Journal of Engineering Mechanics

This paper studies thin‐walled trapezoidal cross‐section beams with lateral cantilevers subjected to a torsional load, considering the distortion of the cross section. Transverse and longitudinal displacements of the walls are governed by six shape functions, which describe bending in the horizontal plane, as well as torsion and distortion of the cross section; the shearing strain in the middle plane of the walls is considered; the shear center is not used. The theorem of minimum total potential energy is used to derive the governing equilibrium equations (and boundary conditions), whose solution is given in closed form. A wide set of constraints is studied (rigid or flexible transverse diaphragms, mixed kinematic and static boundary conditions), from a theoretical point of view and in numerical examples. The following results were obtained: (1) Torsion and distortion are a pair of coupled problems: distortion and warping stresses are always present; in particular, the longitudinal stress at the side cantilevers increases near “critical” loaded or constrained cross sections (the model detects this effect owing to the distinctive hypothesis of piecewise linear warping on the top flange for both nonuniform torsion and distortion); and (2) in some cases a “local” disturbance may affect a large part of the beam, due to the combined effect of shearing strain and distortion of the cross section.

The mean rate of vector processes out‐crossing safe domains is calculated using methods from time‐independent reliability theory. The method is founded on a result for scalar up‐crossing derived by Madsen. The out‐crossing is formulated as a zero down‐crossing of a continuously differentiable scalar process, and the mean crossing rate is obtained as a sensitivity measure of the probability for an associated parallel system domain. The vector process may be Gaussian, non‐Gaussian, stationary or nonstationary, and the failure function defining the boundary of the safe domain may be time‐dependent. A method for calculation of the expected number of crossings in a time interval through the introduction of an auxiliary uniformly distributed variable is presented. For stochastic failure surfaces the ensemble averaged rate is determined. A closed‐form expression for the mean crossing rate of a nonstationary Gaussian vector process crossing into a time‐dependent convex polyhedral set is derived. The method is demonstrated to give good results by examples.

An active control system is designed to induce artificial damping to the isolation structure of a building without increasing its stiffness. The control forces needed are exerted only on the base of the structure, and the system is quite practical. This control system can operate efficiently if the time delay that exists in the actual system does not exceed its critical value. If the delay is long, the steady‐state response will be resonant periodically at its critical delay times, and the transient response will become unstable. The critical delay time depends on both the fundamental frequencies of the structure and the artificial damping introduced by the control system. The higher modes of the system will suffer more of a stability problem than the lower modes, unless the nondimensional damping introduced to them is less than a specific value. A strategy of time‐delay compensation is established in this paper. The instability of the system can be efficiently prevented by a proper choice of artificial damping and artificial stiffness.

The effects of soil‐structure interaction on the form of the control rule and on the effectiveness of active control of the seismic response of structures are examined. The structure is modeled as a uniform shear beam supported on a rigid foundation embedded in an elastic soil. The seismic excitation is represented by vertically incident shear waves. Active control in the form of an absorbing boundary located at the top of the structure is considered. The active absorbing boundary cancels the reflection of waves at the top of the structure and eliminates resonance within the superstructure. It is found that the form of the control rule changes as a result of the rocking of the foundation associated with the kinematic and inertial interaction. However, the effectiveness of this form of active control remains unchanged or is improved when soil‐structure interaction effects are included.

This paper describes the exploratory development of a nondestructive testing (NDT) method for the assessment of dynamic interface shear transfer in concrete based on a frequency analysis. This study focuses on experimental observations of relationships between shear transfer across a concrete interface and the corresponding variations in the response frequencies. The analysis of the data recorded during the tests is performed both in the time and in the frequency domains, and a correlation between these results is presented. The results demonstrate a clear relationship between dynamic interface shear transfer and certain response frequencies. It is concluded that this approach may serve as a basis for an innovative testing approach for assessing interface shear in concrete systems.

