Hybrid strain based three node flat triangular shell elements—I. Nonlinear theory and incremental formulation

Aspects and theories of nonlinear analysis of structures, with special emphasis on structures that are discretized by the finite element method, are discussed. The updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are adopted. The independently assumed fields employed are the incremental displacements and incremental strains. Accordingly, the incremental second Piola-Kirchhoff stress and the incremental Washizu strain are selected as the incremental stress and strain measures. Various schemes for the transformation of the second Piola-Kirchhoff stress to Cauchy stress are included. Two versions of linear and nonlinear element stiffness and mass matrices are considered. These are the director and simplified versions. Variable thickness of the shell is considered so as to account for the ‘thinning effect’ due to large strain. Material nonlinearity studied in this paper is of elasto-plastic type with isotropic strain hardening. Cases in which small elastic but large plastic strain condition applies are considered and the J₂ flow theory of plasticity, in conjunction with Ilyushin's yield criterion, is employed. To simplify the derivation of (small displacement) stiffness matrix and to facilitate the derivation of explicit expressions for the element matrices, the non-layered approach has been applied.     

An extension of 3-D procedure to large strain analysis of shells

A numerical stress integration procedure for general 3-D large strain problems in inelasticity, based on the total formulation and the governing parameter method (GPM), is extended to shell analysis. The multiplicative decomposition of the deformation gradient is adopted with the evaluation of the deformation gradient practically in the same way as in a general 3-D material deformation. The calculated trial elastic logarithmic strains are transformed to the local shell Cartesian coordinate system and the stress integration is performed according to the GPM developed for small strain conditions. The consistent tangent matrix is calculated as in case of small strain deformation and then transformed to the global coordinate system.A specific step in the proposed procedure is the updating of the left elastic Green–Lagrangian deformation tensor. Namely, after the stresses are computed, the principal elastic strains and the principal vectors corresponding to the stresses at the end of time step are determined. In this way the shell conditions are taken into account appropriately for the next step.Some details are given for the stress integration in case of thermoplastic and creep material model.Numerical examples include bulging of plate (plastic, thermoplastic, and creep models for metal) and necking of a thin sheet. Comparison of solutions with those available in the literature, and with solutions using other type of finite elements, demonstrates applicability, efficiency and accuracy of the proposed procedure.     

A robust non-linear solid shell element based on a mixed variational formulation

The paper is concerned with a geometrically non-linear solid shell finite element formulation, which is based on the Hu-Washizu variational principle. For the approximation of the independent displacement, stress and strain fields, the strain field is additively decomposed into two parts. Due to the fact that one part of the strain field is interpolated in the same manner as proposed by the enhanced assumed strain (EAS) method, it is denoted as EAS field. The other strain field is approximated with the same interpolation functions as the stress field. In contrast to the EAS concept the approximation spaces of the stresses and the enhanced assumed strains are not orthogonal. Consequently the stress field is not eliminated from the finite element equations. For the displacements tri-linear shape functions are considered. Shear locking and curvature thickness locking are treated using assumed natural strain interpolations. A static condensation leads to a simple low order hexahedral solid shell element. Numerical tests show that the present model is very robust and allows larger load steps than an EAS solid shell element.     

Improvements in the membrane behaviour of the three node rotation-free BST shell triangle using an assumed strain approach

In this paper an assumed strain approach is presented in order to improve the membrane behaviour of a thin shell triangular element. The so called Basic Shell Triangle (BST) has three nodes with only translational degrees of freedom and is based on a Total Lagrangian Formulation. As in the original BST element the curvatures are computed resorting to the surrounding elements (patch of four elements). Membrane strains are now also computed from the same patch of elements which leads to a non-conforming membrane behaviour. Despite this non-conformity the element passes the patch test. Large strain plasticity is considered using a logarithmic strain–stress pair. A plane stress behaviour with an additive decomposition of elastic and plastic strains is assumed. A hyperplastic law is considered for the elastic part while for the plastic part an anisotropic quadratic (Hill) yield function with non-linear isotropic hardening is adopted. The element, termed EBST, has been implemented in an explicit (hydro-)code adequate to simulate sheet-stamping processes and in an implicit static/dynamic code. Several examples are given showing the good performance of the enhanced rotation-free shell triangle.     

