# Center of mass and rigidity and torsions

I've been researching about this for hours at internet reading dozens of pdfs.. but can't seem to understand the concept. What does it mean if the center of mass and center of rigidity are not coincident, torsions would be produced.. can you give an example of it in more intuitive or using fundamental objects? Thanks.

Simon Bridge
Homework Helper
Why not start by telling us what you understand about the center of mass and center of rigidity - what are they, how are they defined, what is their usefulness? It is likely you just have a slight misunderstanding of the concepts.

haruspex
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Gold Member
Centre of rigidity is somewhat analogous to centre of mass. Mass produces a resistance to acceleration; if the force is through the centre of mass then the resistance from the masses either side of the line of action balance and no rotation occurs. Rigidity (of the building) produces a resistance to movement (of a floor, laterally); if a lateral force is applied through the centre of rigidity then the resistances either side of it balance and the floor does not rotate.
When a seismic movement shifts a building sideways, the floor's inertia acts as a force resisting that movement. This acts through the centre of mass of the floor. If that is not the centre of rigidity then the floor will rotate.
This is all from reading http://www.eng-tips.com/viewthread.cfm?qid=68312. I'd not heard of the concept until reading your post a few minutes ago, so I may have misunderstood.

There is also this gem buried in the middle of the link you posted haruspex

I think it is better to consider the frames as a whole.

I concur with that sentiment entirely.

Centre of rigidity is somewhat analogous to centre of mass. Mass produces a resistance to acceleration; if the force is through the centre of mass then the resistance from the masses either side of the line of action balance and no rotation occurs. Rigidity (of the building) produces a resistance to movement (of a floor, laterally); if a lateral force is applied through the centre of rigidity then the resistances either side of it balance and the floor does not rotate.
When a seismic movement shifts a building sideways, the floor's inertia acts as a force resisting that movement. This acts through the centre of mass of the floor. If that is not the centre of rigidity then the floor will rotate.
This is all from reading http://www.eng-tips.com/viewthread.cfm?qid=68312. I'd not heard of the concept until reading your post a few minutes ago, so I may have misunderstood.

yes.. that's the concept. But I can't quite connect it with the case where there are different elements of different sizes in the building.. during seismic movement.. heavier elements move more.. this would produce more damage than when all elements are symmetrical and movement the same. Let's give an example. Refer to the following picture.

http://img820.imageshack.us/img820/6673/torsionssample.jpg [Broken]

In the 10.16 meter girder between the columns at the middle. The girder is heavier with more width and depth. During seismic movement, it along with the larger columns supporting it would move more. This would pull on the other smaller beams. This may make it more unstable during earthquake. Right guys? Now how do you connect it with the concept of torsions where the center of mass and center of rigidity must be concident to avoid torsions. Where is the approximate center of mass and center of rigidity from the layout plan itself? how do you determine it? note this is not a homework.. but just curious how it works, thanks.

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Simon Bridge
Homework Helper
Center of Rigidity thing is only a guide - a rule of thumb. Kinda assumes the structure is classically rigid. You can comply with getting it in the same place as the com and still have a building that falls down.

You cannot avoid torsions. The floor will try to twist about it's com but can only manage to twist about it's cor. This swings the com around and makes the whole structure more unstable. If you've every tried to carry or rotate something at a point other than the com you'll know the effect.

In the above diagram, the different arts of the building are going to try to move differently - which sets up stresses in the bracing which is trying to hold the building rigid. The closest relationship you'll get will be from treating the heavy supports as off-center com's. Noting where they are tells you where to put extra bracing to get you closer to the rigid-model dynamics.

Center of Rigidity thing is only a guide - a rule of thumb. Kinda assumes the structure is classically rigid. You can comply with getting it in the same place as the com and still have a building that falls down.

You cannot avoid torsions. The floor will try to twist about it's com but can only manage to twist about it's cor. This swings the com around and makes the whole structure more unstable. If you've every tried to carry or rotate something at a point other than the com you'll know the effect.

In the above diagram, the different arts of the building are going to try to move differently - which sets up stresses in the bracing which is trying to hold the building rigid. The closest relationship you'll get will be from treating the heavy supports as off-center com's. Noting where they are tells you where to put extra bracing to get you closer to the rigid-model dynamics.

But according to the concept of Diaphragms. If you pour the entire RC slabs at the same time, it will form one solid floor. So during seismic movement, only the perimeter walls will be affected, the center will become part of the entire frame so there would be no individual movements in the columns at middle in an RC frame with solid poured slabs, contrary to what you described above. Do you agree with me and if not, why.

