The cracks if not detected in time leads to catastrophic failure. As a result techniques of detection and analysis of effect of cracks on the behavior of structures became important in the last six decades. Crack has a paramount effect on the dynamic characteristics, buckling load and dynamic stability of structures. Analysis of cracked members and damage detection are important issues in Civil, Mechanical and Aerospace engineering. In the first decade of the twenty first century the investigation on the cracked members gained more weights. The present author derived a golden derivative of the jump produced in the cracked member displacement at crack position. This derivative paved the way for introduction of new and innovative methods. This book provides a sound basis for these innovative methods and their application in free vibration, buckling analysis and dynamic stability of cracked members.