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Chapter 3
M (^) o 0 18 RB 6(100) 0 RB 33.3 lbf Ans.
F (^) y 0 Ro RB 100 0 Ro 66.7 lbf Ans. RC RB 33.3 lbf Ans.
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3- Body AB : F (^) x 0 RA (^) x RBx F (^) y 0 RA (^) y RBy M (^) B 0 RAy (10) RAx (10) 0 R (^) Ax RAy
Body OAC : M (^) O 0 RAy (10) 100(30) 0 R (^) Ay 300 lbf Ans. F (^) x 0 ROx R (^) Ax 300 lbf Ans. F (^) y 0 ROy RAy 100 0 ROy 200 lbf Ans. ______________________________________________________________________________
0.8 (^) 1.39 kN. O tan 30 R (^) Ans 0.8 (^) 1.6 kN. A sin 30 R (^) Ans
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Step 1: Find RA & RE 4.5 (^) 7.794 m tan 30 0 9 7.794(400 cos 30 ) 4.5(400sin 30 ) 0 400 N.
A E
E
h M R
R Ans
2 2
0 400cos30 0 346.4 N 0 400 400sin 30 0 200 N 346.4 200 400 N.
x Ax Ax y Ay Ay A
F R
R
F R
R
R Ans
Step 2: Find components of RC on link 4 and RD
4
305.4 N.
0 305.4 N
0 ( ) 400 N
C D D x Cx y Cy
M
R
R Ans F R F R
M C 0
1500 R 1 300(5) 1200(9) 0
R 1 (^) 8.2 kN Ans. F (^) y 0 8.2 9 5 R 2 0 R 2 (^) 5.8 kN Ans.
M (^) 1 8.2(300) 2460 N m Ans. M (^) 2 2460 0.8(900) 1740 N m Ans. M (^) 3 1740 5.8(300) 0 checks! _____________________________________________________________________________
3-
F (^) y 0 R 0 (^) 500 40(6) 740 lbf Ans. M 0 0 M (^) 0 500(8) 40(6)(17) 8080 lbf in Ans.
M (^) 1 8080 740(8) 2160 lbf in Ans. M (^) 2 2160 240(6) 720 lbf in Ans.
3 720 1 (240)(6) 0 checks! 2
M
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M B 0
2.2 R 1 1(2) 1(4) 0
R 1 (^) 0.91 kN Ans. F (^) y 0 0.91 2 R 2 4 0 R 2 (^) 6.91 kN Ans.
M 1 (^) 0.91(1.2) 1.09 kN m Ans.
M (^) 2 1.09 2.91(1) 4 kN m Ans. M (^) 3 4 4(1) 0 checks!
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Break at the hinge at B
Beam OB : From symmetry, R 1 (^) VB 200 lbf Ans.
Beam BD : M (^) D 0 200(12) R 2 (10) 40(10)(5) 0 R 2 (^) 440 lbf Ans.
F (^) y 0 200 440 40(10) R 3 0 R 3 (^) 160 lbf Ans.
1 2 1 0 0 0 0 1 0 1 1 0 0 1 2 2 0 0
at 20 in, 0, Eqs. (1) and (2) give
q R x M x x x x V R M x x x x M R x M x x x x V M R
0 ^ 0
0 0 2 0
500 40 20 14 0 740 lbf. (20) 500(20 8) 20(20 14) 0 8080 lbf in.
