Theory of Failure MCQ Quiz - Objective Question with Answer for Theory of Failure - Download Free PDF

Explanation:

Maximum shear stress theory (Guest & Tresca’s Theory):

• According to this theory, failure of the specimen subjected to any combination of a load when the maximum shearing stress at any point reaches the failure value equal to that developed at the yielding in an axial tensile or compressive test of the same material.

Graphical Representation:

• ${\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}\le \frac{{\sigma }_{\mathrm{y}}}{2}$ For no failure
• ${\sigma }_1-{\sigma }_2\le \left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)$ For design
• σ1 and σ2 are maximum and minimum principal stress respectively.
• Here, τmax = Maximum shear stress
• σy = permissible stress
• This theory is justified but a conservative theory for ductile materials. It is an uneconomical theory. 

Additional Information

Maximum shear strain energy / Distortion energy theory / Mises – Henky theory: 

• It states that inelastic action at any point in body, under any combination of stress begging, when the strain energy of distortion per unit volume absorbed at the point is equal to the strain energy of distortion absorbed per unit volume at any point in a bar stressed to the elastic limit under the state of uniaxial stress as occurs in a simple tension/compression test.
• $\frac{1}{2}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]\le {\sigma }_{\mathrm{y}}^2$ for no failure
• $\frac{1}{2}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]\le {\left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)}^2$ For design

• It is the best suitable theory for ductile material.
• It cannot be applied to the material under hydrostatic pressure.​

Maximum principal stress theory (Rankine’s theory):

• According to this theory, the permanent set takes place under a state of complex stress, when the value of maximum principal stress is equal to that of yield point stress as found in a simple tensile test.
• For the design criterion, the maximum principal stress (σ1) must not exceed the working stress ‘σy’ for the material.
• ${\sigma }_{1,2}\le {\sigma }_{\mathrm{y}}$ for no failure
• ${\sigma }_{1,2}\le \frac{\sigma }{\mathrm{F}\mathrm{O}\mathrm{S}}$ for design
• Note: For no shear failure τ ≤ 0.57 σy

Graphical representation:

• For brittle material, which does not fail by yielding but fail by brittle fracture, this theory gives a satisfactory result.
• The graph is always square even for different values of σ1 and σ2.

Maximum principal strain theory (ST. Venant’s theory):

• According to this theory, a ductile material begins to yield when the maximum principal strain reaches the strain at which yielding occurs in simple tension.
• ${ϵ}_{1,2}\le \frac{{\sigma }_{\mathrm{y}}}{{\mathrm{E}}_1}$ For no failure in uniaxial loading.
• $\frac{{\sigma }_1}{\mathrm{E}}-\mu \frac{{\sigma }_2}{\mathrm{E}}-\mu \frac{{\sigma }_3}{\mathrm{E}}\le \frac{{\sigma }_{\mathrm{y}}}{\mathrm{E}}$ For no failure in triaxial loading.
• ${\sigma }_1-\mu {\sigma }_2-\mu {\sigma }_3\le \left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)$ For design, Here, ϵ = Principal strain
• σ1, σ2, and σ3 = Principal stresses   

Graphical Representation:

This theory overestimates the elastic strength of ductile material.

Maximum strain energy theory (Haigh’s theory):

• According to this theory, a body complex stress fails when the total strain energy at the elastic limit in simple tension.
• Graphical Representation:
• $\left\{{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2-2\mu \left({\sigma }_1{\sigma }_2+{\sigma }_2{\sigma }_3+{\sigma }_3{\sigma }_1\right)\right\}\le {\sigma }_{\mathrm{y}}^2$  for no failure
• $\left\{{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2-2\mu \left({\sigma }_1{\sigma }_2+{\sigma }_2{\sigma }_3+{\sigma }_3{\sigma }_1\right)\right\}\le {\left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)}^2$ for design
• This theory does not apply to brittle material for which elastic limit stress in tension and in compression are quite different.

Important Points

• For Brittle material:- Maximum  Principal Stress Theory (Rankine criteria) is used.
• Maximum Shear Stress Theory (Tresca theory), Total strain energy theory, Maximum Distortion Energy Theory (von Mises) useful for a ductile material.
• Tresca's theory fails in the hydrostatic state of stresses.
• All theories will give the same results if loading is uniaxial.

