20.17:
Bending of Curved Members – Strain Analysis
Consider a curved member having a transverse section symmetric with the y-axis. Its upper and lower surfaces intersect with xy-planes along arcs of circles having a center at point C.
If two equal and opposite couples act on a member in the plane of symmetry, then the curvature of the arcs of the section increases, with the new center C'.
The applied couples also result in the reduction of the length of the upper surface and an increase in the length of the lower surface of the member, implying that there exists a surface of unchanged length known as the neutral axis.
The deformation of the arc V' W' that is situated at a distance y from the neutral surface is expressed as a change in the length of the arc.
Using geometry, the deformation can be rewritten in terms of the radius of the curvature of the neutral surface.
The strain is given by dividing deformation by its original length, which shows that the strain varies non-linearly with distance y from the neutral surface.
20.17:
Bending of Curved Members – Strain Analysis
The mechanics of deformation in curved members, such as beams or arches, under bending moments, involve complex responses. When such a member, symmetric about the y-axis and shaped like a segment of a circle centered at point C, is subjected to equal and opposite forces, its curvature and surface lengths change significantly. This alteration results in the shift of the curvature's center from C to C', indicating a tighter curve.
The important part of bending analysis for such a member is the concept of the neutral axis, a hypothetical line within the material whose length remains unchanged despite the bending. This axis does not experience tensile or compressive strain.
The strain at any point on the curved member is influenced by its distance from the neutral axis. The upper surface of the member shortens, and the lower surface elongates due to the bending.
Strain, which measures the deformation per unit length, varies across the member's thickness. This variation is due to the differential change in length between points above and below the neutral axis. Essentially, strain depends on how much more pronounced the curve becomes due to bending, reflecting a non-linear distribution from the neutral axis.