In this thesis, a continuum sensitivity analysis method is presented for calculation of shape sensitivities of an elastic bar. The governing differential equations and boundary conditions for the elastic bar are differentiated with respect to the shape design parameter to derive the continuum sensitivity equations. The continuum sensitivity equations are linear ordinary differential equations in terms of local or material shape design derivatives, otherwise known as shape sensitivities. One of the novelties of this work is the derivation of three variational formulations for obtaining shape sensitivities, one in terms of the local sensitivity and two in terms of the material sensitivity. These derivations involve evaluating (a) the variational form of the continuum sensitivity equations, or (b) the sensitivity of the variational form of the analysis equations. We demonstrate their implementation for various combinations of design velocity and global basis functions. These variational formulations are further solved using finite element analysis. The order of convergence of each variational formulation is determined by comparing the sensitivity solutions with the exact solutions for analytical test cases. This research focusses on 1-D structural equations. In future work, the three variational formulations can be derived for 2-D and 3-D structural and fluid domains.