This paper is concerned with the numerical solution of time‐harmonic transition problems in elasticity, in general, and with soil amplification in inhomogeneous alluvial valleys, in particular. A mixed variational formulation based on Hamilton's principle, involving field equations only within the valley and an integral representation for the surrounding medium, is developed and used to derive a symmetric finite element‐boundary‐element method for this problem. This method, valid for all frequencies, incorporates automatically the displacement and traction interface continuity conditions; therefore, it imposes no boundary constraints on the approximating functions. The BEM‐FEM is applied to the response of semicircular inhomogeneous valleys with linearly increasing shear modulus with depth, due to oblique incident SH waves. Numerical results emphasize the importance of two‐dimensional resonant effects in deep valleys, and the strong effects of varying stiffness on surface motion, including a rapid spatial variation, especially near valley edges. This can have practical implications in design as it suggests that it is possible for two similar structures located near each other to experience different levels of shaking and, thus, different damage levels during the same earthquake.

This paper presents the general, two‐dimensional formulation for the response of free‐standing rigid bodies to base excitation. The formulation assumes rigid body, rigid foundation, and Coulomb friction. The behavior is described in terms of the five possible modes of response (rest, slide, rock, slide‐rock, and free‐flight) and impact between the body and foundation. The governing equations of motion are summarized for the modes: slide, rock, and slide‐rock. Approximate equations are outlined for the rock and slide‐rock modes that are valid for small angles of rotation. A model governing impact from a rock, slide‐rock or free‐flight mode is derived from first principles using classical impact theory. This model assumes a point‐impact, nonzero coefficient of restitution and finite value of friction. The paper presents a complete and consistent formulation, which can be used in its entirety or in part, for investigation of the generalized response using suitable analytical or numerical techniques.

An approximate closed‐form solution is developed for a single‐mode, steady‐state slide‐rock response resulting from a harmonic ground acceleration. The approximate solution is developed using the method of slowly varying parameters and is valid for a rectangular block undergoing small angles of rotation at the frequency of base excitation. Impacts with the foundation are assumed to be perfectly plastic and frictional impulses are included. Periodic solutions are found to exist in general only for relatively high amplitudes of ground acceleration and friction less than the inverse aspect ratio of the block. The rock component of the response is sensitive to changes in aspect ratio and friction and insensitive to changes in ground acceleration. The slide component of response is approximately equal to the amplitude of ground displacement and is insensitive to changes in friction and aspect ratio. Results compare favorably to those obtained by numerical integration. The accuracy of the approximate solution is shown to depend heavily on the magnitude of impulse applied during impact.

A stochastic finite element method is developed to analyze the random responses of geometrically nonlinear beams and frames with combined uncertain material and geometric properties under simultaneous spatial and temporal random excitations. The method is based on an equivalent linearization scheme, combined with the mean‐centered second‐order perturbation technique and the modal expansion approach. The procedure can be straightforwardly extended to incorporate other existing finite elements. Examples include large‐amplitude free vibration and dynamic random response of three simply supported beams and a portal frame with combined uncertain material and geometric properties. For the case of beams and frame with no structural uncertainties, the results obtained are in good agreement with alternative solutions. For the case of beams and frame with structural uncertainties, alternative representative results are also obtained using Monte Carlo simulation to compare and validate the present solution and method.

An approach for the solution of problems of structural mechanics involving material variability is proposed. The material property is modeled as a stochastic process. The Karhunen‐Loeve expansion is used to represent this process in a computationally expedient manner by means of a set of random variables. Further, the well‐established deterministic finite‐element method is used to discretize the differential equations governing the structural response. A spectral expansion of the nodal random variables is introduced involving a basis in the space of random variables. The basis consists of the polynomial chaoses that are polynomials orthogonal with respect to the Gaussian probability measure. The new formulation allows the computation of the probability distribution functions of the response variables in an expeditious manner. Two problems from structural mechanics are investigated using the proposed approach. The derived results are found in good agreement with data obtained by a Monte Carlo simulation solution of these problems.

