Upper-Bound Solutions for Rigid-Plastic Beams and Plates of Large Deflections by Variation Principles

This paper demonstrates deriving upper-bound solutions of geometrically nonlinear problems for beams and plates from rigid perfectly plastic material by the principles of virtual work in general form and stationary of total energy. Presented noncomplicated examples justify that the first is more appropriate when a kinematically admissible displacement field is defined by several generalized displacements. The second can serve as effective means for comparison in accuracy solutions corresponding to different displacement fields playing the same role as the upper-bound theorem in the limit analysis. Procedures of the latter for obtaining upper-bound solutions mainly remain valid. Solutions for a beam and rectangular plate subjected to uniformly distributed load illustrate importance of taking into account transformation forms of displacements in loading process.     

Buckling of Circular Mindlin Plates with an Internal Ring Support and Elastically Restrained Edge

This paper is concerned with the elastic buckling problem of circular Mindlin plates with a concentric internal ring support and elastically restrained edge. In solving this problem analytically, the circular plate is first divided into an annular segment and a core circular segment at the location of the internal ring support. Based on the Mindlin plate theory, the governing differential equations for the annular and circular segments are then solved exactly and the solutions brought together by using the interfacial conditions. New exact critical buckling loads of circular Mindlin plate with an internal ring support and elastically restrained edge are presented for the first time. The optimal radius of the internal ring support for maximum buckling load is also found. An approximate relationship between the buckling loads of such circular plates based on the classical thin plate theory and the Mindlin plate theory is also explored.     

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.     

Spline Semidiscretization Analysis for Orthotropic Plates and Shells

This paper presents a numerical analysis procedure, called spline semidiscretization procedure, for the unified analysis of orthotropic and/or isotropic thin plates and shallow shells of rectangular projection with the two opposite edges in the y direction simply supported. The sine and cosine functions may thus be employed as the displacement trial functions in the y direction. By semidiscretization through dividing plate and shell into N equal subintervals, the B3 spline function, consisting of the (N+3) local B3 spline functions (the first and last three local B3 spline functions have been modified for accommodating to any type of boundary conditions) with respect to the (N+1) points and two extended additional points in the x direction, can then be used as the displacement trial function in the x direction. Governing equations of an orthothopic shallow shell subjected to the distributed, linearly distributed, concentrated loads or their combinations are derived based on its potential energy functional. Unified formulas for the determination of displacements and internal forces of the orthotropic and/or isotropic thin plates and shallow shells are obtained. In comparison to the conventional finite element method, with the displacement trial functions having the good properties with piecewise polynomial as well as orthogonality and decoupling, the present procedure has remarkably fewer unknowns to be solved (more precisely, a term by term analysis involving only much smaller matrices can be conducted), and thus it is computatively more efficient. Likewise, the computational program, with minimal preparation of input data, can be very easily developed through the present formulation. Numerical results indicate that the present method can render a very high accuracy. The fast convergence shown in numerical examples demonstrates the reliability of the results.     

Large-Deflection Mathematical Analysis of Rectangular Plates

A large-deflection mathematical analysis of rectangular plates under uniform lateral loading is presented in this paper. The analysis is based on solving two fourth-order, second-degree, partial differential Von Kármán equations relating the lateral deflections to the applied load. This paper provides a mathematical procedure which benefits from the software and hardware computing capabilities that were unavailable when mathematical modeling was last attempted. The solution is presented in a simple form suitable for direct practical use and can be easily implemented in common spreadsheet packages. Plates with two boundary conditions, namely, simply supported edges and held edges, are considered. Comparisons are held against earlier exact and approximate solutions, including results of finite element analyses. The results show close agreement with other exact analysis methods. The solution is able to produce the same results as other exact solutions, but with a much simpler and a more practical approach.     

