• Structural Engineering and Mechanics
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• v.26 no.6
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• pp.725-739
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• 2007

In recent years there are many plate bending elements that emerged for solving both thin and thick plates. The main features of these elements are that they are based on mix formulation interpolation with discrete collocation constraints. These elements passed the patch test for mix formulation and performed well for linear analysis of thin and thick plates. In this paper a member of this family of elements, namely, the Discrete Reissner-Mindlin (DRM) is further extended and developed to analyze both thin and thick plates with geometric nonlinearity. The Von K$\acute{a}$rm$\acute{a}$n's large displacement plate theory based on Lagrangian coordinate system is used. The Hu-Washizu variational principle is employed to formulate the stiffness matrix of the geometrically Nonlinear Discrete Reissner-Mindlin (NDRM). An iterative-incremental procedure is implemented to solve the nonlinear equations. The element is then tested for plates with simply supported and clamped edges under uniformly distributed transverse loads. The results obtained using the geometrically NDRM element is then compared with the results of available analytical solutions. It has been observed that the NDRM results agreed well with the analytical solutions results. Therefore, it is concluded that the NDRM element is both reliable and efficient in analyzing thin and thick plates with geometric non-linearity.

• Computers and Concrete
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• v.31 no.3
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• pp.223-239
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• 2023

Geometric nonlinear performance simulation and analysis of complex modern buildings and industrial products require high-performance shell elements. Balancing multiple aspects of performance in the one geometric nonlinear analysis element remains challenging. We present a new shell element, flat shell DKMGQ-CR (Co-rotational Discrete Kirchhoff-Mindlin Generalized Conforming Quadrilateral), for linear and geometric nonlinear analysis of both thick and thin shells. The DKMGQ-CR shell element was developed by combining the advantages of high-performance membrane and plate elements in a unified coordinate system and introducing the co-rotational formulation to adapt to large deformation analysis. The effectiveness of linear and geometric nonlinear analysis by DKMGQ-CR is verified through the tests of several classical numerical benchmarks. The computational results show that the proposed new element adapts to mesh distortion and effectively alleviates shear and membrane locking problems in linear and geometric nonlinear analysis. Furthermore, the DKMGQ-CR demonstrates high performance in analyzing thick and thin shells. The proposed element DKMGQ-CR is expected to provide an accurate, efficient, and convenient tool for the geometric nonlinear analysis of shells.

• Steel and Composite Structures
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• v.34 no.2
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• pp.227-239
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• 2020

The aim of this study is to obtain the nonlinear and post-buckling responses of relatively thick functionally graded plates with oblique elliptical cutouts using a new semi-analytical approach. To model the oblique elliptical hole in a FGM plate, six plate-elements are used and the connection between these elements is provided by the well-known Penalty method. Therefore, the semi-analytical technique used in this paper is known as the plate assembly technique. In order to take into account for functionality of the material in a perforated plate, the volume fraction of the material constituents follows a simple power law distribution. Since the FGM perforated plates are relatively thick in this research, the structural model is assumed to be the first order shear deformation theory and Von-Karman's assumptions are used to incorporate geometric nonlinearity. The equilibrium equations for FGM plates containing elliptical holes are obtained by the principle of minimum of total potential energy. The obtained nonlinear equilibrium equations are solved numerically using the quadratic extrapolation technique. Various sets of boundary conditions for FGM plates and different cutout sizes and orientations are assumed here and their effects on nonlinear response of plates under compressive loads are examined.

• Structural Engineering and Mechanics
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• v.66 no.5
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• pp.557-568
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• 2018

An investigation is made in the present work on the post-buckling and geometrically nonlinear behaviors of moderately thick perfect and imperfect rectangular plates made-up of functionally graded materials. Spectral collocation approach based on Legendre basis functions is developed to analyze the functionally graded plates while they are subjected to end-shortening strain. The material properties in this study are varied through the thickness according to the simple power law distribution. The fundamental equations for moderately thick rectangular plates are derived using first order shear deformation plate theory and taking into account both geometric nonlinearity and initial geometric imperfections. In the current study, the domain of interest is discretized with Legendre-Gauss-Lobatto nodes. The equilibrium equations will be obtained by discretizing the Von-Karman's equilibrium equations and also boundary conditions with finite Legendre basis functions that are substituted into the displacement fields. Due to effect of geometric nonlinearity, the final set of equilibrium equations is nonlinear and therefore the quadratic extrapolation technique is used to solve them. Since the number of equations in this approach will always be more than the number of unknown coefficients, the least squares technique will be used. Finally, the effects of boundary conditions, initial geometric imperfection and material properties are investigated and discussed to demonstrate the validity and capability of proposed method.

