Journal of Engineering Mechanics

Dynamic tensile failure of concrete due to cracking is simulated using a new class of numerical methods, referred to as mesh-free methods, that are free of the traditional finite-element node-and-element data structure and are thus well suited to modeling the propagation of isolated tensile cracks. The results from a series of numerical simulations are compared with corresponding results from well-defined laboratory experiments involving the failure of concrete. Analysis of the numerical results, and their comparisons to the laboratory measurements, provide some insight into the subtleties in the experimental strain histories and improve understanding of the dynamic tensile failure of concrete.

A simple analytical solution for the bifurcation buckling of a cylinder under axial loading is provided including finite-strain effects. Thus, the small strain theory result of Batterman is generalized. In addition to the thin shell theory solution (excluding shear deformations), a solution including shear deformation effects is also given. All solutions can be evaluated for either the flow or deformation theory of plasticity. The finite-strain constitutive theory used is one in which small strain type relationships apply between the Jauman rate of the Kirchhoff stress tensor and the deformation rate tensor. The analytical results are compared to finite-element analyses to test the validity of the assumptions made. The solutions are explicit. Starting with a point on the stress-strain curve, one calculates explicitly the diameter-to-thickness ratio D/t for a cylinder that will buckle at that level of stress and strain (repeating this as necessary to generate a plot of wrinkling strains as a function of D/t). Unless the tangent modulus at bifurcation is large compared to the stress, the results clearly indicate that finite strains have an important stabilizing effect, leading to higher bifurcation strains.

In this paper, the problem of a 1D elastic distributed system coupled with a moving linear oscillator, often referred to as the “moving oscillator” problem, is studied. The problem is formulated using a “relative displacement” model, which shows that, in the limiting case of infinite oscillator stiffness, the moving mass problem is recovered. The coupled equations of motion are recast into an integral equation that is amenable to solution by a direct numerical procedure. Both the integral equation and the numerical procedure show that the response of the elastic system at the current time depends only on the time history of its response at the positions of the oscillator. Numerical results are presented for the examples of a string and a simply supported beam and are compared to the moving force solutions. It is shown that the oscillator, with its stiffness suitably tuned, can excite the elastic structure into resonance.

The finite analytic method is employed to solve the vertical, two-dimensional subsurface flow and transport equations in an unsaturated zone. The finite analytic method treats the nonlinear coefficient terms of the governing equations as constants in the element so that linearized partial differential equations can be obtained and solved in each element. The accuracy and limitations of the numerical method are systematically explored. The flow and transport simulations are examined using a one-dimensional laboratory infiltration test and an analytical solution of a two-dimensional subsurface transport problem, respectively. In the advection-dominant, vertical, one-dimensional infiltration problem, nine spatial weighting schemes are proposed to evaluate the averaged unsaturated hydraulic conductivity in a discretized element. Among them, the geometric mean weighting scheme provides the most accurate results as compared with the infiltration data. In verification of the two-dimensional solute transport problem, the nine-node elements are placed in the interior domain, and different layers of five-node elements are placed at the boundaries to investigate if the numerical experiment setup was proper and the algorithm was accurate. The developed numerical model is then applied to an irregular-domain landfill leaching problem to reveal the features of subsurface transport in unsaturated zone. Numerical aspects to be further explored are suggested.

The rectangular grillwork with simply supported boundary condition and periodically distributed internal supports is investigated by using the double U-transformation technique. The equivalent grillwork with cyclic periodicity in two (x and y) directions can be produced by using the image method. The equilibrium equations of nodes of the equivalent grillwork possess cyclic periodicity in the two directions. In the process of analysis for the equivalent grillwork, the double U-transformation is used twice. First, by applying the double U-transformation once, the simultaneous equations for the support reactions are derived. Second, by applying the double U-transformation again, the simultaneous equations can be uncoupled into a set of single-degree-of-freedom equations, thus leading to the exact solution. As an example, a square grillwork with 6 × 6 mesh and 2 × 2 internal supports is worked out by means of the formulas derived in this study. It can be proven that the explicit closed-form results satisfy the nodal equilibrium equations, the support conditions, and the boundary conditions exactly.

