Why Does a Triangular Load on a Beam Require Multiple Moment Calculations?

[nonlocalworlds]

I have a problem that shows a triangular distributed load on a beam (studying for NCEES civil engineering exam). At one end of the triangle we have a force magnitude level of "w" and the other end is labeled "wL/6."

They tell me that a triangular load is equivalent to a concentrated load of (1/2)wL acting at the centroid.

My load is on a simply supported beam on either end. To get the 'correct' answer the moments about one of the simple supports are added together and set equal to zero (non-rotating system). There is an unknown moment on the opposite simple support (that the problem is trying to get you figure out), but then my confusion is that there appears to be two counteracting forces, a moment at the centroid of the triangular load as well as another moment where it tapers off (the triangular load is shown at a maximum w on top of the simple support whose counteracting moment I am trying to find, and tapers to wL/6 at a distance L from the unknown simple support, with the final simple support being 2L away from the unknown location).

I do not understand why all of the load is not accounted with the moment expression for the concentrated load at the centroid of the triangle.

Apparently the correct expression is:

X*2L (unknown moment to calculate) - (wL/6)L (a moment at the point the triangle tapers whose inclusion I don't understand) - (wL/2)(5/3L) (this part is fine, the concentrated load at the centroid) = 0.

 

[SteamKing]

I am confused as well by your description of the triangular load. Try not to use L for the distance to the start of the load or from the end of the load to the support. Generally, L is used to indicate the total length of the beam.

If you could post a sketch of the problem, that would save quite a bit of hand-waving explanation.

 

[nonlocalworlds]

sketch

 

[SteamKing]

The beam in question has a concentrated load of ##w_0L/6## located at x = L from point A. (Total length of the beam = 2L). This concentrated load is not related to the triangular load distributed between 0 < x < L, with ##w_0## @ x = 0 as its load magnitude. The equivalent concentrated load would be ##w_0L/2## located @ x = L/3 from point A.