CHAP. XTV.] THE BRACED AECH. 271 



tion the Bridge over the Mississippi River at St. Louis, by 

 Copt. Eads / one over the Elbe near Hamburg on the Paris- 

 Hamburg It. JR., in which, however, the outward thrust of the 

 arch is balanced by a precisely similar inverted braced arch, 

 or suspension system. Thus the piers have to support a verti- 

 cal reaction only, and the necessity of large and expensive abut- 

 ments of masonry for resisting the horizontal thrust is obviated. 



The strains in the inverted arch of this character are found 

 in a precisely similar manner. The only difference is that the 

 reactions, and therefore the vertical and horizontal components, 

 act now in a direction opposite to the direction for the upright 

 arch, and the strains, though the same in amount, are of re- 

 verse character in each piece. 



The bridge over the Rhine at Coblem is an illustration of 

 the braced arch pivoted at the abutments only. 



Examples of the solid or cast-iron arch of all kinds are 

 common. 



165. Strains due to Temperature. In the first class of 

 braced arches, viz., pivoted at both abutments and crown, there 

 are evidently no strains due to changes of temperature. The 

 arch can accommodate itself to any change of length by rising 

 at the crown and turning at the abutments, and no strains are 

 induced. 



We represent by e the coefficient of linear expansion for one 

 degree [about 0.000012 for iron, for every degree centigrade], 

 and by t, the temperature above or below the mean tempera- 

 ture # , for which no strain exists. 



Then for arch pivoted at abutments only, we have for the in- 

 crease of thrust,* 



__ 2 E I e t sin a _ 

 ~ i* (a 3 sin a cos a + 2 a cos 2 a) + 2 K r* a cos 8 a, 

 where E is modulus of elasticity, I moment of inertia of cross- 



section, and K = -g ; A being area of cross-section, r radius of 

 arch, a the angle of half span, or, approximately, 



= 



(2 a 4 + 15 *) ~ 8 Atf + 15 I' 

 where h is the rise of arch. 



*Lehre von der Ektstioit&t. Winkler. Also Supplement to this chapter, 

 Art. 26. 



