A new discrete Kirchhoff-Mindlin element based on Mindlin–Reissner plate theory and assumed shear strain fields—part II: An extended DKQ element for thick-plate bending analysis

This is the second part of a two-part paper on plate bending elements with shear effects included. This paper presents a new four-node, 12-d.o.f. quadrilateral plate bending element valid for the analysis of thick to thin plates. The element called DKMQ, has a proper rank (contains no spurious zero-energy modes), passes the patch test for thin and thick plates in an arbitrary mesh and is free of shear locking. Very good results have been obtained for thin and thick plates by the element. An extended DKT element for thick-plate bending analysis is evaluated in Part I.¹⁹     

A pure bending exact nodal-averaged shear strain method for finite element plate analysis

An averaged shear strain method, based on a nodal integration approach, is presented for the finite element analysis of Reissner–Mindlin plates. In this work, we combine the shear interpolation method from the MITC4 plate element with an area-weighted averaging technique for the nodal integration of shear energy to relieve shear locking in the thin plate analysis as well as to pass the pure bending patch test. In order to resolve the numerical instability caused by the direct nodal integration, the bending strain field is computed by a sub-domain nodal integration approach based on the Sub-domain Stabilized Conforming Integration and a modified curvature smoothing scheme. The resulting nodally integrated smoothed strain formulation is shown to contain only the primitive variables and thus can be easily implemented in the existing displacement-based finite element plate formulation. Several numerical examples are presented to demonstrate the accuracy of the present method.     

Efficient and accurate four-node quadrilateral C0 plate bending element based on assumed strain fields

Based on the assumed element strain fields and the interrelated displacement-rotation interpolations, a four-node (12 dof) quadrilateral C⁰ finite element, designated as QCCP-2, for the analysis of thick/thin plates is developed in this paper. The four-node C⁰ plate element presented here possesses a linear bending strain field, and the element stiffness matrices are given explicitly. Therefore, the present four-node C⁰ plate element is more efficient and accurate than the existing four-node C⁰ plate elements where the constant strain stiffness matrices are obtained by numerical integration. By the use of the interrelated displacement-rotation interpolations, QCCP-2 is capable of automatically satisfying the Kirchhoff assumption for the case of thin plates. Consequently, QCCP-2 is not only free of shear locking, but also free from the numerical ill-conditioning. Furthermore, QCCP-2 passes the patch test of thin plates. The four-node quadrilateral C⁰ elements presented here can automatically reduce to the corresponding three-node triangular elements. Several numerical examples are given to demonstrate the efficiency and accuracy of the C⁰ plate bending element QCCP-2.     

A new element used in the non-orthogonal boundary plate bending theory—an arbitrarily quadrilateral element

This paper develops an arbitrarily quadrilateral element to analyse bending problems of plates with non-orthogonal boundaries. A second-order Jacobian matrix for the co-ordinate transformation and an explicit form of its inverse matrix are described in detail. A shape function matrix [ N ] for the plate element of arbitrarily quadrilateral configuration, an equivalent load vector { R }, a strain matrix [ B ] and element stiffness matrix are given. Finally, four illustrated examples are given and the results of computation are compared with those from other analytical methods.     

A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation

This communication discusses a 4-node plate bending element for linear elastic analysis which is obtained, as a special case, from a general nonlinear continuum mechanics based 4-node shell element formulation. The formulation of the plate element is presented and the results of various example solutions are given that yield insight into the predictive capability of the plate (and shell) element.     

A new discrete Kirchhoff quadrilateral element based on the third-order theory for composite plates

A new discrete Kirchhoff quadrilateral element based on the refined third-order theory is developed for the analysis of composite plates. The element has seven degrees of freedom per node, namely, the three displacements, two rotations and two transverse shear strain components at the mid-surface. The inplane displacements and the shear strains are interpolated using bilinear interpolation functions and the mid-surface rotations are interpolated using bi-quadratic functions based on the discrete Kirchhoff technique. The element stiffness matrix and the consistent load vector are developed using the principle of virtual work. The finite element formulation is validated by comparing the results for simply-supported plate with the analytical Navier solution. Comparison of the present results with those using other available elements based on the TOT establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The element is free from shear locking     

