UNIT-I-Theories of failures-19072016.pptx

1. Theories of failures
There are number of machine components that are subjected to several
types of loads simultaneously.
Ex:-(1)Power screw is subjected to torsional moment as well as axial
force.(2)Torsional moment induces torsional shear stress,while bending
moment causes bending stresses in the transmission shaft.
the engineer’s have been to trying to predict failure loads for
various structural elements so that they can improve their design. Most
of the structural elements are in complex state of stress and it is not easy
to conduct laboratory tests on models with complex loading. The
procedure followed by an engineers is to conduct uni-axial test on
specimen and get failure condition for the material from a few tests
results arrive at failure criteria for structural elements in complex
condition, use this criteria for predicting failure load in any structural
element.

2. Power screw is a mechanical device used to convert rotary motion
into linear motion and to transmit power

7. Theories of failures
when a part is loaded so that the stress state is uni-axial then the stress
and strength can compared directly to determine the degree of safety
or we can say weather the part will fail or safe, this method is simple
because there is only one stress and one value of strength, it may be
yield strength, ultimate strength, shear strength or what ever is
appropriate.
The problem will become more complicated when the stress state is bi-
axial or tri-axial, In such cases there are multiples of stress but still
only one strength, so how can we say the part is safe or not, if it is
safe, how much safe. Therefore a number of failure theories have been
proposed to answer this question.
The principal theories or major theories of failures for a member
subjected to bi-axial or tri-axial stress state conditions are the
following.

8. Theories of failures
Different Theories of Failure : These are five different theories of
failures which are generally used
(a) Maximum Principal stress theory (Rankine )
(b) Maximum shear stress theory ( Guest - Tresca )
(c) Maximum Principal strain ( Saint - venant ) Theory
(d) max strain energy Theory ( Haigh )
(e) Distortion Energy theory or Max Shear strain energy Theory (Von–
Mises & Hencky )
In all these theories we shall assume.
σYp = stress at the yield point in the simple tensile test.
σ 1, σ 2, σ 3 - the three principal stresses in the three dimensional
complex state of stress systems in order of magnitude.
Note:- the simple tension test gives information about the tensile yield
strength, the ultimate tensile strength and % of elongation.

9. Maximum Principal Stress / Normal Stress / Rankine’s
Theory
According to this theory, the yielding at a point occurs only when the
maximum principle stress in any plane passing through that point
reaches a value equal to the elastic limit as computed in simple tension
test. Thus criteria for failure is
σ Max = σ Yield Point
σ Max ≤ σ Allowable - Safe Condition
Suppose we arrange the three principle stresses for any stress state in
the ordered form σ1 > σ2 > σ3 then this theory predicts the failure
occurs. Whenever σ 1 = σ yt or σ 1 = σ ut which ever is applicable.
This theory considers only the maximum of principal stresses and
disregards the influence of the other principal stresses. the dimensions
of the component are determined by using a factor of safety.
For Tensile Stress σ 1 = σ Yield Ten.Point /F.S or σ Ultimate Ten. Strength / F.S
For Compressive Stress σ 1 = σ Yield Comp. Point / F.S or σ Ultimate Comp. Strength /
F.S

10. Maximum Principal stress theory:-Brittle materials
This theory assume that when the maximum principal stress in a complex
stress system reaches the elastic limit stress in a simple tension,
failure will occur.
Therefore the criterion for failure would be
s1 = syp
For a two dimensional complex stress system s1 is expressed as
Where sx, sy and txy are the stresses in the any given complex stress
system.

11. Maximum Principal Strain / Normal Strain / St. Venant’s
Theory:
According to this theory the yielding occurs at a point when the
maximum principal strain at any point reaches a value equal to the
elastic strain as computed in simple tension test. Thus the criteria for
failure is
ϵmax = ϵ Allowable, ϵ max ≤ ϵ Allowable
ϵ1 =
We know the three principle strains are

13. Maximum Shear Stress/ Guest and Tresca’s Theory:

21. Maximum shear stress theory
This theory states that failure can be assumed to occur when the
maximum shear stress in the complex stress system is equal to the
value of maximum shear stress in simple tension.
The criterion for the failure may be established as given below :

22. Maximum Principal strain theory
This Theory assumes that failure occurs when the maximum strain for a
complex state of stress system becomes equals to the strain at yield
point in the tensile test for the three dimensional complex state of
stress system.

23. Total strain energy per unit volume theory
The theory assumes that the failure occurs when the total strain energy
for a complex state of stress system is equal to that at the yield point a
tensile test.
Therefore, the failure criterion becomes
It may be noted that this theory gives fair by good results for ductile
materials.

24. Maximum shear strain energy per unit volume theory
This theory states that the failure occurs when the maximum shear strain
energy component for the complex state of stress system is equal to
that at the yield point in the tensile test.
Hence the criterion for the failure becomes
As we know that a general state of stress can be broken into two
components i.e,

25. Maximum shear strain energy per unit volume theory
(i) Hydrostatic state of stress ( the strain energy associated with the
hydrostatic state of stress is known as the volumetric strain energy )
(ii) Distortional or Deviatoric state of stress ( The strain energy due to
this is known as the shear strain energy )
As we know that the strain energy due to distortion is given as
This is the distortion strain energy for a complex state of stress, this is to
be equaled to the maximum distortion energy in the simple tension
test. In order to get we may assume that one of the principal stress say
( s1 ) reaches the yield point ( syp ) of the material. Thus, putting in
above equation s2 = s3 = 0 we get distortion energy for the simple test
i.e