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39:55
Welcome everyone. Today's talk is focused mostly on giving kind of an integrated view from the beginning to the end, so that we can find areas of our interest and then dive deeper from the first question that we are going to ask is, why do we need nonlinear modeling when it is generally not really explicitly required by most building codes, or we are not taught? We can mention engineering courses. So that's the first question we will try to answer.
40:24
So let me start with the simplest of the simplest problems. I think nothing can be simpler then is simply supported beam with a single point load and start with that. So let's say if we have a single simply supported beam of this size and span maybe three meter or something, and we apply a point load there, and I run any analysis program to do the analysis and calculate the deflection in the middle. So let's say if I apply a load of, let's say
40:49
5010 and I get a deflection of 2.5mm, if I double the load, I get double the reflection and I get double the load, I get another double deflection from ten to 5 to 10. And if I increase further 8 to 400, I get another deflection. So no matter how many, how much load I apply, I will always get an answer. Even if I apply 1 million Newton here, maybe I will get took 2m or 10m of deflection, which obviously makes no sense because the structure is only three meters long
41:20
and no beam can deflect by 10m or 15m, which means something is not right with this analysis, even though the moment diagram, the shear force diagram, deflection diagram will look all beautiful like this. And you can't tell the difference that this is real or non-real. So which means something is not right. So let's go back and see if we can physically see what will really happen. If I take a beam of the same size and apply load.
41:47
This is a class project from one of my courses, and students are now testing a beam of the same dimensions, and they've observed the first crack. The beam starts to crack very early, and then there are so many cracks coming up, and very soon the beam stops taking any load at 200 newtons. And when you plot that load and deflection curve, it looks something like that. So that means that the beam in reality did not go beyond a certain level.
42:15
And also the deflection did not follow the linear multiplier rule that we saw recently. Which means that if you look at the the deflection here, this is what the linear analysis would have predicted. But the actual response deviated from it quite early and started to go the the red line way. And that we can see clearly that when we load the beam sufficiently, it will start to deviate from the linear analysis and it will go another way,
42:46
and at some point it will fail and then stop taking any load. So that's the that's the reality. So that means we are missing something when we do the linear modeling and linear analysis. So now if you look at the right hand side you can see clearly that at 410 there is no structure to exist. No response exists to calculate the deflection. Even though our analysis gave us a reflection of a certain number. But actually the structure doesn't cannot reach 410, it's already gone.
43:16
Similarly, if you look at 210 level, then the deflection given by the linear analysis and the nonlinear actual response, I should say the real response are very different. That means both the force prediction and deflection predictions are now unreliable when the load is increased beyond a certain level, which means clearly that the linear analysis and linear modeling that we normally typically use has its limits. What are those limits? When can we do it?
43:44
When we cannot do it? That's what we are looking at. So from this simple beam analysis, let's move on to a slightly, slightly more complicated structure of a frame with two columns and one beam, and now move from vertical load to letter load. And we then apply a letter load. And we plot the data deflection. And then also the same do the same thing as we did with the beam. We will get the same response. We will have a similar relationship between lateral load and little deflection
44:12
as we had for the vertical load and vertical deflection. And it will follow what our good friend Robert Hooke said a long time ago. And it follows that the force and displacement are linearly are proportional, and they are related to quantity we now call we now understand the stiffness. So now if you instead of taking that ideal structure, we look at the real real frame which has reinforcement. It's not just concrete. It has bars in the column and beam and all.
44:40
And this those bars have a certain strength. The concrete has a certain strength. It's not a linear elastic model. Only then if we do the same thing, if we look at the the linear response for this structure, at any force deflection you will get f equal to k you following the Newton's law. And then we get a certain displacement. But as we have learned from the previous example, if you do the real structure it will follow a non-linear response which will be the red line.
45:06
And to deviate from this very quickly, even faster probably than the previous beam. So that means now there is a gap between the nonlinear force at a given displacement and the linear force at that same displacement, and that displacement difference. Normally we can call it a delta f. And this delta f is the correction that we must apply to the linear analysis to get our nonlinear response. Because the nonlinear response is now governed by a new equation
45:38
where the corrective force must be applied to find that red response. The problem is that we don't know the red line unless we test it, and if we tested, the structure is already gone. So if we don't know the red line, we also don't know the delta f, which means now this is a highly nonlinear problem where we just have to guess by, you know, we have to draw the red line by, you know, I treated procedures to track and find it. So we are looking for something which we don't see
46:05
by using a creative procedure and convergence and so on. That means it is not guaranteed that we will find the red line unless we find it properly. So it's quite possible that we may or may not find a nonlinear response, or we may even find in a way that it doesn't really represent the real structure. So now that we can see that there is a problem in terms of finding what we know, the equation correction between the linear and nonlinear equation.
46:33
So we can use the nonlinear equation as a starting point. So that means now we have established that to get the red line we need to have a nonlinear analysis and a nonlinear model which can capture the cracking that we saw and the end of its capacity as we saw in the physical test. So with this concept, now let's go back to generalize it a little bit and look at a structure, let's say subjected to Earth quake. And then we have the two floors and we have a base here there.
47:01
And now we plot the base here rather than one force against the deflection, the roof drift or whatever you want to pick it. And now we have the same thing. We have the linear response governed by the gross section. If we start with gross section and then we have the cracked property. That means if we assume that the structure has cracked, the stiffness will reduce. And then we actually can simulate the non-linear starting
47:28
point of the structure by using either the cross section, which we started. But now that we know is going to crack, we can do that. And that crack stiffness corresponds to a stiffness at the force design force level, which could be the example yield strength. So at the yield strength level we can plot on the red curve. And whatever is this defense at that location we can extend that. This is the reduced stiffness that we can use. So at least we recognize that the structure is no longer cross
47:59
at the by the time it reaches a design force level, it would have cracked. So we should use a cracked or a reduced stiffness. That's why the building codes have so many stiffness reduction factors trying to estimate this behavior. So this is all good and well at least up to the design force level. We can we can estimate that either not by the grass by the cracked one. It works very well for a load which is below the design force level for example wind load.
