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.     

Large displacement extension of consistent shell element for static and dynamic analysis

The superior performance of the consistent shell element in the small deflection range has encouraged the authors to extend the formulation to large displacement static and dynamic analyses. The nonlinear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear stiffness matrix and the unbalanced load vector for the consistent shell element is presented in this study. Meanwhile, a simplified method for coding the nonlinear formulation is provided by relating the components for the nonlinear B-matrices to those of the linear B-matrix. The consistent mass matrix for the shell element is also derived and then incorporated with the stiffness matrix to perform large displacement dynamic and free vibration analyses of shell structures. Newmark's method is used for time integration and the Newton-Raphson method is employed for iterating within each increment until equilibrium is achieved. Numerical testing of the nonlinear model through static and dynamic analyses of different plate and shell problems indicates excellent performance of the consistent shell element in the nonlinear range.     

Elastoplastic and geometrically nonlinear thin shell analysis by the semiloof element

This paper extends the use of the Semiloof shell element to geometrically and materially nonlinear situations. For the geometrically nonlinear analysis a moving coordinate process is used. Local coordinate systems are considered, one at each integrating point, which move with the structure and allow deformation and rotation within the element to be taken into account. For elasto-plastic analysis the initial stress method is used and the yield conditions are expressed in terms of nondimensional stress resultants. The element formulation is briefly considered and in particular the definition of the global and local displacements and their first and second derivatives in the non-constrained and constrained form of the element are discussed. Numerical results are presented and comparisons made with other sources where 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.     

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.     

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.     

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.     

Nonlinear Formulations of a Four-Node Quasi-Conforming Shell Element

The quasi-conforming technique was introduced in the 1980’s to meet the challenge of inter-elements conforming problems and give a unified treatment of both conforming and nonconforming elements. While the linear formulation is well established, the nonlinear formulation based on the quasi-conforming technique that includes geometric and material nonlinearity is presented in this paper. The formulation is derived in the framework of an updated Lagrangian stress resultant, co-rotational approach. The geometric nonlinear formulation provides solutions to buckling and postbuckling behaviour while the material nonlinear formulation considers the spread of plasticity within the element while maintaining an explicit construction of element matrices. Aside from the elasto-plastic constitutive relation, formulations on laminate composites and reinforced concrete are also presented. The formulations of laminate composite and reinforced concrete material are present based on the layer concept, the material properties can vary throughout the thickness and across the surface of a shell element. The various failure criteria for laminate composite are included in the formulation which makes it possible to analyses the progressive failure of fibre and matrix. For the reinforced concrete material, the nonlinearities as a result of tensile cracking, tension stiffening between cracks, the nonlinear response of concrete in compression, and the yielding of the reinforcement are considered. The steel reinforcement is modeled as a bilinear material with strain hardening.     

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.     

Linear programming and limit analysis of shell roofs

Limit analysis for cylindrical shell roofs has been formulated as a linear programming problem based on lower bound theorem. The differential equations of equilibrium for a circular cylindrical shell element are transformed into algebraic equations by finite differences. The equilibrium equations and the linearized non-linear yield conditions at various points of the shell are linear functions of the stress resultants. These form the linear constraints of the problem. The load parameter is taken as an objective function and it is maximized using revised simplex method. For a shell of given geometry, stress resultants at various points are obtained to give the optimum collapse load. Thus the versatile technique avoids various trial solutions to achieve best lower bound for complicated shell problems.     

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.     

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.     

A stress analysis technique for plate and shell built-up structures with junctions and its application to minimum-weight design of stiffened structures

A finite element formulation using the penalty function method to analyse exactly the junctions of plate and shell built-up structures is suggested for an isoparametric shell element. The connectivity condition at the junction is added to the potential energy functional by the penalty parameter and the interpolating function of displacements. This formulation yields an integral-type stiffness matrix of the special junction elements, which can directly evaluate the surface tractions at the junction. For applying the technique suggested here to the optimum design of structures with junction parts, a design sensitivity analysis formulation for the adhesive special element is also developed. The technique is applied to the minimum-weight design problems of isotropic and composite laminated plates with a stiffener subjected to stress constraints.     

Postbuckling of multilayer cylindrical and spherical shell panels reinforced with graphene platelet by isogeometric analysis

The present work fills a gap on the postbuckling behavior of multilayer functionally graded graphene platelet reinforced composite (FG-GPLRC) cylindrical and spherical shell panels resting on elastic foundations subjected to central pinching forces and pressure loadings. Based on a higher-order shear deformation theory and the von Kármán’s nonlinear strain–displacement relations, the governing equations of the FG-GPLRC cylindrical and spherical shell panels are established by the principle of virtual work. The non-uniform rational B-spline (NURBS) based isogeometric analysis (IGA), the modified arc-length method and the Newton’s iteration method are employed synthetically to obtain nonlinear load–deflection curves for the panels numerically. Several comparative examples are performed to test reliability and accuracy of IGA and arc-length method in present formulation and programming implementation. Parametric investigations are carried out to illustrate the effects of dispersion type of the graphene platelet (GPL), weight fraction of the GPL, thickness of the panel, radius of the panel and parameters of elastic foundation on the load–deflection curves of the FG-GPLRC shell panels. Some complex load–deflection curves of the FG-GPLRC cylindrical and spherical shell panels resting on elastic foundations may be useful for future references.

     

A method for shell contact analysis

A relatively general and computationally efficient method of shell contact analysis using the discrete Fourier transform is developed for linear and certain types of nonlinear problems. The method predicts the contact boundary and the interfacial pressure distribution. It is illustrated by calculating the road contact pressure predicted by a finite element toroidal shell model of a pneumatic tire.     

Increasing solution stability for finite-element modeling of elasto-plastic shell response

The present article introduces a highly efficient numerical simulation strategy for the analysis of elasto-plastic shell structures. An isoparametric Finite Element, based on a Finite Rotation Reissner–Mindlin shell theory in isoparametric formulation, is enhanced by a Layered Approach for a realistic simulation of nonlinear material behaviour. A general material model including isotropic hardening effects is embedded into each material point. A new, highly accurate integration scheme is combined with consistently linearized constitutive relations in order to achieve quadratic rate of convergence. A global Riks–Wempner–Wessels iteration scheme enhanced by a linear Line-Search procedure was used to trace arbitrary deformation paths. Numerical examples show the efficiency of the present concept.     

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.     

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.     

Nonlinear topology optimization of layered shell structures

Topology stiffness (compliance) design of linear and geometrically nonlinear shell structures is solved using the SIMP approach together with a filtering scheme. A general anisotropic multi-layer shell model is employed to allow the formation of through-the-thickness holes or stiffening zones. The finite element analysis is performed using nine-node Mindlin-type shell elements based on the degenerated shell approach, which are capable of modeling both single and multi-layered structures exhibiting anisotropic or isotropic behavior. The optimization problem is solved using analytical compliance and constraint sensitivities together with the Method of Moving Asymptotes (MMA). Geometrically nonlinear problems are solved using iterative Newton–Raphson methods and an adjoint variable approach is used for the sensitivity analysis. Several benchmark tests are presented in order to illustrate the difference in optimal topologies between linear and geometrically nonlinear shell structures.