Art. 73. OBLIQUE LOADING. 105 



VV and ZZ, some of the area passes into the first and third 

 quadrants (plus) and some passes out of the second and fourth 

 (minus) ; there must be some position of the axes for which the 



plus value of J vzdA. is equal to the minus value and therefore 

 c J 



\ vzdA =-0, and the neutral axis is perpendicular to the plane 



of the bending moment. 



If the following values are calculated for any two re3tangu- 

 lar axes as V :L V 1 and Z^Z^ for example, viz. ( vi 2 dA = 7 Z1 , 

 J zi 2 dA = 7 V1 , and j vizidA = 7 V1Z1 , the j v' 2 dA = 7 Z may be 



found in terms of tb^se quantities and the angle a. Differentiat- 

 ing with respect to a und putting the first differential coefficient 

 equal to zero, it will be found that I z is a maximum or a mini- 

 mum, when a is determined by this equation : 



tan2u=. ? /vlz ; (18) 



J- J zi 



Equation (18) will give two values of 2a differing from each 

 other by 180, or two values of a differing by 90. These two 

 values locate two axes, at right angles to each other, about which 

 the moment of inertia is a maximum or a minimum. These may 

 be distinguished from each other by inspection ; thus in Fig. 76, 

 / z is a maximum and 7 V is a minimum; the corresponding axes 

 are caled principal axes. Every cross section, no matter what 

 its shape may be, has two principal axes. 



Equation (18) shows that when 7 VZ = 0, a =0 and 90, 

 that is t>ie principal axes coincide with the axes of y and Z. 

 Since it I/as been shown that 7 VZ = when one axis is an axis of 

 symmetry, it follows that an axis of symmetry is a principal 

 axis. See Figs. 67, 76 and 77. An axis at right angles to an 

 axis of symmetry is a principal axis, since it has been shown that 

 principal axes make an angle of 90 with each other and that, in 

 this case, 7 VZ = 0. 



For angles of unequal legs, and for Z-bars, the principal 

 axes are located in the Cambria handbook. 



In practice, sections having at least one axis of symmetry ara 

 usually used for beams, and they are so placed that a principal 

 axis lies in the plane of the outer forces. A common exception 

 to this is tho case of a roof purlin, which has an axis of symmetry 



