# A new analytical tool for the elastic/plastic buckling design of pressure vessels subjected to external pressure

Van-Dong Do, Philippe Le Grognec, Philippe Rohart

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Van-Dong Do, Philippe Le Grognec, Philippe Rohart. A new analytical tool for the elastic/plastic buckling design of pressure vessels subjected to external pressure. International Journal of Pressure Vessels and Piping, 2024, 210, pp.105221. 10.1016/j.ijpvp.2024.105221. hal-05340758

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# A new analytical tool for the elastic/plastic buckling design of pressure vessels subjected to external pressure

Van-Dong Doª, Philippe Le Grognecª,*, Philippe Rohartᵇ

## Abstract

This paper deals with the elastic/plastic buckling design of pressure vessels under external pressure loading. The objective is to build a simple analytical tool devoted to the elastic and elastoplastic buckling analysis of pressure equipments. This tool is aimed to provide the critical external pressure of cylindrical pressure vessels with ellipsoidal (or possibly hemispherical) heads. Depending on the geometry of the equipment and material parameters, the first buckling pressure may involve either the cylindrical shell or the closure ends, in the elastic or plastic range. The calculation tool will be thus based on a series of exact or approximate analytical solutions of the critical pressure of each part of the considered equipment. In this paper, the elementary shell structures involved in such pressure equipments are first individually examined. The fundamental critical pressures for each of them are recalled or introduced, both in elasticity and plasticity. The new solutions are validated against reference numerical results obtained through finite element computations. Then, approximate solutions of the critical pressure are derived for the same shells, taking account of specific boundary conditions (and not idealized ones, as before) due to the presence of the remaining part of the equipment. In the end, a VBA Excel software is developed. It is able to provide the mappings of the buckling mode type and the corresponding critical pressures for large ranges of geometric and material parameters in a very efficient way. These analytical solutions are finally confronted to numerical calculations performed on complete pressure equipments and they are shown

Preprint submitted to International Journal of Pressure Vessels and Piping May 16, 2024

<footnote>ªENSTA Bretagne, UMR CNRS 6027, IRDL, F-29200 Brest, France</footnote>

<footnote>ᵇCETIM, Centre Technique des Industries Mécaniques, 5 avenue Félix Louat - BP 80067, F-60304 Senlis Cedex, France</footnote>

<footnote>*Corresponding author.</footnote>

<footnote>Email address: philippe.le_grognec@ensta-bretagne.fr (Philippe Le Grognec)</footnote>
to give quite reliable results. A discussion is also initiated on the use of such
a program to update the current design rules in this scope of application.

***Keywords:*** Elastic/plastic buckling, cylindrical shells, hemispherical/ellipsoidal closure ends, analytical tool, numerical validation, design rules

# 1. Introduction

## 1.1. General background and motivations

Pressure equipments are made up of closed enclosures containing pressurized fluids. The energy stored in such pressure vessels can be very high, due to the internal or external pressure but also to additional mechanical or thermal loadings. Consequently, in the event of rupture or explosion of such equipments, the human, material or ecological risks can be very significant, especially when the nature of the fluid is dangerous (flammable, corrosive, polluting, toxic, etc.). All these factors justify the importance attached to safety in the field of pressure equipments, from the design phase until the installation is dismantled at the end of its life, which manifests itself by the appropriate definition of design codes and the compliance with these standard rules.

