Before continuing with a second example of determining the shear forces and bending moments in a loaded beam, we need to take a moment to discuss the sign associated with the shear force and bending moment. The signs associated with the shear force and bending moment are defined in a different manner than the signs associated with forces and moments in static equilibrium.
In the beam section shown in Diagram 1, we have shown the Shear Force V and Bending Moment M acting in positive directions according to the definitions above.
Notice that there is a possibility for a degree of confusion with sign notation. When
summing forces, the direction of V shown in the diagram is in the negative y-direction,
yet it is a positive shear force. This can lead to some confusion unless we are careful.
We will deal with possible confusion by always working from the left for our beam
sections, and always choosing V & M in a positive direction according to the shear
force and bending moments conventions defined above. That is, we will always select the V
& M directions as shown in Diagram 1. This approach will simplify the sign
conventions, as we will see in the next example.
However before the next example, we will look at the causes of the internal bending moment
in a little greater detail.
In Diagram 2a, we have shown a simply supported loaded beam, and have indicated in an exaggerated way the bending caused by the load. If we then cut the beam and look at a left end section, we have the Diagram 2b.
In this diagram we have, for the sake of clarity, left out the vertical shear force
which develops, but have shown horizontal forces (-Fx and + Fx).
These forces develop since, as the beam bends, the top region of the beam is put into
compression and the bottom region of the beam is put into tension. As a result there are
internal horizontal (x-direction) forces acting in the beam; however for every positive
x-force, there is an equal and opposite negative x-force. Thus the net horizontal
(x-direction) internal force in the beam section is zero. However, even though the
actual x-forces cancel each other, the torque produced by these x-forces is not zero.
Looking at Diagram 2c and mentally summing torque about the center of the beam, we see
that the horizontal x-forces cause a net toque - which we call the internal bending
moment, M. This is the cause of the internal bending moment (torque) inside a loaded beam.
We now continue by proceeding very slowly and carefully through a somewhat extended
example(s). We will also examine an alternate method for determining the bending moments
in a beam.
Please select: Example
1 ; Example 2 ; Example 3
Or:
Topic 4: Beams - Table of Contents
Statics & Srength of
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