# Fixed Beams and end moments

I posted this as a part of a thread on the classical physics forums, but the doubt wasn't resolved, so I thought this might be a more appropriate place...

A fixed beam is considered equivalent to a Simply supported beam (SSB) with a point load at its mid point(and thus having reactions Ra* and Rb* at its ends) and another beam with no external load (as the end reactions are both R in magnitude but opposite in direction), but the forces R on this beam are what cause the end moments Ma and Mb......Now, I know that the function of the end moments Ma and Mb is to neutralise the deflection produced on the beam and thus their directions are shown as in the diagram...

...the problem is that from the direction of the forces 'R' at either end of the beam (which are supposed to cause the end moments), the direction of the moments Ma and Mb do not neutralise the deflection of the SSB......Any idea what might be wrong?(I've picked this up from my book exactly as its given)

The main point of query is as to how a couple is formed by forces acting at the ends of the beam so that it neutralises the deflection at the ends.....perhaps you could look at the diagram and come up with something based on the principles of Physics....?

the problem is that from the direction of the forces 'R' at either end of the beam (which are supposed to cause the end moments), the direction of the moments Ma and Mb do not neutralise the deflection of the SSB......Any idea what might be wrong?(I've picked this up from my book exactly as its given)

Whether a beam is simply supported or fixed, the beam ends do not deflect.

(Yes there is a theory of elastic foundations but this is not the place for it)

The downwards centre load in a beam causes sagging.
The end fixing moments causes hogging.
So the end moments do indeed work in the opposite direction to reduce the deflection. They do not eliminate it completely.

In a simply supported beam there is rotation at the ends.
The end moments cause counter rotation, not counter deflection.
Perhaps this is what your book was describing, but did not make clear.

"The downwards centre load in a beam causes sagging.
The end fixing moments causes hogging.
So the end moments do indeed work in the opposite direction to reduce the deflection. They do not eliminate it completely."

Thanks for stating this clearly,Studiot.....but then could you elaborate more on my original doubt as to why the 'R' forces that are the forces causing end moments act in the directions as shown in my diagram,please?
If required, please tell me in detail as to what you infer from the diagram on the other thread...- I'm referring to post#15 on page 1.

OK I don't like the diagrams from your book so I have drawn some of my own.

A couple acts in a plane and has no (zero) resultant force associated with it. The value and direction of the couple is the same about every point in its plane of action.

You can demonstrate this with a sheet of plywood. Set a few screws into (not through) the plywood at random points so the screw heads are projecting above the surface of the ply. Lay the ply on a flat surface so that it is free to rotate. Then insert a torque measuring screwdriver and turn the assembly by each screw. You will find the torque to overcome friction is the same whichever screw you turn.

A single force (F) has a point of application, say A. This force has a moment about any other point in the plane, say B, which varies according to the distance between points A and B.
If we move F to apply it at another point on a body, say C, it will, in general, exert a different moment about B, although its effect on vertical and horizontal force equilibrium will be nil.
We can account for this by considering the effect of F applied at A is the same as the effect of F applied at C along with an additional couple to allow for the difference in moment about B.
This additional couple is the same for moments about every point in the plane from the new location, C

Before tackling your fixed end beam I have shown a simpler situation in the first 5 sketches.
Take note of this as you will meet it again when studying buckling.

sketch1
Shows a slender strut under axial compression. The dashed line shows the strut bowing or buckling sideways under the action of this compression.

sketch2
Shows why. The load C is never applied perfectly along the centreline. The sketch shows an exaggerated loading eccentricity.

sketch3
If we now introduce two equal and opposite forces on the centreline, we have not changed the the situation overall.

Note we are just considering force C. We are not showing or considering the reaction. None of my arrows are reactions.

sketch4
The upward pointing C on the centreline and the eccentric downward pointing C form a couple. That is they are in vertical force equilibrium, but between them they exert a constant moment everywhere in the plane of the section.

If we cancel these two forces out and replace them by a couple as I have shown, we can place the couple anywhere, including the centreline.

sketch5
So we are left with the other C acting downwards at the centeline plus a couple, which we conveniently ascribe to the centreline as well.

Hopefully these sketches make clear the development of replacing an eccentric force with an inline force plus a couple.

sketch6
Note no reactions are shown.

sketch7
For convenience I have split P into two halves, still acting at the centre.

sketch8
I have added two equal and opposite pairs of forces (p/2) acting at each end.

