• Structural Engineering and Mechanics
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• v.18 no.6
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• pp.807-829
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• 2004

The Element-Based Lagrangian Formulation of a 9-node resultant-stress shell element is presented for the isotropic and anisotropic composite material. The effect of the coupling term between the bending strain and displacement has been investigated in the warping problem. The strains, stresses and constitutive equations based on the natural co-ordinate have been used throughout the Element-Based Lagrangian Formulation of the present shell element which offers an advantage of easy implementation compared with the traditional Lagrangian Formulation. The element is free of both membrane and shear locking behavior by using the assumed natural strain method such that the element performs very well in thin shell problems. In composite plates and shells, the transverse shear stiffness is defined by an equilibrium approach instead of using the shear correction factor. The arc-length control method is used to trace complex equilibrium paths in thin shell applications. Several numerical analyses are presented and discussed in order to investigate the capabilities of the present shell element. The results showed very good agreement compared with well-established formulations in the literature.

• 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.

• Structural Engineering and Mechanics
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• v.8 no.6
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• pp.623-637
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• 1999

This paper presents a four-noded quadrilateral $C^0$ strain plate element for the analysis of thick laminated composite plates. The element formulation is based on: 1) the third-order shear deformation theory; 2) assumed strain element formulation; and 3) interrelated edge displacements and rotations along element boundaries. Unlike the existing displacement-type composite plate elements based on the third-order theory, which rely on the $C^1$-continuity formulation, the present plate element is of $C^0$-continuity, and its element stiffness matrix is evaluated explicitly. Because of the third-order expansion of the in-plane displacements through the thickness, the resulting theory and hence elements do not need shear correction factors. The explicit element stiffness matrix makes the present element more computationally efficient than the composite plate elements using numerical integration for the analysis of thick layered composite plates.

• Advances in aircraft and spacecraft science
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• v.1 no.2
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• pp.125-143
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• 2014

This paper provides a new technique for solving the static analysis of arbitrarily shaped composite plates by using Strong Formulation Finite Element Method (SFEM). Several papers in literature by the authors have presented the proposed technique as an extension of the classic Generalized Differential Quadrature (GDQ) procedure. The present methodology joins the high accuracy of the strong formulation with the versatility of the well-known Finite Element Method (FEM). The continuity conditions among the elements is carried out by the compatibility or continuity conditions. The mapping technique is used to transform both the governing differential equations and the compatibility conditions between two adjacent sub-domains into the regular master element in the computational space. The numerical implementation of the global algebraic system obtained by the technique at issue is easy and straightforward. The main novelty of this paper is the application of the stress and strain recovery once the displacement parameters are evaluated. Computer investigations concerning a large number of composite plates have been carried out. SFEM results are compared with those presented in literature and a perfect agreement is observed.

• Interaction and multiscale mechanics
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• v.6 no.2
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• pp.237-255
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• 2013

In this paper, a meshfree-enriched finite element method (ME-FEM) is introduced for the large deformation analysis of nonlinear path-dependent problems involving contact. In linear ME-FEM, the element formulation is established by introducing a meshfree convex approximation into the linear triangular element in 2D and linear tetrahedron element in 3D along with an enriched meshfree node. In nonlinear formulation, the area-weighted smoothing scheme for deformation gradient is then developed in conjunction with the meshfree-enriched element interpolation functions to yield a discrete divergence-free property at the integration points, which is essential to enhance the stress calculation in the stage of plastic deformation. A modified variational formulation using the smoothed deformation gradient is developed for path-dependent material analysis. In the industrial metal forming problems, the mortar contact algorithm is implemented in the explicit formulation. Since the meshfree-enriched element shape functions are constructed using the meshfree convex approximation, they pose the desired Kronecker-delta property at the element edge thus requires no special treatments in the enforcement of essential boundary condition as well as the contact conditions. As a result, this approach can be easily incorporated into a conventional displacement-based finite element code. Two elasto-plastic problems are studied and the numerical results indicated that ME-FEM is capable of delivering a volumetric locking-free and pressure oscillation-free solutions for the large deformation problems in metal forming analysis.

• Structural Engineering and Mechanics
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• v.69 no.6
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• pp.615-626
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• 2019

The 3-noded metric Timoshenko beam element with an offset of the internal node from the element centre is used here to demonstrate the best-fit paradigm using function space formulation under locking and mesh distortion. The best-fit paradigm follows from the projection theorem describing finite element analysis which shows that the stresses computed by the displacement finite element procedure are the best approximation of the true stresses at an element level as well as global level. In this paper, closed form best-fit solutions are arrived for the 3-noded Timoshenko beam element through function space formulation by combining field consistency requirements and distortion effects for the element modelled in metric Cartesian coordinates. It is demonstrated through projection theorems how lock-free best-fit solutions are arrived even under mesh distortion by using a consistent definition for the shear strain field. It is shown how the field consistency enforced finite element solution differ from the best-fit solution by an extraneous response resulting from an additional spurious force vector. However, it can be observed that when the extraneous forces vanish fortuitously, the field consistent solution coincides with the best-fit strain solution.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.14 no.2
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• pp.67-78
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• 2010

An elastic field composed of symmetric cross-ply laminated material is analyzed in roller guided panel. The plane stress elasticity problem is formulated in terms of two displacement parameters with mixed boundary conditions. The numerical solution for two displacement parameters is obtained using a finite element method considering a panel of glass/epoxy laminated composite. Some components of stress and displacement at different sections of panel are displayed. The results makes sure that the formulation developed in this study can be applied to analyze the characteristics of elastic field made of laminated composite under any boundary conditions.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.22 no.4
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• pp.344-351
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• 2012

A disc spring consists of a conical disc. A load-displacement formula was newly developed in the form of energy method to consider both rigid and friction. The cross section of the disc spring has identical slope angles at the bottom of conical. To solve such a problem, an energy method calculation is proposed. To achieve the goal of this study, the proposed calculation is extended to a disc spring with friction. A firm basis background study based on Almen's work is presented in developing a new numerical approach to predict the available formulation for a disc spring with friction.

• Structural Engineering and Mechanics
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• v.51 no.1
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• pp.163-180
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• 2014

In the displacement based finite element analysis of composite beams that consist of two Euler-Bernoulli beams juxtaposed with a deformable shear connection, the coupling of the displacement fields may cause oscillations in the interlayer slip field and reduction in optimal convergence rate, known as slip-locking. In this study, the B-bar procedure is proposed to alleviate the locking effects. It is also shown that by changing the primary dependent variables in the mathematical model, to be able to interpolate the interlayer slip field directly, oscillations in the slip field can be completely eliminated. Examples are presented to illustrate the performance and the numerical characteristics of the proposed methods.

• Structural Engineering and Mechanics
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• v.13 no.3
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• pp.261-276
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• 2002

A novel, 6-node, two-dimensional mixed finite element (FE) model has been developed to analyze laminated composite beams by using the minimum potential energy principle. The model has been formulated by considering four degrees of freedom (two displacement components u, w and two transverse stress components ${\sigma}_z$, $\tau_{xz}$) per node. The transverse stress components have been invoked as nodal degrees of freedom by using the fundamental elasticity equations. Thus, the present mixed finite element model not only ensures the continuity of transverse stress and displacement fields through the thickness of the laminated beams but also maintains the fundamental elasticity relationship between the components of stress, strain and displacement fields throughout the elastic continuum. This is an important feature of the present formulation, which has not been observed in various mixed formulations available in the literature. Results obtained from the model have been shown to be in excellent agreement with the elasticity solutions for thin as well as thick laminated composite beams. A few results for a cross-ply beam under fixed support conditions are also presented.