A cylinder under internal pressure carries the load as membrane stress, P·R/t, uniform through the wall. A flat plate cannot. Pressure acts normal to the plate, and the plate resists it as a beam: bending stress that varies linearly through the thickness and grows with the square of the span. A rectangular header (plug header, cover-plate header, box header of an air cooler) is four such plates joined at the corners, so both actions are present together.
Membrane: global equilibrium
Cut the box on a plane normal to its length and take a strip of unit length. The pressure on the pair B plates (span h) is P·h per unit length. It is reacted as tension in the two pair A plates (span H), half in each. The same argument with the roles swapped loads pair B. The result does not depend on the stiffness of anything.
Bending: each plate is a beam with elastic ends
Each plate is treated as a straight member of span L with a uniform load w = P per unit width and end moments Ma and Mb from the corners. The engine uses one statics formula for every point on the plate, which is also how it reaches welds and hole rows away from the printed locations.
The end moments come from the frame: the corner rotates until the moments of the two plates meeting there balance. For sketch (1), a box with opposite sides equal, the closed-frame solution is below. The relative stiffness of the two pairs enters through K.
Sketch (1), PTB-4 E4.12.1 (test vector V-PTB-1), corroded gross plate, no holes
| Inputs (in, psi) | H = 7.375, h = 9.5, t1 = t2 = 0.875, P = 400 |
| α = K | 7.375 / 9.5 = 0.77632 |
| F | (1 + 0.77632³) / (1 + 0.77632) = 0.82635 |
| Sm(A) = P·h/(2·t1) | 400 · 9.5 / 1.75 = 2171.4 psi |
| Sm(B) = P·H/(2·t2) | 400 · 7.375 / 1.75 = 1685.7 psi |
| M_Q = P·h²·F/12 | 400 · 90.25 · 0.82635 / 12 = 2485.9 lbf·in/in |
| Sb at the corner Q = 6·M_Q/t² | 6 · 2485.9 / 0.765625 = 19481.6 psi |
| Sb at midspan M (pair B) | −15881.6 psi |
| Sb at midspan N (pair A) | −1830.6 psi |
The fixture holds the same full-precision values (19481.633, −15881.633, −1830.612 psi). At the corner the bending stress is about nine times the membrane stress of the same plate. A cylinder-style check on P·h/(2·t) would miss the governing stress by that factor.
What follows for a header
- Bending stress scales with (span/t)², membrane with span/t. Thickness is the effective lever: halving the span cuts bending by four.
- Stress is linear in P for a fixed geometry (sketches 1 to 3), so the MAWP follows from a ratio of stresses with no iteration.
- The code allows 1.5·S·E on membrane plus bending of a plate section and S·E on membrane alone (13-4(b)). The engine checks both.
- The corners are where the moments peak. In V-PTB-1 the hole row sits at midspan of the pair B plates, where the bending is lower than at the corner but the section is weakened. Article 3 covers that.
Limits. The App 13 equations are a two-dimensional strip per unit width and assume a long box (Lv/span above 4). Corner stress concentration, end-plate restraint and any local load are outside them; short boxes are handled by the J factors in article 2, and the local effects only by the FE cross-check in article 5. The check on this page is the code check; it is not a stress analysis of the actual header.