Area of Moment Formula Homework

[foo9008]

Homework Statement

i'm having problem of understanding the formula of area of moment of uniformly distributed load and uniformly varying load... the shape of graph for moment of uniformly distributed load and uniformly varying load.are similar,right?
http://www.mathalino.com/reviewer/mechanics-and-strength-of-materials/moment-diagrams-by-parts

Homework Equations

The Attempt at a Solution

So, the formula of area of moment should be the same? why for uniformly distributed load , it is (1/6)(w)(L^3),while for uniformly varying load.is (1/24)(w)(L^3) ?

 

[collinsmark]

Homework Statement

i'm having problem of understanding the formula of area of moment of uniformly distributed load and uniformly varying load... the shape of graph for moment of uniformly distributed load and uniformly varying load.are similar,right?
http://www.mathalino.com/reviewer/mechanics-and-strength-of-materials/moment-diagrams-by-parts

They might be "similar" in the lay-person's vernacular of the word. But they are not "similar" in the geometric sense.

Homework Equations

The Attempt at a Solution

So, the formula of area of moment should be the same? why for uniformly distributed load , it is (1/6)(w)(L^3),while for uniformly varying load.is (1/24)(w)(L^3) ?

Before getting into A, let's first explore what M_x is.

For a given value of x, M_x is simply the contribution of the moment to the left of the point defined by x, (I chose "left" because in the figure, the horizontal beam is affixed at the far right).

Let's start with the uniformly distributed load. Pick some arbitrary point on the beam. The value for x is the distance from the leftmost* side of the load to that point. The "x" as it is drawn in the diagram should work nicely; the selected point is a distance x away from the load's leftmost* point. Now, what is the moment around the point defined by x if you only consider the load to the left* of that point (and ignore the load to the right of that point)? That's M_x.

Can you derive that?

*(Again, I say "left" here because the horizontal beam's support is to the right, as depicted in the figure.)

 

[foo9008]

They might be "similar" in the lay-person's vernacular of the word. But they are not "similar" in the geometric sense.
Before getting into A, let's first explore what M_x is.

For a given value of x, M_x is simply the contribution of the moment to the left of the point defined by x, (I chose "left" because in the figure, the horizontal beam is affixed at the far right).

Let's start with the uniformly distributed load. Pick some arbitrary point on the beam. The value for x is the distance from the leftmost* side of the load to that point. The "x" as it is drawn in the diagram should work nicely; the selected point is a distance x away from the load's leftmost* point. Now, what is the moment around the point defined by x if you only consider the load to the left* of that point (and ignore the load to the right of that point)? That's M_x.

Can you derive that?

*(Again, I say "left" here because the horizontal beam's support is to the right, as depicted in the figure.)

that is -(wx / L)(0.5x)(2x / 3 ) ? but , i got -w(x^3) / 3L instead of (1/6)(w)(L^3) ...

 

[collinsmark]

that is -(wx / L)(0.5x)(2x / 3 ) ? but , i got -w(x^3) / 3L instead of (1/6)(w)(L^3) ...

Let's just start with the uniformly distributed load first, since that's the easier one.

What is the total force of the load to the left of the point defined by x?

What is the distance from the centroid of the load to the point defined by x (i.e., where is the center of force if you only consider the load to the left of the specified point)?

 

[foo9008]

What is the total force of the load to the left of the point defined by x?
it's (wx)(0.5x) / L

What is the distance from the centroid of the load to the point defined by x (i.e., where is the center of force if you only consider the load to the left of the specified point)?
(2x/ 3)
is it correct ?

 

[collinsmark]

(I'll come back your most recent post in my next post.)

Ah, I think I see your source of confusion about the uniformly varying load. And I think it's an inconsistency in the coursework. It's a big enough inconsistency to call it a mistake.

The problem is the way the coursework defines w₀. It defines it differently for its A calculation than it does for its M_x calculation. That difference is inconsistent. That's what's wrong.

