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

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707

SAMPLE PROBLEM 11.1

During a routine manufacturing operation, rod AB must acquire an elastic

strain energy of 120 in ? lb. Using E 5 29 3 10

psi, determine the required

yield strength of the steel if the factor of safety with respect to permanent

deformation is to be five.

SOLUTION

Factor of Safety. Since a factor of safety of five is required, the rod

should be designed for a strain energy of

U 5 5

120 in ? lb

5 600 in ? lb

Strain-Energy Density. The volume of the rod is

V 5 AL 5

10.75 in.2

160 in.25 26.5 in

Since the rod is of uniform cross section, the required strain-energy

density is

u 5

600 in ?

5 in

5 22.6 in ? lb/in

Yield Strength. We recall that the modulus of resilience is equal to

the strain-energy density when the maximum stress is equal to s

. Using

Eq. (11.8), we write

u 5

22.6 in ? lb/in

2129 3 10

si2

= 36.2 ksi

◀

Comment. It is important to note that, since energy loads are not

linearly related to the stresses they produce, factors of safety associated with

energy loads should be applied to the energy loads and not to the

stresses.

5 ft

-in. diameter

Modulus of

resilience





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SOLUTION

Bending Moment. Using the free-body diagram of the entire beam,

we determine the reactions

For portion AD of the beam, the bending moment is

For portion DB, the bending moment at a distance v from end B is

a. Strain Energy. Since strain energy is a scalar quantity, we add the

strain energy of portion AD to that of portion DB to obtain the total strain

energy of the beam. Using Eq. (11.17), we write

2EI

dx 1

2EI

dx 1

2EI

6EIL

1a 1 b2

or, since (a 1 b) 5 L,

U 5

6EIL

◀

b. Evaluation of the Strain Energy. The moment of inertia of a

W10 3 45 rolled-steel shape is obtained from Appendix C and the given

data is restated using units of kips and inches.

P 5 40

ips L 5 12

5 144 in

5 3

5 36 in

5 9

t 5 108 in.

E 5 29 3 10

si 5 29 3 10

ksi I 5 248 in

Substituting into the expression for U, we have

U 5

140 kips2

136 in.2

1108 in.2

29 3 10

ksi

248 in

144 in.

U 5 3.89 in ? kips

◀

SAMPLE PROBLEM 11.2

(a) Taking into account only the effect of normal stresses due to bending,

determine the strain energy of the prismatic beam AB for the loading shown.

(b) Evaluate the strain energy, knowing that the beam is a W10 3 45,

P 5 40 kips, L 5 12 ft, a 5 3 ft, b 5 9 ft, and E 5 29 3 10

psi.



From A to D:



rom

708

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PROBLEMS

709

11.1 Determine the modulus of resilience for each of the following

metals:

(a) Stainless steel

AISI 302 (annealed): E 5 190 GPa s

5 260 MPa

(b) Stainless steel 2014-T6

AISI 302 (cold-rolled): E 5 190 GPa s

5 520 MPa

5 230 MPa

11.2 Determine the modulus of resilience for each of the following

alloys:

(a) Titanium: E 5 16.5 3 10

psi s

5 120 ksi

(b) Magnesium: E 5 6.5 3 10

psi s

5 29 ksi

psi s

5 16 ksi

11.3 Determine the modulus of resilience for each of the following

grades of structural steel:

(a) ASTM A709 Grade 50: s

5 50 ksi

(b) ASTM A913 Grade 65: s

5 65 ksi

5 100 ksi

11.4 Determine the modulus of resilience for each of the following alu-

minum alloys:

(a) 1100-H14: E 5 70 GPa s

5 55 MPa

(b) 2014-T6 E 5 72 GPa: s

5 220 MPa

5 150 MPa

11.5 The stress-strain diagram shown has been drawn from data obtained

during a tensile test of an aluminum alloy. Using E 5 72 GPa,

determine (a) the modulus of resilience of the alloy, (b) the modu-

lus of toughness of the alloy.

11.6 The stress-strain diagram shown has been drawn from data obtained

during a tensile test of a specimen of structural steel. Using E 5

29 3 10

psi, determine (a) the modulus of resilience of the steel,

(b) the modulus of toughness of the steel.

(MPa)

600

450

300



150

0.006

0.14 0.18



Fig. P11.5

0.002

0.021 0.2 0.25

100

(ksi)





Fig. P11.6

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710

Energy Methods

0.104

1.85

P (kips)

(in.)



15 in.

Fig. P11.7

11.7 The load-deformation diagram shown has been drawn from data

obtained during a tensile test of a 0.875-in.-diameter rod of an

aluminum alloy. Knowing that the deformation was measured using

a 15-in. gage length, determine (a) the modulus of resilience of

the alloy, (b) the modulus of toughness of the alloy.

11.8 The load-deformation diagram shown has been drawn from data

obtained during a tensile test of structural steel. Knowing that the

cross-sectional area of the specimen is 250 mm

and that the defor-

mation was measured using a 500-mm gage length, determine

(a) the modulus of resilience of the steel, (b) the modulus of tough-

ness of the steel.

