=88. The Truss Element or Triangle of Bracing.=—A number of the preceding formulæ find their applications to bridge-trusses, as well as to beams; hence it is necessary to give attention at least to some simple forms of those trusses. [Illustration: FIG. 18.] [Illustration: FIG. 18_a_.] The skeleton of every bridge-truss properly designed to carry its load is an assemblage of triangles. In other words, the truss element, i.e., the simplest possible truss, is the triangular frame, such as is shown in skeleton in Figs. 18 and 18_a_. These simple triangular frames are sometimes called the King-post Truss. The action of such a triangular frame in carrying a vertical load is extremely simple. In Fig. 18 let the weight _W_ be suspended from the apex _C_ of the triangle. The line _CF_ represents that weight, and if the latter be resolved into its two components parallel to the two upper members of the triangular frame, the two component forces _CG_ and _CD_ will result. If from _D_ and _G_ the horizontal lines _DH_ and _GO_ be drawn, those two lines will represent the horizontal components of the forces or stresses in the two bars _CA_ and _CB_. The force _HD_ will act to the left at the point _A_, and the force _CG_ will act to the right at _B_, and as these two forces are equal and opposite to each other, equilibrium will result. Either of the horizontal forces will represent the magnitude of the tension in _AB_. Both _AC_ and _CB_ will be in compression, the former being compressed by the force _CD_, and the latter by the force _CG_. The manner of drawing a parallelogram of forces makes the triangle _COG_ similar to _CNB_, and _CHD_ similar to _CNA_; hence _HW_ divided by _CH_ will be equal to _AN_ divided by _NB_. But _HW_ is the vertical component of the stress in _CB_, while _CH_ is the vertical component of the stress in _AC_, the latter being represented by the reaction _R_ and the former by the reaction _R′_. It is seen, therefore, that the weight _W_ is carried by the frame to the two abutment supports _A_ and _B_, precisely as if it were a solid beam. In other words, the important principle is established that when weights rest upon a simple truss supported at each end they will produce reactions at the ends in accordance with the principle of the lever, precisely as in the case of a solid beam. In engineering parlance it is stated that the weight _W_ is divided according to the principle of the lever, and that each portion travels to its proper abutment through the members of the triangular frame. If the two inclined members of the triangular frame are equally inclined to a vertical, the case of Fig. 18_a_ results, in which one half of the weight goes to each abutment. The triangular frame, with equally inclined sides, shown in Fig. 18_a_, is evidently the simplest form of roof-truss, constituting two equally inclined members with a horizontal tie.