Chapter 2.4, Problem 8SA

Ideas for further exploration: If the width of the diving board is doubled, how does the

displacement of the diver change? Does it change more or less than if the thickness is doubled? (Both beams have the same mass.) How does the maximum displacement change if the cross- section is circular or annular with the same area as the rectangle? (The area moment of inertia for a circular cross-section of radius r is $I=\pi r^4/4$, and for an annular cross-section with inner radius $r_1$ and outer radius $r_2$ is $I=\pi \left(r\begin{array}{c}4\\ 2\end{array}-r\begin{array}{c}4\\ 1\end{array}\right)/4$. ) Find out the area moment of inertia for I-beams, for example. The Young’s modulus for different materials are also tabulated and available. For example, the density of steel is about $\text{785}0\text{ kg}/{\text{m}}^{\text{3}}$ and its Young’s modulus is about $2\times {10}^{11}$ Pascals. The Euler-Bernoulli beam is a relatively simple classical model. More recent models, such as the Timoshenko beam, take into account more exotic bending, where the beam cross-section may not be perpendicular to the beam’s main axis.