Beam Design Guide — Engineering Reference

AISC 360 Chapter F beam design step-by-step: compact sections, Lp/Lr unbraced length, LTB, Cb factor, shear design, and deflection serviceability checks.

Overview

Steel beam design per AISC 360 Chapter F requires checking flexural yielding, lateral-torsional buckling (LTB), flange local buckling (FLB), and web local buckling (WLB). The designer selects a W-shape or built-up section, classifies it as compact, noncompact, or slender per Table B4.1b, then calculates the available flexural strength considering unbraced length and moment gradient.

Shear design per Chapter G and deflection serviceability checks complete the beam design workflow. For composite beams with concrete slabs, Chapter I governs the shear stud and effective width requirements.

Flexural strength and LTB

The nominal moment capacity depends on unbraced length Lb relative to two transition points:

• Lb <= Lp: Full plastic moment Mp = Fy _ Zx applies (compact sections). Lp = 1.76 _ ry * sqrt(E / Fy).
• Lp < Lb <= Lr: Inelastic LTB zone. Capacity transitions linearly from Mp down to 0.7 _ Fy _ Sx at Lr.
• Lb > Lr: Elastic LTB. Capacity given by the critical buckling stress Fcr using the exact expression from AISC Eq. F2-4 involving Cb, J, Sx, ho, and rts.

The moment gradient factor Cb accounts for non-uniform bending moment distributions. For uniform moment Cb = 1.0; for typical gravity loading on simply supported beams Cb ranges from 1.14 to 1.67 per Eq. F1-1.

Worked example — W18x50 simply supported beam

Given: W18x50, A992 (Fy = 50 ksi), span L = 30 ft, uniform load w_u = 2.5 kip/ft (factored), lateral bracing at midspan only (L_b = 15 ft). Properties: Z_x = 101 in^3, S_x = 88.9 in^3, r_y = 1.65 in., r_ts = 1.98 in., J = 1.24 in^4, h_o = 17.4 in.

1. Check compactness: b_f/(2t_f) = 7.50/(2 x 0.57) = 6.58 < 9.15 (compact). h/t_w = 16.86/0.355 = 47.5 < 90.6 (compact). Section is compact.
2. Plastic moment: M_p = F_y x Z_x = 50 x 101 = 5050 kip-in = 420.8 kip-ft.
3. L_p: L_p = 1.76 x r_y x sqrt(E/F_y) = 1.76 x 1.65 x sqrt(29000/50) = 1.76 x 1.65 x 24.08 = 69.9 in. = 5.83 ft.
4. L_r (from AISC Eq. F2-6): L_r ≈ 16.6 ft (calculated from r_ts, J, S_x, h_o, c).
5. Check LTB: L_b = 15 ft. Since L_p (5.83) < L_b (15.0) < L_r (16.6), inelastic LTB governs.
6. C_b factor: For uniform load with midspan brace, C_b ≈ 1.30 (quarter-point moment method).
7. Nominal moment: M_n = C_b x [M_p - (M_p - 0.7 x F_y x S_x) x (L_b - L_p)/(L_r - L_p)] = 1.30 x [5050 - (5050 - 0.7 x 50 x 88.9) x (15.0 - 5.83)/(16.6 - 5.83)] = 1.30 x [5050 - 1939 x 0.852] = 1.30 x 3398 = 4418 kip-in. But M_n cannot exceed M_p = 5050, so M_n = 4418 kip-in = 368.2 kip-ft.
8. Design strength: phi x M_n = 0.90 x 368.2 = 331.3 kip-ft.
9. Required moment: M_u = w_u x L^2 / 8 = 2.5 x 30^2 / 8 = 281.3 kip-ft. Since 281.3 < 331.3, OK.

Shear design

Web shear capacity per AISC 360 Chapter G: phi_v x V_n = 1.00 x 0.6 x F_y x A_w x C_v1, where A_w = d x t_w. For most rolled W-shapes with h/t_w <= 2.24 x sqrt(E/F_y) = 53.9 (A992), the web shear coefficient C_v1 = 1.0 and no transverse stiffeners are needed.

