theory of structures chapter 5 : three pin...
TRANSCRIPT
by Saffuan Wan Ahmad
THEORY OF STRUCTURESCHAPTER 5 : THREE PIN
ARCH
bySaffuan Wan Ahmad
Faculty of Civil Engineering & Earth [email protected]
For updated version, please click on http://ocw.ump.edu.my
by Saffuan Wan Ahmad
Chapter 5 – Three Pin Arch
• Aims– Determine internal forces, shear forces and bending moments in arch member
• Expected Outcomes :– Able to explain the function of arch– Able to describe the function of arch– Able to determine the reaction at support for three arch structure – Able to determine the internal forces at any point at arch structure – Able to draw shear force, axial force and bending moment diagram
• References– Mechanics of Materials, R.C. Hibbeler, 7th Edition, Prentice Hall – Structural Analysis, Hibbeler, 7th Edition, Prentice Hall– Structural Analysis, SI Edition by Aslam Kassimali,Cengage Learning– Structural Analysis, Coates, Coatie and Kong – Structural Analysis - A Classical and Matrix Approach, Jack C. McCormac and James K.
Nelson, Jr., 4th Edition, John Wiley
by Saffuan Wan Ahmad
hinge
Springingline
crown
5.1 WHAT IS ARCH??
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5.1 WHAT IS ARCH??
by Saffuan Wan Ahmad
• Arches have been used for a very long time to spanlarge distance i.e bridges/building to carrytransverse loading efficiently.
• Arch carries most of the load axially with bendingmoment greatly reduced due to the curvature ofthe arch
5.2 FUNCTIONS OF ARCH
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(a) Fixed arch(a) Two-hinge arch
(a) Three-hinge arch(a) Tied arch
#Due to support system
5.3 TYPES OF ARCH
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h= height of the arch
B (L,0)A (0,0)
P (x,y)C (L/2,h)
x
y
L
5.4 EQUATION OF PARABOLIC ARCH
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• Shear force must be parallel to the cross section surface, whilst the axial force must be perpendicular to the shear force. The positive were shown in figure below.
The axial force, N shear force, Q and bending moment, M in the arch rib
5.5 FORCES IN ARCH
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yh
x
L
yc=h
L/2 L/2
y= kx ( L- x )
When x = L/2 , y = yc sub into enq.
yc=k (L/2) ( L- L/2 ) = kL2/4
Therefore k= 4yc/L2 = 4h/L2 then sub into eqn.
y= (4h/L2 )x ( L- x )
y= (4yc/L2 ) ( Lx- x2 )
y= (4h/L2 )( Lx- x2) simplify
…………Parabolic equation
by Saffuan Wan Ahmad
slopetan Where,
)2(4
84
)4(24Slope
44
44
)(4but
Slope
1-
2
2
2
2
2
2
2
2
2
xLL
hLhx
Lh
Lxh
Lh
dxdy
Lhx
Lhx
Lhx
LhxL
xLLhxy
dxdy
At any point of the arch (parabolic)
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EXAMPLE 1
• Calculate the reaction at support A and B as shown in figure below.
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Solution Example 1…
Draw FBD
kNVV
F
kNVV
M
A
A
y
B
B
A
5.31205.107150140130
0
5.1070)40()15(150)10(140)5(130
0structure wholeheConsider t
kNV
kNV
A
B
5.312
5.107
by Saffuan Wan Ahmad
kNHHence
kNHH
M
A
B
B
C
75.268,
75.2680)8()20(5.107
0
Now, consider segment CB
Solution Example 1…
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EXAMPLE 2
• Determine the internal forces at the points D and E in the three hinge parabolic arch as shown in figure below.
A B
CD E
2.5 m 2.5 m 3 m 2 m
2.5 m
16 kN/m
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Apply equation of equilibrium, consider the whole structure then taking moment about A and B.
