CHAP. VI.] STTPPLEMENT TO CHAP. XIV. 311 



where A is the area and I the moment of inertia of cross-section, and g 

 the radius of gyration. In framed arches this may be taken as approxi- 

 mately equal to the half depth from centre to centre of flanges, 



C. ARCH WITHOUT HINGES FIXED AT ENDS, CONTINUOUS AT CROWN. 



33. Parabolic Arcli Formulae for V, H and M. In this 

 case the reactions no longer follow the law of the lever, and e(j. (71), there- 

 fore, no longer holds good. 



(a) Vertical reaction at unloaded end. 



We have from eq. (28), Art. 14, for the reaction at the right end for a 

 single load, 



v - IP (<* ~ g) 2 (2 a + ) 

 ~4 a 8 



Making P = pds, and integrating between the limits z = a and , we 

 find the reaction for a load coming on from left, 



V = ^- [3 a* - 8 a* z + 6 a* z* - s*~\ . . (76) 

 16 a 3 L J 



for a full load z = a and V = pa, as should be. 



(&) Horizontal thrust. 



In like manner we have for the horizontal thrust at end from (36), Art. 



14, 



15 a 4 - 2 a 2 a* + s 1 



~32 3 A 



Replacing P by p d z, and integrating as before, we obtain directly 



H = -^ - fs of - 15 a' z + 10 a 3 z> - 3 /I . . (77) 

 32 a 3 h \- J 



for a full load z = a, and H = ^r-. 



& n 



(c) Moment at unloaded end. 



In precisely similar manner we have from (37), Art. 14, 



1 (a zY (3 a 2 - 10 a z - 5 a 8 ) 



JVLg *r r 



32 a? 



Putting P = p d z, and integrating, we have for the moment, always at 

 the right end, or for load not extending past the centre, at crown 



M = -JL. |~3 a 4 z - 8 a 3 z* + 6 a 2 s 3 - a 5 ] . . (78) 

 For z = a, this is zero, as should be. For s = 0, or for load extending as 



n ^j2 



far as crown, it is also zero. For z = a, the moment at the end is - . 



2 



A negative moment, as always, denotes tension in lower flange. 



Just as for concentrated load, as shown in Art. 14, as the load comes on, 

 the moment at crown is positive, and increases with increasing load up to 

 a certain point, beyond which any load causes a negative moment, and be- 

 yond which the moment at crown, therefore, decreases, until, when the load 



