Mechanics of Materials 7th Edition Beer Johnson Chapter 4

CCHHAAPPTTEERR 44

PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.

Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted

on a website, in whole or part.

447

M 5 15 kN · m

Dimensions in mm

B

20 40 20

20

20

80

A

PROBLEM 4.1

Knowing that the couple shown acts in a vertical plane, determine

the stress at (a) point A, (b) point B.

SOLUTION

For rectangle: 31

12I bh

Outside rectangle: 31

1(80)(120)

12I

6 4 6 41 11.52 10 mm 11.52 10 mI

Cutout: 32

1(40)(80)

12I

6 4 6 42 1.70667 10 mm 1.70667 10 mI

Section: 6 41 2 9.81333 10 mI I I

(a) 40 mm 0.040 mAy 3

6

6

(15 10 )(0.040)61.6 10 Pa

9.81333 10

AA

My

I

61.6 MPaA

(b) 60 mm 0.060 mBy 3

6

6

(15 10 )( 0.060)91.7 10 Pa

9.81333 10

BB

My

I

91.7 MPaB




448

2 in.

2 in.

1.5 in.

2 in.

2 in.2 in.

A

B

M ! 25 kip · in.

PROBLEM 4.2

Knowing that the couple shown acts in a vertical plane, determine

the stress at (a) point A, (b) point B.

SOLUTION

For rectangle: 31

12I bh

For cross sectional area:

3 3 3 41 2 3

1 1 1(2)(1.5) (2)(5.5) (2)(1.5) 28.854 in

12 12 12I I I I

(a) 2.75 in.Ay (25)(2.75)

28.854

AA

My

I 2.38 ksiA

(b) 0.75 in.By (25)(0.75)

28.854

BB

My

I 0.650 ksiB




449

200 mm

220 mm

12 mm

12 mm

8 mm

C x

y

M

PROBLEM 4.3

Using an allowable stress of 155 MPa, determine the largest bending

moment M that can be applied to the wide-flange beam shown. Neglect the

effect of fillets.

SOLUTION

Moment of inertia about x-axis:

3 21

6 4

3 6 42

6 43 1

1(200)(12) (200)(12)(104)

12

25.9872 10 mm

1(8)(196) 5.0197 10 mm

12

25.9872 10 mm

I

I

I I

6 4 6 41 2 3

6

6 63

56.944 10 mm 56.944 10 m

1with (220) 110 mm 0.110 m

2

with 155 10 Pa

(56.944 10 )(155 10 )80.2 10 N m

0.110x

I I I I

Mcc

I

IM

c

M

80.2 kN mxM




450

200 mm

220 mm

12 mm

12 mm

8 mm

C x

y

M

PROBLEM 4.4

Solve Prob. 4.3, assuming that the wide-flange beam is bent about the y axis

by a couple of moment My.

PROBLEM 4.3. Using an allowable stress of 155 MPa, determine the

largest bending moment M that can be applied to the wide-flange beam

shown. Neglect the effect of fillets.

SOLUTION

Moment of inertia about y axis:

3 6 41

3 3 42

6 43 1

1(12)(200) 8 10 mm

12

1(196)(8) 8.3627 10 mm

12

8 10 mm

I

I

I I

6 4 6 41 2 3

6

6 63

16.0084 10 mm 16.0084 10 m

1with (200) 100 mm 0.100 m

2

with 155 10 Pa

(16.0084 10 )(155 10 )24.8 10 N m

0.100

y

y

I I I I

Mcc

I

IM

c

M

24.8 kN myM




451

M2

M1

0.1 in.

0.2 in.

0.5 in.

0.5 in.

(a)

(b)

PROBLEM 4.5

Using an allowable stress of 16 ksi, determine the largest couple that can be

applied to each pipe.

SOLUTION

(a) 4 4 4 4 3 4

3

(0.6 0.5 ) 52.7 10 in4 4

0.6 in.

(16)(52.7 10 ):

0.6

o iI r r

c

Mc IM

I c

1.405 kip in.M

(b) 4 4 3 4

3

(0.7 0.5 ) 139.49 10 in4

0.7 in.

(16)(139.49 10 ):

0.7

I

c

Mc IM

I c

3.19 kip in.M




452

120 mm

30 mm

30 mmM = 2.8 kN · m

r 5 20 mmA

B

PROBLEM 4.6

Knowing that the couple shown acts in a vertical plane,

determine the stress at (a) point A, (b) point B.

SOLUTION

3 4

6 4

1 1(0.120 m)(0.06 m) 2 (0.02 m)

12 12 4

2.1391 10 mm

I

(a) 3

6 4

(2.8 10 N m)(0.03 m)

2.1391 10 mm

AA

M y

I

39.3 MPaA

(b) 3

6 4

(2.8 10 N m)(0.02 m)

2.1391 10 m

BB

M y

I

26.2 MPaB




453

y

z C

PROBLEM 4.7

Two W4 13 rolled sections are welded together as shown. Knowing that for the steel

alloy used 36 ksiY and 58 ksiU and using a factor of safety of 3.0, determine the

largest couple that can be applied when the assembly is bent about the z axis.

SOLUTION

Properties of W4 13 rolled section.

(See Appendix C.)

2

4

Area 3.83 in

Width 4.060 in.

3.86 inyI

For one rolled section, moment of inertia about axis b-b is

2 2 43.86 (3.83)(2.030) 19.643 inb yI I Ad

For both sections, 42 39.286 in

width 4.060 in.

z bI I

c

all

allall

5819.333 ksi

. . 3.0

(19.333)(39.286)

4.060

U Mc

F S I

IM

c all 187.1 kip in.M




454

y

z

C

PROBLEM 4.8

Two W4 13 rolled sections are welded together as shown. Knowing that for the steel

alloy used 58 ksiU and using a factor of safety of 3.0, determine the largest couple

that can be applied when the assembly is bent about the z axis.

SOLUTION

Properties of W4 13 rolled section.

(See Appendix C.)

2

4

Area 3.83 in

Depth 4.16 in.

11.3 inxI

For one rolled section, moment of inertia about axis a-a is

2 2 411.3 (3.83)(2.08) 27.87 ina xI I Ad

For both sections, 42 55.74 in

depth 4.16 in.

z aI I

c

all

allall

5819.333 ksi

. . 3.0

(19.333)(55.74)

4.16

U Mc

F S I

IM

c all 259 kip in.M

Mechanics of Materials 7th Edition Beer Johnson Chapter 4

Documents