#### Modified thin-wall pipe formula

accounts for external pressure

Wall thickness of an internally pressurized cylindrical vessel is determined by computing the hoop stress. The hoop stress must be less than the maximum allowable stress. If the calculated hoop stress is greater than the allowable stress, the pipe wall thickness must be increased.Jaeyoung Lee, William Rainey, Mark BrunnerAker Engineering

For a pipe-in-pipe design in which the inner pipe is enclosed by an outer casing pipe with the annulus pressurized or a pipe in a marine environment exposed to external hydrostatic head, the external pressure should be considered in the pipe wall thickness determination.

There are two general methods for calculating the hoop stress: a thick wall pipe formula and a thin wall pipe formula.

The thick wall pipe formula gives an exact solution but requires an iterative solution to determine the required pipe wall thickness. The thin wall pipe formula is simpler and easier to use in calculating the pipe wall thickness. It provides reasonably accurate results for thin wall pipes, such as pipe having D/t ratios greater than 20.

The thin wall pipe formula's results are generally less than 5% over the exact solution provided by the thick wall pipe formula, if no external pressure exists. However, the present thin wall pipe formula yields erroneous results in cases where external pressure exists.

### Thick-wall pipe formula

A French Engineer, Lame, derived a thick wall cylinder formula in 1833, using the stress system shown in Figure 1 [10,841 bytes] (Blake, 1990). A pressurized pipe develops both tangential and radial stresses in a two-dimensional cross section. The longitudinal or axial stress is neglected by assuming no constraints at the ends of the pipe. The tangential stress is induced in the circumferential or hoop direction in the pipe wall. The radial stress acts perpendicular to the pipe wall.From the equilibrium of forces (the summation of forces in each direction must be zero) and integration, the tangential or hoop stress (sh) and the radial stress (sr) can be expressed as shown (Shigley, 1983).

The formulas provide "exact" solutions in the elastic range for any cylindrical pipe wall thickness. In these equations, positive stresses indicate tension, and negative stresses indicate compression. By substituting a = Di/2, b = Di+2t, and r = a = Di/2 at the inner pipe surface,Equation 1 [8,595 bytes] can be rewritten as the Equation 3 [19,805 bytes] group.

In the same way, the hoop stress at r = b = D/2 at the pipe outer surface is represented in theEquation 4 [20,087 bytes] group.

When the external pressure is zero, where Po = 0, thenEquation 5 [7,750 bytes] and Equation 6 [7,529 bytes] apply.

In the same way, the radial stresses at the inner and outer surfaces of the pipe wall can be expressed as shown inEquation 7 [4,489 bytes] and Equation 8 [3,741 bytes].

Figure 2 [32,770 bytes] and Figure 3 [20871 bytes] demonstrate the general characteristics of the thick wall pipe stresses along the pipe wall thickness (Equations 1 through 8).

The absolute hoop stress is maximum at the inner wall surface regardless of the relationship of the internal pressure to the external pressure. For this reason, the inside diameter is used for the hoop stress calculation when using the thick wall pipe formula. The hoop stress difference between the inner wall surface and the outer wall surface is the same as the pressure differential, Pi - Po (Equation 3 minus Equation 4).

The hoop stress is shown to be tensile when internal pressure is greater than the external pressure. However, when the internal pressure is equal to the external pressure, the hoop stress becomes compressive (see Figure 3). Equations 3 and 4 show the hoop stress becomes compressive with the internal and external pressure being equal (Pi = Po). This means that the hoop stress transition, from tension to compression, occurs before Pi = Po. In other words, the pipe is already in the compressive region prior to the external and internal pressures being equal. This will be investigated in detail later in this paper.

The radial stress is always compressive.

The absolute values at the inner or outer wall surfaces are the same as the internal or the external pressures. The radial stress differential is the same as the pressure differential between the inside and outside of the pipe. The summation of the hoop stress and the radial stress is always constant across the pipe wall thickness (Summation of Equation 1 andEquation 2 [8,687 bytes]).

As mentioned earlier, zero hoop stress occurs before the external pressure equals the internal pressure. At Pi = Po, the pipe experiences compressive stress which is equal to the ambient pressure (Set Equation 3 equal to zero to solve for the zero hoop stress occurrence point). The equation yieldsEquation 10 [5,609 bytes], the results of which are always less than 1.0.

This means that the zero hoop stress always occur when the external pressure is less than the internal pressure. This is logical if we consider the respective areas contacted by the internal pressure and external pressure.

