184 Mr. Horace Lamb on the 



compared with h. Let the band be divided into rectangular 

 elements by two systems of normal sections respectively 

 perpendicular and parallel to its length. The stresses across 

 any section of the former system will reduce to a tension P, 

 and a couple G, both reckoned per unit length ; whilst across 

 any section of the second system we shall have a tension Q, 

 a couple H, and a shearing force Z in the direction of the 

 normal. All these quantities are functions only of the dis- 

 tance (x, say) from the medial line. Considering the equili- 

 brium of a rectangular element of breadth dx, and resolving 

 parallel to x, we find 



dx U ' 

 and since Q vanishes at the free edges, it follows that 



Q=o (i) 



everywhere. Again, resolving along the normal, we have 



dx p 

 and taking moments about a parallel to the medial line, 



ax 

 Hence 



d 2 H P n 



^~- p -° 0» 



The remaining boundary conditions are evidently H =0, Z = 0, 

 that is 



H =°> sr=° m 



The functions which occur in these equations have now 

 to be expressed in terms of the deformation of the middle 

 surface of the band. If a rectangular plate of thickness 2h 

 undergo extensions <r x , <r 2 , parallel to the edges, the corre- 

 sponding tensions are, per unit length, 



F= x + Z ^ + <T<T > ] ' h 



n 4:{\ + fl)H . , 



w 



