Beer F.P., Johnston E.R., DeWolf J.T., Mazurek D.F. Mechanics of Materials

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537

Problems

8.52 Solve Prob. 8.51, assuming that the magnitude of the 6000-lb force

is reduced to 1500 lb.

8.53 Three steel plates, each 13 mm thick, are welded together to form

a cantilever beam. For the loading shown, determine the normal

and shearing stresses at points a and b.

8.54 Three steel plates, each 13 mm thick, are welded together to form

a cantilever beam. For the loading shown, determine the normal

and shearing stresses at points d and e.

8.55 Two forces are applied to a W8 3 28 rolled-steel beam as shown.

Determine the principal stresses and maximum shearing stress at

point a.

400 mm

60 mm

30 mm

60 mm

75 mm

9 kN

13 kN

150 mm

t  13 mm

Fig. P8.53 and P8.54

8.56 Two forces are applied to a W8 3 28 rolled-steel beam as shown.

Determine the principal stresses and maximum shearing stress at

point b.

90 kips

24 in.

4 in.

W8  28

20 kips

Fig. P8.55 and P8.56

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538

Principal Stresses under a Given Loading

8.57 Two forces P

and P

are applied as shown in directions perpen-

dicular to the longitudinal axis of a W310 3 60 beam. Knowing

that P

5 25 kN and P

5 24 kN, determine the principal stresses

and the maximum shearing stress at point a.

8.58 Two forces P

and P

are applied as shown in directions perpen-

dicular to the longitudinal axis of a W310 3 60 beam. Knowing

that P

5 25 kN and P

5 24 kN, determine the principal stresses

and the maximum shearing stress at point b.

8.59 A vertical force P is applied at the center of the free end of cantilever

beam AB. (a) If the beam is installed with the web vertical (b 5 0)

and with its longitudinal axis AB horizontal, determine the magnitude

of the force P for which the normal stress at point a is 1120 MPa.

(b) Solve part a, assuming that the beam is installed with b 5 38.

0.6 m

1.2 m

75 mm

W310  60

Fig. P8.57 and P8.58

W250  44.8

l  1.25 m



Fig. P8.59



Fig. P8.60

8.60 A force P is applied to a cantilever beam by means of a cable

attached to a bolt located at the center of the free end of the beam.

Knowing that P acts in a direction perpendicular to the longitudi-

nal axis of the beam, determine (a) the normal stress at point a in

terms of P, b, h, l, and b, (b) the values of b for which the normal

stress at a is zero.

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539

Problems

*8.61 A 5-kN force P is applied to a wire that is wrapped around bar AB

as shown. Knowing that the cross section of the bar is a square of

side d 5 40 mm, determine the principal stresses and the maxi-

mum shearing stress at point a.

*8.62 Knowing that the structural tube shown has a uniform wall thick-

ness of 0.3 in., determine the principal stresses, principal planes,

and maximum shearing stress at (a) point H, (b) point K.

*8.63 The structural tube shown has a uniform wall thickness of 0.3 in.

Knowing that the 15-kip load is applied 0.15 in. above the base of

the tube, determine the shearing stress at (a) point a, (b) point b.

*8.64 For the tube and loading of Prob. 8.63, determine the principal

stresses and the maximum shearing stress at point b.

Fig. P8.61

3 in.

2 in.

10 in.

0.15 in.

9 kips

4 in.

6 in.

Fig. P8.62

3 in.

10 in.

15 kips

4 in.

2 in.

1.5 in.

Fig. P8.63

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540

REVIEW AND SUMMARY

␴

min

␴

min

␴

max

␴

max

Fig. 8.28

Fig. 8.29

Principal planes and principal

stresses in a beam

This chapter was devoted to the determination of the principal

stresses in beams, transmission shafts, and bodies of arbitrary shape

subjected to combined loadings.

We first recalled in Sec. 8.2 the two fundamental relations

derived in Chaps. 5 and 6 for the normal stress s

and the shearing

stress t

at any given point of a cross section of a prismatic beam,

(8.1, 8.2)

where V 5 shear in the section

M 5 bending moment in the section

y 5 distance of the point from the neutral surface

I 5 centroidal moment of inertia of the cross section

Q 5 first moment about the neutral axis of the portion

of the cross section located above the given point

t 5 width of the cross section at the given point

Using one of the methods presented in Chap. 7 for the transforma-

tion of stresses, we were able to obtain the principal planes and

principal stresses at the given point (Fig. 8.28).

