On a formulation of a nonlinear theory of plates and shells with applications

A nonlinear theory of plates and shells based on only one consistent kinematical approximation is employed to investigate finite strain and large deformation structural problems. The kinematics, expanded with respect to the thickness parameter, include consistently and adequately higher-order terms to allow fibers initially straight to deform nonlinearly in the large deformation range. A full Lagrangean formulation is considered. Starting from a three-dimensional principle of virtual work, the equilibrium equations and the resultant stresses of the shell together with their incremental form are derived employing an incremental constitutive relation valid for a wide class of rate-independent material. And, to assess the accuracy of the theory adopted, some boundary value problems are solved using a proposed hyperelastic material to model the constitutive behavior of a semi-infinite plate. The analytical and numerical (finite element method) results show good agreements between the exact and the approximate theories.     

A general three-dimensional L-section beam finite element for elastoplastic large deformation analysis

In this paper, the development of a general three-dimensional L-section beam finite element for elastoplastic large deformation analysis is presented. We propose the generalized interpolation scheme for the isoparametric formulation of three-dimensional beam finite elements and the numerical procedure is developed for elastoplastic large deformation analysis. The formulation is general and effective for other thin-walled section beam finite elements. To show the validity of the formulation proposed, a 2-node three-dimensional L-section beam finite element is implemented in an analysis code. As numerical examples, we first perform elastic small and large deformation analyses of a cantilever beam structure subjected to various tip loadings, and elastoplastic large deformation analysis of the same structure under reversed cyclic tip loading. We then analyze the failures of simply supported beam structures of different lengths and slenderness ratios under elastoplastic large deformation. The same problems are solved using refined shell finite element models of the structures. The numerical results of the L-section beam finite element developed here are compared with the solutions obtained using shell finite element analyses. We also discuss the numerical solutions in detail.     

Elastic torsion of thin walled bars of variable cross sections

An application of the finite element method to the theory of thin walled bars of variable cross sections has been presented in this paper. A solution of this problem is based on the linear membrane shell theory with the application of Vlasov's assumptions. A bar is divided into elements along its longitudinal axis and then, a shell mid-surface of the element is approximated by arbitrary triangular Subelements. Nodal displacements of the element are assumed to be polynomials of the third order and the equivalent stiffness matrix is obtained. Calculated nodal displacements enable an analysis of normal and shearing stresses.     

Thermal stress analysis of laminated composite plates and shells

In this paper, Semiloof shell finite element formulation has been extended to thermal stress analysis of laminated plates and shells. The accuracy of the formulation has been verified using sample problems available in the literature. Thermal stresses in cross-ply and angle-ply laminated plates and shells subjected to thermal gradients across the thickness are presented for different boundary conditions, taking into account the temperature dependence of the material properties. The behaviour of laminates under thermal load is found to be different from that under mechanical loads in certain respects.     

A high precision triangular laminated anisotropic shallow thin shell finite element

A high precision triangular laminated shallow thin shell finite element has been developed based on the classical lamination theory. The stiffness matrix is obtained explicitly by pre and post multiplying a few basic matrices. The formulation is almost an order of magnitude faster than those available for similar order elements. The numerical results of the example problems presented demonstrate that both displacements and stresses are predicted accurately with moderately coarse grids. A complete listing of FORTRAN subroutines is presented for users, to ease implementation of the algorithm.     

Comparison of shell theories in the analysis of cylindrical shells with discontinuity in the thickness

Cylindrical shells with discontinuity in the thickness and that are subjected to axisymmetric loading have been analysed. Two types of finite elements are used: the first is based on thin shell theory and the second on thick shell theory. The loadings considered are a uniform internal pressure and a circular ring load at the mid-section. The effect of these loads for various end conditions and various step-ratios in the thickness have been analysed. Numerical results are presented and compared for both the theories. It has been shown that the transverse normal stress acting along the thickness direction is not negligible compared to other stresses at places of discontinuity either in the thickness or in the loading. The weight of the shell is kept constant for various step-ratios.     

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.     

