4. (30 points) For this same frame structure, let E, A and I be constant and...

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4. (30 points) For this same frame structure, let E, A and I be constant and same for all the three elements. Also, supports at A, C and D are built-in, i.e., welded. Let U4, U5 and U be the solution of the generalized displacements at node 2, obtained after a full finite element solution. Compute the bending moment at the midpoint of each of the elements 1, 2 and 3 in terms of the problem quantities given above (i.e., E, A, I, U4, etc.) 2 m B 3 m 2 m Fa 1 Z X 3 Them 4 m The cos e sin 20 -cos e E.A sin 20 sin e -sin 20 - sin 20 sin 0. [K] = he - cos -sin 20 cos be sin 20 - - sin e sin 20 sin O sin 20 *-1-3 (A)+2(A) = 2 3 2 3 3 = (1-)* -x (1) = (#) - (#)' --[(#)**] =3 ' = -x [() 6 -3he -6 -3he ] 2Eele -3he 2h2 3he h [Ke] = h -6 3he 6 3he -3he h 3he 2h 6 Q1 gehe -he Q2 {Fe} = ==== + 12 6 Q3 he Q4 W1 S EES COS sina 0 - sin a cos 0 0 WI 001 S (5.4.6a) 2 cos sina 0 142 W2 0 - sin a cos a 0 W2 52 00 1 S 4. (30 points) For this same frame structure, let E, A and I be constant and same for all the three elements. Also, supports at A, C and D are built-in, i.e., welded. Let U4, U5 and U be the solution of the generalized displacements at node 2, obtained after a full finite element solution. Compute the bending moment at the midpoint of each of the elements 1, 2 and 3 in terms of the problem quantities given above (i.e., E, A, I, U4, etc.) 2 m B 3 m 2 m Fa 1 Z X 3 Them 4 m The cos e sin 20 -cos e E.A sin 20 sin e -sin 20 - sin 20 sin 0. [K] = he - cos -sin 20 cos be sin 20 - - sin e sin 20 sin O sin 20 *-1-3 (A)+2(A) = 2 3 2 3 3 = (1-)* -x (1) = (#) - (#)' --[(#)**] =3 ' = -x [() 6 -3he -6 -3he ] 2Eele -3he 2h2 3he h [Ke] = h -6 3he 6 3he -3he h 3he 2h 6 Q1 gehe -he Q2 {Fe} = ==== + 12 6 Q3 he Q4 W1 S EES COS sina 0 - sin a cos 0 0 WI 001 S (5.4.6a) 2 cos sina 0 142 W2 0 - sin a cos a 0 W2 52 00 1 S