2. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(2)net.hs-eng,com.hs-eng Analysis of
Structures Considering: n = number of parts in the structure. r =
number of reactions. Stability: Structure will be unstable in the
following cases: 1) if r < 3 n 2) if r 3 n but all reactions are
parallel. 3) if r 3 n but all reactions are concurrent. Determinacy
(provided of stability): r = 3 n statically determinate. r > 3 n
statically indeterminate. Degree of indeterminacy = r - 3 n .
3. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(3)net.hs-eng,com.hs-eng Examples of
Unstability partial constraints concurrent reactions parallel
reactions .
4. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(4)net.hs-eng,com.hs-eng Examples of
Determinacy .
5. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(5)net.hs-eng,com.hs-eng Compound beams,
which are composed of pin-connected members could be disassembled.
Applying r = 3n or r > 3n, the resulting classifications are
indicated. . Example 2-3 Classify each of the following beams as
statically determinate or statically indeterminate. If statically
indeterminate, report the number of degrees of indeterminacy. The
beams are subjected to external loadings that are assumed to be
known and can act anywhere on the beams.
6. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(6)net.hs-eng,com.hs-eng Applying r = 3n or
r > 3n, the resulting classifications are indicated. . Example
2-4 Classify each of the pin-connected structures shown in Fig.
2-19a through 2-19d as statically determinate or statically
indeterminate. If statically indeterminate, report the number of
degrees of indeterminacy. The structures are subjected to arbitrary
external loadings that are assumed to be known and can act anywhere
on the structures. r = 7, n = 2, 7 > 6 statically indeterminate
to the first degree. r = 9, n = 3, 9 = 9 statically determinate r =
10, n = 2, 10 > 6 statically indeterminate to the fourth degree.
r = 9, n = 3, 9 = 9 statically determinate
7. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(7)net.hs-eng,com.hs-eng If the members
form closed loop, its necessary to use the method of sections and
"cut" the loop apart. Example 2-5 Classify each of the following
frames as statically determinate or statically indeterminate. If
statically indeterminate, report the number of degrees of
indeterminacy. The frames are subjected to external loadings that
are assumed to be known and can act anywhere on the frames. . r =
9, n = 2 9 > 6 Statically indeterminate to the third degree r =
9, n = 1 9 > 3 Statically indeterminate to the sixth degree This
frame has no closed loops.
8. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(8)net.hs-eng,com.hs-eng The member is
stable since the reactions are non-concurrent and non- parallel.
Its also statically determinate. The compound beam is stable. Its
also indeterminate to the second degree. The member is unstable
since the three reactions are concurrent at B. (b) ! Example 2-6
Classify each of the following structures as stable or unstable.
The structures are subjected to arbitrary external loads that are
assumed to be known.
9. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(9)net.hs-eng,com.hs-eng The beam is
unstable since the three reactions are all parallel. The structure
is unstable since r = 7, n = 3, so that r < 3n, 7 < 9. Also,
this can be seen by inspection, since AB can move horizontally
without restraint. 2-6 (d) . A A
10. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(11)net.hs-eng,com.hs-eng Solution: a) b)
Statically determinate. c) Statically indeterminate to the first
degree. d) Statically indeterminate to the first degree. 2-9
Classify each of the structures as statically determinate,
statically indeterminate, stable, or unstable. If indeterminate,
specify the degree of indeterminacy. .
11. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(11)net.hs-eng,com.hs-eng Solution: ( a) (
) ( ) Statically determinate. (b) ( ) ( ) Indeterminate to ( ) 3.
(c) ( ) ( ) Statically determinate. (d) Lower beam is unstable. .
2-10 Classify each of the structures as statically determinate,
statically indeterminate, or unstable. If indeterminate, specify
the degree of indeterminacy. (Fig. a, b ,c ,d)
12. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(12)net.hs-eng,com.hs-eng Solution: a) n =
1 r = 4 r > 3n Statically indeterminate to ( ) 1o . b) Unstable
since all reactions are parallel. c) n = 1 r = 6 r > 3n
Statically indeterminate to ( ) 3o . d) Unstable since all
reactions are parallel. 2-11. Classify each of the structures as
statically determinate, statically indeterminate, or unstable. If
indeterminate, specify the degree of indeterminacy. .
13. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(13)net.hs-eng,com.hs-eng Solution: (a) ( )
Statically indeterminate to ( ) . (b) ( ) Statically indeterminate
to ( ) . (c) ( ) Statically indeterminate to ( ) . (d) Unstable
since all reactions are parallel. . 2-12 Classify each of the
structures as statically determinate, statically indeterminate, or
unstable. If indeterminate, specify the degree of indeterminacy.
