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Chapter 7 Transformations of Stress and Strain. 7.1 Introduction. Goals: determine: 1. Principal Stresses 2. Principle Planes 3. Max. Shearing Stresses. 3 normal stresses. -- x , y , and z. General State of Stress. - PowerPoint PPT Presentation
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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