1. 1. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999 .26044449hs.com-enginfo@(1)net.hs-eng,com.hs-eng . . . ( 03) . . ( eng-hs.net ) . ) ( ( eng-hs.net ) . Analysis of structures Week 2
2. 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. 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. 4. Oct 19, 2014 ()( )()eng-hs.net 61941842 61862999 .26044449hs.com-enginfo@(4)net.hs-eng,com.hs-eng Examples of Determinacy .
5. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.