48:27
So if you take a wind load or even dead load, you need to multiply it by a load factor to bring it to the design force level. So it's a wind level wind load. You may multiply it by 1.6, for example, which is a service wind load design wind load. And then you multiply it 1.6, bring it to the design force level at which the yielding is expected to occur. So we increase the load by that. So it's fine because we have a control on that.
48:53
And then you can follow the equation beyond that. But for the wind load it may not be necessary because we can we are happy to just reach the design force level. The problem starts that we are looking at the fully yielded structure capacity beyond that one, because now one member structure has so many members, not just one member. So they will yield progressively one by one. Some will have overcapacity, some will have so many things.
49:21
And then gradually the whole structure will have the capacity which will be larger than individual yield, capacity on which the force level is determined. So that means now we have a difference between yield for design yield and also the final structures a fully yielded structures capacity. Now let's go back to another area. Earthquake. Earthquake design. Earthquake level is typically much larger than the yield force level that we have.
49:49
That means the design earthquake level needs to be reduced to come to that level, not increased like wind. So the factor is now the traditional factor is not going to work. We have to come down from a higher level down to the to the estimate of the red line. So that's the reduction in forces which will utilize the nonlinear response. So we can come to the fully yielded structure level. And we can fix the design displacement rather than design force level.
50:16
So now this is a clear difference that certain forces be certain actions. We use the force level control and the yielding point. And for certain we use the displacement level control and then corresponding reduce force from the initial force using the nonlinear expected response. So this gives the sets the basis for most of the code base design and code researcher recognizes for a long time nothing new. So that brings us to the conclusion
50:46
that the linear resumption only works when the material remains. Elastic geometry stays unchanged, boundaries are fixed and do not change, and superposition applies. We can just multiply things and we can add loads with each other as if they were independently applied, and then combined effect would be mediations are valid for nonlinearity. Of course, none of these or some of these may not apply. So if any one of these is not applied, we have seen from the red line
51:14
that we need to do something more. So now going back to our example, which what we can do now is that instead of using the original model, we can now use a nonlinear model of the same structure, which means that we can now look at the interaction diagram of a column, for example for capacity interaction. And then we can look at the moment Croatia relationship at a different load level, zero load level and other load level. We can use the moment curvature curve for the beam for example.
51:40
Now we have. That means we are bringing the non-linearity of the materials and the sections and other things, and we will discuss it in more detail into this rather than the elastic model that we did. Now it's a different model which recognizes the presence of reinforcement, and it recognizes the presence of the capacity in the in the deformation and capacity is no longer unlimited capacity that we can load. When we input these things
52:04
back into the system, we can create, we can get different responses. So this is a comparison for this frame of that kind linear and linear crack then nonlinear nonlinear with with p delta and without p delta. And then so you can see that it keeps on reducing. So as soon as you keep adding more and more nonlinearity, the actual capacity in force keeps reducing. Deformation capacity of course is different. But it's interesting simple structure.
52:33
If you limit the service level, let's say 0.5 diff, you get a certain force vector. Let's say we call it one. And then if you go to 2% drift and then you look at the reduction force vector, it is three. And if you look at 4% drift you get a force reduction factor of eight. So this is you can see now clearly that the deformation or nonlinear response can give us a force reduction factor between three or 2 to 8 in extreme cases for the design of the members.
53:03
Because of that, we are able to capture all kind of non-linearity. And these are three sectors are very similar to the R factor in for frame sections, for well-designed, well detailed frame sections in CI and ASC. So that means this framework is clear and it is somehow included in the four codes design basis. So that's where we have this whole design with our R factors and Q factors and CD factors. And we explained this and another webinar in detail.
53:33
So which means that the codes are already including these these factors indirectly to represent the non-linearity of the structure. So if we follow these we implicitly can capture these ones, but not explicitly because we don't know whether R is correct or not. We don't know whether the Omega is correct or not. We don't know whether D is correct or not because they have been predetermined for a certain structure type, but each structure has unique characteristics. So that is hard.
54:04
So when is nonlinear modeling needed? First of all, when we want to do performance based design that we we need to verify of the performance of the structure, not depend on the code code assumption that the structure will be okay if the safety of the structure has to be a certain, or the performance has to be certain, then nonlinear must be done. Second, one assessment of resisting structures new structures. We can follow the codes and we can design whatever the code requires.
54:30
But existing structures are already existing. You cannot force them to now conform to changing codes. So that means you need to evaluate them and only fix what is necessary, not try to fix every part of the structure and try to enforce the code there. Then we have special irregular structures which the codes may not have considered in the r factors and omega factors, and all because they are based on rather simplistic, you know, to test the regular and certain type of structure.
54:59
And then most importantly, the last one is understanding structural behavior. Because the linear analysis, as you have seen, does not give us any insight into what is really happening. What is this train level? What is the stress level in any location. So if we want to do that, short of testing physical testing, we need to be able to do this nonlinear modeling and analysis to get that insight. Of course, not every project needs it, but more and more people are realizing
55:30
that this is a useful thing, at least, if nothing else, for number four in the list. So this has now given us a very good framework, this nonlinear modeling. And you can see here, you know, it's a very symbolic way to see that we have a material nonlinearity of steel and concrete, bringing to section section non-linearity with material non-linearity coming to moment revision, then the length hinge length effect to moment rotation, applying them to the structure,
55:55
and to be able to link them to the realistic site specific hazard. Nonlinear analysis, nonlinear time analysis, going to performance indicators and realistic performance evaluation. So the framework is very clear and which what is now missing here. One of the things which is missing here is how do you create these nonlinear models which represent the structure realistically? Because nonlinear modeling can lead us in many ways.
