Journal of Engineering Mechanics

In this paper the behavior of concrete subjected to flexural fatigue loading is studied. Notched concrete beams were tested in a three-point bending configuration. Specimens were subjected to quasi-static cyclic and constant amplitude fatigue loading. The cyclic tests were performed by unloading the specimen at different points in the postpeak part of the quasi-static loading response. Low cycle, high amplitude fatigue tests were performed to failure using four different load ranges. The crack mouth opening displacement was continuously monitored throughout the loading process. Crack propagation caused by quasi-static and fatigue loads is described in terms of fracture mechanics. It is shown that the crack propagation in the postpeak part of the quasi-static load response is predicted using the critical value of the mode I stress intensity factor (K_IC). The ultimate deformation of the specimen during the fatigue test is compared with that from the quasi-static test; it is demonstrated that the quasi-static deformation is insufficient as a fatigue failure criterion. It is observed that crack growth owing to constant-amplitude fatigue loading comprises two phases: a deceleration stage when there is a decrease in crack growth rate with increasing crack length, followed by an acceleration stage where the rate of crack growth increases at a steady rate. The crack length where the rate of crack growth changes from deceleration to acceleration is shown to be equal to the crack length at the peak load of the quasi-static response. Analytical expressions for crack growth in the deceleration and acceleration stages are developed, wherein the expressions for crack growth rate in the deceleration stage are developed using the R-curve concept, and the acceleration stage is shown to follow the Paris law. It is observed that the crack length at failure for constant amplitude fatigue loading is comparable to that of the corresponding load in the postpeak part of the quasi-static response. Finally, a fracture-based fatigue failure criterion is proposed.

A discrete three-degree-of-freedom model of a wood frame shear wall has been developed that is suitable for design-type analyses. The model captures the salient features of the wall response, is amenable to exact closed-form solution, and has the flexibility to account for variations in wall geometry, framing and sheathing materials, fastener type, and spacing. Sheathing-to-stud connections are modeled using a linear viscoelastic element; a method is presented for determining the connection properties using the results of full-scale shear wall tests and a closed-form solution for the test excitation. Results show that the model accurately predicts the hysteretic behavior of the wall for low to moderate displacements; at larger displacements the linear model captures the overall behavior (effective stiffness and energy dissipation), but, as would be expected, fails to predict the pinched hysteresis observed in the tests. Finally, a response spectrum analysis is conducted of a single-story wood frame structure to demonstrate how the model can be used for design-type analyses.

An adjoint sensitivity method based on the shallow-water equations is developed for water wave control in river and estuarine systems. The method is used to compute the gradient of a user-defined objective function in the N-dimensional parameter space consisting of system control settings with just one solution of the basic problem and one solution of the associated adjoint problem. Characteristic equations are derived for the adjoint problem and a new formalism is proposed for the sensitivity of shallow-water flow to boundary changes in depth and discharge. New adjoint boundary conditions are developed for river and estuarine forecasting models with open-water inflow and outflow sections. This gives rise to new expressions for sensitivities at these sections. Characteristic analysis of the adjoint and basic problems shows that sensitivities propagate in the reverse time direction along the characteristic paths of the basic problem. The Riemann variables of the adjoint problem are shown to precisely describe the sensitivity of the objective function to changes in depth and discharge at system boundaries. The method is extended to two space dimensions by bicharacteristic analysis.

An elastic-plastic dynamic analysis of simply supported beams with end membrane restraints subjected to impulsive loading is developed. The beam is elastically curved. The model takes into account elastic and plastic deformations and their effect on the stretch force and the distribution of inertia forces. Instantaneous plastification of the midspan zone due to bending and axial actions is considered. The effect of the length of the plastic zone is discussed. Equations of motion are derived by virtual work. Using the yield condition, the number of these equations is reduced by relating the variations of the displacement variables. This analysis is compared to the standard form of Lagrange's equation of motion showing that the latter is not energy conservative for this case where the deflection shape is nonlinearly dependent on the generalized displacements. Test results using a spring-powered apparatus are presented. Strain rate sensitivity is accounted for as plastic damping. The model's results are compared to test and rigid-segment model results. The comparison shows that the tests are sensitive to the curvature of the axis.

