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0:10
And once again, it's a great, great career for me, to be able to speak to our fellow colleagues and also researchers on subjects that are important to structure engineers and students as well. And normally, I'm very interested to talk about a few subjects. Here is one. And modal analysis another one and into performance space design and so on. So this one is kind of close to my heart and also related to the heartbeat. So it makes a little bit sense. have a title
0:41
the presentation and try to link it to the heartbeat of the structures. Just like as the human being. We have the heartbeat. So I think these structures also have kind of a similar property that if we correlate, we can then understand the structure response and the structure itself. And it's, you know, it's it's potential problems. Much better. topic that we have, sort of listed here is that we are going to listen to the heartbeat of the structures
1:08
and understand the model analysis and this presentation, has been prepared with the help of Doctor Najam, one of my colleague and former student here in the city. Currently, he's in the University of British Columbia in Canada. So some of the details, more details. You can also look at his videos and lectures. So let's begin. First of all, we are going to correlate, like I say, these structural help with the, heartbeat of the structure. So looking at the heartbeat of the structure.
1:40
So basically we are assuming or we are saying that every, every structure has a very unique sort of, fingerprint or heartbeat characteristics that we can look into. And if, you know, I hope everybody's healthy and then you don't have to go to see the doctor. But as you get older, you normally have to get your checkup every year or every two years and or even otherwise when you go, I think if you if you might have noticed
2:06
that the first thing they ask you to do is to go and get your, blood pressure and also get your heart rate checked. And after that, they will then proceed to do the evaluation of whatever is the cause. Even the dentist, when you go to have it. Okay, you have a procedure. They will start to check your these indicators, these fundamental indicators. So it's so essential to understanding the health of the human being that they are always checked. And similar is true
2:37
for the, the structures and and the reason the heartbeat is so essential because of the central position of the heart in terms of our, you know, whole health and everything that is derived from there. So the potential sources of any causes, of any illness, they can be detected and by studying the heart rate and, you know, related properties that we will talk about a little bit. So with that in mind, you can you might have seen, and many of you probably are using the fitness trackers
3:10
or the smartwatches these days because, you know, this, this importance of these two indicators has become so prominent that so much devices, wearable devices, have been,
3:23
generated and created in the last one decade that it is now becoming a universal, sort of
3:31
variable thing that we wear like a watch, and a health tracker. And this, these fitness trackers give us a lot of data, ever increasing data and generally the best first thing they measure is the heart rate. And then after that, they can create all these dashboards, weekly, yearly, daily, morning, evening telling you how did you sleep, how did you perform your stress, you're not stressed and so on. So a lot of information is derived from basic measurement
3:59
of a few parameters, the using these health trackers. So if we take that technology forward today, we should be able to do a lot of that for our structures as well.
4:10
So before I, I go to the model response, let me go back to the very basic structural response fundamentals. Before we, you know, drill down to the model response. And I presented this before also. So some of you might have seen this this slide. But it's essential that we but I can't remember where it started. So it started somewhere in the 15th 16th century by Galileo. And he started observing the fundamentals of the, the motion and also of the these structures.
4:45
And then aided by Robert Hooke, he first proposed the proportionality between the displacement and force. And he's, you know, credited with starting the stiffness concept. At that time, it was also, once again, several hundred years ago. And then we have Isaac Newton not so far about, you know, coming in and then giving us the physical connection between the mass and acceleration and force. And then that became very fundamental principles of physics at that still stand.
5:13
You know, of course, Einstein's theory of relativity is one thing, but the within the gravitational field, we still believe that Isaac Newton, Isaac Newton's laws hold. And then we have people like Thomas Young who started to develop correlations between material stress and strain. And from there onwards, everything accelerated. So. us see what was said by Newton, which basically related the mass and acceleration. So he assumed a body in motion.
5:45
But rigid body freely moving not not connected to the to the ground. So he proposed the mass acceleration force relationship, which is fundamental as one of the laws, physical laws. On the other hand, Robert Hooke and others,
6:02
thought of this structure as being fixed and not really moving. And it was fixed to the ground by a stiffness. So it was connected, the mass was connected to the ground. And then when you combine the two together, you got the first equilibrium equation of moving bodies but restrained moving bodies. So stiffness, mass and force were then related in that way varying with time. Of course, because you know, if you are moving, if you're looking at the things which vary
6:28
with time, especially when you're having a consideration in a mass, and then
6:34
that stiffness alone was later modified to include the damping of the bodies that are holding the structure down and the bodies themselves. So, so many types of damping came in. But we are right now we are talking about the basic, you know, structural damping. So we have now okay and damping C and mass. And we have the first complete f equilibrium equation including these key parameters. And then after that we can bring in ground motion.