The supports or connections are to be assumed as stated. (Fig. a, b
,c ,d)
14. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(14)net.hs-eng,com.hs-eng Solution: a) The
frame has no closed loops. Statically indeterminate to 6o . b)
Cutting the loops as shown. ( ) ( ) Statically indeterminate to 14o
. c) Cutting the loops as shown. ( ) ( ) Statically indeterminate
to 27o . Classify each of the frames as statically determinate,
statically indeterminate, or unstable. If indeterminate, specify
the degree of indeterminacy. All internal joints are fixed
connected. . (a) (b) (c)
15. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(15)net.hs-eng,com.hs-eng Solution: a) The
frame has no loops. n = 1 r = 11 r > 3 n Statically
indeterminate to (r 3 n) = 8 . b) n = 1 r = 2 + 1 = 3 r = 3 n
Statically determinate. c) n = 1 r = 9 Statically indeterminate to
(r 3 n) = 6 . 2-14 Classify each of the frames as statically
determinate, statically indeterminate, or unstable. If
indeterminate, specify the degree of indeterminacy. All internal
joints are fixed connected. 2 333 1 T 333 2 .
16. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(16)net.hs-eng,com.hs-eng d) n = 3 r =15
Statically indeterminate to (r 3 n) = 6 . e) n = 2 r = 12
Statically indeterminate to (r 3 n) = 6 . f) n = 2 r = 14
Statically indeterminate to (r 3 n) = 8 . 3 3 33 2 3 3 . 2-14 3 3 2
3 3 1 3 3 3
17. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(17)net.hs-eng,com.hs-eng Solution: a) n =
2 r = 3 + 1 + 2 = 6 r = 3 n Statically determinate. b) Improper
constraints. n = 1 r = 3 r = 3 n But all reactions are parallel
(unstable). c) n = 1 r = 2 + 1 + 1 + 2 = 6 r > 3 n Statically
indeterminate to 3o . 2-16 Classify each of the structures as
statically determinate, statically indeterminate, or unstable. If
indeterminate, specify the degree of indeterminacy. The supports or
connections are to be assumed as stated. .
18. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(18)net.hs-eng,com.hs-eng Ay Ax 45 kN
Solution: ( ) ( ) ( ) 30o . 2-17 Determine the reactions on the
beam, the support at B can be assumed to be roller, Neglect the
thickness of the beam. By 7.5(3) = 22.5 kN 4.5 m1.5 m By
19. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(19)net.hs-eng,com.hs-eng Solution: ( ) ( )
( ) . 2-18 Determine the reactions at the supports A and B. Assume
A is a roller, and B is a pin. ByAy 18 kN 36 kN 36 kN 1.2m 2.1 m
2.1 m 2.1 m1.2m Bx
20. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(21)net.hs-eng,com.hs-eng Solution: ( ) ( )
( ) ( ) ( ) ( ) ( ) 2-19. Determine the reactions on the beam. .
0.90.45 1.350.90.9 Ay F1 By F2 F3 Ax
21. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(21)net.hs-eng,com.hs-eng Ax Ay FB
(30)(7.2) + (0.5)(15)(7.2) = 270 kN 3.6 m 3.6 m 810 kN.m Solution:
( ) ( ) 60o . 2-20 Determine the reactions on the beam.
22. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(22)net.hs-eng,com.hs-eng Solution: ( ) (
)( ) ( ) ( ) ( ) ( ) ( ) ( ) y ( ) ( ) ( ) 2-21 Determine the
reactions on the beam. X X . F2 F1 1.3 3.9 2.6 Ay FB Ax 7.8m
7.2m
23. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(23)net.hs-eng,com.hs-eng By Ay 33.75 kN27
kN 1.2 1.2 Bx 1.2 27 kN 1.2 11.25 kN Solution: ( ) ( ) ( ) ( ) .
2-22 Determine the reactions at the supports A and B. The floor
deck CD, DE, EF and FG transmit their loads to the girder on smooth
support, assume A is a roller and B is a pin.
24. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(24)net.hs-eng,com.hs-eng By 18 kN Cy Cx Cx
24 kN Ay Bx MB Cy 2 m2 m 3 m3 m Solution: Member AC: ( ) ( ) Member
CB: ( ) ( ) . 2-23 Determine the reactions at the supports A and B
of the compound beam. There is a pin at C. 4 kN/m 2-23 Determine
the reactions at the supports A and B of the compound beam. There
is a pin at C. 4 kN/m 18 kN 6 m 2 m 2 m
25. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(25)net.hs-eng,com.hs-eng Ay Ax FC 2.16 kN
0.9 m0.9 m 3 m 30o 1.5 m 2.6 m 30 o C B Solution: ( )( ) ( )( ) ( )
. 2-26 Determine the reactions at the smooth supports C and pinned
support A. Assume the connection at B is fixed connected.
26. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(26)net.hs-eng,com.hs-eng Bx By FA 6 kN 1.5
m 1.5 m 1.5 m 60 o C Solution: ( ) ( ) ( ) ( ) ( ) ( ) . 2-27
Determine the reactions at the smooth support A and pin support B.
