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New method addresses W.T. in offshore pipeline design
Alexander Aynbinder
Fluor Daniel Inc.
A new numerical iterative method for analyzing and designing high-temperature/high-pressure (HT/HP) pipelines provides for considerably greater reduction of wall thickness for these and for high-pressure flow lines than is called for under current design codes.

Such pipelines-operating at up to 350 F. and pressures of more than 5,500 psig-are currently envisioned for much near-future offshore development (OGJ, Mar. 10, 1997, p. 27).

A joint industry project, supported by such major operating companies as Amoco, BP, Shell, and others, was launched in April 1997 to address certain problems that emerge during design, construction, and operation of HT/HP pipelines. The project identified a major need for more-accurate methods for determining pipe wall thickness.

This study focuses only on a single problem in HT/HP pipeline design. Additional investigations within the joint industry project initiative, however, will address such other problems essential for developing HT/HP technology as the effect of cyclic loading on the development of plastic strains and the dependence of the mechanical properties of pipes on temperature.1


Determining wall thickness for restrained pipelines is important for advancing HT/HP technology. Existing engineering codes for oil and gas pipelines-ASME B31.4 and B31.8-have no limitations with respect to the values of temperature differential and pressure.

At the same time, the simplified equations in these design codes for determining hoop, longitudinal, and combined stresses in restrained pipelines are based on the assumption of elastic properties of steel and on the omission of radial stresses distributed across the pipe wall. These assumptions induce significant conservatism into HT/HP pipeline analysis.

A growing number of pipeline engineers in the offshore industry are concerned that these pipeline codes lead to over-conservative designs for high-pressure flow lines2 and therefore to greater cost.

The present study shows these codes require a substantial-and unjustified-increase of the wall thickness of pipelines. More-accurate methods of stress analysis can reduce wall thickness of HT/HP pipelines.

The purpose of this study has been to develop a more-precise and accurate numerical method for determining stresses and strains in HT/HP pipelines. This goal is attained by considering the nonlinear property of pipe materials and by properly handling essential radial and hoop stresses with a specific distribution across the pipe wall.

Thus, the proposed method permits determining the wall thickness by incorporating the conventional criterion for limiting combined stresses in the pipeline or the criterion specifying strain limits.

The method considers the nonlinear behavior of pipe materials, important for stress analysis of HT/HP pipelines because the allowable stress may considerably exceed the limit of proportionality of the stress-strain diagram.

The proposed method can be readily implemented on a personal computer by use of common, commercially available spreadsheet software. And it can be used with both stress and strain criteria, both of which are equally suited for engineering design if an appropriate stress-strain diagram is utilized.


The proposed model of steel describes the corresponding stress-strain relationship quite realistically for a complete range of stresses and thus ensures adequate pipeline analysis for a large range of loading conditions: up to the burst pressure, that is.

Furthermore, the proposed method ensures that the maximum hoop stress for pipes due to internal pressure is located on the external surface of the pipe if stresses across the pipe wall exceed the limit of proportionality of the stress-strain diagram. This result agrees well with experiments that show that bursting of pipes due to internal pressure starts at the external surface of the pipe wall.

Finally, the method can also be applied for analysis of general-purpose pipelines with regular internal pressure and temperature differential. The beneficial effect of the proposed method is considerably less, however: usually on the order of 3-7%.

Solution method

The developed method for evaluating the thickness of HT/HP pipeline walls considers several essential factors that are not included in the method of existing design codes.

Radial stresses are assumed to be significant, and their role in determining pipe-wall thickness is not neglected. Also, the combined stress is calculated by including all three components of the stress tensor: hoop, longitudinal, and radial stresses.

Finally, at the calculation phase, the method considers the inelastic behavior of pipe steel of the stress tensor components' values.

Note that the ASME B31.4 design code provides that for restrained pipelines, the allowable combined stress be up to 90% of the yield stress. At that level of loading, pipe materials exhibit strongly inelastic behavior.

Toward this goal, the finite-element method is used to obtain an approximate distribution of radial and hoop stresses across the pipe wall. The cross-section of the pipe wall is therefore modeled as a multilayered annulus.

The cumulative thickness of the annulus layers is set equal to the thickness of the pipe wall. Further, for every annulus layer, the hoop and radial stresses are assumed to be constant and equal to the corresponding average value.

An iterative computational algorithm has been developed to capture the inelastic behavior of pipeline steel. The algorithm aims at matching the nonlinear stress-strain constitutive relation.

In the first iteration, therefore, the effective modulus of all layers is set equal to the elastic modulus. On subsequent iterations, this value is modified for all layers by calculating the corresponding secant moduli based on the evaluated level of strains associated with every layer.

Note that, for every iteration, the concept of the secant modulus facilitates the analysis of the pipeline as a linear elastic system: On every iteration, the secant moduli are computed based on the nonlinear stress-strain diagram and are treated as elastic constants.3

The total loading-that is, pressure and temperature differential-is completely accounted for on every iteration, and the calculation process is repeated until displacements of modulus of elasticity and the effective Poisson's ratio values at adjoining iterations converge.

