 Beams in simple bending with varying section: notes
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 section 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 ('Options'->'Show formulae' in the menu to see, relationships enforcing boundary conditions not shown).
When present, the button 'Data' will download the vectors of the coefficients (without dimensions) ai,bi,ci as defined in sheet's formulae.
Note that it is up to the user to link the downloaded vectors to the input data: when you download it, the vectors corresponds to the data in the sheet as you view it.