The buckling phenomenon is one of the main failure modes encountered in pressure equipments, due to the thin (shell-like) geometry of such structures and the predominant compressive stresses frequently observed under standard loads such as external pressure. The related geometric instability gives rise to sudden finite deformations which finally lead to local fracture and leakage phenomena and/or implosion of the whole structure. At present, a number of design methodologies exist for these structures with regard to buckling, set out in standard codes such as the ASME Boiler and Pressure Vessel Code (BPVC) [1] or the French CODAP [2]. These rules are either analytical (based on abacuses or formulae) but limited in terms of field of validity, or take the form of recommendations [3] for carrying out finite element analyses in the presence of geometric and/or material non-linearities and possible imperfections [4]. Most recent studies in the literature, dealing with the development of new calculation rules for the buckling design of pressure equipments, are mainly based on numerical finite element computations. However, these calculations require the use of commercial codes and are fairly complex in terms of implementation. As a result, they are often
difficult to master and reproduce in the industrial context, and are therefore
poorly applied in practice, if at all. As a consequence, manufacturers are
currently focusing more and more on the use of analytical solutions. As the
existing rules are often outdated and are likely to be improved in some way,
there is actually a surge in the demand of up-to-date analytical solutions that
may reproduce at best the buckling response of pressure equipments. The
main interest and innovation of the present work is thus to offer a real plastic
buckling analysis of such pressurized equipments, which is not currently the
case in standard codes where the influence of material hardening is taken
into account in a simplified way.

In this framework, the present work deals with the elastic/plastic buck-
ling design of pressure vessels under external pressure loading, with two main
objectives. First of all, the idea is to analyze as a whole the buckling be-
havior of a pressure equipment classically composed of a cylindrical shell
and two ellipsoidal closure ends. It will be shown that several buckling re-
sponses may occur in the pressure vessel, depending on the geometric and
material parameters. The buckling phenomenon primarily concerns either
the cylindrical shell or the ellipsoidal bottom/head. Besides, elastic or plas-
tic buckling may arise, mainly depending on the yield stress of the material,
but also another less well-known phenomenon can be encountered where the
structure buckles at the plasticity threshold, in some particular conditions.
Secondly, the main objective of this work is to propose a dimensioning tool
(in the form of a VBA Excel software) that is as efficient and easy to use as
possible for manufacturers, namely only based on closed-form solutions. The
present new analytical solutions (especially in elastoplasticity), together with
the clear distinction between all the phenomena encountered (elastic, plastic
and threshold buckling) in both the cylindrical shell and ellipsoidal ends,
should make this program an original and interesting tool enabling a pre-
cise and efficient determination of reference critical pressures for supposedly
perfect equipments.

1.2. *State of the art*

Many shell buckling problems have been studied analytically in the litera-
ture so far, involving various geometries such as cylinders, spheres, ellipsoids,
cones or any combination of them. Among the loadings which typically give
rise to geometric instabilities, the external pressure has clearly a prominent
place. The vast majority of these studies deal with elastic buckling and
several analytical solutions are already available in such conditions. Most
investigations are concerned with cylindrical shells which are certainly the most essential components in pressure vessels. For instance, one can mention Bryan [5], who was the first to derive an analytical solution of the elastic critical buckling pressure of long tubes under external pressure, using an energy method. Also, Paimushin [6], Xue [7], and Salahshour and Fallah [8] were interested in the elastic buckling of long cylindrical shells under uniform external pressure, among others. Southwell [9] as for him considered the case of short cylinders under uniform external pressure. Yamaki [10] thoroughly investigated the elastic buckling and post-buckling behavior of circular cylindrical shells. Grigolyuk and Kabanov [11] derived an analytical expression of the critical pressure for thin cylindrical shells with simply-supported end conditions, and Papadakis [12] proposed a new analytical expression of the critical pressure in the case of thick cylindrical shells, including the effects of transverse shear in the analysis. Nguyen et al. [13] also investigated analytically the elastic buckling of cylindrical shells under external pressure, considering a variable thickness.

