Vibration analysis of a layered composite sheet with the help of two-variable theory refined by hierarchical finite element method.

Master's Thesis of Mechanical Engineering-Applied Design

Abstract

In this thesis, classical sheet theories, first-order shear deformation theory and refined two-variable element theory for the free vibration problem, using standard finite element methods and Hierarchical limit is checked. The refined two-variable theory is an equivalent single-layer theory, in which two bending and shearing components are used to express the displacement field, and the bending component has no effect on the shearing forces, while the shearing component has no effect on the bending moments. By removing the shear component, this theory will be similar to the classical sheet theory. Also, this theory considers the shear strain changes along the thickness of the sheet to be parabolic, so there is no need for a shear correction factor. Hierarchical finite element method is an element grid refinement method, in which the number of nodal points of the element does not change by increasing the order of the shape functions used for displacement approximation. In this research, the features of the hierarchical finite element method and the shape functions of the usable one-dimensional element are explained, then the method of using these functions for the two-dimensional element is explained. In the following, the formulation of the standard and hierarchical finite element methods for classical layered sheet theories is obtained and the frequencies are compared for different boundary conditions and by changing the number of layers with exact solution. It can be seen that in the hierarchical finite element method using less degrees of freedom, more accurate answers are obtained than the standard finite element method. Also, the standard and hierarchical finite element formulation for the theory of first-order shear deformation is obtained, and the effect of shear locking is investigated by changing the ratio of length to thickness of the sheet. By using the hierarchical finite element method while avoiding the shear locking effect, the results of the free frequency of the sheet have better accuracy than the standard finite element method. After that, the refined bivariate theory of the sheet is presented and the equations of motion and the formulation of the standard and hierarchical finite element methods are obtained for it. The natural frequencies in this theory for symmetric and asymmetric laminated perpendicular sheets for fundamental and higher modes are obtained by exact solution method, standard and hierarchical finite element methods. By comparing the results, the superiority of the hierarchical finite element method over the standard finite element method is observed. Then, by changing the various parameters of the perpendicular and slanted layered sheet, such as the ratio of the modulus of elasticity, the change of the length-to-thickness ratio, and the change of the length-to-width ratio for different supporting conditions of the sheet, the behavior of this theory is investigated by standard and hierarchical finite element methods and compared with the results of the exact solution of this theory.

Key words: classical theory, first-order shear deformation theory, refined two-variable theory, standard finite element method, hierarchical finite element method

The use of composite materials is widely used in aerospace, automotive and maritime structures. In general, the composite material consists of two parts, field and field. The strands are usually harder and stronger than the base and bear the main load in the composite material, and the base acts as a protector of the strands as well as a means of distributing the load. The matrix and filaments are bonded together at a controlled temperature and pressure to form a composite material that differs from both materials in terms of mechanical properties. Composite materials can be optimized for strength, stiffness, fatigue, and heat and steam resistance by changing fiber orientation. Another characteristic of composite materials compared to ordinary materials is their high strength-to-weight ratio. Structural components such as beams and sheets are created by stacking layers at different angles in order to achieve desired characteristics.

Resonance phenomenon in structural components and mechanical systems reduces the life of equipment and even causes complete and premature failure. Intensification is influenced by the mass and stiffness characteristics of the structure. Modal analysis obtains the vibration modes and their frequencies. This method can be used for simple structures.But when the structure becomes complicated or is subjected to complex loadings, the finite element analysis method is used to obtain the natural frequencies and modes of the system. 1-1 History of solving the free vibration problems of sheets The beginning of the study of the vibration behavior of sheets goes back to the end of the 1800s. When Reilly presented his famous method to investigate the free vibration of structures. [3] Ritz then improved Rayleigh's method in 1909 by considering a set of test shape functions, each with independent amplitude coefficients. In this way, the Rayleigh-Ritz method became one of the widely used approximate methods in the field of investigating the vibration behavior of structures. After that, extensive research was done on the vibration of sheets with different shapes, boundary conditions and different loading. Most of these studies are limited to thin sheets, where the effect of shear deformations is neglected. [8]

Unlike thin sheets, the effect of shear deformation is significant in thick sheets. Ignoring shear effects in these types of sheets leads to a significant increase in the amount of vibration frequencies in the direction of uncertainty. Therefore, first-order shear deformation theories [1] such as Reisner-Mindlin theory and other higher order shear deformation theories [2] have been used by various researchers to investigate the vibration behavior of sheets. Mindlin and his colleagues investigated the vibration of thick rectangular sheets with boundary conditions on four sides of the joint and Levy conditions and presented their analytical solution. They came to the conclusion that three independent modes can be obtained in the four-sided sheets of the joint. Also, the interaction of other modes was studied for a sheet with one pair of free boundaries and another pair of joints.

Noor [9] in 1973 investigated the free vibration of laminated composite sheets. He compared the results of the classical layered sheet theory [3], Mendelin theory and three-dimensional elasticity theory and came to the conclusion that the classical sheet theory is not suitable for estimating the vibration behavior of sheets with a high degree of perpendicularity and a thickness-to-length ratio greater than 0.1. Meanwhile, the results of Mindlin's theory are satisfactory for estimating low vibration frequencies in relatively thick laminated sheets with a thickness-to-length ratio of less than 0.2. The Rayleigh-Ritz method was used in 1980 by Dove and Ranel [10] for the free vibration of Mindlin's sheet. They used Timoshinkov beam functions as shape functions and investigated square sheets with five different combinations of boundary conditions. They also extended this method for a case where the sheet is under in-plane stresses. Based on this method, Liu and his colleagues investigated the vibration of circular and annular sheets for different boundary conditions. [11] This method was also considered in the study of the vibration of parallelogram and triangular sheets with different boundary conditions.

A large number of researchers used the finite element method to study the free vibration of sheets. For example, Rock and Hinton [59] introduced equiparametric quadrilateral bending elements for vibration analysis of thick and thin sheets. Chong and Kwak [12] developed annular and sector-shaped elements to study the free vibration of thick laminated sheets with curved boundaries. Reddy and Kopasamy [13] presented the finite element method based on the three-dimensional elasticity theory for the free vibration of rectangular anisotropic laminated sheets. The finite strip method [4] FSM has also been used by many researchers as one of the widely used methods in the field of solving eigenvalue problems. In reference [14], theories of shear deformation have been used to investigate the problems of free vibration of layered composite sheets. The displacement field and transverse stresses, due to maintaining the conditions of integration and balance, have continuity conditions along the thickness of the sheet. Based on this, various theories have been presented by researchers for the problems of sheets and shells.