THEORIES OF FAILURES
DR. QASIM SHAUKAT KHAN
ASSISTANT PROFESSOR
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CIVIL ENGINEERING DEPARTMENT
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Theories of Failure
 To design using a specific material, it is important to know the upper limit on the state of stress
that defines the material’s failure.
 If the material is ductile, failure is usually governed by the initiation of yielding whereas, if the
material is brittle it is governed by fracture.
 These modes of failures are readily defined if the member is subjected to a uniaxial state of
stress as in case of tension test.
 If the member is subjected to biaxial or triaxial state of stress, the failure criterion becomes
more difficult to be established.
 There are four theories that are often used in engineering to predict the failure of a material
subjected to a multiaxial state of stress.
 No single theory of failure is applicable to a specific material at all times as material may
behave in either ductile or brittle manner depending on the temperature, rate of loading, etc.
 To apply any failure theory, it is necessary to know the normal and shear stresses to determine
the Principal stresses. Since each theory of failure is based on knowing the Principal stresses.
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Theories of Failure (Ductile Materials)
 Maximum Shear Stress Theory or Tresca Yield Criterion
 The most common type of yielding of a ductile material (i.e. steel) is
caused by slipping.
 The slipping occurs along the contact planes of randomly ordered
crystals of the material.
 The slipping causing the yielding of the material can be observed if
highly polished thin strip of material is subjected to tension test as
shown in figure.
 The edges of the planes of the slipping on the surface of the strip are
referred to as Luder Lines.
 Luder lines indicate the slip planes in the strip at approximately 45º
with the axis of the strip.
 The slipping is caused by the shear stress. To show this, take an
element of the material taken from the tension specimen subjected to
the yield stress, σy.
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Theories of Failure
 Ductile Materials
 Maximum Shear Stress Theory or Tresca Yield Criterion
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Theories of Failure (Ductile Materials)
 Maximum Shear Stress Theory or Tresca Yield Criterion
 The maximum shear stress can be determined by drawing Mohr’s circle for the element.
 τmax= σy/2
 The maximum shear stress acts on the planes 45º from the planes of Principal stress and these
planes coincide with the Luder lines on the specimen indicating the failure is due to shear.
 The maximum shear stress theory or Tresca yield criterion can be used to predict the failure
stress of a ductile material subjected to any type of loading.
 The theory states that yielding of the material begins when the absolute maximum shear stress in
the material reaches the shear stress which causes the material to yield when it is subjected only
to axial tension.
 To avoid failure, it is necessary that absolute maximum shear stress (τabs max) in the material is
less than or equal to maximum shear stress (τmax)
Theories of Failure (Ductile Materials)
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 Maximum Shear Stress Theory or Tresca Yield Criterion
 If the two in-plane Principal stresses have the same sign, then failure will occur out of the plane.
 τabs max = σ1/2
 If the two in-plane Principal stresses have the opposite sign then failure occurs within the plane.
 τabs max = σ1-σ2/2
 The maximum shear stress theory for plane stress can be expressed for any two in-plane principal
stresses σ1 , σ2
 ǀ σ1 ǀ = ǀ σ2 ǀ = σy for same signs σ1 , σ2
 ǀ σ1 - σ2 ǀ = σy for opposite signs σ1 , σ2
 If any point of the material is subjected to plane stress & its in-plane Principal stresses
 (σ1 , σ2 ) are plotted on the boundary or outside the hexagonal area, the material will yield at the point
and failure is said to occur.
Theories of Failure (Ductile Materials)
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 Maximum Distortion Energy Theory
 An external loading will deform a material, causing it to store energy internally throughout its volume.
The energy per unit volume of material is called the strain energy density, and if the material is
subjected to uniaxial stress the strain energy density (u ) is
σЄ
 u=
2
 If the material is subjected to triaxial stress then each Principal stress contributes a portion of the total
strain energy density
 u=
σ1Є1 σ2Є2 σ3Є3
+
+
2
2
2
(a)
 Furthermore, if the material behaves in a linear elastic manner then Hooke’s Law is applicable
1
𝐸
1
𝐸
1
𝐸
 Є𝐱 = (σx- v (σy+σz) ; Є𝐲 = (σy- v (σx+σz) ; Є𝐳 = (σz- v (σx+σy) (b)
 Substituting eqn. b in eqn. a, we get
 u=
1
(σ12+ σ22 + σ32 -2v (σ1σ2+ σ1σ3+ σ3σ2) (c)
2𝐸
Theories of Failure (Ductile Materials)
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 Maximum Distortion Energy Theory
=
+
 The strain energy density can be considered as the sum of the energy needed to cause a volume change
of the element with no change in shape and energy needed to distort the element.
 The energy stored in the element as a result of its volume being changed is caused by the application of
average Principal stress (σ1+ σ2 + σ3 )/3 since, Principal stresses cause equal Principal strains in the
material.
 The remaining portion of the stress (σ1- σavg ), (σ2- σavg ), (σ3- σavg ) causes the energy of distortion.
Theories of Failure (Ductile Materials)
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 Maximum Distortion Energy Theory
 The experimental results have show that material subjected to hydrostatic (uniform) stresses (σavg ) do
not yield.
 In 1904, M. Huber proposed that yielding in a ductile material occurs when the distortion energy per
unit volume of the material equals or exceeds the distortion energy per unit volume of the same material
when it is subjected to yielding in a simple tension test.
 The Maximum Distortion Energy Theory was later on redefined by R. Von Mises and H. Hencky.
