Stiffness 7
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Lecture No. : 7
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3
d1d2
C
B
D
A
d3d4
d5
Direct method
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d1
d2
d3
d4
F1
F2
F3
F4
k11F1
F2=
k21
F3 k31
k12
k22
k32
k13
k23
k33
F4 k41 k42 k43
k14
k24
k34
k44
d1
d2
d3
d4
Special CaseLocal
F = K Dl
l
l
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d1d2
d3d4
F1
F2
F3
F4
k11F1
F2=
k21
F3 k31
k12
k22
k32
k13
k23
k33
F4 k41 k42 k43
k14
k24
k34
k44
d1
d2
d3
d4
General CaseGlobal
F = K Dg g g
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a
d c
b1
2
3
4
5
Assembly steps
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a
d c
b1
2
3
4
5
a b1
Kaa1K =
Kba
Kab
Kbb
1 1
1 1
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a
d c
b1
2
3
4
5
Kbb2K =
Kcb
Kbc
Kcc
2 2
2 2
c
b
2
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a
d c
b1
2
3
4
5
Kcc3K =
Kdc
Kcd
Kdd
3 3
3 3
d c3
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a
d c
b1
2
3
4
5
Kaa4K =
Kda
Kad
Kdd
4 4
4 4
a
d
4
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a
d c
b1
2
3
4
5
Kaa5K =
Kca
Kac
Kcc
5 5
5 5
a
c
5
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Kaa1
K=
Kba
Kab
Kbb
1 1
1 1
a b
a
b
Kbb2K = Kcb
Kbc
Kcc
2 2
2 2
b c
b
c
Kcc3K = Kdc
Kcd
Kdd
3 3
3 3
c d
c
d
Kaa
4
K = Kda
Kad
Kdd
4 4
4 4
a da
d
Kaa5K = Kca
Kac
Kcc
5 5
5 5
a ca
c
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a
d c
b1
2
3
4
5
Kaa
K=Kba
Kab
Kbb
Kac
Kbc
Kad
Kbd
Kca Kcb Kcc KcdKda Kdb Kdc Kdd
a b
a
b
c d
c
d
Structure Stiffness Matrix
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a
d c
b1
2
3
4
5
Kaa
K= Kba
Kab
Kbb
Kac
Kbc
Kad
KbdKca Kcb Kcc Kcd
Kda Kdb Kdc Kdd
a b
a
b
c d
c
d
Kaa
Kbb
Kcc
Kdd
Kaa
Kbb
Kcc
Kdd
Kaa
Kbb
Kcc
Kdd
Kaa
Kcc
=1 4 5
+ +
=1
2+
=2
3 5+ +
=3
4
+
Kaa1K =
Kba
Kab
Kbb
1 1
1 1
Kbb2K =
Kcb
Kbc
Kcc
2 2
2 2
Kcc3
K =Kdc
Kcd
Kdd
3 3
3 3
Kaa4K =
Kda
Kad
Kdd
4 4
4 4
Kaa5K =
Kca
Kac
Kcc
5 5
5 5
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K=K
ba
Kab Kac
Kbc
Kad
0
Kca Kcb Kcd
Kda 0 Kdc
a
b
a
b
cd
c
dKdd Kdd
34
+
Kcc Kcc Kcc2
3 5+ +
Kbb Kbb1
2
+
Kaa Kaa Kaa1 4 5
+ +
3
3
2
2
1
1
5
5
4
4
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Kuu
K=
Kru
Kur
Krr
u r
u
r
d5d6
C
B
D
A
d7d8
d1d2
d3d4
Kuu
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d1
d2
d3
d4
F1
F2
F3
F4
F = K Dl
l
l
d1d2
d3d4
F1
F2
F3
F4
F = K Dg g g
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F = K Dl
l
l
d1d2
d3d4
F1
F2
F3
F4
F = K Dg g g
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d1d2
d3d4
F1
F2
F3
F4
lD = T D
g
F = T Fl
g
TransformationMatrix
[ T ]
l l l
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F = K Dl
l
l
F = K Dg g g l
D = T Dg
F = T Fl
g
lT D
T F
l = K
g
lT DT = K
gK Dl
l
TT = KgKl
T-1
TT = KgK
l T
-1
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T-1
TT = KgK
l T
-1
T = KgKl T -1
TK =
g Kl T
-1T =
-1T
T
TK =g K
l T
T
TK =g Kl T
T
m m
T
TTransformation
Matrix
Transposeof Matrix
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After assembly
F = K D
Kuu
Kru
Kur
Krr
Fu
Fr
Du
Dr=
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u : Unrestraint
r : Restraint
d5d6
C
B
D
A
d7d8
d1d2 d3
d4
Kuu
Kru
Kur
Krr
Fu
Fr
Du
Dr=
Fu
Fr
Du
Dr
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u : Unrestraint
r : Restraint
d5d6
C
B
D
A
d7d8
d1d2 d3
d4
Kuu
Kru
Kur
Krr
Fu
Fr
Du
Dr=
= 0Dr
If there is no settlement
Kuu
Kru
Kur
Krr
Fu
Fr
Du
0
=
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Kuu
Kru
Kur
Krr
Fu
Fr
Du
0=
KuuFu Du=
KruFr Du=
Kuu FuDu =-1
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u : Unrestraint
r : Restraint
d5d6
C
B
D
A
d7d8
d1d2 d3
d4
Kuu
Kru
Kur
Krr
Fu
Fr
Du
Dr
=
If there is settlement
= 0Dr
Kuu
Kru
Kur
Krr
Fu
Fr
Du=
Dr
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Kuu
Kru
Kur
Krr
Fu
Fr
Du
Dr
=
KuuFu Du= Kur Dr+
KuuFu Du=Kur Dr-
KuuD
u =
-1
{ }Fu Kur Dr-KruFr Du=
Krr Dr+
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To find internal forces in members
F = K Dl
l
l
m m m
lD = T Dmm
g
lD = T Dmm
gT
gF = K
l
lm m mT D
T
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Summary
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d1d2
d3d4
F1
F2
F3
F4
F = K Dl
l
l
TK =g
K
l
T
T
m m
Drive the member local stiffness matrix
Obtain the member global stiffness matrix
Drive the member transformation matrix
T
Solution Steps :
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Make assembly F = K D
Kuu
Kru
Kur
Krr
Fu
Fr
Du
Dr=
Make partition
Kgm
Kgm
Kgm
Kgm
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KuuKru
KurKrr
FuFr
DuDr
=
Extract the stiffness equation
KuuFu Du= Kur Dr+
KuuDu =-1
{ }Fu Kur Dr-Obtain the deformation
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Find internal forces in members
Calculate the reactions
KruFr Du= Krr Dr+
gF = Kll
m m mT D
T
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Questions
36
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