Bending Members (Beams) under Pure Bending (Couples)

Typical bending members are straight and long with a pure bending couple, M . The figure

below shows various cross-sectional possibilities all of which are symmetrical about the y-axis.

In a Nut Shell: The key assumption for bending members (beams) is that plane sections

normal to the centroidal axis of the cross-section, x, before bending remain plane after bending.

Then the “fibers”above the centroidal axis shrink and “fibers” below the centroidal axis

elongate in direct proportion to their distance from the centroidal axis of the cross-section.

The result is that the longitudinal normal strain at a distance y above the centroidal axis is

ε_x = - y/ρ

where ρ is the radius of curvature of the bending member (beam).

Common units for normal strain are in. / in or μin. / in.

Absence of Shearing Strains and Stresses

The deformation of the beam resulting from the assumption that plane sections remain

plane leads to the following conclusions about the shearing stresses and strains in the beam.

Shearing Strains	γxy = γzx = 0
Corresponding Shearing Stresses	τxy = τzx = 0

Return to Notes on Solid Mechanics

Copyright © 2019 Richard C. Coddington
All rights reserved.