**7. Experimental justification of MMPC**

Loading capacity of cylindrical shells is significantly affected by the unevenness of deformation caused by the ovality ends of the shell [13]. Imperfections in face of shells usually occur as a result of their deformation either under their own weight or during the mechanical handling, storage, as well as installation and assembly of the shells as individual elements. In the case of welded shells, the end face has the form of an oval with *a* and *b* axes and / *a b* compression ratio (or the actual ovality). At the same roundness of the upper end (*a/b*)B may be different from the roundness of the bottom one (*a/b*)*H* because of the conditions introduced by a collection with other elements of the design. In all cases the shape of each end should be within the required tolerances, and roundness introduced by the collection process should not be reduced by more than 0.8.

Another form of imperfections arising from the inaccuracy of the assembly is associated with a weak taper angle characterized by forming the membrane to its axle . A number of studies [14] consider that a small taper with 13 has no significant effect on the magnitude of critical loads of axial compression. However, the results of stability studies of technologically imperfect cylindrical shells based on the multivariate approach [15,16] suggest that an increase of to the value of 3 often leads to a significant change in carrying capacity, and in some cases the interaction with the oval and other factors yields an increase of the critical loads. These studies have shown the need for a more correct approach in establishing the correspondence between the magnitude of these abnormalities and the level of carrying capacity. As a consequence, it is necessary to study the nature of deformation of shells with different ratios of the parameters of roundness and taper.

In order to solve this problem, two-factor second-order experiments on two levels of and *a/b,* and also on two levels of *a/b* at lower and upper ends when *α*° = 1 (when the taper is small the difference between the lower and upper end is missing) have been implemented. Welded specimens with radius *R* = 71.5 *mm* and length *L* = 200 *mm*, made from plate steel of the mark *HN n* 18 9 and with thickness *δ* = 0.25 *mm* have been tested. The use of a multifactor approach allows one to solve correctly the problem of the nonlinear joint influence of defects on the loading capacity of the shell.

Tests on the stability of prototypes carried out on a UME-10TM machine showed that the exhaustion of loading capacity of the shell took place at one stage by reaching a limit point.

The loss of stability of a conical shell with the same low ovality ends (Fig. 9a) is in general related to a form close to its own form of stability loss of oval cylindrical shells under the action of uniform axial compression [17], but shifted to a larger shell butt. On one side of the shell there are two or three belt dents located at the larger end. They cover the smaller curvature of the plate and are shifted to the side of panel larger curvature. Local dents have a relatively large size and do not form a regular closed form buckling.

The increase of taper and roundness of the ends leads to a shift of the zone of wave generation into the longitudinal direction to a lower end (Fig. 9b) while maintaining the overall character of buckling.

20 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**7. Experimental justification of MMPC** 

studies [14] consider that a small taper with

defects on the loading capacity of the shell.

suggest that an increase of

the collection process should not be reduced by more than 0.8.

with a weak taper angle characterized by forming the membrane to its axle

calculation of the stress-strain state parameters to form and direct the shell with a certain step, it is possible to obtain data for plotting the distribution of displacements and stresses.

Loading capacity of cylindrical shells is significantly affected by the unevenness of deformation caused by the ovality ends of the shell [13]. Imperfections in face of shells usually occur as a result of their deformation either under their own weight or during the mechanical handling, storage, as well as installation and assembly of the shells as individual elements. In the case of welded shells, the end face has the form of an oval with *a* and *b* axes and / *a b* compression ratio (or the actual ovality). At the same roundness of the upper end (*a/b*)B may be different from the roundness of the bottom one (*a/b*)*H* because of the conditions introduced by a collection with other elements of the design. In all cases the shape of each end should be within the required tolerances, and roundness introduced by

Another form of imperfections arising from the inaccuracy of the assembly is associated

magnitude of critical loads of axial compression. However, the results of stability studies of technologically imperfect cylindrical shells based on the multivariate approach [15,16]

carrying capacity, and in some cases the interaction with the oval and other factors yields an increase of the critical loads. These studies have shown the need for a more correct approach in establishing the correspondence between the magnitude of these abnormalities and the level of carrying capacity. As a consequence, it is necessary to study the nature of

