Stiffness 9
-
Upload
emad-elhussieny -
Category
Documents
-
view
59 -
download
3
Transcript of Stiffness 9
![Page 1: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/1.jpg)
Lecture No. : 9 ال تاسعةالمحاضرة
![Page 2: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/2.jpg)
F = K Dl l l
T K =g K l T
T
m m
Drive the member local stiffness matrix
Obtain the member global stiffness matrix
Drive the member transformation matrixT
Solution Steps of assembly method :Remember
![Page 3: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/3.jpg)
Make assembly F = K D
Kuu
Kru
Kur
Krr
Fu
Fr
Du
Dr=
Make partition
Kgm K
gm K
gm K
gm
Remember
![Page 4: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/4.jpg)
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
![Page 5: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/5.jpg)
Find internal forces in members
Calculate the reactions
KruFr Du= Krr Dr+
gF = K l lm m mT D
T
Remember
![Page 6: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/6.jpg)
![Page 7: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/7.jpg)
d1
d2
d3
![Page 8: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/8.jpg)
d1
d2
Normal Force doesn’t taken
![Page 9: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/9.jpg)
Drive the member local stiffness matrix
k11F1
F2
=k21
F3 k31
k12
k22
k32
k13
k23
k33
F4k41 k42 k43
k14
k24
k34
k44
d1
d2
d3
d4
d1
d2 d4
d3
![Page 10: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/10.jpg)
First column inLocal Stiffness matrix
d1 =1
6 EI
L2
6 EIL2
12 EIL3
12 EIL3
d3
d4
d2
d2
![Page 11: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/11.jpg)
12 EIL3
F2 =
F3 = -
F4 =
F1 =12 EI
L3
6 EIL3
6 EIL3
6 EI
L2
6 EIL2
12 EIL3
12 EIL3
![Page 12: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/12.jpg)
=k31
12 EIL3
6 EIL2
-12EIL3
6 EIL2
k11
k21
k41
![Page 13: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/13.jpg)
Second column inLocal Stiffness matrix
d2 =1
4 EIL
2 EIL
6 EIL2
6 EIL2
d3
d4
d1
d2
![Page 14: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/14.jpg)
F3 =F1 =
F4 =F2 =
6 EIL2
6 EIL2
-
2 EIL
4 EIL
4 EIL
2 EIL
6 EIL2
6 EIL2
![Page 15: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/15.jpg)
Second column inLocal Stiffness matrix
=
k12
k22
k32
k42
6 EI
L2
4 EIL
6 EIL2-
2 EIL
![Page 16: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/16.jpg)
Third column inLocal Stiffness matrix
d3 =1
6 EIL2
6 EIL2
12 EIL3
12 EIL3
d3
d4
d1
d2
![Page 17: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/17.jpg)
12 EIL3
6 EIL2
6 EIL2
12 EIL3F1 = F3 =
F4 =F2 = - -
-
6 EI
L2
6 EIL2
12 EIL3
12 EIL3
![Page 18: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/18.jpg)
Third column in Local Stiffness matrix
=
k13
k23
k33
k33
12 EIL3
6 EIL2
12 EIL3
6 EIL2
-
-
-
![Page 19: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/19.jpg)
Fourth column inLocal Stiffness matrix
d4 =1
4 EIL
2 EIL
6 EIL2
6 EIL2
d3
d4
d1
d2
![Page 20: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/20.jpg)
F1 = F3=
F2 = F4=
6 EIL2
2 EIL
4 EIL
6 EIL2
-
4 EIL
2 EIL
6 EIL2
6 EIL2
![Page 21: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/21.jpg)
Fourth column inLocal Stiffness matrix
=
k14
k24
k34
k44
6 EI
L2
2 EIL
6 EIL2-
4 EIL
![Page 22: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/22.jpg)
K l
12 EIL3
6 EIL2
6 EIL2
4 EIL
-12 EIL3
=-12 EI
L3
-6 EIL2
-6 EIL2
12 EIL3
-6 EIL2
6 EIL2
2 EIL
6 EIL2
2 EIL
-6 EIL2
4 EIL
![Page 23: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/23.jpg)
K l
12 EIL3
6 EI
L2
6 EI
L2
4 EIL
-12 EIL3
=-12 EI
L3
-6 EIL2
-6 EIL2
12 EIL3
-6 EIL2
6 EI
L2
2 EIL
6 EI
L2
2 EIL
-6 EIL2
4 EIL
![Page 24: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/24.jpg)
K l = EIL3
K l
12 EIL3
6 EIL2
6 EIL2
4 EIL
-12 EIL3
= -12 EIL3
-6 EIL2
-6 EIL2
12 EIL3
-6 EIL2
6 EIL2
2 EIL
6 EIL2
2 EIL
-6 EIL2
4 EIL
12
6 L
-12
6 L
6 L
-6 L
4 L2
2 L2
-12
6 L
12
6 L
-6 L
2 L2
4 L2
6 L
![Page 25: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/25.jpg)
d1
d2
Normal Force doesn’t taken
![Page 26: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/26.jpg)
d1
If Shear is omitted
![Page 27: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/27.jpg)
