Global units
Choose base units for all inputs and results. Change Force/Length first; other unit groups follow automatically.
- Use dot “.” as decimal separator.
- Constraints: 0 ≤ a ≤ L and 0 ≤ x ≤ L.
- Sign convention: shear/deflection positive upward; moments positive when compressing top fibers (sagging).
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Sectional Properties Calculators.
Results
| Parameter | Value |
|---|
| Reaction Force R₁ |
---
lbf |
| Reaction Force R₂ |
---
lbf |
| 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 |
| End Slope Left (θ₁) |
---
radian |
| End Slope Right (θ₂) |
---
radian |
| Deflection @ x (yₓ) |
---
inch |
| Max Deflection (|y|max) |
---
inch |
| Bending Stress @ x (σₓ) |
---
psi |
| Max Bending Stress (σmax) |
---
psi |
About this load case
For a simply supported beam with a concentrated moment M at position a on span L, the support reactions are
R₁ = −M/L and R₂ = +M/L. Using Macaulay brackets ⟨·⟩, the field equations are:
- V(x) = R₁
- M(x) = R₁ x + M ⟨x − a⟩⁰
- θ(x) = θ₁ + R₁ x²/(2 E I) + M ⟨x − a⟩¹ /(E I)
- y(x) = θ₁ x + R₁ x³/(6 E I) + M ⟨x − a⟩² /(2 E I)
End slopes:
- θ₁ = −M(2L² − 6aL + 3a²)/(6 E I L)
- θ₂ = +M(L² − 3a²)/(6 E I L)
Sign convention: shear and deflection are positive upward; bending moments are positive when the top fibers are in compression.