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Cantilever Beam Calculator — Multiple Loads & Moments (Advanced)

Use this advanced cantilever beam calculator when you have combined loading: multiple point loads Pᵢ, applied moments M₀ⱼ, and one or more linearly varying distributed loads wₖ on segments [aₖ, aₖ + bₖ]. Results include reactions, shear, moment, slope, deflection, bending stress, and plots.

Simple cases? For faster setup, use the dedicated cantilever calculators: Point load calculator · Distributed load calculator · Applied moment calculator · Deflection formulas

Global units

lbf/ft

These selections apply to all inputs and results. Choose force/length units first; other units adjust automatically.

Sign convention used on this page

Coordinates are measured from the free end (x = 0) toward the fixed end (x = L). +y is upward. Input loads: P > 0 downward, w > 0 downward. +θ is clockwise rotation (free end rotates downward). Applied moment sign is as shown.

Cantilever beam sign convention: y positive up, slope theta positive clockwise, point load P positive downward, applied moment sign as shown.
Quick check: P>0 (down) → y<0, θ>0, and M<0 near the fixed end.
Point Loads
Cantilever beam with multiple point loads along the span

Input parameters

Point loads (Pᵢ)

Row Pᵢ Unit aₚᵢ (from free end) Unit
1 lbf ft
Concentrated Moments
Cantilever beam with applied bending moments along the span

Applied moments (M₀ⱼ)

Row M₀ⱼ Unit aₘⱼ (from free end) Unit
1 lbf·in ft
Distributed Loads (linearly varying)
Cantilever beam with linearly varying distributed loads on segments [aₖ, aₖ+bₖ]

Distributed loads (wₖ : wₐₖ → w_bₖ on [aₖ, aₖ+bₖ])

Row wₐₖ (at aₖ) w_bₖ (at aₖ + bₖ) aₖ (from free end) bₖ (segment length)
1
Line load unit: lbf/ft
Unit: ft
Unit: ft
Unit: ksi
Unit: inch
Unit: in⁴
  • Use dot “.” as decimal separator.
  • Set all Pᵢ = 0 for a pure distributed / moment case; set all wₖ = 0 and M₀ⱼ = 0 for a pure point-load case, etc.
  • Sign convention: P > 0 acts downward; w > 0 acts downward; shear/deflection positive upward; moments positive when compressing top fibers; slope positive up & to the right.

Need I? Try Sectional Properties Calculators.

Summary of applied loads

All loads below are shown using the selected global units.

Type Value Position / Span

Results — Combined Loading (Pᵢ + M₀ⱼ + wₖ)

ParameterValue
Reaction Force R₁ (free end) --- lbf
Reaction Force R₂ (fixed end) --- lbf
Shear @ x (Vₓ) --- lbf
Max Shear (Vmax) --- lbf
Reaction Moment Left (M₁ @ free end) --- lbf·in
Reaction Moment Right (M₂ @ fixed end) --- lbf·in
Moment @ x (Mₓ) --- lbf·in
Max Moment (Mmax) --- lbf·in
Slope @ x (θₓ) --- radian
Max Slope (θmax) --- radian
End Slope Left (θ₁ @ free end) --- radian
End Slope Right (θ₂ @ fixed end) --- radian
Deflection @ x (yₓ) --- inch
Max Deflection (ymax) --- inch
End Deflection Left (y₁ @ free end) --- inch
End Deflection Right (y₂ @ fixed end) --- inch
Bending Stress @ x (σₓ) --- psi
Max Bending Stress (σmax) --- psi

Charts

Moment, shear, slope, and deflection plots for the combined loading case.

Sign conventions

  • x-axis: x is measured from the free end (x = 0) toward the fixed end (x = L).
  • y-axis: +y is upward.
  • Point load input P: P > 0 acts downward.
  • Distributed load input w: w > 0 acts downward.
  • Slope θ: +θ is clockwise rotation (free end rotating downward).
  • Bending moment M: Moment sign is exactly as shown in the figure; the plots and stress use that same sign.
  • Deflection y: y > 0 upward, y < 0 downward.

With these conventions, downward point loads (Pᵢ > 0) typically produce negative shear and sagging (negative) bending moments near the fixed end, rotations that are positive at the free end, and downward deflections.

Reference – superposition formulas

The calculator uses linear superposition of the contributions from each individual load, together with numerical integration of the curvature κ(x) = M(x) / (E I) to obtain slope and deflection.

  • Shear (global):
    V(x) = − Σ Pᵢ   for x ≥ aₚᵢ (each Pᵢ adds a downward step).
  • Moment (global):
    M(x) = − Σ [ Pᵢ · (x − aₚᵢ) ] (for x ≥ aₚᵢ)  +  Σ [ M₀ⱼ ] (for x ≥ aₘⱼ)
    Each point load adds a linear segment; each applied moment adds a constant jump in M(x) to the right of its position.
  • Curvature:
    κ(x) = M(x) / (E I)
  • Slope and deflection:
    θ(x) = θ(L) − ∫xL κ(ξ) dξ,    y(x) = y(L) − ∫xL θ(ξ) dξ
    Fixed end conditions: θ(L) = 0, y(L) = 0. These integrals are evaluated numerically along the span.
  • Bending stress at distance c from neutral axis:
    σ(x) = |M(x)| · c / I

The “max” values in the results table (Vmax, Mmax, θmax, ymax) correspond to the extreme values by magnitude along the span. σₓ and σmax are shown as |σ| using the selected stress unit.

Frequently Asked Questions

What units can I use?

Any mix from the dropdowns—internally everything is converted consistently. Ensure E, I, lengths and loads align.

How are signs handled?

Point loads Pᵢ are positive when acting downward; applied moments M₀ⱼ are positive when they create sagging (top fibers in compression). Distances are measured from the free end.

Do you include shear deformation?

No—this is Euler–Bernoulli (plane sections remain plane). For deep beams, shear deflection may be non-negligible.

Can I model multiple loads?

Yes—up to 5 point loads, 5 applied moments, and 5 linearly varying distributed loads. The calculator uses linear superposition of their effects.

References
  • Roark’s Formulas for Stress & Strain, Young & Budynas, McGraw-Hill.
  • Machinery’s Handbook, Oberg et al., Industrial Press.
  • Mechanics of Materials, Beer & Johnston, McGraw-Hill.

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