Finite Element MethodProf. C. S. UpadhyayDepartment of Mechanical Engineering Indian Institute of Technology, anpurModule ! " #ecture $ %Intheprevious lecture, wehadlookedat theprincipleof virtual workfor themodel onedimensional problem that we had introduced. We had also introduced concept of the variation ofthe function, where u was our displacement function and corresponding to u we defined thevariation of u as del of u.(Refer Slide ime! ""!#$%We had also introduced a new concept of a functional, which if &ou remember was defined as thefunction of a function' that it would take a function and give us a number. (or different functionswegetdifferent numbers. (romthedefinitionofthisfunctional wereposedourprincipleofvirtualwork asminimi)ationof a functionalandthat functionalwegotcorrespondingto themodel problem of interest turns out to be the total potential energ& corresponding to the s&stem.We had said that the corresponding e*uations in the integral form could be obtained b& taking afirst variation of the functional that is del of pi and setting it to ". hat is, we are looking for thefunction u which minimi)es the functional pi.(Refer Slide ime! "$!+#%,fter we have done all these, we had also introduced the RI- method as one of the techni*uesto get an appro.imate solution to the problem of interest. he *uestion is wh& do we want anappro.imate solution/ ,s we have shown that most of the boundar& value problems that we ma&beinterestedin willnot havea readil& available e.actsolution.hatisgettinga closeformsolution ma& be almost impossible. In that case we would like to obtain an appro.imate solutionto that problem and in the one dimensional setting we would like to introduce all the conceptsthat are needed in order to obtain a good appro.imate solution using the method of our choice.0iven this introduction what we will do is develop the RI- method that we introduced last timeingreaterdetail.,ndwewillappl&ittosomet&pical e.amplescorrespondingtothemodelproblem that we have introduced and we will show how good or bad the RI- method does withrespect to the solution of these model problems.1ssentiall& we are creating an artificial situation where we would like to gauge how good theRI- solution does or how bad it does. 2et us see how we can improve the accurac& of thesolution, and what can be the causes of the solution being bad.(Refer Slide ime! "+!"3%Remember that our model problem was this bar problem where the cross section of the bar wasnon4uniform and it is sub5ected to a distributed bod& force f(.% constrained at the point .6" that isa displacement is set to " at the point .6" and at the point .62 a force 7 is applied. 8nder theaction of this force 7 this distributed load f(.% and the constraint at the point . e*ual to ", I wouldlike to obtain the solution to this problem.(Refer Slide ime! "+!9"%:orresponding to the model problem that we have introduced let us rewrite the functional thatwe are interested in that is I (u% is e*ual to pi of u which is e*ual to integral we will take from.6" to 2, $;# in front 1, u comma . whole s*uared d. minus integral .6" to 2 f u d. minus 7 uevaluated at .62' this is our functional. What we had said as far as the solution to the problemwas concerned' the solution u corresponds to a minimum of this potential pi. We said variation ofpi of u is e*ual to " e*ual to as we have defined, the operation of variation for functions and aswell as for functionals this will be e*ual to 1,u, . variation of u, . d. minus integral .6" to 2 fvariation of u d. minus p variation of u at .62.Remember that we had said the variation of this *uantit& $;# of 1,u, .# was as if I have takenthe derivative of this e.pression with respect to u, and so we end up getting 4 the $;# goes and wewill get integral of 1, u, . the variation of u, . d..(Refer Slide ime! "!?