The Fundamental Structural Problem is specialized to unconstrained rigid systems with lumped elasticity. The governing equations (consisting of congruence, equilibrium and linear elastic law, established in Chap.  6 ) rules the (linear) Elastic Problem. The existence and uniqueness of the solution is proved for this problem. The governing equations are first dealt with according to the Displacement Method, which is aimed to get equilibrium equations expressed in terms of generalized displacements. The Direct Formulation is first discussed, in which all the governing equations are used and properly combined. Alternative formulations, based either on the displacement corollary of the Virtual Work Principle, or on the stationary theorem of the Total Potential Energy, are introduced. These latter use only a part of the governing equations, and replace the remaining part by variational equations. The elastic problem is successively tackled by the Force Method, which leads to compatibility conditions in terms of hyper-static unknowns, and therefore it is applicable only to statically undetermined systems. Once again, the direct and variational formulations are discussed, the latter ones being based on the Complementary Virtual Work Principle (i.e., the virtual stress corollary) and on the stationary theorem of the Elastic Complementary Energy. Finally, self-balanced stress states, triggered by anelastic strains imposed to statically under-determined systems, are addressed. These latter are generated by thermal variations, or viscous effects, or assembly defects. For this class of problems, the Displacement and Force Methods are illustrated, together with variational formulations.