100
CHAPTER 3
A N A LY S I S O F S TAT I C A L LY D E T E R M I N AT E T R U S S E S
FUNDAMENTAL PROBLEMS
F3–1. Determine the force in each member of the truss and
state whether it is in tension or compression.
C
F3–4. Determine the force in each member of the truss and
state whether it is in tension or compression.
40 kN
2k
C
D
3 3m
8 ft
B
A
4m
F3–1
F3–2. Determine the force in each member of the truss and
state whether it is in tension or compression.
B
A
C
D
6 ft
F3–4
F3–5. Determine the force in each member of the truss and
state whether it is in tension or compression.
2m
D
A
C
B
60
2m
2m
6 kN
8 kN
F3–2
F3–3. Determine the force in each member of the truss and
state whether it is in tension or compression.
B
A
10 kN
C
F3–5
2m
F3–6. Determine the force in each member of the truss and
state whether it is in tension or compression.
D
H
3m
G
F
2m
E
A
A
B
3m
2m
F3–3
B
600 N
2m
C
800 N
2m
D
600 N
2m
F3–6
3.4
101
ZERO-FORCE MEMBERS
PROBLEMS
3–5. A sign is subjected to a wind loading that exerts
horizontal forces of 300 lb on joints B and C of one of
the side supporting trusses. Determine the force in each
member of the truss and state if the members are in tension
or compression.
C
3–7. Determine the force in each member of the truss.
State whether the members are in tension or compression.
Set P = 8 kN.
*3–8. If the maximum force that any member can support
is 8 kN in tension and 6 kN in compression, determine the
maximum force P that can be supported at joint D.
300 lb
4m
12 ft
B
13 ft
C
D
5 ft
300 lb
B
60
12 ft
60
E
A
D
13 ft
45
E
A
4m
4m
P
Prob. 3–5
Probs. 3–7/3–8
3–6. Determine the force in each member of the truss.
Indicate if the members are in tension or compression.
Assume all members are pin connected.
3–9. Determine the force in each member of the truss.
State if the members are in tension or compression.
1.5 k
F
2k
G
2k
4k
4k
8 ft
F
E
B
C
H
E
4 ft
A
B
9 ft
2k
D
D
C
A
30
8 ft
8 ft
Prob. 3–6
8 ft
12 ft
12 ft
Prob. 3–9
12 ft
3
102
CHAPTER 3
A N A LY S I S O F S TAT I C A L LY D E T E R M I N AT E T R U S S E S
3–10. Determine the force in each member of the truss.
State if the members are in tension or compression.
8 kN
3k
H
2k
3
*3–12. Determine the force in each member of the truss.
State if the members are in tension or compression. Assume
all members are pin connected. AG = GF = FE = ED.
8 kN
3k
12 ft
F
2m
4 kN
8 kN
F
G
4 kN
E
G
A
A
E
B
10 ft
C
10 ft
10 ft
10 ft
D
C
B
D
4m
4m
Prob. 3–10
Prob. 3–12
3–11. Determine the force in each member of the truss.
State if the members are in tension or compression. Assume
all members are pin connected.
3–13. Determine the force in each member of the truss and
state if the members are in tension or compression.
G
4 kN
F
3m
3m
3m
C
B
E
3m
D
3m
5m
A
F
A
B
2m
C
2m
5 kN
Prob. 3–11
D
E
2m
5 kN
5 kN
5 kN
Prob. 3–13
3.4
3–14. Determine the force in each member of the roof russ.
State if the members are in tension or compression.
103
ZERO-FORCE MEMBERS
*3–16. Determine the force in each member of the truss.
State if the members are in tension or compression.
F
E
3
D
8 kN
4 kN
4 kN
4 kN
4 kN
4 kN
I
J
4 kN
H
K
G
3.5 m
E
F
A
B
C
D
B
30
30
45
45
A
C
30
30
6 @ 4 m 24 m
2m
2m
Prob. 3–14
2 kN
Prob. 3–16
3–15. Determine the force in each member of the roof truss.
State if the members are in tension or compression. Assume
all members are pin connected.
3–17. Determine the force in each member of the roof truss.
State if the members are in tension or compression. Assume
B is a pin and C is a roller support.
