Point of contraflexure; w-V-M relation

Point of contraflexure

A point of contraflexure (inflexion point) is a point on a beam where the bending moment changes sign (passes through zero), so the beam's curvature changes from sagging to hogging or vice versa. The bending moment there is zero, but a zero moment does not always mean contraflexure: the moment must also change sign.

Relation between loading rate, shear force and bending moment

Take an element of length $dx$ of a beam on which the load intensity is $w$ per unit length, acting downward. $V$ and $M$ act on the left face; $V + dV$ and $M + dM$ act on the right face.

          w
   ↓ ↓ ↓ ↓ ↓ ↓
  M ┌─────────┐ M + dM
 V ↑│         │↓ V + dV
    └─────────┘
        dx

Vertical equilibrium

$$V - w\,dx - (V + dV) = 0 \;\Rightarrow\; \frac{dV}{dx} = -w$$

Moment equilibrium about the right face (the term $w\,dx^2/2$ is negligible):

$$M + V\,dx - w\,dx\cdot\frac{dx}{2} - (M + dM) = 0 \;\Rightarrow\; \frac{dM}{dx} = V$$

Combining:

$$\frac{d^2M}{dx^2} = \frac{dV}{dx} = -w$$

Meaning: the rate of change of shear force equals the negative of the load intensity, and the rate of change of bending moment equals the shear force. The moment is a maximum where $V = 0$. A point of contraflexure occurs where $M = 0$ and $M$ changes sign, i.e. where the curvature of the elastic curve ($1/R = M/EI$) changes sign. Where there is no load near that point ($w = 0$), it is simply where the straight-line BMD crosses the axis.