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Nonlinear analysis method
can improve pipeline design

Alexander Aynbinder, Boris Taksa
Gulf Interstate Engineering

Patrick Dalton

A nonlinear engineering method for analyzing pipe stress criteria has been developed and can be used in common spreadsheet software for pipeline design.

Designs based on this method can enhance the operational reliability of pipeline systems because their designs can more accurately determine actual pipe stress and strain.

Most pipeline design codes establish allowable equivalent-stress limits that are higher than the pipe steel's proportional limit (the linear relationship between stress and strain). The limit is approximately 70% of the specified minimum yield strength (SMYS). Therefore, consideration of the nonlinear mechanical properties of the material is reasonable in pipeline stress analysis.

The nonlinear, numerical engineering method proposed is based on small elastic-plastic deformation theories and design data for materials used to manufacture pipe according to industry specifications.

The equations, algorithms, and iterative processes described, when combined with ones already published (OGJ, Feb. 20, 1995, p. 70) concerning the development of stress strain diagrams, can be readily developed into programs within common spreadsheet software.

The method allows for elastic plastic concepts to be easily incorporated in pipeline design. In some cases, this method allows the pipe's design wall thickness to be reduced; in other cases, an increase in the temperature differential can be tolerated.

This method can also be used for calculating the rigidity characteristics of the pipe. The results may be used in pipeline system analysis and such design programs as TRIFLEX or CAESAR.

Stress criteria

All pipeline codes contain at least two design stress requirements: pressure requirements only (including the pipe and piping components) and strength requirements for the pipeline system.

The latter requirement is concerned with stresses that result from all loads and load effects such as, among others, pressure, weight, temperature differentials, frost heave, thaw settlement, and earthquakes, in different combinations.

The pressure design of pipelines is based on determining the hoop stress using Barlow's formula for thin-walled pipe or Lam's formula for thick-walled pipe. This stress must be less than or equal to an allowable stress, usually a fraction of the specified minimum yield strength.

The minimum wall thickness or design pressure is determined from the condition when the hoop stress equals the allowable stress. It should be noted that most codes specify that the minimum wall thickness required for pressure containment may be inadequate to handle stresses resulting from other forces to which the pipeline may be subjected.

Strength design of piping and pipeline systems is limited by a criterion for the allowable longitudinal stresses and by a criterion for the allowable combined stresses (for example bending stresses with torsional stresses).

For unrestrained sections of pipelines, the American codes1-3 and the Canadian code4 establish different levels of allowable longitudinal stresses depending on which loads are combined during calculation of the actual longitudinal stresses.

But now, with increasing frequency, pipeline codes are permitting designs based on an allowable stress for the equivalent stress. (Depending on the code, this stress is also referred to as combined stress or stress intensity.)

To calculate the equivalent stress, some codes prefer Tresca's theory,2 4-6 others prefer Von Mises method,7 8 while some codes permit either theory to be used.3 9-10

When a code does not give a criterion for the equivalent stress but gives only one for the allowable stresses for hoop and longitudinal stresses, calculating allowable equivalent stress is possible with some assumptions. In ASME B31.4, for example, for nonrestrained pipe, the allowable expansion stress (SE,A) and the allowable hoop stress (SH,A) are both equal to 72% of specified minimum yield strength (0.72 SMYS).

If we assume that the expansion stress is equal to the bending stress (that is, the torsional moment = 0) and the axial longitudinal stress resulting from pressure is one-half the hoop stress, then the allowable equivalent stress for the pipeline in compression (compressive stresses being represented in this article by a minus sign) using Tresca's criterion will be as shown in Equation 1 (see accompanying equations box).

Some codes3 6 8 state that an alternative design based on strain may be used. ASME B31.8 does not state the value for the maximum allowable strain but does note that this value depends upon the steel's ductility and the previously experienced plastic strain, that is, consideration of nonlinear characteristics of the material.

Allowable equivalent stress in most codes is equal to 0.9 SMYS or greater. Because the allowable value of the equivalent stress for pipeline systems designed according to most codes exceeds the steel's actual proportional limit for stress or the allowable strain exceeds the proportional limit for strain, there is no reason that the nonlinear, elastic-plastic property of pipe material not be considered in pipeline stress analysis.

