Trusses by joints and sections
A truss is a framework of straight members joined at their ends to form triangles. Because a triangle cannot change shape without a member changing length, a truss is rigid and efficient. In the ideal model, joints are frictionless pins and loads are applied only at joints, so every member carries pure axial force: tension or compression, with no bending. Roof trusses, bridge girders, transmission towers and crane booms are all analysed this way first.
For a plane truss, compare m + r with 2j, where m is the number of members, r the number of reaction components and j the number of joints. If m + r is less than 2j, the truss is unstable (a mechanism). If m + r equals 2j, it is statically determinate provided it is also geometrically stable; if m + r is greater, it is indeterminate and needs deformation methods. The count is a necessary condition only: a truss with the right count can still be unstable if members or reactions are badly arranged, for example three parallel or three concurrent reactions.
The method of joints isolates each pin. Two equilibrium equations (ΣFx = 0, ΣFy = 0) are available per joint, so start at a joint with no more than two unknown members, usually a support after finding reactions. Work joint to joint. Assume every unknown member is in tension (pulling away from the joint); a negative result means compression. Watch for zero-force members: at an unloaded joint with two non-collinear members, both are zero; at an unloaded joint where two members are collinear, the third is zero.
The method of sections cuts the truss through no more than three members of unknown force and treats one side as a free body with three equations. Taking moments about the point where two of the cut members meet gives the third directly. It is the fastest way to get one member force in the middle of a long truss, and it underlies the influence lines used for moving loads on bridge trusses. Real trusses have rigid joints and self-weight, causing small secondary bending, which design codes deal with separately.
Find the reactions first, then pick the joint or cut that isolates the member you need.
Determinacy count (plane truss): m + r = 2j (necessary, not sufficient; check geometric stability)Joints: ΣFx = 0, ΣFy = 0 at each pinSections: ΣFx = 0, ΣFy = 0, ΣM = 0 on the cut partChord force near mid-span ≈ M / h (bending moment over truss depth)
- Ideal truss model: pin joints, loads only at joints, members straight and weightless, so members carry axial force only.
- Real gusset or welded joints add secondary bending; self-weight between joints adds local bending; both are checked separately in design.
- Small-displacement, linear elastic behaviour is assumed; compression members must also be checked for buckling.
Problem. A Pratt truss 24 m long with 6 panels of 4 m and depth 3 m carries 20 kN at each of the 5 interior bottom joints. The diagonals slope down towards mid-span. Find the top and bottom chord forces in the third panel from the left (between 8 m and 12 m).
- Total load 100 kN, so the reactions are 50 kN each by symmetry.
- Cut through the third panel. The cut passes through the top chord, the diagonal (running from the top joint at 8 m down to the bottom joint at 12 m) and the bottom chord.
- Top chord: take moments about the bottom joint at 12 m, where the diagonal and bottom chord meet. Moment of external forces on the left part = 50 × 12 − 20 × (8 + 4) = 600 − 240 = 360 kNm, so the top chord force = 360 / 3 = 120 kN, compression.
- Bottom chord: take moments about the top joint at 8 m, where the top chord and diagonal meet. Moment = 50 × 8 − 20 × 4 = 400 − 80 = 320 kNm, so the bottom chord force = 320 / 3 = 106.7 kN, tension.
- Check with the M / h rule: the bottom chord force follows the moment at the joint opposite it (8 m), the top chord force the moment at 12 m.
Answer. Top chord 120 kN compression; bottom chord about 107 kN tension
Quick quiz
1State the determinacy condition for a plane truss.
m + r = 2j, and the truss must also be geometrically stable.
2In the method of sections, how many unknown members may be cut?
Generally no more than three.
3At an unloaded joint with two non-collinear members, what are their forces?
Both are zero.
References and further reading
Standard texts, listed in no order of merit. We do not copy text, figures, tables or solved problems from any book; our explanations and examples are original.
- F. P. Beer, E. R. Johnston, Vector Mechanics for Engineers: Statics and Dynamics, McGraw Hill, latest edition.
- R. C. Hibbeler, Engineering Mechanics: Statics, Pearson, latest edition.
- S. Timoshenko, D. H. Young, Engineering Mechanics, McGraw Hill, 4th edition, 1956.
Free and official sources
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