A four node quadrilateral shell element for free vibration response of functionally graded spherical shell panels

The present work involves the free vibration analysis of functionally graded spherical shells using a four-node quadrilateral shell element. This four-node, shell element has seven DOF (Degrees of Freedom) per node namely; three displacements, two rotations of mid-plane, and two transverse shear strain com-ponents. The element is developed by modifying the discrete Kirchhoff quadri-lateral shell element based on Reddy’s third-order theory developed earlier by the authors for the cylindrical shell. By transforming all the degrees of freedom from local to global coordinates using a transformation matrix converts the plate bending element into a flat shell element, resulting in the present formulation being suitable to give results for thin shells as well as for thick shells. In the present work functionally graded spherical shell panels with simply supported as well as clamped boundary condition, with various radius to span ratios and with different volume fraction indices are analyzed for free vibration response. The power law property variation through the thickness is considered in this study. To assess the performance of the developed element, the results, of non-dimen-sionalised frequencies are compared with the results presented in the literature. Comparison of results shows that the developed element yields quite accurate results even with the coarser mesh, which indicates the computational efficiency. It is also seen that the percentage difference between the present results and the other results available in the literature is less than 2 in most of the cases.


Introduction
The Functionally graded materials are useful for structural elements that are subjected to high differential temperatures. Laminated composite can also be used for the above applications, but at the layer interface, there is discontinuity of mechanical properties and are susceptible to delamination. FG materials do not suffer from delamination as the material properties vary smoothly across the thickness and thus advantages over the laminated composite materials. FG material is microscopically heterogeneous and usually contains two material phases namely; Ceramic and metal. Ceramic acts as a thermal barrier and metal impart the required ductility. In the literature, various 2D and 3D solutions have been presented by various researchers. In the present study, a four-node quadrilateral shell element is developed based on Reddy's third-order theory which was used earlier by for free vibration response of functionally graded cylindrical shells, since the finite element technique is useful for the analysis of spherical shells with various displacements-based boundary conditions. IOP Publishing doi:10.1088/1757-899X/1236/1/012010 2 Earlier, Dahale et.al. [1] have developed a four node quadrilateral element based on Reddy's theory for the free vibration analysis of functionally graded cylindrical shell panels, where C 1 continuity problem posed by Reddy's third order theory in finite element formulation is successfully circumvented by using improved discrete Kirchhoff interpolation functions. The Present finite element results are compared with the finite element results based on Sander's theory and Donnell's theory of Pandey et.al. [3] and Liu et.al. [4] The present results are also compared with the finite element results based on higher order shear deformation theory with eight-node and nine degrees of freedom per node of Pradyumna et.al. [5] It is seen form the literature that no researcher has so far used the four node quadrilateral plate element using the novel concept of obtaining the transformation matrix, which is used in present formulation for the analysis of functionally graded spherical shell panels.
The present element is observed to yield quite accurate results. The use of two separate coordinate systems for translational and rotational degrees of freedom and the use of the plate theory is the novelty of the present approach.

Displacement Approximation for the Reddy's Third Field Order Theory
Consider a spherical shell panel as shown in ' Figure 1' with thickness 'h'. The shell's middle surface is known to be a reference plane where z is zero and the top surface is at z = 'h/2'and bottom surface is at z = '-h/2'. The displacement field approximation of Reddy's third order theory assumes a third order variation for the in-plane displacement and a uniform transverse displacement across the thickness. The condition of zero shear stress at the top and bottom of the plate imposed and has same number of primary variables as the first order shear deformation theory. Reddy's third order theory does not require the shear correction factor unlike first order shear deformation theory. This theory can yield quite accurate results for thick as well as thin plates.
The displacement field approximation is as follows: Where, Consider a spherical shell panel as shown in ' Figure 1'. The properties changes gradually from ceramic phase to the metal phase in functionally graded material. Voigt's rule of mixtures (ROM) is used for predicting the material property variation, through the thickness as follows.
( ) In above equation n is the power law index which can vary from 0 to ∞. Pb represents the properties of the material at bottom and that of the top of the shell is Pt. Pz is the properties of the material at any z-coordinates. Poisson's ratio is assumed to be constant.

