Nonlinear flexural vibrations of laminated orthotropic plates

Large amplitude free flexural vibrations of laminated orthotropic plates are studied using C⁰ shear flexible QUAD-8 plate element. The nonlinear governing equations are solved using the direct iteration technique. Numerical results are obtained for isotropic, orthotropic and cross-ply laminated plates with simply-supported boundary conditions on immovable edges. It is observed that hardening behaviour is increased for thick plates and orthotropic plates.     

Elasto-plastic analysis of a plane stress problem using linearized yield surface and mixed hardening rule

The objective of this paper is to present the mathematical formulations in the incremental theory of plasticity, which is based on the mixed hardening rule and a linear yield surface. A three-parameter, uniaxial symmetric, linear yield surface suitable for tension-weak as well as equal tension and compression yield stress material is presented. This yield condition, along with the mixed hardening and associated flow rules is used to formulate the constitutive laws for sides and corners of the yield surface. The formulation is based on incremental plasticity with the assumption of small displacements and is suitable for plane stress problems under monotonie and cyclic loading.

The mixed hardening rule, which is mathematically modeled, could be changed to either kinematic or isotropic hardening by a simple change in the model. This hardening rule could handle different degrees of Bauschinger effect, as opposed to kinematic hardening, which assumes only an ideal Bauschinger effect, or isotropic hardening, which does not account for the effect at all. The theory is applied to a ductile material using the finite element method and cyclic loading.     

Elastic stability of orthotropic plates using the conjugate load/displacement method

It is shown that the conjugate load/displacement method which has been used for the elastic stability analysis of isotropic plates can also be used for orthotropic plates. Results are presented for a range of plates of various rectangular configurations and boundary conditions, as well as for different values of the material modular ratios. A range of loadings is covered; in particular the case of concentrated loads is dealt with. For this loading case the conjugate load/displacement method has been found to be particularly useful.     

A finite element model for the nonlinear analysis of reinforced and prestressed masonry walls

A materially nonlinear layered finite element model is proposed for the analysis of reinforced and/or prestressed masonry wall panels under monotonie loadings in the plane and/or out of the plane, capable of evaluating both the serviceability load and the ultimate load. An orthotropic incrementally linear relationship and equivalent uniaxial concept are used to represent the behaviour of masonry under biaxial stresses while a uniaxial bilinear elasto-plastic model with hardening is employed for rebar and the so-called ‘power-formula’ is adopted to describe the stress-strain relationship of prestressing steel.

After cracking, the smeared coaxial rotating crack model is adopted and tension stiffening, reduction in compressive strength and stiffness after cracking, and strain softening in compression are accounted for. The modified Newton-Raphson iteration method is employed to ensure convergency of non linear solution.

The proposed finite element model has been tested by a comparison with experimental data available in literature, both for reinforced and prestressed wall panels. The analysis of results shows good agreement between the values obtained by the proposed model and those obtained experimentally.     

Analysis of cyclic loading of plane RC structures

A numerical procedure for cyclic loading response of planar reinforced concrete structures is presented. A nonlinear orthotropic stress strain law for biaxially loaded plain concrete is developed and compared with experimental results for monotonic biaxial loading and uniaxial cyclic loading. The stress-strain law recognizes strength and ductility changes due to biaxial stress, and strength and stiffness degradation with cycles of loading. The stress strain law is incorporated into a finite element computer program which utilizes isoparametric quadrilaterals with extra non-conforming deformation modes. Numerical and experimental results are presented for a monotonically loaded shear wall-frame system and a cyclically loaded shear wall.     

A linear yield surface in plastic cyclic analysis

A three-parameter, uniaxial symmetric, linear yield surface suitable for tension-weak, as well as equal tension and compression yield stress material, is presented. This yield condition, along with a linear kinematic hardening rule and an associated flow rule, is used to formulate the constitutive laws for sides and comers of the yield surface. The formulation is based on incremental plasticity with the assumption of small displacements and suitable for plane stress problems, under monotonie or cyclic loading. Finite element formulation, numerical solution, and applications are discussed.     

Stability analysis of skew orthotropic plates by the finite strip method

A stability analysis based on the Finite Strip Method is presented for skew orthotropic plates subjected to in-plane loadings. The straight sides of the plate are simply supported and the other two skewed sides are supported with any combination of fixed, free and simply supported boundaries. The plate is divided into strips, in contradistinction to elements in the Finite Element Method, and the displacement function is so chosen that it satisfies the boundary conditions and also the inter-strip compatibility conditions of an elemental strip. The energy expressions required to formulate the stiffness and stability coefficient matrices are formulated using smalldeflection theory. The buckling load intensity factor is evaluated for different aspect ratios of isotropic and orthotropic skew plates and the results of certain rectangular isotropic cases are compared with earlier investigations.     

