Cantilever Beam — Deflection Formulas, Moment & Shear Diagrams

A cantilever beam is a structural member fixed (built-in) at one end and free at the other. The fixed support resists vertical reaction, horizontal reaction, and a moment, which makes the cantilever statically determinate. Cantilevers appear in balconies, canopies, overhanging floor beams, sign supports, and architectural features where a clear span is required at one end.

Because the maximum moment and shear both occur at the fixed support, cantilever design is governed by conditions at the wall or column connection. Deflection is always greatest at the free end and is a common serviceability concern -- cantilevers are significantly more flexible than simply supported beams of the same span.

Notation

Positive bending moment = sagging (tension on bottom). For cantilevers, the dominant moment is negative (hogging) at the fixed support.

Cantilever beam formulas

Case 1: Point load at the free end

              P
              |
              v
[=============|
Fixed         Free

Reactions:   R = P (upward at fixed end)
             M_fixed = P * L (hogging)

Shear force diagram (SFD): Constant positive value V = P from the free end to the fixed support. No variation.

Bending moment diagram (BMD): Linear, zero at the free end, maximum hogging (-PL) at the fixed support. The diagram is a straight line from 0 at the tip to PL at the wall.

Case 2: Uniform distributed load (UDL) over full span

w (force/length)
|||||||||||||||
[=============|
Fixed         Free

Reactions:   R = w * L (upward at fixed end)
             M_fixed = w * L^2 / 2 (hogging)

Shear force diagram (SFD): Linear, zero at the free end, maximum wL at the fixed support. The SFD is a straight line sloping from 0 to wL.

Bending moment diagram (BMD): Parabolic, zero at the free end, maximum hogging (-wL^2/2) at the fixed support. The curvature of the parabola opens upward (concave up when viewed from below).

Case 3: Triangular load (zero at free end, maximum at fixed end)

             w_max
             |||||
             ||||
             |||
             ||
             |
[=============|
Fixed         Free

Reactions:   R = w_max * L / 2 (upward at fixed end)
             M_fixed = w_max * L^2 / 6 (hogging)

Shear force diagram (SFD): Parabolic, zero at the free end, maximum at the fixed support. The load intensity increases linearly toward the wall, so the shear rate of change also increases.

Bending moment diagram (BMD): Cubic curve, zero at the free end, maximum hogging (-wL^2/6) at the fixed support.

Case 4: Applied moment at the free end

             M_0 (CCW moment)
             <--
[=============|
Fixed         Free

Reactions:   R = 0 (no vertical reaction)
             M_fixed = M_0 (hogging)

Shear force diagram (SFD): Zero everywhere (no transverse load).

Bending moment diagram (BMD): Constant hogging moment M_0 along the full span.

Superposition

For combined loading, cantilever responses superpose linearly (as long as the material remains elastic). For example, a cantilever with both a point load P at the free end and a UDL w over the full span:

Total deflection = P*L^3/(3*E*I) + w*L^4/(8*E*I)
Total fixed-end moment = P*L + w*L^2/2
Total reaction = P + w*L

Worked example — Steel cantilever beam

Problem: Design a steel cantilever beam to carry a 5-kip service live point load at the free end. Span L = 8 ft. Use W12x26, ASTM A992 steel. Check moment capacity, deflection, and shear per AISC 360-22 LRFD.

Material properties: A992 Grade 50: Fy = 50 ksi, Fu = 65 ksi, E = 29,000 ksi.

Step 1: Section properties (W12x26)

Compact section check (AISC Table B4.1b): Flange slenderness ratio = bf/(2tf) = 6.490/(20.380) = 8.54. Limit lambda_p = 0.38sqrt(E/Fy) = 0.38sqrt(29000/50) = 9.15. Since 8.54 < 9.15, the flange is compact. Web slenderness ratio = h/tw = (12.22 - 20.380)/0.230 = 49.8. Limit lambda_p = 3.76sqrt(E/Fy) = 3.76*sqrt(29000/50) = 90.6. Since 49.8 < 90.6, the web is compact. Section is compact -- plastic moment controls.

Step 2: Factored loads and moment capacity

Load: Service live load P_L = 5 kips. LRFD factored load: Pu = 1.6 * 5 = 8.0 kips.

