 We have been discussing problems of nonlinear structural mechanics and we are trying to formulate finite element models for that. With that in mind we have reviewed the basic principles of continuum mechanics. So in a linear analysis displacements and strains are small, stress strain relations are linear and equilibrium equations are derived based on undeformed geometry and principle of superposition holds. So these are some of the facts which make linear analysis very simple. Now whereas in a nonlinear analysis displacement and strains need not be small, special efforts need to be expanded to characterize rotations. Then the geometry of the object, the stress strain relations, the boundary conditions could change during process of deformation and that has to be taken into account. The definitions of stress and strain and the formulation of the governing equations also need to take into account the changes in the geometry of the object and constitutive laws and boundary conditions. This necessitates introduction of newer measures of stress and strain. We need to have properly understood definitions of stress and strain. Then principle of superposition which is the bread and butter of linear analysis no longer holds. So if there are multiple loads that act on the structure you cannot perform separate analysis and try to superpose. For example in earthquake response analysis the self weight effect and response due to earthquake loads I cannot be analyzed separately, you need to handle them together. Now we introduce few strain measures. The infinitesimal strains we showed that for body under rigid body rotation the strains would not be 0, so that is we cannot use infinitesimal definitions of the strain. Now new measures are needed so the criteria for establishing these measures we looked at two such requirements that rigid body motions imply zero strains and then for small strains the infinitesimal strain definitions are restored. That helped us to introduce the Green Lagrange strain measure DX is FDX that is the capital DX is the line element in undeformed geometry upon deformation it becomes DX and F is the matrix that relates them and this is a deformation matrix and in terms of deformation matrix we consider the change in square of the lengths of the line segments in deformed and undeformed positions and we obtained this quantity and this quantity E which is half of F transpose F minus I is the Green Lagrange strain measure and this helped us to this is associated with a quantity known as magnification factor as defined here and in terms of the Green Lagrange strain measure this magnification factor is given by this and this is a measure of shearing strain. So if a line element with direction cosines N1, N2, N3 deforms and it undergoes a change in length and this magnification factor is expressed in terms of elements of Green Lagrange tensors. So if the direction if the line segment is aligned with X axis the magnification factor will be E11, if it is aligned along Y axis it will be E22 and so on and so forth. Similarly the shear strain measures if one of the line segment is aligned to X axis and other one aligned to Y axis then gamma AB gives the shearing strain which is epsilon XX and so on and so forth. So we also introduced this is this definition was with respect to the undeformed geometry with respect to the deformed geometry we get another strain measure known as Eulerian strain or Alman C Hamel strain and this is as shown here. Now there are in literature there are different definitions of measures of strain so an engineering strain is defined as change in length by the original length, Green Lagrange is the magnification factor L square minus L naught square by L naught square logarithmic measure is given by this and the so called extensional ratio is given by this. So this I have plotted on X axis L by L naught has been plotted and on Y axis these different strain measures are plotted and as you can see that they show different behavior especially for large strains and when we are reporting results this has to be borne in mind. The stress measures they commonly use stress measures as a Cauchy stress which is defined with respect to the deformed geometry so we define this with respect to the internal forces in the deformed object with respect to area defined with respect to the deformed geometry and that looks to be the most natural way of defining stress. But when we apply this definition to problems of non-linear analysis we will not be knowing the deformed geometry in advance so that this utility of Cauchy stress tensor thus becomes you know difficult and when this tensor becomes it becomes difficult to use this tensor in non-linear analysis so this has necessitated alternative definitions there are two approaches to develop measures of stress. One stress as a measure which conjugates with a measure of strain to produce internal energy there is one way of looking at it, the other one is as a quantity which produces a traction vector in conjunction with a normal vector defined with respect to a surface element. So based on this different measures have been introduced the Cauchy Euler stress based on deformed configuration is I have discussed it is given by this, this is a traction vector on a line element with outward normal N and this is expressed in terms of stress components defined with respect to certain cardinal coordinate systems. First Piola-Kirchhoff stress is based on undeformed configuration