Note again these are not reactions.

sketch9
As in the previous example cancel each of the centre (p/2) forces with one of the end (p/2) forces and introduce a compensating couple. Since I do this twice there are two couples.

sketch10
As before I move the couples to the ends, where it can be seen that they are sagging moments.
The original central load has now been replaced by a downward end load and a sagging end moment at each end of the beam.

sketch11
Finally I get to the reactions.
These are equal and opposite to the end loads and end moments so provide upward support reaction forces and hogging reaction moments.

go well

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Firstly, thankyou Studiot for helping me out in this way....After having read, re-read and re-reread your post, there are still some lingering doubts...

sketch5
So we are left with the other C acting downwards at the centeline plus a couple, which we conveniently ascribe to the centreline as well.
To be honest, I still don't understand the bolded part...I tried proving it on the lines provided by TinyTim(post#5 on the other thread)....my attempt is shown on the diagram attached to this post .....but you see, following this method(of just adding pairs of cancelling forces), we could even create a couple that'd be double in magnitude to the original couple(please see diagram, where I've explained my point better).

Besides, the concept of the moment applied to one point being equal to the moment caused by the same set of (equal and opposite) forces at another point is still vague....

sketch10
As before I move the couples to the ends, where it can be seen that they are sagging moments.
The original central load has now been replaced by a downward end load and a sagging end moment at each end of the beam.
Your diagram and explanation is intuitively clear to me....but it is contradictory to what is in my book (so my book may be wrong)....as you will recall, my original question was as to why the forces causing end moments are in the opposite directions (and not in the same direction as seen in your diagram....please see post #18 on the other thread if it is not clear what I'm referring to.

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Besides, the concept of the moment applied to one point being equal to the moment caused by the same set of (equal and opposite) forces at another point is still vague....
I hope I didn't say that.

A moment is the turning effect of one force and is applied at one point. It varies from point to point. It posseses both a single force and a lever arm.

A couple is a turning effect that applies to the whole plane simultaneously. So we can consider its seat or location at any point. It possesses neither a force nor a lever arm. It is possible but not necessary to make a couple from two forces. I have made one in sketch 4.
In sketch 5 I have used the property that I can ascribe the couple to any point I like to say that it operates at the centreline.
If this was a moment, not a couple, then I could not do this as per note above.

Since both are turning effects they can be set to operate against or with each other and the terms moment and couple are often mixed up or used for both mechanical effects.

we could even create a couple that'd be double in magnitude to the original couple(please see diagram, where I've explained my point better).
Not quite.

1)cancel FR and -GL. This leaves no couple behind as they are in the same line

2)cancel GR and GL. This leaves an clockwise couple = +c

3)There is now an anticlockwise couple of -2c between FL and GR (as you say)

4)Find the resultant couple by adding (2) and (3) = c-2c = -c, the same anticlockwise couple as the original between FR and FL.

I don't see that the circles add anything useful to the analysis?

Incidentally I am only trying to make the method in your book work, it would not be my first choice of approach to fixed (end) moments. It does have the advantage of being an engineer's graphical method.

hi Studiot!

As you can see, it took me a while, but I think I understand why the couple at any point can be shiftes to any other point....your last post-#7 really helped, because now I can apply your logic to any situation and realise how the whole thing works :-)

Btw, while we're still doing this, can I ask another quick question? (I won't ask too many more, promise!)

Well, the question is-

Do the external shear forces applied on a beam (say a simply supported beam) in the form of point load or Uniformly distributed load play any role in determining the magnitude of the bending stresses?

(My thoughts- the internal stresses i.e shear stresses in a beam (that has external shear forces acting on it) act to cancel the residual bending stresses on the beam....and the same shear stresses act to cancel the external shear forces(as a kind of reactional force) ....so the external forces must also have an effect on the bending stresses....)

I don't think it is a good idea to classify loads as shear.

I use the terms axial, transverse and oblique to describe them.

I prefer to reserve term shear for the internal response of the structure.

As your knowledge of the mechanics of materials develops it is a good idea to keep mentally separate the loads and their effects. This will really help when you come to the detail.

Remember loads are forces. The effects are stresses.

The loads only have to obey the three conditions of equilibrium.

The effects have further conditions to obey which are determined by the geometry of the situation.
These extra conditions are known as the conditions of compatibility and it is these that allow us to evaluate structural responses in terms of stresses and strains and deflections.