7:04
And we say okay if the second motion then we can bring it on the right hand side. And then we have the first ground motion equilibrium equation. Once again linear. Because all these quantities, all the changing the time they are, they remain linear during the whole,
7:20
equilibrium cycles. And then we went back to Robert Hooke's law and we discovered, or we could see that this is an infinite law. That means no matter how large is the force, the displacement is always proportional. So if you, for example, for the force as if going to 1 million tons, you can get a displacement of ten kilometers, which obviously is not structurally correct. But the relationship states that way. But the observed response of the structure
7:51
was not this observed response was like this. The nonlinear response of the structure has been people tested them. They found that structure actually deviate and then start to become nonlinear and then collapse or break at some point. So with this one we had the the nonlinear response location, equivalent to the same displacement. And then from there we can we can derive the difference between the linear and nonlinear response.
8:19
And then we can say this is like a critical correction that we should do to the linear response to get the actual response. Now it seems very obvious that if we have, we can find the difference between blue and blue line and the the linear to the point. But the problem is that we don't know the blue line. We never know the blue line. We actually have to search for it. And we have to numerically we can only find by testing, but numerically we can't find it. So we have to search for it.
8:44
So this becomes a highly iterative non-linear problem. And if you take this correction force correction from linear to nonlinear and put it back into the equations that we know and modify and connect the non-linearity there, then we have a nonlinear, equilibrium equation static. And then if we bring back the dynamics then we have the nonlinear, dynamic. And then we can finally go to the ground motion and we have the full ground motion equilibrium situation. There.
9:15
So this gives us the full nonlinear equilibrium system which we can now solve in most of the software that we have IT apps and perform. And whatever task the use to get all of the responses. Then out of that, if you take that whole equation out, it's all varying with time, and you just look at that equation and you try to understand, you can derive a lot of information from there, some analysis by making some assumptions. For example, you could assume like this one,
9:49
we can get the model response from there by ignoring the non-linearity, the damping and the force and the ground motion. We get that and we can also get the equivalent static force by making assumptions about the displacement of the acceleration of the mass at the top, and the force directly through the mass. And then we can get the response spectrum. We can also get the overall nonlinear response by taking the nonlinear portion of the equation.
10:16
And then we can also get the linear time history analysis. So as you can see in this page, we have everything that we normally do to get this structured response. So this is almost all that we need. Almost. Of course there are others. But let's say this is mostly what we need. So let's focus on this part, which is what we are talking about today. let's look at the let us for a single degree of freedom that we showed from the, from the, the, the figures that I showed you.
10:46
But for a real structure, obviously we have matrices, for mass, we have a metaphor for damping, we have matrix for stiffness and so on. And everything is a matrix. So when you do all those matrices, then you the, the relationship is modified like that. And then you can have external force. You have mass acceleration, you have damping velocity and you have stiffness displacement and we have the non-linearity there. So the whole now structural response is now complete
11:13
derive from single degree of freedom all the way to to do matrices.
11:20
Knowns are these and the unknowns are the displacements and the velocity and acceleration the derivatives of displacements. So this is the total problem that we as structure engineers are interested to solve. Okay. So now that we have given the overall background of the structural response, let's pick one of them to go into the detail today. And we look we are picking the modal response today. So what are the responses that you have. None of these. You have no non-linearity.
11:50
You have no external force and you have no damping. And you are left with just only mass and stiffness and displaced mass and acceleration. So derivative second derivative of displacement, stiffness and mass is the only thing left equal to zero, which basically means its initial structure standing by itself, not being being disturbed by anybody, made from ideal materials with no damping. So that's the the situation that we are in right now. So let us see how we can solve this.
12:19
So here the knowns are mass and stiffness only nothing else, which is very very important for today. And then the unknowns are forced. From there we will derive the those parameters that at this point we do not know, do not know.
12:35
So the initial response of this structure is of this form. This is a second order linear homogeneous differential equations and then a lot of math behind that to solve it. And I do not plan to go into that. You can look at other references. I will just summarize briefly. But for the math we we probably need a separate session and you can watch some of the videos. Material format. He has explained this well what we are interested right now
13:01
is to think of this as the heartbeat of the structure, this relationship, and then this indication of stiffness and mass relationship. We will see what can it give us, because it's very basic is the structure by itself. It's intrinsic. It's just the nature property of the structure doesn't depend on the Lord, doesn't depend on anything. So if you look at the, a mass of the stiffness, the one that Newton and other people, not Newton, the, Hooke's law talked about.
13:34
And if you look at that and combine equation in motion, then we can actually see that there's a cyclic motion. We can convert that into linear motion to design and cost. And we assume that there is a frequency of this oscillation omega. And from that we can derive through the first instance second derivative. And we can find the solution. And ultimately we can come up with the omega frequency to be a ratio of stiffness and mass square root of that. And that is very important
14:08
observation important derivation from this. This is a physical, you know, solution. Yeah. That solution, mathematical solution also arrive at the same results but with different assumptions. But I've just presented one of them and the other one also that we will look at later. So having done that, you can see that we can correlate the angular frequency with the translational or cyclic frequency by rotation to the pi two pi.