G
F
3m
H
F
G
E
60
3m
E
A
4m
B
C
D
10 kN
10 kN
10 kN
4m
4m
Prob. 3–15
A
60 60
30
60
D
B
C
2m
4m
2m
2 kN
2 kN
Prob. 3–17
3.6
113
COMPOUND TRUSSES
FUNDAMENTAL PROBLEMS
F3–7. Determine the force in members HG, BG, and BC
and state whether they are in tension or compression.
2k
5 ft
H
A
400 lb
2k
2k
5 ft
F3–10. Determine the force in members GF, CF, and CD
and state whether they are in tension or compression.
5 ft
G
F
E
400 lb
G
400 lb
5 ft
H
400 lb
400 lb
F
5 ft
6 ft
A
B
C
3
6 ft
E
D
C
B
8 ft
F3–7
8 ft
D
8 ft
8 ft
F3–10
F3–8. Determine the force in members HG, HC, and BC
and state whether they are in tension or compression.
F3–11. Determine the force in members FE, FC, and BC
and state whether they are in tension or compression.
4 kN
600 lb
600 lb
600 lb
600 lb
600 lb
2 kN
2 kN
3m
H
I
J
F
G
3 ft
A
C
B
3m
G
F
E
1.5 m
E
D
B
4 ft
4 ft
4 ft
C
1.5 m
4 ft
A
F3–8
F3–9. Determine the force in members ED, BD, and BC
and state whether they are in tension or compression.
E
D
6 kN
D
1.5 m
3m
1.5 m
F3–11
F3–12. Determine the force in members GF, CF, and CD
and state whether they are in tension or compression.
G
H
F
1 ft
2m
E 3 ft
A
B
2m
C
B
C
A
4 ft
4 ft
D
4 ft
4 ft
2m
8 kN
F3–9
500 lb
500 lb
F3–12
500 lb
114
CHAPTER 3
A N A LY S I S O F S TAT I C A L LY D E T E R M I N AT E T R U S S E S
PROBLEMS
3–18. Determine the force in members GF, FC, and CD of
the bridge truss. State if the members are in tension or
compression. Assume all members are pin connected.
G
H
3–21. The Howe truss is subjected to the loading shown.
Determine the forces in members GF, CD, and GC. State if
the members are in tension or compression. Assume all
members are pin connected.
15 ft
F
3
30 ft
E
A
B
C
40 ft
D
40 ft
40 ft
15 k
5 kN
G
40 ft
5 kN
5 kN
10 k
H
Prob. 3–18
3–19. Determine the force in members JK, JN, and CD.
State if the members are in tension or compression. Identify
all the zero-force members.
M
E
B
2m
N
2m
2m
H
20 ft
G
F
A
D
30 ft
20 ft
D
Prob. 3–21
O
C
C
2m
I
L
B
2 kN
A
J
K
3m
F
2 kN
E
20 ft
2k
3–22. Determine the force in members BG, HG, and BC
of the truss and state if the members are in tension or
compression.
2k
Prob. 3–19
*3–20. Determine the force in members GF, FC, and CD
of the cantilever truss. State if the members are in tension or
compression. Assume all members are pin connected.
G
12 kN
12 kN
12 kN
E
F
3m
A
B
C
2m
Prob. 3–20
F
4.5 m
3m
G
2m
H
D
2m
A
E
B
C
D
6 kN
7 kN
4 kN
12 m, 4 @ 3 m
Prob. 3–22
3.6
3–23. Determine the force in members GF, CF, and CD of
the roof truss and indicate if the members are in tension or
compression.
115
COMPOUND TRUSSES
3–25. Determine the force in members IH, ID, and CD of
the truss. State if the members are in tension or compression.
Assume all members are pin connected.
3–26. Determine the force in members JI, IC, and CD of
the truss. State if the members are in tension or compression.
Assume all members are pin connected.
1.5 kN
3
C
10 m, 5 @ 2 m
1.70 m
2 kN
3 kN
0.8 m
D
B
3 kN
3 kN
3 kN
I
H
G
1.5 m
J
K
E
A
H
G
F
E
F
5m
1m
D
C
2m
2m
1.5 kN
B
Prob. 3–23
A
Probs. 3–25/3–26
*3–24. Determine the force in members GF, FB, and BC
of the Fink truss and state if the members are in tension or
compression.
3–27. Determine the forces in members KJ, CD, and CJ
of the truss. State if the members are in tension or
compression.
