For the clamped beam shown in the figure, two design variables are defined, as (I_{1}=b_{1}) and (I_{2}=b_{2}).

Question:

For the clamped beam shown in the figure, two design variables are defined, as $I_{1}=b_{1}$ and $I_{2}=b_{2}$. Using the finite element method and sensitivity analysis, calculate the sensitivity of the vertical displacement $v_{2}$ with respect to $b_{1}$ and $b_{2}$. Use the following values for the current design: $E=30 \times 10^{6} \mathrm{psi}, b_{1}=0.1 \mathrm{in}^{4}$, and $b_{2}=0.05 \mathrm{in}^{4}$. Compare the results with the exact sensitivity.

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