FLAT PLATES 



143 



D 



2a 



fft 



89. Maximum stress in homogeneous rectangular plate under 

 uniform load. In the case of rectangular plates experiment does 

 not indicate so clearly the position of the dangerous section as it 

 does for square plates. It will be assumed in what follows, how- 

 ever, that the maximum stress occurs along a diagonal of the rec- 

 tangle. This assumption is at least approximately correct if the length 

 of the rectangle does not exceed two or three times its breadth. 



Let the sides of the rectangle be denoted by 2 a and 2 b, and the 

 thickness of the plate by h (Fig. 104). Also let d denote the 

 length of the diagonal AC, and c 

 the altitude of the triangle ABC. 

 Now suppose that a diagonal sec- 

 tion AC of the plate is taken, and 

 consider the half plate ABC as a 

 cantilever, as shown in Fig. 104. 

 If w denotes the unit load, the 

 total load on the plate is 4 abw, and 

 consequently the resultant of the 

 reactions of the supports along 

 AB and BC is of amount 2 abw and 



s* 



is applied at a distance - from AC. 



Zi 



Therefore the moment of the sup- 

 porting force about AC is abwc. 



Also, the total load on the triangle ABC is 2 abw, and it is applied 







at the center of gravity of the triangle, which is at a distance of 



o 



from AC. Consequently, the total moment of the load about AC is 

 . Therefore the total external moment M at the section AC is 



FIG. 104 



M = abwc 



2 abwc abwc 







and the maximum stress in the plate is 



abwc h 



Me 

 p = - = 



Zwabc 