600 lb
800 lb
800 lb
F
60
30
10 ft
B
60
10 ft
Prob. 3–24
30
C
10 ft
30 kN 20 kN
10 kN
5 kN
L
A
E
G
A
15 kN 15 kN
K
J
B
D
C
I
E
D
6 @ 3 m 18 m
Prob. 3–27
H
F
5 kN
G
3@1m3m
164
CHAPTER 4
EXAMPLE
INTERNAL LOADINGS DEVELOPED IN STRUCTURAL MEMBERS
4.13
Draw the moment diagram for the tapered frame shown in Fig. 4–17a.
Assume the support at A is a roller and B is a pin.
5k
5k
15 ft
15 ft
C
3k
B
5 ft
5 ft
3k
3k
1k
6 ft
6 ft
4
A
(a)
6k
6k
5k
1k
15 k ft
15 k ft
C
3k
5 ft
3k
B
15 k ft
15 k ft
1k
1k
6k
A
M (kft)
15 ft
3k 3k
3k
3k
(b)
(c)
6k
Fig. 4–17
15
x (ft)
15
SOLUTION
member CB
Support Reactions. The support reactions are shown on the
free-body diagram of the entire frame, Fig. 4–17b. Using these results,
the frame is then sectioned into two members, and the internal reactions at the joint ends of the members are determined, Fig. 4–17c.
Note that the external 5-k load is shown only on the free-body diagram
of the joint at C.
x (ft)
11
6
15
member AC
(d)
M (kft)
Moment Diagram. In accordance with our positive sign convention,
and using the techniques discussed in Sec. 4–3, the moment diagrams
for the frame members are shown in Fig. 4–17d.
4.4
EXAMPLE
165
SHEAR AND MOMENT DIAGRAMS FOR A FRAME
4.14
Draw the shear and moment diagrams for the frame shown in
Fig. 4–18a. Assume A is a pin, C is a roller, and B is a fixed joint.
Neglect the thickness of the members.
SOLUTION
Notice that the distributed load acts over a length of
10 ft 22 = 14.14 ft. The reactions on the entire frame are calculated
and shown on its free-body diagram, Fig. 4–18b. From this diagram the
free-body diagrams of each member are drawn, Fig. 4–18c. The
distributed loading on BC has components along BC and perpendicular
to its axis of 10.1414 k>ft2 cos 45° = 10.1414 k>ft2 sin 45° = 0.1 k>ft
as shown. Using these results, the shear and moment diagrams are
also shown in Fig. 4–18c.
0.1414 k/ft
C
10 ft
B
10 ft
4
A
0.1 k/ft
10 ft
(a)
0.5 k
ft
4
.6
.1
V
62
5
(k
5 kft
ft
)
1.
06
0.
1.77 k
)
x
(f
t)
10
14
1.06 k
Fig. 4–18
0. x (
35 ft
4 )
k
0.1 k/ft
M
(k
1.77 k
1.06 k
5 kft
0.5 k
0.5 k
5
(0.1414 k/ft)(14.14 ft) 2 k
5 kft
x (ft)
2k
x (ft)
2k
–5
5 kft
20 ft
0.5 k
2k
M (kft)
0.5 k
0.5
V (k)
10 ft
0.5 k
5 ft
2k
(c)
(b)
166
CHAPTER 4
EXAMPLE
INTERNAL LOADINGS DEVELOPED IN STRUCTURAL MEMBERS
4.15
Draw the shear and moment diagrams for the frame shown in Fig. 4–19a.
Assume A is a pin, C is a roller, and B is a fixed joint.
80 kN
B
C
40 kN/m
2m
A
4
4m
3m
4m
(a)
80 kN
120 kN
Ax 120 kN
Cy 82.5 kN
36.87
1.5 m
6m
Ay 2.5 kN
2m
(b)
Fig. 4–19
SOLUTION
Support Reactions. The free-body diagram of the entire frame is
shown in Fig. 4–19b. Here the distributed load, which represents wind
loading, has been replaced by its resultant, and the reactions have been
computed. The frame is then sectioned at joint B and the internal
loadings at B are determined, Fig. 4–19c. As a check, equilibrium is
satisfied at joint B, which is also shown in the figure.
Shear and Moment Diagrams. The components of the distributed
load, 172 kN2>15 m2 = 14.4 kN>m and 196 kN2>15 m2 = 19.2 kN>m,
are shown on member AB, Fig. 4–19d. The associated shear and
moment diagrams are drawn for each member as shown in Figs. 4–19d
and 4–19e.
