Analytical calculation of deflection of a planar truss with a triple lattice

Строительная механика

We propose a scheme of a statically determinate truss with straight chord s and four supports, one of which is a fixed hinge, the other movable hinges. The task is to obtain the dependence of the deflection of the middle span of the structure on the number of panels. The problem is solved by induction using operators of the Maple computer mathematics system. The deflection is determined by the Maxwell – Mohr's formula, the forces in the rods are found from the solution of the joint system of the equation of equilibrium of nodes, the unknowns of which include the reaction of the supports. of nodes, whose unknowns include the reaction of the supports. The inclusion of support reactions in the system of equilibrium equations allows us to reveal the external static indeterminability of the structure. Generalizing a number of solutions for trusses with a consistently increasing number of panels, the desired dependence is obtained. To do this, we create recurrent equations that satisfy the terms of the sequences of coefficients in the deflection formula. The resulting homogeneous linear recurrent equations have a degree no higher than the eighth in the case of a problem with a load distributed over the upper chord and the sixth for a concentrated load in the middle of the span. Solving these equations in the Maple environment using the rsolve operator gives expressions of the dependence of the coefficients of the desired formula on the number of panels. The asymptotics of solutions are found. The dependence of the horizontal shift of the mobile support on the action of distributed and concentrated load is also obtained. Formulas are derived for the dependence on the number of panels of forces in some elements in the middle of the span. The obtained solutions can be used for preliminary evaluation of the designed structure and for evaluating the accuracy of numerical solutions.