CHAPTER VIL 



DEFLECTION OF BEAMS AND GIRDERS. 



81. Equation of the Elastic Line. The bending stresses 

 in a beam shorten the fibers on one side, and lengthen them on 

 the other side, of the neutral plane; the result is, the beam is 

 bent, that is, it is deflected from its unstrained position ; its axis 

 forms a curve, called the elastic line. 



Only elastic deflections are of importance, because working 

 stresses are taken well within the elastic limits. It is sometimes 

 important to know the elastic deflection of a beam or girder, 

 although this is quite small in properly designed work. 



Since the bending stresses are determined according to the 

 theory of flexure, the equation of the elastic line must be based 

 on this theory. The origin of coordinates is usually taken at a 

 support, y being the deflection at a distance x from it. The 

 amount of the deflection evidently depends upon the length of 

 the beam, the manner in which it is supported, and the load 

 which it carries; these are taken account of by M. It also de- 

 pends upon the elastic property of the material, and the amount 

 and distribution of material in the beam ; these are taken account 

 of by E and 7 respectively. The equation of the elastic line must 

 therefore be a relation between y, x, M, E, and 7. It is evident 

 too that y, the deflection, increases when M increases, and de- 

 creases when E and 7 increase. / 1 



A general equation of the elastic 

 line is easily derived. Fig. 89 repre- 

 sents a piece of a beam in which the 

 bending caused the sections AB and 

 CD, a distance dx apart, to make an 

 angle d<j>wiih each other. A fiber at 

 a distance v from the neutral plane 

 NN, originally dx long, is increased 

 in length an amount t>d<,and the unit 



