=109. Application to Draw- or Swing-bridges.=—In general the reactions or supporting forces of the beams and trusses of ordinary civil-engineering practice are vertical, and all their points of application are known. Hence there are but two equations of equilibrium, equations (36) and (37), for external forces. These two equations for the external forces and the _n_ - 2 equations derived from the theorem of three moments are therefore always sufficient to determine the _n_ reactions. After the reactions are known all the stresses in the bars or members of the trusses can at once be found. The preceding equations and methods as described are constantly employed in the design and construction of swing- or drawbridges. =110. Special Method for Deflection of Trusses.=—The method of finding the elastic deflections produced by the bending of solid beams has already been shown, but it is frequently necessary to determine the elastic deflections of bridge-trusses or other jointed or so-called articulate frames or structures. It is not practicable to use the same formulæ for the latter class of structures as for the former. The elastic deflection of a bridge- or roof-truss depends upon the stretching or compressions of its various members in consequence of the tensile or compressive forces to which they are subjected. Any method by which the deflection is found, therefore, must involve these elastic changes of length. There are a number of methods which give the desired expressions, but probably the simplest as well as the most elegant procedure is that which reaches the desired expression through the consideration of the work performed in the truss members in producing their elastic lengthenings and shortenings. The general features of this method can readily be shown by reference to Fig. 29. It may be supposed that it is desired to find the deflection of any point, as _J_, of the lower chord produced both by the dead and live load which it carries. It is known from what has preceded that every member of the upper chord will be shortened and that every member of the lower chord will be lengthened; and also that generally the vertical web members will be shortened and the inclined web members lengthened. If there can be obtained an expression giving that part of the deflection of _J_ which is due to the change of length of any one member of the truss independently of the others, then that expression may be applied to every other member in the entire truss, and by taking the sum of all those effects the desired deflection will at once result. While this expression will be found for some one particular truss member, it will be of such a general form that it may be used for any truss member whatever; it will be written for the upper-chord member _BC_ in Fig. 29. [Illustration: FIG. 29.]