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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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*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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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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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

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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.

6–25. 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

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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.

6–26. 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

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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.

6–27. 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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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

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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.

6–25. 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

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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.

6–26. 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

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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.

6–27. 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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