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Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
Transformation of coordinates
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Transformation of coordinates

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  • 1. Structural Analysis and Design II
    • Group Members:
    • Adolfo Aranzales
    • 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

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