Global units
These selections apply to all inputs and results on this page. Choose the line load unit for w here as well.
- 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 = 0, it starts at the free end. If a+b = L, it reaches the fixed support.
.
Numbers and units update together.
Need I for common shapes? Try the sectional properties calculators.
Results
| Parameter | Value |
|---|
| Reaction Force (fixed end) R₂ |
---
lbf |
| Shear @ x (Vₓ) |
---
lbf |
| Max Shear (Vmax) |
---
lbf |
| Reaction Moment (fixed end) M₂ |
---
lbf·in |
| Moment @ x (Mₓ) |
---
lbf·in |
| Max Moment (Mmax) |
---
lbf·in |
| Slope @ x (θₓ) |
---
radian |
| Max Slope (θmax) |
---
radian |
| Free-end slope (θ₁) |
---
radian |
| Fixed-end slope (θ₂) |
---
radian |
| Deflection @ x (yₓ) |
---
inch |
| Max Deflection (ymax) |
---
inch |
| Free-end deflection (y₁) |
---
inch |
| Fixed-end deflection (y₂) |
---
inch |
| Bending Stress @ x (σₓ) |
---
psi |
| Max Bending Stress (σmax) |
---
psi |
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.
Sign convention
Deflection and slope sign convention for a cantilever beam
- x is measured from the free end toward the fixed end.
- w (distributed load) is positive when acting downward.
- Shear V is positive when acting upward on the beam section.
- Bending moment M is positive when it compresses the top fibers (creates “sagging”).
- Slope θ is positive when the beam rotates upward and to the right.
- Deflection y is positive upward.
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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