Factored moment at fixed end:

Mu = Pu * L = 8.0 kips * 8 ft = 64.0 kip-ft

Nominal moment capacity (AISC F2.1):

For compact laterally supported sections:

phi * Mn = phi * Mp = phi * Fy * Zx
phi * Mn = 0.90 * 50 ksi * 37.2 in^3
         = 0.90 * 1860 kip-in
         = 1674 kip-in
         = 139.5 kip-ft

Lateral-torsional buckling check (AISC F2.2): The top (compression) flange at the support is laterally braced by the connection. For the free-end segment, Lb = 8 ft = 96 in.

Lp = 1.76 * ry * sqrt(E/Fy)

ry = sqrt(Iy/A) = sqrt(17.3/7.65) = 1.50 in (from AISC tables).

Lp = 1.76 * 1.50 * sqrt(29000/50) = 1.76 * 1.50 * 24.08 = 63.6 in = 5.30 ft

Since Lb = 8 ft > Lp = 5.30 ft, check LTB range. From AISC Table 3-2 for W12x26:

Since Lp < Lb < Lr, moment capacity with Cb:

For a cantilever with a point load at the free end, the moment diagram is linear from zero at the tip to Mu at the support. Cb per AISC F1:

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

With moment at 1/4 points: MA = 0.25Mu, MB = 0.50Mu, MC = 0.75*Mu, Mmax = Mu:

Cb = 12.5*Mu / (2.5*Mu + 3*0.25*Mu + 4*0.50*Mu + 3*0.75*Mu)
   = 12.5 / (2.5 + 0.75 + 2.0 + 2.25)
   = 12.5 / 7.5
   = 1.67

phi*Mn = Cb * [phi*Mp - (phi*Mp - phi*Mr) * (Lb - Lp)/(Lr - Lp)]  <= phi*Mp
phi*Mn = 1.67 * [139 - (139 - 84.7) * (96 - 63.6)/(184.8 - 63.6)]
       = 1.67 * [139 - 54.3 * 32.4/121.2]
       = 1.67 * [139 - 14.5]
       = 1.67 * 124.5
       = 208 kip-ft > phi*Mp = 139 kip-ft

Capacity capped at phi*Mp:

phi*Mn = 139 kip-ft > Mu = 64.0 kip-ft  --> OK (ratio = 0.459)

Moment capacity check: PASS. The beam has ample reserve (46% utilization).

Step 3: Deflection check

Service live load deflection (unfactored):

P_service = 5.0 kips
L = 8 ft = 96 in
I = 204 in^4
E = 29,000 ksi

delta = P * L^3 / (3 * E * I)
      = 5.0 * 96^3 / (3 * 29,000 * 204)
      = 5.0 * 884,736 / 17,748,000
      = 4,423,680 / 17,748,000
      = 0.249 in

Deflection limits for cantilevers:

Deflection check: PASS. Under live load only, the cantilever deflects 0.249 in, which satisfies the strict L/360 limit.

Step 4: Shear check

Factored shear at support:

Vu = Pu = 8.0 kips

Shear capacity (AISC G2.1): For W shapes with h/tw <= 2.24*sqrt(E/Fy):

h/tw = 49.8
2.24 * sqrt(29000/50) = 2.24 * 24.08 = 53.9

Since 49.8 < 53.9, shear yielding controls:

Aw = d * tw = 12.22 * 0.230 = 2.81 in^2

phi_v * Vn = phi_v * 0.6 * Fy * Aw * Cv1
phi_v = 1.00 (AISC)
Cv1 = 1.0 (compact web)

phi_v * Vn = 1.00 * 0.6 * 50 * 2.81 * 1.0
           = 84.3 kips

phi_v * Vn = 84.3 kips > Vu = 8.0 kips  --> OK (ratio = 0.095)

Shear check: PASS. Shear is far under capacity (9.5% utilization), which is typical for short-span cantilevers with moderate loads.

Design summary

The W12x26 is adequate for this cantilever. Deflection at L/360 governs the design (93% utilization), which is typical -- cantilevers are usually deflection-controlled, not strength-controlled.

Cantilever beam diagram conventions

Understanding the shape of shear and moment diagrams for cantilevers is essential for quick visual checks during design.

Sign convention

SFD and BMD shape summary

Key observation: For all common cantilever loading cases, the maximum moment and maximum shear occur at the fixed support. This is the opposite of simply supported beams, where maximum moment occurs near midspan.

