The Fundamental Structural Problem is formulated for systems made of unconstrained rigid bodies, mutually connected and linked to the soil by deformable devices, and therefore denominated lumped deformability-unconstrained rigid systems. The Fundamental Structural Problem consists in finding the response of the structure (in terms of displacements, strains and stresses) to known active forces, quasi-steadily applied in order not to generate dynamic effects. Three different aspects of the problem are addressed: (i) a geometric problem, in which strains and displacements are related, (ii) a dynamical problem, in which stresses and forces are joined, (iii) a rheological problem, in which stresses and strains are linked, both in the elastic and in the elasto-plastic cases. Kinematic and static classification, the duality properties and the Virtual Work Principle, previously introduced for rigid systems, are restated for lumped-deformability systems. Moreover, the Elastic and Complementary energies, already introduced for the elastic devices, are defined for the assembly. The set of geometric, mechanical and rheological equations rules the Fundamental Structural Problem.