# LRFD Structural Check — 165 ft Round PEB (PRELIMINARY)

**Project:** 165 ft dia circular marquee — Fusion Developers
**Standard:** AISC 360-16 LRFD
**Steel:** A572 Gr 50 — Fy = 345 MPa, Fu = 450 MPa, E = 200,000 MPa
**Prepared:** 04-Aug-2026
**Status:** ⚠️ **PRELIMINARY — final analysis with wind & connections required before construction**

---

## 1. Geometry & Members (confirmed)

| Item | Value |
|---|---|
| Diameter / radius (outer) | 165 ft / 82.5 ft (25.15 m) |
| Eave height | 18 ft (5.486 m) |
| Apex height | 26.25 ft (8.001 m) |
| Cone slope | 1:10 (tan θ ≈ 0.10) |
| Perimeter columns | 20 × ISMB 600, h = 5.486 m |
| Internal columns | 4 × ISMB 600, h = 6.748 m |
| Long rafters | 20 × ISMB 450, L = 12.637 m (pin–pin, perimeter→internal) |
| Short rafters | 4 × ISMB 450, L = 11.424 m (pin–pin, internal→apex) |
| Apex ring | CHS 406.4 × 12.5, 8 ft dia (2.438 m) |

**ISMB 600 section:** Ag = 18,750 mm², Zx = 2820 cm³, rx = 245 mm, ry = 60 mm, 133 kg/m
**ISMB 450 section:** Ag = 13,100 mm², Zx = 1690 cm³, Sx = 1870 cm³, 85.4 kg/m, d ≈ 450 mm, tw ≈ 9.5 mm
**CHS 406.4×12.5:** Ag = 15,468 mm²

**Φ factors:** φ = 0.90 (flexure / axial), φ = 0.75 (shear)

---

## 2. Loads (assumptions stated)

| Load | Value | Notes |
|---|---|---|
| Dead load (roof) | 0.147 kPa = 3.1 psf = 15 kg/m² | steel self-wt + PPGI 0.5 mm (~5) + purlin (~4.3) + misc (~5) |
| Live load (roof, assembly) | 2.87 kPa = 60 psf = 29.3 kg/m² | ASCE 7 assembly occupancy |
| **Total vertical w** | **3.02 kPa = 63.1 psf = 30.8 kg/m²** | DL + LL |
| Snow | 0 | Punjab |
| Wind | 110 mph, Exp. C | **FLAG: governing lateral — NOT analyzed here** (assume rigid diaphragm + perimeter bracing) |

**Unit conversions used:** 1 psf = 0.04788 kPa; 1 kN = 224.8 lbf.
(Sanity: 63.1 psf × 0.04788 = 3.02 kPa ✓; 307 kN × 224.8 = 69,014 lbf.)

---

## 3. Checks & Results

### 3.1 Perimeter Column — ISMB 600
- Tributary plan area per col = π·82.5² / 20 = **1,069 ft² = 99.3 m²**
- Pu = 99.3 m² × 3.02 kPa + self-wt 7.3 kN = **307 kN**
- Slenderness: K = 1.0, Ly = 5486 mm, rmin = 60 mm → **KL/r = 91.4**
- Fe = π²E/(KL/r)² = 236 MPa; λc = (KL/r)·√(Fy/E)/π = **1.21** (λc ≤ 1.5 → inelastic)
- φPn = φ·Ag·Fy·0.658^(Fy/Fe) = 0.9 × 18,750 × 345 × 0.658^1.461 = **3,158 kN**
- **Utilization = 307 / 3158 = 9.7 %** → ✅ **PASS**

### 3.2 Internal Column — ISMB 600
- Tributary = π·41.25² / 4 = **1,336 ft² = 124.2 m²**
- Pu = 124.2 × 3.02 + 9 kN = **384 kN**
- Slenderness: Ly = 6748 mm → **KL/r = 112.5**
- Fe = 156 MPa; λc = **1.49**
- φPn = 0.9 × 18,750 × 345 × 0.658^2.211 = **2,308 kN**
- **Utilization = 384 / 2308 = 16.6 %** → ✅ **PASS**

### 3.3 Long Rafter — ISMB 450 (L = 12.637 m)
- Tributary width = 99.3 m² / 12.637 m = **7.86 m**
- w = 3.02 kPa × 7.86 m = **23.74 kN/m**
- Mmax = wL²/8 = 23.74 × 12.637² / 8 = **473.8 kN·m**
- Vmax = wL/2 = 23.74 × 12.637 / 2 = **150.0 kN**
- φMn = 0.9 × (Zx·Fy) = 0.9 × (1690e3 × 345 /1e6) = 0.9 × 583.0 = **524.7 kN·m**
- φVn = 0.75 × 0.6 × Fy × Aw = 0.75 × 0.6 × 345 × (450×9.5) = **663.7 kN**
- Flexure util = 473.8 / 524.7 = **90.3 %** → ⚠️ **MARGINAL**
- Shear util = 150.0 / 663.7 = **22.6 %** → ✅ **PASS**
- **Governing = flexure 90.3 % → ⚠️ MARGINAL**

