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Formula
| Category | Formula |
|---|---|
| Deflection \( y_{AC} \) | \[ y_{AC} = \frac{x^2}{6EI} \left( R_A x + 3M_A \right) \] |
| Deflection \( y_{CB} \) | \[ y_{CB} = \frac{4R_B L(L-x)^3 - w_0L(L-x)^4}{24EI L} + \frac{-w_0b^2(L-x)\left(bL + 3ab + 6a^2\right)}{48EI L} \] |
| Slope \( \theta_{AC} \) | \[ \theta_{AC} = \frac{x}{2EI} \left( R_A x + 2M_A \right) \] |
| Slope \( \theta_{CB} \) | \[ \theta_{CB} = \frac{-3R_B L(L-x)^2 + w_0L(L-x)^3}{6EI L} + \frac{w_0b^2 \left(bL + 3ab + 6a^2\right)}{48EI L} \] |
| Moment \( M_{AC} \) | \[ M_{AC} = R_A x + M_A \] |
| Moment \( M_{CB} \) | \[ M_{CB} = \frac{2R_B(L-x) - w_0(L-x)^2}{2} \] |
| Shear \( V_{AC} \) | \[ V_{AC} = R_A \] |
| Shear \( V_{CB} \) | \[ V_{CB} = -R_B + w_0(L-x) \] |
| Reaction \( R_A \) | \[ R_A = \frac{w_0b^2 - 2M_A}{2L} \] |
| Reaction \( R_B \) | \[ R_B = \frac{w_0\left(2a + b\right)b + 2M_A}{2L} \] |
| Where \( M_A \) | \[ M_A = \frac{-w_0b^2}{16L^2} \left[2L + b)(L+a) - b^2\right] \] |
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 |