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Lecture No. : 10
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2
F = K Dl
l
l
TK =g
Kl
T
T
m m
Drive the member local stiffness matrix
Obtain the member global stiffness matrix
Drive the member transformation matrix
T
Solution Steps of assembly method :Remember
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3
Make assembly
F = K D
Kuu
Kru
Kur
Krr
Fu
Fr
Du
Dr=
Make partition
Kgm
Kgm
Kgm
Kgm
Remember
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4
Kuu
Kru
Kur
Krr
Fu
Fr
Du
Dr=
Extract the stiffness equation
KuuFu Du= Kur Dr+
KuuDu =-1{ }Fu Kur Dr-
Obtain the deformation
Remember
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5
Find internal forces in members
Calculate the reactions
KruFr Du= KrrDr+
g
F = Kl
l
m m mT DT
Remember
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d1
d2d3
7
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F = K Dll
l
Drive the member local stiffness matrix
d4
d5
d6d1
d2
d3
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Local
F1
F2
F3
F4F5
F6
d4
d5d6
d1
d2d3
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k11
=
k21
k31
k12
k22
k32
k13
k23
k33
k41
k51
k61
k42
k52
k62
k43
k53
k63
k14
k24
k34
k15
k25
k35
k16
k26
k36
k44
k54
k64
k45
k55
k65
k46
k56
k66
F1
F2
F3
F4
F5
F6
D1
D2
D3
D4
D5
D6
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D=1AE
L
AE
L
11
First column inLocal Stiffness matrix
d1 =1
F1 = AELF2 = 0
F4 = - AEL
F6 = 0F3 = 0F5 = 0
d4
d5d6
d1
d2d3
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12
First column in Local Stiffness matrix
k11
=
k21
k31
k41
k51
k61
0
0
0
0
AE
L
AE
L
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13
Second column inLocal Stiffness matrix
d2 =1
DD6 EI
L2
D6 EI
L2
D12 EI
L3D12 EI
L3
d4
d5d6
d1
d2d3
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14
Second column in Local Stiffness matrix
F1 = 0F2 =
F4 = 0
F6 =F3 =F5 =
k12
=
k22
k32
k42
k52
k62
0
0
12 EI
L3
6 EI
L2
6 EI
L2
12 EI
L3
12 EI
L312 EI
L3
6 EIL2
6 EIL2
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15
Third column inLocal Stiffness matrix
d3 =1
q
q4 EI
L
q2 EIL
q6 EI
L2q6 EI
L2
d4
d5d6
d1
d2d3
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Third column in Local Stiffness matrix
F1 = 0F2 =
F4 = 0
F6 =F3 =F5 =
k13
=
k23
k33
k43
k53
k63
0
0
6 EI
L2
6 EI
L2
4 EIL
2 EI
L
6 EI
L2
4 EI
L
6 EI
L22 EI
L
_
_
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Fourth column inLocal Stiffness matrix
d4 =1 D=1AE
L
AE
L
F1 = - AEL
F2 = 0F4 = AE
L
F6
= 0F3
= 0F5 = 0
d4
d5d6
d1
d2d3
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18
Fourth column in Local Stiffness matrix
k14
=
k24
k34
k44
k54
k64
0
0
0
0
AE
L
AE
L
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19
Fifth column inLocal Stiffness matrix
d5 =1
D
D6 EI
L2
D6 EI
L2
D12 EI
L3
D12 EI
L3
d4
d5d6
d1
d2d3
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20
fifthcolumn in Local Stiffness matrix
F1 = 0F2 =
F4 = 0
F6 =F3 =F5 =
k15
=
k25
k35
k45
k55
k65
0
0
-12 EI
L3
-6 EI
L2
-6 EI
L2
12 EI
L3
12 EI
L312 EI
L3
6 EIL2
6 EI
L2
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sixth column inLocal Stiffness matrix
d6 =1
q
q4 EI
Lq2 EI
Lq
6 EI
L2
q6 EI
L2
d4
d5d6
d1
d2d3
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22