we know how the surface element transform through this relation and we define a stress vector acting on element DA with outward normal N which produces force DF, this force is in the deformed geometry and we introduced a quantity capital P in the same way as the Cauchy stress is defined and this capital P is known as the first Piola-Kirchhoff stress and we have seen that this is not symmetric it has nine independent components. The second Piola-Kirchhoff stress eliminates this awkwardness in the stress matrix it produces a symmetric matrix it is based on undeformed configuration we introduce a pseudo force vector DP hat which is F inverse DF this again this transformation is based fashion after the transformation of DX equal to FDX for line elements. So based on this we introduce the second Piola-Kirchhoff stress tensor and we also we saw that this is symmetric and we also establish how these three stress measures are related to each other. Now to develop the finite element formulations we need to have certain clarity on notations for configurations and deformations. So we start with configuration C naught and we consider a point P and a line segment P Q of length DS naught so we introduced a special notation system we use left superscripts and subscripts. So Q J I is to be interpreted like this quantity Q is measured in configuration C J with respect to its value in C I it is how we should interpret that. So here we introduce the coordinate system the left superscript corresponds to the configurations and the right subscripts corresponds to the components of the quantity that we are considering. So this is X1, X2, X3 in configuration 0, 1, 2, 0, 1, 2 and 0. Now configuration is denoted by C0, C1, C2 coordinates of a point X left superscript 0 and so on and so on, 1X, 2X volumes 0V, 1V, 2V areas 0A, 1A, 2A similarly density total displacement 0U, 1U, 2U. So this is configuration C0, this is some intermediate configuration C1 and this is the current configuration. So this U is the incremental displacement from C1 to C2. Now the discussion that I am going to present and the notational system that we are going to use is based on the material available in two books, one is by Professor J N Reddy an introduction to nonlinear finite element analysis and the other one is by KJ Bhattay finite element procedures. Now the material that I am going to use in this lecture is largely based on the coverage given in the book by Professor J N Reddy. The idea here in this lecture is not to develop the complete procedure for nonlinear finite element analysis but instead to give a flavor of how to, what are the issues that one must be aware of in formulating such problems. The detailed formulation of finite elements and their application in specific problems etc are not covered in this lecture. So the objective is to simply introduce you to the basic issues that arise in dealing with nonlinearity and some preliminary indications on how one could proceed with producing a finite element model, okay. So there are a lot more details that I will not be able to touch upon but hopefully this gives you necessary motivation to study this topic further. Now a particle in C0 the position coordinate is X1, X2, X3 with superscript 0, it moves to C1 and it occupies the point X1, X2, X3 at configuration 1 and at C2 it is at X1, X2, X3 configuration 2. So the motion from C0 to C1 the displacement is given by this, U from configuration 0 to 1 is given by, so the components are given as shown here. For motion from C1 to C2 U configuration 1 to 2 is given by this, X2 minus X1. Similarly components are this. Now the law of conservation of mass demands that the mass of the object in different configuration should be the same. So if I have now in second configuration integral over 2V of 2 rho must be equal to integral of 1 rho over D1V and that must be equal to this volume is 1V and this is 0V that you should notice and this is in the undeformed or reference configuration. Now you must be careful in interpreting this quantity it is not D square V, it is D of 2V because we are using left superscripts and subscripts it should not be confused with powers for the preceding quantity. Hopefully that confusion if you carefully understand the flow of logic you will not get into. Now by substituting for X2I in terms of X0I we can show that the law of conservation of mass leads to this requirement between configuration 0 and 2. So this that would mean this 0, 2 Jacobian is given by this determinant and we have this relation between densities at various configurations okay. This if this is satisfied the law of conservation of mass is obeyed. Now let's look at strain tensors for C1 and C2 configurations. The green Lagrange strain tensor we have derived this for configuration from 0 to 1 is given by this, from 0 to 2 it is given by this. If you look at incremental strain tensor that is defined like this it is change in length of the line element in configuration 2 to 1 okay. So we are this is an increment, incremental deformation from C1 to C2. The idea is that as we apply the surface traction and body forces for each increment of the force the body occupies different configurations and while solving the problem we will divide the applied loads into small increments and the increment from say C1 to C2 is affected by a small increment in the load we can linearize the relations when moving from C1 to C2 that is the idea okay, so that you should keep in mind. Now