CHAPTER 6
COMPLEX
STRESS

2
INTRODUCTION
•
The most general state of stress at a
point may be represented by 6
components,
,
,
:
(Note
stresses
shearing
,
,
stresses
normal
,
,
xz
zx
zy
yz
yx
xy
zx
yz
xy
z
y
x
•
Same state of stress is represented by a
different set of components if axes are
rotated.
9.1 Plane Stress Transformation
•
In practice, approximations and
simplifications
are done to reduce the stress components
to a single plane.

3
Plane Stress
•
Plane Stress
state of stress in which two faces
of the cubic element are free of stress. For the
illustrated example, the state of stress is
defined by
.
0
,
,
and
xy
zy
zx
z
y
x
•
State of plane stress occurs in a thin plate
subjected to forces acting in the midplane of the
plate.
•
State of plane stress also occurs on the free
surface of a structural element or machine
component, i.e., at any point of the surface not
subjected to an external force.
9.1 Plane Stress Transformation

4
Plane Stress Transformation
•
The material is then said to be subjected to
plane stress.
•
For general state of plane stress at a pt, we
represent it
via normal-stress components,
x
,
y
and shear-stress
component
xy
.
•
Transforming stress components from one orientation
to the other is similar in concept to how we transform
force components from one system of axes to the
other.
•
Note that for stress-component transformation, we
need to account for
- the magnitude and direction of each stress
component, and
- the orientation of the area upon which each
component acts.
9.1 Plane Stress Transformation

5
Procedure of Analysis
(Method of Equilibrium)
If state of stress at a pt is known for a given orientation of an element of
material, then state of stress for another orientation can be determined.
9.1 Plane Stress Transformation

Transformation of Plane Stress
•
consider the stress element has rotated through an
angle
θ
as shown.
•
As linear equations, the transformed
stresses are given by
145

7
Sign Convention
•
Positive normal stresses,
x
and
y
, acts
outward from all faces
•
Positive shear stress
xy
acts upward on the
right-hand face of the element.

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- Spring '13