174 STRENGTH OF MATERIALS 



that is to say, the average unit stress on any joint must not exceed 

 one twentieth of the ultimate compressive strength of Hit- mt< rmL 



The above conditions for stability can be applied equally as well 

 to a concrete arch by considering the stress on any plane section of 

 the arch ring. 



141. Maximum compressive stress. Let R denote the resultant 

 pressure on any joint, b the width of the joint, /' its area, and > the 

 distance of the jenter of pressure from the center of gravity of the 

 joint. Then, under the assumption of a linear distribution of stress, 



the stress on the joint is due to a uniformly distributed thrust of 

 -p 



amount - l - per unit of area, and a moment M of amount M = Re. 

 JF 



Therefore the unit stress p at any point is given by tin- formula 



R , M. 

 v = * 

 F 1 



where e is the distance of the extreme fiber from the center of gravity, 

 and /is the moment of inertia of the cross section. 



For a section of unit length, F = h- 1 =b, / = ' > and e = ^ . 



Therefore, substituting these values, tin- formula for maximum r 

 minimum stress becomes 



= ^ Rc 



rmvt i im 



mm tr 



For c = - the minimum stress is zero, and if c > - it becomes nega- 

 te 6 



tive, thus restricting the center of pressure to lie within the middle 

 third of the cross section if tensile stress is prohibited (rumpaiv 

 Article 62 and Article 140, 2). 



Combining this result with that of the preceding artiele, the m 

 mum stress calculated by the formula 



= R 6 Re 

 Pm " b b* 



must not exceed -^ , where u c is the ultimate compres- :^[}\ 



of the material. 



142. Location of the linear arch : Moseley's theory. In order to 

 obtain a starting point for the construction of the linear aivh. it is 

 necessary to know the amount, direction, and point of application 



