On Dimensional Adaptivity For Mixed Beam‐Shell‐Structures

A hierarchical model for dimensional adaptivity, using mixed beam‐shell structures, is presented. Thin‐walled beam structures are often calculated on the base of beam theories. Parts of the global structure, like framework corners, are usually analyzed with shell elements in a separate model. To minimize the modeling and calculation expense, a transition element to couple beam and shell structures is used. A dimensional adaptiv algorithm is introduced to automate this the procedure of modeling and calculation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupled multiscale finite element analysis of shell structures

In this paper, a coupled two-scale shell model is presented. A variational formulation and associated linearisation for the coupled global-local boundary value problem is derived. The discretisation of the shell is performed with quadrilaterals, whereas the local boundary value problems at the integration points of the shell are discretised using 8-noded or 27-noded brick elements, or solid shell elements. The coupled boundary value problem is simultaneously solved within a Newton iteration scheme. Solutions for small strain problems are computed within the so-called FE² method. In an important test, the correct material matrix for the stress resultants assuming linear elasticity and a homogeneous continuum is verified. Examples show that the developed two-scale model is able to analyse the global and local mechanical behaviour of heterogeneous shell structures. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Advanced Shell Element Formulations for Coupled Electromechanical Systems

With the significantly increasing applications of smart structures, piezoelectric material is widely used in branches of engineering sciences. Normally, the Finite Element Method is employed in the numerical analysis of these structures [2]. In this contribution, in order to avoid the locking effects and zero energy modes, the Assumed Natural Strain (ANS) Method [4] is implemented into four‐node piezoelectric shallow shell elements, by using the two‐field variational formulation in which displacements and electric potentials serve as independent variables and the three‐field variational formulation in which the dielectric displacement is taken as an independent variable additionally [3]. Moreover, a quadratic variation of the electric potential through the thickness direction is applied in the two‐field formulation. Numerical examples of piezoelectric sensors and actuators are presented, showing the behaviour of the shell elements by using different hybrid finite element formulations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On artificial geometry errors in the adaptive analysis of eigenvalues and eigenmodes of curved shell structures using the finite element method

The standard procedure to compute design loads for shell structures as also proposed in design rules is based on the computation of the limit load taking into account the modification of the so-called stability loads due to geometrical imperfections. The imperfections are mostly chosen affine to the buckling patterns, which are solutions of the eigenvalue-problem for the geometrically perfect structure. Thus, the eigenvalue-problems for stability points have to be solved very accurately. In the present contribution an adaptive h -refinement procedure is taken for the solution using low order shell elements. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Contact interactions and optimization of locally loaded shell structures

Methods for the solution of different problems of contact interaction of elements of shell structures with one another and dies of different types are generalized. We consider schemes of determination of breaking loads using the limit equilibrium theory and critical loads of local stability for shell systems under the indicated loading. Schemes of optimization of the shape of structural elements under local loads are presented. Results of experimental investigations are presented.     

A flexible approach to the simulation of hybrid multibody systems

The present works deals with the incorporation of both flexible beam and shell structures into the realm of flexible multibody dynamics. Geometrically exact beam formulations based on classical Simo-Reissner kinematics are suitable for modelling beam-type flexible components in the context of finite-deformation multibody dynamics. So geometrically exact shell formulations are based on Reissner-Mindlin kinematics. In [2], a flexible framework for dealing with flexible structural elements in a multibody context is described. A specific isoparametric finite element discretization of a shell formulation leads to semi-discrete equations of motions assuming the form of differential-algebraic equations (DAEs). A compatible isoparametric formulation of beams has already been developed in [1]. The uniform DAE framework makes possible the incorporation of alternative finite element formulations. In addition to that, various time-stepping schemes such as energy-momentum methods or variational integrators can be applied. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability Analysis of an Orbiting Plate with Finite Elements

This paper contributes to the analysis of rotating equilibria of non–driven systems (sometimes called relative equilibria), which are characterized by the fact that their angular momentum is conserved and non–zero. Interesting applications are usually found in space dynamics, in particular if large earth orbiting structures such as the space elevator are considered. For flexible structures relative equilibria can be found with Finite Element software packages. However, in this case the stability analysis is a non–trivial task since one usually has only limited account to the internal data of commercial FE solvers. Therefore, in [5] a new finite element based stability test was developed and applied to one–dimensional structural elements. Here, we consider a large–scale flexible plate orbiting around the earth in order to demonstrate that this method works well also for two dimensional shell–elements. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of reinforced concrete structures subjected to dynamic loads with a viscoplastic Drucker–Prager model

