Differential Relations between Load, Shear, and Moment

Differential relations connect the distributed load intensity $q(x)$, shear force $V(x)$ or $Q(x)$, and bending moment $M(x)$. They make it possible to predict diagram shapes and verify results without repeating a full section calculation at every point.

Basic relations

For one common sign convention:

$$\frac{dV}{dx}=-q(x),\qquad \frac{dM}{dx}=V(x).$$

If a different sign convention is adopted for load or shear, the sign of the first equation may change; the underlying geometric relationships remain the same.

Geometric interpretation

Shear gives the slope of the bending-moment diagram, while distributed load controls the rate of change of shear. If $q=0$, shear is constant and moment is linear. If $q$ is constant, shear is linear and moment is quadratic.

Integral form

Over an interval from $x_1$ to $x_2$:

$$V(x_2)-V(x_1)=-\int_{x_1}^{x_2}q(x)\,dx,$$

$$M(x_2)-M(x_1)=\int_{x_1}^{x_2}V(x)\,dx.$$

Thus changes in shear and moment can be read from signed areas of the preceding diagrams.

Practical verification

Zeros of shear correspond to stationary points of moment on smooth intervals. The sign of shear controls whether moment rises or falls, and the sign of load controls how shear changes. These checks quickly reveal inconsistent diagrams.