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
  1. Pinned Roller Shear Moment Diagrams
  2. Fixed Pinned Shear Moment Diagrams
  3. Fixed Fixed Shear Moment Diagrams
    1. Defining the Variables
    2. Uniformly Distributed Load
    3. Concentrated Load (at the center)
    4. Concentrated Load (wild)
  4. Fixed Free Shear Moment Diagrams

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).

Uniformly Distributed Load

Shear Moment Diagram
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)

Shear Moment Diagram
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)

Shear Moment Diagram
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)

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