Finite Element Formulation
The spherical shell shown in ' Figure 1' is discretized using the quadrilateral elements developed by the same authors and presented in Dahale et.al.  Cartesian coordinates (x, y, z) and surface coordinates (ξ1, ξ2, ξ3) are shown in ' Figure 2'. In surface coordinate ξ3 is taken normal to the shell surface at every node of the element. It is assumed that rotation about the normal to the tangent plane would be negligible in the actual shell. Accordingly, rotational degrees of freedom referred to the surface coordinates (ξ1, ξ2) per nodal point are only included in the base system describing the assembled structure by Clough et.al. θ θ ψ ψ are transformed to surface coordinate system (ξ1, ξ2, ξ3), neglecting the contribution to (ξ3). Ui e′ is local and Ui e is the global displacement vector for i th node shown in the following expressions.
Transformation matrix T i for transforming the local translational DOF to global translational DOF for an i th node is given by the following expression.
where λ11, λ12 etc. are the direction cosines of the local (primed) axes with respect to global (unprimed) axes.
The relationship between the local rotations and global rotations for an i th node is given by the following expression.
The relationship between the surface rotations θ0ξ i and θ0 i , for the i th node can be obtained as follows, For a spherical shell, the angle between the global x-axis and tangent ξ1 can be obtained easily as this tangent is actually tangent to the circle x 2 + z 2 = R 2 in the x-z plane. Differentiating this expression for the circle yields, tan α =-x /z. α is the angle between the axes ξ1 and x. In addition, as the axis ξ2 is along the global y-axis, is the tangent to the circle y 2 + z 2 = R 2 in y-z plane, considering the angle between the global y axis and ξ2 is β i.e. tan β = -y/z. Using this, the relationship between 0 i ξ θ and 0 i θ for the i th node can be obtained. Transformation matrix i T ξ for transforming the surface rotational DOF to global rotational DOF for i th node is given by the following expression.
After deriving the expressions for transformation matrix T e , local stiffness matrix K e′ , local mass matrix M e′ and local load vector P e′ , finally the element stiffness matrix K e of size 28×28, element mass matrix M e of size 28 × 28 and element load vector P e of size 28×1 with reference to global axes are given as where , , K M and P are the assembled counterparts of matrices K e , M e , and P e . For synchronous vibration, the P (load vector) is set to zero. The undamped natural frequencies ωn are obtained by solving eigenvalue problem.

Numerical Results
A computer program is developed in FORTRAN for the free vibration analysis based on the formulation in section 2. A computer program and finite element formulation are assessed by comparing the result of non-dimensionalised natural frequencies obtained using developed element with the results in the literature. In all the problems considered in the study, the full shell panel is discretized with N × N mesh of equal size elements. The shell type and material properties considered are specified in ' Table 1'. . It is noted that the present element is not based on shell theory. It is also observed that even though the element is based on plate theory it gives sufficiently accurate results not only for the shallow shell but for deep shells too. The 24 ×24 mesh size shell having 625 total nodes and 4375 total degrees of freedom. It is also to be noted that as the volume fraction increases, the values of the non-dimensionalised frequencies decrease because of the increase in metal part and thereby reducing the stiffness. The non-dimensionalised fundamental frequencies of FGM2 spherical shells are analyzed in ' Table 3' for 'R/a' = 1, 5, and 10 with 'a/h' ratio is equal to 10. The shells of the power law indices IOP Publishing doi:10.1088/1757-899X/1236/1/012010 7 'n' varying from 0, 0.2, 0.5 and 1 are presented in the table. The present results obtained using 12 × 12, 16 ×16, and 24 × 24 mesh are compared with those presented by Pradyumna et.al. [5] obtained using eight-node element having nine degrees of freedom per node. The formulation is based on the higher-order shear deformation theory. It is observed that for deep to shallow shell panels the present element results are quite comparable with those of Pradyumna et.al. [5].

All-round clamped spherical shell panel with square plan-form
The fundamental non-dimensionalised frequencies of all-round clamped square planform FGM1 spherical shells for 'a/h' = 5, 10, and 20 for two different 'R/a' ratios are presented in '  The results for the first six mode shapes for FGM1 shallow shell panel with 'R/a' = 5 and 'a/h' = 5 for volume fraction index = 1 are plotted in Fig 4 and Fig 5 for the all-round clamped and for one edge clamped and other three free (cantilever) boundary conditions respectively. The mesh size of 24 × 24 is used for obtaining mode shapes.These results will serves as a benchmark for the other researchers.