HOMOGENEOUS BEAMS 141 



oblique neutral axis y'-y' makes an angle of 45 with the 

 horizontal. With the angle section with unequal legs and 

 with the Z-bar section, this oblique neutral axis y"-y makes 

 an angle a with a vertical line. Instead of giving this angle 

 direct, it is more convenient to give a value for the trigono- 

 metric tangent of twice the angle, and this is done in the 

 last column of the table. 



The formulas given in the table are long and difficult to 

 use. The values of the properties of various structural- 

 steel sections have therefore been calculated for all the 

 standard sizes of angles, channels, etc. that are ordinarily 

 manufactured. The tables containing these values will be 

 found on page 147, where their use will be explained. 



In the two tables just given, the moment of inertia is 

 taken about the neutral axis. Sometimes, however, it is 

 desirable to find the moment of inertia of the section about 

 some axis other than the neutral axis. This may be accom- 

 plished as follows: 



Let A represent the area of any section; 7, the moment 

 of inertia with respect to an axis through the center of grav- 

 ity of the section; Ix, the moment of inertia with respect to 

 any other axis parallel to the former; and p, the distance 

 between axes. Then, 



EXAMPLE. Determine the moment of inertia of a triangle 

 with respect to its base (see fourth section of the table entitled 

 Elements of Usual Sections). 



SOLUTION. The distance between the base and the neutral 

 axis ci is J d, the area is i bd, and I = fe bd 3 . Then, by the 

 preceding formula, 



bd 3 bd 3 bd 3 



-+--- 



Many sections may be regarded as built up of simpler 

 parts; they are called compound sections. For example, a 

 hollow square consists of a large square and a smaller square; 

 the T shown in the table entitled Elements of Usual Sections 

 consists of two slender rectangles, one horizontal and one 

 vertical. 



