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Cantilever Beam Deflection Calculator

Q: What is a cantilever beam?

A cantilever beam is a structural element fixed at one end and free at the other. The fixed end resists moment, shear, and axial forces. Common examples include balconies, cantilever retaining walls, and aircraft wings.

Q: What is the maximum deflection formula for a cantilever?

For a point load at the free end: δ = PL³/(3EI). For a UDL: δ = wL⁴/(8EI). The deflection varies with the cube or fourth power of the span, making length the most critical parameter.

Q: What is the deflection/span ratio limit?

Typical limits: L/360 for general occupancy (floors), L/240 for roofs, L/180 for industrial buildings. These ensure serviceability and prevent damage to finishes.

Q: How does Young's modulus affect deflection?

Deflection is inversely proportional to E. Steel (200 GPa) deflects 1/3 of aluminum (69 GPa) for the same beam. Timber (10 GPa) deflects 20× more than steel.

Q: What moment of inertia should I use?

Use the moment of inertia about the bending axis. For a cantilever beam under vertical loads, use I about the horizontal neutral axis. Higher I means stiffer beam and less deflection.

Q: Can I use superposition for combined loads?

Yes — for linear elastic beams with small deflections, superposition applies. Calculate each load case separately and sum the deflections and slopes.

Q: What is slope at the free end?

Slope (θ) is the angle of rotation at the free end. For point load: θ = PL²/(2EI). For UDL: θ = wL³/(6EI). Slope is important for connected elements and aesthetic appearance.

Q: How accurate are these formulas?

These are exact elastic beam theory formulas (Euler-Bernoulli) assuming small deflections, homogeneous material, and no shear deformation. For deep beams (span/depth < 5), shear deformation may be significant.

Calculate cantilever beam deflection, slope, and deflection/span ratio for point loads, UDL, UVL, and moment loads. Supports various sections and material presets.

Structural Analysis Structural engineers, civil engineering students Commercial Intent: HIGH

Engineering Formulas

Point Load at Free End

δmax = PL³ / (3EI) at free end
θ = PL² / (2EI) at free end

P: Point load (N)

L: Span length (m)

E: Young's modulus (Pa)

I: Moment of inertia (m⁴)

δ_max: Maximum deflection at free end (m)

θ: Slope at free end (rad)

Point Load at Any Position

δmax = Pa²(3L − a) / (6EI) at free end
θ = Pa² / (2EI)

a: Distance from support to load (m)

Uniformly Distributed Load

δmax = wL⁴ / (8EI) at free end
θ = wL³ / (6EI)

w: UDL intensity (N/m)

Uniformly Varying Load

δmax = wL⁴ / (30EI) at free end
θ = wL³ / (24EI)

w: Maximum UVL intensity (N/m)

Moment at Free End

δ = ML² / (2EI)
θ = ML / (EI)

M: Applied moment (N-m)

Combined Loads (Superposition)

δtotal = Σδi
θtotal = Σθi
Principle of superposition applies for linear elastic beams

δ_total: Total deflection (m)

θ_total: Total slope (rad)

Engineering Notes

For cantilevers, deflection is highly sensitive to span length (δ ∝ L³ or L⁴). A 10% increase in span increases deflection by 33–46%.

For deep beams (L/d < 5), consider shear deformation using Timoshenko beam theory.

Cantilever deflections are typically 3× larger than simply supported beams for the same loading.

For steel beams, use E = 200 GPa. For reinforced concrete, use cracked section properties (Icr ≈ 0.3–0.5 × Ig).

Always check both strength (stress) and serviceability (deflection) in beam design.

Assumptions

• Linear elastic material (Hooke's law)

• Small deflections (Euler-Bernoulli beam theory)

• Plane sections remain plane

• No shear deformation (slender beam assumption)

• Homogeneous isotropic material

• Constant cross-section along span

Common Mistakes

✕ Using wrong formula for load type

✕ Forgetting to convert units — P in N, E in Pa, L in m

✕ Not checking deflection/span ratio against code limits

✕ Confusing UDL total load with intensity (kN/m vs kN)

✕ Using I about the wrong axis

Frequently Asked Questions

What is a cantilever beam?

A cantilever beam is a structural element fixed at one end and free at the other. The fixed end resists moment, shear, and axial forces. Common examples include balconies, cantilever retaining walls, and aircraft wings.

What is the maximum deflection formula for a cantilever?

For a point load at the free end: δ = PL³/(3EI). For a UDL: δ = wL⁴/(8EI). The deflection varies with the cube or fourth power of the span, making length the most critical parameter.

What is the deflection/span ratio limit?

Typical limits: L/360 for general occupancy (floors), L/240 for roofs, L/180 for industrial buildings. These ensure serviceability and prevent damage to finishes.

How does Young's modulus affect deflection?

Deflection is inversely proportional to E. Steel (200 GPa) deflects 1/3 of aluminum (69 GPa) for the same beam. Timber (10 GPa) deflects 20× more than steel.

What moment of inertia should I use?

Use the moment of inertia about the bending axis. For a cantilever beam under vertical loads, use I about the horizontal neutral axis. Higher I means stiffer beam and less deflection.

Can I use superposition for combined loads?

Yes — for linear elastic beams with small deflections, superposition applies. Calculate each load case separately and sum the deflections and slopes.

What is slope at the free end?

Slope (θ) is the angle of rotation at the free end. For point load: θ = PL²/(2EI). For UDL: θ = wL³/(6EI). Slope is important for connected elements and aesthetic appearance.

How accurate are these formulas?

These are exact elastic beam theory formulas (Euler-Bernoulli) assuming small deflections, homogeneous material, and no shear deformation. For deep beams (span/depth < 5), shear deformation may be significant.