What Is the Difference Between Moment and Bending Moment in Beam Calculations?

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Hello, I have a small question about moments and bending moments.

So, if I have a beam with a loading given by q (N/m) which is given as a function of x then what do these calculations get me?

$$\int xq(x) dx$$
$$\int (\int q(x) dx) dx$$

The first integral gives me the moment about a point because I am taking a differential distance and multiplying it with the value of Force there given by q(x) and adding up all these differential moments.

The second integral gives me the bending moment about a point..

And here is what has been troubling me.

Isn't the bending moment the resultant moment on one side of the beam? I mean:

So, isn't the bending moment at a point simply the first equation where you integrate from the point you are interested into the end and then subtract the moment due to the reaction?

So, bending moment = $$(\int_{x}^L xq(x) dx ) - R2x$$

But that that doesn't seem right. Since the bending moment is $$\int (\int q(x) dx) dx$$
and you fill in the boundary conditions and plug in the value of where you want it...

In short, what is the fundamental difference between the bending moment and the resultnt moment on one side of the beam. To me it seems they are the same, the bending moment is the moment the beam is having to apply at that point to counteract the resultant moment on the other side of the beam, but mathematically they seem different.

 

[Studiot]

There are several difficulties with your approach.

Why are you not following convention and working from left to right?

If you do not do this you need a negative sign in you equations since the positive x increasing direction is left to right.

Secondly the integration only works like this with distributed loads, not with point loads.

 

[___]

There are several difficulties with your approach.

Why are you not following convention and working from left to right?

If you do not do this you need a negative sign in you equations since the positive x increasing direction is left to right.

Oh ok, fair enough.

Secondly the integration only works like this with distributed loads, not with point loads.

Yeah. I was looking at beams with point loads but it was too difficult to come up with an expression that described the bending moment (because it was described piecewise with discontinuity in shear forces) so I gave up.


I think I see the problem with my expressions though. Once I take a cut like I have done in the picture, I should still be able to use the force X distance method to work out the moment due to the load and the reaction. But however, since I took my cut this way I need to look at the fact that the loading is described by q as a function of x which begins from the reaction at the left hand side.

So, the expression for moment using the Force X distance becomes a little more complicated since the distance is the distance from the point where I have taken the cut, and the loading is being described through the variable x which is measuring the distance from the left reaction. So I need to find an expression that associates the distance x with the distance from the cut, which is going to be a little difficult.

Thanks for the reply.

 

[Studiot]

Whatever direction conventions you choose you will have to cope with the fact that for many loading conditions the relevant equations, expressed as functions of x, are different on different parts of the beam.

An elegant solution to this is Macaulay's method

A google search will reveal many hits eg

http://www.colincaprani.com/files/notes/SAIII/Macaulay's Method 1011.pdf

There are also some worked examples in recent threads in Physics Forums on this - try a search.

go well

 

[alphy]

Dear reader,

Thank you for your question about moments and bending moments. I am happy to provide you with a response.

Let's start by defining what moments and bending moments are. Moments are a measure of the tendency of a force to rotate an object around an axis. In the case of a beam, the moment is the force applied to the beam multiplied by the distance from the axis of rotation. Bending moments, on the other hand, are a type of moment that occurs in a beam when it is subjected to a bending load. This load causes the beam to bend, resulting in tensile and compressive stresses along its length.

In your question, you have correctly identified that the first integral, \int xq(x) dx, gives the moment about a point. This is because you are integrating the force (q) over the distance (x) to get the moment. However, the second integral, \int (\int q(x) dx) dx, is not the bending moment about a point. This integral actually gives the total bending moment along the entire length of the beam.

To understand the difference between the bending moment and the resultant moment on one side of the beam, we need to look at the forces acting on the beam. In a simple beam, there are two types of forces: external forces (such as the load q) and internal forces (such as the bending moment). The resultant moment on one side of the beam is the sum of all the external forces acting on that side of the beam. This includes the load q and any reaction forces from supports. On the other hand, the bending moment is the internal force that is created by the load q and is responsible for the beam's bending.

In summary, the bending moment and the resultant moment on one side of the beam are two different quantities. The bending moment is the internal force that is responsible for the beam's bending, while the resultant moment on one side of the beam is the sum of all the external forces acting on that side. I hope this helps clarify the difference between these two quantities.

Best regards,