Deflection to BMD

Introduction: In many problems of structural mechanics, the deflection curve of a beam is obtained.
The displacement- based finite element method is one such process that gives deflections/ displacements/
deformations as the first level output, displacement being the field variable. Further quantities of interest
needs to be derived from this basic solution.   Here, I present a working procedure using MATLAB code
for generating the BMD and SFD from deflection curve following the assumptions of pure bending under
small deformations.

Algorithm:

Step1: Fit a polynomial of degree ‘n’ for the deflection data

Step2: Obtain the first derivative of the polynomial fit

Step3: Obtain the second derivative of the polynomial fit. This gives curvature

Step4: Multiply the curvature with flexural rigidity ‘EI’ of the beam to get BMD

MATLAB Code:

 

function [slope,curvature,BMD]=Deflection2BMD(x,y,EI,Order)    f=fittype(['poly' num2str(Order)]);%f = fittype('cubicinterp');    fit1 = fit(x,y,f);    [d1,d2] = differentiate(fit1,x);    subplot(3,2,1);    plot(fit1,x,y);ylabel('Deflection, y');% cfit plot method    subplot(3,2,3);    area(x,d1,'Facecolor','m'); ylabel('Rotation, dy/dx');% double plot method    grid on    legend('1st derivative')    subplot(3,2,5);    area(x,d2,'Facecolor','y');ylabel('Curvature, d^2y/dx^2'); % double plot method    grid on    legend('2nd derivative')    slope=d1;curvature=d2;BMD=d2*EI; end

Illustration:

Let us consider a cantilever steel beam of cross-sectional dimensions 250 mm x 300 mm (Very heavy ofcourse !!!)
 modelled by 3D soild/ brick/ hexhedral elements using Finite Element Analysis, under gravity loading.
The beam is 4 m long.   After completing the static stress analysis of the model under self-weight (gravity),
we can easily obtain the deflection curve of the axis of the beam. This is used as the input for the above
MATLAB function. The following is the result of the analysis.

By multiplying the last figure with EI, w get the BMD. It can be seen that the bending moment at the fixed end
is 481 kN-m (taking E=2E11 Pa). This is almost equal to the actual bending moment of a 4 m long cantilever
beam with a UDL of 6000 N/m (which is the self-weight of the beam taking 8000 kg/m3 as the density of steel).