2 ft

3 ft

in.

Fig. P11.9

0.6

8.6 78 96

500 mm

100

P (kN)

(mm)



Fig. P11.8

11.9 Using E 5 29 3 10

psi, determine (a) the strain energy of the

steel rod ABC when P 5 8 kips, (b) the corresponding strain

energy density in portions AB and BC of the rod.

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711

Problems

11.10 Using E 5 200 GPa, determine (a) the strain energy of the steel

rod ABC when P 5 25 kN, (b) the corresponding strain-energy

density in portions AB and BC of the rod.

11.11 A 30-in. length of aluminum pipe of cross-sectional area 1.85 in

is welded to a fixed support A and to a rigid cap B. The steel rod

EF, of 0.75-in. diameter, is welded to cap B. Knowing that the

modulus of elasticity is 29 3 10

psi for the steel and 10.6 3 10

psi for the aluminum, determine (a) the total strain energy of the

system when P 5 8 kips, (b) the corresponding strain-energy den-

sity of the pipe CD and in the rod EF.

20-mm diameter

1.2 m

0.8 m

2 m

16-mm diamete

Fig. P11.10

11.12 Rod AB is made of a steel for which the yield strength is s

450 MPa and E 5 200 GPa; rod BC is made of an aluminum alloy

for which s

5 280 MPa and E 5 73 GPa. Determine the maxi-

mum strain energy that can be acquired by the composite rod ABC

without causing any permanent deformations.

11.13 A single 6-mm-diameter steel pin B is used to connect the steel

strip DE to two aluminum strips, each of 20-mm width and 5-mm

thickness. The modulus of elasticity is 200 GPa for the steel and

70 GPa for the aluminum. Knowing that for the pin at B the allow-

able shearing stress is t

all

5 85 MPa, determine, for the loading

shown, the maximum strain energy that can be acquired by the

assembled strips.

14-mm diameter

1.6 m

1.2 m

10-mm diameter

Fig. P11.12

1.25 m

0.5 m

5 mm

20 mm

Fig. P11.13

1.8 m

Fig. P11.14

11.14 Rod BC is made of a steel for which the yield strength is s

300 MPa and the modulus of elasticity is E 5 200 GPa. Knowing

that a strain energy of 10 J must be acquired by the rod when

the axial load P is applied, determine the diameter of the rod

for which the factor of safety with respect to permanent defor-

mation is six.

Fig. P11.11

30 in.

48 in.

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712

Energy Methods

11.15 The assembly ABC is made of a steel for which E 5 200 GPa and

5 320 MPa. Knowing that a strain energy of 5 J must be

acquired by the assembly as the axial load P is applied, determine

the factor of safety with respect to permanent deformation when

(a) x 5 300 mm, (b) x 5 600 mm.

11.16 Using E 5 10.6 3 10

psi, determine by approximate means the

maximum strain energy that can be acquired by the aluminum rod

shown if the allowable normal stress is s

all

5 22 ksi.

900 mm

18-mm diameter

12-mm diameter

Fig. P11.15

4 @ 1.5 in.  6 in.

1.5 in.

2.10 in.

2.55 in.

2.85 in.

3 in.

Fig. P11.16

11.17 Show by integration that the strain energy of the tapered rod AB is

U 5

min

where A

min

is the cross-sectional area at end B.

11.18 through 11.21 In the truss shown, all members are made of

the same material and have the uniform cross-sectional area indi-

cated. Determine the strain energy of the truss when the load P

is applied.

Fig. P11.17

Fig. P11.18

60⬚

Fig. P11.20

A A

l l

Fig. P11.21

30°

Fig. P11.19

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713

Problems

11.22 Each member of the truss shown is made of steel and has the

cross-sectional area shown. Using E 5 29 3 10

psi, determine the

strain energy of the truss for the loading shown.

3 in

4 in

20 kips

24 kips

7.5 ft

4 ft

Fig. P11.22

11.23 Each member of the truss shown is made of aluminum and has

the cross-sectional area shown. Using E 5 72 GPa, determine the

strain energy of the truss for the loading shown.

11.24 throug h 11.27 Taking into account only the effect of normal

stresses, determine the strain energy of the prismatic beam AB for

the loading shown.

30 kN

80 kN

2500 mm

2000 mm

2.2 m

1 m

2.4 m

Fig. P11.23

Fig. P11.24

Fig. P11.25

Fig. P11.26

Fig. P11.27

11.28 and 11.29 Using E 5 200 GPa, determine the strain energy

due to bending for the steel beam and loading shown. (Ignore the

effect of shearing stresses.)

Fig. P11.28

180 kN

2.4 m 2.4 m

4.8 m

W360  64

Fig. P11.29

80 kN

W310  74

1.6 m 1.6 m 1.6 m

4.8 m

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714

Energy Methods

11.30 and 11.31 Using E 5 29 3 10

psi, determine the strain energy

due to bending for the steel beam and loading shown. (Ignore the

effect of shearing stresses.)

4 kips

W6  9

8 ft

2 ft

Fig. P11.30

60 in.

15 in. 15 in.