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The effects have further conditions to obey which are determined by the geometry of the situation.
These extra conditions are known as the conditions of compatibility and it is these that allow us to evaluate structural responses in terms of stresses and strains and deflections.
Hmm..meaning that what I call the shear stresses satisfy the two situations as a necessity of one of these conditions of compatibility....but there is no direct link between the two things that the shear stress separately relates to...right?

Compatibility (geometric) conditions might be

The deflection at a support is zero
The slope of the deflected beam shape is zero at midspan
The sum of the angles of a deflected triangle = 180 degrees
The compression shortening in the steel reinforcement = the compression shortening in the concrete

There are many more possibilities depending upon each individual circumstance.

Just to consolidate I have attached two sketches to emphasise the difference between the turning effects we call moments and couples

It is most unfortunate that people who should know better often mix them up.

moment is the turning effect of one force and is applied at one point. It varies from point to point. It posseses both a single force and a lever arm.
Sketch1

Shows any three colinear points, R, S and T distances a and b apart with a>b.
I have shown a force, F applied at T (perpendicular to this line for convenience).
T induces clockwise moments at R and S, as shown. The moments are clearly different and will vary with the locations of S and R.

A couple is a turning effect that applies to the whole plane simultaneously. So we can consider its seat or location at any point. It possesses neither a force nor a lever arm. It is possible but not necessary to make a couple from two forces.
Sketch2

The same three points now have a couple applied at S, equal in magnitude to the moment described in sketch1. To apply this couple I have removed the original force at T and replaced it with a two forces as shown.

The addition of the separate moments induced by each of the two forces show quite clearly that the turning effect is the same at R, S and T (or indeed any other point).
In particular it is the same even at the point of application of one of the forces themselves (ie at T).
Of course in sketch1 the turning effect of F at T is zero.

I suggest this would be a good endpoint for this subject and that we start another about shear and moment in beams rather than mix it up here.

go well

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You know, what really struck me while looking at your sketch is that the Moment of F about R is F(a+b) while the turning effect of F about R is Fb......and the moment of F about different points is different....but its turning effect is the same.....turning effect is thus when you make a couple from a force(like you did in your sketch) whereas moment is when you consider the force alone...so if I had a action-reaction pair(like in a beam), there would be no moment but a turning effect.....did I just get it?? :-)

The points are the same and the turning effect is the same about the chosen point (which is S), but the forces are different in the two sketches.

So there is nothing to suggest
that the Moment of F about R is F(a+b) while the turning effect of F about R is Fb.
I am saying that mechanics shows us that

In the top sketch - The turning effect about S of all forces acting is F(b)

In the bottom sketch - The turning effect about S of all forces acting is F(b)

BUT

In the top sketch - The turning effect about R of all forces acting is F(a+b)

In the bottom sketch - The turning effect about R of all forces acting is F(b)

So the situations are clearly different.
That is why we need two names for turning effects - to account for the different situations.

I like to call the situation in the bottom sketch a couple and the situation in the top sketch a moment - and not to mix them up.

Before posting a new thread about beams, you might like to look through this one and comment on the maths involved. It is about the fixed end moments in more complicated beams than yours.

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Okay...so whenever we have a pair of equal and opposite forces that are not on the same line of action, the turning effect of this pair is the same about every point(forms a couple) whereas when there's one force, without another force to cancel it, it has different turning effects about different points(moment)....

As we saw in this discussion, a couple applied to a beam has the same turning effect about any point and thus we can shift it to any point along the beam....

....then why is it that while using Macaulay's method for a fixed beam (considering a section of the beam at a time like we're supposed to do) we consider only one of the fixed end moments,say Ma and not Mb?? Doesn't Mb have a turning effect on the considered part of the beam??

then why is it that while using Macaulay's method for a fixed beam (considering a section of the beam at a time like we're supposed to do) we consider only one of the fixed end moments,say Ma and not Mb?? Doesn't Mb have a turning effect on the considered part of the beam??
I did say , a couple of posts back, that we should start a new thread about beams and their loadings as this one has too many questions for anyone else to follow.

Further I don't know where you are in your study of beam loadings, so I don't know where to start.

Did you understand how to write the equation I started with in the attachment in the link I gave?
In particular do you understand the process of dividing the beam into two free bodies for the purpose of analysis?

You should note that I haven't included VB either - for the same reason.

