We have been discussing analysis of 2-dimensional continuum, so we will continue with that, so what we did in the previous lecture was we considered a triangular element, so this element has 3 nodes 1, 2, 3, and at each node there are 2 degrees of freedom, so this is a 3-noded element with 6 degrees of freedom, so the field variables are U and V which are functions of X, Y, and T, and we interpolate the field variable in terms of the nodal values, that means U is interpolated in terms of U1, U2, U3, and V is interpolated in terms of V1, V2, V3, and the interpolation functions are derived, so that for example the interpolation function N1 at this node will have value 1, and at these 2 nodes it will have value 0, and in this particular case we use linear interpolation functions. We also discussed rectangular plane stress element, again the field variables here are U and V, this element has 4 nodes 1, 2, 3, 4, and at each node there are 2 degrees of freedom, therefore it's a 8-noded, sorry 4-noded element with 8 degrees of freedom. Now to facilitate the development of the element we made a coordinate transformation, so if you recall the elements of stiffness and mass matrices for the element have to be evaluated by carrying out these integrations, this is stiffness and this is, this is stiffness matrix, this is a mass matrix, so to facilitate this integration we transform this rectangular region to a square region so that integration can be done from minus 1 to plus 1. Now in the examples that we have considered so far it has been possible for us to evaluate these integrals in closed form, but this may not be always possible, for example if you have a quadrilateral element or a triangular element with curved edges or a quadrilateral element with curved edges like this, evaluation of these integrals will not be straightforward, so it becomes necessary to deal with 2 complexities, one is a complexity associated with the geometry and the complexity associated with evaluation of these integrals, so these 2 issues we will try to take a look in today's class. So we will consider to start with a linear quadrilateral element, so this has 4 nodes this is a coordinate system U and V, X and Y and U and V are the displacement fields, U is along X and V is along Y and this is the quadrilateral element with nodes 1, 2, 3, 4 and the nodal coordinates are X1, Y1, X2, Y2, X3, Y3 and X4, Y4. Now we need to now approximate the field variable U and V, variables U and V within this domain. Now what we do is we introduce a transformation so that a point in XY here gets mapped to a new set of coordinates XI and ETA that we have to introduce now what these are and this coordinate transformation is implemented exclusively for the purpose of evaluating these integrals, we will see what transpires when we implement this transformation, the main idea is to introduce this transformation to facilitate evaluation of the integrals, the stiffness matrix is given by this and the mass matrix is given by this, now the transformation that we are doing is from XY is taken to psi ETA, therefore we can consider the transformation in the mathematical form as X is a function of XI and ETA, Y is a function of XI and ETA. Now following the way we represent the field variables, we write now for this X and Y this expression where what we are doing is X of psi, ETA is being evaluated in terms of its nodal values and interpolation functions, similarly Y is interpolated in a similar manner, so what we are going to do is we are going to use the same interpolation function that we use for representing the field variables to represent the geometry also, that is geometry of this transformation. Now if you recall the N1 into N3, N4 for this configuration we have shown that this is of this form and compactly for all the four indices 1, 2, 3, 4 we have obtained this in the previous lecture, so we will use this now here. Now let us consider the line 2, 3, and this line 2, 3, now we want to investigate what is the relationship between the two, so let us consider the line connecting X2, Y2 to X3, Y3, upon making the transformation X, X equal to psi, ETA and Y equal to Y of psi, ETA how does this line gets mapped, or alternatively we can ask if this is a transformation we are using, what happens to this line 2, 3 in this configuration, whichever way you look at we will find the answer to the required question. Now let us therefore consider the line XI equal to 1 connecting 1-1 and 1, 1, that means XI equal to 1 this line, now at one of the vertices the coordinates are 1, minus 1, so what happens to X there, X is X1 into N1 plus X2 into N2, X3, X4, now how does this shape functions behave, except this N2 all other shape functions will be 0, so this becomes X2, similarly you can write the value of Y XI, ETA that means where does this vertex 1, minus 1 in this coordinate system go in the original coordinate system, so X has gone to I mean this, okay let us see, X2 we have found out similarly Y will go to Y2, that means if I am considering 1-1 that means the second point here it is going to 2 here, similarly the other vertex 1, now 1 you can verify that this goes to X3, Y3, now between these two point does the variation remain linear or not, that is the next question we have to see, now consider the line XI equal to 1, which is that line, this line XI equal to 1, now X of psi, ETA is X1, N1, ETA and wherever XI is 1, XI is there I will write it as 1, so I get this expression, okay, I am simply putting for XI 1, then I will use the properties of these functions and I will be able to show that this is given by this expression, it's a linear function of ETA, similarly the Y coordinate also get mapped to another linear function of ETA and I get X and Y to be this, now I can eliminate ETA from this and get the equation in XY plane what it is, so if you do that I get Y- this equal to this- this, so what I am doing is this I am taking to the left side, this to the left side and dividing this 1 plus ETA gets cancelled this is what I get, now this is actually the equation of straight line passing through X2, Y2 and X3, Y3, so we conclude that line 2 3 in figure A is transformed to line 2 3 in figure B, that means 2 3 is mapped to 2 3 here, so we can summarize this, this is the line 2 3 which comes here, psi equal to 1, psi equal to 1 is this line that comes here, ETA equal to 1 is this line it comes here, psi equal to minus 1 is this line it comes here and 1 2 is ETA equal to minus 1 that comes here, so that means that this quadrilateral is getting mapped to this, through the transformation that we are proposing, so these transformations of coordinates that is X, psi, eta in terms of N and nodal values of X and nodal values of Y for Y, this is quite similar to the transformation on field variable, this is what I have been pointing out, this is a field variable, UJ of T are the nodal values which are the degrees of freedom in our finite element model, that is what we conclude from this is the same interpolation functions these NJs are being used to represent both the field variables as well as the geometric variables, the displacement field and the geometry of the element are getting represented through the same set of interpolation functions, the resulting element formulation is called isoparametric formulation, okay, fine. So what we have achieved is we have mapped a quadrilateral to a square, so when we are performing integration instead of performing over a quadrilateral which can be quite you know unwieldy we have now the need to situation of being able to integrate over a square region, but what about the integrand, how does that behave? So let us consider the evaluation of the mass and stiffness matrices by transforming X, Y to psi, eta coordinates, this transformation we will now implement, so we need to digress a bit now to understand what the problem is, so first is we will consider a function F of X, Y in Cartesian coordinates and using this transformation X, I, eta if we map X and Y to X, I and eta what is the rule of transformation, see this is the rule as you may be knowing but how does it really originate? To be able to do that let us consider a point, this point with a position vector R in XY plane, so that is R is I X plus J Y, now psi and X are functions, psi and eta are functions of X and Y, so now I want that is this is a psi and this is a eta some coordinates, okay, so now I want to deal with this situation, so what is dou R by dou psi? It is I dou X by dou psi plus J dou Y by dou Y by dou psi, where I and J, I, J, K are unit vectors along XY and Z respectively, similarly dou R by dou eta is I dou X by dou eta plus J dou Y by dou eta, AB, the line AB if you see is what is this, so this plus this is equal to this, this is a vector sum, so if you are looking at AB it is simply the difference between the two which is this, dou R by dou psi into D psi, similarly if you look AD it is the vector difference of this and this, position vector of D and position vector of A and that AD is this, now the area element DA is given by the triple product of these two position vectors and the K along this, this gives the area, this is a result from vector algebra so you have, you can recall that, and if we do that and now put substitute for dou R by dou psi and dou R by dou eta the quantity that we have just now derived and this itself can be expressed as a determinant, see we have shown just here dou R by dou psi and dou R by dou eta and this is what I am substituting here and implement this cross product, you should know the rule of cross product and the dot product, so if you do that you get this expression. Now this quantity inside this parenthesis can be written as a determinant and the matrix associated with this is the Jacobian