Continuing the example: V_u = w_u x L / 2 = 2.5 x 30 / 2 = 37.5 kip. phi x V_n = 1.00 x 0.6 x 50 x (18.0 x 0.355) x 1.0 = 191.7 kip >> 37.5 kip. Shear is OK by inspection.

Code comparison — beam flexural design

Deflection limits

Beam deflection is a serviceability check using unfactored service loads, not factored LRFD loads:

For the worked example above: service live load w_L = 1.2 kip/ft (assumed), delta = 5 x w_L x L^4 / (384 x E x I_x) = 5 x 0.1 x 360^4 / (384 x 29000 x 800) = 0.95 in. Limit = L/360 = 360/360 = 1.0 in. Since 0.95 < 1.0, deflection OK.

Beam selection tips

• Start with the Z_x required: Z_x,req = M_u / (phi x F_y). This gives the minimum plastic modulus assuming full lateral bracing (C_b = 1.0, compact section). Then verify LTB and adjust upward if needed.
• Use the AISC beam selection tables (Table 3-2) which list beams by Z_x. The table also shows L_p and phi x M_p for quick comparison.
• Deeper beams are more efficient — a W21x44 (Z_x = 95.4) weighs less than a W14x61 (Z_x = 102) for similar capacity, because the deeper section has a larger moment arm.
• Check strong-axis coping — if the beam is coped at the support, the reduced section must be checked for flexural yielding, lateral-torsional buckling of the coped tee, and block shear.

AISC 360-22 beam design procedure -- step-by-step

The following procedure provides a complete beam design workflow per AISC 360-22. Each step references the applicable code section and identifies the typical governing condition.

Step 1: Determine the required moment and shear

Mu = factored moment from LRFD load combinations (kip-ft)
Vu = factored shear from LRFD load combinations (kips)
Ms = service moment for deflection check (kip-ft)

Use the structure's load analysis (gravity, wind, seismic) to determine Mu and Vu at all critical sections. For simple-span beams, the maximum moment is at midspan and maximum shear is at the supports. For continuous beams, check moments at supports (negative) and midspan (positive), and shear at each support.

Step 2: Classify the section (compact, noncompact, slender)

Per AISC 360 Table B4.1b, check the flange and web slenderness ratios:

Flange: lambda = bf / (2 * tf)   (half-flange for I-shapes)
  Compact limit (lambda_p): 0.38 * sqrt(E/Fy) = 9.15 for Fy=50 ksi
  Noncompact limit (lambda_r): 1.00 * sqrt(E/Fy) = 24.1 for Fy=50 ksi

Web: lambda = h / tw
  Compact limit (lambda_p): 3.76 * sqrt(E/Fy) = 90.6 for Fy=50 ksi
  Noncompact limit (lambda_r): 5.70 * sqrt(E/Fy) = 137.3 for Fy=50 ksi

Most standard W-shapes (W4 through W44) are compact for Fy = 50 ksi. Noncompact sections include some very light W-shapes (W6x8.5, W8x10, W12x14) where the flange slenderness ratio exceeds the compact limit. Slender sections are rare among standard rolled shapes but may occur in built-up plate girders.

Step 3: Calculate the plastic moment capacity

For compact sections with adequate lateral bracing (Lb <= Lp):

Mp = Fy * Zx   (kip-in)
phi * Mp = 0.90 * Fy * Zx   (design strength)

The plastic section modulus Zx is found in the AISC Manual Table 1-1 for all standard shapes. For noncompact sections, the capacity is reduced per AISC Chapter F3 (flange local buckling) or F4 (web local buckling).

