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BackCantilever beam - Cosinusoidal distributed load Calculator
Formula
| Quantity | Formula |
|---|---|
| Deflection \(y_{AB}\) | \[ y_{AB} = \frac{-w_0 L}{3\pi^4 EI} \left( 48L^3 \cos \frac{\pi x}{2L} - 48L^3 + 3\pi^3 Lx^2 - \pi^3 x^3 \right) \] |
| Maximum Deflection \(y_{MAX}\) | \[ y_{MAX} = \frac{-2w_0 L^4}{3\pi^4 EI} (\pi^3 - 24) \quad \text{at } x = L \] |
| Slope \(\theta_{AB}\) | \[ \theta_{AB} = \frac{-w_0 L}{\pi^3 EI} \left( 2\pi^2 Lx - \pi^2 x^2 - 8L^2 \sin \frac{\pi x}{2L} \right) \] |
| Slope at B \(\theta_B\) | \[ \theta_B = \frac{-w_0 L^3}{\pi^3 EI} (\pi^2 - 8) \] |
| Moment \(M_{AB}\) | \[ M_{AB} = \frac{-2w_0 L}{\pi^2} \left( \pi L - \pi x - 2L \cos \frac{\pi x}{2L} \right) \] |
| Shear \(V_{AB}\) | \[ V_{AB} = \frac{2w_0 L}{\pi} \left( 1 - \sin \frac{\pi x}{2L} \right) \] |
| Reactions \(R_A\) | \[ R_A = \frac{2w_0 L}{\pi} \] |
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 |