Submit Manuscript | http://medcraveonline.com Introduction Hyperbolic paraboloid shell bounded by straight lines (commonly known as hypar shell) is a good choice as roofng unit to civil engineers due to its aesthetic beauty and capability to allow entry of north light. In this age of advanced materials composite skewed hypars defne a rich area of research. Schwarte 1 worked on free vibration of isotropic rhombic hypar shell and the twisted plates which have structural resemblance with hypar shells received attention from several authors like Kielb, 2 Seshu & Ramamurti 3 and others. Chakravorty et al., 4 in 1998 reported natural frequency and forced vibration response of corner point supported skewed hypar shell. Thus it is evident that most of the work on hypar shells deals with fundamental frequency and frequency for higher modes received limited attention only. Moreover the effect of neglecting tangential and /or rotary inertia on the natural frequencies of hypar shell has not received any attention. The frst four natural frequency of simply supported composite hypar shell and the effect of neglecting tangential and / or rotary inertia on them is presented. Mathematical formulation An eight–noded curved quadratic isoparametric fnite element is used for hypar shell analysis. The fve degrees of freedom taken into consideration at each node are u, v, w, α , β . The strain–displacement relations on the basis of improved frst order approximation theory for thin shell are established which was provided by solution of benchmark problems reported elsewhere 5 and are established as { } { } { } 0 0 0 0 0 T T x y xy xz yz x y xy xz yz T x y xy xz yz z k k k k k ε ε γ γ γ ε ε γ γ γ = + (1) Where the frst vector is the mid–surface strain for a hypar shell and the second vector is the curvature. These are given, respectively, by 0 0 0 0 0 / / / / 2 / / / x y xy xy xz yz u x v y u y v x w R w x w y ε ε γ α γ β γ ∂ ∂ ∂ ∂ ∂ ∂ +∂ ∂ − = +∂ ∂ +∂ ∂                             , / / / / 0 0 x y xy xz yz k x k y k y x k k α β α β ∂ ∂ ∂ ∂ = ∂ ∂ +∂ ∂                             (2) A laminated composite hypar shell of uniform thickness h and twist radius of curvature Rxy is considered. Keeping the total thickness same, the thickness may consist of any number of thin laminae each of which may be arbitrarily oriented at an angle θ with reference to the x–axis of the co–ordinate system. The constitutive equations for the shell are given by { } [ ]{ } F D ε = (3) {} { } T x y xy x y xy x y F N N N M M M Q Q = , [] [][] [] [][][] [] [] [] 0 0 0 0 A B D B D S =           , {} { } 0 0 0 0 0 T x y xy x y xy xz yz k k k ε ε ε γ γ γ = . (4) The stiffness coeffcients are defned as 1 1 ( )( ) np ij ij k k k k A Q z z − = ∑ = − ; 2 2 1 1 1 ( )( ) 2 np ij ij k k k k B Q z z − = ∑ = − ; 3 3 1 1 1 ( )( ) 3 np ij ij k k k k D Q z z − = ∑ = − i,j=1,2,6; 1 1 ( )( ) np ij i j ij k k k k S FF G z z − = ∑ = − i,j=1,2; (5) Material Sci & Eng Int J. 2018;2(4):139‒142. 139 © 2018 Sahoo. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Simply supported composite hypar shells under free vibration–some observations Volume 2 Issue 4 - 2018 Sarmila Sahoo Heritage Institute of Technology, India Correspondence: Sarmila Sahoo, Department of Civil Engineering, Heritage Institute of Technology, Kolkata– 700 107, India, Email sarmila.sahoo@gmail.com Received: November 27, 2017 | Published: August 28, 2018 Abstract A general finite element procedure is presented to model the hypar shells using eight–noded curved quadratic isoparametric elements with five degrees of freedom per node including two inplane displacements and one transverse displacement and two rotations. Problems of twisted cantilever plate, which have structural resemblance with skewed hypar shell, are solved using the present approach and the results are compared with published ones. Having established the exactitude of the present formulation, numerical experiments with simply supported skewed composite hypar shells are conducted for four different types of laminations including four layered symmetric and antisymmetric cross and angle ply laminates. The first four natural frequencies are presented in tabular forms and are studied critically and a set of meaningful conclusions are derived. Keywords: hypar shell, composite, natural frequency Material Science & Engineering International Journal Research Article Open Access