Figure 4 [24,355 bytes] represents Equation 10 in graphical form. At D/t = 20, zero hoop stress occurs when the external pressure is 90.5% of the internal pressure. For example, if the internal pressure is 2,500 psi, zero hoop stress will occur at 2,500 times 0.905, or 2,263 psi external pressure. Above the 2,263 psi external pressure, holding the internal pressure constant, the pipe will experience a compressive hoop stress. The hoop and radial compressive stresses will be 2,500 psi at Pi = Po = 2,500 psi (Figure 3).

The magnitude of the hoop stress is always maximum at the inner wall surface (Figures 2 and 3). Since the pipe is to be designed for the maximum stress across the wall, the thick wall pipe formula with r = a (Equation 3), is used to determine the pipe wall thickness. This equation gives exact maximum hoop stress for any pipe wall thickness and it is called the "thick wall pipe formula".

Equation 11 [7,701 bytes] is applicable for pipes in the elastic region, which is acceptable since the plastic deformation due to internal pressure is normally not allowed in the pipe wall thickness determination. However, the pipe wall thickness determined by Equation 11 must be checked for excessive external buckling collapse pressure. The pipe collapse calculation methods are presented in many industry codes and are not described in this paper.

For a known pipe wall thickness, the hoop stress can be calculated by using Equation 11. However, with an unknown pipe wall thickness, and a known maximum allowable hoop stress (normal practice), the equation requires several iterations to solve for the pipe wall thickness. Therefore, a simpler formula was introduced and is widely used in the pressure vessel industries. The formula is derived for a thin walled pipe, and is thus called the "thin wall pipe formula".

### Thin wall pipe formula

When a long free section of thin wall pipe is subjected to internal pressure, a hoop stress will be induced on the pipe. The hoop forces will be in equilibrium with the Y-component forces of the internal pressure (Figure 5 [6,626 bytes]).Equation 12 [7,278 bytes] is derived from such assumptions as:

(1) Negligible radial stress for a thin wall pipe

(2) Uniform hoop stress across the pipe wall thickness

(3) No external pressure exists.

The radial stress, when Po = 0, is negligible for a thin wall pipe. As shown inFigure 6 [27,382 bytes], the radial stress is less than 10% of the hoop stress for D/t ratios greater than 20. By assuming a negligible radial stress, the hoop stress will be uniform across the wall thickness, since the summation of the hoop stress and radial stress must be constant.

Figure 6 also indicates that the use of the thin wall pipe formula for a thick wall pipe, for example - D/t ratios less than 20, gives inaccurate results because the radial stress is not negligible in this region. As the external pressure increases, the tensile hoop stress decreases. The radial stress at the inner wall surface is equal to the internal pressure (it is not affected by the external pressure), so the ratio of the radial stress to the hoop stress increases as the external pressure increases.

Figure 6 shows that the radial stress to hoop stress ratio becomes 19% at D/t ratio = 20, when Po/Pi = 0.4. Moreover, at Pi = Po, the radial stress is equal to the hoop stress, which means the ratio reaches 100%. As Po/Pi ratios increase, the radial stress contributions to the hoop stress increase exponentially. This shows that the thin wall pipe formula cannot be used for external pressure situations where the radial stress cannot be neglected.

The thin wall pipe formula yields a uniform hoop stress across the wall thickness while the thick wall formula gives a variable hoop stress with a maximum at the inner wall surface. The thin wall pipe formula will give a maximum hoop stress when the pipe outside diameter is used. Figure 7 represents the hoop stress estimated by the thin wall pipe formula using the pipe outside diameter, mean diameter, and inside diameter.

Using the outside diameter gives about 5% over-prediction of the exact solution at 20 D/t ratio. If a mean diameter is used, the thin wall pipe formula slightly under-predicts the exact solution by 0.3-0.1% at D/t ratios of from 20 to 40. The mean diameter may be used if 0.3% or less under-prediction, at D/t ratios greater than 20, is acceptable. However, API and ASME codes use the outside diameter, which is conservative.

Equation 13 [4,867 bytes] provides reasonable results for D/t ratios greater than 20 with no external pressure. To account for the external pressure, most of industry codes such as API 1111 and ASME B31.8 subtract the external pressure from the internal pressure term in Equation 13 (Shown in Equation 14 [4,620 bytes]).

This equation is not correct, as the original equation (Equation 13) is derived from the no external pressure and negligible radial stress assumptions. Therefore, Equation 13 cannot be modified to account for the external pressure effect. This can be proved by inserting Pi = Po in Equation 14. When Pi = Po, Equation 14 shows zero hoop stress, which is contradictory to the thick wall pipe formula. When Pi = Po, the hoop stress must be negative (-) Po, from the thick wall pipe formula (Equation 11).

