Maximum stress intensity Criterion

Maximum Stress Intensity Criterion – Simple Explanation for Piping Engineers

In piping stress analysis, engineers must ensure that pipes can safely handle all applied loads such as pressure, temperature, weight, and external forces. One of the most important concepts used by piping codes like ASME B31 is the Maximum Stress Intensity Criterion.

This article explains the Maximum Stress Intensity Criterion in easy and practical language. It is written for students, fresh engineers, and working professionals who want a clear understanding without complex theory.

Why Stress Evaluation Is Required in Piping

Pipes in process plants carry high-pressure and high-temperature fluids. During operation, pipes experience different types of stresses:

• Internal pressure stress
• Bending stress due to weight and thermal expansion
• Shear stress due to torsion
• Thermal stresses during startup and shutdown

If these stresses are not properly controlled, the pipe may:

• Yield or permanently deform
• Develop cracks
• Fail suddenly and cause safety hazards

To avoid this, piping codes define stress limits using proven failure theories. One such theory is the Maximum Shear Stress Theory, which leads to the Maximum Stress Intensity Criterion.

Basic Concept of Maximum Shear Stress Theory

The Maximum Shear Stress Theory states that a material fails when the maximum shear stress in the material reaches the shear stress at yielding in a simple tensile test.

During a uniaxial tensile test:

• The yield stress is Sy
• The maximum shear stress at yield is Sy / 2

So failure is expected when:

Maximum shear stress ≥ Sy / 2

Piping codes slightly modify this concept to suit complex stress conditions in pipes.

Principal Stresses in a Pipe

At any point in a pipe wall, stresses act in different directions. The three most important stresses are:

• Longitudinal stress (S_L) – along the pipe axis
• Hoop stress (S_H) – around the pipe circumference
• Shear stress (τ) – due to torsion

Using these stresses, we calculate the principal stresses at the outer surface of the pipe.

Principal Stress Equations

The maximum and minimum principal stresses are:

S₁ = (S_L + S_H) / 2 + √[ ((S_L − S_H) / 2)² + τ² ]

S₂ (or S₃) = (S_L + S_H) / 2 − √[ ((S_L − S_H) / 2)² + τ² ]

These equations help us find the maximum shear stress in the pipe wall.

Maximum Shear Stress in the Pipe

The maximum shear stress in a material is given by:

τ_max = (S₁ − S₃) / 2

Substituting the principal stress equations, the expression becomes:

τ_max = √[ (S_L − S_H)² + 4τ² ] / 2

According to the theory, this value must be less than half of the yield stress:

√[ (S_L − S_H)² + 4τ² ] / 2 < Sy / 2

What Is Stress Intensity?

To simplify calculations, piping codes multiply both sides of the inequality by 2. This does not change the safety condition but saves calculation steps.

After multiplying by 2:

√[ (S_L − S_H)² + 4τ² ] < Sy

The term on the left side is called Stress Intensity.

So,

Stress Intensity = √[ (S_L − S_H)² + 4τ² ]

This is the basis of the Maximum Stress Intensity Criterion used in most piping codes.

Why Piping Codes Use Stress Intensity

Pipes are subjected to combined loading. Instead of checking each stress separately, stress intensity:

• Combines longitudinal, hoop, and shear stresses
• Gives a single conservative value
• Ensures protection against yielding

This approach is reliable and simple, which is why it is adopted in ASME B31 codes.

Stress Intensity for Fatigue and Expansion Stress

For fatigue and thermal expansion analysis, only varying stresses are considered.

Pressure-related stresses like hoop stress usually remain constant and are ignored in fatigue calculations.

In such cases, the stress intensity equation becomes:

√( S_b² + 4S_t² ) < S_A

Where:

• S_b = Bending stress (psi)
• S_t = Torsional shear stress (psi)
• S_A = Allowable stress for the load case (psi)

This equation is very similar to the one used in ASME B31.3 for expansion stress range.

Worked Example – Stress Intensity Calculation

Let us understand the concept with a practical example.

Given Data

• Pipe size: 6-inch nominal diameter
• Longitudinal stress (S_L) = 15,547 psi
• Shear stress (τ) = 5,999 psi
• Hoop stress (S_H) = 7,098 psi
• Material yield stress (Sy) = 30,000 psi
• Factor of safety = 2/3

Step 1: Write the Stress Intensity Equation

Stress Intensity = √[ (S_L − S_H)² + 4τ² ]

Step 2: Substitute Values

= √[ (15,547 − 7,098)² + 4 × (5,999)² ]

= 14,674 psi

Step 3: Calculate Allowable Stress Intensity

Allowable = (2/3) × 30,000 = 20,000 psi

Step 4: Compare

14,674 psi < 20,000 psi

✅ The pipe is safe under the given loading conditions.

Practical Importance for Stress Engineers

Understanding stress intensity is important because:

• It forms the base of code compliance
• Used directly in CAESAR II and other stress software
• Helps in judging overstress conditions
• Essential for fatigue and thermal analysis

Without this concept, piping stress evaluation would be incomplete and unsafe.

Conclusion

The Maximum Stress Intensity Criterion is a simplified and reliable method derived from the Maximum Shear Stress Theory. It combines multiple stresses into a single value that can be easily compared with allowable limits defined by piping codes.

By understanding this concept clearly, piping engineers can:

• Design safer piping systems
• Interpret stress software results correctly
• Answer technical interview questions confidently

This concept is not just theoretical—it is used daily in real industrial piping stress analysis.