Castellated or cellular beam
A simply supported W-shape cut and rejoined with hexagonal (castellated) or circular (cellular) openings, noncomposite or composite, symmetric or from two root beams, by AISC Design Guide 31 (2017), Chapter 3, with AISC 360-22. The result is the tee forces at every opening for every load combination, the Vierendeel interaction, web post buckling, horizontal and vertical shear, and deflection. The guide's method is written to the 2016 specification; the equations it uses from Chapters D to H are the same in 2022 for these checks.
This chapter departs from the guide's printed examples on one safety matter: read Verification examples below.
Inputs of the castellated or cellular beam #
- Root beam: the W-shape that is cut. Asymmetric sections take a second root beam below (2.2.6); the spacing is common to both. The steel is the same in both.
- Castellated pattern: post width e, horizontal cut b, tee depth dt. The rest follows: h = d - 2 dt, dg = ho + 2 dt, S = 2e + 2b, cut angle = atan(h / b). The web post equations apply only for angles from 43° to 62°, e/tw from 10 to 30 and 2h/e up to 8. Outside these limits the calculation stops with NOT FEASIBLE.
- Cellular pattern: diameter Do and spacing S. The web post equations apply only for S/Do from 1.08 to 1.5 and dg/Do from 1.25 to 1.75.
- First opening: the distance to its centerline; 0 centers the pattern on the span.
- Loads: line loads over the whole span and up to 6 point loads on the top flange, by load type, with the ASCE 7-22 combinations. Self-weight is the average of the root beams. Gravity loads only.
- Bracing: noncomposite: Lb, 0 for a flange braced by the deck, and Cb. On a composite beam the deck is taken to brace the top flange (3.6).
- Composite: beam spacing and position for the effective width (Eqs. 3-4 to 3-6), f'c, unit weight, rib height hr, concrete above the ribs tc, the studs over the span Ns and Qn, your value with the rib reduction by AISC 360-22 I8.2a. The studs are spread evenly, as the guide's q = Ns Qn / span. Dead load on the steel alone splits D for the deflections only.
- Tee torsional buckling: with the Cw term omitted (the E4 User Note, lower) or with the tee's own Cw (Eq. E4-7 as written). Which you use is your decision; the first is prefilled.
- Web post curve (castellated): the guide's text interpolates Mocr/Mp by e/tw. Its examples use the e/tw = 10 curve, which is lower. The text is prefilled.
- Deflection limits: yours, span / n; 0 skips the check.
Auto-design varies the root beam, lightest first, with optional depth limits, and scales the opening pattern to its depth; the steel when unlocked. A second root beam, the loads, the span and the studs stay as you entered them.
Checks performed #
| Check | Basis |
|---|---|
| Tee force | Pr = Mr / deffec, Eq. 3-1; composite: Eqs. 3-8 to 3-13, the stress block depth iterated at each opening, and To and T1-new where the studs up to that opening fall short |
| Vierendeel moment | Eqs. 3-2, 3-3; composite Eqs. 3-16 to 3-18, the shear net of the deck's Vc, Eqs. 3-14, 3-15; shared by tee area |
| Tee compression | E3, E4, E7 on L = e or Do/2, Kx = 0.65, Ky = 1.0 (3.2.2.1), with the true shear center |
| Bottom tee | Noncomposite: treated as compression, as the guide allows (3.2.2). Composite: tension, D2 |
| Tee flexure | F9 on Lb = e or Do/2: stem in compression, and flange in compression for the reversal over the post |
| Interaction | H1-1a, H1-1b at every opening and midspan, for every combination |
| Concrete | Stress block within tc |
| Web post | Castellated Eqs. 3-19 to 3-29, with φ and Ω by the cut angle (Fig. 3-8), top and bottom separately; cellular Eqs. 3-30 to 3-37. Between 47° and 58° the two equation sets are interpolated on the angle, our reading of 3.4 |
| Horizontal shear | J4.2 on e tw, the thinner web |
| Vertical shear | G2 at the net section, per tee, kv = 1.2; at the gross section, kv = 5.34, on the thinner web |
| Whole beam | Noncomposite: Fy Zx-net, and F2 to F5 on the gross section with Lb and Cb (3.6) |
| Point loads | J10.1, J10.2, J10.3, and J10.4 taking the flange as not restrained against rotation, on the top root beam (3.8). An advisory where a load lies over an opening |
| Deflection | 0.90 × Ix-net, transformed when composite (3.7); composite: live and dead after set against the limits, as the examples; the dead load on the steel alone is reported, for camber |
Limitations of this calculation #
Simple spans and gravity loads only. Not covered: cantilevers, continuous spans, uplift; tapered or curved beams; infills and large copes (2.3); fatigue and dynamic load, where the re-entrant corners of castellated beams are a concern (3.1); fire (2.4.4); vibration (the floor vibration calculation).
