Manfred Bischoff

Shear deformable shell elements for large strains and rotations

Well-known finite element concepts like the Assumed Natural Strain (ANS) and the Enhanced Assumed Strain (EAS) techniques are combined to derive efficient and reliable finite elements for continuum based shell formulations. In the present study two aspects are covered: The first aspect focuses on the classical 5-parameter shell formulation with Reissner–Mindlin kinematics. The above-mentioned combinations, already discussed by Andelfinger and Ramm for the linear case of a four-node shell element, are extended to geometrical non-linearities. In addition a nine-node quadrilateral variant is presented. A geometrically non-linear version of the EAS-approach is applied which is based on the enhancement of the Green–Lagrange strains instead of the displacement gradient as originally proposed by Simo and Armero. In the second part elements are derived in a similar way for a higher order, so-called 7-parameter non-linear shell formulation which includes the thickness stretch of the shell (Buchter and Ramm). In order to avoid artificial stiffening caused by the three dimensional displacement field and termed ‘thickness locking’, special provisions for the thickness stretch have to be introduced.

A unified approach for shear-locking-free triangular and rectangular shell finite elements

A new concept for the construction of locking-free finite elements for bending of shear deformable plates and shells, called DSG (Discrete Shear Gap) method, is presented. The method is based on a pure displacement formulation and utilizes only the usual displacement and rotational degrees of freedom (dof) at the nodes, without additional internal parameters, bubble modes, edge rotations or whatever. One unique rule is derived which can be applied to both triangular and rectangular elements of arbitrary polynomial order. Due to the nature of the method, the order of numerical integration can be reduced, thus the elements are actually cheaper than displacement elements with respect to computation time. The resulting triangular elements prove to perform particularly well in comparison with existing elements. The rectangular elements have a certain relation to the Assumed Natural Strain (ANS) or MITC-elements, in the case of a bilinear interpolation, they are even identical.

On the physical significance of higher order kinematic and static variables in a three-dimensional shell formulation

Abstract In recent years, considerable attention has been given to the development of higher order plate and shell models. These models are able to approximately represent three-dimensional effects, while pertaining the efficiency of a two-dimensional formulation due to pre-integration of the structural stiffness matrix across the thickness. Especially, the possibility to use unmodified, complete three-dimensional material laws within shell analysis has been a major motivation for the development of such models. While the theoretical and numerical formulation of so-called 7-parameter shell models, including a thickness stretch of the shell, has been discussed in numerous papers, no thorough investigation of the physical significance of the additional kinematic and static variables, coming along with the extension into three dimensions, is known to the authors. However, realization of the mechanical meaning of these quantities is decisive for both a proper modeling of shell structures, e.g. concerning loading and kinematic boundary conditions, and a correct interpretation of the results. In the present paper, the significance of kinematic and static variables, appearing in a 7-parameter model proposed by Buchter and Ramm (1992a) are discussed. It is shown, how these quantities ‘refine’ the model behavior and how they can be related to the ‘classical’ variables, such as ‘curvatures’ and ‘stress resultants’. Furthermore, the special role of the material law within such a formulation is addressed. It is pointed out that certain requirements must hold for the variation of kinematic and static variables across the thickness, to ensure correct results. In this context it is found, that the considered 7-parameter model can be regarded as ‘optimal’ with respect to the number of degrees of freedom involved.

Nonlinear shell formulations for complete three-dimensional constitutive laws including composites and laminates

One objective of the present study is to use arbitrary complete 3-dimensional constitutive equations without reduction or manipulation in nonlinear plate and shell analysis. The obvious consequence, namely the extension of a conventional 5-parameter shell formulation with Reissner-Mindlin kinematics to a 6-parameter formulation including the full set of stress and strain state does not solve the problem because a significant error in bending dominated cases occurs. To avoid this error the transverse normal strain is allowed to vary linearly across the thickness. This so-called 7-parameter theory recently proposed in the group of the authors resorts to the Enhanced Assumed Strain concept and preserves the basic features of a displacement formulation.The 7-parameter formulation is extended to the simulation of the response of laminated structures with arbitrarily large displacements and rotations. Following the main idea of the concept, it is sufficient to formulate a complete 3-dimensional material law which considers the layered setup of the shell.Finally, a layer-wise so-called multidirector model is developed which is well suited to grasp local interlaminar effects. In this formulation the displacement interpolation across the thickness is extended to a C0-continuous field described by a layer-wise Reissner-Mindlin kinematics. The purpose of the multidirector formulation is twofold: Firstly the higher order kinematics satisfies the same requirements discussed for the 7-parameter theory and allows also to use complete 3-dimensional material laws. Secondly it is appropriate to simulate laminates with extreme differences of the thicknesses or dissimilar material properties of each layer with sufficient accuracy. These are situations where formulations with C1-continuous displacement fields across the thickness fail.

