Numerical Modeling of Contaminant Transport through Soils: Case Study

This paper presents the development and validation of a numerical model for simulation of the flow of water and air and contaminant transport through unsaturated soils. The governing differential equations include two mass balance equations for the water phase and air phase together with a balance equation for contaminant transport through the two phases. In the model the nonlinear system of the governing differential equations was solved using a finite-element method in the space domain and a finite difference scheme in the time domain. The governing equation of the miscible contaminant transport including advection, dispersion-diffusion and adsorption effects are presented. The mathematical framework and the numerical implementation of the model are described in detail. The model is validated by application to standard experiments on contaminant transport in unsaturated soils. The application of the model to a case study is then presented and discussed. Finally, the merits and limitations of the model are highlighted.     

Buckling and Postbuckling Behavior of Cross-Ply Composite Plate Subjected to Nonuniform In-Plane Loads

This paper presents a study of buckling and postbuckling behaviour of simply supported composite plates subjected to nonuniform in-plane loading. The mathematical model is based on higher order shear deformation theory incorporating von Kármán nonlinear strain displacement relations. Because the applied in-plane edge load is nonuniform, in the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using these stress distributions, the governing equations for postbuckling analysis of composite plates are obtained through the theorem of minimum potential energy. Adopting Galerkin’s approximation, the governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations in the case of postbuckling analysis, and homogeneous linear algebraic equations in the case of buckling analysis. The critical buckling load is obtained from the solution of associated linear eigenvalue problem. Postbuckling equilibrium paths are obtained by solving nonlinear algebraic equations employing the Newton-Raphson iterative scheme. Explicit expressions for the plate in-plane stress distributions within the prebuckling range are reported for isotropic and composite plates subjected to parabolic in-plane edge loading. Buckling loads are determined for three plate aspect ratios (a/b = 0.5, 1, 1.5) and three different types of in-plane load distributions. The effect of shear deformation on the buckling loads of composite plate is reported. The present buckling results are compared with previously published results wherever possible.     

Vibration of Pressure Loaded Nonlinear Rectangular Plates with Cross Stiffener

The resonant frequency response of large static pressure loaded, nonlinear rectangular plates with a cross stiffener have been investigated theoretically. The nonlinear Berger equation was solved by applying the finite-difference method. Replacing the partial differential equation governing the small amplitude vibration of static pressure loaded plates and the boundary conditions by the finite-difference equations approximately, the simultaneous, homogeneous, and algebraic equations are obtained. Under the condition that the determinant of coefficient matrix must be equal to zero, the resonant frequencies are determined. The numerical procedure is simpler than the procedures based on the von Kármán theory, and reasonable results are obtained.     

Two-Dimensional Simulation Model for Contour Basin Layouts in Southeast Australia. I: Rectangular Basins

Contour basin irrigation layouts are used to irrigate rice and other cereal crops on heavy cracking soils in Southeast Australia. In this study, a physically based two-dimensional simulation model that incorporates all the features of contour basin irrigation systems is developed. The model’s governing equations are based on a zero-inertia approximation to the two-dimensional shallow water equations of motion. The equations of motion are transformed into a single nonlinear advection–diffusion equation in which the friction force is described by Manning’s formula. The empirical Kostiakov equation and the quasi-analytical Parlange equation are used to model the infiltration process. The governing equations are solved by using a split-operator approach. The numerical procedure described here is capable of modeling rectangular basins; a procedure for irregular shaped basins is presented in Paper II. The model was validated against field data collected on commercial lasered contour layouts.     

Nonlinear Finite-Element Analysis Software Architecture Using Object Composition

Object composition offers significant advantages over class inheritance to develop a flexible software architecture for finite-element analysis. Using this approach, separate classes encapsulate fundamental finite-element algorithms and interoperate to form and solve the governing nonlinear equations. Communication between objects in the analysis composition is established using software design patterns. Root-finding algorithms, time integration methods, constraint handlers, linear equation solvers, and degree of freedom numberers are implemented as interchangeable components using the Strategy pattern. The Bridge and Factory Method patterns allow objects of the finite-element model to vary independently from objects that implement the numerical solution procedures. The Adapter and Iterator patterns permit equations to be assembled entirely through abstract interfaces that do not expose either the storage of objects in the analysis model or the computational details of the time integration method. Sequence diagrams document the interoperability of the analysis classes for solving nonlinear finite-element equations, demonstrating that object composition with design patterns provides a general approach to developing and refactoring nonlinear finite-element software.     

