Analysis of Stress and Strain

1
Analysis of Stress and Strain
Review
- Torsional shaft
- Axially loaded Bar
p
tnt
sn
q
txy
h
P
P
q
tyx
Questions (1) Is there any general method
to determine stresses on any arbitrary plane
at one point if the stresses at this point
along some planes are known? (2) For an
arbitrary loaded member, how many planes on which
stresses are known are required to
determine the stresses at any plane at one point?
2
Analysis of Stress and Strain
sy
sy
State of stress at one point
tyx
tyx
tyz
txy
txy
txy
Stress element
y
tzy
sx
sx
sx
sz
txz
tzx
tyx
x
sy
z

• Use a cube to represent stress element. It is
  infinitesimal in size.
• (x,y,z) axes are parallel to the edges of the
  element
• faces of the element are designated by the
  directions of their
• outward normals.

Sign Convention

• Normal stresses tension -
  compression.
• Shear stresses the directions
  associated with its subscripts are
• plus-plus
  or minus-minus
• - the
  directions associated with its subscripts are
• plus-minus
  or minus-plus

3
Plane Stress
Definition Only x and y faces are subject to
stresses, and all stresses are
parallel to the x and y axes.
Stresses on inclined planes
txy
q
sx
tyx
sy
Transformation equations for plane stress
4
Transformation Equations
angle between x1 and x axes, measured
counterclockwise
5
Plane Stress Special Cases
Uniaxial Stress
sx
sx
tyx
Pure Shear
txy
txy
tyx
sy
Biaxial Stress
sx
sx
sy
6
Plane Stress
Example 1 A plane-stress condition exists at a
point on the surface of a loaded structure,
where the stresses have the magnitudes and
directions shown on the stress element of the
following figure. Determine the stresses acting
on an element that is oriented at a clockwise
angle of 15o with respect to the original
element.
7
Principal Stresses
Principal stresses maximum and minimum normal
stresses. Principal planes the planes on which
the principal stresses act
The angle defines the orientation of the
principal planes.
8
Principal Stresses
OR
9
Shear Stress
Shear stresses on the principal planes
Example 2 Principal stresses in pure shear case
tyx
txy
txy
tyx
10
Maximum Shear Stresses
11
Plane Stress
Example 3 Find the principal stresses and
maximum shear stresses and
show them on a sketch of a properly oriented
element.
12
Mohrs Circle For Plane Stress Equations of
Mohrs Circle
Transformation equations
(1)2 (2)2
13
Two Forms of Mohrs Circle
14
Construction of Mohrs Circle
Approach 1 For the given state of stresses,
calculate and R. The center Of the circle
is ( , 0) and the radius is R.
15
Construction of Mohrs Circle
Approach 2 Find points corresponding to q 0
and q 90o and then draw a line. The
intersection is the origin of the circle.
16
Applications of Mohrs Circle
Example 4 An element in plane stress at the
surface of a large machine is subjected to
stresses
Using Mohrs circle, determine the
following quantities (a) the stresses acting on
an element inclined at an angle of 40o, (b) the
principal stresses and (c) the maximum shear
stress.
17
Plane Strain
Definition Only x and y faces are subject to
strains, and all strains are
parallel to the x and y axes.
Note Plane stress and plane strain do not occur
simultaneously.
18
Plane Strain
Transformation Equations
Principal Strains