Mr. J. H. Cotterill on Elliptic Ribs. 27 



for the constants are 



*°~ 2 ' rfH ~ U ' dM ~"' 



the abutments being supposed sensibly immoveable. 



- W 

 Therefore, from equations (A), putting F = — -^-, 



H «{2E 1 -P-2S}+M (E 1 -S) = ^«cos/3(R-Q), 



H^E^S} 4-MqE^ y a cos /3R. 



Let now /3 = 60°, in which case the rise of the rib is equal to 

 its span, then these equations become 



136H a + 208M =46Wa, 



208H a + 484M = 85 Wa ; 



whence 



1 4 



M =qW« and H =^W nearly, 



which are the values of the bending moment at the crown and 

 the horizontal thrust of the rib. The bending moment in any 

 other section is given by 



M=W«{- 2 7 T -L s i n ^-^cos^}; 



thus M diminishes to zero as <f> ! increases, becomes negative, 

 attains a maximum when tan<£' = -f^, diminishes again to zero, 

 becomes positive, its final value being (^>' = 90°) T ^g Wa. On 

 substitution of tan -1 f^ for $', the value of M at the negative 

 maximum will be found to be about ^Wa. 



Thus the maximum stress on the rib is at the crown where the 

 bending moment is |W«. 



Let it now be supposed that the same formula which gives 

 the work accumulated in a rib of homogeneous material, will 

 also give the work accumulated in a brick or stone arch-ring, 

 in other words, that this ring would yield under the action of 

 forces acting on it in the same manner as a solid arch-ring 

 would, provided only that the deviation of the centre of pressure 

 is not beyond the limit necessary for the stability of the ring. 

 This supposition has been employed by Professor Rankine in 

 his work on Civil Engineering. 



If now such a semielliptical arch-ring be supposed in equili- 

 brium under any forces consistent with its stability, and H' be 

 the thrust on any section, then, if a weight W be suspended at 

 the crown, the additional bending moment M will be given by 



