Okay. We are live now. So welcome everyone. So welcome to this series of Latin American webinars on physics. Today we have a super interesting seminar. We have a very special one, because it's not about highly physics or as the particles of physics. So we have a very special webinar that I told you. It will be about the modeling COVID-19 on the fly by Professor Roberto Cankel. So Roberto is from the University, from the University of Sao Paulo in Brazil. And so thank you very much Roberto for being here with us and sharing all your expertise. So thank you for this kind invitation. Thank you for watching me actually. So I know the community of the usual attended seminars are very high energy and astroparticle physics. But this will be a low energy down to earth seminar. So just to make, just to introduce myself, I work on what people usually call complex systems or whatever. Nonlinear dynamics with multiple agents and so on and so on and so on. And I work with applications and biological systems and mainly in epidemiology and ecology. So that's the thing I do. I'm over the last 10 or 15 years. Before that I used to be a more theoretical guy. Now I'm more applied theory. So very good. So I will share my screen so that you see my presentation. Let me see. There's always a share screen. Okay, screen share. And here we go. So this seminar will be about modeling coronavirus epidemics, COVID-19 on the fly. On the fly means just as it is happening. And this is very different from doing theoretical work. It is something completely crazy. So the main thing, let's see how all of this works. So say you are at the beginning of an epidemic. And this is an epidemic. It's a new emergent disease. What does it mean? It's a disease that no one is immune. Everybody is susceptible of getting this disease, in the case of COVID-19. But anyway, it would be the same for any emergent disease. So that's in the jargon in epidemiologists that a new emergent disease. So every epidemic of a new disease starts with an exponential growth. So that means that you have a completely susceptible population. Everybody is, let's say, able to catch the disease. I mean, to be infected. And there's a certain number of secondary cases given a primary case. And this number at the beginning of the epidemics is usually constant. So at the very beginning of an epidemic, you could say, OK, I want to model epidemics. And I will talk about that in the second part. So you will want to write down, for instance, equations or models that will explain you all the epidemic curve. The epidemic curve is the number of people that are infected over the time, like this one. So nice. You would like to explain all the epidemic curve. But when you are at the very beginning of the epidemic, you just don't see the epidemic curve. You just see some cases. And this is well approximated by an exponential. So no need of big models at this point. It's very simple. Just an exponential. So what can we do with that? So if you try to take a big model and just fit the model with the first point of your epidemic, you will get an enormous amount of uncertainty. There's no point of doing that at this point of time. So the thing is, at the very beginning, what are the problems that a modular would face? The first thing is that the data is very noisy. Noisy means you don't know, for instance, the date of the first case. You would say, well, OK, I go to the newspaper and say that the first case was they, I don't know, 20-something of February in Brazil at 25 of February, I think. So that's the first case. But is it the relevant case? Because the first cases are all of them important. Somebody coming from abroad bringing the disease. If this person didn't multiply, didn't transmit, it is the first case, but it's irrelevant for the epidemic, actually. So the first uncertainty you have is you don't know the first case. When did your epidemic start? Unknown exact date. More or less, you know it, but so you don't know when the epidemic actually started because of this phenomenon of when really the replication of cases started. So at the first moment, what you try to do is just to characterize what is happening. And you can do this by calculating two numbers. One is the doubling time, which means how long does it take to double the number of cases? So if you have an exponential growth, the doubling time is constant. I mean, constant over time. It's just a geometric series. And so you could try to calculate that and give a characterization of this part. This is the initial part of the... And then another interesting number is the effective reproductive number. It's the number of secondary cases given a primary case, which you can calculate from the doubling time plus some other characteristics of the disease, which I won't go into detail. These are the first things that you can do. And so that's what you do.  Just put it on an exponential logarithmic plot. You got a line and you discover that... And you discover that the number of cases more or less doubles every two and a half or three days. That's everywhere. More or less everywhere in the world except for very special cases like Korea or Taiwan. And so that's the easy part. You see that obviously the points don't lie exactly on a line. You have to do some fitting. You can do this, you'd say least squares or whatever. You can do that better. There are people from statistics that will do this in a very sophisticated way, supposing that you have a personal process behind of this and so on. You can do this in many ways. And you can fit the best line and you can get the even arrow bars and whatever. And then you can also say, okay, we are at the beginning of the epidemic. And the number of people that are infected is very, very small compared to the total population. So we will be for some time at least in the exponential phase. So I can extend this into the future and I can make projections which will be valid for say a week or 10 days or something like that. So you can do this and you actually do it. So the doubling time is important when you are at the very beginning of the epidemic. It's characterized epidemic. So for COVID it's two and a half days or three days more or less. There are lots of uncertainties, but more or less it's like that. So this is the doubling time at the beginning of the epidemic. This has been true for Europe, US, Brazil and so on, Argentina and whatever. So that's more or less characteristic of the COVID-19. And you can also calculate the reproductive number which is the number of secondary infections created by a primary infection. And in Brazil this is a line between two and three. And more or less everywhere in the world and in China it's like 2.7 or something. So this is the characterization of the epidemic at the very first stages. So a way of keeping track of the epidemic at the beginning of the epidemic would be just calculating the doubling time like locally. I have today and I use the last let's say it's five days or seven days and I calculate the doubling time. So I go on and there and so if I'm a really exponential phase this doubling time doesn't depend on time. I mean it's constant, but well there are many in most countries there are isolation measures. People are staying at home and so on. So you expect that the doubling time increases means that the epidemic is going slower and slower. So you can do this and you can follow the doubling time over the time. So here you have an example of what this means. This has been made like a month ago which was March 18. Brazil was only these two points at that time. But imagine that there are countries that are clearly doing something which is Korea in this case which means that the doubling time started somewhere like two or three here. And it increases and I mean I don't know here it's a logarithmic thing but 30 days or something so it's very slow evolution of this case. So that's good. On the other hand you have this country here which is Thailand which is doing very bad. So the doubling time is decreasing means that the pace of the epidemic is speeding up. So that's a way to compare things you usually put today and then you look for some days ago and look how this behaves. So you expect that once you introduce policies like social distancing or contact tracing or whatever things that you do to have the epidemic slowing down you expect to see this on an upload like that with a doubling time that's increasing. This would be a very effective way of keeping track. I'm not talking about modeling at this point. I'm just talking about how to keep track of what is happening. So if this is happening you look at that. So what could go wrong with this? It seems perfect. There are so many things that can go wrong and that's when you as a theoretical physics you at a certain point you want to give up. I can cope with that kind of thing and what kind of thing is. So you don't know the number of cases. At the very beginning you have a certain number of cases being notified every day. Every day there's something that it's in the website of your counter and saying that the number of cases is X. That's it.  So but what happens is that when the number of cases increases you hit barriers and the main barriers. So one barrier is what people call sub notification. Sub notification means not not define. Not keeping track of cases that are potentially positive. So in Brazil this means from March 20 Brazil is only testing people that are in severe cases. So mild cases are not even tested. So these people don't leave a trace on the statistic. Nothing. Anything. There's nothing there. So this is sub sub notification. You only have the severe cases but okay. Then you say okay I have to severe cases like let me follow the severe crazy let me keep track of the severe cases.  Which means that you have a limited number of tests that you can perform per day. So which which means what aren't there sufficient testing tests. Well sometimes you have the test but you don't have the people to perform the test. And then after the people you need machines and you need all the things in the labs and so on. So, there's a maximum ability to perform tests. You cannot perform more than a certain point. So at the very beginning this is not a problem, because you have a constant flux of people of tests going going into the your system your system is not overflow and you get a constant output. But now you have a certain bottleneck here, and a lot of people a lot of tests coming here and through your bottleneck you have only a constant number of outputs. That's a problem. That's a real problem.  