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Static Failure Theories
ch. 5
Shigley
ch. 10
Hibbeler
Static Failure Theories
• Objective is to determine state of stress which causes a component to fail.
• Failure is defined as behaviour that renders a structure unsuitable for its intended function.
• Failure can be broadly classified as excessive distortion (also called yielding) or fracture.
Failure – brittle fracture
• Separation of a loaded member into two or
more parts.
• Fracture is caused due to breakage of bonds,
also called brittle fracture.
• The fracture surface is a shining surface and is
normal to the applied load.
Failure – brittle fracture
• Generally, brittle fracture occurs without any
warning.
• If percentage of elongation at fracture is less
than 5%, the fracture is termed as brittle.
Failure – ductile fracture/yielding
• Failure occurs along slip planes.
• Sliding across these slip planes occurs due to
shear stresses.
• The failure location necks down and then
breaks along a slip plane.
Failure – ductile fracture/yielding
• Generally, ductile failure gives a warning
before fracture.
• If percentage of elongation at failure is more
than 5%, the failure is termed as ductile.
Uni-axial Tensile Loading
F
σ1σ2 = σ3 = 0
Τmax = σ1/2
Mohr’s circle for a sample under uni-
axial loading and plane stress
Static Failure Theories
• If a material can withstand a known stress under uni-axial tension, how highly can it be safely stressed in situations involving bi-axial and tri-axial loading?
• Failure for a multi-axial state of stress is predicted to occur according to the ‘---’ theory whenever the ‘---’ induced by the applied loading is equal to ‘---’ at failure in a uni-axial test.
(--- varies for different failure theories)
Static Failure Theories
• Theories of failure:
– Maximum Normal Stress Theory
– Maximum Shear Stress Theory-MSST (Tresca)
– Distortion Energy/Octahedral Shear Stress Theory
(Von Mises)
Static Failure Theories
• Generally, we will determine the principal
normal stresses and use them as per one of the failure theories.
• Use yield strength (Sy) to predict failure due to yielding (ductile fracture).
• Use ultimate tensile strength (Sut) to predict failure due to brittle fracture.
Max. Normal Stress Theory
• If
• Failure occurs when
1 2 3σ σ σ≥ ≥
1 1
3 3
0
0
for
for
σ σ σ
σ σ
= >
= − <
1 3S or Sσ σ= − = −
y
ut
S S
S
=
=
For yielding (ductile materials)
For brittle fracture (brittle materials)
Max. Normal Stress Theory• This failure theory does not work
well for the following situations:– Hydrostatic compression
– Brittle materials under bi-axial state of stress with principal stresses of opposite signs. See Mohr circle ->
• This failure theory is generally not used for ductile materials.
• However, Max. Normal Stress Theory can be used for failure by brittle fracture (in some cases)
With σ1 slightly
smaller than Sy
Max. Shear Stress Theory (Tresca)
• Failure occurs if the maximum shear stress
induced due to external loading conditions
exceeds shear strength of material.
for
• Failure occurs when
1 3 2 31 2
12 13 23
2 2 2
σ σ σ σσ στ τ τ
− −−= = =
max 13τ τ=
1 3 max max
2
yt
yt ys
SS or or Sσ σ τ τ− = = =
0.5ys yt
S S=Syt – Yield strength (tension)
Sys – Yield strength (shear)
1 2 3σ σ σ≥ ≥ τmax = Radius of biggest circle
on Mohr circle diagram
Max. Shear Stress Theory (Tresca)
Failure locus using MSST (Maximum shear stress theory)
for plane stress
Distortion Energy Theory (Von Mises)
• This is also called the Max. octahedral shear stress theory or the Von Mises theory/criteria.
• Failure occurs when the strain energy per unit volume due to distortion (under loading conditions) is equal to the strain energy per unit volume due to distortion under a uni-axial tensile load.
Distortion Energy Theory (Von Mises)
• Failure occurs when
• σv is called Von Mises stress.
y vS σ=
( ) ( ) ( )
( ) ( ) ( ) ( )
1/22 2 2
1 2 2 3 1 3
1/22 2 2 2 2 2
1
2
16
2
v
x y y z x z xy yz xz
σ σ σ σ σ σ σ
σ σ σ σ σ σ τ τ τ
= − + − + −
= − + − + − + + +
Using Principle
stresses
Using Applied stresses
Failure Theories
• Max. shear stress theory and the Distortion
energy theory are commonly used for ductile
materials.
• Max. shear stress theory is more conservative
than the distortion energy theory.
• Actual failure data fits data from Distortion
energy theory better.
Failure Theories
Comparison - MSST & DET
(Tresca)
(Von Mises)
Failure Theories