Chapter 1-Strain Analysis

8/13/2019 Chapter 1-Strain Analysis

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CHAPTER 1(week 1-3)Strain Analysis


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Plain Strain


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GENERAL EQUATION OF PLANE STRAINTRANSFORMATION

Normal strain ex and e y arepositive if cause elongationalong x and y axisShear strain gxy is positiveif the interior angle AOBbecome smaller than 90 0.q0 will be positive using theright-hand fingers.i.e counterclockwise


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Normal and Shear Strains

In Fig a :

Positive ex occurline d

x elongated e

xd x , which causeline dx toelongated ex d x cos

q.

q

q

sin

cos'

'

dxdy

dxdx


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Positive e y occur line d

y elongated e

yd y , which cause line d y to elongated

e y d y sin q. Assuming dx fix in position. Shear strain g xy changes in angle

between dx and dy. dy displaced g xy dy to the right.

Dx elongateg xy dy cos q

Normal and Shear Strains (cont.)


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Adding all the elongations

qqgqeqee

qqgqqeqqee

e

cossinsincosdx

cos)sindx(sin)sindx(cos)cosdx(dx

x

xy2

y2

xx

'

'xy

'y

'x

x

'

'

x

'

'

'

qgqeqe cosdysindycosdx xyyxx '


q q

sincos

'

'

dxdydxdx


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qqgqeqee cossinsincos x y2y2xx '


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qg

qeeee

e 2sin2

2cos22

xyyxyx

x '

qg

qeeee

e 2sin2

2cos22

x yyxyx

y '

Using trigonometric identities:

qg

q

eeg

2cos2

2sin22

xyyxyx ''


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Principal Strains

Direction axis of principlestrain:

Max in Plane shear strain

Ave shear strain

Direction axis of shearstrain

2

xy

2

yx planeinmax,

222

g

eeg

2

yx

ave

eee

)(2tan

yx

xy p ee

gq

g

eeqxy

yxs2tan

2

xy

2

yxyx2,1 222

g

eeeee


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Construction of the Mohrs Circle PROCEDURE FOR ANALYSIS

The procedure for drawing Mohr's circle for strainfollows the one established for stress.Construction of the Circle. Establish a coordinate system such that the

abscissa represents the normal strain e, withpositive to the right, and the ordinate representshalf the value of the shear strain, g/2, with positivedownward .

Using the positive sign convention for ex, e y, gxy asshown in the Fig. , determine the center of the circleC, which is located on the e axis at a distancee avg= (e x + e y)/2 from the origin.

Plot the reference point A having coordinatesA(e x, g xy/2). This point represents the case forwhich the x' axis coincides with the x axis. Henceq = q .

Connect point A with the center C of the circle andfrom the shaded triangle determine the radius R ofthe circle.

Once R has been determined, sketch the circle.


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The principal strains e1 and e2 are determinedfrom the circle as the coordinates of points Band D, that is where g/2 = 0, Fig. a.

The orientation of the plane on which e1 acts canbe determined from the circle by calculating 2 qP1,using trigonometry. Here this angle is measuredcounterclockwise from the radial reference lineCA to line CB, Fig. a. Remember that the rotationof qP1 must be in this same direction, from the

element's reference axis x to the x' axis, Fig. b.* When e1 and e2 are indicated as being positive as

in Fig. a, the element in Fig. b will elongate in thex' and y' directions as shown by the dashedoutline.

Principal Strain


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The average normal strain and half themaximum in-plane shear strain are determinedfrom the circle as the coordinates of points Eand F, Fig. a.

The orientation of the plane on which gmax in plane and eavg act can be determined from the circleby calculating 2 qs1, using trigonometry. Herethis angle is measured clockwise from the radialreference line CA to line CE, Fig. a. Rememberthat the rotation of qs1, must be in this same

direction, from the element's reference axis xto the x' axis, Fig. c.

Maximum In Plane Shear Strain


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The normal and shear strain components ex, and

g xy for a plane specified at an angle q, Fig. d, canbe obtained from the circle using trigonometry todetermine the coordinates of point P, Fig.a.

To locate P, the known angle q of the x' axis ismeasured on the circle as 2 q. This measurement ismade from the radial reference line CA to theradial line CP. Remember that measurements for2q on the circle must be in the same direction as q for the x' axis!

If the value of e y, is required, it can bedetermined by calculating the e coordinate ofpoint Q in Fig. a. The line CQ lies 180 o away from

CP and thus represents a rotation of 900

of the x'axis.

