leads to the following expression for the pressure at which yielding begins 

 on the outside fiber of the shell plating midway between adjacent ring 

 frames: 



P,3= p.-^^ ,===-=r (31) 



-^&--^JS] 



_ (l-l//2)°( 

 where a = • 



o(+p+{l-p)Fi 



If the uniaxial criterion of Rankine is applied to the circumferential 

 membrane (midfiber) stress "^(bi^. i.e., Equation (16) with ^xbm ^^^ equal 

 to zero, the following expression for the pressure at which yielding has 

 penetrated through the plating thickness is obtained: 



o„h/R 



In deriving the expressions for Pj^j, Pc3> ^.nd Pc5j it has tacitly been 

 assumed that a uniaixial state of stress exists in the shell plating whereas, 



in reality, a biaxial state exists. More realistic criteria for axisymmetric 



17 

 collapse can be derived from the energy-of-distortion theory which grew 



out of the analytical work of Huber, Von Mises, and Hencky. Since the 

 octahedral shear- stress theory gives the same results as the energy-of- 

 distortion theory and permits the use of a more familiar quantity, such as 

 stress, the former theory will be used in what follows. For a biaxial state 

 of stress at midbay, i.e., midway between adjacent ring frames, defined 

 by the principal stresses "^Xm a-^d '^$rn. the octahedral shear-stress is 



33 



