 # Pipe stress I

In this blog I will discuss longitudinal and hoop stress equations given in the Piping code B31.3 and the way that they are derived. Principal stresses (Eq.1) and max shear stress (Eq.2) are derived using stress transformation analysis for two dimensional state. The transformation of plane stress can also be represented in graphical form know as Mohr’s circle. Plane at which principal stress is occur is called principal plane and plane at which maximum shear stress occur is on 45 degree from the principal plane. Please see the figure below. Maximum shear theory states that failure will occur when maximum shear stress exceeds the value obtained for the shear stress at yielding in the uniaxial tensile test. Maximum shear stress in simple tension case occurs at angle 45 with load and it is given by Eq.3. Please see the figure below. Final equitation for longitudinal stress (Eq.4) is obtained combining Eq.2 and Eq.3. Please see the figure below. Longitudinal stress in Piping code B 31.3 is given by Eq.5.Please see the figure below. For sustain load, longitudinal stress is limited to Sh (stress at hot state); for the occasional and sustain load, longitudinal stress is limited to 1.33Sh; and for thermal load longitudinal stress is limited by displacement stress range Sa. In the piping system, there is also hoop stress due to internal pressure, and hoop stress has been limited to Sh (stress at hot state). Pipe hoop stress is
calculated using Barlow’s equation. Please see the figure below. It should be noted that Piping code B 31.3 is focused on limiting each individual component, rather than limiting principle stress. Hoop stress has been limited to Sh, and longitudinal stress has been calculated as per maximum shear theory and limited differently for the certain conditions. Please see the figure below. Note that Tresca theory is applicable for ductile materials only. In the case of brittle material maximum normal stress theory is applied, which states that failure occurs when one of the principal stress reach a yield stress. Ductile materials when exposed to low temperature will lose ductility, and can promote brittle fracture. To remedy this issue, Piping code B 31.3 made requirement for impact test. The photo below shows World War II tanker broken in two by a brittle fracture, despite the normal ductility of used steel. 