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CivilComp Proceedings
ISSN 17593433 CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 38
Hybrid StressStrain Elements based on the FirstOrder SingleLayer and LayerWise Shell Theories G.M. Kulikov and S.V. Plotnikova
Department of Applied Mathematics and Mechanics, Tambov State Technical University, Tambov, Russia G.M. Kulikov, S.V. Plotnikova, "Hybrid StressStrain Elements based on the FirstOrder SingleLayer and LayerWise Shell Theories", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 38, 2004. doi:10.4203/ccp.79.38
Keywords: multilayered shell, fournode curved shell element, rigidbody motion.
Summary
Using the solidshell concept in a finite element (FE) formulation is well established and has been shown to give acceptable results [1,2,3]. In order to develop the
solidshell elements that overcome shear, membrane, trapezoidal and thickness locking, advanced FE techniques were applied. In this light, in some works for construct
ing the solidshell elements only displacements of the face surfaces are used. A main
idea of such approach is that displacement vectors of the face surfaces of the shell
are represented in some global Cartesian basis in order to exactly describe rigidbody motions.
Herein, it is developed a close FE formulation based on the firstorder shell the ory [4,5]. But in our FE development selecting as unknowns the displacements of the face surfaces in a form has a principally another mechanical sense and allows to formulate curved shell elements with very attractive properties, since objective straindisplacement relationships [4], i.e., invariant under rigidbody motions are applied. In approximation (8) the following notations are used: is the displacement vector; are the components of this vector; are the displacement vectors of face surfaces ; are the components of these vectors; and are the orthogonal curvilinear coordinates of the reference surface; is the normal coordinate; and are the tangent unit vectors to the lines of principal curvatures; is the vector normal to the reference surface; are the linear shape functions of the shell. Taking into account that displacement vectors of the face surfaces (9) are represented in the local reference surface basis, the developed FE formulation has computational advantages compared to conventional isoparametric FE formulations be cause it eliminates the costly numerical integration by deriving the stiffness matrix. Besides, our element matrix requires only direct substitutions, i.e., no inversion is needed if sides of the element coincide with the lines of principal curvatures of the reference surface, and it is evaluated by using the full exact analytical integration. The proposed FE formulation is based on a simple and efficient approximation of shells via fournode curved elements. To avoid shear and membrane locking and have no spurious zero energy modes, the assumed stress resultant and strain fields are invoked. In order to circumvent thickness locking, the simplified constitutive stiffness matrix [1,2,4] corresponding to the plane stress state is employed. Note also that fundamental unknowns consist of 6 displacements and 11 strains of the face surfaces of the shell, and 11 stress resultants. Therefore, for deriving element characteristic arrays the HuWashizu variational principle should be applied. The described approach may be readily generalized on the firstorder layerwise shell theory where as unknown functions the displacements of the face surfaces of layers are chosen [6]: where are the linear shape functions of the th layer; is the number of layers. Here, the fundamental unknowns consist of displacements of the face surfaces of layers (11), and strains and stress resultants. The numerical results are presented to demonstrate the efficiency and high accuracy of both developed approaches and to compare their with other stateoftheart FE formulations. For this purpose extensive numerical studies are employed. References
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