Learning topic
Shear-Force and Bending-Moment Diagrams
Learn how to construct shear-force V/Q and bending-moment M diagrams for statically determinate beams, handle jumps, and check key sections.
Shear-force and bending-moment diagrams are used in structural mechanics not only as calculation outputs but also as tools for checking and interpreting beam behavior. The basic section method is supplemented here by structural features that a correct diagram must satisfy.
Characteristic points
Intervals are separated by concentrated forces or moments, the start and end of distributed loads, supports, and internal hinges. Within each interval, the mathematical form of $V(x)$ and $M(x)$ follows from the load distribution.
Jumps and continuity
A concentrated transverse force causes a jump in the shear diagram. A concentrated couple causes a jump in the bending-moment diagram. Without the corresponding concentrated action, the relevant function remains continuous across the point.
Moment extrema
On a smooth interval, a local extremum of $M$ occurs where $V=0$. Zeros of the shear diagram are therefore important control points for the bending-moment diagram.
Diagram checks
Use equilibrium, boundary conditions, and the relationships among load, shear, and moment. At an unloaded internal hinge the bending moment is zero. On an interval with no distributed load, shear is constant and bending moment varies linearly.
Sign convention
Textbooks may use different sign conventions. What matters is consistent use of the selected convention and agreement between the diagram and equilibrium of the isolated structural part.
About this topic
This topic covers V/Q and M diagrams for statically determinate beams, characteristic intervals, jumps due to concentrated actions, and checks at key sections.