# Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/_hp_ Finite Element Model
## 1 Introduction  
During the last several decades, studies of refined shell finite elements have taken importance due to their use in modeling variety of structures, mainly to study complex materials (e.g., laminated and functionally graded materials) and their applications in distinct areas of engineering. Consequently, many researchers have shown a strong interest in new and robust shell formulations, which require no artificial fixes such as reduced integration and stabilization. Through these developments, the performance of shell elements have been enhanced significantly. Continuous improvement of underlying mathematical models and computer implementations have been an area of significant research. Due to the amount of works in literature about shells, a detailed review is not presented here; the review of literature is limited to the present focus, namely, free vibration studies of shells. Through the years, different theories of shells have been proposed and the most used can be classified as follows: Classical Theories (CT), First-order Shear Deformation Theories (FSDT), and High-order Shear Deformation Theories (HSDT). The latest theories have been developed with the intention to avoid locking (of various kinds) present in the numerical solution of some shell theories, to enhance the performance of the elements, and to eliminate limitations (e.g., the thickness inextensibility) of these theories. The classical theories are based on the Kirchhoff-Love's hypothesis in which lines normal to the undeformed mid-surface remain straight and normal to the surface after deformation [1]; the kinematics of these theories is defined using three displacements in each direction of a point located on the mid-surface [2]. The use of formulations based on CT is limited to thin shells, mainly because such assumption underestimates the effects of normal and shear strains in the thickness direction [3]. In addition, often one neglects rotatory inertia in CT and thus overestimates natural frequencies of moderately thick shells [2]. The FSDT is developed [4] to overcome the limitations of CT [2]; nevertheless, FSDT still presents a few limitations and exhibits shear locking (in addition to thickness and membrane locking present in the CT). Here, the term locking refers to the stiffening effects created by the use of low-order \(C^{0}\) interpolation functions or by the kinematic assumptions. Generally, the shell kinematic is described with five parameters, three displacements in each coordinate direction at the mid-surface, and two rotations about the mid-surface coordinates (one can include the drilling degree of freedom or rotation about the transverse normal, as needed). In addition, this theory often is known as theory of five parameters. Limitation of this theory lies in its application to thin shells, need for shear correction factors, and transverse inextensibility. The shear correction factors are difficult to determine since they depend on the shell geometry, boundary conditions, and loads applied [3; 5]. Results obtained are accurate while the shell does not show significant deformation through its thickness [2]. This is due to the fact that the theory is based on zero normal strain [6]. The good accuracy obtained with FSDT, with respect to CT, is a result of taking into account the effects of transverse shear strain and rotatory inertia [7]. The numerical implementation of FSDT exhibits locking phenomena (e.g., shear locking [8; 9]), which is caused by the zero shear strains [10]. Some remedies to avoid locking phenomenon are the use of the method of assumed natural strain (ANS) [8], reduced integration techniques [9], and mixed formulations [11]. In HSDT, the displacement expansions considered are of high-order, and they can be applied to thick shells. These theories are developed using either polynomial or nonpolynomial expansions through thickness coordinate. The HSDT were introduced to avoid the use of shear correction factors needed in the FSDT [3].