Instability analysis of thin plates and arbitrary shells using a faceted shell element with loof nodes

The faceted representation is employed in the paper to derive a 24-dof triangular shell element for the instability analysis of shell structures. This element, without the deficiencies of displacement incompatibility, singularity with coplanar elements, inability to model intersections, and low-order membrane strain representation, which are normally associated with existing flat elements, has previously been found by the authors to perform well in linear static shell analyses. The total Lagrangian approach is used in the nonlinear formulation, and the results of the various numerical examples indicate that its performance is comparable to existing nonlinear shell elements. An extrapolation stiffness procedure, which will improve the convergence characteristics of the constant arc length solution algorithm used here, is also presented.     

A simple and effective element for analysis of general shell structures

A simple flat three-node triangular shell element for linear and nonlinear analysis is presented. The element stiffness matrix with 6 degrees-of-freedom per node is obtained by superimposing its bending and membrane stiffness matrices. An updated Lagrangian formulation is used for large displacement analysis. The application of the element to the analysis of various linear and nonlinear problems is demonstrated.     

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.     

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.     

A generalized geometrically nonlinear formulation with large rotations for finite elements with rotational degrees of freedoms

A generalized geometrically nonlinear formulation using total Lagrangian approach is presented for the finite elements with translational as well as rotational degrees of freedoms. An important aspect of the formulation presented here is that the restriction on the magnitude of the nodal rotations is eliminated by retaining true nonlinear nodal rotation terms in the definition of the element displacement field and the consistent derivation of the element properties based on this displacement field. The general derivation and the formulation steps are applicable to any element with translational and rotational nodal degrees of freedoms. The specific forms of the formulation for axisymmetric shells, two-dimensional isoparametric beams, curved shells, two-dimensional transition elements and solid-shell transition elements can be easily derived by considering the explicit forms of the nonlinear nodal rotations for the element at hand. The specific forms of this formulation have already been well tested and applied to various two- and three-dimensional elements, the results for some of which are presented here. Currently it is being applied to the three-dimensional isoparametric beam elements.     

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.     

Failure analysis of laminated composite cylindrical/spherical shell panels subjected to low-velocity impact

A 4-noded, 48 d.o.f. doubly curved quadrilateral shell finite element based on Kirchhoff–Love shell theory, is used in the nonlinear finite element analysis to predict the damage of laminated composite cylindrical/spherical shell panels subjected to low-velocity impact. The large displacement stiffness matrix is formed using Green's strain tensor based on total Lagrangian approach. An incremental/iterative scheme is used for solving resulting nonlinear algebraic equations by Newton–Raphson method. The damage analysis is performed by applying Tsai–Wu quadratic failure criterion at all Gauss points and the mode of failure is identified using maximum stress criteria. The modes of failure considered are fiber breakage and matrix cracking. The progressive failure analysis is carried out by degrading the stiffness of the material suitably at all failed Gauss points. The load due to low-velocity impact is treated as an equivalent quasi-static load and Hertzian law of contact is used for finding the maximum contact force. After evaluating the nonlinear finite element analysis thoroughly for typical problems, damage analysis was carried out for cross-ply and quasi-isotropic cylindrical/spherical shell panels.     

Geometrically nonlinear analysis of shell structures using a flat triangular shell finite element

Summary This paper presents a state of the art review on geometrically nonlinear analysis of shell structures that is limited to the co-rotational approach and to flat triangular shell finite elements. These shell elements are built up from flat triangular membranes and plates. We propose an element comprised of the constant strain triangle (CST) membrane element and the discrete Kirchhoff (DKT) plate element and describe its formulation while stressing two main issues: the derivation of the geometric stiffness matrix and the isolation of the rigid body motion from the total deformations. We further use it to solve a broad class of problems from the literature to validate its use.     

