Journal of Engineering Mechanics

General analytical equations are developed to predict the overall failure stresses for plain concrete subjected to biaxial stresses. Specifically, failure equations are derived for uniaxial tension, uniaxial compression, biaxial tension, biaxial compression‐tension, and biaxial compression. The general framework of the proposed model is based on the actual events that occur in the material during loading. According to previous experimental investigations, internal microcracks exist in the concrete prior to the application of any external loading. It is shown that the growth and interaction of these preexisting microcracks in the mesostructure of the concrete eventually cause the macroscopic failure of the specimen. Only experimental parameters (concrete mix proportions and coarse aggregate sieve gradation) are required to predict any of the failure stresses. The results generated from the models compare favorably to the experimental data for all loading cases, even though the formulation is not adorned with any unidentifiable constants or parameters.

The spatial variation of seismic ground motion is an important consideration for the seismic design of elongated structures supported on extended foundations or on multiple supports. The temporal and spatial variability of seismic ground motion is effectively described by the frequency‐wave number spectra, which provide information concerning the corresponding correlational characteristics of earthquake motions. Until now, the frequency‐wave number spectra have only been inferred experimentally, for specific seismic events, using data from seismograph arrays. In this work, closed‐form analytic expressions of the frequency‐wave number spectra are established. The corresponding seismic ground motion is produced by a seismic source located in an elastic half‐space and characterized by a double couple. The model used considers the propagation of three‐dimensional waves in an elastic half‐space that are generated at a seismic source and their eventual reflection at the ground surface. Analytic expressions for the frequency‐wave number spectra are then derived for the three displacement components (two horizontal ones and one vertical) at the ground surface.

In this paper, the seismic ground motion is simulated as a stochastic wave according to its analytically derived frequency‐wave number spectra. The simulation is performed using an extension of the spectral representation method, which considers that the simulated wave consists of the superposition of a number of plane waves having amplitudes consistent with the frequency‐wave number spectra. On the other hand, the frequency‐wave number spectra provide all the information concerning the correlational characteristics of ground motion. Therefore, the main characteristics of both the coherent component of ground motion (representation by plane waves) and the incoherent component of ground motion (representation by its cross‐correlational characteristics) are described in the simulated wave. At this stage, the model used to obtain analytically the frequency‐wave number spectra is relatively simple, considering the seismic source as a double couple (point source) located in an elastic half‐space. In addition, the model considers the propagation of three‐dimensional waves in the elastic half‐space that are generated at the seismic source and their eventual reflection at the ground surface.

Statistical energy analysis (SEA) techniques, which are very useful in the dynamic analysis of complex systems, are applied to the study of dynamics of primary‐secondary structural systems. The dynamic response of a system consisting of two spring‐coupled oscillators subject to the filtered white‐noise excitation is examined by means of energy and power flow relations. It is shown that the SEA is conceptually simple and the results obtained from this procedure give useful physical insight into secondary‐system dynamics due to the coupling effect. For example, the mean kinetic energy of the secondary system (indirectly excited oscillator) will always be overestimated if the interaction between the two oscillators is ignored. An evaluation of the coupling effect and response behavior of two coupled multimodal subsystems follows. As numerical examples, two beam‐plate systems, one with point coupling and the other with line coupling, are analyzed. Results of the parametric studies are presented.

Existing analysis techniques for the prediction of response of long‐span bridges to wind and earthquake loads are generally based on a modal synthesis in which the structure is assumed to respond in one or several independent modes. In such analyses, damping generally cannot be estimated with the same approaches as those used for the mass and stiffness, and it is usually added to the modal equations after decomposition as a “known” percentage of critical damping. In the case of bridges, in particular, the total damping, of which part is mechanical or structural, is influenced substantially by the presence of a wind flow around the structure. The total damping is the sum of the mechanical and this aerodynamic damping. This paper presents the results of a study that attempts to separate and estimate the mechanical and aerodynamic damping of a prototype bridge structure using full‐scale ambient vibration and wind measurements, finite element analyses of the bridge structure, and a wind tunnel model of the bridge deck section. The values obtained are compared with those traditionally used in design.

