noc18-me58-Lec 28-Equilibrium Equations for Composite Plates

[Music] [Music] hello welcome to advanced composites today is the fourth day of the ongoing week which is the fifth week of this course and what we have discussed over this week is we have started with force equilibrium equations and we developed these three of these equations for XY and z direction and then yesterday we developed the three moment equilibria mister equations and what we found that while developing the moment equilibrium equation for the six for the third axis that is red axis we found that the fact that tau XY equals tau YX that means n XY equals n YX and that implies directly that the sixth equilibrium equation will be identically satisfied so effectively we have only five equilibrium equations mutually independent equations and we will again rewrite them and process them further so what are these equations del NX over del X plus del n X Y over del y equals 0 this is for the X equilibrium for forces del n X Y and now because NX y is equal to n YX I am just going to write it like that del X plus del and Y over del y equals 0 this is the second force equilibrium equation the third equilibrium equation is Del Q x over del X plus del Q Y over del y plus Q equals zero and then we have the moment equilibrium equations del em x over del X plus del M X Y over del y minus qy no QX equals 0 and finally we have del M X Y over del X plus del my over del Y minus Q y equals 0 so these are the five independent equilibrium equations all of them are partial differential equations in N and M moment resultant and force resultant and what is their order their order is they are first order PDE is first order partial differential equations and if we solve these equations we will get the solutions for n X n X Y n y MX my and QX n Qi so these are the equations now what we will do is we will combine the last three equations into one single equation because that makes things a little bit easier to handle mathematically so from so we will number these so this is a b c d e so we know that from a now from d we know that QX equals del MX over del X plus del M X Y over del Y ok so if we partially differentiate this equation in X we get del QX over del X is equal to second derivative of M X with respect to X plus del 2 m XY over del X del Y so let us call this equation F and then likewise from E what do we have we know that Q y equals Del M X Y over del X plus del my over del Y and if we partially differentiate this whole equation with respect to Y then we get del qy over del y equals cross derivative of M X Y with respect to X and y plus del to my over del Y square so we call this equation G okay so we see that this is del QX over del X and which is same as this and in equation C the second term is partial derivative of Q Y with respect to Y and that is there in equation G so we put put F jee-in see and finally what we get is Del 2 MX over del X square plus twice of del 2 M X Y over del X del y plus del 2 my over del Y square plus Q equals 0 so this is the other equation so we have H so I can either I can use these five equations ABCD any or I can use these three equations okay I'm sorry is equal to Q so did I write Q in the other one so actually Q is on the left side so here what we have done is we have reduced the number of differential equations from five to three but whenever you do such types of reductions what ends up happening is that you raise the order so we have reduced the number of differential equations but we have increased the order of the third differential equation so earlier it was just first order now this is a second order differential equation okay so this is so if we solve these equations we can find out what is so at every point okay so the what is the point so so the aim is to solve these partial differential equations once you solve these partial differential equations we get we get values of n x and y n X Y and what else MX m by m x y at all points in plate so if we are able to solve these partial differential equations then we will be able to calculate n XY and Y and X Y at all the points in the plate once we do that then as a next step what we can do is calculate epsilon naught and care K naught curvatures and midplane strains and midplane curvatures at all points in plate and once we are doable to do that then what do we get out of it calculate the strains at okay strains at points of interest and once we know the strains had different in in different layers because now if we know epsilon naught and K naught then using the relation between epsilon and epsilon naught and K naught we can calculate strains at different layers at different points so then we can calculate stresses in the plate right we can calculate stresses by using cube R matrix and epsilon matrix for the case layer okay and once we know that we can predict the failure of the plate so we can know whether our plate is strong enough or not so this is the overall process you start with partial differential equations solve the partial differential equation you get n X n yn X Y MX M by M X Y once you have knowledge about these moments and force resultants using these force and moment resultant you compute mid plane strains and mid plane curvatures at different points in the plate or actually at all the possible locations once you have that then you can at every cross section you can calculate how strain is varying along the thickness of the plate and that will help you calculate the stresses in each layer of the plate using because then you can use fine epsilon K and Q K Q Bar K and you can calculate stresses in each layer of the plate and then you can transform these stresses to Lt plane right so you can find out Sigma L Sigma T and tau L T and then using the work-energy this maximum work criteria or maximum stress criterion maximum strain criteria you can compute whether the plate is going to fail and if it fails which particular layer is going to fail and so on and so forth so this is the overall process through which you can design composite laminated plates in this entire process there is one thing which we have not discussed till so far that whenever we solve these equations whenever a differential equation is solved then once we solve these equations we will essentially what does that mean when we solve a differential equation we this is a differential equation we integrate it right we integrate it and whenever we integrate a partial differential equation we get some functions so when we solve this we will get a function from here another function from second equation and the third equation is a second-order differential equation so we have to integrate it twice so we will get two functions from the third equation and these are these functions will are unknown so we will have to figure out their values and the way we figure out values is that if we have a plate then on each edge there will be some boundary conditions and if we know these boundary conditions then in because if we know these boundary conditions then in terms of these boundary conditions we can figure out or calculate the integration called constants or integration functions functions which are gotten because of integrating these things and so the boundary conditions will help us determine integration functions so we have to know what kind of boundary conditions are existing on every single edge of the plate so in the next class and this is this requires some detailed discussion so in the next class we will actually discuss what kind of boundary conditions are required what kind of boundary conditions we should know and we have to specify so that we can determine these integration functions so that is what we plan to discuss tomorrow so that's pretty much all for today and I look forward to seeing you tomorrow at the same time thank you [Music] [Music]