*a/b,* and also on two levels of *a/b* at lower and upper ends when *α*° = 1 (when the taper is small the difference between the lower and upper end is missing) have been implemented. Welded specimens with radius *R* = 71.5 *mm* and length *L* = 200 *mm*, made from plate steel of the mark *HN n* 18 9 and with thickness *δ* = 0.25 *mm* have been tested. The use of a multifactor approach allows one to solve correctly the problem of the nonlinear joint influence of

Tests on the stability of prototypes carried out on a UME-10TM machine showed that the exhaustion of loading capacity of the shell took place at one stage by reaching a limit point. The loss of stability of a conical shell with the same low ovality ends (Fig. 9a) is in general related to a form close to its own form of stability loss of oval cylindrical shells under the action of uniform axial compression [17], but shifted to a larger shell butt. On one side of the shell there are two or three belt dents located at the larger end. They cover the smaller curvature of the plate and are shifted to the side of panel larger curvature. Local dents have

a relatively large size and do not form a regular closed form buckling.

deformation of shells with different ratios of the parameters of roundness and taper.

In order to solve this problem, two-factor second-order experiments on two levels of

13 has no significant effect on the

to the value of 3 often leads to a significant change in

. A number of

and

**Figure 9.** Forms of supercritical wave generation of the shell with a small taper and the same low ovality of ends (a); with a large taper, and the same large oval ends (b); with a large taper, and a large oval of the lower extremity (c); with a large taper, and a large oval upper end (d)

At high cone (within a given experiment) increased roundness of the lower end, while maintaining the shape of the upper longitudinal, increases the localization of buckling (Fig. 9c) shifting the dents closer to the lower end, while maintaining the variability in the circumferential direction. Conversely, the prevalence of high cone-roundness of the upper end leads to a significant shift of dents to the end of a large oval (Fig. 9d).

Results of the experiment allow us to derive mathematical models of the form

$$\begin{aligned} \left(\mathbf{a} \,\prime \, b\right)\_{H}^{\diamond} &= \left(\mathbf{a} \,\prime \, b\right)\_{B}^{\diamond} = \left(\mathbf{a} \,\prime \, b\right)^{\diamond} : \, K = 0, 379 + 0, 0029 \alpha^{\diamond} - 0, 012 \left(\mathbf{a} \,\prime \, b\right)^{\diamond} - 0, 014 \alpha^{\diamond} \left(\mathbf{a} \,\prime \, b\right)^{\diamond}; \\\\ \alpha^{\diamond} &= 1 : \, K = 0, 346 - 0, 017 \left(\mathbf{a} \,\prime \, b\right)\_{H}^{\diamond} - 0, 0033 \left(\mathbf{a} \,\prime \, b\right)\_{B}^{\diamond} - 0, 013 \left(\mathbf{a} \,\prime \, b\right)\_{H}^{\diamond} \left(\mathbf{a} \,\prime \, b\right)\_{B}^{\diamond} \end{aligned}$$

Applications of 2D Padé Approximants in Nonlinear Shell Theory:

*Н В ab ab* .

at the beginning of loading. The deformation of the shell in the experiment depends on the character of ends, and imperfections differ significantly on the panels of varying curvature.

comparison between model, single-factor experiment [13] (rhombus) and MMPC calculations (circles) (b).

**Figure 12.** Interferogram envelope with the taper and ovality of the small curvature of the panel (a), the

**Figure 10.** The surfaces of the pair interactions of imperfections

**Figure 11.** Surfaces of pair interactions with imperfections / ,/

joint panel zone (b), and the larger curvature of the panel (c).

Stability Calculation and Experimental Justification 23

and *a b* : (two-factor model (a),

where (...)° is the standardized value; 2 <sup>2</sup> *cr Т K E* is the dimensionless ratio of the critical stress; *cr Т* is the critical compressive load; *E* is the Young's modulus.