matrix and this is the determinant of the Jacobian, so DA is therefore Jacobian of the transformation into D psi D eta, therefore I is integral F of XY DX DY A becomes now minus 1 to plus 1, now that is one first part of our digression, the second is the question of numerical integration, so you may be familiar with Trapezoidal rule, Simpson rule and Newton quotes formula etc., but we will consider what are known as Gauss quadrature rules and I will explain what the rational of those rules are, so let us consider for sake of discussion and evaluation of an integral I cap A to B of X DX, first let us transform X into a new variable U so that the limits of integration are from minus 1 to plus 1, so what I will do, I will substitute U as alpha X plus beta, alpha and beta are the two parameters which I don't know, so when we are at the lower limits I want this to be minus 1 half, so I want U to be minus 1 half when X is A and U to be plus 1 half when X is B, so that helps us to solve for alpha and beta and the transformation I am looking for is given by this, so if you substitute that into this equation I will get minus 1 half to plus half F of this U now B minus A DU, so this is equal to B minus A into I where I is this integral, so I call this F of B minus A U plus A plus B by 2 as phi of U DU, so we will focus our attention on discussing how to evaluate this integral, so now what I wish to do is I have this function phi of U and this is my U axis from minus half to plus half, I want to evaluate this function at some sets of points U1, U2, some UN and I want to attach a weight R1 to phi of U1, R2 to phi of U2 and be able to evaluate this integral, so I am looking for evaluation of I by using this representation, so the effort here will be spent in evaluating phi of U, so any integration method that we use if we should get the ideal integration scheme is get a good approximation for I with minimum number of evaluation of phi of U, or in other words if you evaluate say phi at say 10 points which is the best way that I can combine them, right, for any other combination you know you can use trapezoidal rule and things like that then you can evaluate R, but then you are constraining yourself to place UIs in a particular manner, so the weighting factor also can be optimized, now the question is therefore how to select N that means how many number of points, how to select these weights and where to place these points where I am going to evaluate these functions so that I get a good approximation to I, keeping this UI plus 1 minus UI as a constant that means you sample it at a constant step size that may not be the ideal situation, because that depends on how this function varies, okay, it may not lead to the best solution always, now assuming that phi of U is continuous between minus half and half, we write now a power series expansion, convergent power series expansion in this form, so in this form, now this minus half to plus of a phi of UDU, now for phi of UI substitute this summation, assume summation which is this, 0 to infinity AM, UM, DU and upon performing integral I get this, so I can evaluate these terms and in terms of the unknowns A0, A1, A2, etc. you can get an expansion like this for the integral, now, so this expansion we have got, now we have this power series representation for phi of U, now if I evaluate phi of U at UI this will be the representation, okay, now therefore now I will substitute this into the representation that we are using, so phi of UDU is approximately this, this is what is our proposition and for phi of UI I will write this expansion, this is what I get, so I get now a representation I will interchange the order of summation and I will get a summation I mean representation like this, so in longhand if you expand this R1 into this infinite number of terms, R2 into infinite number of terms so on and so forth, now this I know right hand side what it is and the left hand side I have, I am rearranging the terms now A0 I will collect, coefficient of A0 are collected that is R1 plus R2 plus RN, A1 is this, M is this and this continues, now I will now compare the coefficients of A0, A1, A2, A3 on both sides I will get a set of equations now, for this to be equal coefficient of A0 here is 1 and coefficient of A0 here is this, this should be satisfied and similarly coefficient of 8, this is A0, A1 there is 0 here, so that must be 0, so by writing this we can write as many equations as you want, suppose if you are, if you consider 2N equation there will be 2N unknowns, what are the 2N unknowns? R1, R2, RN first set then U1, U2, UN, so that constitute 2N unknowns, now you can, you select these 2N equations and solve for those 2N unknowns and that gives you the representation that you are looking for, but how do you solve for that? This is a set of nonlinear equations, you can see quadratic terms here, so they are nonlinear equations, but there is a nice result which shows that if phi of U is a polynomial of degree not higher than 2N minus 1 then this U1, U2, UN which are the points where I want to evaluate the function according to the prescription that I just outlined would turn out to be the zeros of Legendre polynomials PN of U, that means these U's will satisfy this equation, so the roots and these roots are real, so once these roots are determined if I know U1, U2, UN then they are remaining N unknowns which are R1, R2, RN can be solved by solving a linear problem, is it okay? So I can find this constant as I wish, so if you are dealing with a polynomial then if you retain adequate number of terms you will be evaluating the integral exactly, there is no approximation at all, okay. Now for example N equal to 3, we have, this is a rule according to which we want to evaluate, so if you expand this now for N equal to 3 and carry out this differentiation I get this equation, so it turns out that the roots of the equations are U equal to 0 and plus minus half square root 3 by 5, so the first point I have to select is this, second one is 0, third point is this, with these weights if I add and evaluate the integral, so 2N minus 1 means 4, 6 minus 1, the 5th order polynomial will be exactly evaluated by these weights, okay, because a function is polynomial. Now I have done a few calculations here for different values of N the roots, these are the roots of the legendary polynomial, this is for N equal to 1, 2, 3, 4 and these are the weights, okay, so if you are implementing this you can use this information to do this. Now we can see how this formula perform, now if I want to evaluate integral minus 1 to plus 1 the answer is 2, so if I take one term it gives the exact answer, so the rule is a polynomial of order P is integrated exactly by employing N number of terms which is smallest integer greater than 0.5 P plus 1, so now X so 1, so P plus 1 is 2 you need at least one term, if you use one term you will get exact solution, but this is of course 0 because it's an order function under symmetric limits there is no problem, but you come here P equal to 2, so 2 plus 1 is 3 1.5 I need at least 2 terms before I can get the value of the integral correctly, so if you see here if I take only one point I get answer as 0 which is wrong, if I get 2 points this is the exact answer, excuse by the whatever you can quickly do this and we get for N equal to the right answer. Similarly 3 of course there is no problem, 4 this is 4 plus 1 2.5 I should have at least 3 terms, so with 1 term I get 0, 2 terms I get 0.22 and this is 0.4, now 6, 6 plus 1, 7, 3.5 I would need at least 4 terms to get that, so 0.2857 is the exact answer, 1 doesn't give a good answer, 2 is this answer, 3 is this, 4 gives the exact answer, so you can quickly verify these things the information needed to check this table are here. Now what we, now we will return to the evaluation of the integrals, this is a mass matrix and using the first result I will be able to write in this form, upon transformation that is our first set of results that we got. Now how do we evaluate this integral is a question, now we propose to evaluate this integral using Gauss quadrature, so for that to do that correctly I should know the order of the function which is this integrand, that means how does N transpose N into determinant of J vary with respect to eta and Xi, if I know that then I will be able to select the correctly the number of integration points, otherwise I will again as you saw here if your integrand is X to the power of 6 and if you happen to take N equal to 3 you will not get the right answer, you have to take N equal to 4, because you know this rule you will be able to make this choice. Now we have to now work through the evaluation of this determinant of the Jacobian, so what we know right now Jacobian is given by this, Jacobian of the transformation X we are writing like this and Y we are writing like this, so given this I can now substitute for these derivatives in terms of NJs and using this expression I will be able to write J as product of two matrices, the first matrix is matrix contains derivatives of N1 and N2 with respect to Xi and eta and second matrix contains the coordinates of the nodes, okay, so all these quantities are known, so we can go ahead and perform the calculation, if you can it's quite straight forward to do this, if you do this I get J to be in this form where E1, E2, F1, F2 and F3 are explained here in terms of nodal coordinates, so moment I have this I can evaluate the determinant of J and it turns out to be this, and interestingly it turns out that the determinant of J in this case is a linear function of X and eta, it is precisely this that I wish to determine, then only I will know which order of integration I need to implement, we knew the order of N transpose N, because