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Oil & Gas Journal, 93:8, 2/20/1995


Alexander Aynbinder
Ventech Engineers Inc.
Pasadena, Tex.

J.T. Powers, Patrick Dalton
Wilcrest/Belmont Engineering Services

A pipeline design method has been developed that can result in thinner walls for pipelines operating in certain conditions and thus reduce capital, construction, and operating costs.

The method employs nonlinear solutions, compared to the more conventional linear approach found in the world's major design codes. It is nonetheless believed to be acceptable to most of those codes, including the widely used U.S. ASME B31.4 and B31.8.

Pipelines transporting hot or temperate products in the Arctic and hot products in other regions can operate with thinner walls, according to this method. If the product is gas, the temperature to which gas must be chilled can be increased, thereby reducing gas-chilling loads and costs.

When the pipeline design method described here is used in conjunction with commercially available software, the proposed engineering methods can be used to predict the internal forces and moments more accurately than those predicted by commercially available software alone.


The feasibility of pipeline projects depends on their economic viability. Among the many factors that affect project economics is the quantity of pipe steel to be used, a quantity that results directly from the pipe's wall thickness.

In the last 10 years, the philosophy of how pipeline wall thickness is determined has been re-examined.

The method of allowable stresses typically used in pipeline design in the U.S. is not used universally. For example, in Norway, The Netherlands, and U.K., codes permit design based on allowable strain. I

Allowable-stress codes generally consider a pipe material's properties to be linear:

A constant relationship between stress and strain. When allowable-strain codes are applied, there is growing support for calculations that use the material's actual nonlinear behavior.

The former Soviet Union's transmission pipeline code takes these considerations one step farther by requiring the designer to consider the nonlinear characteristics of the steel pipe when performing pipeline-stress analysis.

For a variety of reasons, not the least being simplicity, pipeline design according to ASME B31.4 and B31.8, the U.S. oil and gas pipeline codes, has generally used linear material properties."

Because the allowable value of the combined stress for a pipeline system designed in accordance with these codes can exceed the steel's actual proportionality limit, there is every reason to consider the nonlinear property of the material in pipeline stress analysis.


For pipeline systems designed according to ASME B31.4 and B31.8, the allowable hoop stress that results from the system's operating pressure is limited to a fraction (72%) of the specified minimum yield strength (SMYS), depending upon the pipeline system.

The equivalent or combined stress (the maximum allowable stress value due to pressure, other loads, and thermal effects) may be a higher fraction of SMYS. In ASME B31.4 (and in B31.8 for offshore gas-transmission systems), for example, it may equal 90% of SMYS.

The proposed method can be employed in designs that are in accordance with ASME B31.4 and B31.8 for the following reasons:

Paragraph A842.23 does not state the value of 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 the nonlinear characteristics of the material.

Experience has shown that the proportional limit (the linear relationship between the stress and strain) for steel used in pipe fabrication is very close to 70% SMYS.

If we assume that the stress is the maximum allowed by ASME B31.4 (90% SMYS), then a linear relationship between stress and strain would give the following results: for Grade X-52 steel, an elastic strain of approximately 0.17%, and for Grade X-60, an elastic strain of 0.19%.

In reality, the actual strains produced by a stress of 90% SMYS are closer to 0.26% and 0.28% respectively, a difference of nearly 150%.

If the stress is actually equal to SMYS (defined by API SPEC 5L4 as a total strain of 0.5%), the difference increases even farther, approaching twice that predicted with a linear model.

Remembering that increasing the stress beyond the proportional limit initiates a nonlinear relationship between the stress and strain, we note that, in practice, the actual strains in pipes subject to allowable operating conditions are between 0.20 and 0.45%.

Thus, consideration of the nonlinear mechanical properties of the steel is important in pipeline stress analysis and may be incorporated in pipeline design.'

The process begins with the choice of a function to represent the stress-strain diagram in the area of small elastic-plastic deformation and then to use this diagram as the basis for the stress analysis.


Equations (24287 bytes)

Many proposals approximate stress-strain (Or-E) curves. Most of them use a single equation to describe the relationship over an entire strain interval. This approach provides a convenient analytical method for the solution of a wide variety of problems.