Stress-strain diagram

The stress-strain diagram can be written in the form of Equations 1 and 2. (See accompanying box for all equations.)

The effective elastic parameters are calculated by utilizing a stress-strain diagram that contains three sections:


  • The first part of the stress-strain diagram is linear; it is in line with engineering traditions and the methodology of the existing pipeline codes. The corresponding slope is equal to the modulus of elasticity of steel. This part of the diagram is valid up to the limit of proportionality between stresses and strains.
  • The second part of the diagram is described by a curve that continuously and smoothly connects the first and the third linear parts of the diagram.
  • The third part of the diagram is linear. The corresponding slope is equal to hardening modulus. This part of the diagram extends from the yield strength to the ultimate strength.
A comparison of the proposed steel stress-strain diagram with some other approximations and experimental data has been described previously (OGJ, Feb. 20, 1995, p. 70).

The difference between the proposed nonlinear constitutive relation and the corresponding linear model can be seen in Fig. 1 [59,649 bytes]. This figure presents the ratio of strains corresponding to the proposed diagram and the corresponding linear model vs. steel grade specified by API 5L.

The two curves plotted in this figure correspond to two different values of stress equaling 90% and 100% of yield strength, respectively.

Clearly, analysis of the nonlinear behavior of HT/HP pipelines is important for high-loading regimes. It should be noted, therefore, that HT/HP pipeline technology might consider strains as high as 90-100% of SMYS.

Stress determination

A straight restrained pipeline can be modeled as a long cylindrical shell subjected to the axially symmetric loading in the form of internal and/or external pressure and temperature differential. The latter is a result of the change of the temperature conditions during the installation period and the period of pipeline operation.

Since geometry and loading do not change in the longitudinal direction, the strain of the cylinder is symmetrical to its axis. Then, the strain-displacement relations for a restrained long cylinder can be written as shown in Equation 3.

As has been alluded to previously, every step of the iterative procedure for determining the corresponding stresses can be considered as an equivalent elastic problem. The corresponding linear constitutive relation for axi-symmetric elastic isotropic annulus is shown in Equation 4.

The equation4 from static condition of equilibrium in terms of the components of shown in Equation 5.

Substituting Equations 3 and 4 into Equation 5 yields the equation of equilibrium with respect to the displacement (Equation 6).

The solution of Equation 6 with respect to radial displacement is derived as Equation 7.

Finally, the stresses may be obtained by substituting Equation 7 into Equations 3 and 4 and deriving Equation 8.

The obtained solution for the components of the stress tensor is valid for an annulus of arbitrary thickness. It also shows that the radial and hoop stresses and, therefore, the combined stress vary along the wall. The effective modulus and Poisson's ratio depend on the combined stress.

Then, a more-accurate solution may be derived by dividing the annulus into layers so that each layer has such insignificant thickness that corresponding stresses can be assumed constant for every layer.


Note that two constants of Equation 8 (C1 and C2) must be determined for every layer from the boundary conditions and conditions of continuity with respect to stress and displacement. Specifically, the radial stresses at internal and external surfaces are equal to the internal and external pressure for the first and last layers (Equation 10).

Further, the radial stresses and displacement for adjoining layers are equal on the boundary of the layers (Equation 11).

The system of linear algebraic equations can be obtained by combining Equations 10, 11, 7, and the first equation of Equation 8. It can be written in the general form as Equation 12.

The linear system of algebraic Equation 12 can be readily solved yielding constant Ci of Equation 8 for all layers. Note that the effective modulus and Poisson's ratio depend on the combined stress.

Evaluating these mechanical constants can assume that hoop and radial stresses are constant for every layer; they are equal to the average value across the wall of the layer (Equation 13).

Average values for radial, hoop, and longitudinal stresses can be expressed by the equations shown in Equation 14.

The Van-Mises combined stress can be defined based on the maximum distortion energy theory (Equation 15). Upon evaluation of the effective modulus of elasticity by using the combined stress of Equation 15 in conjunction with considering a simplified uniaxial tensile state of steel, strains can be readily evaluated for every iteration (Equation 16).

Note that the modulus of elasticity corresponding to design codes is used on the first iteration.

The iterative process proceeds by evaluating stresses sj,i by using the steel stress diagram of Equation 1. Strains obtained from Equation 16 are used in this regard. Next, the effective modulus of elasticity and the effective Poisson's ratio are computed for the next iteration (Equation 17).

Note that on every step of the iterative process, all loading is considered; the process is repeated until displacements of modulus of elasticity and the effective Poisson's ratio values at adjoining iterations converge.

It should be emphasized that the process converges to the exact solution from above (that is, the result on every subsequent iteration is closer to the exact solution with a reduced margin), providing for the adequate level of conservatism in the obtained solution.

Numerical examples

The proposed method for determining HT/HP pipeline wall thickness can be readily implemented by using digital computers in conjunction with such readily available software products as Lotus or Excel.

Two numerical examples are considered here dealing with restrained pipelines of different dimensions and under quite different loading conditions.

The pipeline walls are partitioned into seven layers of equal thickness for both cases. As the iterative algorithm of the proposed method converges quite rapidly, the presented results were obtained after 6 and 12 iterations for the first and the second numerical example, respectively.