56:24
So we have to make sure that the models that we are creating are realistic representation of the structure that we intend to make or has been made. And of course, on the right hand side, we need know good, realistic site specific hazard analysis, a separate topic to be discussed some other time, and after that we can then correlate the performance with the actual response of the structure. And we can look at the vulnerability hazard consequences
56:51
and go all the way to disaster and risk assessment and disaster resilience. And in fact, I sometimes, you know, joke with my students that structure engineers should be called now disaster risk introduction engineers or ability assessment engineers because that's what we really do vulnerability reduction engineers. And if you do that, you will probably get better jobs in UN because that's where the, you know, the most of the funding goes into resilience. And this is all related to resilience
57:21
and the disaster management, disaster risk reduction and so on. So that's what the ultimate goal is. And if we want to go that way with a proven data with in our hand, we need nonlinear modeling as a starting point. So that's the why of the whole thing. Then the questions are what we behavior do we need to capture. Because then if we decide okay fine I'm going to do nonlinear, then how far do you want to go. What level of accuracy is needed.
57:51
Do you want to be highly accurate or you just want a good estimate? And what are the computational radar? Do you have enough data? Do you have enough tools? Do you have enough time? Did you get paid enough in some cases to be able to do that? So which means there's a trade off between. So we have to look at all of that when we start as as a starting point and make a decision whether we are going to do a simple nonlinear which is possible, a complex one or something which we can interpret.
58:23
For example, pushover is easier to interpret than in the time analysis. So maybe pushover is enough for for some cases. Do we want all non-linearity or some of them. So we make all those decisions along the way. So now we will discuss how many types of nonlinearities are there and what is their nature and what effect they can have. So sources of nonlinearity where does the nonlinearity come from or where does it reside? Two questions seem similar, but they are different in their real context.
58:54
So let's look at that. So nonlinearity can enter at every level of the structure hierarchy. And the modeling question will be at which level do we want to capture it. Where do we want to start. How far you can ignore which level will be enough for a particular application. And then we will discuss these things. So we will discuss the hierarchy sources and the force replacement curve. So this is the response built up of a typical structure.
59:25
The structure as you know in hierarchy is built from materials. Materials make up the sections section make up the members. Members make up the structure and the nonlinearity or the response is similarly built up. So we have material concrete, steel in our case for reinforced concrete structures, steel for steel structures or other materials, wood or whatever you want to do. Carbon fiber doesn't matter. You bring those materials and their non-linearity
59:51
into the material stiffness, and then you add the cross section geometry, and you bring in the cross section nonlinearity into that one, like the moment curvature, for example. And then you bring that on forward and you bring the member geometry and member nonlinearity, for example, be delta and moment rotation and other issues or axial shortening whatever. And then you build up many members and bring them together to form the structure.
1:00:16
Some will fail first, some will fail later, some will be weak, but some will be stronger. Some will have difficulties, some will not have. So the total structural stiffness will be built up and the total nonlinearity will be added. So this is the the response built up hierarchy. And now we can see clearly that nonlinearity exists at every location. It is now our understanding and our decision. Which one. Where do we want to start. Where do we want to ignore?
1:00:44
For example, if you just put and don't do anything else, that's fine. We got one non-linearity in there, which is great for certain cases. Sometimes you put the material non-linearity moment curvature or something, but you didn't. You put p delta. So the decision which nonlinearities you will include as we go ahead will be will determine the response that you will get, the accuracy, the reliability and the depth of results that you get and the tools that you're available.
1:01:12
Now we come to a typical force different mission curve. This exists at every every level that I showed you is similar, whether it's material, section, member or structure, they all look similar. Just the axis changes is force, maybe strain and stress, moment and curvature for rotation and moment and so on. Force and deflection, everything riff and story shear busier and roof displacement. It doesn't matter. They all look similar and they represent a typical relationship
1:01:41
between any of the force stress component and any of the corresponding strain deformation component. And it looks like that. As you can see there's the yielding there's initial stiffness. This is yielding there's strain hardening increase. There's ultimate strength ultimate level. Then we have a drop limit and residual strength. And this we have also degradation because of the cyclic nature. And it's very very complex for particular. And people do a lot of research and testing to
1:02:10
to be able to draw these curves, you must have seen and many of you I'm sure have done research in these areas. So the important consideration for a lot of these nonlinear response especially is for cyclic and earthquake will be the hysteresis behavior, which is more complex than the one directional force deformation. But if you combine this like I showed you in the previous diagram, like light blue loop and you combine this one together, you can create a composite
1:02:39
or backbone curve which includes the degradation and the, you know, one directional monotonic load deformation. And that is what ASC, for example, does as a 41 gives you an idea. This ideal idealized force defamation curve, parameterized and idealized, which includes both the unit directional curve parameters as well as the degradation from the cyclic. And they combine everything together through testing and computations and give this these parametric form of the curve.
1:03:12
Then these curves are available for columns, for beam section for walls, and so on and so forth. So and the response also is measured in the similar manner the input curve for materials and and members and section is very similar to the output curve that we are expecting because as I mentioned to you, the the force defamation for distress is similar to force division curve for the structure. So they can be can be parameterized in the same way. So this parameterization from ASC is very useful
1:03:43
because this becomes a very good tool for us to input or implement in the software without having to draw complex or to generate complex actual force deformation curves. And these are valid that are used by many people, and most of the time they are reliable. They have a lot of data behind that. Invest in parameters and you can choose the tables and tables in ASC and other documents that help us. So this is the input you can say for the nonlinear analysis
1:04:16
where the nonlinearity has been parameterized. Force deflection curve has been parameterized in this way. So using that concept we can go now from materials to sections to members to structures like I showed you earlier. So we can use this same parametric nature of the force deformation input or output. Either way you put it. And then you can model from these structures a different level. And you can include the nonlinearity at any level. Like I said, stress strain,
1:04:47
moment curvature force deformation which here and displacement and all. So the the modeling now will will depend on what data you have and what do you want to include and what software are you using. So let's go to the part three. Now. Now that we have established two things, we have established why we may need nonlinear analysis. And not only a modeling analysis is after that. And what is it that we can do for the nonlinear modeling part.
