Bending

For simple bending of a beam within the linear-elastic model, normal stress varies linearly with distance from the neutral axis:

$$\sigma_x(y)=-\frac{My}{I}.$$

The maximum absolute stress at an extreme fiber is:

$$|\sigma_{\max}|=\frac{|M|y_{\max}}{I}=\frac{|M|}{W},\qquad W=\frac{I}{y_{\max}}.$$

• $M$ — bending moment;
• $I$ — second moment of area about the neutral axis;
• $y$ — distance from the neutral axis;
• $y_{\max}$ — distance to the relevant extreme fiber;
• $W$ — section modulus.

The sign of $\sigma_x$ depends on the sign convention for $M$ and $y$; one side of the section is in tension and the other in compression.

For transverse bending, the shear stress at a point of the cross-section is determined by the Zhuravsky formula:

$$\tau=\frac{QS}{Ib}.$$

• $Q$ — shear force at the section;
• $S$ — first moment of the cut-off part of the area about the neutral axis;
• $I$ — second moment of area of the entire section about the neutral axis;
• $b$ — local section width at the level where $\tau$ is evaluated.

At a free upper or lower boundary of a solid section, $S$ tends to zero and therefore $\tau$ is zero. For a rectangular section, the maximum shear stress occurs at the neutral axis.

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 $M=EI/\rho$.