Mixed-integer linear programming approach for global discrete sizing optimization of frame structures

Abstract

This paper focuses on discrete sizing optimization of frame structures using commercial profile catalogs. The optimization problem is formulated as a mixed-integer linear programming (MILP) problem by including the equations of structural analysis as constraints. The internal forces of the members are taken as continuous state variables. Binary variables are used for choosing the member profiles from a catalog. Both the displacement and stress constraints are formulated such that for each member limit values can be imposed at predefined locations along the member. A valuable feature of the formulation, lacking in most contemporary approaches, is that global optimality of the solution is guaranteed by solving the MILP using branch-and-bound techniques. The method is applied to three design problems: a portal frame, a two-story frame with three load cases and a multiple-bay multiple-story frame. Performance profiles are determined to compare the MILP reformulation method with a genetic algorithm.

Appendices

Appendix A: Mixed-integer linear programming problem

1.1 A.1 Equilibrium equations

The nodal equilibrium is imposed by the equality constraints of (5). In this equation, B _i is a 6 × n _dof binary location matrix that maps the system degrees of freedom to the element degrees of freedom, T _i is a 6 × 6 transformation matrix that accounts for the orientation of the element (Kassimali 1999), and f is the n _dof × 1 nodal load vector. Element loads are taken into account as equivalent nodal loads in the nodal load vector f. R _i is a 6 × 3 matrix giving the relation between the six member end forces as shown in Fig. 1, and the three independent force variables q _ij (see (1)) as follows:

$$ \left[\begin{array}{c} N_{1,ij}\\ V_{1,ij}\\ M_{1,ij}\\ N_{2,ij}\\ V_{2,ij}\\ M_{2,ij} \end{array}\right]=\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & \frac{1}{L_{i}} & \frac{1}{L_{i}}\\ 0 & 1 & 0\\ -1 & 0 & 0\\ 0 & -\frac{1}{L_{i}} & -\frac{1}{L_{i}}\\ 0 & 0 & 1 \end{array}\right]\left[\begin{array}{c} N_{1,ij}\\ M_{1,ij}\\ M_{2,ij} \end{array}\right] $$

(40)

1.2 A.2 Member stiffness relations

In addition to nodal equilibrium, the material law and compatibility conditions are needed in structural analysis. For trusses, Hooke’s law and compatibility conditions can be written as a single equation, because the normal force is the only stress resultant appearing in the members (Rasmussen and Stolpe 2008). As frame members have three (6 in 3D) stress resultants in each node, altogether six (12 in 3D) force-displacement relations are needed. Thus, the relation between the member end forces and the nodal displacements can be written as:

$$ {\mathbf{q}_{ij}=y_{ij}\mathbf{K}_{ij}\mathbf{T}_{i}\mathbf{B}_{i}\mathbf{u}} \qquad \forall\;i\in\mathcal{M},\:\forall\;j\in\mathcal{C}_{i} $$

(41)

where the matrix K _ij assembles the first, third and sixth row of the element stiffness matrix:

$$\mathbf{K}_{ij}=\left[\begin{array}{cccccc} \frac{EA_{ij}}{L_{i}} & 0 & 0 & -\frac{EA_{ij}}{L_{i}} & 0 & 0\\ 0 & \frac{6EI_{ij}}{{L_{i}^{2}}} & \frac{4EI_{ij}}{L_{i}} & 0 & -\frac{6EI_{ij}}{{L_{i}^{2}}} & \frac{2EI_{ij}}{L_{i}}\\ 0 & \frac{6EI_{ij}}{{L_{i}^{2}}} & \frac{2EI_{ij}}{L_{i}} & 0 & -\frac{6EI_{ij}}{{L_{i}^{2}}} & \frac{4EI_{ij}}{L_{i}} \end{array}\right] $$

where E is the Young’s modulus of the material, L _i is the length of member i, and A _ij and I _ij are the section area and second moment of area of profile j for member i, respectively.

Equation (41) ensures that the force variables q _ij become zero when profile j is not selected for member i (y _ij = 0) and q _ij = K _ij T _i B _i u when profile j is selected for member i (y _ij = 1).