Drive the member local stiffness matrix
k11F1
F2
=k21
k12
k22
d1
d2
d1 d2
![Page 28: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/28.jpg)
K l
12 EIL3
6 EIL2
6 EIL2
4 EIL
-12 EIL3
=-12 EI
L3
-6 EIL2
-6 EIL2
12 EIL3
-6 EIL2
6 EIL2
2 EIL
6 EIL2
2 EIL
-6 EIL2
4 EIL
![Page 29: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/29.jpg)
K l4 EI
L=
2 EIL
2 EIL
4 EIL
K l = 2EIL 1
2
2
1
![Page 30: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/30.jpg)
Construct the stiffness matrix for the shown beam where EI is
constant for all members
Example 1:
8 10
A CB
![Page 31: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/31.jpg)
First element : (A-B )Start Joint : A End Joint : B
K l = EIL3
12
6 L
-12
6 L
6 L
-6 L
4 L2
2 L2
-12
6 L
12
6 L
-6 L
2 L2
4 L2
6 L
![Page 32: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/32.jpg)
K l K g=
K l
12 EIL3
6 EI
L2
6 EI
L2
4 EIL
-12 EIL3
=-12 EI
L3
-6 EIL2
-6 EIL2
12 EIL3
-6 EIL2
6 EI
L2
2 EIL
6 EI
L2
2 EIL
-6 EIL2
4 EIL
![Page 33: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/33.jpg)
K l = EIL3
12
6 L
-12
6 L
6 L
-6 L
4 L2
2 L2
-12
6 L
12
6 L
-6 L
2 L2
4 L2
6 L
![Page 34: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/34.jpg)
K l = EI
.023
.0937
.-023
.0937
.0937
.5
.0937
.25
.-023
.0937
.023
.-0937
.0937
.25
.-0937
.5
A
B
�ِA
B
![Page 35: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/35.jpg)
Second element : ( B-c)Start Joint : B End Joint : c
K l
12 EIL3
6 EI
L2
6 EI
L2
4 EIL
-12 EIL3
=-12 EI
L3
-6 EIL2
-6 EIL2
12 EIL3
-6 EIL2
6 EI
L2
2 EIL
6 EI
L2
2 EIL
-6 EIL2
4 EIL
![Page 36: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/36.jpg)
K l = EI
.012
.06
.-012
.06
.06
.4
.-06
.2
.-012
.-06
.012
.-0937
.06
.20
.-06
.4
B C
C
B
![Page 37: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/37.jpg)
Assembly :
K =g1 EI
.023
.0937
.-023
.0937
.0937
.5
.0937
.25
.-023
.0937
.023
.-0937
.0937
.25
.-0937
.5
A B
B
A
![Page 38: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/38.jpg)
EI .06
.-012
.06
.06
.4
.2
.-012
.-06
.012
.-06
.06
.2
.-06
.012
.-06
K =g2
.4
B C
B
C
![Page 39: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/39.jpg)
Ks = EI
.023
.0937
.-023
.093700
.0937
.5
.0937
.2500
.-023
.0937
.035
.-0337.-012.06
.0937
.25
.-0337
.9
.-06
.2
0
0
.-012
.-06
.012.-06
0
0
.06
.2
.-06
.4
A B C
B
A
C
![Page 40: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/40.jpg)
Partition
KuuK =Kru
Kur
Krr
u ru
r
Kuu = EI .90
![Page 41: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/41.jpg)
Example 2:Construct the stiffness matrix for the shown beam where EI is constant for all members
4254
ECB DA
![Page 42: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/42.jpg)
= E I
12/L^3
6/L^2
-12/L^3
6/L^2
6/L^2
2/L
6/L^2-
4/L
12/L^3-
6/L^2-
12/L^3
6/L^2-
6/L^2
-6/L^2
4/L
2/L
K gK l =
K l
![Page 43: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/43.jpg)
First element : (A-B )Start Joint : A
End Joint : BAngle : 0s = sin = 0c = cos = 1
Is conastant EA
LAB = 400 cm
![Page 44: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/44.jpg)
Kl= EI
.1875
.375
.-1875
.375
.375
1
.-375
.5
.-1875
.-375
.1875
.-375
.375
.5
.-375
1
A B
B
A
![Page 45: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/45.jpg)
Second element : B-C( Start Joint : B
End Joint : cAngle : 0s = sin = 0c = cos = 1
Is conastant EA
LBC = 500 cm
![Page 46: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/46.jpg)
EI .24
.-096
.24
.24
.8
.4
.-096
.-24
.096
.-24
.24
.4
.-24
.096
.-24
.8
B C
B
C
K =g1
![Page 47: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/47.jpg)
End Joint : D
Third element : (C-D )
Start Joint : C
Angle : 0s = sin = 0c = cos = 1
LCD = 200 cm
Is conastant EA
![Page 48: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/48.jpg)
= EI
1.5
1.5
-1.5
1.5
1.5
2
-1.5
.2
-1.5
-1.5
1.5
-1.5
1.5
1
-1.5
.2
C D
D
CK l
![Page 49: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/49.jpg)
Fourth element : (D-E )
Start Joint :D
End Joint : EAngle : 0s = sin = 0c = cos = 1
Is conastant EA
LD-E = 400 cm
![Page 50: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/50.jpg)
Kl= EI
.1875
.375
.-1875
.375
.375
1
.-375
.5
.-1875
.-375
.1875