4.4
170 kN m
170 kN m 1.5 kN
B
36.87
B
2.5 kN
1.5 kN 2 kN
36.87 72 kN
96 kN
1.5 kN
170 kN m
2 kN
72 kN
167
SHEAR AND MOMENT DIAGRAMS FOR A FRAME
80 kN
96 kN
A
36.87
B
C
170 kN m
2 kN
2.5 kN
82.5 kN
(c)
4
170 kN m
1.5 kN
80 kN
B 2 kN
14.4 kN/m
170 kN m
5m
x (m)
82.5 kN
19.2 kN/ m
70 kN
2
A
97.5 kN
C
B
2.5 kN
4.86
170.1
V (kN)
V (kN)
170
70
2.5
2
x (m)
82.5
x (m)
M (kN m)
4.86
M (kN m)
170
165
(d)
x (m)
2
(e)
4.5
173
MOMENT DIAGRAMS CONSTRUCTED BY THE METHOD OF SUPERPOSITION
PROBLEMS
4–38. Draw the shear and moment diagrams for each of
the three members of the frame. Assume the frame is pin
connected at A, C, and D and there is a fixed joint at B.
50 kN
1.5 m
*4–40.
Draw the shear and moment diagrams for each
member of the frame. Assume A is a rocker, and D is
pinned.
40 kN
2m
1.5 m
B
4k
C
2 k/ ft
15 kN/ m
B
C
3k
8 ft
4m
4
4 ft
15 ft
6m
A
A
D
D
Prob. 4–40
Prob. 4–38
4–39. Draw the shear and moment diagrams for each
member of the frame. Assume the support at A is a pin and
D is a roller.
4–41. Draw the shear and moment diagrams for each
member of the frame. Assume the frame is pin connected at
B, C, and D and A is fixed.
0.8 k/ft
B
C
6k
3k
8 ft
6k
8 ft
3k
8 ft
0.8 k/ft
0.6 k/ft
16 ft
B
C
15 ft
A
D
D
A
20 ft
Prob. 4–39
Prob. 4–41
174
CHAPTER 4
INTERNAL LOADINGS DEVELOPED IN STRUCTURAL MEMBERS
4–42. Draw the shear and moment diagrams for each
member of the frame. Assume A is fixed, the joint at B is a
pin, and support C is a roller.
*4–44.
Draw the shear and moment diagrams for each
member of the frame. Assume the frame is roller supported
at A and pin supported at C.
1.5 k/ ft
20 k
5
3
4
B
A
0.5 k/ft B
C
10 ft
6 ft
8 ft
2k
4
6 ft
A
6 ft
C
6 ft
Prob. 4–42
Prob. 4–44
4–43. Draw the shear and moment diagrams for each
member of the frame. Assume the frame is pin connected at
A, and C is a roller.
4–45. Draw the shear and moment diagrams for each
member of the frame. The members are pin connected at A,
B, and C.
4 k/ft
B
15 kN
15 k
C
2m
10 kN
2m
2m
4 ft
A
10 k
B
4 ft
6m
A
10 ft
Prob. 4–43
Prob. 4–45
45
4 kN/m
C
4.5
4–46. Draw the shear and moment diagrams for each
member of the frame.
5 kN
10 kN
175
MOMENT DIAGRAMS CONSTRUCTED BY THE METHOD OF SUPERPOSITION
*4–48.
Draw the shear and moment diagrams for each
member of the frame. The joints at A, B, and C are pin
connected.
5 kN
250 lb/ft
B
B
2 kN/m
C
120 lb/ft
A
4m
6 ft
6 ft
8 ft
D
A
3m
2m
2m
4
60
C
3m
Prob. 4–46
Prob. 4–48
4–47. Draw the shear and moment diagrams for each
member of the frame. Assume the joint at A is a pin and
support C is a roller. The joint at B is fixed. The wind load is
transferred to the members at the girts and purlins from the
simply supported wall and roof segments.
4–49. Draw the shear and moment diagrams for each of
the three members of the frame. Assume the frame is pin
connected at B, C, and D and A is fixed.
