Failure Theories and Analysis
Failure Theories for Brittle Materials
Learning Objectives:

Apply various failure theories for to parts under steady loading

Understand the concept of safety factor and how to apply it.

Apply Goodmans and Soderbergs relationships for fatigue failure for nonzero mean stress.
There are five well known theories of elastic failures. These theories explain material under simple or complex loaded condition is elastic in many ways and does not meet a failure but undergoes a failure in one of the following five ways. The material is safe in four remaining ways. For example, cast iron fails under first theory only. Mild steel fails under second theory only.

Failure due to maximum principal stress (Maximum Principal Theory/Rankine’s Theory)

Failure due to maximum shear stress (Maximum Shear Stress Theory/Guest Theory)

Failure due to maximum principal strain (Maximum principal strain theory/St. Venant’s Theory)

Failure due to maximum principal strain energy (Maximum Principal Strain Energy Theory/Haigh Theory)

Failure due to maximum shear strain energy (Distortion Energy Theory/Maximum Shear Strain Energy Theory/ Von MisesHencky Theory)

In any theory, if the word ‘Principal’ exists, that will be applicable to brittle materials only.

In any theory, if the word ‘Shear’ exists, that will be applicable to ductile materials only.

A particular theory is applicable to few specific brittle material(s) or few specific ductile material(s).
MaximumNormalStress Theory:
The theory of failure due to the maximum normal stress is generally attributed to W. J. M. Rankine. The theory states that a brittle material will fail when the maximum principal stress exceeds some value, independent of whether other components of the stress tensor are present. Experiments in uniaxial tension and torsion have corroborated this assumption.
Failure will occur in the multiaxial state of stress when the maximum principal normal stress exceeds the ultimate tensile or compressive strength, \(S_{ut}\), or \(S_{uc}\), respectively.
ColombMohr Theory:
The Mohr Theory of Failure, also known as the CoulombMohr criterion or internalfriction theory, is based on the famous Mohr’s circle. Mohr’s theory is often used in predicting the failure of brittle materials, and is applied to cases of 2D stress.
Mohr’s theory suggests that failure occurs when Mohr’s Circle at a point in the body exceeds the envelope created by the two Mohr’s circles for uniaxial tensile strength and uniaxial compression strength.
Solved Example: 90101
At a certain point in a structural member, there are perpendicular stresses 80 N/mm^{2} and 20 N/mm^{2}, both tensile. What is the equivalent stress in simple tension, according to the maximum principal strain theory? (Poisson's ratio = 0.25) (ISRO Scientist Civil 2020)
A. 0 N/mm$^2$
B. 20 N/mm$^2$
C. 60 N/mm$^2$
D. 75 N/mm$^2$
Correct Answer: D
Solved Example: 90102
If Rankine's criteria is applied for failure of brittle material, then which of the following will be a necessary condition? (CIL MT Civil 2017)
A. Maximum shear stress
B. Maximum principal stress
C. Maximum shear strain energy
D. Maximum strain energy
Correct Answer: B
Solved Example: 90103
Match the following related to theories of failure
A. Max normal stress theory
B. Max shear stress theory
C. Max strain energy theory
D. Max distortion energy theory
1. Von mises theory
2. Haigh's theory
3. Guest and Tresca theory
4. Rankiness theory
(ISRO SDSC Tech Asst Mechanical Apr 2018)
A. A – 4, B – 3, C – 2, D 1
B. A – 3, B – 4, C – 1, D  2
C. A – 4, B – 3, C – 1, D  2
D. A – 3, B – 4, C – 2, D 1
Correct Answer: A
Solved Example: 90104
Region of safety for maximum principal stress theory under biaxial stress is shown by: (UPRVUNL AE ME 2016)
A. Square
B. Hexagon
C. Ellipse
D. Pentagon
Correct Answer: A
Solved Example: 90105
A material may fail if: (SSC JE CE Jan 18 Morning)
A. Maximum principal stress exceeds the direct stress
B. Maximum strain exceeds $\dfrac{\sigma_0}{E}$
C. Maximum shear stress exceeds $\dfrac{\sigma_0}{2}$
D. All options are correct
Correct Answer: D
Solved Example: 90106
Which one of following is NOT correct? (GATE ME 2014 Shift III)
A. Intermediate principal stress is ignored when applying the maximum principal stress theories
B. The maximum shear stress theory gives the most accurate results amongst all the failure theories
C. As per the maximum strain energy theory, failure occurs when the strain energy per unit volume exceeds a critical value
D. As per the maximum distortion energy theory, failure occurs when the distortion energy per unit volume exceeds a critical value.
Correct Answer: B
Failure Theories for Ductile Materials
Learning Objectives:

