## Free Vibrations and Static Deformations of Composite Laminates and Sandwich Plates using Ritz Method

##### Abstract

In this study, Ritz method has been employed to analyze the following problems: free vibrations of plates with curvilinear stiffeners, the lowest 100 frequencies of thick isotropic plates, free vibrations of thick quadrilateral laminates and free vibrations and static deformations of rectangular laminates, and sandwich structures. Admissible functions in the Ritz method are chosen as a product of the classical Jacobi orthogonal polynomials and weight functions that exactly satisfy the prescribed essential boundary conditions while maintaining orthogonality of the admissible functions. For free vibrations of plates with curvilinear stiffeners, made possible by additive manufacturing, both plate and stiffeners are modeled using a first-order shear deformation theory. For the thick isotropic plates and laminates, a third-order
shear and normal deformation theory is used. The accuracy and computational efficiency of formulations are shown through a range of numerical examples involving different boundary conditions and plate thicknesses. The above formulations assume the whole plate as an equivalent single layer. When the material properties of individual layers are close to each other or thickness of the plate is small compared to other dimensions, the equivalent single layer plate (ESL) theories provide accurate solutions for vibrations and static deformations of multilayered structures. If, however, sufficiently large differences in material properties of individual layers such as those in sandwich structure that consists of stiff outer face sheets (e.g., carbon fiber-reinforced epoxy composite) and soft core (e.g., foam) exist, multilayered
structures may exhibit complex kinematic behaviors. Hence, in such case, Cz0 conditions, namely, piecewise continuity of displacements and the interlaminar continuity of transverse stresses must be taken into account. Here, Ritz formulations are extended for ESL and layerwise (LW) Nth-order shear and normal deformation theories to model sandwich structures with various face-to-core stiffness ratios. In the LW theory, the C0 continuity of displacements is satisfied. However, the continuity of transverse stresses is not satisfied in both ESL and LW theories leading to inaccurate transverse stresses. This shortcoming is remedied by using a one-step well-known stress recovery scheme (SRS). Furthermore, analytical solutions of three-dimensional linear elasticity theory for vibrations and static deformations of simply supported sandwich plates are developed and used to investigate the limitations and applicability of ESL and LW plate theories for various face-to-core stiffness ratios. In addition to natural frequency results obtained from ESL and LW theories, the solutions of the corresponding 3-dimensional linearly elastic problems obtained with the commercial finite element method (FEM) software, ABAQUS, are provided. It is found that LW and ESL (even though its higher-order) theories can produce accurate natural frequency results compared to FEM with a considerably lesser number of degrees of freedom.

##### General Audience Abstract

In everyday life, plate-like structures find applications such as boards displaying advertisements, signs on shops and panels on automobiles. These structures are typically nailed,
welded, or glued to supports at one or more edges. When subjected to disturbances such as wind gusts, plate-like structures vibrate. The frequency (number of cycles per second)
of a structure in the absence of an applied external load is called its natural frequency that depends upon plate's geometric dimensions, its material and how it is supported at the
edges. If the frequency of an applied disturbance matches one of the natural frequencies of the plate, then it will vibrate violently. To avoid such situations in structural designs,
it is important to know the natural frequencies of a plate under different support conditions. One would also expect the plate to be able to support the designed structural load without
breaking; hence knowledge of plate's deformations and stresses developed in it is equally important. These require mathematical models that adequately characterize their static and
dynamic behavior. Most mathematical models are based on plate theories. Although plates are three-dimensional (3D) objects, their thickness is small as compared to the in-plane dimensions. Thus, they are analyzed as 2D objects using assumptions on the displacement fields and using quantities averaged over the plate thickness. These provide many plate theories, each with its own computational efficiency and fidelity (the degree to which it reproduces behavior of the 3-D object). Hence, a plate theory can be developed to provide accurately a quantity of interest. Some issues are more challenging for low-fidelity plate theories than others. For example, the greater the plate thickness, the higher the fidelity of plate theories required for obtaining accurate natural frequencies and deformations. Another challenging issue arises when a sandwich structure consists of strong face-sheets (e.g., made of carbon fiber-reinforced epoxy composite) and a soft core (e.g., made of foam) embedded between them. Sandwich structures exhibit more complex behavior than monolithic plates. Thus, many widely used plate theories may not provide accurate results for them. Here, we have
used different plate theories to solve problems including those for sandwich structures. The governing equations of the plate theories are solved numerically (i.e., they are approximately
satisfied) using the Ritz method named after Walter Ritz and weighted Jacobi polynomials. It is shown that these provide accurate solutions and the corresponding numerical algorithms
are computationally more economical than the commonly used finite element method. To evaluate the accuracy of a plate theory, we have analytically solved (i.e., the governing equations
are satisfied at every point in the problem domain) equations of the 3D theory of linear elasticity. The results presented in this research should help structural designers.

##### Collections

- Doctoral Dissertations [14992]

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