this incremental green Lagrange strain tensor is given by this and we can define this with respect to the DS in the original configuration I am adding and subtracting that and this enables me to write this in terms of the green Lagrange tensor from configuration 0 to 2 and 0 to 1 and this by rearranging the terms I define these quantities E ij, eta ij these are linear parts of the incremental strain tensor and the non-linear parts respectively, so this is given by this. So if you use the definitions of green Lagrange tensor capital E's you will be able to show that the linear increment in U i the strain due to linear increment in U i is given by this and this is non-linear increment in U i, so these are incremental green Lagrange tensors this will be needing in our formulation. Then the updated green Lagrange tensor you see that the green Lagrange tensor from configuration 0 to 2 is helpful in total Lagrangian formulations where we refer I mean formulate the problem with respect to quantities in the undeformed configuration. To facilitate updated Lagrangian formulation we introduce another quantity this is 2 epsilon ij see this is epsilon ij with respect to 0 but here for the increment from 1 to 2 I introduce this definition and again using the following results again using the deformation matrix for from moment deformation from 1 to 2 we get all this and we get the increments the linear part and the non-linear part as shown here. Now the Alman C Hamill strain tensor or the Euler strain tensor we can consider if you consider that the body has reached configuration C 1 from C 0 in several increments. The deformation from C 0 to C 1 could be large but we now wish to move to configuration C 2 the increment from C 1 to C 2 is taken to be small and we could refer to strains with respect to C 2, so that means we are looking at current configuration as in the Euler definition Euler strain but we are movement is from C 1 to C 2 and not from C 0 to C 2, so with that certain simplifications become possible and we will be able to introduce what are known as Euler strains this is epsilon 2 to ij, so the same configuration both current and reference configurations are the same therefore this Euler strain tensor. Now this can be again expressed by using definitions of displacement vector we get this and the linear part of this is given by this and this we omit now the two subscripts and simply write it as 2 E ij and this quantity is called the infinitesimal strain tensor, so it's Eulerian strain tensor but the linear incremental part. The strain components these strain components conjugate with Cauchy stress components to produce the expression for internal as these two, it is clear that Eulerian strains are defined with respect to deformed configuration and Cauchy stress is defined with respect to deformed configuration therefore one can expect that they will conjugate to produce the internal work done, internal energy stored. We have Cauchy stress this is the internal force record in the deformed configuration and area record in the deformed configuration, so the configuration C 1 this is 1 sigma ij which is nothing but 1 1 sigma ij, configuration C 2 it is 2 sigma ij which is nothing but 2 2 sigma ij, now we can look at this diagram we have three configurations C 0 C 1 C 2, in C 1 and C 2 we have the Cauchy stress tensor this, this is in the deformed configuration C 1 and C 2. Now the Green Lagrange, no the second Piola-Kirchhoff stress tensor between C 0 to C 2 is this, so this can be written in terms of 0 1 plus an increment that means 0 to 1 there is one Piola-Kirchhoff stress and there is an increment, so this is Kirchhoff stress increment tensor. On the other hand I can move from C 0 to C 1 that is this, this is a second Piola-Kirchhoff stress from 0 to 1 and similarly I get S ij from 1 to 2, this S ij 1 to 2 I can write in terms of the Cauchy stress plus an increment, this increment is with respect to the second Piola-Kirchhoff stress, whereas this increment is due to, is based on the Cauchy stress and this is known as updated Kirchhoff stress. So we have Kirchhoff stress increment tensor and updated Kirchhoff stress, so that becomes important in our discussions in the second Piola-Kirchhoff stress tensor force in C 2 is transformed to C 0 and area is reckoned in C 0, so this is the relation we get between the Piola-Kirchhoff stress and the displacement gradient and the normal. Then updated Kirchhoff stress tensor it is useful in updated Lagrangian formulation, consider the point in P X 1, X 2, X 3 in C 1 the Cauchy stress tensor at P 1, C 1 is this what I am trying to define and here we have summarized, so this is the Piola-Kirchhoff from 1 to 2 in terms of Cauchy stress plus an increment, this is Piola-Kirchhoff from 0 to 2 in terms of the Piola-Kirchhoff at configuration 1 plus an increment, so this is the Cauchy stress in the second configuration and we know these relations I am simply putting it in the indicial notation, so this is the relationship between Cauchy stress and the second Piola-Kirchhoff stress and this is the relationship between the second Piola-Kirchhoff stress and the Cauchy stress. Now since we conservation of mass we have seen that this relation must hold good we can also express these relations, instead of writing the Jacobian I can write the ratios of masses and I get this relation, this is same as this except that Jacobian is written in terms of ratios of masses. Now what is relationship between