310 SUPPLEMENT TO CHAP. XIV. [CHAP. VI. 



We have, as before, simply to make P=pdz, and then integrate between 

 the limits z a and z indeterminate. 

 We thus find at once 



(78) 



64a*Al 

 For * = a, this reduces to zero, as it should. For z = a, the load covers 



2 



the whole span, and we have H=^ . For = 0, the load reaches from 



2 h 



j 



the left as far as the crown, and H =^ rT . The formulae is simple, and re- 



4 n 



quires no table. Numerical values may be easily inserted. 



33. Horizontal Thrut Circular Arch. As already noticed, 

 the vertical end reaction for this case has been given in eq. (71). It re- 

 mains to determine the thrust. We have, as before, simply to insert pdx = 

 p r cos d in place of P in the expression for the thrust for concentrated 

 load of Art. 10, and then integrate between the limits /3 = a and /3 inde- 

 terminate. 



We have thus similarly to that Art. 



where H is the value of H when terms containing * are neglected, or 



The quantities Ai, Bi, A a and B 2 are as follows : 



Ai = 7 sin 3 a + 8 a cos a 3 sin a 6 a cos a sin* a 6 sin* a sin /3 

 + 2 sin 3 j8 3 cos a 9 cos a sin /3 cos /3 + 12 cos 2 a sin /3 

 + 12 a cos a sin a sin ft 6 ft cos a sin 2 /3. 

 A = 3 [2 a cos a sin* a + a cos a sin a cos* a 4 a COS a sin a sin ft 



+ 2 /3 cos a sin* /3 cos a + cos a sin 3 cos j8]. 

 BI = a 3 sin a COS a + 2 a COS* a. B a = 2 a COS 2 a. 



These expressions can be tabulated as in Art. 10, or developed into 

 series as in that Art., and the formula thus made practically available. 



For /3 = a, we have H zero, as should be. For /3 = a, we have the 

 load covering the entire span. 

 For this case we have 



__1 rina-8(l- 8sln2a)(idntt-acogtt) -3icco8a(a+2aBln rinaOOBa) 

 ' ~" 6^ f -a + 2 a cos* o 8 sin a cos + 3 K cos* a 



For the SEMI-CIRCLE, this reduces simply to 



H = p r = 0.424 p r. 

 8 IT* 



In any case where exact results are desired, eq. (74) must be used, and a 

 table calculated for the central angle a. We have approximately by 

 series also, more especially for small central angles, or for a large in respect 

 to A, for total load over whole span : 



.pa* 8 A A* _pa* 8 h* 

 ~ 