Let's concentrate on the total force to the left of the point defined by x (and let's only discuss the uniformly varying load case):

In the case of the M_x calculation, the total force (to the left) is:

F = \frac{1}{2} w_0 \frac{x}{L}

The \frac{1}{2} comes from the fact that the force is triangle shaped (i.e., 1/2 the force if it was rectangle [uniform] shaped). Then, w_0 is scaled by \frac{x}{L} because the triangle is smaller the smaller x is. So in this case, w_0 is defined such that if x = L, the total force is simply \frac{1}{2} w_0.

But the case of the A calculation, the total force (to the left) is:

F = \frac{1}{2} w_0 \frac{x}{L} x

Like before, the \frac{1}{2} comes from the fact that the force is triangle shaped. Also like before, w_0 is scaled by \frac{x}{L} because the triangle is smaller the smaller x is. However, under this definition this resulting force density is then multiplied by x to get the total force. So in this case, w_0 is defined such that if x = L, the total force is \frac{1}{2} w_0 L.

So it's really the definition of w_0. It's a matter if whether w_0 is defined as a force, or a force density. Arguably, both are fine choices in how one defines w_0. But the fact that it's used inconsistently in the same example is a problem. I'd call that a mistake in the coursework.

 

[collinsmark]

What is the total force of the load to the left of the point defined by x?
it's (wx)(0.5x) / L

In the case of the uniformly distributed load, I'd define w_0 such that if x = L the total force is F = w_0 L. That's the way I've seen it defined before and I'm pretty sure that's the way your coursework defines w_0, at least for the case of the uniformly distributed load case.

So the force to the left of an arbitrary point defined by x is simply F = w_0 x.

At least your coursework is consistent with this definition in the case of the uniformly distributed load example.

What is the distance from the centroid of the load to the point defined by x (i.e., where is the center of force if you only consider the load to the left of the specified point)?
(2x/ 3)
is it correct ?

It's almost correct. It's a distance of \frac{2}{3}x from the leftmost point of the load, but how far is that from the point defined by x?

So what is the moment about the point defined by x? (Multiplying the two answers together, with a possible negative sign indicating into or out of the page)

 

[foo9008]

In the case of the uniformly distributed load, I'd define w_0 such that if x = L the total force is F = w_0 L. That's the way I've seen it defined before and I'm pretty sure that's the way your coursework defines w_0, at least for the case of the uniformly distributed load case.

So the force to the left of an arbitrary point defined by x is simply F = w_0 x.

At least your coursework is consistent with this definition in the case of the uniformly distributed load example.It's almost correct. It's a distance of \frac{2}{3}x from the leftmost point of the load, but how far is that from the point defined by x?

So what is the moment about the point defined by x? (Multiplying the two answers together, with a possible negative sign indicating into or out of the page)

ok , i got it . I assume it 's 2x/3 because i am taking moment about the leftmost end... x/3 is taking moment about point x . Thanks!

 

[foo9008]

(I'll come back your most recent post in my next post.)

Ah, I think I see your source of confusion about the uniformly varying load. And I think it's an inconsistency in the coursework. It's a big enough inconsistency to call it a mistake.

The problem is the way the coursework defines w₀. It defines it differently for its A calculation than it does for its M_x calculation. That difference is inconsistent. That's what's wrong.

Let's concentrate on the total force to the left of the point defined by x (and let's only discuss the uniformly varying load case):

In the case of the M_x calculation, the total force (to the left) is:

F = \frac{1}{2} w_0 \frac{x}{L}

The \frac{1}{2} comes from the fact that the force is triangle shaped (i.e., 1/2 the force if it was rectangle [uniform] shaped). Then, w_0 is scaled by \frac{x}{L} because the triangle is smaller the smaller x is. So in this case, w_0 is defined such that if x = L, the total force is simply \frac{1}{2} w_0.

