Art. 40 



METHODS OF APPLICATION. 



47 



and is, therefore, applicable for finding reactions only. These are 

 evidently independent of the shape of the structure, and may 

 often be conveniently gotten by considering one or more result- 

 ants of the loads, in place of the loads themselves. 



The conditions at the supports must be known. If a beam, 

 supported at its two ends, for example, is not free to move longi- 

 tudinally at one support, the reactions are statically indetermi- 

 nate, and it is useless to write equations of equilibrium; but the 

 assumption of such freedom is usually on the safe side. 



The second method evidently applies only to trusses but both 

 stresses and reactions (in certain cases) may be gotten by its 

 means. The forces dealt with are concurrent, their lines of action 

 meeting at the center of the joint. In this method only two of the 

 equations are independent as explained in Art. 35. For an ex- 

 planation as to which equations to use, see Art. 41. 



The third method is called the method of sections, and is 

 applicable in finding stresses in all sorts of structures and the 

 only method for finding stresses in simple beams and girders. 



If a structure is cut in two by a plane or curved section, 

 either part must be in equilibrium, if the forces cut off are con- 

 sidered like external forces. This method, then, reduces to the 

 first method, the only difference being that in one case the whole 

 structure is considered and in the other a part of it. It may also 

 be considered to reduce to the second method when the part cut 

 off includes but a single joint. 



It is very important to leave out of consideration altogether 

 any forces acting on that part of the structure which has been 

 discarded, and to include all those acting on the other part loads, 

 reactions and stresses. To avoid confusion the beginner should 

 invariably erase the discarded part, or place a piece of paper over 

 it when writing the equations of equilibrium. 



It is evident that, since a section may be taken through any 

 part of the structure, as many sections as are necessary to find 

 all stresses desired may be made successively. It is useless, how- 

 ever, to consider any section cutting more than three unknowns, 

 because there are but three independent equations. 



41. Which Equations of Equilibrium to Use. A choice may 

 be made of any of the following combinations of the equations 

 of equilibrium in solving for three unknowns. 



