List of top Solid Mechanics Questions on Energy methods (Castigliano’s Theorems)

A simply-supported beam, with a point load \(P = 150 \, kN\) at a distance of \(L/3\) from the left end, is shown in the figure. The elastic-strain energy (U) of the beam is given by: \[ U = \frac{2}{243} \cdot \frac{P^2 L^3}{EI} \] where the section modulus, \(EI = 16.66 \times 10^5 \, Nm^2\), and the length of the beam \(L = 1 \, m\).

The deflection at the loading point is .......... mm (rounded off to two decimal places).

A spherical rigid ball of mass $10 \, kg$ is moving with a speed of $2 \, m/s$ in the direction shown. The ball collides with a rigid frictionless wall and rebounds at an angle $\alpha$ with a speed $v$, as shown. The coefficient of restitution is $0.9$. The angle $\alpha$ (in degrees) is ........... (rounded off to one decimal place).

A naturally curved steel beam AB having Young's modulus 208 GPa, area moment of inertia \( I = 26.7 \, \text{cm}^4 \) and radius \( R = 2 \, \text{m} \) is subjected to a vertical load \( P = 1000 \, \text{N} \) at B. The end A at \( \theta = 90^\circ \) is rigidly fixed. The bending strain energy of the beam (in Nm, rounded off to two decimal places) is \(\underline{\hspace{2cm}}\).
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