This form uses the classical theory of beams in bending as represented by the equation M=-EJy''
with an added axial tensile load.
The solution is based onto a polynomial approximation formula obtained from strain energy V minimization (hit the 'Formulas' button to see, relationships enforcing boundary conditions not shown).
Note also that the principle of superposition is only valid in a limited way for such beams: the axial load must be considered as if it was an element of the supporting scheme. In other words the effect of various transverse loadings may be combined by summation, but the same axial load must be present with the same value in all the single superposed load conditions.