Journal of Engineering Mechanics

The natural boundary conditions for corners of a certain type of twodimensional problem are derived from the variational principle. A plate problem formulated by the principle of minimum potential energy belongs to this type of problem. The physical problem requires that the first variation of the total potential energy vanish, which involves a functional integral with second partial derivatives. The first variation of a functional with second partial derivatives for a general 2‐D problem is studied for a closed region bounded by piecewise smooth curves. A line integral along the edge boundaries is obtained from the integration by parts of the integral. Natural boundary conditions along smooth edges are obtained directly from the line integral, and the natural boundary condition at a corner is obtained by considering the discontinuity of the slopes of the two edges at the corner. Curvilinear coordinates are considered. However, the boundary curves may or may not coincide with the coordinate lines.

A particle model for brittle aggregate composite materials such as concretes, rocks, or ceramics is presented. The model is also applicable to the behavior of unidirectionally reinforced fiber composites in the transverse plane. A method of random computer generation of the particle system meeting the prescribed particle size distribution is developed. The particles are assumed to be elastic and have only axial interactions, as in a truss. The interparticle contact layers of the matrix are described by a softening stress‐strain relation corresponding to a prescribed microscopic interparticle fracture energy. Both two‐ and three‐dimensional versions of the model are easy to program, but the latter poses, at present, forbidding demands for computer time. The model is shown to simulate realistically the spread of cracking and its localization. Furthermore, the model exhibits a size effect on: (1) The nominal strength, agreeing with the previously proposed size effect law; and (2) the slope of the post‐peak load‐deflection diagrams of specimens of different sizes. For direct tensile specimens, the model predicts development of asymmetric response after the peak load.

An equivalent linearization technique and Monte Carlo simulation are used to investigate the statistics of the hysteretic energy dissipated by SDOF inelastic hysteretic systems under random dynamic loading. It is shown that the equivalent linearization predicts well both the standard deviation and coefficient of variation of the hysteretic energy dissipated. Under stationary conditions the standard deviation of the hysteretic energy dissipated increases with the square root of time, while its coefficient of variation decreases with the square root of time. It is also observed that the mean and coefficient of variation of the hysteretic energy dissipated depend mostly on the ductility demands of the loading, the bandwidth of the response process, and the elastic frequency of the oscillator. The first two moments of the energy dissipated by hysteresis appear to behave much in the same manner as the first two moments of cumulative damage parameters based on linear damage accumulation.

In this paper the creep effects in structures subjected to successive structural system changes are analyzed. The problem is related to structures that develop their structural system from statically determinate to indeterminate, with a gradual increase in the number of redundant restraints introduced at different times. Assuming a linear creep law, the stress redistribution caused by creep in the structure for each stage is evaluated. Writing the compatibility equation at the last restraint introduced, a recurrent formula for the history of redundant reaction is derived. The general solution is proposed as the sum of elastic quantities multiplied by integral terms (which take the creep effects into account) that are the same as the simple case where the additional restraints are introduced all together. The method presented demonstrates that, at all stages, the reaction at the last restraint does not depend on the introduction times of the previous restraints. This reaction is determined by a particular Volterra integral equation solved by numerical procedures of the general method.

A fast and accurate method of generating realizations of a homogeneous Gaussian scalar random process in one, two, or three dimensions is presented. The resulting discrete process represents local averages of a homogeneous random function defined by its mean and covariance function, the averaging being performed over incremental domains formed by different levels of discretization of the field. The approach is motivated first by the need to represent engineering properties as local averages (since many properties are not well defined at a point and show significant scale effects), and second to be able to condition the realization easily to incorporate known data or change resolution within sub‐regions. The ability to condition the realization or increase the resolution in certain regions is an important contribution to finite element modeling of random phenomena. The Ornstein‐Uhlenbeck and fractional Gaussian noise processes are used as illustrations.

