WORLD FOR CIVIL

Topic 4.2: Shear Forces and Bending Moments II

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.

• The Shear Force is positive if it tends to rotate the beam section clockwise with respect to a point inside the beam section.
  
• The Bending Moment is positive if it tends to bend the beam section concave facing upward. (Or if it tends to put the top of the beam into compression and the bottom of the beam into tension.)

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 (-F_x and + F_x). 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