-5- 



If the forces and moments are expressed through the deformations, and 

 these through the displacements u, v, and w, we have in equations I to III the ne- 

 cessary equations for the determination of u, v, and w, 

 3. Expression for Displacements. 



Since the equations I to' III become linear and homogeneous after the in- 

 troduction of u, V, and w, and do not contain the coordinates x, yezplicitly, we 

 can substitue sine and cosine terms for the dependent variables, u, v, w in a well 

 known manner. Let us write 



u = A sin n opsin ^s^ v = B cos nqpcos -^-^ 



— 'a — ' a 



(11) 



w = sin na>cos -^-S 

 — 'a 



The change of sine and cosine is governed by the equations themselves 



as we will see below. The term n can be used to signify a real integral number, 



and the termora quantity such that ^^ makes an uneven integral number, where / 



an 



is the free length of the tube. For only if this expression is an uneven integer 

 are the radial displacements w for x = + Z. equal to zero. It is sufficient for us 

 to consider the case in which the uneven integral number has the value 1 (one), 

 since all other cases (multiple buckling) are then easily explained. We have then 



CT=«ZL (12, 



If we substitute equation (11) in equations (1), (4), and (6), we get 



a-2 



a: 



C^ + 3) 



Let us introduce these values in equations (J), (5), and (7). For sim- 

 plification we will write (l-o-)(nBfl) =■ C, nB + 1 - crA = D. 

 Then 



aJ^- ={-^i{l-aXn'l>-^'')-C(o^-cc')]ur -—(15) 



e,-l^['''+<'''-'*^^]'^ 



