Historical. The germ of graphical statics. dates back to Stevinus, 'Beghinselen der Waag konst) (1585), who experimented with loaded cords. But the first systematic geometric treat ment was published by Varignon, 'Nouvelle micanique ou (1725). For may years following, graphics was utterly neglected; indeed, Lagrange boasted that his great 'Mecanique (17::), a work of al most 800 pages, contained not a single diagram. Poinsot, 'Statique graphique) (1804), Lame and Clapeyron (1826), Poncelet, Tours de mecanique industrielle) (1828), and Mains, 'Lehrbuch der Statik) (1837) were the only important writers after Varignon. The studies of Maxwell, 1864, and Cremona, 1872, cited above, were the first of the modern investiga tions. To Culmann, whose appeared in 1866, belongs the honor of having produced the first great treatise on graphical statistics as a branch of technical mechanics. Among recent treatises Levy, 'Statique graph ique) (1886-88), Henneberg, der starren Systeme) (1886), Miiller-Breslau, 'Die graph. Statik der Baukonstruktionen) (5th ed., 1912) and Mohr, 'Abhandlungen aus dem Gebiete der technischen Mechanik) (1905) are perhaps the most important. Mohr's researches, extending over a period of 40 years, are the most note worthy since those of Maxwell and Culmann he has enriched nearly every department of structural mechanics.
- -- Frames. The principal statical systems to be studied by the aid of graphics occur in such structures as bridges and buildings. A frame is a structure built up of rods or links so as to form a geometric net work. It is jointed or pin-connected if its members are hinged by a single pin or rivet at each joint. If the joints are made rigid the frame is statically indeter minate but its internal reactions can be ob tained by assuming elastic deformation to take place. The forces which arise in this case are called secondary stresses. Engesser and Man derla made the first successful studies of them in 1878 and 1879; for the literature see Gehler, eisener Fachwerkbrficken' (1910). Consult also the books by Grimm and Johnson cited below under graphical solutions. Since the jointed triangle is the simplest frame it is usually chosen as the structural unit. It was used in ancient Roman and Egyptian con structions, and by the 16th century fairly com plicated structures had been built. The external reactions on a frame cannot be found if they contain more than three unknown elements be cause there are only three conditions for the equilibrium of the frame as a whole. For this reason the fastenings of a structure to its abutments (supports) must be designed to secure rigidity without introducing more than three unknowns, and to allow slight motion due to deformations caused by elastic yielding and temperature changes without permitting the structure appreciably to change its position. Without going into the details of actual con struction, we shall divide them into two types: hinge support to permit turning, and flat abut ment to allow sliding.
Statically determinate structures. A struc structure is statically determinate when all ex ternal and internal forces can be found from the conditions for equilibrium of a rigid body with out regard to elastic or temperature deforma tions. The idea is due to Mfibius. For each link of a pin-connected frame there are three equa tions of equilibrium and for each pin two equa tions. (See MECHANICS). Hence a frame con taining 1 links and p pins will yield 2,p + 31 equations. Although the frame as a whole will give three more equations, containing, however, only external reactions, they are superfluous because they can be derived from the 2p + 31 equations by eliminating the internal forces. Or from another point of view, the whole frame is in equilibrium if every part is. Consider now the number of unknown quantities. The reaction at a hinge or pin is unknown in both magnitude and direction, or else consists of two forces unknown in magnitude but given in direction. That is, since each link can be held in equilib rium by two forces at each end, I links require 41 unknown external magnitudes, the opposites of which act on the nins. In addition to these there are in general three unknown external elements: total, 41 -r 3. As the number of equations must equal the number of unknowns 1 = 2p 3, which is the criterion to be satisfied by a statically determinate frame. One link with three pins counts as three links, being statically equivalent to a collapsed triangle; likewise, one link with n pins counts as 2n-3. By similar reasoning we find that I links will rigidly connect f frames if 1--=3f 3. Observe that 2p 3 is odd. If n links meet at every pin, 2 p = 2p 3 because each link connects two pins; since p is a positive integer, n = 2 or 3 and /=3 or 9: see Fig. 7. A frame is re dundant or over-rigid if it has more than enough links, and deformable if it has less. Some frames which satisfy the criterion may be indeterminate and vice-versa; this is due to special elements in their configurations. For example in Fig. 7, i, ii and iii are all con structed in about the same way. But the forces in ii and iii cannot be determined because instantaneous centers (see Knremierics) can be found for the parts above the section lines S. On the other hand v should be indeterminate but is not; vi is interesting because it contains no triangles at all.
Graphical solutions. Graphical methods of solving problems in statics depend on several powerful theorems: I. A body on which forces act at only two points transmits force along the line joining those points. The theorem of four forces is a special case, the points being the two inter sections.
FIG. 7.
II. Three forces in equilibrium are either concurrent or parallel. This is the basis of the funicular construction.