Finite Element Anslysis (Solved Problems)

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    02-Oct-2015

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Finite Element Anslysis

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ASSIGNMENT 6-1 Find the shape function for the following element using zero method. 1 =234672(1)3(1)4(1)6(1)7(1) (1) { = 27. = 9 1 =(9 + )(9 )(27 )()(9 )(9 27)(9 + 27)(27 + 27)(9)(9 + 9) 1 =(81 2)(27 )()(9 )5668704 1,4,6,9 =15668704. (81 2)(27 + )(9 + ) 2 =134671(2)3(2)4(2)6(2)7(2) () { = 9. = 9 2 =(27 + )(9 )(27 )()(9 )(27 9)(9 + 9)(27 + 9)(9)(9 + 9) 2 = (729 2)(9 )()(9 )1889568 2,8,3,7 =11889568. (729 2)(9 + )(9 + ) Z1 = 27 + , Z2 = 9 + , Z3 = 9 Z4 = 27 3 = +9 2 = 9 12 11 10 6 7 8 9 5 4 1 7 6 5 4 3 2 1 7 = 9 6 = = 9+ = 27 = +27 =123 71( )2( )3( ) ( )7( ) () { = +27. = 0 =(27 + )(9 + )(9 )(9 + )(9 )(27 + 27)(9 + 27)(9 27)(9)(9) = (27 + )(81 2 )()(9 )2834352 ,10 = 12834352. (81 2)(27 + )(9 + ) remains constant in node 5 and 10 Therefore the negative sign remains as well in the generalized form. 11 =134 71(11)3(11)4(11) (11)7(11) (11) { = 9. = 0 11 =(27 + )(9 )(27 )(9 + )(9 )(27 9)(9 + 9)(27 + 9)(9)(9) 11 =(9 )(729 2)(81 2 )944784 11,12 =1944784. (9 + )(729 2)(81 2 ) ASSIGNMENT 6-3 Find the shape function for the following element: 1 =23 62(1)3(1) (1)6(1) (1) { = 0.75. = 2 1 =(10.75 )()(2 )(0.75)(10.75 + 0.75 )(2)(2 + 2)1 =(10.75 )()(2 )4 1, =14. 2 0.75 1, =14. 2 + 0.75 2 =13 61(2)3(2) (2)6(2) () { = 0. = 2 2 =(0.75 + ) (10.75 )()(2 )(0.75)(10.75 + 0 )(2)(2 + 2)2 = 18. ()(18 )(0.75 + ) (0.75 ) 2,6 =18. (18 + )(0.75 + ) (0.75 ) remains with the same value t in node 2 and 6 Therefore there is no need to generalize solution for value. Z1 = 0.75 + , Z2 = , Z3 =10.75 3 6 7 2 = 0, = 0 = +2 5 = 0 = 2 8 4 1 6 5 4 3 2 1 = 0.75 = 0 = 6 = 9 = 4 = 9+ 4 =23462(4)3(4) (4)6(4) () { = 0.75. = 0 4 = (10.75 )(2 + )(2 )(0.75)(10.75 + 0.75 )(2)(2)4 = (10.75 )(4 2 )2 4 = 12. (0.75 )(18 2 ) 8 =13462(4)3(4) (4)6(4) (8) { = 0. = 0 8 =(0.75 + ) (10.75 )(2 + )(2 )(0.75)(10.75 + 0 )(2)(2)8 =(0.75 + ) (10.75 )(4 2 )4 8 =14.(0.75 + )(0.75 )(4 2) ASSIGNMENT 6-4 Find the shape function for the following element using Zero method 1 =23 672(1)3(1) (1)6(1) (1) { = 9. = 9 1 =()(19 )(3 + )(3 )(9 )(9)(19 + 9 )(3 9)(3 + 9)(9 + 9)1 =()(9 )(9 )(9 2 )648 1, =1648. (9 )(9 )(9 2 ) 1,6 =1648. (9 2 )(9 + ) (9 ) 2 =13 671(2)3(2) (2)6(2) () { = 0. = 9 2 =(9 + ) (19 )(3 + )(3 )(9 )(9)(19 + 0 )(3 9)(3 + 9)(9 + 9)2 = (9 + )(9 )(9 2 )(9 )1296 2,7 = 11296. (9 + )(9 2 ) (9 + )(9 ) Z1 = 9 + , Z2 = , Z3 =19 = +9 7 = 9 5 = +3 6 = 3 = 9 4 8 7 3 2 1 6 5 4 3 2 1 = 9 = 0 = 6 = 3 = 3+ 4 = 9+ 7 4 =234672(4)3(4)4(4)6(4)7(4) () { = 9. = 3 4 = (19 )(9 + )(3 )(9 )(9)(19 + 9 )(9 3)(3 + 3)(9 + 3)4 = (19 )(3 )(81 2 )216 4 = 1216. (81 2 ) (3 )(9 ) 4,5 = 1216. (81 2 ) (3 + )(9 ) ASSIGNMENT 6.5 Find the shape function for the following element using zero method? 1 = 18 + 4 = 9 + 7 = 9 2 = = 8 = 3 =118 6 = 9 9 = 9 = +9 = 0 = 9 = 0 = + 9 6 17 11 19 13 7 15 9 5 16 10 14 8 18 12 1 4 2 3 = -18 = 0 = = - 9 3 2 1 6 5 4 1 =23 6892(1)3(1) (1)6(1)8(1)9(1) (1) { = 18 = 9 = +91 =()(118 )()(9 )()(9 + )(18)(118+ 18 )(9)(18)(9)(9 + 9)1 =(18 )()(9 )()(9 + )13122 1, =113122. (18 )()(9 )(9 + ) 1,6,14,18 = 113122. (18 )()()(9 + )(9 + ) 2 =13 6891(2)3(2) (2)6(2)8(2)9(2) (2) { = 0 = 9 = +91 =(18 + )(118 )()(9 )()(9 + )(18)(118 )(9)(18)(9)(9 + 9)2 =(18 + )(118 )()(9 )()(9 + )26244 1 = 126244(18 + )(118 )()(9 )()(9 + ) 2,7,1 ,17 =126244.(18 + )(18 ) ()()(9 + )(9 + ) 4 =2346892(4)3(4)4(4)6(4)8(4)9(4) (4) { = 18 = 0 = +94 = (118 )(9 + )(9 )()(9 + )(18)(118 + 18 )(9)(9)(9)(9 + 9)4 = (118 )(9 + )(9 )()(9 + )6561 4 = 16561. (18 )(81 2 )()(9 + ) 4,16 = 16561. (18 )(81 2 )()(9 + ) =1346891( )3( )4( )6( )8( )9( ) (5) { = 0 = 0 = +95 =(18 + )(118 )(9 + )(9 )()(9 + )(18)(118 )(9)(9)(9)(9 + 9)5 =(18 + )(118 )(9 + )(9 )()(9 + )13122 5 =113122. (18 + 18 )(81 2 )()(9 + ) ,17 =113122. (18 + 18 )(81 2 )()(9 + ) 10 =2346792(10)3(10)4(10)6(10)7(10)9(10) (10) { = 18 = 0 = 010 =(118 )(9 + )(9 )(9 )(9 + )(18)(136 )(9)(9)(9)(9)10 =(18 )(81 2 )(81 2 )3280.5 10 = 26561. (18 )(81 2 )(81 2 ) 11 =1346791(11)3(11)4(11)6(11)7(11)9(11) (11) { = 0 = 0 = 011 =(18 + )(118 )(9 + )(9 )(9 )(9 + )(18)(118 )(9)(9)(9)(9)11 =(18 + )(18 ) (81 2 )(81 2 )6561 11 =16561. (18 + )(18 ) (81 2 )(81 2 ) 8 =2346792(8)3(8)4(8)6(8)7(8)9(8) (8) { = 18 = 9 = 08 =()(118 )( )(9 )(9 )(9 + )(18)(118 + 18 )(9)(18)(9)(9)8 =(18 )( )(9 )(81 2)6561 8,12 = 16561. (18 )( )(9 + )(81 2) 9 =1346791(9)3(9)4(9)6(9)7(9)9(9) (9) { = 0 = 9 = 09 =(18 + )(118 )( )(9 )(9 )(9 + )(18)(118 )(9)(9 + 9)(9)(9)9 =(18 + )(18 ) ( )(9 )(81 2)13122 9,13 =113122. (18 + )(18 )( )(9 + )(81 2) ASSIGNMENT 6-8 Find Jacobi matrix for the following matrix. Shape functions of the above element are as follow: 1 =14(1 )(1 ) 2 =14(1 + )(1 ) 3 =14(1 + )(1 + ) 4 =14(1 )(1 + ) [.] = [.] [.] [] = [11 12.21 22] X (27, -1) 2 1 3 4 (7,-1) (7, 18) (27, 18) Y -Y 1= 14(1 ) 1= 14(1 ) 2= 14(1 ) 2= 14(1 + ) 3= 14(1 + ) 3= 14(1 + ) 4= 14(1 + ) 4= 14(1 ) 11= 14(1 )(7) + 14(1 )(27) + 14(1 + )(27) 14(1 + )(7) = 74(1 ) + 274(1 ) + 274(1 + ) 74(1 + ) =204(1 ) + + 204(1 + ) = 5(1 ) + + 5(1 + ) = 5 5 + 5 + 5 = 10 12= 14(1 )(1) + 14(1 )(1) + 14(1 + )(18) 14(1 + )(18) =14(1 ) 14(1 ) + 92(1 + ) 92(1 + ) = 0 21= 14(1 )(7) 14(1 + )(27) + 14(1 + )(27) + 14(1 )(7) = 74(1 ) 274(1 + ) + 274(1 + ) + 74(1 ) = 0 22= 14(1 )(1) 14(1 + )(1) + 14(1 + )(18) + 14(1 )(18) =14(1 ) +14(1 + ) + 184(1 + ) + 184(1 ) =194(1 ) +194(1 + ) =194194 +194+194 =194+194 =192 [] = [10 0 .0 ] Therefore, det [J] = 95 19 20 Dimension X= 27-7 = 20 Y= 18-(-1) = 19 A = 20 x 19 = 380 Therefore, J = 4=3804= 95 ASSIGNMENT 6.9: Determine the Jacobi matrix for the following element? Shape functions for the above element in each node are as follow: 1 =18(1 )(1 )(1 + ) 2 = 18 (1 + )(1 )(1 +) 3 =18(1 )(1 + )(1 +) 4 =18(1 + )(1 + )(1 +) =18(1 )(1 )(1 ) 6 =18(1 + )(1 )(1 ) 7 =18(1 )(1 + )(1 ) 8 =18(1 + )(1 + )(1 ) [ ]=[ ] [ ]X Z Y 8 7 6 5 413 2 1 Point (X, Y, Z) Point 1 (-9,-18, 27) Point 2 (18,-18, 27) Point 3 (-9, 27, 27) Point 4 (18, 27, 27) Point 5 (-9,-18, 0) Point 6 (18,-18, 0) Point 7 (-9, 27, 0) Point 8 (18, 27, 0) [] =[ 11 12 1321 22 2331 32 33] 1= 18(1 )(1 + ) 2=18(1 )(1 +) 3= 18(1 + )(1 +) 4=18(1 + )(1 +) 5= 18(1 )(1 ) 6=18(1 )(1 ) 7= 18(1 + )(1 ) 8=18(1 + )(1 ) 1= 18(1 )(1 + ) 2= 18(1 + )(1 +) 3= 18(1 )(1 +) 4= 18(1 + )(1 +) 5= 18(1 )(1 ) 6= 18(1 + )(1 ) 7= 18(1 )(1 ) 8= 18(1 + )(1 ) 1=18(1 )(1 ) 2= 18(1 + )(1 ) 3=18(1 )(1 + ) 4= 18(1 + )(1 + ) = 18(1 )(1 ) 6= 18(1 + )(1 ) 7= 18(1 )(1 + ) 8= 18(1 + )(1 + ) 11 = = 18(1 )(1 + )(-9)+ 18(1 )(1 +)(18) 18(1 + )(1 +)(-9)+ 18(1 + )(1 +)(18) 18(1 )(1 )(-9)+ 18(1 )(1 )(18) 18(1 + )(1 )(-9)+ 18(1 + )(1 )(18) = 98(1 )(1 + )+ 94(1 )(1 +)+98(1 + )(1 + )+ 94(1 + )(1 +)+ 98(1 )(1 ) +94(1 )(1 )+ 98(1 + )(1 )+ 94(1 + )(1 ) = 94(1 + ) [12(1 ) + (1 ) +12(1 + ) + (1 + )] +94(1 ) [ 12(1 ) + (1 ) + 12(1 + ) + (1 + )] = 94(1 + ) [32(1 ) +32(1 + )] +94(1 ) [ 32(1 ) + 32(1 + )] = 94(1 + ) [3232 +32+32] +94(1 ) [3232 + 3232] = 94(1 + ) [32+32] +94(1 ) [32+32] = 94(1 + )[3] +94(1 )[3] = 274(1 + ) +274(1 ) = 274+274 +274274 = 274+274 = 44 12 = 13 = = 18(1 )(1 + )(27)+ 18(1 )(1 +)(27) 18(1 + )(1 +)(27)+ 18(1 + )(1 +)(27) 18(1 )(1 )(0)+ 18(1 )(1 )(0) 18(1 + )(1 )(0)+ 18(1 + )(1 )(0) = 278(1 )(1 + ) + 278(1 )(1 +) 278(1 + )(1 +)(27)+ 278(1 + )(1 +) 0 0 0+ 0 = 0 21 = = 18(1 )(1 + )(-9) 18(1 + )(1 +)(18)+ 18(1 )(1 +)(-9)+ 18(1 + )(1 +)(18) 18(1 )(1 )(-9) 18(1 + )(1 )(18)+ 18(1 )(1 )(-9)+ 18(1 + )(1 )(18) =98(1 )(1 + ) 94(1 + )(1 +) 98(1 )(1 +) + 94(1 + )(1 +) +98(1 )(1 )94(1 + )(1 ) 98(1 )(1 )+ 94(1 + )(1 ) = 0 = 18(1 )(1 + )(-18)+ 18(1 )(1 +)(-18) 18(1 + )(1 +)(27)+ 18(1 + )(1 +)(27) 18(1 )(1 )(-18)+ 18(1 )(1 )(-18) 18(1 + )(1 )(27)+ 18(1 + )(1 )(27) = 94(1 )(1 + ) 94(1 )(1 +)278(1 + )(1 + )+ 278(1 + )(1 +)+ 94(1 )(1 ) 94(1 )(1 ) 278(1 + )(1 ) + 274(1 + )(1 ) = 0 22 = = 18(1 )(1 + )(-18) 18(1 + )(1 +)(-18)+ 18(1 )(1 +)(27)+ 18(1 + )(1 +)(27) 18(1 )(1 )(-18) 18(1 + )(1 )(-18)+ 18(1 )(1 )(27)+ 18(1 + )(1 )(27) =94(1 )(1 + )+94(1 + )(1 +)+ 278(1 )(1 +) + 278(1 + )(1 +) +94(1 )(1 )+94(1 + )(1 )+ 278(1 )(1 )+ 278(1 + )(1 ) =94(1 + ) [(1 ) + (1 + ) + 32(1 ) + 32(1 + )] +94(1 ) [(1 ) + (1 + ) + 32(1 ) + 32(1 + )] =94(1 + ) [ 52(1 ) + 52(1 + )] +94(1 ) [ 52(1 ) + 52(1 + )] =94(1 + ) [ 5252 + 52+52] +94(1 ) [ 5252 + 52+52] = 94(1 + ) [ 52+ 52] +94(1 ) [ 52+ 52] = 94(1 + ) [ 102] +94(1 ) [ 102] = 94(1 + )(5) +94(1 )(5) = 454(1 + ) +454(1 ) = 454+ 454 +454454 = 454+454 = 904 23 = = 18(1 )(1 +)(27) 18(1 + )(1 +)(27)+ 18(1 )(1 +)(27)+ 18(1 + )(1 +)(27) 18(1 )(1 )(0) 18(1 + )(1 )(0)+ 18(1 )(1 )(0)+ 18(1 + )(1 )(0) = 278(1 )(1 + ) 278(1 + )(1 +)+278(1 )(1 +)+ 278(1 + )(1 +) = 0 31 = =18(1 )(1 )(9) + 18(1 + )(1 )(18) +18(1 )(1 + )(9) + 18(1 + )(1 + )(18) 18(1 )(1 )(9) 18(1 + )(1 )(18) 18(1 )(1 + )(9) 18(1 + )(1 + )(18) =98(1 )(1 ) + 94(1 + )(1 ) +98(1 )(1 + ) + 94(1 + )(1 + ) 98(1 )(1 ) 94(1 + )(1 ) 98(1 )(1 + ) 94(1 + )(1 + ) = 0 32 = =18(1 )(1 )(18) + 18(1 + )(1 )(18) +18(1 )(1 + )(27) + 18(1 + )(1 + )(27) 18(1 )(1 )(18) 18(1 + )(1 )(18) 18(1 )(1 + )(27) 18(1 + )(1 + )(27) = 94(1 )(1 ) 94(1 + )(1 ) +278(1 )(1 + ) + 274(1 + )(1 + ) +94(1 )(1 ) +94(1 + )(1 ) 278(1 )(1 + ) 274(1 + )(1 + ) = 0 33 = =18(1 )(1 )(27) + 18(1 + )(1 )(27) +18(1 )(1 + )(27) + 18(1 + )(1 + )(27) 18(1 )(1 )(0) 18(1 + )(1 )(0) 18(1 )(1 + )(0) 18(1 + )(1 + )(0) =278(1 )(1 )(27) + 278(1 + )(1 )(27) +278(1 )(1 + )(27) + 278(1 + )(1 + )(27) 0 =278(1 )[(1 ) + (1 + )] + 278(1 + )[(1 ) + (1 + )] =278(1 )[1 + 1 + )] + 278(1 + )[1 + 1 + )] =278(1 )[1 + 1)] + 278(1 + )[1 + 1] =278(1 )(2) + 278(1 + )(2) =548(1 ) + 548(1 + ) =548+548 +548548 =548+548 =272 [] =[ 272 0 00 4 2 00 0 272 ] Therefore, || = 4100.625 Volume of cube element = (XYZ) =27 4527 = 32805 =8=328058= 4100.625 45 Cube element dimensions: X= 18-(-9) =27 Y= 27-(-18) = 45 Z= 27-0=27

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