Geometrically non‐linear analysis of thin plates and shells using a generalized displacement method

SummaryThe Field/Boundary Element method is used to solve problems involving static and dynamic analysis of thin elastic plates and shallow shells undergoing finite delections. The method is developed through the use of the weak form of the appropriate governing differential equations and boundary conditions. The resulting equations are then linearized and incremental solution strategies are discussed. In the case of problems involving limit points, a procedure which is a variation of an arc-length method is described. Numerical results of both plate and shallow shell problems are given to demonstrate the computational efficiency of the method.

This paper shows applications of a recently developed shell element to the analysis of thin shell and membrane structures. The element is a three node triangle with only translational DOFs (rotation free) that uses the configuration of the three adjacent elements to evaluate the strains. This allows to compute (constant) bending strains and (linear) membrane strains. A total Lagrangian formulation is used. Strains are defined in terms of the principal stretches. This allows to consider rubber materials and other type of materials using the Hencky stress-strain pair. An explicit central difference scheme is used to integrate the momentum equations. Several examples, including inflation and deflation of membranes show the excellent convergence properties and robustness of the element for large strain analysis of thin shells and membranes.

PurposeThe purpose of this paper is to evaluate the capabilities of the generalized finite element method (GFEM) under the context of the geometrically nonlinear analysis. The effect of large displacements and deformations, typical of such analysis, induces a significant distortion of the element mesh, penalizing the quality of the standard finite element method approximation. The main concern here is to identify how the enrichment strategy from GFEM, that usually makes this method less susceptible to the mesh distortion, may be used under the total and updated Lagrangian formulations.Design/methodology/approachAn existing computational environment that allows linear and nonlinear analysis, has been used to implement the analysis with geometric nonlinearity by GFEM, using different polynomial enrichments.FindingsThe geometrically nonlinear analysis using total and updated Lagrangian formulations are considered in GFEM. Classical problems are numerically simulated and the accuracy and robustness of the GFEM are highlighted.Originality/valueThis study shows a novel study about GFEM analysis using a complete polynomial space to enrich the approximation of the geometrically nonlinear analysis adopting the total and updated Lagrangian formulations. This strategy guarantees the good precision of the analysis for higher level of mesh distortion in the case of the total Lagrangian formulation. On the other hand, in the updated Lagrangian approach, the need of updating the degrees of freedom during the incremental and iterative solution are for the first time identified and discussed here.

An eighteen‐node, three‐dimensional, solid element with 54 degrees of freedom is presented for the finite element analysis of thin plates and shells. The element is based on the Hellinger‐Reissner principle with independent strain. The assumed independent strain is divided into higher and lower terms. The stiffness matrix associated with the higher order independent strain plays the role of stabilization matrix. A modified stress‐strain relation decoupling inplane and normal strain is used to incorporate thin shell behaviour. Numerical results demonstrate that, with a properly chosen set of assumed strain, this element is effectively free of locking even for very thin plates and shells.

A boundary point method solution for the analysis of thin Kirchhoff plates with internal rigid supports is presented in this paper. The model is capable of handling supports of different shapes and layouts. The rigidity condition used extensively in this work refers to zero deflection over all the patched supported area. A typical application to the aforementioned is the analysis of floor slab systems supported with columns or walls, in which the stiffness of the columns and walls is considered infinitely large. To accurately model the plate–support interaction, each patched support area is divided to a group of cells and the compatibility conditions between the plate and the supported patched areas are satisfied. Two numerical examples are presented and their results are compared with the finite element methods (FEM) as well as with the available results in the literature. The comparison verifies the accuracy of the proposed solution.

A refined non-conforming triangular plate/shell element for linear and geometrically nonlinear analysis of plates and shells is developed in this paper based on the refined non-conforming element method (RNEM). A conforming triangle membrane element with drilling degrees of freedom in Cartesian coordinates and the refined non-conforming triangular plate-bending element RT9, in which Kirchhoff kinematic assumption was adopted, are used to construct the present element. The displacement continuity condition along the interelement boundary is satisfied in an average sense for plate analysis, and the coupled displacement continuity requirement at the interelement is satisfied in an average sense, thereby improving the performance of the element for shell analysis. Selectively reduced integration with stabilization scheme is employed in this paper to avoid membrane locking. Numerical examples demonstrate that the present element behaves quite satisfactorily either for the linear analysis of plate bending problems and plane problems or for the geometrically nonlinear analysis of thin plates and shells with large displacement, moderate rotation but small strain.

