Fixed-Pinned 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-Pinned Beam with a Distributed Load
| R1 = V1 | = \frac{3wl}{8} |
| R2 = V2 = Vmax | = \frac{5wl}{8} |
| Vx | = R_1 - wx |
| Mmax | = \frac{wl^2}{8} |
| M1 (at x = 0.375*l) | = \frac{9}{128}wl^2 |
| Mx | = R_1x - \frac{wx^2}{2} |
| Δmax (@ x = 0.4215*l) | = {{wl^4}\over{185EI}} |
| Δx | = {{wx}\over{48EI}}(l^3-3lx^2+2x^3) |
Concentrated Load (at the center)
Fig. 2: Fixed-Pinned Beam with a Concentrated Load @ Center
| R1 = V1 | = \frac{5P}{16} |
| R2 = V2 = Vmax | = \frac{11P}{16} |
| Mmax (@ fixed end) | = \frac{3Pl}{16} |
| M1 (@ pt of load) | = \frac{5Pl}{32} |
| Mx (when x < 0.5*l) | = \frac{5Px}{16} |
| Mx (when x > 0.5*l) | = P\left(\frac{l}{2} - \frac{11x}{16}\right) |
| Δmax (at x = 0.4472*l) | = \frac{Pl^3}{48EI\sqrt{5}} |
| Δx (at pt. of load) | = \frac{7Pl^3}{768EI} |
| Δx (when x < 0.5*l)) | = \frac{Px}{96EI}(3l^2-5x^2) |
| Δx (when x > 0.5*l) | = \frac{P}{96EI}(x-l)^2(11x-2l) |
Concentrated Load (wild)
Fig. 3: Fixed-Pinned Beam with a Wild Concentrated Load
| R1 = V1 | = \frac{Pb^2}{2l^3}(a+2l) |
| R2 = V2 | = \frac{Pa}{2l^3}(3l^2-a^2) |
| M1 (@ pt. of load) | = R_1a |
| M2 (@ fixed end) | = \frac{Pab}{2l^2}(a+l) |
| Mx (when x < a) | =R_1x |
| Mx (when x > a) | = R_1x - P(x-a) |
| Δmax (when a < 0.414*l @ x = l\frac{l^2 + a^2}{3l^2-a^2} ) | = \frac{Pa}{3EI} \frac{(l^2-a^2)^2}{(3l^2-a^2)^2} |
| Δmax (when a > 0.414*l @ x = l\sqrt{\frac{a}{2l+a}} ) | = \frac{Pab^2}{6EI} \sqrt{\frac{a}{2l+a}} |
| Δa (@ pt. of load) | = \frac{Pa^2b^3}{12EIl^3}(3l+a) |
| Δx (when x < a) | = \frac{Pb^2x}{12EIl^3}(3al^2-2lx^2-ax^2) |
| Δx (when x > a) | = \frac{Pa}{12EIl^3}(l-x)^2(3l^2x-a^2x-2a^2l) |