The behavior of reinforced concrete structures subjected to dynamic loads is analyzed. The concrete material is modelled by an elasto-viscoplastic law, whose inviscid counterpart is the Drucker–Prager model. A viscous regularization is introduced in order to avoid the mesh dependency effects that usually appear when strain softening occurs. The model is implemented in a general finite element computer code for fast transient analysis of fluid-structure systems, based on an explicit central difference scheme. The model is activated to both continuum elements and layered shell elements. So, realistic numerical analyses of complex 3-D engineering problems are simple and efficient. Three examples, two of which are modelled with layered shell elements, are presented below.     

Solid-Shell Finite Elements for Tesselated Geometries

Sandwich structures made of sheets of composite materials are in widespread use, particularly in the transportation industry. Finite Element simulation of the thin, tesselated structures with complex, three-dimensional material behaviour is a challenging task for the underlying element technology. In particular, frequently used linear isoparametric approaches exhibit unphysical stiffening phenomena. Recent developments in solid-shell finite element technology aim to overcome these undesirable effects. Here, they are applied to an example of a corrugated sandwich core under transversal compression. A study of convergence is conducted with respect to commercially available shell and solid finite elements, and their ability to reproduce the bending dominated deformation state. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dual-mixed variational formulation and hp finite element method for axisymmetric shell problems in elastodynamics

A new dimensionally reduced axisymmetric shell model is presented briefly for modeling time-dependent problems. This is based on the extended version of the three-field dual-mixed variational formulation of elastostatics [1, 2] to linear elastodynamics, the independent fields of which are the non-symmetric stress tensor, the displacement- and the rotation vector. An important property of the related shell model is that the classical kinematical hypotheses regarding the deformation of the normal to the shell middle surface are not used, i.e., unmodified three-dimensional constitutive equations are applied. The computational performance of the new h- and p-version axisymmetric shell finite elements is tested through a representative cylindrical shell problems. The development presented in this paper has been motivated by the fact that efficient dual-mixed hp plate and shell finite elements were managed previously to be developed for elastostatics by [1-5]. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of thin structured sheet metals by using FEM

Thin structured sheet metals promise high potential concerning lightweight design in industrial applications regarding the classical mechanical engineering and vehicle construction as well as the aeronautics. Compared to flat, unstructured sheet metals the component stiffness and buckling behavior can significantly be improved by structuring especially in out of plane direction. To be able to calculate the elastic behavior of large structures from structured sheet metals a mechanical surrogate model is developed which describes effectively average material parameters based on processes of homogenization. For the surrogate properties symmetry and antisymmetry boundaries and periodic boundaries respectively are contemplated on elementary cells whose structural mechanical behavior is decisive. By using an energetic approach [3] the stiffnesses of large plate and shell structures can be determined by a cooperatively small amount of finite elements. By means of these material properties elastic behavior can easily be calculated. With it an efficient numerical design is guaranteed. This explained analysis can be applied to other periodically built up plate structures. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geometrical super-elements for elasto-plastic shells with large deformation

This paper presents the application of the so-called Geometrical Elements Method (Lukasiewicz and Szyszkowski, 1974; Pogorelov, 1967) to the solution of elasto-plastic problems of shells. The approach is based on the observation that, during large deformations, the shell structure deforms in a nearly isometrical manner. Therefore, its deformed shape can be determined and analysed making use of the Gauss theorem according to which the Gaussian curvature of the isometrically deformed surface remains unchanged. The shell structure is subdivided into elements of two kinds: purely-isometrically deformed elements and quasi-isometrically deformed elements. The equilibrium of the whole structure is defined by the stationary value of the Hamiltonian function which requires the calculation of the strain energy in the elements. This can easily be obtained if we recognize that the isometrically deformed elements contain only bending energy. Using the method described, we are able to significantly the number of unknown values defining the shape of the deformed structure. The problem is reduced to the numerical evaluation of the minimum of a function of many variables. The elasto-plastic state of stress of the plastic material in the structure canbe determined by using the deformation theory of plasticity or the theory of plastic flow. Also, the strains and stresses in the plastic regions are the only functions of the assumed displacements field. The corresponding energy of the plastic deformation can easily be evaluated and added to the minimized functionals. For example, the elasto-plastic behaviour of a spherical shell under a concentrated load is studied. The solution obtained defines the large deformation behaviour and the motion of the plastic zones on the surface of the shell.     

