Journal of Engineering Mechanics

In second‐order reliability methods the failure surface in the standard normal space is approximated by a parabolic surface at the design point. The corresponding probability is computed by asymptotic formulas and by approximation formulas. In this paper the probability content of the parabolic failure domain is computed exactly by inversion of the characteristic function for the parabolic quadratic form. Also, the exact result for the probability content of the failure domain obtained from the full second‐order Taylor expansion of the failure function at the design point is presented. The approximating parabola does not depend on the formulation of the failure function as long as this preserves the original failure surface. This invariance characteristic is in general not shared by the approximation obtained using the full second‐order Taylor expansion of the failure function at the design point. The exact results for the probability content of the approximating quadratic domains significantly extend the class of problems that can be treated by approximate methods.

The effect of randomness in the rapture process at the source on the near‐field seismic ground motions is examined. Time histories and response spectra are determined at an observation point in the vicinity of a vertical, strike‐slip fault. The seismic motions resulting from a deterministic, kinematic source model with uniform slip are compared with the motions obtained from stochastic models, in which the rupture parameters vary randomly along the length of the fault. The analysis shows that the random variability in the source parameters (namely, the dislocation amplitude, the rise time, the rupture velocity, and the fault width) enriches the frequency content of the motions. When the random variability in each rupture parameter is investigated separately, it is found that randomness in the rise time and the rupture velocity enhances the frequency content of the motions, whereas the variation in the dislocation amplitude and the fault width produces motions similar to the deterministic ones.

Elastic flexural buckling formulas for columns under an end‐concentrated load and axial uniformly distributed load (or selfweight) found in the literature or newly derived by the writers, are summarized herein. However, these formulas overestimate the critical buckling loads when the columns are short or consist of struts connected by lacing bars or batten plates, since the effect of shearing force on their deflections becomes significant. This paper investigates the stability of such columns, with a high shear flexibility factor, under a general combined end‐concentrated load and axial distributed load. Stability criterion design curves, obtained from the finite element approach, are presented for a wide range of column end conditions. The influence of the shear flexibility factor on the critical loads is also examined. Moreover, some simple shear‐flexural buckling formulas for columns under combined end‐concentrated load and axial trapezoidal load are proposed.

A nonlinear fracture‐mechanics‐based cracking model for the finite element analysis of crack propagation in concrete materials is proposed. Since the method of evaluating macroscopic mechanical properties of composites, specifically the modified Voigt‐Reuss model, is utilized in its development, the proposed model is more general than the existing ones. In fact, the well‐known models such as the crack band model and the composite damage model are shown to be the special cases of the proposed model. By solving a benchmark problem, the validity of the proposed model is verified. It is also found through the present numerical studies that, while the realistic behavior of a notched concrete beam is obtained only by the nonlinear fracture‐mechanics concept, the influence of lateral stress on compression‐induced crack growth in a brittle solid predicted by nonlinear fracture mechanics is in good agreement with that due to linear‐elastic fracture mechanics.

A simple delta function form of the velocity distribution function is proposed for smooth, uniform, slightly inelastic spheres in rapidly sheared granular flow. The collisional stresses are computed for simple‐shear flow and compared to existing theories. Good agreement is observed with more elaborate kinetic theories, numerical simulations, and shear cell data. The kinetic formulation is placed in the physically transparent form proposed by Bagnold. This formulation is used to show the sources of the discrepancy in results between existing data and the theory of Shen and Ackermann, which is based on Bagnold's formulation. It is shown that the discrepancies can be substantially alleviated by incorporating more detail into Shen and Ackermann's phenomenological picture of collisions, and by improving their estimate of the mean‐free path. Thus, it is shown that using the proper form for averaging has a much more significant effect on the magnitude of the predicted stresses than the form of the velocity distribution function. The delta function form can be employed to simplify calculations while retaining an acceptable level of accuracy.

The dynamic tear (DT) test is often used to obtain key dynamic fracture parameters. Previous work indicates that the first impact waves arriving at the test specimen notch may not create a stress state critical for fracture. In this article, a finite element program for the transient dynamic simulation of an actual DT test is first compared with an analytical solution for the time interval prior to the arrival of wave reflections from the specimen surfaces. The program, which uses a uniform spatial mesh size and does not require singular elements, is then used to determine the transient notch stress field at any time. It is found that the arrival of the first reflected waves generate the stress field necessary for fracture. A critical dynamic stress intensity factor value for the DT test is then estimated. This estimate shows good agreement with other experimentally based results.

A span‐wise variation of nonlinear hydrodynamic pressures on dams during earthquakes was studied using a finite‐difference approximation of Euler's equations coupled with the nonlinear free‐surface boundary condition. The pressure distribution was sensitive to both the longitudinal and lateral components of the ground motion. The dynamic effect of reservoir banks would increase with the dam width. The inward motion of a reservoir bank produced higher pressures on one side while the outward motion of the other bank resulted in a reduction in hydrodynamic pressures. The combined effect can be effectively obtained when the geometric parameters of a dam/reservoir system, the changes in water surface, and the earthquake force are cast in the equations of motion. In this case, the time‐dependent water surface is transformed to the horizontal plane, and the flow field is mapped onto a cubic region, making it convenient to model the complex reservoir geometry and the wavy water surface by the finite‐difference method.

A new numerical method for solving stationary free surface flow problems with floating bodies in two dimensions is presented. The fluid flow is assumed to be incompressible and irrotational. Velocity components, discretized with isoparametric finite elements, are the basic unknowns. The discrete equations are obtained via a modified least‐squares functional. The free surface geometry is described with Hermitian cubic polynomials passing through so‐called geometry points with which certain design parameters are associated. A cost function optimum design procedure with sensitivity derivatives is employed to iterate the free surface and the floating body to the final positions. Numerical results are given for flow over a Crump weir without a floating body, and for two floating body examples. The approach presented has considerable advantages, especially with respect to the flexibility in applying boundary conditions over conventional velocity potential or stream function formulations.

