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Structural: Shear & Moment Diagrams and Equations

What are Shear and Moment Diagrams?

Shear and Moment Diagrams are used to help you understand what the shear and moment curve look like for different generic types of beams and loading conditions. Basically it's a shortcut!! No longer will you have to use statics to find a reaction then back up a shear and moment curve. Also with the law of superposition it is very useful to have these so they can be combined and find even more complex systems (like a distributed load with a concentrated load, e.g. a concrete floor with a car driving over it).

The Breakdown
  1. Pinned Roller Shear Moment Diagrams
  2. Fixed Pinned Shear Moment Diagrams
  3. Fixed Fixed Shear Moment Diagrams
  4. Fixed Free Shear Moment Diagrams

Normally Shear and Moment Diagrams will provide the following:

An Example:

Shear Moment Diagram
Fig. 1: Simply Supported Beam with a Distributed Load

This is the most basic of Shear Moment Diagrams.

Notice the following:

It is important to note that::

V = \int F
-M = \int V

Notice that if you integrate the shear diagram you will get your moment curve. In my examples you actually get the negative of the moment curve but you could just as easily get the positive of the moment curve depending on how you've set up your axis'. It's up to your own preference (I prefer for my shear curves to start in the negative). If you need to brush up on your math, here is more information on derivatives.

The equations:

For each moment diagram you will be provided with a grouping of equations. Here are the applicable equations for the simply supported distributed load diagram I've provided in Figure 1 above:

R = V = \frac{wl}{2}
V_x = w\left(\frac{l}{2}-x\right)
M_{max} = \frac{wl^2}{8}
M_x = \frac{wx}{2}\left(l-x\right)
{Δ_{max}} (@ center) = {{5wl^4}\over{384EI}}
{Δ_x} = {{wx}\over{24EI}}{\left(l^3-2lx^2+x^3\right)}

where:

R = Reactions
Vx = Shear value at a distance 'x' along the beam
Mx = Moment value at a distance 'x' along the beam
Δx = Deflection value at a distance 'x' along the beam
Vmax = Maximum Shear Value
Mmax = Maximum Moment Value
Δmax = Maximum Deflection Value

I'm sure there is more, but that's enough for now. Now feel free to go back to The Breakdown at the top of the page and see what this shortcut has to offer.

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Page last modified on March 16, 2011