Solved Problems

Problem: A solid steel bar consists of two segments. The upper segment has a length of $2 \text{ m}$ and an area of $500 \text{ mm}^2$. The lower segment has a length of $1 \text{ m}$ and an area of $250 \text{ mm}^2$. An axial tensile load of $50 \text{ kN}$ is applied at the bottom. Calculate the total strain energy stored in the bar. Assume $E = 200 \text{ GPa}$.

Problem: A $200 \text{ kg}$ block is dropped from a height of $50 \text{ mm}$ onto a rigid collar at the bottom of a vertical steel rod. The rod has a length of $3 \text{ m}$, a diameter of $20 \text{ mm}$, and an elastic modulus $E = 200 \text{ GPa}$. Determine the maximum dynamic stress developed in the rod. (Use $g = 9.81 \text{ m/s}^2$).

Problem: A train car of mass $15,000 \text{ kg}$ is rolling at a speed of $2 \text{ m/s}$. It collides with a spring bumper at the end of the track. If the bumper spring must absorb all the kinetic energy without compressing more than $150 \text{ mm}$, what is the minimum required stiffness ($k$) of the spring?

Problem: A simply supported uniform beam of length $L$ carries a concentrated load $P$ at its center. Using Castigliano's Theorem, determine the deflection of the beam at the point of load application. Assume the flexural rigidity $EI$ is constant.