Navier

In section 4 of Navier's 1826 Leçons he tackles the problem of the redundant truss. The example used is that of a weight II supported from the ground by a number of bars, fig. 7.1, and the problem is to determine the forces in the bars. Navier states that if the number of bars is more than two in the same plane, or more than three not in the same plane, then the equations of equilibrium do not determine the values of the bar forces. Navier shows how the problem may be solved, using the three-bar plane-truss example of fig. 7.1.

There is first a short digression in which Navier attempts to estimate limits within which the bar forces must lie. If, for example, all bars are removed from a plane truss of the type sketched in fig. 7.1, except for the two necessary to carry the load, then the forces in those two bars may be found from the equations of statics. By considering different arrangements of bars to produce such statically determinate trusses, a greatest load may be found for a particular bar; the stability of that bar against buckling may then be checked, using the ‘Euler’ theory of a previous section of the Leçons. (These observations are, of course, incorrect. Even in the absence of the load II, a turnbuckle tightened in bar A'C will produce compression in the two outer bars, and buckling of one or the other will eventually occur. The ability of a redundant truss to sustain self-stress was certainly known to Maxwell (1864).)

Navier then lays out clearly the three groups of equations required for the elastic solution of the problem.