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