mom2e chap2a

8/10/2019 MOM2E chap2A

1/16

2005 Pearson Education South Asia Pte Ltd

2. Strain

1

CHAPTER OBJECTIVES

Define concept of normal

strain

Define concept of shear

strain

Determine normal andshear strain in

engineering applications


2/16


2. Strain

2

CHAPTER OUTLINE

1. Deformation2. Strain


3/16


2. Strain

3

Deformation

Occurs when a force is applied to a body

Can be highly visible or practically unnoticeable

Can also occur when temperature of a body is

changed

Is not uniform throughout a bodys volume, thus

change in geometry of any line segment within

body may vary along its length

2.1 DEFORMATION


4/16


2. Strain

4

To simplify study of deformation

Assume lines to be very short and located in

neighborhood of a point, and

Take into account the orientation of the line

segment at the point

2.1 DEFORMATION


5/16


2. Strain

5

Normal strain

Defined as the elongation or contraction of a linesegment per unit of length

Consider lineABin figure below

After deformation, schanges to s

2.2 STRAIN


6/16


2. Strain

6

Normal strain

Defining average normal strainusing avg(epsilon)

As s 0, s 0

2.2 STRAIN

avg=s s

s

=

s s

s

lim

BA along n


7/162005 Pearson Education South Asia Pte Ltd

2. Strain

7

Normal strain

If normal strain is known, use the equation to

obtain approx. final length of a shortline segment

in direction of nafter deformation.

Hence, when is positive, initial line will elongate,if is negative, the line contracts

2.2 STRAIN

s (1 + )s



2. Strain

8

2.2 STRAIN

Units

Normal strain is a dimensionless quantity, asits a ratio of two lengths

But common practice to state it in terms ofmeters/meter (m/m)

is small for most engineering applications, sois normally expressed as micrometers permeter (m/m)where1m = 106

Also expressed as a percentage,e.g.,0.001 m/m = 0.1 %



2. Strain

9

2.2 STRAIN

Shear strain

Defined as the change in anglethat occurs

between two line segments that were originally

perpendicularto one another

This angle is denoted by (gamma) andmeasured in radians (rad).



2. Strain

10

2.2 STRAIN

Shear strain

Consider line segmentsABandACoriginating

from same pointAin a body, and directed along

the perpendicular nand taxes

After deformation, lines become curves, such thatangle between them atAis



2. Strain

11

2.2 STRAIN

Shear strain

Hence, shear strain at pointAassociated with n

and taxes is

If is smaller than /2, shear strain is positive,otherwise, shear strain is negative

nt=

2

lim

BA along n

C A along t



2. Strain

12

Cartesian strain components

Using above definitions of normal and shear strain,

we show how they describe the deformation of the

body

2.2 STRAIN

Divide body into smallelements with

undeformed dimensions

ofx,yandz



2. Strain

13


Since element is very small, deformed shape of

element is a parallelepiped

Approx. lengths of sides of parallelepiped are

(1 + x) x (1 + y)y (1 + z)z

2.2 STRAIN



2. Strain

14


Approx. angles between the sides are

2.2 STRAIN

2 xy

2 yz

2 xz

Normal strains cause a change in its volume

Shear strains cause a change in its shape

To summarize, state of strain at a point requires

specifying 3 normal strains; x, y, zand 3 shear

strains of xy,yz,xz



2. Strain

15

Small strain analysis

Most engineering design involves applications

for which only small deformationsare allowed

Well assume that deformations that take place

within a body are almost infinitesimal, so normalstrainsoccurring within material are very small

compared to 1, i.e.,


16/162005 P Ed ti S th A i Pt Ltd

2. Strain

16

Small strain analysis

This assumption is widely applied in practical

engineering problems, and is referred to as

small strain analysis

E.g., it can be used to approximate sin = , cos= and tan = , provided is small

2.2 STRAIN

mom2e chap2a

Documents