What is stress linearization?

Stress linearization is the separation of stresses along a section into constant (membrane) and varying (bending) stress components. The stresses obtained through this method are often referred to as linearized Stresses.

Why are stresses linearized?

The stress linearization method has its roots in pressure vessel design and analysis. Pressure vessel design predates the birth of numerical analysis techniques and finite element analysis software. The design and qualification of pressure vessels can be fairly complicated, especially for geometries that have complex geometric features. Over the years, and in the absence of sufficient analytical (theoretical) and numerical methods, engineers resorted to observations which were formulated into design rules. Later generation of engineers refined, and formalized these design rules based on linear-elastic finite element analysis. These rules were based on the separation of a stress field along the vessel cross section into average and varying components.

Definition of terms

Stress Classification Line

A Stress Classification Line or SCL is a straight path along the cross section, from the inside to the outside of a pressure vessel. It is a line drawn perpendicular to both the inside and outside surfaces. In a finite element analysis (FEA), the points along this path are represented by nodes. A FEA simulation provides the stress results at the nodes. The stress linearization tool in the software converts the nodal data for the stress along this line into membrane and bending components.

Linearized stresses are classified in three ways:

1. Primary or Secondary
2. General or Local
3. Membrane, Bending or Peak

There is plenty of material available on these stress classifications in books, design codes, technical papers and websites. All the information can be confusing and overwhelming. In this article, we will attempt to clarify some of the concepts related to stress linearization.

Primary vs Secondary Stress

If you do some research, you will come across several definitions for primary and secondary stresses. Usually, the various definitions are implying the same facts, though this may not be immediately obvious to the average reader. The following are some of the statements one comes across with regards to these stresses.

• Primary stress is the stress necessary to satisfy the law of equilibrium
  This statement is often made but can be misleading. All stress fields that develop in a material obey the laws of static equilibrium! A better definition is the concept of self-equilibration, which is what is implied by this definition.
  
• Secondary stress is self-equilibrating while Primary stress is not
  What do we mean by self-equilibrating in the context of stress? Self-equilibrating means that force and moment equilibrium for the material under stress is satisified entirely by internal forces. Secondary stresses exist without the application of external forces. In other words primary and secondary stresses differ in how equilibrium is satisified for the stressed material.
  
• Secondary stress is self-limiting while primary stress is not
  This is due to self-equilibration of stress. Static equilibrium is entirely satisfied by internal forces, and there are stresses associated with these forces. In other words, some of the balancing forces may manifest themselves as high stresses and local yielding. You could also think of this as stress redistributing within the material (as opposed to continuous stress increase based on the typical stress-strain response) as the source of the stress is intensified.
  
  Thermal stresses and residual stresses are two examples of secondary stresses. All other mechanical stresses such as those originating from weight or pressure are examples of primary stresses.

General vs Local Stress

General stress is something that refers to a global phenomenon. By global we imply that stress gradient is more or less smooth across a cross section.

General stress fields are at areas far from geometric discontinuities, and exclude the effects of stress concentrations.

Local stress field can be near geometric discontinuities and can include the effects of stress concentrations.

Membrane, Bending and Peak Stress

Membrane stress is the constant, average component of the total stress along the cross section, or stress classification line.

Bending Stress is the varying component of the total stress.

Peak stress is the summation of primary and secondary stress, and includes the effect of stress intensification factor (SIF). There are two things to note regarding peak stresses:

• Peak stress is NOT just the maximum raw stress observed.
• Stress Intensity Factor should not be confused with Stress Concentration or Stress Singularity. You may read about SIF in detail here.

Protection against failure

ASME code specifies allowable limits for various stresses obtained through stress linearization. The stress levels are limited to prevent various types of failures:

• Primary Stress is limited to prevent gross plastic deformation under static loads
• Primary plus Secondary Stress is limited to prevent ratcheting (of plastic strains) under cyclic load
• Primary plus Secondary plus Peak (Total) Stress is limited to prevent failure due to fatigue

A note on Elasto-Plastic analysis

It must be reiterated that stress linearization was born out of a need to safely design pressure vessels in the absence of any other suitable techniques. As such:

• Stress linearization is a primitive analysis technique
• This technique only makes sense with linear elastic analysis
• It is ideally suited to pressure vessel designs, and is not necessarily a good fit for other applications
• There is generally no good reason to perform stress linearization, if you are able to determine the true stress profile in a component using elasto-plastic analysis.

In this day and age, engineers have the know-how, software have the functionality, hardware has the capability, and (many) companies and organizations have the budget to facilitate and promote elasto-plastic analysis.

Elasto-Plastic (EP) analysis is where the true stress strain response of materials is captured and plasticity is modeled. When compared to linear elastic (LE) analyses, elasto-plastic analyses are more complicated (when modeling plasticity and collapse), require more skill (often a lot more skill), and are generally more time consuming. Because of these reasons, a lot of engineers are afraid to dabble with it. Which is quite unfortunate!

Elasto-Plastic analyses provide a level of accuracy which cannot be matched by elastic analysis. As a result, the kind of design optimizations that can be performed with EP analyses are not achievable with LE. Yes, it takes a lot of skill, effort and time to simulate a complex system with EP FEA, but in my opinion the gains (think cost saving by preventing overkill) are certainly worth it. If you would like more information on EP analysis, take a look at the Material Plasticity Section of this website.