Engineering
Cantilever Beam Tip Deflection, Rotation Angle & Root Bending Stress
Calculate Euler-Bernoulli cantilever beam tip deflection delta = F*L^3 / (3*E*I), end slope theta = F*L^2 / (2*E*I), and fixed-wall bending stress sigma = F*L*c / I.
An ideal uniform cantilever with a transverse tip load
These Euler-Bernoulli expressions assume a slender, straight, uniform beam, a perfectly fixed root, linear elastic material and small deflection and rotation. Shear deformation, distributed self-weight, dynamic loading and support flexibility are omitted.
Use force magnitude in newtons, length and outer-fiber distance in meters, Young's modulus in pascals and second moment of area in m^4. I is the area moment about the relevant bending axis, not a mass moment of inertia. The displayed outputs are deflection magnitude in mm and nominal bending-stress magnitude in MPa.
For the same tip load and unchanged E and I, doubling length gives eight times the tip deflection, four times the small-angle rotation and twice the nominal root stress within this model. This calculation does not check yield, buckling, stress concentrations or design-code requirements.
| Cantilever Structural Metric | Formula | 2 m Steel Beam (6 kN Tip Load) | Doubled Span L = 4 m |
|---|---|---|---|
| Tip Deflection delta (mm) | ((F * L^3) / (3 * E * I)) * 1000 | 2.00 mm | 16.00 mm (8x sag!) |
| Fixed-Root Stress sigma (MPa) | ((F * L * c) / I) / 1e6 | 30.00 MPa | 60.00 MPa (2x stress) |
| Tip Rotation Slope (degrees) | DEGREES((F * L^2) / (2 * E * I)) | 0.0859 degrees | 0.3438 degrees (4x slope) |
Cantilever Free-End Tip Deflection delta_max in Millimeters
F is a nonnegative load magnitude. Length, E and I must be positive. The formula converts meters to millimeters after evaluating FL^3/(3EI).
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<0,B3<=0,B4<=0,B5<=0),"Invalid inputs",ROUND(((B2 * (B3 ^ 3)) / (3 * B4 * B5)) * 1000, 2)))Clamped-Root Maximum Bending Stress sigma_max in MPa
c is the largest relevant distance from the neutral axis to an outer fiber, not necessarily half the overall depth for an asymmetric section. FLc/I is nominal bending stress; it does not include local clamp stress concentrations.
=IF(COUNT(B2:B3,B5:B6)<>4,"Invalid inputs",IF(OR(B2<0,B3<=0,B5<=0,B6<=0),"Invalid inputs",ROUND(((B2 * B3 * B6) / B5) / 1000000, 2)))Small-angle tip rotation in degrees
Euler-Bernoulli small-slope theory identifies slope with rotation in radians to first order. DEGREES converts that small-angle estimate; it is not an exact large-rotation solution.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<0,B3<=0,B4<=0,B5<=0),"Invalid inputs",ROUND(DEGREES((B2 * (B3 ^ 2)) / (2 * B4 * B5)), 4)))Try the example
Use the original values or change them to test your understanding.