| 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. | 