The main objective of this research is to investigate the stability analysis of the composite laminated plate-Pasternak-type (two-parameter) foundation interaction and the influence of the second parameter on the critical buckling loads. Two-parameter foundation models are more accurate than the one-parameter (Winkler) foundation model. A functional for thin laminated composite plates is obtained with the proper dynamic and geometric boundary conditions using the Gâteaux differential via potential operator concept. The functional is linearized by the incremental formulation, and the necessary steps of the stability analysis for the mixed finite-element method are given. As a special case if the second parameter is neglected, the mechanical modeling of the foundation using the Pasternak formulation converges to the Winkler formulation. A four-noded quadrilateral, isoparametric, C⁰ class element with 4 × 9 degrees of freedom is generated. The independent variables per node are three translations, three membrane forces, and three moments. The element is verified by the numerical studies existing in the literature. It is observed that the influence of the second parameter on the critical buckling loads is noticeable, and the results are closely related to the foundation modeling.

An approach using mechanics of saturated porous media is presented to model strongly coupled hydromechanical effects in concrete. Fracture mechanisms of the matrix are taken into account by introducing a tensorial damage variable, which makes it possible to describe orthotropic damage states as well as their effects on hydromechanical parameters (permeability and Biot tensor). An experimental procedure, allowing simultaneous control of pore pressure and applied stresses in a concrete specimen, leads to the identification of material parameters introduced in the constitute model. This model is implemented in the finite-element code CASTEM 2000; numerical simulations of a hydraulic fracture test are then performed and show that the damage-dependence of hydraulic parameters has significant influence on the global response of the structure.

The first part of this two-part study presents a new improved microplane constitutive model for concrete, representing the fourth version in the line of microplane models developed at Northwestern University. The constitutive law is characterized as a relation between the normal, volumetric, deviatoric, and shear stresses and strains on planes of various orientations, called the microplanes. The strain components on the microplanes are the projections of the continuum strain tensor, and the continuum stresses are obtained from the microplane stress components according to the principle of virtual work. The improvements include (1) a work-conjugate volumetric deviatoric split—the main improvement, facilitating physical interpretation of stress components; (2) additional horizontal boundaries (yield limits) for the normal and deviatoric microplane stress components, making it possible to control the curvature at the peaks of stress-strain curves; (3) an improved nonlinear frictional yield surface with plasticity asymptote; (4) a simpler and more effective fitting procedure with sequential identification of material parameters; (5) a method to control the steepness and tail length of postpeak softening; and (6) damage modeling with a reduction of unloading stiffness and crack-closing boundary. The second part of this study, by Caner and Bažant, will present an algorithm for implementing the model in structural analysis programs and provide experimental verification and calibration by test data.

This paper represents Part II of a two-part study in which a new improved version of the microplane constitutive model for damage-plastic behavior of concrete in 3D is developed. In Part II, an explicit numerical algorithm for model M4 is formulated, the material parameters of model M4 are calibrated by optimum fitting of the basic test data available in the literature, and the model is verified by comparisons with these data. The data in which strain localization must have occurred are delocalized, and the size effect is filtered out from the data where necessary. Although model M4 contains many material parameters, all but four have fixed values for all types of concretes. Thus the user needs to adjust only four free material parameters to the data for a given concrete, for which a simple sequential identification procedure is developed. If the user's data consist only of the standard compression strength and the strain at uniaxial stress peak, the adjustment is explicit and immediate. Good agreement with an unusually broad range of material test data is achieved.

