# Light LRFD Structural Check — 165 ft Round PEB (A36 Redesign)

**Standard:** AISC 360-16 LRFD
**Steel:** ASTM A36/A36M — `Fy = 250 MPa`, `Fu = 400 MPa`, `E = 200000 MPa`
**Φ factors:** `Φ = 0.9` (axial / flexure), `Φ = 0.75` (shear)
**Status:** ⚠️ **PRELIMINARY** — wind (110 mph, Exp C) and connection checks NOT yet analyzed.

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## Verdict: **PASS** (PRELIMINARY)

- All individual member demand/capacity ratios are **< 1.0**.
- **Peak utilization = 74.5 %** → long rafter flexure (ISMB 600).
- No member FAILS. The long rafter is the governing member and sits in **MARGINAL-adjacent** territory because the axial thrust interaction and wind are not yet included (see Flags).

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## 1. Perimeter Columns — ISMB 600 (`Ag = 18750 mm²`, `r_min = 60 mm`, `h = 5.486 m`)

| Quantity | Value |
|---|---|
| `KL/r` (K=1, Ly=5486) | 91.43 |
| `λc = (KL/r)·√(Fy/π²E)` | 1.029 |
| `λc > 0.8` ⇒ `ΦPn = Φ·Ag·Fy·0.658^λc²` | 2708 kN |
| `Pu` (trib 99.3 m² @ 3.02 kPa) | 307 kN |
| **Flexural-buckling util.** | **11.3 %** ✅ |

## 2. Internal Columns — ISMB 600 (`h = 6.748 m`)

| Quantity | Value |
|---|---|
| `KL/r` (Ly=6748) | 112.47 |
| `λc` | 1.266 |
| `ΦPn` | 2158 kN |
| `Pu` (trib 124.2 m² @ 3.02 kPa) | 384 kN |
| **Flexural-buckling util.** | **17.8 %** ✅ |

## 3. Long Rafters — ISMB 600 (20 off, `L = 12.637 m`, pinned-pinned)

| Check | Capacity | Demand | **Util.** |
|---|---|---|---|
| Flexure `ΦMn = Φ·Zx·Fy = 0.9·2820e3·250` | 634.5 kN·m | Mmax = wL²/8 = 473 kN·m | **74.5 %** ✅ (governing) |
| Shear `ΦVn = Φ·0.6·Fy·Aw = 0.75·0.6·250·4275` | 480.9 kN | V = wL/2 = 150 kN | **31.2 %** ✅ |

`w = 3.02 kPa × 7.86 m trib width = 23.7 kN/m`

## 4. Short Rafters — ISMB 600 (4 off, `L = 11.424 m`)

| Check | Demand | **Util.** |
|---|---|---|
| Flexure (M = 23.7·11.424²/8 = 386 kN·m) | 386 kN·m | **60.8 %** ✅ |
| Shear (V = 23.7·11.424/2 = 135 kN) | 135 kN | **28.1 %** ✅ |

## 5. Apex Ring — CHS 406.4 × 12.5 (`A = 15468 mm²`, `122 kg/m`, `length 7.66 m`)

| Quantity | Value |
|---|---|
| `A = π/4·(406.4² − 381.4²)` | 15468 mm² |
| `ΦPn(comp) = Φ·A·Fy` | 3480 kN |
| Roof thrust into ring (≈ V/tanθ, θ≈0.1) | ~1500 kN |
| **Util. if 100 % to ring** | **43.1 %** ✅ (thrust shared among segments ⇒ actual local util. lower) |

Adequate on a simple hoop check; distributed 3D ring analysis pending.

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## Utilization Summary

| Member | Governing mode | Util. | Result |
|---|---|---|---|
| Perimeter column | Flex-buckling | 11.3 % | PASS |
| Internal column | Flex-buckling | 17.8 % | PASS |
| Long rafter | Flexure | **74.5 %** | PASS (MARGINAL-adjacent) |
| Short rafter | Flexure | 60.8 % | PASS |
| Apex ring | Hoop comp | 43.1 % (if all thrust) | PASS |

**Peak utilization = 74.5 %** (long rafter flexure).

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## Flags / Open Items (why PRELIMINARY)

1. **Long-rafter axial thrust not in interaction.** Rafter thrust (~1500 kN class) is carried by the apex ring but also induces axial compression in the rafters themselves; the flexural check above is moment-only. With axial + moment interaction (`Φ·Pn/ΦPn + ...`), the 74.5 % flexural util. will rise — likely into MARGINAL (80–100 %) range. Full interaction check required.
2. **Wind (110 mph, Exposure C) not analyzed.** Windward/leeward pressure, uplift, and asymmetric load cases can govern the ring and rafters; could raise peak util. materially.
3. **Connections not checked** — column base plates, rafter apex welds/bolts, ring-to-rafter splices, and CHS brace/ring connectivity pending.
4. **Local buckling / slenderness of ISMB 600 web & flange** under combined axial+flexure not verified (compact-section assumption to be confirmed for A36).
5. **Snow = 0, LL = 60 psf** assumed; confirm serviceability (deflection L/240 rafter) separately.

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## Conclusion

**PASS** on a light LRFD screen — every member ratio is under unity with a comfortable reserve except the **long rafter at 74.5 %**, which is the controlling element. Treat the result as **PRELIMINARY**: once axial-thrust interaction and wind are folded in, the long rafter is expected to approach (and could exceed) the MARGINAL threshold, and a full 3D analysis of the ring/rafter interaction is needed before final sign-off.

*Generated by light LRFD screen — calculation only, no web references. Verify all inputs against the controlling design drawings before construction.*