140 RESISTANCE OF MATERIALS 



For the beam AS of length 2 a the maximum external moment is 



R^a. Also, since AB is assumed to be of unit width, /= and e = - 

 Hence the maximum stress p 1 in AB is 



Similarly, the maximum stress p n in CD is 



2 h 2 

 Consequently, 



J = 4' 



7? 73 



or, since from equation (197) - = , 



~LL n a 



p"~ a 2 ' 



By hypothesis a >b. Therefore^" >p f , that is to say, the maxi- 

 mum stress occurs in the strip CD (that is, in the direction of the 

 shorter axis of the ellipse). In an elliptical plate, therefore, rupture 

 may be expected to occur along a line parallel to the major axis 

 a result which has been confirmed by experiment. 



87. Maximum stress in homogeneous elliptical plate under uniform 

 load. The method of finding the maximum stress in an elliptical 

 plate is to consider the two limiting forms of an ellipse, namely, 

 a circle and a strip of infinite length, and express a continuous 

 relation between the stresses for these two limiting forms. The 

 method is therefore similar to that used in Article 54 in obtaining 

 the modified form of Euler's column formula. 



Consider first an indefinitely long strip with parallel sides, 

 supported at the edges and bearing a uniform load of amount 

 w per unit of area. Let the width of the strip be denoted by 2 b 

 and its thickness by h. Then, if this strip is cut into cross strips 

 of unit width, each of these cross strips can be regarded as an 

 independent beam, the load on one of these unit cross strips 



being 2 bw and the maximum moment at the center being - 



o 