Simon Bridge
Homework Helper
I said they will "try to move differently". In your example that will stress the floor as it tries to maintain it's rigidity. This is the diaphragm action - basically another kind of bracing. In a biggish shock the floor cracks (breaks, collapses etc) because the different bits want to move differently in response.

Details are tricky - in the CCTV building in CHCH the outer walls gave way first and the floors collapsed onto each other... breaking away from the heavy structures.

See: R.B. Fleischman and K.T. Farrow; On the Seismic Behavior and Design of Long Span Precast Concrete Diaphragms; Pacific Conference on Earthquake Engineering (2003).
... for some detail on the behavior of diaphragm floors in Earthquakes.

I said they will "try to move differently". In your example that will stress the floor as it tries to maintain it's rigidity. This is the diaphragm action - basically another kind of bracing. In a biggish shock the floor cracks (breaks, collapses etc) because the different bits want to move differently in response.

Details are tricky - in the CCTV building in CHCH the outer walls gave way first and the floors collapsed onto each other... breaking away from the heavy structures.

See: R.B. Fleischman and K.T. Farrow; On the Seismic Behavior and Design of Long Span Precast Concrete Diaphragms; Pacific Conference on Earthquake Engineering (2003).
... for some detail on the behavior of diaphragm floors in Earthquakes.

I'm not talking about precast concrete but one where fresh concrete are poured into the rebars with formwork all over.

Anyway. Are you saying that in concrete floor slabs that were poured at the same time and set at the same time. The unequal sized columns underneath it would try to move differently.. causing the diaphragms portions above it to move differently? In other words, the columns and slabs both contribute to the torsion movements... or seismic forces can transfer forces to both columns and slabs with different proportions? The past week I sat dinner with 3 structural engineers and ask many questions. What I told you was what he told me. That the slabs would become one and it is the whole RC frame or building that would twist.. not individuality. It's like this. Imagine you have a dinner plate put on top of matchboxes. The dinner plate becomes one unit and move at the same time. It doesn't depends on each of the matchboxes positions. Why aren't concrete slabs like this too as they put on top of beams and columns?

Simon Bridge
Homework Helper
Your engineer at dinner tells you, as part f informal dinner conversation, that the diaphragm makes the structure move as a whole. Naturally he has not gone into great detail about how each of the parts contribute to this and what happens under different circumstances.

I have told you here that the different parts of the structure will try not to - under lots of stress they will break apart because of this.

I have agreed with the engineer.

In your dinner-plate analogy, the plate is not attached to the matchboxes like a floor is attached to columns. Also the plate is very rigid compared with the stresses you placed on the system.

If you looked under the plate at the matchboxes you'd see that they shifted around. If you had different mass matchboxes they would have moved differently. If the matchboxes were attached to the plate, then the different ways they try to move (but are restrained by the rigidity of the plate) place stresses on the plate.

I don't know why this is so hard to understand: if I twirl a ball on the end of string, the ball will try to move off in a line but the string stops it - so there is tension in the string.

Your engineer at dinner tells you, as part f informal dinner conversation, that the diaphragm makes the structure move as a whole. Naturally he has not gone into great detail about how each of the parts contribute to this and what happens under different circumstances.

I have told you here that the different parts of the structure will try not to - under lots of stress they will break apart because of this.

I have agreed with the engineer.

In your dinner-plate analogy, the plate is not attached to the matchboxes like a floor is attached to columns. Also the plate is very rigid compared with the stresses you placed on the system.

If you looked under the plate at the matchboxes you'd see that they shifted around. If you had different mass matchboxes they would have moved differently. If the matchboxes were attached to the plate, then the different ways they try to move (but are restrained by the rigidity of the plate) place stresses on the plate.

I don't know why this is so hard to understand: if I twirl a ball on the end of string, the ball will try to move off in a line but the string stops it - so there is tension in the string.

So in such cases, the stress would be on the floor with strong seismic movement breaking them apart.. but I wonder what happens to the rc beams themselves. Imagine a column both connected to a 5 meter beam and another 10 meter beam on the opposite side of it (see the previous picture). If a strong seismic wave would hit from say the north portion, is it possible the middle column moving south more due to its more mass with respect to the rear column just stresses or breaks the column-beam joint connection (although partially restrained by the floor)? What do you think?

Simon Bridge
Homework Helper
As with everything: depends.

As with everything: depends.