14:38
And so this relationship we can actually this is supposed to be an animation. But unfortunately we couldn't run it. But anyway, so this is how it works. So you can correlate them through sine and cos and through that through cycles. And if you go further So we can now correlate this oscillation with the, the frequency. And we can say if it can reach two pi in one cycle that is cycle of oscillation. And if we have two of those then we have two cycles and so on.
15:08
So this omega one, two and three gives us an indication of how many times this structure can oscillate in one second. So that becomes our basis for deriving three important parameters that will define this whole model response. And this is based on the assumption of the equation that we started with. So there we are. So the natural frequencies and periods we have three fundamental quantities here. The omega which is the, the rotational or cyclic frequency and which is
15:45
a ratio square root of the ratio between stiffness and mass, nothing else. So that means if we know these two things this can be computed. And the, the frequency, is given right from there by dividing eight by two pi as we see before. And then if period or time period some people call it is then the reverse inverse of that. So now we have a relationship for the time period or the period as two pi square root of mass and stiffness. And that's it.
16:15
This is all we need for a single degree of freedom in fact. And then we can use that to actually now work on many things. So we know that there's a stiffness mass relationship. So if you want to change or control the natural frequency or period, all we need to do is change this ratio, not the absolute absolute values. It doesn't matter. The ratio is important. So a very heavy mass with very heavy stiffness or very light mass with very light stiffness may result in the same natural frequency.
16:46
And this is very important.
16:49
So let us look at some of the so so this is a plot between various mass and stiffness ratios. So this is a mass is about four times the original mass on the left hand side. And then stiffness is same. Then if we continue then they will oscillate at different time. They have a different period. This has a period of two two seconds. Let's say the first one has a period of one, and this one has a period of 0.5 where the stiffness is large.
17:14
And similarly on the right hand side we have some proportion. And you can see this will oscillate first with a longer period, then shorter period and then shorter period because the stiffness is increasing as the stiffness increases, the initial period reduces and frequency increases. So that's the basic relationship between the mass and stiffness ratio. And there we are. So we can extend that to a multi degree freedom system.
17:42
And we can solve the equation mathematically using it as an eigenvalue problem. And then finding the eigenvectors and eigenvalues. And they give us the natural frequencies and the mode shapes corresponding displacement patterns for each of the modes. So this is how the programs use it. They used multiple ways to to determine determine these. But eigenvalue analysis is one of them, which is, you know, the basic one, the primary one. There are of course, other other methods too.
18:11
So this gives us the tools now in the programs to do that. Okay. Let's go to the. multi degree freedom system. So you can you can see here that this is a tall structure with three storeys. And then it will vibrate first because it's a cantilever. Then the second one and then the first one because each case the mass stiffness ratios are different. And here they are. And then you will see that now they start to oscillate in higher modes as we call them.
18:38
And that is the first one for the first mass three mass system. And then after that you will see the second one will start to vibrate. And then they back to the further higher modes with more complicated complex mode shapes. So then we have this third, you know, third, fourth more shift. So you can see the masses are now moving in different patterns. And each pattern is a pattern doesn't have real values. These are scaled later to find real values.
19:08
So we have now a complete picture of the nature of frequency ordinates. Look period versus the mode shape derived from step by step from single to multi degree freedom. okay. So now we are jumping to the full structures and trying to understand what do they mean and what do they tell us and what they can tell us. So this one So if you play this video this is a tall building, 40 storey something. And it can show you three of its primary modes here.
19:40
The first mode, the second mode and the third mode. And you can clearly see how the building is oscillating in these modes. And they will be much higher in many, many modes to come. But these are let's say, the first primary modes we are looking for in each direction and so on. So it's a symmetrical building. So it is oscillating like that. So you can clearly see that the first mode is the it has the smallest stiffness associated with that.
20:03
With the largest mode movement the relative second mode will be it will be higher stiffness, smaller movement, higher frequency faster moving. Third mode will be very fast moving with small amplitude and very large stiffness. So that's how we we can understand them in this way. And then if we have a ground motion then you get the total response of the structure for the ground motion. All of the responses for many of the modes will be added together, combined
20:33
with the ground motion, in a, you know, the way that is structured moving. So it's a complex thing but but physically So if you can see that these are the, the some of the modes will give us the actual vibration of the structure. Because those modes are only independent. But when they are combined together they added together depending upon the ground motion you will get a certain movement. And that's how you can derive that from that.
20:56
And we will talk about the ground motion causes, you know, additive and all of that stuff.
