Why does a cantilever beam deflect 9.6 times more than a simple span?

For a uniformly loaded cantilever, maximum deflection is wL^4/(8EI) compared to 5wL^4/(384EI) for a simple span. The ratio is 9.6. This dramatic difference means cantilevers often require deeper sections or shorter spans than simple spans for equivalent deflection performance. In the worked example, the 8 ft cantilever W16x26 deflects 0.105 in. versus only 0.011 in. if simply supported.

How is lateral bracing handled for cantilever beam bottom flanges?

Cantilever bottom flanges are in compression and typically unbraced because the slab is on top. AISC 360 Appendix 6 requires a bracing force of 0.02 x Mu/(h0 x phi) at each brace point. For the 8 ft W16x26 cantilever, the required bracing force is approximately 1.0 kip per cantilever. A perimeter spandrel beam at the tips provides the lateral restraint.

What deflection limits apply to cantilever balconies?

ASCE 7-22 Appendix C recommends L/240 for total load deflection on cantilevered pedestrian areas. The worked example achieves L/914 for total deflection and L/1,579 for live load only, both well within limits. Vibration performance (natural frequency 9.66 Hz) also exceeds the 3 Hz minimum per AISC Design Guide 11 for pedestrian comfort.

Cantilever Beam Design Worked Example — AISC 360 Lateral Bracing Deflection

AISC 360-22 Cantilever Beam Design — Worked Example

Complete step-by-step flexural design of a cantilever steel beam per AISC 360-22. Covers section classification, lateral-torsional buckling with bottom flange bracing, shear capacity, deflection (cantilever vs. simple span comparison), and serviceability limits. All calculations in US customary units.

Related pages: AISC Beam Design Guide | Deflection Guide | Beam Capacity Calculator | Lateral Torsional Buckling

Problem Statement

A steel-framed balcony extends 8 ft from the face of the building. The cantilever beams are W16x26 at 6 ft spacing, supporting a 4-1/2 in. normal-weight concrete slab on 2 in. composite steel deck (total 6-1/2 in.). The building is a 3-storey office located in a moderate wind region.

Design data:

Cantilever span: L = 8 ft
Beam spacing: s = 6 ft c/c
Beam: W16x26, ASTM A992 (Fy = 50 ksi, Fu = 65 ksi)
Deck: 2 in. deep, 20-gauge composite deck with 4-1/2 in. NWC topping
Loads: Dead = 58 psf (slab + deck + MEP + ceiling), Live = 100 psf (balcony, per ASCE 7-22 Table 4.3-1), Superimposed dead = 15 psf (railing + finishes)
Deflection limit: L/240 for total load (ASCE 7-22 Appendix C, pedestrian comfort)
Lateral bracing: Bottom flange unbraced (exposed), tip bracing via perimeter beam at 8 ft

Step 1 — Load Determination

Service loads (per beam):

Tributary width per beam = 6 ft

w_D = (58 + 15) psf x 6 ft = 438 plf = 0.438 klf (dead + superimposed)
w_L = 100 psf x 6 ft = 600 plf = 0.600 klf (live)

Factored load combination (ASCE 7-22, LRFD):

wu = 1.2 x 0.438 + 1.6 x 0.600 = 0.526 + 0.960 = 1.486 klf

Step 2 — Design Actions (Cantilever)

For a uniformly loaded cantilever of length L = 8 ft:

Maximum factored moment (at support): Mu = wu x L^2 / 2 = 1.486 x 8.0^2 / 2 = 1.486 x 32.0 = 47.6 kip-ft

Maximum factored shear (at support): Vu = wu x L = 1.486 x 8.0 = 11.9 kips

Step 3 — Section Properties (W16x26, A992)

Property	Value	Unit
Depth d	15.7	in.
Flange width bf	5.50	in.
Flange thickness tf	0.345	in.
Web thickness tw	0.250	in.
Plastic modulus Zx	44.2	in^3
Elastic modulus Sx	38.4	in^3
Moment of inertia Ix	301	in^4
Radius of gyration ry	1.12	in.
Torsional constant J	0.262	in^4
Warping constant Cw	565	in^6

Section classification per AISC Table B4.1b: flange lambda_f = 7.97 < lambda_pf = 9.15 (Compact). Web lambda_w = 60.0 < lambda_pw = 90.5 (Compact). Section is Compact for flexure.