Unfortunately, this method does not very accurately describe the true relationship over the elastic interval and the part of the curve between the points of the proportional limit and the yield stress.

One well known and often used equation for approximating uniaxial stress-strain curves is the Ramberg-Osgood formulation 6 shown in Equation 1 (24287 bytes) in the accompanying equations box.

This equation uses four parameters: the modulus of elasticity (E), yield strength (oys), the plastic strain at the yield stress (Epy), and a plasticity exponent (Np).

Fig. la (47687 bytes) presents a stress-strain curve generated with Equation 1 (24287 bytes) for three values of the parameter Np. This parameter may be found from test data. In the initial phases of pipeline design, however, this is normally not possible.

Equation 2 (24287 bytes) describes the uniaxial stress-strain curve with only three parameters.' The parameters are the elastic limit stress (oy), the modulus of elasticity (E), and the strain (h) at which the ultimate tensile strength is reached in a uniaxial tensile test.

Fig. 1b (47687 bytes) presents a stress-strain curve generated with Equation 2 (24287 bytes) for three values of the parameter h. Fig. lb illustrates that stress-strain curves at strains greater than 0.50% depend heavily on the parameter h.

The engineering solution being proposed for pipeline design begins with the division of the u-E curve into the three stress sections described in Table 1. (13286 bytes)

In Section 2 of Table 1 (13286 bytes), the section of the diagram with curvature, AU/AE has the boundary conditions Ao/AE = Eo, at the beginning of the curve and Ao/AE = E1 at the end.

Because the approximations in this article use only material data from API SPEC 5L and the ASME B31 codes E. and El can be determined.

The equations (Equation 3) for calculating the stresses (a-) from the strains (E) are found in Table 2. (19684 bytes)

Fig. 2 (21685 bytes) presents a stress-strain diagram generated from the equations in Table 2 (19684 bytes) f01 actual steel-pipe specimens-with the properties described in Table 3 (15096 bytes).

Fig. 2 illustrates the qualitative agreement between the theory and experience.

In Fig. 2, the two symbols represent one of the author's pipe test-sample results from his experience while working at the Russian Research Institute for Pipeline Construction.8 The solid and dashed lines result from the mathematical model described previously.

It can be seen that the model tracks experimental data well.

Fig. 3 (20022 bytes) presents stress-strain curves generated by Equations 1 through 3 (24287 bytes) for Grade X-70 steel.

This comparison shows that the proposed equation (Equation 3) that uses only specification input values (from API SPEC 5L and ASME B31) may be used for pipeline engineering design calculations in the initial design phase.


In the region of small elastic-plastic strains, it is possible to use the generalized Hook's law for isotropic materials by replacing the constant elastic modulus (E,,) and Poisson's ratio (y.) for the linear relationship between uniaxial stress and linear strain with an effective-modulus (E) and effective-Poisson's ratio (p), which are determined with actual stress values.

If the actual stress (oact) is known (the combined stress as defined in ASME B31.8 or the equivalent tensile stress as defined in ASME B31.4, that is, the stress intensity) at any point in a pipe section, one can use the nonlinear equations found in Equation 3 (24287 bytes) to determine the strain

Once the strain (F-) is determined, it may be used for calculating the effective-elastic modulus according to the equations found in Table 4 (15333 bytes).

The value of the effective Poisson's ratio may be calculated with the effective elastic modulus with a

The variables and coefficients in Equations 5 (Table 4 ) (15096 bytes) and 6 are the same as those defined and used in Equation 3 (Table 2) (19684 bytes).

The ratios E/E. and v/vo, may be plotted against the allowable stress range o(r) = (oact/OYS. Conversely, for the range of allowable stress, the corresponding ratios may be chosen.

Fig. 4 (20705 bytes) illustrates an example that uses Grade X-60 steel. The values of E,) 27,900 ksi and vo = 0.3 are taken from ASME B31.3 and B31.4, and the SMYS and SMYS values from API SPEC 5L.

If the actual stress reaches an allowable limit of 90% of SMYS (oact = 0.9oys), then the elastic modulus is reduced approximately 30%, and Poisson's ratio increases about 19%.

