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# Transformation of coordinates

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### Transformation of coordinates

1. 1. Structural Analysis and Design II <ul><li>Group Members: </li></ul><ul><li>Adolfo Aranzales </li></ul><ul><li>Jon Deacon </li></ul><ul><li>Brian Spake </li></ul><ul><li>Enea Mushi </li></ul><ul><li>Rachel </li></ul><ul><li>Alvin Sosa </li></ul><ul><li>Jack </li></ul>
2. 2. TRANSFORMATION OF COORDINATES [K] global =[  ] T * [k] elemental * [  ]
3. 3. ROTATIONAL MATRIX [ L ]= 3 x 3 [L] 0 0 0 [  ] = 0 [L] 0 0 0 0 [L] 0 0 0 0 [L]
4. 4. [L] MATRIX <ul><ul><ul><ul><ul><li>L 11 L 12 L 13 </li></ul></ul></ul></ul></ul><ul><li>[L] = L 21 L 22 L 23 </li></ul><ul><ul><ul><ul><ul><li>L 31 L 32 L 33 </li></ul></ul></ul></ul></ul>
5. 5. A, B, & K COORDINATES <ul><li>[L] = 3 x 3. Directional Cosines </li></ul><ul><li>X A X B X K </li></ul><ul><li>Y A Y B Y K </li></ul><ul><li>Z A Z B Z K </li></ul>x A K B y z
6. 6. FIRST ROW <ul><li>L 11 =(X B -X A )/AB </li></ul><ul><li>L 12 =(Y B -Y A )/AB </li></ul><ul><li>L 13 =(Z B -Z A )/AB </li></ul><ul><li>Where AB is the length of the member </li></ul>
7. 7. THIRD ROW <ul><li>Z X = (Y B -Y A ) (Z K -Z A ) - (Z B -Z A ) (Y K -Y A ) </li></ul><ul><li>Z y = (Y B -Y A ) (Z K -Z A ) - (Z B -Z A ) (Y K -Y A ) </li></ul><ul><li>Z z = (Y B -Y A ) (Z K -Z A ) - (Z B -Z A ) (Y K -Y A ) </li></ul><ul><li>Where Z = Z X 2 + Z Y 2 + Z Z 2 and where: </li></ul><ul><li>L 31 = Z X /Z </li></ul><ul><li>L 32 = Z Y /Z </li></ul><ul><li>L 33 = Z Z /Z </li></ul>
8. 8. SECOND ROW <ul><li>Y X = L 13 *L 32 – L 12 *L 33 </li></ul><ul><li>Y Y = L 11 *L 33 – L 13 *L 31 </li></ul><ul><li>Y Z = L 12 *L 31 – L 11 *L 32 </li></ul><ul><li>Where Y = Y X 2 + Y Y 2 + Y Z 2 and where: </li></ul><ul><li>L 21 = Y X / Y </li></ul><ul><li>L 22 = Y Y / Y </li></ul><ul><li>L 23 = Y Z / Y </li></ul>
9. 9. 6.4 EXAMPLE OF A 3D FRAME X Y Z MEMBER 3 MEMBER 1 MEMBER 2 1 2 3 4 COORDINATES NODE 1: (15, 0, 15) NODE 2: (0, 0, 15) NODE 3: (15, 15, 15) NODE 4: (180, 0, 0) (0, 0, 0)
10. 10. 6.4 EXAMPLE OF A 3D FRAME <ul><li>MEMBER #1 </li></ul>2 1 z x y (0, 0, 0) NODE 1 IS THE LEFT NODE X A = 15, X B = 0, X K = 0 Y A = 0, Y B = 0, Y K = 0 Z A = 15, Z B =15, Z K = 0 A B
11. 11. MEMBER #1 [L] MATRIX <ul><ul><ul><ul><ul><li>-1 0 0 </li></ul></ul></ul></ul></ul><ul><li>[L] = 0 0 -1 </li></ul><ul><ul><ul><ul><ul><li>0 -1 0 </li></ul></ul></ul></ul></ul>
12. 13. Defining the Problem <ul><li>3, 4 number of member and nodes </li></ul><ul><li>1, 2, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>0, 0, 0 </li></ul><ul><li>1, 3, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>15, 0, 0 </li></ul><ul><li>1, 4, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>0, 0, 0 </li></ul><ul><li>15, 0, 15, 0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 15, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>15, 15, 15, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>15, 0, 0, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>0, 0, 0, 0, 0.41, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul>
13. 14. Defining the Problem <ul><li>3, 4 </li></ul><ul><li>1, 2, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>--Member 1, starts at node 1 and ends at node 2 </li></ul><ul><li>--29000 is the modulus of elasticity </li></ul><ul><li>--11150 is the shear modulus </li></ul><ul><li>--7.08 is the area </li></ul><ul><li>--18.3 in the moment of inertia about the Y-axis </li></ul><ul><li>--82.3 is the moment of inertia about the Z-axis </li></ul><ul><li>--0.35 is the polar moment of inertia </li></ul><ul><li>*(note the “huge” difference between the inertias, due to the element shape) </li></ul><ul><li>0, 0, 0 </li></ul><ul><li>1, 3, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>15, 0, 0 </li></ul><ul><li>1, 4, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>0, 0, 0 </li></ul><ul><li>15, 0, 15, 0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 15, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>15, 15, 15, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>15, 0, 0, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>0, 0, 0, 0, 0.41, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul>
