PLANE STRAIN TRANSFORMATION. 2 Apply the stress transformation methods derived in Chapter 9 to similarly transform strain Discuss various ways of measuring

CHAPTER 10PLANE STRAIN TRANSFORMATION

CHAPTER OBJECTIVES

2

• Apply the stress transformation methods derived in Chapter 9 to similarly transform strain

• Discuss various ways of measuring strain

• Develop important material-property relationships; including generalized form of Hooke’s law

• Discuss and use theories to predict the failure of a material

CHAPTER OUTLINE

1. Plane Strain2. General Equations of Plane-Strain

Transformation3. Strain Rosettes4. Material-Property Relationships

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10.1 PLANE STRAIN

As explained in Chapter 2.2, general state of strain in a body is represented by a combination of 3 components of normal strain (x, y, z), and 3 components of shear strain (xy, xz, yz).

Strain components at a pt determined by using strain gauges, which is measured in specified directions.

A plane-strained element is subjected to two components of normal strain (x, y) and one component of shear strain, xy.

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10.1 PLANE STRAIN The deformations are shown graphically

below. Note that the normal strains are produced

by changes in length of the element in the x and y directions, while shear strain is produced by the relative rotation of two adjacent sides of the element.

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10.1 PLANE STRAIN Note that plane stress does not always

cause plane strain. In general, unless = 0, the Poisson effect

will prevent the simultaneous occurrence of plane strain and plane stress.

Since shear stress and shear strain not affected by Poisson’s ratio, condition of xz = yz = 0 requires xz = yz = 0.

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10.2 GENERAL EQNS OF PLANE-STRAIN TRANSFORMATION

Sign Convention To use the same convention as

defined in Chapter 2.2. With reference to differential

element shown, normal strains xz and yz are positive if they cause elongation along the x and y axes

Shear strain xy is positive if the interior angle AOB becomes smaller than 90.

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Sign Convention Similar to plane stress, when measuring the

normal and shear strains relative to the x’ and y’ axes, the angle will be positive provided it follows the curling of the right-hand fingers, counterclockwise.

Normal and shear strains Before we develop the

strain-transformation eqn for determining x;, we must determine the elongation of a line segment dx’ that lies along the x’ axis and subjected to strain components.

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Normal and shear strains Components of line dx and dx’ are elongated

and we add all elongations together.

From Eqn 2.2, the normal strain along the line dx’ is x’ =x’/dx’. Using Eqn 10-1,

cossincos' dydydxx xyyx

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210cossinsincos 22' - xyyxx


Normal and shear strains To get the transformation equation for x’y’,

consider amount of rotation of each of the line segments dx’ and dy’ when subjected to strain components. Thus, sincossin' dydydxy xyyx

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Normal and shear strains Using Eqn 10-1 with = y’/x’,

As shown, dy’ rotates by an amount .

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310sincossin 2 - xyyx


Normal and shear strains Using identities sin ( + 90) = cos ,

cos ( + 90) = sin ,

Thus we get

410sincoscossin2 22

''

-

xyyx

yx

12

2

2

cossincos

90sin90cos90sin

xyyx

xyyx


Normal and shear strains Using trigonometric identities sin 2 = 2 sin

cos, cos2 = (1 + cos2 )/2 and sin2 + cos2 = 1, we rewrite Eqns 10-2 and 10-4 as

5102sin2

2cos22' -

xyyxyx

x

6-102cos2

2sin22

''

xyyxyx

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Normal and shear strains If normal strain in the y direction is required,

it can be obtained from Eqn 10-5 by substituting ( + 90) for . The result is

6102sin2

2cos22' -

xyyxyx

y

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Principal strains We can orientate an element at a pt such

that the element’s deformation is only represented by normal strains, with no shear strains.

The material must be isotropic, and the axes along which the strains occur must coincide with the axes that define the principal axes.

Thus from Eqns 9-4 and 9-5,

8102tan -yx

xyp

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Principal strains

Maximum in-plane shear strain Using Eqns 9-6, 9-7 and 9-8, we get

910222

22

2,1 -

xyyxyx

1110222

22plane-in

max

-

xyyx

16

10102tan -

xy

yxs


Maximum in-plane shear strain Using Eqns 9-6, 9-7 and 9-8, we get

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12102

-avgyx


IMPORTANT The state of strain at the pt can also be

represented in terms of the maximum in-plane shear strain. In this case, an average normal strain will also act on the element.

The element representing the maximum in-plane shear strain and its associated average normal strains is 45 from the element representing the principal strains.

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EXAMPLE 10.1

A differential element of material at a pt is subjected to a state of plane strain x = 500(10-

6), y = 300(10-6), which tends to distort the element as shown. Determine the equivalent strains acting on an element oriented at the pt, clockwise 30 from the original position.