Cauchy stress in C 2 and updated Kirchhoff stress, so that means I want now 2 sigma ij in terms of updated Kirchhoff stress, so that if we consider now the second Piola-Kirchhoff from 1 to 2 this is given by this as we have seen and this is the relationship between the Cauchy stress in the second configuration with the S from 1 to 2. Now this PK 2 stress in different configurations we can write by using this relation explicitly just for reference I have given, then relations between incremental stresses, so S ij 0 and S 1 P Q I mean this is how it is related, so this and this how they are related is what I am explaining, I mean these are fairly simple if you observe I mean sit with a pen and paper you can easily understand but all these relations would be needed as we proceed. Now for the discussion in this lecture we are limiting our attention to linear relations between conjugate stress trend pairs, material behavior is elastic that means constitutive behavior is function of current state of deformation and loading unloading path will be identical upon removal of the load the original configuration will be restored. Then relationship between Piola-Kirchhoff stress and Green Lagrange strain is defined through this quantity C ij kl where C is the material elasticity tensor, the stress strain relations in incremental form we can write in terms of this, this is the Kirchhoff stress increment tensor and epsilon kl this is the incremental strain and this is the C matrix. Similarly updated Kirchhoff stress increment and Green Lagrange strain increment can be related through this relation. Now C in different configurations can be written in this form, this is C in first configuration related to zeroth configuration this is the other way the relation other way. Now the formulation of finite element in this case will be based on principle of virtual displacements what it says is some of virtual external work done on a body and the virtual work stored in the body should be 0, so if you consider configuration C2 the virtual work is the Cauchy stress into the Euler strain virtual Euler strain integrated over the volume in the second configuration and this is the contribution from virtual displacement from the surface, this is body forces this is surface traction. Now this in the indicial notation it is given in this form. Now we cannot use this equation directly see if you see here we have used the Cauchy stress and the Eulerian strain that means both of them correspond to the deformed configuration so it looks nice but as I already said the definition of the configuration C2 would not be known, so we cannot use this equation directly. So the configuration keeps changing as the deformation evolves but in a linear analysis you can quickly recall in the assumption made in the linear analysis that the configuration of the body does not change so that the equations can be formulated based on undeformed geometry is not valid in nonlinear analysis. This calls for introduction of measures of stress and strain which take into account changes in configuration during deformation, this point I have been making couple of times already, I have made up this point couple of times already, this enables the evaluation of integrals in the expression for the internal work done over known configuration by introducing the alternative measures of stress and strain what we will be doing is we will be able to evaluate these integrals with respect to volumes in known configurations, so that is a you know another way of expressing the need for alternative definitions of stress and strain. So in our lecture we will use stress is the second pyrologic of stress tensor and the strain is the green Lagrange strain tensor, they are the conjugate in the sense that they produce the internal work done. This I have mentioned in the previous one of the previous lecture in the total Lagrangeian formulation the base and reference configuration coincide and is taken to remain fixed and this is a current configuration but all equations are formulated with respect to the reference configuration, whereas in updated Lagrangeian approach the reference configuration is updated with each increment in the load and we analyze the incremental displacement with respect to a reference state that keeps evolving as the loads are incremented. So let's take a look at the total Lagrangeian approach, so here all quantities are reckoned with respect to undeformed configuration, so the virtual work statement leads to these terms these are the terms that are present in the virtual work statement if you see here we have to analyze each one of them now, so this is the Cauchy stress into the virtual Eulerian strain and now this is second pyrologic of stress and the virtual green Lagrange strain, and this is similarly these are identities that they should be equal is an identity, so nothing is gained or lost, okay. So this is the terms involving body forces, this is terms involving surface traction and this is the virtual work statement of the virtual work principle in terms of second pyrologic of stress and green Lagrange strain measure and all volumes are now in the first configuration, this is 0V and this is surface in the 0th configuration. Now our idea is to now you know work with this and develop a weak form based on which we can develop a finite element model. Now as I already said we are going to take an incremental loading approach, so as the load is incremented by a small