Okay, so you're doubling time. That doesn't mean you don't know anymore. And then still, once you say okay I have now I hired more people like I bought more tests and everything. And still is a lag between the results of the tests and typing them into the database. So this is really down down to real world things and and you have to cope with that. If you want to do something relevant for the for informing people that take decisions. Okay, so and all these legs between for instance legs between having the result and typing them into the system are not constant. Maybe just on Sunday people don't work. They don't have anything on on Sunday, maybe a better take a moving average or whatever so you need a lot of statistical machinery to cope with this kind of problem. And so, most importantly is if a lab can only test a certain fixed number of cases. It means that the number of confirmed cases of COVID-19 or whatever you're talking about but take COVID-19 the number of confirmed cases will not increase exponentially it will increase linearly if I can only put out say 100 tests per day. And I have a proportion of certain number say 80% is positive COVID I will get 80 positive tests of COVID-19 per day which is linear increase. However, the epidemic is actually probably increasing still exponentially so there's no relation between confirmed cases and actual actual number of cases because you have a bottleneck with your. With your testing apparatus. So that's, that's really, really, really the end of the war. Okay, so you cannot the thing which seems so so nice that's keeping track of the state of the epidemic if, if our measures of our isolation policies are doing okay or not. Just keeping track of the doubling time doesn't work like that. And I know that this is really problematic because we have a site we have a we have a website. So, when I say we there's a, it's a bunch of people that work together. And, and we had this doubling times and, and at the very beginning and looking at that and lots of journalists or whatever. Asking us how is the thing going on so on and a certain point. We saw an enormous increase in the doubling time as if the epidemics were just going well, I mean, if the effects of the isolation measures were super super super strong. And just look at that and well, it's not true. It's just because we are not testing enough. So the bottleneck of having a number of tests, very important, and, and spoils your analysis. So, very big problem. So what can you do in this case. So depends, depends. The first thing is what data you have access. So if you have access to honor to the data that is usually made public by most of the government, which is the number of confirmed cases per day, you cannot do much. But just just a minute. But sometimes you can be lucky enough or you can be smart enough to get access to more detailed data. So if you get access to more detailed data, not only the number of confirmed case. But you can try to do the now casting of cases. So now casting of cases is the following. I had, I had need to have access to a full data of cases, which are suspected cases. I know, when is the data of the first symptom. I know the day the test has been performing. How long does it take to have the result of the test. How long does it take to have the test on the database and so on. I have a statistics of that. I can use the present situation and using the knowledge that there are set of number of people that are a number of tests that are being performed and so on. I can project from this from the past to the present and estimate the number of cases I have right now. So to do this, you need more specific data as the public data means that you have to work with authority. And this is very difficult. This is, well depends on the country and depends on your position and whatever. So in Brazil, my group and the people that have been working with me, we have access only to the municipality of Sao Paulo, not the state Sao Paulo, not not to speak of Brazil, the country. But we have access to data of the municipality of Sao Paulo. You have a spreadsheet where you have each line as a case and this data has been anonymous. They just took out a name and then the occasion of the person. But it has all the dates of all occurrences like first symptoms, the person went to the doctor, the person got tested and so on and so on. And so you can do some now casting of data. So now the casting of data means this. You see, that's what it's in Portuguese, but I think you will understand it. So that's the absurd number of cases here, which is actually looking like it's leveling. It's not increasing very much. But here you have the now casting of the day. How many cases we have right now? Well, with some simple model, you can hear in the case we have a logistic model, but you can have access to financial model and so on. So that's not the worst case scenario. It's a relatively mild scenario, which is adjusting a logistic curve here, which is an S-shaped curve and to the observed data. So you can do some projection for some days here. And you could do it with other models instead of the logistic, you could have some exponential model and it would give you the worst case scenario. So doing now casting allows you to have a better view of what is actually happening. And you are not being, I mean, dumped. You are not being fooled by the fact that the number of observed cases is like becoming constant. It's becoming constant because you don't have the necessary number of tests. But knowing what is coming on in line for your test, maybe you can get a better estimate of what is happening right now. You see there are error bars. All of this is a lot of statistics. It's Bayesian statistics and so on and a lot of things behind that. So now casting of days is the way to keep track of what is happening right now. And just to know the situation, right? We are not talking about models at this point. So next, okay, looking ahead, now you say, okay, you have the now casting and you have an idea of how many cases you have. So what will happen next? So models in epidemiology are all of them based on all of them, maybe 90% of them are based on dividing the population into classes, susceptible individuals, infected individuals, recovered individuals, exposed individuals, hospitalized people, infected but not infectious, infectious but not symptomatic, whatever you want, lots of possibilities. So, and you have fluxes between these classes, which will be connected to contact rates between the classes. So I'm not writing down the equation for that and it's no point here to analyze equations. So you divide the populations and so what kind of models do people in epidemiology work with? So what I like most is differential equation is deterministic equations for the number of people in each class, susceptible, infected, recovered or whatever other classes you have in your model. So this is a nice way to do the things. Then you could also do stochastic dynamics, suppose you have a marker process and you go on and you have like reactions, okay, and you do stochastic dynamics and you could also do individual based modeling. Individual based modeling means that you have individuals that can be in any of these classes and then they do something, they may move around, they may interact or whatever depends on the model. And then you see how the epidemic spreads in the population. So, what we are working on right now is it's the Brazilian version of what it's called COMO model, COMO is construction of models, which is based at the University of Oxford, and which has all these classes. I don't want you to, I mean, I don't expect you to now to look at all the classes you have infected with no symptoms, mild symptoms, hospitalized and then lots of things can happen if you need hospitalization, maybe you'll find a bed for you, but maybe you don't find, maybe you need an ICU, but you have an ICU or don't you have an ICU and so on. And there are probabilities of all from going from one class to the other classes, if you want rates or probabilities, if you want, and then you get, at a certain point you get uninfected or dead. And most people now take for granted that most models take that immunity is there, you are immune after having the symptom. So this is a big model, and this generates this kind of kind of a plot. So don't pay too much attention on the numbers because this is a preliminary results, this has been done doing, I mean, this is the result of two days ago, so it's not still something at the level of being published and so on. But the important thing about models is, and this is very, very different from what you have usually in, what you have usually in, with systems and physics, models are not made to make really precise predictions. Because why? Why? Because uncertainties are enormous. As I told you, you don't know even your initial condition, which is the number of infected people, you don't know this, you have to estimate this and you have an enormous error bars, and you know, these error bars go on multiplying and so on, and it's a hell, and there are a lot of parameters and so on. So what do you do with the models? So what you do with the models is to build scenarios, and what's the scenario? You have a baseline scenario, which means that you don't do anything. Just let the epidemic go on, epidemic go on. What will happen? Like in Brazil it would be like almost two million of people dead. But then you know that even the most crazy government in the world will not do anything.  So there are interventions, and you can model the interventions in your model and see how effective they will be. So say intervention, just making people wash their hands a lot of time. This has a very small effect, but you can try to quantify this into a model. Then you say, OK, isolation, OK, voluntary isolation, or mandatory isolation, then you say quarantine, or lockdown, full lockdown. So you can trace scenarios and try to see what will happen according to the assumptions you made and for how long you do this and this, and so on and so on. So I don't have results for Brazil at this point. I'm working on that actually, but I think one of the only groups that work on that and so these results are not final, but there are results for other groups in the war. And this is a paper from yesterday. So it's just an idea how this thing is evolving a month ago as eternity. So this is a paper from a Harvard group from Mark Lipzig, which is one of the leading at the emergency ward. So, and he was looking at scenarios, not for the new future like one month or two, but what will be the issue of this, all of this, and so there are two main factors which are completely anonymous in order to make more long time scenarios. One factor is, is this seasonal or not? No idea. Is it depending on temperature? We don't know. If you look at the archives, there's a specific archive for this kind of thing, which is not the archive, it's called math archive, and there's a section of epidemiology, which has a lot of paper, more than 10 papers per day on COVID. And there are people saying, yeah, temperature has effect and other people say, temperature has no effect. So we don't know the seasonal behavior of the epidemic of this disease. And the second thing, we don't know anything about immunity. Is immunity going to be long term? So we know something about immunity, which means that if you're at COVID now, you don't have it next day. But how long will this last? Can I have a lifetime immunity, lifelong immunity? A lifetime immunity will be nice, but maybe you have a veining immunity, immunity over some time. So impossible to know. So you have to build up scenarios and try to figure out what is happening, and then as the epidemic progresses, you will get more and more results. But maybe we'll select which scenario you are. So here, just as an example, this paper appeared yesterday in Science. So this is a long term thing, you see, until the year 2025, five years. What happens is, so this is taking into account, you have seasonality. So this is with US data, and it's usually assumed for temporary climate. So you have strong seasonality, and in winter you have more infections than in the summer, which may not be the case in many of our countries, not in America. So this has to be adapted. But what you see is, here are several, there's a big model, differential equation model. And you see that if you have immunity for 40 weeks, which is a very short time, it's not even one year. You will have recurrent epidemic, just like