A simply supported beam of length $10\text{ m}$ carries a central point load of $100\text{ kN}$. Determine the internal normal force, shear force, and bending moment at a point exactly $3\text{ m}$ from the left support.

A cantilever beam of length $L$ is fixed at the left end ($x=0$) and free at the right end ($x=L$). It carries a uniform distributed load $w$ over its entire length. Derive the equations for shear force $V(x)$ and bending moment $M(x)$ as a function of the distance $x$ from the free end.

A beam is subjected to a linearly varying distributed load that starts at $0$ and reaches $w_0$ at the end. Explain mathematically the shape of the resulting Shear Force Diagram (SFD) and Bending Moment Diagram (BMD).

A structural engineer uses Bending Moment Diagrams (BMD) extensively when designing reinforced concrete beams. Explain conceptually why the shape and sign (positive vs. negative) of the BMD dictate where the steel rebar must be placed within the concrete beam.

Consider a heavy piece of machinery resting on a small area of a concrete floor slab. What specific feature on the Shear Force Diagram indicates a high risk of failure, and what kind of failure is this called?