Keywords: multilayered shell, four-node curved shell element, rigid-body motion.

Using the solid-shell 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 solid-shell elements that overcome shear, membrane, trapezoidal and thickness locking, advanced FE techniques were applied. In this light, in some works for construct- ing the solid-shell 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 rigid-body motions.

Herein, it is developed a close FE formulation based on the first-order 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 strain-displacement relationships [4], i.e., invariant under rigid-body 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 curvi-linear 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 four-node 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 Hu-Washizu variational principle should be applied.

The described approach may be readily generalized on the first-order layer-wise 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 state-of-the-art FE formulations. For this purpose extensive numerical studies are employed.

1

M.F. Ausserer, S.W. Lee, "An eighteen-node solid element for thin shell analysis", International Journal of Numerical Methods in Engineering, 26, 1345-1364, 1988. doi:10.1002/nme.1620260609

2

K.Y. Sze, S. Yi, M.H. Tay, "An explicit hybrid stabilized eighteen-node solid element for thin shell analysis", International Journal for Numerical Methods in Engineering, 40, 1839-1856, 1997. doi:10.1002/(SICI)1097-0207(19970530)40:10<1839::AID-NME141>3.0.CO;2-O

3

K.Y. Sze, W.K. Chan, T.H.H. Pian, "An eight-node hybrid-stress solid-shell element for geometric non-linear analysis of elastic shells", International Journal for Numerical Methods in Engineering, 55, 853-878, 2002. doi:10.1002/nme.535

4

G.M. Kulikov, S.V. Plotnikova, "Simple and effective elements based upon Timoshenko-Mindlin shell theory", Computer Methods in Applied Mechanics and Engineering, 191, 1173-1187, 2002. doi:10.1016/S0045-7825(01)00320-6

5

G.M. Kulikov, S.V. Plotnikova, "Investigation of locally loaded multilayered shells by a mixed finite-element method. 1. Geometrically linear statement", Mechanics of Composite Materials, 38, 397-406, 2002. doi:10.1023/A:1020978008577

6

G.M. Kulikov, "Non-linear analysis of multilayered shells under initial stress", International Journal of Non-Linear Mechanics, 36, 323-334, 2001. doi:10.1016/S0020-7462(00)00017-2

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