Strain based triangular finite element for plate bending analysis

ABSTRACT

In this work, the formulation of a new triangular finite element is presented for static and free vibration of plate bending. The developed element which contains the three essential external degrees of freedom at each of the three corner nodes is based on the Reissner/Mindlin theory and the strain-based approach. This element is based on the linear variation of the three bending strains and constant transverse shear strains. The present element performances are evaluated through several tests related to moderated thick and thin plates with various shapes where it is found to be numerically more efficient than the bilinear element.     

A low‐order,hexahedral finite element for modelling shells

A thin, eight‐node, tri‐linear displacement, hexahedral finite element is the starting point for the derivation of a constant membrane stress resultant, constant bending stress resultant shell finite element. The derivation begins by introducing a Taylor series expansion for the stress distribution in the isoparametric co‐ordinates of the element. The effect of the Taylor series expansion for the stress distribution is to explicitly identify those strain modes of the element that are conjugate to the mean or average stress and the linear variation in stress. The constant membrane stress resultants are identified with the mean stress components, and the constant bending stress resultants are identified with the linear variation in stress through the thickness along with in‐plane linear variations of selected components of the transverse shear stress. Further, a plane‐stress constitutive assumption is introduced, and an explicit treatment of the finite element's thickness is introduced. A number of elastic simulations show the useful results that can be obtained (tip‐loaded twisted beam, point‐loaded hemisphere, point‐loaded sphere, tip‐loaded Raasch hook, and a beam bent into a ring). All of the gradient/divergence operators are evaluated in closed form providing unequivocal evaluations of membrane and bending strain rates along with the appropriate divergence calculations involving the membrane stress and bending stress resultants. The fact that a hexahedral shell finite element has two distinct surfaces aids sliding interface algorithms when a shell folds back on itself when subjected to large deformations. Published in 2004 by John Wiley & Sons, Ltd.     

A simple method for including shear deformations in thin plate elements

A method is described for including the effect of shear deformations in existing thin plate finite elements, and thereby extending their range of application to include moderately thick plates. The method does not add extra degrees of freedom to the final element, so the thick and thin plate elements can be used interchangeably, and the thick plate solution is not appreciably more expensive than the thin plate solution. It is assumed that the shear deformations are constant over the element and, to account for this, two extra internal shear strain variables are added to the element. Various methods for eliminating these internal variables are examined but it is shown to be impossible to simultaneously satisfy both the constant bending moment and constant shear patch tests, except for parallelograms. However, one method gives elements which pass the constant shear patch test and, although failing the constant bending moment patch test for arbitrary geometries, gives errors which are small enough to be neglected in most engineering applications. This method has been applied to a triangular plate element and it is shown that the results obtained with this element converge (for all practical purposes) to the correct thick plate results.     

非协调实体退化厚/薄板单元

在八节点三维实体等参元基础上,将板理论基本假定引入弹性应力-应变矩阵,并为局部坐标系下的挠度位移分量附加非协调位移项,构造了非协调实体退化板单元。板假定的引入用以克服单元厚度泊松自锁,非协调项的引入可缓解单元剪切自锁。通过修正常应变矩阵,确保单元通过分片检验。分域积分方法的采用,使单元可用于层合板结构分析。算例表明:单元对等厚度的厚/薄板均适用。     

A six-node finite element for plate bending

A six-node plate bending element has been developed by employing mixed formulation based on a modified Hellinger–Reissner principle and the Reissner–Mindlin plate bending theory. The numerical result indicates that, among the types of assumed independent transverse shear strains considered, a combination of 2α version with either 5α version or 6α version is free of spurious kinematic modes and leads to accurate and reliable solutions even for very thin plates.     

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

This is the first of a two-part paper on plate bending elements with shear effects included. This paper presents a new three-node, nine-d.o.f. triangular plate bending element valid for the analysis of thick to thin plates. The element, called DKMT, 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 DKQ element for thick-plate bending analysis is evaluated in Part II.²⁴     

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     

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.     