Besides, pressure vessels generally include closure ends of various shapes, which may be hemispherical, ellipsoidal, torispherical or toriconical, which also have to sustain reliably extreme design loads and thus require a detailed buckling analysis. However, when compared to cylindrical shells, only a few analytical studies deal with such shell geometries, and very few, if any, when it comes to elastoplastic buckling. Zoelly [14] was the first to derive an analytical solution for the critical external pressure of elastic buckling for a perfect spherical shell. Hutchinson [15, 16] analyzed the post-buckling behavior of spherical shells under external pressure within the context of Koiter's general theory and showed the significant imperfection sensitivity of such a problem, even in elasticity. Palusamy and Lind [17] developed a consistent theory for the limit analysis of spherical shells under radial pressure. The elastic (and plastic) buckling of ellipsoidal vessel heads was investigated early on by Alix and Roche [18] using experimental results together with empirical formulae. More recently, one can mention, for instance, the review paper from Krivoshapko [19] dealing with static, dynamic and stability analyses of complete ellipsoidal shells (and also ellipsoidal heads and bottoms), specially focusing on ellipsoids of revolution and building applications. A few approximate solutions for the critical external pressure are particularly listed with their respective domain of validity. Otherwise, one only finds some numerical (or experimental) studies concerning the buckling of ellipsoidal structures, such as the finite element analysis of super ellipsoidal shells under uniform
pressure [20], the buckling design of externally pressurized ellipsoidal heads
with variable wall thicknesses [21], and the recent non-linear buckling anal-
ysis of a semi-elliptical dome [22].

The buckling problem is much more difficult to analyze when considering
thicker structures, for which plasticity may occur before buckling, and one
finds even less papers dealing with this particular aspect in the literature.
As regards shell plastic buckling analyses, the case of axially compressed cir-
cular cylindrical shells was certainly one of the most considered, which is
further limited to axisymmetric modes. In this context, one can mention the
pioneering works of Batterman [23, 24] who derived first adequate formulae
for the critical axial compressive stress in a cylindrical shell with various
boundary conditions. The elastoplastic buckling of a cylindrical shell under
external pressure was far less commonly studied. The pioneering works on
this subject are certainly due to Chakrabarty [25, 26]. He examined first
the general conditions under which bifurcation may occur in a shell under
pressure loading beyond the elastic limit, and considered then the particular
case of cylindrical shells subjected to external fluid pressure. More recently,
Takla investigated numerically both material and geometric instability phe-
nomena (such as necking, bulging, column buckling and section collapse) in
elastoplastic cylindrical shells under internal/external pressure and/or axial
tension/compression [27–30]. One can also find some rare experimental stud-
ies on this subject, such as the one from Zhu et al. [31], which considers a
very specific constitutive law and clamped ends. Besides, unlike the case of
cylindrical shells, to the best authors’ knowledge, there is no plastic buckling
analysis of spherical shells in the literature, at least analytically speaking.

Apart from a few exceptions, no exact or even approximate closed-form
expressions are available in the literature for the plastic buckling pressure
of a cylindrical, spherical or ellipsoidal shell. Such expressions have only
recently been provided by the authors [32] in the case of a simply-supported
cylindrical shell under external pressure and axial compression (due to the
pressure end thrust) and a spherical shell under external pressure.

To go further and be more precise, attention should be paid to the equip-
ment as a whole. In particular, the presence of hemispherical or ellipsoidal
heads may influence the buckling response of a cylindrical shell under external
pressure. One can find in the literature several papers dealing on the buck-
ling behavior of pressure vessels, most often composed of a cylindrical shell
and closure ends of various shapes. Let us mention, among others, Zhang
et al. [33] who investigated the buckling of cold-stretched cylindrical vessels
(manufactured with two ellipsoidal heads) under external pressure, from an
experimental and numerical point of view. Galletly and Aylward [34] ana-
lyzed the influence of realistic boundary conditions on the buckling pressure
of cylinders with hemispherical or torispherical ends. Jasion and Magnucki
[35] investigated the buckling behavior of clothoidal-spherical shells under
external pressure. Magnucki et al. [36] focused on the edge effect at the
joint between a cylindrical pressure vessel and a dished head on the buckling
analysis under internal and external pressure. Finally, Zingoni and Enoma
[37] explored the buckling under external pressure of the original assembly of
a spherical and a conical shell. There are many more papers on this subject,
but all are based on numerical (finite element) computations.