1:05:17
So let's look at the implementation, some of the implementation details. And for this one we will use small RC building to show some of the implementation details. Once again, as I as I mentioned earlier, I won't be able to go into showing every possible data import for it as a SAP and there are so many good sources available for that. I will just show you generally where it is located in the program and roughly what data is needed, but I won't be showing how the data
1:05:46
is calculated or entered because that requires considerable amount of time. So for sources of nonlinearity, coming back material is where you need to to model the physical behavior. For example, stress trend, curve yield, steel strength, concrete cracking, crushing, and many other things. So we can put all of that into the material models. And this is one of the more difficult areas where software have limitations programs, for example, and and others,
1:06:18
they can do far greater depth of material modeling than other programs can. So depending upon the data available, depending on factor of type of material we are using, we can model the material in multiple ways. Second one is sections. And then if we look at the sections physical behavior is we are looking at moment Croatia, we can PM interaction surfaces and all and we need to model through hinges and fiber sections and all, which I will talk about in a minute.
1:06:46
Next one is the geometric and geometric is physical behavior p delta effects large p delta which is related to these sway, and local p delta, which is the local deformation of the members between the two ends, and member buckling, and so on, and large displacement and other things. So that would be geometric, especially useful when we are doing the construction sequence analysis. This this needs to be included. And then you have the analysis settings. And so this is more on a load case level.
1:07:14
So material and section is typically handled at member level. And geometric is handled at the structure and load cases level. And then we come to the final. The most difficult one is the boundary. Normally as you know and I talked about this before. Also engineers are very good at presenting the boundary by a very nicely drawn line with, with hatches calling it a fixed foundation or a pin or a triangle and say it's a pin or a roller. But in reality none of that is true.
1:07:46
As you know, there is no fixed foundation ever. There is no roller or pin unless it is a real steel structure with really constructed pin and roller. Most concrete buildings will have a footing or a foundation going all the way down to the to the soil, deep into the ground, or a large mats or complex foundation systems. So where do you end? The boundary? Is the boundary at the column base of the column, or is the boundary
1:08:12
of the structure at the base of the raft, or is it the boundary of the structure at the bottom of the pile or the side of the pile? So how far we go to define what is structure and what is boundary will change entirely how the boundary is modeled. So boundary typically is modeled by link elements. I will talk about that. There are gaps friction isolators, damper soil structure interaction, very complex nonlinearity exist that area and often ignored or often
1:08:47
not explicitly modeled because we lack data on that side. Okay. So this is the the four types of nonlinearity that we are going to talk about in the following slides. So first of all nonlinear modeling approach is for the first two that means material and the the section level nonlinearity can be handled by using either a lump plasticity model or a fiber modeling or a distributed plasticity model. So lump plasticity means that we assume that entire nonlinearity
1:09:21
is concentrated at some location, and remaining part of the structure or the member is linear, which is not the reality, but it could be a good starting point to simplify the problem. Plastic hinge approach is one of them and then localized interaction in action, member level input and so on. So
1:09:43
can be handled by this concept. Fiber modeling says that okay the material. So for lump plasticity we separate the material from the plasticity model at that location. And it is derived from material separately. Fiber mode on the other hand includes the materials and the section together into one single integrated model. And it has a more. I will explain to you why this is better in some cases, and why it can be useful compared to the lump fund, and distributed plasticity is where we can
1:10:20
distribute it over an area, over in length or over a surface, and it's not really focus on one location or one point or one section. So that's the three approaches typically we can use. I'm talking about structural engineering in other fields. There are other kinds of approaches as well. So this is roughly how we can see that. So plastic hinge is a first category which means you choose the lumped plasticity model or lumped that means non-linearity. You put the lonely and assume it to be
1:10:53
focused at one point, or distributed over a certain small length. And then second one is a fiber section. You assume that the cross section of the section entirely is modeled at multiple locations, and the cross section materials are part of the model that we are going to consider. So there prescribed curve there is nonlinearity is included in this model. Whereas in the first one that is excluded that's an input, but it's not part of the analysis done once not modified later.
1:11:25
Finite element is on the other hand, is that the finite element the elements is becomes nonlinear. For example, we have a nonlinear element in SAP and that means the nonlinearity can be can be distributed anywhere, wherever you can model anything by shell. You can also make it nonlinear. Typically it's linear, but you can say, okay, this slab I want nonlinear, this wall I want nonlinear or this beam, I want to model by by a shell element or the depth and want to make it nonlinear and whatever.
1:11:53
So this is a really powerful distributed nonlinearity that you can model. And what you see below is expensive in both in terms of time and in terms of effort required to do that. Of course, with software always trying to make it easier, these things become easier and easier as days go go by. We have more parametric definitions. We have more tools to model these things easier. So that's the three primary tools we have for the section and member level.
1:12:25
Together those nodes non-linearity can be handled this way. So lumped non-linearity we can put where we can put on the beams and columns. And based of the supports we can put them at the point of maximum bending moment. We can put at the mid span. For example, for shear hinging we can put them at the within the joint regions. So we can use this in many ways. And it's quite useful and it's very fast and it's quick.
1:12:53
We can model moment curvature with that. We can model shear force and a future deformation with with those we can model axial force and displacement with that. Very good for modeling struts, which we will see in a minute where we apply that. And then we can also mortar talk and twist which nonlinearity is a bit hard, but it is possible to do that if we have enough modeling there. But as you can see these are mostly uncoupled. That means they are looking at one action, one deformation at a time.
