Principle stresses and planes

Principal Stresses
and Planes

1. Topic :- principal STreSSeS
and planeS

2. 7.1 Introduction
General State of Stress
3 normal stresses
3 shearing stresses -- t
xy
, t
yz
, and t
zx
-- s
x
, s
y
, and s
z
Goals: determine:
1. Principal Stresses
2. Principle Planes
3. Max. Shearing Stresses

3. 2-D State of Stress
Plane Stress condition
Plane Strain condition
A. Plane Stress State:
B. Plane Stress State:
s
z
= 0, t
yz
= t
xz
= g
yz
= g
xz
= 0
e
z
¹ 0, t
xy
¹ 0
e
z
= 0, t
yz
= t
xz
= g
yz
= g
xz
=
0
s
z
¹ 0, t
xy
¹ 0

4. Examples of Plane-Stress Condition:

5. Max. s
x
& s
y

Max. t
xy

(Principal stresses)

6. 7.2 Transformation of Plane Stress

7. 0
0
' ': ( cos )cos ( cos )sin
( sin )sin ( sin )cos
x xyx x
y xy
F A A A
A A
s s q q t q q
s q q t q q
S = D - D - D
- D - D =
0
0
' ' ': ( cos )sin ( cos )cos
( sin )cos ( sin )sin
x xyy x y
y xy
F A A A
A A
s s q q t q q
s q q t q q
S = D + D - D
- D + D =

8. 2 2
2
' cos sin sin cos
x y xyx
s s q s q t q q= + +
2 2
' '( )sin cos (cos sin )
x y xyx y
t s s q q t q q=- - + -
2 2
2 2 2sin sin cos , cos cos sinq q q q q q= = -
2 21 2 1 2
2 2
cos cos
cos , sin
q q
q q
+ -
= =
After rearrangement:
(7.1)
(7.2)
Knowing

9. 2 2
2 2
' cos sin
x y x y
xyx
s s s s
s q t q
+ -
= + +
2 2
2
' ' sin cos
x y
xyx y
s s
t q t q
-
=- +
2 2
2 2
' cos sin
x y x y
xyy
s s s s
s q t q
+ -
= - -
'
ys
Eqs. (7.1) and (7.2) can be simplified as:
(7.5)
(7.6)
Can be obtained by replacing q with (q + 90
o
) in Eq.
(7.5)
(7.7)

10. 1. s
max
and s
min
occur at t = 0
2. s
max
and s
min
are 90
o
apart. t
max
and t
min
are 90
o
apart.
3. t
max
and t
min
occur half way between s
max
and s
min

11. 7.3 Principal Stresses: Maximum Shearing Stress
Since s
max
and s
min
occur at t
x’y’
= 0, one can set Eq. (7.6) = 0
2 2 0
2
' ' sin cos
x y
xyx y
s s
t q t q
-
=- + =
2
2tan
xy
x y
t
q
s s
=
-
1 2
2 2
2
2
4
/
( )/
cos
( ) /
x y
x y xy
s s
q
s s t
-
= ±
é ù- +
ë û
1 2
2 2
2
4
/
sin
( ) /
xy
x y xy
q
s s t
t
=±
é ù- +
ë û
(7.6)
It follows,
Hence,
(a)
(b)

12. Substituting Eqs. (a) and (b) into Eq. (7.5) results in s
max
and
s
min
:
2 2
2 2
max,min
( )
x y x y
xy
s s s s
s t
+ -
= ± +
2

x y
ave
s s
s
+
=
2
2
( )
x y
xy
R
s s
t
-
= +
This is a formula of a circle with the center at:
and the radius of the circle as:
(7.14
)
(7.10
)

13. Mohr’s Circle

14. The t
max
can be obtained from the Mohr’s circle:
Since t
max
is the radius of the Mohr’s circle,
2
2
max( )
( )
x y
in plane xy
R
s s
t t
-
-
= = +

15. Since t
max
occurs at 2q = 90
o
CCW from s
max
,
Hence, in the physical plane t
max
is q = 45
o
CCW from s
max
.
¨¨ In the Mohr’s circle, all angles have been doubled.

16. 7.4 Mohr’s Circle for Plane Stress

17. Sign conventions for shear
stresses:
CW shear stress = Å and is plotted above the s-
axis,
CCW shear stress = ⊝ and is plotted below the s-
axis

19. 7.5 General State of Stress – 3-D
cases
Definition of Direction Cosines:
cos , cos , cosl
x x y y z z
m nl q l q l q= = = = = =
2 2 2 2 2 2
1 1 l
x y z or m nl l l+ + = + + =
with

21. 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
s s l l t l l t l l
t l l s l l t l l
t l l t l l s l l
S = D - D - D - D
- D - D - D
- D - D - D =

22. Dividing through by DA and solving for s
n
, we have
2 2 2
2 2 2
n x x y y z z xy x y yz y z zx z x
s s l s l s l t l l t l l t ll= + + + + +
2 2 2
= + +
n a a b b c cs s l sl sl
(7.20
)
We can select the coordinate axes such that the RHS of Eq.
*7.20) contains only the squares of the l’s.
(7.21
)
Since shear stress t
ij
= o, s
a
,
s
b
, and s
c
are the three
principal stresses.

23. 7.6 Application of Mohr’s Circle to
the 3-D Analysis of Stress
s
A
> s
B >
s
C
1 1
2 2
max max min A C
t s s s s= - = -
= radius of the Mohr’s circle