Alternate stress and conjugate strain measures,and mixed variational formulations involving rigid rotations,for computational analyses of finitely deformed solids,with application to plates and shells—I: Theory

Attention is focused in this paper on: (i) definitions of alternate measures of “stress-resultants” and “stress-couples” in a finitely deformed shell (finite mid-plane stretches as well as finite rotations); (ii) mixed variational principles for shells, undergoing large mid-plane stretches and large rotations, in terms of a stress function vector and the rotation tensor. In doing so, both types of polar decomposition, namely rotation followed by stretch, as well as stretch followed by rotation, of the shell midsurface, are considered; (iii) two alternate bending strain measures which depend on rotation alone for a finitely deformed shell; (iv) objectivity of constitutive relations, in terms of these alternate strain/“stress-resultants”, and “stress-couple” measures, for finitely deformed shells. To motivate these topics, and for added clarity, a discussion of relevant alternate stress measures, work-conjugate strain measures, and mixed variational principles with rotations as variables, is presented first in the context of three-dimensional continuum mechanics.Comments are also made on the use of the presently developed theories in conjunction with mixed-hydrid finite element methods. Discussion of numerical schemes and results is deferred to the Part II of the paper, however.     

含初应力复合材料柱壳结构的双稳态特性

为分析初应力对复合材料圆柱壳结构双稳态特性的影响,采用经典板壳理论建立复合材料圆柱壳力学模型,基于层合结构本构关系推导用双参数表达的系统应变能公式;根据最小势能原理得到双稳态产生的条件和稳态时的曲率表达式。利用Abaqus软件构建圆柱壳的有限元模型,通过附加边界弯矩对柱壳稳态跃迁过程进行模拟。理论计算结果与有限元结果的对比验证理论模型的正确性。分析结果表明：当初应力满足一定条件时,复合材料柱壳结构在其变形过程中有2个稳定平衡位置,并且在稳定平衡位置结构都不产生扭转变形;2个稳定平衡位置的曲率方向可以相同或相反,这与无初应力时反对称复合材料柱壳双稳态曲率方向只能相同的情况有区别。     

Numerical evaluation of the asymptotic energy behavior of intermediate shells with application to two classical benchmark tests

We consider the class of shell problems which are neither purely bending neither purely membrane dominated. In such cases the asymptotic energy norm behavior (which is useful not only because it represents the structure stiffness, but also for numerical comparison purposes) is not a priori known. In this work we apply a numerical procedure in order to estimate the energy behavior of a general shell problem. In order to test its reliability, the method is applied to various problems for which the theoretical energy behavior is known and the results can be compared. Among the problems tested, we have two classical engineering shell benchmarks which are neither bending neither membrane dominated, and for which an analytical evaluation has been obtained in a recent work. All the energy behavior estimates obtained with the numerical method are in perfect agreement with the theoretical values.     

Surface loading of a thin-walled toroidal shell

Series solutions for elastostatic shell problems can provide results for particular loading cases economically, and give a basis of comparison with finite element method (FEM) solutions. For toroidal shells series solutions have been given by several authors, using mainly a stress approach. In the present study a displacement-based solution is given using the linear Sanders shell theory. This theory is considered to be one of the most accurate first-order theories. The governing equations are first developed in toroidal coordinates. The loading case of a pad of normal pressure on the shell surface is then considered in detail, and series expansions are written for the load, displacement and stress terms. Results are computed using the shell theory for a sample problem, and these results are compared with results obtained using the FEM. The results given provide practical information of interest to designers and furthermore give information about the shell theory and FEM solution characteristics.     

Viscoplasticity based on additive decomposition of logarithmic strain and unified constitutive equations: Theoretical and computational considerations with reference to shell applications

In contrast to multiplicative models of finite strain plasticity and viscoplasticity, a framework of additive nature is developed in this paper. The theory is based on the additive decomposition of the logarithmic strain tensor. The stress conjugate to the logarithmic strain then plays the role of the thermodynamically driving force. The approach in this paper is motivated by the search for numerically accessible structures which can be extended to incorporate anisotropy as well. Specifically in this region, multiplicative formulations become extremely tedious. The evolution equations are of the unified type due to Bodner and Partom, and are modified so as to fit into the theoretical framework adopted. The numerical treatment of the problem is fully developed. Specifically, the algorithmic aspects of the approach are discussed and various applications to shell problems are considered. A shell theory with seven degrees of freedom, together with a four-node enhanced strain finite element formulation, is used. A central feature of the shell formulation is its eligibility to the application of a three-dimensional constitutive law.     

