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Solutions (8th ed structural analysis) chapter 6

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6–1. Draw the influence lines for (a) the moment at C,
(b) the reaction at B, and (c) the shear at C. Assume A is
pinned and B is a roller. Solve this problem using the basic
method of Sec. 6–1.

A
10 ft

146

B

C
10 ft

10 ft



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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–2.

Solve Prob. 6–1 using the Müller-Breslau principle.

A
10 ft

147

B

C
10 ft

10 ft


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–3. Draw the influence lines for (a) the vertical reaction
at A, (b) the moment at A, and (c) the shear at B. Assume
the support at A is fixed. Solve this problem using the basic
method of Sec. 6–1.
A

B
5 ft

148

5 ft


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.


*6–4.

Solve Prob. 6–3 using the Müller-Breslau principle.

A

B
5 ft

149

5 ft


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–5. Draw the influence lines for (a) the vertical reaction
at B, (b) the shear just to the right of the rocker at A, and
(c) the moment at C. Solve this problem using the basic
method of Sec. 6–1.

B

A
6 ft

150

C
6 ft

6 ft


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–6.

Solve Prob. 6–5 using Müller-Breslau’s principle.
B

A
6 ft

151

C
6 ft

6 ft


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–7. Draw the influence line for (a) the moment at B,
(b) the shear at C, and (c) the vertical reaction at B. Solve
this problem using the basic method of Sec. 6–1. Hint: The
support at A resists only a horizontal force and a bending
moment.

A

4m

152

B

C

4m

4m


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

*6–8.

Solve Prob. 6–7 using the Müller-Breslau principle.

A

4m

153

B

C

4m

4m


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–9. Draw the influence line for (a) the vertical reaction at
A, (b) the shear at B, and (c) the moment at B. Assume A is
fixed. Solve this problem using the basic method of Sec. 6–1.
A

B
2m

154

1m


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–10.

Solve Prob. 6–9 using the Müller-Breslau principle.

A

B
2m

155

1m


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–11. Draw the influence lines for (a) the vertical reaction
at A, (b) the shear at C, and (c) the moment at C. Solve this
problem using the basic method of Sec. 6–1.

B

A
6 ft

156

C
6 ft

3 ft

3 ft


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

*6–12.

Solve Prob. 6–11 using Müller-Breslau’s principle.
B

A
6 ft

157

C
6 ft

3 ft

3 ft


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–13. Draw the influence lines for (a) the vertical reaction
at A, (b) the vertical reaction at B, (c) the shear just to the
right of the support at A, and (d) the moment at C. Assume
the support at A is a pin and B is a roller. Solve this problem
using the basic method of Sec. 6–1.

A

2m

158

B

C

2m

2m

2m


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–14.

Solve Prob. 6–13 using the Müller-Breslau principle.
A

2m

159

B

C

2m

2m

2m


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–15. The beam is subjected to a uniform dead load of
1.2 kN>m and a single live load of 40 kN. Determine (a) the
maximum moment created by these loads at C, and (b) the
maximum positive shear at C. Assume A is a pin. and B is
a roller.

6m
A

C

40 kN

1
(MC) max = 40 kN (3 m) + 1.2 kN>ma b(12 m)(3 m) = 141.6 kN # m
2

Ans.

1
1
1
1 1
(VC) max = 40 a b + 1.2 kN>mca b a - b (6) + a b(6)d = 20 kN
2
2
2
2 2

Ans.

160

6m
B


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

*6–16. The beam supports a uniform dead load of
500 N͞m and a single live concentrated force of 3000 N.
Determine (a) the maximum positive moment at C, and (b)
the maximum positive shear at C. Assume the support at A
is a roller and B is a pin.

A

1m

Referring to the influence line for the moment at C shown in Fig. a, the maximum
positive moment at C is
1
(Mc) max (+) = 0.75(3000) + c (4 - 0)(0.75) d(500)
2
= 3000 N # m = 3 kN # m

Ans.

Referring to the influence line for the moment at C in Fig. b, the maximum positive
shear at C is
1
1
(Vc) max (+) = 0.75(3000) + c (1 - 0)( - 0.25) d(500) + c (4 - 1)(0.75) d(500)
2
2
= 2750 N = 2.75 kN

Ans.

161

B

C

3m


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–17. A uniform live load of 300 lb>ft and a single live
concentrated force of 1500 lb are to be placed on the beam.
The beam has a weight of 150 lb>ft. Determine (a) the
maximum vertical reaction at support B, and (b) the
maximum negative moment at point B. Assume the support
at A is a pin and B is a roller.

A
B

20 ft

Referring to the influence line for the vertical reaction at B shown in Fig. a, the
maximum reaction that is
1
(By) max (+) = 1.5(1500) + c (30 - 0)(1.5) d(300 + 150)
2
= 12375 lb = 12.4 k

Ans.

Referring to the influence line for the moment at B shown in Fig. b, the maximum
negative moment is
1
(MB) max (-) = -10(1500) + c (30 - 20)(-10) d(300 + 150)
2
= -37500 lb # ft = -37.5 k # ft

Ans.