Unilateral Contact Problem for Aging Viscoelastic Beams

The frictionless unilateral contact problem of a viscoelastic Bernoulli–Euler beam resting on a viscoelastic soil is studied. The mathematical formulation involves equilibrium equations, compatibility equations, and constitutive laws, with an aging integral-type form. The unilateral nature of the contact is imposed through a compatibility inequality, which allows the determination of the contact imprint at each time. Further, the governing integro-differential equation for the unknown contact pressure is derived. As special cases, the elastic Winkler-type soil and the rigid soil conditions are discussed. A numerical approach is presented, which employs the finite difference method along space and an adaptive step-by-step algorithm along time. The procedure allows for time discontinuities of both external loads and contact pressure. Several selected numerical examples are presented and the influence of the most important material and geometrical parameters are shown. For the simplest situations, it was possible to compare the results obtained with known analytical solutions.     

Continuum Bed Model for Estuarine Sediments Based on Nonlinear Consolidation Theory

The goal of this note is to examine a continuum theory that describes the evolution of sediment beds when subjected to time-dependent shearing forces resulting from surface water movement. The bed was conceptualized as a medium with continuously varying properties such as shear strength and void ratio. The nonlinear equation describing finite strain consolidation, and the complicated nature of the shearing forces acting on top of the bed, preclude the possibility of analytical solutions. Ramifications of linearizing the governing flow equations were explicitly evaluated for applications in bed modeling. Numerical solutions were obtained for the linear and nonlinear models under transient boundary conditions. Model results indicated that the linear model typically predicts lower void ratios, and consequently underestimates the amounts of sediment eroded from the bed as compared to the nonlinear model.     

Uniform Shear Buildings under the Effect of Gravity Loads

Gravity loads play an important role in the linear and nonlinear behavior of buildings during earthquakes. They can also be the cause of ultimate collapse of the structures. In this study, the governing equation of continuous uniform shear buildings under the effects gravity loads is derived and eigenfrequencies, displacement, and drift mode shapes are obtained by eigenanalysis. It is shown that how the geometric properties of the structure affect the fundamental oscillation period and response of the building. Inclusion of the effects of the gravity loads makes the solution of the governing differential equation dependent on the Bessel functions of the first and second kind. The modal load and mass equations are solved using the orthogonality relations of Bessel functions. Effects of gravity on displacement and drift behavior of shear buildings on soft soils and rock subjected to limited near-fault earthquake excitations are shown.     

Random Vibration of Systems with Viscoelastic Memory

The equation of motion of linear dynamic systems with viscoelastic memory is usually expressed in a integrodifferential form, and its numerical solution is computationally heavy. In two recent papers, the writers suggested that the system memory be accounted for through the introduction of a number of additional internal variables. Following this approach, the motion of the system is governed by a set of first-order, linear differential equations, whose solution is quite easy. In this paper, the approach is extended to single-degree-of-freedom systems subjected to random, nonstationary excitation. The equations governing the time variation of the second-order statistics are derived, and an effective step-by-step solution procedure is proposed. Numerical example shows the accuracy of the procedure for white and nonwhite excitations.     

Forced Vertical Vibration of Rigid Circular Disc on a Transversely Isotropic Half-Space

Vertical vibration of a rigid circular disc attached to the surface of a transversely isotropic half-space is considered in such a way that the axis of material symmetry is normal to the surface of the half-space and parallel to the vibration direction. By using Hankel integral transforms, the mixed boundary-value problem is transformed to a pair of integral equations termed dual integral equations in the literature, which generally can be reduced to a Fredholm integral equation of the second kind. With the aid of complex variable or contour integration the governing integral equation is numerically solved in the general dynamic case. The reduced static case of the dual integral equations is solved analytically and the vertical displacement, the contact pressure, and the static impedance/compliance function are explicitly solved. The dynamic contact pressure under the disc and the impedance function are numerically evaluated, and it is shown that the singularity that exists at the edge of the disc is the same as the one obtained for the static case. In addition, the impedance functions evaluated here are identical to the solution given by Luco and Mita for the isotropic domain. To show the effect of different material anisotropy, the numerical evaluations are given for some different transversely isotropic materials and compared.     