Plastic-Buckling of Rectangular Plates under Combined Uniaxial and Shear Stresses

This paper is concerned with the plastic-buckling of rectangular plates under uniaxial compressive and shear stresses. In the prediction of the plastic-buckling stresses, we have adopted the incremental theory of plasticity for capturing the inelastic behavior, the Mindlin plate theory for the effect of transverse shear deformation, the Ramberg-Osgood stress–strain relation for the plate material, and the Ritz method for the bifurcation buckling analysis. The interaction curves of the plastic uniaxial buckling stress and the plastic shear buckling stress for thin and thick rectangular plates are presented for various aspect ratios. The effect of transverse shear deformation is examined by comparing the interaction curves obtained based on the Mindlin plate theory and the classical thin plate theory.     

Analysis of Laminated Sandwich Plates Based on Interlaminar Shear Stress Continuous Plate Theory

The bending response of sandwich plates with stiff laminated face sheets is studied by a six-noded triangular element having seven degrees of freedom at each node. The element formulation is based on a refined higher-order plate theory having all the features for an accurate modeling of sandwich plates with affordable unknowns. The refined plate theory is quite attractive but suffers from a problem concerned with an interelement continuity requirement when it is used in finite element analysis. The problem has been dealt satisfactorily in this new element, which is applied to the analysis of sandwich plates of different kinds.     

Orthotropic Plate Model for Estimating Deflections in Filled Grid Decks

Concrete filled grid bridge decks exhibit orthogonal elastic properties and significant two-way bending action enabling orthotropic plate theory to determine structural response for these elements. Current American Association of State Highway and Transportation Officials load and resistance factor design (LRFD) specifications employ an orthotropic plate model to predict live load moment in concrete filled grid bridge decks but provide no guidance for computing displacement, a potentially important serviceability consideration. This paper presents equations to approximate the maximum deflection in concrete filled grid bridge decks based on orthotropic plate theory, multiple patch loads, LRFD design truck and tandem load cases, the influence of multiple spans, and the two most common deck orientations.     

Thermomechanical Postbuckling of Cross-Ply Laminated Rectangular Plates

Postbuckling analysis is presented for shear deformable cross-ply laminated composite rectangular plates subjected to the combination of in-plane edge compressive mechanical loading and thermal loads due to a linearly varying temperature across the thickness. The formulation is based on the first-order shear deformation theory and von-Karman-type nonlinearity. The analysis uses a quadratic extrapolation technique for linearization and Chebyshev polynomials for spatial discretization. An incremental iterative approach is employed to estimate the critical load. The boundary conditions consisting of clamped, simply supported, free edge, and their combinations are considered. The effects of the thinness ratio, aspect ratio, lamination scheme, the number of layers, and the modulus ratio on the critical load/limit load and postbuckling behavior are studied.     

Analysis of Hybrid Laminated Composite Plates

Hybrid laminated composite plates are analyzed using a nine‐noded isoparametric plate finite element based on Mindlin's theory. The shear flexibility is included in the finite element modeling. Shear flexibility is of importance, especially when different materials are used in the laminate design. Hybrid laminates consisting of graphite∕epoxy and kevlar∕epoxy plies are considered for illustration. The study indicates that hybrid laminates provide stiffnesses that are intermediate to the values obtained for single‐material laminates. The minimum deflection is achieved at different fiber orientation for thick plates compared to thin plates. The deflection behavior of hybrid laminates seems to be less affected by outer‐ply stiffness in the case of thick plates. Thick plates show less variation in the first natural frequency with fiber orientation but hybridization changes the natural frequency considerably. The first natural frequency of the hybrid laminate can be made higher than the stiffer single‐material laminate.     

Vibration of Thick and Thin Plates Using a New Triangular Element

A triangular element based on Reissner–Mindlin plate theory is developed and it is applied to free vibration analysis of plates in different situations. The element has three corner nodes, three mid-side nodes and an internal node at the element centroid where each node contains three usual degrees of freedom (transverse displacement and bending rotations). To make the element free from the shear locking problem, the formulation is done in an efficient manner taking transverse displacement and transverse shear rotations as the field variables. The degrees of freedom of the internal node are condensed out to improve the computational elegance. As the condensation cannot be done with a consistent mass matrix, a lumped mass matrix having no mass contribution at the internal node is used. In this context two mass lumping schemes are proposed where the effect of rotary inertia is considered in one of these schemes. All these features have made the element quite elegant, which is tested with numerical examples to show its performance.     

