correlation structure are considered. The effect of changes in member strength variance on zero‐time failure probability and outcrossing rate is described. The rate of convergence of the simulation results is noted.

Previous buckling analyses for three‐dimensional frames based on discrete element formulations have been criticized for the lack of equilibrium at angled joints in the buckling state. While this problem can be overcome through the introduction of the so‐called semitangential rotations and moments, this approach suffers from the drawback that the semitangential definitions are rather artificial. In this paper, it is demonstrated that the so‐called lack of equilibrium for rotated angled joints is not a physical reality, but a result of overlooking compatibility conditions for interconnected elements in the deformed position. One feature of such compatibility conditions is that they are irrelevant to any elastic deformations. In a finite element formulation, they can only be included through the external virtual work terms. In this paper, the procedure for calculating the element forces in a step‐by‐step nonlinear analysis is also discussed. The adequacy of the present approach is confirmed in the numerical study.

Equations of motion for a plane, nonlinear prestressed truss are expressed in terms of six generalized coordinates. Three coordinates define a dominant centrally symmetric vibration mode. The dominant coordinate of that mode, $q_1\text{,}$ is a centrally symmetric rotation of the compressive struts about the supports. The equation for coordinate $q_1$ is of the cubic hardening type. For two common types of applied loads, the centrally symmetric mode is excited only through the $q_1$ coordinate. $q_1$ is excited in an additive way by antisymmetric forces and in a parametric way by symmetric forces. For a broad range of excitation frequencies, two coordinates of the centrally symmetric mode respond statically, and the mode can be defined by only the $q_1$ coordinate. With such a simplification, the equation for the $q_1$ coordinate excited by antisymmetric forces is of the Duffing type, while the equation for the $q_1$ coordinate excited by symmetric forces is of the nonlinear Mathieu type.

Results of a bifurcation analysis of a geometrically nonlinear cable truss are reported. Additive antisymmetric harmonic excitation and parametric symmetric harmonic excitation are used. For antisymmetric excitation, responses have significant odd superharmonic components over a broad frequency range. Odd superharmonic components become resonant at excitation frequencies much smaller than those that cause the fundamental resonance. Therefore such superharmonic components may be important for wind design of nonlinear cable systems. There are regions of multiple steady‐state responses. The frequency range over which they occur is affected by the coupling between coordinates. Bifurcations to a torus do occur, but the resultant motions do not seem to have marked increases in maximum responses. No period‐doubling bifurcations are observed for the parameter space studied. For reasonable magnitudes of symmetric harmonic excitation, it is very probable that the only steady‐state response will be a stable equilibrium.

The seismic performance of fixed‐base and base‐isolated concentrically braced and special moment‐resisting steel frames are determined. The fixed‐base frames are designed in accordance with the 1990 Structural Engineers Association of California (SEAOC) Recommended Lateral Force Requirements and Commentary, while the base‐isolated frames are designed to 100%, 50%, and 25% of the SEAOC‐recommended lateral force levels. Nonlinear time‐history analyses, for 54 ground motion records, are then conducted. The number of yielded or collapsed frames, number of yielded elements, total and relative roof displacement, and isolation‐bearing displacement are statistically evaluated. Comparable or better performance is achieved for the base‐isolated braced frame designed to 50% of the SEAOC‐recommended lateral force. Comparable performance is achieved for the base‐isolated moment frame at various design force levels, depending on the performance criteria.

The response of a nonlinear marine riser model to both parametric and external excitation is examined using Markov methods. Moment‐closure techniques are used to predict the response of the riser in a random seaway. An approximate nonlinear polynomial representation of the hydrodynamic damping term is developed in order to implement Gaussian and non‐Gaussian moment‐closure schemes. These methods are applied to solve the case of pure parametric excitation and the results are compared with those obtained using stochastic averaging. It is shown that as the parametric excitation level is varied, the response predictions from different techniques exhibit the same linear trends. However, the non‐Gaussian closure scheme provides improved response estimates that are closer to stochastic averaging results. Moment‐closure techniques are then extended to consider a riser under combined parametric and external excitation. For a fixed level of parametric excitation, the response statistics are found to vary nonlinearly with external excitation levels. The Gaussian closure scheme is found to overpredict structural response for the case of pure parametric excitation and to underpredict structural response for the case of pure external excitation.