Sizing Stormwater Infiltration Structures

A design aid is presented for sizing stormwater infiltration structures. The proposed procedure is based on the hydrological storage equation for an infiltration structure coupled with the Green and Ampt infiltration equation. For the filling process, the two equations are solved simultaneously using a numerical method, and the results are presented in chart form. These charts are useful to determine the maximum water depth in an infiltration structure. For the emptying process, the governing equations are integrated analytically resulting in an algebraic equation that can be solved for the emptying time explicitly. A practical application is included to demonstrate the ease of the suggested procedure.     

Free Out-of-Plane Vibrations of Masonry Walls Strengthened with Composite Materials

The natural frequencies and the out-of-plane vibration modes of one-way masonry walls strengthened with composite materials are studied. Due to the inherent nonlinear behavior of the masonry wall, the dynamic characteristics depend on the level of out-of-plane load (mechanical load or forced out-of-plane deflections) and the resulting cracking, nonlinear behavior of the mortar material, and debonding of the composite system. In order to account for the nonlinearity and the accumulation of damage, a general nonlinear dynamic model of the strengthened wall is developed. The model is mathematically decomposed into a nonlinear static analysis phase, in which the static response and the corresponding residual mechanical properties are determined, and a free vibration analysis phase, in which the dynamic characteristics are determined. The governing nonlinear differential equations of the first phase, the linear differential eigenvalue problem corresponding to the second phase, and the solution strategies are derived. Two numerical examples that examine the capabilities of the model and study the dynamic properties of the strengthened wall are presented. The model is supported and verified through comparison with a step-by-step time integration analysis, and comparison with experimental results of a full-scale strengthened wall under impulse loading. The results show that the strengthening system significantly affects the natural frequencies of the wall, modifies its modes of vibration, and restrains the deterioration of the dynamic properties with the increase of load. The quantification of these effects contributes to the understanding of the performance of damaged strengthened walls under dynamic and seismic loads.     

Interaction of Stream and Sloping Aquifer Receiving Constant Recharge

An analytical solution and finite-element numerical solution of a linearized and nonlinear Boussinesq equation, respectively, were obtained to describe water table variation in a semi-infinite sloping∕horizontal aquifer caused by the sudden rise or fall of the water level in the adjoining stream. Transient water table profiles in recharging and discharging aquifers having 0, 5, and 10% slopes and receiving zero or constant replenishment from the land surface were computed for t = 1 and 5 days by employing analytical and finite-element numerical solutions. The effect of linearization of the nonlinear governing equation, recharge, and slope of the impermeable barrier on water table variation in a semi-infinite flow region was illustrated with the help of a numerical example. Results suggest that linearization of the nonlinear equation has only a marginal impact on the predicted water table heights (with or without considering constant replenishment). The relative errors between the analytical and finite-element numerical solution varied in the range of ?0.39 to 1.59%. An increase in slope of the impermeable barrier causes an increase in the water table height at all the horizontal locations, except at the boundaries for the recharging case and a decrease for the discharging case.     

Nonlinear Response of Fluid-Filled Membrane in Gravity Waves

This paper describes a time-domain model for the nonlinear response of fluid-filled membranes in gravity waves. A formulation based on the principle of virtual work provides an integral governing equation for membrane deformation that fully accounts for geometric nonlinearity, which is known to be important even for relatively small deformation. The incident wave amplitude and membrane deformation are considered to be small, to allow linearization of the hydrodynamic problems. The potential flows inside and outside the membrane are solved by two boundary element models, which are coupled to the finite element model of the membrane. An iterative scheme based on Newmark’s method integrates the resulting nonlinear equations of motion in time. The computed results for a bottom-mounted fluid-membrane system show favorable agreement with available experimental and numerical data. Membrane geometric nonlinearity increases the system stiffness due to strain-stiffening and gives rise to hysteresis response at some frequencies.     

