188 STRENGTH OF MATERIALS 



series of vertical sections taken at equal intervals along the rib. Thus, 

 since ds in this case is constant, 



r = 



DE-z 



I 



in which the functions under the summation signs are to be calculated 

 for each section separately, and their sum taken. After r has been 

 found in this way the linear arch is obtained by decreasing the 

 ordinates of the equilibrium polygon in the ratio r : 1, and the stress 

 can then be calculated as explained in Article 149. 



This method of determining the linear arch is due to Ewing. 



154. Temperature stresses in two-hinged arched rib. When the 

 temperature of an arched rib changes, the length of the rib also changes, 

 and consequently stresses called temperature stresses are produced in 

 the rib (compare Article 19). .To calculate the amount of this stress 

 let L denote the coefficient of linear expansion and T the change in 

 temperature in degrees. Then each element of the rib of length //.% 

 changes its length by the amount LTds, the horizontal projection <>f 

 which is LTdx. Therefore the total change / in the length of the rib is 



= r C 



Jo 



/ 



Jo 



where 2 c is the span. From Article 152, the total change in length 

 of the span is given by the expression 



/= / =*>. 



Therefore 



= C 



"Jo EI 



jf: 



To simplify this expression assume that the modulus of elasticity 

 E is constant throughout the rib, and that the moment of inertia / 



ds 



increases towards the abutments hi the ratio Under this assump- 



ds 

 tion / = / , where J denotes the moment of inertia at the crown, 



and the above equation becomes 



1 r 2c 



Mzdx = 2 cLT. 