The resulting models are adequate to the experimental data by Fisher criteria at the 5% significance level. The presence of significant second-order terms indicates a significant nonlinearity of the relationship between the parameters, and, therefore, incorrect to separate consideration of the parameters and the placement of single-factor experiments.

Let us investigate the derived mathematical models. The corresponding surface of the pair interactions are shown in Fig. 10 and 11. They demonstrate good agreement between calculation results of MMPC and experimental data.

Analysis of the surface in Fig. 10 shows that the increase in single imperfections significantly reduces the carrying capacity of the shell. In addition, in these limits roundness has a greater impact on the setting than the taper. This is consistent with the single-factor experiments reported in [13,14]. But the analysis of Fig. 10 also shows that the simultaneous increase in taper and ovality can lead to an increase in carrying capacity to a level corresponding to the defect of a free shell. This is essentially a nonlinear effect, which could not be found by single-factor experiments.

Further study of the nonlinear interaction of defects (Fig. 11) showed that in the developed cone-of-roundness of the lower shell end has a more significant impact on the setting of critical effort than the roundness of the upper end. The joint increase in roundness of ends leads to an increase in carrying capacity, which is also an essentially nonlinear effect and is in good agreement with the results shown in Fig. 10.

Subcritical deformation has been studied in thin-walled shells with an oval on the lower and upper end being equal to 0.84 and 0.96, respectively, and taper equal to 0o56' and 2o16', respectively. The selected values , *a b*/ , / *<sup>В</sup> a b* and / *<sup>Н</sup> a b* correspond to characteristic points of the models [13,14].

A qualitative analysis of the effect of displacement fields on the results of the holographic experiment suggests an important role played by the strain state of shells under nonuniform roundness in the district and in the longitudinal direction (Fig. 12).

With the increase of up to 2o16' heterogeneity of the radial deflection is shifted to the lower end. A comparison of interferograms obtained at different load levels shows that an increase in the last number of fringes decreases with equal values of the additional load, and this indicates the hardening of structures, possibly caused by high deformability of the shell at the beginning of loading. The deformation of the shell in the experiment depends on the character of ends, and imperfections differ significantly on the panels of varying curvature.

22 Nonlinearity, Bifurcation and Chaos – Theory and Applications

where (...)° is the standardized value; 2 <sup>2</sup>

calculation results of MMPC and experimental data.

in good agreement with the results shown in Fig. 10.

respectively. The selected values

With the increase of

characteristic points of the models [13,14].

*ab ab ab <sup>K</sup>* 0,379 0,0029 0,012 / 0,014 / ;

 1 : 0,346 0 017 / 0 0033 / 0,013 / / , , , *Н ВН <sup>В</sup> K a b a b a <sup>b</sup> <sup>a</sup> <sup>b</sup>*

*cr Т*

*E*

The resulting models are adequate to the experimental data by Fisher criteria at the 5% significance level. The presence of significant second-order terms indicates a significant nonlinearity of the relationship between the parameters, and, therefore, incorrect to separate

Let us investigate the derived mathematical models. The corresponding surface of the pair interactions are shown in Fig. 10 and 11. They demonstrate good agreement between

Analysis of the surface in Fig. 10 shows that the increase in single imperfections significantly reduces the carrying capacity of the shell. In addition, in these limits roundness has a greater impact on the setting than the taper. This is consistent with the single-factor experiments reported in [13,14]. But the analysis of Fig. 10 also shows that the simultaneous increase in taper and ovality can lead to an increase in carrying capacity to a level corresponding to the defect of a free shell. This is essentially a nonlinear effect, which could not be found by

Further study of the nonlinear interaction of defects (Fig. 11) showed that in the developed cone-of-roundness of the lower shell end has a more significant impact on the setting of critical effort than the roundness of the upper end. The joint increase in roundness of ends leads to an increase in carrying capacity, which is also an essentially nonlinear effect and is

Subcritical deformation has been studied in thin-walled shells with an oval on the lower and upper end being equal to 0.84 and 0.96, respectively, and taper equal to 0o56' and 2o16',

A qualitative analysis of the effect of displacement fields on the results of the holographic experiment suggests an important role played by the strain state of shells under non-

lower end. A comparison of interferograms obtained at different load levels shows that an increase in the last number of fringes decreases with equal values of the additional load, and this indicates the hardening of structures, possibly caused by high deformability of the shell

uniform roundness in the district and in the longitudinal direction (Fig. 12).

