Short Answer:

The Maximum Normal Stress Theory is also known as the Rankine’s Theory. It states that failure of a material begins when the maximum principal stress in a complex stress system reaches the ultimate or yield stress of the material in a simple tensile test.

In simple words, according to this theory, a material fails when the largest normal stress acting on it becomes equal to the strength of the material under simple tension or compression. It is mainly applicable to brittle materials, such as cast iron, glass, and concrete, where failure occurs due to normal stress and not due to shear stress.

Detailed Explanation :

Maximum Normal Stress Theory

The Maximum Normal Stress Theory, proposed by William John Macquorn Rankine, is one of the earliest and simplest theories of failure. It is based on the concept of principal stresses, which are the maximum and minimum normal stresses acting on the material when the shear stress is zero.

This theory assumes that failure of a material occurs when the maximum principal stress (tensile or compressive) in a complex stress condition reaches the value of the yield stress or ultimate stress obtained from a simple tension or compression test.

Since brittle materials generally fail by sudden fracture when the normal stress exceeds their ultimate limit, this theory is suitable for them. It is not suitable for ductile materials, which fail mainly due to shear deformation rather than direct tension or compression.

Definition

The Maximum Normal Stress Theory can be defined as:

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

In other words, the material will begin to yield or fracture when:

where,

= maximum principal stress in the system,
= yield or ultimate stress of the material obtained from a simple tension test.

Thus, the theory relates the failure condition directly to the maximum normal stress acting within the material.

Mathematical Form of the Theory

Let the principal stresses in a three-dimensional stress system be , where is the maximum and is the minimum.

According to the Maximum Normal Stress Theory, the material fails when:

for tensile failure, or

for compressive failure,

where,

= ultimate tensile stress of the material,
= ultimate compressive stress of the material.

This means that the maximum normal stress in any direction should not exceed the yield or ultimate limit of the material in tension or compression.

For Two-Dimensional Stress System

In a plane stress condition, where only two principal stresses and act, the failure conditions become:

For tensile failure:

For compressive failure:

Hence, the safe condition can be written as:

In practice, the design stress is kept below these limiting values by using a factor of safety (FOS) to ensure the material remains within the elastic limit.

Graphical Representation

In a principal stress diagram ( vs ), the Maximum Normal Stress Theory is represented by a square or rectangle whose sides correspond to the tensile and compressive limits ( and ).

Points inside the rectangle represent safe stress conditions.
Points on or outside the rectangle represent failure.

This graphical view makes it clear that failure occurs whenever the maximum normal stress in any direction exceeds the material’s strength.

Physical Meaning

According to this theory:

Failure is governed entirely by the maximum normal stress (tensile or compressive).
Shear stresses do not contribute to failure.
The material is considered to fail by cracking or cleavage due to direct normal stresses.

This behavior matches brittle materials, which typically fail suddenly and without significant plastic deformation when subjected to excessive normal stress.

Suitability and Applications

Applicable to:

Brittle materials, such as cast iron, glass, ceramics, concrete, and stones.
Materials that fail suddenly due to tensile rupture or compressive crushing.

Not applicable to:

Ductile materials, such as mild steel, copper, or aluminum, because these materials yield due to shear stress, not normal stress.

Advantages

Simple to apply:
It uses only the maximum principal stress and is easy to compute.
Useful for brittle materials:
Accurately predicts failure in materials where fracture is caused by tension or compression.
Direct relation to material properties:
It directly compares stress with the yield or ultimate stress from a simple tension test.

Limitations

Not valid for ductile materials:
Ductile materials fail due to shear, which this theory neglects.
Neglects the effect of shear stress:
Since it only considers normal stress, it gives unsafe results for materials where shear is significant.
Experimental mismatch:
Does not match experimental results for combined loading conditions.
Not suitable for complex stress states:
In multiaxial stress systems, this theory becomes inaccurate.

Example (Conceptual)

If the maximum principal stress in a component is 90 MPa, and the ultimate tensile strength of the material is 100 MPa, then according to the Maximum Normal Stress Theory:

Hence, the component is safe.

However, if , then , and the material will fail according to Rankine’s criterion.

Comparison with Other Theories

Theory	Basis	Suitable For
Maximum Normal Stress Theory (Rankine)	Maximum normal 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

This shows that the Rankine theory is the most suitable for brittle materials, while ductile materials require energy or shear-based theories.

Conclusion

The Maximum Normal Stress Theory (Rankine’s Theory) states that failure occurs when the maximum principal stress in a material reaches the yield or ultimate strength observed in a simple tensile test. This theory provides a simple and conservative approach for analyzing brittle materials, where fracture occurs due to direct tension or compression. However, it is not suitable for ductile materials, which fail primarily due to shear stress. Despite its simplicity, it remains an important foundation for understanding material failure under normal stress conditions.