Natural vibrations and buckling of a spatial lattice structure using a continuous model derived from an energy approach

Lattice structures composed by parallel members named chords (or legs for vertical configurations) and connected by diagonals are very common among steel constructions in Civil and Mechanical Engineering and in particular, in the telecommunications industry. In the present study, a continuous model of a typical spatial lattice structure is derived. The legs configure a triangular cross-section and the diagonals are arranged in a zig-zag pattern. The differential system is derived from the potential and kinematic energies of the discrete model as the sums of each component contribution. Then, after accepting that the number of diagonals is large enough, the sums are approximated in the limit with classical integrals. Thus, the discrete system is replaced with a continuous formulation. The natural vibration problem of a lattice mast with a zig-zag diagonal pattern is studied using the proposed model. Also, the axial load influence is also accounted for through the second-order effect allowing to solve the buckling problem. Static deflection problems are also addressed. The Hamilton principle application yields the governing differential system in terms of nine unknown displacements. Several examples are solved numerically and the results are compared with the outcomes of a finite element spacial model. It is shown that there is an excellent agreement. The proposed continuous model can represent adequately the spatial lattice with a strong reduction in the degrees of freedom and the time of computation of the solution in comparison with a finite element approach.