An indentation problem of infinite specially orthotropic laminates on an elastic foundation is solved. The applied load is taken to be uniformly distributed over a finite but small rectangular area. Friction between the laminate and foundation is neglected. The Fourier transform technique is used to formulate and solve the problem in the Fourier domain. The inverse transformation is performed by using the Gaussian integration scheme, in which symmetry of the transformed responses due to material orthotropy is employed. Distributions of interlaminar normal and shear stresses in the vicinity of the applied load are found and the variation of the stresses through the thickness in the interior and at the layer interfaces is displayed. Dependence of the stress components on the material orthotropy is found. The effect of the foundation stiffness on the interlaminar normal stress is also investigated.

Simple finite strain versions of the Maxwell and Kelvin‐Voigt viscoelastic solids are investigated within the framework of continuum mechanics. Three‐dimensional constitutive equations are examined with two different measures of the strain‐rate tensor: the standard Eulerian strain rate and the Jaumann rate of the logarithmic strain tensor. The analysis centers on examining the basic deformation patterns of simple shear and uniaxial tension. A detailed comparison is made among the theoretical predictions obtained from the different material models. These are based on exact analytical solutions of the field equations. The choice of the strain‐rate measure is shown to have an appreciable effect on the stress field in simple shear, particularly for large strains. Also discussed is the role of elastic compressibility in uniaxial tension, including the area relaxation phenomena for the Maxwell solids under constant stretch. A further example is given for nonlinear viscous materials—in conjunction with the pure power law—in simple shear. The results of this study indicate the possible applicability of various rates of strain tensors in constructing viscoelastic models.

Center‐notched mortar plate specimens are loaded in tension. A multiple sensitivity vector holographic setup is developed to record several deformation stages during the stable crack propagation range. The three‐sensitivity vector setup enables the calculation of both crack opening displacements and strain fields around the crack trajectories. An image analysis system is used to isolate the interferometric effect from the sandwich holograms, resulting in fringe patterns with perfect contrast. Image analysis is also used as a faster, more accurate, and more consistent method for fringe count. After evaluation of the holograms, the existence of tensile forces transmitted through the crack faces is associated with the presence of tensile strain behind the crack tip. A definition of the fracture process zone (FPZ) is proposed based on the difference between experimentally observed and linear elastic fracture mechanics (LEFM) strain fields. Deviations from the linear elastic solution show a relatively small zone of nonlinearity in front of the crack tip and a wake fracture process zone (WFPZ) behind the crack tip. A cohesive crack type of model with a bilinear closing process zone is used to predict the experimental observations.

A generalized microplane model, which was previously developed to describe tensile cracking and nonlinear triaxial response of brittle‐plastic materials in compression and shear, is implemented in a finite element code. To limit localization instabilities due to strain softening and the consequent spurious mesh‐sensitivity, the recently proposed concept of nonlocal continuum with local strain (nonlocal damage) is adopted and combined with the microplane model. An effective numerical algorithm permitting large loading steps is developed by applying the idea of exponential algorithms previously used for creep. Problems due to non‐symmetry of the tangential stiffness matrix are avoided by using the initial elastic stiffness matrix in the incremental force‐displacement relations. Numerical results demonstrate that the microplane model, which previously has allowed an excellent description of the test data on nonlinear triaxial behavior of concrete as well as unidirectional and multidirectional crack or crack shear, is endowed (in its nonlocal generalization) with the capability of also modeling tensile fracture. The model yields the correct transitional size effect observed in concrete and agrees with the recently proposed size effect law. The formulation is applicable to brittle‐plastic materials in general.