.-375
.375
.5
.-375
1
D "ُE
D
E
![Page 51: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/51.jpg)
K =s
.1875 .375.375 1
.-1875 .-375
.375 .5
0 0
0 0
0 0
0 0
.-096 .-24.24 .4
0 00 0
0
0
0 00
00
0 0
0 0
0
0
0
0
0
0 00
0.-1875 .375.-375 .5
.2835 .-135.-135 1.8
0000
.-096
.-24
.24.4
1.596 1.261.26 2.8
-1.5 1.51.5 1
0 00 0
00
-1.5 1.51.5 1
1.687 -1.125-1.125 3.-1875.375
.-375.5
0
00
0
0
0
.-1875 .375.-375 .5
.1875.-375 1
.-375
A B C D E
AB
CDE
![Page 52: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/52.jpg)
KUU1.8 EI0.4 EI
0
0.4 EI2.8 EI
EI
0EI
3 EI=
1.80.4
0
0.42.8
1
01
3EI=
![Page 53: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/53.jpg)
Example 3:Calculate the deformation of the shown beam where EI = 105 kN.m2 for all members
4254DCB
EA 30 kNm50 kNm60 kNm
![Page 54: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/54.jpg)
KUU
1.8 EI0.4 EI
0
0.4 EI2.8 EI
EI
0EI
3 EI=
1.80.4
0
0.42.8
1
01
3EI
The stiffness equation
F = K D
Stiffness matrix From Exampl (2)
![Page 55: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/55.jpg)
F1
F2
=-60
-50
F3 30
= d1
d2
-60
-50
1.8
.4
.4
2.8
F = K D
30 d3
0
1
0 1 3
D =
B
C
D
![Page 56: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/56.jpg)
D = K-1 F- 60- 50
30=
1.80.4
0
0.42.8
1
01
3
B
C
D
1EI
-1
- 60- 50
30=
7.4-1.2
.4
-1.25.4
-1.8
.4-1.8
4.88
B
C
D
1105
112.84
=
- 0.290- 0.196
0.165
X10-3
rad
![Page 57: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/57.jpg)
Example 4:Draw B.M.D for the shown beam where EI = 105 kN.m2 for all members
4
A
B
E
5 2 4C D
60 kNm 50 kNm 30 kNm
![Page 58: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/58.jpg)
From the previous example
=
B
C
D
- 0.290
- 0.196
0.165
X10-3
rad
![Page 59: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/59.jpg)
4
AB
E
5 2 4C D
=
B
C
D
- 0.290- 0.196
0.165
X10-3
rad
EI = 105 kN.m2
For member AB
LAB
MAB = == - 14.5 kN.m
2 EILAB
MBA= (2 B + A ) = 5x104(2X-0.029)x10-3
5x104(0-0.29)x10-3(2 A + B )2 EI
= - 29 kN.m
![Page 60: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/60.jpg)
4
AB
E
5 2 4C D
=C
D
- 0.290- 0.196
0.165
X10-3
rad
B
EI = 105 kN.m2
For member BC
LBC
MBC = (2 B + C ) =
2 EI
LBC
MCB= (2 C + B ) =
4x104(2x-0.29-0.196)x10-3
= - 31 kN.m
4x104(2x-0.196-0.029)x10-3
= - 27.3 kN.m
![Page 61: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/61.jpg)
MAB = - 14.5 kN.mMBA= - 29 kN.m
MBC = - 31 kN.mMCB= - 27.3 kN.m
MCD = - 22.7 kN.mMDC= 13.5 kN.m
MDE = 16.5 kN.mMED= 8.3 kN.m
A B B C C D D E
14.5 29 31 27.3 22.7 13.5 16.5 8.3
![Page 62: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/62.jpg)
A B B C C D D E
14.5 29 31 27.3 22.7 13.5 16.5 8.3
14.531
27.3
22.713.5
16.5
B.M.D
![Page 63: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/63.jpg)
14.5 13.5
16.527.3
22.731
29
A
B
E
C D
60 kNm 50 kNm 30 kNm
![Page 64: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/64.jpg)
Example 5:Draw B.M.D for the shown beam where EI is constant for all members
2
A B C D
1 2 2 1 2
100 kN 200 kN 150 kN
![Page 65: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/65.jpg)
Solution Steps of assembly method :
Drive the member local stiffness matrixLocal
k11F1
F2
=k21
F3 k31
k12
k22
k32
k13
k23
k33
F4k41 k42 k43
k14
k24
k34
k44
d1
d2
d3
d4
![Page 66: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/66.jpg)
d3
d1
d2 d4
First column inLocal Stiffness matrix
6 EI
L2
6 EI
L212 EI
L3
![Page 67: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/67.jpg)
F1 = 12 EIL3
F2 = 6 EI
L2
6 EI
L2
F3 = 12 EIL3
-
F4 =
first column in Local Stiffness matrix
![Page 68: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/68.jpg)
=
k31
k41
12 EIL3
6 EIL2
-12 EIL3
6 EIL2
k11
k21
![Page 69: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/69.jpg)
Second column inLocal Stiffness matrix
d2 =1
4 EIL
2 EIL
6 EIL2
6 EIL2
![Page 70: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/70.jpg)
F3 =F1 =
F4 =F2 =
6 EIL2
6 EIL2
-
2 EIL
4 EIL
![Page 71: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/71.jpg)
Second column inLocal Stiffness matrix
=
k12
k22
k32
k42
6 EI
L2
4 EIL
6 EIL2
- 2 EIL
![Page 72: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/72.jpg)