300 lb/ ft
6k
C
3.5 ft
30
500 lb/ft
3.5 ft
B
8 ft
3k
8 ft
8 ft
0.8 k/ft
B
C
7 ft
15 ft
7 ft
A
D
A
Prob. 4–47
Prob. 4–49
176
CHAPTER 4
INTERNAL LOADINGS DEVELOPED IN STRUCTURAL MEMBERS
4–50. Draw the moment diagrams for the beam using the
method of superposition. The beam is cantilevered from A.
600 lb
600 lb
600 lb
A
4–54. Draw the moment diagrams for the beam using
the method of superposition. Consider the beam to be
cantilevered from the pin support at A.
4–55. Draw the moment diagrams for the beam using
the method of superposition. Consider the beam to be
cantilevered from the rocker at B.
3 ft
3 ft
30 kN
4 kN/m
1200 lbft
3 ft
80 kN m
Prob. 4–50
C
A
B
4
8m
4–51. Draw the moment diagrams for the beam using the
method of superposition.
80 lb/ft
12 ft
4m
Probs. 4–54/4–55
*4–56.
Draw the moment diagrams for the beam using
the method of superposition. Consider the beam to be
cantilevered from end C.
12 ft
30 kN
4 kN/m
80 kN m
600 lb
C
A
B
Prob. 4–51
8m
4m
Prob. 4–56
*4–52. Draw the moment diagrams for the beam using
the method of superposition. Consider the beam to be
cantilevered from end A.
4–53. Draw the moment diagrams for the beam using the
method of superposition. Consider the beam to be simply
supported at A and B as shown.
4–57. Draw the moment diagrams for the beam using the
method of superposition. Consider the beam to be simply
supported at A and B as shown.
250 lb/ft
150 lb ft
150 lbft
200 lb/ft
100 lbft
A
B
100 lbft
A
B
20 ft
20 ft
Probs. 4–52/4–53
Prob. 4–57
190
CHAPTER 5
CABLES AND ARCHES
PROBLEMS
5–1. Determine the tension in each segment of the cable
and the cable’s total length.
5–3. Determine the tension in each cable segment and the
distance yD.
A
yD
A
D
7m
4 ft
D
7 ft
B
2m
B
5
2 kN
C
C
50 lb
4 ft
5 ft
4m
3 ft
5m
3m
4 kN
100 lb
Prob. 5–1
Prob. 5–3
5–2. Cable ABCD supports the loading shown. Determine
the maximum tension in the cable and the sag of point B.
*5–4. The cable supports the loading shown. Determine the
distance xB the force at point B acts from A. Set P = 40 lb.
5–5. The cable supports the loading shown. Determine the
magnitude of the horizontal force P so that xB = 6 ft.
xB
A
5 ft
D
A
yB
2m
B
8 ft
C
C
2 ft
B
1m
3m
4 kN
3
0.5 m
6 kN
Prob. 5–2
D
5
4
3 ft
30 lb
Probs. 5–4/5–5
P
5.3
5–6. Determine the forces P1 and P2 needed to hold the
cable in the position shown, i.e., so segment CD remains
horizontal. Also find the maximum loading in the cable.
191
CABLE SUBJECTED TO A UNIFORM DISTRIBUTED LOAD
*5–8. The cable supports the uniform load of
w0 = 600 lb>ft. Determine the tension in the cable at each
support A and B.
B
A
1.5 m
E
A
B
1m
C
D
15 ft
10 ft
5 kN
P1
4m
2m
P2
5m
w0
4m
5
25 ft
Prob. 5–6
Prob. 5–8
5–7. The cable is subjected to the uniform loading. If the
slope of the cable at point O is zero, determine the equation
of the curve and the force in the cable at O and B.
5–9. Determine the maximum and minimum tension in the
cable.
y
y
10 m
A
10 m
B
B
A
8 ft
O
2m
x
x
500 lb/ ft
15 ft
15 ft
Prob. 5–7
16 kN/m
Prob. 5–9
192
CHAPTER 5
CABLES AND ARCHES
5–10. Determine the maximum uniform loading w,
measured in lb>ft, that the cable can support if it is capable
of sustaining a maximum tension of 3000 lb before it will
break.
5–13. The trusses are pin connected and suspended from
the parabolic cable. Determine the maximum force in the
cable when the structure is subjected to the loading shown.
50 ft
6 ft
D
E
14 ft
6 ft
w
K
J
I
16 ft
A
Prob. 5–10
C
F
G H B
5k
4 @ 12 ft 48 ft
5–11. The cable is subjected to a uniform loading of
w = 250 lb >ft. Determine the maximum and minimum
5
tension in the cable.
