CHAP. XV.] THE STONE AKCH. 31 9 



when the neutral axis touches the cross-section, or P acts in 

 the kernel, the strain S is twice as great as when P passes 

 through the centre of gravity of the joint surface and is uni- 

 formly distributed. 



As P passes beyond the kernel, the neutral axis, as we have 

 seen, enters the joint area, and on the side away from P occa- 

 sions, or would occasion in a beam, tensile strains. But as the 

 assumption is that the joint (neglecting mortar) cannot resist 

 tensile strains, we may remove all that portion on the opposite 

 side of the neutral axis without increasing the pressure on the 

 other side. 



In this case, then, the central ellipse is not that for the whole 

 joint area, but only for that portion on the same side as P, and 

 P is upon the kernel for that portion. 



This portion can be determined directly for a certain posi- 

 tion of P only in a few individual cases ; generally, it must be 

 found by trial. We must first find for the central ellipse of 

 the entire joint area the neutral axis corresponding to given 

 position of P, and then draw a parallel cutting off somewhat 

 more of the area. Then determine the central ellipse of the 

 cut-off portion, and see if the pole lies symmetrically to the 

 pole of the cutting line. 



The parallelogram is one of the areas in which we can de- 

 termine directly the amount cut off when P acts at a point 

 upon the line joining the centres of two opposite sides. For if 

 we cut off by a parallel to these sides a portion so that P is at 

 d of the line joining the centres of the opposite sides of the 

 new parallelogram, then P lies upon the kernel for this new 

 area. The proof is easy. The moment of inertia of the 

 parallelogram is -^ b A 3 , with reference to the diameter b. The 

 square of the radius of gyration <z a is then ^ A". The distance 

 of the point of application of P from one of the sides is 



a' 

 i + m = i + . Hence 



^ 



The half height of the kernel is, then, -J-th the height of the 

 parallelogram, or the kernel occupies the inner third. (See Fig. 

 36 ; also Woodbury : Theory of the Arch, p. 328, Art. 3.) 



Fcr any given position of P 3 then, three times its distance 



