D-01 — free body diagrams

before the beam, there's the node.

equilibrium checker

build a system in equilibrium.

Two forces act on this node. Add a third that brings it into equilibrium — adjust F3's x and y components until both sums hit zero.

F3 components
F3x 0.0
F3y 0.0
—
equilibrium status
ΣFx = 5.0   ΣFy = -8.0
not in equilibrium
F1 = 8 kips down F2 = 5 kips right F3 node

coordinate convention: right = +x, up = +y

live equation
SFx = 0 + 5 + 0.0 = 5.0 kips SFy = -8 + 0 + 0.0 = -8.0 kips
adjust F3 until both sums equal zero.
ΣFx = All horizontal forces must sum to zero for equilibrium
ΣFy = All vertical forces must sum to zero for equilibrium
explained
Equilibrium means the sum of all forces in every direction equals zero. If either sum is nonzero, the node accelerates — it is not in balance. Every structural analysis starts here.
check your work

what did you find?

kips
kips
key concepts
overview What a free body diagram is

A free body diagram isolates a structure and shows every force acting on it — the first step in any structural analysis.

the starting point of everything Every calculation begins with an FBD

Every structural calculation begins with a free body diagram. Isolate the body you want to analyze — cut it free from its surroundings. Replace every connection, support, and contact with the forces and moments they exert. Draw all applied loads. Now you have a complete picture of every force acting on that body. Without an FBD, you're guessing. With one, equilibrium gives you the answer.

three rules Include all forces, correct reactions, assume directions

Three rules for a correct FBD: (1) Include ALL forces — applied loads, self-weight, reactions at supports, internal forces at cuts. Missing one makes the whole analysis wrong. (2) Show the correct number of reactions per support type — pin gives 2 (Rx, Ry), roller gives 1 (Ry), fixed gives 3 (Rx, Ry, M). (3) Assume directions for unknowns and let the math tell you if you're right — a negative answer just means the force acts opposite to your assumed direction.

the concept Isolate, then solve

An engineer's first move is always the same: isolate the thing you're analyzing. Cut it free from everything around it. Replace every connection, every support, every contact point with an arrow representing the force at that location. What you're left with is a free body diagram — the object, floating alone, with every force that acts on it shown explicitly.

This sounds abstract. It isn't. If you're standing still, gravity pulls you down. The floor pushes you up. Those two forces are equal and opposite. You are in equilibrium. A node in a structure is the same idea — except instead of two forces, there might be three or four, and they might not be vertical. The FBD makes sure you account for all of them.

the math Two equations — that's it

For a 2D problem with no rotation, equilibrium means two things are simultaneously true:

SFx = 0 — all horizontal forces sum to zero.
SFy = 0 — all vertical forces sum to zero.

That's the whole framework. Every force has an x-component and a y-component. You add them up separately. If both totals are zero, nothing is moving, nothing is accelerating. The system is in static equilibrium.

In the sandbox above, F1 pulls down (-8 in y) and F2 pushes right (+5 in x). For the node to stay still, something has to push up with +8 and pull left with -5. That's what you just found.

why it matters Equilibrium is the assumption everything else is built on

When a structural engineer analyzes a building, they assume it is not moving. Not accelerating. Not about to tip over or slide sideways. That assumption — static equilibrium — is what makes the math solvable.

Every beam, every column, every connection in a structure is assumed to satisfy SFx = 0 and SFy = 0 (and SM = 0, which you'll meet in the statics module). If any of those equations aren't satisfied, the structure is either moving or failing.

This is why FBDs matter. You can't check equilibrium on something you haven't isolated. You can't find the forces if you haven't drawn them. The diagram isn't busywork — it's the only way to make sure you haven't missed anything.

practice Check your equilibrium answer

Use the sandbox above to find the equilibrium force, then enter your values in the "check your work" section to verify. If you found F3x = -5 kips and F3y = +8 kips, the node is in equilibrium.