21:02
Okay. So model response now tells us that there is an initial period and there is a more chip. And both of them come together as a as a as a pair, as a set. And there are as many modes in a structure as there are degrees of freedom, because each degree of freedom can move independently. When excited with, with a particular causing a particular mode chip to be generated. So if there is a structure with 100,000 degrees of freedom, theoretically speaking, there are 100,000 nature periods
21:33
or modes or more chips in that structure. Not all of them are important. We will talk about that and then local modes are also present there. We will differentiate between global modes and local modes in a in a while. And then we have to make sure that we understand which is local, which is global. Each has its own purpose. And then we have this concept of higher mode effects on tall buildings specially, which we will also focus. Then model analysis should be
22:02
should be run before anything else. We always recommend our students and our colleagues here that if you are creating a model, you need to have whatever software using even before running. Detect load. Run the model first. Make sure that the properties are right. Make sure the structure is right. Make sure the the model is right. If the model is right and the properties are not good, that means structure is not good. So we need to go step by step to make sure that we have
22:28
what we are dealing with is correct, reasonably correct. Before we start applying loads and doing all of the rest of the thing. So model analysis is actually kind of a precursor to most of the analysis that we will do later. Response spectrum, time, history and everything even for wind and so on. So we will look into that. And as you know, as you know, these animations are always fun to watch. From in depth you can get a lot of data and they keep adding these scores
22:56
of, you know, before they only used to be three rows. Now there are five, six, seven rows. They keep adding more information that can be extracted from the model analysis. And there are some interesting things here added recently which lead to the the higher mode computations. So you can see here that from this data you can get the basic, you know, factors like natural period frequency and so on. The participation factor, load participation factor, mass participation factor and so on.
23:26
So on this slide two and when you get the nodes then you can go into the detail of what is the purpose of each of these analysis. And you can also read the documentation from it. And there's a lot of discussion about this there. So the point is that there's a lot of information insight into the model analysis alone.
23:48
Before we go into the other ones. So more chips to me are fundamental to understand the structure. Of course network frequency number is one. But the more chip gives us far more information, far more insight into the structure. If we know how to read it, just like, you know, the doctors, if they they know how to read the EKG or then they can quickly see what's wrong with the patient. Same way we should be able to to see the more chips and just
24:13
see what's wrong with the structure or it's good or not good. So we can detect the symmetry the and symmetry. We can detect the local modes. We can detect so many things. So let's look into those one by one. First of all is your model trustworthy. Because once you created the model, how do you know that the model is good and then you haven't made any cross mistakes. And I have so many examples over the years where people have made cross
24:39
mistakes in the model, did not detect it, and they went as far as to the side. There was one case particularly which is of note that structure was designed and the foundations are very heavy. And the contractor complained that this foundation seems to be too heavy. It's not possible the structure. And
24:59
then to pick it up. And when they went back, they found that they had applied the steel mask by mistake to the concrete elements. So everything mass was very heavy. Had they checked the initial periods, they would have immediately determined that this is not possible anyway. So you can check the missing stiffness, missing mass, mass increase in accuracy, boundary condition, inaccuracy, which is quite common. And we can also look at the the connectivity errors,
25:27
structural irregularity and symmetry validation. So so much can be done by looking at these two parameters. Let's look at them. So you can see here that this 4045 storey building and we are looking at various more shapes. And then the second one there's a second mode. As you can see there's a torsion there. And then the third one And you can see now the higher modes are coming in. The structure is deforming in higher modes So you can watch now the modes.
25:56
And there's a lot of information in each mode which we will analyze one by one. So just want to give you from a real building how you can look at those modes of the know, the lower modes and higher modes. Okay. So let's look at the let's start with a very simple structure symmetrical structure. It's a 20 storey building. It's within the frame rigid frame structure range. It's a little bit flexible but I've started that. It's not too high. It's not too short.
26:24
And yes it doesn't have a sheer wall, doesn't have anything. But it is within the general limits of, of multi of rigid frames. So moment is in frame. So we start with a very symmetrical model of this one reasonably sized and see what happens. So if you look at the initial periods you will find here that we have the first period, initial period equal to 12.2 six seconds, little bit longer than it should be because structure is not constrained by shared walls.
26:52
Second one place and then it's moving in one direction. And the second one is moving in the other direction. But also both of them are translational. And the third one you will see I is moving in rotation is is rotating. This is the most typical response we should expect from a symmetrical building, first two modes being translational each in the weak and weaker and the stronger direction. And then the cardinal mode has a third one.
27:21
So this should be the order that we are typically looking for in buildings. Not always to get it, but this is what we should be looking at. Okay. Let's look at the same modes in plan. So it's more clear. So it's moving in its direction as a first mode.
27:35
Just complete translational second one please. And that is moving in the right direction completely translational. And the third one is moving in rotation which is pure rotation about the center. So from the rotational mass movement there won't be any mass participation, much mass participation coming because the traditional other two should be giving the the the higher mass participations. In this case. And we can also look at those modes in the in the sequence.
28:04
So in the sequence like for example the first mode along x which is the first mode of this structure also. Then second one is the second mode along x, but it's the fourth mode in the order. And that is the you know, you can see. And then the next one is the third mode of in the x direction. And it's the seventh mode in the order. And the 10th mode and the fourth mode in the x direction. So you can see as we go higher, the modes are becoming more complex. The movements are more complex.
