Three Dimensional Elasticity Analyses for Isotropic and Orthotropic Composite Cylinders

The demand for using shell theories comes from its efficiency in computational and analytical cost. On another side, new materials that are orthotropic and/or anisotropic in nature are discovered and broadly used in many fields. Many advanced shell theories are developed for these new materials, particularly in the recent decades. A study about the accuracy of these shell theories is very meaningful to build confidence in them for further applications. This study requires a precise benchmark against which shell theories can be tested. This is the main research subjective in this dissertation: to build a set of solutions using the three dimensional (3D) theory of elasticity against which shell theories can be tested for accuracy. The contents of this dissertation to support this research include a comprehensive literature review for the shell theories and recent usage and to find the gaps which need to be filled. These gaps include, among others, the lack of studies on the accuracy of the theories used and the absence of results using the 3D theory, particularly for orthotropic materials. Some of these studies are conducted here. The deficiency of some commercial finite element packages is discussed here. The reasons for the absence of accurate results are investigated. The 3D theory and analyses of isotropic and orthotropic materials of hollow cylinders is investigated here for reliable results

Some results on thermal stress of layered plates and shells by using Unified Formulation

This work presents some results on two-dimensional modelling of thermal stress problems in multilayered structures. The governing equations are written by referring to the Unified Formulation (UF) introduced by the first author. These equations are obtained in a compact form, that doesn't depend on the order of expansion of variables in the thickness direction or the variable description (layer-wise models and equivalent single layers models). Classical and refined theories based on the Principle of Virtual Displacements (PVD) and advanced mixed theories based on the Reissner Mixed Variational Theorem (RMVT) are both considered. As a result, a large variety of theories are derived and compared. The temperature profile along the thickness of the plate/shell is calculated by solving the Fourier's heat conduction equation. Alternatively, thermo-mechanical coupling problems can be considered, in which the thermal variation is influenced by mechanical loading. Exact closed-form solutions are provided for plates and shells, but also the applications of the Ritz method and the Finite Element Method (FEM) are presented

Preliminary studies of the time-dependent shear and uniaxial tensile behaviour of oriented polymers

Summary The work reported in this memo is the initial stages of an investigation of the time-dependent behaviour of certain anisotropic polymers. In the first instance low density polyethylene with a transversely isotropic symmetry is being examined. Different degrees of anisotropy have been induced by cold drawing and the time dependent material parameters necessary to describe the stiffness of the anisotropic polyethylene have been determined. This involved the measurement of uniaxial tensile creep: lateral contraction creep, and torsional creep under conditions of constant load at 20°C ± 0.5°C. The tensile creep and contraction creep apparatus has been described elsewhere (Darlington (a) 1968) and only the principle of the apparatus is discussed here. The torsional creep apparatus is described in detail. Analysis of the experimental data is not yet complete. The data is tabulated in section 5 and a preliminary analysis is presented in section G. Details of proposed future work are discussed in section 7

Uniqueness of analytic solutions for stationary plate oscillations in an annulus

AbstractThe equations governing the harmonic oscillations of a plate with transverse shear deformation are considered in an annular domain. It is shown that under nonstandard boundary conditions where both the displacements and tractions are zero on the internal boundary curve, the corresponding analytic solution is zero in the entire domain. This property is then used to prove that a boundary value problem with Dirichlet or Neumann conditions on the external boundary and Robin conditions on the internal boundary has at most one analytic solution

Dynamic stiffness matrix of a rectangular plate for the general case

The dynamic stiffness matrix of a rectangular plate for the most general case is developed by solving the biharmonic equation and finally casting the solution in terms of the force-displacement relationship of the freely vibrating plate. Essentially the frequency dependent dynamic stiffness matrix of the plate when all its sides are free is derived, making it possible to achieve exact solution for free vibration of plates or plate assemblies with any boundary conditions. Previous research on the dynamic stiffness formulation of a plate was restricted to the special case when the two opposite sides of the plate are simply supported. This restriction is quite severe and made the general purpose application of the dynamic stiffness method impossible. The theory developed in this paper overcomes this long-lasting restriction. The research carried out here is basically fundamental in that the bi-harmonic equation which governs the free vibratory motion of a plate in harmonic oscillation is solved in an exact sense, leading to the development of the dynamic stiffness method. It is significant that the ingeniously sought solution presented in this paper is completely general, covering all possible cases of elastic deformations of the plate. The Wittrick-Williams algorithm is applied to the ensuing dynamic stiffness matrix to provide solutions for some representative problems. A carefully selected sample of mode shapes is also presented

Vibration frequencies of whirling rods and rotating annuli

Static Whirling Rods: Past researchers suggested that “static instabilities” exist at certain rotational speeds of whirling rods. This thesis shows these instabilities are an artefact of the material constitutive laws that are being used well outside their range of applicability. An alternative approach is developed where strains due to rotation are separated from the superimposed vibration. This enables the generally predicted lowering of longitudinal natural frequencies with rotational speed shown to be simply a result of the bulk changes in the geometry of whirling rods. Steady state equations of whirling rods are formulated in Lagrangian coordinates. Due to the non-linear nature of the governing equations, an original numerical method is applied to solve the problem. Numerical results are compared with analytical results obtained from the linearized uniaxial model. There is a close agreement between these two models at low angular velocities. However, at high angular velocities, discrepancies between them arise, confirming that the nonlinear strain-displacement relationship has significant effect on the results and the inferred “static instabilities”. This approach first solves the “static” problem of the deformed geometry of a highly strained whirling rod before longitudinal natural modes are determined by classical methods. Furthermore, conditions for existence and uniqueness of solutions are derived. Dynamic Rotating Annuli: In-plane modes of vibration of annular plates are investigated. Two different models of equations one from Bhuta and Jones and the other from Biezeno and Grammel that govern the rotational motions of annuli will be studied. Since Biezeno and Grammel’s model was originally derived in Eulrian coordinates, their model will be transformed to the Lagrangian coordinates for the purpose of comparison with Bhuta and Jones’ model.The solutions of the equations assume small oscillations of vibration being superimposed on the steady state of the annulus while it is in rotation. Exact and approximate solutions are obtained for the Bhuta and Jones’ model, where the approximate solutions on in-plane displacements and natural frequencies are acquired by ignoring the Coriolis effect. A proposed numerical scheme is implemented to solve the governing equations coupled with radial and circumferential displacements. Uniqueness of solutions will be mentioned although it will not be rigorously derived because it is out of the scope of this thesis. Approximate analytical results show that both radial and circumferential natural frequencies are decreasing when the rotational speed of an annulus is increasing. The exact and numerical results on both models that take the Coriolis effect into account show that radial natural frequencies are increasing and circumferential natural frequencies are decreasing when the rotational speed of an annulus is increasing