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BackCantilever beam - Uniform load partially distributed at fixed end Calculator
Formula
| Category | Formula |
|---|---|
| Deflection (\( y_{AC} \)) | \[ y_{AC} = \frac{-w_0}{24EI} \left( 6a^2x^2 - 4ax^3 + x^4 \right) \] |
| Deflection (\( y_{CB} \)) | \[ y_{CB} = \frac{-w_0a^3}{24EI} (4x - a) \] |
| Maximum Deflection (\( y_{\text{MAX}} = y_B \)) | \[ y_{\text{MAX}} = y_B = \frac{-w_0a^3}{24EI}(4L - a) \] |
| Slope (\( \theta_{AC} \)) | \[ \theta_{AC} = \frac{-w_0}{6EI}(3a^2x - 3ax^2 + x^3) \] |
| Slope (\( \theta_{CB} = \theta_C = \theta_B \)) | \[ \theta_{CB} = \theta_C = \theta_B = \frac{-w_0a^3}{6EI} \] |
| Moment (\( M_{AC} \)) | \[ M_{AC} = \frac{-w_0}{2}(a - x)^2 \] |
| Moment (\( M_{CB} = M_C = M_B \)) | \[ M_{CB} = M_C = M_B = 0 \] |
| Maximum Moment (\( M_{\text{MAX}} = M_A \)) | \[ M_{\text{MAX}} = M_A = \frac{-w_0a^2}{2} \] |
| Shear (\( V_{AC} \)) | \[ V_{AC} = w_0(a - x) \] |
| Shear (\( V_{CB} = V_C = V_B \)) | \[ V_{CB} = V_C = V_B = 0 \] |
| Reaction (\( R_A \)) | \[ R_A = w_0a \] |
Definitions
| Symbol | Physical quantity | Units |
|---|---|---|
| E·I | Flexural rigidity | N·m², Pa·m⁴ |
| y | Deflection or deformation | m |
| θ | Slope, Angle of rotation | - |
| x | Distance from support (origin) | m |
| L | Length of beam (without overhang) | m |
| M | Moment, Bending moment, Couple moment applied | N·m |
| P | Concentrated load, Point load, Concentrated force | N |
| w | Distributed load, Load per unit length | N/m |
| R | Reaction load, reaction force | N |
| V | Shear force, shear | N |