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:
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).
Uniformly Distributed Load
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) |
Concentrated Load (wild)
Fig. 3: Fixed-Fixed Beam with a Wild Concentrated Load
| R1 = V1 (=Vmax if a < b) | = \frac{Pb^2}{l^3}(3a+b) |
| R2 = V2 (=Vmax if a > b) | = \frac{Pa^2}{l^3}(a+3b) |
| M1 (max when a < b) | = \frac{Pab^2}{l^2} |
| M2 (max when a > b) | = \frac{Pa^2b}{l^2} |
| Ma (@ pt. of load) | = \frac{2Pa^2b^2}{l^3} |
| Mx (when x < a) | = R_1x - \frac{Pab^2}{l^2} |
| Δmax (when a > b @ x = \frac{2al}{3a+b} ) | = \frac{2Pa^3b^2}{3EI(3a+b)^2} |
| Δa (at pt. of load) | = \frac{Pa^3b^3}{3EIl^3} |
| Δx (when x < a) | = \frac{Pb^2x^2}{6EIl^3}(3al - 3ax -bx) |