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محاضرات تالتة مدنى د.عاطف العراقى
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d1
d2d3
x
yz
d4
d5
d6
x
yz
x
y
z
x
yz
1
AEL
AEL
AEL
AEL
d=1
x
yz
1
AEL
AEL
AEL
AEL
d=1
x
yz
y
z
y6EIzyL2
6EIzyL2
12EIzyL3
12EIzyL3
x
yz
d=1
z
y
x
6EIzL2
12EIzL3
6EIzL2
12EIzL3
x
yz
z
y
6EIzyL2
12EIzyL3
12EIzyL3
6EIzyL2
y
x
yz
d=1
z
y
x
6EIzL2
12EIzL3
6EIzL2
12EIzL3
x
yz
z
z
y
6EIyzL2
12EIyzL3
6EIyzL2
12EIyzL3
x
yz d=1
z
y
x
6EIyL2
12EIyL3 6EIy
L2
12EIyL3
x
yz
z
y
z
6EIyzL2
12EIyzL3
6EIyzL2
12EIyzL3
x
yz
d=1
z
y
x
6EIyL2
12EIyL3 6EIy
L2
12EIyL3
x
yz
z
y z
4 EIzz
L
6 EIzz
L2
6 EIzz
L2
2 EIzz
L
x
yz
d=1
z
y
x
d=1
4 EIz
L
6 EIz
L2
6 EIz
L2
2 EIz
L
x
yz
z
y
z2 EIzz
L
6 EIzz
L2
6 EIzz
L2
4 EIzz
L
x
yz
d=1
z
y
x
d=1
4 EIz
L
6 EIz
L2
6 EIz
L2
2 EIz
L
x
yz
y
6 EIyy
L2
2 EIyy
L
6 EIyy
L2
4 EIyy
L
z
y
x
yz
d=1
z
y
x
6 EIy
L2
2 EIy
L
6 EIy
L2
4 EIy
L
x
yz
y
6 EIyy
L2
4 EIyy
L
2 EIyy
L
6 EIyy
L2
z
y
x
yz
d=1
z
y
x
6 EIy
L2
2 EIy
L
6 EIy
L2 4 EIy
L
x
yz
x
G Ixx
L
G Ixx
L
x
yz
d=1 z
y
x
G Ix
L
G Ix
L
x
yz
x
z
yx
G Ixx
L
G Ixx
L
x
yz
d=1
z
y
x
G Ix
L
G Ix
L
Example 2:Draw all diagrams for the shown space frame where E = 1200 kN/cm2, G = 500 kN/cm2 and the sections are shown in figure
8 m
10 m
C
30x80 cm
30x4
0 cm
20x60 cm
100 kN200 kN
40 kN
10 m
D
B
A
z
y
xz
y
x
zy
x
30x80 cm
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4 30x4
0 cm
20x60 cm
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
Sections Properties
C
D
A
d1d2
d3
d4d5
d6
xy
z
Modeling
k11
K =
k21
k31
k12
k22
k32
k13
k23
k33
Stiffness matrix
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
C
D
B
A
C
D
B
A
d1=1
xy
z
First column in Stiffness matrix
z
y
x
z
y
x
zy
xxy
z
d 1=1
d 1=1
d 1=1
z
y
xz
y
x
z y
x
xy
z
d 1=1
d 1=1
d 1=1
AEL12EIy
L3
6 EIy
L2
12EIy
L3
6 EIy
L2
z
y
xz
y
x
z y
x
xy
z
d 1=1
d 1=1
d 1=1
AEL12EIy
L3
6 EIy
L2
12EIy
L3
6 EIy
L2
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
E = 1200 kN/cm2
z
y
x
xy
z
d 1=1
AEL
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
E = 1200 kN/cm2
1200X1200
800
1,800
z
y
x
xy
z
d 1=1
12EIy
L3
6 EIy
L2
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
E = 1200 kN/cm2
12x1200X180,000
10003
2.6
6x1200X180,000
10002
1,296
z y
x
xy
z
d 1=1
12EIy
L3
6 EIy
L2
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
E = 1200 kN/cm2
12x1200X90,000
10003
1.3
6x1200X90,000
10002
648
xy
z
1.3648
2.6
1,296
1,800
d1d2
d3
d4d5
d6
k11
=
k21
k31
k41
k51
k61
1,803.9
0
0
0
- 648
-1,296
C
D
B
A
xy
z
Second column in Stiffness matrix
d2=1
z
y
x
z
y
x
z y
x
xy
z
d2=1
d2 =1
d2 =1
z
y
xz
y
x
z y
x
xy
z
d2=1
d2 =1
d2 =1
AEL
12EIz
L3
12EIy
L3
6 EIy
L2
6 EIz
L2
z
y
xz
y
x
z y
x
xy
z
d2=1
d2 =1
d2 =1
AEL
12EIz
L3
12EIy
L3
6 EIy
L2
6 EIz
L2
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
E = 1200 kN/cm2
z
y
x
xy
z
d2=1
12EIy
L3
6 EIy
L2
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
E = 1200 kN/cm2
12x1200X40,000
8003
1.1
6x1200X40,000
8002
450
z
y
x
xy
z
d2 =1
AEL