flu, which is really a problem, a real problem. If you have longer time epidemic, for instance, if you have 104 weeks immunity, you will have biannual. Outbreaks every two years. And obviously, if you have complete immunity sometimes, at a certain point, the thing will go away. And this scenario here is something intermediate, where you have small outbreaks all the time. So with seasonality and final duration of immunity, you can have several scenarios, but the main thing is you expect this to come back unless you have lifetime immunity. So bad news. Obviously, if there's a vaccine, then everything changes. We hope to have a vaccine actually, because there's no reason that we should be very pessimistic for a vaccine. But in the absence of a vaccine, you see the human having partial immunity and not lifelong immunity will bring back the epidemic. Now I think that these are the same people have studied. Yes. Okay, let's see there's no seasonality just to make the things easier and let's do one time interventions. It's a long time scenario. It's an intermediate time scenario. What happens is we only have one time intervention, which is, is the temptations of many of governments and, and, and, well, in Latin America, and not only there, so you say, okay, enough with enough with interventions. It should be okay. So what you see here in blue is the time of interventions we have. And, and the many curves you see is the fact of intervention so no intervention and how effective the interventions are. So you see that the only optimistic case is to maintain an intervention for all time. It's obviously not possible. In this case, if the intervention is strong enough you have a curve here with a very small mortality that these are mortality curve here. So you see the secondary peaks here in this cases. In all cases you have secondary ways. Okay.  What are our best, I mean, best hopes, it's, it's vaccination, which can happen. And well, and something which is not shown here in this, in this thing is to have a long time testing and isolation, which would be more or less corresponding to this case here. Okay. So then the ventures doesn't mean necessarily that you are at home all the time for the rest of your life. The ventures can be very effective. Also, if people are tested intensively test everybody like that, and isolate people that the tests that test the positive, and everybody that has been in contact with these people over the last week. For instance, this would generate a curve of low mortality like the one that we see here. So this result from the Harvard group is means that just short time interventions will generate second wave and that's it. You need long time interventions and this can be of several kinds. I'm not saying you need to stay at home forever, but you need to have some way of having having interventions over the time and the one way of having interventions. Testing a large majority of the population and isolate people that test the positive. So that's it. And just so there are a lot of people that have been working on several aspects. So you see the number of names here is impossible to read. There are people from physics from biology from computation and medicine and working and we have we have an initiative. It's called the COVID 19 Brazil observatory. And so these people are doing a lot of things which I didn't mention everything that people are doing. And so that's it. And so thank you. Thank you for your attention. Thank you very much Roberto for the super nice talk. Are there problems? Yeah, the virus is a problem at the moment. I do have one question. Problems that are a lot of problems. Yeah, problems we do have Nicholas. Thank you for your webinar. I have a question that maybe you can comment without going into many technicalities, but I was wondering like, how should we read those error bars? Like, more or less like, for me that I'm not an expert modeling or something. What type of things are keeping into account whenever you have these error bars when you present something. Two things. Two things. Two main things. First, your parameters are not precise. Okay, you have a model as parameters and activity and time to time of recovery and so on. All of this is our probability distributions not not not really constant. So this generates the four error bars probability distribution. Second and more difficult than just just looking for the sensitivity to your parameters is the fact that you don't know your state your initial state is unknown. And that that's a problem. That's a problem you don't know your initial state so if you if you go for a problem for things with differential equations or even other other kind of models. So you have two sources. One, you don't know the parameters. Second, you don't know the initial state. I mean, it's not that you don't know anything about it. You know, but you have a probability distribution for your initial state also. So, you have, you have this two sources are of of errors. They are a problem. And that's why it's so this is a little bit like meteorology. Okay, you want to it's it's not it's not a. It's not a key it exists in that system with time dependent precision and with time dependency being whatever you want. And, and the problem is, at a certain point you make prediction with error bars that says that you could have a say 1000 or a million of people infected, which means nothing, nothing. This is not informative for any, any people, any person in it at the government or trying to take a decision and the person wants to know, will I have a lot of people dead or not. If I say you have 10 people that they say okay, that's normal. You say I have a million people that it's very important for this person. And if your model just says between 10 and one million in the model says nothing. I don't have questions. I actually have one. If you, Robert, if you could go back to your slide. Okay. Share screen. Okay. Go back where maybe to slide back. I think slide 12. I don't know the number of the that one. So, in your first scenario, the, the one course point to 40 weeks. There's a case where the number of cases increases with time.  Yes, there, there are, there are cases that can increase again because there's a replenishment of susceptibles due to the birth of people. That are given birth. Okay, that's like the worst case scenario, right? This is the scenario. Yeah, that's, I don't think I mean this, this is a little bit the frightening scenario in the sense that this is pretty much like, like, like influenza like like flu. It's, it's not that it's different in the sense that you had, you don't have long time immunity because because the, the, the, the virus has such a high, high mutation rate that you everybody actually does not have immunity for more than one year. So then you get this kind of scenario. Okay, so this would be the flu scenario like a thing for, for COVID but you know, COVID is 20 or 30 times more deadly than, than flu, which would be a disaster. But I mean, I'm not saying that this is necessarily the only scenario. I mean, this is one possible scenario but not necessarily the one that will happen. And we, but, but we are really in the dark about immunity. Nobody knows because we have this epidemic for three months. So people don't know if the people will continue to be immune or not. No idea. No idea though. Okay, thanks. So there's a question from public, I think from Diego Restrepo. So he's asking in practice how well your models have been predicting the evolution of the epidemic in Brazil. So that's a point that we, we have not completed the work that that's why I didn't show you the full things. Okay. This kind of model. I didn't show you the full results. Because, you know, with this kind of model with this in certainties and parameters and initial conditions so fitting model is difficult. So it's easy to say my, my model is a good model because my predictions are consistent with the, with, with what to observe it because of the error bars, but well this is, your error bars are very, very, very large and anything is consistent. So, we're still working on that on it's not our models and I mean, all models are more like that, more or less like that. Okay. And so we don't have the results at this point. And then you have to ask yourself for which locality are you doing the model because we are now working for the municipality of some power. So the first results of the model are pretty okay. But let's see because I mean we can set the model to what is known now but let's see what the predictions will be confirmed or not. So we need some months and months are true to know if the predictions will be confirmed and so on. And this is for the specific case of the municipality of some public why why not why not apply this to anywhere, because if you apply it on into the confirmed case, which is the public data, which you have for anywhere in Brazil or any state in Brazil, this will give you nothing because the confirmed cases are not the representative of the number of actual cases so you need the now casting procedure in order to initialize your model. So and we only have this first citizen Tom. Okay, thanks. Another question from the audience, this time from Omar Suarez is asking how our environment and the environmental variables included in the simulation of these models. So they are not included explicitly. But you know when you do the fitting variables like the transmissibility of the disease or the transmissibility could depend on the environment. Okay. And so, but transmissibility is fitted, because you don't have the absolute value of that. So you are fitting the environmental variable in this case. Now, for the model that I showed from the Harvard School, from Harvard Medical School, the last model. They assume seasonality, which is an environmental bear. But I mean, that's like, like the scenario like, let's say it's seasonal. Nobody knows because we didn't have even one season. Okay. You cannot know with certainty you can try to infer from other diseases. Like there are all other kinds of coronavirus and they're called beta coronaviruses, which circulated and around and so on, which has some seasonality, not very strong. So but just this coronavirus will it be seasonal or not. No way to know. No way to know because you don't have any not even one season you have three months. For three months that that's it. So you don't know the effect of the environment. Yeah. Maybe last question. So, first of all, very nice to talk is very explaining how how to model and all the difficulties that has, in fact, to try to make any prediction on the in this case, but I was wondering in the because this is a stuff that most of the government are using like an excuse to why not to make a full down, because it's in that if everybody make a lockdown, the, the, all these epidemics should pass quite fast, let's say, just like a nightly speaking. But most of the government, they all the time they use the excuse that is the because of the economical impact. Is that a way or is it a way to to try to include in the model this economic impact of to have these people out of the community because government they just based on no it's the economy so all the people has to go outside and we don't have to get infected because the economy has to continue. This is the kind of the version of many point that you gave was like this month until the Prime Minister got infected kind of they were pushing to to here in Chile sometime there is a mix opinion for the government making like with like a blind test to make politician politics