## 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)

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)

Main