Nine node and seven node thick shell elements with large displacements and rotations

In the present paper we discuss the total Lagrangian formulation for shell elements under large displacements and rotations to perform nonlinear geometrical analyses. This formulation is applied to nine node and seven node quadratic shell elements initially developed for small strain elasto-plastic analyses. The formulation we use is based on a three dimensional continuum approach in which we introduce a linear dependence of displacements with respect to thickness and a plane stress hypothesis. The measure of deformation we take is that of Green–Lagrange related to the second Piola–Kirchhoff tensor for the stresses by a linear material law. Linear buckling is treated as a limit case of the nonlinear geometrical analysis.     

Crippling analysis of composite stringers based on complete unloading method

This paper addresses a nonlinear finite element method for the crippling analysis of composite laminated stringers. For the finite element modeling, a nine-node laminated shell element based on the first order shear deformation theory is used. Failure-induced stiffness degradation is simulated by the complete unloading method. A modified arc-length algorithm is incorporated in the nonlinear finite element method to trace the post-failure equilibrium path after a local buckling. Finite element results show excellent agreement with those of previous experiment. A parametric study is performed to assess the effect of the flange-width, web-height, and stacking sequence on the buckling, local buckling, and crippling stresses of stringers.     

Nonlinear analysis of an axisymmetric shell using three noded degenerated isoparametric shell elements

An updated Lagrangian formulation of a quadratic degenerated isoparametric shell element is presented for geometrically nonlinear elasto-plastic shell problems. A finite rotation effect is included in the formulation by adopting a co-rotational scheme. The load stiffness matrix has been derived for the treatment of a pressure load. For elasto-plastic behavior, the layered element model is used. The Newton-Raphson iteration method is employed to solve incremental nonlinear equations. For tracking of post-buckling behavior, the work control method is taken into account. Verification of the present technique is obtained by analyzing the available reference problems. Good correlations between the computed results and referenced data can be drawn.     

Geometrically non-linear transient analysis of laminated composite and sandwich shells with a refined theory and C finite elements

A C⁰ continuous finite element formulation of a higher order shear deformation theory is presented for predicting the linear and geometrically non-linear, in the sense of von Karman, transient responses of composite and sandwich laminated shells. The displacement model accounts for the non-linear cubic variation of the tangential displacement components through the thickness of the shell and the theory requires no shear correction coefficients. In the time domain, the explicit central difference integrator is used in conjunction with the special mass matrix diagonalization scheme which conserves the total mass of the element and includes effects due to rotary inertia terms. Numerical results for central transverse deflection and stresses are presented for composite and sandwich laminated shells with various boundary conditions subjected to different types of loads and are compared with the results from other sources. Some new results are also included for future reference.     

On the finite element formulation of bifurcation mode of creep buckling of axisymmetric shells

In this paper, a finite element formulation is given in detail for the creep buckling of an axisymmetric shell. A special emphasis is placed on the bifurcation mode of creep buckling. A bifurcation point is determined by examining the shape of the potential energy in the vicinity of an axisymmetric equilibrium state obtained from a creep deformation analysis in the prebuckling stage. To illustrate the capability of the finite element formulation, a numerical example is presented for the creep buckling of a shallow spherical shell subjected to a uniform external pressure. In this analysis, not only the axisymmetric snap-through type but also the asymmetric bifurcation one are considered as buckling modes.     

Collapse of long, noncircular, cylindrical shells under pure bending

This paper describes an extension of a method developed in a previous paper to determine the moment carrying capacity of elastoplastic noncircular cylindrical shells with infinite length by the finite element method. As a result of the shape change in the cross section of a shell during deformation, the bending moment reaches a global maximum value and then decreases as the bending curvature further increases. The shell would consequently collapse at the maximum moment. However, a bifurcation buckling may occur before the maximum moment can be developed. This bifurcation buckling could induce collapse of the shell under a moment less than the maximum. Determination of the likelihood that the bifurcation buckling would generate shell collapse may be made from the initial post-buckling behavior. An initial post-buckling analysis based on the J₂ deformation theory of plasticity has been developed in this paper. The finite element method with one spatial variable is used to locate the bifurcation point as well as to analyze the initial post-buckling behavior. Numerical examples of cylindrical shells with various cross-sectional shapes are shown. In particular, for a shell of square cross section, the moment at the bifurcation is much lower than the maximum value; however, the initial post-buckling analysis reveals that the state of equilibrium is still stable. Deep post-buckling analysis is required to determine the moment carrying capacity of a shell with such cross section.     

