142 



RESISTANCE OF MATERIALS 



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 

 dangerous 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 consider either half as a canti- 

 lever, as shown in Fig. 103. Then the total 

 load on the plate is 4wa 2 , and the reac- 

 tion of the support under each edge is wa 2 . 

 If d denotes the length of the diagonal AC, the resultant pres- 

 sure on each edge of the plate is applied at a distance from AC, 



ft 

 and therefore the moment of these resultants about AC is 2 (wa?} -, 



2, 



or - The total load on the triangle ABC is 2 wa\ and its result- 



2i 



ant is applied at the center of gravity of the triangle, which is at a 



distance of - from AC. Therefore the moment of the load about 

 b , , 



AC is (2wa 2 )-, or - Therefore the total external moment M 

 b o 



at the section A C is 



M= 



wa z d 



236 

 Hence the maximum stress in the plate is 



wa?d h 



2 



from which 



(201) p = i 



The maximum stress in a square plate of side 2 a is therefore the 

 same as in a circular plate of diameter 2 a. 



* Bach, Elasticitat und Festigkeitslehre, 3d. ed., p. 561. 



