Draw free-body diagrams and use the equilibrium equations to solve 2D problems for cable tensions and support reactions.
Why it matters
Statics is the study of bodies that are not accelerating: a bridge, a shelf bracket, a crane holding a load still. Almost every structural or machine calculation starts with a statics step, because before you can ask "is this member strong enough?" you have to know what force it carries. Those forces come from one idea: if a body is in equilibrium, the forces and moments on it add up to zero.
The equilibrium equations
For a particle (a point where forces meet, such as a ring where cables join), only the forces matter:
ΣFx = 0
ΣFy = 0
For a rigid body (a beam, a bracket), forces can also make it rotate, so a third equation is added. The moment of a force about a point is the force times its perpendicular distance from that point (units N·m or kN·m):
ΣFx = 0
ΣFy = 0
ΣM about any point = 0
In 2D that gives at most three independent equations per body, so you can solve for at most three unknowns. A body with more unknown support reactions than equations is statically indeterminate and needs strength-of-materials methods, which come later in a degree.
The free-body diagram comes first
- Isolate the body: draw it alone, cut free from its supports.
- Replace each support with the reactions it can provide. A pin resists movement in both directions (two unknowns, Ax and Ay). A roller resists movement only perpendicular to the surface (one unknown). A fixed support resists both directions and rotation (three unknowns).
- Draw every applied load at its real location, with its direction.
- Add dimensions, then choose axes and a sign convention (for example, up and right positive, counter-clockwise moments positive).
If you guess a reaction's direction wrong, the algebra simply returns a negative number. That is fine; just report it in the opposite direction.
Worked example 1: a sign on two cables
A 1,000 N sign hangs from a ring held by two cables. Cable 1 pulls up and to the left at 30° above horizontal; cable 2 pulls up and to the right at 45°. Find the tensions.
ΣFx = 0: −T1·cos30° + T2·cos45° = 0 → T1 = T2 · (0.7071 / 0.8660) = 0.8165·T2
ΣFy = 0: T1·sin30° + T2·sin45° − 1000 = 0
0.8165·T2·0.5 + 0.7071·T2 = 1000 → 1.1154·T2 = 1000
T2 ≈ 896.6 N, T1 ≈ 732.1 N
The steeper cable carries more of the load. Here neither tension exceeds the weight, but if both cables were nearly horizontal the tensions would grow very large: a flat cable has only a small vertical component to hold the load with.
Worked example 2: reactions on a simply supported beam
A 6 m beam rests on a pin at A (left end) and a roller at B (right end). A 12 kN load acts 2 m from A. Ignore the beam's own weight.
Take moments about A, so the unknowns at A drop out:
ΣMA = 0: RB·6 − 12·2 = 0 → RB = 4 kN
ΣFy = 0: RA + RB − 12 = 0 → RA = 8 kN
ΣFx = 0: Ax = 0 (no horizontal loads)
The support nearer the load carries more of it. A quick check: take moments about B instead. RA·6 − 12·4 = 0 gives RA = 8 kN, the same answer.
Choosing where to take moments
Pick the point where the most unknown forces act. Their moment arms are zero, so they vanish from that equation and you solve for one unknown at a time. This trick saves more algebra than any other in statics.
Common mistakes
- Forgetting a force. The weight of the body itself, or a reaction at a support you forgot to draw. Check every place the body touches something.
- Mixing units. Keep kN with m (giving kN·m), or N with mm (giving N·mm). Do not mix kN with mm.
- Using the wrong distance. A moment arm is the perpendicular distance to the line of action of the force, not the distance to where the force is drawn.
- Including internal forces. Forces between parts of the same body cancel and do not appear on its free-body diagram.
A note on real design
These are idealised textbook supports and loads. Real structures carry combinations of dead, live, wind, snow and seismic loads defined by building codes, and real connections are never perfect pins or rollers. Use these methods to learn and to sanity-check, not to size a real structure: that work is done to code and reviewed by a licensed professional engineer.
Recap
- Equilibrium means ΣF = 0 and, for rigid bodies, ΣM = 0.
- Always start with a free-body diagram showing supports as reactions.
- Take moments about the point with the most unknowns.
- A negative answer means the force acts the other way.
Try it
On the Do tab, finish a Python function that finds the support reactions of a simply supported beam carrying several point loads.