Numerical results for mimetic discretization of Reissner-Mindlin plate problems

A low-order mimetic finite difference (MFD) method for Reissner-Mindlin plate problems is considered. Together with the source problem, the free vibration and the buckling problems are investigated. Full details about the scheme implementation are provided, and the numerical results on several different types of meshes are reported

Laminated Beam Analysis by Polynomial, rigonometric, Exponential and Zig-Zag Theories

A number of refined beam theories are discussed in this paper. These theories were obtained by expanding the unknown displacement variables over the beam section axes by adopting Taylor's polynomials, trigonometric series, exponential, hyperbolic and zig-zag functions. The Finite Element method is used to derive governing equations in weak form. By using the Unified Formulation introduced by the first author, these equations are written in terms of a small number of fundamental nuclei, whose forms do not depend on the expansions used. The results from the different models considered are compared in terms of displacements, stress and degrees of freedom (DOFs). Mechanical tests for thick laminated beams are presented in order to evaluate the capability of the finite elements. They show that the use of various different functions can improve the performance of the higher-order theories by yielding satisfactory results with a low computational cost

ISOGEOMETRIC ANALYSIS AND PATCHWISE REPRODUCING POLYNOMIAL PARTICLE METHOD FOR PLATES

Isogeometric analysis (IGA) ([8, 16, 27]) is designed to combine two tasks, design by Computer Aided Design (CAD) and Finite Element Analysis (FEA), so that it drastically reduces the error in the representation of the computational domain and the re-meshing by the use of “exact” CAD geometry directed at the coarsest level of discretization. This is achieved by using B-splines or non-uniform rational B-splines (NURBS) for the description of geometries as well as for the representation of unknown solution fields. In order to handle the singularities arising in the PDEs, Babu?ska and Oh [7] introduced mapping techniques, called the Method of Auxiliary Mapping (MAM), into conventional p-version of Finite Element Methods (FEM). In a similar spirit to MAM, it is possible to construct a novel NURBS geometrical mapping that generates singular functions resembling the singularities. The proposed mapping technique is concerned with constructions of unconventional novel geometrical mappings by which push-forward of B-spline functions defined on the parameter space generates singular functions in a physical domain that resemble the given point singularities. In other words, the pull-back of the singularity into the parameter space by the non standard NURBS mapping becomes highly smooth. However, the mapping technique is not able to handle in the framework of IGA. Thus, we consider how to use the proposed mapping method in IGA of elliptic prob- lems and elasticity containing singularities without changing the design mapping. For this end, we embed the mapping method into the standard IGA that uses NURBS basis functions for which h - p - k-refinements are applicable for improved computational solution. In other words, the mapping method will be used to enrich NURBS basis functions around neighborhood of singularities so that they can capture singular behaviors of the solution to be approximated. Finally, Reproducing Polynomial Particle Method (RPPM) is one of meshless methods that use meshes minimally or do not use meshes at all. In this disserta- tion, the RPPM is employed for free vibration and buckling of the first order shear deformation model (FSDT), called the Reissner-Mindlin plate, and for analysis of boundary layer of the Reissner-Mindlin plate. For numerical implementation, we use flat-top partition of unity functions, introduced by Oh et al, and patchwise RPPM in which approximation functions have high order polynomial reproducing property and shape functions satisfying the Kronecker delta property. Also, we demonstrate that our method is more effective than other existing methods in dealing with Reissner- Mindlin plates with various material properties and boundary conditions

Development of the DKMQ element for buckling analysis of shear-deformable plate bending

In this paper the discrete-Kirchhoff Mindlin quadrilateral (DKMQ) element was developed for buckling analysis of plate bending including the shear deformation. In this development the potential energy corresponding to membrane stresses was incorporated in the Hu-Washizu functional. The bilinear approximations for the deflection and normal rotations were used for the membrane stress term in the functional, while the approximations for the remaining terms remain the same as in static analysis. Numerical tests showed that the element has good predictive capability for thin plates. For thick plates, however, the element tends to give a slightly lower solution

Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory

In this paper, we present an effectively numerical approach based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) for geometrically nonlinear analysis of laminated composite plates. The HSDT allows us to approximate displacement field that ensures by itself the realistic shear strain energy part without shear correction factors. IGA utilizing basis functions namely B-splines or non-uniform rational B-splines (NURBS) enables to satisfy easily the stringent continuity requirement of the HSDT model without any additional variables. The nonlinearity of the plates is formed in the total Lagrange approach based on the von-Karman strain assumptions. Numerous numerical validations for the isotropic, orthotropic, cross-ply and angle-ply laminated plates are provided to demonstrate the effectiveness of the proposed method

Buckling and vibration analysis of laminated composite plate/shell structures via a smoothed quadrilateral flat shell element with in-plane rotations

This paper presents buckling and free vibration analysis of composite plate/shell structures of various shapes, modulus ratios, span-to-thickness ratios, boundary conditions and lay-up sequences via a novel smoothed quadrilateral flat element. The element is developed by incorporating a strain smoothing technique into a flat shell approach. As a result, the evaluation of membrane, bending and geometric stiffness matrices are based on integration along the boundary of smoothing elements, which leads to accurate numerical solutions even with badly-shaped elements. Numerical examples and comparison with other existing solutions show that the present element is efficient, accurate and free of locking

Free vibrations of laminated composite elliptic plates

The free vibrations are studied of laminated anisotropic elliptic plates with clamped edges. The analytical formulation is based on a Mindlin-Reissner type plate theory with the effects of transverse shear deformation, rotary inertia, and bending-extensional coupling included. The frequencies and mode shapes are obtained by using the Rayleigh-Ritz technique in conjunction with Hamilton's principle. A computerized symbolic integration approach is used to develop analytic expressions for the stiffness and mass coefficients and is shown to be particularly useful in evaluating the derivatives of the eigenvalues with respect to certain geometric and material parameters. Numerical results are presented for the case of angle-ply composite plates with skew-symmetric lamination