Deformations of a planar multi-span arch truss: analytical solutions

The object of research. A scheme of a statically determinable indefinite truss with arched spans is proposed. The upper belt of the structure is rectilinear. The side supports of the truss are fixed hinges, the intermediate ones are movable. The truss is subjected to a vertical load, evenly distributed over all nodes of the upper chord of the truss or concentrated in the middle of the middle span. The derivation of formulas for calculating the dependence of the deflection of the middle span on the load, dimensions and number of spans is given. Analytical expressions are derived for the forces in the critical elements for an arbitrary number of spans of the structure. Method. The calculation of the forces in the elements and the reaction of supports of an externally statically indeterminate structure is carried out by cutting out all nodes from the solution of a joint system of equilibrium equations for nodes. The expression for the deflection is in symbolic form according to the Maxwell - Mohr formula. The generalization of a series of solutions to an arbitrary number of spans is carried out by induction. Results. The dependences found for the forces in the rods and deflections have a compact form and allow one to give simple estimates of the solutions. It is noted that the forces in all the rods of the upper chord, except for the side ones, in the case of a uniform load are equal to zero for arbitrary truss sizes and the number of spans. The dependence of the deflection on the number of spans has a jumplike character. All necessary transformations and analysis of solutions are performed in the Maple symbolic mathematics system. Linear asymptotics of the solutions for the deflection are derived.