In a regular finite element analysis, the global stiffness matrix K is assembled by replacing q _ij in (5) with the expression given by (41). The resulting equilibrium equation can not be reformulated as a linear system of equations in terms of the design variables since the global stiffness matrix depends on the binary decision variables. Therefore, the linear nodal equilibrium, (5), and the member stiffness relation, (41), are adopted as separate constraints.

The member stiffness relation in (41) is nonlinear in terms of the design variables but it can be equivalently reformulated as a set of linear inequality constraints by introducing artificial upper and lower bounds (Rasmussen and Stolpe 2008) of (6). In this equation, the force variables become equal to q _ij = K _ij T _i B _i u when profile j is selected for member i (y _ij = 1). When profile j is not selected for member i (y _ij = 0), the force variables do not become zero but are bounded by $\mathbf {K}_{ij}\mathbf {T}_{i}\mathbf {B}_{i}\mathbf {u}-\bar {\mathbf {q}}^{\prime }_{ij}\leqslant \mathbf {q}_{ij}\leqslant \mathbf {K}_{ij}\mathbf {T}_{i}\mathbf {B}_{i}\mathbf {u}-{\mathbf {\underline {q}}^{\prime }_{ij}}$. In order to ensure that the force variables become zero when profile j is not selected for member i, additional constraints given by (7) are introduced.

The artificial upper and lower bounds $\bar {\mathbf {q}}^{\prime }_{ij}$ and ${\mathbf {\underline {q}}^{\prime }_{ij}}$ ensure that, when profile j is not selected for member i, the nodal displacements are not bounded by (6). Each element k of the artificial bound vectors is calculated as follows (Stolpe and Svanberg 2003):

$$\begin{array}{@{}rcl@{}} &&{{\underline{q}}^{\prime}_{k,ij}}=\underset{\mathbf{u}}{\min}\qquad \mathbf{k}_{r,ij}\mathbf{T}_{i}\mathbf{B}_{i}\mathbf{u}\\ &&~~~~~~~~~~~~~~ \textrm{s.t.} \qquad \mathbf{\underline{\mathbf{u}}} \leqslant\mathbf{u}\leqslant\mathbf{\overline{\mathbf{u}}} \end{array} $$

(42)

$$\begin{array}{@{}rcl@{}} &&\bar{q}^{\prime}_{k,ij}=\underset{\mathbf{u}}{\max}\qquad \mathbf{k}_{r,ij}\mathbf{T}_{i}\mathbf{B}_{i}\mathbf{u}\\ &&~~~~~~~~~~~~~~\textrm{s.t.} \qquad \mathbf{\underline{\mathbf{u}}} \leqslant\mathbf{u}\leqslant\mathbf{\overline{\mathbf{u}}} \end{array} $$

(43)

where k _{r, ij} represents row $r\in \left \{1,3,6 \right \}$ of the element stiffness matrix K _ij , and $\mathbf {\underline {\mathbf {u}}}$ and $\mathbf {\overline {\mathbf {u}}}$ are the prescribed minimum and maximum allowed nodal displacements, respectively. Equations. (42) and (43) are linear optimization problems with bound constraints, that can be solved without effort (Stolpe and Svanberg 2003).

1.3 A.3 Displacement vectors

$$\begin{array}{@{}rcl@{}} \mathbf{D}_{i}^{u}(x) & =& \left[\begin{array}{cccccc} 1-\frac{x}{L_{i}} & 0 & 0 & \frac{x}{L_{i}} & 0 &0\end{array}\right] \end{array} $$

(44)

$$\begin{array}{@{}rcl@{}} \mathbf{D}_{i}^{v}(x) & =& \left[\begin{array}{cccc} 0 & 1-\frac{3x^{2}}{{L_{i}^{2}}}+\frac{2x^{3}}{{L_{i}^{3}}} & x\left( 1-\frac{x}{L_{i}}\right)^{2} & \cdots\end{array}\right.\\ && \left.\begin{array}{ccc} 0 & \frac{3x^{2}}{{L_{i}^{2}}}-\frac{2x^{3}}{{L_{i}^{3}}} &x\left( -\frac{x}{L_{i}}+\frac{x^{2}}{{L_{i}^{2}}}\right) \end{array}\right] \end{array} $$

