Deformations in a Symmetric Member in Bending

When a symmetric prismatic member bends under equal and opposite couples, the only non-zero stress component is normal stress, creating uniaxial stress conditions at any point. This uniform stress distribution occurs because the member deforms into circular arcs of constant curvature, allowing stress to vary predictably across the cross-section based on distance from the neutral surface.

The neutral surface exists parallel to the upper and lower faces of the member, where both longitudinal strain and stress components are zero. This surface serves as the reference point for measuring deformation, with the distance from the neutral surface to the center of curvature designated as ρ. All deformation calculations are based on distance from this neutral surface.

Longitudinal normal strain varies linearly with distance y from the neutral surface. By comparing arc lengths at different distances from the center of curvature and dividing by the neutral surface arc length, the strain relationship becomes proportional to distance from the neutral surface, with negative values indicating upward concavity in positive bending.

Dividing the member into tiny cubic elements reveals that normal stress is the only significant stress component, confirming uniaxial stress conditions throughout the member. This elemental analysis demonstrates how stress and strain distribute uniformly across the cross-section and validates the linear relationship between strain and distance from the neutral surface.

Deformation is calculated as the difference between the arc length at distance y from the neutral surface and the neutral surface arc length itself. Expressing these lengths in terms of radius and angle subtended, then dividing by the neutral arc length, yields the longitudinal normal strain. This method quantifies how material fibers extend or compress based on their position relative to the neutral surface.

Point C is the center of curvature around which the bent member's originally straight lines transform into circular arcs of constant radius. The distance from the neutral surface to Point C is designated as ρ and serves as the reference for all deformation calculations. This geometric center is essential for understanding how the member's shape changes under bending loads.

Hooke's Law relates stress and strain in elastic materials, allowing stress to be determined at any point based on its distance from the neutral surface. Since longitudinal strain varies linearly with distance y, applying Hooke's Law produces a corresponding linear stress distribution across the member's depth, enabling accurate prediction of stress at any location.