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I did say , a couple of posts back, that we should start a new thread about beams and their loadings as this one has too many questions for anyone else to follow.
Could we finish off this question on this thread and then if I have any further queries, I'll start off a new thread,please?.....Actually, I have only one chapter left in my syllabus, so I might not have too many other queries :-)

Further I don't know where you are in your study of beam loadings, so I don't know where to start.
We've done theory of pure bending (with problems involving calculation of bending stresses and moment of resistance), shear stresses in beams, and now I'm doing deflection of beams(of which the last topic is fixed beams)...there's one more unit to go-- involving columns, struts and cylinders(both thick and thin).

Did you understand how to write the equation I started with in the attachment in the link I gave?
Well...I've done derivations of formulae of this nature while doing calculation of deflection by double integration method...

In particular do you understand the process of dividing the beam into two free bodies for the purpose of analysis?
Yes....I've seen this done in books like Beer and Johnson...(for e.g I've seen it used to show that to an externally applied point load,there is an equal and opposite reaction at each point in the beam,provided there are no other loads in between,ofcourse)

OK here is a quick tour of beams.

It is important to fully understand what is happening in Figs 1 to 6 as this forms the basis for much of mechanics of meaterials

Fig1
I've started with a simple beam supporting a single point load somewhere towards the right hand end.

Fig2
I have broken the beam into two free bodies a C and D. In terms of external loads the right hand FB carries the load and one of the support reactions. The left hand FB only carries the left hand support reaction.

Fig3
I'd like to concentrate only on the left hand free body. As shown this only has one force acting on it and so is not in equilibrium.

Fig4
In order to maintain vertical force equilibrium something must be exerting a downward force on the free body. The only candidate is the right hand free body so it must be exerting a downward force I have shown as VC at the break.

Fig5
However the left hand free body is still not in full equilibrium because the reaction RA exerts a clockwise moment at C so to maintain full equilibrium the right hand FB must also exert an anticlockwise moment at C. I have shown this as MC.

Fig 6
It should be realised that we have not used the loading on the right hand free body in any way to determine VC and MC. They are completely determined by the external forces acting on the left hand free body.
This is useful as we could replace the right hand end of the beam with a totally differnt one, so long as it provides the required vertical force and moment at the junction.

Note also that I have not at this juncture specified a sign convention. Due to the laws of mechanics, the forces and moments shown on the left hand free body must act in the directions shown, regardless of whether we call them positive or negative.
In what follows I have chose to call them positive. I do not know your testbook and I apologise if they use the opposite sign convention.

Fig 7
Having established the idea of working our way from the left hand end of the beam and not considering any forces until we encounter them I illustrate further with a more complicated beam ABCD. This time I have put some numbers in to make the presentation simpler.

This beam has three distinct sections, AB, BC and CD. If we wrote ordinary expressions for bending moment as before different ordinary expressions would be needed for the different parts of the beam.
I have shown these expressions in the attachment.
Note the first equation is like out first example, it contains only the left hand reaction RA
As we move right RA still affects our left hand free body but the point load at B must also be considered so the second equation contains two terms.
Similarly the effect of the distributed load comes into effect as we pass C, still moving rightwards with our exploratory section and the left hand free body.

We never include the moment effect of the right hand reaction RB because when the section reaches the right hand support RB passes through it and has no moment about itself.

Macaulay looked over those three equations and noted that the third one incorporates the other two so long as we only consider appropriate values of x for each term. Each term in the equation is a product of a force and a distance. He realised that we could do this if we disregarded the terms when the values made the distance part of each term negative.

So if we disregard the second term when x<3 and the third term when x<6 we effectively have one equation we can integrate for the whole length of the beam.

Finally Fig8

This is your question about the end moments revisited in the light of the foregoing.

Starting at the left hand end we have RA and MA only acting.
So at section C we will not encounter any terms including the centre load L.
However at section D will will have a term for the left hand supports and the centre load L.
There are no further loads on the beam so this two term equation will extend to the right hand end.
As before we do not add the right hand moment.

go well

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Firstly, thankyou for your efforts to make me understand this...I really appreciate this :-)!!

Fig 5
However the left hand free body is still not in full equilibrium because the reaction RA exerts a clockwise moment at C so to maintain full equilibrium the right hand FB must also exert an anticlockwise moment at C. I have shown this as MC.
Okay, I understand the necessity of there being an anticlockwise moment Mb, but this moment has to be caused by some force...the only force capable of creating an anticlockwise moment is Rb (support reaction at B)...no?

Moreover, is this moment Mb the internal moment of the beam? If so, I read that the internal moment has to be equal to external moment (the one that causes bending stress)..in the chapter on pure bending)...so the internal moment Mb is different all along the beam,since the external load does not have equal effect all along the beam,right?