to try to So, yeah. There are attempts to include economical variables into epidemiological models. It's not trivial and we have been discussing this with this consortium this co mold. And that's more variables and more unknown parameters. Yeah, I know. But what for sure. If you are at the start of an exponential growth. If you do anything. You get a million of people dead in Brazil in some, some, some weeks so so that whatever the economic car is. You have to, I mean, you have to take action. Now, serious debate, which is not obviously the case of Brazil because you know, situations. But seriously, but for instance in Europe like in Germany or follow model as the German situation is about obviously how do we get out of this thing. Okay, so the population will still be susceptible because we are in home we didn't have the infection. I'm still susceptible and I think just lose out the all I mean just lose the the the the interventions I go how I go on the streets and so on. The buyers will say okay nice new, new people for me. Okay, so we'll be a mess and will be just really bad. Okay, so how how should we go out of the thing. Well, the favorite policies that I've been discussed it is like the Korean way the Korean way is you just test people. But I mean this means intensive testing, lots of tests, putting up a system of testing people. It's a kind of very intensive thing. And then you have and then also issuing immunity cars like I mean if you have had that you have been tested, you will have mark on your on your car. Okay, positive, negative, positive, you are immune and so on, which would then decide what what kind of activities you would be allowed to do and so on. But it's, you know, this is this is a very complicated thing. It's not an easy thing to do to put up. And in countries like Brazil, this is seems seems crazy because you know, what the time being in Brazil if you want that we are in a soft lockdown and like in some problems, Rio de Janeiro and other cities.  Okay, but the commerce is closed so maybe you don't go out because there's nothing to do outside but you can go out. Now, in such a situation, there has been no attempt to build a way of getting out of this situation so we are stuck with the situation. We are stuck and economically this is obviously very, very, very difficult situation. But you have to build a way of getting out without having a super epidemic, because if you just say now, okay, it's over. You go out, you have an epidemic, you have again a million of people there. So that's the situation. Yeah, it's complicated. No, it's very complicated. No, please go ahead. One more question because since in my case, I live in a small city from respect with the capital of Chile that is where most of the cases are but we do have cases here locally. You know, how this, this mobilization in terms of the points that when you were presenting the different models, sIR model and so on, should be also made locally in each city, I guess, no, not because sometimes I can take the whole country like all the population is mixed together, kind of. So, yes. Actually, collaboration with some of those people that were on my list on the at the end, arguing network theory and not with the airplanes but with roads. So, to assess vulnerability of cities because maybe you are a small city which is not connected to anything, then you are not very vulnerable. If you're connected to a lot of things you are more vulnerable. So, using network theory, network metrics can be a way to access vulnerability. The second layer of analysis is which is how many hostile bad you have. So, maybe you are in a place that it's likely to be to take some time for have the epidemic. But maybe you don't have bad. So then there's the second layer of vulnerability says okay you're in that situation. There are these two things and one thing is this is the spread distribution of the spacious distribution through the road and two movement of people in a certain region so we have made this kind of analysis for the status and power and for the north eastern part of Brazil. So, you'll see the hubs and so on and where are the next cities that you're expecting to have the epidemic and so on. You can get others and all of that you can do this. And then you have the second layer of okay. There will be a certain city, but okay this city has a lot of bad. So, you don't expect to have an overwhelming problem there, but could be also the contrary. For instance, in the St. Paulo State we have discovered that some cities are very far apart from the capital, and which have no cases at all at this moment of time. Well, if nobody does anything or there should be cases because, well, some, unless you just shut down the, I mean, completely the transportation system. And then there will be a case and the case will be transmitted. It's highly effective. And then then you look at this places and I say okay but this places can be a problem because even if it will take time to get the epidemic there. It will have very bad, very high, very strong consequences because there are no, no hospital. Okay, thanks. Let me take a very last question. This one comes from Mario. He's asking about vaccination and whether they're treated as immunity in the model. I didn't understand whether there are treated as immunity in the people that got immune to the virus in the model. So, in the model we are working for, Brazil immunity is lifelong, but obviously the model is meant to model the next months. So if you want to go for a long time, thank you have to have something like the Harvard model, then you can say okay, I don't know immunity. Okay, so it is completely open, the natural immunity of people. How long will it take? Okay, then I have a vaccine. Okay, if I have a vaccine like the ones that we have for the child disease and so on, then you are okay, then you are immune and that's fine, that's problem solved. But maybe you have a vaccine with a certain percentage of effectiveness and maybe immunity conferred by the vaccine is not lifelong. So, if you look, if you talk about, for instance, vaccines today are miscellaneous.  The disease never in their life. It's fabulous. But this is also a result of many years of research from going from something which has an effectivity of say 60%, 70% and gives you immunity over five years to something which actually has a very good coverage and lifelong immunity. So there's this research to be done. It's kind of not the kind of research that we do because there's this research in microbiology and in biology. But this is likely to take time. And it's not that there will not be a vaccine. Maybe there's a vaccine next year. What kind of vaccine? We have a vaccine, lifelong immunity. How long will the immunity be built? So you will have to have tests and you know, in order to access, just to make an assessment of how long is the immunity conferred by a vaccine. And you have to follow a cohort of people over a lot of years. And why are you doing this? You are completely in the black and you don't know. Unknown. And that's the way it is. And it's not only for COVID, obviously. Okay, thank you very much, Roberto. Welcome and thank you for the opportunity to talk to you. And so one more thing. So next week we will not have a webinar, but in two weeks time we'll have a look at it in early talking, maybe about action physics, I think. So we're, we're back to, to, to hire this piece. But okay. Thank you very much, Roberto. We can create a day. Thank you for the, I mean, for the, for the same. Thank you. All the audience and for the invitation. Thank you very much. Goodbye. Bye. Hasta la vista.


Thank you, Jim. Let me at the very outset thank Terry for giving me the opportunity of being here today and Let me on behalf of all of us here and perhaps on behalf of all the people in the world who care Thank Terry and his team were brought together in this wonderful City of Marrakesh a Galaxy of global leaders and thinkers Who have devoted the last three days of their time to find solutions? Which we confront in common the grave challenges Which all of us face in common which make This world a challenged world. I'm a hopeful person by choice and by nature but all hope and all initiatives towards realizing our hopes must begin with a candid acceptance That we are living in a very troubled very challenged Conflict driven and violent world. We are living in a world Which is for a very large number of our people Neither just nor stable nor peaceful Nor a world that gives hope to future generations But what do we do? Do we put our hands up and say that all is lost? certainly not we know from Carlisle's theory of history that The story of human civilization and its progress through the ages Could be measured in terms of the contributions of genesis and great world leaders Who came for a limited Time on this earth, but to change the course of history for the better What we see today is not all gloom what we see today is a vastly Superior world in the sense of the fastest and the and the most abundant growth of material prosperity We see a world in which there is hope for the very sick people because of technological advancement We see a world Which has endless possibilities for better life for a very vast majority of our people But who has got to realize that potential? When I come to the conclusion of my presentation, I will talk about the way forward but but today Let me at this point of time Talk about the fears that I have My first and foremost fear is that The trouble spots The potentially volcanic trouble spots in this world are growing We discussed about the situation in the Middle East and North Korea Yemen Libya Afghanistan Pakistan Iran Qatar Saudi Arabia. We've discussed all these and these are these are potentially growing areas of conflict the South and East China Sea and I think These conflicts are going to exacerbate in the future Depending upon the proclivity of the rising powers to assert their expansionist programs We see this phenomena We recognize it as well, but sometimes We are diplomatic enough not to assert it in the manner that we ought to as a lawyer I'm extremely worried at the impotency of the international legal order We recently had the Hague tribunals judgment About East China territorial claims and China great power in fact in some ways a responsible power to openly defied international legal order and said it does not accept the judgment of the tribal I Asked myself the question What is the point of Repeatedly asserting that we must live by the International legal order if we have in our midst not one but several powers Who if they chose to defy the international Rules of the game cannot be held accountable for defiance of the law Therefore, I think one of the foremost challenges is for global leaders to sit together And to devise rules of engagement that would not depend upon their enforceability upon the sufferance of one or two or three powerful states and I think as somebody rightly pointed out international morality and law are Important without the backing of power Power is the only constant Which defines the contours of international relationships? This has been the theory for as long as we have read it. We have known from history that at the end of the day The week must yield what they must and the strong take what they can Is that the world we want our children to live in? No We need a just fair law-based international order and that and the establishment of that international order Is one of the foremost challenges for statesmen of the world the second point that I want