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
E = 1200 kN/cm2
1200X2400
1000
2,880
z y
x
xy
z
d2 =1
12EIz
L3
6 EIz
L2
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
E = 1200 kN/cm2
12x1200X160,000
10003
2.3
6x1200X160,000
10002
1,152
xy
zE = 1200 kN/cm2
2,880
2.31,152
1.1
450
k12
=
k22
k32
k42
k52
k62
2,883.4
0
0
0
1,152
450d1d2
d3
d4d5
d6
C
D
B
A
xy
z
Third column in Stiffness matrix
d3=1
z
y
x
z
y
x
z y
x
xy
z
d3=1
d3=1
d3=1
z
y
x
z
y
x
z y
x
d3=1
d3=1
d3=1
AEL
12EIz
L3
6 EIz
L2
12EIz
L3
6 EIz
L2
xy
z
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
E = 1200 kN/cm2
z
y
x
d3=1
12EIz
L3
6 EIz
L2
xy
zA =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
E = 1200 kN/cm2
12x1200X360,000
8003
10.1
6x1200X360,000
8002
4,050
z
y
x
d3=1
12EIz
L3
6 EIz
L2
xy
zA =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
E = 1200 kN/cm2
12x1200X1,280,000
1000318.4
6x1200X1,280,000
10002 9,216
z y
x
d3=1AEL
xy
z
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
E = 1200 kN/cm2
1200X1200
1000
1,440
xy
z
1,440
18.4
9,216 10.1
4,050
k13
=
k23
k33
k43
k53
k63
0
0
1,468.5
- 4,050
9,216
0d1d2
d3
d4d5
d6
C
D
B
A
xy
z
Fourth column in Stiffness matrix
d4=1
z
y
x
z
y
x
zy
xxy
z
d 4=1
d 4=1
d 4=1
z
y
x
z
y
x
zy
xxy
z
d 4=1
d 4=1
d 4=1 GIx
L
4 EIz
L6 EIz
L2
4 EIz
L6 EIz
L2
z
y
x
z
y
x
zy
x
xy
z
d 4=1
d 4=1
d 4=1 GIx
L
4 EIz
L6 EIz
L2
4 EIz
L6 EIz
L2
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
E = 1200 kN/cm2
z
y
x
xy
z
d 4=1
GIx
L
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
G = 500 kN/cm2
500X126,435
80079,021.9
z
y
x
xy
z
d 4=1
4 EIz
L6 EIz
L2
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
E = 1200 kN/cm2
6x1200X1,280,000
10002
9,216
4x1200X1,280,000
1000
6,144,000
zy
x
xy
z
d 4=1
4 EIz
L6 EIz
L2
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
E = 1200 kN/cm2
6x1200X160,000
10002
1,152
4x1200X160,000
1000
768,000
xy
z
1,152768,0009,216
6,144,000
79,021.9
k14
=
k24
k34
k44
k54
k64
1,152
0
9,216
0
6,991,022
0d1d2
d3
d4d5
d6
C
D
B
A
xy
z
Fifth column in Stiffness matrix
d5=1
z
y
x
z
y
x
zy
xxy
z
d5 =1
d5 =1
d5 =1
z
y
x
z
y
x
zy
xxy
z
d5 =1
d5 =1
d5 =1
GIx
L
4 EIy
L
6 EIy
L2
4 EIz
L6 EIz
L2
z
y
x
z
y
x
zy
x
d5 =1
d5 =1
d5 =1
GIx
L
4 EIy
L
6 EIy
L2
4 EIz
L6 EIz
L2
xy
z
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
E = 1200 kN/cm2
z
y
xd5 =1
4 EIz
L6 EIz
L2
xy
z A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
6x1200X360,000
8002
4,050
4x1200X360,000
800
2,160,000
E = 1200 kN/cm2
z
y
xd
5 =1
GIx
L
xy
z A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
G = 500 kN/cm2
500X550,180
1000
275,090
zy
x
d5 =1
4 EIy
L
6 EIy
L2
xy
zA =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
E = 1200 kN/cm2
6x1200X90,000
10002
648
4x1200X90,000
1000
432,000
xy
z
d1d2
d3
d4d5
d6
275,090
4,050
2,160,000
648 432,000
k15
=
k25
k35
k45
k55
k65
0
- 648
- 4,050
0
0
2,867,090
C
D
B
A
xy
z
Sixth column in Stiffness matrix
d6=1
z
y
x
z
y
x
zy
xxy
z
d6=1
d6=1d6=1
z
y
x
z
y
x
zy
xxy
z
d6=1
d6=1d6=1
GIx
L
4 EIy
L
6 EIy
L2
4 EIy
