Curvature of a Beam
A beam in pure bending will have constant curvature, and a beam in nonuniform bending will have varying curvature.
curvature, κ a measure of how sharply a beam is bent
radius of curvature, ρ
for small deflections, ds = dx
sign convention for curvature
anticlastic curvature
Longitudinal Strains in Beams
assumptions
- beam initially has a straight longitudinal axis (x axis)
- cross section is symmetric about the y axis
Cross sections of a beam in pure bending remain plane. (based on symmetry only!)
For positive curvature, the lower portion of the beam is in tension and the upper portion is in compression. (mn shortens & nq elongates creating normal strains εx)
neutral surface
where no strain (or stress) occurs in the beam (ss stays the same length)
neutral axis
the intersection of the neutral surface with any cross-sectional plane (z axis)
The longitudinal strains in a beam in pure bending are proportional to the curvature and vary linearly with the distance from the neutral surface regardless of the shape of the stress-strain curve of the material. If the x axis lies along the neutral surface of the undeformed beam,
