This form uses the classical theory of beams in simple bending as represented by the equation M=-EJy'', where J is taken as J(x)=JA(1+Kx)n where n may typically take one of the values 1,2,3,4.
Also the resistance modulus Z is taken as Z(x)=ZA(1+kx)m, where m is often, but not always, equal to n-1.
The following examples illustrate different values of n and m.
Beam with rectangular solid section, width varying linearly with x: n=1 and m=1
Beam with rectangular hollow section and constant thickness, width varying linearly with x: n=1 and m=1 (approximate)
I beam with constant thicknesses, width of flanges varying linearly with x: n=1 and m=1 (approximate)
Beam with rectangular solid section, depth varying linearly with x: n=3 and m=2
Beam with rectangular hollow section and constant thickness, depth varying linearly with x: n=2 and m=1 (approximate)
I beam with constant thicknesses, depth varying linearly with x: n=2 and m=1 (approximate)
Round solid bar, radius varying linearly with x: n=4 and m=3
Round hollow bar with constant thickness, outer radius varying linearly with x: n=3 and m=2 (approximate)
Round hollow bar with constant section area, outer radius varying linearly with x: n=2 and m=1 (approximate)
Note also that Z is used in the calculation only to derive the stresses; all other results remain valid if the variation of Z is not according to the value of m indicated in the sheet.
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).