6 column in Local Stiffness matrix
F1 = 0F2 =
F4 = 0
F6 =F3 =F5 =
k16
=
k26
k36
k46
k56
k66
0
0
6 EI
L2
6 EI
L2
2 EIL
4 EI
L
6 EI
L2
4 EI
L
6 EI
L22 EI
L
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23
=Kl
EA
L 0 0
-EA
L 0 0
012EI
L3
6EI
L20
-
12EI
L3
6EI
L2
06EI
L2
4EI
L0
-6EI
L2
2EI
L
-EA
L0 0
EA
L0 0
0
-
12EI
L3
-6EI
L20
12EI
L3
-6EI
L2
0 6EIL2
2EIL
0 -6EIL2
4EIL
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24F1
F2
F4
F5
g
g
g
g
Drive the member transformation matrix
q
F3
F6
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25
F1
F2g
g
qq
F1 cosq
F2 cosq F1 sinq
F2 sinq
F1
= F1
cos q F2
sin q
F2 = F1 sin q + F2 cos q
g
g
F3
F3 = F3
g
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26
F4
F5g
g
qq
F4 cosq
F5 cosq F4 sinq
F5 sinq
F4
= F4
cos q F4
sin q
F5 = F5 sin q + F5 cos q
g
g
F6
F6 = F6
g
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27
F1g
F2g
F3g
F4g
F5g
F6g
=
cos q - sin q 0 0
sin q cos q
0 0
0 0 cos q - sin q
0 0 cos q
0 0
0
1
0
sin q 010 0
0 0 0
000
0
0
000
x
F1
F2
F3
F4
F5
F6
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28
T =
cos q - sin q 0 0
sin q cos q
0 0
0 0 cos q - sin q
0 0 cos q
0 0
0
1
0
sin q 010 0
0 0 0
000
0
0
000
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29
TK =g K
l
TT
m m
Obtain the member global stiffness matrix
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Example 1:
Draw N.F, S.F & B.M.Ds for the shown frame
where E = 106 kN/m2
8
A B
3
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
50 kN
12 kN/m
2
30
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2EI/L4EI/L6EI/L212EI/L3EA/LSCLmemb
5000100001875469750000108AB
2000400012004808000010905CB
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First element : ( AB)
=Kl
EA
L 0 0
-EA
L 0 0
012EI
L3
6EI
L20
-12EI
L3
6EI
L2
06EI
L2
4EI
L0
-6EI
L2
2EI
L
-EA
L0 0
EA
L0 0
0
-12EI
L3
-6EI
L20
12EI
L3
-6EI
L2
0 6EIL2
2EIL
0 -6EIL2
4EIL
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33
=
75000
0
0
-75000
0
0
469
0
1875
-469
0
1875
1875
0
10000
-1875
0
5000
0
-7500
0
0
75000
0
-469
0
-1875
469
0
-1875
1875
0
5000
-1875
0
10000
lK1
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34
T =
cos q - sin q 0 0
sin q cos q
0 0
0 0 cos q - sin q
0 0 cos q
0 0
0
1
0
sin q 010 0
0 0 0
000
0
0
000
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35
=T
1
0
00
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
01
0
0
1
0
0
0
0
1
Obtain the member global stiffness matrix
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TK =g K
l T
T
m m
Obtain the member global stiffness matrix
Kg1 =
Kg1
=
m
KL1
75000
0
0
-75000
0
0
469
0
1875
-469
0
1875
1875
0
10000
-1875
0
5000
0
-7500
0
0
75000
0
-469
0
-1875
469
0
-1875
1875
0
5000
-1875
0
10000
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37
BA
B
A
Kg1
=
75000
0
0
-75000
0
0
469
0
1875
-469
0
1875
1875
0
10000
-1875
0
5000
0
-7500
0
0
75000
0
-469
0
-1875
469
0
-1875
1875
0
5000
-1875
0
10000
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second element : ( CB )
=K2l
EA
L 0 0
-EA
L 0 0
012EI
L3
6EI
L20
-12EI
L3
6EI
L2
06EI
L2
4EI
L0
-6EI
L2
2EI
L
-EA
L0 0
EA
L0 0
0
-12EI
L3
-6EI
L20
12EI
L3
-6EI
L2
06EI
L2
2EI
L0
-6EI
L2
4EI
L
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39
=K
80000
0
0
-80000
0
0
480
0
1200
-480
0
1200
