FLAT PLATES 



141 



which is the required formula for the maximum stress p in a homo- 

 geneous elliptical plate of thickness h and semi-axes a and b. 



Problem 143. A cast-iron manhole cover 1 in. thick is elliptical in form, and 

 covers an elliptical opening .3 ft. long and 18 in. wide. How great a uniform pres- 

 sure will it stand ? 



120. Maximum stress in homogeneous square plate under uniform 

 load. In investigating the strength of square plates the method of 

 taking a section through the center of the plate and regarding the 

 portion of the plate on one side of this section as a cantilever is used, 

 but experiment is relied upon to determine the position of the dan- 

 gerous section. From numerous experiments on flat plates, Bach 

 has found that homogeneous square plates under uniform load always 

 break along a diagonal.* 



Consider a homogeneous square plate of thickness h and side 

 2 a, which bears a uniform load w per unit of area. Suppose that 

 a diagonal section of this plate is taken, 

 and - ( .n>i.l.-r either half as a cantilever, 

 as shown in Fig. 99. Then the total load 

 on the plate is 4wa*, and the reaction of 

 the support under each edge is wa 2 . If d 

 denotes the length of the diagonal AC\ tin- 

 resultant pressure on each edge of the plate 



is applied at a distance from AC, and 

 therefore the moment of these resultants 



about AC is 2 (wo?) - , or - - The total load on the triangle ABC 

 4 "2 



is 2 wa*, and its resultant is applied at the center of gravity of the 



triangle, which is at a distance of - from AC. Therefore the mo- 



b 



iin-Mt of the load about AC is (2 wo?) - or - - Therefore the tota] 



b 6 



external moment M at the section A C is 



Fio. 99 



M 



wa?d 



* Bach, Elasticitdt u. Festigkeitslehre, 3d ed., p. .561. 



