Trusses

Why it matters

A truss is a frame of straight members joined at their ends to form triangles. Roof trusses, bridge trusses, crane booms, transmission towers and bicycle frames all use the idea. Triangles are used because a triangle cannot change shape without changing the length of a side, so a truss carries load mostly by stretching or squashing its members rather than bending them. That makes trusses light and stiff, and it makes them pleasant to analyse.

The idealisation

Textbook truss analysis assumes:

1. Members are straight and connected only at their ends, by frictionless pins.
2. Loads and reactions are applied only at the joints.
3. The members' own weight is ignored, or shared out to the joints.

Under these assumptions every member is a two-force member: it carries a single axial force along its length, either tension (pulling on the joints, the member is being stretched) or compression (pushing on the joints, the member is being squashed). The usual sign convention: tension positive, compression negative.

A planar truss is statically determinate when m + r = 2j, where m is the number of members, r the number of support reactions and j the number of joints.

The method of joints

1. Find the support reactions from the equilibrium of the whole truss (previous lesson).
2. Pick a joint with no more than two unknown member forces.
3. Draw that joint's free-body diagram. Assume every unknown member force is tension, so it points away from the joint, along the member.
4. Apply ΣFx = 0 and ΣFy = 0 at the joint. Solve the two unknowns.
5. Move to the next joint that now has at most two unknowns, and repeat.

Because each joint is a particle, there is no moment equation at a joint: two equations, two unknowns.

Worked example: a triangular roof truss

Joints: A (pin) at (0, 0), B (roller) at (4, 0) and apex C at (2, 1.5), in metres. Members AB, AC and BC. A 10 kN load acts downward at C.

Check determinacy: m = 3, r = 3, j = 3, so m + r = 6 = 2j. Good.

Reactions. By symmetry, Ay = By = 5 kN up and Ax = 0.

Geometry of member AC. It runs 2 m across and 1.5 m up, so its length is 2.5 m (a 3-4-5 triangle). The unit vector from A toward C is (0.8, 0.6).

Joint A. Unknowns: F_AC (along (0.8, 0.6)) and F_AB (along (1, 0)).

ΣFy = 0:  5 + 0.6·F_AC = 0          →  F_AC = −8.33 kN  (compression)
ΣFx = 0:  F_AB + 0.8·F_AC = 0       →  F_AB = +6.67 kN  (tension)

Joint C (check): BC is the mirror image of AC, so F_BC = −8.33 kN. Vertical forces at C: the two rafters push up on C with 2 × 8.33 × 0.6 = 10 kN, balancing the 10 kN load. The bottom chord AB is in tension, tying the feet of the rafters together so they do not spread apart. That is exactly why the bottom chord of a roof truss is called a tie.

Solving a joint as a 2 × 2 system

At any joint you can write the two equations as a linear system:

ux1·F1 + ux2·F2 = −Px
uy1·F1 + uy2·F2 = −Py

where (ux, uy) are the unit vectors along the two unknown members pointing away from the joint and (Px, Py) is the sum of the known forces at the joint. Cramer's rule solves it:

det = ux1·uy2 − ux2·uy1
F1  = (−Px·uy2 + Py·ux2) / det
F2  = (−Py·ux1 + Px·uy1) / det

A large truss is the same idea scaled up: computer programs assemble every joint's equations into one big matrix and solve them together, which is the linear algebra you met in the prerequisite course.

Zero-force members

At an unloaded joint where only two non-collinear members meet, both carry zero force. At an unloaded joint where three members meet and two are collinear, the third carries zero force. Spotting these first shortens the work. Zero-force members are still needed: they brace other members against buckling and carry load when the loading changes.

The method of sections

When you only need the force in one member in the middle of a long truss, cut the truss through that member (and at most two others), and apply the three rigid-body equilibrium equations to one side. Taking moments about the point where the other two cut members meet gives the force you want in one line.

Common mistakes

• Sign confusion. Always assume tension (arrow pointing away from the joint). Then a negative answer simply means compression.
• Starting at a joint with three unknowns. You only have two equations per joint.
• Wrong angles. Get member directions from coordinates (Δx/L, Δy/L), not by eyeballing the sketch.
• Forgetting buckling. A compression member can fail by buckling long before its material yields. Truss design checks slender compression members for this, which is a design-code topic.

Recap

• Ideal truss members are two-force members: pure tension or compression.
• Method of joints: reactions first, then joints with at most two unknowns, ΣFx = ΣFy = 0.
• Each joint is a 2 × 2 linear system.
• Method of sections: cut and take moments to get one member quickly.

Try it

On the Do tab, finish a small Cramer's-rule solver and use it to solve joint A of the roof truss.