THEORIES OF FAILURE

CLASS NOTES FOR MECHANICAL

ENGINEERING

 Failure means breaking or deforming.

Failure of ductile material is entirely different

from the failure of a brittle material under

same type of load. There are five well

known theories of elastic failures. These

theories explain material behavior under

simple or complex loaded

condition. It is elastic under five different

criterions. It does not meet a failure in four

of the five. Failure occurs in one of the

remaining criterions. The material is safe

and elastic in four remaining ways. For

example, cast iron fails under first theory

only. Mild steel fails under second theory

only.

• Maximum Principal Theory (Rankine Theory)

             Failure due to maximum principal stress

• Maximum Shear Stress Theory (Guest Theory)

              Failure due to maximum shear stress

• Maximum principal strain theory (Saint Venant’s Theory)

             Failure due to maximum principal strain

• Maximum Principal Strain Energy Theory (Haigh Theory)

           Failure due to maximum principal strain energy

• Distortion Energy Theory (Maximum Shear Strain Energy Theory OR Von Mises-Hencky Theory)

          Failure due to maximum shear strain energy

Theory

Assumed in all the theories

Firstly σ₁  is tensile

Secondly σ₂ is tensile

Thirdly σ₃ is tensile

Fourthly σ₁  > σ₂ > σ₃

FOR DUCTILE MATERIALS

σ_allow =σ_yp/FOS

FOR BRITTLE MATERIALS

σ_allow =σ_utm/FOS

ALLOWABLE VALUES for a 2-dimensional stress system is σ_allow, σ_allow, and 0. It is because in 3-dimensional stress system, one of the stresses becomes zero . It leads to 2-dimensional stress system.

Maximum allowable shear stress

= (σ_allow— σ_allow)/2,

(σ_allow—o)/2,     (0–σ_allow)/2

We know Stresses actually present in a 2-dimensional stress system is σ₁, σ₂ and σ₃=0

Actually present maximum shear stress

(σ₁– σ₂)/2, (σ₂-σ₃)/2 and (σ₃– σ₁)/2

Fig. MAXIMUM PRINCIPAL STRESS THEORY (RANKINE THEORY)

 

MAXIMUM PRINCIPAL STRESS THEORY (RANKINE THEORY)

ALLOWABLE stress VALUES σ_allow, σ_allow, and 0

Stresses actually present σ₁, σ₂ and σ₃=0

Finally Maximum Principal Stress Actually present≤ σ_ult/FOS

MAXIMUM SHEAR STRESS THEORY (Guest Theory or Tresca Theory)

Fig. Maximum Shear Stress Theory (Guest Theory)

We know Maximum shear stress from allowable stress

(i) (σ_allow— σ_allow)/2,

(ii) (σ_allow—o)/2,

(iii) (0–σ_allow)/2

Principal Stresses actually present σ₁, σ₂ and σ₃=0

Actually present maximum shear stress

(i) (σ₁– σ₂)/2

(ii) (σ₂-σ₃)/2

(iii) (σ₃– σ₁)/2

Actually present maximum shear stress≤ allowable shear stress

MAXIMUM PRINCIPAL STRAIN THEORY

(SAINT VENANT’S THEORY)

Fig. Maximum Principal Strain Theory (st. Venant’s Theory)

ALLOWABLE stress VALUES σ_allow, σ_allow, 0

Allowable strains  (1/E) (σ_allow-µ σ_allow),

(1/E)(σ_allow–0) and     (1/E)(0–σ_allow)

Principal Stresses actually present σ₁, σ₂ and σ₃=0

Actual strains present

(i) (1/E)(σ₁– µσ₂)

(ii) (1/E)(σ₂–µ σ₃)

(iii) (1/E)(σ₃— µσ₁)

If Actually present≤ Allowable strain, it is safe

MAXIMUM PRINCIPAL STRAIN ENERGY THEORY-HAIGH THEORY

Fig. Maximum Principal Strain Energy Theory (Haigh Theory)

ALLOWABLE stress VALUES σ_allow, σ_allow, 0

Allowable strain energy

u=(1/2E)( σ²_allow+ σ²_allow–2µ σ_allow σ_allow)