1200
0
4000
-1200
0
2000
0
-80000
0
0
80000
0
-480
0
-1200
480
0
-1200
1200
0
2000
-1200
0
4000
2
l
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40
=K2
l
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41
T =
cos q - sin q 0 0
sin q cos q
0 0
0 0 cos q - sin q
0 0 cos q
0 0
0
1
0
sin q 010 0
0 0 0
000
0
0
000
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42
=T
0
1
0
0
0
0
0
-1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
00
1
0
0
-1
0
0
0
1
l T
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43
010000
0
-1
0
0
0
0
0
0
1
0
0
0
000
0
0
0
0
0
0010
0
-1
0
0
0
1
8000000
-8000000
480
0
1200
-480
0
1200
1200
0
4000
-1200
0
2000
0
-80000
0
0
80000
0
-480
0
-1200
480
0
-1200
1200
0
2000
-1200
0
4000
0-10000
0
1
0
0
0
0
0
0
1
0
0
0
000
0
0
0
0
0
00-10
0
1
0
0
0
1
TK =g K
l T
T
m m
80000
0
0
-80000
0
0
-480
01200
4800
1200
-1200
04000
12000
2000
0
-800000
080000
0
480
0-1200
-4800
-1200
-1200
02000
12000
4000
0-10000
0
1
0
0
0
0
0
0
1
0
0
0
000
0
0
0
0
0
00-10
0
1
0
0
0
1
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44
Ks =
75480
0
1200-480
1200
80469
0
-1875
0
0
-1875
1200
14000
2000
-1200
0
-480
-1200
-1200
480
0
1200
2000
4000
-1200
d1 d2 d3 d7 d9
d1
d2
d3d7
d9
Obtain the member global stiffness matrix
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45
TK =g K
l T
T
m m
Obtain the member global stiffness matrix
Kg2
480
0
-1200
-480
0
-1200
80000
0
0
-80000
0
0
0
-1200
4000
0
1200
2000
0
-480
1200
0
480
1200
-80000
0
0
80000
0
0
0
-1200
2000
0
1200
4000
=
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46
BC
B
C
K
g
2
480
0
-1200
-480
0
-1200
80000
0
0
-80000
0
0
0
-1200
4000
0
1200
2000
0
-480
1200
0
480
1200
-80000
0
0
80000
0
0
0
-1200
2000
0
1200
4000
=
Assembly :
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47
Assembly :
0 80469
0
K =s
AB
A
B
0 75000
014000-1875 0
-750001200
-75000 0
-18751200
C
0 0
50001875-469 -1875
00
C
5000 0
018750 -1875
-64901875 10000
1875469
75480
-1200 0-480 0
2000 0
001200 0
-800000
0
-1200
0
-480
0 0
20000-80000 0
12000
0
0
0 0
00
480 0 -1200
-1200
0
0 4000
0800000
0
0
0
0
0
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48
Modeling
A
B
C
d1
d2d3
d9d7
48
d4
d5d6
d8
C
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4949
0 80469
0
K =s
B
A
B
0 75000
014000-1875 0
-750001200
-75000 0
-18751200
C
0 0
50001875-469 -1875
00
C
5000 0
018750 -1875
-64901875 10000
1875469
75480
-1200 0-480 0
2000 0
001200 0
-800000
0
-1200
0
-480
0 0
20000-80000 0
12000
0
0
0 0
00
480 0 -1200
-1200
0
0 4000
0800000
0
0
0
0
0
A
d1 d2 d3 d4 d5 d6 d7 d8 d9
d1d2
d3
d4
d5
d6
d7
d8
d9
Force vector
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50
Force vector
8
A B
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
50 kN
12 kN/m
3
2
Force vector
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51
Force vector
8
A
12 kN/m
B
64 kNm12x82
1264 kNm
48 kN 48 kN
Force vector
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52
Force vector
B
C
50 kN
3
2
24 kNm
50x2x32
52
36 kNm
32.4 kN
50x3x22
52
17.6 kN
Force vector Fixed End
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53
Force vector
A B
64 kNm 64 kNm
48 kN 48 kN B
C
24 kNm
36 kNm
32.4 kN
17.6 kN
Fixed EndReaction
(FER)
53
Fi d E d
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54
