Hello, friends in this article I am going to discuss with you

**theories of failure**in which you will know what is the theory of failure and their types, application, etc.The design of machine parts subjected to combined load is done with the help of theories of failure which help us to determine the safe dimensions of machine parts.

But before knowing the theories of failure first of all we have to know what a mechanical failure such that we can better understand the theory of failure.

So without wasting time let's get started.

## What is Mechanical Failure?

Machine parts fail when the stresses induced by external forces exceed their strength.

In machine design, an element is said to have failed if it ceases to perform its function.

There are basically two types of mechanical failure :

- Yielding
- Fracture

### Yielding

Yield refers to an indication of the maximum stress that can be developed in a material without plastic deformation.

### Fracture

Fracture is the field of mechanics concerned with the study of the propagation of cracks in materials in which the separation of a body or materials into two or more pieces under the action of maximum stress.

### Some important points to remember before knowing the theories of failure :

- The limiting strength for ductile materials is yield point stress. Ductile materials usually fail by yielding which means permanent deformation occurs in the materials.
- The limiting strength for brittle materials is the ultimate stress point. Brittle materials usually fail by fracture.
- A two-dimensional state of stress in which only two normal stress are present is called biaxial stress.

## What is Theories of Failure?

Theories of failure are those theories that are helpful to determine the safe dimensions of machine components which is subjected to combined stresses due to various loads acting on it during their operation and working.

Some examples of such components are as follows :

- I.C. engine crankshaft
- Bolted and welded joints are used under eccentric loading.
- A shaft which is used in power transmission
- Ceiling fan rod

## Types of Theories of Failure

There are several theories of failure. Many of these were originally developed for brittle materials but were later applied to ductile materials.

Some of the more common Theories of failure which we will be discussing are as follows :

- Maximum Principal Stress Theory
- Maximum Shear Stress Theory
- Maximum Principal Strain Theory
- Maximum Total Strain Energy Theory
- Distortion Energy Theory

### Maximum Principal Stress Theory

This theory of failure is given by the British engineer, W.J.M. Rankine so this theory is also called Rankine's Theory.

" Maximum principal stress theory states that the failure of the mechanical component, subjected to bi - axial or tri - axial stresses, occurs when the maximum principal stress reaches the yield or ultimate strength of the material. "

So according to this theory, taking the Factor of safety ( FOS ) into consideration, the maximum principal or normal stress (σ ₁ ) in a bi-axial stress system is given by

**σ₁ = σy/FOS**

For ductile materials where σy is yield point stress from simple tension tests.

**σ₁ = σu/FOS**

For brittle materials where σu is the ultimate stress from a simple tension test.

This theory of failure is generally used for brittle materials.

### Maximum Shear Stress Theory

This theory of failure is given by C.A. Coulomb, H. Tresca, and J.J. Guest that's why this theory is also known as Coulomb, Tresca, and Guest's Theory.

" According to this theory, the failure of a mechanical component subjected to bi - axial or tri axial stresses occurs when the maximum shear stress at any point in the component becomes equal to the maximum shear stress in the standard specimen of the tension test when yielding starts ."

The maximum shear stress ( τ max ) in a bi-axial stress system is given by

**τmax = τy/F.O.S.**

where τy is shear stress at yield point from simple tension test.

The shear stress at the yield point in a simple tension test is equal to one-half the yield stress in tension.

τy = σy/2

So,

**τ max = σy/(2 × F.O.S)**

This theory is generally used for a ductile material.

### Maximum Principal Strain Theory

This theory states that the failure or yielding occurs at a point in a member when the maximum principal ( or normal ) strain in a bi-axial stress system reaches the limiting value of strain which means strain at the yield point as determined from a simple tensile test.

This theory is also called St. Venant's Theory.

The Maximum principal ( or normal ) strain ( Σmax ) in a bi-axial stress system is given by

Σ max = 1/E ( σ₁ - µ σ₂)

According to this theory,

Σmax = Σy

**1/E ( σ₁ - µ σ₂) = σy/(E × F.O.S)**

Where,

σ₁ and σ₂ = Maximum and minimum principal stress

µ = Possions ratio

Σy = Yield point strain

This theory is generally applicable to ductile material.

Maximum Total Strain Energy Theory

Maximum total strain energy theory states that the failure or yielding occurs at a point in a member when the strain energy per unit volume in a bi-axial stress system reaches the limiting strain energy ( i.e. strain energy at the yield point ) per unit volume as determined from simple tension test.

This theory is also called Haigh's Theory.

The strain energy per unit volume in a bi-axial stress system

U₁ = 1/2E [σ₁ ² + σ₂² - µ ( 2σ₁×σ₂ ) ]

And limiting strain energy per unit volume for yielding in a simple tensile test

U₂ = 1/2E(σy/FOS)^2

According to the theory,

U₁ = U₂

1/2E [σ₁ ² + σ₂² - µ ( 2σ₁×σ₂ ) ] = 1/2E(σy/FOS)^2

**[σ₁ ² + σ₂² - µ ( 2σ₁×σ₂ ) ] = (σy/FOS)^2**

This theory is generally used for a ductile material.

### Distortion Energy Theory

It is also known as Huber von mises and Hencky's theory.

Distortion energy theory is the difference between the total strain energy and strain energy due to uniform stress.

It is also called the maximum shear strain energy theory.

" According to this theory the failure of the mechanical component subjected to bi - axial or tri axial stresses occurs, when the distortion energy per unit volume at any point in the component, becomes equal to the distortion energy per unit volume in a standard specimen of tension - test when yielding starts. "

**σ₁ ² + σ₂² - 2σ₁×σ₂ = (σy/FOS)^2**

This theory is applicable to ductile material.

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