But the case of the A calculation, the total force (to the left) is:

F = \frac{1}{2} w_0 \frac{x}{L} x

Like before, the \frac{1}{2} comes from the fact that the force is triangle shaped. Also like before, w_0 is scaled by \frac{x}{L} because the triangle is smaller the smaller x is. However, under this definition this resulting force density is then multiplied by x to get the total force. So in this case, w_0 is defined such that if x = L, the total force is \frac{1}{2} w_0 L.

So it's really the definition of w_0. It's a matter if whether w_0 is defined as a force, or a force density. Arguably, both are fine choices in how one defines w_0. But the fact that it's used inconsistently in the same example is a problem. I'd call that a mistake in the coursework.

sorry , why the force not equal to w(x^2) / L ?
I think the the force = w(x^2) / L because wx / L is the force per unit length of beam , so Force = force per unit length x length = (wx / L) x (x) = w(x^2) / L ...

One more thing , why the area of moment is -(w)(L^3) / 24 ?

 

[collinsmark]

sorry , why the force not equal to w(x^2) / L ?
I think the the force = w(x^2) / L because wx / L is the force per unit length of beam , so Force = force per unit length x length = (wx / L) x (x) = w(x^2) / L ...

You'll still need the factor of 1/2 in there somewhere, because force is distributed in the shape of a triangle. The factor of 1/2 wouldn't be necessary if the force distribution was rectangular shaped, but since it's a triangle, it gets a factor of 1/2.

That's the way the coursework describes it in the case case of its A calculation. In that particular case, the force to left of the point defined by x is F = \frac{1}{2} w_0 \frac{x}{L}x = \frac{w_0}{2L} x^2.

One more thing , why the area of moment is -(w)(L^3) / 24 ?

First let's calculate the moment. The moment M_x is the cross product of the force F and the displacement from the point defined by x (since they are perpendicular, it's just multiplying them together with a possible negative sign). Calculate that first.

To find the area, integrate the moment M_x over x from 0 to L.

[Edit: By the way, keep in mind, the coursework treats w_0 differently when it calculates its M_x. So when you calculate your own M_x above, it's not going match your coursework's M_x. That's because of the inconsistency I was talking about earlier.]

 

[foo9008]

You'll still need the factor of 1/2 in there somewhere, because force is distributed in the shape of a triangle. The factor of 1/2 wouldn't be necessary if the force distribution was rectangular shaped, but since it's a triangle, it gets a factor of 1/2.

That's the way the coursework describes it in the case case of its A calculation. In that particular case, the force to left of the point defined by x is F = \frac{1}{2} w_0 \frac{x}{L}x = \frac{w_0}{2L} x^2.

Why the moment is -(w)(x^2)/ 6L ? the moment is force x distance , right ? since the force is F = \frac{1}{2} w_0 \frac{x}{L}x = \frac{w_0}{2L} x^2 , then moment should have x^3 , right ? why moment has x^2 , not x^3 ?

 

[collinsmark]

Why the moment is -(w)(x^2)/ 6L ? the moment is force x distance , right ? since the force is F = \frac{1}{2} w_0 \frac{x}{L}x = \frac{w_0}{2L} x^2 , then moment should have x^3 , right ? why moment has x^2 , not x^3 ?

In our calculations here (consistent with your coursework's A calculation) The force has an x^2 in it, not the moment. In order to get the moment M_x, you need to multiply the force times the distance \frac{x}{3}

So, multiply force \frac{w_0}{2L}x^2 by the distance \frac{x}{3} to get your M_x, with a negative sign to indicate direction (into or out of the page). That will have an x^3 in it like you suggest.

(Note that this M_x is different than your coursework has listed for M_x. That's because your coursework used inconsistent definitions for w_0. In it's own M_x calculation it treated w_0 as a force magnitude itself, not as a force density [force per unit length] as we are doing here. So you can expect the M_x calculation you are doing here to be different than the one in the coursework by a multiple of x).

Once you find your M_x, integrate that to find the "area under the curve."

 

[collinsmark]

Just to be clear, I'd like to reiterate that there is a "mistake" in your coursework. The mistake is an inconsistent definition of what w_0 means.

