Cold-formed steel purlin

A line of C or Z purlins over one to six spans, with bolted laps at the supports. Girts follow the same equations. To AISI S100-16 with S2-20: Chapters F, G, H and Section I6, laid out by regions as the MBMA Roof Framing Design Guide (2nd Ed., 2024) examples are.

Inputs of the purlin #

• Section: from a list named as the AISI Design Manual names its sections, or custom dimensions, which is what a manufacturer's purlin usually needs. End bays and interior bays each take a section, of one depth and profile: the webs lap. Lip length is measured along the lip from the outer flange line. A Z's lip angle is 50° on the Manual's sections.
• Web: punched or unpunched. Purlins are usually unpunched, and that is prefilled. The sheet states which. S100's web hole provisions in the Effective Width Method are for C-sections, so a punched C is checked (F3.1.2, F4.2, G3, G6) and a punched Z is not feasible here. Design that one by the Direct Strength Method or tests.
• Spans and laps: per span, its length and how far a lap runs into it from each support. Ends are pinned. With no lap, the two spans must be one section, continuous over the support.
• Laps bolted for continuity: you confirm it. The analysis takes it.
• Roof system: through-fastened or standing seam.
• kφ (through-fastened): the panels' rotational restraint for distortional buckling, from tests or the manufacturer. The prefilled 0.300 kip-in./rad/in. is the MBMA example's, not a value for your panel.
• R for through-fastened uplift: Table I6.2.1-1, with conditions (i) to (n) confirmed by you, or entered from full-scale tests by K2.1.
• R (standing seam): gravity and uplift, end and interior bays, from AISI S908 base tests of your panel, clip and purlin. The prefilled values are the MBMA example's.
• Effective length factor, Z-section Fcre, Cb, β: the code's choices. Eq. F2.1.3-2 and Cb of 1.67 are prefilled, as the Guide uses them, and both are on the safe side. β is by Eq. 2.3.3.3-3, or 1.0.
• Supports: bearing length and whether the flanges are fastened, or web plates or clips, which take away G5 and H3. Past 1.3 for thicker to thinner, H3 takes the thicker section at 1.3 t or twice the thinner, as the Guide allows.
• Loads: on this line, parallel to the web, with self-weight in the dead load. Wind acts as entered: a negative row is uplift, a positive row inward pressure, each a case of its own. ASCE 7 gives them as separate pressures, so neither sign is assumed from the other.
• Patterns: L by ASCE 7-22 4.3.3 and snow by 7.5.1 are on. Lr by 4.3.3.1 is off: that section asks for it where the roof live load is reduced below 20 psf, so turn it on if yours is.
• Deflection limits: your entries. IBC Table 1604.3's values for roof members with no ceiling are prefilled.
• Anchorage: I6.4.1. Purlin count and spacing, top flange direction, roof slope, the purlins with a device, its stiffness and strength, the panel area and diaphragm stiffness.

Auto-design varies the listed sections of the given profile and depth, end bays and interior bays. The spans, laps, loads, steel, roof and its R values and the anchorage are kept. A custom section is kept as entered.

Checks performed #

A negative moment region next to an interior support takes Chapter F, since S100 says I6.2.1 does not apply there. A negative moment region not next to one takes the R method. That is the geometry of the diagram, whichever combination made it.

Limitations #

• Lapped regions rest on assumptions. S100 gives no method of analysis for laps. Ix over a lap is the sum of the two sections, the strength in a lap is the sum of their strengths, and a lap is taken as restrained against lateral-torsional and distortional buckling. These are the MBMA Guide's assumptions and they are yours to accept. The results state them.
• The section list is not from a source on hand: confirm your section against the maker's data. The list of depths and flange widths follows the AISI Design Manual's names from memory, and its tables are paid. The check results and the sheet say so whenever a listed section is used. The lips of the C-sections at x085 and x105 have nothing on hand to check them at all. Custom dimensions take your maker's numbers and carry no such line.
• The section list's lip lengths are not all printed in what we have. The AISI Design Manual's tables are paid. Three are printed in free reports, three are read back from printed gross properties and land on all of them, and C-section lips at x085 and x105 are carried on from the others with nothing to check them. Metal building purlins differ from the Manual's in any case: for a real purlin, enter its dimensions.
• Not built: S100 I6.1, the analytical route for standing seam. Discrete bracing of a standing seam roof (the Appendix A alternative). Overhangs and cantilevers. Point loads and loads on part of a span. Axial load in strut purlins (I6.2.3, I6.2.4). Mid-point and third-point anchorage over several spans: those lines average three bays and have rows of their own, and the check says NOT FEASIBLE rather than guess. I6.4.2 torsional bracing.
• H3 past the limits of Eq. H3-3: S100 gives no equation for nested Zs out there. This calculation uses the stricter Eq. H3-1 on the summed strengths, and says so. That choice is ours.
• R by Table I6.2.1-1 with bays varying more than 20 percent: simple span values on all bays, with r on them. S100 names the adjacent bays. Taking all of them is on the safe side.
• Bolts in the laps and at the supports, anchorage devices, panels and their fasteners, and I6.3 are not designed.

Verification examples #

Verification status: 183 values compared in 12 examples (12 published, 0 hand calculations from the cited equations): 173 within tolerance, 10 recorded deviations. Code basis: the provisions it uses were read in the target edition.

From the free MBMA Roof Framing Design Guide, 2nd Ed. (2024), with printed page numbers. The Guide takes section properties from the paid Design Manual; here they are computed and compared with what it prints.

• Example 3.3.1.1, pp. 64 to 79: four span Z, through-fastened, ASD. Section properties, every moment, shear and reaction of the non-prismatic analysis, F3, F4 with kφ, F2.1.3, the laps, shear, H2, G5, H3, I6.2.1.
• Example 3.3.2.1, pp. 93 to 104: the same line, standing seam, ASD.
• Example 3.3.2.2, pp. 106 to 118: four span C, standing seam, LRFD.
• Examples 5.4.1, 5.4.2, 5.4.3, pp. 193 to 204: anchorage at third points and at the supports. Example 5.4.4, pp. 205 to 207: opposed C-sections, on the library, in the behaviour tests, as its tributary widths are unequal.
• AISI RP20-5 p. 5, RP21-01 pp. 5, 19: gross properties and lips.
• Hand calculations: the non-prismatic analysis against the three-moment equation, the Z column of Table 2.3.1.3-1 against its C column at 90°, the R table, r and the 20 percent rule.

Where they part from the Guide, each is recorded plainly on the Verification page:

• The half-wavelength of a Z-section comes out about 2 percent longer than the Manual's table. The buckling stress lands on the printed value.
• The Guide's Example 3.3.2.2 torsional stress leaves the square off the polar radius, 452 ksi for about 116 ksi. Here the elastic stress is 262 ksi for its 542 ksi. No strength changes in that example.
• The Guide's LRFD uplift takes 1.6 wind though it names ASCE 7-16. Here it is ASCE 7-22, 1.0 wind.
• The Guide's Example 3.3.1.1 H3 prints 1.00 at or below 0.97 and calls it OK. It is not. Here that check, with the Guide's twice-the-thinner choice, fails at 1.03. With the thicker section at 1.3 t, which the Guide also allows and which is prefilled, it passes.
• Not verified by any published value: S100 F2.1.3 Eq. -1, the 20 percent rule and r, a punched C, deflection (no printed value), anchorage at mid-span, top flange movement at third points, and the partial snow cases.