Short Answer:

The Maximum Principal Strain Theory states that failure of a material begins when the maximum principal strain in a complex stress system reaches the same value as the strain at yield point in a simple tensile test. This theory assumes that yielding occurs when the maximum strain, not stress, exceeds the elastic limit.

In simple terms, this theory predicts material failure based on the largest deformation (strain) experienced by the material. It is mainly applicable to ductile materials under small stress conditions but is not widely used because it gives less accurate results compared to energy-based theories.

Detailed Explanation :

Maximum Principal Strain Theory

The Maximum Principal Strain Theory is also known as Saint-Venant’s Theory. It is one of the classical theories of failure used in strength of materials to predict when a material will begin to yield or fail under complex loading. This theory is based on the concept of strain, rather than stress or energy. It assumes that a material fails when the largest principal strain in a multidirectional stress condition equals the strain at yield in a simple tension test.

This theory was developed before more advanced theories like the Maximum Shear Stress Theory (Tresca) and Maximum Shear Strain Energy Theory (Von Mises). Although simple, it is not widely used for ductile materials because it does not accurately represent how these materials actually fail, as ductile materials yield primarily due to shear distortion, not because of normal strain.

Definition

The Maximum Principal Strain Theory can be defined as:

“A material is said to fail when the maximum principal strain in the material reaches the limiting strain observed in a simple tension test at the yield point.”

In other words, yielding or failure begins when the maximum extension or compression strain in a loaded member equals the strain at yield during a uniaxial tensile test.

Basic Concept

When a body is subjected to stresses in different directions, it experiences strains in each direction due to the combined effect of these stresses. The deformation in the direction of the principal stress is known as the principal strain.

The maximum of these strains is the maximum principal strain, which this theory uses to determine the onset of failure.

Since strain is related to stress through Hooke’s Law, the theory expresses the strain in terms of principal stresses and material properties (Young’s modulus and Poisson’s ratio).

Mathematical Expression

Let a material element be subjected to three mutually perpendicular principal stresses .

The corresponding principal strains in the three directions are given by:

where,

= Young’s modulus of the material,
= Poisson’s ratio.

According to this theory, yielding begins when the maximum principal strain () reaches the strain at yield () from a simple tensile test.

For a uniaxial stress condition (simple tension test),

where = yield stress in tension.

Thus, the failure condition is:

Simplifying, we get the failure criterion as:

This is the mathematical expression for the Maximum Principal Strain Theory.

For Two-Dimensional Stress System

In most practical engineering problems, stresses act in two directions, so . The above equation becomes:

This is the yield condition according to the Maximum Principal Strain Theory for a 2D state of stress.

Graphical Representation

In the principal stress diagram (σ₁ vs σ₂), the yield surface defined by this theory appears as an ellipse, similar to the Von Mises theory, but its orientation and curvature differ due to the Poisson’s ratio term.

The ellipse encloses the region of safe stress states:

Points inside the ellipse → Safe condition.
Points on the ellipse → Material yields (failure starts).
Points outside the ellipse → Material fails.

However, because of the Poisson’s effect, the results given by this theory differ from experimental data, especially for ductile materials.

Comparison with Other Theories

Theory	Basis	Best For	Remarks
Maximum Principal Stress Theory (Rankine)	Maximum normal stress	Brittle materials	Simple but conservative
Maximum Principal Strain Theory (Saint-Venant)	Maximum normal strain	Ductile materials (approx.)	Not very accurate
Maximum Shear Stress Theory (Tresca)	Maximum shear stress	Ductile materials	Good accuracy
Maximum Shear Strain Energy Theory (Von Mises)	Distortion energy	Ductile materials	Most accurate

Thus, while the Maximum Principal Strain Theory provides a simple conceptual understanding, it is less reliable than Von Mises or Tresca for real ductile materials.

Applications

Although rarely used in modern engineering, the theory has limited use in:

Preliminary stress analysis, where only approximate strain behavior is required.
Elastic deformation studies, to determine strain limits in components under low load conditions.
Design of brittle materials, in which normal stress and strain are more critical than shear distortion.

However, for ductile materials like steel, this theory tends to overestimate the safe load, leading to unsafe designs.

Limitations

Not accurate for ductile materials — these fail mainly due to shear, not normal strain.
Neglects distortion effects — does not consider the shape change (shear strain energy).
Not experimentally verified — predictions differ from actual test results.
Unsuitable for combined loading — gives inconsistent results when stresses act in multiple directions.
Less reliable for design — replaced by more accurate energy-based theories.

Conclusion

The Maximum Principal Strain Theory or Saint-Venant’s Theory states that failure occurs when the maximum principal strain in a material equals the strain at yield in a simple tensile test. Although it provides a simple approach to predict failure, it is not widely used today because it does not accurately represent the real behavior of ductile materials, which fail primarily due to shear. The theory is mainly of theoretical importance and has been replaced by more accurate theories such as the Maximum Shear Strain Energy Theory (Von Mises) for modern design applications.