 To obtain the distortion energy per unit volume, the stresses (σ1- σavg ), (σ2- σavg ), (σ3- σavg ) are
substituted for σ1, σ2 , σ3 respectively into eqn. c.
 As average stresss (σavg ) is (σ1+ σ2 + σ3 )/3, after simplification we get
 u𝐝 =
1+v
[(σ1-σ2)2 + (σ2-σ3)2 + (σ3-σ1)2 )]
6𝐸
(d)
Theories of Failure (Ductile Materials)
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 Maximum Distortion Energy Theory
 In the case of Plane stress, σ3 = 0 is reduced to
 u𝐝 =
1+v 2
[σ1 - σ1σ2 + σ22 ] (e)
3𝐸
 For a uniaxial tension test, σ1=σy , σ2=σ3 =0, so
 (u𝐝)𝐲 =
1+v 2
[σ ]
3𝐸 y
(f)
 Since the maximum distortion energy theory requires u𝐝 = (u𝐝)𝐲 , then for the case of plane or biaxial
stress, we have (eqn e = eqn f)
 [σ12 - σ1σ2 + σ22 ] = [σy2] (g)
 The eqn. (g) is the equation of an ellipse. Thus, if a point in the material is stressed such that (σ1, σ2) is
plotted on the boundary or outside the shaded area, the material is said to fail.
Theories of Failure (Ductile Materials)
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 Comparison of Ductile Failure Theories
 A comparison of Maximum Shear Stress Theory and Maximum Distortion Energy Theory failure
criteria exhibits that both theories give the same results when the Principal stresses are equal i.e. σ1=
σ2=σy=0 or when the one of the principal stress is zero and the other has a magnitude of σy.
Theories of Failure (Ductile Materials)
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 Comparison of Ductile Failure Theories
 If the material is subjected to pure shear (τ) then the theories have the largest discrepancy in predicting
the failure.
 The stress co-ordinates of these points on the curves can be determined by considering the element as
shown in the Figure.
Theories of Failure (Ductile Materials)
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 Comparison of Ductile Failure Theories
 From the Mohr’s circle for the given state of stress,
 σ 1= τ ; σ 2= -τ
 Thus with σ 1 = -σ 2, the Maximum Shear Stress Theory gives (σ y/2, - σ y/2) and the Maximum
Distortion Energy Theory gives (σ y/ 3, - σ y/ 3).
 In the torsion tests, a condition of pure shear in a ductile specimen have shown that Maximum
Distortion Energy Theory gives more accurate results than the Maximum Shear Stress Theory.
 σ y/ 3 / σ y/2 = 1.15, the shear stress for yielding of the material as given by the maximum distortion
energy theory is 15% more accurate than that given by Maximum Shear Stress Theory.
Theories of Failure (Brittle Materials)
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 Maximum Normal Stress Theory
 Brittle materials (cast iron) tend to fail suddenly by fracture with no apparent yielding.
 In tension test, the fracture occurs when the normal stress reaches the ultimate stress (σ ult).
 A brittle fracture occurs in a torsion test due to tension since the plane of fracture for an element is at 45
deg to the shear direction. The fracture surface is helical.
Theories of Failure (Brittle Materials)
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 Maximum Normal Stress Theory
 Experiments have exhibited that during Torsion the materials strength is somewhat unaffected by the
presence of the associated Principal compressive stresses being at right angle to the Principal tensile
stresses.
 As a result, the tensile stress needed to fracture a specimen during a torsion test is approximately the
same as that needed to fracture a specimen during simple tension test.
 Based on this the Maximum Normal Stress Theory states that a brittle material will fail when the
maximum tensile stress σ1, in the material reaches a value that is equal to the ultimate normal stress the
material can sustain when it is subjected to simple tension test.
 If the material is subjected to plane stress then
ǀσ1ǀ = σult
: ǀσ2ǀ
= σult
 If the stress coordinates (σ1, σ2) at a point in the material fall
 on the boundary or outside the shaded area, the material is said to have a fracture
Theories of Failure (Brittle Materials)
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 Maximum Normal Stress Theory
This theory was proposed by W. Rankine and experimental results have exhibited that this theory gives
good predictions if the behavior of the brittle materials have stress-strain diagrams similar in tension and
compression.
Mohr’s Failure Criterion
 If brittle materials have different tension and compression properties then Mohr’s failure criterion can
be used to predict the failure.
 To apply Mohr’s failure criterion first three tests are required to be performed.
 A uniaxial tensile test and uniaxial compressive test are performed to determine the ultimate tensile
stress (σ ult)t and ultimate compressive stress (σ ult)c, respectively.
 A torsion test is performed to determine the material’s ultimate shear stress (τ ult).
 Mohr’s circle for each of these stresses (σ ult)t , (σ ult)c, (τ ult) is then plotted.
Theories of Failure (Brittle Materials)
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Mohr’s Failure Criterion
 Mohr’s circle for each of these stresses (σ ult)t , (σ ult)c, (τ ult) is then plotted. These three circles are
contained in a failure envelope indicated by the extrapolated color curve that is drawn tangent to all
three circles.
 If a plane stress condition at a point is represented by a circle that has a point of tangency with the
envelope, or if it extends beyond the envelope’s boundary then failure is said to occur.
Theories of Failure (Brittle Materials)
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Mohr’s Failure Criterion
 The failure criterion on a graph of Principal stresses, σ 1 , σ 2. Here the failure occurs when the absolute
value of either one of the Principal stresses reaches a value equal to or greater than the ultimate stress in
compression and tension (σ ult)t , (σ ult)c
 In general if the state of the stress defined by the stress coordinates is plotted on the boundary or outside
the shaded area.