WRC 107 Stress Analysis of a Nozzle in a Cylinder

WRC 107 Stress Analysis of a Nozzle in a Cylinder Intergraph CAS Ray Delaforce Welding Research Bulletin 107 (WRC 107) was developed around 1965 to meet the needs of the engineering fraternity who needed a method of quickly analyzing the local stresses at the junction of a nozzle and a shell when the nozzle (or attachment) was subjected to external loads. The bulletin contained numerous dimensionless graphs that could be used to estimate the local stress in four locations on the inside surface and four locations on the outside surface making eight locations in all. Though the results are not as accurate as those obtained by other means, such as FEA, the method lent itself to computerization. Engineers have been using the WRC 107 method for years. Stress Categories In stress analysis, stresses have traditionally been characterized by the origin of the stresses. For example, do we have a membrane stress, or a bending stress? Each type of stress can have a limiting or allowable stress depending on its nature. For many years, the ASME Section VIII, Division 2 code has been the source for defining the stress Categories. There are 4 stress categories – ignoring stresses that give rise to fatigue. Pm PL Pb Q

General primary membrane stress Local primary membrane stress General primary bending stress Secondary membrane and bending stresses

Exists everywhere in the structure. Characterized failure when yield stress is reached Exists everywhere in the structure. Characterized failure when 1.5 x yield stress is reached Exists everywhere in the structure. Characterized failure when 1.5 x yield stress is reached Exists at very small region in the structure characterized by failure when 2 x yield is reached. These stresses are derived usually from thermal expansion and are strain controlled.

An example of a general primary membrane stress is the stress in a cylinder wall resulting from pressure (the load). It exists everywhere in the cylinder. Another example, this time of a general primary bending stress is that of a cantilever with a weight on the end. In both these cases, a catastrophic failure would occur is the applied load is too great. Stresses are traditionally considered in groups for convenience. And the WRC 107 method follows this grouping regimen: Stress grouping Pm Pm+(Pb)+PL Pm+(Pb)+PL+Q

Allowable stress S 1.5S 3S or 2Sy

Derivation of Pm Pm is is the general primary membrane stress generated from internal pressure. There is a Pm in the hoop direction, and Pm in the axial or longitudinal direction. The hoop stresses is different on the outside surface of a shell from that generated on the inside of the shell. They are derived from thick cylinder theory – known as the Lamé Theorem. Let us consider a typical example. Consider the following situation:

Let us summarise these stresses:

Now, if we look at the printout from PV Elite look at the lines typed in bold font: WRC 107 Stress Summations: Vessel Stress Summation at Attachment Junction -----------------------------------------------------------------------Type of | Stress Values at Stress Int. | (MPa ---------------|-------------------------------------------------------Location | Au Al Bu Bl Cu Cl Du Dl ---------------|-------------------------------------------------------Circ. Pm (SUS) | 99 101 99 101 99 101 99 101 Circ. Pl (SUS) | 27 27 71 71 36 36 38 38 Circ. Q (SUS) | 53 -53 131 -131 148 -148 168 -168 -----------------------------------------------------------------------Long. Pm (SUS) | 49 49 49 49 49 49 49 49 Long. Pl (SUS) | 29 29 45 45 48 48 50 50 Long. Q (SUS) | 108 -108 219 -219 86 -86 96 -96 -----------------------------------------------------------------------Shear Pm (SUS) | 0 0 0 0 0 0 0 0 Shear Pl (SUS) | 0 0 0 0 0 0 0 0 Shear Q (SUS) | 3 3 3 3 3 3 3 3 ------------------------------------------------------------------------

Now, the worst primary membrane stress (Pm) is in the hoop direction and is this one shown in red font: -----------------------------------------------------------------------Type of | Max. S.I. S.I. Allowable | Result Stress Int. | MPa ---------------|-------------------------------------------------------Pm (SUS) | 101.01 137.90 | Passed Pm+Pl (SUS) | 172.97 206.85 | Passed Pm+Pl+Q (TOTAL)| 315.04 413.70 | Passed ------------------------------------------------------------------------

Remember, this stress is generated by the internal pressure only, and the greatest stress is the hoop stress existing everywhere on the inside surface. Now, The local stress Pm+PL has to summed up in the same direction, and we have to find the greatest stress in that direction. Here they are: Vessel Stress Summation at Attachment Junction -----------------------------------------------------------------------Type of | Stress Values at Stress Int. | (MPa ---------------|-------------------------------------------------------Location | Au Al Bu Bl Cu Cl Du Dl ---------------|-------------------------------------------------------Circ. Pm (SUS) | 99 101 99 101 99 101 99 101 Circ. Pl (SUS) | 27 27 71 71 36 36 38 38 Circ. Q (SUS) | 53 -53 131 -131 148 -148 168 -168 -----------------------------------------------------------------------Long. Pm (SUS) | 49 49 49 49 49 49 49 49 Long. Pl (SUS) | 29 29 45 45 48 48 50 50 Long. Q (SUS) | 108 -108 219 -219 86 -86 96 -96 -----------------------------------------------------------------------Shear Pm (SUS) | 0 0 0 0 0 0 0 0 Shear Pl (SUS) | 0 0 0 0 0 0 0 0 Shear Q (SUS) | 3 3 3 3 3 3 3 3 -----------------------------------------------------------------------Pm (SUS) | 99.0 101.0 99.0 101.0 99.0 101.0 99.0 101.0 -----------------------------------------------------------------------Pm+Pl (SUS) | 126.1 128.1 171.0 173.0 135.5 137.5 137.1 139.1 ------------------------------------------------------------------------

Pm+Pl+Q (Total)| 189.2 104.1 315.0 166.9 283.8 22.3 305.3 33.4 ------------------------------------------------------------------------

Now, we simply have to sum Pm+PL+Q: -----------------------------------------------------------------------Type of | Stress Values at Stress Int. | (MPa ) ---------------|-------------------------------------------------------Location | Au Al Bu Bl Cu Cl Du Dl ---------------|-------------------------------------------------------Circ. Pm (SUS) | 99 101 99 101 99 101 99 101 Circ. Pl (SUS) | 27 27 71 71 36 36 38 38 Circ. Q (SUS) | 53 -53 131 -131 148 -148 168 -168 -----------------------------------------------------------------------Long. Pm (SUS) | 49 49 49 49 49 49 49 49 Long. Pl (SUS) | 29 29 45 45 48 48 50 50 Long. Q (SUS) | 108 -108 219 -219 86 -86 96 -96 -----------------------------------------------------------------------Shear Pm (SUS) | 0 0 0 0 0 0 0 0 Shear Pl (SUS) | 0 0 0 0 0 0 0 0 Shear Q (SUS) | 3 3 3 3 3 3 3 3 -----------------------------------------------------------------------Pm (SUS) | 99.0 101.0 99.0 101.0 99.0 101.0 99.0 101.0 -----------------------------------------------------------------------Pm+Pl (SUS) | 126.1 128.1 171.0 173.0 135.5 137.5 137.1 139.1 -----------------------------------------------------------------------Pm+Pl+Q (Total)| 189.2 104.1 315.0 166.9 283.8 22.3 305.3 33.4 ------------------------------------------------------------------------

The calculation of Q and PL is complicated, and is the method shown in WRC 107