Roark's Formulas for Stress and Strain | |||||
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Shear, moment, slope, deflection of elastic straight beams-table 8.1 | |||||

W=load (N), w=unit load (N/mm), M_{0}=applied couple (Nmm), θ_{0}=angular displacement (rad), |
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R_{A} and R_{B} are the vertical end reactions at left and right respectively (N), |
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M_{A} and M_{B} are the reaction end moments at left and right respectively (Nmm) |

2a. Left end free, right end fixed (cantilever) | |||||
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R_{A} (N): |
RA | R_{B} (N): |
RB | ||

M_{A} (Nmm): |
MA | M_{B} (Nmm): |
MB | ||

σ_{A} (MPa): |
σA | σ_{B} (MPa): |
σB | ||

θ_{A} (degrees): |
θA | θ_{B} (degrees): |
θB | ||

deflection y_{A} (mm): |
yA | deflection y_{B} (mm): |
yB |

If a=0 and w_{l} = w_{a} (uniform load on the entire span) |
|||||
---|---|---|---|---|---|

Max M = M_{B} (Nmm): |
maxM | ||||

Max σ = σ_{B} (MPa): |
maxσ | ||||

Max θ = θ_{A} (degrees): |
Maxθ | ||||

Max y = y_{A} (mm): |
maxY |

If a=0 and w_{a} = 0 (uniformly increasing load) |
|||||
---|---|---|---|---|---|

Max M = M_{B} (Nmm): |
maxM | ||||

Max σ = σ_{B} (MPa): |
maxσ | ||||

Max θ = θ_{A} (degrees): |
Maxθ | ||||

Max y = y_{A} (mm): |
maxY |

If a=0 and w_{l} = 0 (uniformly decreasing load) |
|||||
---|---|---|---|---|---|

Max M = M_{B} (Nmm): |
maxM | ||||

Max σ = σ_{B} (MPa): |
maxsigma | ||||

Max θ = θ_{A} (degrees): |
Maxθ | ||||

Max y = y_{A} (mm) |
maxY |

2b. Left end guided, right end fixed | |||||
---|---|---|---|---|---|

R_{A} (N): |
RA | R_{B} (N): |
RB | ||

M_{A} (Nmm): |
MA | M_{B} (Nmm): |
MB | ||

σ_{A} (MPa): |
σA | σ_{B} (MPa): |
σB | ||

θ_{A} (degrees): |
θA | θ_{B} (degrees): |
θB | ||

deflection y_{A} (mm): |
yA | deflection y_{B} (mm): |
yB |

If a=0 and w_{l} = w_{a} (uniform load on the entire span) |
|||||
---|---|---|---|---|---|

Max -M = M_{B} (Nmm): |
max-M | ||||

Max -σ = σ_{B} (MPa): |
max-σ | ||||

Max +M = M_{A} (Nmm): |
max+M | ||||

Max +σ = σ_{A} (MPa): |
max+σ | ||||

Max y = y_{A} (mm): |
maxY |

If a=0 and w_{a} = 0 (uniformly increasing load) |
|||||
---|---|---|---|---|---|

Max -M = M_{B} (Nmm): |
max-M | ||||

Max -σ = σ_{B} (MPa): |
max-σ | ||||

Max +M = M_{A} (Nmm): |
max+M | ||||

Max +σ = σ_{A} (MPa): |
max+σ | ||||

Max y = y_{A} (mm): |
maxY |

If a=0 and w_{l} = 0 (uniformly increasing load) |
|||||
---|---|---|---|---|---|

Max -M = M_{B} (Nmm): |
max-M | ||||

Max -σ = σ_{B} (MPa): |
max-σ | ||||

Max +M = M_{A} (Nmm): |
max+M | ||||

Max +σ = σ_{A} (MPa): |
max+σ | ||||

Max y = y_{A} (mm): |
maxY |

2c. Left end simply supported, right end fixed | |||||
---|---|---|---|---|---|

R_{A} (N): |
RA | R_{B} (N): |
RB | ||

M_{A} (Nmm): |
MA | M_{B} (Nmm): |
MB | ||

σ_{A} (MPa): |
σA | σ_{B} (MPa): |
σB | ||

θ_{A} (degrees): |
θA | θ_{B} (degrees): |
θB | ||

deflection y_{A} (mm): |
yA | deflection y_{B} (mm): |
yB |

If a=0 and w_{l} = w_{a} (uniform load on the entire span) |
|||||
---|---|---|---|---|---|

R_{A} (N): |
RA | ||||

R_{B} (N): |
RB | ||||

Max -M = M_{B} (Nmm): |
max-M | ||||

Max -σ = σ_{B} (MPa): |
max-σ | ||||

Max +M (Nmm): | max+M | at x (mm) = | x | ||

Max +σ (MPa): | max+σ | ||||

Max θ = θ_{A} (degrees): |
maxθ | ||||

Max y (mm) | maxY | at x (mm) = | x |

If a=0 and w_{a} = 0 (uniformly increasing load) |
|||||
---|---|---|---|---|---|

R_{A} (N): |
RA | ||||

R_{B} (N): |
RB | ||||

Max -M = M_{B} (Nmm): |
max-M | ||||

Max -σ = σ_{B} (MPa): |
max-σ | ||||

Max +M (Nmm): | max+M | at x (mm) = | x | ||

Max +σ (MPa): | max+σ | ||||

Max θ = θ_{A} (degrees): |
maxθ | ||||

Max y (mm) | maxY | at x (mm) = | x |

If a=0 and w_{l} = 0 (uniformly decreasing load) |
|||||
---|---|---|---|---|---|

