Exact vibrational analysis of prismatic plate and sandwich structures

TLDR: In this paper, a member stiffness determinant is defined as the determinant of the member stiffness matrix when the member is sub-divided into an infinite number of identical submembers.

Abstract: Transcendental stiffness matrices for vibration (or buckling) analysis have long been available for a range of structural members. Such stiffness matrices are exact in the sense that they are obtained from an analytical solution of the governing differential equations of the member. Hence, assembly of the member stiffnesses to obtain the overall stiffness matrix of the structure results in a transcendental eigenproblem that yields exact solutions and which can be solved with certainty using the Wittrick-Williams algorithm. Convergence is commonly achieved by bisection, despite the fact that the method is known to be relatively slow. Quicker methods are available, but their implementation is hampered by the highly volatile nature of the determinant of the structure's transcendental stiffness matrix, particularly in the vicinity of the poles, which may or may not correspond to eigenvalues. However, when the exact solution exists, the member has a recently discovered property that can also be expressed analytically and is called its member stiffness determinant. The member stiffness determinant is a property of the member when fully clamped boundary conditions are imposed upon it. It is then defined as the determinant of the member stiffness matrix when the member is sub-divided into an infinite number of identical sub-members. Each sub-member is therefore of infinitely small length so that its clamped-ended natural frequencies are infinitely large. Hence the contribution from the member stiffness matrix to the Jq count of the W-W algorithm will be zero. In general, the member stiffness determinant is normalised by dividing by its value when the eigenparameter (i.e. the frequency or buckling load factor) is zero, as otherwise it would become infinite. Part A of this thesis develops the first two applications of member stiffness determinants to the calculation of natural frequencies or elastic buckling loads of prismatic assemblies of isotropic and orthotopic plates subject to in-plane axial and transverse loads. A major advantage of the member stiffness determinant is that, when its values for all members of a structure are multiplied together and are also multiplied by the determinant of the transcendental overall stiffness matrix of the structure, the result is a determinant which has no poles and is substantially less volatile when plotted against the eigenparameter. Such plots provide a significantly better platform for the development of efficient, computer-based routines for convergence on eigenvalues by curve prediction techniques. On the other hand, Part B presents the development of exact dynamic stiffness matrices for three models of sandwich beams. The simplest one is only able to model the flexural vibration of asymmetric sandwich beams. Extending the first model to include axial and rotary inertia makes it possible to predict the axial and shear thickness modes of vibration in addition to those corresponding to flexure. This process culminates in a unique model for a three layer Timoshenko beam. The crucial difference of including axial inertia in the second model, enables the resulting member dynamic stiffness matrix (exact finite element) to be included in a general model of two dimensional structures for the first time. Although the developed element is straight, it can also be used to model curved structures by using an appropriate number of straight elements to model the geometry of the curve. Finally, it has been shown that considering a homogeneous deep beam as an equivalent three-layer beam allows the beam to have additional shear modes, besides the flexural, axial and fundamental shear thickness modes. Also for every combination of layer thickness, the frequencies of the three-layer beam are less than the corresponding frequencies calculated for the equivalent beam model with only one layer, since it is equivalent to providing additional flexibility to the system. However, a suitable combination of layer thicknesses for any mode may be found that yields the minimum frequency. It is anticipated that these frequencies would probably be generated by a single layer model of the homogeneous beam if at least a third order shear deformation theory was incorporated. Numerous examples have been given to validate the theories and to indicate their range of application. The results presented in these examples are identical to those that are available from alternative exact theories and otherwise show good correlation with a selection of comparable approximate results that are available in the literature. In the latter case, the differences in the results are attributable to many factors that vary widely from different solution techniques to differences in basic assumptions.