1:13:23
Then we have coupled also where you can couple the nonlinearities. For example PM where the two bending moments and one axial force are coupled together, and then we get moment and moment curvature at different load levels, different angles. And I will tell you in a minute why this can be replaced by other means, because it's very, very difficult to generate all the possible data for such coupling to be properly captured. Decoupling that is generally not considered is the axial moment
1:13:55
PM and the shear. And we will talk a little bit about that towards the end. So that's the types of basic nonlinear modeling tool in terms of the hinges that we can use. So modeling in it maybe this slide may be better. So we can see that we can select this section. You can go to frames frames you can put the you can select the frame section. And that you can add this hinge over there. And you can see here that that it has an sap have like ten, 10 or 12 types of hinges that you can choose
1:14:29
all of these modeling one or more nonlinearities. Uncoupled hinges require action deformation for one action, one deformation and then coupled which can do more like m2 m3, p and m2 m3 and pm the full one. So all of them are similar and they are most commonly we use the axial, the shear, the single moment and the couple. And then we also have the ones below that which are the fiber ones which are coupled already. Now let me go back to another. Normally we are we are accustomed to
1:15:01
most of us are thinking about nonlinearity in buildings. But bridges are another area where nonlinearities is extremely important. And it's a very different kind of nonlinearity compared to buildings, because most of the non-linearity in bridges comes from disconnection of the members and the gaps between the members and the bearings provided there, and the soil structure interaction and the abutments and so on. So the girders are just resting on bearings so bearings can slide.
1:15:29
There's expansion joints. There is a seismic strainer. There's so many things. So bridge archly become have a very different kind of nonlinearity. And mostly this nonlinearity requires a different understanding. Hinges is one thing that you can see here like in the column or the bent or the piles, but we also need soil structure, types of nonlinearity and handling these gap type of types which henges cannot handle. So that leads us to the very powerful tool
1:15:57
in Sep 2008, ABS, which is the non-linear links. I think these are extremely powerful tools and often not used as much as they could be, because look at them. The first one on the left hand side is a single link that can represent any of any or all of the six degrees of freedom together. So you can have two joints connected by a link, and you can choose one, two, three, or any one or all six of the degrees of freedom and represent their nonlinearity. This is a very powerful thing.
1:16:30
The other one is a little bit called complex advanced links, which have the nonlinearity in terms of the gap, damper and hook combined with the linear portion. So they provide and the one on the right are the one which will be very useful in bridge modeling, where you can model the limited movement and then non-linearity kicks in. So which means that and you can also add the soil structure p y curves into that and make them really complex. So
1:16:58
this these models link models can be used effectively for when it's in fact one of the students in eight. Currently he's finishing his thesis on modeling of abutments. Using these combination of links, he has spent two years trying to just create reliable nonlinear models of just the abutment and the interaction between the girders and the and the piles and the back soil and all of that. So it takes really long time to include this kind of thing.
1:17:29
So just abutment modeling with the sliding bearings with the gaps and the earthquake risk trainers and everything. It's a very complex system, requires multiple uses of linear and non-linear links. So if you look at the comparison between the internal hinge that we normally use and the simulated hinge is like this. So left hand side is a lump. Plus is the hinge that you can model by using moment curvature or whatever. So that means the cross section is treated as one entity.
1:18:00
You convert that into a stiffness relationship and then you put it there. On the other hand on the right hand side you can actually disconnect. So that is an internal hinge inserted in the beam on the right hand side. You break the beam up and put a small gap there. And then you reconnect the beam by using either a single link or a multiple links. And this is the way this is where highly advanced or complicated behavior can be modeled.
1:18:30
Even some interactions that the plastic hinges or plastic hinges do not allow. We can model here, and other programs in research programs have more capability to model this kind of thing, including the contacts and slippage and many things. So this picture on the right hand side is a very advanced way to to include nonlinearity anywhere, by disconnecting and reconnecting by a nonlinear behavior by that one. So that's the application or difference between the two types of models
1:19:02
that you could use. And this is in all linear link property. As you can see here, you select that there are so many types of nonlinear links there. And then you have six degrees of freedom. Each one can be independently nonlinear. There are also couplings. There are so many things. So if you understand nonlinear link properties and their use, I think you can handle almost any kind of nonlinearity that we normally encounter in in our structures. So
1:19:30
what is the suggestion here? We can look at simple intermediate or advanced level of modeling. Now after understanding these these things so we can look at just linear plus delta plus default edges and pushover. We can look at pushover with defined hinges with actual properties. The right the five from the existing structures not using the the default engines from ASC. And we can use the advanced one in which we can use nonlinear responses tree
1:19:58
with distributed plasticity and whatever not so that can be used for research. The intermediate is used for PD and then simple could be used by code compliance and evaluation and verification for existing structures. So depending upon the type you can go with a simple intermediate or advanced level of modeling. Okay, so now that we have covered the why of the the problem, we also covered what is available to do this. If we decide to do this.
1:20:27
And now we can just look at one demonstration building. This is a I chose a very simple building here intentionally. It's actually our favorite building in our office. And it's a school building that we we designed because it's a three story building, because I think it's a good demonstration of many basic behavior, and also because a lot of the time the school building evaluation for artworks is retrofit is a common project in many countries. You know, we have projects in Nepal.
1:20:56
We had a project in Philippine that so people are looking at the safety of school buildings quite a lot because they can be used as as safe havens after an earthquake and so on. So there and also their children there, so that that means there are good models to look at both for design and evaluation. So we have a typical building that we typically use for our. So so now what I'm going to do is very quickly to show you where those nonlinearities in this
1:21:24
simple building can exist and how IT apps can go there. So we are going to look at this is the demonstration building. It has three floors. It has state walls and walls and so on. And then let's go on. That's the structural system in this one. Those are the foundation. That is the floor plan. And then we will look at this. And then we say so typically in a SAP or apps, the workflow from the for the typical model is like this. You start a new model. You model the stiffness.
1:21:54
To model the stiffness you have the materials and sections. You draw the elements. Then you have loads define. Then you assign the loads to the members that you have drawn and then you run the load cases. Then you get the analysis and from there you can determine so many results. So this is the typical workflow for creating models in many software and especially in ABS and separate housing. And this is taken from one of his lectures. So you can get more details about him on his channel.
