FRAME ANALYSIS USING THE STIFFNESS METHOD · PDF fileFrame-Member Global Stiffness Matrix FRAME ANALYSIS USING THE STIFFNESS METHOD. 2 Simple Frames. 3 Frame-Member Stiffness Matrix

1

! Simple Frames! Frame-Member Stiffness Matrix! Displacement and Force Transformation Matrices! Frame-Member Global Stiffness Matrix

! Special Frames! Frame-Member Global Stiffness Matrix

FRAME ANALYSIS USING THE STIFFNESSMETHOD

2

Simple Frames

3

Frame-Member Stiffness Matrix

0 0 00 - AE/LAE/L

4EI/L - 6EI/L2 2EI/L6EI/L2 00

6EI/L2 - 12EI/L3 6EI/L212EI/L3 00

0

2EI/L

-6EI/L20

- 6EI/L2

12EI/L30

-6EI/L2

4EI/L

0

6EI/L2

-12EI/L3AE/L

0

0

-AE/L

0

0

m

xy

i

j

3

5

6

2

4

1

3 62 41 5

[k]

456

12

3

6EI/L24EI/L

6EI/L24EI/L

AE/L

AE/L

6EI/L2

6EI/L2

12EI/L312EI/L3

2EI/L

d 1 = 1

AE/L

AE/L

d 2 = 1

12EI/L312EI/L3

d 3 = 1

2EI/L

6EI/L2

6EI/L2

6EI/L26EI/L2AE/L

2EI/L

6EI/L26EI/L2

d 6 = 1

6EI/L26EI/L2

2EI/L

6EI/L2

12EI/L312EI/L3

4EI/L4EI/L12EI/L3

12EI/L3

6EI/L2

6EI/L2AE/L

6EI/L2d 4 = 1

d 5 = 1AE/L

AE/L

4

m

i

j

m

i

j

xy

x

y

Displacement and Force Transformation Matrices

12

3

45

6

yx

4

5

6

1

2

3

5

xq4 = q4 cos x - q5 cos yq5 = q4 cos y + q5 cos xq6 = q6

y

i

jy

x

y

x

45

6

1

2

3

y

x

m

i

j

x

y

12

3

45

6

Force Transformation

Lxx ij

x

=

Lyy ij

y

=

=

6'

5'

4'

6

5

4

10000

qqq

qqq

xy

yx

=

6'

5'

4'

3'

2'

1'

6

5

4

3

2

1

1000000000000000010000000000

qqqqqq

qqqqqq

xy

yx

xy

yx

[ ] [ ] [ ]'qTq T=

6

[q] = [T]T[q]

= [T]T ( [k][d] + [qF] )

= [T]T [k][d] + [T]T [qF]

[q] = [T]T [k][T][d] + [T]T [qF] = [k][d] + [qF]

Therefore, [k] = [T]T [k][T]

[qF] = [T]T [qF]

[q] = [T]T[q]

[d] = [T][d]

[k] = [T]T [k][T]

7

[q] = [T]T[q]= [T]T ( [k][d] + [qF] ) = [T]T[k][d] + [T]T[qF] = [T]T [k][T][d] + [T]T [qF]

Frame Member Global Stiffness Matrix

[k] [qF][ k ] = [ T ]T[ k ][T] =

Ui

Vi

Mi

Uj

Vj

Mj

Vj Mj

- iy6EIL2

ix6EIL2

2EIL

jy6EIL2

- jx6EIL2

4EIL

Ui Vi Mi

- iy6EIL2

ix6EIL2

4EIL

jy6EIL2

jx6EIL2

-

2EIL

Uj

AEL

- ixiy)(12EIL3

AEL

iy2 + 12EIL3

ix2 )(

ix6EIL2

ix6EIL2

AEL

iyjx - 12EI

L3ixjy)-(

AEL

ixjx + 12EIL3

iyjy)-(

jy6EIL2

AEL

jx2 + 12EIL3

jy2 )(

jy6EIL2

jx6EIL2

-

AEL

- jxjy)(12EIL3

- jx6EIL2

AEL

ixjy -12EIL3

iyjx)-(

AEL

- ixiy)(12EIL3

- iy6EIL2

- iy6EIL2

AEL

ixjx + 12EIL3

iy jy)-(

)(AEL

ix2 + 12EIL3

iy2

AEL

iyjy + 12EIL3

ix jx )-(

AEL

iyjy +12EIL3

ixjx)-(

12EIL3

jx2 )AEL

- ( ixjy- iyjx )12EIL3

AEL

iyjx - 12EIL3

ixjy)-(

AEL

- jxjy)(12EIL3

AEL jy

2 + (

8

5 kN

6 m

6 m

AB

C

Example 1

For the frame shown, use the stiffness method to:(a) Determine the deflection and rotation at B.(b) Determine all the reactions at supports.(c) Draw the quantitative shear and bending moment diagrams.E = 200 GPa, I = 60(106) mm4, A = 600 mm2

9

5 kN

6 m

6 m

AB

C kN/m666.667(6m)