A model for process‐zone growth and crack propagation is discussed and the model predictions are compared to experimental results. Critical crack tip opening displacement $\left({\mathrm{CTOD}}_c\right)$ is used as a traction‐free crack extension criterion. It is assumed, as a first approximation, that ${\mathrm{CTOD}}_c$ has two critical values, initial and secondary. Based on the parameter analysis, comparison to experimental results, and findings of other researchers, it is concluded that ${\mathrm{CTOD}}_c$ changes during crack propagation and depends on specimen size and crack length. Model predictions are compared successfully to experimental results obtained on concrete beams in a broad range of sizes and initial crack lengths from initial loading through nonlinear prepeak and postpeak softening response to failure.

Predictions for the seismic response of the Lotung, Taiwan, large‐scale containment model are compared with the response recorded during the Taiwan earthquake of May 20, 1986 $(M=6.5)\text{.}$ The predictions involve the use of three sets of theoretical models (A, B, and C) and include the effects of soil‐structure interaction, The first set of models (A) is based on basic geotechnical data while the second set (B) is based on more complete geotechnical and geophysical information. The third set (C) is based on the additional knowledge of the response of the structure during forced vibration tests. It was found that the predictions contain most of the recorded response's characteristics. In particular, the predicted horizontal peak accelerations at the top of the structure differ from the data by less than 15%. Some localized differences observed in the floor response spectra may be associated with differences between the backfill and the outside soil, or may have resulted from underestimation of the soil degradation resulting from the seismic strains.

Steel‐laminated elastomeric bearings are widely used in bridges. In order to impose the smallest possible force on the substructure, the bearing should have the smallest plan dimensions and the greatest height possible. Both are controlled by stability. However, buckling of the bearing differs significantly from that of a conventional column in that it is strongly influenced by shear flexibility and axial deformations. Existing theories ignore the latter and become very conservative for bearings with the relatively low profiles typically used today. Extensions to the theory were developed to account for the influence of axial shortening as well, and experiments were conducted. The general form of the modified theory matches the pattern of the test results, and in particular they both show that bearings with a height‐to‐width ratio below a certain limit will never buckle. Perfect correlation could not be achieved because of difficulties in establishing reliable flexural stiffnesses for the bearings. Design equations are proposed based on the form of the theoretical equations, calibrated against the test results.

The dynamic interaction between nonlinear hydrodynamic pressures and the vibration of a concrete gravity dam is analyzed by coupling Euler's equations with a finite element model of the dam. The earthquake force components in the horizontal and vertical directions are cast as the forcing functions in the finite difference equations for fluid motion. The results are compared with those associated with a rigid dam motion. The pressure distributions are very sensitive to dam vibration and surface wave. For El Centra ground motion recorded on October 15, 1979, the calculated rise of the water surface on Pine Flat Dam was 8 ft (2.7 m), indicating a strong nonlinear effect on hydrodynamic pressures and the potential hazards of surface waves in the reservoir.

One of the recurring aerodynamic concerns of both existing and projected new and flexible bridges, typically those of long span, is the possibility that the deck may be set into periodic oscillation by cross winds of relatively low velocity. This general phenomenon, regardless of the specific details of the local flow about the structures, has been termed “vortex‐induced oscillation.” In this paper, prediction of the expected vortex‐induced response of flexible bridges is treated as a two‐step process. In the first step, the parameters of a mathematical model, used to describe the response of a rigid, elastically mounted, scaled section model of a bridge are identified. A technique for estimating these parameters using wind‐tunnel test results is proposed. In the second step, model parameters are used in the equation of motion of the prototype bridge to obtain its response. A method that accounts for the effect of imperfect spanwise correlation of aerodynamic forces on flexible‐bridge response, based on measurements on rigid section models, is developed. Design applications are discussed and an example is provided.

The understanding and modeling of the complete failure of structures made of brittle heterogeneous materials that exhibit distributed damage due to cracking has recently become a subject of keen interest in many countries, especially the United States and France. The importance of the subject for concrete structures, foundations, underground excavations and failure of soils, rocks, or ice, etc., as well as for the design of modern ceramics and composites, is generally accepted. The central issue is the localization of strain and energy dissipation due to strain‐softening damage and the inherent size effect on the failure loads as well as ductility of structures. To stimulate progress, an international research workshop, gathering the foremost experts from the United States and France, along with guests from other European countries and Japan, was held at E.N.S. de Cachan, France, under the joint sponsorship of French and U.S. governmental agencies. The lectures from the workshop, whose full texts appear in a separate proceedings volume, are reviewed in the present paper and the current status of knowledge is appraised from the perspective of the workshop. Considerable progress has been registered in tensile cracking and fracture, and the description of the size effect. However, further research is required on compression and shear fractures, rate effects and time‐dependence, effect of microstructural characteristics and micromechanics of damage, numerical modeling of instabilities and bifurcations, and the probabilistic treatment.

A method for the synthetic generation of curvograms (curvature versus time) associated with strong earthquake shaking is presented. It is shown that small radii of curvative of ground motion, in the range of several hundred meters, are possible only for very large accelerations (1–3 ge/u), near epicenter, and in soft geologic and soil environment, with shear wave velocities in the range near 50–100 m/s. Therefore, the pseudostatic bending of long foundation structures by ground waves may be expected only for large accelerations and for short wavelengths, in the vicinity of shallow or surface faults. Similar conditions are possible in isolated areas, where the three-dimensional geometry of surface layers leads to focusing, and thus large amplification of incident wave amplitudes.