*K*

consideration of the parameters and the placement of single-factor experiments.

stress; *cr Т* is the critical compressive load; *E* is the Young's modulus.

 

*a b a b*

is the dimensionless ratio of the critical

, *a b*/ , / *<sup>В</sup> a b* and / *<sup>Н</sup> a b* correspond to

up to 2o16' heterogeneity of the radial deflection is shifted to the

 / // : *Н В*

single-factor experiments.

**Figure 10.** The surfaces of the pair interactions of imperfections and *a b* : (two-factor model (a), comparison between model, single-factor experiment [13] (rhombus) and MMPC calculations (circles) (b).

**Figure 11.** Surfaces of pair interactions with imperfections / ,/ *Н В ab ab* .

**Figure 12.** Interferogram envelope with the taper and ovality of the small curvature of the panel (a), the joint panel zone (b), and the larger curvature of the panel (c).

Applications of 2D Padé Approximants in Nonlinear Shell Theory:

were perturbed with a natural small parameter. It is shown that the application of PAs provides them with sufficient accuracy in the studied area. This paper shows the advantage

Calculations of nonlinear deformation and stability of elastic flexible circular cylindrical shell under uniform external pressures and of free oscillations of simply supported stringer

The methodology and results of a holographic experiment with thin low-conical shells

having oval ends are presented. They show good agreement with calculation results.

*Institute of General Mechanics, RWTH Aachen University, Templergraben, Aachen, Germany* 

*Lodz University of Technology, Department of Automation and Biomechanics, Stefanowski Str.,* 

J. Awrejcewicz input to this chapter was supported by the Alexander von Humboldt Award.

[1] Baker GA Jr, Graves-Morris P **(**1996) Padé Approximants. Encyclopedia of Mathematics and Its Applications. Cambridge: Cambridge University Press, 2nd ed., v. 59. [2] Vavilov VV, Tchobanou MK, Tchobanou PM (2002) Design of multidimensional recursive systems through Padé type rational approximation, Nonlinear Analysis:

[3] Wasov W (1965) Asymptotic Expansions for Ordinary Differential Equations. New

[4] Obraztsov IF, Nerubaylo BV, Andrianov IV (1991) Asymptotic Methods in Structural

[5] Adomian G (1989) A review of the decomposition method and some recent results for

[6] Abassy TA, El-Tawil MA, Saleh HK (2007) The solution of Burgers' and good Boussinesq equations using ADM–Padé technique. Chaos, Solitons and Fractals*,* 32:

Mechanics of Thin-Walled Structures. Moscow: Mashinostroyenie.

nonlinear equations. Comp. Math. Appl., 21: 101-127.

*Ukrainian State Chemistry and Technology University, Gagarina av., 8, UA-49070,* 

of approximations which were obtained based on the MMPC.

**Author details** 

Igor Andrianov

Jan Awrejcewicz

Victor Olevs'kyy

**9. References** 

1008-1026.

*Dnipropetrovs'k, Ukraine* 

**Acknowledgement** 

Modelling and Control, 7(1): 105-125.

York: John Wiley & Sons.

*Lodz, Poland* 

shell demonstrated the efficiency and accuracy of the proposed method.

Stability Calculation and Experimental Justification 25

**Figure 13.** The deformation of the shell with ovality and taper. Solid curves correspond to the middle of panels: black – large curvature, gray – small curvature, dash – forming at the junction of the panels; positive direction goes toward the center of curvature

The change of / *<sup>Н</sup> a b* from 0.96 to 0.84 significantly (1.2-1.4 fold) increases compliance of the membranes, while maintaining the overall picture of the distribution of displacements in the circumferential direction and increasing heterogeneity in the longitudinal direction.

The field of displacements was explained semi-automatically on the basis of the above algorithm. The forms of the radial deflection of some shell generatrixes are shown in Fig. 13.