We investigated the distribution of the principal stresses in a

narrow rectangular cantilever beam subjected to a concentrated load

P at its free end and found that in any given transverse section—

except close to the point of application of the load—the maximum

principal stress s

max

did not exceed the maximum normal stress s

occurring at the surface of the beam.

While this conclusion remains valid for many beams of non-

rectangular cross section, it may not hold for W-beams or S-beams,

where s

max

at the junctions b and d of the web with the flanges of

the beam (Fig. 8.29) may exceed the value of s

occurring at points

a and e. Therefore, the design of a rolled-steel beam should include

the computation of the maximum principal stress at these points.

(See Sample Probs. 8.1 and 8.2.)

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541

Review and Summary

In Sec. 8.3, we considered the design of transmission shafts sub-

jected to transverse loads as well as to torques. Taking into account

the effect of both the normal stresses due to the bending moment M

and the shearing stresses due to the torque T in any given trans-

verse section of a cylindrical shaft (either solid or hollow), we found

that the minimum allowable value of the ratio Jyc for the cross

section was

A2M

1 T

max

(8.6)

In preceding chapters, you learned to determine the stresses in

prismatic members caused by axial loadings (Chaps. 1 and 2), tor-

sion (Chap. 3), bending (Chap. 4), and transverse loadings (Chaps. 5

and 6). In the second part of this chapter (Sec. 8.4), we combined

this knowledge to determine stresses under more general loading

conditions.

For instance, to determine the stresses at point H or K of the

bent member shown in Fig. 8.30, we passed a section through these

points and replaced the applied loads by an equivalent force-couple

system at the centroid C of the section (Fig. 8.31). The normal and

shearing stresses produced at H or K by each of the forces and cou-

ples applied at C were determined and then combined to obtain the

resulting normal stress s

and the resulting shearing stresses t

and

at H or K. Finally, the principal stresses, the orientation of the

principal planes, and the maximum shearing stress at point H or K

were determined by one of the methods presented in Chap. 7 from

the values obtained for s

, t

, and t

Design of transmission shafts

under transverse loads

Stresses under general

loading conditions

Fig. 8.31

Fig. 8.30

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542

REVIEW PROBLEMS

8.65 (a) Knowing that s

all

5 24 ksi and t

all

5 14.5 ksi, select the most

economical wide-flange shape that should be used to support the

loading shown. (b) Determine the values to be expected for s

, and the principal stress s

max

at the junction of a flange and the

web of the selected beam.

8.66 Determine the smallest allowable diameter of the solid shaft

ABCD, knowing that t

all

5 60 MPa and that the radius of disk B

is r 5 80 mm.

8.67 Using the notation of Sec. 8.3 and neglecting the effect of shear-

ing stresses caused by transverse loads, show that the maximum

normal stress in a circular shaft can be expressed as follows:

max

31M

1 M

1 1M

1 M

1 T

max

8.68 The solid shaft AB rotates at 450 rpm and transmits 20 kW from

the motor M to machine tools connected to gears F and G. Know-

ing that t

all

5 55 MPa and assuming that 8 kW is taken off at

gear F and 12 kW is taken off at gear G, determine the smallest

permissible diameter of shaft AB.

12 ft 6 ft

1.5 kips/ft

Fig. P8.65

150 mm

T  600 N · m

150 mm

Fig. P8.66

150 mm

225 mm

60 mm

225 mm

100 mm

60 mm

150 mm

Fig. P8.68

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543

Review Problems

8.69 Two 1.2-kip forces are applied to an L-shaped machine element

AB as shown. Determine the normal and shearing stresses at

(a) point a, (b) point b, (c) point c.

12 in.

6 in.

1.2 kips

1.0 in.

0.5 in.

1.8 in.

1.0 in.

Fig. P8.69

8.70 Two forces are applied to the pipe AB as shown. Knowing that the

pipe has inner and outer diameters equal to 35 and 42 mm, respec-

tively, determine the normal and shearing stresses at (a) point a,

(b) point b.

8.71 A close-coiled spring is made of a circular wire of radius r that is

formed into a helix of radius R. Determine the maximum shearing

stress produced by the two equal and opposite forces P and P9.

(Hint: First determine the shear V and the torque T in a transverse

cross section.)

8.72 Three forces are applied to a 4-in.-diameter plate that is attached to

the solid 1.8-in. diameter shaft AB. At point H, determine (a) the prin-

cipal stresses and principal planes, (b) the maximum shearing stress.

8 in.

2 in.

6 kips

2.5 kips

6 kips

Fig. P8.72

75 mm

45 mm

1500 N

1200 N

45 mm

20 mm

Fig. P8.70

P P

Fig. P8.71

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544

Principal Stresses under a Given Loading

8.73 Knowing that the bracket AB has a uniform thickness of

in., deter-

mine (a) the principal planes and principal stresses at point K, (b) the

maximum shearing stress at point K.

