5. Bending of Lightweight Circular Tubes—Optimal Design

The necessary condition for for lightweight design of a thin-walled tube subject to bending is defined by Eq.  3.​14. By applying this equation, we obtain Equation  5.28 represents a necessary condition, i.e. it contains the Pareto-optimal set plus other non-optimal solution. In the next subsections, the non-optimal solutions will be identified and removed. Let us consider the available room constraint defined in Eq.  5.3. The outer room constraint is related to \(R_{max}\). The inner room constraint refers to \(R_{min}-2t_{max}\). In the design variables domain, the room constraints are described by a rectangle with vertices \(P_{t_{min},R_{min}}\), \(P_{t_{min},R_{max}}\), \(P_{t_{max},R_{min}}\) and \(P_{t_{max},R_{max}}\) (see Fig.  5.8).

\(P_{t_{min},R_{min}}\) and \(P_{t_{max},R_{max}}\) represent, respectively, the solution with minimum mass and the solution with minimum compliance. According to the definition of the Pareto-optimal solution (Definition 1.​11 in Sect.  3.​1), such two points, if feasible, are the extreme points of the Pareto-optimal set. Let us consider such two points \(P_{t_{min},R_{max}}\) and \(P_{t_{max},R_{min}}\) and If \(c'=\overline{c}\), by equating Eqs.  5.31 and 5.33, the following relationship between \(k_{R}\) and \(k_{t}\) can be obtained. By replacing Eq.  5.35 into Eq.  5.30 and considering Eqs.  5.32 and 5.34, we obtain Conversely, if \(m'=\overline{m}\) the relationship between \(k_{R}\) and \(k_{t}\) reads By replacing Eq.  5.37 into Eq.  5.30 and considering Eqs.  5.31 and 5.33, we obtain From the definition of Pareto-optimal solution (Definition 1.​11 in Sect.  3.​1), Eqs.  5.36 and 5.38 imply that the point \(\left( c',m'\right) \) cannot dominate the point \(\left( \overline{c},\overline{m}\right) \). The corresponding point in the design variable domain \(P_{t_{max},R_{min}}\) cannot dominate the point \(P_{t_{min},R_{max}}\). But, from Eq.  5.13, one and only one of the two points must be part of the Pareto-optimal set. The only point that can be part of the Pareto-optimal set is \(P_{t_{min},R_{max}}\), while \(P_{t_{max},R_{min}}\) is not part of such set. For continuity, excluding \(P_{t_{max},R_{min}}\) from the Pareto-optimal set implies that also the boundaries \(R=R_{min}\) and \(t=t_{max}\) do not belong to the Pareto-optimal set. So in Eq.  5.28 the expressions \(t=t_{max}\) and \(R=R_{min}\) can be dropped. Only the two expressions \(R-R_{max}=0\) and \(t-t_{min}=0\) remain in Eq.  5.28 to describe the available room constraint. If a sufficiently large design domain is considered, buckling and structural safety constraints become active. For any value of t, the two constraints (Eqs.  5.5 and 5.7) can be rewritten as The two constraints are continuous and, from Eqs  5.39 and 5.40, it can be observed that for \(R\rightarrow 0\) the constraint on the admissible stress is active, while for \(R\rightarrow \infty \), the buckling constraint is active. Therefore, a switching point exists where the active constraints switches from the maximum admissible stress to the buckling limit. This point can be computed by equating eq 5.39 to Eq.  5.40 and reads By substituting Eq.  5.39 into the expressions of the mass and of the compliance, the following equation representing the maximum stress constraint in the objective functions domain can be obtained Equation  5.42 is a straight line with positive angular coefficient, so the stress constraint is not part of the Pareto-optimal set. The intersection between stress and buckling constraints is the extreme point of the Pareto-optimal set and is the point with the minimum value of mass. The only active constraint in the Pareto-optimal set can be the buckling constraint which is active up to the maximum acceptable stress level. Considering only the buckling constraint, the L-matrix of Eq.  3.​7 reads By applying Eq.  5.11, the analytical expression of the Fritz John necessary condition when the buckling constraint is active can be derived. This expression is coincident with the expression of the buckling itself (Eq.  5.40). By substituting Eq.  5.44 into the expressions of the mass and of the compliance, the following equation representing the buckling constraint in the objective functions domain can be obtained It can be easily checked that this function is decreasing and convex for any value of c. Therefore, the points belonging to the buckling constraint condition belong to the Pareto-optimal set [90]. From Eqs.  5.42 and 5.45, it can be observed that for \(c \rightarrow 0\) the admissible stress condition violates the buckling condition and for \(c \rightarrow \infty \) the buckling condition violates the admissible stress condition. Thus, the buckling constraint condition is limited for high values of compliance by the stress constraint condition. The limit point corresponds to the switch between the two conditions reads