Mechanics of particles and systems Definition

Truss

Definition

Let $d \in \set{2, 3}$ , let $G = (V, E)$ be a finite graph, and let $p \colon V \to \R^d$ place the vertices so that $p(i) \ne p(j)$ for every edge $ij$ . A truss (or pin-jointed framework) is such a placement, read as follows.

Each vertex $i$ is a frictionless pin joint at $p(i)$ , and each edge $ij$ is a straight bar from $p(i)$ to $p(j)$ .
External forces $f_i \in \R^d$ act only at the joints.
Each bar $ij$ carries only an axial force $t_{ij} = t_{ji} \in \R$ , positive in tension and negative in compression.

The truss is in equilibrium under $f$ when each free joint $i$ satisfies $\sum_{j : ij \in E} t_{ij} \, \frac{p(j) - p(i)}{\norm{p(j) - p(i)}} + f_i = 0.$ Here a free joint is one that is not a support, and the denominator is the Euclidean distance of the two joints.

Remarks

Example: three bars forming a triangle in the plane with joints $p(1) = (0, 0)$ , $p(2) = (2, 0)$ , $p(3) = (1, 1)$ , pinned at joint $1$ , on a roller at joint $2$ , with a downward load $f_3 = (0, -F)$ . At joint $3$ the two bars point along $(-1, -1)/\sqrt{2}$ and $(1, -1)/\sqrt{2}$ , and the horizontal balance gives $t_{13} = t_{23}$ . The vertical balance gives $-\sqrt{2}\, t_{13} - F = 0$ , so both inclined bars are in compression with $t_{13} = t_{23} = -F/\sqrt{2}$ .

Non-example: a portal frame whose columns and beam are welded rigidly at the corners. Its members carry bending moments, so clause 3 fails, and it is a frame rather than a truss.

Reading path

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Euclidean distance concept
Graph concept
Truss definition

Tag	`QT4EJ0Q136`
URI	`m2://QT4EJ0Q136`
Kind	definition
Writing	W0 stub
MSC	70C20, 74K10, 52C25
Defines	Truss
Uses	Graph, Euclidean distance
Changed	3 November 2026

Truss

Definition

Remarks

Reading path

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