Numerical Analysis for the Nonlinear Free Vibration of Shallow Shells

A variational approach for the nonlinear free vibration of shallow shells having a quadrilateral boundary is presented in this paper. Natural coordinates ξ and η are used to map the prescribed geometry in the x–y plane. Displacement fields corresponding to u, v, w, β1, and β2 are expressed in terms of the product of two algebraic functions, the form of which is so chosen that the displacement boundary condition can be imposed by manipulating the coefficients. In arriving at the stiffness matrix, no simplification is applied to the nonlinear strains and the variation of the complete energy equation is considered. For the plate problems numerical results are obtained and compared with approximate analytical results by other researchers. Numerical results for the shallow shells are also presented and their characteristics are found to be significantly different from the results for the plates.     

Stress Analysis of Orthotropic Plate on Elastic Foundation with Transverse Point Load

An infinite orthotropic plate on an elastic foundation subjected to a transverse point load is studied. A three‐dimensional stress distribution in the vicinity of the applied load is sought without considering the friction between the plate and the foundation. Based on the assumption of a uniform stress distribution for the applied load, a double Fourier transform technique is employed to solve the problem in the transform domain. The Gaussian integration scheme is used to carry out the inverse transformation to obtain the real stress components. Symmetry of the transform response due to the material orthotropy has been used to reduce the effort involved in performing the integration. Distribution of various stress components around the point of load application is presented for a typical orthotropic layer. A comparison of stress distribution for orthotropic and nearly isotropic material is also presented.     

Tilting Analysis of Rectangular Elastic Layers Bonded between Rigid Plates

A theoretical approach to determine the tilting stiffness of a rectangular elastic layer bonded between two rigid plates is presented. On the basis of two kinematics assumptions, the governing equation for the mean pressure is derived from the equilibrium equations. Using the approximate shear boundary condition, the mean pressure is solved and the tilting stiffness of the bonded rectangular layer is then established in an explicit single-series form. Whereas the finite element method can be applied to calculate the stiffness, the series solution provides a convenient way for parametric studies. Through the obtained pressure expressions, the horizontal displacements are derived from the corresponding equilibrium equations, from which the shear traction on the bonding surface can be found. The error of using the approximate shear boundary condition is negligible for the tilting stiffness, but becomes significant for the horizontal displacements and bonding shear stresses near the edges of the rectangular layers.     

Ritz Buckling Analysis of Rectangular Plates with Internal Hinge

This paper presents the Ritz method for buckling analysis of rectangular plates with an internal line hinge. The Ritz method involves the domain decomposition method to cater for the discontinuity of slope at the hinge line. The correctness of the Ritz formulation and solutions is confirmed by the exact solutions derived using the Levy method for plates with two opposite sides simply supported. Based on the Ritz method, buckling factors are generated for rectangular plates of various aspect ratios, hinge locations, and support and loading conditions.     

Novel Formulation to Study Thermal Postbuckling of Circular Plates with Edges Elastically Restrained against Rotation

A novel formulation is used to study the thermal postbuckling behavior of circular plates, with the edges supported to not have lateral deflection and elastically restrained against rotation. The elastic restraint is mathematically represented by an elastic rotational spring. The circular plate is subjected to a uniform edge compressive radial load, developed because of a uniform temperature rise. The formulation is on the basis of on the radial tensile load developed in the plate because of the large deflections of the plate with edges immovable in the plane normal to the edge and the linear buckling load corresponding to the uniform edge radial compressive load. The developed radial tensile load is obtained by using Berger’s approximation. The numerical results obtained from the present investigation in terms of the ratios of the postbuckling to the buckling loads for several rotational spring stiffness values compare well with those obtained by using the versatile finite-element analysis.     

