Journal of Engineering Mechanics

The stability of a shear layer inside a granular material in a gravity field is studied experimentally and numerically. A shear cell is built of transparent acrylic to visualize the motion of the granular material. This shear cell consists of two concentric cylinders containing layers of uniform spheres in the annular space between the cylinders. The shearing motion of the spheres is produced by rotating the bottom boundary of the cell. Friction of the cylinder walls resists the shear motion, thus creating a single shear layer adjacent to the bottom boundary, while the rest of the layers above move with constant speed as a solid body. As the rotation speed of the bottom boundary increases, two layers adjacent to the bottom boundary begin to shear. This shearing zone quickly thickens and dilates as the rotational speed increases. The transition of this shear motion from a single layer to many layers of shearing is studied by video recording. The initiation of this transition is observed to depend on the material properties and the number of layers overlaying the shear layer. A one-dimensional numerical model is constructed to bring insight into this transitional phenomenon.

In this paper, an improved probabilistic linearization approach is developed to study the response of nonlinear single degree of freedom (SDOF) systems under narrow-band inputs. An integral equation for the probability density function (PDF) of the envelope is derived. This equation is solved using an iterative scheme. The technique is applied to study the hardening type Duffing's oscillator under narrow-band excitation. The results compare favorably with those obtained using numerical simulation. In particular, the bimodal nature of the PDF for the response envelope for certain parameter ranges is brought out.

A family of invariant stress surfaces with a cubic dependence on the deviatoric stress components is expressed as a linear combination of the second and third deviatoric stress invariants. The general format is described in terms of two functions of the mean stress. A simple geometric derivation demonstrates the convexity of the contours in the deviatoric plane. An explicit representation of the deviatoric contours in terms of a size and a shape parameter is given. The shape parameter effects a continuous transition from a triangle to a circle in the deviatoric plane. An explicit format in terms of the triaxial compression and tension generators is derived, and the plane stress contour is given in explicit form. Many previously proposed failure and yield criteria are obtained as special cases.

A probabilistic creep analysis is made of an underground shaft in salt in support of performance-assessment activities of the Waste Isolation Pilot Plant (WIPP) Project. The finite-element method, coupled with an advanced reliability technique, is used to perform the analysis. Randomness due to uncertainties in creep properties is addressed. The results clearly demonstrate the utility of the advanced reliability approach in making complex underground probabilistic creep analyses.

Under extreme environmental loadings such as winds, waves, and earthquakes, civil engineering structures are known to undergo large vibration responses, often resulting in severe damage. Recent developments in active structural control have focused on limiting the amplitude of oscillation of a structure to be within safe, allowable bounds. In this paper, a nonlinear control approach that is capable of limiting extreme dynamic responses of a structure at a reduced expense in control effort is studied. To derive the nonlinear control law, a performance index that is quartic in the states and quadratic in the control is defined. Utilizing a tensor expansion solution procedure for the resulting nonlinear optimal control problem, general polynomial representations of the nonlinear control law are obtained in illustrative problems. Numerical examples of a seismically excited, controlled structure are presented to demonstrate the ability of selected nonlinear control strategies to expend a smaller control effort than an equivalent linear control strategy while limiting the extreme displacement responses.

Three-dimensional axisymmetric calculations of penetration of concrete targets have been conducted with the CALE (Arbitrary Lagrangian Eulerian) code. All the constitutive models studied accounted for the compressibility of concrete and employed either solid or porous equation of state models; the influence of these models on the penetration calculations is studied and presented. The size of the plastic zone in front of the penetrator is estimated from the calculations for both compressibility models and is compared with the available experimental data. Comparison between the calculated values of the extent of this zone and the experimental data indicates that the porous equation of state model is more realistic in representing the elastic-plastic deformation in concrete. Effects of the initial porosity of concrete on penetration calculations are also studied and discussed. The application of constant yield-strength and pressure-dependent yield-strength models for the concrete penetration analysis is studied, and the results of the calculations are presented and discussed. Crater profiles calculated with the pressure-dependent yield model showed good agreement with the available experimental penetration depths and the profiles of the tunnel portion of the crater.

Granular soil is represented in this work by a three-dimensional random array of rough quartz spheres using the discrete element method. Three-dimensional simulations have previously been performed to study the dynamic small-strain behavior as well as the large-strain behavior of sand at macroscopic level. The three-dimensional small-strain simulation presented herein includes the determination of constrained moduli for isotropically and anisotropically loaded random arrays of spheres at microscopic level. The results are in reasonable agreement with the experiments performed on the large triaxial cubical specimen at the University of Texas. It is found that the change of the distribution of contact forces plays a very important role in the above phenomena. During the simulations, the observed macroscopic response is related to the microscopic phenomena at the interparticle contact level.

The paper presents an improvement of the microplane model for concrete—a constitutive model in which the nonlinear triaxial behavior is characterized by relations between the stress and strain components on a microplane of any orientation under the constraint that the strains on the microplane are the projections of the macroscopic strain tensor. The improvement is achieved by a new concept: the stress-strain boundaries, which can never be exceeded. The advantage of this new concept is that various boundaries and the elastic behavior can be defined as a function of different variables (strain components). Thus, whereas for compression the stress-strain boundaries are defined on the microplanes separately for volumetric and deviatoric components, for tension an additional boundary is defined in terms of the total normal strains. This is necessary to achieve a realistic triaxial response at large tensile strains. For microplane shear, a friction law with cohesion is introduced. The present model is simpler than the previous microplane model. Finally, the microplane model is generalized to finite, but only moderately large, strains. Verification and calibration by test data are left to a subsequent companion paper.

