טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentEmuna Nir
SubjectBending Theory of Functionally Graded Beams
DepartmentDepartment of Aerospace Engineering
Supervisor Professor Emeritus David Durban
Full Thesis textFull thesis text - English Version


Abstract

This work presents a unified outlook on the bending of functionally graded beams. A new analytical method is suggested for plane bending analysis of linear elastic orthotropic functionally graded beams under continuous mechanical and thermal loads.

Available methods center on expanding the Airy stress function in powers of the axial coordinate, or on the standard Fourier series solution. Point of departure in the present work is upon representing the stress function as a series in increasing order of derivatives of the bending moment. The coefficients of each term are self-induced functions of the transverse coordinate that transcend any particular load distribution.

A simple, consistent, straightforward recursive procedure is detailed for finding the transverse functions, thus providing the solution for stress function at increasing levels of accuracy. Compliance with stress boundary conditions over the long faces of the beam leads to the complete analytical solution for stress components.

Turning to beam kinematics, we show how edge data, at the beam supports, is accounted for in an averaged Saint-Venant type fashion. In particular, we derive a universal relation for first and second order bending rigidities and show comparison with both Euler-Bernoulli and Timoshenko beam theories.

A few sample examples are illustrated and accuracy is assessed against finite elements and Fourier solutions. Also, an estimate of decay rates of end effects is discussed, shedding light on the validity of Saint-Venant principle in functionally graded beams.

The initial motivation for the present work was to develop elasticity solutions for two dimensional functionally graded elastic strips. However, it has emerged that application of the method for deriving enhanced beam theories, anisotropic and/or non-homogenous is surprisingly useful. It is argued that the present method determines leading terms of beam bending rigidity by efficient and compact relations in terms of cross section averages of elastic moduli.