#### Abstract

The main subjects of the article are theoretical strength and buckling analyses of a sinusoidal velaroidal shell subjected to its self-weight and uniformly distributed load in geometric nonlinearity. A short history of finite element applications to shell buckling is given. The finite element method is used in its matrix formulation. The elastic stiffness matrixes in the local coordinate system of the membrane element are defined in their general form. The out-of-plane geometric stiffness matrix for the plate along the same lines is derived

Velaroidal surface is a surface of translation on the flat rectangular plan with a generating curve of variable curvature [1],[2]. Thus, the surface is limited by four mutually orthogonal contour straight lines (kx = ky = 0) lying in the same plane. A sinusoidal velaroid generates by two families of half waves of the sinusoids lying in mutually perpendicular planes and facing by convexities into the same side [3]. Each set of sinusoids has the identical period. Sinusoidal velaroid is limited by a flat rectangular contour. The history of finite element applications to shell buckling is extensive going back to the work of Clough and Johnson (1968). The natural mode contribution of Argyris et al. (1977) was a major addition to shell theory. It was recently modified to include elastoplastic effects (Argyris et al. 2000). Horrigmoe and Bergan (1978) used the co-rotational method for nonlinear analysis and Bathe and Ho (1980), Hsiao (1987), Mohan and Kapania (1997), Peng and Crisfield (1992) improved element performance along those lines. The 3-D elasticity "degenerate" element of Ahmad et al. (1970) was followed by, among others, Bathe and Balourchi (1980), Hughes and Lui (1981), Dvorkin and Bathe (1984), and Buechter and Ramm (1992). In an excellent review, Ibrahimbegovic (1997) addresses the various approaches and the complex issues involved. Here the derivation of the geometric stiffness matrix is somewhat different but consistent with the approach used throughout this text. The linear equilibrium equations for a flat triangular shell element in its local coordinates system are first perturbed to yield the in-plane geometric stiffness matrix. Then out-of-plane considerations that involve the effect of rigid body rotations on member forces yield an out-of-plane geometric stiffness matrix. The shell element that was chosen for that purpose combines the constant stress triangle (CST) flat triangular membrane element (Zienkeiwicz (1977)) and of the discrete Kirchhoff theory (DKT) flat triangular plate element (Batoz et al. (1980)). Let’s consider a sinusoidal velaroidal shell [4],[5] with the inner radius r0 = 0, the outer variables radii from 10m to 20m and the number of waves n= 8 (Fig. 1). The Geometric Stiffness Matrix of Triangular Element Shells. The local geometric stiffness matrix of the shell element is split into three distinct matrices: , (1) where the first, second and third terms on the R.H.S. of Eq.1. represent the in-plane geometric stiffness matrix of the membrane, the in-plane geometric stiffness matrix of the plate and the out-of-plane geometric stiffness matrix of the shell element respectively. The total, 'tangential' stiffness matrix for use in nonlinear analysis will include, in addition, the linear elastic stiffness matrices of a plane stress triangular element (membrane) and that of a triangular plate element. The geometrically nonlinear triangular shell element has eighteen local degrees of freedom (DOF's): 3 displacements and 3 rotations at each node. The membrane element contributes to nine displacement DOF's only. The basic three noded constant stress triangular flat element has only six local displacement DOF's that are shown in Figure 2. The out-of-plane contribution (the normal stiffness) of the membrane element to the basic local shell element is a displacement DOF in the direction normal to the plane of the element. The flexural element contributes to eighteen local DOF's. The basic three noded plate triangular flat element has only nine DOF's that are shown in Figure 3. The in-plane contribution to the basic local element adds two displacement DOF's in the plane of the element. The out-of-plane contribution adds a rotation DOF in the direction normal to the plane of the element. In-Plane Contribution of the Triangular Membrane Element. The elastic stiffness matrix in the local coordinate system of the membrane element has the general form: . (2) The local in-plane contribution of the membrane element to the geometric stiffness matrix is the gradient of the nodal force vector: . ((3) In-Plane Contribution of the Triangular Plate Bending Element. The general form of the elastic stiffness matrix of the DKT triangular plate element may be written as: , (4) where and are the usual area coordinates and Bplate, which appears explicitly in Batoz et al. (1980) as a function of is slightly more complex than the Bmem of the membrane. Here again the local in-plane contribution of the plate element to the geometric stiffness matrix will be derived as the gradient for fixed M, of the element nodal force vector which is given as: (5) where , (6) , (7) . (8) The in-plane geometric stiffness matrix may be written symbolically as: (9) where . (10) The expressions for the individual terms of the geometric stiffness matrix were obtained in closed form using symbolic algebra. When applied to the plate, chain rule differentiation with respect to the coordinates yields (11) where dxr =ur ; dyr =vr ; r=i,j,m. The first expression on the R.H.S. of Eq. l1 returns the elastic stiffness matrix whereas the second expression becomes the geometric stiffness matrix. It is left to clarify what is meant by Mfixed in Eq. 11. The moment vector M, of Eq. 6 contains three components that are functions of and and defined as: (12) , (13) , where (14) , (15) , (16) . (17) When the components of M (Eqs. 12-14) are inserted into Eq. 11 it is the values of these components at the nodes (Eqs. 15-17) that are held fixed. Out-of-Plane Contribution to the Shell Geometric Stiffness Matrix. This section will derive the out-of-plane geometric stiffness matrix for the plate along the same lines and subsequently present a combined out-of-plane contribution. The derivation starts with the change in a vector G, due to a small rotation that is given by Goldstein (1950) as , (18) where is the rigid body rotation vector due to changes in the geometry. In terms of joint displacements with respect to the local coordinate system, two components and -, of the rigid body rotation are obtained from Figure 4. The third component is chosen for the plate arbitrarily, as the local z-rotation of node i. Recall that for the membrane this component is included in the in-plane contribution: ; , (19) where is the displacement in the local (primed) z-direction of node r. At each node, the forces, moments and rotations may be written as ; ; for r = i, j, m and the changes in the force and moment vectors are given in more detail as , (20) . (21) The out-of-plane stiffness contribution of the plate is now obtained as: , (22) where ; . (23) Now can be described in terms of the displacement vector as: , (24) where (25) , (26) , (27) and ; . (28) Carrying out Eq. 22 for the plate and adding the out-of-plane contribution to the geometric stiffness matrix, in the local coordinate system more explicitly (adjusted to 18 DOF's) for the membrane, results in the following out-of-plane shell geometric stiffness matrix with respect to the local coordinate system: (29) where (30); (31) , (32) Conclusions: The theoretical approach of the buckling and strength analyses of the sinusoidal velaroidal shell is worked out using the finite element method. The in-plane and the out-of-plane shell geometric stiffness matrices with respect to the local coordinate system are obtained. This result gives a possibility of further numerical strength and buckling analyses of the sinusoidal velaroidal shells.

### MATHIEU GIL-OULBE

RUDN University

**Author for correspondence.**

Email: gil-oulbem@hotmail.com

Russia 117198, Moscow, Miklukho-Maklaya street, 6 PhD, Assoc. Prof.

### Svetlana Lvovna SHAMBINA

RUDN University

Email: shambina_sl@mail.ru

Russia 117198, Moscow, Miklukho-Maklaya street, 6 PhD, Assoc. Prof.

### TYEKOLO DAU

RUDN University

Email: tiek.d@hotmail.com

Russia 117198, Moscow, Miklukho-Maklaya street, 6 PhD, Assis. Prof.

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