190 STRENGTH OF MATERIALS 



J2c 

 z*dx and integrating, 



we have 



Consequently for a parabolic arched rib the horizontal reaction of 

 the abutments due to a change in temperature of T degrees is 



Ph ' 16c-/r 8 A 2 



15 



155. Continuous arched rib fixed at both ends. For a continuous 

 arched rib fixed at both ends the problem <>t constructing the 

 librium polygon is subject to a threefold indetermination, >imv 

 of the three conditions necessary for its deter mi nation 

 The theoretical solution of the question by the principle ,.!' lc; ( >t 

 work is as follows. 



Let the vertical reaction R lf the horizontal reaction /',, and the 

 bending moment M Q at the left support be chosen as the three 

 unknown quantities necessary to determine the linear arch. 1 

 system of concentrated loads the moment M at any section of the 

 rib distant x from the left support is 



M = M + Rx - P h z - 



in which d is the distance of any load /' from the left support, and 

 the summation is to be extended over all the loads between the left 

 support and the point under consideration. Similarly, for a uniform 

 load of amount w per unit of length, 



t -- - 



Now, from Article 72, the work of deformation W is given by the 



expression 



in which M has the value given by one or the other of the 

 expressions, depending on whether the loading is concent rat el or 

 uniform. To apply Castigliano's theorem to this expression it is 



