Chapter 20: Bending

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20.18:

Bending of Curved Members - Neutral Surface

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Mechanical Engineering

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JoVE Core Mechanical Engineering

Bending of Curved Members – Neutral Surface

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20.17: Bending of Curved Members – Strain Analysis

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For a curved member with a uniform cross-section, the strain equation along the section shows that it varies non-linearly with the distance from the neutral axis. Applying Hooke's law gives the stress produced in the curved member. Similar to strain, stress also varies non-linearly, and the plot of stress versus the distance from the neutral axis results in a hyperbola. Here, all the elementary forces acting on a section are statically equivalent to the bending couple, and its sum over the transverse z-axis gives the equation for the moment. Substituting the value of stress and simplifying the equations gives the relation that gives the distance from the center of curvature C to the neutral surface. It shows that the neutral surface of the curved member under bending does not pass through the centroid of that section. Rewriting the expression for the moment in terms of the stress and performing the integration shows that the neutral surface for a curved member is always located between the centroid and the radius of curvature, regardless of its shape.

20.18:

Bending of Curved Members - Neutral Surface

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.