The formulation of microplane model M4 in Parts I and II is extended to rate dependence. Two types of rate effect in the nonlinear triaxial behavior of concrete are distinguished: (1) Rate dependence of fracturing (microcrack growth) associated with the activation energy of bond ruptures, and (2) creep (or viscoelasticity). Short-time linear creep (viscoelasticity) is approximated by a nonaging Maxwell spring-dashpot model calibrated so that its response at constant stress would be tangent to the compliance function of model B3 for a time delay characteristic of the problem at hand. An effective explicit algorithm for step-by-step finite-element analysis is formulated. The main reason that the rate dependence of fracturing must be taken into account is to simulate the sudden reversal of postpeak strain softening into hardening revealed by recent tests. The main reason that short-time creep (viscoelasticity) must be taken into account is to simulate the rate dependence of the initial and unloading stiffness. Good approximations of the rate effects observed in material testing are achieved. The model is suitable for finite-element analysis of impact, blast, earthquake, and short-time loads up to several hours duration.

The formulation of the microplane model for concrete and development of model M4 in the three preceding companion papers in this study is here extended to large strains. After giving examples of certain difficulties with the second Piola-Kirchhoff stress tensor in the modeling of strength and frictional limits on weak planes within the material, the back-rotated Cauchy (true) tensor is introduced as the stress measure. The strain tensor conjugate to the back-rotated Cauchy (or Kirchhoff) stress tensor is unsuitable because it is nonholonomic (i.e., path-dependent) and because its microplane components do not characterize meaningful deformation measures. Therefore Green's Lagrangian tensor is adopted, even though it is not conjugate. Only for this strain measure do the microplane components of the strain tensor suffice to characterize the normal stretch and shear angle on that microplane. Using such nonconjugate strain and stress tensors is admissible because, for concrete, the elastic parts of strains as well as the total volumetric strains are always small, and because the algorithm used guarantees the energy dissipation by large inelastic strains to be nonnegative. Examples of dynamic structural analysis are given.

This paper investigates the thermoelastic stability of two bonded half planes consisting of different materials, using a perturbation method. It is found that the conduction of heat across an interface between two dissimilar elastic bodies can cause thermoelastic instabilities within the system. This type of instability has been shown to occur for the case where two frictionless half planes are in contact. The characteristic equation governing the thermoelasticity stability of the two bonded half planes takes the same form as that of the two frictionless half planes, thereby having, qualitatively, the same stability boundaries. The significance of this investigation relates to civil and marine engineering applications in which sandwich or bimetallic plate construction may be applicable.

The main objective of this study is to develop an analytical model to predict the pullout load versus end slip relationship of a smooth fiber having an end anchorage embedded in a matrix. The resisting pullout load of the fiber is composed of a component due to interfacial bond at the fiber-matrix interface and a component due to mechanical anchorage at the embedded end of the fiber. The concept of a relationship between the bond shear stress and the slip at the fiber-matrix interface is used to obtain the force component due to the interfacial bond. To account for the mechanical anchorage resistance at the embedded end, a spring component at the embedded end of the fiber is used. The constitutive property of the spring is assumed to be nonlinear. Based on these concepts and assumptions, a set of analytical solutions to predict the pullout load vesus end slip relationship is derived and then solved by an iterative procedure. Examples predicting the pullout load versus slip curve of a smooth fiber with and without mechanical anchorage at the embedded end are shown and compared. Different modeling aspects of the fiber end anchorage effect and its influence on the pullout load versus slip response are investigated and discussed, with particular emphasis on the tensile force and the bond shear stress distribution along the fiber-matrix interface.

In this study we examine the dynamic strength enhancement and the regularization properties of viscoplasticity with regard to strain softening in tension. To this end we extend the elastoplastic concrete model by Kang and Willam to the viscoplastic Duvaut-Lions overstress formulation. Our main focus is (1) the difference of dynamic strength increase in tension, compression, and shear when only a single viscosity parameter is introduced in the form of the relaxation time, and (2) the regularization of localized failure when we compare elastoviscoplastic with elastoplastic strain softening. Thereby, it is understood that the brittle/ductile modes of failure in tension, compression, and shear depend not only on the loading path, but also on the loading rate at high-speed impact.