Global units
These selections apply to all inputs and results on this page. Other load cases live on separate pages.
- Use dot “.” as decimal separator.
-
a = 0 places the moment at the free end.
Increase a to move it toward the fixed support.
-
x is the distance from the free end where results are evaluated.
Use x = 0 at the free tip and
x = L at the fixed support.
Values marked “@ x” in the Results table are computed at this location.
(a locates the applied moment; x locates the reporting position.)
-
Tip: Enter c in its own unit
(often inches) regardless of the length unit for
L, a, x.
Numbers and units update together.
Need I for common shapes? Try the sectional properties calculators.
Results
| Parameter |
Value |
| Reaction Force (fixed end) R₂ |
---
lbf
|
| Reaction Moment (fixed end) M₂ |
---
lbf·in
|
| Shear @ x (Vₓ) |
---
lbf
|
| Max Shear (Vmax) |
---
lbf
|
| Moment @ x (Mₓ) |
---
lbf·in
|
| Max Moment (Mmax) |
---
lbf·in
|
| Slope @ x (θₓ) |
---
radian
|
| Max Slope (θmax) |
---
radian
|
| Fixed-end slope (θ₂) |
---
radian
|
| Free-end slope (θ₁) |
---
radian
|
| Deflection @ x (yₓ) |
---
inch
|
| Max Deflection (ymax) |
---
inch
|
| Fixed-end deflection (y₂) |
---
inch
|
| Free-end deflection (y₁) |
---
inch
|
| Bending Stress @ x (σₓ) |
---
psi
|
| Max Bending Stress (σmax) |
---
psi
|
About this load case
Cantilever beam with a single applied bending moment
of magnitude M at position
a measured from the free end
(set a = 0 for a moment at the free tip).
The fixed end is at x = L. Coordinates are
measured from the free end.
Sign conventions
Deflection and slope sign convention for a cantilever beam
- Coordinates: x is measured from the free end
(x = 0) toward the fixed end
(x = L), positive to the right.
y is positive upward.
- Applied moment M (input): A positive value of
M acts in the direction of the red arrow
in the diagram and produces a positive bending moment
inside the beam (top fibers in compression) for
x > a.
- Reactions / shear V: Positive forces act upward;
negative downward. For this pure-moment load case the net shear
is zero everywhere, so V(x) = 0.
- Bending moment M(x): Positive when the top fibers
are in compression (sagging), negative when the bottom fibers are
in compression (hogging).
- Deflection y(x): Positive = upward;
negative = downward.
- Slope θ(x): Positive when dy/dx > 0
(centerline rises as x increases); negative when dy/dx < 0.
- Stresses: Bending stresses are reported as
magnitudes |σ| (sign is not shown).
With these conventions, a positive applied moment
M > 0 produces a positive internal
bending moment M(x) = M for
x > a, zero shear everywhere, and a
curved deflected shape between a and
L that satisfies
y(L) = 0 and
θ(L) = 0 at the fixed end.
Formula summary (single moment at position a)
Origin at the free end (x = 0).
Fixed end at x = L. Constant
E and I.
- End reactions:
R₁ = 0,
M₁ = 0,
R₂ = 0,
M₂ = M,
θ₂ = 0,
y₂ = 0.
- Shear diagram:
V(x) = 0, 0 ≤ x ≤ L
A pure couple does not create net transverse force.
- Moment diagram:
M(x) = 0, 0 ≤ x ≤ a
M(x) = M, a < x ≤ L
- Slope (EI constant):
θ₁ = θ(0) = − M (L − a) / (E I)
θ(x) = θ₁, 0 ≤ x ≤ a
θ(x) = M (x − L) / (E I), a < x ≤ L
- Deflection:
y₁ = y(0) = M (L² − a²) / (2 E I)
y(x) = y₁ + θ₁ x, 0 ≤ x ≤ a
y(x) = y₁ + θ₁ x + M (x − a)² / (2 E I),
a < x ≤ L
y(x) = M (L − x)² / (2 E I), a < x ≤ L
- Extreme bending stress at a section:
σ(x) = |M(x)| · c / I
In the results table, the “max” values (Vmax, Mmax, θmax, ymax)
represent the extreme values along the span. Their sign follows the
conventions above, while σₓ and σmax are reported as magnitudes
|σ|.
Related calculators & 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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