Application of ADINA and hole drilling method to residual stress determination in weldments

Residual stress distributions with depth at weld toes in controlled shot-peened, high strength steel weldments were determined through the measurement of relaxed surface strains by the incremental hole drilling method. The stress states were determined from the relaxed surface strains by a novel elasto-plastic interpretation of the strain readings, using the finite element software, ADINA. Residual stress distributions determined from the theory of elasticity, using experimental strain gauge readings from the hole drilling method, gave values much greater than the 0.2% offset yield stress of 613 MPa. The stress states were corrected through elasto-plastic finite element modeling using ADINA. After the corrections, the maximum residual stress was less than 613 MPa. The corrected stress distributions were applied to determine the effect of controlled shot peening on residual stress distributions with depth. The distributions within 0.8 mm below the material surface was used as an indicator of shot-peening depth and the effect of control parameters.     

A general isoparametric finite element program SDRC SUPERB

SDRC SUPERB is a general purpose finite element program that performs linear static, dynamic and steady state heat conduction analyses of structures made of isotropic and/or orthotropic elastic materials having temperature dependent properties. The finite element library of SUPERB contains isoparametric plane stress, plane strain, flat plate, curved shell, solid type curved shell and solid elements in addition to conventional beam and spring elements. Linear, quadratic and cubic interpolation functions are available for all isoparametric elements. Independent parameters such as displacements and temperatures are obtained from SUPERB using the stiffness method of analysis. The remaining dependent parameters, such as stresses and strains, are evaluated at element gauss points and extrapolated to nodal locations. Averaged values are given as final output. The graphic capabilities of SUPERB consists of geometry and distorted geometry plotting, and stress, strain and temperature contouring. Contours are plotted at user defined cutting planes for solids and at top, middle or bottom surfaces for plate and shell types of structures.In the first part of this paper, the program capabilities of SUPERB are summarized. Extrapolation techniques used for determining dependent nodal parameters and for contour plotting are explained in the second part of the paper. Behavior of standard, wedge and transition type isoparametric elements and the effect of interpolation function orders on accuracy are discussed in the third part. The results of several illustrative problems are included.     

Simple and effective elements based upon Timoshenko–Mindlin shell theory

The simple and effective mixed models are developed for the analysis of multilayered anisotropic Timoshenko–Mindlin-type shells. The effects of the transverse shear and transverse normal strains, and laminated anisotropic material response are included. The precise representation of rigid body motions in the displacement patterns of curved shell elements is considered. This consideration requires the development of the strain–displacement equations of the Timoshenko–Mindlin-type theory with regard to their consistency with the rigid body motions. The fundamental unknowns consist of six displacements and eleven strains of the face surfaces of the shell, and 11 stress resultants. The element characteristic arrays are obtained by using the Hu–Washizu mixed variational principle. Numerical results are presented to demonstrate the high accuracy and effectiveness of the developed mixed models and to compare their performance with other finite-element models reported in the literature.     

A posteriori error estimator and adaptive procedures for computation of shakedown and limit loads on pressure vessels

Adaptive finite element procedures are presented for the computation of upper bounds estimates of limit and shakedown loads for pressure vessels. The method consists of an h-type adaptive mesh refinement strategy based upon an a-posteriori error estimator measured by the energy norm. The problem is formulated in a kinematic approach using Koiter's shakedown theorem. A constitutive model, for elastic-perfectly plastic materials, relates the plastic strains increments and curvatures to plastic multipliers through the flow law associated with a shell piecewise-linear yield surface (hexagonal prism). A consistent relationship between nodal displacements and nodal plastic multipliers is enforced by minimizing the strain residual between the total strain and the plastic strain increments, which is measured with respect to the energy norm. Discretization of the shell into finite elements allows the reduction of the problem to a minimization problem which is solved by linear programming.     