The pre-composite stage is not checked: run a noncomposite calculation on the wet load and construction load, with the unbraced length then in force. The first web post at each end is not checked: the guide's examples assume the connection keeps it from buckling. Erection bracing is usually set by the manufacturer (2.4.3). The guide's opening spacing has no match with the stud spacing or the deck: the studs are spread evenly. J10 assumes a continuous web at the load: over an opening, design the infill.
Verification examples #
Verification status: 263 values compared in 11 examples (11 published, 0 hand calculations from the cited equations): 228 within tolerance, 35 recorded deviations. Code basis: the provisions it uses were read in the target edition.
AISC Design Guide 31 Examples 4.1 to 4.4, LRFD and ASD: section properties, forces at the openings, tee strengths, interaction, web posts, shear, deflection, and Table 4-19 for partial composite action. Run on the examples' e/tw = 10 web post curve.
Tee compression: not matched on purpose (DG31-FTB). Examples 4.1 and 4.2 say the Cw term is omitted, yet their printed Fez still has π²E/L² in it, as if Cw were 1 in.⁶, and their top tees measure yo from the stem tip, not from the shear center at the flange. Example 4.3's bottom tee does it correctly.
| Example | Printed Fez | Module Fez, Cw omitted | Printed available Pc | Module available Pc | Printed largest ratio | Module, Cw omitted | Module, the tee's own Cw |
|---|---|---|---|---|---|---|---|
| 4.1 LRFD | 3,440 ksi | 89.5 ksi | 64.9 kips | 51.6 kips | 0.741 | 0.936 | 0.823 |
| 4.1 ASD | 3,440 ksi | 89.5 ksi | 43.2 kips | 34.3 kips | 0.815 | 1.03 | 0.904 |
| 4.2 LRFD | 678 ksi | 76.5 ksi | 65.6 kips | 51.7 kips | 0.759 | 0.975 | 0.911 |
| 4.2 ASD | 678 ksi | 76.5 ksi | 43.7 kips | 34.4 kips | 0.835 | 1.07 | 1.00 |
The printed strengths are about 26 % high, on the unsafe side. Under ASD, both beams the guide shows as adequate fail on tee compression here with the Cw term omitted; under LRFD they pass. With the tee's own Cw, Example 4.1 passes either way and Example 4.2 under ASD sits at 1.00. There is no option to use the printed values. It is recorded for the owner to raise with AISC; check for guide errata before relying on this reading.
Example 4.1's total deflection also reads as a small failure here: its own 2.68 in. is above span / 180 = 2.67 in., which it rounds to o.k.
Other recorded differences: H in Example 4.1 (DG31-H41); cellular web post ratio, about 6 % lower here, unrounded (DG31-ROUND); a printed horizontal shear in Example 4.4 (DG31-44-ASD); Example 4.4's Ix-net and composite Ix, 2,078 and 5,518 in.⁴ here against 1,710 and 5,100 printed, from the net section's own values (DG31-44-I); and the gross shear in its summary (DG31-44-VG).
By hand only, no published example: J10 point loads, the whole-beam LTB with Lb, angles between 47° and 58°, and the tee's own Cw.