Simplified Braiding through Integration Points Model for Triaxially Braided Composites

A simplified methodology has been developed for modeling two-dimensional triaxially braided composite plates impacted by a soft projectile using an explicit nonlinear finite-element analysis code LS-DYNA. The fiber preform architecture is modeled using shell elements by incorporating the fiber preform architecture at the level of integration points. The soft projectile was modeled by an equation of state. An arbitrary Lagrangian–Eulerian formulation is used to resolve numerical problems caused by large deformation of the projectile. The computed results indicate that this numerical model is able to simulate a triaxially braided composite undergoing a ballistic impact effectively and accurately, including the deformation and failure with a reasonable level of computational efficiency.     

Hygrothermal Effects on the Nonlinear Bending of Shear Deformable Laminated Plates

The influence of hygrothermal effects on the nonlinear bending of shear deformable laminated plates subjected to a uniform or sinusoidal load is investigated using a micro-to-micromechanical analytical model. The material properties of the composite are affected by the variation of temperature and mositure, and are based on a micromechanical model of a laminate. The governing equations of a laminated plate are based on Reddy’s higher-order shear deformation plate theory with von Kármán-type kinematic nonlinearity, and including hygrothermal effects. A perturbation technique is employed to determine the load-deflection and load-bending moment curves. The numerical illustrations concern nonlinear bending behavior of antisymmetric angle-ply and symmetric cross-ply laminated plates under different sets of environmental conditions. The results presented show the effects of temperature rise, the degree of moisture concentration, and fiber volume fraction on the nonlinear bending behavior of the plate.     

Nonlinear Buckling of Composite Shells of Revolution

By adopting the energy method, a method of calculating the stability of the rotational composite shell is presented that takes into account the influence of nonlinear prebuckling deformations and stresses on the buckling of the shell. The relationships between the prebuckling deformations and strains are calculated by nonlinear Karman equations. The numerical method is used to calculate the energy of the whole system. The nonlinear equation is solved by combining the gradient method and the amended Newton iterative method. A computer program is also developed. Examples are given to demonstrate the accuracy of the method presented in this paper.     

Analysis of Soft Fibers with Kinematic Constraints and Cross-Links by Finite Deformation Beam Theory

This paper presents a hybrid analytical–computational mechanics formulation for an arbitrarily curved Timoshenko beam undergoing planar finite deformation and subjected to kinematic constraints in the form of fixed displacement and cross-linking. On the basis of an analytical reduction of the governing equations, the system reduced to a single nonlinear differential equation coupled with integral equations associated with translational constraints. An effective numerical formulation of the problem with general distributed and pointwise constraints is shown to be possible by using a simple finite-element procedure. To illustrate the efficiency and accuracy of the method, several examples are introduced to study both stable and bifurcation problems and a system of interacting fibers with different types of cross-link constraints. Because of the reduction of discretization error and the dimension of the matrix system, the proposed formulation is likely to be an attractive computational platform for modeling large-scale multifiber problems, as in fibrous microstructure simulations and other applications.     

Predicting Leakage through Composite Landfill Liners

Leakage through composite landfill liners having various characteristics was analyzed using existing analytical and numerical models developed for the study. Three-dimensional numerical models were used to analyze leakage through circular defects and two-dimensional numerical models were used to analyze leakage from defective seams. Leakage rates predicted with the numerical models were compared to leakage rates predicted using existing equations and analytical models currently being used. These comparisons show that existing equations and analytical models all have limitations and no universal equation or method is available for predicting leakage rates. To overcome some of the deficiencies in the existing equations and models, new equations were developed based on results from the numerical models. Recommendations are made for using the new equations, existing equations, and analytical models to predict leakage rates in thick composite liners having a geomembrane overlaying a compacted soil liner and thin composite liners having a geomembrane overlaying a geosynthetic clay liner.     

Optimal Channel Cross Section with Composite Roughness

For channels with composite roughness, an equivalent uniform roughness coefficient and flow geometric elements are used in an optimal design method using the Manning equation. The optimal design problems are formulated in a nonlinear optimization framework with the objective function being a cost function per unit length of the canal. Constraints are the Manning equation, positive values for design variables, and specified values of side slopes or top width. The constrained problem is transformed into an unconstrained problem using the Lagrangian multipliers. To obtain an optimal solution for the resulting unconstrained problem, the first-order necessary conditions for optima are applied. The resulting simultaneous nonlinear equations are solved using the computational methodology developed. This technique is applied to illustrative numerical examples. The evaluations establish the potential applicability of the developed computational methodology for optimal design of open channel cross sections with composite roughness.