1.3. *Outline of the paper*

In this paper, using the reliable formulae defined earlier for the elastic
and plastic buckling pressures of the simple shell structures constituting the
different components of a pressure equipment, one aims at examining the
buckling response of the whole pressure vessel in an analytical way. The
general case of a cylindrical shell with ellipsoidal (possibly hemispherical)
heads is considered. In Section 2, the main theoretical foundations and hy-
potheses of the present elastoplastic buckling analysis are recalled, and the
useful closed-form solutions derived in [32] for the perfect cylindrical and
spherical shells are specified. New approximate solutions are then proposed
for the elastic but also plastic buckling pressure of an ellipsoidal shell, relying
on the spherical case. Next, in Section 3, the influence of each component
on the other one is investigated and little changes are made to the previ-
ous solutions so as to take into account the possible interaction between the
cylindrical shell and the heads. In Section 4, an analytical tool is proposed
in the form of a VBA Excel software, whose objective is to determine, for
an arbitrary geometric and material configuration, the buckling localization
(cylinder or head) and the elastic or elastoplastic nature of the buckling phe-
nomenon encountered, after comparison of many numerical values deriving
from the previous formulae. It will be shown that the problem is a little bit
complicated by the presence of another phenomenon that is likely to occur
frequently, which will be called threshold buckling. Owing to its very high
efficiency, such a program will enable, in a very short time, to provide a
general overview of the buckling scenarii and of the corresponding critical
values of such pressure vessels in the form of maps, depending on the geo-
metric and/or material parameters. Finally, the analytical tool is confronted
to numerical finite element computations performed on complete pressure
equipments for validation purposes (Section 5), and the new analytical re-
sults are also compared to the solutions obtained through the current rules
from standard codes as the French CODAP (Section 6). The ultimate objec-
tive of this contribution is to enrich and simplify simultaneously the current
calculation rules in standard codes with all these new explicit solutions and
efficient calculation tools, enabling thus an ever more efficient dimensioning
with both a better accuracy and a wider range of applications.

## 2. Buckling results for classical shells under external pressure

The buckling analysis of pressure vessels will be further carried out in
an analytical way. Pressure equipments can be easily divided into simple
idealized shells, whose buckling behavior can be first investigated individu-
ally. In the present paper, one considers only pressure vessels made up of a
cylindrical shell and ellipsoidal (possibly hemispherical) heads. Therefore, a
particular attention needs first to be paid to the buckling response of cylin-
drical, spherical and ellipsoidal shells. In the sequel, the main theoretical
foundations and hypotheses which are used in the present study to analyze
the elastic and plastic buckling of such shells are emphasized. Then, the
closed-form analytical solutions derived in a previous paper from the authors
[32] for the cylindrical and spherical cases are briefly recalled. Lastly, the
more complicated case of an ellipsoidal shell is examined, and approximate
closed-form solutions for the critical pressure are also obtained in this case,
both in the context of elastic and plastic buckling.

### 2.1. *Theoretical framework*

A generic method was presented in [32] devoted to the calculation of
elastic or plastic buckling pressures of arbitrary shells. It is based on the 3D
elastoplastic bifurcation theory and uses the so-called bifurcation equation
[38]. The J₂ flow theory of plasticity is employed with the von Mises yield
criterion and a linear isotropic hardening. First, it makes it easier to find
explicit solutions for sufficiently simple shell configurations, and it is in line
with the choices made in most standard codes. According to the geomet-
ric and loading configuration, the plane stress hypothesis is enforced and a
pre-critical biaxial stress state is assumed. The so-called shallow shell the-
ory is retained so as to describe the kinematics. It has been shown to be
a good compromise in that it allows for simple and yet reliable closed-form
expressions of the sought critical loads in a straightforward way. A particular attention is given to the follower nature of the external pressure, which turns out to be a non-conservative loading and may modify the bifurcation equation and the critical pressures. In the sequel, elastic and plastic buckling critical values will be given for various shell structures. It should be remembered the existence, in the case of elastoplasticity, of continua of bifurcation points for all successive buckling modes, which were emphasized early on by Cimetière [39] in the case of compressed rectangular plates. Each continuum means that, for each mode, the structure may theoretically buckle at different critical values, depending on the extent of the unloading zone at the bifurcation time. It is worth mentioning that, for the time being, we will only be talking about the so-called tangent modulus critical load, which corresponds to the minimum possible elastoplastic buckling load for the considered mode, at which the structure may buckle in practice (with only incipient unloading). For more details on the theoretical formulation, the reader may refer to the previous paper [32].

