engineering fundamentals Theory: Beam Equation
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Resultants
Force resultants are a convenient means for tracking the important stresses in a beam. They are analogous to the moments and forces of statics theory, in that their influence is felt thoughout the beam (as opposed to just a local effect). Their convenience lies in them being only functions of x, whereas stresses in the beam are functions of x and y.

If we were to cut a beam at a point x, we would find a distribution of direct stresses s(y) and shear stresses sxy(y),

     

Each little portion of direct stress acting on the cross section creates a moment about the neutral plane (y = 0). Summing these individual moments over the area of the cross-section is the definition of the moment resultant M,

where z is the coordinate pointing in the direction of the beam width (out of the screen). Summing the shear stresses on the cross-section is the definition of the shear resultant V,

There is one more force resultant that we can define for completeness. The sum of all direct stresses acting on the cross-section is known as N,

N(x) is the total direct force within the beam at some point x, yet it does not play a role in (linear) beam theory since it does not cause a displacement w. Instead, it plays a role in the axial displacement of rods and bars.

By inverting the definitions of the force resultants, we can find the direct stress distribution in the beam due to bending,

Note that the bending stress in beam theory is linear through the beam thickness. The maximum bending stress occurs at the point furthest away from the neutral axis, y = c,

What about the other non-linear direct stresses shown acting on the beam cross section? The average value of the direct stress is contained in N and does not contribute to beam theory. The remaining stresses (after the average and linear parts are subtracted away) are self-equilibriating stresses. By a somewhat circular argument, they are self-equilibriating precisely because they do not contribute to M or N, and therefore they do not play a global role. On the contrary, self-equilibriating loads are confined to have only a localized effect as mandated by Saint-Venant's Principle.

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