4k
4 @ 12 ft 48 ft
Prob. 5–13
50 ft
5–14. Determine the maximum and minimum tension in
the parabolic cable and the force in each of the hangers. The
girder is subjected to the uniform load and is pin connected
at B.
6 ft
w
Prob. 5–11
5–15. Draw the shear and moment diagrams for the pinconnected girders AB and BC. The cable has a parabolic
shape.
*5–12. The cable shown is subjected to the uniform load w0.
Determine the ratio between the rise h and the span L that
will result in using the minimum amount of material for the
cable.
E
9 ft
1 ft
L
D
h
2 k/ ft
10 ft
C
A
w0
Prob. 5–12
10 ft
B
30 ft
Probs. 5–14/5–15
5.3
*5–16. The cable will break when the maximum tension
reaches T max = 5000 kN. Determine the maximum uniform distributed load w required to develop this maximum
tension.
193
CABLE SUBJECTED TO A UNIFORM DISTRIBUTED LOAD
5–19. The beams AB and BC are supported by the cable
that has a parabolic shape. Determine the tension in the cable
at points D, F, and E, and the force in each of the equally
spaced hangers.
5–17. The cable is subjected to a uniform loading of
w = 60 kN/m. Determine the maximum and minimum
tension in cable.
E
D
3m
3m
F
100 m
9m
12 m
A
C
B
5 kN
3 kN
w
5
2m 2m 2m 2m 2m 2m 2m 2m
Probs. 5–16/5–17
Prob. 5–19
5–18. The cable AB is subjected to a uniform loading of
200 N>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.
*5–20. Draw the shear and moment diagrams for beams
AB and BC. The cable has a parabolic shape.
y
B
60
E
D
3m
3m
F
9m
A
30
x
A
C
B
200 N/ m
15 m
Prob. 5–18
3 kN
5 kN
2m 2m 2m 2m 2m 2m 2m 2m
Prob. 5–20
196
CHAPTER 5
CABLES AND ARCHES
EXAMPLE 5.4
The three-hinged open-spandrel arch bridge like the one shown in the
photo has a parabolic shape. If this arch were to support a uniform
load and have the dimensions shown in Fig. 5–10a, show that the arch
is subjected only to axial compression at any intermediate point such
as point D. Assume the load is uniformly transmitted to the arch ribs.
y
500 lb/ft
x
B
D
y
A
25
x2
(50)2
25 ft
C
50 ft
25 ft
25 ft
5
(a)
Fig. 5–10
SOLUTION
Here the supports are at the same elevation. The free-body diagrams
of the entire arch and part BC are shown in Fig. 5–10b and Fig. 5–10c.
Applying the equations of equilibrium, we have:
50 k
B
Ax
Cx
50 ft
50 ft
Cy
Ay
(b)
Entire arch:
d+ ©MA = 0;
Cy1100 ft2 - 50 k150 ft2 = 0
Cy = 25 k
5.5
197
THREE-HINGED ARCH
Arch segment BC:
d+ ©MB = 0;
25 k
-25 k125 ft2 + 25 k150 ft2 - Cx125 ft2 = 0
Cx = 25 k
Bx
B
+ ©F = 0;
:
x
Bx = 25 k
25 ft
By
Cx
+ c ©Fy = 0;
By - 25 k + 25 k = 0
25 ft
25 ft
Cy
By = 0
(c)
A section of the arch taken through point D, x = 25 ft,
y = -2512522>15022 = -6.25 ft, is shown in Fig. 5–10d. The slope of
the segment at D is
5
tan u =
dy
-50
=
x`
= -0.5
dx
15022 x = 25 ft
u = -26.6°
12.5 k
Applying the equations of equilibrium, Fig. 5–10d we have
+ ©F = 0;
:
x
25 k - ND cos 26.6° - VD sin 26.6° = 0
+ c ©Fy = 0;
-12.5 k + ND sin 26.6° - VD cos 26.6° = 0
d+ ©MD = 0;
MD + 12.5 k112.5 ft2 - 25 k16.25 ft2 = 0
25 k
6.25 ft B
MD
D
26.6
26.6
ND
VD
ND = 28.0 k
Ans.
VD = 0
Ans.
MD = 0
Ans.
