Mohrs Circle Revisited

1
Mohrs Circle Revisited

• Mohrs circle for plane stress

2
Combined Shear and Normal Stresses
s S sin y
Plane
Normal
S
Shear
t S cos y
y
s S sin2 y t S sin y cos y
3
Consider an axial stress S 200 MPa, and let y
vary from 00 to 900 .
t
t 200 sin (150) cos (150) 50 MPa s 200 sin2
(150) 13.4 MPa
s
4

• The previous slide shows the basis for Mohrs
  Circle for Stress.
• Mohrs Circle is used to determine
• The State of Stress in any direction
• The Principal Stresses at any point in a
  stressed body
• The Maximum Shear Stress at any point.
• Sign Convention
• Normal Stress - Tension positive
• Shear Stress - Clockwise Positive

s
Ve
t
5
2D-Mohrs Circle for Stress
t
t xy
s
t
Shear Stress
s
s
s
Normal Stress, s
t xy
6
Principal Stresses
t max
smax, and smin are Principal Stresses - They
act on planes with zero shear Stress
t xy
2 qs
smax
smin
s
s
2 qp
The maximum shear stress is tmax.
t xy
7
Try It!
At the web /tension-flange interface of an I
beam in bending there exists a normal stress of
80 MPa and a shear stress of 30 MPa.
Determine the maximum Principal Stress and the
maximum shear stress at this location.
30 MPa
80 MPa
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Solution
t max 50 MPa
t (MPa)
(80, 30)
o
x
o
V face
smin - 10 MPa
smax 90 MPa
(40,0)
s (MPa)
o
(0, -30)
y
H face
All Stresses MPa
9
Solution
t max
t (MPa)
(80, 30)
o
x
o
2qs
smax
(40,0)
s (MPa)
2qp 90 53.13 143.130 qp 71.560
o
(0, -30)
y
2qs tan-1(40/30) 2qs 53.130 qs 26.560
H face
10
Results
18.44º
11
Results, contd
12

1. Mohrs circle works only when there is a
single shear stress acting on the faces of
the cube. The front face in the 2-D view of
the element must be free of shear stress i.e.,
it must be a Principal Plane. 2. Because
there are three principal stresses, there are
always three circles. The 2-D element you
choose to examine may not be the critical
one.
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In the explanation in the previous section, two
important issues were skipped

• Mohrs circle works only when there is a single
  shear stress acting on the faces of the cube. The
  front face in the 2-D view of the element must be
  free of shear stress i.e., it must be a
  Principal Plane.
• Because there are three principal stresses, there
  are always three circles. The 2-D element you
  choose to examine may not be the critical one.

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Example
For the following state of stress, find the
principal and critical values.
80 MPa
y
50 MPa
120 MPa
Tensor shows that sz 0 and t xz t yz 0
x
15
The other 2 faces
x
80 MPa
y
0 MPa
0 MPa
z
z
16
3-D Mohrs Circles
t max 77 MPa
Shear Stress, MPa