Third column in Local Stiffness matrix
d3 =1
12 EIL3
12 EIL3
6 EIL2
6 EIL2
![Page 73: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/73.jpg)
Third column in Local Stiffness matrix
=
k13
k23
k33
k33
-12 EIL3
-6 EIL2
12 EIL3
-6 EIL2
![Page 74: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/74.jpg)
12 EIL3
-6 EIL2
-6 EIL2
F4 =
F3 = F1 = -12 EIL3
F2 =
![Page 75: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/75.jpg)
Fourth column inLocal Stiffness matrix
6 EIL2
6 EIL2
4 EIL
2 EIL
![Page 76: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/76.jpg)
F1 = F3=
F2 = F4 =
6 EIL2
2 EIL
-6 EIL2
4 EIL
![Page 77: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/77.jpg)
K l
12 EIL3
6 EIL2
6 EIL2
4 EIL
-12 EIL3
=
-12 EIL3
-6 EIL2
-6 EIL2
12 EIL3
-6 EIL2
6 EIL2
2 EIL
6 EIL2
2 EIL
-6 EIL2
4 EIL
![Page 78: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/78.jpg)
First element : (A-B )
Start Joint : A
End Joint : BAngle : 0s = sin = 0c = cos = 1EA Is conastant
LAB = 300 cm
![Page 79: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/79.jpg)
K l K g=
K l = E I
12/L^3
6/L^2
-12/L^3
6/L^2
6/L^2
2/L
6/L^2-
4/L
12/L^3-
6/L^2-
12/L^3
6/L^2-
6/L^2
-6/L^2
4/L
2/L
![Page 80: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/80.jpg)
K l = EI
0.44
0.67
-0.44
0.67
0.67
1.33
-0.67
0.67
-0.44
-0.67
0.44
-0.67
0.67
0.67
-0.67
1.33
A B
"ِ"ِA
B
![Page 81: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/81.jpg)
Second element : ( B-c)Start Joint : B End Joint : c
= E I
12/L^3
6/L^2
-12/L^3
6/L^2
6/L^2
2/L
6/L^2-
4/L
12/L^3-
6/L^2-
12/L^3
6/L^2-
6/L^2
-6/L^2
4/L
2/LK l
![Page 82: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/82.jpg)
K l = EI
0.1875
0.375
-0.1875
0.375
0.375
1
-0.375
0.5
-0.1875
-0.375
0.1875
-0.375
0.375
0.5
-0.375
1
B C
C
B
![Page 83: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/83.jpg)
Second element : ( C-D )Start Joint : C End Joint : D
= E I
12/L^3
6/L^2
-12/L^3
6/L^2
6/L^2
2/L
6/L^2-
4/L
12/L^3-
6/L^2-
12/L^3
6/L^2-
6/L^2
-6/L^2
4/L
2/LK l
![Page 84: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/84.jpg)
K l = EI
0.44
0.67
-0.44
0.67
0.67
1.33
-0.67
0.67
-0.44
-0.67
0.44
-0.67
0.67
0.67
-0.67
1.33
C D
�ِC
D
![Page 85: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/85.jpg)
K
g = EI
0.44
0.67
-0.44
0.67
0.67
1.33
-0.67
0.67
-0.44
-0.67
0.44
-0.67
0.67
0.67
-0.67
1.33
A B
�Aِ
B
![Page 86: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/86.jpg)
K g= EI
0.1875
0.375
-0.1875
0.375
0.375
1
-0.375
0.5
-0.1875
-0.375
0.1875
-0.375
0.375
0.5
-0.375
1
B C
C
B
![Page 87: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/87.jpg)
K g= EI
0.44
0.67
-0.44
0.67
0.67
1.33
-0.67
0.67
-0.44
-0.67
0.44
-0.67
0.67
0.67
-0.67
1.33
C D
�ِC
D
![Page 88: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/88.jpg)
Ks = EI
0.44
.0937
0.67
-0.44
0.67
0
0
0.67
1.33
-0.67
0.67
0
0
0.67
0.67
-.0.295
2.33
-0.375
0.5
0
0
-0.1875
.-375
.2525
0.295
0
0
0.375
0.5
0.295
2.33
0
0
.-44
.-67
A B C
B
A
C
K l K g=
00 0 .-44 .-67 .-67
0 0 0 .67 0.67 1.33
D
D
-0.44
-0.67
.0.6275
-0.295
-0.1875
0.375
0
0
0
0
.67
0.67
0
0
0
0
.-67
0.44
![Page 89: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/89.jpg)
Partition
KuuK =Kru
Kur
Krr
u ru
r
Kuu = EI 2.33
2.33
0.5
0.5
![Page 90: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/90.jpg)
Force vectorTransformation from member forces to Joint forces
L
P
8LP
8LP
90
P a b2
L2
L
P
a b P b a2
L2
![Page 91: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/91.jpg)
2
A B C D
1 2 2 1 2
100 kN 200 kN 150 kN
100 kN 150 kN
200 kN
100 kNm100 kNm
44.4 kNm
22.2 kNm
66.7 kNm
33.3 kNm
Fixed End Reaction
(FER)
![Page 92: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/92.jpg)
2
A B C D
1 2 2 1 2
100 kN 200 kN 150 kN
100 kN 150 kN
200 kN
100 kNm100 kNm
44.4 kNm
22.2 kNm
66.7 kNm
33.3 kNm
Fixed End Action (FEA)
![Page 93: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/93.jpg)
2
A B C D
1 2 2 1 2
100 kN 200 kN 150 kN
100 kNm
44.4 kNm 66.7 kNm
100 kNm
33.3 kNm55.6 kNm
![Page 94: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/94.jpg)
2
A B C D
1 2 2 1 2
100 kN 200 kN 150 kN
33.3 kNm55.6 kNm
F1
F2
=-55.6
33.3