to make in this regard Is that we are witnessing a Shore retreat from multilateralism President Trump's walking out of even UNESCO Leave aside the TPP or live aside the other arrangements, but all the Paris Accord, but even from UNESCO shows an utter contempt for what has been achieved thus far It is not my place to pass judgment on an individual leader of a great country But it is time for us to interrogate ourselves whether Collective judgments and democracies are always the right judgements of course Some may argue that the collectivity has spoken and Democracy is about the rule of majority and so wait. Yes. This is certainly a point of view But does the majority always get it right? It's the question that we have to ask and the terms of Engagement in the debate between right and wrong is what we all know instinctively to be right and what we all know Instinctively to be wrong or bad the second take away that We can all claim from this conference is that whether we like it or not Globalization is in retreat. Is it necessarily a bad thing? I Think the jury is out. We will have to await the verdict of history But the fact remains that globalization has not necessarily led to a just society or a just world order In an age of rights a great International lawyer professor Louis Hinkum Hinkum who has described our age as the age of rights What is it that we actually confront we confront the sad and the tragic reality of Three billion people by 2020 Having to live and subsist in less than two dollars a day Seven hundred and ninety-three million people facing hunger eleven million children dying of hunger every year and 155 million children being undernourished Globalization leading to imports from China into the US Led to job losses in America to the extent of two point four million jobs Surely Surely Americans had the right to challenge The conventional wisdom and the established order we know that drugs and human trafficking continue to be a Grave reality and all the laws in the world and all our collective resolve and resolve of leaders in the world has failed to check that menace Something is therefore not right and I would like to compliment theory for Organizing this conference because as Goeth reminded us we must proclaim from time to time what we believe in and reject what we must condemn And I think Conferences and platforms like this Or to be used to repeat what we believe to be right and to reject what we believe to be wrong That will be I think the lasting service of conferences of this service that conferences of this kind will make to humanity Fundamentalism a rise of identity politics break down of interfaith dialogue And as somebody said the failure of happy modernity are inescapable realities of the present world and my very grave Doubt about our capacity to we generate Failing democracies Troubles me a lot. I come from a country which has provided itself in the defense of human rights and In this irrevocable commitment to the fundamentals of constitutional democracy democracies But what do I see? I See my country rejecting lock stock and barrel The plea of Rohingya's for asylum in our country I can understand any government's right to reject one two five ten hundred five hundred people on grounds of Suspected involvement into a receptivity, but how can you dub a whole class of helpless people as? Potential terrorists and deny them to humanitarian relief of asylum and this is a country Associated with the Mahatma and the Buddha Who taught us the first principles of humanitarianism as the country that give to the world the slogan of Vasudeva Kutumbakam That that is that the whole world is one family and I see this happening We have a stellar record of human rights Jurisprudence in our country which has been borrowed by many many countries not only in a neighborhood But even in the advanced countries I myself fighting a case for the Rohingya's in the Supreme Court of India and on a number of issues I'm myself a public and presbyterian and I find that the rise of a dominant person perhaps in reaction to things that were not so positive does lead to the weakening of the institutions of libertarian democracy This is the lesson of history whether it is in the United States of America or India or Or in any other democracy and I think it is absolutely necessary for us to insist on Diffused power structures whether that is represented by multilateralism on the global stage or by institutions of constitutional democracy in In in democracies across the world a diffused power structure Yeah, one a diffuse power structure is something that we have to struggle for I will just take one or one minute more technology and ethics Yes, technology Has been a great saver in some respects, but technology has Defied our humanity. It is invading our humanity the right to privacy the right to dignity the right to reputation which are Absolute casualties of fake news is something that we have to be concerned about the monster of social media is threatening Our family lives it is threatening our social existence and we need to ask ourselves whether in the name of technology we can allow all of that to happen and Finally, I Have to say do we need to barter our freedom to qualify as patriots? This is about the assault on liberal democracies are we our freedom and nationalism mutually exclusive Do we need to pronounce the demise of reason to appease radical populist demagogues? Is it fair to invoke moral relativism to eliminate the testing measure of power and can we fork and can we afford to forget the caution? Sounded by James medicine that there are more instances of abridgement of freedom of the people by the gradual and silent encroachments of those in power then by violent and sudden recipients Jim I would like to thank you for giving me the opportunity, but let me conclude by what? Toin B my favorite historian Reminded us. He said civilization moves forward on the basis of challenge and response and Each age has to write its own history and Martin Luther King told us There is no such thing as tomorrow and he said We are confronted with a fierce urgency of now in the unfolding of life and history and let me tell you ladies and gentlemen We are all individually and collectively Duty-bound not once not twice, but over and over again to repeat what we Believe is right because as Dante Cautioned us the hottest place is in hell The hottest places in hell said Dante are reserved for those Who in a period of moral crisis which we face today preserved their neutrality. I thank you for your attention. Thank you very much


We have been discussing energy methods for structural stability analysis, so in the previous lecture we introduced the axioms, if we consider a system with n generalized coordinates and if we focus attention on statically loaded structures there are two axioms, first axioms is a stationary value of the total potential energy with respect to the generalized coordinates is necessary and sufficient condition for the equilibrium state of the system. This provides the condition for establishing the equilibrium, the axiom 2 helps us to examine the stability of that equilibrium, so a complete relative minimum of the total potential energy with respect to the generalized coordinates is necessary and sufficient for the stability of an equilibrium state of the system. So based on this we analyzed a few problems and basically we considered simple systems to bring out the central ideas of the analysis, so we considered the beam column problem and in this case when p equal to, for small values of p, y equal to 0 is the equilibrium position and as p is increased we find that a neighboring equilibrium position becomes possible and that we discussed as buckling due to bifurcation of equilibrium. So the load deflection path has this feature, there are infinite number of buckling loads and the lowest one obviously is of the fundamental interest, so as the load is increased at some point when p coincides with the first critical or the first buckling load the zero position becomes unstable and two new branches emerge. In another type of problem what is known as limit load buckling example for which is shown here there are two rigid lengths supported through springs as shown here and it is loaded by a force P and as the force P is increased the deflection of the system and the load increase together and at some point both of them reach a critical stage when a faraway equilibrium position becomes possible and this is known as limit load buckling. Here for we start with here at p equal to 0 this will be the deflection and as load is increased it will go the load deflection path rises and at this point it snaps and reaches this position and this is a load deflection path and at some value for certain values of P the load deflection path is multi-valued but although it is multi-valued there are certain portion of this load deflection path is not physically realizable because those positions are unstable. We also started discussing about approximate methods for stability analysis wherein we considered the expression for the V that is total strain energy as is shown here and we postulated the displacement field in terms of trial functions where A is a generalized coordinate and we evaluated V as a function of A and then applied the two axioms to get the critical load. Here we started with the total potential energy whereas in Galerkin's method we could as well start with the governing differential equation and assume the solution in this form where phi N of X are need to be admissible and we substitute this into the governing equation and we get a residue and we demand that this residue is orthogonal to the trial functions and we get a set of capital N number of equations and this leads to the equilibrium equation as shown here and for non-trivial solutions we need to do an eigenvalue analysis to determine the critical load. Now the question now that we should consider is how to generalize these two more a difficult class of problem, for example if beam is multi-span or we consider a plate with a hole with complex geometry and a hole and things like that or a building frame which carries lateral loads and as well maybe there could be vertical loads as well, how do we perform the stability analysis of this structure, for example if we consider this type of problems suppose this is the loading pattern that is acting on the structure we would like to know if the keeping the entire loading pattern the same we increase the magnitude of these loads, so at some point we ask the question can the structure lose stability, okay, so that is the type of questions we wish to answer. So we use to tackle this problem the finite element method, in the Rayleigh-Ritz and Galerkin's method we constructed trial functions which are globally valid and as we have seen in the discussion as in the discussion of vibration problems constructing trial functions which are globally valid for these type of built up structures is not possible and what we do is we divide the domain of the structure into subdomains and within a subdomain we interpolate the field variables in terms of the nodal values of the field variables and that is that finite element method and that provides a powerful alternative to tackle problems of stability analysis as well, so as we will see shortly this leads to equilibrium equation of this form where J is the stability matrix and P into J is the geometric stiffness matrix and K minus P J is the total stiffness matrix and we can either solve the response of the structure by allowing for effect of