Nonlinear displacement-based finite-element analyses of composite shells—A new total Lagrangian formulation

Adaptive Total Lagrangian Eulerian SPH for high-velocity impacts

Donnell’s thin shell theory and basic equations based on the wave propagation method discussed in detail here, is used to investigate the natural frequencies of thin finite length circular cylindrical shells under various boundary conditions. Mode shapes are drawn to explain the circumferential mode numbern and axial mode numberm, and the natural frequencies are calculated numerically and compared with those of FEM (finite element method) to confirm the reliability of the analytical solution. The effects of relevant parameters on natural frequencies are discussed thoroughly. It is shown that for long thin shells the method is simple, accurate and effective.

A novel approximate theoretical solution is presented in this paper for the boundary disturbance of thin elastic shells of revolution with a nearly spherical middle surface and constant thickness. The indicated solution is a procedure of successive corrections, which was originally used by S. D. Poisson in handling non‐linear differential equations and introduced into thin shell analysis by E. F. Burmistrov.

We analyse the stability of generic spherically symmetric thin shells to linearized perturbations around static solutions. We include the momentum flux term in the conservation identity, deduced from the ‘ADM’ constraint and the Lanczos equations. Following the Ishak–Lake analysis, we deduce a master equation which dictates the stable equilibrium configurations. Considering the transparency condition, we study the stability of thin shells around black holes, showing that our analysis is in agreement with previous results. Applying the analysis to traversable wormhole geometries, by considering specific choices for the form function, we deduce stability regions and find that the latter may be significantly increased by considering appropriate choices for the redshift function.

The present study aims at the investigation of free vibration analysis of thin cylindrical shells surrounded by an elastic medium, which is distributed over a particular length of the shell. The Love’s shell theory is utilized along with the Winkler foundation to obtain the governing equations of motion. An exact series expansion method of solution is employed for any arbitrary boundary conditions. The excellent accuracy of the obtained results is validated by comparing to the available literature. Moreover, several case studies are performed to provide an insight into the influence of some practically important parameters on free vibrations of circular cylindrical shells, including stiffness and length of the partial foundation. Finally, it is concluded that the proposed method can accurately determine the exact solution to the free vibration of shells partially surrounded by elastic foundations with any boundary conditions.

The paper begins by noting that the practical and efficient numerical analysis of thin walled shells is far from a reality. Groundwork for the investigation starts with an examination of existing sufficiency conditions for convergence of the finite element method of analysis with refinement of mesh size; new and more practical conditions are then given specifically for shells. Working formulae of a suitable first approximation theory for the linear small deflexion behaviour are then given for arbitrary shells in lines of curvature and in geodesic coordinates. A variational principle is introduced which is more general than that for the well known assumed stress hybrid finite element model; its purpose is to provide a means to overcome the excessive rank deficiency which is sometimes encountered in the derive element stiffness matrix. , The formulae are next specialized to general developable shells for they are tne simplest to analyse and frequently occur in technology. Emphasis is given to the derivation of general formulae governing inextensional deformation, membrane action and rigid body movement because these constitute important factors in any adequate numerical analysis. . . , Specific application is made to circular cylindrical shells by first considering the interpolation of the kinematic continuity conditions along an arbitrary geodesic line. Details and numerical examples are provided for the first known fully compatible lines of curvature rectangular finite element which directly recovers arbitrary rigid body movements as well as inextensional deformations and membrane actions. The paper concludes with details and numerical examples of an arbitrarily shaped triangular finite element which employs the above mentioned variational principle m conjunction with linearly varying stress fields. All the rigid body movements are directly recovered as well as inextensional deformations and membrane actions. It is anticipated that this finite element and its derivatives will find widespread application.

A presentation is made of the numerical results obtained in a stress–strain analysis of thin and nonthin orthotropic shells with due regard for the physical nonlinearity and small and nonsmall shear stiffness of composites. A spherical shell with a circular hole is used as an example to analyze how the above-mentioned factors affect the distribution of stresses and strains depending on the shell thickness for adopted deformation models (the Kirchhoff–Love and Timoshenko hypotheses). Generalized conclusions are drawn from which it is possible to decide which of the composite properties and shell models should be given more priority.

An analysis of metal forming processes using large deformation elastic-plastic formulations