R Ans R M M
Ans
0 8 : 740 lbf, 740 8080 lbf in 8 14 : 740 500 240 lbf 740 8080 500( 8) 240 4080 lbf in 14 20 : 740 500 40( 14) 40 800 lbf 740 8080
x V M x x V M x x x x V x x M x
500( x 8) 20( x 14) 2 20 x^2 800 x 8000 lbf in Plots of V and M are the same as in Prob. 3-6. ______________________________________________________________________________
3- 1 1 1 1 1 2 0 0 0 1 2 1 1 1 1 2
q R x x R x x V R x R x x M R x x R x x
^ ^ ^
at x = 3.2+, V = M = 0. Applying Eqs. (1) and (2),
Solving Eqs. (3) and (4) simultaneously,
1 2 1 2 1 2 1 2
R R R R
R R R R
R 1 = -0.91 kN, R 2 = 6.91 kN Ans. 0 1.2 : 0.91 kN, 0.91 kN m 1.2 2.2 : 0.91 2 2.91 kN 0.91 2( 1.2) 2.91 2.4 kN m 2.2 3.2 : 0.91 2 6.91 4 kN 0.91 2(
x V M x x V M x x x x V M x x
1.2) 6.91( x 2.2) 4 x 12.8 kN m Plots of V and M are the same as in Prob. 3-7. ______________________________________________________________________________
1 1 1 0 0 1 1 2 3 0 0 1 1 0 1 2 3 1 1 2 2 1 1 2 3 1
0 at 8 in 8 400(
q R x x R x x x R x V R x R x x x R x M R x x R x x x R x M x R
^ ^ ^
8 4) 0 R 1 200 lbf Ans. at x = 20+, V = M = 0. Applying Eqs. (1) and (2), 2 3 2 3 2 2 2
200(20) 400(16) (10) 20(10) 0 440 lbf.
R R R R
R R A
3 600 440 160 lbf^.
ns R Ans
0 4 : 200 lbf, 200 lbf in 4 10 : 200 400 200 lbf, 200 400( 4) 200 1600 lbf in 10 20 : 200 400 440 40( 10) 640 40 lbf 200 400( 4)
x V M x x V M x x x x V x x M x x
440( x 10) 20 x 10 2 20 x^2 640 x 4800 lbf in
Plots of V and M are the same as in Prob. 3-8. ______________________________________________________________________________
3-13 Solution depends upon the beam selected. ______________________________________________________________________________
3- (a) Moment at center,
2
c
c
x l^ a
M l^ l a l^ l a
^ ^ ^
w wl
At reaction, Mr wa^2 a = 2.25, l = 10 in, w = 100 lbf/in
2
100(10) (^10) 2.25 125 lbf in 2 4 100 2. 253 lbf in. 2
c
r
M
M Ans
^
(b) Optimal occurs when M (^) c Mr
(c)
2 2 1 2
(^24 10) 17 kpsi 2 (^24 10) 7 kpsi 2 7 6 9.22 kpsi 17 9.22 26.22 kpsi 17 9.22 7.78 kpsi
C
CD
R
(^1 90) tan 1 7 69.7 ccw p 2 6
^ ^
^
1 9.22 kpsi s 69.7^45 24.7^ ccw
R
^ ^
(d)
2 2 1 2
(^12 22) 5 kpsi 2 (^12 22) 17 kpsi 2 17 12 20.81 kpsi 5 20.81 25.81 kpsi 5 20.81 15.81 kpsi
C
CD
R
^
(^1 90) tan 1 17 72.39 cw p 2 12
^ ^
^
1 20.81 kpsi s 72.39^45 27.39^ cw
R
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(c)
2 2 1 2
(^12 4) 4 MPa 2 (^12 4) 8 MPa 2 8 7 10.63 MPa 4 10.63 14.63 MPa 4 10.63 6.63 MPa
C
CD
R
(^1 90) tan 1 8 69.4 ccw p 2 7
^ ^
^
1 10.63 MPa s 69.4^45 24.4^ ccw
R
^ ^
(d)
2 2 1 2
(^6 5) 0.5 MPa 2 (^6 5) 5.5 MPa 2 5.5 8 9.71 MPa 0.5 9.71 10.21 MPa 0.5 9.71 9.21 MPa
C
CD
R
(^1) tan 1 8 27.75 ccw p 2 5.