)m10)(60mkN1012(20012

3

462

6

3 =

=

LEI

kN/m200006m

)mkN10)(200m10(600 2

626

=

=

LAE

kN2000(6m)

)m10)(60mkN106(2006

2

462

6

2 =

=

LEI

mkN80006m

)m10)(60mkN104(2004

462

6

=

=

LEI

mkN40006m

)m10)(60mkN102(2002

462

6

=

=

LEI

Global :

AB

C

1

2

78 9

4

6 5

12 3

10

Global :

AB

C

1

2

78 9

4

6 5

12 3

A B14

5 6

1

2 3

Local :

5

2

1 3

4

6

2

Using Transformation Matrix:

Member Stiffness Matrix

[ ]

=

LEILEILEILEILEILEILEILEI

LAELAELEILEILEILEILEILEILEILEI

LAEAE/L

/4/60/2/60/6/120/6/120

00/00//2/60/4/60/6/120/6/120

00/00

k'

22

2323

22

2323

Mi

VjMj

Vi

Nj

Ni

i j ji ji

11

A B14

5 6

1

2 3

Local :

[q] = [q]

-> [k]1 = [k]1

Stiffness Matrix: Member 1

Global:

AB

C

1

2

78 9

4

6 5

12 3

4 6 5 1 2 34

6

5

1

2

3

20000

0

0

-20000

0

0

0

666.667

2000

0

-666.667

2000

0

2000

8000

0

-2000

4000

-20000

0

0

20000

0

0

0

-666.667

-2000

0

666.667

-2000

0

2000

4000

0

-2000

8000

[k]1 =

12

Local:

5

2

1 3

4

6

2

[q]2 = [ T ]T[ q]2

q1

q3

q2

q4q5q6

q1

q3

q2

q7q8q9

[T]T

Stiffness Matrix: Member 2

=

123789

40000

0-1

5000100

6000001

10

0-1

000

2100000

3001000

90o

jx = cos (-90o) = 0jy = sin (-90o) = -1

ix = cos (-90o) = 0iy = sin (-90o) = -1

Global:

AB

C

1

2

78 9

4

6 5

12 3

13

[k]2 = [ T ]T[ k ]2[ T ]

1 2 3 4 5 61

2

3

4

5

6

20000

0

0

-20000

0

0

0

666.667

2000

0

-666.667

2000

0

2000

8000

0

-2000

4000

-20000

0

0

20000

0

0

0

-666.667

-2000

0

666.667

-2000

0

2000

4000

0

-2000

8000

[k]2 =

1 2 3 7 8 9

666.667 20001

2

3

7

8

9

2000

0

0

-666.667

-666.667

0

0

2000

2000

20000 0

0

0

0

-20000

-20000

0

0

8000 -2000

-2000

0

0

4000

4000

666.667 0

0

-2000

-2000

20000 0

0 8000

[k]2 =

14

[k]14 6 5 1 2 3

4

6

5

1

2

3

20000

0

0

-20000

0

0

0666.667

20000

-666.667

2000

02000

80000

-2000

4000

-200000

020000

0

0

0-666.667

-20000

666.667

-2000

02000

40000

-2000

8000

1 2 3 7 8 9666.667 20001

2

3

7

8

9

2000

00

666.667

-666.667

0

0

2000

2000

20000 0

0

0

0

-20000

-20000

0

0

8000 -2000-2000

0

0

4000

4000

666.667 0

0

-2000

2000

20000 0

0 8000

[k]2

Global Stiffness Matrix:

20000

0

-20000

0

0

0

8000

0

-2000

4000

0

-2000

0

4000

-20000

0

0

20666.667

-2000

2000

-2000

16000

20666.667

0

2000

[K]

4

5

1

2

3

4 5 1 2 3

Global:

AB

C

1

2

78 9

4

6 5

12 3

15

AB

C

Q4 = 0Q5 = 0

Q1 = 5Q2 = 0

Q3 = 0

D4D5D1D2D3

+

0 0

0 0

0

D4D5D1D2D3

=

0.01316 m

0.01316 m9.199(10-4) rad

-9.355(10-5) m

-1.887(10-3) rad

5 kN

6 m

6 m

AB

C

1

2

Global:

1

2

78 9

4

6 5

12 35 kN

=

4

5

1

2

3

4 5 1 2 3-20000 0 0

20666.667 00

2000

2000

20666.667 -2000

-2000 16000

0-2000

4000

08000 0 -2000 4000

-200000

20000

0

0

[Q] = [K][D] + [QF]

16

D4 = 0.01316

D5 = 9.199(10-4)

D6 = 0

D1 = 0.01316

D2 = -9.355(10-5)

D3 = -1.887(10-3)

15 kN

6 m

6 m

AB

C

2

q4

q5

q6

q1

q2

q3

0

0

-1.87

0

1.87

-11.22

Member 1

A B11

2 3

4

6 5

A B1

1.87 kN 11.22 kNm1.87 kN

4 6 5 1 2 3

4

6

5

1

2

3

20000

0

0

-20000

0

0

0

666.667

2000

0

-666.667

2000

0

2000

8000

0

-2000

4000

-20000

0

0

20000

0

0

0

-666.667

-2000