Apply various failure theories for brittle materials to parts under steady loading.
MaximumShearStress Theory: (Tresca)
As per this theory, yielding in the material is governed by the maximum shear stress developed in the material under the action of the principal stresses. The maximum shear stress due to the principal stresses should not exceed the maximum shear stress developed in a simple tensile test, that is, at yield point.
When Yielding occurs in any material, the maximum shear stress at the point of failure equals or exceeds the maximum shear stress when yielding occurs in the tension test specimen. \[\tau_{max} = \dfrac{\sigma_1  \sigma_2}{2}\] The theory applies to ductile materials only, because it is based on yielding.
This yield criterion gives good agreement with experimental results for ductile materials; because of its simplicity, it is the most often used yield theory.
The main objection to this theory is that it ignores the possible effect of the intermediate principal stress, \(\sigma_2\). However, only one other theory, the maximum distortional strain energy theory, predicts yielding better than does the Tresca theory, and the differences between the two theories are rarely more than 15%.
Failure is predicted when any of the three shear stresses corresponding to the principal stresses, \(\sigma_{1,2}\), equals or exceeds the shear stress corresponding to the yield strength, \(\sigma_{y}\), of the material in uniaxial tension or compression.
Distortion Energy Theory:
According to the Von Mises’s theory, a ductile solid will yield when the distortion energy density reaches a critical value for that material.
Or, a more formal statement of this theory would be,
When Yielding occurs in any material, the distortion strain energy per unit volume at the point of failure equals or exceeds the distortion strain energy per unit volume when yielding occurs in the tension test specimen.
Again, the theory applies to ductile materials only, because it is based on yielding. \[\left[\dfrac{(\sigma_1  \sigma_2)^2 +(\sigma_2  \sigma_3)^2 +(\sigma_1  \sigma_3)^2}{2}\right]^{\dfrac{1}{2}}\geq S_y\] The effective or VonMises stress is given by, \[\sigma' = \left(\sigma_A^2  \sigma_A \sigma_B + \sigma_B^2\right)^{\dfrac{1}{2}}\] or, \[\sigma' = \left(\sigma_x^2  \sigma_x \sigma_y + \sigma_y^2 + 3\tau_{xy}^2\right)^{\dfrac{1}{2}}\] where \(\sigma_A\) and \(\sigma_B\) are the two nonzero principal stresses and \(\sigma_x\), \(\sigma_y\) , and \(\tau_{xy}\) are the stresses in orthogonal directions.
Shear Strain Energy Theory:
This theory is also known as the Von MisesHencky theory. Detailed studies have indicated that yielding is related to the shear energy rather than the maximum shear stress.
Strain energy is energy stored in the material due to elastic deformation. The energy of strain is similar to the energy stored in a spring. Upon close examination, the strain energy is seen to be of two kinds : one part results from changes in mutually perpendicular dimensions , and hence in volume, with no change angular changes: the other arises from angular distortion without volume change. The latter is termed as the shear strain energy , which has been shown to be a primary cause of elastic failure.
Solved Example: 90201
All the theories of failure, will give nearly the same result when:
A. When one of the principal stresses at a point is large in comparison to the other.
B. When shear stresses act.
C. When both the principal stresses are numerically equal.
D. For all situations of stress.
When one of the principal stresses at a point is large in comparison to the other, the situation resembles uniaxial tension test. Therefore all theories give nearly the same results.
Correct Answer: A
Solved Example: 90202
The state of stress at a point is given as $\sigma_x$ = 100 N/mm$^2$ $\sigma_y$ = 40 N/mm$^2$ and $\tau_{xy}$= 40 N/mm$^2$. If the yield strength $S_y$ of the material is 300 MPa, the factor of safety using maximum shear stress theory will be: (VIZAG MT Mechanical 2015)
A. 3
B. 2.5
C. 7.5
D. 1.25
Correct Answer: B
Solved Example: 90203
According to Tresca, yield locus is a/an: (BPSC AE Paper 5 Civil 2012)
A. Rectangle
B. Circle
C. Hexagon
D. Ellipse
Correct Answer: C
Solved Example: 90204
For ductile materials the most appropriate failure theory is: (KPSC JE 2017)
A. Maximum shear stress theory
B. Maximum Principal stress theory
C. Maximum Principal strain theory
D. Shear strain energy theory
Correct Answer: D
Solved Example: 90205
Which theory of failure will you use for aluminium components under steady loading? (RRB JE ME CBT II Aug 2019)
A. Maximum principal stress theory
B. Maximum shear stress theory
C. Maximum strain energy theory
D. Maximum principal strain theory
Correct Answer: B
Variable Loading Failure Theories
Learning Objectives:

