AMESWEB

Cantilever Beam — Distributed Load Calculator

This page solves a cantilever beam with an intermediate distributed line load: the load varies linearly from wa at position a to wb at position a+b (end of the loaded segment). If a+b = L, the segment reaches the free end.

Cantilever beam with a line load on the segment [a, a+b], varying from w_a at x=a to w_b at x=a+b; diagram shows reactions R1, R2, end moments M1, M2, slopes θ1, θ2, and deflections

Global units

These selections apply to all inputs and results on this page. Choose the line load unit for w here as well.

Input parameters

Unit: lbf/in
Unit: same as wa
Unit: ft
Unit: ft
Unit: ft
Unit: ft
Unit: ksi
Unit: inch
Unit: in⁴
  • Use dot “.” as decimal separator.
  • Positive w acts downward (as drawn).
  • Load varies linearly from wa at x = a to wb at x = a + b. The loaded region spans [a, a+b]. If a+b = L, it reaches the free end.
Numbers and units update together.

Need I for common shapes? Try the sectional properties calculators.

Results

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

Charts

Moment, shear, slope, and deflection plots update after calculation.

About this load case

This calculator treats a cantilever beam with a line load that begins at position a and extends a length b. The intensity varies linearly from wa at x = a to wb at x = a + b. Positive w acts downward.

Formulas used (Euler–Bernoulli)

x is measured from the free end (beam is fixed at x = L). The distributed load acts only on the segment [a, a+b] and varies linearly.

  • Load intensity (for a ≤ x ≤ a+b): w(x) = wa + (wb − wa) · (x − a) / b
  • Shear in the loaded zone (a ≤ x ≤ a+b): V(x) = R1 − wa(x − a) − [(wb − wa)/(2 b)] · (x − a)2
  • Moment in the loaded zone (a ≤ x ≤ a+b): M(x) = M1 + R1 x − (wa/2) · (x − a)2 − [(wb − wa)/(6 b)] · (x − a)3
  • Before the load (x < a): V(x) = R1, M(x) = M1 + R1 x
  • After the load (x > a+b): V(x) = R1 − b · (wa + wb) / 2, M(x) = M(a+b) + V(a+b) · (x − a − b)
  • Bending stress: σ(x) = M(x) · c / I
  • Slopes & deflections: θ'(x) = M(x)/(E · I), y'(x) = θ(x). The calculator uses the corresponding closed-form polynomials internally for the plots.

Assumptions & limits

  • Linear elastic, small deflection, prismatic beam; constant E and I.
  • Load varies linearly over the loaded segment; sign per convention.
  • Sign convention: moments positive when compressing top fibers; shear/deflection positive upward; slope positive up & right.

FAQ

Where does the distributed load act?
Only on the segment [a, a+b]. If a+b = L it reaches the free end; otherwise it is an intermediate load.
What units should I use for w?
Use force per length (e.g., lbf/in, N/mm). The calculator converts internally and keeps inputs/results consistent with the Global units.
What sign convention is used?
Positive w acts downward. Moments are positive when compressing top fibers; shear/deflection positive upward; slope positive up & right.

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