This paper complements the theoretical study of the same subject. The aim is to verify the theory formulated in the companion paper. Free vibration of large‐span, self‐supported, lightweight roofs backed by cavities with wall openings is experimentally investigated in the laboratory using two model roofs: a circular membrane roof and a flexible, circular plate behaving in a flexural way. Similarity requirements of the roof‐air system are reviewed. The influence of wall openings on the frequencies and total damping of the structure is investigated. Comparison of the experimental results with those of the theoretical analysis formulated by the writers is made. Excellent agreement between the theory and experiments is obtained. The notions of a leaking kettledrum and a Helmholtz oscillator with a flexible wall are confirmed. The air movement through the wall openings as well as the acoustical damping associated with the motion of both the roof and the air mass at the openings have significant effects on roof natural frequencies and modal damping. The volume of the enclosure under the roof has also a marked effect.

Consistent (algorithmic) tangent moduli for the generalized Duvaut‐Lions viscoplasticity model are derived in this work. The derivations are based on consistent linearization of the residual functions associated with two alternative unconditionally stable constitutive integration algorithms; namely, the implicit backward Euler and the “full integration” algorithms. This “consistent linearization” procedure is equally applicable to the Perzyna‐type viscoplasticity formulations. In particular, the von Mises isotropic/kinematic hardening viscoplasticity model is chosen as a model problem for demonstration. Consistent viscoplastic tangent moduli for other choices of (single or multiple) loading surfaces can be derived in a similar fashion provided that consistent elastoplastic (inviscid) tangent moduli are available. It is noted that since continuum tangent moduli do not exist at all for viscoplasticity, use of the proposed consistent tangent moduli is not only desirable but necessary in the Newton‐type finite‐element computations. In addition, due to the difference in the two constitutive integration algorithms used, the corresponding consistent tangent moduli are not the same even when time steps are small. Numerical examples are also presented to illustrate the remarkable quadratic performance of the proposed consistent tangent moduli for the generalized Duvaut‐Lions viscoplasticity model.

A finite element model is developed for the numerical simulation of a deep‐well wet‐oxidation reactor. The temperature distribution in the well‐earth system is investigated. The governing equations involved in the analysis are the conductive heat equation for the earth, and an energy balance equation describing convective heat transfer and reaction in the reactor tubes. The two equation sets are coupled by the continuity of the temperature and heat flux at the interface between the earth and the reactor tubes. Proper scaling is carried out for the dimensionless forms of these equations. A Galerkin finite element formulation is used for the spatial discretization of the heat equation in the earth. A Petrov‐Galerkin finite element formulation is employed for the convection‐reaction equation in the reactor tubes. The resultant set of ordinary differential equations is solved by a predictor/multi‐corrector algorithm. A numerical test is performed for a model deep‐well reactor. Compared to our previously published work, this formulation is more accurate and consumes less CPU time. It can be used in the design of a deep‐well reactor for oxidation of aqueous sludge. It can also be employed to test control strategies for the operating reactor system.

Structures usually exhibit nonlinear hysteretic behavior under dynamic loading, and this behavior may in general have aftereffects or memory. In this paper we study the properties and the constitutive law of a hysteretic structure with memory. When the memory is null or finite, the hysteretic characteristics are Markovian. Otherwise, they are non‐Markovian. Based on the distributed element model, we deduce the constitutive law for a general hysteretic structure, which unifies the current hysteretic models conceptually. In particular, for the Markovian hysteretic structure the constitutive law is described by the ordinary multidimensional nonlinear state equation in which additional state variables may be involved to express the memory. This constitutive law, in terms of the nonlinear state equation, is demonstrated through several Markovian hysteretic models, including the soft‐hardening hysteretic model, the maximum displacement‐dependent model, and the energy‐dependent model.

