Portal Frame Worked Example — Rigid Steel Frame per AISC 360 LRFD

Problem: Design a rigid steel portal frame for a 50 ft wide × 120 ft long industrial warehouse. The frame has a clear height of 20 ft at the eave and a 4-on-12 roof pitch (18.4°). Design the columns and rafters for gravity loads (dead + live), wind loads, and seismic loads per AISC 360-22 LRFD and ASCE 7-22. Frames are spaced at 25 ft on center.

Step 1: Frame Geometry and Loading

Step 2: Gravity Loads

Dead loads:

• Metal roof panel + insulation: 2.5 psf
• Purlin + girt system: 2.5 psf
• Collateral (MEP, ceiling): 3.0 psf
• Self-weight of frame: assumed 15 psf initially
• Total dead load (D): 8.0 psf (excluding frame self-weight)

Tributary width = 25 ft

Uniform dead load on rafter (horizontal projection):

w_D = 8.0 psf × 25 ft = 200 plf (0.200 kip/ft)

Roof live load (Lr): 20 psf

w_Lr = 20 psf × 25 ft = 500 plf (0.500 kip/ft)

For LRFD: wu = 1.2 × 0.200 + 1.6 × 0.500 = 0.240 + 0.800 = 1.040 kip/ft

Vertical reaction at column base from gravity: V_gravity = 1.040 × 50 / 2 = 26.0 kips

Step 3: Preliminary Member Sizing

Rafter: Use the simply-supported moment as an initial estimate:

M_gravity = w × L² / 8 = 1.040 × 50² / 8 = 325 kip·ft

Required Zx ≈ M / (ϕb × Fy) = 325 × 12 / (0.90 × 50) = 86.7 in³

Try W21x44 (Zx = 95.4 in³, Ix = 843 in⁴) for rafters.

Column: Use W12x65 (A = 19.1 in², Zx = 96.8 in³, rx = 5.28 in, ry = 3.02 in). AISC Manual Table 4-1 for W12x65, KL = 20 ft → ϕcPn ≈ 614 kips (Fy = 50 ksi).

This is preliminary — frame action will produce moments at the column top that must be checked.

Step 4: Wind Loads (ASCE 7-22 Directional Procedure)

For a 120 mph (Risk II, Exposure C), the simplified wind calculation:

Velocity pressure: qh = 0.00256 × Kz × Kzt × Kd × V²

For Exposure C, at mean roof height h ≈ 24 ft: Kz = 0.85 (Table 26.10-1) Kzt = 1.0, Kd = 0.85

qh = 0.00256 × 0.85 × 1.0 × 0.85 × 120² qh = 0.00256 × 0.85 × 0.85 × 14,400 qh = 26.6 psf

Windward wall pressure (MWFRS):

p_windward = qh × G × Cp = 26.6 × 0.85 × 0.80 = 18.1 psf

Leeward wall pressure:

p_leeward = 26.6 × 0.85 × (-0.30) = -6.8 psf

Roof pressure:

For 18.4° slope, Cp ≈ -0.70 (leeward side) and -0.90 (windward near edge)

Simplified: Roof net pressure ≈ 26.6 × 0.85 × (-0.70) = -15.8 psf (uplift)

Wind load on frame (per frame line at 25 ft spacing):

Column wind load (windward): 18.1 × 25 / 1000 = 0.453 kip/ft (distributed along column)

Column wind load (leeward): -6.8 × 25 / 1000 = -0.170 kip/ft

Roof uplift: -15.8 × 25 / 1000 = -0.395 kip/ft along sloped length

Step 5: Frame Analysis Moments (Approximate)

Using a rigid-jointed frame analysis with the combined gravity + wind:

Gravity-only case (1.2D + 1.6Lr):

• Midspan rafter moment: 325 kip·ft (max positive moment)
• Column top moment: 190 kip·ft (fixed-end action from rafter continuity)
• Column base moment: 95 kip·ft (assuming pinned base — design intent)
• Column axial: 26.0 kips

Gravity + wind case (0.9D + 1.0W):

Wind load produces lateral drift that induces additional moments. The portal frame action (column + rafter forming a rigid closed frame) distributes wind moments as follows:

• Windward column moment at knee: 240 kip·ft (governs — wind adds to gravity)
• Leeward column moment: 185 kip·ft
• Rafter moment at knee: 240 kip·ft
• Midspan rafter moment: 280 kip·ft (wind uplift reduces gravity moment)

Second-order check: The frame must satisfy AISC 360 Chapter C stability requirements. For Δ_2nd/Δ_1st < 1.5, the Direct Analysis Method (Appendix 7) with reduced stiffness can be used.

Step 6: Check W21x44 Rafter

At knee (combined moment + axial):

Mu = 240 kip·ft, Pu = 18.0 kips (reduced axial due to uplift)

Check AISC H1-1 Interaction:

Pr/Pc = 18.0 / (ϕcPn) — but note the rafter is in TENSION due to wind uplift in this combination.