28:35
The initial period is is dropping. Frequency is increasing because they have higher stiffness attached to them. And because of high stiffness, they also generate higher forces because, as you know, force and displacement are proportional. So by looking at the modes this is a ideal order one and then four and seven and ten for the right direction. And the other direction would be two and so on. And then torsional would be the same way.
29:00
So we can look at about the 12 modes for this building should be enough. Okay. Let's see. We make it slightly irregular. We add one shear wall on one one end of the building and then see what happens. And then you can see here we are looking from the top and we will see that the first mode does not change because the wall in this direction is just out of plane, does not add any resistance, and it almost did nothing. So in out of plane the wall is as bad as anything, so it didn't do anything.
29:30
But when the second mode y direction came, then the wall held on it did not go move together with the frame. So the whole building now revolving or rotating about the wall. So the stiffness center shifted toward the wall. Mass centers remained the same. We have the eccentricity. And then we can see And then this one is rotating about the one. The third mode is still torsion which is rotating all about the center kind of. So this is a two sort of indication of a and symmetrical building
30:02
where the second mode or the first mode will go into torsional or rotational. So if any one of the first two modes go into rotational means that we have a regularity, regularity in the structure. Let us see. We have used a vertical regularity. We put two walls and we put one wall shorter than the other. This will create a highly complex dynamic response, because not part of the structure up to its floor is symmetrical. Invite erection, but it's not symmetrical. above that.
30:37
So it will not shift the vertical stiffness entered from the center to the right and so on. So it's now we are we are creating highly
30:47
irregular structure. And if we look at the the model response for this one, So you can see here, the first one is like that because the both walls are moving And it's rotating now
31:01
and then the next one, and then third one, the last one is becomes becoming very complicated. Some parts are moving this way some part and moving this way. There's no pattern because the raw part tends to be symmetrical. Upper part is not. So the movements are not becoming highly unpredictable kind of. And the forces force distribution cannot be predicted by a typical 2D analysis or the typical sort of, structure response. It would be more complicated than that.
31:29
So verticality, regularity and building of this kind will be more, probably more harmful or more difficult to manage than the planned regularity which we can see clearly. So vertical regularity let let let's look at this video of the different modes. So we are looking at the first mode. So I put all the modes here so we can watch them as they transform. So after that the first mode is goes to the second mode. And then from that And then we go to the third mode rotation.
32:03
And then you can see here and then we have the third, the next mode fourth mode.
32:11
And then we have higher fifth mode. These continue. Yeah. Like that. You know you can see the structure is deforming and very irregular. We get it. And the last mode that you can see is even more complex. So now some part of the structure is moving some is not moving. So the forces displacements everything will be highly unpredictable in this guy, especially when combined with ground motion which are coming from different directions.
32:37
And they will activate the structure modes in different different contributions and different manner. So here you can see very clearly that verticality, regularity of a building like this is going to make the response pretty complicated. Okay. So different stiffness along x and y. Now this may not seem to be a major problem because the structure is symmetric about symmetrical about x and y independently. But the problem is that the two stiffness are now very different.
33:07
And the structure responds when the real or ground motions will be on at any given angle or even wind, then this structure, which appears to be symmetrical about each axis, is going to be performing not so well when we have x excitations at an angle. So let's go to the next one and let's look at the model response. The first mode is like this. But look at the initial period is two point something. And then the next one is one point something.
33:40
So which means the second direction is very stiff along y and the x direction is very weak stiffness wise. So if you have a ordered one full
33:50
order of difference between the two modes that mean the stiffness is very different in two directions, and then we have inclined excitation, this response will not be predictable because the two directions are not close to each other. So the lesson from here is that the two directional response for the first modes, almost most of the modes, we should try to keep it similar, not identical, difficult, but similar of the same order not two times or 50 and 50% different
34:22
because that will create once again and regularity, directionality, regularity which is often missed in a structure of this kind. The we do not because the structure is very long. For example, and one direction we didn't provide enough walls and then we only rely on the frame. The direction we have was. Then the structure becomes specially structure which are long in plan and have different stiffness, are more susceptible to irregular dynamic and wind response. So next we come to that.
34:57
Not all modes are created equal. That means we have to filter out the noise. Because when you make a structure you have all kinds of components there and everything is vibrating. So it's quite possible that some think vibrating before the structure is vibrating, or they are vibrating in between the modes and they will disturb or disrupt your main structure of response. Computations. So that's why we need to be vigilant to detect the modes which are noise
35:29
or which are which are important, but not from the global point of view. They are important in their own right. So let us look at that. So finding the real story among the vibration is to make sure that when you look at the initial period, it should be reasonable in a certain number. And you look at the mode shape and it should be in a certain order. So if the order is out or the frequencies are not progressing systematically
35:58
in a certain order on a certain sequence, that means something is wrong in between. So. So left hand side is a normal building. And the second one I just put a small sort of canopy on top of that. And here you will see that the canopy or the top, you know structure will start to vibrate, move before the main structure or will oscillate in, in nature. So it will attract the first mode and second mode to itself. And the structure is now stiffer than this one.
