We see that the strain varies linearly from the neutral surface.  If we assume that the material stays within the linear, elastic region of its behavior, then

 

             s = E e = - E y / r.

 

So stress varies linearly.  The maximum stress or strain occurs at the point farthest from the neutral surface.  Imagine this distance is to the top and call it “c”.  Now looking at a plot of s versus y,

 

         s_max = - E c / r

 

So we could write

 

         s = s_max y / c

 

This stress distribution must be statically equivalent to the forces on the beam cross section.  Therefore,

 

         P =  ∫  s dA = 0

 

          ∫ (s_max / c) y dA = 0

 

Since s_max and c are constants

 

                  ∫  y dA = 0

 

We remember that this integral is used to locate the centroid.  Therefore,

 

                  ∫  y dA = A y = 0

 

This tells us that the neutral surface must pass through the centroid of the cross section.  So the neutral axis is a centroidal axis.

 

Also, the stress distribution must be equivalent to the moment on the cross section.

 

             M = -  ∫  s y dA

 

The negative sign is included because M goes in the opposite direction caused by a tensile stress.  Substituting the equation for stress,

 

             M = -  ∫ (s_max / c) y² dA  =  - (s_max / c)  ∫  y² dA

 

Note that   ∫  y² dA = I, the moment of inertia of the cross section about the neutral axis.

 

 

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