Beam Curvature, Bending Moment, and Flexural Rigidity

In classical linear-elastic bending theory, curvature of the deflected beam axis is related to bending moment by:

$$\kappa=\frac{1}{\rho}=\frac{M}{EI},$$

where the sign of the right-hand side depends on the adopted conventions for bending moment and deflection.

For small rotations, curvature is approximately:

$$\kappa\approx w''(x).$$

Therefore, the differential equation of the elastic curve is often written as:

$$EI\,w''(x)=M(x),$$

or with a minus sign according to the chosen sign convention.

• $E$ — Young's modulus;
• $I$ — second moment of area;
• $EI$ — flexural rigidity;
• $w$ — transverse deflection;
• $\rho$ — radius of curvature.

The calculator below uses the equivalent relation $M=EI/\rho$.