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BackFixed-fixed beam - Uniform load partially distributed Calculator
Formula
| Quantity | Formula |
|---|---|
| Deflection \(y_{AC}\) | \[ y_{AC} = \frac{x^2}{6EI}(3M_A + R_A x) \] |
| Deflection \(y_{CD}\) | \[ y_{CD} = \frac{-1}{24EI} \left[ w_0 (x - a)^4 - 4R_A x^3 - 12M_A x^2 \right] \] |
| Deflection \(y_{DB}\) | \[ y_{DB} = \frac{3(M_B + L R_B)x^2 - R_B x^3}{6EI} + \frac{L^2(3M_B + LR_B) - 3(2M_B + LR_B)Lx}{6EI} \] |
| Slope \(\theta_{AC}\) | \[ \theta_{AC} = \frac{x}{2EI}(2M_A + R_A x) \] |
| Slope \(\theta_{CD}\) | \[ \theta_{CD} = \frac{-1}{6EI} \left[ w_0(x - a)^3 - 3R_A x^2 - 6M_A x \right] \] |
| Slope \(\theta_{DB}\) | \[ \theta_{DB} = \frac{-1}{2EI} \left[ R_B x^2 - 2(M_B + LR_B)x + L(2M_B + LR_B) \right] \] |
| Moment \(M_{AC}\) | \[ M_{AC} = M_A + R_A x \] |
| Moment \(M_{CD}\) | \[ M_{CD} = R_A x + M_A - \frac{w_0 (x - a)^2}{2} \] |
| Moment \(M_{DB}\) | \[ M_{DB} = M_B + R_B (L - x) \] |
| Shear \(V_{AC}\) | \[ V_{AC} = R_A \] |
| Shear \(V_{CD}\) | \[ V_{CD} = R_A - w_0 (x - a) \] |
| Shear \(V_{DB}\) | \[ V_{DB} = -R_B \] |
| Reactions \(R_A\) | \[ R_A = \frac{w_0(2c + b)b - 2M_A + 2M_B}{2L} \] |
| Reactions \(R_B\) | \[ R_B = \frac{w_0(2a + b)b + 2M_A - 2M_B}{2L} \] |
| Where \(M_A\) | \[ M_A = \frac{-w_0 b}{24L^2} \left[ b^2(2L - 6c - 3b) + (6a + 3b)(2c + b)^2 \right] \] |
| Where \(M_B\) | \[ M_B = \frac{-w_0 b}{24L^2} \left[ b^2(2L - 6a - 3b) + (6c + 3b)(2a + b)^2 \right] \] |
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 |