Equations for pressure design of straight pipe are provided in para. 304.1. The minimum thickness of the pipe selected, considering manufacturer’s minus tolerance, must be at least equal to rm, defined as
thickness. Therefore, the wall thickness can be directly calculated when the outside diameter is used in the equation.
Equation (4.2) is an empirical approximation of the more accurate and complex Lame equation (ca. 1833). The hoop or circumferential stress is higher toward the inside of the pipe than toward the outside. This stress distribution is illustrated in Fig. 4.1. The Lame equation, provided below, can be used to calculate the stress as a function of location through the wall thickness. Equation (4.2) is the Boardman equation. Although it has no theoretical basis, it provides a good match to the more accurate and complex Lame equation for a wide range of diameter-to-thickness ratios. It becomes increasingly conservative for lower D/t ratios (thicker pipe) if Y is held constant.
The Lame equation for hoop stress on the inside surface of pipe follows. Note that for internal pressure, the stress is higher on the inside than the outside. This is because strain in the longitudinal direction of the pipe must be constant through the thickness, so that any longitudinal strain caused by the compressive radial stress (due to Poisson effects and considering that the radial stress on the inside surface is equal to the surface traction of internal pressure) must be offset by a corresponding increase in hoop tensile stress to cause an offsetting Poisson effect on longitudinal strain. The Lame equation is
Simple rearrangement of the above equation and substitution of SE for ct), leads to Eq. (3a) of the code [Eq. (4.2) here). Further, inside-diameter-based formulas add 0.6 times the thickness to the inside radius of the pipe rather than subtract 0.4 times the thickness from the outside radius. Thus, the inside-diameter-based formula in the pressure vessels codes and Eqs. (3a) and (3b) of ASME B31.3, the Piping Code, are consistent. The additional consideration in Eq. (3b) of ASME B31.3 is the addition of the allowances (internal corrosion increases the inside diameter in the corroded condition). With this additional consideration, Eq. (3b) of ASME B31.3 based on inside diameter provides the same required thickness as Eq. (3a) based on outside diameter.
A comparison of hoop stress calculated using the Lame equation versus the Boardman equation (4.2) is provided in Fig. 4.2. Remarkably, the deviation of the Boardman equation from the Lame equation is less than 1% for Dll ratios greater than 5.1. Thus, the Boardman equation can be directly substituted for the more complex Lame equation.
For thicker wall pipe, ASME B31.3 provides the following equation for the calculation of the Y factor in the definition of Y in para. 304.1.1. Use of this equation to calculate Y results in Eq. (4.2), matching the Lame equation for heavy wall pipe as well:
The factor Y depends on temperature. At elevated temperatures, when creep effects become significant, creep leads to a more even distribution of stress across the pipe wall thickness. Thus, the factor Y increases, leading to a decrease in the calculated required wall thickness (for a constant allowable stress).Three additional equations were formerly provided by the Code, but two were removed to be consistent with ASME B31.1 and simplify the Code. They may continue to be used. The first of the removed equations is
This equation is the simple Barlow equation, which is based on the outside diameter and is always conservative. It may be used, because it is always more conservative than the Boardman equation, which is based on a smaller diameter (except when Y = 0). The second removed equation is
This equation is the Lame equation rearranged to calculate thickness. Although it is not specifically included, it could be used, in accord with para. 300(c)3. However, it should not make a significant difference in the calculated wall thickness. The following optional equation remains in ASME B31.3, Eq. 3b.
Equation (4.9) is the same as (4.2), but with (d + 2c + 21) substituted for D and rearranged to keep thickness on the left side. #Little_PEng