The paper deals with coupled flexural-torsional vibrations of straight prismatic elastic bars made of a linearly elastic isotropic and homogeneous material. One of the aims is to develop an effective method of modelling vibrations of train rails of cross-sections being mono-symmetric, taking into account warping due to torsion as well as transverse shear deformations. The Librescu-Song 1D model has been appropriately adapted to the above research aims by incorporating all the inertia terms corresponding to the kinematic hypotheses. The finite element(FE) program has been written and its correctness has been verified. The results concerning natural vibrations compare favourably with those predicted by 3D FE modelling using dense meshes. The paper proves that neglecting warping due to torsion leads to omitting several eigen-modes of vibrations, thus showing that the popular Timoshenko-like models are useless for the vibration analysis of bars of mono-symmetric cross sections.

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