1.1 Research Background
Plate structures are very common in engineering practice, and are extensively used in civil, mechanical, and aeronautical engineering, such as concrete floor slab and aircraft skin components. Their vibrational behaviors have caught great interests of many researchers.
Stress singularities mean that infinite stresses exist at some points in the domain under consideration, and are often encountered in plate problems. Three main reasons causing stress singularities are: (1) discontinuousness of geometry, such as cracks in the domain or sharp re-entrant angles at the boundary; (2) concentrated loads, such as point forces or moments; (3) suddenly change of material properties, such as composite material. When the stress singularity behaviors exist in the domain under consideration, it is necessary to find the asymptotic solutions, which can exactly describe the stress singularities, for obtaining accurate solutions for static or vibration problems. However, as the demand for engineering structures is improving, a singularity problem is unavoidable in engineering analysis.
Vibrations of V-notch plates are concerned with stress singularity problems due to the sharp vertex. Such notches may be generated intentionally in the plate for clearance or other reasons. This thesis utilizes the well-known Ritz method to analyze the vibration of rectangular plates with a V-notch based on the classical plate theory. Two
sets of admissible functions are used in the analysis method simultaneously: (1) algebraic polynomials, which form a complete set of functions; (2) corner functions, which are the general solutions of bi-harmonic equation, duplicate the boundary conditions along the edges of the notch, and describe the stress singularities at the sharp vertex of the V-notch exactly. The rectangular plates under consideration are completely free and cantilevered, respectively. The effects of the asymptotic solutions on the convergence of numerical solutions are demonstrated through convergence studies. The efftects of the V-notch on the vibration behaviors of rectangular plates also are discussed in detail.
1.2 Literature Review
On the topic of plate vibrations, at least 2000 research papers have been published. Leissa (1967) summarized the methods of analysis and numerical results found in 500 references on the free vibration of plates published before 1967 in his classical monograph. Since then, research and publication on this subject has been at an increasing rate. In these studies, vibration of cracked plates is a problem of greatest interests, which combines the fields of vibration analysis and stress singularity.
Only a few papers about this problem are published. Most of them are based on the classical theory and are reviewed below.
Most of the published works considered the cracked rectangular plates with simply supported at all sides or at two opposite sides. Because analytical solutions exists for such plates with no crack, semi-analytical solutions can be constructed for such plates with cracks along a straight
line perpendicular to the simply supported edge. To investigate the vibrations of simply supported rectangular plates with cracks, Lynn and Kumbasar (1967) used Green’s function approach to obtain the solutions for Fredholm integral equations of the first kind, while Stahl and Keer (1972) formulated the problem as dual series equations and reduced to homogeneous Fredholm integral equations of the second kind. Aggarwala and Ariel (1981) used Stahl and Keer’s approach to analyze the vibration of a plate with various crack configurations along the symmetry axes of the plate. Solecki (1983) constructed a solution for vibrations of a cracked plate by using Navier form of solution along with finite Fourier transformation of discontinuous functions for the displacement and slope across the crack. Recently, Khadem and Rezaee (2000) used so called modified comparison functions constructed from Levy’s form of solution as the admissible functions of the Ritz method to analyze a simply supported rectangular plate with a crack having an arbitrary length, depth and location parallel to one side of the plate.
To study the vibration behaviors of cracked rectangular plates with two opposite edges simply supported, Hirano and Okazaki (1980) used Levy’s form of solution and matched the boundary conditions by means of a weighted residual method, while Neku (1982) modified Lynn and Kumbasar’s approach by establishing the needed Green’s function using Levy’s form of solution.
To consider the vibrations of a cracked rectangular plate with arbitrary boundary conditions, a numerical method has to be used. Qian et
al. (1991) developed a finite element solution by deriving the stiffness
matrix for an element including the crack tip from the integration of the stress intensity factor. Yuan and Dickinson (1992) decomposed a rectangular plate under consideration into several domains and introduced artificial springs at the joints between the domain so that the Ritz method with regular admissible functions can be easily applied to find the solutions. Krawczuk (1993) proposed a finite element solution similar to that of Qian et al., except that the stiffness of an element including the crack tip was expressed in a closed form. Liew et al. (1994) developed a domain decomposition method for the vibrations of cracked rectangular plates with various boundary conditions.
In the above-mentioned literature, the solutions, except for the finite element solutions, by no means considered the characteristic of the stress singularities. In the present thesis, the Ritz method is used to analyze the vibrations of rectangular plates with a V-notch. It is more suitable for solving the problem than a traditional finite element approach. Based on the classical plate theory, a finite element approach needs C1 type elements, which are much more complicated than C0 type elements, and are difficult to establish. The asymptotic solutions derived by Williams (1952) are used along with suitable polynomials as admissible functions in the present problem. Similar analysis procedure has been used to determine the natural frequencies and mode shapes for sectorial plates and circular plates with V-notches by Leissa et al. (1993a, 1993b). It is demonstrated here by obtaining extensive results for frequencies and mode shapes of rectangular plates having various notch angle, depths and locations. The present results serve not only to improve the understanding
the vibration behavior of a V-notched plate, but also as benchmark data against those from other numerical methods or experiments.
1.3 Contents in the Thesis
The contents in the thesis are mainly divided into five chapters. The contents in the following chapters are introduced briefly below. Chapter 2 shows the derivation of asymptotic solutions, and discusses the stress singularities at a corner. Chapter 3 analyzes the vibration of completely free rectangular plate with a V-notch, where stress singularities occur at the vertex of the V-notch. Chapter 4 analyzes the cantilevered rectangular plates with a V-notch. Finally, conclusions and recommendations for this study are presented in Chapter 5.