Even though arbitrary geometries can be considered and other loading conditions could be introduced in the method, it has been first used in the classical cases of cylindrical and spherical shells, as these are the most encountered shapes in practical pressure vessels. In the particular case of cylindrical shells, an axial compressive force was added to the external pressure so as to take into account the well-known pressure end thrust.

## 2.2. Cylindrical shells

Let us consider first the problem of a cylindrical shell under external pressure, defined in the reference configuration by its length $L$, its average radius $R$ and thickness $t \ll R$ (see Figure 1). The cylinder is simply-supported at both ends, which means that the radial and circumferential displacements are fixed, whereas the longitudinal displacement and the rotation about the circular edges are let free. These boundary conditions enable one to obtain closed-form solutions and correspond at best to the real conditions in the presence of closure ends. The external pressure $p$ gives rise first to a uniaxial compressive hoop stress (in the circumferential direction) of $-\frac{pR}{t}$. The presence of top and bottom heads, also subjected to external pressure, results in a given axial compression stress in the cylindrical shell, independent of their shape. This axial stress is half of the circumferential stress, namely it is equal to $-\frac{pR}{2t}$.
<figure><img src="image_2.png" /><figcaption>Figure 1: Cylindrical shell under external pressure and axial compression [32]</figcaption></figure>

Without going into details, the buckling modes satisfying the boundary conditions applied were found to take the following form:

$$ \left\{ \begin{array}{l} U = U_0 \cos\left(\frac{m\pi x}{L}\right) \cos\left(\frac{ny}{R}\right) \\ V = V_0 \sin\left(\frac{m\pi x}{L}\right) \sin\left(\frac{ny}{R}\right) \\ W = W_0 \sin\left(\frac{m\pi x}{L}\right) \cos\left(\frac{ny}{R}\right) \end{array} \right. \qquad (1) $$

where $U$, $V$ and $W$ represent the longitudinal, circumferential and radial components, respectively (see Figure 1), and $m$ and $n$ stand for the longitudinal half-wave and circumferential wave numbers.

The following analytical solution was then obtained for the elastic critical pressure of a cylindrical shell under such a biaxial stress state [32]:

$$ p_e^{cyl-head} = \left( \frac{Et}{Rn^2} \frac{\left(\frac{m\pi R}{L}\right)^4}{\left[\left(\frac{m\pi R}{L}\right)^2 + n^2\right]^2} + \frac{D}{R^3 n^2} \left[ \left(\frac{m\pi R}{L}\right)^2 + n^2 \right]^2 \right) \frac{n^2}{n^2 + \frac{1}{2}\left(\frac{m\pi R}{L}\right)^2} \quad (2) $$

Equation (2) is derived in the context of an isotropic linear elastic material, with Young's modulus $E$ and Poisson's ratio $\nu$, and $D = \frac{Et^3}{12(1-\nu^2)}$ is the associated classical elastic bending stiffness.
A closed-form expression was also derived in [32] in the case of plastic buckling, which writes:

