Analytical Solution of a Spacer Beam Truss Deflection with an Arbitrary Number of Panels
The object of research is a planar statically definable truss with straight belts, a Sprengel-type lattice, and two fixed hinge supports that create external static indeterminability. The purpose of this work is to derive a formula for the dependence of the deflection of the structure in the middle of the span on the number of panels, dimensions, and load. We consider a load evenly distributed over the nodes of the upper belt. Method. To output the calculation formula, the induction method is used. The forces in the rods simultaneously with the four reactions of the supports are determined by cutting out the truss nodes from the solution of the system of equilibrium equations in symbolic form. The deflection is found by the Maxwell-Mohr formula. A series of solutions for trusses with successive increases in the number of panels give sequences of coefficients whose common terms are determined from the solution of homogeneous linear recurrent equations of the ninth order, compiled in the Maple computer mathematics system. Results. The solution for coefficients is polynomial in the number of panels. It is noted that for an odd number of panels, the determinant of the system of equilibrium equations turns to zero, which corresponds to the instantaneous kinematic variability of the structure. Given an appropriate diagram of the possible speeds of the nodes. The graph of the dependence of the dimensionless deflection on the number of panels shows significant jumps in the deflection values, which decrease with the increase in the number of panels. The dependence of the deflection on the ratio of the vertical dimensions of the spangle part and the entire truss significantly depends on the parity of the number of panels. Using Maple, we obtained a linear asymptotic solution for the number of panels, inversely proportional to the span.