L
6 EIy
L2
z
y
x
z
y
x
zy
x
d6=1
d6=1d6=1
GIx
L
4 EIy
L
6 EIy
L2
4 EIy
L
6 EIy
L2
xy
z
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
E = 1200 kN/cm2
z
y
xd6=1
4 EIy
L
6 EIy
L2
xy
zA =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
E = 1200 kN/cm2
6x1200X40,000
80024x1200X40,000
800450
240,000
z
y
x
d6=1
4 EIy
L
6 EIy
L2
xy
z A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
E = 1200 kN/cm2
4x1200X180,000
1000
6x1200X180,000
10002
1,296
864,000
zy
x
d6=1
GIx
L
xy
z
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
G = 500 kN/cm2
500X194,385
1000
97,192.5
xy
z
1,296
864,000
97,192.5
450240,000
k16
=
k26
k36
k46
k56
k66
450
-1,296
0
0
0
1,201,192.5d1d2
d3
d4d5
d6
k16
=
k26
k36
k46
k56
k66
450
-1,296
0
0
0
1,201,192.5
k15
=
k25
k35
k45
k55
k65
0
- 648
- 4,050
0
0
2,867,090
k14
=
k24
k34
k44
k54
k64
1,152
0
9,216
0
6,991,022
0
k13
=
k23
k33
k43
k53
k63
0
0
1,468.5
- 4,050
9,216
0
k12
=
k22
k32
k42
k52
k62
2,883.4
0
0
0
1,152
450
k11
=
k21
k31
k41
k51
k61
1,803.9
0
0
0
- 648
-1,296
450
-1,296
0
0
0
1,201,192.5
0
- 648
- 4,050
0
0
2,867,090
1,152
0
9,216
0
6,991,022
0
0
0
1,468.5
- 4,050
9,216
0
2,883.4
0
0
0
1,152
450
=K
1,803.9
0
0
0
- 648
-1,296
8 m
10 m
C
30x80 cm
30x4
0 cm
20x60 cm
100 kN200 kN
40 kN
10 mD
B
A
Force vector
200 kN
40 kN
100 kN
20
20
50
250100
100
250
50
50
100
100
Fixed End Reaction(FER)
200 kN
40 kN
100 kN
20
20
50
250100
100
250
50
50
100
100
Fixed End Action(FEA)
200 kN
40 kN
100 kN
20
20
50
250100
100
250
50
50
100
100
d1d2
d3
d4d5
d6
F1
=
F2
F3
F4
F5
F6
20
0
- 150
- 250
100
- 50
Stiffness Equation F = K D
20
0
- 150
- 250
100
- 50
450
-1,296
0
0
0
1,201,192.5
0
- 648
- 4,050
0
0
2,867,090
1,152
0
9,216
0
6,991,022
0
0
0
1,468.5
- 4,050
9,216
0
2,883.4
0
0
0
1,152
450
1,803.9
0
0
0
- 648
-1,296
d1
=
d2
d3
d4
d5
d6
Stiffness Equation D = K-1 F
20
0
- 150
- 250
100
- 50
450
-1,296
0
0
0
1,201,192.5
0
- 648
- 4,050
0
0
2,867,090
1,152
0
9,216
0
6,991,022
0
0
0
1,468.5
- 4,050
9,216
0
2,883.4
0
0
0
1,152
450
1,803.9
0
0
0
- 648
-1,296
d1
=
d2
d3
d4
d5
d6
-1
Deformations
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
Internal Forces Normal Force
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
AEL
N =
Internal Forces Normal Force
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
z
y
x
d 1=1
AEL
1,800
NBC = 1,800x.0011027 = 2 kN compression
Internal Forces Normal Force
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
z
y
xd
2 =1AEL
2,880
NBA = 2,880x.000035 = 0.1 kN tension
Internal Forces Normal Force
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
NBD = 1,440x0.103072 = 148.4 kN compression
z y
x
d3=1AEL
1,440
Internal Forces
B
C
B
A
B
D
148.4
148.4
0.1
0.1
2
2
Normal Force
C
D
A
Internal Forces Normal Force
148.4
0.12
Internal Forces Torsional Moment
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
IxGL
T t=
Internal Forces Torsional Moment
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
IxGL
T t=
d 4=1
GIx
L79,021.9
TBC = 79,021.9x1.0012x10-4 = 7.9 kNm