^ ^
1 9.71 MPa s^45 27.75^ 17.25^ cw
R
^ ^
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(a)
2 2 1 2
(^12 6) 9 kpsi 2 (^12 6) 3 kpsi 2 3 4 5 kpsi 5 9 14 kpsi 9 5 4 kpsi
C
CD
R
(^1) tan 1 4 26.6 ccw p 2 3
^ ^ ^
1 5 kpsi s^45 26.6^ 18.4^ ccw
R
^ ^
(b)
2 2 1 2
(^30 10) 10 kpsi 2 (^30 10) 20 kpsi 2 20 10 22.36 kpsi 10 22.36 32.36 kpsi 10 22.36 12.36 kpsi
C
CD
R
(^1) tan 1 10 13.28 ccw p 2 20
^ ^
1 22.36 kpsi s^45 13.28^ 31.72^ cw
R
^ ^
(a)
2 2 1 2 3
(^80 30) 55 MPa 2 (^80 30) 25 MPa 2 25 20 32.02 MPa 0 MPa 55 32.02 22.98 23.0 MPa 55 32.0 87.0 MPa
C
CD
R
^
1 2 2 3 1 3
(^23) 11.5 MPa, 32.0 MPa, 87 43.5 MPa 2 2
(b)
2 2 1 2 3
(^30 60) 15 MPa 2 (^60 30) 45 MPa 2 45 30 54.1 MPa 15 54.1 39.1 MPa 0 MPa 15 54.1 69.1 MPa
C
CD
R
1 3
1 2
2 3
39.1 69.1 (^) 54.1 MPa 2 39.1 (^) 19.6 MPa 2 69.1 (^) 34.6 MPa 2
(c)
2 2 1 2 3
(^40 0) 20 MPa 2 (^40 0) 20 MPa 2 20 20 28.3 MPa 20 28.3 48.3 MPa 20 28.3 8.3 MPa z 30 MPa
C
CD
R
1 3 1 2 2 3
48.3 (^30) 39.1 MPa, 28.3 MPa, 30 8.3 10.9 MPa 2 2
^
(d)
2 2 1 2 3
(^50) 25 MPa 2 (^50) 25 MPa 2 25 30 39.1 MPa 25 39.1 64.1 MPa 25 39.1 14.1 MPa z 20 MPa
C
CD
R
1 3 1 2 2 3
64.1 (^20) 42.1 MPa, 39.1 MPa, 20 14.1 2.95 MPa 2 2
^
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(a) Since there are no shear stresses on the stress element, the stress element already represents principal stresses. 1 2 3
10 kpsi 0 kpsi 4 kpsi
x
y
1 3
1 2
2 3
10 ( 4) (^) 7 kpsi 2 (^10) 5 kpsi 2 0 ( 4) (^) 2 kpsi 2
3-20 From Eq. (3-15),
3 2 2 2 2 2 2 3
^2
^
Roots are: 21.04, 5.67, –26.71 kpsi Ans. 1 2
2 3
max 1 3
21.04 5.67 (^) 7.69 kpsi 2 5.67 26.71 (^) 16.19 kpsi 2 21.04 26.71 (^) 23.88 kpsi. 2 Ans
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From Eq. (3-15)
3 2 2 2
(^2 2 )
3 2
^2
^
^
Roots are: 60, 20, –40 kpsi Ans.
1 2
2 3
max 1 3
(^60 20) 20 kpsi 2 (^20 40) 30 kpsi 2 (^60 40) 50 kpsi. 2
Ans
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From Eq. (3-15)
3 2 2 2 ^2
2 2 2 3 2
^
^
Roots are: 90, 0, 0 MPa Ans.
2 3 1 2 1 3 max
(^90) 45 MPa. 2 Ans
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2
6
1 6
(^15000) 33 950 psi 34.0 kpsi. 4 0.
33 950 60 0.0679 in. 30 10 0.0679 (^) 1130 10 1130. 60
F (^) Ans A FL L (^) Ans AE E
Ans L
From Table A-5, v = 0.
2 1 6 6 2
330 10 (0.75) 248 10 in.
v A d d An
ns s
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2
6
6 1
(^3000) 6790 psi 6.79 kpsi. 4 0.
6790 60 0.0392 in. 10.4 10 0.0392 (^) 653 10 653. 60
F (^) Ans A FL L (^) Ans AE E
Ans L
From Table A-5, v = 0.
2 1 2 6 6
217 10 (0.75) 163 10 in.
v Ans d d An s