The connection at C is fixed. 3 m
27. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(27)net.hs-eng,com.hs-eng Bx By Ay 25 kN 35
kN 50 kN 10 kN 2.5 kN 4 m 3 m 3 m 4 m 3 m Solution: ( )( ) ( )( ) (
)( ) ( )( ) ( ) . 2-31 Detremine the reactions at supports A and B
of frame. Assume that support at A is a roller.
28. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(28)net.hs-eng,com.hs-eng Bx By Ay 20 kN 30
kN 40 kN 84 kN 8 m6 m 3.5 m 3.5 Solution: Bx (7 * 12) Bx 84 kN (7 *
12 * 3.5) + (30 * 8) + (20 * 14) (Ay * 8) 0 Ay 101.75 kN Ay + By 40
30 20 0 By 11.75 kN 2-33. Determine the horizontal and vertical
components of reaction at the supports A and B. the joints at C and
D are fixed connections. .
29. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(29)net.hs-eng,com.hs-eng Ey Ex Ay By 25 kN
180 kN C D Ax Solution: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( )( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2-34. Determine
the reactions at the supports A, B, and E. Assume the bearing
support at B is a roller. .
30. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(31)net.hs-eng,com.hs-eng Ay By Cy Dy 180
kN120 kN 40 kN FE Solution: ( ) ( )( )( ) ( )( ) ( )( ) ( ) ( ) (
)( ) ( )( )( ) ( )( ) ( )( ) ( ) : ( )( )( ) ( )( ) ( )( )( ) ( )(
) ( )( ) ( ) ( )( ) ( )( ) 2-35 Determine the reactions at the
supports A, B, C, and D. .
31. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(31)net.hs-eng,com.hs-eng Cy Ay Ax Dx 67.5
k(30)(3) = 90 kN 1.81.51.51.5 B B Solution: ( ) ( ) ( )( ) ( ) ( )(
) ( )( ) ( ) 2-38 Determine the reactions at the supports A, C, and
D. B is pinned. .
32. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(32)net.hs-eng,com.hs-eng Cy 25 kN Ay 80 k
2 m2 m 2 .5 m 2 .5 m B Solution: ( )( ) ( )( ) ( ) : . 2-39
Determine the reactions at supports A and C. Assume that support at
A is a roller, B is a fixed-connected joint, and C is a pin.
Cx
33. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(33)net.hs-eng,com.hs-eng Ay Ax 50 kN By
350 kN 2 .5 m2 .5 m2 m 2 m 75 kN C Solution: ( ) ( ) ( )( ) ( )( )
. 2-40 Determine the reactions at supports A and B. Assume that
support at B is a roller, C is a fixed-connected joint.
34. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(34)net.hs-eng,com.hs-eng By Bx Ay Ax MA 45
kN27 kN 67.5 kN C D Solution: ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) (
)( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) . 2-41
Determine the reactions at the supports A and B.
35. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(35)net.hs-eng,com.hs-eng FB Ay Ax 30 kN 4
m 1.5 m 2 m1.5 m 4 3 5 A C D B Solution: ( ) ( ) ( ) ( ) ( ) . 2-44
Determine the reactions at supports A and B. The joints C and D are
fixed connected.
36. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(36)net.hs-eng,com.hs-eng Solution: ( ) (
)( ) ( ) ( )( ) ( ) ( )( ) ( )( ) 2-45 Determine the horizontal and
vertical coomponents of reaction at supports A and B. . XX 100 kN
2.5 m2.5 m 3 m 3 m XX XX 4 m By Bx Ay Ax 48kN X X D A B C
37. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(37)net.hs-eng,com.hs-eng Cy Cx 30 kN 8 kN
6 kN 8 kN Ay Ax 1 m2.5 m1.5 m 2 m 2 m 1.5 m B Solution: ( )( )( ) (
)( ) ( )( )( ) ( )( ) ( )( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )( )
2-47 Determine the reactions at the supports A and C. The frame is
pin connected at A, B, and C and the two joints are fixed
connected.
38. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(38)net.hs-eng,com.hs-eng 1.8 m Bx By Cx B
A c Ax Ay C Solution: Member AB: ( ) ( ) Member BC: ( ) ( ) ( )( )
. 2-48 Determine the horizontal and vertical components of force at
the connections A, B and C. Assume each of these connections is a
pin. (Fig. 2-48). Cy 18 kN By Bx B 1.2 m 0.9 m1.8 m 13.5 kN 1.8 m
21.6 kN
39. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999
.26044449hs.com-enginfo@(39)net.hs-eng,com.hs-eng Ax Ay Cy Cx By MC
Solution: ( ) ( )( ) ( )( )( ) ( ) ( )( ) ( )( )( ) ( )( ) ( )( ) (
) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) y . 2-50.
Determine the horizontal and vertical components of reaction at the
supports A, B, and C. Assume that the frame is pin connected at A,
B, D, E, and F, and there is a fixed connected joint at C.