• Structural Engineering and Mechanics
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• v.68 no.5
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• pp.603-619
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• 2018

In this research, an effective computational technique is carried out for nonlinear and post-buckling analyses of cracked imperfect composite plates. The laminated plates are assumed to be moderately thick so that the analysis can be carried out based on the first-order shear deformation theory. Geometric non-linearity is introduced in the way of von-Karman assumptions for the strain-displacement equations. The Ritz technique is applied using Legendre polynomials for the primary variable approximations. The crack is modeled by partitioning the entire domain of the plates into several sub-plates and therefore the plate decomposition technique is implemented in this research. The penalty technique is used for imposing the interface continuity between the sub-plates. Different out-of-plane essential boundary conditions such as clamp, simply support or free conditions will be assumed in this research by defining the relevant displacement functions. For in-plane boundary conditions, lateral expansions of the unloaded edges are completely free while the loaded edges are assumed to move straight but restricted to move laterally. With the formulation presented here, the plates can be subjected to biaxial compressive loads, therefore a sensitivity analysis is performed with respect to the applied load direction, along the parallel or perpendicular to the crack axis. The integrals of potential energy are numerically computed using Gauss-Lobatto quadrature formulas to get adequate accuracy. Then, the obtained non-linear system of equations is solved by the Newton-Raphson method. Finally, the results are presented to show the influence of crack length, various locations of crack, load direction, boundary conditions and different values of initial imperfection on nonlinear and post-buckling behavior of laminates.

• Structural Engineering and Mechanics
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• v.45 no.5
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• pp.677-691
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• 2013

A semi-analytical finite strip method is developed for analyzing the post-buckling behavior of rectangular composite laminated plates of arbitrary lay-up subjected to progressive end-shortening in their plane and to normal pressure loading. In this method, all the displacements are postulated by the appropriate harmonic shape functions in the longitudinal direction and polynomial interpolation functions in the transverse direction. Thin or thick plates are assumed and correspondingly the Classical Plate Theory (CPT) or Higher Order Plate Theory (HOPT) is applied. The in-plane transverse deflection is allowed at the loaded ends of the plate, whilst the same deflection at the unloaded edges is either allowed to occur or completely restrained. Geometric non-linearity is introduced in the strain-displacement equations in the manner of the von-Karman assumptions. The formulations of the finite strip methods are based on the concept of the principle of the minimum potential energy. The Newton-Raphson method is used to solve the non-linear equilibrium equations. A number of applications involving isotropic plates, symmetric and unsymmetric cross-ply laminates are described to investigate the through-thickness shearing effects as well as the effect of pressure loading, end-shortening and boundary conditions. The study of the results has revealed that the response of the composite laminated plates is particularly influenced by the application of the Higher Order Plate Theory (HOPT) and normal pressure loading. In the relatively thick plates, the HOPT results have more accuracy than CPT.

• Coupled systems mechanics
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• v.8 no.5
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• pp.439-457
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• 2019

In this paper, a new application of a four variable refined plate theory to analyze the nonlinear bending of functionally graded plates exposed to thermo-mechanical loadings, is presented. This recent theory is based on the assumption that the transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces, and similarly, the shear components do not contribute toward bending moments. The derived transverse shear strains has a quadratic variation across the thickness that satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The material properties are assumed to vary continuously through the thickness of the plate according to a power-law distribution of the volume fraction of the constituents. The solutions are achieved by minimizing the total potential energy. The non-linear strain-displacement relations in the von Karman sense are used to derive the effect of geometric non-linearity. It is concluded that the proposed theory is accurate and simple in solving the nonlinear bending behavior of functionally graded plates.

• Steel and Composite Structures
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• v.29 no.6
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• pp.785-802
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• 2018

In this paper, two different computational methods, called Rayleigh-Ritz and collocation are developed to estimate the ultimate strength of composite plates. Progressive damage behavior of moderately thick composite laminated plates is studied under in-plane compressive load and uniform lateral pressure. The formulations of both methods are based on the concept of the principle of minimum potential energy. First order shear deformation theory and the assumption of large deflections are used to develop the equilibrium equations of laminated plates. Therefore, Newton-Raphson technique will be used to solve the obtained system of nonlinear algebraic equations. In Rayleigh-Ritz method, two degradation models called complete and region degradation models are used to estimate the degradation zone around the failure location. In the second method, a new energy based collocation technique is introduced in which the domain of the plate is discretized into the Legendre-Gauss-Lobatto points. In this new method, in addition to the two previous models, the new model named node degradation model will also be used in which the material properties of the area just around the failed node are reduced. To predict the failure location, Hashin failure criteria have been used and the corresponding material properties of the failed zone are reduced instantaneously. Approximation of the displacement fields is performed by suitable harmonic functions in the Rayleigh-Ritz method and by Legendre basis functions (LBFs) in the second method. Finally, the results will be calculated and discussions will be conducted on the methods.