amount we wish to deal with linearized stress strain relations and to formulate that we have to get into many details, so the in this approach we will not formulate a nonlinear set of equations and then develop an independent solution strategy. The solution strategy is embedded into the development of the finite element model itself. So we have this statement, now we start working with the second pyrologic of stress from 0 to 2 is expressed in terms of PK2 from 0 to 1 and the Kirchhoff stress increment. Now so we will consider each one separately and then you know we will look at these quantities in terms of increments and certain reference values. So this 0 to 2 green lagrange strain tensor is written in terms of 0 to 1 plus these incremental quantities and therefore the virtual strains you use a delta operator and you will get this. Now delta of Eij01 is 0 because this does not depend on the unknown displacements, the unknown displacements are the increments therefore this will be 0 and I get this. Now we have the expression for this quantity E0ij that is this and therefore I will be able to compute the variations and for linear component and nonlinear component, okay. So this matter of derivations that you have to absorb and similarly this, the contribution from body force and surface traction is denoted as delta of 0 R2 and that is given by this. Now let's consider this part and write the 0 to PK in terms of this and we will expand this and this virtual strain also is written in terms of the increment and as we multiply we get one of the terms as Sij01 delta of 0 Eij D0V. This quantity represents the virtual internal energies stored in the body in configuration C1. Now by applying virtual work principle to body in configuration C1 we can show that this quantity is given by the body force and surface traction contributions in configuration 1. Equipped with that we have now the statement that we will need for formulating the finite element model. Now this is the expression we get. Now we can again summarize what these terms are. The first term is change in virtual strain energy due to virtual incremental displacement UI between configuration C1 and C2. The next term is the virtual work done by forces due to initial stress 01 Sij. This arises due to change in geometry between the two configurations. Now we need to introduce the constitutive relations. So the constitutive relation we use in the incremental form as shown here and that leads to the equation that forms the basis for developing the finite element model. Now before we jump into that we can simplify this further. If we assume the displacement UI to be small that means the increments are small, if load increments are small the displacement increments are also small. If we assume that then I can simplify and remove certain nonlinear terms and the constitutive law can be expressed as shown here and consequently the virtual work statement gets further simplified to this form. Now I have not talked about inertia forces till now but if inertial forces and I am not going to discuss this but at least in this part of the lecture but we can quickly you know see how we can deal with this if such terms were to be present. So the identity that we will use is the total inertial force in configurations 0 and 2 would be equal and this is the statement that is the identity. Then equation of motion while formulating the principle of virtual work now we need to consider inertial forces also and we get this statement. So this is a new term that has to be handled if you are dealing with dynamics. Now for purpose of illustration we can consider 2D element and develop the total Lagrangian formulation. Suppose this is configuration C0, this is configuration C1 and this is C2 and these are the coordinates the 2D element therefore the third direction is not taken and for sake of simplifying the notation we call X1, X2 and 0th configuration as XY and displacement 01 U1 that is this components as UV and the increments as U bar V bar, so this is UV, this is U bar V bar and these coordinates are XY. So this is the weak form that we will have to use to formulate that. Now what follows is you know we have to now convert all the expressions to the expressions in terms of incremental displacements and the incremental displacement need to be interpolated in terms of nodal values and once that interpolation form is established we return to this equation and get the finite element model that is the story. So we have 0E as this and in terms of incremental displacements it is given by a linear part and a nonlinear part, this is still linear in U bar so we get this equation, this is in the matrix form we introduce D and DU and this is U bar. So this is a relationship between strain and the displacements and now if I substitute that into the terms in the virtual work statement we can simplify the first term in terms of this representation and we get this term simplifies to this, where this matrix is the elastic C matrix that will stress and strain we are assuming in this formulation that it is an orthotropically elastic material. Now the other terms from eta, increments eta that again if we look at eta this is nonlinear terms so we write in this form U bar V bar is D of U bar because you can see here U bar the U bars are here this is nonlinear and the variation is obtained in this form so this is the way you know written in this form where we introduce certain notations as shown here. Actually