Ok. I just remembered seismic wave moves thru the ground. For some few unlucid moments, I was thinking of them as shock waves in the air from say supersonic jets moving or nuclear bomb detonation sending shock waves in air. In this case, the diaphagms would be initially affected then the columns.

But then I don't think I have to worry about nuclear. Just seismic from ground. I'll have meeting with the 4th structural engineer tomorrow. I plan to make every columns symmetric... but then the lot is irregular.. so maybe would build it in portions... if this scheme won't make it worse.

Simon Bridge
Homework Helper
Oh you are trying to design a building to withstand an earthquake without being an engineer?

I've been responding just for understanding - the structural engineer will have math and knowledge of the regulations. I can only do the general from here.

When discussing diaphragm you need to have a proper understanding of what the structural engineer means. There are two different and distinct uses for the term.

On the one hand a diaphragm refers to a thin plate or membrane that is constrained at the edges and develops internal resisting stresses to transverse loads in the plane of of the plate or membrane. These are sometimes known as diaphragm stresses and may be found in standard volumes such as Roark.
Mechanical engineers often use this version of the term.

On the other hand diaphragms are also thick slabs or beams at right angles to floor or deck slab or wall ie a thick plate. They resist the transverse loads imposed on the plate and some transfer these loads to the plate supports. They are not the plate itself, which may have many diaphragms.

Oh you are trying to design a building to withstand an earthquake without being an engineer?

I've been responding just for understanding - the structural engineer will have math and knowledge of the regulations. I can only do the general from here.

I'm not trying to design a building but just want to be seismic aware.. my architect didn't even understand what was Torsion or center or mass or rigidity.. when queried about best structural positionings... he just replied he was an architect and didn't know any about structural.. hence he didn't prioritize on symmetrical lots.. all his designs were assymetrical... this was the reason why I parted ways with him and looked for a new architect last week.

Well. I just sat a few hours with the 4th structural engineer to come up with the best symmetrical design. We kept on discussing about seismic wave that would come from the vertical and horizonal. We forgot to talk about seismic wave that comes from the slant or other angles besides them. According to Studiot here... it would make the different portions move differently causing torsions too even on symmetrical lots especially in columns with individual footing. This means the most important is not so much to make the lot perfectly symmetrical but to make the connections much stronger? Maybe time to consult a 5th structural engineer on this. What I learnt was not all structural engineers have same knowledges. They have their own biases and opinions.

We kept on discussing about seismic wave that would come from the vertical and horizonal. We forgot to talk about seismic wave that comes from the slant or other angles besides them. According to Studiot here... it would make the different portions move differently causing torsions too even on symmetrical lots especially in columns with individual footing. This means the most important is not so much to make the lot perfectly symmetrical but to make the connections much stronger?

Well you have absorbed some of what I said but look back at it again.

There are two waves in a seismic event. One is vertical.
One is horizontal.
They arrive at different times.

There is no 'slant wave' betwen these two.

However as each wave passes along the ground it does does in a line we call the wavefront.
The wave arrives at every point along this line at the same time.
However, this line can be slanted at any angle to the building so it can arrive so that it meets the two front columns at the same time or different times. If the arrival times are different that will introduce a twist or torsion into the effect on the building.
Since there are many, many ways it can arrive slanted and only one way it can arrive paralle to the building line torsion is most likely.

The effect on pad foundations and importance of connections are correctly noted.

Well you have absorbed some of what I said but look back at it again.

There are two waves in a seismic event. One is vertical.
One is horizontal.
They arrive at different times.

There is no 'slant wave' betwen these two.

However as each wave passes along the ground it does does in a line we call the wavefront.
The wave arrives at every point along this line at the same time.
However, this line can be slanted at any angle to the building so it can arrive so that it meets the two front columns at the same time or different times. If the arrival times are different that will introduce a twist or torsion into the effect on the building.
Since there are many, many ways it can arrive slanted and only one way it can arrive paralle to the building line torsion is most likely.

The effect on pad foundations and importance of connections are correctly noted.

I think you may be using the term Torsion differently than Seismic Engineers. For example. Read this article.

http://www.scribd.com/doc/50184844/Buildings-twist-during-earthquakes

The torsions they were describing were what occurs when you kicked you bed at the side of the foot, the whole bed moves. This was what they meant by Torsions. I can't find any reference about your claim of one column being rised with respected to others especially in individual column footing foundations (where you shared it in an illustration).

