Principal Stress-Strain & Theories of Failure

# Principal Stress-Strain & Theories of Failure | Strength of Materials (SOM) - Mechanical Engineering PDF Download

 Table of contents What is Principal Stress? Principal Stress Theory What is Maximum and Minimum Principal Stress? Maximum Principal Stress Theory Principal Stress and Principal Strain What are the Theories of Failure?

## What is Principal Stress?

Principal stress is the maximum or minimum normal stress which may be developed on a loaded body. It is classified as major principal stress and minor principal stress. On the plane of principal stress shear stress value is termed as zero.
When principal stress gets on the major principal plane, it is called major principal stress and when it is found on the minor principal plane, it is known as minor principal stress. These are denoted as σand σ2, respectively.

## Principal Stress Theory

Shear stress is a sloping applied force that causes deformation due to lateral load. In any stress block surface, there is applied shear stress over the plane, but to stabilize the body there, we need to apply complementary shear stress. This is known as complementary shear stress.
The principle of complementary shear stress is when the same intensity but opposite direction shear stress is applied over the surface of the body, it creates a couple and this couple stabilizes the body. This balancing couple mechanism is known as the principle of complementary shear stress.

## What is Maximum and Minimum Principal Stress?

Principal stress is calculated on the principal plane as maximum stress is called major principal stress, and minimum stress on the principal plane is called minor principal stress.
Here;

• σx = Stress in x direction
• σY = Stress in the Y direction
• τn = Normal shear stress
• Ө = inclination angle of stress to the principal axis
• X and Y are the axes of the plane.

Where

• σis the maximum principal stress, and
• σis the minimum principal stress. it is also termed as σMaxand σMin.

Maximum shear stress (τmax) = (Maximum principal stress- Minor principal stress)/2 =R

Here maximum shear stress is the magnitude of that point with an equal radius in a mohr circle, and principal stress is the end point of the circle’s diameter.
Here we take only two-dimensional theory in which we consider σ1 ( the maximum principal stress ) and σ2 (the minimum principal stress). But in the case of three-dimensional theory, we take σ1, σand σ3. In this case, we decide on the major or minor principal stress by using principal stress theory or Rankine, lames or maximum principal stress theory.

## Maximum Principal Stress Theory

When the applied load needed to be calculated under the design criteria, the total applied load should be less than the ultimate yield capacity of material divided by the factor of safety. This theory is also a part of principal stress theories and its also known as Rankine’s theory, lame’s theory and reciprocal theory.
According to the maximum principal stress theory-
σ1 + σ2 + σ3 ≤ yield strength/Factor of safety

## Principal Stress and Principal Strain

Principal stress is related to the existing principal plane available stresses as when stress is line on maximum principal planes as positive and negative stress value higher in magnitudes called principal stress. Stress is a tensor because it follows the transformation equation.
In principal strain the maximum and minimum normal strains are obtained by differentiating x, and y coordinates and the orientation of the planes of the strains is determined there are two roots, p1 and p2, for 1 and 2 directions. Principal strains are the nominal strains with higher magnitude.

## What are the Theories of Failure?

Various load types or combinations act over the components while under operating conditions. So, the theories of failure help to give a safe dimension to machine components. Before learning about Theories of Failure, let us briefly learn about a machine’s failure.

### Failure of Machine

The meaning failure of the machine is the non-ability of the component or a machine to perform its function as per requirement. Failure of a machine or any mechanical components does not mean a total breakdown or not operating. It can operate but not as per our requirements.

### Theories of Failure Diagram

The diagram shown below shows a graphical representation of various Theories of Failure for ductile material:

For example, A mechanical shaft transmits the power, but due to increased load, it gets bent slightly. Still, it can transmit power but not that efficiently. In this case, we can say that failure of the shaft happens.

### The factor of Safety (FOS)

• The ratio of ultimate to allowable load or stress is known as a factor of safety, i.e., the ratio of the material strength or failure stress to the allowable or working stress.
• The factor of safety must always be greater than unity. It is easier to refer to the ratio of stresses since this applies to material properties.
• FOS = Failure Stress/Working or Allowable Stress

### Types of Theories of Failure

• Maximum principal stress theory or Rankine’s Theory
• Maximum principal strain theory or St. Venant’s Theory
• Maximum Shear Stress Theory or Guest and Tresca’s Theory
• Maximum Strain Energy Theory or Haigh’s Theory
• Maximum Shear Strain Energy theory or Von-Mises and Henckey Theory

Maximum Principal Stress Theory or Rankine’s Theory

In this case of complex loading, when the maximum principal stress value reaches the ultimate stress or yield stress of the material, the component will get failed.

