Singularity Functions for Beam Bending

[tangodirt]

There is a beam of width 10cm, and vertical reaction loads on each end (x1 = 0cm, x2 = 10cm). Starting from the left end of the beam, we have a vertical distributed load of 2,000 N/m spanning from 0cm to 5cm. Finally, we have a 1,000 N point load located 7.5cm from the left end of the beam.

Through statics, it can be said that the left most reaction load (x1 = 0cm) is of magnitude 325 N while the right most reaction load (x2 = 10cm) has a magnitude of 775 N.

My singularity function for this system is shown below:

$$V = 325<x - 0>^{0} - 2000<x - 0>^{1} + 2000<x - 0.05>^{1} - 1000<x - 0.075>^{0} + 775<x - 0.1>^{0}$$

Which, when plotted (my end goal here), works perfectly and as it should. My issue comes when I switch the shear (V) singularity function to a moment function by increasing the exponents by one (as I've been told).

Through integration of the shear singularity function, the moment equation then becomes:

$$M = 325<x - 0>^{1} - 2000<x - 0>^{2} + 2000<x - 0.05>^{2} - 1000<x - 0.075>^{1} + 775<x - 0.1>^{1}$$

Which doesn't work quite as well. The moment function falls completely apart, but from every source I've read so far, it shouldn't. Also, if I draw the moment equation by hand (through the "area under the curve" approach), it hardly matches the output of the moment singularity equation.

Any ideas?

 

[nvn]

tangodirt: But for n ≥ 0, w*integral[(<x - a>^n)*dx] = w*[1/(n+1)]*<x - a>^(n+1), not w*<x - a>^(n+1). Therefore, don't the second and third terms of your M equation contain a typographic mistake? See if this resolves the problem.

 

[Mene]

I would suggest taking a closer look at the integration process to ensure that it is being done correctly. It is possible that there may be a mistake in the integration or in the application of the boundary conditions. I would also recommend consulting with other experts in the field or conducting further research to see if there are any alternative methods for determining the moment function that may produce more accurate results. Additionally, it may be helpful to experiment with different values for the exponents and see if that improves the accuracy of the moment function. Overall, it is important to carefully analyze and check all steps in the process to ensure the accuracy of the results.