1:22:23
And non-linearity can come in all those places. So in definition of materials, definition of sections, assignment of the nonlinearity and also on the load cases and running the analysis. So non-linearity can come into play in all of those steps. So let's look at the beam first. Very simple one. So in this case we have two hinges. Actually you can have three inches two for the moment and one for the shear in the middle. So you put those hinges there.
1:22:49
And then you can see from here that that hinge that I showed you earlier. The lumps plasticity is derived from the section which has the materials stress stretched in curve and that is converted into moment curvature curve. And that is you know if you're not using it directly you can use the AC which is derived on that. And then you put those, those ASC kind of definition back into the hinge properties. And then for the literal cyclic load where the loads are being reversed,
1:23:21
you also bring in the cyclic response degradation into that ASC. So combining traditional moment that AC hinge is defined you use that, put it back and you're ready to model the beam. So that's the general accumulation of non-linearity in this case. And then you can model yielding plastic rotation acceptance criteria hysteresis type and many things. So like I said many of these slides you can get the when we post this recording you can look into that.
1:23:54
And then you can study these slides in more detail more detail. This is mostly for reference. So this is what we can do through the beam hinges. And then to define it in it apps you once again like I mentioned to you, you basically define a section. Then you pick the actual hinge like in here we are picking the moment M3 hinge. You take that hinge, you give the property and you give the load points on that AC. Basically you give the points on that as a curve.
1:24:23
Those, those A, B, C, d and then you define them in the table. As you can see here, both for negative and positive direction which could be same or different. If for example, if it's a section with only negative reinforcement and nothing but the bottom two direction moment will be different, not the same. Similarly, at the middle, if the top enforcement is smaller than the middle reinforcement, the two direction will be not symmetrical. So it all depends on your section,
1:24:48
and you should realistically put those numbers in this curve. And there are tables. There are guidelines on that on the essay book you can follow from there. So you put everything together and then you run it. This is a short video showing that once again it's the same, same thing that I just showed. But I just want to show you how you go there. From there you select that and then you add that, and from there you select the hinge, and then you enter the data,
1:25:15
and then you assign this one, and then you select the beam there, you want to put it. And then you locate the hinge over there by the offset. From this the start point one hinge two three hinges as many as you want. And then you assign them together on that one. And then the hinges will come here and there you are. So which means the beam beam nonlinearity is now modeled for this one. Using the M3 hinge alone you could add a sheer hinge, typically with that if you want to.
1:25:46
Normally we we rely on shear to be elastic or linear and check the capacity not model explicitly. But we could then columns. Let's let's look at the column column is a little bit more complicated stuff because column has interaction. So we have a moment and we have the moment which is changing. So we have a low moment and large large axial force. We have a low axial force large moment and so on. So there is a balance condition. So there's a tension there's compression.
1:26:16
So there's a whole thing going on as you know in PM interaction. And the problem now is this one. Just look at the next one I'll come back to this one here. So this is the problem here on the left hand side you can see is a moment curvature curve for a typical column. This will depend on the angle of bending. It will depend on the presence of axial load. It will depend on the cross section and material. So that means the moment curvature non-linearity is not section dependent
1:26:45
only it is direction dependent, unlike beam where it just depends on the section in a column. It depends on the direction of bending as well as on the presence of axle load. The color diagram that you see in the middle in an interaction between the axial load versus the curvature capacity in all round. The section, as you can see, is very costly and reduces grossly. That means this whole complex behavior of the column needs to be modeled.
1:27:14
If you want the nonlinearity to be captured, and which obviously 1 p.m. hinge alone will be hard to do. So that's why the fiber section is comes into play, where we can model this thing rather than a single hinge with predetermined curvature or hinge hinge data. We define the cross section material and it is. The computation is done on the fly. So this is what will happen. You take the cross column cross section and then you divide it into coarse fibers. This is not going to
1:27:48
give you the same kind of moment richer that a full integration will give you. But it's good enough. So from there, you put that hinge over the length so that you can and based on the actual bending moment, actual direction. So on this is the the hinge properties are the stiffness properties are calculated on the fly non-linear properties and then used back in the analysis. So this is a good way to do that because it otherwise you will require thousands of points of
1:28:21
moment rotation curve for each direction and each load value. And we can still do that in SAP. You can actually input that whole interaction if you want.
1:28:32
So this is how you assign a fiber hinge is similar. But now you go back and you define it at the at the level of the fiber rather than the previous way that we short video where you enter the low deflection curve directly. And this is just another short video where you can see the same thing that we did for the beam. You go there, you start select. From there, you select the fiber hinge now and you enter the data same way that we mentioned earlier.
1:28:59
This time you will have to add the material properties there too. And then you select where you want it. And then you select the column. You assign the fiber hinge and then PM hinge. So similar video, similar way we can assign the fiber hinge. This one is presenting the beam hinge assignment similar to the beam that we saw earlier. So let's get back to now. Shear wall modeling. Shear one modeling is is obviously more complicated,
1:29:27
although it could it has previously been simplified as a large column model. So but you can see you can understand that the the shear mall behavior is more complicated because of the section is is not is the linear assumption of the strain, although shown here may not work very well. We have we have the confinements. You know, we have the distribution of reinforcement uneven and so on. So the wall modeling and it is also varies along the height.
1:30:01
So we can see that modeling the wall by just a pure column may not be justified in all cases. So which means there are other ways to model this one. So we have layered shell approach fiber. We have white column and we have nonlinear shell. Shear springs can also be used. So in this building we will we can show the this example. So this is the first of all the fiber model the same that we used for column. So this is very similar. So you can like just like the column fiber model.