![Page 95: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/95.jpg)
F = K Dk11F1
F2
=k21
k12
k22
d1
d2
-55.6
33.3=
7/3
0.5 7/3
0.5EI
B
C
B
C=
1EI
7/3
0.5 7/3
0.5-1
-55.6
33.3
![Page 96: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/96.jpg)
B
C=
1EI
7/3
0.5 7/3
0.5-1
-55.6
33.3
B
C=
1EI
-28.18
20.31
![Page 97: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/97.jpg)
Internal forces in beam elements2 EI
LMAB= ( 2 + )
MBA=2 EI
L( + 2 )
MBA=2 EI
L( + 2 ) M(FER) BA +
MAB= ( 2 + ) M(FER) AB +2 EI
L
![Page 98: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/98.jpg)
2
A B CD
1 2 2 1 2
100 kN 200 kN 150 kN
100 kN 150 kN
200 kN
100 kNm100 kNm
44.4 kNm
22.2 kNm
66.7 kNm
33.3 kNm
Fixed End Reaction
(FER)
![Page 99: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/99.jpg)
B
C=
1EI
-28.18
20.31
2 EILMAB= ( 2 + ) M(FER) AB +
MBA=2 EI
L( + 2 ) M(FER) BA +
100 kN
44.4 kNm22.2 kNm
= 22.2 + 2/3 (-28.18) = 3.4
= -44.4 + 2/3 (2x-28.18) = - 82
A B
![Page 100: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/100.jpg)
B
C=
1EI
-28.18
20.31
2 EILMBC= ( 2 + ) CM(FER) BC +
MCB=2 EI
L( + 2 ) CM(FER) CB +
= 100 + 2/4 (2x-28.18+20.31) = 82
= -100 + 2/4 (-28.18+2x20.31) = - 93.8
200 kN
100 kNm100 kNm
B C
![Page 101: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/101.jpg)
B
C=
1EI
-28.18
20.31
2 EILMCD= ( 2 + )C DM(FER) CD +
MDC=2 EI
L( + 2 )C DM(FER) DC +
= 66.7 + 2/3 (2x20.31) = 93.8
= -33.3 + 2/3 (20.31) = - 19.8
150 kN
33.3 kNm66.7 kNm
C D
![Page 102: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/102.jpg)
MAB= 3.4MBA= -82
MBC= 82MCB= -93.8
MCD= 93.8MDC= -19.8
3.4
82
19.8
93.8
B.M.D
![Page 103: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/103.jpg)
3.4
82
19.8
93.8
B.M.D
2
A B C D
1 2 2 1 2
100 kN 200 kN 150 kN
55.887.9
69.1
66.7
10.9
200
112.1
100
30.9
![Page 104: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/104.jpg)
Example 6:Draw B.M.D for the shown beam where EI is shown in figure
3
A B C D
5 5 2
100 kN 200 kN 100 kN
3 2
250 kN
III2 I2 I2 I
E
![Page 105: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/105.jpg)
Solution Steps of assembly method :
Drive the member local stiffness matrixLocal
k11F1
F2
=k21
F3 k31
k12
k22
k32
k13
k23
k33
F4k41 k42 k43
k14
k24
k34
k44
d1
d2
d3
d4
![Page 106: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/106.jpg)
d3
d1
d2 d4
First column inLocal Stiffness matrix
6 EI
L2
6 EI
L212 EI
L3
![Page 107: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/107.jpg)
F1 = 12 EIL3
F2 = 6 EI
L2
6 EI
L2
F3 = 12 EIL3
-
F4 =
first column in Local Stiffness matrix
![Page 108: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/108.jpg)
=
k31
k41
12 EIL3
6 EIL2
-12 EIL3
6 EIL2
k11
k21
![Page 109: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/109.jpg)
Second column inLocal Stiffness matrix
d2 =1
4 EIL
2 EIL
6 EIL2
6 EIL2
![Page 110: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/110.jpg)
F3 =F1 =
F4 =F2 =
6 EIL2
6 EIL2
-
2 EIL
4 EIL
![Page 111: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/111.jpg)
Second column inLocal Stiffness matrix
=
k12
k22
k32
k42
6 EI
L2
4 EIL
6 EIL2
- 2 EIL
![Page 112: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/112.jpg)
Third column in Local Stiffness matrix
d3 =1
12 EIL3
12 EIL3
6 EIL2
6 EIL2
![Page 113: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/113.jpg)
Third column in Local Stiffness matrix
=
k13
k23
k33
k33
-12 EIL3
-6 EIL2
12 EIL3
-6 EIL2
![Page 114: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/114.jpg)
12 EIL3
-6 EIL2
-6 EIL2
F4 =
F3 = F1 = -12 EIL3
F2 =
![Page 115: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/115.jpg)
Fourth column inLocal Stiffness matrix
6 EIL2
6 EIL2
4 EIL
2 EIL
![Page 116: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/116.jpg)
F1 = F3=
F2 = F4 =
6 EIL2
2 EIL
-6 EIL2
4 EIL
![Page 117: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/117.jpg)
K l
12 EIL3
6 EIL2
6 EIL2
4 EIL
-12 EIL3
=
-12 EIL3
-6 EIL2
-6 EIL2
12 EIL3
-6 EIL2
6 EIL2
2 EIL
6 EIL2
2 EIL
-6 EIL2
4 EIL
![Page 118: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/118.jpg)
First element : (B-C )
Start Joint : B
End Joint : CAngle : 0s = sin = 0c = cos = 1EA Is conastant
LAB = 1000 cm
