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).