Cylindrical shells have wide engineering applications such as in the construction and study of aircraft, spacecraft and nuclear reactor, tanks for liquid and gas storage and pressure vessels, etc. The analyses of the buckling of cylindrical shells under various loading conditions have been made in the past years and both theoretical and experimental studies have been considered just as in  and . Earlier studies on the buckling of shells were done by     and , while Amazigo and Frazer , studied the buckling under external pressure of cylindrical shells with dimple-shaped initial imperfection. It would be recalled that Budiansky and Amazigo , investigated the buckling of infinitely long imperfect cylindrical shells subjected to static loads and Ette and Onwuchekwa , 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 , 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. , 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    and  and Ette     and .
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  and Budiansky and Amazigo . 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.
As in , the general Karman-Donnell equation of motion and the compatibility equation governing the normal deflection and Airy stress function for cylindrical shell, of length L, radius R, thickness h, bending stiffness (where E and are the Young’s modulus and Poisson’s ratio respectively), mass per unit area , subjected to external pressure per unit area P, are
where, X and Y are the axial and circumferential coordinates respectively and , 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
and , is the two-dimensional bi-harmonic operator defined 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  and , we now introduce the following non-dimensional quantities.
where, is Poisson’s ratio and 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 .
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 is the non-dimensional load amplitude. The first terms on the right hand sides of (2.7a) and (2.7b), are pre-buckling approximations, while the parameter , shall take the value , if pressure contributes to axial stress through the ends, otherwise , if pressure only acts laterally.
Substituting (2.7a) and (2.7b) into (2.1 and (2.2), using (2.5a) and (2.5b), and simplifying results to
3. Classical Buckling Load
The classical buckling load is the load that is required to buckle the associated linear perfect structure and its equations, from (2.8) and (2.9) are
The solution to (3.1)-(3.3) is a superposition of the form
where, and is an inconsequential phase.
On substituting (3.4) into (3.1), using (3.3) and after lengthy simplification, we get
Substituting (3.5) into (3.2) and simplifying, yields
Thus, if n is the critical value of k that minimizes , then, the value of at was taken as the classical buckling load . Thus, in this case, we get
Therefore, corresponding to ,we see that (3.6) is now equivalent to
Usually, m and n take the values and
We recall that  had assumed that k varies continuously, and so, minimized with respect to k. If is the nontrivial values of m and we let , then, we have
The corresponding displacement and Airy Stress function are
4. 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 of , the following equations are obtained
We seek solutions to (4.5)-(4.7) in the form
and now assume
As earlier obtained, we shall need the following simplifications
so that, if
then, we have
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 and integrating with respect to y from 0 to 2π and with respect to x from 0 to π, we note that for , we easily get
Similarly, by multiplying the resultant equation through by , and integrating with respect to y from 0 to 2π and with respect to x from 0 to π, and for , we obtain
In the same manner, substituting (4.8) and (4.9) into (4.12), assuming (4.10a), (4.10b) and (4.10c), thereafter multiplying the resultant equation by and integrating with respect to x from 0 to π and y from 0 to 2π, and for , we get
On substituting for from (4.14) and from (4.10d) in (4.15) and simplifying, yields
Next, multiplying the resultant equation by and integrating with respect to x and y from 0 to π and 0 to 2π, respectively for , and simplifying, we get
We therefore expect from (4.8) that for
Solution of Equations of Second Order Perturbation
Equations of the second order to be solved are from (4.6), namely
Evaluating the symmetric bi-linear functions on the right hand sides of (4.20) and (4.21), substituting the same and simplifying, we get (after simplifying trigonometric terms)
Next we substitute (4.8) and (4.9) into (4.22), assuming (4.10a), (4.10b) and (4.10c), for . Thereafter, we multiply the resultant equation through by and integrate with respect to x and y, to get
Similarly, we next multiply the resultant equation by and integrate as usual, and for , we get
Next, we substitute (4.8) and (4.9) into (4.23), using (4.10a), (4.10b) and (4.10c), for , and then multiply the resultant equation through by and integrate, as usual for to get
In the same manner, multiply the resultant equation by and integrate with respect to x and y, for , and simplify to get
Therefore, we observe from (4.8), and for , that
On substitution in (4.29) using (4.24) and in the first part of (4.26), we get
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) : , , and .
However, of the four modes, it is only the mode in the shape of that is in the shape of the imperfection. We shall consider this mode and the additional mode in the shape of .
We now substitute (4.8) and (4.9) into (4.33), using (4.10a), (4.10b) and (4.10c), for , and thereafter, multiply the resultant equation through by and integrate and for , , to get
In the same way, we multiply the resultant equation by , integrate and for , , to get
We next substitute (4.8) and (4.9) into (4.34), using (4.10a), (4.10b) and (4.10c), for . Thereafter, we multiply the resultant equation by , integrate and note that for, , , we get
Similarly, we multiply through by and note that for , we get
Thus, of the four non-zero buckling modes of this order and their respective Airy stress function, the ones we shall consider are
As a summary so far, we can write the displacement and its respective Airy stress functions as
Equation (4.40) determines the displacement and the corresponding Airy stress functions.
5. 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 . In this respect, we neglect the displacements of order in (4.40) so that the displacement becomes
When , we get the exact displacement that is purely in the shape of imperfection, but when , we get the resultant displacement incorporating the modes and .
Since the displacement w in (5.1) depends on x and y then, the conditions for maximum displacement are as follows
We now let and be critical values of x and y respectively at maximum displacement.
From (5.1), using (5.2), we see that for maximum displacement,
where (5.3) are the least nontrivial values of and .
6. 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
Meanwhile, (6.1) can be recast as
7. Static Buckling Load
The static buckling load, according to   and , is obtained from the maximization
The usual procedure (as in   and ), is to first reverse the series (6.2) in the form
By substituting (6.1) into (7.2) and equating the coefficients of powers of orders of , we get
The maximization in (7.1) easily follows from (7.2) where is now being substituted for w to yield, after some simplifications,
On substituting into (7.4), using (6.3) and simplifying, we get
This determines the static buckling load of the circular cylindrical shell structure, and the determination is implicit in the load parameter ,
8. Results and Discussion
The result (7.5) is asymptotic in nature. The results of the classical buckling load , 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 , is as shown in (7.5). A computer program in MATLAB gives the relationship between the static buckling load , and the imperfection parameter , at or , and where we have fixed the following as , , , , , and is as shown in Table 1.
A careful appraisal of the graph of Figure 1, shows that static buckling load , decreases with increased imperfection parameter . This is expected. In other words, static buckling load increases with less imperfection. The value of static buckling load is higher when the buckling mode is a combination of the modes in the shape of imperfection ( ) and shape of other geometric form ( ), i.e. ( ) compared to the case when the buckling mode is in the shape of imperfection ( ) (i.e. ).
Table 1. Relationship between static buckling load, and imperfection parameter, for some values of of the circular cylindrical shell structure using Equation (7.5).
Figure 1. Graph of Static buckling load , as a function of imperfection parameter, for some values of of the circular cylindrical shell structure using Table 1.
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.
This paper is dedicated to the memory of late EZIKE OGBUEHI FELIX IKENNA OZOIGBO, who died on 20thday of July, 2019 and to be buried on 22nd day of November, 2019.
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