structure analysis isite.iugaza.edu.ps/marafa/files/chapter-5-2019.pdf · sfd bmd. example 6: the...
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![Page 1: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/1.jpg)
Structure Analysis IChapter 5
Structure Analysis IChapter 5
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Cables & Arches
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Cables
• Cables are often used in engineering structure to
support and or transmit loads from one member to
another
Structural Analysis IDr. Mohammed Arafa
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Cables subjected to concentrated
loads
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Example 1Determine the tension in cables and what is the dimension h ?
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kNT
TT
M
CD
CDCD
A
79.6
0)4(8)2(35.52
0
54
53
o
BC
BC
BCBCy
BCBCx
T
TF
TF
3.32
82.4
0sin8)(79.60
0cos)(79.60
54
53
![Page 7: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/7.jpg)
o
BA
BA
BABA
o
y
BABA
o
x
T
TF
TF
8.53
90.6
0sin3)3.32(sin82.40
0cos)3.32(cos82.40
mh 74.28.53tan2
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Cables subjected to a Uniform
Distributed Load
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Cables subjected to a Uniform
Distributed Load
![Page 10: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/10.jpg)
Analysis Procedure
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0
0
0 cos cos 0
0 sin sin 0
0 cos sin 02
Dividing each eq. by Δx and taking the limit as Δx 0
and hence y 0, 0 and T 0
x
y
o
F T T T
F T w x T T
xM w x T y T x
Analysis Procedure
Structural Analysis IDr. Mohammed Arafa
![Page 12: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/12.jpg)
0
cos0 (1)
sin(2)
tan (3)
d T
dx
d Tw
dx
dy
dx
Analysis Procedure
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H
0
0
at x=0 T=F
Integrate eqs. 1 where T=F at x=0
cos (4)
Integrate eqs. 2 where Tsin =0 at x=0
sin (5)
dividing 4 by 5
tan =
H
H
H
T F
T w x
w xdy
dx F
Analysis Procedure
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2
0 0
2
0
2
2
2
,2
H H
H
w x w xdyy
dx F F
w Lat x L y h F
h
hy x
L
Analysis Procedure
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max
2 2
max
0
222
max 0 0
cos
cos
is at where is maximum
( )
In our Case at x=L
12
H
H
H
H
T F
FT
T
at x L
T F V
V w L
LT F w L w L
h
Analysis Procedure
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2
2
2
0
2 2
22 2 2
max max 0
2
cos
H
HH
H H
hy x
L
w LF
h
FT F V
T F V F w L
Summary
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Suspension Bridge
![Page 18: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/18.jpg)
Cable Stayed Bridge
![Page 19: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/19.jpg)
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Example 2Determine the tension of the cable at points A, B, C
Assume the girder weight is 850 lb/ft
Structural Analysis IDr. Mohammed Arafa
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2 2 2 2
max
500 307500 Ib=7.5k
2 2
7.031 7.5 10.280
A B
A B
WLV V
T T T H V
Example 3Determine the tension of the cable at points A, B
2
500 157031.25
2 8HF
2
0
2H
w LF
h
HF
AVBV
HF
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Example 4
![Page 27: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/27.jpg)
![Page 28: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/28.jpg)
![Page 29: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/29.jpg)
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The beams AB and BC are supported by the cable that
has a parabolic shape. Determine the tension in the cable
at points D and E.
Example 5
![Page 31: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/31.jpg)
Referring to the FBD of member BC, Fig. b,
Then from the FBD of member AB, Fig. a,
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![Page 33: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/33.jpg)
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Draw the shear and moment diagrams for members
AB and BC.
Example 5 -continue
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100
35.035.020.0
-5.0-5.0
SFD
BMD
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Example 6: The cable AB is subjected to a uniform loading of
200kN/m. If the weight of the cable is neglected and the slope angles at
points A and B are 30 and 60, respectively, determine the curve that
defines the cable shape and the maximum tension developed in the
cable.
![Page 37: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/37.jpg)
0 0 1
Integrate the equations
(1)
(2)
cos0 cos
sinsin
H
d TT F
dx
d Tw T w x C
dx
1
0
Substitute at 0
sin 30 tan 3
30cos cos30
eq.(2) will be:
sin tan
0
(3)30
A
H H
H
A H
F FT T
T
x
C T
F
F
w x
![Page 38: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/38.jpg)
0
2
tan 30tan
10.2 tan 30
60
1tan 60 0.2 15 tan 30
Divide
2.6
1 10.2 tan 30 0.2 2.
(3) by (1)
Substitute
6 0.5772.60
0
at 15
(Equation of thecablecurv.0385 0.577 e)
H
H
H
H
H
H
H
H
H
w x F
F
dyx F
dx F
FF
F kN
dyx F x
dx F
x x
x
y
![Page 39: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/39.jpg)
max
cos
2.6
cos cos
At 0 30
2.63.0
cos3
Substitute in (1)
0
At 15 60
2.65.2
cos 60
H
H
A
B
T F
FT
x
T kN
x
T kN T
![Page 40: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/40.jpg)
When the weight of a cable becomes important in the force analysis, the
loading function along the cable will be a function of the arc length s rather
than the projected length x.
Cable Subjected to its own Weight
w = w(s)
![Page 41: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/41.jpg)
To perform a direct integration, it is necessary to replace dy/dx by ds/dx.
one obtains the following relationships
2 2
2 2
1 1
sd dx dy
dy ds ds dy
dx dx dx dx
![Page 42: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/42.jpg)
Separating the variables and integrating we obtain
The two constants of integration, say C1 and C2, are found using
the boundary conditions for the curve.
2
1/22
2
1
11
H
ds dy
dx dx
dsw s ds
dx F
1/2
2
2
11
H
dsx
w s dsF
![Page 43: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/43.jpg)
continued on next slide
Example 7
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![Page 45: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/45.jpg)
![Page 46: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/46.jpg)
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![Page 49: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/49.jpg)
Arches
![Page 50: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/50.jpg)
Arches Types
![Page 51: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/51.jpg)
![Page 52: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/52.jpg)
Example of Fixed Arch
![Page 53: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/53.jpg)
Three Hinge Arches
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Example 5
Determine the internal forces at Section D
kNA
kNA
F
kNB
BM
kNB
BM
y
x
x
x
xRightC
y
yA
93
0F
86
0
86
0)20(67)8(60)6(0
67
0)28(60)10(100)40( 0
y
![Page 55: Structure Analysis Isite.iugaza.edu.ps/marafa/files/Chapter-5-2019.pdf · SFD BMD. Example 6: The cable AB is subjected to a uniform loading of 200kN/m. If the weight of the cable](https://reader030.vdocuments.net/reader030/viewer/2022040115/5e6aa9ccc6b04f61c443246e/html5/thumbnails/55.jpg)
86
93
86 cos20
93 sin20
86 sin20
93 cos20
242
242
112.6
58
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Example 6
Determine the internal forces at Section D
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y
0 25
F 0
25
0 25 (50) (25) 25 (25) 0
25
0
25
A y
y
B xRight
x
x
x
M C kips
A kips
M C
C kips
F
A kips
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yF 0
0
0
25
y
x
x
B
F
B kips
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2
2
2
1
2
25
25
50
252
50
25tan 2 25 0.5
50
26.6
x
o
y x
dyx
dx
dy
dx
25cos 12.5sin 27.95
25 12.5cos 0.0
25(6.25) 12.5(12.5) 0
D
D
D
N
V sin
M
26.6o
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Example 7
Structural Analysis IDr. Mohammed Arafa
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Problem 3Determine the internal forces at B