ANALYSIS OF STRESS IN BEAMS 65 



60. Antipole and antipolar. The theorems in the preceding para- 

 graph prove that if the point of application of an eccentric load lies 

 outside, on, or within the inertia ellipse, the corresponding neutral 

 cuts this ellipse, is tangent to it, or lies wholly outside it. 

 This relation is analogous to that of poles and polars in analytical 

 geometry, except that in the present case the point and its corre- 

 sponding line lie on opposite sides of the center instead of on the 

 same side. For this reason the point in the present case is called 

 till* antipole, and its corresponding line the antipolar. 



The following theorem is analogous to a well-known theorem of 

 poles and polars. 



If the antipole moves along a fixed straight line, the antipolar 

 revolves about a fixed point. Conversely, if the antipolar revolves 

 about a fixed point, the antipole moves along a fixed straight line. 



If tin- antipole moves to infinity, the antipolar, or neutral axis, 

 passes through the center of gravity of the section, which is the 

 ordinary case of pure bending strain. The bending moment in this 

 case can be considered as due to an infinitesimal force at an infinite 

 distance from the center of gravity. 



If tin- anti{K)le coincides with the center of gravity, the neutral 

 axis lies at infinity, which means that the stress is uniformly dis- 

 tributed over the cross section. 



e the stresses on opposite sides of the neutral axis are of oppo- 

 site siuMi. if the neutral axis cuts the cross section, stresses of both 

 signs occur (Le. both ten>ion and compression), whereas if the neutral 

 axis lies outside the cross section, the stress on the section is all of 

 the same sign (Le. either all tension or all compression). 



61. Core section. Let it be required to find all positions of the 

 point of application of an eccentric load such that the stress on 

 the cross section shall all be of the same sign. From the preceding 

 article, the condition for this is that the neutral axis shall not cut 

 the cross section. If. then, all i>ossible lines are drawn touching the 

 cross section or having one point in common with it, and the anti- 

 poles of these lines are found, the locus of these antipoles will form 

 a closed figure, called the core section. 



a point within or on the boundary of the core section the neu- 

 tral axis lies entirely without the cross section, or, at most, touches it, 