Computation of the Three-Dimensional Stress State in Composite Shell Structures with Mixed Finite Elements

Being able to compute the complete three-dimensional stress state in layered composite shell structures is essential in order to examine complicated interlaminar failure modes such as delamination. We lay out a mixed finite element formulation with independent displacements, rotations, stress resultants and shell strains. A mixed hybrid shell element with 4 nodes and 5 or 6 nodal degrees of freedom is developed, so that the element formulation can also be used for problems with shell intersections. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical modeling of nonlinear deformation and buckling of composite plate-shell structures under pulsed loading

Nonlinear three-dimensional problems of dynamic deformation, buckling, and posteritical behavior of composite shell structures under pulsed loads are analyzed. The structure is assumed to be made of rigidly joined plates and shells of revolution along the lines coinciding with the coordinate directions of the joined elements. Individual structural elements can be made of both composite and conventional isotropic materials. The kinematic model of deformation of the structural elements is based on Timoshenko-type hypotheses. This approach is oriented to the calculation of nonstationary deformation processes in composite structures under small deformations but large displacements and rotation angles, and is implemented in the context of a simplified version of the geometrically nonlinear theory of shells. The physical relations in the composite structural elements are based on the theory of effective moduli for individual layers or for the package as a whole, whereas in the metallic elements this is done in the framework of the theory of plastic flow. The equations of motion of a composite shell structure are derived based on the principle of virtual displacements with some additional conditions allowing for the joint operation of structural elements. To solve the initial boundary-value problem formulated, an efficient numerical method is developed based on the finite-difference discretization of variational equations of motion in space variables and an explicit second-order time-integration scheme. The permissible time-integration step is determined using Neumann's spectral criterion. The above method is especially efficient in calculating thin-walled shells, as well as in the case of local loads acting on the structural element, when the discretization grid has to be condensed in the zones of rapidly changing solutions in space variables. The results of analyzing the nonstationary deformation processes and critical loads are presented for composite and isotropic cylindrical shells reinforced with a set of discrete ribs in the case of pulsed axial compression and external pressure.Scientific Research Institute of Mechanics, Lobachevskii Nizhegorodsk State University, N. Novgorod, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 757–776, November–December, 1999.     

Fraeijs de Veubeke variational principle-based cylindrical shell finite element model

In this paper a dimensionally reduced cylindrical shell model based on the dual-mixed variational principle of Fraeijs de Veubeke will be presented. The fundamental variables of this variational principle are the not a priori symmetric stress tensor and the skew-symmetric rotation tensor. The tensor of first-order stress functions is applied to satisfy translational equilibrium. A shell model derived in this way makes the application of the classical kinematical hypotheses unnecessary, and enables us to use unmodified three-dimensional constitutive equations. On the basis of this shell model, a new dual-mixed cylindrical shell finite elements capable of both h- and p-approximation can be derived. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling and optimal control of coupled structural acoustic systems with piezoelectric elements

We construct a model of a shell with piezoelectric elements (patches) that take into account the mutual influence of deformations and electric fields. Coupled problems for the shell with piezoelectric patches and an acoustic field, are studied and results on the existence and the uniqueness are obtained. For this system we consider an optimal control problem on noise attenuation and obtain results on the existence, the uniqueness, necessary and sufficient conditions of optimality. Copyright © 2003 John Wiley & Sons, Ltd.     

A Four-node Finite Element for Piezoelectric Shell Structures in Convective Co-ordinates

In this contribution, an isoparametric piezoelectric shell element is presented which is based on convective coordinates and which allows for the analysis of arbitrary shell geometries. A two-field variation formulation [1, 2] is used in which the displacements and the electric potentials serve as independent variables. Especially, for thin-walled structures under certain boundary conditions and load cases, the displacement based element tend to shear and membrane locking. In order to avoid this poor behaviour, the Assumed Natural Strain (ANS) method [3] is introduced into the piezoelectric shell element. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of the EAS-concept to a p-version of the finite shell formulation to avoid volumetric locking

A modification of the enhanced assumed strain concept is proposed to avoid volumetric locking phenomena in finite elements using higher-order p-shell element formulations based on Lagrangean polynomials and a linear finite shell kinematics. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)