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 | |||||||||||||
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)
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-Free Beam with a Distributed Load
| R = V | = wl |
| Vx | = wx |
| Mmax (@ fixed end) | = \frac{wl^2}{2} |
| Mx | = \frac{wx^2}{2} |
| Δmax (at free end) | = \frac{wl^4}{8EI} |
| Δx | = \frac{w}{24EI}(x^4-4l^3x+3l^4) |
Triangular Distributed Load
Fig. 2: Fixed-Free Beam with a Triangular Load
| R = V | = W |
| Vx | = W\frac{x^2}{l^2} |
| Mmax (at fixed end) | = \frac{Wl}{3} |
| Mx | = \frac{Wx^2}{3l^2} |
| Δmax (at free end) | = \frac{Wl^3}{15EI} |
| Δx | = \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 |
| Mmax (@ fixed end) | = Pl |
| Mx | = Px |
| Δmax (@ free end) | = \frac{Pl^3}{3EI} |
| Δx | = \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 |
| Mmax (@ fixed end) | = Pb |
| Mx (when x > a) | = P(x-a) |
| Δmax (@ free end) | = \frac{Pb^2}{6EI}(3l-b) |
| Δa (@ pt. of load) | = \frac{Pb^3}{3EI} |
| Δx (when x < a) | = \frac{Pb^2}{6EI}(3l-3x-b) |
| Δx (when x > a) | = \frac{P(l-x)^2}{6EI}(3b-l-x) |