![Page 119: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/119.jpg)
K l K g=
K l =E I
12/L^3
6/L^2
-12/L^3
6/L^2
6/L^2
2/L
6/L^2-
4/L
12/L^3-
6/L^2-
12/L^3
6/L^2-
6/L^2
-6/L^2
4/L
2/L
![Page 120: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/120.jpg)
K l = EI
0.024
0.12
-0.024
0.12
0.12
0.8
-0.12
0.4
-0.024
-0.12
0.024
-0.12
0.12
0.4
-0.12
0.8
B C
�ِB
C
![Page 121: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/121.jpg)
First element : (C-D )
Start Joint : C
End Joint : DAngle : 0s = sin = 0c = cos = 1EA Is conastant
LAB = 500 cm
![Page 122: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/122.jpg)
K l K g=
K l = E I
12/L^3
6/L^2
-12/L^3
6/L^2
6/L^2
2/L
6/L^2-
4/L
12/L^3-
6/L^2-
12/L^3
6/L^2-
6/L^2
-6/L^2
4/L
2/L
![Page 123: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/123.jpg)
K l = EI
0.096
0.24
-0.096
0.24
0.24
0.8
-0.24
0.4
-0.096
-0.24
0.096
-0.24
0.24
0.4
-024
0.8
C D
C
D
![Page 124: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/124.jpg)
Assembly :
K g= EI
0.024
0.12
-0.024
0.12
0.12
0.8
-0.12
0.4
-0.024
-0.12
0.024
-0.12
0.12
0.4
-0.12
0.8
B C
�ِB
C
![Page 125: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/125.jpg)
K g= EI
0.096
0.24
-0.096
0.24
0.24
0.8
-0.24
0.4
-0.096
-0.24
0.096
-0.24
0.24
0.4
-024
0.8
C D
C
D
![Page 126: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/126.jpg)
Ks = EI
0.024
.0937
0.12
-0.024
0.12
0
0
0.12
0.8
-0.12
0.4
0
0
-0.024
-0.12
0.12
0.12
-0.096
0.024
0.12
0.4
0.12
1.6
-0.024
0.4
0
0
-0.096
-0.24
0.096
-0.24
0
0
0.24
0.4
-0.24
0.8
B C D
C
B
D
K l K g=
![Page 127: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/127.jpg)
Partition
KuuK =Kru
Kur
Krr
u ru
r
Kuu = EI 0.8
0.4
0
0.4
1.60.4 0.8
0.4
0
![Page 128: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/128.jpg)
Force vectorTransformation from member forces to Joint forces
L
P
8LP
8LP
128
P a b2
L2
L
P
a b P b a2
L2
![Page 129: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/129.jpg)
100 kN
100 kN200 kN
250 kNm250 kNm
300 kNm
200 kNm
Fixed End Reaction (FER)
3
A B C D
5 5 2
100 kN 200 kN 100 kN
3 2
250 kN
E
250 kN180 kNm 120 kNm
![Page 130: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/130.jpg)
100 kN
100 kN200 kN
250 kNm250 kNm
300 kNm
200 kNm
Fixed End Action (FEA)
3
A B C D
5 5 2
100 kN 200 kN 100 kN
3 2
250 kN
E
250 kN180 kNm 120 kNm
![Page 131: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/131.jpg)
250 kNm250 kNm
300 kNm
200 kNm
3
A B C D
5 5 2
100 kN 200 kN 100 kN
3 2
250 kN
E
180 kNm 120 kNm
50 kNm 70 kNm 80 kNm
![Page 132: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/132.jpg)
3
A B C D
5 5 2
100 kN 200 kN 100 kN
3 2
250 kN
E
50 kNm 70 kNm 80 kNm
F1
F2 =
50
70
F3 - 80
![Page 133: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/133.jpg)
F = K DThe stiffness equation
=
50
70- 80
0.80.4
0
0.41.6
0.4
00.4
0.8EI
B
C
D
=
50
70- 80
0.80.4
0
0.41.6
0.4
00.4
0.8
B
C
D
1EI
-1
=1EI
27.08
70.83- 135.42
![Page 134: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/134.jpg)
=
B
C
D
1EI
27.08
70.83- 135.42
![Page 135: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/135.jpg)
Internal forces in beam elements
2 EILMAB= ( 2 + )
MBA=2 EI
L( + 2 )
M(FER) AB +
M(FER) BA +
![Page 136: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/136.jpg)
100 kN
100 kN200 kN
250 kNm250 kNm
300 kNm
200 kNm
Fixed End Reaction (FER)
3
A B C D
5 5 2
100 kN 200 kN 100 kN
3 2
250 kN
E
250 kN180 kNm 120 kNm
![Page 137: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/137.jpg)
2 E(2I)LMBC= ( 2 + ) CM(FER) BC +
MCB=2 E(2I)
L( + 2 ) CM(FER) CB +
= 250 + 4/10 (2x27.08+70.83) = 300
= -250 + 4/10 (27.08+2x70.83) = - 182.5
200 kN
250 kNm250 kNm
B C
=
B
C
D
1EI
27.08
70.83- 135.42
![Page 138: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/138.jpg)
2 E(2I)LMBC= ( 2 + ) CM(FER) BC +
MCB=2 E(2I)
L( + 2 ) CM(FER) CB +
= 250 + 4/10 (2x27.08+70.83) = 300
= -250 + 4/10 (27.08+2x70.83) = - 182.5
200 kN
250 kNm250 kNm
B C
=
B
C
D
1EI
27.08
70.83- 135.42
![Page 139: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/139.jpg)
MBC= 300MCB= -182.5
MCD= 182.5MDC= -200
300200
B.M.D
A
B C D
E
182.5
![Page 140: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/140.jpg)
