Failure and Failure Theories:

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Failure and Failure Theories
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Stress-Analysis is performed on a component to
determine

• The required size or geometry (design)
• an allowable load (service)
• cause of failure (forensic)
• For all of these, a limit stress or allowable
  stress value for the component material is
  required.
• Furthermore, a Failure-Theory is needed to define
  the onset of failure.

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Failure

• Component can no longer function as intended.
• Failure Mode
• yielding a process of global permanent plastic
  deformation.
• Change in the geometry of the object.
• low stiffness excessive elastic deflection.
• fracture a process in which cracks grow to the
  extent that the
• component breaks apart.
• buckling the loss of stable equilibrium.
  Compressive loading
• can lead to bucking in columns.
• creep a high-temperature effect. Load carrying
  capacity drops.

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Failure Theories

• 1. The Tresca Criterion.
• also known as the Maximum Shear Stress criterion.
  
• yielding when the shear stress reaches its
  maximum value.
• In a tensile test, this occurs when the diameter
  of the largest
• Mohrs circle is equal to the tensile yield
  strength.
• If the principal stresses are ordered such that
  s1 gt s2 gt s3,
• then the Tresca criterion is expressed as

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2. The maximum principal stress criterion.

• states that a tensile yield (fracture) will
  occur in a
• previously un-cracked isotropic material when
  the
• maximum principal stress reaches a critical
  value.
• The critical value is usually the yield
  strength, Sy, or
• the ultimate tensile strength, Su.
• This criterion does not characterize fracture in
  brittle
• materials with cracks.

smax Sy (or Su)
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3. Von-Mises Criterion

• Also known as the Maximum Energy of Distortion
  criterion
• based on a more complex view of the role of the
  principal stress
• differences. In simple terms, the von Mises
  criterion considers the
• diameters of all three Mohrs circles as
  contributing to the
• characterization of yield onset in isotropic
  materials.
• When the criterion is applied, its relationship
  to the the
• uniaxial tensile yield strength is

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Von Mises

• For a state of plane stress (s30)
• It is often convenient to express this as an
  equivalent stress, se

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And the von-Mises failure criterion becomes se
Sy
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Plane Stress Biaxial Failure Envelopes
Von-Mises
sc critical value of stress so yield
stress sut ultimate tensile stress
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4. Mohrs failure criterion

• Applies to brittle materials much stronger in
  compression
• than in tension.
• Data from tension and compression tests
  establish limiting
• Mohrs circle envelope.
• For any given stress state, failure will not
  occur if the
• largest Mohrs circle lies within the failure
  envelope.

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Try it!
Determine if failiure will occur for the
following Complex stress state, given the
material has a tensile yield strength of 250 MPa
and an ultimate tensile strength of 300 MPa.
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Material Behaviour in Tension
Stress
Ultimate Su 300 MPa
300
Yield Sy 250MPa
200
100
Strain
0 0.002 0.010
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Solution
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
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The other 2 faces
x
80 MPa
y
0 MPa
0 MPa
z
z
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3-D Mohrs Circles
t max 77 MPa
Shear Stress, MPa
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s1 0 MPa s2 45 MPa s3 154 MPatmax 77
MPa.
1. Tresca Criterion Sy / 2 250/2 125 MPa
t max 77 lt 125 MPa, SAFE! FS 1.62 2.
Maximum Principal Stress Criterion Su
350MPa smax s3 154 lt 350 MPa, SAFE! FS
2.27 3. Von-Mises Criterion
1/?2(0-45)2 (45-154)2 (154-0)2 1/2 137
MPa lt 250 MPa, SAFE! FS 250/1371.82
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BUCKLING!
Load, P
I 2nd Moment of Area about weak axis. E
Youngs Modulus
Le
Deflected shape
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The effective length, Le, depends on the Boundary
Conditions
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Try It!
Find the Buckling load for a pin-ended
aluminum column 3m high, with a rectangular
x-section as shown
P
Weak axis Iyy 100 (50)3/12 1.04x106 mm4
100 mm
50 mm
82246 N
P