Short Answer:

The Maximum Stress Theory, also known as the Rankine’s Theory, states that failure of a material occurs when the maximum principal stress in a complex stress condition reaches the ultimate or yield stress obtained from a simple tensile test. In other words, the material fails when the highest normal stress acting on it equals its strength limit.

This theory is mainly applicable to brittle materials such as cast iron, glass, and concrete, which fail due to normal stress and not by shear stress. It is one of the simplest and earliest failure theories used in mechanical engineering for predicting material failure.

Detailed Explanation :

Maximum Stress Theory (Rankine’s Theory)

The Maximum Stress Theory or Rankine’s Theory was proposed by William John Macquorn Rankine. It is one of the basic and most straightforward theories of material failure. The main principle of this theory is that a material will fail when the maximum principal stress in a complex stress system becomes equal to the yield or ultimate tensile stress of that material obtained from a simple uniaxial tension test.

This theory assumes that failure occurs due to normal stress only and that shear stresses have no influence on the failure. It is particularly useful for brittle materials, which fail suddenly without much plastic deformation when the normal stress exceeds their ultimate strength.

Definition

According to the Maximum Stress Theory (Rankine’s Theory):

“Failure of a material occurs when the maximum principal stress in the material reaches the limiting stress obtained from a simple tensile or compressive test.”

Mathematically, the condition for failure is expressed as:

where,

•  = maximum principal stress in the complex stress condition
•  = yield or ultimate stress in a simple tensile test

This means that if the maximum principal stress in any part of the material exceeds the yield or ultimate value, the material will fail.

Mathematical Expression

Let the principal stresses at a point in a material be , where  is the maximum and  is the minimum.

According to the Maximum Stress Theory, failure will occur when:

For tensile failure:

For compressive failure:

where,

•  = ultimate tensile stress
•  = ultimate compressive stress

Hence, the condition for safe design can be written as:

This ensures that the stresses in the material remain below the yield or ultimate limit, preventing failure.

For Two-Dimensional Stress System

In most practical cases, materials are subjected to stresses in two perpendicular directions (plane stress condition).

Let the two principal stresses be  and . Then the failure conditions become:

For tension failure:

For compression failure:

Hence, for safe design:

This provides the safe working stress range for the material.

Graphical Representation

In the principal stress diagram (σ₁–σ₂ plane), the Rankine’s theory is represented by a rectangle or square.

• The horizontal and vertical sides correspond to the limiting tensile and compressive stresses ( and ).
• Points inside the rectangle represent safe conditions.
• Points on or outside the boundary represent failure.

This makes it easy to visualize the safe stress combinations for a brittle material.

Assumptions of the Theory

1. Failure occurs when the maximum normal stress reaches the limiting value obtained from a simple test.
2. Shear stresses do not affect the failure.
3. The material behaves elastically up to the point of failure.
4. The effect of strain or distortion is neglected.

These assumptions simplify calculations but make the theory less accurate for ductile materials.

Applicability

• This theory is mainly used for brittle materials, such as:
  • Cast iron
  • Concrete
  • Glass
  • Stone
  • Ceramics

Brittle materials usually fail by cracking under normal stresses without undergoing large deformation, which justifies this theory’s assumption.

However, this theory is not suitable for ductile materials such as mild steel, copper, or aluminum, because these materials fail by shear yielding, not normal stress.

Advantages

1. Simple to apply:
   The theory involves only principal stresses and does not require complex calculations.
2. Good for brittle materials:
   Predicts fracture accurately for materials that fail under normal stress.
3. Useful for preliminary design:
   Provides a quick estimation of safe stress limits for brittle materials.

Limitations

1. Not suitable for ductile materials:
   Ductile materials fail due to shear stress, which this theory ignores.
2. Neglects shear stress effects:
   In combined loading, shear stress can be significant, but this theory does not consider it.
3. Inaccurate for multiaxial stress conditions:
   Does not match experimental results for components under combined or triaxial stress states.
4. No consideration of distortion:
   Shape changes due to distortion energy are ignored.

Comparison with Other Theories

Theory	Basis	Suitable For
Maximum Stress Theory (Rankine)	Maximum principal stress	Brittle materials
Maximum Shear Stress Theory (Tresca)	Maximum shear stress	Ductile materials
Maximum Strain Energy Theory	Total strain energy	Ductile materials
Maximum Shear Strain Energy Theory (Von Mises)	Distortion energy	Ductile materials

From this comparison, it is clear that the Rankine’s theory is most applicable to brittle materials, whereas energy and shear-based theories are better for ductile ones.

Practical Example

If a cast iron bar has an ultimate tensile stress of 120 MPa and ultimate compressive stress of 400 MPa, then according to the Rankine theory, failure will occur if the maximum principal tensile stress reaches 120 MPa or if the compressive stress reaches 400 MPa.
Therefore, for safe design, the working stresses should always remain below these limits.

Conclusion

The Maximum Stress Theory (Rankine’s Theory) states that failure occurs when the maximum principal stress in a material reaches the yield or ultimate stress from a simple tension or compression test. It is most suitable for brittle materials, which fail by normal stress and not by shear. Although simple and easy to use, this theory is not accurate for ductile materials, where shear stress governs failure. Still, Rankine’s theory is very useful for preliminary design and understanding failure in brittle materials like cast iron and concrete.