Failure Theories

1
Failure Theories Static Loads
Static load a stationary load that is gradually
applied having an unchanging magnitude and
direction
2
Ductile and Brittle Materials
A ductile material deforms significantly before
fracturing. Ductility is measured by elongation
at the fracture point. Materials with 5 or more
elongation are considered ductile.
3
Failure Theories Ductile Materials
Yield strength of a material is used to design
components made of ductile material

• Maximum shear stress theory (Tresca 1886)

4
Failure Theories Ductile Materials

• Distortion energy theory (von Mises-Hencky)

(total strain energy) (strain energy due to
hydrostatic stress) strain energy due to
angular distortion gt strain energy obtained from
a tension test at the yield point ? failure
5
Failure Theories Ductile Materials
The area under the curve in the elastic region is
called the Elastic Strain Energy.
U ½ ?e
Strain energy
6
Failure Theories Ductile Materials
Distortion strain energy total strain energy
hydrostatic strain energy
Ud UT Uh
7
Failure Theories Ductile Materials
Strain energy from a tension test at the yield
point
?1 Sy and
?2 ?3 0
Substitute in equation (2)
(2)
To avoid failure, Ud lt Utest
8
Failure Theories Ductile Materials
9
Failure Theories Ductile Materials
Pure torsion, ? ?1 ?2
(?12 ?2 ?1 ?22) Sy2
10
Design Process
Maximum shear stress theory
Distortion energy theory

• Select material

• Choose a safety factor

The selection of an appropriate safety factor
should be based on the following

• Degree of uncertainty about loading (type,
  magnitude and direction)

• Degree of uncertainty about material strength

• Uncertainties related to stress analysis

• Consequence of failure human safety and economics

• Type of manufacturing process

• Codes and standards

11
Design Process

• Use n 1.2 to 1.5 for reliable materials
  subjected to loads that can be determined with
  certainty.

• Use n 1.5 to 2.5 for average materials
  subjected to loads that can be determined. Also,
  human safety and economics are not an issue.

• Use n 3.0 to 4.0 for well known materials
  subjected to uncertain loads.

12
Design Process - Static load Ductile material
Sy ,
Su

• Select material ?

• Choose a safety factor, n

• Choose a cross section round, rectangular,
  hollow, I-beam, C channel,

• Formulate the maximum stresses in the component
  in terms of size ?x, ?xy

• Determine the principal stresses, maximum shear
  stress and von Mises stress in terms of the size
  ?1, ?2 , ?max and ?'

• Optimize for weight, size, or cost.

13
Failure Theories Brittle Materials
Characteristics of brittle materials
Perform two tests, one in compression and one in
tension, draw the Mohrs circles for both tests.
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Failure Theories Brittle Materials
Modified Coulomb-Mohr theory
?2 or ?3
?1
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Failure Theories Brittle Materials
?2
Zone I
Sut
I
Sut
?1
II
-Sut
III
Suc