## 8 Application FVIM to Solve Structural Engineering Design Problems
### Experimental Setup
#### 8.1.2 The Welded Beam Design Problem  
The Welded Beam Design Optimization Problem is a fundamental engineering challenge that focuses on designing an optimal welded beam structure under specific constraints while minimizing costs [65]. The problem involves determining the optimal dimensions of the beam, such as the length, width, height, and thickness of the weld, to ensure the structure's safety and functionality. Engineers must balance various factors, including material strength, beam deflection, shear stress, and bending stress, ensuring the design adheres to industry standards and safety regulations. The objective often includes minimizing the cost associated with the material and fabrication while maximizing the beam's performance under expected loads. This optimization problem is typically addressed using computational methods, including numerical optimization techniques and finite element analysis, to explore the design space efficiently and identify the most cost-effective and reliable beam design. Through this problem, engineers enhance their understanding of how different design variables impact the structural integrity and cost-efficiency of welded beams, contributing to more effective and economical structural design practices in engineering. The objective here is to minimize the manufacturing cost of a welded beam, involving four optimized variables and seven constraints. These include the thickness of the weld (\(x_{1}\)), the length of the clamped bar (\(x_{2}\)), the height of the bar (\(x_{3}\)), and the thickness of the bar (\(x_{4}\)) and constraints like shear stress, bending stress in the beam, buckling load, end deflection of the beam, normal stress, and boundary considerations. The design of the welded beam is depicted in Figure 5.  
The mathematical formulation of the problem is as follows:  
Let \(LB=[lb_{1},lb_{2},lb_{3},lb_{4}]\) and \(UB=[ub_{1},ub_{2},ub_{3},ub_{4}]\) denote the lower and upper bounds of the design variables, respectively. Here, \(lb_{i}\) and \(ub_{i}\) represent the lower and upper bounds of the \(i\)th design variable. _Vio_ denotes any violation conditions, and _Obj_ represents the objective function to be minimized. The global minimum achieved is denoted by \(GloMin\). The objective function, denoted as \(z\), is defined as:  
\[z=1.10471\cdot x_{1}^{2}\cdot x_{2}+0.04811\cdot x_{3}\cdot x_{4}\cdot(14+x_{ 2})\]  
where \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\) are the design variables representing certain characteristics of the welded beam. Figure 5: The welded beam design problem. Constraints \(g\) are defined as follows:  
\[g_{1}(x) =x_{2}\cdot x_{3}-x_{1}\geq 0\] \[g_{2}(x) =\frac{p}{\sqrt{2}\cdot x_{1}\cdot x_{2}}-1.5\geq 0\] \[g_{3}(x) =\frac{p}{\sqrt{2}\cdot x_{1}\cdot x_{2}}-1.5\geq 0\] \[g_{4}(x) =\frac{6\cdot p\cdot l}{x_{4}\cdot x_{3}^{2}}-30000\geq 0\] \[g_{5}(x) =\frac{4\cdot p\cdot l^{3}}{E\cdot x_{3}^{3}\cdot x_{4}}-10000\geq 0\] \[g_{6}(x) =\frac{p}{2\cdot\sqrt{2}\cdot x_{1}\cdot x_{2}}-13600\geq 0\] \[g_{7}(x) =\sqrt{(\frac{p}{2\cdot\sqrt{2}\cdot x_{1}\cdot x_{2}})^{2}+ \frac{m\cdot x_{2}}{2\cdot r}+(\frac{m}{j})^{2}}-30000\geq 0\] \[z=1.10471\cdot x_{1}^{2}\cdot x_{2}+0.04811\cdot x_{3}\cdot x_{4}\cdot(14+x_{ 2})\]0 \[z=1.10471\cdot x_{1}^{2}\cdot x_{2}+0.04811\cdot x_{3}\cdot x_{4}\cdot(14+x_{ 2})\]1 \[z=1.10471\cdot x_{1}^{2}\cdot x_{2}+0.04811\cdot x_{3}\cdot x_{4}\cdot(14+x_{ 2})\]2  
where \(g_{i}(x)\) represents the \(i\)th constraint function, and \(x_{1}\), \(x_{2}\), \(x_{3}\), and \(x_{4}\) are the design variables. \(p\) represents the load, \(l\) represents the length, \(E\) represents the modulus of elasticity, \(m\) represents the bending moment, \(r\) represents the radius of gyration, and \(j\) represents the section modulus. \(p_{c}\) represents the critical load. The lower and upper bounds for the design variables are defined by \(LB\) and \(UB\), respectively, and any other conditions such as violation conditions are defined by _Vio_.