$$ p_p^{\text{cyl-head}} = \left( \begin{aligned} & tE \left[ \frac{1}{2}t^2\pi^8m^8R^8\left(E + \frac{E_T}{3}\right)^2(1+\nu) + \pi^4m^4R^6L^2 \right. \\ & \left. \quad \left(E_T^2\left[\frac{8}{9}n^2t^2m^2\pi^2\left(\nu + \frac{5}{8}\right) - \frac{32}{3}L^2\left(\nu - \frac{1}{2}\right)\left(\nu + 1\right)\left(\nu + \frac{1}{2}\right)\right] \right. \\ & \left. \quad + 8EE_T\left[\frac{1}{3}n^2t^2m^2\pi^2\left(\nu + \frac{3}{4}\right) + L^2(1+\nu)\right] + E^2\pi^2m^2t^2n^2 \right) \\ & + \frac{4}{3}E_T L^4 n^4 t^2 \pi^4 m^4 R^4 \left[E_T(3\nu + 1) + E(\nu + 3)\right] \\ & + \frac{4}{3}E_T n^6 \pi^2 t^2 L^6 m^2 R^2 \left[\frac{1}{3}E_T(8\nu + 5) + E\right] + \frac{8}{9}E_T^2 n^8 L^8 t^2(1+\nu) \right] \\ & / \left( 6L^2R^3\left(\frac{1}{2}\pi^2R^2m^2 + L^2n^2\right)\left[E + \frac{1}{3}E_T(1 - 4\nu^2)\right](\nu + 1) \right. \\ & \left. \quad \left[m^4\pi^4R^4\left(E + \frac{1}{3}E_T\right) + \frac{8}{3}\pi^2R^2E_Tm^2n^2L^2 + \frac{4}{3}E_Tn^4L^4\right] \right) \end{aligned} \right) \quad (3) $$

where $E_T$ is the elastoplastic tangent modulus which is related to the Young's modulus $E$ and the (constant) isotropic hardening modulus $H$ by $\frac{1}{E_T} = \frac{1}{E} + \frac{1}{H}$.

The critical pressures depend on $m$ and $n$ which have to be adequately identified so as to find the minimum critical pressure, which alone is of practical interest. It has been shown that, whether in elasticity or plasticity, the longitudinal half-wave number $m$ minimizing the critical pressure was always equal to 1 in practice. Conversely, the circumferential wave number $n$ giving rise to the minimum pressure depends on the material and above all geometric parameters.

### 2.3. Spherical shells

Let us now consider the case of a spherical shell under external pressure. The full sphere is defined in the reference configuration by its average radius $R$ and thickness $t \ll R$ (see Figure 2). The external pressure gives rise here to a compressive equi-biaxial stress state, with a compressive stress in any arbitrary direction in the shell plane which is equal to $-\frac{pR}{2t}$.

In this new configuration, the buckling modes are still sinusoidal in both directions. Owing to the high wave numbers observed in practice, the minimum critical pressure can be easily obtained here, with a very good accuracy, by minimizing the general expression of the buckling pressure, considering the wave numbers as being real. It leads to the following well-known expression of the elastic critical pressure, initially derived by Zoelly [14]:

$$ p_e^{\text{sph}} = \frac{2E}{\sqrt{3(1-\nu^2)}} \left(\frac{t}{R}\right)^2 \quad (4) $$
<figure><img src="image_3.png" /><figcaption>Figure 2: Spherical shell under external pressure [32]</figcaption></figure>

which corresponds to many multiple modes with as many pairs of wave numbers.

An original expression for the minimum plastic buckling pressure of a spherical shell was also found in [32]:

$$p_p^{sph} = \frac{2}{3} \frac{\sqrt{6} E E_T}{\sqrt{E_T(\nu + 1)(-2E_T\nu + E + E_T)}} \left(\frac{t}{R}\right)^2 \quad (5)$$

## 2.4. Ellipsoidal shells

### 2.4.1. Analytical solutions

Closure ends of a pressure vessel are sometimes hemispherical, but most often ellipsoidal, so that it is interesting to analyze as well the elastic and plastic buckling behavior of an ellipsoidal shell under external pressure. For the time being, let us consider a complete ellipsoid of revolution, whose equation takes the following form in an appropriate orthonormal Cartesian basis ($e_x, e_y, e_z$) with $e_z$ as axis of revolution:

$$\frac{x^2 + y^2}{a^2} + \frac{z^2}{b^2} = 1 \quad (6)$$
In Equation (6), a and b describe the semi-axes of the ellipsoid. Two types of ellipsoids of revolution may be encountered in practice, namely prolate and oblate, depending on whether a < b or b < a, respectively. In the particular case where a = b, the ellipsoid reduces to a sphere.

