Related Module(s): materialSelectionFilter module

How to Construct Performance Metrics Equation for materialSelectionFilter Module

What is Performance Metrics?

Performance metrics quantify the relationship between functional requirements, geometric parameters, and material properties to evaluate the effectiveness of a design. These metrics provide a basis for comparing materials and geometric configurations for optimized performance.

In mathematical terms, performance can be expressed as:

$P = \left[ f_1(F), f_2(G), f_3(M) \right]$

where: \begin{itemize} \item $F$ : Functional requirements (e.g., applied force). \item $G$ : Geometric parameters (e.g., dimensions). \item $M$ : Material properties (e.g., density, elastic modulus). \end{itemize}

Problem Example: Bench Design

Consider designing a bench subjected to a load F = 100 kg and a length L = 1 m. The deflection ( $y_{\text{max}}$ ) must not exceed $5 \, \mathrm{mm}$ . The governing equation is:

$y_{\text{max}} = \frac{F L^3}{48EI}$

where: $I = \frac{1}{12} b h^3$ is the moment of inertia. $E$ is the material’s elastic modulus. Substituting $I$ and rearranging, the stiffness constant $k$ becomes:

$k = \frac{4 E b h^3}{L^3}$

The total mass $m$ is:

$m = \rho b h L$

Thus:

$P = \frac{k \cdot L^4}{4 h^2} \cdot \frac{\rho}{E}$

For material selection, we consider:

$f_1(F) = \frac{k}{4}, \quad f_2(G) = \frac{L^4}{h^2}, \quad f_3(M) = \frac{\rho}{E}$

Set y(x) to Minimize Metrics To minimize the mass m, aim to reduce f3(M)= rho/E

This is achieved by maximizing the material property ratio:

$M = \frac{E}{\rho}$

Plot the material property relationship as:

$E(\rho) = M \cdot \rho$

This relationship guides the material selection process [plot y(x) = M *x equation to materialSelectionFilter module], prioritizing materials with high M with trial and error input of M.

Narrow the Search with Minimum E

To satisfy the deflection limit $y_{\text{max}} \leq 5 \, \mathrm{mm}$ , substitute values:

$E > \frac{F L^3}{48 \cdot y_{\text{max}} \cdot I}$

Given $b = 100 \, \mathrm{mm}$ and $h = 20 \, \mathrm{mm}$ , calculate:

$I = \frac{1}{12} b h^3 = \frac{1}{12} \cdot 0.1 \cdot (0.02)^3$

Substitute $F = 100 \cdot 9.81 \, \mathrm{N}$ , $L = 1 \, \mathrm{m}$ , and $y_{\text{max}} = 0.005 \, \mathrm{m}$ to find:

$E > 5 \, \mathrm{GPa}$

Validate the Strength

Finally, validate the material selection by checking its strength against the maximum stress:

$\sigma_{\text{max}} = \frac{F}{A}$

where $A = b \cdot h$ . The material must have a yield strength $\sigma_y$ greater than $\sigma_{\text{max}}$ . For example, if $A = 0.1 \cdot 0.02 \, \mathrm{m}^2$ :

$\sigma_{\text{max}} = \frac{100 \cdot 9.81}{0.1 \cdot 0.02}$

Plot

$\sigma_{\text{min}} = 36 Mpa$

on materialSelectionFilter to Ensure $\sigma_{\text{max}} \leq \sigma_y$ for the selected material. By following these steps, the performance metric equation helps filter materials systematically, optimizing designs for weight, deflection, and strength.

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Related Module(s): materialSelectionFilter module

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Author: Caesar Wiratama