The new microplane model developed in the preceding companion paper is calibrated and verified by comparison with test data. A new approximate method is proposed for data delocalization, i.e., decontamination of laboratory test data afflicted by localization of strain-softening damage and size effect. This method, applicable more generally to any type of constitutive model, is based on the series-coupling model and on the size-effect law proposed by Bažant. An effective and simplified method of material parameter identification, exploiting affinity transformations of stress-strain curves, is also given. Only five parameters need to be adjusted if a complete set of uniaxial, biaxial, and triaxial test data is available, and two of them can be determined separately in advance from the volumetric compression curve. If the data are limited, fewer parameters need to be adjusted. The parameters are formulated in such a manner that two of them represent scaling by affinity transformation. Normally only these two parameters need to be adjusted, which can be done by simple closed-form formulas. The new model allows good fit of all the basic types of uniaxial, biaxial, and triaxial test data for concrete.

For a wide class of near-integrable systems with additive or multiplicative noise the mean zero upcrossing rate for the stochastic system's Melnikov process ${\tau }_{\mathit{u}}^{-1}$, provides an upper bound for the system's mean exit rate, ${\tau }_{\mathit{e}}^{-1}$. Comparisons between ${\tau }_{\mathit{u}}^{-1}$ and ${\tau }_{\mathit{e}}^{-1}$ show that in the particular case of additive white noise this upper bound is weak. For systems excited by processes with tail-limited distributions, the stochastic Melnikov approach yields a simple criterion guaranteeing the nonoccurrence of chaos. This is illustrated for the case of excitation by square-wave, coin-toss dichotomous noise. Finally, we briefly review applications of the stochastic Melnikov approach to a study of the behavior of wind-induced fluctuating currents over a corrugated ocean floor; the snap-trough of buckled columns with continuous mass distribution and distributed random loading; and open-loop control of stochastically excited multistable systems.

The symbolic manipulator Mathematica is used to model the nonlinear dynamic behavior of closed, elastic toroidal shells. Transverse shears are neglected and the nonlinearities are of the Von Kármán type. Two fiber-reinforcing schemes are considered: reinforcement with fibers along the major direction of the torus, and reinforcement with fibers along the minor direction of the torus. These schemes result in orthotropic material characteristics. Differential geometry is used to derive the nonlinear kinematic relationships, and a combination of the Rayleigh-Ritz technique and the method of harmonic balance is used to approximate the nonlinear natural frequencies of the tori. Numerical examples show that the linear natural frequency increases as the fiber volume fraction increases for any radii ratio. On the other hand, the nonlinear analysis of some reinforcing schemes shows a competition between the geometric and material parameters of the tori. This competition has a significant effect on the qualitative behavior of the torus and demarks the borders separating shell-like behavior and ring like behavior.

Thin shells are prone to fail by buckling. In most of the practical situations, shell structures have membrane stresses as well as bending stresses and the response of these shells becomes nonlinear. Linearization of the nonlinear equilibrium equations gives rise to an algebraic eigenvalue problem, solving which, buckling load is obtained. Eigenvalue buckling analysis is computationally much cheaper than nonlinear analysis involving tracing the load-deflection path and finding the corresponding collapse load. But buckling loads obtained by eigenvalue buckling analysis are always overestimated, and for systems with large prebuckling rotations this approach may give highly unconservative results. For better prediction of the actual buckling load of a structure, a new methodology involving the proper combination of eigenvalue buckling analysis and geometric nonlinear analysis is used here. This method is computationally cheaper than nonlinear buckling analysis but more reliable than linear buckling analysis. The methodology is used to calculate the buckling load of shells of revolution. The conical frustum shell element with two nodal circles is used in the present study. Also discussed is how to include the effect of initial geometric imperfection in buckling analysis.

A method is presented for a simplified seismic analysis of nonlinear multidegree-of-freedom structures. The method represents an improvement over an earlier attempt to extend the use of the conventional response-spectrum method to nonlinear structures, and is also based on the use of a modal superposition in combination with nonlinear response spectra. It involves the computation of their natural frequencies and mode shapes on the basis of their initial elastic properties, the calculation of a yield deformation for each of their modes using formulas proposed here, and readings from a nonlinear response spectrum. The procedure is formulated for plane rigid frames, but limited to elastoplastic force-deformation behavior. Its application is illustrated by means of numerical examples with a three-story shear building and a six-story plane frame. In a comparison with solutions obtained with a step-by-step integration method, the proposed approximate method predicts the maximum displacements of these two structures with an average error of about 5%.

The mode-localization phenomenon is here proposed as a mechanism for suppressing undesired oscillations of trusses. Sample trusses consisting of repeated unit cells are designed to have weak intermediate member(s) in addition to geometric disorders. These irregularities lead to the confinement of motion to a limited part of the structure. The results of a finite-element analysis show the natural frequencies and the relative amplitudes of mode shapes as functions of a disorder parameter.