Stress & Strain

Chapter Objectives

To show how to use the method of sections for determining the internal loadings in a member.

To introduce the concepts of normal and shear stress.

To analyse and design members subject to axial load and direct shear.

To define normal and shear strain, and show how they can be determined for various types of problems.

Copyright 2011 Pearson Education South Asia Pte Ltd

Reading Quiz

Applications

Normal Stress

Shear Stress

Normal Strain

Shear Strain

Cartesian Strain Components

Concept Quiz

In-Class Activities


READING QUIZ

1. What is the normal stress in the bar if P=10

kN and 500 mm?

a) 0.02 kPa

b) 20 Pa

c) 20 kPa

d) 200 N/mm

e) 20 MPa


READING QUIZ (cont)

2. What is the average shear stress in the

internal vertical surface AB (or CD), if F=20

kN, and AAB

=ACD

=1000 mm?

a) 20 N/mm

b) 10 N/mm

c) 10 kPa

d) 200 kN/m

e) 20 MPa


READING QUIZ (cont)

3. The center portion of the rubber balloon has

a diameter of d = 100 mm. If the air pressure

within it causes the balloons diameter to become d = 125 mm, determine the average

normal strain in the rubber.

a) 0.2

b) 0.25

c) 0.25

d) 1.25


READING QUIZ (cont)

4. What is the unit of strain?

a) mm

b) mm/m

c) Micron

d) no unit


APPLICATIONS


Where does the average normal stress occur?

APPLICATIONS (cont)


Will the total shear force over the anchor length be equal to

the total tensile force tensile A in the bar?

APPLICATIONS


AVERAGE NORMAL STRESS


Will the total shear force over the anchor length be equal to the total tensile force tensile A in the bar?

A

P

NORMAL SHEAR STRESS


A

FzA

z

0lim

A

F

A

F

y

Azy

x

Azx

0

0

lim

lim

EXAMPLE 1


The bar in Fig. 7-14a has a constant width of 35 mm and a thickness of 10 mm. Determine the maximum average normal stress in the bar when it is subjected to the loading shown.

EXAMPLE 1 (cont)


Graphically, the normal force diagram is as shown.

By inspection, different sections have different internal forces.

Solution

EXAMPLE 1 (cont)


By inspection, the largest loading is in region BC,

Since the cross-sectional area of the bar is constant, the largest average normal stress is

Solution

kN 30BCP

(Ans) MPa 7.8501.0035.0

1030 3

A

PBCBC

DESIGN OF SIMPLE CONNECTION


For normal force requirement

For shear force requirement

allow

PA

allow

VA

EXAMPLE 2


The rigid bar AB shown in Fig. 129a is supported by a steel rod AC having a diameter of 20 mm and an aluminum block having a cross sectional area of 1800 mm2. The 18-mm-diameter pins at A and C are subjected to single shear. If the failure stress for the steel and aluminum is and , respectively, and the failure shear stress for each pin is , determine the largest load P that can be applied to the bar. Apply a factor of safety of FS=2.

MPa 680failst

MPa 70failal

MPa 900fail

EXAMPLE 2 (cont)


The allowable stresses are

There are three unknowns and we apply the equations of equilibrium,

Solution

MPa 4502

900

..

MPa 352

70

..

MPa 3402

680

..

SF

SF

SF

fail

allow

failal

allowal

failst

allowst

(2) 075.02 ;0

(1) 0225.1 ;0

PFM

FPM

BA

ACB

EXAMPLE 2 (cont)


We will now determine each value of P that creates the allowable stress in the rod, block, and pins, respectively.

For rod AC,

Using Eq. 1,

For block B,

Using Eq. 2,

Solution

kN 8.10601.010340 26 ACallowstAC AF

kN 171

25.1

28.106P

kN 0.631018001035 66 BallowalB AF

kN 168

75.0

20.63P

EXAMPLE 2 (cont)


For pin A or C,

Using Eq. 1,

When P reaches its smallest value (168 kN), it develops the allowable normal stress in the aluminium block. Hence,

Solution

kN 5.114009.010450 26 AFV allowAC

kN 183

25.1

25.114P

(Ans) kN 168P

NORMAL STRAIN


s

ssavg

'

ss 1'

s

ss

nAB

'lim

along

SHEAR STRAIN


'lim2

along along

tACnAB

nt

CARTESIAN STRAIN COMPONENTS


The approximate lengths of the sides of the parallelepiped are

The approximate angles between sides, again originally defined by the sides x, y and z are

Notice that the normal strains cause a change in volume of rectangular element, whereas the shear strain cause a

change in shape

zyx zyx 1 1 1

xzyzxy

2

2

2

CARTESIAN STRAIN COMPONENTS (cont)


EXAMPLE 3


The slender rod creates a normal strain in the rod of

where z is in meters. Determine (a) displacement of end B

due to the temperature increase, and (b) the average normal

strain in the rod.

2/131040 zz

EXAMPLE 3 (cont)


Part (a)

Since the normal strain is reported at each point along the rod, it has a deformed length of

The sum along the axis yields the deformed length of the rod is

The displacement of the end of the rod is therefore

Solution

dzzdz 2/1310401'

m 20239.010401'2.0

0

2/13 dzzz

(Ans) mm39.2m00239.02.020239.0 B

EXAMPLE 3 (cont)


Part (b)

Assumes the rod has an original length of 200 mm and a change in length of 2.39 mm. Hence,

Solution

(Ans) mm/mm 0119.0200

39.2'

s

ssavg

EXAMPLE 4


Due to a loading, the plate is deformed into the dashed shape

shown in Fig. 26a. Determine (a) the average normal strain along the side AB, and (b) the average shear strain in the plate

at A relative to the x and y axes.

EXAMPLE 4 (cont)


Part (a)

Line AB, coincident with the y axis, becomes line AB after deformation, thus the length of this line is

The average normal strain for AB is therefore

The negative sign indicates the strain causes a contraction of AB.

Solution

mm 018.24832250' 22 AB

(Ans) mm/mm 1093.7240

250018.248' 3

AB

ABABavgAB

EXAMPLE 4 (cont)


Part (b)

As noted, the once 90 angle BAC between the sides of the plate, referenced from the x, y axes, changes to due to the displacement of B to B.

Since then is the angle shown in the figure.

Thus,

Solution

'2

xy xy

(Ans) rad 121.02250

3tan 1

xy

CONCEPT QUIZ

1) The thrust bearing is subjected to the loads

as shown. Determine the order of average

normal stress developed on cross section

through B, C and D.

a) C > B > D

b) C > D > B

c) B > C > D

d) D > B > C


CONCEPT QUIZ (cont)

2) The rectangular membrane has an

unstretched length L1 and width L2. If the

sides are increased by small amounts L1 and L2, determine the normal strain along the diagonal AB.


2221

2

2

2

1

21

21

2

2

2

1

2

2

2

1

2

2

1

1

D) C)

B) A)

LL

LL

LL

LL

LL

LL

L

L

L

L

CONCEPT QUIZ (cont)

3) The rectangular plate is subjected to the

deformation shown by the dashed line.

Determine the average shear strain xy

of the

plate.


22 2001503

200

150

200

3

150

3

D) C)

B) A)

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Stress & Strain