Analytical modeling of regular-type truss loading conditions

Abstract

The scheme of a statically determinate flat truss with a complex lattice is suggested. The truss structure includes two supports: one with a movable hinge and the other with a fixed hinge. The truss upper belt’s uniform loading conditions are considered. It is assumed that the rods used in the belts and lattice have different stiffness. A dimensionless stiffness redistribution parameter is introduced. The problem of determining the truss deflection as a function of truss size, load, and number of panels is stated and solved. Unlike standard girders with parallel belts and triangular-type lattices, the considered truss structure does not allow forces arising in a single rod or in a group of rods to be calculated independently of the forces arising in the other rods or their groups. The system of equilibrium equations for all nodes shall in any case be analyzed in order to calculate the forces in the system. As a result, serious difficulties are encountered in calculating trusses containing a large number of panels, where numerical methods are inefficient in connection with the well-known «curse of dimensionality», which leads to accumulation of computational errors. To overcome the encountered problems, it is proposed to use the induction method with engaging the Maple computer algebra system. The deflection was determined using the Maxwell–Mohr formula under the assumption of elastic rod behavior. The sequence of solutions obtained for 16 trusses having different numbers of panels was analyzed, and a general formula for calculating the deflection was derived based on the analysis results. For solving the problem, the operators rgf_findrecur and rsolve from the genfunc package of recurrent equations available in the Maple system were involved. Some specific features of the solution are noted. Thus, a minimum in the deflection versus the height truss curve is revealed. It is also shown that there exists a critical truss height value below which the redistribution of belt stiffness values in favor of the lower one causes the deflection to decrease and vice versa. The limiting characteristic of the “deflection versus the number of panels” dependence is found, according to which the deflection grows as a cubic function of the number of panels. It also follows from the analysis of the obtained solution that at certain combinations of truss sizes, the determinant of the system of linear equations becomes zero irrespective of the load and number of panels.