Common cantilever applications

Connection considerations

The fixed end of a cantilever must resist the full moment and shear. Common connection types include:

• Welded end plate: Full-penetration groove weld of beam flanges to end plate, fillet weld of web. End plate bolted to supporting column with high-strength bolts in tension and shear. Design per AISC Design Guide 4 (End-Plate Moment Connections).
• Bolted flange plates: Top and bottom flange plates bolted or welded to column, web shear plate. Moment transfer through flange plates in tension and compression.
• Directly welded: Beam flanges groove-welded to column flange, web fillet-welded. Requires careful welding and inspection. Use of weld access holes per AISC Section M2.2.
• Seat and top angle: Simple connection for light cantilevers. Top angle resists the horizontal couple, seat angle bears vertical reaction. Limited moment capacity.

Deflection limits for cantilevers

Deflection limits for cantilevers use the span L (the cantilever length from support to free end). Some codes recommend using 2L as the equivalent simple-span length for comparison with standard beam deflection ratios, but the limit below is expressed against the actual cantilever span L.

Practical guidance

• Always check live load deflection separately from total deflection. Many engineers check only total and miss the live load criterion, which can govern for cantilevers with light dead loads.
• Use L/360 for cantilevers supporting brittle finishes (plaster, masonry, glass). The IBC permits L/180 as a minimum, but L/360 is standard practice when cracking is a concern.
• Camber does not fully solve cantilever deflection. Unlike simply supported beams where camber offsets dead load deflection, camber in cantilevers reverses the direction of dead load deflection. Use it carefully and only for dead load.
• Consider vibration for long cantilevers. Cantilevers with spans over 6 ft in occupied floors should have their natural frequency checked. A fundamental frequency below 3 Hz is perceptible and usually unacceptable.

Cantilever vs. simply supported beam comparison

A cantilever carrying the same point load at its tip as a simply supported beam at midspan will deflect 16 times more (PL^3/3EI vs PL^3/48EI) and carry 4 times the moment (PL vs PL/4). This is why cantilever spans are typically shorter than simply supported spans in the same structure.

Frequently asked questions

What is the maximum span for a steel cantilever beam?

There is no fixed maximum span -- it depends on the section size, loading, and deflection limits. In practice, steel cantilevers in buildings typically range from 3 ft to 12 ft. Spans beyond 12 ft are possible with heavier sections (W16, W18, W21) but deflection and vibration become governing concerns. For comparison, a W12x26 cantilever at 8 ft with a 5-kip point load deflects about L/385, while a W16x36 at 12 ft with the same load deflects about L/290.

Why is the moment negative (hogging) in a cantilever?

A cantilever deflects downward under gravity loads. This puts the top fibers in tension and the bottom fibers in compression at the fixed support. By convention, hogging (tension on top) is negative moment. The entire moment diagram for a gravity-loaded cantilever is in the negative (hogging) region.

How do I check lateral-torsional buckling of a cantilever?

Cantilevers are vulnerable to lateral-torsional buckling (LTB) because the compression flange (the bottom flange near the support for a downward-loaded cantilever) may not be laterally braced along its length. AISC Chapter F provides the LTB check. The effective length factor for a cantilever is typically taken as 1.0 or higher, and Cb for a cantilever with a point load at the free end is approximately 1.67. If LTB governs, adding a bottom flange brace near the support significantly increases capacity.

Can I use back-to-back channels for a cantilever?

Yes. Back-to-back channels (e.g., 2C8x11.5) bolted or welded together at regular intervals behave as a closed or partially closed section with much higher torsional stiffness than a single channel. This is common for canopy framing and sign supports where torsion from eccentric loads is a concern. Design per AISC Chapter F and H for combined bending and torsion.

What is the difference between a cantilever and a propped cantilever?

A cantilever has one fixed end and one free end (zero moment and zero shear at the free end). A propped cantilever has a fixed end and a roller/pin support at the other end, making it statically indeterminate to the first degree. The propped cantilever is stiffer and develops smaller moments and deflections than a simple cantilever of the same span and loading.

Do I need stiffeners at the fixed end of a cantilever?

Stiffeners are required when the concentrated force from the support reaction or the applied moment causes web local yielding, web crippling, or web compression buckling per AISC Chapter J (Sections J10.2 through J10.5). For light to moderate loads on W12 and larger sections, transverse stiffeners are often not needed. For heavy moments on shallow sections (W8, W10), full-depth transverse stiffeners at the support are common.

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Related references

• Beam Bending Moment Formulas
• Deflection Limits
• Beam Sizes Reference
• Beam Design Guide
• Lateral-Torsional Buckling
• Serviceability
• Load Combinations
• Steel Grades
• Section Comparison

Disclaimer

This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against AISC 360-22, the AISC Steel Construction Manual, and the governing project specification. The site operator disclaims liability for any loss arising from the use of this information.