Bending of Curved Members - Strain Analysis

When equal and opposite couples act on a curved member in its plane of symmetry, the curvature of the arcs increases, shifting the center of curvature from point C to C'. This indicates the member bends into a tighter curve. The applied couples simultaneously reduce the upper surface length and increase the lower surface length, demonstrating the material's response to bending moments.

The neutral axis is a surface within the bent member whose length remains unchanged despite deformation. While the upper surface shortens and the lower surface elongates under bending, the neutral axis experiences neither tensile nor compressive strain. This unique property makes it a reference line for analyzing strain distribution across the member's thickness.

Strain is determined by dividing the deformation of an arc by its original length. For an arc at distance y from the neutral surface, the deformation is expressed geometrically in terms of the radius of curvature of the neutral surface. This calculation reveals that strain varies non-linearly with distance from the neutral surface, increasing as you move away from it.

Strain varies non-linearly with distance from the neutral axis because the deformation per unit length changes differentially above and below this axis. Points farther from the neutral surface experience greater changes in arc length relative to their original length. This differential response to bending creates a non-linear strain profile that depends on the member's curvature change.

Under bending, the upper surface shortens while the lower surface elongates, indicating compression and tension respectively. The neutral axis separates these regions, experiencing no length change itself. This differential length change across the member's depth is fundamental to understanding deformations in a symmetric member in bending and how strain distributes through the material.

The radius of curvature of the neutral surface is essential for expressing deformation geometrically and calculating strain at any point. When couples are applied, the radius decreases, indicating increased curvature. The strain at distance y from the neutral surface depends directly on this radius and the distance y, making curvature change the key geometric parameter in strain analysis.

A member symmetric about the y-axis with circular arc surfaces centered at point C provides a predictable deformation pattern under equal and opposite couples. This symmetry ensures the neutral axis remains centered and strain distribution is symmetric about the y-axis. The symmetric geometry simplifies analysis of deformations in a transverse cross section and allows direct calculation of strain variation.