the resulting format of these terms is still not in a form suitable for development of FE solution, we need to reorganize these terms so that requires certain you know introduction of certain vectors and we write the stress matrix in terms of a vector like this and these coefficients in this form and we multiply them out and if we do that we will be able to get the expressions in this form some of these details you need to you know absorb once you are done with this we are now ready to launch the FE formulations. So what we do is the incremental displacement U bar V bar is interpolated in terms of N nodal values, sizes are the interpolation functions and this is written like this and the total displacement is also represented in the same form which is this. So I have psi capital psi matrix and delta bar vector and delta vector as shown here and then we get into the principle of virtual work statement and we consider each term and submit these substitutions and then assemble all of them and obtain the final finite element equation. So we start with this so we have now already derived the terms this delta E naught in terms of U bar as here and that we are using here and here I am now making the substitution for U bar in terms of the assumed shape functions and that is this and we you know introduce a matrix BL which is D plus DU into psi. Similarly the other terms we simplify and we get BNL and so on and so forth and body force and surface traction terms are also simplified in the same vein and that leads to this is the weak form and this is the finite element equation. This KL is this, KNL is this and these are the forcing terms you can you know spend some time and go through that. Some of the observation that we can make is that this K matrix is symmetric this is because these matrices are symmetric the material constitutive matrix is symmetric and this S01 is also symmetric and also as already said this is an incremental formulation. So the stiffness matrix here is the tangent stiffness matrix that means we already that linearization is implied in our formulation okay. So but however we can still note that if you were to do out and out a linear analysis this delta bar will be simply delta and this quantity will be 0 and KNL will be 0. So this reduces to the linear stiffness matrix if we are not including nonlinear behavior. So we can go through the details of these formulations and derive the expressions for different components of these B matrices as shown here and we can proceed and you know write these equations in a way that the elements of these matrices can be evaluated. So I am not going to get into these details I will stop the discussion on total Lagrangian with this. We can quickly take a look at this updated Lagrangian approach. Now here all quantities are recant with respect to the latest known configuration that is C1. So the identities are now the Cauchy stress in configuration 2 and the Eulerian stress strains they are related the increments these are related to this and this is the relation for body force this is relation for surface traction. So here this Epsilon IJ1 to 2 is updated Green-Lagrange tensor this F1 to FI is a body force referred to in C1 this is surface traction referred to in C1. So we can now rewrite the virtual work statement in this form and now using the definitions of the strains we can derive the virtual strains for linear and nonlinear parts and the Paolo-Kirchhoff stress from 1 to 2 is decomposed into Cauchy stress in configuration 1 plus updated Kirchhoff stress. So that if we substitute and simplify we get the, we are aiming at getting the weak form appropriate for this model. So this is the weak form now we are going to use the virtual strain expressions and if we consider the equilibrium of body in C1 this is the principle of virtual work applied to that and from this if we now use the constitutive relations we get this equation and as we did in the total Lagrangeian approach if we now linearize we get these equations and this leads to the required weak form that forms a basis for application of development of financial model. The Cauchy stress here are evaluated using this relation where this Epsilon KL1 to 1 are the Alman C strain components are the Eulerian strain components and that is defined through these terms. Now I have not given all the steps but we can use updated Lagrangeian formulation and derive the finite element model for the 2D element just based on the procedure that we use for total Lagrangeian except now that you have to take care of the fact that we are dealing with all quantities in configuration 1 instead of 0. So I have reproduced here the basic equations and the various matrices that arise in this formulation. As I said in the beginning of this lecture the objective of this discussion has been basically to provide a flavor of how to carry out non-linear analysis what issues arise as geometry of the body changes during deformation why we need to introduce newer definitions of stresses and strains and consequently what is the role played by these newer definitions in the formulation of the FE model. Now with this the we have come to the conclusion of this course so in the next lecture what I wish to do is to quickly review what we have done so far and where we can go from what we have learnt till now. Now the material that is covered in the course you know from the coverage that we have already achieved in this course we can move on further to