As brittle material is weak in tension, this is most suitable for the brittle material

In simple tension, σ1= σyt
In compression, σ2= σyc
Where,

• σyt– Ultimate tensile stress
• σyc – Ultimate compressive stress
• σ1– Maximum principal stress

Maximum Principal Strain Theory or St. Venant’s Theory

According to this failure theory, components are safe if the maximum principal strain at a critical point in the loaded member will not exceed the strain under the yielding condition when subjected to uniaxial loading.
It is the most suitable for all types of material, like ductile material, brittle material, and material under hydrostatic conditions.
ε1≤σyt/E
σ1−[ μ ( σ2+ σ3)]≤σyt/E
Where,

• σyt– Ultimate tensile stress
• μ– Poisson ratio
• σ1, σ2, and σ3– Stresses along the x, y, and z-axis

Maximum Shear Stress Theory or Guest and Tresca’s Theory

According to this theory, the material will be failed in the complex loading system at any place in components, and the magnitude of shear stress induced in the material is greater than or equal to the maximum shear stress of the material. Therefore, it is the most suitable theory of failure to experiment on ductile material, especially in the case of a shaft subjected to torque.

The limitation of maximum shear stress theory is that it will not give proper results for a pure shear loading as well as triaxial tensile or compressive loading having approximately equal magnitude (for example body submerge in water)

Maximum Strain Energy Theory or Haigh’s Theory

Per the maximum strain energy theory, failure occurs when the total strain energy in the fundamental component exceeds strain energy under yield conditions. According to this theory, strain can be fully recoverable up to the elastic limit. However, this theory will not give results in the case of triaxial loading having almost equal magnitude.

The Strain energy is developed and stored in the material after the elastic deformation. This energy will also be the work done during the elastic transformation. This theory says the developed strain energy will be responsible for the failure of the strain energy under yield conditions.

Maximum Shear Strain Energy Theory or Von-Mises and Henckey Theory

It is also known as the maximum distortion energy theory. As it was introduced by the scientists Von-Mises and Hickey, it is also known as von Mises-Hencky’s theory. When loads are applied to the component, it will deform, due to which the size and shape of the components will change. The energy responsible for changes in shape and volume is known as distortion energy.
Maximum shear strain theory state that distortion energy associated with the unit volume of the member exceeds the distortion energy at uniaxial loading per unit volume.

The document Principal Stress-Strain & Theories of Failure | Strength of Materials (SOM) - Mechanical Engineering is a part of the Mechanical Engineering Course Strength of Materials (SOM).
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## FAQs on Principal Stress-Strain & Theories of Failure - Strength of Materials (SOM) - Mechanical Engineering

 1. What is the significance of principal stress-strain in material behavior?
Ans. Principal stress-strain refers to the maximum and minimum stress and strain values experienced by a material. It is significant because it helps determine the material's strength, stiffness, and ability to withstand external loads. Understanding principal stress-strain allows engineers to design structures and materials that can handle the expected stress and strain levels.
 2. How do we calculate the principal stresses from a stress-strain curve?
Ans. To calculate the principal stresses from a stress-strain curve, we need to determine the maximum and minimum stress values. The maximum stress is the peak value on the curve, while the minimum stress is the lowest point. These two values represent the principal stresses experienced by the material under the given loading conditions.
 3. What are the theories of failure in material science?
Ans. Theories of failure in material science are mathematical models used to predict the failure of materials under different loading conditions. Some commonly used theories include the Maximum Shear Stress Theory, Maximum Normal Stress Theory, and the Distortion Energy Theory. These theories help engineers assess the safety and reliability of structures and materials by determining the conditions under which failure may occur.
 4. How does the Maximum Shear Stress Theory determine failure in materials?
Ans. The Maximum Shear Stress Theory states that failure occurs in a material when the maximum shear stress exceeds the shear strength of the material. Shear stress is calculated using the maximum and minimum principal stresses. If the maximum shear stress exceeds the shear strength, the material is predicted to fail. This theory is commonly used for brittle materials like ceramics.
 5. What is the relationship between principal stress-strain and the theories of failure?
Ans. Principal stress-strain values are essential inputs for the theories of failure. The theories of failure rely on the principal stress values to determine the likelihood of material failure under different loading conditions. By analyzing the principal stress-strain data, engineers can apply the appropriate theory of failure to assess the safety and reliability of materials and structures.

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