36:26
So if you look at the first two modes, those modes belong to the canopy, not to the structure. All the structure is moving a little bit too. So now the mass participation ratios will be different. And now you are looking at if you're normally in a building like this left hand side, the first verse mode will probably carry about 8,580% of the mass in the right hand building the first mode will carry maybe 4% 5% of the mass. So suddenly your
36:52
your understanding of the first, second, third mode will not be different. And if you're doing your analysis, for example, then if you by default select the first mode, then it won't be right because first mode on the right hand side will only select the top canopy and not the structure, and it will be the target displacement and everything will be incorrect. So you have to be careful to see that the modes are global for the global response. For local response, we could look at those separately.
37:22
So for a case like this probably is unnecessary to model the canopy in this global structural system and just apply it as loads. Is this structure good enough to check the pose. So that means you need to. Now we have come to the kind of a conclusion that we need to make sure that the natural period numbers are reasonable. For example, tall buildings, people say, the natural frequency should be the number of storeys divided by ten, which is not always true.
37:49
Actually, it is slightly less than that. And a lot of data has been collected. And there are many rules and many formulas to check that observed. In general, observed buildings have have a shorter period than computed because nonstructural, you know, the stiffness comes in and there are other things that we could not estimate properly. And we have to use checking factors and things like that, that we will talk later. So we need to make sure that we estimate it. Well.
38:18
Then this to me is the most important slide of today's talk that we need to arrange the modes in a certain sequence to understand what's going on and get the full picture. That means we should not look at the numbers of the modes in this sequence. They are reported by the program, but in the sequence that we are looking for in each direction, and as soon as the sequence is out or the drops from one to the other, other level is is not reasonable.
38:50
Something is wrong with this with the model or the structure. So for example, like I showed you earlier, the first mode in the weaker direction should be one, four and then seven. And if that is not the case, that means there is a vertical irregularity somewhere that is pulling the modes away. And here in this case the orders are not the same. So there is something in the building that altered that. Similarly, the second row should also be in sequence. If it is not, that means we have something
39:18
in the structure that is altering the sequence. Same is the case with the last one that we should look at. How much mass is participating in each mode? The first mode first three modes in each direction should generally accumulate a lot of mass. If not, that mean the higher modes and the local modes are are corrupting the results. Because generally speaking, for the building of this kind,
39:41
nine modes 12 modes generally should accumulate enough mass, but sometimes you have to go to 30 modes or 40 modes to capture all the mass, which means there is a lot of local modes are also coming in, or there are far more higher modes with low low stiffness which are also coming in. So we need to study. We need to arrange the modal response in a table like this. Put the next period, put the mode number, put the the mass participation everything, and then study this.
40:11
Because this picture can tell you a lot about the structure.
40:16
And we normally always do that in the projects. And also ask the students to create this for their class projects.
40:24
So so far we talked about earthquake and modal response. But mode response is also important for a lot of other things. If you are going to put base isolation tuning, modal response is important. You mass dampers model response is model damage detection. I will talk about that is important. Monitoring this truck health monitoring modal response is important when engineering a very big relationship between the initial periods and the
40:53
the event forces generated, specially the crosswind response is very important. So event in tuning of the event by mass dampers or whichever we attach is very much related to the initial period. So this is some of the things other than that we can do. the spectrum analysis, which is a derivative or is derived from the modal response. And it's one step next to that. So let's spend a little bit time on that. So from the ground motion equation that I showed you earlier
41:25
for a single degree of freedom or this system, we can actually compute. Now the response that is the displacement for a given damping given mass given stiffness. For those three things given, we can for a given ground motion acceleration, we can determine the displacement and its derivatives. And that's the thing. So if we do that let's say that there's a ground motion at the top. We can then use the damping ratio. Let's say a certain damping ratio. We put that in the equation.
41:57
We put the mass, we put the stiffness and we can get that. And from then we change the damping ratio change initial period frequency. And then we get a second one. We change the frequency again and then we get the third one. And then we pick the maximum ordinate of each of the d. The displacement from each of these different initial period, computations. And we plot them on the graph and that becomes our response spectrum. I think everybody knows that. So that response spectrum becomes our key
42:28
to determine the structure response. But let me mention this, Professor Wilson, who had developed this one when I met him a few a couple of years ago, and he has said it many times in his blogs that he started this long time ago. And he says that I regret starting the response spectrum method, at that time, because it has so many, sort of inaccuracies or so many conceptual, you know, sort of compression of the things that it becomes. It's always used a lot, but there's a lot of debate
43:01
about the relevance of this in the current day and state. But anyway, assuming that we still use it, we we can go ahead.