we know entries in N so we know what order that product would be, but we didn't know what was the order of this J is, I mean we didn't even know whether it is linear or not, first of all. So if we now look into this we see that it is a bi-coderatic or a bi-cubic function depending on which terms you are looking at, so using the rule that we are using a 2 by 2 array of integration points would be adequate to perform this integration, so mass matrix can be evaluated exactly by using Gauss quadrature with 2 by 2 Gauss quadrature point mesh, how about stiffness matrix, so I need here B transpose DB into the Jacobian, so again I need to perform some calculations here, so this D matrix has to operate now, so I need this information dou by dou psi is dou X by dou psi into dou by dou X plus dou Y by dou psi into dou by dou Y, etc., and this if you put in the matrix form I have this vector of gradient dou by dou psi and dou by dou eta, and this is the Jacobian matrix this, so I can either write this or the inverse of this I can express dou by dou X and dou by dou Y in terms of dou by dou psi and dou by dou eta through J inverse, now recall B was this for this problem, so we get after multiplication I get this and it turns out that the quantity that we are looking for is given in this form, okay, so elements of B transpose DB into determinant of J are ratios of bi-coderatic functions and linear functions, see there is a inverse here, so therefore if you look at individual terms they won't be polynomials, they will be ratios of polynomials, okay it turns out they are ratios of bi-coderatic functions and linear functions, so this would mean Gauss quadrature won't be able to evaluate elements of KE in an exact manner, so if you use Gauss quadrature to evaluate this we are introducing certain approximation, but Gauss quadrature is so convenient we will make some discussion on this subsequently, but right now the recommendation based on experience is used 2 by 2 Gauss quadrature, we will come to this slightly later once again, okay, so summary is mass matrix gets evaluated exactly and stiffness matrix there will be an approximation. Now this is, I have used now a four-noded quadratic element, now the question is if you are using discretization, if you are modeling any structural behavior how do you improve upon the accuracy? One way is stick to four-noded quadrilateral elements and use more of them, that means define the mesh, introduce more number of elements and do that, the other way is use a higher order polynomial to interpolate the field variables, if you want to do that it amounts to increasing the number of nodes in an element, so instead of having four-noded element we can have eight-noded element, okay, so then when I interpolate the interpolation functions will be of the higher order polynomials, they won't be what we saw for a four-noded element, because now there are additional nodes that come into our formulation. Suppose I take this, I will as before I will retain these three nodes and introduce additional nodes at the midpoint of each of these elements, again we will transform this to a square region and evaluate the elements of stiffness and mass matrices using Gauss quadrature, so this is a eight-noded quadrilateral element with 16 degrees of freedom. Now the field variable now needs to be represented in terms of the nodal values and there are eight nodal values for U, so if you go back here for a U1, U2, U3, U4, U5, U6, U7, U8, so I have to utilize all of that to interpolate U of X, Y, T you know to facilitate the development of this element. Now similarly V is represented like this, now using the logic that we already discussed we need not have to go into all the details every time, for we get these interpolation functions which have this coronacal delta property, for nodes 1 to 4 I get this is the interpolation function, for 5 and 7 these are the interpolation functions and 6 and 8 we get this. Now I would not like to get into all the details of the formulation of these integrants but if you indeed perform all that you will be able to notice that you need to use 3 by 3 Gauss integration points, with a 3 by 3 mesh mass matrix is evaluated exactly but the stiffness matrix continues to be evaluated approximately. Now how about this is a rectangular element, this discussion is not for this element, this discussion is for a rectangular element, for a quadrilateral element the same formulary works except now that the X and Y also need to be interpolated using the same shape functions, okay. So in this case it turns out we need to use 4 by 4 Gauss integration points in evaluating stiffness and mass matrices, and if you use 4 by 4 mesh the mass matrix is evaluated exactly, so the mesh size that you need to select should be such that the mass matrix elements of mass matrix matrix are evaluated exactly, see mass matrix elements turn out to be polynomials because N transpose N remains as polynomial, whereas here because of the Jacobian also of course would play a role, but here because of this you know various operations involved here the K matrix there will be difficulties as we saw while before. Now let us return to the shear wall and earth diameter example that we considered in the previous lecture, so the problem was there is a shear wall of thickness 0.2286 meters and these are the dimensions and the problem was to find, first find natural frequencies and normal modes. Now analytical estimates of natural frequencies using Timoshenko beam theory where obtained, pointed out these were their numbers, these are frequencies in hertz. Now we discuss this model in the previous lecture we use 64 elements with first order elements, that means we used four nodded quadrilateral elements, actually they were rectangular elements here and the degrees of freedom that the model had was 160. Now what I have done is we have retained the same mesh and same element configuration, but now the elements are modified to be second order elements, that means they are now eight nodded quadrilateral elements, so the degree of freedom increases to now 448 and the natural frequencies from 5.03 I come to 4.95 and these numbers change and you can see here they are reasonable, if this is believed to be correct because it's an analytical solution and the structure, structural geometry satisfies the requirements for the application of Timoshenko beam theory, so consequently we can test this result, so and this analytically derived, so we see that the first natural frequency is much better approximated than this, similarly the other frequencies also get represented better. So these are the mode shapes, first four mode shapes are shown here, so you can see that now the mode shapes are smoother because they are represented with greater detail within an element, this is the tenth mode, higher modes become representation of higher modes would not be so smooth as it is seen here, if you had used the first order element this mode would not be as smooth as it is seen here. Now this earth dam problem it was a triangular wedge model using plane strain approximation and these were the properties Poisson's ratio 0.45 this is the density and this is the angst modulus, with triangular elements with 42 elements and 42 degrees of freedom we got these natural frequencies and if you use the shear beam model and this is analytical estimate of the shear beam and natural frequency this is available in terms of Bessel's functions and then exact solution to this is available so based on that these numbers have been computed. Now this we have done in the last class, now what I have done now is this analysis is with first order triangular elements, now this analysis is second order triangular although I didn't discuss the formulation of second order quadrilateral element the logic is conceptually quite similar to what we briefly mentioned for quadrilateral elements, so if I use this same number of elements but the order has increased so degrees of freedom will increase, so the frequencies now change and the first frequency is if you have to believe this this is 1.25 this is 1.227 and this seems to be moving towards that, but mind you this is also an approximation to this behavior of this wedge, so the underlying assumptions of shear beam behavior must be satisfied by this structure, so it is difficult to pass a clear judgment on which of these two are exact. Now these are the mode shapes from a higher order element, this is the first mode and fifth mode, this matches with what we did with the Corsair element, I mean first order element, now in this illustration I am using a combination of triangle and quadrilateral elements, this is quadrilateral element, this here we are using triangular elements, so this is again first order elements, so there are 32 elements 56 degrees of freedom and we get natural frequencies like this, so this is just for illustration, and these are the mode shapes that we obtain from this model. Now what I do, I will repeat this exercise using the same mesh, instead of first order element I will use second order element, so this quadrilateral element here has 8 nodes and 8 degrees of freedom, the triangular elements here will have 6 nodes and 12 degrees of freedom, so number of elements remain the same, there are 210 degrees of freedom and I get different sets of estimates for natural frequency, so these are the mode shapes for the two first and the fifth mode. Now so far what we have discussed is basically free vibration problems, now if we have to consider first vibration response we need to worry about how the equivalent forces are computed, so let me just the basic rule for that is this, okay, so to explain this what we will do is we will consider a, suppose there is a edge 2, 3 in a triangular