On the Maximum Displacement and Static Buckling of a Circular Cylindrical Shell

The static buckling load of an imperfect circular cylindrical shell is here determined asymptotically with the assumption that the normal displacement can be expanded in a double Fourier series. The buckling modes considered are the ones that are partly in the shape of imperfection, and partly in the shape of some higher buckling mode. Simply-supported boundary conditions are considered and the maximum displacement and the static buckling load are evaluated nontrivially. The results show, among other things, that gener-ally the static buckling load, S λ decreases with increased imperfection and that the displacement in the shape of imperfection gives rise to the least static buckling load.

equally studied the static buckling of an externally pressurized finite circular cylindrical shell using asymptotic method. In this regard, mention must be made of Lockhart and Amazigo [11], who used perturbation method to investigate the dynamic buckling of finite circular cylindrical shells with small arbitrary geometric imperfections under external step-loading. In the same way, Bich et al. [12], by using analytical approach, investigated the nonlinear static and dynamic buckling behaviour of eccentrically shallow shells and circular cylindrical shells based on Donnell shell theory. Relevant studies on the buckling analysis were investigated in [13] [14] [15] and [16] and Ette [17] [18] [19] [20] and [21].
In this study, we consider a statically loaded imperfect finite circular cylindrical shell and aim at determining the maximum displacement and the static buckling load for the case where the displacement is partly in the shape of imperfection and partly in some other buckling mode. The analysis is purely on the use of asymptotic expansions and perturbation procedures.
This analysis is organised as follows. We shall first write down the governing equations as in Amazigo and Frazer [8] and Budiansky and Amazigo [9]. Using the techniques of regular perturbation and asymptotics, we shall analytically determine a uniformly valid expression of the displacement which is followed determining by the maximum displacement. Lastly, we shall reverse the series of maximum displacement and determine the static buckling load.

Formulation
As in [11], the general Karman-Donnell equation of motion and the compatibility equation governing the normal deflection (where E and υ are the Young's modulus and Poisson's ratio respectively), mass per unit area ρ , subjected to external pressure per unit area P, are 4 , where, X and Y are the axial and circumferential coordinates respectively and ( ) , W X Y , is a continuously differentiable stress-free and time independent imperfection. In this work, an alphabetic subscript placed after a comma indicates partial differentiation while S is the symmetric bi-linear operator in X and Y given by here, we shall neglect both axial and circumferential inertia and shall similarly assume simply-supported boundary conditions and neglect boundary layer effect by assuming that the pre-buckling deflection is constant.
As in [11] and [22], we now introduce the following non-dimensional quantities.
where, υ is Poisson's ratio and ε is a small parameter which measures the amplitude of the imperfection while L is the length of the cylindrical shell which is simply-supported at 0, x = π .
We shall neglect boundary layer effect by assuming that the pre-buckling deflection is constant so that we let ( ) where, P is the applied static load and λ is the non-dimensional load amplitude. The first terms on the right hand sides of (2.7a) and ( , , 0 at 0, , 0 , 0 2 , 0 1,

Classical Buckling Load
The classical buckling load C λ is the load that is required to buckle the asso-Journal of Applied Mathematics and Physics The solution to ( Thus, if n is the critical value of k that minimizes λ , then, the value of λ at k n = was taken as the classical buckling load C λ . Thus, in this case, we get Therefore, corresponding to k n = ,we see that (3.6) is now equivalent to Usually, m and n take the values We recall that [23] had assumed that k varies continuously, and so, minimized λ with respect to k. If The corresponding displacement and Airy Stress function are

Static Theory
In this section, we shall derive the equations satisfied by the displacement and Airy stress functions when the static load is applied.
Similar to (2.8) and (2.9), the structure satisfies the following equations at static loading ( ) ( ) We now assume the following asymptotic expansions Substituting (4.4) into (4.1) and (4.2), and equating the coefficients of orders etc.
We seek solutions to (4.5)-(4.7) in the form 1 , 0 1 2 cos sin sin As earlier obtained, we shall need the following simplifications We shall use the fact that