For this reason, Equation 14 over-predicts the hoop stress at high external pressures. Figure 8 shows how much Equation 14, which is called "original thin wall pipe formula," over-estimates the hoop stress, compared to the thick wall pipe formula. If no external pressure exists, when Po = 0, and for D/t ratios greater than 20, the original thin wall pipe formula over-estimates the hoop stress a maximum of 5%, compared to the thick wall pipe formula.

The 5% over-estimate is caused by neglecting the radial stress and it is acceptable in terms of conservative designs. However, at high external pressure such as Po/Pi = 0.4 at D/t = 20, the original thin wall pipe formula predicts 13% higher hoop stress than the thick wall pipe result. The over-prediction is increased to 80% at Po/Pi = 0.8 and D/t = 20.

Figure 8 demonstrates how overly conservative and inaccurate the original thin wall pipe formula is in high external pressure cases. As pipe wall thickness and the external pressure increase, the differences of the hoop stress ratios between the original thin wall formula to the thick wall formula increases significantly. This indicates that the original thin wall formula does not predict the hoop stress correctly, especially for high external pressure cases.

### Modified formula

To find a more realistic formula to account for the external pressure, we need to go back to the thick wall pipe formula (Equation 11), which provides an exact solution. In Equation 11, assuming a very thin thickness (as t approaches zero), we have Equation 15 [5,765 bytes].Substituting Equation 15 into Equation 11 producesEquation 16 [6,147 bytes].

The left hand side of the Equation 15 is always smaller than D, thus Equation 16 always yields higher wall thickness requirements than the thick wall pipe formula. Note that Equation 16 has an additional minus (-) Po term compared to the original thin wall pipe formula (Equation 13). Equation 16 is called the "modified thin wall pipe formula."

When the external pressure is equal to the internal pressure, Equation 16 gives compressive stresses equivalent to the external pressure, which agrees with the thick wall pipe theory. The modified thin wall pipe formula predicts the hoop stress more accurately than the original thin wall pipe formula as presented in Figure 9.

For D/t ratios greater than 20 at Po/Pi =0.4, the original thin wall pipe formula over-estimates by 13% or less (Figure 8), while the modified thin wall pipe formula over predicts the hoop stress by 6% (Figure 8), both are compared to the thick wall pipe formula. As the pipe wall thickness and the external pressure increase, the difference in error between the original and the modified thin wall formulas becomes more significant.

### Recommendation

As the external pressure increases, for example as the water depth increases for a marine pipeline, the thin wall formula predicts a higher hoop stress than the exact value. This is caused by using the thin wall pipe theory replacing the Pi term with (Pi - Po) to account for the external pressure effect. This is not correct as the original thin wall pipe equation is derived from the "no external pressure" and "negligible radial stress" assumptions.Based on the "exact" thick wall pipe theory, the original thin wall pipe formula should have an additional minus external pressure term as shown in Equation 16, to more accurately account for the external pressure effect.

The proposed modified thin wall pipe formula (Equation 16) will provide reasonably accurate estimates, less than 10% over the exact values, for D/t ratios greater than 20. For D/t ratios less than 20, using the thick wall pipe formula is recommended.

Several industry codes do not account for the external pressure effect, these should be modified for deep water application by accounting for the external pressure, either using the modified thin wall pipe formula or the thick wall pipe formula.

### References

API 1111, "Design, Construction, Operation, Maintenance of Offshore Hydrocarbon Pipelines," 1993, API, Washington, D.C.ASME B31.8, "Gas Transmission and Distribution Piping Systems," 1992, The American Society of Mechanical Engineers, New York.

Blake, Alexander, "Practical Stress Analysis in Engineering Design," 1990, Marcel Dekker, Inc., New York.

Shigley, Joseph E. and Mitchell, Larry D., "Mechanical Engineering Design," 1983, McGraw-Hill Book Company, New York.

**Nomenclature**aPipe inside radiusbPipe outside radiusDPipe outside diameterDiPipe inside diameterdrChange in pipe radiusPiInternal design pressurePoExternal pressurerPipe radius at any point along the pipe wall thicknesstPipe nominal wall thicknessshHoop stress = circumferential stress = tangential stress, exact value from thick wall pipe formulashmHoop stress from the modified thin wall pipe formulashoHoop stress from the original thin wall pipe formulasho'Hoop stress from the original thin wall pipe formula without considering external pressuresrRadial stress

*Copyright 1998 Oil & Gas Journal. All Rights Reserved.*