Theory of Failure MCQ Quiz in मल्याळम - Objective Question with Answer for Theory of Failure - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Explanation:

Maximum strain energy theory (Haigh’s theory)

According to this theory, a body complex stress fails when the total strain energy at 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 is quite different.

Yield Locus is Ellipse

Important Points

• Maximum Principal stress theory was postulated by Rankine. It is suitable for brittle materials.
• Maximum Principal strain theory was postulated by St Venant. This theory is not accurate for brittle and ductile materials both.
• Maximum Shear stress theory was postulated by Tresca. This theory is suitable for ductile materials. Its results are the safest.
• Maximum shear strain energy theory was postulated by Von-mises. Its results in case of pure shear are the accurate for ductile materials

Explanation:

The following chart shows the ratio of shear yield stress to the tensile yield stress 

Concept:

Relationship between maximum shear strain (γ_max) and principal strains is:

$\left(\frac{{\gamma }_{max}}{2}\right)=\frac{{ϵ}_1-{ϵ}_2}{2}$

Calculation:

Given:

ϵ₁ = 100 x 10-6 and ϵ2 = -200 x 10-6

$\left(\frac{{\gamma }_{max}}{2}\right)=\frac{\left(100-\left(-200\right)\right)\times {10}^{-6}}{2}$

γmax = 300 × 10^-6

τ_max =  $\frac{{\sigma }_{max}-{\sigma }_{min}}{2}$

${\left(\frac{\gamma }{2}\right)}_{max}=\frac{{ϵ}_{max}-{ϵ}_{min}}{2}$

γ_max = ϵ_max - ϵ_min

Concept:

Rankine’s theory is the Maximum Normal Stress Theory according to which a material yields if:

$max\left({\sigma }_1,{\sigma }_2\right)=\frac{S_{yt}}{FOS}$

Calculation:

Given:

σ₁ = σ₂ = 200 MPa and S_yt = 400 MPa

$max\left({\sigma }_1,{\sigma }_2\right)=\frac{S_{yt}}{FOS}$

$200=\frac{400}{FOS}$

∴ FOS = 2.

Explanation:

Shear Strain Energy Theory ​(Von-Misses Theory):

• It is also known as the Maximum Energy of Distortion theory.
• The Von-Mises criterion considers the diameters of all three Mohr’s circles as contributing to the characterization of yield onset in isotropic materials. 
• This theory provides the best agreement between experiment and theory and, along with the Tresca theory, is very widely used.

Additional Information

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.

Concept:

Factor of safety (FOS):

It is defined as the ratio of ultimate/yield stress of the component material to the working stress. It denotes the additional strength of the component than the required strength.

FOS = $\frac{\mathrm{Y}\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{d}/\mathrm{U}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}}{\mathrm{W}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}}$

If a material yields before failure, then FOS is calculated on yield stress.

Calculation:

Given:

Working stress = 100 N/mm²; Yield stress = 150 N/mm2; Ultimate stress = 200 N/mm²;

FOS = $\frac{\mathrm{Y}\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{d}\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}}{\mathrm{W}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}}$ = ${\displaystyle \frac{150}{100}}$

FOS = 1.5 

Thus, option 1 is the correct answer.

Explanation:

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.

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.

Maximum principal stress theory (Rankine’s theory)

According to this theory, 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 design criterion, the maximum principal stress (σ₁) must not exceed the working stress ‘σ_y’ 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 σ_y

Graphical representation

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

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

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.

Calculation:

σ2 = 1000 kg/cm2 and σy = 1500 kg/cm2

${\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}\le \frac{{\sigma }_{\mathrm{y}}}{2}$

$\frac{{\sigma }_1-(-{\sigma }_2)}{2}\le \frac{{\sigma }_{\mathrm{y}}}{2}$

1000+σ₂ = 1500

σ2 = 500 kg/cm2