| Kinematics describe how the beam's deflections
are tracked. We've already mentioned the out-of-plane displacement
w, the distance the beam's neutral plane moves from its
resting (unloaded) position. Out-of-plane displacement is usually
accompanied by a rotation of the beam's neutral plane, defined as
q, and by a rotation of the beam's cross
section, c,
What we really need to know is the displacement in the
x-direction across a beam cross section,
u(x,y), from which we can find the direct
strain e(x,y) by the
equation,
To do so requires that we make a few assumptions on just how a
beam cross section rotates. For the Euler beam, the assumptions were
given by Kirchoff and dictate how the "normals" behave
(normals are lines perpendicular to the beam's neutral plane
and are thus embedded in the beam's cross sections). | 
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