Consider RHS or LHS, take moment about C is equal to zero
kNVkNV BA 20,60
kNH 40
Solution Example 2…
by Saffuan Wan Ahmad
• Consider segment CD
5.0
))5.2(210(10
)5.2(4
)2(4875.1
)5.210(10
)5.2)(5.2(4
)(4
2
2
2
2
xLLhSlope
m
xLLhxyD
kNQQ
o
003.00)57.26cos(40)57.26cos(60)57.26sin(40
direction Qin Resolving57.26
)5.0(tan 1
by Saffuan Wan Ahmad
• Consider segment CD
5.0
))5.2(210(10
)5.2(4
)2(4875.1
)5.210(10
)5.2)(5.2(4
)(4
2
2
2
2
xLLhSlope
m
xLLhxyD
kNNN
o
72.440)57.26sin(40)57.26sin(60)57.26cos(40
direction Nin Resolving57.26
)5.0(tan 1
by Saffuan Wan Ahmad
• Consider segment CD
5.0
))5.2(210(10
)5.2(4
)2(4875.1
)5.210(10
)5.2)(5.2(4
)(4
2
2
2
2
xLLhSlope
m
xLLhxyD
kNmM
M
D
D
o
25
0)875.1(40)5.2(60)25.2(40
0Mmoment Bending
57.26)5.0(tan
D
1
by Saffuan Wan Ahmad
EXAMPLE 3
• Determine the bending moment at 25 m fromthe right hand support B and axial and shearforce at point D and E.
30 m
15 m
A B
C
10 kN/m
30 m
8 kN
5 kN
10 m 10 m
D E
VB
HB
VA
HA
20 m
by Saffuan Wan Ahmad
• Calculate YE
m
yE
33.8
)1060(60
)10)(15(42
kNVV
F
kNVV
M
A
A
y
B
B
A
36.1070)20(10864.100
0
64.1000)33.8(5)2010)(20(10)10(8)60(
0
• Consider whole structure
Solution Example 3…
by Saffuan Wan Ahmad
• Consider RHS of the arch, taking moment at C is equal to zero.
10 m 10 m 10 m
15 m8.33 m
10 kN/m
5 kN
100.64 kN
HB
kNHH
F
kNHH
M
A
A
x
B
B
C
72.1700572.165
0
72.1650)15()33.815(5)5)(10(10)30(64.100
0
by Saffuan Wan Ahmad
• Calculate the bending moment at 25 m from support , say point F
m
yF
58.14
)2560(60
)25)(15(42
kNmM
MM
F
F
F
45.560)5.2)(5(10)33.858.14(5
)58.14(72.165)25(64.1000
by Saffuan Wan Ahmad
• At point D (with point load)
kNN
N
kNQ
Q
16.1970)67.33sin(8
)67.33sin(36.107)67.33cos(72.170direction Nin Resolving
03.120)67.33cos(8
)67.33cos(36.107)67.33sin(72.170direction Qin Resolving
by Saffuan Wan Ahmad
• At point F (with point load)
kNN
N
kNQ
Q
87.1970)67.33cos(5
)67.33cos(72.165)67.33sin(64.100direction Nin Resolving
96.100)67.33sin(5
)67.33sin(72.165)67.33cos(64.100direction Qin Resolving
by Saffuan Wan Ahmad
EXAMPLE 4
Determine the reactions at supports and bending momentunder the load :
by Saffuan Wan Ahmad
mymr
r
rr
ll
A 1241641640
20
1
1
2
1
2
1
my
y
xLLhxy
A
A
A
12
)1040(40
)10)(16(4
)(4
2
2
Or…
Solution Example 4…
by Saffuan Wan Ahmad
• Consider the whole structure. Taking moment at A is equal zero
)(.........................240012300)12()30()20(100)5(80
0
iHVHV
M
BB
BB
A
• Consider RHS, taking moment at C is equal to zero
)...(....................100016200)16()20()10(100
0
iiHVHV
M
BB
BB
C
by Saffuan Wan Ahmad
• Resolving by using calculator
kNHkNV
B
B
75110
kNHHF
kNVV
F
BA
x
A
A
y
750
70010080110
0
• Apply static equation
by Saffuan Wan Ahmad
• Consider LHS, taking moment under load 80kN
mym
xLLhxy
3121515
)1540(40
)15)(16(4
)(4
'80
2
280
kNmMMM
1250)3(75)5(70
0
80
80
80
by Saffuan Wan Ahmad
• Consider RHS, taking moment under load 100kN
m
xLLhxy
12
)1040(40
)10)(16(4
)(4
2
2100
kNmMM
M
2000)10(110)12(75
0
100
100
100
by Saffuan Wan Ahmad
THANKS
by Saffuan Wan Ahmad
Author Information
Mohd Arif Bin SulaimanMohd Faizal Bin Md. Jaafar
Mohammad Amirulkhairi Bin ZubirRokiah Binti Othman
Norhaiza Binti GhazaliShariza Binti Mat Aris