The number of load‐dependent Ritz vectors required for an accurate dynamic analysis, using the mode superposition method, depends on both the spatial distribution and the spectral (frequency) content of the loading. In this paper we demonstrate that, by monitoring two modal truncation criteria for spectral content and spatial representation of the loading, in the progress of algorithm we can predetermine the accuracy of subsequent dynamic analysis. In the case of earthquake excitation, the loading spatial representation truncation criterion is a participating mass. The spectral content truncation criterion exploits the analogy to the static correction method and unique properties of the load‐dependent vector algorithm. The numerical effort required for the calculation of modal truncation criteria is slight compared to the numerical effort of generating the Ritz vectors. Therefore, monitoring truncation criteria should become a standard step in the load‐dependent vector algorithm. In addition, it provides a possibility for automated truncation of Ritz vector basis in the modal dynamic analysis of complex structures.

To permit more efficient use of the experimental data in the end‐notched flexure (ENF) test, a mixed finite element method is used to model the macromechanical situation with a small number of four‐noded quadrilateral elements. This small number of elements is made possible by the development of two types of specialty finite elements (constant shear‐strain bending element and delamination bending element) within the Hu‐Washizu variational principle. With these elements it is possible to satisfy exactly the kinematic boundary conditions of the ENF test—especially at the point of zero slope. Assuming linear elastic fracture mechanics, the critical interlaminar fracture toughness $\left(G_{\mathrm{II}c}\right)$ is found at one critical load by calculating the difference between the strain energy at the initial crack length and that for an incremental increase in crack length. For a 32‐ply, unidirectional, linear transversely isotropic laminate of varying thickness, the values for $G_{\mathrm{II}c}$ calculated by this finite element model compared well with the results obtained from a previously developed mixed experimental and structural model.

According to Drucker's postulate, materials should become unstable when the second increment of plastic work is negative. A series of conventional triaxial tests on fully saturated and partly saturated specimens is performed to study regions of stable and unstable behavior. Experiments on specimens exhibiting dilation or compression show that, during drained conditions, stress paths directed in the region where Drucker's postulate is violated do not produce unstable behavior. However, for specimens that compress and have degrees of saturation higher than critical, there is a change in drainage conditions causing the stress path to be directed within the region where plastic loading occurred with a negative second increment of work and instability. Previously stated stability conditions for materials with nonassociated flow are analyzed in view of the experimental results on saturated and partly saturated sands.

Based on the condition that a stable material is able to sustain a small perturbation in load, the stability of an elasto‐plastic granular material with non‐associated flow is analyzed, and necessary conditions for mechanical stability of saturated and partly saturated specimens are proposed. Using experimental data obtained from drained tests, the local yield and plastic potential surfaces are established. Their position shows the existence of a wedge where plastic loading occurs, while the second increment of plastic work is negative. Under drained conditions where strains are free to develop without restraint, instabilities cannot develop, and the material will be stable, even when Drucker's postulate is violated. For an elasto‐plastic material under undrained conditions with full or partial saturation, the proposed conditions show that instability may occur when the yield surface opens in the outward direction of the hydrostatic axis, and the material tends to compress. The experimental data supported the suitability of the proposed stability conditions to predict the instabilities created under undrained conditions.

A numerical method is derived for finding the slope of a function from its tabulated values. This method requires that the interpolation function directly approximate the slope in the mean-square sense. In this way, a formal differentiation of the interpolation function is avoided. As a result of the mode of approximation, it turns out that numerical integration, not numerical differentiation, is what is needed for finding the slope. An advantage of this feature is the possibility of using the Gaussian quadrature to derive a formula for nonequidistant sampling points.

This work deals with the overall flexural instability of an angle section column under axial load, taking into account deformations of the cross section as buckling occurs. A nonlinear differential formulation with two variables is considered. The perturbation method is used to follow the initial postbuckling behavior from the bifurcation point. Deformations of the cross section do not modify the Euler critical load, but they may significantly affect the nature of the postcritical path. Unlike the well-known behavior of columns with solid cross sections, the angle section may exhibit unstable postbuckling deformations typical of the light-gauge sections, as reported in previous studies on thin circular arc sections.