Note that the speed of seismic wave is so fast that in one second.. many waves would pass thru the lot.. therefore there may be no time to each column to rise with respect to another. They smear one into simply lateral seismic waves.

Are you an structural engineer? Can you share reference on what you were saying? Thanks.

Thank you for telling me what torsion is.

Unlike diaphragm, Torsion has only one definition which involves rotation about the main geometric axis.

This distinguishes it from bending which involves rotation about a geometric axis at right angles to the main axis.

In the case of kicking one corner of your bed at the floor, the main axis is along the bed's length and the torsion axis is vertical so you induce a slight twist in any horizontal plane.
This is the effect of the horizontal seizmic wave.

My earlier picture described the effect of the vertical seismic wave, which is easier to describe.

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Thank you for telling me what torsion is.

Unlike diaphragm, Torsion has only one definition which involves rotation about the main geometric axis.

This distinguishes it from bending which involves rotation about a geometric axis at right angles to the main axis.

In the case of kicking one corner of your bed at the floor, the main axis is along the bed's length and the torsion axis is vertical so you induce a slight twist in any horizontal plane.
This is the effect of the horizontal seizmic wave.

My earlier picture described the effect of the vertical seismic wave, which is easier to describe.

the question is how significant are vertical seismic wave. During earthquakes, do you feel being accelerated up and down? it's more about horizonal movement, isn't it?

anyway, after firing my architect for not knowing about any seismic design, I plan to get another architect but this time I need to suggest the framing after consultation with structural engineers before he starts the architectural design. After talking with numbers of structural engineers, they still can't decide how much or how significance is torsion in the following case versus the perfectly symmetrical case (at bottom):

http://img109.imageshack.us/img109/3074/scheme1.jpg [Broken]

Due to certain setback requirements, My lot length is shorter by 2 meters compared to the original. Now notice the center column is same distance front to back (6.965m).. but on the left and right columns.. they are of different lengths.

Now the following has identical lengths in the beams and lot perfectly symmetrical

http://img440.imageshack.us/img440/7199/scheme2.jpg [Broken]

But if I use this design. I'd lose about 16 square meters of space. Now my big question is... are the seismic torions and distorions in the first unsymmetrical layout significant enough that I should lose about 16 square meters to make the bottom straight and making the lot symmetrical??

Hope someone here with structural background can comment or others with good common sense. I'd consult another structural engineer for this tomorrow but in case others can share please do because the other engineers I have consulted are not sure or can't tell for sure and the architect is totally clueless. Thanks.

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Simon Bridge
Homework Helper
Subjectively: a small earthquake feels like shaking: - large ones can feel either side-to-side rocking or like standing on an undulating surface.
(Speaking from experience you understand. Earthquake stuff is taught in school here.)
The earthquake in Christchurch NZ was mostly side-to-side inside buildings: you can watch video from store cameras.

It looks like you want to use all the available section space for your building but your engineers are telling you you are not allowed to - I'm guessing due to some earthquake part of the building code. You are now consulting more engineers until you find one who will sign off on the project anyway.

Building something novel to regulations tends to be expensive because you have to prove that the spirit of the regs is being kept to (while it is possible to obey the letter of them and still build badly.) This is where the top firms are worth their fees. I know this feels stifling, but look at it from their POV:

In NZ, the engineers who signed off on the CCTV building in Christchurch (which collapsed) are finding themselves under suspicion - possible criminal negligence charges you understand? Their defense is that they stuck to the regs ... had they gone with something non-standard or a bit off they'd be having a harder time. For this sort of reason your engineers are likely to be cautious.

I'm puzzled though - is there some reason you cannot have the symmetric pillar spacing and the trapezoidal floor shape? This gives you a triangular bit off the bottom (about 3m for the bottom right corner) that you support like a balcony - I'm sure you have seen structures like that. You'll probably have to pay for another couple of columns... so your cost/benefit boils down to if the floor space is worth the cost of supporting it! I'd bet that it's cheaper than the extra engineers.

As for if extra torsions are worth the extra floor-space: you don't really want to be doing calculations which pit floor-space against the number of (projected) additional deaths and injuries risked over the lifetime of the building now do you?

Subjectively: a small earthquake feels like shaking: - large ones can feel either side-to-side rocking or like standing on an undulating surface.
(Speaking from experience you understand. Earthquake stuff is taught in school here.)
The earthquake in Christchurch NZ was mostly side-to-side inside buildings: you can watch video from store cameras.

