The paper concerns the problem of minimization of the total potential energy of trusses subjected to static loads in the presence of prescribed displacements of selected supporting nodes. The positions of the internal (free) nodes are fixed and the supporting nodes are imposed, the member stiffnesses being design variables, while the truss volume represents the cost of the design. Due to the assumption of the stiffnesses being non-negative, the problem is reduced to a problem of optimization of structural topology. Upon eliminating all the design variables analytically the optimum design problem is eventually reduced to the two mutually dual problems expressed either in terms of member forces or in terms of displacements of free nodes. The problem setting concerning the case when the prescribed displacements of supports are the only loads applied (i.e. kinematic loads) assumes a particularly simple form. A specific numerical method of solving the stress-based auxiliary problem has been developed for the selected 2D and 3D optimal designs. The study is the first step towards topology optimization of trusses with distortions