## Fixed-Free Beams (Shear & Moment Diagrams)

Fixed-Pinned beams are common around the edges of a building. One side will retain no moment, and the other will be able to carry a moment force. Since a fixed connection is stronger than a pinned connection a majority of the force will attempt to travel in the direction of the fixed connection (this connection is stiffer) as is evident by the shear diagrams.

The Breakdown
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### Defining the Variables

The variables for the Shear and Moment are defined below. We have also provided common units they will be given in.
*where:*

**R**= Reactions (lbs, kips, kg)

**V**= Shear value at a distance 'x' along the beam (lbs, kips, kg)

_{x}**M**= Moment value at a distance 'x' along the beam (lb-ft, kip-ft, kip-in, kg-m)

_{x}**Δ**= Deflection value at a distance 'x' along the beam (in, ft, m)

_{x}**V**= Maximum Shear Value

_{max}**M**= Maximum Moment Value

_{max}**Δ**= Maximum Deflection Value

_{max}**P**= The force of the concentrated load (kips, lbs, kg)

**W**= The total load acting on the beam (kips, lbs, kg)

**w**= The unit load acting on the beam (lbs/ft, kg/m)

**l**= the length of the beam (ft, m)

**x**= a distance along the beam from the designated end (ft, m)

**E**= the modulus of elasticity of the beam (ksi)

**I**= the Moment of Inertia of the beam (in

^{4})

**Note:** Check your units! You don't want to use ft for length and then ksi for your modulus of elasticity (your answer will be off).

### Uniformly Distributed Load

**Fig. 1: Fixed-Free Beam with a Distributed Load**

R = V | = wl |

V_{x}
| = wx |

M_{max} (@ fixed end)
| = \frac{wl^2}{2} |

M_{x}
| = \frac{wx^2}{2} |

Δ_{max} (at free end)
| = \frac{wl^4}{8EI} |

Δ_{x}
| = \frac{w}{24EI}(x^4-4l^3x+3l^4) |

### Triangular Distributed Load

**Fig. 2: Fixed-Free Beam with a Triangular Load**

R = V | = W |

V_{x}
| = W\frac{x^2}{l^2} |

M_{max} (at fixed end)
| = \frac{Wl}{3} |

M_{x}
| = \frac{Wx^2}{3l^2} |

Δ_{max} (at free end)
| = \frac{Wl^3}{15EI} |

Δ_{x}
| = \frac{W}{60EIl^2}(x^5-5l^4x+4l^5) |

### Concentrated Load (at the free end)

**Fig. 3: Fixed-Free Beam with a Concentrated Load @ the free end**

R = V | = P |

M_{max} (@ fixed end)
| = Pl |

M_{x}
| = Px |

Δ_{max} (@ free end)
| = \frac{Pl^3}{3EI} |

Δ_{x}
| = \frac{P}{6EI}(2l^3-3l^2x+x^3) |

### Concentrated Load (wild)

**Fig. 4: Fixed-Free Beam with a Wild Concentrated Load**

R = V | = P |

M_{max} (@ fixed end)
| = Pb |

M_{x} (when x > a)
| = P(x-a) |

Δ_{max} (@ free end)
| = \frac{Pb^2}{6EI}(3l-b) |

Δ_{a} (@ pt. of load)
| = \frac{Pb^3}{3EI} |

Δ_{x} (when x < a)
| = \frac{Pb^2}{6EI}(3l-3x-b) |

Δ_{x} (when x > a)
| = \frac{P(l-x)^2}{6EI}(3b-l-x) |