Bends were required to have, after bending, a wall thickness at least equal to the minimum required wall thickness for straight pipe in para. 304.2.1. However, this was changed in the 2000 Addendum. The Lorenz equation (ca. 1910) was included; it provides a means of calculating the required wall thickness. Note that the prior requirement that simply stated the thickness should be the same as required for straight pipe was deleted. The new requirement is more conservative for the intrados (inside curves) of bends and less conservative for the extrados (outside curve).
The Lorenz equation is basically the equation for a toroid. If the intrados and extrados had the same wall thickness, the inside would be subjected to higher hoop stress than straight pipe and the outside would be subjected to lower hoop stress than straight pipe. A simple way to envision this is that the inside has less metal over the curve and the outside has more metal over the curve. The Lorenz equation for an elbow or bend is given by ASME B31.3, Eq. (3c)
where the terms are as defined in Section 4.2 for Eq. (4.2), except for /, which is a stress index that accounts for the difference in hoop stress due to internal pressure in bends versus straight pipe. On the inside curve of the bend, the intrados, we have ASME B31.3, Eq. (3d)
where R1 = radius of bend
On the outside of the bend, or the extrados, we have ASME B31.3, Eq. (3e)
On the side of the elbow, or the crown, / = 1.0 (i.e., the hoop stress is the same as in straight pipe). The thickness variation from the intrados to the extrados is required to be gradual, and the requirements are stated to apply at the midspan of the bend. The thickness at the ends is required to satisfy the required thickness for straight pipe per para. 304.1.
The normal process of making a bend by bending straight pipe produces this type of thickness variation. Part of the reason for providing these new rules is due to the practice of fabricating elbows by forming two “clamshells” out of plate and welding them together. This produces a bend of uniform thickness, and the thickness on the intrados would be too thin if it simply satisfied the required thickness for straight pipe.
Elbows in accordance with listed standards, or qualified by para. 304.7.2, are also permitted. #Little_PEng.