104 



OBLIQUE LOADING. 



Art. 73 



moments about this axis is zero ; the sum of the moments of the 

 stresses about this axis must also be zero, that is, 



CsdAz = Q 



Substituting the value of s from equation (9), 

 :=0or 



C 

 I 



(17) 



When equation (17) holds, the plane of the bending moment 

 and that of a moment equivalent to the sum of the moments of 

 the bending stresses are coincident. In Fig. 67, there is, in the 

 second quadrant, an element of area for every one, similar!/ 

 located, in the first quadrant. The product vz is plus in the first 



quadrant and minus in the second, hence j vzdA, for these 

 quadrants is zero. Similar reasoning applies for the third and 

 fourth quadrants; therefore, when the plane of the bending mo- 

 ment cuts the cross section of a beam in an axis of symmetry, the 



I vzdA = ; this is also true lohen there are similar areas 



similarly situated in the first and fourth quadrants, 

 and in the third and second quadrants, as shown in 



Fig. 76, because j vzdA for these quadrants cancel 



J s* 7 



each other, and therefore I vzdA = 0. If the plane of 



the bending moment corresponds with ZZ, we have 

 the case of symmetry. For these two cases the neutral 

 axes are at right angles to each other. The similar 



axes for a single angle of equal legs are shown in 

 Fig. 77. 



A Z-bar cross section does not -have an axis 

 of symmetry, but there are also two axes, at right 

 angles to each other, each of which will be the 

 neutral axis when the other is the trace of the 

 plane of the bending moment. For these axes 

 J vzdA = as will be seen by ref- 

 erence to Fig. 78. For the axes 

 F^ and Z^Z 19 Cv^dA^O for''* 



the web, since they are axes of 

 symmetry; for the flanges this is 

 not true because v^ is minus for 

 both of them. Now if the axes be 

 turned so as to take some position 



76 - 



--z 



