Thin element applications of a new formulation for C structural elements

A new formulation was proposed recently for the removal of the shear and membrane locking mechanisms from the finite element equations of the structural C⁰ shell, plate and beam elements. The use of full integration with the proposed formulation does not allow development of the zero energy modes or the softening effects, usually associated with the use of the technique of reduced integration in C⁰ plate and shell element applications. In the present paper a beneficial side effect of the new formulation is presented with regard to the development of the purely machine dependent locking. Questions concerning the introduction of softening effects by the new formulation in some flat C⁰ plate/shell element applications are addressed.     

Application of a new stress finite element to analysis of shell structures

A new stress finite element for analysis of shell structures of arbitrary geometry and loading has been introduced in Ref. [1]. The purpose of the present paper is to demonstrate the versatility of the proposed element with respect to all kinds of shell structures.     

基于假定应变法的协同转动四边形复合材料壳单元

为对复合材料层合板壳结构进行精确的大变形数值模拟,提出一种采用假定应变法的能分析层合结构大转动问题的协同转动四边形壳单元.该方法在建立有限元公式时引入假定应变法以克服膜闭锁和剪切闭锁的不利影响.与其他能分析大转动问题的复合材料壳单元相比,在新的协同转动框架中采用矢量型转动变量,可大大降低在非线性增量求解过程中更新转动变量的难度,且能得到对称的单元切线刚度矩阵,提高单元的计算效率.分析两个典型算例,并与其他学者的结果进行对比,结果表明在计算层合结构大转角问题时拥有较好的精度和收敛性.     

A conoidal shell analysis by modified isoparametric element

The quadratic isoparametric element is modified for applications in thin shell analysis. Four extra nonconforming displacement modes are added to transverse displacement then, later, they are eliminated through static condensation. A conoidal shell under uniform pressure is analyzed using the new element and the results compared with previous works.     

Efficient sensitivity analysis and optimization of shell structures by the ABAQUS code

In this paper, a new efficient sensitivity analysis procedure is presented for the optimization of shell structures without access to the finite element source code. It is devised as a general interface tool to extend existing finite element systems from pure structural analysis to design capability. The implementation is performed based on the ABAQUS code. Kirchhoff flat shell elements are taken into account in the study with the element thickness as design variables. To ensure the performance and the validity of the proposed procedure, satisfactory sensitivity and optimization results are illustrated for numerical examples.     

New quadratic solid–shell elements and their evaluation on linear benchmark problems

This paper is concerned with the development of a new family of solid–shell finite elements. This concept of solid–shell elements is shown to have a number of attractive computational properties as compared to conventional three-dimensional elements. More specifically, two new solid–shell elements are formulated in this work (a fifteen-node and a twenty-node element) on the basis of a purely three-dimensional approach. The performance of these elements is shown through the analysis of various structural problems. Note that one of their main advantages is to allow complex structural shapes to be simulated without classical problems of connecting zones meshed with different element types. These solid–shell elements have a special direction denoted as the “thickness”, along which a set of integration points are located. Reduced integration is also used to prevent some locking phenomena and to increase computational efficiency. Focus will be placed here on linear benchmark problems, where it is shown that these solid–shell elements perform much better than their counterparts, conventional solid elements.     

Isogeometric shell analysis: The Reissner–Mindlin shell

A Reissner–Mindlin shell formulation based on a degenerated solid is implemented for NURBS-based isogeometric analysis. The performance of the approach is examined on a set of linear elastic and nonlinear elasto-plastic benchmark examples. The analyses were performed with LS-DYNA, an industrial, general-purpose finite element code, for which a user-defined shell element capability was implemented. This new feature, to be reported on in subsequent work, allows for the use of NURBS and other non-standard discretizations in a sophisticated nonlinear analysis framework.     