1.5 in.

3 in.

2 kips

Fig. P11.31

11.32 Assuming that the prismatic beam AB has a rectangular cross sec-

tion, show that for the given loading the maximum value of the

strain-energy density in the beam is

max

5 15

where U is the strain energy of the beam and V is its volume.

11.33 The ship at A has just started to drill for oil on the ocean floor at

a depth of 5000 ft. The steel drill pipe has an outer diameter of

8 in. and a uniform wall thickness of 0.5 in. Knowing that the top

of the drill pipe rotates through two complete revolutions before

the drill bit at B starts to operate and using G 5 11.2 3 10

psi,

determine the maximum strain energy acquired by the drill pipe.

11.34 Rod AC is made of aluminum and is subjected to a torque T applied

at C. Knowing that G 5 73 GPa and that portion BC of the rod is

hollow and has an inner diameter of 16 mm, determine the strain

energy of the rod for a maximum shearing stress of 120 MPa.

Fig. P11.32

5000 ft

Fig. P11.33

24-mm diameter

400 mm

500 mm

Fig. P11.34

11.35 Show by integration that the strain energy in the tapered rod AB

U 5

min

where J

min

is the polar moment of inertia of the rod at end B.

Fig. P11.35

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715

Problems

11.36 The state of stress shown occurs in a machine component made

of a grade of steel for which s

5 65 ksi. Using the maximum-

distortion-energy criterion, determine the factor of safety associated

with the yield strength when (a) s

5 116 ksi, (b) s

5 216 ksi.

11.37 The state of stress shown occurs in a machine component made

of a grade of steel for which s

5 65 ksi. Using the maximum-

distortion-energy criterion, determine the range of values of s

for

which the factor of safety associated with the yield strength is equal

to or larger than 2.2.

11.38 The state of stress shown occurs in a machine component made of

a brass for which s

5 160 MPa. Using the maximum-distortion-

energy criterion, determine the range of values of s

for which

yield does not occur.

8 ksi

14 ksi

Fig. P11.36 and P11.37

75 MPa

100 MPa

20 MPa

Fig. P11.38 and P11.39

11.39 The state of stress shown occurs in a machine component made of

a brass for which s

5 160 MPa. Using the maximum-distortion-

energy criterion, determine whether yield occurs when (a) s

145 MPa, (b) s

5 245 MPa.

11.4 0 Determine the strain energy of the prismatic beam AB, taking into

account the effect of both normal and shearing stresses.

*11.41 A vibration isolation support is made by bonding a rod A, of radius

, and a tube B, of inner radius R

, to a hollow rubber cylinder.

Denoting by G the modulus of rigidity of the rubber, determine

the strain energy of the hollow rubber cylinder for the loading

shown.

Fig. P11.40

(b)

(a)

Fig. P11.41

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716

Energy Methods

11.7 IMPACT LOADING

Consider a rod BD of uniform cross section which is hit at its end B

by a body of mass m moving with a velocity v

(Fig. 11.22a). As the

rod deforms under the impact (Fig. 11.22b), stresses develop within

the rod and reach a maximum value s

. After vibrating for a while,

the rod will come to rest, and all stresses will disappear. Such a

sequence of events is referred to as an impact loading (Photo 11.2).

In order to determine the maximum value s

of the stress

occurring at a given point of a structure subjected to an impact load-

ing, we are going to make several simplifying assumptions.

First, we assume that the kinetic energy T 5

of the strik-

ing body is transferred entirely to the structure and, thus, that the

strain energy U

corresponding to the maximum deformation x

(11.37)

This assumption leads to the following two specific requirements:

1. No energy should be dissipated during the impact.

2. The striking body should not bounce off the structure and

retain part of its energy. This, in turn, necessitates that the

inertia of the structure be negligible, compared to the inertia

of the striking body.

In practice, neither of these requirements is satisfied, and only

part of the kinetic energy of the striking body is actually transferred

to the structure. Thus, assuming that all of the kinetic energy of the

striking body is transferred to the structure leads to a conservative

design of that structure.

We further assume that the stress-strain diagram obtained from

a static test of the material is also valid under impact loading. Thus,

for an elastic deformation of the structure, we can express the maxi-

mum value of the strain energy as

(11.38)

In the case of the uniform rod of Fig. 11.22, the maximum

stress s

has the same value throughout the rod, and we write

5 s

Vy2E. Solving for s

and substituting for U

from Eq.

(11.37), we write

(11.39)

We note from the expression obtained that selecting a rod with a

large volume V and a low modulus of elasticity E will result in a

smaller value of the maximum stress s

for a given impact

In most problems, the distribution of stresses in the structure

is not uniform, and formula (11.39) does not apply. It is then

convenient to determine the static load P

, which would produce

the same strain energy as the impact loading, and compute from P

the corresponding value s

of the largest stress occurring in the

structure.

(a)

Area  A

v  0

(b)

Fig. 11.22 Rod subject to impact loading.

Photo 11.2 Steam alternately lifts a weight

inside the pile driver and then propels it

downward. This delivers a large impact load to

the pile that is being driven into the ground.

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