① the finite derwent meThod Ir position of boundary conditions(olis plcewent) We had obtained the aesewbed finite element eeve Kik2∞ Kht K Ki 2 for the f lowing problem and wo del The global stiffness etrix Is singular, which ts an indication tnst we have not imposed Hh ch boundary conditions(the varitional appo oada does no enforce then aulowattcall apropriste boundary Condition in this coGe s U4=0
Replacing in our systeM of equstions K!+KK2|〈(2Y=〈2 Ki Eliwinstng the first row ond cowan IKa+ k2 Kn u2(s)2beL L K2 K11U114 Where K,,= K=K=2AE K 2 12 =K=-2AE [31{8}-+{3 38A4」 R Sob port: replacing solion in tirst epn
-2AEU2=RA Note: this Is en appoximate velue. Why?ca1 proximate alisplecewemns are A)=「中(+=0+23Ae=4 2 8EA +4坚1里3+2=2 BEA K-4 E= du e=1(x dx 4EA 32⊥8poL e=2 Lex<L 8EA 4 PoL 8毛A
‖G=EE e=1 4+=2 The exact sdution for this broblem is given by ‖(xL×1_ EA 2L ∈A 、=(L-×) A the exact solution at nodes 2 and 3 以/3如2了,(L=42L 8EA 2EA which coincides with Hhe finite elemnet sowton! Is Hhis trye for all valves of x q9 cas