Statics

❏ 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 (that is, we do not replace it with a statically equivalent system).

❏ Free-body diagram

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

We do not draw the supports themselves on the FBD; instead, we replace them with their effect on the beam (forces and moments).

Use qualitative equilibrium analysis to try to deduce the directionality of each reaction. 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. Please note that if the loading isn't complicated, you can often predict the directionality of the reaction forces (and moments). However, if the loading is complicated, a qualitative equilibrium analysis is not reasonable. In that case, make a prediction, and draw the vectors on the FBD accordingly.

❏ Equations of equilibrium

We can use planar analysis procedures to solve beam reactions, because they lie in a plane. We can simply elevate (or project) the front face of the beam; we do not need to draw the cross-section.

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. In general, this will usually be 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

When you use the E.o.E., a final answer that is positive confirms that the direction you assumed for the vector is correct. For instance, say that you predicted that Ay was upward. Then, you calculated a positive value for Ay, thereby confirming that 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 ↑

Here is the best way to handle a negative answer. Let's say we assumed Ay was upwards, but in our final answer we got a negative sign. That means Ay is downwards. You would simply write the following:

Ay = -5 kN ↑ ∴  5 kN ↓

Important. Never go back, erase, and correct the incorrect assumption you drew on the FBD. That makes it impossible to understand what you did. Instead, if you would like to redraw the FBD with the arrows pointing in the correct directions, you may do that as the very last step.