14. 15. Defining the Problem <ul><li>3, 4 </li></ul><ul><li>1, 2, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>0, 0, 0 K node for element 1 </li></ul><ul><li>1, 3, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>15, 0, 0 K node for element 2 </li></ul><ul><li>1, 4, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>0, 0, 0 K node for element 3 </li></ul><ul><li>15, 0, 15, 0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 15, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>15, 15, 15, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>15, 0, 0, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>0, 0, 0, 0, 0.41, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul>
15. 16. Defining the Problem <ul><li>3, 4 </li></ul><ul><li>1, 2, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>0, 0, 0 </li></ul><ul><li>1, 3, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>15, 0, 0 </li></ul><ul><li>1, 4, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>0, 0, 0 </li></ul><ul><li>15, 0, 15, 0, 0, 0, 0, 0, 0 Coordinate of Node 1 in ft </li></ul><ul><li>0, 0, 15, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>15, 15, 15, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>15, 0, 0, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>0, 0, 0, 0, 0.41, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul>
16. 17. Defining the Problem <ul><li>3, 4 </li></ul><ul><li>1, 2, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>0, 0, 0 </li></ul><ul><li>1, 3, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>15, 0, 0 </li></ul><ul><li>1, 4, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>0, 0, 0 </li></ul><ul><li>15, 0, 15, 0, 0, 0, 0, 0, 0 Zero for no-bound </li></ul><ul><li>0, 0, 15, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>15, 15, 15, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>15, 1, 1, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>0, 0, 0, 0, 0.41, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul>
17. 18. Defining the Problem <ul><li>3, 4 </li></ul><ul><li>1, 2, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>0, 0, 0 </li></ul><ul><li>1, 3, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>15, 0, 0 </li></ul><ul><li>1, 4, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>0, 0, 0 </li></ul><ul><li>15, 0, 15, 0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 15, 1, 1, 1, 1, 1, 1 “One” for bound </li></ul><ul><li>15, 15, 15, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>15, 0, 0, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>0, 0, 0, 0, 0.41, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul>
18. 19. Defining the Problem <ul><li>3, 4 </li></ul><ul><li>1, 2, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>0, 0, 0 </li></ul><ul><li>1, 3, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>180, 0, 0 </li></ul><ul><li>1, 4, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35 </li></ul><ul><li>0, 0, 0 </li></ul><ul><li>180, 0, 180, 0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 180, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>180, 180, 180, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>180, 0, 0, 1, 1, 1, 1, 1, 1 </li></ul><ul><li>0, 0, 0, 0, 0.41, 0 Linear forces, translations </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul><ul><li>0, 0, 0, 0, 0, 0 </li></ul>
19. 20. Understanding the problem <ul><li>member = 1 </li></ul><ul><li>Member Stiffness Matrix in Global coordinates </li></ul><ul><li>1 2 3 4 5 6 </li></ul><ul><li>1 140.6666667 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 AE/L = (29000*7.08)/180 Equation 6.1 Page 111 </li></ul><ul><li>2 00.0000000 01.0919753 00.0000000 00.0000000 00.0000000 -98.2777778 The resulting Member Stiffness Matrix is equivalent with what </li></ul><ul><li>3 00.0000000 00.0000000 04.9407407 00.0000000 444.6666667 00.0000000 the Example 6.4 (page 113) says they are. </li></ul><ul><li>4 00.0000000 00.0000000 00.0000000 21.6805556 00.0000000 00.0000000 </li></ul><ul><li>5 