Analytical calculation of the deflection of the lattice truss

Abstract

Abstract. An algorithm is given for deriving the dependence of the deflection of a planar statically determinate beam truss on the number of panels, dimensions and load. Three load cases are considered: uniform load on the lower belt, upper belt and vertical force in the middle of the span. By induction, generalizing a series of solutions for trusses with a consecutively increasing number of panels, the desired formula is obtained for the deflection and horizontal displacement of the mobile support of the truss. All transformations are performed in the system of symbolic mathematics Maple. For a sequence of coefficients of the desired formula, using the special Maple operators, homogeneous recurrent equations are constructed and solved. The coefficients found are in the form of polynomials in the number of panels. The asymptotic property of the solution is found. On the graphs of the dependence of the deflection on the number of panels and on the height, extreme points are found. The solution can be used to test the calculations obtained numerically.

Analytical calculation of the combined suspension truss

“…A rare feature of this solution is the presence of the number of panels n in the denominator of the formula. All known solutions [10][11][12][13][15][16][17][18][19][20] have the form of polynomials of different degrees. The calculation of a series of trusses reveals sequences of coefficients for 3 a , 33    is obtained.…”

“…To calculate the deflection of the truss and obtain an analytical form of the solution, it is necessary to obtain expressions for the forces in the rods as well in symbolic form. To do this, using the program [10][11][12][13], you need to enter data about the coordinates of the nodes of the truss and the order of connecting the rods. We number nodes and rods, choosing the origin of coordinates in the left movable hinged support ( Fig.…”

“…The matrix G of the system of equilibrium equations GS B  for nodes consisting of equations in the projections on the axis is filled with the direction cosines of the forces. In odd rows, projections are recorded on the horizontal axis, in even lineson the vertical [13], Sis the vector of all forces in the rods. The displacement is obtained from the Mohr's integral 3 (…”

“…Among other things, such formulas make it possible to find a solution for trusses with a very large number of rods, without a loss of accuracy characteristic of numerical methods. In [10][11][12][13][14], the method of induction of generalization of a series of particular solutions for trusses with different (consecutive) number of panels was used to calculate flat trusses. Operators of the Maple system were used to find symbolic expressions for the stresses in the rods and to compile recurrence equations for the coefficients of the required formulas with their subsequent solution.…”

The Simplified Analytical Algorithm to the Time Effect of the Simple-Supported Steel and Concrete Composite Beam

“…In the equation, w s 0 and w st are the deflections [ 25 – 27 ] of the steel beam before and after the creep. It can be seen from equation ( 13 ) that the stiffness of the steel beam is constant before and after the creep of the composite beam, and the deflection of the steel beam is proportional to the bending moment of the steel beam.…”