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EXAMPLE 10.1 (SOLN)

Since is clockwise, then = –30, use strain-transformation Eqns 10-5 and 10-6,

6'

6

6

6

'

10213

302sin210200

302cos102

300500

102

300500

2sin2

2cos22

x

xyyxyxx

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EXAMPLE 10.1 (SOLN)

Since is counterclockwise, then = –30, use strain-transformation Eqns 10-5 and 10-6,

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6''

6

''

10793

302cos210200

302sin2

300500

2cos2

2sin22

yx

xyyxyx

EXAMPLE 10.1 (SOLN)

Strain in the y’ direction can be obtained from Eqn 10-7 with = –30. However, we can also obtain y’ using Eqn 10-5 with = 60 ( = –30 + 90), replacing x’ with y’

6'

6

6

6'

104.13

602sin210200

602cos102

300500

102

300500

y

y

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EXAMPLE 10.1 (SOLN)

The results obtained tend to deform the element as shown below.

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EXAMPLE 10.2

A differential element of material at a pt is subjected to a state of plane strain defined by x = –350(10-6), y = 200(10-6), xy = 80(10-6), which tends to distort the element as shown. Determine the principal strains at the pt and associated orientation of the element.

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EXAMPLE 10.2 (SOLN)

Orientation of the elementFrom Eqn 10-8, we have

Each of these angles is measured positive counterclockwise, from the x axis to the outward normals on each face of the element.

9.8514.4

,17218028.828.82

)10(200350

)10(802tan

6

6

and

thatsoandThus

p

p

yx

xyp

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EXAMPLE 10.2 (SOLN)

Principal strainsFrom Eqn 10-9,

6

26

1

66

6226

22

2,1

1035310203

109.277100.75

102

802

2003502

10200350

222

xyyxyx

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EXAMPLE 10.2 (SOLN)

Principal strainsWe can determine which of these two strains deforms the element in the x’ direction by applying Eqn 10-5 with = –4.14. Thus

6'

6

66

'

10353

14.42sin2

1080

14.4cos102

20035010

2200350

2sin2

2cos22

x

xyyxyxx

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EXAMPLE 10.2 (SOLN)

Principal strainsHence x’ = 2. When subjected to the principal strains, the element is distorted as shown.

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EXAMPLE 10.3

A differential element of material at a pt is subjected to a state of plane strain defined by x = –350(10-6), y = 200(10-6), xy = 80(10-6), which tends to distort the element as shown. Determine the maximum in-plane shear strain at the pt and associated orientation of the element.

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EXAMPLE 10.3 (SOLN)

Orientation of the elementFrom Eqn 10-10,

Note that this orientation is 45 from that shown in Example 10.2 as expected.

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9.1309.40

,72.26118072.8172.812

1080

102003502tan

6

6

and

thatsoandThus,

s

s

xy

yxs

EXAMPLE 10.3 (SOLN)

Maximum in-plane shear strainApplying Eqn 10-11,

The proper sign of can be obtained by applying Eqn 10-6 with s = 40.9.

6

622

22

10556

102

802

200350

222

plane-in

max

plane-in

max

xyyx

plane-in

max

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EXAMPLE 10.3 (SOLN)

Maximum in-plane shear strain

Thus tends to distort the element so that the right angle between dx’ and dy’ is decreased (positive sign convention).

6''

6

6

''

10556

9.402cos2

1080

9.402sin102

200350

2cos2

2sin22

yx

xyyxyx

plane-in

max

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EXAMPLE 10.3 (SOLN)

Maximum in-plane shear strainThere are associated average normal strains imposed on the element determined from Eqn 10-12:

These strains tend to cause the element to contract.

66 1075102

2003502

yx avg

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*10.3 MOHR’S CIRCLE: PLANE STRAIN

Advantage of using Mohr’s circle for plane strain transformation is we get to see graphically how the normal and shear strain components at a pt vary from one orientation of the element to the next.

Eliminate parameter in Eqns 10-5 and 10-6 and rewrite as

22

avg

22

2avg

222

where

13-102

xyyxyx

xyx

R

R

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*10.3 MOHR’S CIRCLE: PLANE STRAIN

Procedure for AnalysisConstruction of the circle Establish a coordinate system such that the

abscissa represents the normal strain , with positive to the right, and the ordinate represents half the value of the shear strain, /2, with positive downward.

Using positive sign convention for x, y, and xy, determine the center of the circle C, which is located on the axis at a distance avg = (x + v)/2 from the origin.

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Documents

PLANE STRAIN TRANSFORMATION. 2 Apply the stress transformation methods derived in Chapter 9 to similarly transform strain Discuss various ways of measuring