Better understanding of the multiaxial low‐cycle fatigue strength of structures and/or components is important for the design of structures under severe repeated loading. Significant discrepancies exist between experimental results and currently available analytical predictions. This paper presents a method to evaluate the low‐cycle fatigue strength of structural metals based on a plastic work concept. The maximum principal tensile stress is incorporated into the evaluation as a second parameter to account for the progressive failure characteristics. This proposed approach narrows the scatter band of low‐cycle fatigue life predictions, and is convenient for engineering applications. The two necessary parameters in this evaluation method, the plastic work per cycle and the maximum tensile principal stress, can be obtained by experiments or by using theoretical constitutive modeling. In this study, they have been obtained through theoretical calculations and validated by experimental data. Also illustrated in this paper is a relatively simple design procedure using the proposed method.

The governing nonlinear vector differential equations for large deflections of an end‐loaded, three‐dimensionally deformed, initially straight, axisymmetric elastica are developed and simulated numerically without decomposition. All simulations are based on repeated applications of truncated Taylor's expansions to advance along a deformed elastica. Both initial value problems, and two‐point boundary value problems with a corresponding shooting method are discussed. A number of examples are solved and compared with corresponding theoretical solutions, and with a finite element solution. An elastica solution with at least six independent equilibrium configurations satisfying a single set of boundary conditions and not heretofore treated in the literature is also presented.

Numerical simulations show that two‐dimensional granular solids in an inclined chute can flow as a steady, uniform stream or as intermittent groups, or slugs. Slugs, or groupings of the flowing solids, were found to be enhanced with decreases in the coefficient of restitution of the granular particles and with decreases in the bulk number density or concentration of solids. A statistical parameter was developed to describe the degree of uniformity of the distribution of the granular particles along the chute. This parameter was found to have a predictable value at a well‐defined transition point when the flow changed from a uniform to a sluglike behavior. An explanation is presented describing the significance of this parameter in terms of the mechanics of the flow process.

An approach is developed for approximate random vibration analysis of multidegree hysteretic structures based on the concept of energy balance, i.e., the excitation energy is balanced with the energy the structure dissipates in hysteresis and viscous damping. All response measures are estimated based on random vibration analyses of an associate linear system. Inelastic response measures, such as cumulative plastic drift at the local level, are modeled based on Markov process theory. The approach is developed for elastoplastic and trilinear hysteretic multidegree structures. Analytical results are compared to those based on simulation for the distribution of the mean peak ductility ratio over the structure's height for several multistory buildings subjected to stochastic seismic excitation. The focus is on studying the effect of earthquake intensity and the presence of a soft story on the distribution of ductility ratio versus height. Insights gained from such analyses should be helpful in evaluating seismic design strategies of buildings.

An appealing approach to formulate constitutive models for characterizing distributed damage due to microcracks and voids is continuum damage mechanics with the concepts of effective stress and strain equivalence. In that approach, in which damage is imagined to characterize the reduction of the net stress‐transmitting cross‐section area of the material, the constitutive model is separated into two independent parts, one for damage and the other for elastic and inelastic behavior (rheology) other than damage, which, if combined appropriately, give the overall constitutive behavior. However, the existing multidimensional formulations for damage are quite complex, and practical implementations capable of fitting experimental data are hard to obtain. The microplane models, by contrast, provide conceptual simplicity and close fits of multiaxial test data for concrete, soils, etc., although, as formulated in the past, various kinds of physical phenomena were mixed in the definition of the microplane stress‐strain curves. In this work the microplane theory is reformulated in a manner that separates damage from rheology and makes the formulation fit the basic framework of continuum damage mechanics. Aside from a kinematic constraint between macrostrains and micro‐strains, the model satisfies a static constraint such that the effective microstresses are the resolved components of the effectiveness macrostresses.