A quadrilateral hybrid-Trefftz plate bending element for the inclusion of warping based on a three-dimensional plate formulation

A plate formulation, for the inclusion of warping and transverse shear deformations, is considered. From a complete thick and thin plate formulation, which was derived without ad hoc assumptions from the three-dimensional equations of elasticity for isotropic materials, the bending solution, involving powers of the thickness co-ordinate z, is used for constructing a quadrilateral finite plate bending element. The constructed element trial functions, for the displacements and stresses, satisfy, a priori, the three-dimensional Navier equations and equilibrium equations, respectively. For the coupling of the elements, independently assumed functions on the boundary are used. High accuracy for both displacements and stresses (including transverse shear stresses) can be achieved with rather coarse meshes for thick and thin plates.     

A degenerate solid element for linear fracture analysis of plate bending and general shells

A general curved element of arbitrary shape for both thick and thin shells is proposed for the linear fracture analysis of a through crack in a shell or a plate. The element is derived from a degenerate 20-noded solid isoparametric element using reduced integration technique. The 1/√(r) singularity of the strains is obtained by the same procedure proposed earlier for two- and three-dimensional problems,^1,2 viz. by placing the mid-side nodes near the crack at the quarter points. Several illustrated examples ranging from classical solutions to practical problems are given to assess the accuracy of solution attainable.     

Crack tip stress fields for thin,cracked plates in bending,shear and twisting: A comparison of plate theory and three-dimensional elasticity theory solutions

A three-dimensional finite element study of crack tip fields in thin plates under bending, shearing, and twisting loads is carried out to study the relation of the plate theory crack tip fields to the actual, three dimensional crack tip fields. In the region r>0.5h the Kirchhoff theory is a good approximation of the three dimensional stress fields for symmetric plate bending. The Reissner theory gives a good approximation in the region r<0.1h. Similar results are found for the shear and twisting problems, although for pure shear loading, the Kirchhoff theory is a good approximation somewhat farther r>h from the crack tip than in the bending problem. In the case of shear loading the near tip out-of-plane shear stresses do not vary quadratically through the thickness as in plate theory, but are nearly constant, except in the neighborhood of the free surface. Quadratic variation, as predicted by plate theory, is observed for r>h. Energy release rates based on the Kirchhoff and Reissner theories agree well with those computed by means of three dimensional finite element analyses.     

A finite element formulation based on Nadai's deformation theory for elasto-plastic analysis

A numerical method of the Newton-Raphson type is presented for elasto-plastic analysis using the finite element method. The method is developed from Nadai's deformation theory and Hooke's law. Numerical examples are used to show that the method provides very rapid solution convergence.     

An efficient FE model based on combined theory for the analysis of soft core sandwich plate

An efficient C⁰ continuous finite element (FE) model is developed based on combined theory (refine higher order shear deformation theory (RHSDT) and least square error (LSE) method) for the static analysis of soft core sandwich plate. In this (RHSDT) theory, the in-plane displacement field for the face sheets and the core is obtained by superposing a global cubically varying displacement field on a zig-zag linearly varying displacement field with a different slope in each layer. The transverse displacement assumes to have a quadratic variation within the core and it remains constant in the faces beyond the core. The proposed model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the sandwich plate. The nodal field variables are chosen in an efficient manner to circumvent the problem of C¹ continuity requirement of the transverse displacements. In order to calculate the accurate through thickness transverse stresses variation, LSE method has been used at the post processing stage. The proposed combine model (RHSDT and LSE) is implemented to analyze the laminated composites and sandwich plates. Many new results are also presented which should be useful for future research.     

