1. Initial Deflection Formula: [cite: 23, 25]
For a simply supported beam with a central point load $W$, the maximum deflection $\delta$ at the center is given by:
$$\delta = \frac{WL^3}{48EI}$$
Where $E$ is the Young's modulus and $I$ is the area moment of inertia. [cite: 23, 25]
2. Relation to Beam Dimensions: [cite: 23, 25]
The moment of inertia for a rectangular cross-section is $I = \frac{BH^3}{12}$. [cite: 23, 25]
Substituting this into the deflection formula:
$$\delta \propto \frac{1}{I} \implies \delta \propto \frac{1}{BH^3}$$ [cite: 23, 25]
3. Interchanging Dimensions: [cite: 23, 25]
If we interchange width ($B$) and depth ($H$), the new width becomes $B' = H$ and the new depth becomes $H' = B$. [cite: 23, 25]
The new deflection $\delta'$ will be:
$$\delta' \propto \frac{1}{H \cdot B^3}$$ [cite: 23, 25]