NapkinCalc

Statics — Forces & Equilibrium

A truss joint — method of joints

continues from lesson 4 — values defined earlier in the course stay live here

ELI5: a truss is a triangle of straight members, each only ever pulled or pushed along its length. At any joint the forces must balance — so a single pin loaded downward, braced by a diagonal at 45°, lets you solve both member forces by hand.

Pjoint:=10kNP_{joint} := 10 kN = 10 kN downward load at the joint
ϕ:=45deg\phi := 45 deg = 45 deg diagonal member angle
Fdiag=Pjointsin(ϕ)F_{diag} = \frac{P_{joint}}{\mathrm{sin}\left(\phi\right)} = 14.142 kN diagonal force ≈ 14.1 kN (tension)
Fchord=Fdiag⋅cos(ϕ)F_{chord} = F_{diag} \cdot \mathrm{cos}\left(\phi\right) = 10 kN the horizontal member ≈ 10 kN (compression)
✓ pass abs(Fdiag∗sin(ϕ)−Pjoint)<1e−9kNabs(F_{diag} * sin(\phi ) - P_{joint}) < 1e-9 kN vertical balance at the joint holds

Real-world hook: this exact bookkeeping, joint after joint, sizes every member of a roof truss, a pylon, or a pedestrian bridge.

Try it yourself: a joint carries a 6 kN downward load on a diagonal at 30°. What diagonal force is needed? (F = P / sin θ.)

Fdiagyou:=6kN/sin(30deg)Fdiag_{you} := 6 kN / sin(30 deg) = 12 kN ✏️ Your turn: divide the 6 kN load by sin(30°) to get the diagonal force.
✓ pass abs(Fdiagyou−6kN/sin(30deg))<1e−6kNabs(Fdiag_{you} - 6 kN / sin(30 deg)) < 1e-6 kN green when your diagonal force is correct