It looks like you want to use all the available section space for your building but your engineers are telling you you are not allowed to - I'm guessing due to some earthquake part of the building code. You are now consulting more engineers until you find one who will sign off on the project anyway.

Building something novel to regulations tends to be expensive because you have to prove that the spirit of the regs is being kept to (while it is possible to obey the letter of them and still build badly.) This is where the top firms are worth their fees. I know this feels stifling, but look at it from their POV:

In NZ, the engineers who signed off on the CCTV building in Christchurch (which collapsed) are finding themselves under suspicion - possible criminal negligence charges you understand? Their defense is that they stuck to the regs ... had they gone with something non-standard or a bit off they'd be having a harder time. For this sort of reason your engineers are likely to be cautious.

I'm puzzled though - is there some reason you cannot have the symmetric pillar spacing and the trapezoidal floor shape? This gives you a triangular bit off the bottom (about 3m for the bottom right corner) that you support like a balcony - I'm sure you have seen structures like that. You'll probably have to pay for another couple of columns... so your cost/benefit boils down to if the floor space is worth the cost of supporting it! I'd bet that it's cheaper than the extra engineers.

As for if extra torsions are worth the extra floor-space: you don't really want to be doing calculations which pit floor-space against the number of (projected) additional deaths and injuries risked over the lifetime of the building now do you?

In web sites that show torsions, they always give examples of seismic wave coming from the vertical or horizontal, but for those with other angles.. do symmetrical lots really have advantage? Remember when the corner column is hit first for horizontal seismic force.. would symmetry in the entire lot help?

Well. My building will have cantilever second to third floor, that is why your idea of balcony like support at front with symmetrical pillar spacing won't work because the cantilever front needs to be supported continuous to the back beams. Unless I don't have to use cantilever. In this option I'd prefer some loss of space if I can find proof that a little unsymmetrical lots like mine would be significantly more hazardous than perfectly symmetrical one... remember the beam difference is only about 1.4 meters or so. Why, do you or others live and work in perfectly symmetrical pillar spacing house or building?

Subjectively: a small earthquake feels like shaking: - large ones can feel either side-to-side rocking or like standing on an undulating surface.
(Speaking from experience you understand. Earthquake stuff is taught in school here.)
The earthquake in Christchurch NZ was mostly side-to-side inside buildings: you can watch video from store cameras.

It looks like you want to use all the available section space for your building but your engineers are telling you you are not allowed to - I'm guessing due to some earthquake part of the building code. You are now consulting more engineers until you find one who will sign off on the project anyway.

Building something novel to regulations tends to be expensive because you have to prove that the spirit of the regs is being kept to (while it is possible to obey the letter of them and still build badly.) This is where the top firms are worth their fees. I know this feels stifling, but look at it from their POV:

In NZ, the engineers who signed off on the CCTV building in Christchurch (which collapsed) are finding themselves under suspicion - possible criminal negligence charges you understand? Their defense is that they stuck to the regs ... had they gone with something non-standard or a bit off they'd be having a harder time. For this sort of reason your engineers are likely to be cautious.

I'm puzzled though - is there some reason you cannot have the symmetric pillar spacing and the trapezoidal floor shape? This gives you a triangular bit off the bottom (about 3m for the bottom right corner) that you support like a balcony - I'm sure you have seen structures like that. You'll probably have to pay for another couple of columns... so your cost/benefit boils down to if the floor space is worth the cost of supporting it! I'd bet that it's cheaper than the extra engineers.

As for if extra torsions are worth the extra floor-space: you don't really want to be doing calculations which pit floor-space against the number of (projected) additional deaths and injuries risked over the lifetime of the building now do you?

Btw.. in my place and experience. Structural engineers don't suggest or tells you what is allowed and not allowed. They always follow the architect's plan. For example if the architect requires 15 meters span, then they would build massive columns or pillars for it. They won't suggest splitting them into three 5 meter span to create more footings and absorb more of the seismic energy. It is only during earthquakes that they are tested. The ChristChurch NZ earthquakes give engineers more ideas about seismic design. This means the knowledge is still evolving as shown in this interested article:

http://www.nzsee.org.nz/db/SpecialIssue/44(4)0239.pdf

Anyway. I talked to the architect and insisted on putting the two columns underneath the front of the second story overhang. This would make both symmetrical pillar spacing and trapezoidal shape possible. I think this is the only solution. Thanks for the last minute tips which I integrated into my discussions with the architect a while ago.

Besides. Constructions would be easier with similarly sized beam depths instead of different sizes which may just confuse the construction crews.