I did a web search. It seems that what you are referring to is McCaulay Notation. It appears at a glance to be just a way to referring to the differential equations that can describe the load/deflection of an element. If I'm not mistaken... it merely allows you to write one equation that is understood differently "left" of a load point and "right" of the same load point. In the grand scheme of things though... I don't think it is really what you "Need"... though... it will do the trick if you want to use it.

The method I suggested is a little simpler ( understanding wise anyway ). Given a beam of length L. Put a point load, P, on it at a distance "a" from the left end. Let "b" equal the distance from the load to the right end.

The fixed end moment at the left is.. Ml= Pab^2/L^2. The fixed end moment at the right is.. Mr = Pa^2b/L^2. These formulas are in most books and can also be derived fairly easily. I like to use the conjugate beam method myself to derive them.

NOW... take the same beam and put a uniform load on it of "w" ( units of force/length ).

Pick some point in the beam that is "x" away from the left end of the beam. Take a thin slice of this load that is dx wide. This slice has a total load on the beam of "w*dx".

Due to just this one little slice of the load... the small moment at the left end of the beam ( from the point load formula above ) is...

d(Ml) = (w*dx)*x*(L-x)^2/L^2

Now.. just integrate:

Ml = Integral(from x=a to x=b) of d(Ml)

where a = distance to the start of the uniform load

where b = distance to the end of the uniform load

If your loading is not uniform... you have to derive a formula for how "w" changes as a function of "x". That will make your integration a little more difficult, however, intirely doable. ie.. you don't have to be a Calculus whizz. What's tough is cleaning up the algebra so that you have clean final equations. The actual integration is generally quite simple.

I didn't get this method from any book anywhere and I don't know if anyone else has ever done similar derivations. Several years back... I wanted to do the same thing you are currently doing. So.. I had to come up with the same formulas you are now seeking ( and in a way that made sense to me ). I used this technique to derive several formulas. I even derived the formula for the fixed end forces due to distributed moment loadings. I did extensive checking and verification on the formulas I derived and they all passed all the test.

Feel free to send me email directly if you wish.

Dan

dan@dtware.comwww.dtware.com