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Structural Analysis and Design II
Group Members: Adolfo Aranzales Jon Deacon Brian Spake Enea Mushi Rachel Alvin Sosa Jack
TRANSFORMATION OF COORDINATES
[K] global=[]T * [k]elemental * []
ROTATIONAL MATRIX
[L] 0 0 0
[] = 0 [L] 0 0
0 0 [L] 0
0 0 0 [L]
[ L ]= 3 x 3
[L] MATRIX
L11 L12 L13
[L] = L21 L22 L23
L31 L32 L33
A, B, & K COORDINATES
• [L] = 3 x 3. Directional Cosines
• XA XB XK
• YA YB YK
• ZA ZB ZK
x
A
K By
z
FIRST ROW
• L11=(XB-XA)/AB
• L12=(YB-YA)/AB
• L13=(ZB-ZA)/AB
Where AB is the length of the member
THIRD ROW
• ZX = (YB-YA) (ZK-ZA) - (ZB-ZA) (YK-YA)• Zy = (YB-YA) (ZK-ZA) - (ZB-ZA) (YK-YA)• Zz = (YB-YA) (ZK-ZA) - (ZB-ZA) (YK-YA)
Where Z = ZX2+ ZY
2+ ZZ2 and where:
L31= ZX/Z
L32= ZY/Z
L33= ZZ/Z
SECOND ROW
• YX = L13*L32 – L12*L33
• YY = L11*L33 – L13*L31
• YZ = L12*L31 – L11*L32
Where Y = YX2+ YY
2+ YZ2 and where:
L21= Y X/ Y
L22= Y Y/ Y
L23= Y Z/ Y
6.4 EXAMPLE OF A 3D FRAME
X
Y
ZMEMBER 3
MEMBER 1MEMBER 2
12
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)
6.4 EXAMPLE OF A 3D FRAME
• MEMBER #1
2
1
z
x
y
(0, 0, 0)
NODE 1 IS THE LEFT NODE
XA = 15, XB = 0, XK = 0
YA = 0, YB = 0, YK = 0
ZA = 15, ZB =15, ZK = 0
A
B
MEMBER #1 [L] MATRIX
-1 0 0
[L] = 0 0 -1
0 -1 0
Defining the Problem
3, 4 number of member and nodes1, 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
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, 0.3515, 0, 01, 4, 29000#, 11150#, 7.08, 18.3, 82.8, 0.350, 0, 015, 0, 15, 0, 0, 0, 0, 0, 00, 0, 15, 1, 1, 1, 1, 1, 115, 15, 15, 1, 1, 1, 1, 1, 115, 0, 0, 1, 1, 1, 1, 1, 10, 0, 0, 0, 0.41, 00, 0, 0, 0, 0, 00, 0, 0, 0, 0, 00, 0, 0, 0, 0, 0
Defining the Problem 3, 41, 2, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35
0, 0, 0 K node for element 11, 3, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35
15, 0, 0 K node for element 21, 4, 29000#, 11150#, 7.08, 18.3, 82.8, 0.35
0, 0, 0 K node for element 315, 0, 15, 0, 0, 0, 0, 0, 00, 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, 10, 0, 0, 0, 0.41, 00, 0, 0, 0, 0, 00, 0, 0, 0, 0, 00, 0, 0, 0, 0, 0
Defining the Problem3, 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
Defining the Problem3, 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-bound0, 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
Defining the Problem3, 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
Defining the Problem3, 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, translations0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0
Understanding the problemmember = 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 1112 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 1112 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
Understanding the ProblemBeta 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
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.
3D FRAMES
[F] = [K] * [U]
[k] local ELEMENTAL STIFFNESS
MATRIX 12X12
P9, 9
P12, 12P6, 6
P3, 3
P5, 5
P2, 2P10, 10P7, 7
P8, 8
P11, 11
P1, 1P4, 4