Force vector
A B
64 kNm 64 kNm
48 kN 48 kN B
C
24 kNm
36 kNm
32.4 kN
17.6 kN
Fixed EndAction(FEA)
A
B
C
d1
d2d3
d9 d7
F =
F1 -17.6
=
F2 - 48
F3 40
F7 -32.4
F9 36 54
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55
F = K D
d1
d2
-17.6
=
- 48
40
-32.4
36
75,480
1200
0
- 480
1200
0
80,469
-18750
0
1200
-1875
14,000-1,200
2,000
1200
0
2,000-1,200
4,000
- 480
0
-1,200480
-1,200
d3
d7
d9
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56
D = K-1 F
d1
d2
-17.6
=
- 48
40
-32.4
36
75,480
1200
0
- 480
1200
0
80,469
-1875
0
0
1200
-1875
14,000
-1,200
2,000
1200
0
2,000
-1,200
4,000
- 480
0
-1,200
480
-1,200
d3
d7
d9
-1
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D = K-1 F
d1
d2
- 0.0007
=
- 0.0008
- 0.0088
- 0.2244
- 0.0538
d3
d7
d9
57
Internal forces
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Internal forcesd1d2
- 0.0007
=
- 0.0008
- 0.0088
- 0.2244
- 0.0538
d3
d7
d9
A
B
C
d1
d2d3
d9 d7
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
EIAB
= 2x104
EIBC
= 5x103
58
Force vector Fixed End
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Force vector
A B
64 kNm 64 kNm
48 kN 48 kN B
C
24 kNm
36 kNm
32.4 kN
17.6 kN
Fixed EndReaction
(FER)
59
64 kNm 64 kNm d 0 0007
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A B48 kN 48 kN
d1d2
- 0.0007
=
- 0.0008
- 0.0088
- 0.2244
- 0.0538
d3
d7
d9
d1
d2d3
d9 d7
60
Find internal forces in members
FERm
gm
Tmm FDTKF
=
Member (AB)
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61
Fm
l =
75000 0 -750000
0 469 1876 0
0
1875
0 1875 10000 0
-75000 0 0 75000 0 0
0 -469 -1875 0 469 -1875
0 5000
5000
0 -1875 10000
0
-469
-1876
1875
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
- 0.0007- 0.0008
- 0.0088
048
+
0
0
0
0
48
64
-64
052 5
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62
m
l
F = +
Fml =
048
48
64
0
-64
52.5-16. 125
-42.5-52.516. 25-86.5
52.531.875
21.5-52.5
64.125
150.5
d1 0 000724 kNm dd2d3
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d1d2
- 0.0007
=
- 0.0008
- 0.0088
- 0.2244
- 0.0538
d3
d7
d9
B
C
24 kNm
36 kNm
17.6 kN
d1d3
d9 d7
63
Member (CB)
g
F = K
l
l
m m mT D
T
Member (CB)
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64
( )
64
Fm
l =
80000 0 -800000
0 480 1200 0
0
1200
0 1200 4000 0
-80000 0 0 80000 0 0
0 -480 -1200 0 480 -1200
0 2000
2000
0 -1200 4000
0
-480
-1200
1200
0 1 0 0 0 0
-1 0 0 0 0 0
0 0 1 0 0 0
0 0 0 0 1 0
0 0 0 -1 0 0
0 0 0 0 0 1
- 0.0007- 0.0008
- 0.0088
17.60
+
0
0
-36
24
- 0.22440
- 0.0538
64
-
8/2/2019 Stiffness 10
65/70
65
m
l
F =17.6
0
+ -36
24
0
0
Fl
m=
64
32.25
36-64-32.25
126
6432.25
046.4-32.25
150
21 75 kN 150 kN
-
8/2/2019 Stiffness 10
66/70
A
12 kN/m
B
21.75 kNm 150 kNm
32 kN 64kN
B
C
50 kN
150 kNm
0 kN
50 kN
3
28
66
21 75 kN 150 kN
-
8/2/2019 Stiffness 10
67/70
A
12 kN/m
B
21.75 kNm 150 kNm
32 kN 64 kN B
C
50 kN
150 kNm
50 kN
64 kN50 kN
67
21 75 kN 150 kN
-
8/2/2019 Stiffness 10
68/70
A
12 kN/m
B
21.75 kNm 150 kNm
32 kN 64 kN B
C
50 kN
150 kNm
50 kN
64 kN50 kN
N.F.D
50
64
-
-
68
21 75 kNm 150 kNm
-
8/2/2019 Stiffness 10
69/70
A
12 kN/m
B
21.75 kNm 150 kNm
32 kN 64 kN B
C
50 kN
150 kNm
50 kN
64 kN50 kN
S.F.D
64
50
50
32
-
+
+
69
12 kN/m21.75 kNm 150 kNm
-
8/2/2019 Stiffness 10
70/70
A
12 kN/m
B32 kN 64 kN
B
C
50 kN
150 kNm
50 kN
64 kN
50 kN
B.M.D
-
150
21.7515085.875
96
10.125
--
+