Internal Forces Torsional Moment
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5IxGL
T t=
d5 =1
GIx
L275,090
TBA = 275,090x-1.0823x10-4 = - 29.77 kNm
Internal Forces Torsional Moment
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
IxGL
T t=
d6=1
GIx
L97,192.5
TBD = 97,192.5x-2.9715x10-5 = - 2.89 kNm
Internal Forces
B
C
B
A
B
D
2.89
2.89
29.77
29.77
7.9
7.9
Torsional Moment
C
D
A
Internal Forces
2.89
29.77 7.9
Torsional Moment
Internal Forces
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
Bending momentIn Plan
d3=1
6 EIz
L24,050
4,050
Internal Forces
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
Bending momentIn Plan
d5 =1
4 EIz
L2,160,000
1,080,000
2 EIz
L
Internal Forces
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
Bending momentIn Plan
100 kN
100
100
Internal Forces0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
Bending momentIn Plan
d3=1
4,050
4,050
d5 =1 2,160,000
1,080,000
100
100FER
B
C
MBC = 100+4,050 d3 – 2,160,000 d5
MBC = 100+4,050x-.103072– 2,160,000x-1.0823x10-4
MBC = - 83.66 kNm
Internal Forces0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
Bending momentIn Plan
d3=1
4,050
4,050
d5 =1 2,160,000
1,080,000
100
100FER
B
C
MCB = -100+4,050 d3 – 1,080,000 d5
MCB = -100+4,050x-.103072– 1,080,000x-1.0823x10-4
MCB = - 400.55 kNm
Internal Forces Bending momentIn Plan
83.66
400.55B
C100 kN
50 + (400.55+83.66)/8110.53
50 - (400.55+83.66)/810.53
Internal Forces
d1d2
d3
d4d5
d6 Bending momentIn Plan
d3=1
6 EIz
L2
9,216
9,216
d 4=14 EIz
L6,144,000
3,072,000
200 kN
250
250FER
MBA = 250+9,216 d3 + 6,144,000 d4
MBA = 250+9,216x-.103072+ 6,144,000x1.0012x10-4
MBA = - 84.77 kNm
B
A
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
Internal Forces
d1d2
d3
d4d5
d6 Bending momentIn Plan
d3=1
6 EIz
L2
9,216
9,216
d 4=14 EIz
L6,144,000
3,072,000
200 kN
250
250FER
MAB = -250+9,216 d3 + 3,072,000 d4
MAB = -250+9,216x-.103072+ 3,072,000x1.0012x10-4
MAB = - 892.3 kNm
B
A
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
Internal Forces
d1d2
d3
d4d5
d6 Bending momentIn Plan
84.77
892.3B
A200 kN
100 + (892.3+84.77)/10197.71 100 - (892.3+84.77)/10
2.29
Internal Forces
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
Bending momentIn Plan
d2 =1
6 EIz
L2
1,152
1,152
d 4=14 EIz
L
768,000
384,000
B
D
MBD = 1,152 d2 + 768,000 d4
MBD = 1,152x-.000035+ 768,000x1.0012x10-4
MBD = 76.85 kNm
Internal Forces
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
Bending momentIn Plan
d2 =1
6 EIz
L2
1,152
1,152
d 4=14 EIz
L
768,000
384,000
B
D
MDB = 1,152 d2 + 384,000 d4
MDB = 1,152x-.000035+ 384,000x1.0012x10-4
MDB = 38.41 kNm
Internal Forces
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
Bending momentIn Plan
B
D
76.85
38.41
(76.85+38.41)/10
11.53
11.53
Internal Forces Bending momentIn Plan
83.66
400.55B
C100 kN
110.53
10.53
84.77
892.3B
A200 kN
197.71
2.29 B
D
76.85
38.41
11.53
11.53
C
D
A
C
D
A
Internal Forces
d1d2
d3
d4d5
d6 Bending momentOut of Plan
d2 =1
6 EIy
L2
450
