SECTION V 



DEFLECTION OF CANTILEVER AND SIMPLE BEAMS 



36. General deflection formula. By a simple beam is meant one 

 which is simply supported at the ends. The only external forces 

 acting on it in addition to the loads are, then, the two vertical reac- 

 tions at the supports. A cantilever is a beam which overhangs, or 

 projects outward from the support, the loads on it being equili- 

 brated by the moment at the support and by the vertical reaction 



at this point. The results of 

 applying the general deflec- 

 tion formula, derived below, 

 to these two classes of beams 

 will be made the basis of the 

 treatment of continuous and 

 restrained beams in the sec- 

 tions which follow. 



Taking a vertical longi- 

 tudinal section of a beam, 

 the line in which this plane 

 intersects the neutral-fiber 

 surface is called the elastic 

 curve. Any small segment, 

 Az, of the elastic curve may be considered as a circular arc with 

 center at some point (Fig. 61). This point is therefore called 

 the center of curvature for the arc A#. The radius of curvature is 

 not constant, but changes from point to point along the beam. 

 Evidently the radius of curvature is least where the beam is curved 

 most sharply. 



Any two adjacent plane sections, AB and DH (Fig. 61), origi- 

 nally parallel, intersect after flexure in the center of curvature 0. 

 Let KC = A# denote the original length of the fibers, and draw 



60 



H 



FIG. 61 