8.74 Three forces are applied to the machine component ABD as

shown. Knowing that the cross section containing point H is a

20 3 40-mm rectangle, determine the principal stresses and the

maximum shearing stress at point H.

8.75 Knowing that the structural tube shown has a uniform wall thick-

ness of 0.25 in., determine the normal and shearing stresses at the

three points indicated.

30

2 in.

5 in.

3 kips

2.5 in.

Fig. P8.73

50 mm

150 mm

160 mm

40 mm

3 kN

0.5 kN

2.5 kN

20 mm

Fig. P8.74

2.75 in.

1500 lb

5 in.

6 in.

3 in.

600 lb

20 in.

0.25 in.

3 in.

Fig. P8.75

300 mm

600 N

40 mm

60 mm



Fig. P8.76

8.76 The cantilever beam AB will be installed so that the 60-mm side

forms an angle b between 0 and 908 with the vertical. Knowing

that the 600-N vertical force is applied at the center of the free

end of the beam, determine the normal stress at point a when

(a) b 5 0, (b) b 5 908. (c) Also, determine the value of b for which

the normal stress at point a is a maximum and the corresponding

value of that stress.

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545

COMPUTER PROBLEMS

The following problems are designed to be solved with a computer.

8.C1 Let us assume that the shear V and the bending moment M have

been determined in a given section of a rolled-steel beam. Write a computer

program to calculate in that section, from the data available in Appendix C,

(a) the maximum normal stress s

, (b) the principal stress s

max

at the junc-

tion of a flange and the web. Use this program to solve parts a and b of the

following problems:

(1) Prob. 8.1 (Use V 5 45 kips and M 5 450 kip ? in.)

(2) Prob. 8.2 (Use V 5 22.5 kips and M 5 450 kip ? in.)

(3) Prob. 8.3 (Use V 5 700 kN and M 5 1750 kN ? m)

(4) Prob. 8.4 (Use V 5 850 kN and M 5 1700 kN ? m)

8.C2 A cantilever beam AB with a rectangular cross section of width b

and depth 2c supports a single concentrated load P at its end A. Write a

computer program to calculate, for any values of xyc and yyc, (a) the ratios

max

and s

min

, where s

max

and s

min

are the principal stresses at point

K(x, y) and s

the maximum normal stress in the same transverse section,

(b) the angle u

that the principal planes at K form with a transverse and a

horizontal plane through K. Use this program to check the values shown in

Fig. 8.8 and to verify that s

max

exceeds s

if x # 0.544c, as indicated in the

second footnote on page 517.

8.C3 Disks D

, D

, . . . , D

are attached as shown in Fig. 8.C3 to the

solid shaft AB of length L, uniform diameter d, and allowable shearing stress

all

. Forces P

, P

, . . . , P

of known magnitude (except for one of them)

are applied to the disks, either at the top or bottom of its vertical diameter,

or at the left or right end of its horizontal diameter. Denoting by r

the

radius of disk D

and by c

its distance from the support at A, write a com-

puter program to calculate (a) the magnitude of the unknown force P

(b) the smallest permissible value of the diameter d of shaft AB. Use this

program to solve Prob. 8.18.



max





min

Fig. P8.C2

Fig. P8.C3

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546

Principal Stresses under a Given Loading

8.C4 The solid shaft AB of length L, uniform diameter d, and allowable

shearing stress t

all

rotates at a given speed expressed in rpm (Fig. 8.C4).

Gears G

, G

, . . . , G

are attached to the shaft and each of these gears

meshes with another gear (not shown), either at the top or bottom of its

vertical diameter, or at the left or right end of its horizontal diameter. One

of these gears is connected to a motor and the rest of them to various

machine tools. Denoting by r

the radius of disk G

, by c

its distance from

the support at A, and by P

the power transmitted to that gear (1 sign) or

taken off that gear (2 sign), write a computer program to calculate the

smallest permissible value of the diameter d of shaft AB. Use this program

to solve Probs. 8.27 and 8.68.

8.C5 Write a computer program that can be used to calculate the normal

and shearing stresses at points with given coordinates y and z located on

the surface of a machine part having a rectangular cross section. The inter-

nal forces are known to be equivalent to the force-couple system shown.

Write the program so that the loads and dimensions can be expressed in

either SI or U.S. customary units. Use this program to solve (a) Prob. 8.45b,

(b) Prob. 8.47a.

Fig. P8.C4

Fig. P8.C5

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