### 3.4 Short Rafter — ISMB 450 (L = 11.424 m)
- Same w = 23.74 kN/m, same section
- Mmax = 23.74 × 11.424² / 8 = **387.2 kN·m**
- Vmax = 23.74 × 11.424 / 2 = **135.6 kN**
- φMn = 524.7 kN·m; φVn = 663.7 kN
- Flexure util = 387.2 / 524.7 = **73.8 %** → ✅ **PASS**
- Shear util = 135.6 / 663.7 = **20.4 %** → ✅ **PASS**
- **Governing = flexure 73.8 % → ✅ PASS**

### 3.5 Apex Ring — CHS 406.4 × 12.5 (FLAGGED PRELIMINARY)
- Roof thrust per rafter at apex (task approximation): H = (wL/2) / tan θ = Vmax / 0.10
  - Long rafter: H ≈ 150.0 / 0.10 = **1,500 kN**
  - Short rafter: H ≈ 135.6 / 0.10 = **1,356 kN**
- Ring section compression capacity (low λ assumed): φPn = 0.9 × Ag × Fy = 0.9 × 15,468 × 345 = **4,803 kN**
- **Verdict: ⚠️ FLAGGED — PRELIMINARY.** The CHS 406.4×12.5 has ample axial capacity (4,803 kN) vs. an individual rafter thrust (~1,500 kN), and the actual distributed hoop demand is a fraction of the simple sum of thrusts. **Full 3D ring analysis (hoop force, local bending, ring/rafter connection) is pending.** Note: the 1:10 cone slope used here is approximate; the local rafter rise (eave 18 ft → apex 26.25 ft over a 41.25 ft run) is closer to 1:5, which would reduce the thrust magnitude — confirm in final analysis.

---

## 4. Summary Table

| Member | Check | Demand | Capacity | Util % | Verdict |
|---|---|---|---|---|---|
| Perimeter column ISMB 600 | Axial comp. | 307 kN | 3,158 kN | **9.7 %** | ✅ PASS |
| Internal column ISMB 600 | Axial comp. | 384 kN | 2,308 kN | **16.6 %** | ✅ PASS |
| Long rafter ISMB 450 | Flexure | 473.8 kN·m | 524.7 kN·m | **90.3 %** | ⚠️ MARGINAL |
| Long rafter ISMB 450 | Shear | 150.0 kN | 663.7 kN | 22.6 % | ✅ PASS |
| Short rafter ISMB 450 | Flexure | 387.2 kN·m | 524.7 kN·m | **73.8 %** | ✅ PASS |
| Short rafter ISMB 450 | Shear | 135.6 kN | 663.7 kN | 20.4 % | ✅ PASS |
| Apex ring CHS 406.4×12.5 | Hoop comp. | ~1,500 kN/raf | 4,803 kN | — | ⚠️ FLAGGED |

**Peak utilization across all members: 90.3 % (long rafter flexure).**

---

## 5. Conclusions

- **Columns (perimeter & internal):** comfortably PASS (≤ 16.6 %). Significant reserve.
- **Short rafter:** PASS at 73.8 % flexure — acceptable but not generous.
- **Long rafter:** ⚠️ **MARGINAL at 90.3 % flexure.** This is the governing member. Note this light check treated the rafter as pure flexure + shear; the real rafter also carries **axial compression from roof thrust**, which reduces flexural capacity via AISC Chapter H interaction — the long rafter may effectively approach or exceed 100 % in the final analysis. **Recommend:** upsize long rafter (e.g. ISMB 500/ISMB 600) OR reduce tributary (additional internal column ring) OR confirm lower governing live load.
- **Apex ring:** FLAGGED for full analysis; section capacity appears adequate but hoop force distribution unverified.

### ⚠️ Mandatory caveats
1. **PRELIMINARY** — final analysis with **wind (110 mph Exp. C, governing lateral)**, connection design, and 3D ring behaviour required before construction.
2. Rafter axial thrust / P-δ effects not included in flexure checks → long rafter likely conservative-worst in final.
3. Snow = 0 assumed (Punjab); revise if site snow load applies.
4. Cone slope taken as 1:10 per brief; local rafter geometry ~1:5 — confirm and re-run thrust.