Use a conservative approach: For the 1.2D + 1.6Lr case:

Mu = 325 kip·ft, Pu_rafter = 8.7 kips (compression)

ϕbMp = 0.90 × 50 × 95.4 / 12 = 357.8 kip·ft

Mu/ϕbMn = 325/357.8 = 0.91 → OK (91% utilized)

Check Lb: Rafter is braced by purlins at 5 ft spacing. For W21x44, Lp = 6.47 ft (from Manual). Lb = 5 ft < Lp → Plastic moment capacity is achieved.

Deflection check (D+L):

Δ_vertical = 5 × w × L⁴ / (384 × E × I) = 5 × 0.700 × (50×12)⁴ / (384 × 29,000 × 843) = 5 × 0.700 × (5.184 × 10⁹) / (9.396 × 10⁹) = 1.93 in

L/240 = 50 × 12 / 240 = 2.50 in → 1.93 < 2.50 → OK

Step 7: Check W12x65 Column

At knee (combined):

Pu = 26.0 kips (gravity) + wind contribution = 32.0 kips Mu_column_top = 240 kip·ft

KL = 20 ft, ϕcPn from AISC Table 4-1 ≈ 614 kips (for KL = 20 ft, W12x65)

Pr/Pc = 32.0/614 = 0.05 < 0.20 → Use AISC Equation H1-1b:

Pr/(2Pc) + (Mrx/Mcx + Mry/Mcy) ≤ 1.0

Mrx/Mcx = 240 / 357.8 = 0.67

0.05/2 + 0.67 = 0.03 + 0.67 = 0.70 → OK (70% utilized)

Drift check (service wind):

For a rigid frame with pinned column bases, lateral drift can be significant. Using approximate portal method:

Δ_lateral = H × h³ / (12 × E × I_column × number_of_bays)

For windward column load of 0.453 kip/ft over 20 ft = 9.06 kips total wind shear:

Simplified: Δ ≈ 9.06 × (20×12)³ / (3 × 29,000 × 533 × 2 columns) = 9.06 × 13,824,000 / (92,742,000) = 1.35 in

Drift index = 1.35 / (20×12) = 1/178

ASCE 7-22 limits: H/400 = 0.6 in for building content damage (service wind). At 1.35 in, the frame drift exceeds the recommended limit. Options:

• Increase column size to W12x87 (Ix = 740 in⁴)
• Add a knee brace
• Use fixed column bases (requires moment-resisting base plates)
• Use a braced bay at intermediate column line

Step 8: Connection Design

Knee connection (rafter-to-column):

The critical connection must transfer Mu = 240 kip·ft and Vu = 26.0 kips.

Option: Bolted end-plate connection or welded flange plates.

For a welded connection (field weld with backup bars):

• Top flange: Full-penetration groove weld — capacity equals flange strength
• Bottom flange: Full-penetration groove weld
• Web: Shear tab with 3/16 in fillet welds

Flange force = M / (d - tf) = 240 × 12 / (21.1 - 0.450) = 2,880 / 20.65 = 139.5 kips

Flange thickness check (W21x44): bf = 6.50 in, tf = 0.450 in Flange yield capacity: ϕFy × bf × tf = 0.90 × 50 × 6.50 × 0.450 = 131.6 kips

131.6 kips < 139.5 kips → Flange stress exceeds yield near the connection. Use stiffeners or doubler plates, or increase the rafter to W21x50 (tf = 0.535 in, ϕ = 156.5 kips → OK).

Base plate connection (column to foundation):

Designed as a pinned base (no moment transfer at base). Use a base plate with slotted holes to minimize moment transfer.

Final Member Summary

Try the Calculator

Use the Portal Frame Calculator to analyze and design rigid steel frames for your own span, height, loading, and code requirements. Supports AISC, AS 4100, EN 1993, and CSA S16.

Frequently Asked Questions

How do I account for frame stability in portal frame design? AISC 360 Chapter C requires stability design per the Direct Analysis Method (Appendix 7) or Effective Length Method (Appendix 8). The Direct Analysis Method is preferred: it applies a 0.80 factor to member stiffness (EI* = 0.80EI) and adds notional lateral loads (0.002Yi) to account for initial imperfections. Second-order analysis (P-Δ and P-δ) is then required.

What is the typical drift limit for rigid portal frames? ASCE 7-22 Table CC-1 recommends H/400 for building content damage (wind) and 0.020hsx for seismic (life safety). For industrial buildings with overhead cranes, the crane manufacturer typically specifies stricter limits: H/500 to H/1000 for horizontal crane runway deflections.

When should I use pinned versus fixed column bases for portal frames? Pinned bases are simpler, cheaper, and avoid moment transfer to the foundation. They are common for industrial buildings where drift is not critical. Fixed bases reduce lateral drift by 40-60% but require larger foundations and moment-resisting base plates. For buildings with overhead cranes, fixed bases are often necessary to meet drift limits.

How do I handle second-order (P-Δ) effects in portal frame design? All frames with significant gravity loads must be checked for P-Δ effects. The AISC stability coefficient B2 = 1/(1 - ΣP×Δoh/ΣH×L) amplifies the first-order drift and moments. If B2 > 1.5, the frame is too flexible and must be stiffened. Direct second-order analysis (geometric nonlinearity) is the most accurate method and is required by AISC 360 for the Direct Analysis Method.

See Also

• Column Capacity Calculator
• Column Buckling Guide
• Column Design Guide
• Effective Length Reference