The former Soviet Union (FSU) code requires the designer to consider the nonlinear characteristic of steel pipe when performing pipeline stress analysis.7

Nonlinear properties

Experience has shown that the proportional limit for steel used for pipe fabrication is very close to a value of 0.7 SMYS.

If the equivalent stress is 0.9 SMYS, then the linear relationship between stress and strain would give an elastic strain of approximately 0.19% for Grade X-60 steel (modulus of elasticity = 27,900 ksi). The actual strain produced by a stress of 0.9 SMYS is closer to 0.28%, a difference of 1.5 times.

If the stress is actually equal to SMYS (defined by API SPEC 5L as a total strain of 0.5%11), the difference is even larger, approaching twice that predicted using a linear model. In practice, the actual strain in the pipe for typical operating conditions is between 0.20 and 0.45%.

Thus, it is reasonable to consider the nonlinear mechanical properties of the material in pipeline stress analysis and incorporate these considerations in the designs of pipelines. This idea was also discussed by Palmer who states the "design which allows a very small amount of plastic strain leads to substantial economies and does not appear to threaten the safety of the pipeline."12

The first step is to examine the stress-strain diagram in the area of small elastic-plastic deformation and then to use this diagram as the basis for stress analysis.

Previously, comparisons were made of different approximations for stress-strain diagrams with experimental data (OGJ, Feb. 20, 1995, p. 70). A series of equations (Number 3 in Table 2 of that article) was proposed based on dividing the stress-strain curve into the three stress sections. These input data for the series of equations used only material data from API SPEC 5L11 and ASME codes.

Fig. 1 [23413 bytes] presents a stress-strain diagram generated from equations for steel Grade X-42.

For the simple case of stresses in a straight restrained pipeline as a result of pressure and temperature differentials and using the stress criteria in ASME B31.4, analytical solutions are possible with a spreadsheet developed for determining either the wall thickness or the temperature differential.

Fig. 2 [31453 bytes] illustrates the change in wall thickness with an increasing positive temperature differential using both the linear and nonlinear solution for 24-in. pipe assuming a design pressure 1,440 psig for X-42 and X-60 steel grades.

By use of nonlinear considerations, the proposed pipeline design method can result in thinner walls for pipelines transporting hot or temperate products in the Arctic and hot products in other regions.

Stresses due to force, moments

Traditionally, the linear relationship between uni-axial stress and linear strain in Hook's generalized law for isotropic materials is characterized by a constant elastic modulus and constant Poisson's ratio. In the method discussed here, the constant elastic modulus and constant Poisson's ratio are replaced with nonlinear modulus and Poisson's ratio terms that may be used for the more complicated task of calculating the effective modulus and effective Poisson's ratio for cross sections of pipe.

In these cross sections, the distribution of longitudinal stresses is not uniform across the section. Then, with these nonlinear considerations in the forms of the effective modulus and Poisson's ratio, the stresses over the cross section can be determined.

Assume the force and moments acting on a pipe section are known. The internal axial force and moments may be calculated with a variety of available programs for pipeline stress analysis such as TRIFLEX or CAESAR, in which the solutions are based on linear properties.

For a statically indeterminate system, the distribution of force within the system depends on the stiffness of the section.

Examining the force and moments in the pipeline section requires definition of the variables, all of which comprise Equation 2.

Because the pipe has a vertical and horizontal axis of symmetry, for simplicity assume that the resultant bending moment is M =

(Mz2 + My2)1/2 and define a new coordinate system uow with a transformation angle w = arctan (My/Mz) as shown in Fig. 3 [58802 bytes].

The basic kinematic assumption of this solution is that the plane sections through the beam (normal to its axis) remain planar after the beam is subject to the stress and vary linearly (proportionately) with their distances from the neutral axis. Because the stress and elastic modulus are interrelated, the problem must be solved using iterative, numerical analysis.

The pipe's cross section is an annulus divided into n areas using i as an index ranging from 1 to n (Fig. 3 [58802 bytes]). The location of any ith area in the uow coordinate system may be determined by the equations that comprise Equation 3.