Explanation:

Designing a Transmission Shaft Subjected to Bending and Torsional Moments:

• When designing a transmission shaft that is subjected to both bending and torsional moments, it is crucial to ensure the structural integrity of the shaft under the combined loading conditions. The correct design theory to be applied in this scenario is Maximum Shear Stress Theory, which is also known as the Tresca Criterion.

Maximum Shear Stress Theory (Tresca Criterion):

• The Maximum Shear Stress Theory states that failure occurs in a material when the maximum shear stress induced in the material exceeds the shear stress at yield point in pure shear. This theory is widely used for ductile materials, such as steel, which are commonly used in transmission shafts.

Mathematical Representation:

The maximum shear stress in a material subjected to combined loading can be determined using the principal stresses. If σ₁ and σ₃ are the maximum and minimum principal stresses, respectively, the maximum shear stress is given by:

τ_max = (σ₁ - σ₃) ÷ 2

For a transmission shaft, both bending and torsional moments contribute to the stress distribution. The bending moment generates normal stresses, while the torsional moment induces shear stresses. Therefore, the combined effect must be analyzed to ensure the shaft does not fail under the applied loading conditions.

Concept:

According to the Maximum Shear Stress Theory (Guest’s or Tresca’s theory), the maximum shear stress in a shaft subjected to combined bending and torsion is given by:

${\tau }_{max}=\frac{1}{2}\sqrt{{\sigma }_b^2+4{\tau }^2}$

To ensure safety, this is set equal to half the yield strength in simple tension:

${\tau }_{max}=\frac{{\sigma }_y}{2}$

Calculation:

Given:

Bending moment, $M=16kN\cdot m=16\times {10}^{3}N\cdot m$

Torque, T = $12kN\cdot m=12\times {10}^{3}N\cdot m$

Elastic limit in tension, ${\sigma }_y=160MPa=160\times {10}^{6}Pa$

The combined equivalent moment is calculated using:

$M_e=\sqrt{M^2+T^2}=\sqrt{(16{)}^2+(12{)}^2}\times {10}^3=\sqrt{256+144}\times {10}^3=20\times {10}^{3}N\cdot m$

Using the formula:

$\frac{32M_e}{\pi d^3}={\sigma }_y\Rightarrow d^3=\frac{32M_e}{\pi {\sigma }_y}$

$d^3=\frac{32\cdot 20\times {10}^3}{\pi \cdot 160\times {10}^6}=\frac{640\times {10}^3}{502.65\times {10}^6}=1.273\times {10}^{-3}{m}^3$

$d=(1.273\times {10}^{-3}{)}^{1/3}=0.108m=108mm$

Concept:

According to **shear strain energy theory (Von Mises and Hencky’s theory)**, the failure criterion is based on the distortion energy. The equivalent stress is given by:

${\sigma }_{eq}=\sqrt{\frac{({\sigma }_1-{\sigma }_2{)}^2+({\sigma }_2-{\sigma }_3{)}^2+({\sigma }_3-{\sigma }_1{)}^2}{2}}$

where:

• ${\sigma }_1=2x(tensile)$
• ${\sigma }_2=x(tensile)$
• ${\sigma }_3=\frac{x}{2}(compressive)$

The material will fail when the equivalent stress equals the **elastic limit stress**:

${\sigma }_{eq}={\sigma }_y$

where ${\sigma }_y=200$ N/mm².

Calculation:

Step 1: Compute equivalent stress

${\sigma }_{eq}=\sqrt{\frac{(2x-x{)}^2+(x-(-x/2){)}^2+(-x/2-2x{)}^2}{2}}$

${\sigma }_{eq}=\sqrt{\frac{(x{)}^2+(3x/2{)}^2+(5x/2{)}^2}{2}}$

${\sigma }_{eq}=\sqrt{\frac{x^2+9x^2/4+25x^2/4}{2}}$

${\sigma }_{eq}=\sqrt{\frac{4x^2+9x^2+25x^2}{8}}$

${\sigma }_{eq}=\sqrt{\frac{38x^2}{8}}=\sqrt{\frac{19x^2}{4}}=\frac{x\sqrt{19}}{2}$

Step 2: Equating to Elastic Limit

$\frac{x\sqrt{19}}{2}=200$

$x=\frac{400}{\sqrt{19}}$

Concept:

Given:

The circular shaft is subjected to twisting moment only which results in maximum shear stress of 90MPa.