A hybrid‐mixed four‐node quadrilateral plate element based on sampling surfaces method for 3D stress analysis

The hybrid‐mixed assumed natural strain four‐node quadrilateral element using the sampling surfaces (SaS) technique is developed. The SaS formulation is based on choosing inside the plate body N not equally spaced SaS parallel to the middle surface in order to introduce the displacements of these surfaces as basic plate variables. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the thickness direction permits the presentation of the plate formulation in a very compact form. The SaS are located at Chebyshev polynomial nodes that allow one to minimize uniformly the error due to the Lagrange interpolation. To avoid shear locking and have no spurious zero energy modes, the assumed natural strain concept is employed. The developed hybrid‐mixed four‐node quadrilateral plate element passes patch tests and exhibits a superior performance in the case of coarse distorted mesh configurations. It can be useful for the 3D stress analysis of thin and thick plates because the SaS formulation gives the possibility to obtain solutions with a prescribed accuracy, which asymptotically approach the 3D exact solutions of elasticity as the number of SaS tends to infinity. Copyright © 2016 John Wiley & Sons, Ltd.     

A generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates based on isogeometric analysis

This paper presents a generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates. We exploit a higher-order shear deformation theory in each layer such that the continuity of the displacement and transverse shear stresses at the layer interfaces is ensured. Thanks for enforcing the continuity of the displacement and transverse shear stresses at an inner-laminar layer, the minimum number of variables is retained from the present theory in comparison with other layerwise theories. The method requires only five variables, the same as what obtained from the first- and higher-order shear deformation theories. In comparison with the shear deformation theories based on the equivalent single layer, the present theory is capable of producing a higher accuracy for inner-laminar layer shear stresses. The free boundary conditions of transverse shear stresses at the top and bottom surfaces of the plate are fulfilled without any shear correction factors. The discrete system equations are derived from the Galerkin weak form, and the solution is obtained by isogeometric analysis (IGA). The discrete form requires the C¹ continuity of the transverse displacement, and hence NURBS basis functions in IGA naturally ensure this condition. The laminated composite and sandwich plates with various geometries, aspect ratios, stiffness ratios and boundary conditions are studied. The obtained results are compared with the 3D elasticity solution, the analytical as well as numerical solutions based on various plate theories.     

Development of new spatially curved frame finite element for crash analysis part 1: Formulation and validation

This paper deals with the development of a new space curved frame finite element to be used for crash analysis (non-linear). The formulation has been validated for problems of large deflection and rotation, and for problems involving initially curved members. Based on the validation performed, it is expected that crash problems may be modeled using a single element per member thus retaining computational efficiency while performing an accurate analysis.The support of this work by the FAA, under a grant to the Center of Excellence for Computational Modeling     

A posteriori error analysis and adaptive processes in the finite element method: Part I—error analysis

This is a paper presented in two parts dealing respectively with error analysis and adaptive processes applied to finite element calculations. Part I contains the basic theory and methods of deriving error estimates for second-order problems. Part II of the paper deals with the strategy for adaptive refinement and concentrates on the p-convergent methods. It is shown that an extremely high rate of convergence is reached in practical problems using such procedures. Applications to realistic stress analysis and potential problems are presented.     

A contact algorithm for 3D discrete and finite element contact problems based on penalty function method

A contact algorithm in the context of the combined discrete element (DE) and finite element (FE) method is proposed. The algorithm, which is based on the node-to-surface method used in finite element method, treats each spherical discrete element as a slave node and the surfaces of the finite element domain as the master surfaces. The contact force on the contact interface is processed by using a penalty function method. Afterward, a modification of the combined DE/FE method is proposed. Following that, the corresponding numerical code is implemented into the in-house developed code. To test the accuracy of the proposed algorithm, the impact between two identical bars and the vibration process of a laminated glass plate under impact of elastic sphere are simulated in elastic range. By comparing the results with the analytical solution and/or that calculated by using LS-DYNA, it is found that they agree with each other very well. The accuracy of the algorithm proposed in this paper is proved.     

A boundary element formulation based on the three-dimensional elastostatic fundamental solution for the infinite layer: Part II—Three-dimensional examples

This work applies the specialization of the integral identities used in the boundary element method to the numerical solution of three-dimensional elasticity problems involving geometries containing two parallel planar surfaces (see Part I this issue). Two three-dimensional problems are numerically analysed by using the above procedure. These are the problems of pressurized circular and elliptical holes in infinite plates of uniform thickness. For the circular hole problem, the accuracy of our scheme is established by direct comparison of our results with the available analytical solution. For the ellipse problem, with an aspect ratio of 4:1, the boundary element results are compared with those of a finite element calculation.