On the layout of a least weight multiple span structure with uniform load

This paper focuses on identifying an analytical description for the layout of a minimum volume pin-jointed structure (truss) that is uniformly loaded and supported on equally spaced locations. The methodology utilized takes advantage of Maxwell’s Theorem of Load Paths and the properties of reciprocal diagrams (as described by the authors in previous papers). The approach followed in this work provides considerable insight in the properties of the optimal layout. In addition, the solution to the problem is presented in terms of a structure and its dual. The dual truss associated with the force diagram of the original structure has the same total load path as the original structure itself but completely different geometry and corresponds to a different, but related, load condition. Therefore, multiple optimal layouts are available to the practicing engineer for the problem considered. Analytical expressions describing the geometry and the volume of the optimal truss layout and its dual are provided showing that simple formulas describe the geometric properties of the solution. The solutions are presented for equal magnitude tensile and compressive stress limits first and then the effect of unequal stress limits is explored. Comparison with numerical results confirms the validity of the solutions presented.