c2_strainanalysis

7/25/2019 C2_StrainAnalysis

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SME 2123

MECHANICS OF SOLIDS IIDr. Zaini AhmadB.Eng (Hons)(Mech), M.Sc (Structures &Crashworthiness), Ph.D (Structural Impact)Department of Applied Mechanics & Design,Faculty of Mechanical Engineering,

Universiti Teknologi Malaysia,

81310 Skudai, Johor, Malaysia.

Tel : (+607) 553 4647(Office)Tel : (+6012) 717 0581 (Mobile)Email: [email protected]


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Mechanics of Solids II

Strain Analysis


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Strain Analysis

(Plane Strain)

General state of strain at a point in a body is represented

by 3 normal strains and 3 shear strains .

The normal and shear strain componentswill vary

according to the orientation of the element

zyx

zyx


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Plane strainone dimension is very large compared to the other

two (z= 0, xz= 0 and yz= 0) (e.g. tunnel, dam, retaining wall, etc)

Strain Analysis

(Plane Strain)

Retaining wall Earth dam


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Strain Analysis

(Plane Strain)

cossinsincos

cossincos

sin

cos

22

'

'

'

'

xyyxx

xyyx dydydxdxdydy

dxdx


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Strain Analysis

(Plane Strain Transformation)

Strain transformation equations:-

2cos2

2sin22

2sin

2

2cos

22

2sin2

2cos22

''

'

'

xyyxyx

xyyxyx

y

xyyxyx

x


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Strain Analysis


Principal strain:-

Maximum In-Plain Shear Strain - the associated average

normal strain:-

22

2,1222

2tan

xyyxyx

yx

xy

p

2,

222

2tan

22

plane-inmax yx

avg

xyyx

xy

yx

S


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A differential element of material at a point is subjected to a state of plane strain

defined by which tends to distort the

element. Determine:-i) The principal strain at the point and the associated orientation of the element

and show the state of strain on the principal plane

ii) The maximum in-plane shear strain at the point and the associated orientation

of the element and sketch the distorted element.

iii) The equivalent strains acting on an element oriented at the point clockwise

60ofrom the original position

x 350 106 , y 200 106 , xy 80 106

Strain Analysis


Example 1:-


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Orientation of the element:-

72.1711808.28-and28.82

)10(200350

)10(802tan

6

6

P

yx

xy

P

Strain Analysis


85.9and14.4P

22

2,1222

xyyxyx

6

226

102

80

2

200350

2

10200350

)10(353

)10(203

6

2

6

1

Substitute = -4.14ointo strain transformation eq.

to determine the deformation of the element


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Orientation of the

element:-

For maximum in-plane shear strain,

311and9.40

80

2003502tan

s

xy

yx

s

6planeinmax

22

planeinmax

10556

222

xyyx

Strain Analysis


)10(75)10(2

200350

2

66

yx

avg


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Strain Analysis


86.177

2sin2

2cos22

2sin22cos22

16.258)60(2cos2

80)60(2sin

2

200350

2

2cos2

2sin22

86.27)60(2sin2

80)60(2cos

2

200350

2

200350

2sin2

2cos22

'

'

'

''

''

'

'

y

xyyxyx

y

xyyxyx

y

ooyx

xyyxyx

oo

x

xyyxyx

x

(* if = -60ois used)

(* if = -60o+90o=30ois used)


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Strain Analysis

(Mohrs Circle - Plane Strain)

Solve problems involving the transformation of strain usingMohrs circle

It has a center on the axis at point C(avg, 0) and a radius R


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Example 2:-

The state of plane strain at a point is represented by the components

Determine the maximum in-plane shear strains and the orientation of an

element.

Strain Analysis


666 10120,10150,10250 xyyx


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From the coordinates of point E, we have

To orient the element,we can determine the clockwise

angle,

6planeinmax''

6planeinmax''

10418

108.2082

yx

yx

7.36

35.829050250

60tan902

1

1

1

s

s

Strain Analysis


6622

66

108.208102

120

2

)150(250

1050102

)150(250

R

avg


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666 10750,10760,10520 xyyx

Example 3:-The state of strain at the point on the gear tooth has the components .

Use the strain transformation equations to determine:-

a) The in-plane principal strains

b) The maximum in-plane shear strain and average normal strain

In each case, specify the orientation of the element and show how the strains

deform the element within the x-y plane

Strain Analysis


Ans (1= 622, 2= -862, avg=-120)


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Strain Analysis


Absolute maximum shear strain

Circle with the largest radius

It occurs on the element oriented 45oabout the axis

2

minmax

minmaxmaxabs

avg


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Strain Analysis


For plain strain:-

maxmax''maxabs

zx minmaxmax''maxabs yx


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Example 4:-

The state of plane strain at a point is represented by the strain components,

Determine the maximum in-plane shear strain and the absolute maximum shear

strain.

Strain Analysis


x 400 106 , y 200 106 , xy150 106


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From the strain components, the centre of the circle is on the axis at

Since , the reference point has coordinates

66 10100102

200400

avg

A400 106 ,75 106

Thus the radius of the circle is 9622 103091075100400

R

610752

xy

Strain Analysis


Computing the in-plane principal strains, we have

66

min

66

max

1040910309100

1020910309100


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From the circle, the maximum in-plane shear strain is

66minmaxplaneinmax 1061810409209max abs

10409,0,10209 6minint

6

max

From the above results, we have

Thus the Mohrs circle is as follows,

Strain Analysis



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Strain Analysis

(Strain Rosettes)

Normal strain in a tension-test specimen can bemeasured using an electrical-resistance strain gauge

strain rosettes.

