ELISHA MITCHELL SCIENTIFIC S0CII:TY. '-U 



It is admitted, however, that it is difficult to cut the 

 stones with the exactness demanded, and in addition, there 

 'will be a slight )-ielding of the centers, though the stones 

 can easily be cut to bear along the whole length of joint 

 when placed in position on the centers after they have 

 yielded somewhat, as it only requires a close fit of the key- 

 stone after the other stones are in place. 



If thin cement mortar joints be used, that are allowed to 

 harden perfectly before the centers are removed, the arch 

 ring is assimilated completely to a solid arch, except that 

 in the theory the successive blocks of cement and arch 

 stones with their different moduli of elasticity must be con- 

 sidered, making the solution very complex. It would 

 seem though that for very thin joints the theory pertain- 

 ing to a homogeneous arch of stone should approximate 

 sufficiently near to the truth to give results of practical 

 importance. 



For thick mortar joints of common mortar or for brick 

 arches the theory proposed may be a rude guide, but it is 

 not pretended that it can be anything but a rough approxi- 

 mation to the truth, so that the depth of key for such 

 arches had better be increased empirically over the depths 

 given by the theory above for a homogeneous solid arch. 



The theory of the solid arch supposes immovable abut- 

 ments and it requires three conditions to be fulfilled when 

 the a?r/i ring is under stress from its own weight and the 

 weight of backing, roadway, etc., and any loads that may 

 be placed on it in any position: 



1. The end tangents, at the springing, to the center line 

 of the arch ring, must remain fixed in direction; 



2. The deflection of one end of the arch ring below the 

 other, due to its deformation under stress, must be zero; 



3. There must be no change in span due to the deforma- 

 tion of the arch rinsf. 



