AECHES AND ARCHED RIBS 



173 



stress over the joints, the center of pressure is restricted to lie within 

 the middle third of any joint (compare Article 62). 



Thus, in Fig. 130 (A), if ABCD represents the distribution of 

 pressure on any joint AD, the resultant ft must pass through the 

 center of gravity of the trapezoid ABCD. Consequently, when the 

 compression at one edge becomes zero, as shown in Fig. 130 (B), 

 the resultant R is applied at a point dis- 



tant - from A, and cannot approach any 



u 



nearer to A without producing tensile 

 stress at D. Therefore, the criterion for 

 stability against overturning is that the 

 center of pressure on any joint shall not 



approach nearer to either edge than - > 



o 



where b is the width of the joint ; or, in 

 other words, that the linear arch must lie 

 irj thin the middle third of the arch ring. 



3. Failure by crushing can only occur 

 when the maximum stress on any joint 

 exceeds the ultimate compressive strength 

 of the material. To guard against this 

 kind of failure, 10 is universally chosen 

 as the factor of safety. Hence, if u c denotes the ultimate compressive 

 strength of the material, and p mat the maximum unit stress on any 

 joint, the criterion for stability against crushing is 



FIG. 130 



m " 10 



From Fig. 130 (B), the maximum unit stress is twice the average. 

 Therefore, if F denotes the area of a joint, and p a the average unit 

 stress on it, ^ 



p a =- and p mAT =2p a . 



Consequently the criterion for stability against crushing can be 

 expressed in the more convenient form 



F 20' 



