Statics

Did you notice the notation in the flipbook? For engineering connections, we typically label reaction forces with the joint designation (A, B, C, etc.) and a directional subscript (x, y, or z), like this:

❏ Loading diagram

First, sketch the beam's loading diagram. Include the supports (pins, pin-rollers, and fixed connections) and the loading (point loads, line loads, etc.). On this diagram, we show the actual loading (please do not replace it with a statically equivalent system).

❏ Free-body diagram

Your second picture is the FBD. On the FBD, you may replace the actual loading with a statically equivalent system if you like. For instance, if you have have a distributed load (a line load -- units of force per distance, or force intensity), you can replace it with a statically equivalent resultant force (or forces).

Be sure not to draw the support symbols on the FBD. Instead, replace them with their effect on the beam (forces and moments).

On the FBD, please draw the reactions in the direction that you think is correct, based on qualitative equilibrium analysis and/or your own intuition. If the loading isn't complicated, you can often predict the directionality of the reaction forces (and moments). If the loading is complicated, a qualitative equilibrium analysis is not reasonable. In that case, please simply make an assumption as to the direction of the reaction vectors.

It is perfectly OK to assume the incorrect directions for any (or all) of the unknown forces (and moments). The act of making a prediction will help you build intuition and become a better engineer. That said, the math will always correct you if your prediction turns out to be incorrect.

❏ Equations of equilibrium

We use the planar E.o.E. to solve beam reactions. We simply elevate (or project) the side view of the beam; we do not need to draw the cross-section that was so important when we learned about the centroid of an area.

The planar E.o.E. are:

• the summation of forces in the x-direction must equal zero

• the summation of forces in the y-direction must equal zero

• the summation of moments about any z-axis must equal zero

Select an equation that allows you to immediately solve for one of the unknowns. This is usually the moment equilibrium equation. 

Be strategic: sum moments about a point (really, an axis) that is coincident to one or more unknown forces.

Use the remainder of the E.o.E. to solve for the remainder of the unknowns.

❏ Reporting final answers

Take a minute to revisit this example problem for solving for an unknown force (or moment). Then, read below.

When you use the E.o.E., a final answer that is positive confirms that the direction you assumed for the vector is correct. A final answer that is negative contradicts your assumption.

For instance, say that you assumed that Ay was upward, and then calculated a positive value for Ay. The positive assumption is validated by the positive sign; your assumption was correct. In your final answer, use an arrow to emphatically communicate that you understand the true direction of the reaction on the beam, like this:

Ay = +5 kN ∴ 5 kN ↑

What do you do if you get a negative answer? Read this carefully. Let's say we assumed Ay was upwards, but got a negative sign in our final answer. The negative sign tells us that Ay points downwards. You would simply write the following:

Ay = -5 kN ↑ ∴  5 kN ↓

Please do not use any double negatives, such as Ay = -5 kN ↓. They just cause confusion.

Important. Do NOT erase and correct the incorrect assumption you drew on the original FBD. We cannot fairly evaluate your work when we don't understand your original assumptions. It is OK to leave the FBD with the arrow drawn the long way as long as you are effectively communicating the final answer per above. If you would like to redraw the FBD with the arrows pointing in the correct directions, you may make a NEW drawing at the bottom of your solution, as the very last step.