ARCHES AND ARCHED RIBS 



189 



The only forces which tend to resist the change in length of the rib 

 due to temperature stresses are the horizontal reactions P h of the 

 abutments. Therefore the external moment at any section of the rib 

 with ordinate z is M = P h z, and substituting this value in the above 

 integral, it becomes 



whence 



This expression is easily evaluated in any given case, thus determin- 

 ing P h and consequently the linear arch. The temperature stresses 

 can then be calculated by the methods explained above, and combined 



_ 2 EIjLT 



^ i. 



FIG. 136 



with those due to the given loading. For a rise in temperature above 

 that for which the arch was designed, T is positive and the horizontal 

 reactions P h of the abutments act inwardly ; for a fall in temperature 

 T is negative and the reactions P h act outwardly. 



To illustrate what precedes, the above formula will now be applied 

 to a parabolic arched rib, which on account of its simplicity is the 

 form ordinarily assumed in designing. Let h denote the rise of 

 the arch, 2 c its span, and x, z the coordinates of any point A on 

 the rib (Fig. 136). Then, from the intrinsic property of the parabola, 



it follows that 



h - z _ (c xf . 



~~ ~~ 



whence 



hx 



- 



