CHAP, m.] SUPPLEMENT TO CHAP. XIV. 287 



CHAPTER III 



AEOH HINGED AT ABUTMENTS ONLY - CONTINUOUS AT OEOWN. 

 A. PARABOLIC ABC CONSTANT CROSS-SECTION CONCENTRATED LOAD. 



8. Horizontal Tlirut. We can apply here directly the results of 

 Art. 5. Thus, in equations (22) for x = 0, A <(>' = A ^>" = 0, hence A' = A". 

 If then in eq. (V.) of that Art. we put A' for A", and then for A A' ita 

 value from (I.), we have at once 



This is the formula which we have given in Art. 159 of the text, without 

 demonstration. The thrust is greatest when the load is at the crown. We 



have then z = and H = fi P ^. The value of V is given in Art. 3. 



n 



9. Intersection Curve. Denote the ordinate of the curve cdeik 

 (PL 23, Fig. 91), taken above the line A B by y. Then we see from the 



V 



Fig. that y = A N tang, d A N = (a 2) ^. The value of H is given 



above, that of V has already been found in Art. 3, eq. (16). viz., 



V = P - . Hence we have, after reduction, 

 a 



(28) 



which is the equation already given in Art. 159, and from (18) and (19) 

 the values of the table in that Art. have been calculated. 



The above values of H and V are simple and of easy application, not in- 

 volving much calculation in any special case. Hence we can readily com- 

 pute H and V, and thus check the accuracy of our method of construction 

 given in Chap. XIV. 



B. CIRCULAR ARC CONSTANT CROSS-SECTION CONCENTRATED LOAD. 



1O. Horizontal Thrust. Here we can apply directly the results of 

 Art. 6. Thus, inserting in eq. (TV.) of that Art. A A' for A A*, and 

 taking the value of A A' from (I.), we have an equation for the deter- 

 mination of H. This, after reduction, becomes 



. sln'o rin?|3 + 2 cos a (COB ft cos a) 2(l + <e) cos a (a sin a /3 sin ft) 

 2 [a 3 sin a cos a + 3 (1 + K) a catf a] 



which is the equation given in Art. 159 (2) of the text. 

 For the semi-circle, a = 90, sin a = 1, and cos a = 0, and this becomes 