Dealing now with the buckling of an ellipsoid under external pressure, two quite distinct behaviors may happen depending on the type of ellipsoid. Unlike the case of a sphere, the radius of curvature throughout the surface of the shell is not uniform, which leads to an heterogeneous stress state in the structure during the pre-critical stage. Consequently, it can be shown that the buckling modes of an oblate ellipsoid induce more deformations at the poles whereas the ones of a prolate ellipsoid are more likely to take shape near the equator, as illustrated in Figure 3. As ellipsoidal heads in pressure vessels are invariably oblate, only this particular case will be dealt with below.

The heterogeneous compressive stress state in the shell makes it impossible to derive an exact solution for the critical pressure, even in elasticity. Instead, an approximate solution is proposed here, as in [19], based on the simple observation that the buckling modes actually take shape in a very small area located very close to the poles. At the poles, the two principal radii of curvature are similar and equal to:

$$R' = \frac{a^2}{b} \qquad (7)$$

The approximation consists then in considering that the ellipsoidal shell can be viewed locally as a spherical shell of radius $R'$, submitted to a local equi-biaxial compressive stress equal to $-\frac{pR'}{2t}$ (see Figure 4), and the same buckling pressures as derived in the case of a spherical shell can therefore be used.

The two following expressions will be thus retained in practice for the elastic and plastic buckling pressures of an oblate ellipsoidal shell, respectively:

$$p_e^{\text{ellip}} = \frac{2E}{\sqrt{3(1-\nu^2)}} \left(\frac{tb}{a^2}\right)^2 \qquad (8)$$

$$p_p^{\text{ellip}} = \frac{2}{3} \frac{\sqrt{6EE_T}}{\sqrt{E_T(\nu+1)(-2E_T\nu + E + E_T)}} \left(\frac{tb}{a^2}\right)^2 \qquad (9)$$

### 2.4.2. Numerical validation

The critical pressures recalled previously in the cases of cylindrical shells (Equations (2,3)) and spherical shells (Equations (4,5)) were already vali-
<figure><img src="image_4.png" /><figcaption>Figure 3: Linearized buckling modes of ellipsoids</figcaption></figure>

dated in [32]. The new formulae retained for the case of an ellipsoidal shell
will be now confronted to the results of numerical computations in the same
way. Finite element computations are thus performed using Abaqus software.
In elasticity, one can resort to linearized buckling analyses which provide di-
rectly the successive critical values (and first and foremost the minimum one)
and the corresponding buckling modes. Conversely, in plasticity, incremental
calculations are needed so as to take into account the material non-linearities,
and use is made of arc-length approaches (Riks method) so as to deal with
non-monotonous equilibrium curves. As for the sphere problem, there is no
need for additional initial defects in the model so as to trigger the buckling
phenomenon during the incremental calculations, due to the extreme imper-
<figure><img src="image_5.png" /><figcaption>Figure 4: Ellipsoidal shell under external pressure</figcaption></figure>

fection sensitivity of this geometric and loading configuration. From these incremental results, the first critical pressure can be identified as the limit admissible pressure and the associated mode can be estimated through the observation of the immediate post-critical deformed shapes.

Dealing now with the finite element method, a shell finite element model is retained, made of eight-node (quadratic) elements with five degrees-of-freedom per node and reduced integration (S8R5 elements in Abaqus). Only one eighth of the total ellipsoid is modeled so as to reduce the computation times and a specific regular quadrangular mesh is retained after a short mesh convergence study. The elastic properties used are as follows: $E = 200000$ MPa and $\nu = 0.3$. A follower external pressure is finally applied all over the shell structure.

