Understanding Bending Stress in Curved Beams

When a beam is subjected to bending, internal stresses are developed within its cross-section to resist the applied bending moment. For straight beams, the bending stress is zero at the neutral axis and increases linearly with the distance from the neutral axis, reaching maximum values at the outermost fibres.

However, the behavior of curved beams under bending is different from that of straight beams. Due to the initial curvature, the distribution of bending stress across the section is not linear. This non-linear distribution arises because the fibres closer to the center of curvature are shorter, and those further away are longer, even before bending occurs. When subjected to a bending moment, the strain distribution is still generally assumed to be linear across the depth, but the corresponding stress distribution becomes hyperbolic.

Location of Maximum Bending Stress in Curved Beams

In curved beams, the neutral axis, where the bending stress is zero, does not coincide with the centroidal axis. Instead, the neutral axis shifts towards the center of curvature. The bending stress ($\sigma_b$) in a curved beam section at a distance 'y' from the neutral axis is given by a formula, often based on the Winkler-Bach theory. A simplified representation of the stress distribution shows that the magnitude of the stress is inversely proportional to the distance from the center of curvature.

Consequently, the fibres closer to the center of curvature (the inside fibres) experience a higher magnitude of stress compared to the fibres further away from the center of curvature (the outside fibres), for the same magnitude of strain difference from the neutral axis. This effect is more pronounced for beams with a larger curvature (smaller radius of curvature relative to the section depth).

Therefore, even for a symmetrical cross-section, the maximum bending stress (in terms of magnitude) under a given bending moment will occur at the fibre located closest to the center of curvature, which is the inside fibre.

Analyzing the Options

• Neutral axis: In a curved beam, the neutral axis is where the bending stress is zero, not maximum. This option is incorrect.
• Centroidal axis: In a curved beam, the neutral axis shifts away from the centroidal axis (towards the center of curvature). The stress is generally not maximum at the centroidal axis. This option is incorrect.
• Inside fibre: The inside fibre is the fibre closest to the center of curvature. Due to the non-linear stress distribution in curved beams, the stress magnitude is highest at the inside fibre. This option is correct.
• Outside fibre: The outside fibre is the fibre furthest from the center of curvature. While it is an extreme fibre, the stress magnitude at the outside fibre is less than at the inside fibre in a curved beam. This option is incorrect.

Based on the principles of curved beam theory, the maximum bending stress in a curved beam with a symmetrical section always occurs at the inside fibre.

Revision Table: Curved vs. Straight Beams

Additional Information on Curved Beam Stress

The analysis of stress in curved beams is more complex than in straight beams. Key concepts include:

• Neutral Axis Shift: The neutral axis in a curved beam is located inward from the centroidal axis, closer to the center of curvature. The amount of shift depends on the shape and dimensions of the cross-section and the radius of curvature.
• Winkler-Bach Formula: This is the classical formula used to calculate bending stress in curved beams. It accounts for the non-linear stress distribution. The formula for bending stress $\sigma$ at a distance $y$ from the neutral axis is typically given by $\sigma = \frac{My}{A \cdot e \cdot (R_n + y)}$, where $M$ is the bending moment, $A$ is the cross-sectional area, $e$ is the distance between the centroidal and neutral axes, $R_n$ is the radius of the neutral axis, and $y$ is measured from the neutral axis (positive outwards, negative inwards). The maximum stress occurs at the extreme values of $y$ corresponding to the inner and outer fibres.
• Stress Concentration: In addition to bending stress, curved beams can experience hoop stress and radial stress, especially in thick sections or under specific loading conditions. However, the question specifically asks about maximum bending stress.

Understanding the shift of the neutral axis and the hyperbolic nature of the stress distribution is crucial to correctly identifying the location of maximum bending stress in a curved beam.