Thin shell and surface crack finite elements for simulation of combined failure modes

In this study we present a new approach to analyse cracked shell structures subjected to large geometric changes. It is based on a combination of a rectangular assumed natural deviatoric strain thin shell finite element and an improved linespring finite element. Plasticity is accounted for using stress resultants. A power law hardening model is used for shell and linespring material. A co-rotational formulation is employed to represent nonlinear geometry effects. With this, one can carry out nonlinear fracture mechanics assessments in structures that show instabilities due buckling (local/global), ovalisation and large rigid body motion. By numerical examples it is shown how geometric instabilities and fracture compete as governing failure mode.     

A conical shell finite element

In the analysis of rocket and missiles structures one frequently encounters cylindrical and cornica' shells. A simple finite element which fits the above configuration is obviously a conical shell finite element. In this paper stiffness matrix for a conical shell finite element is derived using Novozhilov's strain-displacement relations for a conical shell. Numerical integration is carried out to ge. the stiffness matrix. The element has 28 degrees-of-freedom and is nonconforming. An eigenvalue analysis of the stiffness matrix showed that it contains all the rigid body modes (six in this case) adequately, which is one of the convergence criteria. An advantage of this element is that a cylindrical shell, an annular segment flat plate, a rectangular flat plate elements can easily be obtained as degenerate cases. The effectiveness of this element is shown through a variety of numerical examples pertaining to annular plate, cylindrical shell and conical shell problems. Comparison of the present solution is made with the existing ones wherever possible. The comparison shows that the present element is superior in some respects to the existing elements     

Large deflection analyses of skew plates using hybrid/mixed finite element method

A new hybrid/mixed shell element is developed using oblique coordinate systems to analyze the large deflection behavior of skew plate with various skew angles, length to width ratios, thicknesses and supported edges under uniformly distributed and concentrated loads. The results obtained from the new element are compared with available theoretical and numerical solutions. An excellent agreement is achieved even for coarse meshes. The accuracy and efficiency of the proposed element are demonstrated.     

Isoparametric axisymmetric shell elements with temperature gradients for heat conduction

This paper presents a finite element formulation for axisymmetric shell heat conduction where temperature gradients through the shell thickness are retained as primary nodal variables. The element geometry is constructed using the coordinates of the nodes lying on the middle surface of the shell and the middle surface nodal point normals. The element temperature field is approximated in terms of element approximation functions, the nodal temperature, and the nodal temperature gradients. The weak formulation of the two-dimensional Fourier heat conduction equation in cylindrical coordinate system is constructed. The finite element properties of the shell element are then derived using the weak formulation and the element temperature field approximation. The formulation permits linear temperature gradients through the shell thickness. Distributed heat flux as well as convective boundaries are permitted on all four faces of the element. Furthermore, the element can also have internal heat generation as well as orthotropic material properties. The superiority of the formulation in terms of efficiency and accuracy is demonstrated. Numerical examples are presented and a comparison is made with the theoretical results.     

On finite element large displacement and elastic-plastic dynamic analysis of shell structures

Finite element procedures for nonlinear dynamic analysis of shell structures are presented and assessed. Geometric and material nonlinear conditions are considered. Some results are presented that demonstrate current applicabilities of finite element procedures to the nonlinear dynamic analysis of two-dimensional shell problems. The nonlinear response of a shallow cap, an impulsively loaded cylindrical shell and a complete spherical shell is predicted. In the analyses the effects of various finite element modeling characteristics are investigated. Finally, solutions of the static and dynamic large displacement elastic-plastic analysis of a complete spherical shell subjected to external pressure are reported. The effect of initial imperfections on the static and dynamic buckling behavior of this shell is presented and discussed.     

Solid to shell element geometric transformation

The geometric transformation applied to three-dimensional solid elements producing the so-called degenerate shell element is reviewed. An alternative to generating the shell element directly is proposed. The solid element is integrated first, with appropriate modifications to the material matrix. Then the shell element is generated from this matrix in a manner resembling ordinary geometrical transformations. The net result is the same stiffness matrix with reduced effort and simpler coding.     