The scattering of elastic waves by a subsurface crack in a fiber‐reinforced composite half‐space immersed in a liquid is analytically studied in this paper. The composite half‐space is subjected to a bounded acoustic beam, propagating at an arbitrary angle of inclination. To solve this problem, two new Green's functions are developed for unit loads acting in horizontal and vertical directions in a flawless submerged orthotropic half‐space. The governing equations along with boundary, regularity, and interface conditions are reduced to a coupled set of singular integral equations in terms of the unknown crack opening displacement (COD) by using representation theorem along with Green's function. The solution of these equations is obtained by expanding the unknown COD in terms of Chebychev polynomials. The problem is first solved in the frequency domain. Time histories are then obtained by using fast Fourier transform (FFT) routines.

A two‐part analytical model is developed to assist in seismic response analysis of highway bridges that have abutments supported on earth embankments. In the first part of the model, an expression is developed for identification of the fundamental transverse frequency of the earth embankment. In the second part, two simple equivalent springs are used to represent the transverse and vertical stiffness of a monolithic abutment system. The springs are based on linear elastic plane‐strain analysis of a typical trapezoidal‐shaped embankment cross section. The spring stiffnesses are related to the cross‐sectional dimensions and the soil properties of the earth embankment, which supports the bridge abutments. These stiffnesses can be incorporated into a detailed dynamic model of the superstructure as the equivalent stiffnesses of the bridge abutments. The frequency and the translational stiffnesses predicted by the model are in good agreement with results from a more complicated analytic theory and from finite element analysis of the abutment soil system. A companion paper illustrates an application of the model and provides comparisons to recorded earthquake responses.

Strong‐motion earthquake data obtained on the Meloland Road Overpass during the 1979 Imperial Valley earthquake are used to examine the suitability of simple models for representing the transverse and vertical stiffnesses of the abutment‐embankment systems on highway bridges. Information on the seismic responses of both the abutment‐embankment systems and the bridge superstructure is obtained by applying techniques of system identification to the strong‐motion records. Analytic models, as developed in a companion paper, that account for the transverse and vertical stiffness characteristics of the bridge abutment systems are used to calculate equivalent abutment stiffnesses for incorporation into finite element models of the Meloland bridge. Predicted earthquake responses of these models are compared to the measured seismic responses and to optimal modal parameters obtained from the system identifications. Results of the various analyses show that the simplified models work quite well in simulating the stiffnesses of the abutments and are able to predict with good accuracy the frequencies and mode shapes of the first symmetric transverse and first symmetric vertical modes of the bridge. These two modes are the major contributors to the seismic response of this bridge. Use of the abutment stiffness models also provides insight into the damping ratios that have been observed for the bridge response.

A procedure based on the theory of generalized functions is applied to the calculation of impulse response (Green's) functions of a structural network with rigid body motion. The system considered may be viewed as representation of certain large space structures and consists of torsional members with no shear and bending capacity, and Timoshenko beams with no torsional capacity. All members are rigidly connected at the network joints. Equations yielding impulse response functions are derived, and results of numerical calculations based on these equations are presented. The results confirm the practical feasibility of using the procedure for the calculation of impulse response functions for relatively complex structural networks.

The utility of the Hu-Washizu variational principle as basis for finite elements is illustrated by application to vibrations of beams. Within the framework of Timoshenko beam theory a finite element is formed in which transverse displacement and cross-section rotation vary linearly with the spatial coordinate. The differential-difference equations for a uniform element-length model of a uniform beam are developed. Fundamental natural frequencies of simply-supported beams, as predicted by the spatially discretized model, are compared with those deduced from the partial differential equations of the Timoshenko theory. Agreement is excellent even when the beam is modeled by fewer than ten elements and when the element-length is significantly greater than the depth of the beam. Comparison is also made with the results of finite-element modeling where the strain-displacement equations are satisfied a priori. Not surprisingly this displacement-compatible model produces poor results, owing to the excess stiffness generated by the strain energy of shear in conjunction with the linearly varying displacements.