## 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)
Vx = Shear value at a distance 'x' along the beam (lbs, kips, kg)
Mx = Moment value at a distance 'x' along the beam (lb-ft, kip-ft, kip-in, kg-m)
Δx = Deflection value at a distance 'x' along the beam (in, ft, m)
Vmax = Maximum Shear Value
Mmax = Maximum Moment Value
Δmax = Maximum Deflection Value
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 (in4)

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).

Fig. 1: Fixed-Fixed Beam with a Distributed Load
 R = V = \frac{wl}{2} Vx = w \left(\frac{l}{2} - x\right) Mmax (@ ends) = \frac{wl^2}{12} M1 (@ center) = \frac{wl^2}{24} Mx = \frac{w}{12}(6lx-l^2-6x^2) Δmax (@ center) = \frac{wl^4}{384EI} Δx = \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} Mmax (@ center & ends) = \frac{Pl}{8} Mx (when x < 0.5*l) = \frac{P}{8}(4x-l) Δmax (@ center) = \frac{Pl^3}{192EI} Δx (when x < 0.5*l) = \frac{Px^2}{48EI}(3l-4x)