geometric stiffness or find out the critical conditions where the structure loses stability we can do a static analysis or we can do a dynamic analysis, so we will see some of this as we go along. So let's start with as we did when we discussed vibration problems Euler-Bernoulli beam problems and then we will generalize to more complicated elements, so the total potential energy for Euler-Bernoulli beam we have derived it is in this form, so we considered Euler-Bernoulli beam element carrying an axial load P it has two nodes and it has two degrees of freedom displacement and rotation at each degree of freedom and we assume the field variable displacement field in this form, where UIs are the nodal values and phi I of X are the trial functions, so as we have seen in the context of vibration problems we, you know the cubic polynomials are appropriate to handle this problem and these are the polynomials that we have used in the dealing with vibration problems, so see it's very clear there are two nodes and there are two degrees of freedom because the functional here the highest derivative is 2 therefore the nodal degrees of freedom are V and DV by DX there are two nodes so we need a polynomial with 4 coefficient that would mean we need to use a cubic polynomial. So we substitute now that into the expression for the total potential energy and we get now the strain energy or total potential energy in terms of the nodal coordinates, so these are once we substitute for this these being polynomials they can easily be evaluated and for equilibrium we have dou U by dou UJ equal to 0 and with that we get this equation and we define K IJ as EI phi double prime, phi J double prime X DX and J IJ as this, for equilibrium this must be equal to 0 for I equal to 1, 2, 3, 4 and this I can arrange in a matrix form and this K is the elastic stiffness matrix that we have seen earlier and this is a new thing J is the stability matrix and P is the axial load, J IJ if you notice is independent of elastic constants and loads on the system so that is why it called, that's why the name geometric stiffness. So we can evaluate those integrals and I get the elastic stiffness matrix this we have seen earlier and this is a new matrix that we have derived. Now to examine the critical condition we construct a Hessian and this is the Hessian is a matrix of these elements which is KJK-PJK and this is a Hessian matrix, for equilibrium I need this equation to be satisfied and for the critical state the determinant of the Hessian must be 0. So we can make few observations, K is the elastic stiffness matrix which has been derived earlier, KG is equal to PJ is the geometric stiffness matrix, it is called as a consistent geometric stiffness matrix since the shape functions which have been used in evaluating K have also been used in evaluating this matrix, the same shape functions have been used, J is called the stability matrix. Now KT which is K-PJ is called the total stiffness matrix, so it modifies the elastic stiffness matrix by contribution from the geometric stiffness term. Now KG which is P into J is independent of EI of the beam and depends upon only the length of the beam and hence the name geometric stiffness. Now we can also see the structure of U, so if we write this and expand we can see that U using KIJ notations KIJ and JIJ this can be written as half U transpose K-PJ into U, now based on this we can consider few examples so that we get an idea on how to use this, so we start with a proper cantilever, it has 1 degree of freedom namely U4 of T, U1, U2 and U3 are 0, so this is one of the simplest problem that we can think of, so for equilibrium I have 1, 2, 3 as 0 and U4 is to be determined and this is again 0, 0, 0, U4 and I get the equilibrium equation to be given by this, that would mean U4 equal to 0 is the equilibrium state, is this stable? Now you differentiate this with respect to U4 and demand that that derivative is greater than 0, so for that to happen P critical, P should be less than some number and the critical value is 30 EI by L square, exact Euler buckling load for this case is 20 point, around 20.1997 EI by L square and the buckling mode shape according to this theory is given by 5 4 of X which is shown here, so this is a very simple application of the idea of elastic stiffness and geometric stiffness matrix. We can now consider a single span simply supported beam discretized as a single element, so there are 4 nodal degrees of freedom, 2 of which will be 0, U1 and U3 will be 0, the equilibrium equation now in the, this is K into U and in U I have zeros here and this is a geometric stiffness matrix, now extracting the equations for U2 and U4 I get the equilibrium equation as this, now the critical condition is given by determinant of SCN matrix to be 0 and that leads to this condition, so this is this, finding this condition in fact is equivalent to performing an eigenvalue analysis of K and KG, that means I have, we need to perform this eigenvalue analysis, so we can do that and we get the equation to be this and from this I get lambda to be PI square by 30, I mean by substituting for lambda this number and carrying lambda will be the eigenvalue parameter, I get the modal matrix to be this and the two eigenvalues to be this, so the smallest of these eigenvalues is helpful in determining the lowest critical load and using that we arrive at the solution that critical load is 12EI by L square, whereas the exact one is 9.869EI by L square and this is the mode shape, okay, so we can see how the mode shape look like, exact one is a sine, half sine wave and the blue one is the approximate using these two cubic polynomials. So this is a model for, simple model for critical condition, simple model for establishing critical condition for a single span, how to improve upon the results, so in the first example suppose if we introduce an additional node, okay, so now what happens we have two elements and procedure for assembling matrices and transforming into global coordinates, etc. is quite similar to what we have done in dealing with vibration problem, so that story need not be repeated now. So we have two elements and the boundary conditions will be U1 and U2 are 0 here and U5 is 0, so degrees of freedom are 3, 4 and 6, so for the first element I formulate the elastic stiffness and geometric stiffness matrix, for second element I form this and now using the fact that 3, 4, 6 are the degrees of freedom we can assemble them, either you can form the A matrices for the two elements and assemble them through matrix operations, or if you are working with pen and paper you can assemble them by observations, so we get the 3 by 3 structure, reduced structure elastic stiffness matrix and reduced structure geometric stiffness matrix. For equilibrium it is KU is equal to PJU and a critical condition is this, so this is a equilibrium equation and again by introducing lambda as PL square by 30EI now I get a 3 by 3 eigenvalue problem associated with the 3 by 3 matrix and the minimum eigenvalue if you calculate will turn out to be 0.1726 from which I get P critical as this, exact we have seen to be this, and now we have improved on the earlier solution, note that in this case I have taken element length to be L, whereas in the first example the span of the beam was taken as L, therefore we need to do a slight adjustment in comparing the results. Similarly for a simply supported beam if we now improve upon accuracy we have now degrees of freedom will be U2, U3, U4 and U6, so 2, 3, 4, 6 are the degrees of freedom, so once we assemble the structure stiffness matrix, elastic stiffness matrix and geometric stiffness matrix we get these two matrices, and hence the eigenvalue problem to be solved is now 4 by 4 because there are 4 degrees of freedom and lambda which is now this parameter P by 3EI, if I solve this problem I get these are the 4 eigenvalues and similarly by using the smallest of these eigenvalues the P critical using this method turns out to be 9.438EI by L square, L is 2L here, and this compares much better with the exact solution because we have increased the degree of freedom. Now how do we tackle problems such as this, for example there is an axial load P here and an axial load 3P here, these are externally applied, so what we do is first we analyze the structure for finding the membrane forces which are the axial thrust here, so by simple logic we can see that the reaction at this end will be 4P and the axial thrust diagram will be this, so again I will divide this beam into two elements and while forming the geometric stiffness for element 1 I will use 4P as axial force, whereas when I form the geometric stiffness for the second element P will be the axial load, so that is the only difference, so for first element the geometric stiffness is 4P by 30L, for second element 1 it is P by 30L, all other details remain the same as far as elastic stiffness matrix is concerned, so we get the eigenvalue problem like this and lambda is again obtained as this and P critical in this case turns out to be this. Now what we are doing here is if we see when I increase P the two loads are increased in same proportion, for example if this load was Q and this was P then what I will do is I will increase P by alpha P and Q also by alpha Q, that means as I increase the load all the loads increase by the same factor, there could be transverse loads and other loads as well, so I will find out the value of alpha at which critical condition is reached and that value of alpha is known as load factor, so in problems of elastic stability analysis we aim to find the load factor, by that what we mean is for a given structural configuration carrying certain specially distributed load keeping the loading pattern the same I will increase the entire loading pattern by a constant amount, so that is alpha, so the question we ask is how far alpha should be increased before the structure reaches a critical state, so this example is illustrative of that procedure that we will follow. Now this is the axial thrust diagram of the structure when it is on the verge of buckling, so this is a critical loading condition as far as axial loads are concerned, so some more examples suppose we have encastered beams I take two elements and introduce a node at the mid-span and here again we can perform the analysis and I get 40EI by 2L square as against 39.4784, so these are some simple examples that you can run through and convince yourself that these steps are correct. Now one more example, suppose there is a cantilever beam which is mounted on a spring and it is load P is applied, so we discretize there are two elements now, the first element is a beam and the second element is a spring and the first element is discretized using single element with two nodes and U1 and U2 are 0 and I have basically degrees of freedom 3 and 4, 3 is also the degree of freedom shared by K, so I construct the elastic stiffness matrix and the geometric stiffness matrix and when we assemble I use a notation alpha as KL cube by EI and I get this equation, so clearly the critical condition, critical loads will be function of K, so that parametric study has been done when alpha equal to 0 it becomes a cantilever beam and when alpha tends to infinity it becomes a property cantilever, so for K equal to 0 and K equal to infinity we know the exact solutions, so here when alpha equal to 0 this is the exact solution and when alpha tends to infinity this is exact solution, so now as alpha is varied for alpha equal to 0 I get to this factor as 2.486 that has to be compared