318 THE STONE ARCH. [CHAP. XV: 



From a drop a perpendicular upon A C, and we obtain the 

 re litre of action for axis at A. A similar construction for other 

 axes, as A B, B C, etc., give us other points, and we thus find 

 the small central parallelogram, which is the kernel or locns 

 of the centres of action of the moment forces for all positions 

 of the axis enveloping the parallelogram A B, C D. A similar 

 construction gives us the kernel for the other figures. 

 We have from Art. 60 



a' 



m = -r-> 

 * 



where in = the distance of the resultant P of the forces acting 

 upon the cross-section from the centre of gravity, and a the 

 semi-diameter of the central curve, and i the distance of the 

 neutral axis from the parallel diameter of the central curve. 



If we call c the distance of an outer fibre from this diameter 

 measured on the side of P, its distance from the neutral axis 

 is i + c. If the strain in this fibre is S, we have 



p 



S : t + c ::: ^, 



where A is the area of the cross-section. Hence 



If P acts at the centre of gravity of the cross-section, * = oo 



p 



(Art. 60), the neutral axis is infinitely distant, and S = y. If 



P moves away from the centre of gravity, the neutral axis 

 approaches, and is always parallel to the conjugate diameter 

 in the central ellipse. When P readies the perimeter of the 

 kernel, the neutral axis touches the perimeter of the cross- 

 section, and at least, then, in one point of this perimeter, the 

 pressure is zero. If P passes beyond the kernel, the neutral 

 axis enters the cross-section, and tensile strains enter on one 

 side to balance the cornpressive strains on the other. The ker- 

 nel then forms a limit beyond which the resultant P must not 

 act. , 



172. Position of Kernel for different Cross-sections. 

 If the cross-section is symmetrical with reference to the cen- 



c P 



tre of gravity, we have = 1, and therefore S = 2 - ; that i&, 



