**1. Introduction**

The widest class of shells used in the civil and mechanical engineering is the class of shells with developable principal surface. The stress-strain state of shell structures under loads, which corresponds to buckling, is inhomogeneous, significantly bended, and nonlinear. Permanent interest of researchers in the problem of inhomogeneous compression of shells of zero Gaussian curvature has not led so far to a correct solution. Therefore, there is a need for the development and application of new methods that allow considering the problem in a complex setting, the most appropriate to study real behavior of structures.

The approximate analytic integration of nonlinear differential equations of the theory of flexible elastic shells in most practical cases is based on the method of continuation of solution on the artificially introduced parameter. They can be satisfactorily applied only with an effective method of summation. The most natural analytical continuation method is that using Padé approximants (PAs). PAs effectively solves the problem of analytical continuation of power series, and this is a basis of their successful application in the study of applied problems. Currently, the method of PAs is one of the most promising non-linear methods of summation of power series, and the localization of its singular points. Recently, the method of PAs for single-variable functions has been successfully extended to the approximation of two variable functions (2D PAs).

A method that provides polynomial asymptotics of the exact solution of the general form and its meromorphic continuation based on 2D Padé approximants is proposed in this work. Several examples of displacements, stability and vibration calculations for inhomogeneous loaded shells with developable principal surface are presented. The accuracy of 2D PAs theoretical results are confirmed by experiments with stainless steel

<sup>© 2012</sup> Olevs'kyy et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

specimens based on holographic interferometry. It is shown that the application of PAs provides sufficient accuracy in the studied area that confirms the advantage of our proposed approach.

Applications of 2D Padé Approximants in Nonlinear Shell Theory:

1 2 1 2

 

0 0 0 0 (, ) , , , 1, *n n m m*

*i j i j*

be the class of rational functions, i.e. the ratio of 2-D polynomials whose degrees do not exceed 1 2 *n nn* , and 1 2 *m mm* , for each variable. It may be written briefly as

Each rational function *r Rnm* (, ) may be identified with its power series that converges in some neighborhood of the origin. It should be mentioned that *r pq R* depends on

The set of integer points <sup>2</sup> *I nm Z* , for fixed 1 2 *n nn* , and 1 2 *m mm* , is called the

2. 11 22 *n m n m I nm* ,0 , 0, , (this property guarantees that in the case when 1*z* 0 (or 2 *z* 0 ) one would have the classical 1-D rational approximation of Padé type),

4. if 1 2 *k k I nm* , , then 0, , *k I nm* , where <sup>2</sup>

*I nm ij* <sup>1</sup> , , :

0 ,0 *in jn n in m jm i n jn m* 1 2 1 11 2 2 22 1 ,0 0, 1 ,

*I nm i j* <sup>2</sup> , , :

0 ,0 *i n j n i mn j n m n i n mj* 1 2 0 ,1 12 2 2 1 1 1 1 , 0,

The generalized PAs for given 1 2 *n nn* , and 1 2 *m mm* , are defined as the rational function , *nm F Rnm* for which 0 *ij nm TFF* for all *ij I nm* , , , where *ij T*

Taylor's coefficients of the power series for function *F*. The rational function *nm F* is called the 2-D PAs 11 22 *nm nm* , /, of 1 2 *Fz z*, which corresponds to the determinative set

As in the 1-D case, the existence and uniqueness of PAs (in the sense of the above given definition) for <sup>2</sup> *C* require special type of analysis. It should be mentioned that PAs do not

a. Two and only two possible variants of these sets satisfying requirements are:

*nn mm* parameters (the coefficients of *p* and *q*).

determinative (interpolation) set, if it has the following properties:

*<sup>p</sup> Rnm r p pzz q q zz q q*

deg( ) ,deg( ) *pn qm* .

1. dim , *nm I nm*

3. 1 2 *n n n I nm* , , ,

– the rectangle rule,

*ij I nm* , , .

5. 1 12 *n m m Inm* , (, ) or 12 2 *m n m Inm* , (, ) .

always exist in the sense of the given definition.

12 1 2 ( 1)( 1) ( 1)( 1) 1 *nm*

, 1 2 1 2 00

*i i j j ij ij*

Stability Calculation and Experimental Justification 3

1 2 0, , : 0 , 1,2 *j j k ss Z s kj*

are
