Analytical dependence of the deflection of the spatial truss on the number of panels

Строительная механика

A statically definable girder truss consists of two flat side trusses with parallel belts and descending braces, connected along the upper and lower belt by horizontal ties. The truss at the corners has four supports, the vertical reaction of one of which is selected from the equilibrium condition of the structure as a whole. Several types of external load are considered: the load in the middle of the span, vertical, evenly distributed over the nodes of the upper or lower belt, and uniform horizontal (wind), applied to the nodes of the upper belt. The forces in the rods and supports are determined by the method of cutting out the nodes. The deflection is found by the Maxwell-Mohr formula under the assumption of linear elasticity of the rods. The dependence of the deflection on the number of panels is obtained from solving problems for trusses with a consistently increasing number of panels. Generalization of these solutions to an arbitrary number of panels is found using special operators of the Maple computer mathematics system. Using the Maple operators, we derive and solve homogeneous linear recurrent equations that satisfy the coefficients of the desired formula. The found dependence has the form of a polynomial in the number of panels. Curves of the deflection dependence on the number of panels, the load and the size of the structure are constructed.  Some asymptotic properties of the obtained solution are found. Formulas for the dependence of forces in the most stretched and compressed rods on the size of the structure, the load, and the number of panels that can be used to analyze the strength and stability of the structure are derived.