element and the loads will be in the plane of the element, the plane stress problem, suppose at any point here there is a load PX and load, sorry this load PY, now when we formulate the finite element model we also need to assemble the nodal forces, first of all we should evaluate the equivalent nodal forces, so what we do is we have this over the surface of the element delta U, delta V transpose PX, PY into DS, now we also have the approximation UV which is N into UV, so now we write delta WE for the nodal, in terms of the equivalent nodal forces these are the forces that are acting on the surface I want to represent them in terms of equivalent nodal forces, so we demand that equivalence of these two will give me the expression FE equal to, see I have to consider the edge 2, 3, I am assuming if you recall we are considering force on the edge 2, 3, so this has to be evaluated along that edge, so if I do this we will get FE for the example that we have considered as half length of 2, 3 into 0, 0, 0, PX, PY, PX, PY transpose, so what we are getting is at the two nodes half of the load comes here, half of the load goes there as per this formulation, so this is what the equivalent nodal forces, the logic for computing equivalent nodal forces, this can be done for quadrilateral elements and rectangular and quadrilateral elements as well, so at some point we will illustrate this with some examples, now in the subsequent lectures what we will do now, so far what we have done is we have dealt with a 3D beam element with two nodes and 6 DOF for node, now we have now considered 2D continuum, we have derived different elements, now the generalization for further analysis, now there are three things that we can do, this beam theory will get extended to plate and shell, this 2D continuum we can consider now three dimensional problems, that means when conditions of plane stress and plane strain are not satisfied, here another 2D model becomes possible that is axisymmetric solids, and then 3D solids, so what in the remaining next few lectures what we will do is we will develop the structural matrices for these problems and these problems, and at the end of it I will, at the end of these developments I will make a few remarks on choice of shape functions and their role on how the answers converge, now we will consider the question of how to compute equivalent nodal forces, we will quickly recall what we did for a beam element, so this is a beam element say 2 dimensional beam element deforming in its own plane, this is a distributed load F of X, T, so discretization of this element we had two nodes and two degrees of freedom per node U1, U2, U3, U4 were the four degrees of freedom for the element, the problem here is see the displacement field is represented in terms of nodal displacement and these interpolation functions, so how do we represent this distributed load in terms of equivalent nodal forces, so how to find P1, P2, P3, P4 in a way that these form an equivalent, there is an equivalence between this load and this set of loads, so what we considered was a virtual displacement, and that we represented in terms of virtual nodal displacements multiplied by the interpolation functions and the work done was equated, so the work done by virtual displacements on these set of forces and the virtual displacement on this, by this set of forces are equated, and by manipulating the terms we determined the nodal forces to be given by these expressions, so we are going to do something similar for even two-dimensional elements, so suppose PX and PY are the distributed loads along the edge 2, 3, now I want to find out what is the equivalent nodal forces because of these forces, so PX and PY are distributed along this edge 2, 3, so the virtual work is given by this expression, and the displacement field we are interpolating in terms of nodal degrees of freedom as shown here, now along this edge, so this virtual work can be written in terms of virtual displacement and applied forces as shown here, and this is our representation, so the virtual work which is this is also equal to nodal displacements into these forces, and after we manipulate this we get this expression, and we should note that this trial functions need to, here we need to evaluate this along the edge 2, 3, now this is the virtual nodal displacements into the nodal forces, and by equating these two we get the, by equating this and this we get the expression for equivalent nodal forces which is shown here, now here we have assumed that PX and PY are constant along the edge, otherwise there will be a quadrature that has to be implemented, the subscript L2, the quantity L2, 3 represent length of the edge 2, 3, this length, so if you know the nodal coordinates you can derive that length in a straightforward manner. Now we can make some remarks, first is the quadrature rule that we discussed was in the context of a scalar integral, this can be, the same formulation can also be extended to two dimensional integrals and one gets quadrature rule of this form. Now in this calculation of equivalent nodal forces we considered surface tractions, but if there are body forces within the element we can denote them by FX, FY transpose, suppose if that body force is constant within the element how we can find the equivalent nodal forces, but again we have to use the same formulation, I leave this as an exercise, this will be the equivalent nodal forces and here again we need to evaluate this integral and we need to examine the order of terms in the integrand and decide upon the quadrature rule, but this needs to be evaluated again using Gauss quadrature. Now a small exercise, so let's consider a four-noded linear rectangular element, now we consider the displacements in the discussion so far we considered this part, now to this I will now add two more terms alpha 1, 1-xi square, alpha 2, 1-eta square and similarly for V I will add this. Now you can see here that at the nodes which are xi equal to plus minus 1 and eta equal to plus minus 1 these terms are 0, so this is like a bubble function so they don't affect the behavior of U near the nodes, at the nodes and V is given by this similarly. So now again these trial functions are same as what we have used for a linear rectangular element, now these alpha 1, alpha 2 and alpha 3, alpha 4 are not nodal displacements now, they are generalized coordinates, so there will be now 12 degrees of freedom for the element, 4 nodal displacement U1, U2, U3, U4 and 2 generalized coordinates alpha 1 and alpha 2, this is 6 for U, similar 6 quantities for V. Now the exercise that is being suggested is use the energy expressions and obtain the 12 by 12 stiffness matrix and eliminate this alpha 1, 2, 3, 4 in terms of the nodal degrees of freedom using static condensation and obtain a 8 by 8 stiffness matrix, and also examine are the displacement fields continuous across the element boundaries, so this exercise can be carried out, so in using this element you may notice that we can continue to use the consistent mass matrix that was derived using only this approximation, kinetic energy can be characterized using only this, and whereas the displacement fields will be, the strain energy will be computed by an alternative representation. Now in the subsequent lectures now we will now consider problems of 3 dimensional elasticity, we have briefly discussed this earlier, so we will quickly recall, so stress, state of stress is now described in terms of 6 stress components, either it can be arranged as a matrix or a vector like this, these are the strain components, and for an isotropic linearly elastic material the stress and strain are related through this relation and this is the matrix D, and we have the expression for strain energy and kinetic energy, this is sigma transpose epsilon DV naught is the expression integral of that over volume element into half is a strain energy, and this is a kinetic energy. Now to compute strain energy we can now what we are doing is we are expressing sigma in terms of strain, and the strain subsequently we will express in terms of displacement as we have been doing, this is a strain displacement relation for small deformations, and substituting that we will get displacements as N into U E, where U E are the nodal displacements, and consequently epsilon is obtained in this form where B is equal to this matrix into N, so given that we will have the expression for strain energy and kinetic energy in terms of assumed displacement fields as given here, and we need to now carry out this integration over the volume elements. Now the volume elements in a 3 dimensional solid becomes more diverse, we can have a tetrahedron or a rectangular hexahedron or a pentahedron or an isoparametric hexahedron, and we can also have isoparametric versions of tetrahedron and pentahedron, and these can be first order elements or higher order elements, so there is a great diversity of elements, so we can examine some of them as we go along. Subsequently in 3 dimensional problems we can also consider solids of revolution, these are called axisymmetric problems, we will be doing that, and we will develop at least one element for this type of applications, and this would be followed by a discussion on a plate bending element, so this is a thin say lamina which carries now transverse loads in addition to the inline loads, so the action of this transverse loads on this lamina is to, the plate is to induce bending and about X and Y axis, and we need to formulate the correct expressions for strain energy and kinetic energy and develop the element, so we will take up these exercises in the following lectures, and may be some examples on force response analysis, so with this we will conclude this today's lecture.


