Basic
How to find a Shear Force Diagram (SFD) of a Simple Beam
In this tutorial, we will look at calculating the shear force diagram of a simple beam. A shearing force occurs when a perpendicular force is applied to static material (in this case a beam). Think of a knife cutting through a carrot. Imagine the beam is the carrot and a point load is the knife. As the knife applies a downward force, it cuts (or shears) the carrot. These forces occur along numerous points of a beam, and it is important to determine where these shears are at the greatest points as this may be where a beam fails.
To calculate the shear forces of a beam, follow the following simple steps:
1. Start with the far left side of the beam
If there is an upward force (i.e a support) then the SFD will start at this force above the x-axis. If there is a downward point load and no support, than the shear force diagram will start as a negative at the value of the point load.
2. Move across the beam
As you come across loads, you simply add (or subtract) these loads from the value you already have, keeping a cumulative total.
t is much easier to understand when considering an example of a how to calculate a Shear Force Diagram. So let's consider the following example to calculate the shear force diagram of a beam:
Step 1: After calculating the reactions at A and B, start the Shear Force Diagram at the first value of the force acting on the beam. In this case it is a +10kN due to the reaction at point A:
Step 2: Keep moving across the beam, stopping at every load that acts on the beam. When you get to a load, add to the Shear Force Diagram by the amount of the force. In this case we have come to a negative 20kN force, so we will minus 20kN from the existing 10kN. i.e. 10kN - 20kN = -10kN.
Step 2 (repeated): Moving across the beam again, we come to another force; a positive 10kN reaction at support B. Again, add this +10kN to the shear force diagram (which is currently at -10kN) which will bring us to a shear force of 0. Since we are at the end of the beam, we will go no further and we have our final Shear Force Diagram (SFD):
Things to keep in mind:
- The area under the SFD above the x axis should equal the area between the x-axis and the SFD below the x axis. i.e the area should sum to zero. Check this is true in our above example.
- Any points where the SFD cross the x-axis, will be a max or min Bending Moment
- The SFD should always equal zero at both ends
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Example 4-2
a cantilever beam AB subjected to a
linearly varying distributed load as shown, find
the shear force V and the bending moment M
q = q0 x / L
Fy = 0 - V - 2 (q0 x / L) (x) = 0
V = - q0 x2 / (2 L)
Vmax = - q0 L / 2
M = 0 M + 2 (q0 x / L) (x) (x / 3) = 0
M = - q0 x3 / (6 L)
Mmax = - q0 L2 / 6