Note: If the arch had a different shape or if the load were nonuniform, then the internal
shear and moment would be nonzero. Also, if a simply supported beam were used to
support the distributed loading, it would have to resist a maximum bending moment of
M = 625 k # ft. By comparison, it is more efficient to structurally resist the load in direct
compression (although one must consider the possibility of buckling) than to resist the
load by a bending moment.
12.5 ft 12.5 ft
(d)
198
CHAPTER 5
CABLES AND ARCHES
EXAMPLE 5.5
The three-hinged tied arch is subjected to the loading shown in
Fig. 5–11a. Determine the force in members CH and CB. The dashed
member GF of the truss is intended to carry no force.
20 kN
15 kN
H
G
20 kN
15 kN
15 kN
15 kN
F
1m
1m
C
B
D
4m
A
E
5
3m
3m
3m
Ax
E
A
3m
3m
3m
3m
3m
Ay
Ey
(b)
(a)
Fig. 5–11
SOLUTION
The support reactions can be obtained from a free-body diagram of
the entire arch, Fig. 5–11b:
20 kN
15 kN
d+ ©MA = 0;
Ey112 m2 - 15 kN13 m2 - 20 kN16 m2 - 15 kN19 m2 = 0
0
C
Cy
5m
FAE
3m
3m
Cx
+ ©F = 0;
:
x
+ c ©Fy = 0;
Ey = 25 kN
Ax = 0
Ay - 15 kN - 20 kN - 15 kN + 25 kN = 0
Ay = 25 kN
The force components acting at joint C can be determined by considering the free-body diagram of the left part of the arch, Fig. 5–11c.
First, we determine the force:
25 kN
d+ ©MC = 0;
(c)
FAE15 m2 - 25 kN16 m2 + 15 kN13 m2 = 0
FAE = 21.0 kN
5.5
201
THREE-HINGED ARCH
PROBLEMS
5–21. The tied three-hinged arch is subjected to the
loading shown. Determine the components of reaction at
A and C and the tension in the cable.
5–23. The three-hinged spandrel arch is subjected to the
loading shown. Determine the internal moment in the arch
at point D.
8 kN 8 kN
15 kN
6 kN 6 kN
3 kN
3 kN
2m 2m 2m
4 kN
4 kN
2m 2m 2m
10 kN
B
2m
B
A
D
A
C
2m
2m
5m
3m
C
5
1m
3m
0.5 m
5m
8m
Prob. 5–21
Prob. 5–23
5–22. Determine the resultant forces at the pins A, B, and
C of the three-hinged arched roof truss.
*5–24. The tied three-hinged arch is subjected to the
loading shown. Determine the components of reaction
A and C, and the tension in the rod.
4 kN
3 kN
2 kN
4 kN
5k
3k
4k
5 kN
B
B
15 ft
5m
A
C
3m
3m
1m1m
2m
Prob. 5–22
3m
2m
C
A
6 ft
6 ft
8 ft
10 ft
Prob. 5–24
10 ft
202
CHAPTER 5
CABLES AND ARCHES
5–25. The bridge is constructed as a three-hinged trussed
arch. Determine the horizontal and vertical components of
reaction at the hinges (pins) at A, B, and C. The dashed
member DE is intended to carry no force.
*5–28. The three-hinged spandrel arch is subjected to the
uniform load of 20 kN兾m. Determine the internal moment
in the arch at point D.
5–26. Determine the design heights h1, h2, and h3 of the
bottom cord of the truss so the three-hinged trussed arch
responds as a funicular arch.
20 kN/m
60 k
40 k 40 k
20 k 20 k
D 10 ft E
B
B
100 ft
h2
h1
3m
A
A
5
5m
D
h3
C
C
3m
30 ft 30 ft 30 ft 30 ft 30 ft 30 ft 30 ft 30 ft
5m
8m
Probs. 5–25/5–26
Prob. 5–28
5–27. Determine the horizontal and vertical components
of reaction at A, B, and C of the three-hinged arch. Assume
A, B, and C are pin connected.
5–29. The arch structure is subjected to the loading
shown. Determine the horizontal and vertical components
of reaction at A and D, and the tension in the rod AD.
4k
B
B
2 ft
3k
5 ft
A
8 ft
3 ft
2 k/ft
3k
3 ft
A
D
C
4 ft
7 ft
10 ft
Prob. 5–27
5 ft
E
C
8 ft
4 ft
Prob. 5–29
4 ft
6 ft