Stresses actually present σ₁, σ₂ and σ₃=0

Strain energy ACTUALLY present

u=(1/2E)( σ₁²+ σ₂²–2µ σ₁ σ₂)

FINAL IS

( σ₁²+ σ₂²–2µ σ₁ σ₂)= σ²_allow

MAXIMUM SHEAR STRAIN ENERGY THEORY (VON MISES HENCKY THEORY)

 

ALLOWABLE stress VALUES σ_allow, σ_allow, 0

Allowable shear strain energy

u_SSE =(1/12G)[( σ_allow—σ_allow)² +(σ_allow–0)² + (0- σ_allow)²

u_SSE =(1/6G) σ²_allow

Stresses actually present σ₁, σ₂ and σ₃=0

Fig. Maximum Shear Strain Energy Theory (Von Mises Hencky Theory )

ACTUAL SHEAR STRAIN ENERGY PRESENT

u_SSE =(1/12G)[( σ₁— σ₂)² +(σ₂–σ₃)² + (σ₃– σ₁)²]

Actually present≤ Allowable

FINAL SHEAR STRAIN ENERGY THEORY

[( σ₁—σ₂)² +(σ₂–0)² + (0- σ₁)² = 2 σ²_allow

GRAPHICAL REPRESENTATION OF VARIOUS THEORIES OF FAILURE

Fig. Theories of Failure (Combined)

ABCD Max Principal Stress Theory Boundary

EBFGDHE is Max. Shear Stress Boundary

IJKLI is Max. Principal Strain Boundary

Dark Ellipse  Max. Shear Strain Energy Theory Boundary

Dotted Ellipse Max. Principal Strain Energy Theory Boundary

ULTIMATE FINAL RESULTS

• Maximum principal theory

Largest of σ₁, σ₂ and σ₃ =σ_yp/FOS

• Maximum shear stress theory

Largest of (σ₁– σ₂)/2, (σ₂–σ₃)/2 and (σ₃– σ₁)/2 = σ_yp/2 FOS

• Maximum principal strain theory

σ₁– µσ₂ = σ_Yp/FOS

• Maximum Principal Strain energy theory

( σ₁²+ σ₂²–2µ σ₁ σ₂)= (σ_yp/FOS)²

• MAXIMUM SHEAR STRAIN ENERGY THEORY

OR

MAXIMUM DISTORSION ENERGY THEORY

[( σ₁—σ₂)² +(σ₂–0)² + (0- σ₁)²  = 2 (σ_yp /FOS)²      

Conclusions

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

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

(iii) A particular theory is applicable to few specific brittle material(s) or few specific ductile material(s).

(iv) Different theories won’t be applicable to the same particular brittle or same ductile material.

PROBLEM

A bolt is under an axial thrust of 9.6 k N together with a transverse force of 4.8 k N. Calculate the diameter according to:

1. Maximum principal stress theory

2. Maximum shear stress theory

Given factor of safety =3,

yield strength of the material=270 N/mm²

Poisson’s ratio=0.3

SOLUTION

Let A be the area of the bolt.

σ =9600/A

τ = 4800/A

We know      σ_1,2 =σ/2 + (1/2)[(σ/2)²+4τ²]^0.5

Firstly   σ_1,2=9600/2A/2)±[(9600/2A)²+4x(4800/A)²]^0.5

Secondly σ₁=11588.2/A

Thirdly σ₂ =–1988.2/A

Allowable stress =σ_yp/FOS=270/3=90

• As per maximum principal stress theory

11588.2/A=90

A=128.76

(π/4)d²=128.76

d=12.8 mm ANS

• As per maximum shear stress theory

[11588.2/A—(–1988.2/A)] =90

A = 150.8

(π/4)d²=150.8

= 13.8 mm ANS

Final answer  d=13.8 mm

https://www.mesubjects.net/wp-admin/post.php?post=4215&action=edit  MCQ FOS

https://www.mesubjects.net/wp-admin/post.php?post=4378&action=edit             MCQ theories of elastic failure