Yeah, we on boss talk one-on-one one-on-one. Yeah, we're gonna Tapping in all of my People man. It's so nice to you know, get back to the solo getting you guys informed on Everything that we've done in 2023 man. Thank you guys for liking and subscribing to the channel I know miss Jamaica been totally on you guys about Membership subscribing to the membership and you guys have done it We've seen so many different people come in show us so much love when it comes down to the membership So I just want to say thank you guys Also, I want to set just really set the record straight man on what we went after this year this year was so dope because This was the second year at the end of the year. We went over a hundred thousand subscribers This year looks like we're about 72 set 172 173 thousand Subscribers, so we did 73,000. It's a little bit under what I wanted. I wanted a hundred thousand this year I wanted to be at 200,000. We're almost there. We just have to keep banging away at it man I just want to say thank you everybody that's been following us like and subscribe to the channel man But top of the year man was so live man We had so such a good time when when you think of the people that we sit down with and we was so You know happy to get with the people who love boss talk 101 Starting the year off man I was so excited when we was able to sit down with silk the shocker So if the shocker gave us all type of things that was going on some of the most exclusive content back during that time I can remember Master P and little Romeo having their issues and he kind of expounded on that And I think that was just something that I will people needed to hear because you hadn't heard from a family member except for Master P and Little Romeo or Romeo will call him a little no more, but at any rate you you've seen what was really going down when it when everything started to unfold Silk silk the shocker was very very dope and answering that question and just being a real cool Brother that was trying to keep the pieces and family members and basically saying he wouldn't have even never done it in public at The end of the day I keep it consistent So at the end they don't have to respect it because it's consistency.  Shout out to the Miller family man. Shout out to Master P I know you guys rock with boss talk 101. It's just been a live live Introduction when they come down to what we do so thank you guys man It's out to be a court who always hold us down who always get me certain interviews man And just always rocking out with us what but I did have the dream of music and see I signed with P I got with no limit in 95. Yeah, so I graduated high school in 94. Whoa What song what rap what what was peeved? Oh, what was he doing? That was right that was I signed with P right with 99 ways the die came out I just want to say thank you to him and he went over a hundred thousand subscribers as well Thanks for the big shout out man. You guys are live You know already it's big love between us man when it comes down to what we do man So hey man family always but the next interview we went up to New Jersey So we was in LA at that time and we interviewed a lot more people as well And if I didn't call your name doesn't mean I didn't enjoy the interview doesn't mean we didn't do our thing We did great great interviews out there, but I'm just speaking on a few of them We went down to New Jersey after that we sat down with Ice-tea this probably this one of them boys man that you just don't it don't it don't happen all the time So man, thank you so much for inviting us into your home, man Definitely definitely, you know Kenny said I want to get boss talk over there to the crib You know because we wanted to do promotion for the book man, and I said well, you know We can do it at my house. So, uh, you know you guys were invited and come to find out your friends of my man Big boy and holding corn so his love is love, you know, I like I told you as you were setting up I kind of try to pick and choose a lot of these podcasts out of control, you know So I'm happy to be here with you and I'm uh, you know Tell you what you want to know We sit down with Ice-tea man We talked about so many different variables all of the different movies all of the different things that he'd done in law and order running Actor oh 500 episodes. I'm the longest running actor, right the longest running actor in television history But Mariska is the longest running I call her Olivia Olivia They she started she got maybe 25 episodes, right? That's what I was trying to figure out I'm the longest running. Okay, okay Just just a dope interview man this guy man. He showed us the utmost respect He we we actually did the interview in his home He just showed us love and I just want to say thank you to Ice-tea man I hope you're enjoying your uh, your End of the year your new years your holidays you guys all should be enjoying holidays right now. Um So Ice-tea was so dope man. It was so and literally we was in new jersey We was in new york We had a great time We killed it and I just want to say man everybody that came on those interviews as well Shout out to pepper Ken shout out to HHL. What they did I'm pimp this I'm pimp that they tried to take they tried to create it and say that it was them So, you know, you see this, you know from my perspective, you know, send me so I can imagine how these young people feel because I felt the same way You know send me so that's why I it had I said the only way I'm gonna beat these niggas I got to get in the game and when I get in the game, they can't beat me I swear to god. Ain't nothing's got more. I got more game.  So they know that they know I'm coming. They see me You know what I'm saying? They see the hip hop attorney. They see me signing artists They see me about to make my move because the niggas too stingy. They don't want to give up the game They don't want to let niggas in Um, my family like I said me and pepper can we we always been working together And we always trying to figure out ways to do something to get great content great interviews to our people And pepper can his family as well. So thank you so much pepper can You've been a dope as said to boss talk 101 and we always gonna applaud you. Um We also sit down with I want to say hustle mom And I'm gonna be honest with you. She was the loudest one man. She gave us so many dope stories, man See this is what everybody needs to know A woman Cannot I repeat Cannot be a pimp and I just I had to say her name because she was in New Jersey as well That was when we first met we had talked before but that's when we linked up man, and it was just incredible. Um Then we brought it on back to the south man shout out to bum b man We went down to we went down to uh, shreveport. We linked up with bum b. We talked about trail burgers We talked about ugk We talked about a lot of different things that whole history It was just man. We had a great great time And I I I'll never forget the fact of how everything came together with us man. We hung out Uh bum b told stories about jz and the whole time we was talking he was writing to the song In his head That's cold wrote a 16 But get throwed is Twelves the verses of twelves so he had actually wrote a 16 and he was shooting at another rapper on that verse To get throwed The good the get throw song was something else He talked about how the preparation was for that song. He also shouted out too short. He also shouted out Um, just all of the people that helped them Even big time records from even the very beginning. So Man shout out to bum b man. You know, i'm adult ugk fan for life I never never stopped rocking with you your music man and just the movement We was also able to give him an award So we gave him an award man, and I just man that was uh, I just Valued valued that interview. It was big for me. Um Next we did all type of things we came We went up to chicago. We hung out with larry hoover jr.  Okay. Okay. So almost everywhere wherever sugar was You know, I'm there today. I'll be there tomorrow, you know, we went to larry hoover jr's birthday party turn 50 so We were up there and we had a splendid time, man And I got to hang out with ak with do or die man. Our kelly was He was letting the song say what it was gonna say And then went behind it and said I need you as the camera person to line it back up A crucial conflict and all of the different, you know, uh Amen, we hung out with a lot of people Shout out to myra. Shout out to uh, my boy at four one four. Uh, this guy here, man This guy was good. You know, to be honest with you, me and him had a great time and just Shout out to myra's son as well. So we killed it In chicago chicago was live chicago was dope.  I had a great time, man And like I said, there's been so many different interviews this year in 2023. I'm just giving you a A sum of what happened. I couldn't tell all of the things that happened to you because Man from c4s. Hey, we got c4s in the building. The biggest I already know what time it is quit playing man.  She done got a little drunk She done got a little tips To all of our regulars rainwater killed it. You see that man right there? Tupac. You see him. Yeah, young boy got it You know, I heard the statement.  Young boy got more plaster Rainwater came on the show as usual and gave these concepts all these different things that was going on in his mind Shout out rainwater Half pint and you got to understand I was very proud of you when you got Yellow beesy on there, you know me. I've been asking about yellow bees. I Ever since I started boss talk ever since you sitting that chair It's been a yellow beesy conversation between me and you. You know what I'm saying? Yeah, he don't really I see him the other night We was out and same little rounds around him.    I got another love for y'all Little runny. Lf is in the building in the building Say man, and I've been using that what a boss is talking. I go hard with that. Yeah I don't see a lot of bosses come talk with y'all Say man. It's been some crazy things happening, man Like we we probably went viral like five six times on some I'll be going down. I'll be like what the heck just happened Say I'll be watching. So I don't know which ones I didn't watch I do know which ones I didn't watch It be going on like crazy niggas be mad. It be a lot of them And y'all don't even Y'all ain't Hey, man, I'm just here. Shout out to Django Derek's. We added that Uh that selection as well in so Django Derek's is one of the new additives where we're bringing in the cooking side of basically the good cuisine that the Jamaican The really the entrees and the food that these guys are are giving us man And I'm getting a little fat messing around with uh, Django Derek's man, but dope dude, man Big shout out to Django Derek's.   