Quadratic NMS Mindlin‐plate‐bending element

The development of a robust and efficient quadratic Mindlin‐plate‐bending elements mainly by the use of non‐conforming displacement modes is presented in this paper. A brief review on the previous efforts to develop efficient non‐conforming Mindlin plate bending elements is also given. The behaviour of the newly proposed plate element is further improved by the combined use of nonconforming displacement modes, the selectively reduced integration scheme, and the assumed shear strain fields. Thus, the newly developed element has been designated as ‘NMS‐8P’. The improvement achieved may be attributable to the fact that the merits of these improvement techniques are merged in the formation of the new element in a complementary manner. The proposed 8‐node element passes the patch tests, does not show spurious mechanism, and does not produce shear locking phenomena even with distorted meshes. It is also shown that the element produces reliable solutions through numerical tests for standard benchmark problems. Copypright © 1999 John Wiley & Sons, Ltd.     

《薄壁Timoshenko梁弯扭耦合振动的动态有限元法》

通过直接求解单对称均匀薄壁Timoshenko梁单元弯扭耦合振动的运动微分方程,推导了其精确的动态刚度矩阵。在本文研究中考虑了弯扭耦合、翘曲刚度、转动惯量和剪切变形的影响。针对某弯扭耦合的薄壁梁算例,应用本文推导的动态刚度矩阵,采用自动Muller法和结合频率扫描法的二分法求解频率特征方程,计算了该薄壁梁的固有特性,并讨论了翘曲刚度、剪切变形和转动惯量对该弯扭耦合薄壁梁的固有频率和模态形状的影响。数值结果验证了本文方法的精确性和有效性,并指出随着模态阶次的增加,剪切变形、转动惯量和翘曲刚度对薄壁梁的固有特性的影响更加显著。     

A discrete shear triangular nine D.O.F. element for the analysis of thick to very thin plates

This paper deals with the formulation and the evaluation of a new three node, nine d.o.f. triangular plate bending element valid for the analysis of thick to thin plates. The formulation is based on a generalization of the discrete Kirchhoff technique to include the transverse shear effects. The element, called DST (Discrete Shear Triangle), has a proper rank and is free of shear locking. It coincides with the DKT (Discrete Kirchhoff Triangle) element if the transverse shear effects are not significant. However, an incompatibility of the rotation of the normal appears due to shear effects. A detailed numerical evaluation of the characteristics and of the behaviour of the element has been performed including patch tests for thin and thick plates, convergence tests for clamped and simply supported plates under uniform loading and evaluation of stress resultants. The overall performance of the DST element is found to be very satisfactory.     

An assumed-stress hybrid 4-node shell element with drilling degrees of freedom

An assumed-stress hybrid/mixed 4-node quadrilateral shell element is introduced that alleviates most of the deficiencies associated with such elements. The formulation of the element is based on the assumed-stress hybrid/mixed method using the Hellinger-Reissner variational principle. The membrane part of the element has 12 degrees of freedom including rotational or ‘drilling’ degrees of freedom at the nodes. The bending part of the element also has 12 degrees of freedom. The bending part of the element uses the Reissner-Mindlin plate theory which takes into account the transverse shear contributions. The element formulation is derived from an 8-node isoparametric element by expressing the midside displacement degrees of freedom in terms of displacement and rotational degrees of freedom at corner nodes. The element passes the patch test, is nearly insensitive to mesh distortion, does not ‘lock’, possesses the desirable invariance properties, has no hidden spurious modes, and for the majority of test cases used in this paper produces more accurate results than the other elements employed herein for comparison.     

A C0 triangular plate element with one-point quadrature

A triangular plate element of the Mindlin type, which uses linear fields for rotations and transverse deflections, is developed. Success is achieved, in contrast to previous failures with this element, by decomposing the displacements into well-defined bending and shear modes which are associated with bending and shear energies, respectively. The element only requires a single quadrature point, which is very attractive for nonlinear analysis. Results are presented for a variety of plate problems; the accuracy is similar to the 4-node bilinear Mindlin plate and its rate of convergence is approximately of order h².     

