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# Transformation of coordinates

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### Transcript

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