Strains on Arbitrary Plane


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We can not measure stresses within a structuralmember,

Instead we can measure strains and from them

the stresses can be computed Even so, we can only measure strains on the

surface Also in view of the very small changes in

dimensions, it is difficult to achieve accuracy inthe measurements

Strain Gauge and Rosette


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In practice, electrical strain gage provide a moreaccurate and convenient method of measuring strains.

A typical strain gauge is shown below. The gage shown can measure normal

strain in the local plane of the surface inthe direction of line PQ, which is parallelto the folds of paper.

This strain is an average value of forthe region covered by the gage, ratherthan a value at any particular point.

Strain Gauge and Rosette (cont)


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The strain gauge is not sensitive to normal strain in the direction perpendicular to

PQ, nor does it respond to shear strain.

Therefore, in order to determine the state of strain ata particular small region of the surface, we need morethan one strain gage.

To define a general two dimensional state of strain, weneed to have three pieces of information, such as

ex , e y and gxy We therefore need to obtain measurements from threestrain gages.

These three gages must be arranged at differentorientations on the surface to from a strain rossette.


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A group of three gages arranged in a particular fashion iscalled a strain rosette. Rosette is mounted on the surface of the body, where the

material is in plane stress, therefore, the transformationequations for plane strain to calculate the strains in variousdirections.

The axes of the three gauges are arranged at the angles of qa,qb, qc.

If the reading of ea, eb, ec taken, ex, e y, gxy can be defined. Value of ex, e y, gxy are determined by solving these equations.

ccxyc2

yc2

xc

b bxy b2

y b2

x b

aaxya2

ya2

xa

cossinsincos

cossinsincos

cossinsincos

qqgqeqeeqqgqeqeeqqgqeqee


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45 o or Rectangular Rosette

0

0

0

90

45

0

c

b

a

q

q

q

The equation become:

ca bxycy

ax

2 eeegeeee

Example of 45 o strain rosette


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60 0 Strain Rosette

c bxy

ac by

ax

32

2231

eeg

eeee

ee

0c

0 b

0a

120

60

0

q

q

q

The equation become:


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Example


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Exercise

Figure shows 60 strain rosette attached on the mechanicalcomponent to measure surface strains. The reading of thestrains measured by this gauge is as follows:

a = 1000,

b = 750,c = -650Determine:(i) the principal strains,

(ii) the maximum shearing strain and(iii) the principal angles


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Solution Applying Transformation equation

Solving the above equations , we get ex=1000 , e y=-266.7 and gxy=-1616.6

ccxyc2

yc2

xc

b bxy b2

y b2

x b

aaxya2

ya2

xa

cossinsincos

cossinsincos

cossinsincos

qqgqeqeeqqgqeqeeqqgqeqee

geegee

gee

60cos60sin60sin60cos650



xy2

y2

x

xy2

y2

x

xy2

y2

x

l ( d )


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Solution (Contd.) Applying the principal strain equation or using Mohrs strain

circle, we get

e1=1394 and e2-660 and gxy=-1616.6 Max Shear Strain is

= 2050

Direction of principal planes

i.e., q1 =-25.9 or 64.1 andq2 =154.1(bcas 2 q2=180+2q1)

2xy

2yxyx

2,1 222

g

eeeee

2xy

2yxmax

222

g

eeg

)(2tan

yx

xy p ee

gq


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Stress Strain Relationship If a material subject to triaxial

stress ( s x, s y, s z), associatednormal stress( ex, e y, ez)developed in the material.

When s x is applied in x-direction, the element elongatedwith ex in x direction.

Application on s y cause theelement to contract with astrain e x in the x direction.

Application Of s z cause the

element to contract with astrain ex in the x direction. z

x

y x

x x

s e

s e

s e

'''

''

'


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The same result can bedeveloped for the normalstrain in the y and zdirection.

Final results can bewritten as..

yxzz

zxyy

zyxx

1

1

1

ssse

ssse

ssse

Stress Strain Relationship (cont.)


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Applying only shear stress, y to the element.If to apply shear stress, t y to the element.t xy will only cause deformation to gxy.t xy will not cause deformation to g yz.and gxzt yz and t xz will only cause deformation to

g yz and gxz respectively. Hooke Law for shear stress and shearstrain written as:

xz xz

yz yz

xy xy

G

G

G

t g

t g

t g

1

1

1

Element subjected to normalstresses only Shear stress applied to theelements



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Modulus of elasticity, E isrelated to shear modulus, G.

Dilatation (the change involume per unit volume orvolumetric strain,e .

Bulk Modulus (volumemodulus of elasticity), k .

12 E

G

zyxE21 sssue


213 E

k


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Chapter 1-Strain Analysis