Step 4: Check lateral-torsional buckling (LTB)

Determine the unbraced length Lb (distance between lateral bracing points) and the transition lengths:

Lp = 1.76 * ry * sqrt(E/Fy)
Lr = 1.95 * rts * sqrt(E/(0.7*Fy)) * sqrt((J*c/(Sx*ho)) + sqrt((J*c/(Sx*ho))^2 + 6.76*(0.7*Fy/E)^2))

Then compare Lb to Lp and Lr:

• If Lb <= Lp: phiMn = phiMp (full plastic capacity, no LTB reduction)
• If Lp < Lb <= Lr: phiMn = phiCb*[Mp - (Mp - 0.7FySx)*(Lb-Lp)/(Lr-Lp)], limited to phi*Mp
• If Lb > Lr: phiMn = phiCb*(pi^2E/(Lb/rts)^2)sqrt(1 + 0.078Jc/(Sxho)(Lb/rts)^2)*Sx

The moment gradient factor Cb is calculated per AISC Eq. F1-1:

Cb = 12.5*Mmax / (2.5*Mmax + 3*MA + 4*MB + 3*MC)

Where MA, MB, MC are moments at the quarter-point, midpoint, and three-quarter point of the unbraced segment. Cb >= 1.0 for all cases.

Step 5: Check shear capacity

Per AISC Chapter G:

phi*Vn = 1.00 * 0.6 * Fy * Aw * Cv1    (kips)

Where Aw = d * tw and Cv1 depends on the web slenderness. For most rolled W-shapes with h/tw <= 2.24*sqrt(E/Fy) = 53.9 (A992), Cv1 = 1.0 and shear almost never governs the design of rolled shapes. Shear may govern for:

• Very short, heavily loaded beams (high Vu, low Mu)
• Plates girders with thin webs (Cv1 < 1.0)
• Beams with coped ends or web openings

Step 6: Check deflection (serviceability)

Using unfactored service loads, calculate the maximum deflection and compare to the applicable limit:

delta_max = 5 * w * L^4 / (384 * E * Ix)   (uniform load, simple span)
delta_limit = L / 360 (floor live load), L / 240 (total load), etc.

Deflection is often the governing criterion for longer-span beams. A beam that passes the strength check (steps 3-5) may fail the deflection check, requiring a heavier section with a larger moment of inertia Ix.

Step 7: Check web yielding, web crippling, and sidesway web buckling

Per AISC Chapter J10, verify the web at concentrated force locations (column reactions, point loads, beam framing connections):

Web local yielding (J10.2):  phi*Rn = 1.00 * Fy * tw * (N + 5*k)
Web crippling (J10.3):       phi*Rn = 0.75 * 0.80 * t_w^2 * [1 + 3*(N/d)*(t_w/t_f)^1.5] * sqrt(E*Fy*t_f/t_w)

Where N = bearing length, k = distance from outer face of flange to web toe of fillet. When the applied concentrated force exceeds the web capacity, bearing stiffeners or transverse stiffeners are required.

Quick sizing rules of thumb

Rules of thumb provide rapid preliminary member selection before detailed calculations. These are based on industry practice and provide a starting point for design verification.

Depth-to-span ratios

Weight estimation by span and loading

For preliminary cost estimating, the beam weight (lb/ft) can be estimated from the span and load. For typical office floor loading (100 psf total, composite construction):

Steel weight estimation for entire floors

For cost estimating of the complete structural steel frame:

Preliminary beam selection table by span and load

The following table provides direct preliminary beam selections for common design conditions. All beams are A992 (Fy = 50 ksi), simply supported, with composite action (where noted), and adequate lateral bracing.

Non-composite beams, uniformly distributed load

Composite beams (with shear studs), uniformly distributed load

Composite construction typically allows a reduction of 15-30% in beam weight compared to non-composite beams at the same span and loading. The savings come from the increased moment capacity provided by the concrete slab acting compositely with the steel beam through shear studs.

Worked example overview -- W21x44 floor beam

Given: W21x44, A992 (Fy = 50 ksi), span L = 28 ft, simply supported. Service loads: dead load = 0.75 kip/ft, live load = 1.00 kip/ft. Lateral bracing at midspan only (Lb = 14 ft). Composite with 3.25 in. lightweight concrete on 1.5 in. metal deck.

LRFD factored loads:

wu = 1.2 * 0.75 + 1.6 * 1.00 = 0.90 + 1.60 = 2.50 kip/ft
Mu = 2.50 * 28^2 / 8 = 245.0 kip-ft
Vu = 2.50 * 28 / 2 = 35.0 kips

Section properties: Zx = 95.4 in^3, Sx = 81.6 in^3, Ix = 843 in^4, ry = 1.26 in., J = 0.77 in^4, ho = 20.7 in.