R_{A} (N): |
RA | ||||

R_{B} (N): |
RB | ||||

Max -M = M_{B} (Nmm): |
max-M | ||||

Max -σ = σ_{B} (MPa): |
max-σ | ||||

Max +M (Nmm): | max+M | at x (mm) = | x | ||

Max +σ (MPa): | max+σ | ||||

Max θ = θ_{A} (degrees): |
maxθ | ||||

Max y (mm) | maxY | at x (mm) = | x |

2d. Left end fixed, right end fixed | |||||
---|---|---|---|---|---|

R_{A} (N): |
RA | R_{B} (N): |
RB | ||

M_{A} (Nmm): |
MA | M_{B} (Nmm): |
MB | ||

σ_{A} (MPa): |
σA | σ_{B} (MPa): |
σB | ||

θ_{A} (degrees): |
θA | θ_{B} (degrees): |
θB | ||

deflection y_{A} (mm): |
yA | deflection y_{B} (mm): |
yB |

If a=0 and w_{l} = w_{a} (uniform load on the entire span) |
|||||
---|---|---|---|---|---|

Max -M = M_{A}=M_{B} (Nmm): |
max-M | ||||

Max -σ = σ_{A}= σ_{B} (MPa): |
max-σ | ||||

Max +M (Nmm): | max+M | at x (mm) = | x | ||

Max +σ (MPa): | max+σ | ||||

Max y (mm) | maxY | at x (mm) = | x |

If a=0 and w_{a} = 0 (uniformly increasing load) |
|||||
---|---|---|---|---|---|

R_{A} (N): |
RA | ||||

M_{A} (Nmm): |
MA | ||||

σ_{A} (MPa): |
σA | ||||

R_{B} (N): |
RB | ||||

Max -M = M_{B} (Nmm): |
max-M | ||||

Max -σ = σ_{B} (MPa): |
max-σ | ||||

Max +M (Nmm): | max+M | at x (mm) = | x | ||

Max +σ (MPa): | max+σ | ||||

Max y (mm) | maxY | at x (mm) = | x |

2e. Left end simply supported, right end simply supported | |||||
---|---|---|---|---|---|

R_{A} (N): |
RA | R_{B} (N): |
RB | ||

M_{A} (Nmm): |
MA | M_{B} (Nmm): |
MB | ||

σ_{A} (MPa): |
sigmaA | σ_{B} (MPa): |
σB | ||

θ_{A} (degrees): |
θA | θ_{B} (degrees): |
θB | ||

deflection y_{A} (mm): |
yA | deflection y_{B} (mm): |
yB |

If a=0 and w_{l} = w_{a} (uniform load on the entire span) |
|||||
---|---|---|---|---|---|

R_{A} (N): |
RA | R_{B} (N): |
RB | ||

Max M (Nmm): | maxM | when x (mm) = | x | ||

Max σ (MPa): | maxσ | ||||

Max θ = θ_{B} (degrees) |
maxθ | ||||

Max y (mm): | maxY | when x (mm)= | x |

If a=0 and w_{a} = 0 (uniformly increasing load) |
|||||
---|---|---|---|---|---|

R_{A} (N): |
RA | R_{B} (N): |
RB | ||

Max M (Nmm): | maxM | when x (mm) = | x | ||

Max σ (MPa): | maxσ | ||||

Max θ = θ_{A} (degrees) |
maxθ | Max θ = θ_{B} (degrees) |
maxθ | ||

Max y (mm): | maxY | when x (mm)= | x |

2f. Left end guided, right end simply supported | |||||
---|---|---|---|---|---|

R_{A} (N): |
RA | R_{B} (N): |
RB | ||

M_{A} (Nmm): |
MA | M_{B} (Nmm): |
MB | ||

σ_{A} (MPa): |
σA | σ_{B} (MPa): |
σB | ||

θ_{A} (degrees): |
θA | θ_{B} (degrees): |
θB | ||

deflection y_{A} (mm): |
yA | deflection y_{B} (mm): |
yB |

If a=0 and w_{l} = w_{a} (uniform load on the entire span) |
|||||
---|---|---|---|---|---|

Max M = M_{A}(Nmm): |
maxM | ||||

Max σ = σ_{A}(MPa): |
maxσ | ||||

Max θ = θ_{B} (degrees) |
maxθ | ||||

Max y = y_{A}(mm): |
maxY |

If a=0 and w_{a} = 0 (uniformly increasing load) |
|||||
---|---|---|---|---|---|

Max M = M_{A}(Nmm): |
maxM | ||||

Max σ = σ_{A}(MPa): |
maxσ | ||||

Max θ = θ_{B} (degrees) |
maxθ | ||||

Max y = y_{A}(mm): |
maxY |

If a=0 and w_{l} = 0 (uniformly decreasing load) |
|||||
---|---|---|---|---|---|

Max M = M_{A}(Nmm): |
maxM | ||||

Max σ = σ_{A}(MPa): |
maxσ | ||||

Max θ = θ_{B} (degrees) |
maxθ | ||||

Max y = y_{A}(mm): |
maxY |