STRAIN! !

Deformation Strain! ! !

Normal Strain Shear Strain Cartesian Strain Components

1

Deformation

C C B A u(A) Deformed body Undeformed body A B

2

Strain

Normal Strain y

y

x z y y 1 x 1 x 1 1 z y

x

x

3

Shear Strain y y

2

x y y 1 xy

xy

x

x 1 x

4

Example 1 The slender rod shown subjected to an increase of temperature along its axis, which creates a normal strain in the rod of z = 40(10-3)z1/2, where z is given in meters. Determine (a) the displacement of the end B of the rod due to the temperature increase, and (b) the average normal strain in the rod.

A z dz 200 mm

B

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A z dz 200 mm

(a) the displacement of the end B of the rod due to the temperature increasedz ' = [40(10 3 ) z1/ 2 ]dz

B =

0.2 m

03

[40(10 3 ) z1/ 2 ]dz0.2 m 0

B

2 = 40(10 )( z 3 / 2 ) 3

= 0.00239 m

(b) the average normal strain in the rod.s ' s 2.39 mm = = 0.0119 mm / mm s 200 mm

avg =

6

Example 2 A force acting on the grip of the lever arm shown causes the arm to rotate clockwise through an angle of = 0.002 rad. Determine the average normal strain developed in the wire BC.

C 2L

B L A

7

2L C 2L C 2L L A Solution B B

Lsin

B

A

Lcos

L

Since is small, the stretch in wire CB, L sin L , is using Eq.2-6. The average normal strain in the wire is then

avg =

CB 'CB L 0.002 = = = = 0.001 CB 2L 2 2

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Example 3 The plate is deformed into the dashed shape shown. If in this deformed shape horizontal lines on the plate remain horizontal and do not change their length, determine (a) the average normal strain along the side AB, and (b) the average shear strain in the plate relative to the x and y axes. y 3 mm B 250 mm A C x 2 mm

300 mm

9

y 3 mm B B 250 mm A y 3 mm B 2 mm 250 mm A B 2 mm

(a) the average normal strain along the side ABAB ' = ( 250 2 ) 2 + ( 3) 2 = 248 . 018 mm( AB ) avg = AB ' AB 248.018 250 = AB 250

= 7.93(10 3 ) mm/mm

(b) the average shear strain in the plate relative to the x and y axes.

xy = xy = tan 1 (

2

'

xy 300 mm C x

3 ) = 0 . 0121 rad 250 2

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Example 4 The plate shown is held in the rigid horizontal guides at its top and bottom, AD and BC. If its right side CD is given a uniform horizontal displacement of 2 mm, determine (a) the average normal strain along the diagonal AC, and (b) the shear strain at E relative to the x, y axes. y A 150 mm E B 150 mm C 2 mm D x

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y A 150 mm E D

x 76 mm 76 mm A 75 mm 75 mm B 150 mm C 2 mm B D

E C

(a) the average normal strain along the diagonal ACAC = ( 0 . 150 ) 2 + ( 0 . 150 ) 2 = 0 . 21213 m AC ' = ( 0 . 150 ) 2 + ( 0 . 152 ) 2 = 0 . 21355 m( AC ) avg =AC ' AC 0.21355 0.21213 = = 0.00669 mm/mm AC 0.21213

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y A 150 mm E D

x 76 mm 76 mm A 75 mm 75 mm B 150 mm C 2 mm B D

E C

(b) the shear strain at E relative to the x, y axes.

' 76 tan( ) = 2 75 ' = 90 . 759 o = xy = ( 90 . 759 o ) = 1 . 58404 rad o 180 1 . 58404 = 0 . 0132 rad 2

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