Assumed strain formulation for the four-node quadrilateral with improved in-plane bending behaviour

A new assumed strain quadrilateral element with highly accurate in-plane bending behaviour is presented for plane stress and plane strain analysis. The basic idea of the formulation consists in identification of various modes of deformation and then in proper modification of the strain field in some of these modes. In particular, the strain operator corresponding to the in-plane bending modes is modified to simulate the strain field resulting from the assumptions usually made in structural mechanics. The modification of the strain field leads to the assumed strain operator on the element level. As a result, the so-called shear and membrane locking phenomena are alleviated. The element exhibits remarkable success in bending- dominated problems even when severely distorted and high aspect ratio meshes are used. Another advantage of the present assumed strain element is that locking for nearly incompressible materials is also mitigated. While this assumed strain element passes the patch test only for the parallelogram shapes, the element provides convergent solutions as long as the initially general form of the element approaches a parallelogram shape with the refinement of the mesh.     

Modelling edge effects with the C0 plate bending elements: Part 2. The performance of the reformulated four-node element

The reformulated four-node element has exhibited an excellent behaviour in static finite element analyses being free of shear and machine locking problems and of zero energy modes and modelling accurately relatively thick plates. In the present paper, the element under consideration is shown to exhibit an analogous excellent performance in modelling highly demanding boundary layer plate bending problems.     

面剪切振动模式弛豫铁电单晶换能器

[011]极化方向、zxt-45°切型的PIN-PMT-PT单晶因其高剪切压电应变常数、高机电耦合系数和高柔顺系数等特点,在水声换能器中存在广阔的应用前景.通过设计中间质量块的方法,将单晶产生的剪切振动转换为换能器的纵向振动,并利用辐射头的弯曲振动和圆环尾质量块振动的耦合拓宽工作频带.通过有限元仿真分析,研究了结构参数...     

A general finite element for beams or beam-columns with or without an elastic foundation

A general finite element is derived for beams or beam-columns with or without a continuous Winkler type elastic foundation. The need to discretize members into shorter elements for convergence towards an ‘exact’ solution is eliminated by employing in the derivation of the element exact shape functions obtained from the equation of the elastic line. Inter-nodal values of deflections, bending moments and shear forces are obtained using the exact shape functions and trigonometric series. The effect of heavy compressive or tensile axial forces on bending stiffness is treated as a linear problem by considering the axial force as a constant parameter affecting the stiffness. FORTRAN subroutines to compute the stiffness matrix, equivalent nodal forces, deflected shape, bending moments and shear forces are provided and verified by an example.     

Large deflections and large-amplitude free vibrations of straight and curved beams

This paper is concerned with the large-deflection (bending and free vibration)analysis of thin elastic curved beams by conventional and mixed finite element methods. The conventional finite element method is based on the total potential energy expression, whereas the mixed method is based on a Reissner-type variational statement and involves the bending moments and deflections as primary dependent variables. Incremental and direct formulations are presented for both methods. The nonlinearity is included in the numerical method via an iterative procedure (i.e. the in-plane force due to the large deflection is not treated as constant but included under the integral in the energy expression). In the case of vibrations, the transverse shear and rotary inertia effects are also included. A number of numerical examples of beams with various edge conditions are analysed for deflections and natural frequencies, and the results are compared with those reported in the literature.     

Deformation and failure of blast-loaded square plates

Numerical results for clamped, thin square steel plates subjected to blast loading are presented. The numerical analysis is based on a finite element formulation, which includes the nonlinear effects of geometry and material as well as strain rate sensitivity. A phenomenological interactive failure criterion comprising bending, tension and transverse shear is proposed to predict the various modes of failure. A node release algorithm is developed to simulate the progression of plate rupture from the boundary. The analysis is continued in the post-failure phase to account for the free flight deformation of the torn plate. The predicted failure modes for a blast-loaded plate are presented and compared with previously published experimental data.     

A generalized layerwise finite element for multi-layer damping treatments

This paper presents a 4-node facet type quadrangular shell finite element, based on a layerwise theory, developed for dynamic modelling of laminated structures with viscoelastic damping layers. The bending stiffness of the facet shell element is based on the Reissner–Mindlin assumptions and the plate theory is enriched with a shear locking protection adopting the MITC approach. The membrane component is corrected by using incompatible quadratic modes and the drilling degrees of freedom are introduced through a fictitious stiffness stabilization matrix. Linear static tests, using several pathological tests, showed good and convergent results. Dynamic analysis evaluation is provided by using two eigenproblems with exact analytical solution, as well as a conical sandwich shell with a closed-form analytical solution and a semi-analytical ring finite element solution. The applicability of the proposed finite element to viscoelastic core sandwich plates is assessed through experimental validation.