Compact check: bf/(2tf) = 6.525/(2*0.450) = 7.25 < 9.15. h/tw = 18.83/0.350 = 53.8 < 90.6. Compact OK.

Lateral-torsional buckling check:

Lp = 1.76 * 1.26 * sqrt(29000/50) = 53.4 in. = 4.45 ft
Lr ~ 12.5 ft (calculated per Eq. F2-6)

Lb = 14 ft > Lr = 12.5 ft --> Elastic LTB zone
Cb = 1.30 (uniform load, midspan brace)

phi*Mn = 0.90 * Cb * Fcr * Sx
Fcr = [pi^2*E/(Lb/rts)^2] * sqrt(1 + 0.078*J*c/(Sx*ho)*(Lb/rts)^2)

Calculating with rts = 1.49 in.:

Lb/rts = 168/1.49 = 112.8
Fcr = [pi^2*29000/112.8^2] * sqrt(1 + 0.078*0.77*1.0/(81.6*20.7)*112.8^2)
    = 22.52 * sqrt(1 + 0.463) = 22.52 * 1.208 = 27.2 ksi

phi*Mn = 0.90 * 1.30 * 27.2 * 81.6 / 12 = 0.90 * 1.30 * 185.0 = 216.4 kip-ft

Wait -- Mu = 245.0 kip-ft > phi*Mn = 216.4 kip-ft. Non-composite beam fails.

But this is a composite beam, so Chapter I applies. The composite moment capacity
is significantly higher than the non-composite capacity. For a W21x44 with
full composite action (NA in the slab):
  phi*Mn,comp ~ 350-400 kip-ft (depending on stud count and slab effective width)
  >> Mu = 245.0 kip-ft. Composite beam OK.

Deflection check (service loads):

w_service = 0.75 + 1.00 = 1.75 kip/ft
delta = 5 * 1.75/12 * (28*12)^4 / (384 * 29000 * 843)
      = 5 * 0.1458 * 13,585,367,040 / (9,338,880,000)
      = 0.843 in.  (non-composite Ix)

For composite section, I_eff ~ 1.5-2.0 * Ix = 1,265 - 1,686 in^4
delta_comp = 0.843 / (1.5 to 2.0) = 0.42 to 0.56 in.
Limit = L/360 = 336/360 = 0.93 in.

0.42 - 0.56 in. < 0.93 in. --> Deflection OK.

This example shows how composite action transforms a non-composite beam that fails the LTB check into an efficient design with significant margin. The concrete slab provides continuous lateral bracing to the top flange, eliminating the LTB concern entirely for positive moment regions.

Common mistakes to avoid

• Using Z_x when the section is noncompact — noncompact flanges or webs require reduced capacity per AISC F3 or F4. Some lighter W shapes (e.g., W12x14, W6x8.5) have noncompact flanges for F_y = 50 ksi.
• Ignoring the C_b factor — conservatively using C_b = 1.0 for all cases can oversize beams by 15-40%. For a simply supported beam with uniform load, C_b = 1.14. For a beam with end moments only, C_b can reach 2.27 for single curvature, significantly increasing the LTB capacity.
• Checking deflection with factored loads — deflection limits apply to unfactored service loads. Using LRFD factored loads overstates deflections by 40-60% and leads to unnecessarily heavy beams.
• Not verifying web crippling and web yielding — at concentrated load points (columns, hangers, point loads), the beam web must be checked for local yielding (J10.2), crippling (J10.3), and sidesway buckling (J10.4). These checks often require bearing stiffeners.
• Assuming full lateral bracing when deck is not attached — metal deck provides lateral bracing to the top flange only when properly attached with puddle welds or screws. During construction (before deck placement), the unbraced length equals the beam span or the distance between temporary bracing.

Run this calculation

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• Section Properties Database

Related references

• Beam Sizes
• Beam Formulas
• How to Verify Calculations
• Lateral-Torsional Buckling
• Deflection Limits
• Composite Beam Design
• steel beam capacity calculator
• Beam Web Design
• Steel Crane Girder

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