Maximum Shear stress is given by:

${\tau }_{max}=\frac{16T}{\pi d^3}$

∴ It is a case of pure shear. And the Mohr circle for pure shear is represented by:

• Here, the compressive stress will be equal to the maximum shear stress.
• In the case of pure shear, τ_xy = τ and σ_x = σ_y = 0

σ₁ = τ , σ2 = -τ

Centre of Mohr circle = 0

Radius = $\frac{{\sigma }_1-{\sigma }_2}{2}$ = $\frac{\tau -\left(-\tau \right)}{2}=\tau$

Hence, the maximum compressive stress in the material is 90MPa.

Concept:

1) Maximum principal stress theory (Rankine theory/ Lame’s theory):-

As per this theory, for no failure maximum principal stress should be less than yield stress under uniaxial loading.

i.e. σmax < fy

For design, ${\sigma }_{max}<\frac{f_y}{F.O.S}$

This theory is most suitable for brittle materials, not applicable to ductile materials, not applicable to pure shear case because as per this theory τ should be less than fy

2) Maximum Principle strain theory (Saint-Venant theory):-

As per this theory, for no failure maximum principal strain should be less than strain under uniaxial loading when the stress is fy.

$i.e{\epsilon }_{max}<\frac{f_y}{E}$

For Design, ${\epsilon }_{max}<\frac{\left(f_y/FOS\right)}{E}$

This theory is satisfactory for brittle material, not suitable for hydrostatic stress conditions.

This theory is not suitable for the pure shear case.

3) Maximum shear stress Theory (Tresca theory/ Guest Theory/ Coulomb Theory):-

As per this theory, for no failure absolute maximum shear stress should be less than maximum shear stress under uniaxial loading, when the stress is fy.

Maximum shear stress under the uniaxial condition when the stress is fy is given as fy/2

$\therefore {\tau }_{abs}max<\frac{f_y}{2}$

For Design,

${\tau }_{abs}max<\frac{\left(f_y/FOS\right)}{2}$

4) Maximum strain energy theory (Beltrami-Haigh Theory):-

As per this theory, for no failure maximum strain energy per unit volume should be less than strain energy per unit volume under uniaxial loading when the stress is fy.

$\frac{1}{2E}\left[{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2-2\mu \left({\sigma }_1{\sigma }_2+{\sigma }_2{\sigma }_3+{\sigma }_3{\sigma }_1\right)\right]<\frac{f_y^2}{2E}$

Where, σ1, σ2, σ3 are principal stresses.

This theory is applicable for ductile material, not suitable for brittle material, and not suitable for the pure shear case.

5) Maximum shear strain energy theory (Von mises/ Distortion energy theory):-

As per this theory, for no failure, maximum shear strain energy per unit volume should be less than maximum shear strain energy per unit volume under uniaxial loading.

$\frac{1}{12G}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]<\frac{f_y^2}{6G}$

This theory is in perfect agreement with the result of the test in case of pure shear.

This theory is the most suitable theory of failure for a ductile material.

Conclusion:

• All theories of failure give similar results in the case of uniaxial loading.

Explanation:

Yield loci for two yield criteria in-plane stress as per Von Mises and Tresca Criteria is given below:

∴ According to Tresca, the yield is a hexagon.

 ​

For brittle material

For Ductile material

Concept:

Maximum principal stress theory (Rankine’s theory)

According to this theory, the permanent set takes place under a state of complex stress, when the value of maximum principal stress is equal to that of yield point stress as found in a simple tensile test.

For the design criterion, the maximum principal stress (σ1) must not exceed the working stress ‘σy’ for the material.