Nonlinear shell problem formulation accounting for through-the-thickness stretching and its finite element implementation

We discuss a theoretical formulation of shell model accounting for through-the-thickness stretching, which allows for large deformations and direct use of 3d constitutive equations. Three different possibilities for implementing this model within the framework of the finite element method are examined: one leading to 7 nodal parameters and the remaining two to 6 nodal parameters. The 7-parameter shell model with no simplification of kinematic terms is compared to the 7-parameter shell model which exploits usual simplifications of the Green–Lagrange strains. Two different ways of implementing the incompatible mode method for reducing the number of parameters to 6 are presented. One implementation uses an additive decomposition of the strains and the other an additive decomposition of the deformation gradient. Several numerical examples are given to illustrate performance of the shell elements developed herein.     

Membrane locking and assumed strain shell elements

An explanation of membrane locking behaviour in shell elements and also the use of reduced and selective integration is described. To overcome the conflict between the locking and mechanism problems the author proposed the degenerated shell elements with assumed transverse shear and membrane strains. The location of sampling points for the assumed strain fields is given in the present work. In the formulation of the new elements, assumed transverse shear strains in the natural coordinate system are used to overcome the shear locking problem. Also, assumed membrane strains in the orthogonal curvilinear coordinate system are applied to avoid membrane locking behaviour. Some numerical tests are presented to illustrate the good performance of the assumed strain shell elements.     

A hybrid strain technique for finite element analysis of plates and shells

A specialization of the Hu-Washizu [1] functional wherein strains and displacements are taken as independent variables is employed in the formulation of ‘hybrid’ elements. Both the strains and displacements are independently interpolated with the strains being eliminated at the element level, leaving displacement variables only to be assembled into the global system of equations. This distinguishes such elements as ‘hybrid’, in contrast to ‘mixed’ wherein the global system of equations contains all the discretized variables. Applications including ‘thick’ plate and shell elements are considered. In many applications the hybrid strain technique appears more natural than the hybrid stress technique since stress discontinuities are accommodated quite conveniently.     

Vibration analysis of laminated anisotropic shells of revolution

An efficient computational procedure is presented for the free vibration analysis of laminated anisotropic shells of revolution, and for assessing the sensitivity of their response to anisotropic (nonorthotropic) material coefficients. The analytical formulation is based on a form of the Sanders-Budiansky shell theory including the effects of both the transverse shear deformation and the laminated anisotropic material response. The fundamental unknowns consist of the eight stress resultants, the eight strain components, and the five generalized displacements of the shell. Each of the shell variables is expressed in terms of trigonometric functions in the circumferential coordinate and a three-field mixed finite element model is used for the discretization in the meridional direction.The three key elements of the procedure are: (a) use of three-field mixed finite element models in the meridional direction with discontinuous stress resultants and strain components at the element interfaces, thereby allowing the elimination of the stress resultants and strain components on the element level; (b) operator splitting, or decomposition of the material stiffness matrix of the shell into the sum of an orthotropic and nonorthotropic (anisotropic) parts, thereby uncoupling the governing finite element equations corresponding to the symmetric and antisymmetric vibrations for each Fourier harmonic; and (c) application of a reduction method through the successive use of the finite element method and the classical Bubnov-Galerkin technique.The potential of the proposed procedure is discussed and numerical results are presented to demonstrate its effectiveness.     

Hybrid strain based three node flat triangular shell elements—II. Numerical investigation of nonlinear problems

In a companion paper [M. L. Liu and C. W. S. To, Comput. Struct. 54, 1031–1056 (1995)] theories and incremental formulation of nonlinear shell structures discretized by the finite element method are discussed. The updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are adopted. The independently assumed fields employed are the incremental displacements and incremental strains. Based on the theory and incremental formulation explicit element stiffness and mass matrices of three node flat triangular shell finite elements are derived. In the present paper the derived element matrices are applied to nine examples. The latter include static and dynamic response analysis of shell structures with geometrical, material, and geometrical and material nonlinearities. The formulation adopted and element matrices derived are found to be accurate, flexible and applicable to various types of shell structures with geometrical and material nonlinearities.