(45)

$$\begin{array}{@{}rcl@{}} \mathbf{D}_{i}^{\varphi}(x) & =& \left[\begin{array}{cccc} 0 & \frac{-6x}{{L_{i}^{2}}}+\frac{6x^{2}}{{L_{i}^{3}}} & 1-\frac{4x}{L_{i}}+\frac{3x^{2}}{{L_{i}^{2}}} & \cdots\end{array}\right.\\ && \left.\begin{array}{ccc} 0 & \frac{6x}{{L_{i}^{2}}}-\frac{6x^{2}}{{L_{i}^{3}}} &\frac{-2x}{L_{i}}+\frac{3x^{2}}{{L_{i}^{2}}} \end{array}\right] \end{array} $$

(46)

1.4 A.4 Stress vectors

$$\begin{array}{@{}rcl@{}} \mathbf{S}_{ij}^{\sigma_{\mathrm{t}}}(x) &=&\left[\begin{array}{cccc} -\frac{E}{L_{i}} & \frac{6EI_{ij}}{{L_{i}^{2}}W_{\mathrm{t},ij}}-\frac{12xEI_{ij}}{{L_{i}^{3}}W_{\mathrm{t},ij}} & \frac{4EI_{ij}}{L_{i}W_{\mathrm{t},ij}}-\frac{6xEI_{ij}}{{L_{i}^{2}}W_{\mathrm{t},ij}} & \cdots\end{array}\right.\\ && \left.\begin{array}{ccc} \frac{E}{L_{i}} & \frac{12xEI_{ij}}{{L_{i}^{3}}W_{\mathrm{t},ij}}-\frac{6EI_{ij}}{{L_{i}^{2}}W_{\mathrm{t},ij}} & \frac{2EI_{ij}}{L_{i}W_{\mathrm{t},ij}}-\frac{6xEI_{ij}}{{L_{i}^{2}}W_{\mathrm{t},ij}}\end{array}\right] \end{array} $$

(47)

$$\begin{array}{@{}rcl@{}} \mathbf{S}_{ij}^{\sigma_{\mathrm{b}}}(x) & =&\left[\begin{array}{cccc} -\frac{E}{L_{i}} & \frac{12xEI_{ij}}{{L_{i}^{3}}W_{\mathrm{b},ij}}-\frac{6EI_{ij}}{{L_{i}^{2}}W_{\mathrm{b},ij}} & \frac{6xEI_{ij}}{{L_{i}^{2}}W_{\mathrm{b},ij}}-\frac{4EI_{ij}}{L_{i}W_{\mathrm{b},ij}} & \cdots\end{array}\right.\\ && \left.\begin{array}{ccc} \frac{E}{L_{i}} & \frac{6EI_{ij}}{{L_{i}^{2}}W_{\mathrm{b},ij}}-\frac{12xEI_{ij}}{{L_{i}^{3}}W_{\mathrm{b},ij}} & \frac{6xEI_{ij}}{{L_{i}^{2}}W_{\mathrm{b},ij}}-\frac{2EI_{ij}}{L_{i}W_{\mathrm{b},ij}}\end{array}\right] \end{array} $$

(48)

$$\begin{array}{@{}rcl@{}} \mathbf{S}_{ij}^{\tau}(x) & =&\left[\begin{array}{cccccc} 0 & -\frac{12ES_{ij}}{{L_{i}^{3}}b_{ij}} &-\frac{6ES_{ij}}{{L_{i}^{2}}b_{ij}} & 0 & \frac{12ES_{ij}}{{L_{i}^{3}}b_{ij}} &-\frac{6ES_{ij}}{{L_{i}^{2}}b_{ij}}\end{array}\right] \end{array} $$

(49)

Appendix B: Detailed results for the portal frame

Table 3 Constrained displacements evaluated at the optimum design

Fig. 11

Deformation and internal force diagrams at the optimum design

Table 4 Constrained stresses evaluated at the optimum design

Appendix C: Detailed results for the three-bay three-story frame

Fig. 12

Deformation and internal force diagrams at the optimum design

Table 5 Constrained deflections evaluated at the optimum design

Table 6 Constrained interstory drifts evaluated at the optimum design

Table 7 Constrained stresses evaluated at the optimum design