Solution of Equations of First Order Perturbation
The equations necessary here, from (4.5), are Substituting (4.8) and (4.9) into (4.11), using (4.10a), (4.10b) and (4.10c), multiplying the resultant equation through by cos sin ny mx and integrating with respect to y from 0 to 2π and with respect to x from 0 to π, we note that for , p n k m = = , we easily get In the same manner, substituting (4.8) and (4.9) into (4.12), assuming (4.10a), Journal of Applied Mathematics and Physics and integrating with respect to x from 0 to π and y from 0 to 2π, and for , p n k m = = , we get On substituting for ( ) Next, multiplying the resultant equation by cos cos ny mx and integrating with respect to x and y from 0 to π and 0 to 2π, respectively for , p n k m = = , and simplifying, we get ( ) We therefore expect from (4.8) that for Similarly, we next multiply the resultant equation by sin sin ny mx and integrate as usual, and for , p n k m = = , we get In the same manner, multiply the resultant equation by sin sin ny mx and integrate with respect to x and y, for , p n k m = = , and simplify to get Therefore, we observe from (4.8), and for 2 i = , that On substitution in (4.29) using (4.24) and in the first part of (4.26), we get w ny mx f ny mx

Solution of Equations of Third Order Perturbation
The actual equations of the third order are from (4.7), namely Evaluating the symmetric bi-linear functions on the right sides of (4.31) and (4.32) and substituting the same and simplifying, yields We observe from the simplifications on the right hand sides of (4.33) and (4.34) that there will be four buckling modes generated with their respective Airy stress functions ( ) . These buckling modes correspond to the following terms on the right hand sides of (4.33) and (4.34) : sin 3 sin ny mx , sin sin ny mx , cos 2 sin 3 mx ny and cos 2 sin mx ny .
However, of the four modes, it is only the mode in the shape of sin sin ny mx that is in the shape of the imperfection. We shall consider this mode and the additional mode in the shape of sin 3 sin ny mx .

Values of Independent Variables at Maximum Displacement
The analysis henceforth will be concerned with the displacement components that are partly in the shape of the imperfection or partly in the shape of sin sin 3 mx ny .
In this respect, we neglect the displacements of order 0 Ω = , we get the exact displacement that is purely in the shape of imperfection, but when 1 Ω = , we get the resultant displacement incorporating the modes sin sin 3 mx ny and sin sin mx ny .
Since the displacement w in (5.1) depends on x and y then, the conditions for maximum displacement are as follows We now let a x and a y be critical values of x and y respectively at maximum displacement.

Maximum Displacement
The maximum displacement is obtained from (5.1) at the critical values of x and y where w has a maximum value. Hence, the value of w at these values becomes  , ,

Results and Discussion
The result (7.5) is asymptotic in nature. The results of the classical buckling load C λ , and that of the cylindrical shell structure are as seen in (3.9), whereas, the corresponding displacement and Airy Stress function of the structure are as in (3.10). Similarly, the static buckling load S λ , is as shown in (7.5). A computer program in MATLAB gives the relationship between the static buckling load S λ , and the imperfection parameter ε , at 0 Ω = or  Table 1. A careful appraisal of the graph of Figure 1, shows that static buckling load S λ , decreases with increased imperfection parameter ε . This is expected. In other words, static buckling load S λ increases with less imperfection. The value of static buckling load S λ is higher when the buckling mode is a combination of the modes in the shape of imperfection ( sin sin mx ny ) and shape of other geometric form ( sin sin 3 mx ny ), i.e. ( 1 Ω = ) compared to the case when the buckling mode is in the shape of imperfection ( sin sin mx ny ) (i.e. 0 Ω = ).

Conclusion
This analysis has analytically determined the maximum of the out-of-plane normal displacement of a finite imperfect cylindrical shell trapped by a static load. We have used the techniques of perturbation and asymptotics to derive an implicit formula for determining the static buckling load of the cylindrical shell investigated. The formulation contains a small parameter depicting the amplitude of the imperfection and on which all asymptotic series are expanded. Such an analytical approach can be duplicated for other structures including toroidal shell segments and plates.

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.