Chapter 7

Transformations of Stress and Strain

7.1 Introduction

General State of Stress

3 normal stresses

3 shearing stresses -- xy, yz, and zx

-- x, y, and z

Goals: determine:

1. Principal Stresses

2. Principle Planes

3. Max. Shearing Stresses

2-D State of Stress

Plane Stress condition

Plane Strain condition

A. Plane Stress State:

B. Plane Stress State:

z = 0, yz = xz = yz = xz = 0z 0, xy 0

z = 0, yz = xz = yz = xz = 0z 0, xy 0

Examples of Plane-Stress Condition:

Thin-walled Vessels

In-plane shear stress

Out-of-plane shear stressShear stress

Max. x & y

Max. xy

(Principal stresses)

7.2 Transformation of Plane Stress

0

0

' ': ( cos )cos ( cos )sin

( sin )sin ( sin )cos

x xyx x

y xy

F A A A

A A

0

0

' ' ': ( cos )sin ( cos )cos

( sin )cos ( sin )sin

x xyy x y

y xy

F A A A

A A

2 2 2' cos sin sin cosx y xyx

2 2' ' ( )sin cos (cos sin )x y xyx y

2 22 2 2sin sin cos , cos cos sin

After rearrangement:

(7.1)

(7.2)

2 21 2 1 22 2

cos coscos , sin

Knowing

2 22 2

' cos sinx y x yxyx

2 22

' ' sin cosx yxyx y

2 22 2

' cos sinx y x yxyy

Eqs. (7.1) and (7.2) can be simplified as:

(7.5)

(7.6)

'y Can be obtained by replacing with ( + 90o) in Eq. (7.5)

(7.7)

1. max and min occur at = 0

2. max and min are 90o apart. max and min are 90o apart.

3. max and min occur half way between max and min

7.3 Principal Stresses: Maximum Shearing Stress

Since max and min occur at x’y’ = 0, one can set Eq. (7.6) = 0

2 2 02

' ' sin cosx yxyx y

22tan xy

x y

1 22 2

22

4/

( ) /cos

( ) /

x y

x y xy

(7.6)

It follows,

Hence, 1 22 22

4/sin

( ) /

xy

x y xy

(a)

(b)

Substituting Eqs. (a) and (b) into Eq. (7.5) results in max and min :

2 2

2 2max, min ( )x y x yxy

2 x y

ave

2

2( )x y

xyR

This is a formula of a circle with the center at:

and the radius of the circle as:

(7.14)

(7.10)

Mohr’s Circle

The max can be obtained from the Mohr’s circle:

Since max is the radius of the Mohr’s circle,

2

2max ( ) ( )x yin plane xyR

Since max occurs at 2 = 90o CCW from max,

Hence, in the physical plane max is = 45o CCW from max.

In the Mohr’s circle, all angles have been doubled.

7.4 Mohr’s Circle for Plane Stress

Sign conventions for shear stresses:

CW shear stress = and is plotted above the -axis,

CCW shear stress = ⊝ and is plotted below the -axis

7.5 General State of Stress – 3-D cases

Definition of Direction Cosines:

cos , cos , cosx x y y z zm n

with2 2 2 2 2 21 1x y z or m n

0

0

: ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

n n x x x xy x y xz x z

yx y z y y y yz y z

zx z x zy z y z z z

F A A A A

A A A

A A A

Dividing through by A and solving for n, we have

2 2 2 2 2 2n x x y y z z xy x y yz y z zx z x

2 2 2 n a a b b c c

(7.20)

We can select the coordinate axes such that the RHS of Eq. *7.20) contains only the squares of the ’s.

(7.21)

Since shear stress ij = o, a, b, and c are the three principal stresses.

7.6 Application of Mohr’s Circle to the 3-D Analysis of Stress

A > B > C

1 12 2max max min A C = radius of the Mohr’s circle

7.9 Stresses in Thin-Walled Pressure Vessels

1

prhoop stress

t

0 2 2 0: ( ) ( )z lF t x p r x

(7.30)

Hoop Stress 1

Longitudinal Stress 2

220 2 0: ( ) ( )xF rt p r

2 2prt

Solving for 2 (7.31)

Hence 1 22

Assuming the end cap or the fluid inside takes the pressure

Using the Mohr’s circle to solve for max

2

12 4max( )in plane

prt

2 2max( )out of plane

prt

1 2

1 2 2

pr

t

1

12 4max( )out of plane

prt

1 2 02( )in plane

7.8 Fracture Criteria for Brittle Materials under Plane stress

2

b d, 0 and (u )6

Ya Y Y G

2 2 2 a a b b Y

2 2 2 a a b b Y

b a U U

lpr

t

2

1

2 12

prttr

' ' 2 ' ' 2 ' ' 2

' ' ' '

( ) ( ) ( )

2( )( )cos( )2 xy

AB AC C B

AB BC

2 2 2 2

2 2

( ) [1 ( )] ( ) (1 )

( ) (1 )

2( )(1 )( )(1 )cos( )2

x

y

x y xy

s x

y

x y

( )cos y=( s)sinx s

cos( ) sin2 xy xy xy

7.10 Transformation of Plane Strain

2 2( ) cos sin sin cosx y xy

1(45 ) ( )

2OB x y xy

2 ( )xy OB x y

' cos2 sin 22 2 2

x y x y xy

x

' cos2 sin 22 2 2

x y x y xy

y

' ' x yx y

7.11 Mohr’s Circle for Plane Strain

' sin 2 cos22 2 2

x y x y xy

OB

' ' ( )sin 2 cos2x y xyx y

' ' ( )sin 2 cos2

2 2 2x y x y xy

2 2 and R= ( ) ( )2 2 2

x y x y xyave

max min and ave aveR R

7.12 3-D Analysis of Strain

tan 2 xyp

x y

2 2max(in plane) 2 ( )x y xyR

max max min

a ba E E

a bb E E

( )c a bE

7.13 Measurements of Strain : Strain Rosette

1( )a b a bE

( )1c a b

2 ( )xy OB x y

2 21 1 1 1 1

2 22 2 2 2 2

2 23 3 3 3 3

cos sin sin cos

cos sin sin cos

cos sin sin cos

x y xy

x y xy

x y xy

'

1 cos2 1 cos2sin 2

2 2

x y xyx

1

2

1

2

1

2

1

2

1

2

2 2max ( )

2

x y

xy

'

2

x y

ave

'max 2

P

RA

max,min Tc

RJ

0 : ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) 0

n n x x x xy x y

xz x z yx y z y y y

yz y z zx z x zy z y

z z z

F A A A

A A A

A A A

A

' ' x yx y

' ' '2 2 2 2( ) ( )

2 2

x y x y

xyx x y

2 and ( )2 2

x y x y

ave xyR

'2 2 2( ) ave xyx

R

2tan 2

xy

px y

max min and ave aveR R

2 2max,min ( )

2 2

x y x y

xy

cos2 sin 2 02

x y

s xy s

tan 22

x y

sxy

2 2 2 2

2 2

n x x y y z z xy x y

yz y z zx z x

2 2 2 n a a b b c c

max max min

1

2

a Y b Y

a b Y

2 21( )

6 d a a b buG