In the example, Uniform Varying Load, it has

A = \frac{1}{24} w_0 L^3
which is consistent with w_0 being a force density, in other words a force per unit length. This is consistent with what we've been discussing in the last couple of posts.

But it also has

M_x = \frac{w_0}{6L}x^2
which is consistent with w_0 being a force magnitude, independent of unit length.

Arguably, either interpretation of w_0 is okay on its own, but the fact that both interpretations are used, each in a different section of the same example, is a mistake in my opinion. So either way your look at it, one way or the other, one of them is wrong; the inconsistency being the mistake in the coursework.

[Edit: corrected another typo.]

[Another edit: if the example in the coursework had just said, M_x = -\frac{w_0}{6L} x^3, that would have cleared up the inconsistency.]

 

[foo9008]

e mistake is an inconsistent d

Just to be clear, I'd like to reiterate that there is a "mistake" in your coursework. The mistake is an inconsistent definition of what w_0 means.

In the example, Uniform Varying Load, it has

A = \frac{1}{24} w_0 L^3
which is consistent with w_0 being a force density, in other words a force per unit length. This is consistent with what we've been discussing in the last couple of posts.

But it also has

M_x = \frac{w_0}{6L}x^2
which is consistent with w_0 being a force magnitude, independent of unit length.

Arguably, either interpretation of w_0 is okay on its own, but the fact that both interpretations are used, each in a different section of the same example, is a mistake in my opinion. So either way your look at it, one way or the other, one of them is wrong; the inconsistency being the mistake in the coursework.

[Edit: corrected another typo.]

ok , let us make some correction here ... Mx = (1/6)(w)(x^3)/ L , so , A should be moment multiply by the location of centroid , am i right ? so Area of moment = (1/6)(w)(L^3)(L/5) = (1/30)(w)(L^4) ??

 

[collinsmark]

ok , let us make some correction here ... Mx = (1/6)(w)(x^3)/ L , so , A should be moment multiply by the location of centroid , am i right ? so Area of moment = (1/6)(w)(L^3)(L/5) = (1/30)(w)(L^4) ??

Um, I not sure understand your approach. I think you just need to integrate.

Using our new moment, M_x = -\frac{w_0}{6L}x^3, find the area under the curve.

A = \int_0^L M_x dx

 

[collinsmark]

I see now that your coursework gives you an alternative way to find the area, saying that

A = \frac{1}{n+1}bh

Noting that the Uniformly Varying Load example is third degree.

That will give you the same answer for simple load distribution functions like this one. I like the integral approach better since you can use it for arbitrarily shaped load distributions. The method in your coursework will work for load distributions that meet certain criteria, but you can't use it for any old load distribution.

But for here, use the one in the coursework if you like; it gives you the same answer. If you'd rather integrate over M_x, use that instead; it is more powerful and more flexible.

[Edit: either way, you should end up with the coursework's answer of A = \frac{1}{24}w_0 L^3.]

 

[foo9008]

I see now that your coursework gives you an alternative way to find the area, saying that

A = \frac{1}{n+1}bh

Noting that the Uniformly Varying Load example is third degree.

That will give you the same answer for simple load distribution functions like this one. I like the integral approach better since you can use it for arbitrarily shaped load distributions. The method in your coursework will work for load distributions that meet certain criteria, but you can't use it for any old load distribution.

But for here, use the one in the coursework if you like; it gives you the same answer. If you'd rather integrate over M_x, use that instead; it is more powerful and more flexible.

[Edit: either way, you should end up with the coursework's answer of A = \frac{1}{24}w_0 L^3.]

by intergrating Mx with dx i get A = \frac{1}{24}w_0 L^4, not L^3

 

[collinsmark]

by intergrating Mx with dx i get A = \frac{1}{24}w_0 L^4, not L^3

Don't forget there is already an L in the denominator.

Recall, M_x = -\frac{w_0}{6 \color{red}{L}}x^3