B.M.D
3
A B C D
5 5 2
100 kN 200 kN 100 kN
3 2
250 kN
E
300200
A
B C D
E
182.5241.25
500
258.75
189.5
300
110.5
![Page 141: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/141.jpg)
Example 7:Draw B.M.D for the shown beam where EI is constant for all members
A B
5 5
240 kN
C
5 5
120 kN
![Page 142: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/142.jpg)
K l
12 EIL3
6 EIL2
6 EIL2
4 EIL
-12 EIL3
=
-12 EIL3
-6 EIL2
-6 EIL2
12 EIL3
-6 EIL2
6 EIL2
2 EIL
6 EIL2
2 EIL
-6 EIL2
4 EIL
4 EIL
![Page 143: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/143.jpg)
First element : (A-B )
EA conastant
LAB = 10 m
12 EIL3
=0.012EI
6 EIL2
4 EIL
2 EIL
=0.06 EI
=0.4 EI
=0.2 EI
![Page 144: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/144.jpg)
K l =
0.012 EI
0.012 EI 0.012 EI
0.012 EI
0.06 EI
0.06 EI
0.06 EI0.06 EI
0.06 EI
0.06 EI
0.06 EI
0.06 EI
0.4 EI
0.4 EI
0.2 EI
0.2 EI
A
A
B
B
![Page 145: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/145.jpg)
Second element : ( B-c)
EI conastant LAB = 10 m
12 EIL3
=0.012EI
6 EIL2
4 EIL
=0.06 EI
=0.4 EI
=0.2 EI2 EI
L
![Page 146: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/146.jpg)
K l =
0.012 EI
0.012 EI 0.012 EI
0.012 EI
0.06 EI
0.06 EI
0.06 EI0.06 EI
0.06 EI
0.06 EI
0.06 EI
0.06 EI
0.4 EI
0.4 EI
0.2 EI
0.2 EI
A
A
B
B
![Page 147: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/147.jpg)
K gK l =
Assembly :
=
K gg1
0.012 EI 0.06 EI 0.012 EI 0.06 EI
0.06 EI 0.4 EI 0.06 EI 0.2 EI
0.012 EI 0.06 EI 0.06 EI0.012 EI
0.06 EI 0.2 EI 0.4 EI0.06 EI
![Page 148: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/148.jpg)
2=
K gg
0.012 EI
0.012 EI 0.012 EI
0.012 EI
0.06 EI
0.06 EI
0.06 EI0.06 EI
0.06 EI
0.06 EI
0.06 EI
0.06 EI
0.4 EI
0.4 EI
0.2 EI
0.2 EI
A B
A
B
![Page 149: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/149.jpg)
=
0.012
-0.012
-0.012
0.06
0.06
-0.06 0.06
0.06
-0.06
0.06
-0.06
0.4
0.4
0.2
0.2
A
A
B
B
K s
0.0
0.8
0.024
0.0
0.0
0.0
0.0 0.0
0.0
0
0.0
0.0
0.012
-0.012
-0.012
0.06
-0.06
0.2
0.2
-0.06
-0.06
EI
c
c
![Page 150: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/150.jpg)
Partition
KuuK =Kru
Kur
Krr
u ru
r
![Page 151: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/151.jpg)
Force vectorTransformation from member forces to Joint forces
L
P
8LP
8LP-
![Page 152: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/152.jpg)
240 kN300 kNm300 kNm
Fixed End Reaction
(FER)
A B
5 5
240 kNC
5 5
120 kN
120 kN150 kNm150 kNm
![Page 153: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/153.jpg)
300 kNm300 kNm
A B
5 5
240 kNC
5 5
120 kN
150 kNm150 kNm
120120 6060
![Page 154: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/154.jpg)
F1
F2
=
120
-300
F3
F6
F5
F4
180
150
60
150
![Page 155: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/155.jpg)
120
-300
180
150
60
150
0.012
-0.012
-0.012
0.06
0.06
-0.06 0.06
0.06
-0.06
0.06
-0.06
0.4
0.4
0.2
0.2
0.0
0.8
0.024
0.0
0.0
0.0
0.0 0.0
0.0
0
0.0
0.0
0.012
-0.012
-0.012
0.06
-0.06
0.2
0.2
-0.06
-0.06
d1
d2
d3
d4
d5
d6
= EI
![Page 156: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/156.jpg)
120
-300
180
150
60
150
-0.012 0.06
-0.06
0.06
-0.06
0.06
-0.06
0.4
0.4
0.2
0.2
0.0
0.8
0.024
0.0
0.0
0.0
0.0 0.0
0.0
0.0
0.0
0.0
0.012
-0.012
-0.012
0.06
-0.06
0.2
0.2
-0.06
-0.06
d1
d2
d3
d4
d5
d6
=EI
![Page 157: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/157.jpg)
=B
C
150
150
0.8
0.2 0.4
0.2EI
B
C=
1EI
-1150
150
0.8
0.2 0.4
0.2
![Page 158: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/158.jpg)
B
C=
1EI
107.14
321.43
B
C=
1EI
-1150
150
0.8
0.2 0.4
0.2
![Page 159: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/159.jpg)
Internal forces in beam elements
2 EILMAB= ( 2 + )
MBA=2 EI
L( + 2 )
M(FER) AB +
M(FER) BA +
![Page 160: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/160.jpg)
240 kN300 kNm300 kNm
Fixed End Reaction
(FER)
A B
5 5
240 kNC
5 5
120 kN
120 kN150 kNm150 kNm
![Page 161: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/161.jpg)
2 EILMAB= ( 2 + )
MBA=2 EI
L( + 2 )
M(FER) AB +
M(FER) BA +
= 300 + 2/10 (107.14) = 321.4
= -300 + 2/10 (2x107.14) = - 257.1
B
C=
1EI
107.14
321.43
240 kN
300 kNm300 kNm
A B
