272 THE BKACKD ABCH. [CHAP. XIV. 



For the moment at any point, then, due to change of temper- 

 ature, we have 



M = H r (cos ft cos a), 



/9 being the angle from vertical to that point. 



This moment, if positive, causes tension in outer and com- 

 pression in inner flanges, and we can, as before, easily find the 

 corresponding strains either by diagram or calculation. 



For an arch fixed at ends and continuous at crown, we have 



H = 



2 E I t (1 + K) a sin a 



V* 1(1 + K) (a? -}- a sin a cos a) 2 sin 2 a]' 

 or, approximately, 



45 E I e t 45E!Ae* 



= i* (a 4 + 45 K ) ~ 4 A # + 45 1* 



But this thrust does not act at the abutment, since, if it did, 

 there would be no moment there. It must be considered as 

 acting at a distance for rise of temperature, below the crown of 



a+ K) a sin a 

 _ L _ at 



(I + ic) a 



or, at a distance above the end abutment of h e^, 

 Approximately, we have 



_ a 2 + 6 K _ ( A a 2 + 6 I) h 

 6 3Aa 3 ' 



a being the half span. 



This thrust and its point of application being known, we can 

 easily find the moment, and hence the strains at any point. 

 "We see that the horizontal thrust is about six times as great as 

 for the case of an arch pivoted at both ends. 



The constant moment acting at the abutment, which may be 

 considered as acting at every point, is 



/sin a \ TT * 



A = I -- cosal Hr; * 



it acts to cause compression in outer and tension in inner 

 flanges at abutment. If we find this moment, we can then con- 

 sider H as acting at the end, and then we have for the moment 

 at any point 



Capt. Bads' Report to the lUinois and St. Louis Bridge Co., May, 1868. 



