mTTx 

 u(x,e)s Asinnocos 



v(x,e)= Bcosne sin (54) 



L 



w(x,e)r C sinne sin 



where n and m are real integers denoting the number of full waves around 

 the circumference and the number of half waves in the longitudinal direc- 

 tion, respectively, which form when the cylinder buckles. The buckling 

 shape, Equation (54), may be considered as being the first terms in a more 

 general double trigonometric- series solution. This general type solution 

 will be discussed later in this section, but for our present purposes, 

 attention will be restricted to the solution of, Equation (54). Substituting 

 these expressions in Equations (52), we obtain for A, B, and C three 

 homogenous linear algebraic equations. For a solution other than the 

 trivial one (A = B = C = 0), the determinant formed by the coefficients of 

 A, B, and C in these three- algebraic equations must vanish. Expanding 

 this 3x3 determinant and after some simplifications where only the linear 

 pressure terms are retained, the resulting equation for calculating the 

 critical values of pressure can be put in the following form: 



^1 ^^2°^''^3'''l "^^4'''2 ^^^^ 



in which 



2 4 

 C2 = (n2+x.2)4 _ 2|z/X^+3x4n^+ (4-i/)X^n^+n J + 2(2-1/ )X^n^ + n^ (56) 

 C3 : n2(n2+x2)2 _ (n^+SX^n^) 



47 



