Using Singularity Functions

Problem 1: Concentrated Load, simply supported beam...This is the same problem that was presented in the last two classes so you can make comparisons...A 6.6 m long beam is simply supported at the ends and carries a concentrated load of 5 kN at its center. Write the equations defining the shear and bending moment as functions of x.

Solution: From the symmetry, the reactions are equal and are 2.5 kN. Next, I mentally draw free body diagrams using a cut near A and another near B. That is, I take very short free body diagrams from each end, keeping in mind that the V and M obtained from the right hand free body have to be reversed. Remember that, by convention, V is positive down and M is positive counterclockwise, on a free body taken from the left end.

From the mentally drawn free body near A, I conclude that the shear very close to A is equal to the reaction, or 2.5 kN down, and that there is no moment there. I write

and, realizing that this is valid only for x = 0 to 3.3 m, add a singularity function that applies over the remaining length:

Since the singularity function is zero for x greater than or equal to 3.3, this yields the correct value over the first half of the beam. For the second half, the shear is driven down by 5 kN to -2.5 kN and this agrees with the plots previously obtained in earlier problems (see below). We can check this (and should!) using a free body close to point B, where we do indeed get a shear of -2.5 kN.

Now dM/dx = V. Integrating, I get

When applying the lower limit, remember that the singularity function is zero for x = 0. It is always a good idea to check the function for known values; in this case, we know that the moments are zero at A and B. At x = 0 (point A), this function is clearly zero. At x = 6.6 m, the singularity function becomes ordinary parentheses, and M = 0 again. So it checks!

Another value we may be able to get from this is the maximum value of the bending moment. Taking the derivative and equating that to zero, we can sometimes find the point where the moment is a maximum:

For these functions, we are unable to find the maximum using a derivative. That's because the maximum occurs where the slope of the moment does not exist!

The plots are: