Details of Sustained and Displacement Stresses-Core of Pipe Stress Analysis

Piping flexibility analysis as per B31.3code requirements is dealing with two types of stress named as Sustained Stress and Displacement stress.  Both types of stress must be considered separately because sustained stresses are associated with sustained forces while displacement stresses are associated with fixed displacements.

SUSTAINED STRESSES: Sustained Stresses are stresses caused by loads that are not relieved as the piping system deflects. See the diagrammatic representation below to get an idea about this. 

The weight of the valve placed at the end of the cantilevered pipe induce Stress at the T-Joint.

Regardless of the magnitude of the displacement (∆), the magnitude of the load (the weight of the valve) which causes the stress is unchanged.  Therefore, to avoid catastrophic failure, the magnitude of any sustained stress must not exceed the yield strength of the material.  Another example of a sustained stress is the hoop and longitudinal stresses induced by the internal pressure inside the pipe, details are covered in the previous Post.

The loadings, which induce sustained stresses, are termed sustained loadings.

The sustained stress principle is expressed as a Code requirement. The sum of the longitudinal stresses due to pressure, weight, and other sustained loadings S_L must not exceed the hot allowable stresses S_h.

S_L   ≤   S_h

Now,  Code Allowable Stress Value  S_h is available from the code, How to find out S_L.

It can be tabulated by the following relation

                    Where 

                           F_A=Axial force

                           A_W=Cross-sectional area of corroded pipe wall

DISPLACEMENT STRESSES: Displacement stresses are caused by fixed displacements, as the piping system deforms these types of stresses will relive. 

Consider the Cantilever arrangement shown below,

Displacement Stress

Imagine the beam end is displaced within the Elastic limit and the elastic range is δ.  As long as any displacement cycle is within the elastic range of the beam, no yielding will take place.

Now Consider the same beam displaced from its original position to "A" as shown below

Representation of Displacement Range

Here the beam will Yield as the Elastic limit is exceeded.But as long as " D" does not exceed "δ", no further yielding of the beam will take place provided all successive displacement cycles are within the displacement range D.

If the beam is made of a relatively ductile material, yielding only in the first half cycle will not cause failure of the beam.  Therefore, fixed displacements can be allowed to cause displacement stresses that exceed the yield strength of the material as long as the elastic range of the material is never exceeded.

Let us consider the stress induced by thermal expansion in a pipe, referer the figure for more details.

                 Restrained & unrestrained piping

Right side image illustrates how an increase in temperature of the restrained piping system is equivalent to displacing the unrestrained hot pipe from position (F₂) to position (F₁).  Therefore, stresses caused by thermal expansion are displacement stresses.

Any yielding or permanent strain, with attendant relaxation or reduction of stress in the hot condition, leads to the creation of a stress reversal when the piping system returns to the cold position.  This reversal of stress referred as self-springing.

As per the code, the calculated stress range SE (also known as the expansion stress range) must not exceed the allowable stress range SA.

SE   ≤     SA.

If a Piping System satisfies the above condition, it is judged to be adequately flexible against thermal expansion and restraint displacement. Because the elastic range of the system will never be exceeded even though the system may yield.

Since the inherent flexibility in most piping systems is provided by changes in direction, the code considers only bending and torsional stresses significant in the calculation of  SE  and gives the following equation for its computation.

                                               SE   = (S_b ²  + 4S_t 2) ^½

These stress Values, nowadays calculate with the help of Finite Element analysis assisted Software packages and compare with Code allowable stresses to determine the adequacy of piping systems.  The Code differentiates between stresses caused by pressure and other sustained loads and stresses caused by displacement strains.  These allowable values are a function of the basic allowable stresses.

The allowable stress for sustained loads is the basic allowable stress at the maximum metal temperature, S_h.

The computed Displacement stress range S_E in a piping system shall not exceed the allowable displacement stress range S_A. 

S_A = f (1.25 S c + 0.25 S_h)

We discussed that S_L is compared with S_H for Sustained Stress check, When S_h is greater than S_L, the difference between them may be added to the term 0.25S_h in the equation of Stress Range. In that case, the allowable stress range is calculated by the following equation.

S A = f [ 1.25 (S_c + S _h) - S_L]

This value of Stress range is popularly called as “Liberal Allowable Stress”.

In summary, we can say that two types of stresses are the major concern of piping flexibility analysis.Sustained Stress and Expansion stresses might be limited by the allowable stress conditions.

Systems that do not comply with sustained allowable conditions may experience Brittle failure.Due to this critical nature of the failure, Sustained stresses are also called as "Primary Stresses". Displacement stresses are less critical in comparison with Sustained stress and usually call as "Secondary Stresses".