Um, like I said, there's so many different people that we interviewed this year. Um I just can't begin to go back and just reflect on all of them. I wish I could But man, you guys are dope. Thanks everybody for coming on boss talk 101 Shout out to all the people who linked up with me Shout out to trio talk no pill talk shout out shout out to mama scott. I went out and spoke with her function as well Um, shout out to all of the people in dallas that uh, the row Shout out to the road the road music he comes on and he shows us mad love every time man Top 10 on the urban choice at a time and I was just like damn. It's time for a remake I hit diddy. I hit him with a text mixer. I just text him. I was like, hey, bro I reckon get big Top 10 right now. We trying to put the remix together. I really want you on it Man, he hit me right back. He was like sending to me sent it to him He heard it. He was like, I'm gonna have it back to you tomorrow and and he sent me back to Now first he told he said I'm gonna have it back to tomorrow.  You bubba dubb. We're here with boss talk 101 I'm talking about boss talk 101 boss of our bosses some of y'all bosses, but y'all taking losses We ain't taking no losses boss talk 101 pull up when you see a drop on youtube Make sure you tune in get your popcorn get your soda and it's going down If y'all ain't watching boss talk 101 y'all trash Uh showed us so much love and and I'm gonna be real which I had a great time dealing with him Uh bubba dubb has grown so much man, and I'm just seeing him He's gonna blow up this year coming forth in 2024. Watch what I tell you be on look out for bubba dubb trash is a whole another level Also, man big shout out to face on love face on love always your boss talk 101 So much love man. So we ended up giving him an award as well Second award i'm giving out the photo end of the year like this one We gave out a lot of awards this year and 2023 has been good and we're giving out I believe one more this is this may be the last one that's some crazy happen, but we got one more award winner It's crazy. It's here. I can't wait to see how it came. Oh, yeah Oh, yeah Look out. Oh, okay man, oh look Bam face on love.   He's been supporting our channel Even when he just when we didn't even have nobody coming to the channel He came through and showed me love man. Shout out to face on love. You deserve this man. My god Boss talk 101. We did it again We had to do it this amazing award It says thank you face on love and recognition of your many accolades in comedy and acting Him and bambi got the same award. We just started giving these uh awards out. It was dope man Before I get into your history and so forth, you know, we can't start our podcast without doing this to you right now Okay. Okay. Look at him. Yeah. Yeah We have this right here. You said doing this to my touch out for No, my wife and me man It was something we wanted to do man because we never gave this award out to nobody nobody has one like this Man, so it came in I'm boxing is real man. This it baby Yeah, it's going down man. I never thought I'd be doing this. This guy mean a lot to me man I couldn't go see him empty handed. So I went on all of them the new play we're gonna be giving out He's right here different man.  This thing's kind of heavy, man Man, look at that thing Man bambi man You guys mean so much to me man. I rock with him man. Boss talk 101 man It's here. This is what everything is. That's why we do do this man. This guy right here. Man, Menor, Freeman man Check it man. Congratulations. Best wishes man. I'm gonna make it. Don't boss talk man. I'll let your boy man Boss talk where the bosses talk. Bambi So we're gonna present you with this award because you know how much we're fans of you and UGK And it says thank you. Bernard Freeman Bumbi It says in recognition of your many accolades including star new gk with pimp see I enjoyed it. I ain't gonna lie to you phase on man. He showed us so much love down in austin He sold the show out. It was crazy. I won't forget All the things that we've done down there man had a great time man toe down came and Rocked out with us and it was just a great event man um When you think about all the people like I said that we talked to and all the people that we hung out with We had a great time this year, man I wouldn't trade a thing man. Shout out to chico bean Chico bean came and then he invited us to so many different shows Yeah, we on boss talk tv. Shout out to ihita reason you see me We doing numbers y'all looking pv. Y'all looking hungry. We looking greedy. Remember back then we were pope Nice and unique hustle come shop at the store. We don't never sleep. We don't go 10 11 interviews all in a row He showed us so much love just like My boy bubble dub chico bean showed up and showed us crazy mad respect and love um Through carlos miller and it was just awesome time who I met through dunk master.  They show me mad mad love Sound like i'm from new york when I say mad love man. Shout out to new york boys up there But it's just crazy that We never seen this coming boss talk 101 love Communicating and connecting with all of the people that rock with us And even if you don't man, you need to get with us, man And I want to say a special shout out to shone cotton too because he came I'm saying but at the end of the day, it's like man, you want to see him get a get through it No, I said that in our last interview, but I only speak on the things that boosie speaks on on velad That's all so I don't go into the personal shit I only ask some questions about what boosie said about him on velad Outside of that if it's not on the internet, I don't you don't got to tell me Wow So I just make sure I keep it everything that's on the internet on the internet And he pretty much came and showed us love. He's been on the show three different consecutive times. He's coming back Uh, we're we're definitely going to be uh linking back up with shone cotton Uh shone cotton is one of the he is the biggest platform here in texas. There is no bigger than say cheese, man And the way he's killing the game you got to give him his respect and his props, man Say cheese tv is dope Also gutter tv you got to mention him. Okay say the one without the other those guys iron sharpen and iron out there So all of these guys have came on the show and they've showed us so much love and we try to give that same love back My houston guys y'all already know a shout out to my boy ad and la man for flying in coming on Talk 101 showing us so much love ad man. Listen, man.  I also want to say shout out to uh my boy Uh clink payback sayings Just a lot man. I'm going back in there. I'm getting in my bag I'm steadily getting in my bag trying to figure out who else who else came and showed us love It was a bunch of them. If I didn't say your name, it's called it. It ain't because I don't love you.  We're already moving and grooving toward different, uh An analogies man. We also wanted a most sincere interviews that we done I gotta go back into my bag was esg esg ended up getting hit by two cars Esg ended up having his leg to be decapitate. He had to cut it off amputated then they had to turn around And uh, also he had was fighting counselor during the same time So esg man, listen, let me just be real with you, man. This guy showed us so much Love man. He still continues to call us. He still continues to Show us just so much respect and show us so much Just he even did a song he throw boss talk 101 in that thing man esg is my guy. I don't even play about it, man Um, I definitely respect him and I cherish what he done And and it's just real man. Thank god for esg and all the stuff that he went through He still stayed true to boss talk 101 and we did that in the hospital The first time I ever did one at the hospital But he asked us to come down and we came down there. We had a great great time man It wasn't a good situation But it became something great because we was able to be there with him Be it we was able to sit there with him and hang out with him and it was just a blessing man And I gotta say one more. I gotta talk George George was a guy That called me and at first I wasn't gonna do the interview. I knew it was gonna be a touchy situation George was I want to say Steven's brother Steven was the one who was pretty much uh He was gunned down at a club allegedly they tried to say see murder done it But George wanted uh wanted closure George came on and spoke deeply about his brother and what he felt I pretty much told him how I felt about what he felt about his brother passing and about the legalities of it And it was just a touchy situation But all in all George wanted to make sure that he basically George wanted to make sure that he nobody he felt like he never was shown remorse for losing this brother He would had it in for Kim Kardashian. She it ended up on TMZ and it was just crazy so I just want to say man. I hope George called he definitely Uh went through some things and and him and his mother and his father Um were able to get closure behind that interview I'm not gonna go into much detail if you want you to know he'll tell you but That that interview uh changed his life Changed his life in a in a big way They was able to get closure to some things that they've been trying to get closure to for over 20,000 20 for over 20 years Sorry, but um Also, uh, big shout out to regi right who's in la. I hung out with regi right and regi right has thrown us some heck of Hell marries and and he supports the channel as well There's just some dope people that support boss talk 101 But he's one of those guys that every time I talk with him it lights up the show man Um, I told him off. I had told him off is one of the guys that always flying in and And just love the show og percy like we got our people that come and do boss talk 101 Man, I also man want to send a special shout out to man mama scott Man, we love you boss talk 101 um dope Keep on pushing fast cnt also man. Hey my god trill talk no pill talk Trill talk no pill talk man. You know what you do. You know how you come Bring the pain when it comes down to boss talk 101. Thank you Happy 2024. It's going down man big love Keep on putting a putting out everything you need to understand what's going on in these streets Another thing we want to do is uh also I want to shout out Anita Jawa Anita Jawa killed it as well She keeps coming and she keeps praying over boss talk 101 And my boy Corey man Corey cloud man keep going man. All my east texas people man stop playing man 2024 we kicking in the door. It's going down one more time Man, this was something man when you think about it. You gotta you gotta understand man Twisted black have been locked up for 17 years And he came home And when they called me I was ready and we pretty much did a killer interview And this guy man, he did his thing man. I can't even lie man. He came home He even put out a album And it was a fresh album and we did a recap on that album as well When I said recap we just before it came out. We came and did our thing man. Um Man shout out to cc man cc man.  