43:09
So for the ground motion, for each ground motion we can generate, response spectrum like this. For example, this is a this is a different or different intensity. Different. We can generate this one for the maximum credible earthquake. The design this is earthquake, surface level earthquake. And those response spectrum then can be used back into the analysis. So that's the story here. And the second spectrum is affected by the soil type of soil. Because the ground motion from the rock
43:42
coming all the way to structure is heavily influenced by that. And in Bangkok, recent earthquake has shown that this is a very important, magnification or that we have to consider.
43:55
Then think visual, because the default spectrum is derived from a damping ratio and initial period as the primary input. Because mass and stiffness being constant, these two can change. And you can see here that the change of the damping ratio significantly changes the response spectrum. So an assumed wrong damping can heavily underestimate or overestimate the response. In fact, one of the thing that, you know, a lot of researchers are saying that this 5% damping that we assume is too high.
44:29
Real structures, especially the new ones, have much less damping, maybe 2%, maybe 2.5, maybe 1%. So if that is the case, then you can see that the response spectrum that we use for to compute the forces will be far less than the actual response of the structure for a for a low damping. And the the difference is very big, as you can see from here, from 2 to 5 and 5 to 10, from 2 to 5, the error is too big. So if we assume 5% and the structure
44:59
only have 2%, the response experience by the structure, especially at a longer period, will be very big difference. So this is this is something that has been observed and it's now being sort of focused a lot, especially since the recent collapse of the building in Bangkok. Okay. So that brings me to the higher more effects, the higher modes that, tall buildings have. They behave very differently than the first modes.
45:28
Typically, most of the design methods and design procedures are based on the first mode response. But if you go to a tabs recently, they have added a new output. The last column, which gives us the modal separation seismic response using this one spectrum analysis, by giving you factors that you can use to multiply with the individual modes and get individual mode response, rather than let the program combine the modes by some combination method. So when you separate the modes separately,
46:03
you can get the response for each mode independently. And that's how you do that. And once again, I will not go into that much detail, but more information is available on how to do that. So I've added this a few years back when the hard mode, you know, separation became important.
46:22
And from that you can now see that for building of this kind, you can separate the response for each mode. And you can see that the patterns of the forces generated story, moments, story here are very different. They are not the typical one that we normally assume. So this drift is very different for each mode, even though the some maybe didn't like that. But the individual modes give it very different story. And similarly the moment and shear are very different.
46:55
And you can see here that there are the top of the building where we assume a small shear. Typically that's how we think gets a peak, and woman also gets a peak in the middle or somewhere. So depending on the building height characteristics, the higher modes can alter the way the story moments and so restrictions and the forces on the shear wall are acting especially on shear wall. So the assumption that we can have more reinforcement
47:23
and more thickness at the base and we can reduce as we go up, actually may not be correct because we may need higher, thickness and reinforcement at some portions in the wall and upper stories.
47:36
So that's one of the observation from this, this, this. Thank you. Now higher modes once again you can see us. You change in this one. You have we have gone to mode modes that have been first three. And you can once again see the patterns are very different. Now the response vector method may be able to add those modes and give you the total response, which may be correct. But the problem starts here. When we go to the nonlinear response. Because the other factor
48:11
that we use to reduce the response,
48:15
strictly speaking, is really available in the first mode, because that's where the yielding is occurring at the base of the wall, and it's not really available at the higher modes. And hybrid modes have a much higher stiffness. So you think with that would be not there. So
48:32
with that in mind, when you compare the nonlinear response of of a structure, the higher modes are contributing a lot. And in the traditional method we reduce the total response spectrum by, reducing all the modes by the same factor. But in fact the higher modes should not be reduced by the same factor, because there's no shielding or no ductility available for them to be reduced in the same proportion, so that that leads us to believe that the actual shear in the especially the shear,
49:05
if you look at the shear in the structure, is much higher than what the, divided, you know, total response divided by OD gives us. And this is something that could be cause of a lot of difficulties in the tall buildings. So that's why many methods are being devised using these multimode analysis. But there's partial or different for each mode. So this is another subject that we can talk about sometime. Also the response spectrum and the
49:36
frequency model, the response or the frequency ratio period have a very close relationship. So this is a this is this is showing two different to this spectrum here. One is from Philippine, one is from Bangkok. And you can see here that their response spectrums are very different. The Bangkok one peaks at two seconds. Manila peaks at this. It goes to zero seconds. This is typical of a solid rock base and, and soft base basin,
50:06
type. And but the important thing is that let's say it's a tall building, 50 storey building. The first model period will be around 5.5 second for this one. And it has 50% mass mass participation. And the ordinate is very low. So it will have very low contribution from the response spectrum. The higher mode, the second mode is about 2.5 second 30% mass participation. It will attract a very large force, even though the percentage of the mass
50:33
is less, mass participation less, but the response spectrum ordinate is high, so you get higher and then 10%, but the ordinate is much higher. And remember I said these two modes may actually not hurt. So they should not be divided by art. So their contribution becomes much more significant in that way.
