טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentGrebshtein Michael
SubjectAnalytical Formulation and Solutions of Anisotropic
Non-Homogeneous Beams
DepartmentDepartment of Aerospace Engineering
Supervisor Professor Emeritus Omri Rand


Abstract

The dissertation presents a closed-form formulation for uncoupled monoclinic non-homogeneous beams that are subjected to tip and axially non-uniform distributed surface and body loads .

The presented analysis applies to both micro- and macro- problems . In the micro- case, the analysis can be referred to a typical composite material, where a matrix material surrounds a fiber.  In the macro- case, the plane analysis presented in the study serves as a major component of the corresponding non-homogeneous beam analysis .

The entire reasoning behind the analytical approach in this investigation is founded on the St. Venant's semi-inverse method of solution. The class of solutions presented is based on a set of initially assumed expressions for the stress components that contain series of unknown coefficients, and generalized harmonic and biharmonic stress functions. By employing the governing equations, the field equations, the boundary and interface conditions for the stress functions are established, and the involved parameters and functions are determined .

The above field equations, the boundary and interface conditions are associated with various types of boundary value problems in a non-homogeneous domain. In the present study, the generalized biharmonic and Neumann type boundary value problems in non-homogeneous domain are commonly expressed .

The work presents a correct formulation for anisotropic non-homogeneous domain of a special kind of biharmonic boundary value problem, the so called Auxiliary Plane Strain Problems. Its application in the analysis of anisotropic non-homogeneous beams that undergo both tip and axially non-uniform distribution of surface and body loads is discussed.

The solution of the beams under axially non-uniform distribution of surface and body loads is based on the Michell-Almansi Method. Then, the solution of a three dimensional problem is executed by solutions of a number of boundary value problems of plane deformations in different levels .

For the first time, the solution approach for beams under axially non-uniform distribution of surface and body loads, based on the above Michell-Almansi method, is presented in an analytical symmetric form .

For symbolic exactness of the expressions , the entire methodology is documented, verified and proved to be exact by a suitable system of symbolic programs written in Maple symbolic software environment .

This research also includes solution methodologies for boundary value problems which are oriented to global polynomial analysis in order to keep the discussion generic enough and capable of handling many and different domain shapes. Some approximation methodologies are also presented along with adequate illustrative examples .