They support boss talk 101 Oh man in a heck of a way. So we want to make sure that we point that out as well. Um Man Be real with you. There are so many many different people that came on man from gator man to All of these different people that came on and and just showed us so much So much love man from solo luci to there was this incredible movement man with all type of people that came on the show and just Man killed the game for us man. The doc Doc came on man. Listen man. That was a huge one for me. It's because I'm a big doc fan And me and me and doc talk actually and I'm trying to get him come back through because we got more to talk about And I was like man, I gotta get doc back on the show Because that was just so much stuff that we had to go back and get into so I I he has a history him and dr. Dre him and uh yellow ren nwa Shug knight All kind of stuff man, you know so much stuff is going on in 2023 I don't really know what's going to happen 2024 But I can tell you right now 2023 I was saying at the first of the year is what it's going to be and we killed it I can promise you that it was not a game man Carl crawford shout out man the guy who supports us man who wears the gear who always supporting what we're doing Shout out to his artists. He keep pushing man Carl crawford dope man. I can't lie. He killed the game this year. He continued to invest in artists and in music So it's crazy, but it's going down for real man And and I can promise you we did people that you guys wouldn't even think of man from we did take money Take money came through killed the game with us So that was just a lot of different a litany of people that came through Take money had a lot of energy man. We're going to get up back to in 2024 Ain't going to be no game when it comes down to her and uh shout out to rap economics too, man DJ nightcap man that boy at press man. It's going down man.  So shout out to goddess man And I want to say shout out to a special shout out to columbus short as well Um, thank you that you it was your third time on the show this year Man you've showed us so much love ever since we started And you came and you pretty much killed the game when it come down to Always being there for us and answering the phone when I pick when I call, you know um Propane you're another one man. Uh shout out to propane shout out to uh All of those guys king noir all of those guys man Like I said you guys you guys are the bomb man, and I just want to say man. Thank you guys from Everything that you guys done man from just basically rocking out with us and Hey, thank you. Um Shout out to the radio people to in dallas the ones that rock with boss talk 101 Uh j cruise we cruising Vita loca. Hey, man. Thank you guys for rocking out with uh with the channel You guys are radio people that That stop by stop what you're doing and come through and rock out with us, man And I'm like I said, well, I don't even know where to stop It's so many Nina Loretta, I could go on and on There's just a litany of people that have been on boss talk 101 and Just killed the game and kenya out of man. Thank you. Thank you so much for supporting us motivate me You haven't been talking to me lately though. You need to pick that phone up and call me sometime But we're gonna get back on track man. Your brother been picking up where you hadn't been talking So it's love man, but when we come to la you always invite us here to show us the utmost respect and love man, um shout out to bobby sessions. Well, thank you guys, man Um, there's just like I said, I could keep naming them I just want to come back and do a part two of the recap. Make sure you guys like and subscribe to the channel Make sure you guys join our membership Thank you guys for being so, you know dedicated to what we do here At boss talk 101. Yeah, we on boss talk say they're ready for me to come back down there We're coming back to new islands in 2024 the lords say the same la We plan to be out there as well. We'll be out there for the Super Bowl in Las Vegas Um, there's so many different things that we'll be doing. Um Can't wait for you guys to see man Wear this thing heads from here. Um, I want to stop for a moment. We lost a true soldier this year Big D the mobile my guy We talked today before he died He uh, I will end up going to I end up going to his funeral. It was a sudden death.  You can go on this page and see them But they they they would they they are they are they're incredible kids and The mother is strong. She's hanging in there with everything she got Um guys keep that family in your prayers because that was a big loss for them He was 39. Um, I remember the day he passed away. He was supposed to be doing, uh, his 100k plaque His 100k Subscribers um plaque celebration And he didn't get to do it, but we we definitely did it for me and my heart.  Um Rest easy big guy. Um, thank you guys so much man got called up there for a second, but I'm back Thank you Thank you. Thank you. Thank you for supporting our gentleman. It's been another great segment of boss talk one on one One of bosses talk. Yeah, we on boss talk one on one. One on one. Yeah, we're gonna talk.  We'll be lily. It's a unique husband


You see, hypocrisy is projecting one thing in a certain direction and doing totally opposite.  That can also be described as progress. We saw the minister of wealth going to the hospitals and he was told in his face because he wanted to show the country as usual in these times they do with the media that you are going to see where the medicine and she met people with prescriptions who are going to buy medicine to say madam what you are saying in your office is not true. We don't have medicine even when you are finding us we are going to buy medicine. What does that show you? Isn't that failure? That is failure in someone's face and when we have said probably the minister has mispressed the misunderstanders to think we are just talking. Where? We are saying look do the right thing.  Here is a prescription I am going to buy medicine. There is no medicine in the hospital. And you want to still just tell people what you have in your mind. It doesn't work. You are bringing doctors to be on the phones. Look who doesn't know that these gadgets you have are not about papers gadgets. It's not all the time that anyone who is on the phone is doing propaganda or just gossiping. No, doctors, teachers now can use this phone to do real time research. They can consult their colleagues on a challenge that affects with all the phone. So you as an informed minister you can't go and start branding them as lazy people for having gadgets. Just do what is right. And this thing of intimidating public servants must come to an end. Those are public servants who are in these jobs because they deserve them. They got educated to get these jobs. So we, all politicians will be there and will come and go. And like she was saying, yes you will go, you are there minister of health and you will leave. There will be someone else who will come. But those doctors that you have found will remain there to save citizens. So don't intimidate them by thinking you are a super human, you know. They have existed, they have yielded people, they have worked after the assistance for so many years. It's the same with those nurses that you are trying to intimidate. Just do that thing, do your work and stop the bramkel.


r 9 layers if you can coordinate and lock that down for a couple days seriously you can control for weeks afterwards it might end up working that way in just snowballs you might be right about that I just think it's going to be so hard to get around the clock and not having anybody being able to contest them especially since it's not international servers like there were before on private servers it's a different thing so yeah also how do you know you're on all the layers that's another good point because you never know how many layers there are active at a time there might be another layer that you don't know about and what a lot of people don't know is that layers aren't just generated on a population basis it's not like 2000 people generating next layer 4000 people generating next layer that's part of the equation but the layering technology has understood it you can also spin up new layers new layers will also spin up they haven't been disclosed by Blizzard and I think it has to do with bugs like something on a server crashes internally or something like that it can spin up a new layer which is why a lot of people found themselves layer hopping without choice during even the late stages of the beta there's some weird layering Illuminati going on behind the scenes so there might be 20 layers there might be 30 layers who knows if we're talking like a 20,000 pop server at launch you might have 30 layers we don't know I think it's going to be really interesting and there's going to be a whole lot of unknowns let's do this let's take one last question before we call it a day guys also, if you haven't already please please please follow StaySafeTV, follow TipsOutBaby follow our friend Joanna who joined us early on unfortunately a tornado came through his area and it cut out his power and everything so everything's fine with him we got an update from him but that's why Joanna couldn't join us for the rest of the show so make sure to go follow Joanna as well absolutely, he got put in a different layer so that's another thing we were talking about the concerns about him on release, if his internet and stuff isn't sorted out then his power and all that stuff because who knows how much damage the tornado did his house is fine but in general in the area so yeah do we know if the layers have their own auction house or all the auction house is linked? layering only affects the people that you see in the world around you but as far as auction house as far as chat as far as all this other stuff it is wide open everybody else is playing on the same server layering is essentially like a dynamic they're like dynamic server clusters that are constantly merging and unmerging but the difference is that all the names are linked all the auction houses are linked but there's one auction house all the names are the same chat channels are all linked together it's all one big chat channel so from that respect you're not really fracturing the economy or taking people out of the game in that regard it's only just basically to handle lag what's going on in the world directly around you kind of you read general chat as layered have you guys heard the general chat as layered? I