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).

An overhanging beam is supported at $A$ ($x=0$) and $B$ ($x=8.00\text{ m}$), with an overhang extending to $C$ ($x=10.0\text{ m}$). A point load of $40.0\text{ kN}$ is applied at $x=4.00\text{ m}$, and a point load of $20.0\text{ kN}$ is applied at the free end $C$. Determine the internal shear and moment at $x=6.00\text{ m}$.

A cantilever beam of length $5.00\text{ m}$ is fixed at the left end ($x=0$). A concentrated clockwise moment of $60.0\text{ kN}\cdot\text{m}$ is applied at the free end ($x=5.00\text{ m}$). Determine the internal shear and moment at $x=2.50\text{ m}$.

A simply supported beam of length $6.00\text{ m}$ carries a triangular load that is $0$ at the left support $A$ and increases linearly to $30.0\text{ kN/m}$ at the right support $B$. Find the maximum shear force in the beam.

A rigid L-shaped frame consists of a vertical member $AB$ ($3.00\text{ m}$ tall, $A$ is the fixed base) and a horizontal member $BC$ ($4.00\text{ m}$ long). A downward point load of $50.0\text{ kN}$ acts at the free end $C$. Determine the internal forces and moment at the midpoint of member $AB$ (height $y=1.50\text{ m}$ from $A$).

A solid circular shaft is subjected to an internal bending moment of $1.50\text{ kN}\cdot\text{m}$ and an internal torsional moment (torque) of $2.00\text{ kN}\cdot\text{m}$ at a specific cross-section. Calculate the equivalent internal moment $M_e$ and equivalent internal torque $T_e$ used in the design of such shafts, using the formulas $M_e = \frac{1}{2}(M + \sqrt{M^2 + T^2})$ and $T_e = \sqrt{M^2 + T^2}$.

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?

In beam analysis, locating the points where the shear force diagram crosses the zero axis is a critical step. Explain why these points are significant when determining the design capacity required for a beam.

A long continuous beam over multiple supports is sometimes designed with an internal hinge inserted at a specific location along its span. Explain how an internal hinge alters the internal forces at that specific point.