${\sigma }_{1,2}\le {\sigma }_{\mathrm{y}}$ for no failure

${\sigma }_{1,2}\le \frac{\sigma }{\mathrm{F}\mathrm{O}\mathrm{S}}$ for design

Note: For no shear failure τ ≤ 0.57 σy

Graphical representation

For brittle material, which does not fail by yielding but fail by brittle fracture, this theory gives a satisfactory result.

The graph is always square even for different values of σ1 and σ2.

Maximum principal strain theory (ST. Venant’s theory)

According to this theory, a ductile material begins to yield when the maximum principal strain reaches the strain at which yielding occurs in simple tension.

${ϵ}_{1,2}\le \frac{{\sigma }_{\mathrm{y}}}{{\mathrm{E}}_1}$ For no failure in uniaxial loading.

$\frac{{\sigma }_1}{\mathrm{E}}-\mu \frac{{\sigma }_2}{\mathrm{E}}-\mu \frac{{\sigma }_3}{\mathrm{E}}\le \frac{{\sigma }_{\mathrm{y}}}{\mathrm{E}}$ For no failure in triaxial loading.

${\sigma }_1-\mu {\sigma }_2-\mu {\sigma }_3\le \left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)$ For design, Here, ϵ = Principal strain

σ1, σ2, and σ3 = Principal stresses   

Graphical Representation

This theory overestimates the elastic strength of ductile material.

Maximum shear stress theory

(Guest & Tresca’s Theory)

According to this theory, failure of the specimen subjected to any combination of a load when the maximum shearing stress at any point reaches the failure value equal to that developed at the yielding in an axial tensile or compressive test of the same material.

Graphical Representation

${\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}\le \frac{{\sigma }_{\mathrm{y}}}{2}$ For no failure

${\sigma }_1-{\sigma }_2\le \left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)$ For design

σ1 and σ2 are maximum and minimum principal stress respectively.

Here, τmax = Maximum shear stress

σy = permissible stress

This theory is well justified for ductile materials.

Maximum strain energy theory (Haigh’s theory)

According to this theory, a body complex stress fails when the total strain energy at the elastic limit in simple tension.

Graphical Representation.

$\left\{{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2-2\mu \left({\sigma }_1{\sigma }_2+{\sigma }_2{\sigma }_3+{\sigma }_3{\sigma }_1\right)\right\}\le {\sigma }_{\mathrm{y}}^2$  for no failure

$\left\{{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2-2\mu \left({\sigma }_1{\sigma }_2+{\sigma }_2{\sigma }_3+{\sigma }_3{\sigma }_1\right)\right\}\le {\left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)}^2$ for design

This theory does not apply to brittle material for which elastic limit stress in tension and in compression are quite different.

Maximum shear strain energy / Distortion energy theory / Mises – Henky theory.

It states that inelastic action at any point in body, under any combination of stress begging, when the strain energy of distortion per unit volume absorbed at the point is equal to the strain energy of distortion absorbed per unit volume at any point in a bar stressed to the elastic limit under the state of uniaxial stress as occurs in a simple tension/compression test.

$\frac{1}{2}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]\le {\sigma }_{\mathrm{y}}^2$ for no failure

$\frac{1}{2}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]\le {\left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)}^2$ For design

It cannot be applied for material under hydrostatic pressure. 

All theories will give the same results if loading is uniaxial.

Mistake Points  Maximum shear stress theory gives uneconomic design whereas maximum shear strain energy theory gives economic design. therefore, the maximum shear strain energy theory is most suitable.

Concept:

Maximum shear stress theory:

The Maximum Shear Stress theory states that failure occurs when the maximum shear stress from a combination of principal stresses equals or exceeds the value obtained for the shear stress at yielding in the uniaxial tensile test.