The first numerical example addresses a restrained pipeline of 10.75 in. OD made of X-60 steel. The thickness of its walls is selected to equal 0.58 in. The loading conditions involve the internal pressure of 5,000 psig and the temperature differential of 200 F.

Note that at this level of loading, the maximum value of combined stresses is approximately equal to 90% of yield strength.


The proposed method of analysis of HT/HP pipelines is compared with the code method of ASME B31.4 and with the method built upon the linear Lame's solution.6 The code method is therefore examined in conjunction with the Tresca equation for determination of combined stress considering only hoop and longitudinal stresses.

Some pertinent results corresponding to this numerical example are plotted in Fig. 2 [47,558 bytes], Fig 3 [55,225 bytes], and Fig. 4 [54,462 bytes]. Note that the difference between the maximum and minimum values of the combined strain and the combined stress for linear and nonlinear methods is approximately equal to 14 and 17%, respectively.

This discrepancy between the linear and nonlinear methods of HT/HP pipeline analysis originates from the changes of the modulus of elasticity and the Poisson's ratio (Fig. 4) associated with the nonlinear stress-strain diagram. Also, the results of pipeline analyses corresponding to the three methods of interest in HT/HP design are summarized in Table 1 [10,907 bytes].

Therefore, the required wall thickness has been calculated based on the existing B31.4 criterion; that is, the allowable combined stress is 90% of yield strength. The comparison of results shown in Table 1 clearly demonstrates the advantages of the proposed nonlinear method of HT/HP pipeline analysis.

Thus, the proposed method yields much lower maximum combined stress. Correspondingly, the thickness of pipeline walls can be reduced almost by 50% in comparison with the current ASME B31.4 code.

This method may be applied to pipelines subjected only to internal and/or external pressure. A substantial number of theoretical and experimental studies address such loading conditions.

The example of calculations was made for pipe with parameters listed in Table 1 for which the actual internal burst pressure was obtained by tests performed by Shell E&P Technology Co. in 1995-1996.2

The pipe characteristics are shown in Fig. 5 [54,772 bytes] and Fig. 6 [52,703 bytes]. The calculations were provided for internal pressure of 16,600 and 25,200 psig. The combined stress is calculated in accordance with Tresca equation by considering hoop and radial stresses. That is, the tensile longitudinal stresses due to pressure are not considered.

Plotted in Fig. 5 is one of the calculations results, the hoop stresses distribution across the wall due to pressure of 16,600 psig. The maximum hoop stress as a result of this pressure, by use of nonlinear solution, is approximately 47 ksi; that is, the required wall thickness may be 0.78 in. compared with 1.02 in. per B31.4 method to satisfy existing code hoop-stress criteria.

Note that the linear Lame's solution yields the maximum hoop stress located on the internal surface of the pipe. Alternatively, the maximum hoop stress is on the external surface for the nonlinear model of steel.


The same qualitative result of hoop-stress distribution was obtained previously using analytical solution for simplified bilinear approximation of the stress-strain diagram (OGJ, Apr. 29, 1996, p. 57). This result, which is obtained by using nonlinear models of steel, agrees with experiments which show that bursting of pipe from internal pressure starts at the external surface of the pipe wall.

The suggested method of calculation allows for predicting burst pressure. The combined and hoop stresses distribution across the pipe wall due to a burst pressure obtained by the test described by Langer2 are shown in Fig. 6.

The maximum combined stress due to burst pressure of 25.2 ksi is approximately 86 ksi. The ultimate tensile strength of tested pipes is close to the value of combined stress.

A first step

It should be strongly emphasized that if combined stress is up to 90% of the yield strength, equal to the allowable combined stress of B31.4 code, the actual strain that produces this stress may significantly exceed the elastic strain.

Therefore, the nonlinear stress analysis should be applied for the successful development of the HT/HP technology. The proposed method can be considered as the first step in the direction of improving the design of HT/HP pipelines.


  1. Beckman, J., "Erskine performance indicator for other high temperature lines," Offshore, September 1997, pp. 101-24.
  2. Langer, C.G., and Shah, B.C., "Code Conflicts for High Pressure Flowlines and Steel Catenary Risers," OTC Paper 8494, May 1997.
  3. Desai, S., and Abel, J., Introduction to the Finite Element Method, A Numerical Method for Engineering Analysis, Van Nostrand Reinhold Co., 1972.
  4. Timoshenko, S., Strength of Materials, 3rd Edition, Part II, Kriegas, Huntington, New York.

The Author

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Alexander Aynbinder is a senior project engineer at Fluor Daniel Co., Houston. Previously, he was in the civil engineering department of Gulf Interstate Engineering, Houston. Before emigrating to the U.S. in 1990, Aynbinder was a lead research scientist in the Russian State Research Institute for Pipeline Construction. He is a graduate of the Moscow Civil Engineering University and received a PhD in civil engineering from the Central Research Institute of Civil Structures, Moscow. Aynbinder is a member of ASME.

Copyright 1998 Oil & Gas Journal. All Rights Reserved.