Chapter ObjectivesChapter Objectives

Understand the concept of normal and shear strainUnderstand the concept of normal and shear strain

Apply the concept to determine the strains for Apply the concept to determine the strains for various types of problemsvarious types of problems

1.1. Normal StrainNormal Strain

2.2. Shear StrainShear Strain

3.3. Cartesian Strain ComponentsCartesian Strain Components

In-class ActivitiesIn-class Activities

Copyright © 2011 Pearson Education South Asia Pte Ltd

NORMALNORMAL STRAINSTRAIN

s

ssavg

'

ss 1'

s

ssnAB

'lim

along

SHEAR STRAINSHEAR STRAIN

'lim2

along along

tACnAB

nt

CARTESIAN STRAINCARTESIAN STRAIN• The approximate lengths of the sides The approximate lengths of the sides

of the parallelepiped areof the parallelepiped are

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

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

zyx zyx 1 1 1

xzyzxy

2

2

2

READING QUIZ READING QUIZ

1)1) The center portion of the rubber balloon has a The center portion of the rubber balloon has a diameter of d = 100 mm. If the air pressure within diameter of d = 100 mm. If the air pressure within it causes the balloon’s diameter to become d = it causes the balloon’s diameter to become d = 125mm, determine the average normal strain in 125mm, determine the average normal strain in the rubberthe rubber.

a)a) 0.20.2

b) 0.25 π

c)c) 0.250.25

d)d) 1.251.25Copyright © 2011 Pearson Education South Asia Pte Ltd

READING QUIZREADING QUIZ

2) What is the unit of strain?What is the unit of strain?

a)a) mmmm

b)b) mm/mmm/m

c)c) MicronMicron

d)d) no unitno unit

Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 1

dzzdz 2/1310401'

EXAMPLE 1 SOLUTION EXAMPLE 1 SOLUTION

Part (a)Part (a)• Since the normal strain is reported at each point along Since the normal strain is reported at each point along

the rod, it has a deformed length ofthe rod, it has a deformed length of

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

• The displacement of the end of the rod is thereforeThe displacement of the end of the rod is therefore

dzzdz 2/1310401'

m 20239.010401'2.0

0

2/13 dzzz

(Ans) mm39.2m00239.02.020239.0 B

EXAMPLE 1 Solutions (cont)

Part (b)Part (b)• Assumes the rod has an original length of 200 Assumes the rod has an original length of 200

mm and a change in length of 2.39 mm. Hence,mm and a change in length of 2.39 mm. Hence,

(Ans) mm/mm 0119.0200

39.2'

s

ssavg

EXAMPLE 2

Due to a loading, the plate is deformed into the Due to a loading, the plate is deformed into the dashed shape shown in Fig. 2–6dashed shape shown in Fig. 2–6aa. Determine (a) . Determine (a) the average normal strain along the side the average normal strain along the side ABAB, and , and (b) the average shear strain in the plate at (b) the average shear strain in the plate at A A relative to therelative to the and y axesand y axes..

EXAMPLE 2 EXAMPLE 2 Solutions (cont)(cont)

Part (a)• Line AB, coincident with the y axis, becomes line after

deformation, thus the length of this line is

• The average normal strain for AB is therefore

• The negative sign indicates the strain The negative sign indicates the strain causes a contraction of causes a contraction of ABAB..

mm 018.24832250' 22 AB

( ) ( ) (Ans) mm/mm 10 -7.93=250

250-248.018

AB

AB'AB 3-

avgAB ==-

ε

EXAMPLE 2 EXAMPLE 2 Solutions Solutions (cont)(cont)

CONCEPT QUIZ CONCEPT QUIZ

1) The rectangular membrane has an unstretched The rectangular membrane has an unstretched length L1 and width L2. If the sides are increased by length L1 and width L2. If the sides are increased by small amounts small amounts ΔΔL1 and L1 and ΔΔL2, determine the normal L2, determine the normal strain along the diagonal AB. strain along the diagonal AB.

2

221

22

21

21

21

22

21

22

21

2

2

1

1

D) C)

B) A)

LL

LL

LL

LL

LL

LL

L

L

L

L

L1

Δ L2

ΔL1

L2

CONCEPT QUIZ (cont) CONCEPT QUIZ (cont)

2) The rectangular plate is subjected to the The rectangular plate is subjected to the deformation shown by the dashed line. deformation shown by the dashed line. Determine the average shear strain Determine the average shear strain γγxyxy of the of the plate.plate.

22 200150

3

200

150

200

3

150

3

D) C)

B) A)

Thank YouThank You