 ${\tau }_{max}\le \frac{{\tau }_y}{FOS}$

$\frac{{\sigma }_{max}-{\sigma }_{min}}{2}\le \frac{{\sigma }_y}{2\times FOS}$

Calculation:

Given:

σ_x = 100 N/mm², σ_y  = 40 N/mm², τ_xy = 40 N/mm² σ_y = 300 MPa

${\sigma }_1=\left(\frac{{\sigma }_x+{\sigma }_y}{2}\right)+\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$

${\sigma }_1=\left(\frac{100+40}{2}\right)+\sqrt{{\left(\frac{100-40}{2}\right)}^2+{40}^2}=120\frac{N}{m{m}^2}$

${\sigma }_2=\left(\frac{{\sigma }_x+{\sigma }_y}{2}\right)-\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$

${\sigma }_2=\left(\frac{100+40}{2}\right)-\sqrt{{\left(\frac{100-40}{2}\right)}^2+{40}^2}=20\frac{N}{m{m}^2}$, ${\sigma }_3=0\frac{N}{m{m}^2}$

$\frac{{\sigma }_1-{\sigma }_2}{2}=50,\frac{{\sigma }_2-{\sigma }_3}{2}=10,\frac{{\sigma }_1-{\sigma }_3}{2}=60$

$\frac{{\sigma }_{max}-{\sigma }_{min}}{2}\le \frac{{\sigma }_y}{2\times FOS}=\frac{120-0}{2}=\frac{300}{2\times N}$

N = 2.5

Explanation:

Theory of failure:

The following theory of failure are given below:

(1) Maximum principal stress theory (Rankine's theory):

• According to this, the failure of the material occurs when the maximum principal stress reaches the elastic limit stress of the material in simple tension.

(2) Maximum principal strain theory (St. Venant's theory)

(3) Maximum shear stress theory (Guest & Tresca's theory)

(4) Maximum strain energy theory (Haigh's theory)

(5) Maximum shear strain energy theory (Mises-Henky theory)

Buckling:

Buckling is defined as the sudden change in the shape (lateral deformation) of a structural component (member) under axial load.

Principal plane:

• At any point within a stressed body, there always exist three mutually perpendicular planes on each of which the resultant stress is normal stress. 
• These mutually perpendicular planes are called principal planes, and the resultant normal stresses acting on them are called principal stresses.
• The principal plane is subjected to zero shear stress it is only subjected to only normal stress.

Slabs:

• Slabs are primarily designed for bending (sagging and hogging bending moments) and deflection not for torsion.

Explanation:

Total strain energy in the three-dimensional system is:

Total strain energy per unit volume:

U = $\frac{1}{2E}[({\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2)-2\mu ({\sigma }_1{\sigma }_2+{\sigma }_2{\sigma }_3+{\sigma }_3{\sigma }_1)]$

Total strain energy can be split into two strain energy:

• The strain energy of distortion
• The strain energy of uniform compression, tension or energy of dilation.

The energy of dilation:

U_v = $\frac{3(1-2\mu )}{2E}{\left(\frac{{\sigma }_1+{\sigma }_2+{\sigma }_3}{3}\right)}^2$

Strain energy of distortion:

Distortion energy = (total Strain energy) - (energy of dilation)

U_d = $\frac{1+\mu }{6E}[({\sigma }_1-{\sigma }_2{)}^2+({\sigma }_2-{\sigma }_3{)}^2+({\sigma }_3-{\sigma }_1{)}^2]$

on simplifying above expression we get

$\frac{1+\mu }{3E}\left[{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2-\left({\sigma }_1{\sigma }_2+{\sigma }_3{\sigma }_2+{\sigma }_1{\sigma }_3\right)\right]$

Explanation:

Maximum Shear Stress Theory: 

• Applied satisfactorily to ductile materials.
• This theory states that the 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 relation between the theories of failure, their suitable materials, and graphical representation is given below.

Explanation:

Maximum principal strain theory (ST. Venant’s theory)

• According to this theory, a ductile material begins to yield when the maximum principal strain reaches the strain at which yielding occurs in simple tension.

${ϵ}_{1,2}\le \frac{{\sigma }_{\mathrm{y}}}{{\mathrm{E}}_1}$ For no failure in uniaxial loading.

$\frac{{\sigma }_1}{\mathrm{E}}-\mu \frac{{\sigma }_2}{\mathrm{E}}-\mu \frac{{\sigma }_3}{\mathrm{E}}\le \frac{{\sigma }_{\mathrm{y}}}{\mathrm{E}}$ For no failure in triaxial loading.

