Page 288 - Introduction to Continuum Mechanics

272 Pure Bending of a Beam

where a/ are constants of integration. In fact, a 4, a 5 , a 6 define an overall rigid body
translation of the bar and a\, a-i, a^ being constant parts of the antisymmetric part of the
displacement gradient, define an overall small rigid body rotation. For convenience, we let all
the a,- = 0 [ note that this corresponds to requiring u = 0 and (Vw^ = 0 at the origin ]. The
displacements are therefore,

Considering the cross-sectional plane x\ - constant, we note that the displacement perpen-
dicular to the plane is given by

Since u\ is a linear function of x$, the cross-sectional plane remains plane and is rotated about
the*2 axis (see Fig. 5.14) by an angle

In addition, consider the displacement of the material that is initially along the x\ axis
Cr 2=.T3 = 0)

The displacement of this material element (often called the neutral axis or neutral fiber) is
frequently used to define the deflection of the beam. Note that since