with 2.4674 and as this number is increased this model converges to 29.993 this factor whereas it should really converge to 20.1997, but this difference is basically due to the simplicity of this model, this can be improved by introducing more number of elements and discretizing the structure with finer detail, okay. Let's make some remarks now based on what we have seen, if you recall when we found natural frequencies we solved this eigenvalue problem, this was associated with the structure elastic stiffness matrix and the mass matrix, whereas to determine buckling loads we are again doing an eigenvalue analysis and one of the matrix here is the same that is elastic stiffness matrix, there is a new matrix KG which is a geometric stiffness matrix. The eigenvalue here where the natural frequencies whereas here they correspond to the load factors or the critical loading conditions there will be N such situation in N degree of freedom model and lambda N eigenvalues provide that. Now one important difference that we have to notice between these two problems is P critical is a function of the applied load, the same structure can have different P critical for different applied loads, see for example we considered this problem because there is a load 3P the same property cantilever will have a different natural frequency than when suppose this 3P was not there only P was there you will get another load factor. In this aspect the buckling load is different from natural frequency of a structure whereas natural frequency of the structure unless you include geometric stiffness effects also into that is not dependent on applied loads. So load factor is a constant factor by which a given loading pattern needs to be increased so that the structure would be on the verge of buckling. If axial stresses are tensile the stiffnesses of the structure increases this effect is called stress stiffening, so in a more general problem this will automatically get accounted for based on the sign of the axial thrust, so that automatically will be included in the analysis. So a small exercise would be to improve upon the result that we obtained for this cantilever beam supported on the spring by introducing one more element and you can redo this exercise for different values of K and see whether we get improved solutions. Now after this point we can take a traditional route and for example compute geometric stiffness matrix for a 3D beam, so 3D beam has bending along two planes twisting as we have seen earlier that we can derive geometric stiffness for that, then we can move on to a plate and other elements, but instead of taking each of these problems one by one we can since given that we already have experience in formulating these elements we can take a look at the general formulation as applied for a 3 dimensional solid and then we can specialize it to specific structures of interest, so with that in mind we will first start with the preliminaries as you have seen it is important to include non-linear strain displacement relations in our formulations for stability analysis, so to be able to do that in a systematic manner we will quickly review the notion of green lagrange strain tensor, so we will consider a Cartesian coordinate system X1, X2, X3 in which there is an object which is supported and loaded by surface structures and body forces in certain manner and our interest is this is the structural configuration before deformation and this is the structural configuration after deformation. We are focusing our attention on a line element PQ which is in the undeformed configuration and it will occupy a position P star Q star in the deformed configuration, so this length is DS and this length is DS star, so we want to examine properties of these two line elements which will help us to define the strain, so we will consider P coordinates is X1, X2, X3, Q which is X1 plus DX1 plus X2 plus DX2 and X3 plus DX3, similarly P star and Q star have these coordinates, PQ length is DS, P star Q star is DS star, now the displacement field is clearly given by X1 star is X1 plus U1, X2 star is X2 plus U2 and X3 star is X3 plus U3, so these relations define the displacements U1, U2 and U3, so this is a standard definition of displacement field. Now components of DS are DX1, DX2, DX3, components of DS star, DX1 star, DX2 star, DX3 star, similarly components of PP star, this is a displacement field, P moves to P star and that is U1, U2, U3, so if you now find length, square of the length DS square this is given by DX1 square plus DX2 square plus DX3 square, similarly length DS star square is given by this, now you find the difference I get this equation, now each point in the deformed configuration is a function of the initial coordinates, so I have XI star as XI star of X1, X2, X3, so we are selecting X1, X2, X3 as a coordinate system, so the coordinate system is with respect to the undeformed geometry and that's what we are doing, so we can differentiate this with respect to, we can find the DX1 star, DX2 star, DX3 star using this relation and I get DX star as J into DX, and if I now write a equation for X1 star which is X1 plus U1, X2 star is X2 plus U2 so on and so forth, using that if I evaluate elements of J I get them to be in terms of displacement, gradients of displacement field to be this. So now we will look at the change in this quantity DS star square minus DS square, so I can write in this form DS star square minus DS square by 2, I can put it in this form, so what I have to do is I have for DX star in this expression should be written in terms of U1, U2, U3 using this relation, so it involves quite a bit of an algebra at the end of which I will be able to write this quantity DS star square minus DS square by 2 in terms of quantities known as epsilon IJ and DXI DXJ, and if I divide by DS I get epsilon IJ in the summand we have DXI by DS, DXJ by DS and this is nothing but the direction cosines of the line segment so I call them as NINJ, so this is epsilon IJ NINJ, what are epsilon IJ that has to be synthesized by actually performing these calculations and it turns out that these epsilon's are given by these relations. So you immediately recognize that the quantities shown in the black are the equations represent the linear strain displacement relations, whereas the quantities that are shown in the red are the contributions due to nonlinear effects and this is a complete set of nonlinear strain displacement relations, so we call this quantity epsilon IJ as strain components and the quantity DS star square minus DS square by 2DS square lambda PQ is called the strain of the line segment PQ, and that is given by epsilon IJ NINJ, so this quantity epsilon IJ that is the matrix epsilon IJ is called the Green Lagrange strain tensor at P, it is a tensor therefore the rules of coordinate transformation is epsilon prime in the transform coordinate system is C epsilon C transpose. Now if DS is equal to DS star the magnification factor lambda PQ is 0 and all epsilon IJs will be 0, so in tensorial notation we can write this in this form where repeated index implies summations and the indices run from 1, 2, 3. Now if you now take, let us focus on this expression lambda PQ NJ, now if I take N1 to be 0, 1 and N2 to be 0 and N3 to be 0, on the right hand side I will be left with only one term which is epsilon 11, so what it means is if you are taking a line segment which is parallel to X axis here, and epsilon 11 represents the magnification of that line element due to deformation, that is the definition of strain that we are interested in, similarly 0, 1, 0 if you take it is epsilon 2, 2 that means a line segment along Y axis will get amplified by epsilon 2, similarly the line segment along Z axis gets amplified by this. Now we would also like to evaluate the final direction of the line segment, so direction cosine of PQ is N1, N2, N3 which is given by this, and direction cosine of P star, Q star is N1 star, N2 star, N3 star, so now we wish to evaluate N1 star, N2 star, N3 star in terms of strains and displacements, so again we use this relation XI star is this and if I differentiate with respect to SS star I get this and this is nothing but Ni star. Now lambda PQ is given by this and from this I can manipulate and obtain DS by DS star to be given by this, so DXI star by DS therefore is obtained in this form, by writing for displacements we can write the three equation DX1 star by DS, DX2 star by DS etc. in this form, and consequently by rearranging these terms I get the direction cosines of the line segment in the deformed configuration to be given by the direction cosines in the undeformed configuration through this matrix. This is important in the definition of shearing strain, so to define shearing strain what we do?  Now cos theta, using this I can write it as N1, M1 plus N2, M2 plus N3, M3 and cos theta star is given by this, so now using the relation between the direction cosines of the line segment in the deformed configuration with those in the undeformed configuration which we derived a while before for the two line segments I write this and if we now evaluate cos theta star in terms of U's and N1, N2, N3 it can be shown that this factor, there will be these factors so that if we absorb and take it to the cos theta star on the left hand side I get this quantity to be cos theta plus 2 epsilon i j N i M j. Now I define this quantity gamma AB as a shearing strain, so why that is shearing strain, for example if theta equal to 90 degrees that means the two line segments are at 90 degrees, one of the line segment is aligned with X axis, other one is with Y axis, if you compute gamma AB it turns out to be 2 epsilon 1 2, which is in the engineering this is gamma 1 2 or gamma XY if you use XYZ notation, so similarly if you take a line segment along X axis and a line segment along Z axis, gamma AB turns out to be 2 epsilon 1 3. Now if theta is small we can approximate theta star as pi by 2 and cos theta star as pi by 2 minus theta star, and in which case gamma AB turns out to be pi by 2 minus theta star which is the definition of the shearing strain for a small deformation. So the traditional definition of strain is recovered, so based on this we accept that gamma AB given by this quantity is a measure of shear strain, so this is some basics of Green Lagrange strain tensor. Now a few observations, all displacements are measured with respect to the stationary reference coordinate system, which is the original coordinate system for the structure in its undeformed configuration, this is true no matter how large the rotation of the structure, rotations of the structure are, now the Green Lagrange strain components are equal to 0 for all rigid body motions, the rigid body motions can be smaller large. The stationary coordinates are also called material coordinates, this approach to solving non-linear problems is called total Lagrangian approach, so there are alternative notations in tensorial notation with repeated indices all these relations can be compactly written in this form where repeated indices imply summation and comma implies differentiation, and in terms of XYZ coordinates we can write this expression instead of X1, X2, X3 I will write XYZ and I get this. Now with this brief introduction to non-linear strain displacement relations we will now return to the problem of a general formulation of stability in non-linear solids, so let us consider 3D solid, let an initial stress sigma naught exist in the body, okay, now assume that the body as the body strains the stress sigma naught remains constant, the work