Assume the hoop and torsional stresses are independent of the elastic modulus and are calculated by the equations which together comprise Equation 4.

At the beginning of the iteration process, it is assumed that the effective-elastic modulus and effective-Poisson's ratio are the same as for the linear solution, that is, E = Eo and y = yo. Based on these assumptions, the pipe's axial strain efo and curvature ko are shown in Equation 5.

On the jth iteration involving the ith section, the longitudinal stress (SLji) may be calculated with Equation 6 in which all parameters with index j-1 were determined on the previous iteration.

Then, using this iteration's solution, calculating the equivalent stress is possible, as shown in Equation 7.

The uni-axial strain that corresponds with stress Seji (Equation 7) may be found by Equation 8.

The stress produced by the above strain can then be calculated with the set of equations from Equation 3 in Table 2 of OGJ, Feb. 20, 1995, p. 70, that describes the relationship between stress and strain near where the elastic range ends and the elastic-plastic area begins.

This stress is then used to calculate the effective modulus and effective Poisson's ratio for the next iteration using the equations shown is Equation 9.

Then, for the entire cross section, in the jth iteration, the tension-compression rigidity, the flexural rigidity, and the static moments are determined by the three equations that comprise Equation 10.

The resultant force and moments for the whole section are determined by Equation 11.

Substituting the value SLji from Equation 6 and the values of efo and ko from Equation 5 into Equation 11 and considering the conditions from Equation 10 yield a system of two linear equations with efj and kj unknown (Equation 12). Solutions to this system of equations are found in Equation 13.

The next step is to use these values of efj and kj in Equation 6 for the next iteration. The result of final iteration may be used in the design.

The authors have written a spreadsheet program for internal use that performs this method of stress-analysis on pipe. This was one of the stress-analysis methods used for the PGT-PG&E pipeline expansion project in 1991 to analyze stresses in pipeline bends and multibends laid in peat.

An accompanying box presents sample input and output data.

The output data consist of all the components of the stresses that must have their allowable limits checked to stay in compliance with the ASME B31 Codes. The output data also include the effective-modulus and effective-Poisson's ratio for the section which may be used in a program for calculating the internal force and moments.

In this example, the elastic modulus was reduced 11% and Poisson's ratio was increased 7%.

Fig. 4 [25189 bytes] shows the distribution of the effective modulus and effective Poisson's ratio over the pipe's cross-section. At the outermost compressive fiber, the elastic modulus is reduced approximately 30% and Poisson's ratio is increased approximately 20%.

Fig. 5 [27542 bytes] shows the stress distribution for the linear (elastic) and nonlinear (plastic) solutions for a pipe with input data listed in the example box. In this example, the nonlinear solution for the maximum compressive bending stress is 18% less and the equivalent stress 3% less than the linear solution.


1. ASME B31.3-90, "Chemical Plant and Petroleum Refinery Piping."

2. ASME B31.4-92, "Liquid Transportation Systems for Hydrocarbons, Liquid Petroleum Gas, Anhydrous Ammonia, and Alcohols."

3. ASME B31.8-92, "Gas Transmission and Distribution Piping Systems."

4. CSA-Z662-94, "Oil & Gas Pipeline Systems."

5. BS 8010:Section 2.8: 1992, "Code of Practice for Pipelines-Section 2.8 Steel for Oil and Gas."

6. BS 8010:3: 1993, "Code of Practice for Pipelines - Part 3. Pipelines subsea; design, construction and installation."

7. SNIP 2.05.06-85, "Transmission Pipelines."

8. ISO TC 67 SC2, "Draft of the International Organization for Standardization: Pipeline Transportation Systems for the Petroleum and Natural Gas Industries."

9. NEN 3650-1992, "Requirements for Steel Pipeline Transportation Systems."

10. CEN/TC 234 WG 3-1993, "Draft of the European Committee for Standardization: Pipelines for Gas Transmission."

11. API SPEC 5L-1995, "Line Pipe."

12. Palmer, A., "Limit State Design of Pipelines and Its Incorporation in Design Codes," The Society of Naval Architects and Marine Engineers Design Criteria and Codes Symposium, 1991, p. XIII-1.