The method is specially appropriate when the frame, although just rigid, is not “simple” in the sense of § 6, and when accordingly the method of reciprocal figures is not immediately available. To avoid the intricate trigonometrical calculations which would often be necessary, graphical devices have been introduced by H. Müller-Breslau and others. For this purpose the infinitesimal displacements of the various joints are replaced by finite lengths proportional to them, and therefore proportional to the velocities of the joints in some imagined motion of the deformable frame through its actual configuration; this is really (it may be remarked) a reversion to the original notion of “virtual velocities.” Let J be the instantaneous centre for any bar CD (fig. 12), and let s1, s2 represent the virtual velocities of C, D. If these lines be turned through a right angle in the same sense, they take up positions such as CC′, DD′, where C′, D′ are on JC, JD, respectively, and C′D′ is parallel to CD. Further, if F1 (fig. 51) be any force acting on the joint C, its virtual work will be equal to the moment of F1 about C′; the equation of virtual work is thus transformed into an equation of moments.

Fig. 12.Fig. 51.
Fig. 52.

Consider, for example, a frame whose sides form the six sides of a hexagon ABCDEF and the three diagonals AD, BE, CF; and suppose that it is required to find the stress in CF due to a given system of extraneous forces in equilibrium, acting on the joints. Imagine the bar CF to be removed, and consider a deformation in which AB is fixed. The instantaneous centre of CD will be at the intersection of AD, BC, and if C′D′ be drawn parallel to CD, the lines CC′, DD′ may be taken to represent the virtual velocities of C, D turned each through a right angle. Moreover, if we draw D′E′ parallel to DE, and E′F′ parallel to EF, the lines CC′, DD′, EE′, FF′ will represent on the same scale the virtual velocities of the points C, D, E, F, respectively, turned each through a right angle. The equation of virtual work is then formed by taking moments about C′, D′, E′, F′ of the extraneous forces which act at C, D, E, F, respectively. Amongst these forces we must include the two equal and opposite forces S which take the place of the stress in the removed bar FC.

The above method lends itself naturally to the investigation of the critical forms of a frame whose general structure is given. We have seen that the stresses produced by an equilibrating system of extraneous forces in a frame which is just rigid, according to the criterion of § 6, are in general uniquely determinate; in particular, when there are no extraneous forces the bars are in general free from stress. It may however happen that owing to some special relation between the lengths of the bars the frame admits of an infinitesimal deformation. The simplest case is that of a frame of three bars, when the three joints A, B, C fall into a straight line; a small displacement of the joint B at right angles to AC would involve changes in the lengths of AB, BC which are only of the second order of small quantities. Another example is shown in fig. 53. The graphical method leads at once to the detection of such cases. Thus in the hexagonal frame of fig. 52, if an infinitesimal deformation is possible without removing the bar CF, the instantaneous centre of CF (when AB is fixed) will be at the intersection of AF and BC, and since CC′, FF′ represent the virtual velocities of the points C, F, turned each through a right angle, C′F′ must be parallel to CF. Conversely, if this condition be satisfied, an infinitesimal deformation is possible. The result may be generalized into the statement that a frame has a critical form whenever a frame of the same structure can be designed with corresponding bars parallel, but without complete geometric similarity. In the case of fig. 52 it may be shown that an equivalent condition is that the six points A, B, C, D, E, F should lie on a conic (M. W. Crofton). This is fulfilled when the opposite sides of the hexagon are parallel, and (as a still more special case) when the hexagon is regular.

Fig. 53.

When a frame has a critical form it may be in a state of stress independently of the action of extraneous forces; moreover, the stresses due to extraneous forces are indeterminate, and may be infinite. For suppose as before that one of the bars is removed. If there are no extraneous forces the equation of virtual work reduces to S·δs = 0, where S is the stress in the removed bar, and δs is the change in the distance between the joints which it connected. In a critical form we have δs = 0, and the equation is satisfied by an arbitrary value of S; a consistent system of stresses in the remaining bars can then be found by preceding rules. Again, when extraneous forces P act on the joints, the equation is

Σ(P·δp) + S·δs = 0,

where δp is the displacement of any joint in the direction of the corresponding force P. If Σ(P·δp) = 0, the stresses are merely indeterminate as before; but if Σ (P·δp) does not vanish, the equation cannot be satisfied by any finite value of S, since δs = 0. This means that, if the material of the frame were absolutely unyielding, no finite stresses in the bars would enable it to withstand the extraneous forces. With actual materials, the frame would yield elastically, until its configuration is no longer “critical.” The stresses in the bars would then be comparatively very great, although finite. The use of frames which approximate to a critical form is of course to be avoided in practice.

A brief reference must suffice to the theory of three dimensional frames. This is important from a technical point of view, since all structures are practically three-dimensional. We may note that a frame of n joints which is just rigid must have 3n − 6 bars; and that the stresses produced in such a frame by a given system of extraneous forces in equilibrium are statically determinate, subject to the exception of “critical forms.”