00.0000000 00.0000000 444.6666667 00.0000000 53360.0000000 00.0000000 </li></ul><ul><li>6 00.0000000 -98.2777778 00.0000000 00.0000000 00.0000000 11793.3333333 </li></ul><ul><li>7 -1140.6666667 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>8 00.0000000 -01.0919753 00.0000000 00.0000000 00.0000000 98.2777778 </li></ul><ul><li>9 00.0000000 00.0000000 -04.9407407 00.0000000 -444.6666667 00.0000000 </li></ul><ul><li>10 00.0000000 00.0000000 00.0000000 -21.6805556 00.0000000 00.0000000 </li></ul><ul><li>11 00.0000000 00.0000000 444.6666667 00.0000000 26680.0000000 00.0000000 </li></ul><ul><li>12 00.0000000 -98.2777778 00.0000000 00.0000000 00.0000000 5896.6666667 </li></ul><ul><li>7 8 9 10 11 12 </li></ul><ul><li>1 -1140.6666667 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 -AE/L = - (29000*7.08)/180 Equation 6.1 page 111 </li></ul><ul><li>2 00.0000000 -01.0919753 00.0000000 00.0000000 00.0000000 -98.2777778 </li></ul><ul><li>3 00.0000000 00.0000000 -04.9407407 00.0000000 444.6666667 00.0000000 </li></ul><ul><li>4 00.0000000 00.0000000 00.0000000 -21.6805556 00.0000000 00.0000000 </li></ul><ul><li>5 00.0000000 00.0000000 -444.6666667 00.0000000 26680.0000000 00.0000000 </li></ul><ul><li>6 00.0000000 98.2777778 00.0000000 00.0000000 00.0000000 5896.6666667 </li></ul><ul><li>7 1140.6666667 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>8 00.0000000 01.0919753 00.0000000 00.0000000 00.0000000 98.2777778 </li></ul><ul><li>9 0.0000000 00.0000000 04.9407407 00.0000000 -444.6666667 00.0000000 </li></ul><ul><li>10 00.0000000 00.0000000 00.0000000 21.6805556 00.0000000 00.0000000 </li></ul><ul><li>11 00.0000000 00.0000000 -444.6666667 00.0000000 53360.0000000 00.0000000 </li></ul><ul><li>12 00.0000000 98.2777778 00.0000000 00.0000000 00.0000000 11793.3333333 </li></ul>
20. 21. Understanding the Problem <ul><li>Beta matrix </li></ul><ul><li>1 2 3 4 5 6 </li></ul><ul><li>1 -01.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>2 00.0000000 00.0000000 -01.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>3 00.0000000 -01.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>4 00.0000000 00.0000000 00.0000000 -01.0000000 00.0000000 00.0000000 </li></ul><ul><li>5 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 -01.0000000 </li></ul><ul><li>6 00.0000000 00.0000000 00.0000000 00.0000000 -01.0000000 00.0000000 </li></ul><ul><li>7 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>8 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>9 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>10 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>11 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>12 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>7 8 9 10 11 12 </li></ul><ul><li>1 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>2 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>3 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>4 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>5 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>6 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>7 -01.0000000 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>8 00.0000000 00.0000000 -01.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>9 00.0000000 -01.0000000 00.0000000 00.0000000 00.0000000 00.0000000 </li></ul><ul><li>10 00.0000000 00.0000000 00.0000000 -01.0000000 00.0000000 00.0000000 </li></ul><ul><li>11 00.0000000 00.0000000 00.0000000 00.0000000 00.0000000 -01.0000000 </li></ul><ul><li>12 00.0000000 00.0000000 00.0000000 00.0000000 -01.0000000 00.0000000 </li></ul>
21. 22. Defining beta ( β ) <ul><li>When the local x-axis is parallel to the global Y-axis, as in the case of a column in a structure, the beta angle is the angle through which the local z-axis has been rotated about the local x- axis from a position of being parallel and in the same positive direction of the global Z-axis. </li></ul>
22. 23. 3D FRAMES [F] = [K] * [U]
23. 24. [k] local ELEMENTAL STIFFNESS MATRIX 12X12 P 9 ,  9 P 12 ,  12 P 6 ,  6 P 3 ,  3 P 5 ,  5 P 2 ,  2 P 10 ,  10 P 7 ,  7 P 8 ,  8 P 11 ,  11 P 1 ,  1 P 4 ,  4