The deflection of a structural beam of constant cross section subjected to a transverse loading is related to the internal bending moment (in-lb) as: d2v
dx2
where E (psi) is the modulus of elasticity for the material of which the beam is made, I (in^4) is the moment of inertia of the cross-section geometry, and v (in) is the vertical deflection of the centroidal axis of the beam. The equation v(x) describing the position of the centroid of the cross section as a function of x is known as the elastic curve. Calculating the internal bending moment is useful for subsequent calculations to determine the internal stresses in the beam and ensure it will not fail under load.
For this problem, you'll work with measured deflection data to determine the bending moment for a steel I-beam subjected to unknown transverse loading forces. Write a function to do this calculation for a W12x22 steel I-beam (E = 29.0E6 psi for structural steel, and I = 156 in^4 for the W12x22 geometry) as follows. Your function should receive a single input that is a two column matrix with evenly spaced data for axial position, x, in the first column and the corresponding lateral deflection measurements, v, in the second column. Use three different approaches to compute the second derivative of the data that you will need to compute M(x):
Use fourth order accurate centered finite difference formulas for the second derivative to compute the derivative directly from the data. Use second order forward difference formulas for the first two points and second order backward difference formulas for the last two points.
Use MATLAB's gradient function twice on the data.
Use MATLAB's gradient function twice on interpolated data generated with a clamped spline with a derivative set to 0 at both ends and 0.5 inch increments in x.
Your function should have the following three column vector outputs (in order):
The bending moment values M(x) calculated using 4th order centered finite difference formulas.
The bending moment values M(x) calculated using the gradient function on the data.
The bending moment values M(x) calculated using the gradient function on the clamped spline interpolation of the data.
Note: The first three test cases check each of the three approaches respectively.