We never include the moment effect of the right hand reaction RB because when the section reaches the right hand support RB passes through it and has no moment about itself.
I understand this...but when we're considering fixed beams, it's a couple, not moment that causes the end turning effects(that's just what I thought)...if you remember my sketches in which I showed the treatment given to fixed beams in my book, you'll remember 2 equal and oposite forces 'R' at the two ends of the beam...constituing a couple....

Thanks again, Studiot!

I'm sure I've said this somewhere along the line.

It is important to distinguish between loads forces and internal response forces.
So far we have only discussed external loads. You are corect in noting that they have different effects in differnt parts of a loaded body.

Okay, I understand the necessity of there being an anticlockwise moment Mb, but this moment has to be caused by some force...the only force capable of creating an anticlockwise moment is Rb (support reaction at B)...no?
But where did I mention MB?

I also said that you have to understand the decomposition of the beam into two free bodies before you can make progress.

This free body idea is absolutely fundamental to mechanics of all descriptions, not just beams.

That is why I made so much fuss in Figs 2 - 5 of developing all the forces acting on the left hand free body.

The left had free body is subject to one, and only one, external load (the left had reaction).
This load produces effects at the beak section C.

I understand this...but when we're considering fixed beams, it's a couple, not moment that causes the end turning effects
A reaction force is a single force. As such it exerts a turning effect about any point not on its line of action. That is to say it exerts a moment that depends upon its distance form the point.

Built in beams have a reaction force and a couple applied at the support.
This couple is what is meant by a fixed end moment.
Since it is a couple this moment is the same at all sections along the beam ie at all values of x.

But where did I mention MB?.
I'm sorry, I meant the Mc on your diagram...now this Mc is an anticlockwise moment...which force causes this? Only a force acting in the direction same as that of Rb can cause this moment...

Built in beams have a reaction force and a couple applied at the support.
This couple is what is meant by a fixed end moment.
Since it is a couple this moment is the same at all sections along the beam ie at all values of x.
I've noted that in fixed beams with unsymmetrical loading, the Bending moment diagrams have different values of Ma and Mb (which are referred to as fixed end moments at the beginning...and it varies all along the beam..then obviously, the couple that you're referring to isn't what's plotted on the B.M diagram...

(Sorry if I'm trying your patience...but I feel I'm getting kinda close to understanding this and it'll just require a little more input from you....if possible, please suggest a book on basic mechanics(statics and dynamics) so that I can get this stuff on moments straight...)

I'm sorry, I meant the Mc on your diagram...now this Mc is an anticlockwise moment...which force causes this? Only a force acting in the direction same as that of Rb can cause this moment...
Okay, I read up on the part in Beer and Johnson where they describe how to split a beam into two parts and hence analyse the beam, but they just mention that there's a moment to oppose that on the left/right portion due to the support reactions, (the same as Vc and Mc in Studiot's diagrams)....but I'm still not aware of where this moment comes from (as I am currently under the impression that every moment is caused by a force somewhere...)

I've noted that in fixed beams with unsymmetrical loading, the Bending moment diagrams have different values of Ma and Mb (which are referred to as fixed end moments at the beginning...and it varies all along the beam..then obviously, the couple that you're referring to isn't what's plotted on the B.M diagram...
The current confusion is as to how Ma and Mb in a fixed beam can be different values if there is indeed a pair of equal and opposite forces (denoted by 'R' in the diagram that I took from the book) that is responsible for them...

I'm not forgetting this thread, I've been rather tied up with other things at the moment.

Okay, I read up on the part in Beer and Johnson where they describe how to split a beam into two parts and hence analyse the beam, but they just mention that there's a moment to oppose that on the left/right portion due to the support reactions, (the same as Vc and Mc in Studiot's diagrams)....but I'm still not aware of where this moment comes from (as I am currently under the impression that every moment is caused by a force somewhere...)
Let us suppose for an instant that my left hand free body is actually a cantilever.
Now there is only support at one end.
Since any reaction force, horizontal or vertical passes through the support point it has no moment about the support.
Yet we have agreed that there must be a moment at the support, or the cantilever would tip over.

Since you are so concerned about the source of the internal forces and moments acting at our section here is a foretaste of how it works. Look ahead in your textbook to 'horizontal shear'. This is where MC comes from internally. It is not only vertical forces that can provide moments.

The action of a beam is to turn vertical forces into horizontal ones and back again. This must be so if the loads are in the middle and the supports at the ends.

I have no time for diagrams at the moment, but we can return to this.