162

10 ft


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–18. The beam supports a uniform dead load of 0.4 k͞ft,
a live load of 1.5 k͞ft, and a single live concentrated force of
8 k. Determine (a) the maximum positive moment at C,
and (b) the maximum positive vertical reaction at B.
Assume A is a roller and B is a pin.

B
10 ft

Referring to the influence line for the moment at C shown in Fig. a, the maximum
positive moment is
1
1
(MC) max (+) = 5(8) + c (20 - 0)(5) d(1.5) + c (20 - 0)(5) d(0.4)
2
2
1
+ c (35 - 20)(-7.5) d(0.4)
2
= 112.5 k # ft

Ans.

Referring to the influence line for the vertical reaction at B shown in Fig. b, the
maximum positive reaction is
1
1
(By) max (+) = 1(8) + c (20 - 0)(1) d(1.5) + c (20 - 0)(1) d(0.4)
2
2
1
+ c (35 - 20)(-0.75) d (0.4)
2
= 24.75 k

Ans.

163

A

C
10 ft

15 ft


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–19. The beam is used to support a dead load of 0.6 k͞ft,
a live load of 2 k͞ft and a concentrated live load of 8 k.
Determine (a) the maximum positive (upward) reaction
at A, (b) the maximum positive moment at C, and (c) the
maximum positive shear just to the right of the support
at A. Assume the support at A is a pin and B is a roller.

A
10 ft

Referring to the influence line for the vertical reaction at A shown
in Fig. a, the maximum positive vertical reaction is
1
(Ay) max (+) = 1.5(8) + c (30 - 0)(1.5) d(2)
2
1
1
+ c (30 - 0)(1.5) d(0.6) + c (35 - 30)(-0.25) d(0.6)
2
2
Ans.

= 70.1 k

Referring to the influence line for the moment at C shown in Fig. b, the maximum
positive moment is
1
1
(Mc) max (+) = 5(8) + c (30 - 10)(5) d(2) + c (10 - 0)( -5) d(0.6)
2
2
1
1
+ c (30 - 10)(5) d(0.6) + c (35 - 30)(-2.5) d(0.6)
2
2
= 151 k # ft

Ans.

Referring to the influence line for shear just to the right of A
shown in Fig. c, the maximum positive shear is
1
(VA + ) max (+) = 1(8) + c (10 - 0)(0.5) d (2)
2
1
+ c (30 - 10)(1) d(2)
2
1
1
+ c (10 - 0)(0.5) d(0.6) + c (30 - 10)(1) d(0.6)
2
2
1
+ c (35 - 30)(-0.25) d (0.6)
2
= 40.1 k

Ans.

164

B

C
10 ft

10 ft

5 ft


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

*6–20. The compound beam is subjected to a uniform
dead load of 1.5 kN͞m and a single live load of 10 kN.
Determine (a) the maximum negative moment created by
these loads at A, and (b) the maximum positive shear at B.
Assume A is a fixed support, B is a pin, and C is a roller.

C

B

A

5m

10 m

1
(MA) max = 1.5 a b (15)( -5) + 10(-5)
2
= -106 kN # m

Ans.

1
(VB) max = 1.5 a b (10)(1) + 10(1)
2
= 17.5 kN

Ans.

6–21. Where should a single 500-lb live load be placed on
the beam so it causes the largest moment at D? What is this
moment? Assume the support at A is fixed, B is pinned, and
C is a roller.

A

B

C

D
8 ft

At point B: (MD) max = 500(-8) = -4000 lb # ft = -4 k # ft

Ans.

165

8 ft

20 ft


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–22. Where should the beam ABC be loaded with a
300 lb͞ft uniform distributed live load so it causes (a) the
largest moment at point A and (b) the largest shear at D?
Calculate the values of the moment and shear. Assume the
support at A is fixed, B is pinned and C is a roller.

A

B
D
8 ft

(a) (MA) max =

1
(36)( -16)(0.3) = -86.4 k # ft
2

(b) (VD) max = c(1)(8) +

C

8 ft

20 ft

Ans.

1
(1)(20) d(0.3) = 5.40 k
2

Ans.

6–23. The beam is used to support a dead load of 800 N͞m,
a live load of 4 kN͞m, and a concentrated live load of 20 kN.
Determine (a) the maximum positive (upward) reaction
at B, (b) the maximum positive moment at C, and (c) the
maximum negative shear at C. Assume B and D are pins.

A

C
4m

Referring to the influence line for the vertical reaction at B, the maximum positive
reaction is
1
1
(By) max (+) = 1.5(20) + c (16 - 0)(1.5) d(4) + c (16 - 0)(1.5) d(0.8)
2
2
Ans.

= 87.6 kN

Referring to the influence line for the moment at C shown in Fig. b, the maximum
positive moment is
1
1
(Mc) max (+) = 2(20) + c (8 - 0)(2) d (4) + c (8 - 0)(2) d(0.8)
2
2
1
+ c (16 - 8)( -2) d (0.8)
2
= 72.0 kN # m

Ans.

