Main Body
Beam Reactions and Diagrams
Diagrams
Learning Objectives
At the end of this chapter you should be able to:
- Determine the reactions of simply supported, overhanging and cantilever beams
- Calculate and draw the shearing force and bending moment diagrams of beams subject to concentrated loads, uniform distributed loads and combinations of the two.
Beams are structural elements with various engineering applications like roofs, bridges, mechanical assemblies, etc. In general, a beam is slender, straight, rigid, built from isotropic materials, and most important, subjected to loads perpendicular to their longitudinal axis. If instead of perpendicular loads the same structural member would be subjected to longitudinal loads it would be called column or post. If the same member would be subjected to a torque, it would be called and treated as a shaft. Therefore, when identifying mechanical or structural components, consideration of the manner of loading is very important.
Note that when it comes to orientation, beams can be horizontal, vertical or any inclination in between (like submerged plates analyzed in fluid mechanics)… provided the loading is perpendicular to their major axis.
Beam supports:
Beam Loads[1]:
Beam types:
| Types | Diagram | Examples | Covered |
| Simple beams, or simply supported |
|
Yes | |
| Overhanging beams |
|
Yes | |
| Cantilever beams |
|
Yes | |
| Compound beams |
|
No | |
| Continuous beams |
|
No |
When solving for reactions, the following steps are recommended:
- Draw the beam free body diagram
- Replace the uniform distributed load (if any) with the equivalent point load
- Solve ΣMA = 0 (sum of moments about support A). This will give you RB (reaction at support B).
- Solve ΣMB = 0. This will give you RA.
- Using RA and RB found at steps 3 and 4 check if ΣV = 0 (sum of all vertical forces) is satisfied.
- Note that steps 4 and 5 can be reversed.
- For a cantilever beam use ΣV = 0 to find the vertical reaction at the wall and ΣMwall = 0 to find the moment reaction at the wall. There is no other equation to validate your results.
Please note:
“Shearing forces are internal forces developed in the material of a beam to balance externally applied forces in order to secure equilibrium of all parts of the beam.
Bending moments are internal moments developed in the material of a beam to balance the tendency for external forces to cause rotation of any part of the beam.” [3]
The shear force at any section of a beam may be found by summing all the vertical forces to the left or to the right of the section under consideration.
Similarly, the bending moment at any section of a beam may be found by adding the moments from the left or from the right of the section considered. The moment’s pivot point is the location under consideration.
By convention, internal shearing forces acting downward are considered positive. They counteract upward external forces. Therefore, when representing the shear forces you can draw them in t