LELATIO1 



r EEN MOMENT AND SHEAR. 



107 



by preventing deflection in the direction of the axis of Z\ this 

 cancels Msina&nd then equation (10) is applicable. 



It is evident that the position of the neutral axis is not 

 needed in getting the maximum fiber stress, but it may easily be 

 found from equation (19), if the moments of inertia about the 

 principal axes are known (7 V and Z z ). If the coordinates, v' and 

 2', of a fiber on the neutral axis are substituted in (19), s must 



be zero and sine, -- = cosa-r- 



or 



^ tana -j-=tan($ 



(20) 



in which (3 is the angle between the neutral axis and the axis of Z. 

 In the above case, Fig. 80, tan /? = 0.5 - = 9.28 and 



/? = S3 51'. /? determines the direction of the neutral axis 

 and the deflection must be in a direction perpendicular to it. 



When the direction of the neutral axis is not known, it may, 

 in certain cases, be doubtful which are the extreme points; this 

 is easily determined by calculating the stresses from equation 

 (19) 1 . 



74. Relation Between the Bending Moment and the Shear 



at Any Cross Section 

 of a Beam. A very 

 important and very 

 T simple relation exists 



^ between the bending 

 ' moment and the 

 shear. Fig. 81 shows 

 Fig. 81. a beam supported at 



its ends and carrying concentrated loads, but the same relation 

 holds in beams loaded and supported in any manner. R is the 

 resultant of the forces to the left of the section pq. 



R = S X = R^-yP and M x = Rl) = JB^-SJP (xa) 

 For the section at a distance dx from pq, 

 = R (b+dx) = 



and the increase of the moment is dM x = S K dx. The rate of in- 

 crease of the moment along the beam is therefore, 



S, (21) 



dx 



*For a more general discussion of this general case of bending, with 

 examples, see Johnson's Modern Framed Structures p. 154, or German 

 works on technical mechanics or the statics of construction. 



