How to Draw a Bending Moment Diagram (BMD): Calculus, Areas & Zero-Crossings
Master the construction of Bending Moment Diagrams (BMD). Understand the fundamental calculus link between shear and moment, calculate inflection points, and locate peak stresses.
Key Engineering Takeaways
- The slope of the Bending Moment Diagram is equal to the Shear Force: dM/dx = V(x).
- The change in bending moment between two coordinates equals the area under the shear force curve.
- Maximum or minimum bending moments occur where the shear force crosses the zero axis (V = 0).
- Inflection points (contraflexure) occur where M(x) = 0 and curvature changes from sagging to hogging.
1. Definition of Bending Moment
The Bending Moment $M(x)$ is the internal reaction couple that develops within a beam cross-section to resist external bending actions. Bending moments create normal flexural stresses according to the elastic flexure formula:
2. The Calculus Connection: dM/dx = V(x)
Because the derivative of the bending moment is the shear force:
- Where $V(x) > 0$, the BMD has a positive upward slope.
- Where $V(x) < 0$, the BMD has a negative downward slope.
- Where $V(x) = 0$, the BMD has a horizontal tangent slope ($dM/dx = 0$), which marks a local peak moment.
- Under a uniform load $w$, the SFD is degree 1 (linear), which makes the BMD degree 2 (parabolic).
3. Points of Contraflexure (Inflection Points)
A point of contraflexure is a location where the bending moment vanishes ($M = 0$) and the beam curvature changes sense. In continuous or overhanging beams, identifying inflection points is vital for determining where reinforcement steel must transition from the bottom face (sagging zone) to the top face (hogging zone).
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