Nonlinear Analysis of Angle‐Ply Composite and Sandwich Laminates

A refined higher order shear deformation theory for linear and geometrically nonlinear behavior of fiber‐reinforced angle‐ply composite and sandwich laminates is established. Laminae material is assumed to be linearly elastic, homogeneous and isotropic/orthotropic. The theory accounts for nonlinear quadratic variation of transverse shear strains through the thickness of the laminate and higher order terms in Green's strain vector in the sense of von Karman. A simple C0 finite‐element formulation of this theory is then presented with a total Lagrangian approach, and a nine node Lagrangian quadrilateral element is chosen with nine degrees of freedom per node. Numerical results are presented for linear and geometrically nonlinear analyses of multilayer angle‐ply composite and sandwich laminates. The theory is shown to predict displacements and stresses more accurately than first‐order shear deformation theory. The results are compared with available closed‐form and numerical solutions of plate theories and three‐dimensional finite‐element solutions. New results are also generated for future evaluations.     

Postbuckling Analysis of Plates Resting on a Tensionless Elastic Foundation

The buckling and large deflection postbuckling behavior of plates laterally constrained by a tensionless foundation and subjected to in-plane compressive forces are investigated. A nonlinear finite-element formulation based on Marguerre’s nonlinear shallow shell theory, modified by Mindlin’s hypothesis, is employed to model the plate response. To overcome difficulties in solving the plate–foundation equilibrium equations together with the inequality constraints due to the unilateral contact condition, two different approaches are used: (1) the unilateral constraint is accounted for indirectly by a bilinear constitutive law and (2) the problem is formulated as a mathematical programming problem with inequality constraints from which a linear complementarity problem is derived and solved by the Lemke algorithm. To obtain the nonlinear equilibrium paths, the Newton–Raphson algorithm is used together with path-following strategies. Plate–foundation interaction leads to interesting deformation sequences, characterized by the variation of the contact and noncontact zones along the postbuckling path, leading sometimes to sudden changes in the deformation pattern. The results have a remarkable dependence on the plate aspect ratio, foundation stiffness, and buckling shape. The effects of geometric imperfections on the nonlinear response of the plate are also investigated. From these results, a number of insightful conclusions regarding the behavior of such plate–foundation systems are drawn.     

Large Deflection Stability of Slender Beam-Columns with Semirigid Connections: Elastica Approach

The large-deflection elastic analysis of slender beam-columns of symmetrical cross sections with semirigid connections under end loads (forces and moments) including the effects of out-of-plumbness is developed in a classical manner. The classical theory of the “Elastica” and the corresponding elliptical functions are utilized in the proposed method which can be used in the large-deflection stability analysis of slender beam-columns with rigid, semirigid, and simple connections under any combination of end loads (conservative and nonconservative). The proposed method consisting of a closed-form solution of the Elastica can also be utilized in the large deflection analysis of beam-columns whose connections suffer from flexural degradation or, on the contrary, flexural stiffening. The main limitation of the Elastica is that only flexural strains are considered (the effects of axial and shear strains are neglected). Therefore results from the proposed method are theoretically exact from small to very large curvatures and transverse and longitudinal displacements for plane beam-columns under bending actions. The large-deflection analysis of a beam-column with flexible connections at both ends becomes a complex problem requiring the simultaneous solution of at least two highly nonlinear equations with elliptical integrals. The solution of this problem becomes even more complex when the end connections are nonlinear or the direction of the applied end load changes (like “follower” loads). The validity and effectiveness of the proposed method and equations are verified against available solutions of very large deflection elastic analysis of beam-columns. Four comprehensive examples are included for verification and easy reference.     

Simple Formulation for the Flexure of Plates on Nonlinear Foundation

Plates resting on an elastic medium are normally analyzed in a simplified way using the linear Winkler foundation approach. Nevertheless, plates resting on layered medium with vast differences in their moduli exhibit nonlinear behavior under pressure. The present technical note deals with a nonlinear finite-element procedure to analyze plates with linear strain displacement relations resting on a nonlinear elastic media. The coupled problem is formulated using the total potential energy (TPE) concept. The nonlinear foundation stiffness matrices have been derived using the Taylor expansion of the TPE at equilibrium and a symbolism of grouping the energy contributions. The nonlinear foundation stiffness matrices derived in the present technical note have been demonstrated to yield results that agree well with published results in the literature. A brief parametric study on the effects of nonlinearity of the foundation is also presented using the proposed foundation stiffness matrices.