50:52
For for a 20 storey building, 35 storey building, the Bangkok response is actually very, very critical because it peaks at that point. So the first mode will give a higher than the higher mode. So the relationship between the the response spectrum of the the location and the and the the ground motion that derived it, and the nature period is very critical. And this has been shown recently in the Bangkok earthquake. That is the data that we got from Professor Benton's and,
51:26
on, so this one is shows clearly that the response spectrum from the earthquake was very different. And it sort of kind of remained peaked at this range of the. So every building, any building that has an initial period between that was heavily influenced by this response spectrum from this particular ground motion, very long flat area in the middle
51:51
and very small in the two story, one story range. So what are the things that effect the model response? what response is influenced by, of course, mass and stiffness? Those are the only two things that can influence it the structure, the attachment and the occupants, the influence, the mass and the stiffness will be influenced by the tower, podium, basement, foundation and soil. So all these stiffness contributions will be needed.
52:17
And then as you can, if you look at the stiffness, stiffness is contributed by the material, by cross-section, by members and by foundation. So we need to make sure that we include all this stiffness correctly. And many of these differences are actually non-linear because after the damage occurs they become the change which means that the nature period will change when this damage occurs. Such new building after few few years, after an earthquake, after a rainstorm,
52:45
the stiffness will change because of the damage. So all of that can occur. And so we need to be able to determine the initial period of a structure at a given time. At a given situation.
52:59
And that brings us to structure monitoring and model properties. So which means that over the period over the life of the structure, we should monitor the change of the initial period of the building.
53:12
And there are a lot of ways to measure that, because this will not be the same, that we assume the damping will change and stiffness will change. So the whole response of the structure can change. So we can see key concept that we have to measure the frequency of the structure because it changes over time. And then we need a long term plan to measure them. And that that's why in almost all major structures these days, bridges, especially tall buildings, are those the sensors are put in
53:43
to measure the natural vibrations and acceleration and so on. And stresses and strains.
53:49
And this is very similar to going back to the health, metaphor. This is a stress test I have. Young people do not need to do that. But people of my age, when you go to the hospital, they ask us to stand on the, on the treadmill. And then they attach a lot of sensors and they measure the change in the heart rate and the pattern as you run or walk. And then they take you to the peak as fine as you can go until you cannot take it anymore. Then you bring it. You bring you back.
54:18
So they say, okay, how much stress the body will take. And then they tell you, okay, you have ready for marathon or not ready for marathon and so on. And same is true for structures. If you put just put the sensors, we can monitor the same way. And we can always know where the structure is going and how it will respond when it is stressed.
54:39
So there's good frequencies like I said is a pulse. So we need to measure the pulse as we are measuring the pulse of the human being by trackers and by health things. So it's just and we can do that by measuring. There are many ways to measure the, the, the initial periods of the structure. I don't have the time to do that. We'll discuss that some other time. We can detect the damage by change of the metric period. One of the most common way to detect the damage is by measuring the initial
55:07
period today, and then after one year and after three years and so on. So if we measure it, if there is a change in the period and the mass hasn't changed, that means stiffness has changed. Stiffness has changed because damage has occurred.
55:21
And this is a model that we propose that we can link the buildings to our sensor system and to the whole cloud system. It's you know, we proposed that in Philippines a while ago because they have sensors, acceleration sensors in each building. So we did some studies, got some data. I'm just going to go through quickly to show you what's possible. So after that you can get the data from the site. You can measure the initial periods. You can plot all these things.
55:45
You can get all the data from that. You can link it back to the x and y direction pre damage and post damage, post earthquake damage. You can determine those numbers and you can say okay this must aggregate degradation or damages occurred because in metric period has changed. So it's up to that. You can see where it happened and you can do the whole nonlinear stuff. So now which means full cycle is possible by putting the sensors by by recording the next period, measuring it over time,
56:12
measuring it after an earthquake, having the model we can complete the whole evaluation immediately without checking the building. We can see what will happen. Now let's wrap up then.
56:26
So as I was saying, fitness trackers model analysis, they are similar.
56:32
You have not. Everybody's so concerned about the fitness. We should be equally concerned about the model analysis and structural response. So let's put some trackers on the structure, either physical or even metaphorical. You can say it by looking at these properties. And this is a kind of a one on one correlation between what the tracker is giving about the human health and what the model response can give us about the structure health.
56:58
Once again, I leave it for you to read it when we send the slides to you, because it's just saying that there is a kind of a rough correlation between almost everything that the tracker reads. And the model response similarly, can help us find.
57:13
So let us conclude this by saying that estimating, understanding and including dynamic response from this in in the design is a key to the performance of structures water, rain and earthquake, and also for vibration through vibration, walking and all that kind of stuff. Even blast. But it all starts with the model analysis. So I think with that we come to the end. Thank you so much everyone,
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