In curved beams, unlike straight beams, the stress distribution across the cross-section is not uniform due to the beam's curvature. This non-uniformity arises because the neutral axis, where stress is zero, does not align with the centroid of the section. In a curved beam, the strain varies along the section as a function of the distance from the neutral axis.

Consider the curved member described in the previous lesson. According to Hooke's law, which relates stress to strain within the material's elastic limits, the stress also varies non-linearly, resulting in a hyperbolic stress distribution from the neutral axis. The bending moment in a curved beam is calculated by integrating these stress distributions across the beam's cross-section as shown in Equation 1.

The elementary forces acting on any section sum up to create a bending couple equivalent to the moment. This cumulative effect of stress results in the moment equation, which is essential for determining the beam's behavior under load. Analysis reveals that the neutral surface, where longitudinal stress is zero, does not align with the centroid but shifts toward the center of curvature. Regardless of the beam's shape, the neutral axis always lies between the centroid and the radius of curvature.