A three-dimensional nonlinear analysis of cross-ply rectangular composite plates

The results of a three-dimensional, geometrically nonlinear, finite-element analysis of the bending of cross-ply laminated anisotropie composite plates are presented. Individual laminae are assumed to be homogeneous, orthotropic and linearly elastic. A fully three-dimensional isoparametric finite element with eight nodes (i.e. linear element) and 24 degrees of freedom (three displacement components per node) is used to model the laminated plate. The finite element results of the linear analysis are found to agree very well with the exact solutions of cross-ply laminated rectangular plates under sinusiodal loading. The finite element results of the three-dimensional, geometrically nonlinear analysis are compared with those obtained by using a shear deformable, geometrically nonlinear, plate theory. It is found that the deflections predicted by the shear deformable plate theory are in fair agreement with those predicted by three-dimensional elasticity theory; however stresses were found to be not in good agreement     

Equivalent orthotropic properties of corrugated sheets

The analysis of corrugated shells (plates) is based on the assumption that they can be analyzed as thin, equivalent orthotropic shells of uniform thickness. The analytical expressions for the equivalent rigidities of orthotropic thin shells given in the literature are reviewed. The results of a finite element analysis of a corrugated sheet subjected to constant strain states reveals an inadequacy in some of the classical expressions in use today. These equivalent orthotropic properties are improved with the derivation of new expressions. In addition, expressions for the localized stress concentrations developed at the ridges of the corrugations are derived.     

Approximation of the critical buckling factor for composite panels

This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented.     

Nonlinear axisymmetric static analysis of orthotropic thin circular plates with elastically restrained edge

This study deals with the nonlinear axisymmetric static analysis of elastic orthotropic thin circular plates with elastically restrained edges for rotation as well as in-plane displacement. Von-Karman equations have been employed and spatial discretisation has been done by using the method of orthogonal point collocation. Numerical results have been presented for orthotropic plates with hinged edge, simply supported edge, movable clamped edge and immovable clamped edge. Results are given for the whole range of elastic edge restraints from the movable simply supported edge to the immovable clamped edge. The effect of a prescribed inplane force or an inplane displacement at the edge, on the static response under a uniformly distributed transverse load has been investigated. Several new results are presented. The present results are in good agreement with the available results.     

Post-buckling of cylindrically orthotropic linearly tapered circular plates by finite element method

The post-buckling behavior of tapered circular plates with cylindrically orthotropic material properties is studied in this paper through a finite element formulation. The results in the form of an empirical formula for the radial load ratios are presented for various values of the taper parameter and orthotropy parameter. Both simply supported and clamped boundary conditions are considered.     

Design of energy dissipating elastoplastic structures under cyclic loads using topology optimization

A topology optimization approach for designing structures with maximum energy absorption capacity under cyclic loads is proposed. To simulate the Bauschinger effect in materials under cyclic loads, Prager and the Armstrong-Frederick kinematic hardening rules are considered together with the von Mises plasticity in the optimization process. Path-dependent sensitivities are derived analytically using the adjoint method, which are further verified by the central difference method. Effectiveness of the proposed approach is demonstrated on several examples. Results show that the optimized designs with kinematic hardening are remarkably different from the ones obtained with isotropic hardening and are highly dependent on the loading patterns.     

Vibration analysis of orthotropic rectangular plates using superelements

In recent years, development of the superelements has significantly influenced the analysis of structural dynamics. One of the common implementation of this element has been in the analysis of plates and shells. In this paper, application of superelements in the vibration analysis of orthotropic rectangular plates with different boundary conditions is considered, in which bending and in-plane displacements are also included. Results of this study indicate that there is a good agreement between single superelement and 625 elements using conventional methods     

Large deflection of clamped skew plates

A theoretical analysis is presented for the large deflection elastic behaviour of clamped, uniformly loaded orthotropic skew plates. The governing nonlinear partial differential equations are transformed into a set of nonlinear algebraic equations. For this purpose a network of discrete points over the domain is considered, and the integral equation of beams along the skew directions is used with appropriate boundary conditions. The resulting nonlinear algebraic equations are then solved by a Newton-Raphson procedure.A convergence study of the solution has been made, and numerical results for a few cases of orthotropy and skew angles are presented in graphical form. The results obtained by this method are compared with available results of other investigators.     

Estimation of dynamic buckling for composite columns with open cross-section

Local, global and interactive dynamic responses of thin-walled columns with open cross-sections subjected to pulse compressive loading of different shape are discussed.An analytical–numerical method has been proposed to analyse the dynamic buckling problem. The applied method is based on the asymptotic Koiter’s theory for conservative systems in the second-order approximation. In order to obtain equations of column walls, the non-linear theory of orthotropic thin-walled plates has been modified in such a way that additionally it accounts for all components of inertia forces.The results of calculations obtained with the analytical–numerical method have been compared with those attained from the finite element method. The results are in good agreement.     

Static analysis of orthotropic curved bridge decks

The static analysis of orthotropic curved bridge decks with two radial edges simply supported and other two curved edges free are analysed, using a combination of Fourier series and finite difference technique. The governing fourth order partial differential equation of orthotropic plates in polar co-ordinates is first converted to an ordinary differential equation and subsequently solved by finite difference method. In order to estimate the accuracy and reliability of the present formulation an example of curved plate is solved and the results are compared with the existing solutions.     

Post-buckling behaviour of cylindrically orthotropic annular plates with uniform internal radial load

A finite element formulation for the study of the post-buckling behaviour of cylindrically orthotropic annular plates with uniform internal radial load is presented in this paper. The results for radial load ratios are presented in the form of empirical formulae in terms of the central deflection of the annulus to the thickness ratios of the plates for the first time in the literature for various values of the orthotropic parameters and radii ratios.