Large deformation in-plane analysis of elastic beams

A large deformation theory for in-plane beam problems, analogous to Budiansky's non-linear shell theory, is formulated. The formulation results in objective equations. A finite element representation of displacements, using cubic interpolating functions, is combined with the virtual work form of these equations, in order to obtain numerical solutions. The capability of the formulation is demonstrated by computing the displacements associated with a thin cantilever strip, subjected to pure moment, until it forms a complete circle. A solution of the elastica problem illustrates a potential of the formulation in solving ‘post-buckling’ problems.     

A three-dimensional state space finite element solution for laminated composite cylindrical shells

On the basis of the theory of three-dimensional elasticity, this paper presents a state space finite element solution for stress analysis of cross-ply laminated composite shells. This is a continuation of the authors’ previously published work on laminated plates [Compos. Struct. 57 (1–4) (2002) 117; Comput. Methods Appl. Mech. Engrg. 191 (37–38) (2002) 4259]. Once again a state space formulation is introduced to solve for through-thickness stress distributions, while the traditional finite elements are used to approximate the in-surface variations of state variables. A three-dimensional laminated shell element is established in an arbitrary orthogonal curvilinear coordinate system, while the application of the element is shown by calculating stresses in laminated cylindrical shells. Compared with the traditional finite element method, the new solution provides accurate continuous through-thickness distributions of both displacements and transverse stresses.     

Plastic deformation of a boron/aluminum tube under multi-axial loadings

A recently developed constitutive theory, encoded in the form of a finite-element program, has been utilized to study the elastic-plastic deformation of a metal-matrix composite tube (shell). Numerical results have been obtained for a unidirectional boron/aluminum cylindrial tube specimen, integrally attached to steel end-fittings and subjected to a combined loading of tension, torsion and internal pressure. Of particular interest are the effect of load-step size on the accuracy of stresses in the plastic range, influence of plasticity on the “boundary-layer effect” and uniformity of the stress field in the central region of the fiber reinforced cylindrical tube when loaded in the plastic range under combined loading conditions.     

A high precision triangular laminated anisotropic cylindrical shell finite element

The stiffness matrix for a high precision triangular laminated anisotropic cylindrical shell finite element has been formulated and coded into a composite structural analysis program. The versatility of the element's formulation enables its use in the analysis of multilayered composite plate and cylindrical shell type structures taking into account actual lamination parameters. The example applications presented demonstrated that accurate predictions of stresses as well as displacements are obtained with modest number of elements.     

Element free method for static and free vibration analysis of spatial thin shell structures

The implementation of the element free Galerkin method (EFG) for spatial thin shell structures is presented in this paper. Both static deformation and free vibration analyses are considered. The formulation of the discrete system equations starts from the governing equations of stress resultant geometrically exact theory of shear flexible shells. Moving least squares approximation is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometry. Discrete system equations are obtained by incorporating these interpolations into the Galerkin weak form. The formulation is verified through numerical examples of static stress analysis and frequency analysis of spatial thin shell structures. For static load analysis, essential boundary conditions are enforced through penalty method and Lagrange multipliers while boundary conditions for frequency analysis are imposed through a weak form using orthogonal transformation techniques. The EFG results compare favorably with closed-form solutions and that of finite element analyses.     

Geometrically nonlinear analysis of cylindrical shells using surface-parallel quadratic elements

A moderately thick cylindrical shell isoparametric element that is capable of accurately modeling cylindrically curved geometry, while also incorporating appropriate through-thickness kinematic relations is developed. The analysis accounts for fully nonlinear kinematic relations so that stable equilibrium paths in the advanced nonlinear regime can be accurately predicted. The present nonlinear finite element solution methodology is based on the hypothesis of linear displacement distribution through thickness (LDT) and the total Lagrangian formulation. A curvilinear side 16-node element with eight nodes on each of the top and bottom surfaces of a cylindrical shell has been implemented to model the transverse shear/normal deformation behavior represented by the LDT. The BFGS iterative scheme is used to solve the resulting nonlinear equations. A thin-shallow clamped cylindrical panel is investigated to test the convergence of the present element, and also to compare the special case of the present solution based on the KNSA (von Karman strain approximation) with those computed using the available faceted elements, discrete Kirchhoff constraint theory (DKT) and classical shallow shell finite elements, spanning the entire computed equilibrium path.