Postbuckling analysis is presented for a simply supported, shear deformable laminated plate subjected to a uniform lateral pressure and thermal loading, and resting on an elastic foundation. The temperature fields considered are associated with a nonuniform tentlike and parabolic distribution over the plate surface. The material properties are assumed to be independent of temperature. The lateral pressure is first converted into an initial deflection, and the initial geometric imperfection of the plate also is taken into account. The formulations are based on Reddy's higher-order shear deformation plate theory and include the plate-foundation interaction and thermal effects. The analysis uses a mixed Galerkin-perturbation technique to determine the thermal postbuckling equilibrium paths. The numerical illustrations concern the thermal postbuckling behavior of preloaded antisymmetric angle-ply laminated plates under a tentlike temperature field and symmetric cross-ply laminated plates under a parabolic temperature field resting on Pasternak-type or softening nonlinear elastic foundations from which the results for Winkler elastic foundations are obtained as a limiting case. The effects played by foundation stiffness, fiber orientation, transverse shear deformation, the plate aspect ratio, thermal load ratio, and initial geometric imperfection as well as initial lateral pressure are studied.

An improved formulation for spatial stability of thin-walled curved beams with nonsymmetric cross sections is presented based on the displacement field considering both constant curvature effects and the second-order terms of finite-semitangential rotations. By introducing Vlasov's assumptions and invoking the inextensibility condition, the total potential energy is derived from the principle of linearized virtual work for a continuum. In this formulation, all displacement parameters and the warping function are defined at the centroid axis so that the coupled terms of bending and torsion are added to the elastic strain energy. Also, the potential energy due to initial stress resultants is consistently derived corresponding to the semitangential rotation and moment. Analytical solutions are newly derived for in-plane and lateral-torsional buckling of monosymmetric thin-walled curved beams subjected to pure bending or uniform compression with simply supported conditions. In a companion paper, finite-element procedures for spatial buckling analysis of thin-walled circular curved beams under arbitrary boundary conditions are developed by using thin-walled straight and curved beam elements with nonsymmetric sections. Numerical examples are presented to demonstrate the accuracy and the practical usefulness of the analytical and numerical solutions.

In a companion paper, for spatial stability of nonsymmetric thin-walled curved beams, a general formulation was derived based on a displacement field considering the second-order terms of semitangential rotations. Closed-form solutions were newly derived for in-plane and out-of-plane buckling of simply supported curved beams with monosymmetric sections subjected to pure bending or uniform compression. In this paper, to get numerical solutions for the buckling of thin-walled curved beams subjected to general loadings, finite-element procedures are developed using thin-walled curved beam elements and straight frame elements with nonsymmetric sections. Numerical examples for the spatial buckling of doubly symmetric, monosymmetric, and nonsymmetric thin-walled circular beams are presented and compared with previously published solutions to illustrate the accuracy and the practical usefulness of the analytical solutions and numerical procedures.

The fluid loads on some hydraulic structures and the free-surface profiles of the flow need to be determined for design purposes. This is a difficult task because the governing equations have nonlinear boundary conditions. The goal of the present work is to develop a suitable and accurate numerical procedure for the computation of free-surface profiles, velocity and pressure distributions, and flow rate for a 2D gravity fluid flow through a conduit in the pattern of a free jet. The problem involves two highly curved unknown free surfaces and arbitrary curve-shaped boundaries. These features make the problem more complicated than the flow under a sluice gate or over a weir. A combination of a variable domain and a fixed domain finite-element method is used to solve the problem. The results of the calculations show good agreement with previous flow solutions for the water surface profiles and pressure distributions throughout the flow domain and on the gate. Results are also confirmed by conducting a hydraulic model test.