Apply Goodmans and Soderbergs relationships for fatigue failure for nonzero mean stress.
Mean and Alternating Stress:
A purely reversing or cyclic stress means when the stress alternates between equal positive and negative peak stresses sinusoidally during each cycle of operation.

Mean stress \[\sigma_m = \dfrac{\sigma_{max} + \sigma_{min}}{2}\]

Alternating stress \[\sigma_a = \dfrac{\sigma_{max}  \sigma_{min}}{2}\]
Endurance Strength:
The maximum completely reversing cyclic stress that a material can withstand for indefinite (or infinite) number of stress reversals is known as the fatigue strength or endurance strength (S\(_e\)) of the part material.
Soderberg’s Theory:

If a part only contains the steady part of the stress \(\sigma_m\), (that is \(\sigma_a\)=0) then to prevent failure: \[\sigma_m < \dfrac{S_{y}}{K*N},\] where,
K= geometric stress concentration factor, and
N= factor of safety.
Usually parts subjected to fatigue loading are made of ductile material, and for steady stress, we learned that the geometric stress concentration factor can be neglected. Thus the limiting condition is: \[\sigma_m < \dfrac{S_{y}}{N}\] Which means that \(\sigma_m\) can go up to \(\dfrac{S_{y}}{N}\) when \(\sigma_a\) = 0 
Similarly, when there is only reversing stress \(\sigma_a\) present, then for safe design: \[\sigma_a < \dfrac{S_e}{N*Kf},\] where, K\(_f\)= is the fatigue stress concentration factor. Which means \(\sigma_a\) can go up to \(\dfrac{S_e}{N*Kf}\), when \(\sigma_m\)=0
If we plot steady stress (\(\sigma_m\)) along x axis and the range stress (\(\sigma_a\)) along y axis, then the two extreme stress conditions (i) and (ii) described above, constitute two point on x and y axis. Soderberg Line is obtained by joining these two points. When in a machine part, both types of stress are present simultaneously, if the stress combination (\(\sigma_m\) and \(\sigma_a\)) is contained in the shaded area defined by the Soderberg’s line, then the part should be safe. Any stress combination falling above the Soderberg’s line would be unsafe. Using intercept form of the equation of straight line, i.e., \[\dfrac{x}{a}+ \dfrac{y}{b}=1,\] the safe design area is the shaded area.
\[\dfrac{\sigma _{a}}{S_{e}} + \dfrac{\sigma _{m}}{S_{y}} \geq 1,\ \sigma_m \geq 0\]
Goodman’s Line:
Because of brittle nature of failure, Goodman proposed the safe design stress for steady stress should be extended to \(\dfrac{S_u}{N}\) instead of \(\dfrac{S_{y}}{N}\) in Soderberg’s equation. This resulted in the safe design space as shown and the resulted in Goodman Design equation:
Goodman Equation can be obtained from Soderberg equation by replacing S\(_{y}\) by S\(_u\). However, in the safe area defined by Goodman line, when the magnitude of steady stress \(\sigma_m\) becomes more than \(\dfrac{S_{y}}{N}\), the part may fail from yielding from plastic deformation. The area is shown as unsafe region.
Modified Goodman Theory:
To eliminate this shortcoming, a line with 45\(^\circ\) angle from the \(\dfrac{S_{y}}{N}\) point on the x axis. Mathematically this Modified Goodman space is equivalent to satisfying the following two equations, simultaneously.
\[\dfrac{\sigma_{a}}{S_{e}} + \dfrac{\sigma_{m}}{S_{ut}} \geq 1\]
\[\dfrac{\sigma_{max}}{S_y} \geq 1\]