Trihedral lattice towers with optimal cross-sectional shape

The article discusses a new design of triangular lattice supports and suggests a technique for optimizing their cross-sectional shape. Optimization is carried out from the condition of the cross-section being equally stable in two planes. Such relationships are found between the geometric parameters of the section, at which the moments of inertia take the maximum value.


Introduction
This article will consider a new rational type of trihedral lattice structures used in the construction of tower structures for various purposes. The structure under study (patent for invention RU2584337 [1]) ( Figure 1) contains belts 1 of a polyhedral closed section, lattice bars 2, attached to sheet gussets 3. The purpose of this work is to determine the optimal parameters b, h, l of the pentagonal section of the belts (Figure 2). The l1 size is assigned for design reasons to ensure the required length of the weld, and we will consider it as given. An axial moment of inertia relative to the main central axis y is selected as the objective function. A limitation is introduced in order to ensure the equal stability of the belt in two planes. Thus, optimization is performed from the stability condition. In the process of optimization, the perimeter of the section without taking into account the gussets is taken to be constant, which corresponds to the constancy of material consumption.

Methods
For simplicity, we will assume that the section is thin-walled and We represent the crosssection shown at fig. 2 as the set of seven rectangles ( Figure 3). .
Let us choose the coordinate system zC1y1 as the auxiliary one. The coordinates of the figures 1-7 centers of gravity are determined by the formulas:  2 , where A is the total cross-sectional area.
The moments of inertia of the entire section relative to the main central axes y and z are determined by the formulas: Finally, the expression for the moment of inertia Jz takes the form: The expression for the moment of inertia Jy is not presented here due to its cumbersomeness. Formula (6) can be presented in a simplified form if we neglect the terms that include the quantities δ 3 and δ 2 .
In the problem under consideration, the objective function and constraints are nonlinear, therefore, to solve it, it is necessary to apply nonlinear optimization methods.
We carried out the solution in the Matlab environment using the Optimization Toolbox package. The function fmincon was used, which determines the minimum of the nonlinear objective function 1 y J  with nonlinear constraints. The interior point method is selected as the nonlinear optimization method. Table 1 shows the optimal values of the ratios b / L, h / L and l / L depending on the ratio l1 / L. No solution was found for l1 / L > 0.125. It probably does not exist for such relations between l1 and L. Note that the pentagonal section without gussets at l1 = 0 can act as a replacement for square tubes in truss chords of the Molodechno type [2][3][4][5]. Compared to a square tube, a pentagonal tube with optimal parameters b, h, l for the same cross-sectional area has 5.2% higher moments of inertia. The optimum angle α is 31.46º. When designing real structures, for convenience, you can take α = 30º.

Results and Discussion
The equality of the moments of inertia Jy and Jz is necessary to ensure equi-stability in the case of the same fixation of the rod in the xOz and xOy planes. At different reduced lengths in two planes, it becomes necessary to find the optimal section parameters for which the ratio Jy / Jz is specified. The technique proposed by the authors and the developed program in the Matlab environment allows to do this. Table 2 shows the optimal values of b / L, h / L and l / L for various ratios Jy / Jz at l1 = 0.  (Fig. 4). The values b / L, h / L and l / L obtained from the condition of equi-stability and the maximum moment of inertia, depending on the ratio l1 / L, are presented in table. 3.

Summary
The optimal ratios between the pentagonal section sides sizes of the trihedral lattice supports are determined, providing their maximum rigidity and uniform stability at a given material consumption.
The optimal angle at the top was close to 120 degrees, and this value can be used in the design for simplification. In the future, it is advisable to consider the issues of local stability of the support elements with the proposed cross section.