Referring to the influence line for the shear at C shown in, the maximum negative
shear is
1
(VC) max (-) = -0.5(20) + c (4 - 0)( -0.5) d(4)
2
1
1
+ c (16 - 8)( -0.5) d(4) + c (4 - 0)( -0.5) d(0.8)
2
2
1
1
+ c (8 - 4)(0.5) d(0.8) + c (16 - 8)( -0.5) d(0.8)
2
2
= -23.6 kN

Ans.

166

4m

D

E

B
4m

4m


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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

*6–24. The beam is used to support a dead load of
400 lb͞ft, a live load of 2 k͞ft, and a concentrated live load of
8 k. Determine (a) the maximum positive vertical reaction at
A, (b) the maximum positive shear just to the right of the
support at A, and (c) the maximum negative moment at C.
Assume A is a roller, C is fixed, and B is pinned.

B

A
10 ft

Referring to the influence line for the vertical reaction at A shown in
Fig. a, the maximum positive reaction is
1
(Ay) max (+) = 2(8) + c (20 - 0)7(2) d(2 + 0.4) = 64.0 k
2

Ans.

Referring to the influence line for the shear just to the right to the support
at A shown in Fig. b, the maximum positive shear is
1
(VA + ) max (+) = 1(8) + c (10 - 0)(1) d (2 + 0.4)
2
1
+ c (20 - 10)(1) d(2 + 0.4)
2
= 32.0 k

Ans.

Referring to the influence line for the moment at C shown in Fig. c, the
maximum negative moment is
1
1
(MC) max (-) = -15(8) + c (35 - 10)(-15) d(2) + c (10 - 0)(15) d(0.4)
2
2
1
+ c (35 - 10)(-15) d (0.4)
2
= -540 k # ft

Ans.

167

10 ft

C
15 ft


© 2012 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–25. The beam is used to support a dead load of 500 lb͞ft,
a live load of 2 k͞ft, and a concentrated live load of 8 k.
Determine (a) the maximum positive (upward) reaction
at A, (b) the maximum positive moment at E, and (c) the
maximum positive shear just to the right of the support
at C. Assume A and C are rollers and D is a pin.

A
B

E
5 ft

Referring to the influence line for the vertical reaction at A shown in Fig. a, the
maximum positive vertical reaction is
1
(Ay) max (+) = 1(8) + c (10 - 0)(1) d(2 + 0.5) = 20.5 k
2

Ans.

Referring to the influence line for the moment at E shown in Fig. b, the maximum
positive moment is
1
(ME) max (+) = 2.5(8) + c (10 - 0)(2.5) d (2 + 0.5)
2
= 51.25 k # ft

Ans.

Referring to the influence line for the shear just to the right of support C, shown in
Fig. c, the maximum positive shear is
1
(VC + ) max (+) = 1(8) + c (15 - 0)(1) d(2 + 0.5)
2
1
+ c (20 - 15)(1) d(2 + 0.5)
2
= 33.0 k

Ans.

168

5 ft

C
5 ft

D
5 ft


© 2012 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–26. A uniform live load of 1.8 kN͞m and a single
concentrated live force of 4 kN are placed on the floor beams.
Determine (a) the maximum positive shear in panel BC of
the girder and (b) the maximum moment in the girder at G.
B

A
0.5 m

Referring to the influence line for the shear in panel BC shown in Fig. a, the
maximum positive shear is
1
(VBC) max (+) = 1(4) + c (1 - 0.5)(1) d (1.8) + [(2.5 - 1)(1)](1.8) = 7.15 kN Ans.
2
Referring to the influence line for the moment at G Fig. b, the maximum negative
moment is
1
(MG) max (-) = -1.75(4)c (1 - 0.5)(-0.25) d(1.8)
2
1
+ e (2.5 - 1)[-0.25 + ( -1.75)] f(1.8)
2
= -9.81 kN # m

Ans.

169

G

C

0.5 m
0.25 m 0.25 m

E

D
0.5 m

F
0.5 m


© 2012 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

6–27. A uniform live load of 2.8 kN͞m and a single
concentrated live force of 20 kN are placed on the floor
beams. If the beams also support a uniform dead load of
700 N͞m, determine (a) the maximum positive shear in
panel BC of the girder and (b) the maximum positive
moment in the girder at G.

B

A
1.5 m

1
(VBC) max (+) = 0.6(20) + c (7.5 - 1.875)(0.6) d(2.8)
2
1
1
+ c (1.875 - 0)( -0.2) d(0.7) + c (7.5 - 1.875)(0.6) d(0.7)
2
2
Ans.

Referring to the influence line for the moment at G shown in Fig. b, the maximum
positive moment is
1
(MG) max (+) = 1.35(20)c (1.5 - 0)(1.05) d(2.8 + 0.7)
2
1
+ c (3 - 1.5)(1.05 + 1.35) d(2.8 + 0.7)
2
1
+ c (7.5 - 3)(1.35) d(2.8 + 0.7)
2
= 46.7 kN # m

Ans.

170

C

0.75 m 0.75 m

Referring to the influence line for the shear in panel BC as shown in Fig. a, the
maximum position shear is

= 17.8 kN

G

1.5 m

F

E

D
1.5 m

1.5 m


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