This form uses the theory of beams in bending under transverse loads and in tension under an axial load generated by the ends being not allowed to get closer when loads are applied. The behaviour is no more linear under such conditions, and the principle of superposition does not apply: that's why a uniform load is allowed to be superimposed to the main loading.
The solution is based onto a polynomial approximation formula obtained from strain energy minimization (hit the 'Formulas' button to see, relationships enforcing boundary conditions not shown).
When the area of beam's cross section is zero, the elementary solution for a beam in bending is obtained.