Three dimensional curved shell finite elements for heat conduction

This paper presents a finite element formulation for three dimensional curved shell heat conduction where nodal temperatures and nodal temperature gradients through the shell thickness are retained as primary variables. The three dimensional curved shell geometry is constructed using the coordinates of the nodes lying on the middle surface of the shell and the nodal point normals. The element temperature field is defined in terms of the element approximation functions, nodal temperatures and nodal temperature gradients. The weak formulation of the three dimensional Fourier heat conduction equation is constructed in the Cartesian coordinate system. The properties of the curved shell elements are then derived using the weak formulation and the element temperature approximation. The element formulation permits linear temperature distribution through the element thickness.Distributed heat flux as well as convective boundaries are permitted on all six faces of the element. The element also has internal heat generation as well as orthotropic material capability. The superiority of the formulation in terms of applications, efficiency and accuracy is demonstrated. Numerical examples are presented and comparisons are made with theoretical solutions.     

Resultant-stress degenerated-shell element

In this paper, a resultant-stress degenerated-shell element is described and a variety of numerical examples, including the post-buckling analysis of an axially loaded perfect cylinder, are presented. The general degenerated nonlinear shell theory of Hughes and Liu is employed in deriving this resultant-stress degenerated-shell element.Contrary to the traditional integration through the thickness approach, which assumes no coupling between the in-plane and transverse material and structural response matrices, the present approach can permit use of arbitrary, three-dimensional (3-D) nonlinear constitutive equations. Furthermore, explicit expressions of the element matrices for a 4-node shell element are developed. This rank-sufficient 4-node shell element, termed the resultant-stress degenerated-shell (RSDS) element, avoids the need for the costly numerical quadrature function evaluations of the element matrices and force vectors. And thus there are large increases in computational efficiency with this method. The comparisons of this RSDS element with six other shell elements are also given in this paper.     

A critical survey of the 9-node degenerated shell element with special emphasis on thin shell application and reduced integration

Since the publication of Ahmad's degenerated thick shell element in 1970, a large number of authors have studied this simple type of element and made their proposals for improving its behaviour. Special emphasis was placed on thin plate and shell applications.

The present paper discusses special aspects of the 9-node Lagrangian element. Interesting facts are presented-in particular, a mathematical explanation of why reduced shear integration leads to improved results. Another topic deals with thin plate and shell applications. A modification of the stiffness is proposed which allows the application of the element like a Kirchhoff-type model to any plate problem.

In the final section a case study of a variety of proposed element models is presented, and the accuracy is shown for various plate and shell problems.     

SHB8PS––a new adaptative,assumed-strain continuum mechanics shell element for impact analysis

This paper presents the formulation of a new adaptative shell element. This is an eight-node brick element in which one chooses a special direction denoted “thickness”. The element is an assumed-strain element based on the formulation first proposed by Belytchko and Bindeman. It is integrated with a set of Gauss points through the thickness and with only one point in the other directions. A new assumed-strain formulation for the element is proposed and its development is carried out in elastoplasticity. The element adjusts to the physical situation automatically in the sense that the stabilizing forces are chosen to be proportional to the mean tangent modulus of the material across the thickness. This is a most important assumption because if the stabilizing forces were calculated without regard to the physical state of the element they would turn out to be too large. The element is then tested on simple academic cases and applied in a more complex situation.     

A nine-node assumed-strain finite element for composite plates and shells

A new finite element for modeling fiber-reinforced composite plates and shells is developed and its performance for static linear problems is evaluated. The element is a nine-node degenerate solid shell element based on a modified Hellinger-Reissner principle with independent inplane and transverse shear strains. Several numerical examples are solved and the solutions are compared with other available finite solutions and with classical lamination theory. The results show that the present element yields accurate solutions for the test problems presented. Convergence characteristics are good, and the solution is relatively insensitive in element distortion. The element is also shown to be free of locking even for thin laminates.