An expression is derived for the probability of crack growth in a material with penny cracks of random size and orientation. It is assumed that crack growth occurs when the maximum local Mode I stress intensity factor, $K_l^l\text{,}$ exceeds $K_{\mathrm{lc}}\text{,}$ the criterion is generalized for the three‐dimensional problem. The location along the crack front and the magnitude of $K_l^l$ are determined conditional on a penny crack of prescribed size and orientation. The probability distribution function of $K_l^l$ is then computed using a derived probability density function for the crack size. The probability of crack growth is expressed as a function of four dimensionless parameters: The expected number of cracks in a volume; the expected value of the Mode I stress intensity factor for a crack normal to the direction of stress, divided by $K_{\mathrm{lc}}\text{,}$ the probability density function of a normalized crack size; and the Poisson ratio. A study that quantifies the effects of the four dimensionless parameters is presented.

Dense two‐phase flows occur in many material processing situations. In these flows, particle collision presents a mechanism for internal resistance. In the literature, restitution coefficients are given for collisions between dry objects. Little is known about two colliding objects placed in a fluid. Thus, although two‐phase flow models exist for dense mixtures, realistic values of the required restitution coefficient are not available. Recent theoretical and experimental analyses of colliding spheres in a viscous fluid show that collisional rebound depends greatly on the interstitial fluid. At very high viscosity, contact may not occur. This paper presents an experimental study of collisional restitution between steel and nylon, and a range of Newtonian viscous fluids. The fluids have a viscosity range from $27{\mathrm{mm}}^2/s$ to $2\text{,}600{\mathrm{mm}}^2/s\text{.}$ The normal impact velocity ranges from 0.015 m/s to 0.75 m/s. Preliminary results show that the behaviour of the impact and the resulting restitution coefficient depend on the interstitial fluid. These results will be useful to the mathematical modeling of dense two‐phase flows.

A constitutive model based on the hierarchical single surface (HISS) approach and the elastoplasticity theory is proposed here to describe the behavior of interfaces subjected to static and cyclic loading conditions. In the hierarchical scheme, a model describing basic characteristics of the interfaces is modified to include increasingly complex behavior of the interfaces. The proposed model can simulate associative, nonassociative, and strain‐softening behavior during monotonic as well as cyclic loading. The parameters influencing interface behavior are identified using data from laboratory simple shear tests on sand‐steel and sand‐concrete interfaces. A parameter called the interface roughness ratio, R, is defined to quantify roughness for different interfaces. Similarly, a cyclic parameter Ω is introduced to simulate the cyclic volumetric behavior of the interfaces. The proposed model is verified with respect to comprehensive test data on interfaces with different roughnesses, normal loads, initial densities and type of sand, and quasi‐static and cyclic loadings.

This paper presents a method for the dynamic analysis of linear systems with uncertain parameters to stochastic excitation. The applied forcing function is modeled as a modulated Gaussian white‐noise process in time. A procedure to derive the random state‐space equation for the response covariance matrix of the system is presented. The covariance equation is then integrated in time and the response variability is computed. The proposed method is applied to the random response of a five‐degree‐of‐freedom primary‐secondary system. Five different values of the ratio of the fundamental natural frequency of the primary to secondary system are considered along with two different ratios of primary‐ to secondary‐system mass. Two types of uncertainty are considered: uncertain primary‐system stiffness and damping, and uncertain secondary‐system stiffness and damping. Results are reported for the effect of system uncertainty on both the relative displacement and absolute acceleration response of the secondary system. It is concluded that uncertainty in the stiffness of the primary and secondary systems has a very strong influence on the response of the secondary system and its reliability, and must be properly accounted for in the analysis of such systems.

A general, high‐order theory based on variational principles is presented for the bending behavior of a sandwich beam with a core that is vertically flexible. The theory embodies a rigorous and systematic approach to the analysis of such structurs, which have high‐order effects caused by the nonlinearity of the longitudinal and the transverse deformations of the core through the height. As such, it improves on the available classical and superposition theories. Beam construction consists of the upper and lower skin, metallic or composite laminated symmetric, with nonidentical mechanical and geometrical properties, and a soft core made of foam or honeycomb. The formulation uses a beam theory for the skins and a two‐dimensional elasticity theory for the core. The behavior is presented in terms of internal resultants and displacements in skins, peeling and shear stresses in skin‐core interfaces, and stress and displacement fields in the core, even in the vicinity of concentrated loads. The method is applicable to any type of loading exerted on the skins and to any type of boundary or continuity conditions, including cases in which at the same section the conditions at the upper skin are different from those at the lower. Some typical cases are studied numerically.

In this technical note, power flow between two oscillators with spring and damping (nonconservative) couplings is introduced, and its relationship with energies stored in the two oscillators is established. It is shown analytically as well as through numerical examples that power flow and energies are appropriate response variables of a primary-secondary (PS) system, and that they both reflect the tuning characteristics of system and provide information on optimum design of the S-system.