done as strain epsilon occurs is given by epsilon transpose into sigma naught, the sigma naught is a multi-axial state of stress, in the beam problem if you recall what this assumption means, so when we analyzed beam we found out the state of axial stress under this load P and P by A was the sigma naught, sigma XX was P by A and all other stress components were 0, so as the body strain, this strain this load did work on the deformation, that is what we are using in computing total potential, now in a 3 dimensional solid the generalization of that is there exist a multi-axial state of stress in the body and this remains constant as the body strain, thus here the axial load in the beam problem was not changing as a structure deformed, that was held constant, so the same assumption if you make it means sigma naught is constant as the body strains, so the work done is given by this, now stress components there are 6 stress components initial stress components, 3 normal stresses and 3 shear stresses, and the strain components are 6, so now we compute the work done this epsilon transpose into sigma naught by including the non-linear strain displacement relations, okay, suppose if you do that I will have to put all this now into this equation, sigma XX naught will get multiplied by epsilon XX which is entirely all these terms, not just the first term, all these terms, so it is the interaction between sigma XX naught and these terms that create the issues that are related to buckling or loss of stability. Similarly I get other terms, now I will split this work done into a component U1 and U sigma, the U1 component is on the linear part of the strain displacement relations, and the other part we call it as U sigma is the contribution that is crucial for the analysis of stability of the structure. Now this U1 if you consider what it does, there is an initial state of stress in a finite element analysis so how will you take that into account, it will become simply set of equivalent nodal forces, and this is what leads to that, so to see that let us consider U1 to be this, here I am retaining only the linear part of the strain displacement relations, next I will approximate UVW using finite element approximation, so there are say S degrees of freedom, N1 and these N's are the interpolation functions I represent like this, and by assembling UVW in a matrix N and in terms of nodal degrees of freedom I write in this form and the epsilon linear is given by this acting on NE this is BUE, so if you now consider U1 this will be UV transpose B transpose sigma naught DV, so this is nothing but UV transpose B transpose sigma naught DV, this is nothing but, now for sigma naught I will write D into epsilon naught where it is a matrix of elastic constants, this is essentially the equivalent nodal forces due to presence of an initial stress, so this becomes a load vector that has to be handled. Now how about the remaining terms, that is U sigma terms, so all the terms that are present here are gradients, dou U by dou X, dou U by dou W by dou X, etcetera, etcetera, so I declare this gradients of UVW with respect to XYZ, the nine coordinates at delta, so this delta is related to UVW through this matrix, okay, so this I call it as capital XI UVW, where capital XI is this a matrix, this is similar to the matrix that we formulate when we write the elastic you know stiffness or in this case the effect of initial stress on linear part of strain displacement relation, so now I have U sigma to be this, and I am now making the substitution delta is capital XI into U, and again UVW I will interpolate in terms of that of interpolation functions and nodal coordinates, and delta consequently becomes a capital XI into N into UV, this matrix capital XI into N I call it as capital G, so this is G into UV, this is exactly similar to B into UV, whereas B was this matrix of these operators delta I have here capital XI to be this, so consequently this complicated expression can be put neatly in this form, U sigma is half delta transpose into this matrix of initial stresses delta DV, S is a stress matrix due to initial stresses, so U sigma is this, now for delta I will now make this substitution G into UV and I will get U sigma as U E transpose K sigma UV, and K sigma is this matrix, this G transpose into this stress matrix into G DV, this is the geometric stiffness matrix in this general form, so when you have to apply this theory to specific structural elements you have to simply identify this G matrix, so for example if you are dealing with Euler-Bernoulli beam, Mortimer-Schenko beam, etc. you will know the assumed displacement form from which you will be able to derive the strain in displacement relations, and you will have to evaluate K sigma with that, so let us consider a simple example of a axially deforming bar, suppose I have axially deforming bar in XZ plane, so U and W are the displacement fields, so this is the undefined configuration and these are deformed configuration, so this point now will have U2W2 as the degrees of freedom, it is not necessary that this should deform in its own axis, along its own axis, initial stresses we will assume that in this case all components will be 0 except sigma XX naught, now U and W I approximate using, there are two nodes and linear interpolation function we will use, N1 is L-X by L, N2 is X by L, so this is the representation for the field variables, there are two field variables U and W, now G matrix in this case turns out to be this, and you can do this simple calculation and find out this integral of G transpose S into G ADX, it turns out that A is, the GK sigma matrix is given by this, so this is the geometric stiffness associated with axial deformation, where P is A into sigma XX naught, now if A is taken to be much greater than P we can show that this matrix reduces to this, so if you see text books, this K sigma for axial deforming bar sometimes will be given in this form, sometimes in this form, both are valid if you accept this assumption, otherwise this is a more general form which we can use. Now we will consider a general 2D beam element and this is a beam element carrying an axial load, and this is a 2-noded beam element with 3 degrees of freedom at each node, U1 and U4 are the axial degrees of freedom, and U2, U3, U5, U6 are the flexural degrees of freedom. Following the procedure that we have developed it can be shown that the geometric stiffness matrix for this type of element can be shown to be given by this matrix, I leave this as an exercise, the kind of context in which this element becomes useful is depicted here, for example if you consider a member like this it is subjected to combined action of axial deformation as well as bending, and when we formulate the geometric stiffness matrix for this type of situation we use this matrix, the other steps involving coordinate transformation assembly etc remains similar to what we have already done in previous occasions. The rules for coordinate transformation and assembly all that is similar, so this we have seen already I will flash these slides, you can go back to the earlier lectures for details, so this is how we transform the nodal degrees of freedom in global coordinates relate the global degrees of freedom with local degrees of freedom through this relation, and this is the transformation matrix and I write U bar as T naught transpose into U where T naught is this etc, so all this we have seen, now I can substitute that in the expression for total potential and the elastic stiffness matrix in the global coordinate system we already derived, similarly the stability matrix can also be derived using this formulation, so K bar and J bar are the element stiffness and stability matrices in the transform coordinate system, so this we need not again get into all the details, now the next question is how do we solve this type of problems, so now we are equipped ourselves with analysis of 2D beam element, so we can discretize this frame into different elements as shown here, and for each element I can use the 2D beam element that we have 2-noded beam element with 3 degrees of freedom per node for which I have the elastic stiffness matrix and the geometric stiffness matrix, so first thing what we have to do is we have to perform an elastic stress analysis and find out the axial thrust in the structure, so the first step would be don't include geometric stiffness and analyze this problem, you will find the initial stress that is what we have to first determine, once the initial stress is determined for every element I know the state of initial stress, then the question that we are asking is this loading pattern that exists will be now multiplied by a factor alpha, and as this alpha is increased the initial state of stress is not I mean assumed to be constant, so the question that we are asking is as the structured strains the initial stress won't change that is the assumption, we want to find out the value of alpha at which the structure as a whole loses its stability, so this alpha is known as load factor, so the procedure for this would be to first of course we have to assemble the K matrix and perform the elastic stress analysis and find out the initial state of stress, then you formulate KG matrix, then we have the eigenvalue problem to be solved and we evaluate this, those values actually this is alpha, so I will write this in this form, number alpha that needs to be increased so that this quantity becomes 0 is a critical value of alpha at which structure loses its stability, so what we will do is we will close the lecture, this lecture at this stage, in the next class we will consider some details about this. Now following this we need to consider how this formulation can be extended to three-dimensional situations, now one way to assume that is when we developed a general 3D beam element assuming that cross section is symmetric, what we did was we analyzed each of the mode of deformation separately, axial deformation separately, twisting along the longitudinal axis differently, bending in one plane and bending in other plane, these four entities were analyzed separately and they were all subsequently assembled, so there was no qualitatively new feature associated with the built up stiffness matrix, it was only that the amount of computational effort increased, there was no new concept that entered our formulation, but whereas when we come to doing the same problem for geometric stiffness analysis that is not so, it is not just that the axial loads interact with bending, the other interaction can also take place, the initial state of stress is multi-axial, so in the initial state of stress there are torques, there are axial stresses, there are two bending moments, so there are four forms of initial stress resultants in the system, they interact, so we need to include that, I mean that is fairly obvious that that is not going to be simple by looking at these expressions, there is complex interaction between sigma XX naught and these slopes, it is not just that the axial load interacts with bending, so depending on to what extent we are able to capture these interactions, there are various forms of non-linear analysis possible, and to be able to perform a reasonable stability analysis we should capture at least a few important interactions, and we will see some of these details in the next lecture, and that discussion on beams will be followed up by discussion on plates, and let's see how it goes, we will conclude this lecture at this stage.


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