Beams in bending with large deflections: notes
This form uses the theory of beams in simple bending as represented by the equation M=-EJ/r , where r is the local radius of curvature of the deformed axis. The common linearizing approximation 1/r=y'' is here replaced by the closer approximation 1/r²=(1-3y'²)y''² , 1/r²=y''²/(1+y'²)³ being the exact formulation.
This correction is, for beams, usually of low significance, however for very flexible beams it may help in representing a closer approximation of the deflections.
Note also that the behavior of the beam is no more linear: the superposition principle does not hold.
The solution is based onto a polynomial approximation formula obtained from strain energy V minimization (choose 'Show formulae' under 'Options' to see, relationships enforcing boundary conditions not shown).
The solution is obtained by means of an iterative procedure: if the deflections are very large (of the order of 1/10th of beam length), the calculation will not converge and odd results will be displayed.
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.