Triaxial Stress State

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Triaxial Stress State
?
?
(ve sense shown)
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3D Principal Triaxial Stress
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3D Stress Principal Stresses
The three principal stresses are obtained as the
three real roots of the following equation
where
I1, I2, and I3 are known as stress invariants as
they do not change in value when the axes are
rotated to new positions.
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Stress Invariants for Principal Stress
Zero shear stress on principal planes
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Mohrs Circle?

• There is no Mohrs circle solution for problems
  of triaxial stress state
• Solution for maximum principal stresses and
  maximum shear stress is analytical
• Either closed form solution or numerical solution
  (or computer program) are used to solve the
  eigenvalue problem.

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Maximum Shear Stresses
Absolute max shear stress is the numerically
larger of
tyz, tabs max
txy
tyz
s3
s1
s2
Normal Stress, s
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3D Mohrs Circle Plane Stress
A Case Study The two principal stresses are of
the same sign
s3
s1
s2
-s
t
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3D Mohrs Circle Plane Stress
A Case Study The two principal stresses are of
opposite sign
s3
s1
s2
s
t
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Example
For the following state of stress, find the
principal and critical values.
y
Tensor shows that sz 0 and t xz t yz 0
x
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The other 2 faces
x
y
z
z
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3-D Mohrs Circles
t max 77 MPa
Shear Stress, MPa
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Example triaxial stress state, not plane stress

• Determine the maximum principal stresses and the
  maximum shear stress for the following triaxial
  stress state. (ve values as defined in slide 1)

s
MPa
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Solution
s

MPa
20 30 10 40 MPa
-3025 MPa
89500 MPa
Solve
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Open the Exel Spreadsheet triaxial stress.xls
for a template to solve the cubic eqn.
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Results
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Mohrs circles
Shear (MPa)
tyz, tabs max58.5
s226.5
s363.5
s1 -51.8
Normal Stress, s (MPa)
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Safety Factor?

• If the stress state was determined on a steel
  crankshaft, made of forged SAE1045 steel with a
  yield strength of 300 MPa, what is the factor of
  safety against yield?
• Tresca Criterion tmax 58.5 MPa

2. Max Principal Stress Criterion smax 63.5 MPa
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3. Von Mises Criterion
103.31 MPa