Nonlinear elastoplastic dynamic analysis of single-layer reticulated shells subjected to earthquake excitation

A nonlinear dynamic finite element technique is developed to analyze the elastoplastic dynamic response of single-layer reticulated shells under strong earthquake excitation, in which the nonlinear three-dimensional beam elements are employed. An elastoplastic tangent stiffness matrix of three-dimensional beam element is derived by using the updated Lagrangian formulation, in which the isotropic hardening model, the Von-Mises yield criterion and the Prandtl-Reuss flow relations are applied to this study. This procedure considers both geometric and material nonlinearities. In this paper, several condensation and reduction techniques in matrices and degrees of freedom are used to simplify the analysis. An incremental-iterative technique based on the Newmark direct integration method and the modified Newton-Raphson method is employed for obtaining the solutions of the nonlinear dynamic equilibrium equations. Moreover, an accurate method is developed to compute the large rotations of space structures. As a numerical example, the elastoplastic dynamic response of a single-layer reticulated shell under strong seismic excitation is investigated. It is shown through the numerical example that the method developed in this paper is efficient for the nonlinear dynamic response analysis and plastic design of space structures.     

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.     

Non-linear finite element analysis of composite beams with interlayer slips

This article presents a new non-linear finite element formulation for the analysis of two-layer composite plane beams with interlayer slips. The element is based on the corotational method. The main interest of this approach is that different linear elements can be automatically transformed to non-linear ones. To avoid curvature locking that may occur for low order element(s), a local linear formulation based on the exact stiffness matrix is used. Five numerical applications are presented in order to assess the performance of the formulation.     

Large deflection analysis of plates and shells by discrete energy method

The discrete energy method—a special form of finite difference energy approach—is presented as a suitable alternative to the finite element method for the large deflection elastic analysis of plates and shallow shells of constant thickness. Strain displacement relations are derived for the calculation of various linear and nonlinear element stiffness matrices for two types of elements into which the structure is discretized for considering separately energy due to extension and bending and energy due to shear and twisting. Large deflection analyses of plates with various edge and loading conditions and of a shallow cylindrical shell are carried out using the proposed method and the results compared with finite element solutions. The computational efforts required are also indicated.     

A layered cylindrical quadrilateral shell element for nonlinear analysis of RC plate structures

This paper proposes a simple and accurate 4-node, 24-DOF layered quadrilateral flat plate/shell element, and an efficient nonlinear finite element analysis procedure, for the geometric and material nonlinear analysis of reinforced concrete cylindrical shell and slab structures. The model combines a 4-node quadrilateral membrane element with drilling or rotational degrees of freedom, and a refined nonconforming 4-node 12-DOF quadrilateral plate bending element RPQ4, so that displacement compatibility along the interelement boundary is satisfied in an average sense. The element modelling consists of a layered system of fully bonded concrete and equivalent smeared steel reinforcement layers, and coupled membrane and bending effects are included. The modelling accounts for geometric nonlinearity with large displacements (but moderate rotations) as well as short-term material nonlinearity that incorporates tension, cracking and tension stiffening of the concrete, biaxial compression and compression yielding of the concrete and yielding of the steel. An updated Lagrangian approach is employed to solve the nonlinear finite element stiffness equations. Numerical examples of two reinforced concrete slabs and of a shallow reinforced concrete arch are presented to demonstrate the accuracy and scope of the layered element formulation.     

A doubly curved quadrilateral finite element for the analysis of laminated anisotropic thin shells of revolution

The details of development of the stiffness matrix for a doubly curved quadrilateral element suited for static and dynamic analysis of laminated anisotropic thin shells of revolution are reported. Expressing the assumed displacement state over the middle surface of the shell as products of one-dimensional first order Hermite polynomials, it is possible to ensure that the displacement state for the assembled set of such elements, is geometrically admissible. Monotonic convergence of total potential energy is therefore possible as the modelling is successively refined. Systematic evaluation of performance of the element is conducted, considering various examples for which analytical or other solutions are available.     