1:30:32
You can take the wall and you can divide it into steel fiber and concrete fiber. Concentrate them like this one in zones get their their cumulative properties some them or average them. And then so so ten bars are resented by one fiber for example. And similarly large concrete areas represented by one concentrated zone. So we reduced the
1:30:56
number of fibers much smaller than originally required. Then modeling every bar because we can based on the size, we can see that most of the bars will be in a similar state of strain and stress. So we use that and then we can come back to use this in the model modeling. So this is one way. So this is the fiber section assignment to the wall. And you can see from here that you can divide the wall into multiple fibers with the coordinates and the properties
1:31:25
and the materials and degradation and everything you can assign there. And then assign this to the peer section. Next one is the layered shell layer. Shell is a very good way to model complex behavior of walls, because layered shell can model can divide this section of the shell or the wall into as many layers as you want. For this wall shown here, typically we will use seven layers. For example. In this case the outer cover will be. Each side will be one cover, horizontal bar, one
1:31:59
layer and vertical bar on one side, one layer, middle zone, one layer, and so on. So you can count seven layers that you can represent one section and each layer can have independent properties, independent non-linearity. So you can horizontal bars have a yielding on the horizontal direction. Vertical bars are building on the vertical direction. You can specify the angle of the five, the nonlinearity and so on. So this is a very good way to model the walls as well as slabs.
1:32:28
When we want a behavior in plain you know for first transfer. Otherwise cannot be model non-linearity of slabs especially in diaphragms and so on can be modeled. Also we can model deep beams by using this approach, because you can divide the depth of the beam into, you know, model it as a, as a shell and then make it non-linear. So this is a very powerful tool that can be used. So this is how you would enter the data in it ABS or SAP. So you can select from there you can define the layers
1:32:59
each layer you can give the properties a lot of properties can be given. And then the program will use them as a composite non-linearity when it does the analysis. Messy walls. This is something that we often miss in our analysis and we say okay, we I don't need to consider the message world. I can ignore them. I can use the bare frame analysis because I'm making making a safe assumption by not including the wall. But that's not really true because masonry walls can really disrupt
1:33:27
the the seismic performance of structures. They can attract the forces in a long way. They can break unexpectedly. They can, you know, shift the forces away. And so many, you know, it's something that we don't have time to discuss. But normally mystery walls, especially the missing walls, which are framed by beams and columns, infill should be modeled for both for linear and non-linear analysis. So this is just the the anomaly. The easiest way to do is is to use an equivalent structure.
1:34:01
That means just replace the wall by a diagonal, and then it can be assigned the properties, as you will see here.
1:34:10
So we can you can for example, if these are the walls here, there are very detailed guidelines in AC here that you can use to model these walls in multiple ways. Single double construct. And these trucks have a have adverse effect on the columns here. That can also be modeled by shifting the the location of the struts. And so these are the formulas. So struts can be modeled in by two ways. One is that you convert them into tension compression limit members, which means that you say okay,
1:34:45
this wall will fail after this much compressive force. So you can simulate the behavior that the wall is not broken. In the beginning it had a stiffness after reaching its capacity is gone, so it is either there or not there nothing in between. So that's one way to modulate by using tension compression limits on these struts. And the formula is given. Their second one is to model the behavior as true nonlinear axial hinges P and deformation hinges.
1:35:13
And you have these curves available once again from ASC. And you can see for enforced and unreinforced Missouri, the hinge can be generated. And it can be assigned as a center point as a p hinge or only. And those axial hinge we already saw in it and saps, we have that ability. And we put this curve there. And then we can just go ahead with the nonlinear analysis. So this is the the way that you can do that. And we must make sure that there is no tension capacity only compression.
1:35:47
So here we can model only on one side. So these kind of things have to be clear that when we model them then we come to the the structure level which we are done with the, the cross section and material level at the structure level, P delta is something I think you all know large displacement and a small one local. Both should be considered generally. So this is not difficult now these days to do that. By default most analysis should switch on the delta analysis in the programs
1:36:19
because they should capture the the structures. Nonlinear non-linearity in geometry. In any case, because especially for tall buildings, for slender structures, they can affect your results significantly without doing much work. There's not much data involved in that one. Let's look at now the including some of the special element like for example, buckling restraint braces. Working strain bases are very good tool. You know available now these days has damper as well as stiffness
1:36:48
as well as a nonlinear energy dissipation devices. So this you know, you can put them here. You can see in the structure here you can use them especially for retrofitting. And once again they can be modeled as using an axial hinge similar to representing the mystery wall we talked about. And then you assign this genesis curve and curve to this one. This normally is taken from manufacturer's data. So you would get it from the manufacturer. And then you input the data.
1:37:15
And now both it adds in SAP have a way to model the BRP effectively. So this is this can be done. Isolators coming now to the boundary modeling. Once again this advice topic we will generally discuss briefly. So we have LED rubber bearings. We have friction pendulum. We have high temping rubber bearings. So many types of isolators are there. And and sometimes they are placed at the base of the foundation. Sometimes they are placed.
1:37:43
Now many people are placing them mid story height rather than especially for the retrofitting because it's hard to do that. So people are just exploring whether we can place them in the mid story height. So these are various types of base isolation that can be placed, as you can see at the base or at the mid story. And they can be modeled using non-linear links as I mentioned earlier. So non-linear links have the ability to model almost all kinds of isolation system, even very complex ones.
1:38:13
So once again, once we have the data, we take the data. And it's not easy because getting the data, converting it into the input requires considerable understanding. And once again, one of the this is currently being done is the modeling of base isolators for school building that I mentioned earlier. So you put all this data. Unfortunately now we are running out of time and then we come to the running and interpreting results. So once you have created the model that's only part of the story.
1:38:44
Once you have done the modeling next is what analysis to run. And that can range from so many types of analysis in eatables and SAP and other programs. So choosing the right connecting this model to the load patterns, to load cases, to analysis types is another step and which is very critical. We have to match the analysis type with the type of non-linearity that has been modeled. So we have the nonlinear static pushover most common one simple one. And we have the nonlinear dynamic
1:39:15
or dementia response history, whatever you call it. So analysis model should match the question. Push over for capacity time history for demand under specific ground motion. So once again pushover is is easy. Not that difficult. But it may not be applicable to all the structure. But it's always good to do push over, you know, in certain direction just to see how the structure is responding. Time is obviously general, but it requires a lot more data and time to run. So pushover analysis.