450
d6=1
4 EIy
L240,000
120,000
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
B
C
MBC = 450 d2 + 240,000 d6
MBC = 450x-.000035 + 240,000x-2.9715x10-5
MBC = - 7.15 kNm
Internal Forces
d1d2
d3
d4d5
d6 Bending momentOut of Plan
d2 =1
6 EIy
L2
450
450
d6=1
4 EIy
L240,000
120,000
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
B
C
MCB = 450 d2 + 120,000 d6
MCB = 450x-.000035 + 120,000x-2.9715x10-5
MCB = - 3.58 kNm
7.15
3.58
B
C
(7.15+3.58)/8
1.34
1.34
Internal Forces Bending momentOut of Plan
Internal Forces
d1d2
d3
d4d5
d6 Bending momentOut of Plan
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
MBA = 50-1,296 d1 + 864,000 d6
MBA = 50-1,296x.011027 + 864,000x-2.9715x10-5
MBA = 10.04 kNm
d 1=1
6 EIy
L2
1,296
1,296
d6=1
4 EIy
L
864,000
432,000
B
A
40 kN
50
50
Internal Forces
d1d2
d3
d4d5
d6 Bending momentOut of Plan
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
MAB = -50-1,296 d1 + 432,000 d6
MAB = -50-1,296x.011027 + 432,000x-2.9715x10-5
MAB = - 77.13 kNm
d 1=1
6 EIy
L2
1,296
1,296
d6=1
4 EIy
L
864,000
432,000
B
A
40 kN
50
50
77.13
10.04
B
C
26.71
40 kN
20+(77.13-10.04)/10
13.29
20-(77.13-10.04)/10
Internal Forces Bending momentOut of Plan
Internal Forces
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
Bending moment
d 1=1
6486 EIy
L2
648
d5 =1
4 EIy
L432,000
216,000
B
D
MBD = 648 d1 - 432,000 d5
MBD = 648x.011027 - 432,000x-1.0823x10-4
MBD = 53.9 kNm
Out of Plan
Internal Forces
d1d2
d3
d4d5
d6
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
Bending moment
d 1=1
6486 EIy
L2
648
d5 =1
4 EIy
L432,000
216,000
B
D
MDB = 648 d1 - 216,000 d5
MDB = 648x.011027 - 216,000x-1.0823x10-4
MDB = 30.52 kNm
Out of Plan
53.9
30.52
8.44(53.9+30.52)/10
8.44
Internal Forces Bending momentOut of PlanB
D
Internal Forces Bending moment
B
C
B
A
B
D
Out of Plan
53.9
30.528.44
8.44
77.13
10.04
26.71
40 kN
13.29
7.15
3.58
1.34
1.34
C
D
A
C
D
A
C
D
A
Internal Forces Bending moment
53.9
8.44
10.04
13.29
7.15
1.34
84.77
2.29
76.85
11.53
10.53
83.660.1
148.4
2
29.77
2.89
7.9
Internal Forces
B
C
B
A
B
D
2.89
2.89
29.77
29.77
7.9
7.9
Torsional Moment
Summary
d1
d2
d3
AEL
AEL
AEL
AEL
6 EI
L2
6 EIL2
12 EIL3
12 EIL3
6 EIL2
6 EIL2
12 EIL3
12 EIL3
3 EIL2
3 EIL3
3 EIL3
3 EIL2
3 EIL3
3 EIL3
4 EIL
2 EIL
6 EI
L2
6 EI
L2
4 EIL
2 EIL
6 EI
L2
6 EI
L2
3 EIL 3 EI
L2
3 EI
L2
3 EIL
3 EI
L2
3 EI
L2
6EI sinL2
EA cos2 L
12EI sin2L3
+
EA cos sin
L-12EI sincos
L3
6EI sinL2
6EI sinL2
EA cos2 L
12EI sin2L3
+
EA cos sin
L-12EI sincos
L3
Horizontal Deformation
6EI cosL2
EA sin2 L
12EI cos2L3+
EA sincos
L-12EI cossin
L3
6EI cosL2
EA sincos
L-12EI cossin
L3
EA sin2 L
12EI cos2L3+
6EI cosL2
-
Vertical Deformation
4 EIL
6 EI sin L2
6 EI cos L2
2 EIL
-
6 EI sin L2
6 EI cos L2
4 EIL
Rotational Deformation
( )cos + ( )sin
bxax
- byay
-AEL
N =
a
b
bx
by
ax
ay
[ ]
Internal forces
Normal Force
Internal forces
Bending moment
MAB= ( 2 + ) M(FER) AB +2 EI
L
3 L
-
MBA=2 EI
L( + 2 )
M(FER) BA +3 L
-
Questions
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