study problems of material non-linear I have not addressed this issue in most of the discussion that we had. So to study this you need to first understand theory of plasticity and continuum mechanics has to be generalized to allow for non-linear material constitutive laws and that requires newer preparation. Now in our discussion of stability analysis again we have included geometric non-linearity alright but there are different layers of sophistication that is possible in carrying out stability analysis we could deal with the modified definitions of stresses and strains and the updated Lagrangian total Lagrangian approaches and investigate the problem of stability once again and see what we did earlier without being aware of all these refinements what was that we achieved that can be reexamined. Now in the problems of stability analysis what would happen if we consider material non-linearity that is one issue that we have not taken into account there is another topic what is known as hybrid testing in recent years the sophistication in experimental hardware and computing have led to newer testing procedures in a laboratory they are known as hybrid testing procedures this hybrid the word hybrid here connotes indicates the combined use of computational and experimental methods. So there are methods especially in earthquake engineering these methods are being developed so I will be briefly touching upon in the next class the idea here is to model that part of the structural behavior which is reasonably well understood through computational schemes and to resolve to experimentation only to complement what is not clearly understood in computational modeling and there are complicating issues that an analysis computational modeling and its analysis of a resulting model and the testing of the structure need to go hand in hand many times in real time so that includes I mean that introduces newer challenges. I also briefly talked about problems of structural health monitoring so in structural health monitoring we make finite element models for existing structures typically and we wish to combine the predictions from the computational model with what we actually observe the observations that we make on structural performance. So there exist a need to combine the experimental observations with computational models and a framework for that as I already mentioned is so called Bayesian filtering methods. So here the finite element methods have to be combined with experimentally observed you know measurements and a newer entity which is essentially a mathematical model that has assimilated the observations made on a existing structure need to be developed that new entity can be used to you know prognosize the future behavior of the structure. One of the question that we have not discussed in this course is what happened to uncertainty we know that loads like earthquake, wind, guideway and even as waves etc are uncertain in nature and they are typically modeled using theory of probability and random processes. Now when structural behavior is captured through finite element methods either the loads or the constitutive relations or inertial properties etc if they are all modeled as random processes or random variables how do we combine the finite element formulation with the probabilistic modeling of structural behavior and how do you specify inputs how do you compute the responses and what are they expected response descriptors. So this is another area and one of the computational tools that is available in the literature is the so called stochastic finite element method. So what we have learnt in the course can be a framework to pursue this study. Another area where we have what we have studied could be of application in structural engineering is analysis of structures under fire loads. So the fire loads create time varying temperature environment in which the structure need to perform and stress analysis and heat transfer analysis need to be carried out to understand the behavior of the structure. They can be carried out in an uncoupled manner that is first we analyze the heat, perform the heat transfer analysis and find out the thermal characterized heat distribution, temperature distribution in the body and if constitutive laws are described as a function of temperature then that temperature dependence need to be accounted for in our formulation of structural behavior. Typically in problems of practical interest one need to consider both geometric and material nonlinearity in problems of analyzing structures under fire loads. So that offers immense challenge to analyst in terms of being able to handle the combined problem of thermomechanical stress analysis, treatment of uncertainties, treatment of material and geometric nonlinearities etc. Many structural elements display anisotropy so that is another feature that we need to develop that is we have basically in most of our studies we have assumed isotropy except in one or two small examples but many structural components like composites etc. inherently anisotropic in nature and inclusion of these features into mathematical modeling again offers challenges and some of this need to be taken up if we have to take the subject forward. So what I will do is we will close this lecture at this point, in the next lecture I will briefly review the whole content of the course and again revisit some of these questions with slightly more details. So we will close this lecture at this point.