Transformation of coordinates

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