On the mathematical foundation of the (1,1,2)-platemodel

Abstract The plate problem of three-dimensional (3-D), linearized elastostatics is considered inthe framework of the hierarchical modelling with the help of the energy projection method. It isshown that the (1,1,2)-bending model is the simplest asymptotically correct model in thehierarchical family, i.e. the distance between the solution of the 3-D problem and themodel-solution in the energy norm tends to zero for vanishing plate thickness h whileusing unmodified, 3-D material laws. The formulation of the (1,1,2)-bending model and the mechanical significance of itsingredients are discussed. We present error estimates for the deviation of the (1,1,2)-solutionfrom the Kirchhoff solution as well as from the 3-D-solution, by using the (1,1,2)-energy normand the 3-D-energy norm, respectively. The analysis leads to h as order of convergence. The results are illustrated by a numerical example.

MAGNETOSTRICTIVE BENDING OF AN ANISOTROPIC FREE CRYSTAL SUBSTRATE : ANALYTICAL AND NUMERICAL SOLUTIONS

A model to calculate the effective elastic modulus of a single crystal cantilever used as a substrate for magnetostrictive measurements of thin films is given. Calculations are presented for Si (110), sapphire (1100) (M plane) and (1210) (A plane) cantilevers. Numerical calculations using the finite element method for sapphire show a very high accuracy of the proposed model.

Algorithmic aspects of deformation dependent loads in non‐linear static finite element analysis

The present study focusses on algorithmic aspects related to deformation dependent loads in non‐linear static finite element analysis. If the deformation dependency is considered only on the right hand side, a considerable increase in the number of iterations follows. It may also cause failure of convergence in the proximity of critical points. If in turn the deformation dependent loading is included within the consistent linearization, an additional left hand side term emerges, the so‐called load stiffness matrix. In this paper several numerical test cases are used to show and quantify the influence of the two different approaches on the iteration process. Consideration of the complete load stiffness matrix may result in a cumbersome coding effort, different for each load case, and in certain cases its derivation is even not practicable at all. Therefore also several formulations for approximated load stiffness matrices are presented. It is shown that these simplifications not only reduce the additional effort for linearization and implementation, but also keep the iterative costs relatively small and still allow the calculation of the entire equilibrium path.

The Skeleton of the Sand Dollar as a Biological Role Model for Segmented Shells in Building Construction: A Research Review

Concrete double-curved shell constructions have been used in architectural design and building constructions since the beginning of the twentieth century. Although monolithic shells show a high stiffness as their geometry transfers loads through membrane forces, they have been mostly replaced by the more cost-efficient lattice systems. As lattice systems are covered by planar glass or metal panes, they neither reach the structural efficiency of monolithic shells, nor is their architectural elegance reflected in a continuous curvature. The shells of sand dollars’ – highly adapted sea urchins – combine a modular and multi-plated shell with a flexible, curved as well as smooth design of a monolithic construction. The single elements of the sand dollars’ skeleton are connected by calcite protrusions and can be additionally supported by organic fibres. The structural efficiency of the sea urchin’s skeleton and the principles behind them can be used for innovations in engineering sciences and architectural design while, at the same time, they can be used to illustrate the biological adaptations of these ecologically important animals within their environments. The structure of the sand dollar’s shell is investigated using modern as well as established imaging techniques such as x-ray micro-computed tomography (μCT), scanning electron microscopy and various optical imaging techniques. 3D models generated by μCT scans are the basis for Finite Element Analysis of the sand dollar’s shell to identify possible structural principles and to analyse their structural behaviour. The gained insights of the sand dollar’s mechanical properties can then be used for improving the state-of-the-art techniques of engineering sciences and architectural design.

Numerical approaches to stability analysis of cylindrical composite shells based on load imperfections

Purpose – Current procedures of buckling load estimation for thin-walled structures may provide very conservative estimates. Their refinement offers the potential to use structure and material properties more efficiently. Due to the large variety of design variables, for example laminate layup in composite structures, a prohibitively large number of tests would be required for experimental assessment, and thus reliable numerical techniques are of particular interest. The purpose of this paper is to analyze different methods of numerical buckling load estimation, formulate simulation procedures suitable for commercial software and give recommendations regarding their application. All investigations have been carried out for cylindrical composite shells; however similar approaches are feasible for other structures as well. Design/methodology/approach – The authors develop a concept to apply artificial load imperfections with the aim to estimate as good as possible lower bounds for the buckling loads of shel...

APPLICATIONS OF VARIATIONALLY CONSISTENT SELECTIVE MASS SCALING IN EXPLICIT DYNAMICS

Abstract. The aim of Selective Mass Scaling (SMS) in context of non-linear structural mechanics is to increase the critical time-step for explicit time integration without substantial loss in accuracy in the lower modes. The Conventional Mass Scaling (CMS) adds artificial mass only to diagonal terms of the lumped mass matrix and thus preserves diagonal format of mass matrix. It is usually applied in little number of small or stiff elements, like spot-welds in car crash, whose high eigenfrequencies limit time-step. However, translational and rotational inertia of the structure increases, which may cause non-physical phenomena. SMS technique adds artificial terms both to diagonal and non-diagonal terms, which results in non-diagonal mass matrix, but at least allows preservation of translational mass. Thus SMS can be used uniformly in domain with less non-physical artifacts. The previous works on SMS rely on algebraically constructed mass scaling matrices or stiffness proportional mass scaling. These approaches provide very small choice of mass scaling templates and they lack rigorous variational formulation. The goal of this paper is to develop variational basis for SMS with consistent discretization of inertial term and to assess efficiency of the proposed approach.