Shear and Moment in Beams - Examples & Applications

Problem: A simply supported beam of length $L = 4.00 \text{ m}$ is subjected to a central point load $P = 50.0 \text{ kN}$. Determine the maximum shear force and maximum bending moment.

Problem: A simply supported beam of length $L = 6.00 \text{ m}$ carries a uniformly distributed load of $w = 20.0 \text{ kN/m}$. Determine the maximum shear and maximum bending moment.

Problem: A cantilever beam of length $L = 4.00 \text{ m}$ is fixed at the left end ($A$) and carries a point load $P = 10.0 \text{ kN}$ at the free end ($B$). Determine the internal reactions at the support.

Problem: A cantilever beam of length $L = 3.00 \text{ m}$ is fixed at the right end ($B$) and subjected to a uniformly distributed load $w = 15.0 \text{ kN/m}$ over its entire length. Find the maximum shear and moment.

Problem: A simply supported beam of length $L = 5.00 \text{ m}$ has supports at $A$ ($x=0$) and $B$ ($x=5.00 \text{ m}$). A point load $P = 40.0 \text{ kN}$ is applied at $x = 2.00 \text{ m}$. Calculate the maximum bending moment.

Problem: A beam of length $L = 6.00 \text{ m}$ is simply supported at its ends $A$ and $B$. A concentrated clockwise couple moment $M_0 = 30.0 \text{ kN}\cdot\text{m}$ is applied at the midspan. Determine the shear and moment distribution.

Problem: A beam with supports at $A$ ($x=0$) and $B$ ($x=6.00 \text{ m}$) has an overhang $BC$ of $2.00 \text{ m}$. It carries a uniform load of $10.0 \text{ kN/m}$ over the entire length ($8.00 \text{ m}$). Determine the maximum positive and negative moments.

Problem: A simply supported beam of length $L = 6.00 \text{ m}$ carries a triangular load starting from zero at the left support $A$ to a maximum of $w_0 = 30.0 \text{ kN/m}$ at the right support $B$. Determine the maximum bending moment.

Problem: Why is it mathematically essential to locate the point of zero shear when analyzing beam designs for maximum bending stresses?

Problem: How does the type of applied load dictate the mathematical shape of the shear and bending moment diagrams?

Problem: How does the introduction of an internal hinge in a continuous beam affect the internal forces and the overall determinacy of the structure?

Problem: What are the physical differences between sagging (positive) and hogging (negative) bending moments, and how do they influence reinforced concrete design?