${\sigma }_1-\mu {\sigma }_2-\mu {\sigma }_3\le \left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)$ For design, Here, ϵ = Principal strain

where σ1, σ2, and σ3 = Principal stresses   

Graphical Representation

• This theory overestimates the elastic strength of ductile material.

Additional Information

Maximum principal stress theory (Rankine’s theory)

• According to this theory, the permanent set takes place under a state of complex stress, when the value of maximum principal stress is equal to that of yield point stress as found in a simple tensile test.
• For the design criterion, the maximum principal stress (σ1) must not exceed the working stress ‘σy’ for the material.

${\sigma }_{1,2}\le {\sigma }_{\mathrm{y}}$ for no failure

${\sigma }_{1,2}\le \frac{\sigma }{\mathrm{F}\mathrm{O}\mathrm{S}}$ for design

• Note: For no shear failure τ ≤ 0.57 σy

Graphical representation

• For brittle material, which does not fail by yielding but fail by brittle fracture, this theory gives a satisfactory result.
• The graph is always square even for different values of σ1 and σ2.

Maximum shear stress theory (Guest & Tresca’s Theory)

• According to this theory, failure of the specimen subjected to any combination of a load when the maximum shearing stress at any point reaches the failure value equal to that developed at the yielding in an axial tensile or compressive test of the same material.

Graphical Representation

${\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}\le \frac{{\sigma }_{\mathrm{y}}}{2}$ For no failure

${\sigma }_1-{\sigma }_2\le \left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)$ For design

where σ1 and σ2 are maximum and minimum principal stress respectively, τmax = Maximum shear stress and σy = permissible stress

• This theory is well justified for ductile materials.

Maximum strain energy theory (Haigh’s theory)

• According to this theory, a body complex stress fails when the total strain energy at the elastic limit in simple tension.
• Graphical Representation.

$\left\{{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2-2\mu \left({\sigma }_1{\sigma }_2+{\sigma }_2{\sigma }_3+{\sigma }_3{\sigma }_1\right)\right\}\le {\sigma }_{\mathrm{y}}^2$  for no failure

$\left\{{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2-2\mu \left({\sigma }_1{\sigma }_2+{\sigma }_2{\sigma }_3+{\sigma }_3{\sigma }_1\right)\right\}\le {\left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)}^2$ for design

• This theory does not apply to brittle material for which elastic limit stress in tension and in compression are quite different.

Maximum shear strain energy / Distortion energy theory / Mises – Henky theory.

• It states that inelastic action at any point in body, under any combination of stress begging, when the strain energy of distortion per unit volume absorbed at the point is equal to the strain energy of distortion absorbed per unit volume at any point in a bar stressed to the elastic limit under the state of uniaxial stress as occurs in a simple tension/compression test.

$\frac{1}{2}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]\le {\sigma }_{\mathrm{y}}^2$ for no failure

$\frac{1}{2}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]\le {\left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)}^2$ For design

• 
• It cannot be applied for material under hydrostatic pressure.
• All theories will give the same results if loading is uniaxial.

Conclusion:

• For Brittle material:- Maximum  Principal Stress Theory (Rankine criteria) is used.
• Maximum Shear Stress Theory (Tresca theory), Total strain energy theory, Maximum Distortion Energy Theory (von Mises) useful for a ductile material.
• Tresca theory fails in hydrostatic state of stresses.

Explanation:

Maximum shear stress theory (Guest & Tresca’s Theory)

According to this theory, failure of the specimen subjected to any combination of a load when the maximum shearing stress at any point reaches the failure value equal to that developed at the yielding in an axial tensile or compressive test of the same material.

Graphical Representation

${\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}\le \frac{{\sigma }_{\mathrm{y}}}{2}$ For no failure

${\sigma }_1-{\sigma }_2\le \left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)$ For design

σ1 and σ2 are maximum and minimum principal stress respectively.

Here, τmax = Maximum shear stress

σy = permissible stress

This theory is well justified for ductile materials but gives over safe result and hence called uneconomical theory.

Maximum strain energy theory (Haigh’s theory)

According to this theory, a body complex stress fails when the total strain energy at the elastic limit in simple tension.

Graphical Representation.

