Formula used
Timoshenko theory adds shear deformation to classic bending. Total maximum deflection is: δ = δb + δs.
- δb uses beam bending with E and I.
- δs uses shear term with k G A.
- If ν is provided: G = E / [2(1+ν)].
Implemented maximum-deflection formulas:
Simply supported, midspan point: δb = P L³/(48 E I), δs = P L/(4 k G A)
Simply supported, UDL: δb = 5 w L⁴/(384 E I), δs = w L²/(8 k G A)
Cantilever, end point: δb = P L³/(3 E I), δs = P L/(k G A)
Cantilever, UDL: δb = w L⁴/(8 E I), δs = w L²/(2 k G A)
Simply supported, UDL: δb = 5 w L⁴/(384 E I), δs = w L²/(8 k G A)
Cantilever, end point: δb = P L³/(3 E I), δs = P L/(k G A)
Cantilever, UDL: δb = w L⁴/(8 E I), δs = w L²/(2 k G A)
How to use this calculator
- Select a load case that matches your support and loading.
- Enter beam length and material stiffness values.
- Choose a section shape and provide dimensions.
- Optionally adjust the shear correction factor k.
- Click Calculate to see deflection and stress checks.
Example data table
| Case | L (m) | E (GPa) | Shape | Dims | Load | Expected trend |
|---|---|---|---|---|---|---|
| Simply supported, midspan point | 2.0 | 200 | Rect | b=50 mm, h=100 mm | P=1000 N | Shear increases for short, thick beams |
| Cantilever, end point | 1.0 | 70 | Circ | d=40 mm | P=500 N | Large deflection sensitivity to I |
| Simply supported, UDL | 3.0 | 210 | Hollow circ | Do=60 mm, Di=40 mm | w=1.0 kN/m | Bending dominates for slender beams |
Run the calculator to generate precise numeric outputs.