![Page 162: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/162.jpg)
2 EILMBC= ( 2 + ) C
MCB=2 EI
L( + 2 ) C
M(FER) BC +
M(FER) CB +
= 150 + 2/10 (2x107.14+321.43) = 257.1
= -150 + 2/10 (107.14+2x321.43) = 0
120 kN
150 kNm150 kNm
B C B
C=
1EI
107.14
321.43
![Page 163: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/163.jpg)
MAB= 321.4MBA= -257.1
MBC= 257.1MCB= 0
321.4257.1
B.M.D
![Page 164: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/164.jpg)
93.8
128.55
300
171.45
A B
5 5
240 kNC
5 5
120 kN
321.4257.1
B.M.D
289.25
600
310.75
![Page 165: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/165.jpg)
Beams with settlement
6 EIL2
6 EIL2 12 EI
L3
12 EIL3
Fixed End Reaction
(FER)
![Page 166: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/166.jpg)
Beams with settlement
6 EIL2
6 EIL2
12 EIL3
12 EIL3
Fixed End Reaction
(FER)
![Page 167: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/167.jpg)
Beams with settlement
3 EIL2 3 EI
L3
3 EIL3
Fixed End Reaction
(FER)
![Page 168: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/168.jpg)
Beams with settlement
3 EIL2
3 EIL3
3 EIL3
Fixed End Reaction
(FER)
![Page 169: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/169.jpg)
169
![Page 170: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/170.jpg)
Example 8:Draw B.M.D for the shown beam due to the shown loads and vertical downward settlement at support B (2000/EI) and at support C (1000/EI) where EI is constant for all members
A B
5 5
240 kN
C
5 5
120 kN
![Page 171: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/171.jpg)
=0.8
0.2 0.4
0.2K EI
From example 7 :
![Page 172: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/172.jpg)
240 kN300 kNm300 kNm
Fixed End Reaction
(FER)
A B
5 5
240 kNC
5 5
120 kN
120 kN150 kNm150 kNm
ForLoads
![Page 173: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/173.jpg)
Fixed End Reaction (FER)
A B
5 5
C
5 5
Forsettlement
1000EI
2000EI
6 EIx1000102 EI
6 EIx1000102 EI
6 EIx2000102 EI
6 EIx2000102 EI
120
120 6060
![Page 174: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/174.jpg)
300 kNm300 kNm
A B
5 5
240 kNC
5 5
120 kN
150 kNm150 kNmFor
Loads
Forsettlement 120 120 60 60
Fixed End Reaction
(FER)
210 kNmTotal 90 kNm
420 180 90 210
![Page 175: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/175.jpg)
A B
5 5
240 kNC
5 5
120 kN
Fixed End Reaction
(FER)210 kNm90 kNm
Fixed End Action (FEA)
210 kNm90 kNm
![Page 176: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/176.jpg)
A B
5 5
240 kNC
5 5
120 kN
Fixed End Action (FEA)
210 kNm90 kNm
F1
F2
=90
210
![Page 177: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/177.jpg)
F = K Dk11F1
F2
=k21
k12
k22
d1
d2
=B
C
B
C=
1EI
-1
90
210
90
210
0.8
0.2 0.4
0.2EI
0.8
0.2 0.4
0.2
![Page 178: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/178.jpg)
B
C=
1EI
-21.43
535.71
B
C=
1EI
-190
210
0.8
0.2 0.4
0.2
![Page 179: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/179.jpg)
Internal forces in beam elements
2 EILMAB= ( 2 + )
MBA=2 EI
L( + 2 )
M(FER) AB +
M(FER) BA +
![Page 180: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/180.jpg)
300 kNm300 kNm
A B
5 5
240 kNC
5 5
120 kN
150 kNm150 kNmFor
Loads
Forsettlement 120 120 60 60
Fixed End Reaction
(FER)
210 kNmTotal 90 kNm
420 180 90 210
![Page 181: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/181.jpg)
2 EILMAB= ( 2 + )
MBA=2 EI
L( + 2 )
M(FER) AB +
M(FER) BA +
= 420 + 2/10 (-21.43) = 415.7
= -180 + 2/10 (2x-21.43) = - 188.6
180 kNm420 kNm
A B B
C=
1EI
-21.43
535.71
![Page 182: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/182.jpg)
2 EILMBC= ( 2 + ) C
MCB=2 EI
L( + 2 ) C
M(FER) BC +
M(FER) CB +
= 90 + 2/10 (2x-21.43+535.7) = 188.6
= -210 + 2/10 (-21.43+2x535.71) = 0
210 kNm90 kNm
B C B
C=
1EI
-21.43
535.71
![Page 183: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/183.jpg)
MAB= 321.4MBA= -257.1
MBC= 257.1MCB= 0
415.7188.6
B.M.D
![Page 184: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/184.jpg)
93.8
94.3
300
205.7
A B
5 5
240 kNC
5 5
120 kN
415.7188.6
B.M.D
302.15
600
297.85
![Page 185: Stiffness 9](https://reader031.fdocuments.net/reader031/viewer/2022020101/5403662ddab5ca4b658b4764/html5/thumbnails/185.jpg)