Truss Determinacy from Members and Joints

For a planar 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. A pin supplies 2 reactions and a roller 1, so r = 3. Then m + r = 12 + 3 = 15 and 2j = 14. Because 15 exceeds 14 by one, there is one more unknown than equilibrium equations: the truss is statically indeterminate to the first degree. Equality (m + r = 2j) would be determinate, and m + r < 2j would be a mechanism. This count is necessary but not sufficient for stability; a poorly arranged truss can still collapse even when the numbers balance.