engineering fundamentals Theory: Beam Equation
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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).

Kirchhoff Assumptions
 1. Normals remain straight (they do not bend)
 2. Normals remain unstretched (they keep the same length)
 3. Normals remain normal (they always make a right angle to the neutral plane)
With the normals straight and unstretched, we can safely assume that there is neglible strain in the y direction. Along with normals remaining normal to the neutral plane, we can make the x and y dependance in u(x,y) explicit via a simple geometric expression,

With explicit x dependance in u, we can find the direct strain throughout the beam,

Finally, again with normals always normal, we can tie the cross section rotation c to the neutral plane rotation q, and eventually to the beam's displacement w,

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