The unsteady wake of a long square cylinder has been numerically analyzed in the present study. Velocity signals at selected locations in the near-wake and the instantaneous forces on the cylinder have been recorded from the numerical model at various Reynolds numbers. These form the basis of investigating the dynamic behavior of the flow system. Results of the present work show the following. Flow past a square cylinder undergoes a sequence of transitions from a steady pattern up to a Reynolds number of 40 to a chaotic one around a Reynolds number of 600. The transition to chaos is manifested through a quasi-periodic route that includes the frequency-locking phenomenon. The quasi-periodicity is seen to set in with two or more Hopf bifurcations. The transition to chaos in the wake of a bluff object is related to the three-dimensionality of the flow. In a 2D simulation, this appears in the form of new harmonics in the velocity traces. The quasi-periodic route to chaos has been established through different characterization tools, such as the spectra, autocorrelation function, time-delay reconstruction, and the Poincaré section. Chaotic behavior is quantified through the calculation of Lyapunov exponent and fractal dimension.

The hydrodynamic performance of a perforated wall breakwater with an internal horizontal plate is studied. It is suggested that a horizontally submerged plate be installed inside the wave chamber to enhance the stability of the structure. Based on the linear wave theory, the 2D problem is formulated to analyze the wave reflection with different porosity, physical dimensions, and wave conditions. The method of matched eigenfunction expansions is used to obtain the solution. Generally, the hydrodynamic performance of a wave chamber is similar with or without an internal horizontal plate. However, the minimum reflection occurs at some particular ratios of the length of the wave chamber to the wavelength, which are less than the corresponding ratios for a wave chamber without the plate, because the waves become shorter over the submerged plate. Thus, the size of the wave chamber can be reduced. It is also found that a moderate porosity is optimal to dissipate the wave energy. By investigating the wave-induced force and moment, such breakwaters with an internal horizontal plate can be designed and constructed with a higher degree of confidence and reliability.

A mathematical model for water migration in concrete that changes from a homogeneous to a nonhomogeneous material due to its loading history is developed in detail. In the proposed model, concrete is assumed to be composed of aggregate, cement paste, water, a crack band, and an interfacial crack between the aggregate and cement paste. The discontinuities for displacement and hydraulic gradient on cracks are considered on the microlevel. Then the governing equation for water migration in concrete as a homogeneous and a nonhomogeneous material is developed by coupling the mass conservation law and the force equilibrium. Therefore, it is possible to apply the developed model not only to cement-based material but also to porous permeable material like soil material and rock. However, the applicability of the developed model must be confirmed by comparing the experimental results.

The material parameters to express the degree of discontinuity of hydraulic gradient in the developed model (Part I) are calibrated based on the sensible analysis of the experimental results, and then the applicability of the calibrated parameters is discussed. As a result, the material parameters are calibrated by the sensible analysis, and then its relevance can be confirmed by another experiment. Moreover, it can be seen that the normalized permeability of concrete as a nonhomogeneous material by the initial permeability (homogeneous material) can be regressed with three straight lines in the crack width axis with the logarithm. Then the intersection point at the crack width axis shows the critical width at which leakage will not occur. The intersection point at the last two straight lines shows the critical Reynolds number at which flow becomes laminar to turbulent according to the experimental result.

A mathematical model for water migration in concrete as a homogeneous material (one that does not contain cracks) is developed in detail. In the proposed model, concrete is assumed to be composed of aggregate, cement paste, and water, for which the compressibility is taken into account. Then, the governing equation for water migration in concrete as a homogeneous material is developed by coupling the mass conservation law and the force equilibrium equation.

Stretching for fluid materials slowly flowing within a circular cavity is investigated in the present study. Two belts, which are performed in periodic rotations on the boundary, are used to drive the flow motions. It is shown that the stretching values of fluid filaments are very sensitive to the boundary conditions for which different periods and different angle-phase-shifts are used. Flows generated by the rotations with longer periods and/or flows generated by the two belts rotating out-phase will create greater stretchings, and possibly better stirrings. Furthermore, under these situations the stretching values associated to fluid filaments are much uniformly distributed.