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

Fixed-Fixed beams are common in the interior section of a building (not around the edges). Since both sides of the beam is capable of retaining a moment, this beam is significantly stronger that the Simply Supported Beams you've seen earlier. For example the max moment for a fixed-fixed connection can be found by taking \frac{wl^2}{12} vs \frac{wl^2}{8} for a simply supported beam (a 50% increase in strength). These diagrams will also be useful for getting approximations for moment distribution.

The Breakdown
| |||||||||||||

### 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-Fixed Beam with a Distributed Load**

R = V | = \frac{wl}{2} |

V_{x}
| = w \left(\frac{l}{2} - x\right) |

M_{max} (@ ends)
| = \frac{wl^2}{12} |

M_{1} (@ center)
| = \frac{wl^2}{24} |

M_{x}
| = \frac{w}{12}(6lx-l^2-6x^2) |

Δ_{max} (@ center)
| = \frac{wl^4}{384EI} |

Δ_{x}
| = \frac{wx^2}{24EI}(l-x)^2 |

### Concentrated Load (at the center)

**Fig. 2: Fixed-Fixed Beam with a Concentrated Load @ Center**

R = V | = \frac{P}{2} |

M_{max} (@ center & ends)
| = \frac{Pl}{8} |

M_{x} (when x < 0.5*l)
| = \frac{P}{8}(4x-l) |

Δ_{max} (@ center)
| = \frac{Pl^3}{192EI} |

Δ_{x} (when x < 0.5*l)
| = \frac{Px^2}{48EI}(3l-4x) |

### Concentrated Load (wild)

**Fig. 3: Fixed-Fixed Beam with a Wild Concentrated Load**

R_{1} = V_{1} (=V_{max} if a < b)
| = \frac{Pb^2}{l^3}(3a+b) |

R_{2} = V_{2} (=V_{max} if a > b)
| = \frac{Pa^2}{l^3}(a+3b) |

M_{1} (max when a < b)
| = \frac{Pab^2}{l^2} |

M_{2} (max when a > b)
| = \frac{Pa^2b}{l^2} |

M_{a} (@ pt. of load)
| = \frac{2Pa^2b^2}{l^3} |

M_{x} (when x < a)
| = R_1x - \frac{Pab^2}{l^2} |

Δ_{max} (when a > b @ x = \frac{2al}{3a+b} )
| = \frac{2Pa^3b^2}{3EI(3a+b)^2} |

Δ_{a} (at pt. of load)
| = \frac{Pa^3b^3}{3EIl^3} |

Δ_{x} (when x < a)
| = \frac{Pb^2x^2}{6EIl^3}(3al - 3ax -bx) |