:;:; i 



Ml!, I,. N. (1. FILON ON THE RESISTANCE TO 



1 fi 



and cos $ being always < 2, S A is always numerically greater than S B for small 



values of a. This confirms the usual rule, which we should expect, since the section 

 approximates in this case to a flat rectangular section. 



12. Discussion of the Variation* in these Stresses. 



The variations in the stresses S A , S B are shown in fig. 9 (p. 340). This figure gives, 

 not the values of the stresses themselves, but the ratio of the stresses to the maximum 

 stress S in the circular section of equal area. This ratio is plotted as ordinate to the 

 various values of a as abscissae. Of course, S A , as given by the expressions (21) 

 being negative throughout, the curve shows the ratio ( S A /S ) and not S A /S . 



The diagram is comparatively rough, especially near the origin, owing to the very 

 limited number of points which I could calculate. The value of ft selected was 



ft = IT/6. 



When a is small, it is not difficult to show that, for ft = -ir/6 



SB/S O = 1 799 X x/, S A /S = - 2-563 X v 7 ", 



and when a is large 



SB/S O = 5-196e-, S A /S = - 7831. 



These last enable us to see the form of the curves near the origin and at a great 

 distance from it. They are perpendicular to the axis of a at the origin, and at first 

 the curve of S B lies below that of S A . At some point between a = and a ='5 they 

 cross. This corresponds to the case of the square for rectangular cross-sections. S A 

 is now less than S B , but instead of its remaining so, as we should have expected, the 

 curves cross again near the value of a = 1 7. The curve of S A /S now tends to 

 become practically a straight line parallel to the axis, at a distance 7831 from it, and 

 the curve of S B /S approaches the axis asymptotically. 



The values of S A /S , S B /S are given in the tables below. 



TABLE of S A /S . 