A new approach to geometric nonlinearity of cable structures

The basic structural principles surrounding nonlinear behaviour of cable networks are explained through the example of a two-link structure. The nonlinear static response to load for this structure is then derived explicitly using the proposed simple approach, and results are compared with those obtained from a general two-dimensional non-linear bar element (derivation given), and to results quoted in the literature. The proposed approach to geometric nonlinearity is then tested on three three-dimensional cable networks and the results compared with those obtained by three other techniques, namely geometric stiffness matrix, dynamic relaxation and general minimum energy. The proposed technique has been found to be comparable to established techniques in accuracy, stability and speed of solution while at the same time exhibiting the key features of separation of the numerical computation from the underlying structural mechanics, and the requirement of understanding only the most elementary of structural mechanics. The proposed technique is thus also most suitable for introducing cable structures to undergraduate courses.     

Linear and nonlinear stability analysis of cylindrical shells

The paper describes the application of a curved isoparametric shell element to large displacement analyses including instability phenomena. A total Lagrangian formulation has been adopted using the standard incremental/iterative solution procedure. The linear stability analyses usually performed for the initial position were repeated at several advanced fundamental states on the nonlinear prebuckling path. Thus a current estimate of the final failure load is given. The method has been applied to several perfect and imperfect cylindrical shells under uniform pressure or wind load. Finally the example of a cylindrical panel under one concentrated transverse load is discussed.     

A rectangular laminated anisotropic shallow thin shell finite element

This report contains the details of the development of the stiffness matrix for a rectangular laminated anisotropic shallow thin shell finite element. The derivation is done under linear thin shell assumptions. Expressing the assumed displacement state over the middle surface of the shell as products of one-dimensional first-order Hermite interpolation polynomials, it is possible to insure that the displacement state for the assembled set of such elements, to be geometrically admissible. Monotonic convergence of the total potential energy is therefore possible as the modelling is successively refined. The element is systematically evaluated for its performance considering various examples for which analytical or other solutions are available.     

p-Version axisymmetric shell element for geometrically nonlinear analysis

This paper presents a p-version geometrically nonlinear (GNL) formulation based on total Lagrangian approach for a three-node axisymmetric curved shell element. The approximation functions and the nodal variables for the element are derived directly from the Lagrange family of interpolation functions of order p_ξ and p_η. This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and η directions for the three- and one-node equivalent configurations that correspond to p_ξ + 1 and p_η+ 1 equally spaced nodes in the ξ and η directions and then taking their products. The resulting element approximation functions and the nodal variables are hierarchical and the element approximation ensures C⁰ continuity. The element geometry is described by the coordinates of the nodes located on the middle surface of the element and the nodal vectors describing top and bottom surfaces of the element.

The element properties are established using the principle of virtual work and the hierarchical element approximation. In formulating the properties of the element complete axisymmetric state of stresses and strains are considered, hence the element is equally effective for very thin as well as extremely thick shells. The formulation presented here removes virtually all of the drawbacks present in the existing GNL axisymmetric shell finite element formulations and has many additional benefits. First, the currently available GNL axisymmetric shell finite element formulations are based on fixed interpolation order and thus are not hierarchical and have no mechanism for p-level change. Secondly, the element displacement approximations in the existing formulations are either based on linearized (with respect to nodal rotation) displacement fields in which case a true Lagrangian formulation is not possible and the load step size is severely limited or are based on nonlinear nodal rotation functions approach in which case though the kinematics of deformation is exact but additional complications arise due to the noncummutative nature of nonlinear nodal rotation functions. Such limitations and difficulties do not exist in the present formulation. The hierarchical displacement approximation used here does not involve traditional nodal rotations that have been used in the existing shell element formulations, thus the difficulties associated with their use are not present in this formulation.

Incremental equations of equilibrium are derived and solved using the standard Newton method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the preset formulation. The results obtained from the present formulation are compared with those available in the literature.