Beams fail mainly in two ways: shear and flexure. A flexure failure occurs when bending forces cause the beam to snap, like if you bow a toothpick until it breaks. A shear failure occurs when up-and-down forces cause the beam to slice off, like a slab of glacier ice calving into the sea.
So far we know how to identify loads, how to distribute them to a beam, and how to calculate reactions at the beam’s supports. But these forces are all external to the beam. To find out if the beam will survive, you need to look at what’s going on inside. In particular, the maximum shear force in the beam tells you if a shear failure will occur (duh), and the maximum bending moment tells you if a flexure failure will occur (not so duh).
To determine the shear force in a beam, start with a diagram of the loads (including reactions) and “read” the beam from left to right. Keep a running total of all the loads you’ve passed so far: upward loads increase the shear force while downward loads decrease it. The resulting plot is called a shear diagram. This process makes intuitive sense – a beam carries weight from one point to another, and shear force reflects how much weight the beam carries.
To determine the bending moment in a beam, start with a shear diagram and apply the same procedure as before. Working left to right, keep a running total of all the shear forces you encounter until you complete your moment diagram. The unit of moment is the foot-pound, equaling a shear force of one pound held over a distance of one foot. (So, you have 50 pounds of shear held over 5 feet? That’s 250 foot-pounds.) Moment increases with positive shear and decreases with negative shear.
Once you’ve drawn your shear and moment diagrams, find the locations of maximum shear force and maximum bending moment, and compare them to the beam’s capacity in shear and in flexure. BUT WAIT, how do you find out the capacity?! That my friends is a tale for another time.