**5. Calculation of nonlinear deformation and stability of shells**

The proposed MMPC method has been applied to calculate the deformation and stability of a long flexible elastic circular cylindrical shell of radius *R* with half the central angle 0 in the case of cylindrical bending under uniform external pressure with a simple support of the longitudinal edges. The corresponding system of resolving equations in the normal form is given in [8]. Dependences of "dimensionless intensity of pressure *P* – deflection / *w R* " for

the top cross-section of the shell at different angles and dimensionless flexibility 4 *С* 10 are shown in Fig. 4. The dependence of the dimensionless intensity of limit load \* *P* on the size of half angle 0 is shown in Fig. 5.

Applications of 2D Padé Approximants in Nonlinear Shell Theory:

2

The proposed method can be used in a combination with the known asymptotic method. Consider free vibrations of a flexible elastic circular cylindrical shell of radius *R* , thickness *h* and length *L* , backed by a set of uniformly distributed stringers having a simple support

The calculation is based on mixed dynamical equations of the theory of shells after splitting them in powers of natural small parameters [9]. The shape of radial deflection *w* satisfies

1 11 22 2 11 *w f t sx sx f t sx* ( )sin cos ( )sin .

12 2 <sup>2</sup> 2 1 *f R sf* 0,25 ,

, are the parameters characterizing the wave generation along the

Here 1 2 *f* , *f* functions depend on time and are related by the condition of continuity of

The governing equations can be reduced by the Bubnov – Galerkin method to the Cauchy

1 1 *t tB R* 

2 3 5 12 3

 

, 0: , 0

*AA A*

 

> 

*f* / *R* on <sup>2</sup>

*d*

 

> 1 1

*t f dt*

at the ends.

displacements

where <sup>1</sup>

with [9])

1 2 *s ml s n* 

problem with respect to 1

generator and directrix, respectively.

 

**Figure 5.** The dependence of limit loads *<sup>b</sup> P* versus 0

the boundary conditions given in the form

Stability Calculation and Experimental Justification 15

(all symbols are taken in accordance

(22)

0,

 

(1 - data [8], 2 - calculation).

**Figure 3.** Approximate solutions of Eq. (18) for = 0.2 (1 – three terms ADM, 2 – <sup>1</sup> *z* for ADM, 3 – *<sup>x</sup> z* for ADM, , 4 – three terms HAM, 5 – *x z* for HAM, 6 – <sup>1</sup> *z* for HAM and MMPC, 7 – *<sup>x</sup> z* and 2-D Padé for MMPC).

For comparison, Fig. 4b also shows the dependence of the critical loads for inextensible shell obtained by S.P. Timoshenko [8]. We see that dependences are in good agreement, while consideration of deformation of the longitudinal axis substantially affects the value of critical loads of the construction.

**Figure 4.** The dependence of the intensity of pressure *P* versus deflection *w R*/ for different values of 0 (the value of 0 is indicated by curves).

The proposed method can be used in a combination with the known asymptotic method. Consider free vibrations of a flexible elastic circular cylindrical shell of radius *R* , thickness *h* and length *L* , backed by a set of uniformly distributed stringers having a simple support at the ends.

14 Nonlinearity, Bifurcation and Chaos – Theory and Applications

is shown in Fig. 5.

**Figure 3.** Approximate solutions of Eq. (18) for

critical loads of the construction.

2-D Padé for MMPC).

0 

(the value of 0

is indicated by curves).

*<sup>x</sup> z* for ADM, , 4 – three terms HAM, 5 – *x z* for HAM, 6 – <sup>1</sup> *z*

of half angle 0

the top cross-section of the shell at different angles and dimensionless flexibility 4 *С* 10 are shown in Fig. 4. The dependence of the dimensionless intensity of limit load \* *P* on the size

For comparison, Fig. 4b also shows the dependence of the critical loads for inextensible shell obtained by S.P. Timoshenko [8]. We see that dependences are in good agreement, while consideration of deformation of the longitudinal axis substantially affects the value of