Step 4 — Flexural Capacity (AISC 360 Chapter F)

Plastic moment (AISC Eq. F2-1): Mp = Fy x Zx = 50 x 44.2 = 184.2 kip-ft, phi_b x Mp = 165.8 kip-ft

Lateral-Torsional Buckling (AISC Section F2.2): The cantilever bottom flange is in compression and unbraced between support and tip brace. Lb = 8 ft = 96 in.

Lp = 1.76 x ry x sqrt(E/Fy) = 1.76 x 1.12 x 24.08 = 47.5 in. = 3.96 ft Lr approximately 12.2 ft for W16x26. Lp < Lb < Lr — inelastic LTB governs.

phi_b x Mn = Cb x [phi_b x Mp - (phi_b x Mp - 0.7 x Fy x Sx) x (Lb - Lp)/(Lr - Lp)]

For cantilever, Cb = 1.0 per AISC Manual Part 3.

phi_b x Mn = 1.0 x [165.8 - (165.8 - 112.0) x (8.0 - 3.96)/(12.2 - 3.96)] = 165.8 - 53.8 x 0.490 = 139.4 kip-ft

Flexural utilisation: Mu / (phi_b x Mn) = 47.6 / 139.4 = 0.341 (34.1%)

Step 5 — Shear Capacity (AISC 360 Section G2)

h/tw = 60.0. 2.24 x sqrt(E/Fy) = 53.9 < 60.0 — check shear buckling. kv = 5.34 for unstiffened web. 1.10 x sqrt(kv x E/Fy) = 61.2 > 60.0 — Cv = 1.0, shear yielding governs.

Aw = d x tw = 3.93 in^2, Vn = 0.6 x Fy x Aw x Cv = 117.9 kips, phi_v x Vn = 106.1 kips

Vu = 11.9 kips << 106.1 kips — OK, utilisation = 11.2%

Step 6 — Deflection Check (Serviceability)

Cantilever deflection: delta = w x L^4 / (8 x E x I)

w_total = 1.038 klf = 0.0865 kips/in. delta_total = 0.105 in. = L/914

Simple span comparison (same length, same load):

delta_simple = 5 x w x L^4 / (384 x E x I) = 0.0110 in. = L/8,760 Ratio: delta_cantilever / delta_simple = 0.105 / 0.0110 = 9.55 — consistent with theoretical 9.6x.

Live load deflection: w_L = 0.600 klf. delta_LL = 0.0608 in. = L/1,579

Deflection limits: L/240 (total) = 0.400 in. — OK (0.105 in.) L/360 (live) = 0.267 in. — OK (0.061 in.)

Step 7 — Lateral Bracing Detail

A perimeter spandrel beam (W12x22) at cantilever tips provides lateral brace points at 8 ft. Bracing force per AISC Appendix 6:

P_brace = 0.02 x Mu x Cd / (h0 x phi) = 0.02 x (47.6 x 12) / (15.4 x 0.75) = 0.99 kips per cantilever

With 10 bays: total bracing force = 9.9 kips. A 2x2x1/4 angle brace is adequate.

Step 8 — Vibration Check

Natural frequency: f_n = (1/(2 x pi)) x sqrt(g/delta_total) = 0.1592 x sqrt(386.4/0.105) = 9.66 Hz > 3 Hz — OK per AISC DG 11.

Summary of Checks

Limit State	Reference	Capacity	Demand	Utilisation
Flexure (LTB)	AISC F2-2	139.4 kip-ft	47.6 kip-ft	34.1%
Shear yielding	AISC G2-1	106.1 kips	11.9 kips	11.2%
Deflection (total)	L/240	0.400 in.	0.105 in.	26.3%
Deflection (live)	L/360	0.267 in.	0.061 in.	22.8%
Vibration	AISC DG 11	>3 Hz	9.66 Hz	OK

Conclusion: The W16x26 cantilever beam at 8 ft span satisfies all AISC 360-22 limit states, with flexure at 34% utilisation being the controlling strength check. The beam is deflection-limited — a shallower section would increase deflections and reduce vibration frequency.