Calculating BMD for a hollow pipe beam

I have tested steel pipe (as a beam) under lateral distributed loads. Longitudenal strain gages are on the top and bottom lines of the pipe. Could any one suggest to me how can I derive the bending moment distribution along the pipe. I have E, I , t (thickness, and r(radius) for the pipe.

 I have found a formula from litrature as follows : M = K.E.I

Where K = curvature = 0.5 * (ε_tension side - ε_compression side) / C (i.e the Half Section Depth, or pipe radius)

So, if  σ = MI/r and since curvature, K = 1/ρ = ε / C, where is the link so that we end up with taking the " HALF " the difference of the strains at both sides ?

Help in this regard is very much apprciated