The strain-transformation equation for each gauge:-

ccxycycxc

bbxybybxb

aaxyayaxa

cossinsincos

cossinsincos

cossinsincos

22

22

22


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Strain Analysis

(Strain Rosettes)

Two types of typical strain measurement using strainrosettes45oand 60orosettes


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Strain Analysis

(Strain Rosettes)

Example 5:-

The state of strain at pointA on the bracket is measure using the strain rosette

as shown. Due to the loadings, the readings from the gauge give a= 60(10-6),

b= 135(10-6) and c= 264(10

-6). Determine the in-plane principal strains at the

point and the directions in which they act.


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Measuring the angles counter-clockwise,

By substituting the values into the 3 strain-transformation equations, we have

120and60,0 cba

666 10149,10246,1060 xyyx Using Mohrs circle, we have A(60(10-6), 74.5(10-6)) and center C (153(10-6), 0).

3.19,108.33,10272

101.119105.7460153

p2

6

2

6

1

6622

R

Strain Analysis

(Strain Rosettes)


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Strain Analysis

(Strain Rosettes)

Example 6:-

The 45ostrain rosette is mounted near the tooth of the wrench. The following readings

are obtained for each gauge give a= 800(10-6), b= 520(10

-6) and c= -450(10-6).

Determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain

and associated average normal strain. In each case, show the deformed element due

to these strains.

Ans (1= 889, 2= -539, max/2 =1428, avg=175)


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Strain Analysis

(Material Property Relationship)

Generalized Hookes Law:-

For a tri-axial state of stress, the general form for

Hookes law is as follow:

Valid only for a linearelasticmaterials

Hookes law for shear stress and shear strain is

written as

xzxzyzyzxyxyGGG

1

1

1

yxzzzxyyzyxx vEvEvE 1

,1

,1


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Strain Analysis


Relationship involving E, vand G:-

Bulk modulus - a measure of the stiffness of a volumeof material

vE

G

12

v

Ek

213


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Strain Analysis


Example 7:-

The copper bar is subjected to a uniform loading along its edges. If

it has a = 300 mm, b = 50 mm, and t = 20 mm before load is applied,

find its new length, width and thickness after application of the load.

Take 34.0,GPa120 cucu

vE


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Strain Analysis


From the loading we have

The associated normal strains are determined from the generalized Hookes law,

0,MPa500,MPa800 zyx

000850.0,00643.0,00808.0 yxzzzxyyzyxx Ev

EE

v

EE

v

E

The new bar length, width, and thickness are therefore

mm98.1920000850.020'

mm68.495000643.050'

mm4.30230000808.0300'

t

b

a


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Strain Analysis


Example 7: Final exam question

A cantilevered T-structure made from 40

mm-diameter bars, supports forces 1 kN and

3 kN. The material is mild steel which the

elastic modulus is 200 GPa, the shear

modulus is 75 GPa and Poissons ratio is 0.3.Point E which is on the side of the bar AB, is

to be analyzed.

i) Find the principal stresses at point E and

sketch the orientation and deformation

of the element in this condition

ii) Find the in-plane maximum shear stress

at point E and sketch the orientation anddeformation of the element in this

condition.

iii) At point E, calculate the normal strain in

the direction 30oto the axial direction as

shown in this figure


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Three strain gauges were fixed onto a surface of a automotive frame.

Under loading, the readings of the gauges were:-

a) By sketching Mohrs circle and using trigonometry method,

determine the value of the principal strains and maximum shearstrain, and draw the orientation and the element under these

conditions. Verify your answers using the strain transformation

equations.

b) Calculate x, y and xyusing material properties relation. By

plotting Mohrs circle on graph paper, determine the values of

the principle stresses and maximum shear stress, and draw the

orientation and the element under this condition. Also, verify your

answers using the stress transformation equations (E = 200

GPa, G = 75 GPa, v= 0.35)

c) Determine the values of the normal stress, shear stress, normal

strain and shear strain at direction P4o(clockwise) from the x-

axis and draw the orientation of element for each condition

Strain Analysis

Assignment 1:-

P1a

P2b

P3c

P5o

25o

a

bc

x

y

Due date for submission: 11 Oct 2013


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Strain Analysis

Group 1P1= 220

P2= -130

P3= 100

P4= 25o

P5= 65o

Group 2P1= -220

P2= 130

P3= 150

P4= 20o

P5= 45o

Group 3P1= 150

P2= -180

P3= -110

P4= 38o

P5= 60o

Group 4P1= 190

P2= -100

P3= -95

P4= 30o

P5= 53o

Group 5P1= 150

P2= -130

P3= -100

P4= 32o

P5= 50o

Group 6

P1= -300

P2= -180

P3= 110

P4= 30o

P5= 75o

Group 7

P1= 280

P2= 100

P3= -150

P4= 40o

P5= 55o

Group 8

P1= -190

P2= 130

P3= -120

P4= 55o

P5= 70o

Group 9

P1= -300

P2= -200

P3= -95

P4= 35o

P5= 80o

Group 10

P1= 320

P2= -230

P3= -130

P4= 15o

P5= 77o

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