$\left\{{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2-2\mu \left({\sigma }_1{\sigma }_2+{\sigma }_2{\sigma }_3+{\sigma }_3{\sigma }_1\right)\right\}\le {\sigma }_{\mathrm{y}}^2$  for no failure

$\left\{{\sigma }_1^2+{\sigma }_2^2+{\sigma }_3^2-2\mu \left({\sigma }_1{\sigma }_2+{\sigma }_2{\sigma }_3+{\sigma }_3{\sigma }_1\right)\right\}\le {\left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)}^2$ for design

This theory does not apply to brittle material for which elastic limit stress in tension and in compression are quite different.

Maximum normal stress theory (Rankine’s theory)

According to this theory, the permanent set takes place under a state of complex stress, when the value of maximum principal stress is equal to that of yield point stress as found in a simple tensile test.

For the design criterion, the maximum principal stress (σ1) must not exceed the working stress ‘σy’ for the material.

${\sigma }_{1,2}\le {\sigma }_{\mathrm{y}}$ for no failure

${\sigma }_{1,2}\le \frac{\sigma }{\mathrm{F}\mathrm{O}\mathrm{S}}$ for design

Note: For no shear failure τ ≤ 0.57 σy

Graphical representation

For brittle material, which does not fail by yielding but fail by brittle fracture, this theory gives a satisfactory result.

The graph is always square even for different values of σ1 and σ2.

Maximum shear strain energy (Distortion energy theory) or Von-Mises – Henky theory.

It states that inelastic action at any point in body, under any combination of stress begging, when the strain energy of distortion per unit volume absorbed at the point is equal to the strain energy of distortion absorbed per unit volume at any point in a bar stressed to the elastic limit under the state of uniaxial stress as occurs in a simple tension/compression test.

$\frac{1}{2}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]\le {\sigma }_{\mathrm{y}}^2$ for no failure

$\frac{1}{2}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]\le {\left(\frac{{\sigma }_{\mathrm{y}}}{\mathrm{F}\mathrm{O}\mathrm{S}}\right)}^2$ For design

It is best suitable theory for ductile material

Concept:

Hooke’s law: As per Hooke’s law, stress is proportional to strain

i.e. σ ∝ ϵ or, σ = ϵ × E

Where, E =  Modulus of Elasticity

For Hooke’s law to be valid:

(a) The material should be homogenous.

(b) The material should be isotropic.

(c) The material should behave in a linearly elastic manner.

Thus for a plane bar with Area ‘A’, Length ‘L’ and Modulus of Elasticity ‘E’,

$\sigma =\frac{\mathrm{P}}{\mathrm{A}}\mathrm{a}\mathrm{n}\mathrm{d}ϵ=\frac{\delta \mathrm{L}}{L}$

Generalised Hooke's law:

${ϵ}_1=\frac{{\sigma }_1}{\mathrm{E}}-\mu \frac{{\sigma }_2}{\mathrm{E}}-\mu \frac{{\sigma }_3}{\mathrm{E}}$

${ϵ}_2=\frac{{\sigma }_2}{\mathrm{E}}-\mu \frac{{\sigma }_1}{\mathrm{E}}-\mu \frac{{\sigma }_3}{\mathrm{E}}$

${ϵ}_3=\frac{{\sigma }_3}{\mathrm{E}}-\mu \frac{{\sigma }_1}{\mathrm{E}}-\mu \frac{{\sigma }_2}{\mathrm{E}}$

ϵ_equ = ϵ₁ + ϵ₂ + ϵ₃    

Calculation:

Given:

Two perpendicular tensile stress

σ₁ = 80 N/mm2, σ₂ = 20 N/mm2, μ = 0.25

Generalised Hooke's law for two stresses

ϵequ = ϵ1

$\frac{{\sigma }_{equ}}{E}=\frac{{\sigma }_1}{\mathrm{E}}-\mu \frac{{\sigma }_2}{\mathrm{E}}$

${\sigma }_{\mathrm{e}\mathrm{q}\mathrm{u}}={\sigma }_1-\mu {\sigma }_2$

${\sigma }_{\mathrm{e}\mathrm{q}\mathrm{u}}=80-0.25\times 20=75\mathrm{N}/\mathrm{m}{\mathrm{m}}^2$