In this dissertation, two-dimensional torsion analysis of bars with circular and rectangular cross sections having multiple cracks and cavities reinforced by Functionally graded materials coatings has been performed. It is assumed that cracks and cavities are located in the rectangular and circular bars and then the bars are reinforced by the desired coating. The bars are made of isotropic homogeneous materials and considered to be straight. The material properties of the coating layer change exponentially in the direction perpendicular to the axis of the bar. Also, the connection between the coating layer and the bar is assumed to be perfectly bonded. The analysis method is based on employing Dislocations to model cracks and cavities. For this purpose, at first the torsion solution of a screw dislocation is obtained by making use of finite Fourier transform. Following distributed dislocation approach, multiple defects with arbitrary patterns are considered and the effects of them on the torsional rigidity and stress distribution of the bars have been studied. Cracks and cavities are considered two dimensional and extended throughout the axis of the bar. The distributed dislocation solution is accomplished to derive a system of Cauchy singular integral equations that are solved numerically. The bars under torsional loading and having various types of cracks and cavities such as one inclined edge crack, one circular crack, two circular cracks, one elliptical cavity and two embedded cracks and … have been investigated and stress intensity factors at the crack tips and hoop stress around the cavities have been calculated. The influences of FG coating layer on the stress intensity factors of crack tips and torsional rigidity are studied and the results show that increasing the FG normalized parameter results in decreasing the normalized stress intensity factor and increasing the normalized torsional rigidity. Also, it is shown that an FG coating layer with sharply varyi