1:39:45
Normally this is the procedure. You must start with gravity lock the gravity loads first define the load pattern, set the displacement control, and then enable the delta and large displacement, and then run the pushover. So that's the most typical way to do that. We have to select a particular load pattern normally a model shape or whatever. And then you make sure that you monitor the correct displacement. And then it's so you know that we make sure that we are monitoring
1:40:14
the displacement compatible with the load pattern that we chose. Trimester analysis. We need ground motion selection field by itself. We need the integration methods the time step selection and damping models. These are the some of the key decisions to be made in the time. And and they all obviously need deeper understanding of dynamics as well as earthquake engineering. So few just common things here. Issues could be certain strength loss. Brittle hinges can fail when you run this one.
1:40:47
You can have the negative stiffness effects. Sometimes that that will happen. Convergence will not happen. The step size is too large. You skip over some important event and so on. So you know we have to try many things. First time. It will be unlikely that you succeed once you run the nonlinear analysis the first time. So the last step is comment. You should see that not not all convergence failures are indicate that there is a bad model.
1:41:15
It actually means that the structure may self be be bad, and your analysis may be just telling you that this structure is that the response is. So we have to be careful not to to quantify a non convergent model to bad analysis and look at the structure itself for the source of that instability. So finally we look at the hinge status output whether pushover or time we can we can finally go and go to look at the the status of the output in terms of the level of performance
1:41:50
and that they define, you know, in terms of yielding and immediate occupancy collapse, ultimate and, and definite failure and so on. So depending upon the code you're using, standard you're using, you will have to come up with a way to assess the output from the non-linear model and analysis, which is ultimately the reason that we started this whole thing for. So we have to look at many things during the results, global response as well as the local response story,
1:42:21
drift profiles, storage, shear profiles, energy dissipation, residual drift. These are the common ones for tall buildings, and there are others for other types of structures. So common pitfalls and checks that we should do first most often is forgetting the gravity loads. Gravity loads sometimes I've seen many, you know, if they just apply the lateral load directly, but the gravity load has to be applied and locked first and start from there.
1:42:47
This is one wrong hinge locations not matching with where you expect the the large
1:42:54
forces to occur for yielding. Ignoring the shear failure and not modeling it or not checking it. Either way over interpreting the precision we are trying to get very, very, you know, fine results. Third 42nd degree of decimal is useless in a non-linear analysis. We are trying to just look at the overall response and then we have to use it. The check of the basic stiffness and sequence. Everything should make sense. It should not.
1:43:24
We cannot expect a structure to be giving you results that make no sense. So common sense should prevail, and those sanity checks should confirm that some advanced considerations very quickly. One is because I expect I expect some some questions on that. So I'll just just a few of them. BM engineer interaction. People always worry about that, that we normally do not consider that that's true. Most hinge models in software do not consider this interaction.
1:43:53
And typically the way to do that is to separately put a share hinge and maybe a non-linear and. Of course interaction won't be there, but the most behavior in real structure should avoid shear failure or, or shear so large that it will start affecting the PM. So that is controlled by sheer shear stress ratios and shear force versus capacity versus demand ratios. So those codes normally control that. So this is an area of research.
1:44:25
In research you can but in most practice it has to be done independently PM has nonlinear shear as linear as a capacity check force. Check out by adding both of them together. Whichever fails first should stop the model. Second one splices and encourage slippage in non-linear models, because when we assume a nonlinear hinge, we are assuming that there's. The Anchorage is perfect and this prices are perfect and they can actually fail first.
1:44:54
Once again, the following the good detailing rules and following the proper, you know, Anchorage splicing is a way to prevent that from happening. And this is a failure that we should not expect, so we should prevent it. Once again, nonlinear analysis should be should be performed with the assumption that these will be prevented by proper detailing and proper design models are available. You can model. You can use non-linear links to model many of these,
1:45:21
but there are highly complicated and advanced good for research, not so good for practical usage. And many of these code methods already include these in their caudal hinges. These like strain rate effects, cyclic regulation, fracture mechanics, damping. So many things are talked about in nonnative analysis, but they are not actively used in practice because they are good in terms of research, verification of lab results with models
1:45:51
and developing of new models that very that that are compatible with lab results. We need to include all of these things. But normally in practice their application is still complicated. So lastly about the tools. Normally we have two variety of tools. One are used for research and want to use for design. So we have abacus and so on and open seas. Very good for research. They can do a lot of things that we can't do in a typical design,
1:46:20
because often they are not required or they are too difficult to model. So we can still do them by using those tools. On practice we use normally Sep 2000 eTap form 3D and others and they as I showed you, they captured most of the practical range of nonlinearities and cases. But if you are looking for something more, then you go to the left side. Practical guidelines, a couple of them. I think this will mostly be useful as a reading material. So first thing is what behavior do I need to capture?
1:46:52
This is very fundamental question why are you doing it? Do you need it? If you do need it, why? What do you want to capture? What level of accuracy? I started with this one in the beginning and what resources are available? What software do you have? How many train people do you have? What do you have? Because nonlinear analysis requires deep understanding, lot of time and a lot of good resources. So if you have all that you can go to advance advice you go to or you start simple.
1:47:23
And these are some of the references that you can use when you start. I think you can use that slide later. So with that, I think I'll come to the final slides, which this one is, you know what to use when, for example RC beams, when to use moment, when to use PM hinge and so on. So these slides will be useful I plan I have already given some of these slides in a, in a, in a document that I shared on LinkedIn, and I can put some of these slides again for reference in that document.
1:47:55
So with that I think we come to the end. Thank you so much.
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