Fig. 5 (18241 bytes) illustrates how the ratios E/Eo and v/y, vary with SMYS when the same codes and specifications are used for input variables and the allowable stress is 0.9 SMYS. The curve shows that with increasing SMYS, the plastic properties of the steel are reduced.


In ASME B31.4, the longitudinal and hoop stresses due to the effects of temperature differential and fluid pressure for restrained pipelines are computed with the expressions shown in Equation 7. (24287 bytes)

The first expression is from the generalized Hook's law for an isotropic material in a plane-stress state shown in Equation 8. (24287 bytes)

To be in accordance with ASME B31.4 the stresses must satisfy requirements (Equation 9) shown in Table 5. (10801 bytes)

Because it is possible to remain in accordance with ASME B31.4 when the allowable stress is equal to 0.9oys (which is greater than the proportional limit for the material, or -70% of yield stress), we may use nonlinear solutions for the elastic modulus (E) and Poi,,son's ratio y, as described previously.

These principles may be used to accomplish the following three tasks:

  1. If the actual wall thickness (t,ct) is the same as that calculated due to internal pressure (t,), can be Equation 10 used.

    Then, using Equations 7 (24287 bytes) and 9 (Table 5) (10801 bytes), the allowable temperature differentials may be calculated with Equations 11 and 12.

  2. If the actual wall thickness (t,,t) is greater than the wall thickness determined by Equation 10 (24287 bytes), then the allowable temperature differential may be calculated with Equations 13 and 14. (24287 bytes)
  3. If it is known that the actual positive or negative temperature differential (absolute value; ATact) will be greater than that determined by Equations 11 or 12 (24287 bytes), then the allowable wall thickness may be calculated from Equations 15 and 16.

Fig. 6 (52785 bytes) illustrates the change in wall thickness with an increasing positive temperature differential using both the linear and nonlinear solutions for different steel grades.

Fig. 6a (52785 bytes) plots steel Grades X-42 and X-60 with the same base values as for the analysis performed in ASME B31 . 8-92 .3 It should be noted that the results from both analyses are in close agreement.

Fig. 6b (52785 bytes) plots steel Grades X-52 and X-70 with the real properties and parameters from an actual pipeline project.

Samples of the analysis input and output are presented in an accompanying box.

Fig. 7 (22477 bytes) presents an analysis of how the allowable temperature differential increases with rising steel grade. It also illustrates how, when the steel grade is increased, the influence of the nonlinear solution is reduced.

For restrained piping with a bending moment (Mb) to be in accordance with ASME B31.4, the beam bending stress (ob; positive for tensile) must be included in the longitudinal stress. Therefore, Equation 7 (24287 bytes) may be re-written as Equation 17.

Using those same stress criteria described in Equation 9, the allowable temperature differential may be calculated using Equations 19-22.

The allowable wall thickness can be calculated by Equations 23 and 24.

As demonstrated in this section, by using the nonlinear solutions as opposed to the linear solution (the method directly from B31.4), it is possible to reduce the allowable wall thickness or increase allowable temperature differential.

Editor's note: The work described in this article was performed while the authors were employed at Gulf Interstate Engineering, Houston.


  1. Klever, F.J., Palmer, A.C., and Kyriakides, S. "Limit-State Design of High-Temperature Pipelines," 1994 OMAE (ASME), Pipeline Technology, Vol. 5, P. 77-92.
  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. API SPEC 5L-92, "Specification for Line Pipe."
  5. Palmer, A., "Limit State Design of Pipelines and Its Incorporation in Design Codes," Society of Naval Architects and Marine Engineers Design Criteria and Codes Symposium, 1991.
  6. Ramberg, W., and Osgood, W. R., "Description of Stress-Strain Curves by Three Parameters," Technical Note 902, National Advisory Commission for Aeronautics, July 1943, Washington.
  7. Aynbinder, A., Tabakman, Y., Powers, J., and Dalton, P., "FSU Pipeline Design Code Compared to U.S. Codes," OGJ, Mar. 7, 1994, P. 67.
  8. Aynbinder, A. B., "Calculation of Strength and Stability Field and Transmission Pipelines," Nedra (Moscow, 1991), P. 287.

Copyright 1995 Oil & Gas Journal. All Rights Reserved.