Visualization of behaviour of the FSD algorithm for the model problem taking into account nonlinearities

The stress ratio algorithm to achive the fully stressed design is analyzed geometrically on a test problem of optimizing the symmetrical three-bar truss with taking into account physical nonlinearity. The paper analyzes the behavior of an algorithm from the field of optimal structural design. The problem of minimizing the mass of a statically loaded structure having a given shape is considered under constraints from above on the stress levels in rods and the constraints from below on the values of design parameters (cross-sectional thickness values), under one loading case. To solve this problem, a rather well-known heuristic algorithm is considered, intended to build fully stressed design (FSD). This algorithm known as the stress-ratio algorithm which belongs to the "optimality criteria methods". The optimality criteria techniques have the iterative process consisting in successive multiplications, and they have the behavior less clear than that of the mathematical programming methods whose iterations based on additive increments. In addition, the FSD heuristic concept meets the minimal mass not in all cases. Therefore, it would be useful to visualize the operation of the algorithm on a simple test model. As a test structure, we consider a symmetrical three-bar truss made of two materials (one in the central rod, the other in the side ones), loaded statically at the free node by a tensile force directed along the symmetry axis of the truss. For this model, the problem of mass minimization is considered:
$$ \begin{cases} m&\rightarrow\underset{F_i}{\min}\\ \sigma_i &< \bar\sigma_i, \quad \quad i\in\{c,s\}\\ F_i &\ge \bar F_i \end{cases} $$
where $F_c$ and $F_s$ are design parameters (sectional areas of the central and side rods, respectively), $\sigma_c$ and $\sigma_s$ are appropriate stresses in the rods, the top bar indicates the limiting permissible values specified in the constraints. The materials used take into account the property of plasticity (nonlinearity of the relationship between stress and deformation), which takes place under load level close to te breaking load. For the truss under consideration, the problem of mass minimization is a linear programming problem, even taking into account physical and geometrical nonlinearity. The algorithm for building the FSD has the form $ F_i^{(k+1)}=\max\Bigl(F_i^{(k)}\frac{\sigma_i^{(k)}}{\bar\sigma_i},\;\bar F_i\Bigr), \quad i\in\{c,s\}, \quad k=0,1,\dotsc $ where the design corresponding to the initial approximation (for $k=0$) is specified arbitrarily, with non-zero values of the design parameters. The presented article demostrates that for the truss under consideration, the workflow of the stress-ratio algorithm in the space of design variables can be visualized geometrically, using the triangles similarity. As a result of the corresponding geometric constructions, it turns out that, regardless of the initial approximation, the first step of the algorithm under study results in a design lying on the straight line connecting the projects $\left\{F_c=0, F_s = \frac{P}{2\sigma_s\cos\alpha}\right\}$ and $\left\{F_c = \frac{P}{\sigma_c}, F_s = 0\right\}$, which is beyond the constraint region. The next steps of the algorithm go along this straight line towards the design in which have the strength material concentrated in the element (-s) that has the active stress constraint, and maximally removed from elements that are underloaded according to the strain compatibility condition.