heard that it was not I don't think so I was under the impression that everything was that everything was together wait if general chat is layered then how are you going to get invited to the asmen layer if you're spamming a general chat to get invited to the asmen layer who's going to invite you oh no dude I was under the impression that the chats are not layered unless they changed it I'm so happy I'm not a lion's dude I'm so happy are cities exempt from layering and this is something else let's talk about this real quick before I finish do you I kind of wish that cities were on their own layer because I don't think it works this way but cities were on their own layer and if you are in that city you are in that city and you see everybody else who's in town like whether it's stormwind, iron forge, ogre mar under city, whatever it is you see everybody in town but the issue is is let's say you have a raid coming into your city your entire raid goes into that city's layer I'll get a city raid situation and then you don't see them and then all of a sudden you see the tunnel, iron forge you got the statues, all this stuff and then you're just sitting out here minding your own business you would be watching some people duel and stuff outside then all of a sudden you see about like 200 tauren rushing in at you and here's the thing, I think that is such a city raids don't happen every day but you are in the capital cities every day from my perspective if I had to choose between the two I would rather have the cities be on their own layer in town, you're in town and you can see everybody that's how I feel here's the big thing I don't even think with the way layering works right now they can do that because I remember we asked them hey what if you only layered the 1 to 20 zones or the 1 to 30 zones then everything above 30 or everything above 20 was not layered and it was free for all you could see everyone and they were like no with the way layering works it has to be all or nothing it's the entire continent or it's not the entire continent so I don't even think they can take an area or a zone or a city or whatever and just pick it out of the layer thing and make its own deal I don't even think they can do that I don't know I don't know if that's something that they can't do at all or if that's just not how it works right now though that's what I'm saying I'm saying in an ideal situation and not everything is ideal that's just what I would hope that they could do is that they could have the cities on their own layers so that everybody is in town at the same time I think it would be cool but then of course you have like you said the city raid thing but I think if you wear the pros and cons I mean who knows right who knows how it would actually be approached what do you think this I think at the end of the day when it comes to layering I think it's such a hard question to answer because we don't know how layering works 100% I think at the end of the day like what Stay Safe said certain zones can be layered I don't know I don't have enough information to give a clear answer to be honest so I mean like this is my understanding of it if you have certain zones that are layered you just have shards again because the way sharding works in retail and how it has worked for several expansions you might have an elwyn forest shard and a duskwood shard and a red ridge mountain shard that might reappear or up here as you hop between the zones and if there's too many people in one zone in elwyn forest for example you might have a gold shire shard you might have a fargo deep mine shard and you might have a hogger area shard and as you travel between those smaller zones inside of elwyn forest you're seeing things reappear and disappear so the second you start having like like a zone wide layer or a town wide layer whatever you want to call it you're just back to sharding kind of yeah yeah it just it's really annoying and I think at this point whatever we saw in the beta is probably what it's going to be so I just hope it turns out for the best and it doesn't really take away from the game so yeah the overall the net result of the equation of everything else that's put into this thing is overall positive so that's what the goal would be I guess that's what the hope would be well I mean like with this recent blue post yesterday or two days ago whatever it was they once again they said hey it's something we're only doing the first couple weeks to deal with launch it's only launch it's for launch dates to help with launch etc so they said that multiple times and also they said we're getting rid of layering before phase two to say that again so they haven't backed down on that they reaffirmed it at least the message they're putting out is that they want to get rid of ASAP that's my impression will that actually happen will they get rid of ASAP I don't know but they've said it multiple times I guess we just sort of have to take their word for it and cross their fingers and hope it it's gone soon and we will start to find out how all this stuff is going to roll here in just under eight days by the time this is posted on YouTube about seven days so it's going to be really really exciting to see you guys thank you so much for joining us for Classic Cast Joanna was with us for a little bit unfortunately he couldn't stay, we would love to have Joanna on another time and get the chance to talk with him some more he's a really really great guy make sure you guys go follow Joanna also make sure to follow StaySafeTV TipsOutBaby check out their YouTube channels as well this is going to be posted on YouTube just like all the Classic Cast and everything else going on on YouTube also for myself, for my own front I'm looking at having somebody else manage my YouTube channel here for Classic Launch so that I can get more YouTube content and stuff out to you guys and trying to get all that stuff sorted out so those of you guys watching on YouTube you guys have something to look forward to a lot more consistent content from me hopefully if everything works out come Classic Launch so that's YouTube.com YouTube TipsOutBaby and YouTube StaySafeTV so anyways guys thank you so much for joining us continue the stream I will be playing Dauntless today and we will see you guys next time yeah thanks chat take it easy boys see you either on Classic Cast before launch or we'll see you in game 7 days let's go!


This is TFNN, The Tiger, Financial News Network Update. Good morning everyone, Basel Chapman here on this Monday the 17th of April, we're looking at on Marathon Day in Boston, we're looking at the Dow at this 10 a.m. Tiger Financial News Network Market Update, the Dow is up 62, 33,949, it's getting into this area of strong resistance in a leg D, we're going to be watching this closely why, because the S&P also is getting into resistance areas, got this long leg at Doji from Friday. This is leg E in the Chapman Way methodology, up 5 at 41, 43, there's a lot to discuss today, when I get to my show in a few minutes time, the Tiger technicians are, we're also looking at QQQ, now the QQQ is up 4 cents at 318, 68, you can see how it's stalled here, all the technicals in three indices I've spoken about are extremely strong right now, so the price is holding, it's not breaking out but it's holding. The IWM Russell 2011 77 72 is up 1, doing okay, ah, here's the story, the gold contract had a sharp slide, fried ends, unusual to have a second slide without a big rally today's down 7 at 2008 and why is that, I think it's because as I've discussed with my subscribers to my opening call in my over an hour long video on Saturday for the market overview, the dollar is acting well and it's coming in right at the left side, right at price time at 100.82 was the low in February and on Friday the low was 100.79, it's ready for a bit of a bounce, I think that's what we're looking at and if you're looking at bonds, bonds are down against, yields are moving up, still within a range, an acceptable range and if you're looking at crude oil, crude oil is holding, it's had that peak see above the 200-beard moving average, holding steady at 82.03 down 50 cents, you'll see whether it breaks to the 84s or it slides under a 79 support, looking at the volatility index which is at 17.41, it's actually up 34 cents today, we'll see that last, I do expect this week to be a week of consolidating and I wouldn't be surprised if there's a bit of slippage in the next couple of days, I'll be back for the timecare technicians hour, as I said a lot to discuss, check out my opening call and we're also talking about natural gas yes when I get back, see you in a few moments.


What's going on everybody got Cal here relating sports cards and we are ripping one box of prism draft football This one is for Ken s. Good luck. Mr. Ken 2023 prism draft See what we got. Let's see the guy on the cover mr. Richardson luck mr. Ken snakeskin They're sweet Silver rookie Isaiah Foskey And there he is called it 230 we got signing day auto anthony richardson boom Called it Signing day richardson that one is number 24 out of 30 There we go Ken The little uh little gators orange color match their orange pulsar Signing day of rich nice one there to 30 Very nice one Ken called it Called it There's a red Peyton Manning Peyton Manning till 299 and snakeskin rookie rakeem Jarrett. That's a uh bucks rookie rakeem Jarrett Prism break of levis and trophy hunters of burrow Hey, no worries. Can I got you my friend?  How about that? Yeah, I got you Ken. I got you. I'm gonna get it right after we finish here There's a silver sauce And auto is gonna be michael wilson. That's a rookie um rookie receiver for the cardinals michael wilson got zeke on the blue wave to 249 got a um Muhammad ibrahim red wave not numbered Tebow and a hype of jaylon carter. Hey, let's go ken. Let's go My man got two turnaround cards. Wonder what's going on there This is an auto auto is james skalski. It's james skalski out of clemson gonuls james skalski out of clemson there on the um prism Got a silver will mcdonald jets And this one is going to be another auto. We got a prism red auto of christopher smith out of georgia To 199 christopher smith blue of ken walker To 199 and a dion neon dion purple ice to 149 Last pack let's go ken We're to london. Let's go Thank god man that guy. Thank god. Let's go jake I've been waiting on a touchdown man, man. We got another auto. This one is signing day of matt landers for um the uh razorbacks razorbacks matt landers signing day Got a red of a manual forps to two two ninety nine got gear at wilson red wave not numbered Chuck woodson and charbonate There we go ken. Let's go. Let me get that um richerson magda for you, sir We'll get that right out to you as always my friend. Congrats We called that one very nice, sir