**Figure 4.** The dependence of the intensity of pressure *P* versus deflection *w R*/ for different values of

= 0.2 (1 – three terms ADM, 2 – <sup>1</sup> *z*

for HAM and MMPC, 7 – *<sup>x</sup> z* and

for ADM, 3 –

The calculation is based on mixed dynamical equations of the theory of shells after splitting them in powers of natural small parameters [9]. The shape of radial deflection *w* satisfies the boundary conditions given in the form

$$w = f\_1(t)\sin s\_1 \mathbf{x}\_1 \cos s\_2 \mathbf{x}\_2 + f\_2(t)\sin^2 s\_1 \mathbf{x}\_1 \dots$$

Here 1 2 *f* , *f* functions depend on time and are related by the condition of continuity of displacements

$$f\_2 = 0.25R^{-1} 
 s\_2^2 f\_1^{22}$$

where <sup>1</sup> 1 2 *s ml s n* , are the parameters characterizing the wave generation along the generator and directrix, respectively.

The governing equations can be reduced by the Bubnov – Galerkin method to the Cauchy problem with respect to 1 *f* / *R* on <sup>2</sup> 1 1 *t tB R* (all symbols are taken in accordance with [9])

$$\begin{aligned} \ddot{\xi} + \alpha \xi \Big[ \left( \dot{\xi} \right)^2 + \xi \ddot{\xi} \Big] + A\_1 \xi + A\_2 \xi^3 + A\_3 \xi^5 &= 0, \\ \left( \stackrel{\cdot}{\phantom{\cdot}} \Big) \equiv \frac{d \left( \stackrel{\cdot}{\phantom{\cdot}} \right)}{dt\_1}, t\_1 = 0 : \underline{\xi} = f, \dot{\underline{\dot{\varphi}}} = 0 \end{aligned} \tag{22}$$

**Figure 5.** The dependence of limit loads *<sup>b</sup> P* versus 0 (1 - data [8], 2 - calculation).

The application of the proposed method of parameter continuation to the Cauchy problem (22) gives approximation of the second order for the artificial parameter for frequency of nonlinear oscillations in the form

Applications of 2D Padé Approximants in Nonlinear Shell Theory:

automated processing of the results of the holographic research has been proposed, which eliminates the above drawbacks [12]. Next, we have extended this technique to study the motion of shell structures of zero Gaussian curvature which is based on modern means of an interactive data processing. The surface of zero Gaussian curvature can be approximated with sufficient accuracy with respect to the system of flat rectangular panels whose sides are segments close to the case which occurred during the analysis of generators. To determine all components (points) of the displacement vector, three holograms of a circuit design interferometer based on a reference beam is used. The interferometer is shown schematically

**Figure 7.** The scheme of the interferometer (1 – laser generator, 2 – mirror, 3 – expanding lens, 4 –

After registering the two exposures, i.e. unloaded and loaded state of the object, we get a flat image of the interference pattern corresponding to the observation of points , , *Mi <sup>н</sup><sup>i</sup> <sup>н</sup><sup>i</sup> <sup>н</sup><sup>i</sup> xyz* , *i* 1,3 . Let us enter the order line using a computer in the following manner. The photos of interferograms are scanned and entered into the computer memory in the form of graphic files with the extension, for example, jpg, which is the most popular choice of compression of graphic information on all platforms, or equivalently in other file formats. Next, the file is displayed on the screen in a specially designed box on the toolbar image processing. The information produced is removed by a successive mouse click on the corresponding image points at the request of a specially created database. Algorithms for further processing of the data are widely described in [12]. In the *XOY* coordinate system (Fig. 8) associated with the imaging plate, base point *MBBB x y* , and a segment of the *OY* axis of the *XOY* coordinate system, whose direction coincides with the vertical axis of the projection, are given. Further calculations are performed in the *XOY* system in which the entered coordinates of the points of lines

, cos sin , cos sin *B B xx y xyy x y*

in Fig 7.

studied object, 5 – camera)

of equal order are transformed by the formulas

Stability Calculation and Experimental Justification 17

$$
\Omega = \sqrt{\frac{1 + f^2 \left(A\_2 \mid A\_1\right) + f^4 \left(A\_3 \mid A\_1\right)}{\left(1 + af\right)}} \dots
$$

It is seen that the oscillations are not isochronous. This agrees well with previous results reported in reference [9] (Fig. 6). However, our approach allows for a significant reduction of the computation time (in [9] to obtain similar results the approximation of the fourth order is taken).

**Figure 6.** Amplitude of the initial disturbance versus oscillation frequency of stringer shell (1 – according the proposed method, 2 – data [9]).
