Module 4 · Lesson 4.1
Assumptions and determinacy
What we pretend about trusses, and whether equilibrium can solve them.
Why this matters
Trusses look complicated and are actually the easiest structures to analyse by hand — but only because of some bold assumptions. Knowing what those assumptions are tells you when to trust the answer.
By the end of this lesson you should be able to
- List the truss assumptions
- Apply the determinacy check
- Identify zero-force members
Truss analysis rests on three assumptions. The joints are frictionless pins, so they cannot carry moment. Loads are applied only at the joints. And the members are straight and light enough that self-weight can be ignored or split between the joints.
Put those together and each member can only pull or push along its own length. No bending, no shear — just axial force. That is why trusses are efficient and why they are easy to solve.
What it calculates: Whether equilibrium alone can find every member force and reaction.
- m
- Number of members
- r
- Number of reaction components
- j
- Number of joints
This assumes
- A plane, pin-jointed truss
- The members are arranged so the truss is actually stable
In plain terms: Each joint gives two equations, so there are 2j equations available. If unknowns match equations the truss is determinate. If m + r is less, it is a mechanism; if more, it is indeterminate. Passing the count does not prove stability — a badly arranged truss can pass and still collapse.
Predict first
At an unloaded joint where exactly two members meet and they are not in line with each other, what are the member forces?
Members like that are zero-force members. They are not useless: they often brace another member against buckling, or they carry load only under a different load case. But for the load case in front of you they carry nothing, and spotting them first saves a lot of work.
Practice
A plane truss has 11 members, 7 joints and 3 reaction components. What is m + r − 2j?
Practice
A plane truss has 14 members, 8 joints and 3 reaction components. To what degree is it statically indeterminate?
Practice
A plane truss has 9 members, 7 joints and 3 reaction components. What is m + r − 2j, and what does that tell you? (Enter the number.)
Summary
- Pinned joints, loads at joints, light straight members → axial force only
- m + r = 2j is a necessary but not sufficient determinacy check
- Zero-force members carry nothing under this load case but may still be needed
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint