Structural optimization problem
Published:
As part of my personal goal to improve my writing skill, I will be posting several notes (or documentation) I used in learning some tools or concepts I utilize in my current research. These topics will be mostly about structural reliability, optimization and disaster risk. Along the way, I will be able to deepen my understanding on the fundamental principles and at the same time it will be good exercise to articulate ideas in a clear and concise manner.
For this post, my topic will be on structural optimization.
Optimization as an engineering tool
I have learned about optimization during my master’s degree studies. I was extremely fascinated by how mimicking the behavior of flocking animals can be used to solve engineering problems. It was very rewarding that we were tasked to code manually our own particle swarm optimizer.
I see optimization techniques as a tool that engineers can (or should) use in design tasks or decision making. One of the main task of a structural engineer is to decide the building specifications, that is, materials to be used, dimension and cross-sectional areas of structure and many more. Due to many design choices to make: what materials to use? what should the dimension of the columns, beams, etc.? Structural design is best framed as an optimization problem (OP). Mathematically, it is stated as
\[\begin{aligned} \text{Find} \: & x^{\ast} \: \text{that} \\ \min_{x} \quad & f(x) \\ \text{subject to} \quad & g_i(x) \le 0, \quad i = 1,\dots,m \\ & h_j(x) = 0, \quad j = 1,\dots,p \end{aligned}\]The goal is to find the best design or optimal design $x^{\ast}$ that will minimized the objective function $f(\cdot)$ and satisfy the constraint functions, $g_i(\cdot)$ and $h_j(\cdot)$. This is a standard constrained optimization problem. In structural optimization, the objective function to minimize is often the construction or material costs of the building while the constraints are related to deformation or stress thresholds. The requirements (geometric, deformation and strength) set by building codes are usually the constraints used in structural optimization.
Solution to the OP stated above are extensively described in [1] and [2]. The primary hurdle in using optimization in engineering design would be the computational cost, that is, either evaluating the objective function or the constraint functions. For example, if the design you are doing requires performing a nonlinear analysis then doing the iterative nature of optimization solutions can be prohibitive.
A simple illustrative problem
A simple problem that demonstrates the application of the optimization framework to structural problems is hereby explained. The figure shown below is a simple truss structure with three members and an applied load at the free end of each member [3].
Using principle of structural analysis, the stresses ($\sigma_i, i=1,2,3$) for each truss member can be easily calculated as
\[\begin{aligned} \sigma_1 & = \frac{\sqrt{2} A_1 + A_2 }{\sqrt{2}A_1^2 + 2A_1 A_2} P \\ \sigma_2 & = \frac{1 }{\sqrt{2}A_2 + A_1 } P \\ \sigma_3 & = \frac{A_2 }{\sqrt{2}A_1^2 + 2A_1 A_2} P \end{aligned}\]As a structural designer, one has to balance safety and economy. For a given force $P$, you should decide what are the values of the cross sections $A_1, A_2, A_3$ of each truss member subject to some minimum allowable thresholds on the stresses developed to each member. For example, if an allowable stress threshold, $\bar{\sigma} = 20 : \text{MPa}$ can be set. The same with the load force $P$ and threshold $\bar{\sigma}$, these values are normally set by some building code or technical guideline. The objective is to minimize the cost of the structure which is usually proportional to the volume of the materials ($\text{Volume}: = (2\sqrt{2}A_1 + A_2) \times L$) to be used in the construction. Thus, the optimization problem can be stated as
\[\begin{aligned} \min_{x} \quad & f(x) = (2\sqrt{2}A_1 + A_2) \times L \\ \text{subject to} \quad & \sigma_1 = \frac{\sqrt{2} A_1 + A_2 }{\sqrt{2}A_1^2 + 2A_1 A_2} P \le 20 \: \text{MPa} \\ & \sigma_2 = \frac{1 }{\sqrt{2}A_2 + A_1 } P \le 20 \: \text{MPa} \\ & \sigma_3 = \frac{A_2 }{\sqrt{2}A_1^2 + 2A_1 A_2} P \le 20 \: \text{MPa} \\ \end{aligned}\]where $x = [A_1,A_2 ]$. In this example, it is shown how a structural engineering problem is viewed in an optimization framework. Since there are a lot of available tools to solve optimization problems, the problem above can easily be solved.
Reference
- Nocedal, J., & Wright, S. J. (2006). Numerical optimization (2nd ed). Springer.
- Martins, J. R. R. A., & Ning, A. (2022). Engineering design optimization. Cambridge University Press. https://doi.org/10.1017/9781108980647
- Ray, T., & Saini, P. (2001). Engineering design optimization using a swarm with an intelligent information sharing among individuals. Engineering Optimization, 33(6), 735–748. https://doi.org/10.1080/03052150108940941