A few geometries are considered for the numerical validation in the elastic case. An equatorial radius $a$ of 500 mm is systematically considered, whereas the semi-minor axis $b$ and the thickness $t$ take on different values. The associated geometric ratios are indicated in Table 1, together with the corresponding analytical and numerical critical pressures. The relative errors are also listed. The analytical results are shown to be generally in good agreement with the numerical ones. When the aspect ratio $a/b$ is bigger, the ellipsoid differs more from a sphere (it is more flattened), and the error
is naturally bigger, due to the strong assumptions made on the pre-critical stress state. However, the maximum relative error encountered in the practical cases examined does not exceed 10%, and all the analytical values are found to be conservative (less than their numerical counterpart). It is worth mentioning that all these critical values correspond to the first buckling mode of an oblate ellipsoid, namely to a modal deformed shape similar to the one in Figure 3(a).

<table><caption>Table 1: Ellipsoidal shell - Comparison between analytical and numerical first critical values in elasticity for various aspect ratios and radius-to-thickness ratios</caption><thead><tr><th rowspan="3">a/t</th><th colspan="3">a/b = 1.25</th><th colspan="3">a/b = 2</th></tr><tr><th>Analytical</th><th>FE</th><th>Relative error</th><th>Analytical</th><th>FE</th><th>Relative error</th></tr><tr><th>p<sup>ellip</sup><sub>e</sub><br>(MPa)</th><th>p<sup>ellip</sup><sub>e-num</sub><br>(MPa)</th><th><sup>p<sup>ellip</sup><sub>e</sub>-p<sup>ellip</sup><sub>e-num</sub></sup>&frasl;<sub>p<sup>ellip</sup><sub>e</sub></sub><br>(%)</th><th>p<sup>ellip</sup><sub>e</sub><br>(MPa)</th><th>p<sup>ellip</sup><sub>e-num</sub><br>(MPa)</th><th><sup>p<sup>ellip</sup><sub>e</sub>-p<sup>ellip</sup><sub>e-num</sub></sup>&frasl;<sub>p<sup>ellip</sup><sub>e</sub></sub><br>(%)</th></tr></thead><tbody><tr><td>100</td><td>15.494</td><td>16.183</td><td>-4.45</td><td>6.0523</td><td>6.6371</td><td>-9.66</td></tr><tr><td>200</td><td>3.8735</td><td>4.0083</td><td>-3.48</td><td>1.5131</td><td>1.6176</td><td>-6.91</td></tr><tr><td>500</td><td>0.6198</td><td>0.6374</td><td>-2.84</td><td>0.2421</td><td>0.2535</td><td>-4.69</td></tr></tbody></table>

Next, a particular geometric configuration is retained and a few incremental calculations are performed for various hardening moduli, so as to validate the analytical plastic buckling expression. The initial yield stress of the material has been defined in such a way that plastic strains appear before buckling (at least in the vicinity of the poles), but still sufficiently high in order to avoid excessive pre-critical deformations. The geometric and material parameters are indicated in Table 2 as well as the analytical and numerical critical pressures. The relative errors between the analytical values and the numerical limit pressures derived from the incremental curves (as the one illustrated in Figure 5) do not exceed 4% (the smallest aspect ratio has been retained here), which proves again the good reliability of the theoretical hypotheses and resulting expressions. Additionally, Figure 5 displays the immediate post-buckling deformed shape and a more advanced post-critical state (which shows a localization of the deformations around the poles) in a particular configuration.
<table><caption>Table 2: Ellipsoidal shell - Comparison between analytical and numerical first critical values in plasticity for various hardening moduli (a = 500 mm, b = 400 mm, a/t = 50)</caption><thead><tr><th rowspan="2">H</th><th>Analytical (plastic)</th><th>FE (incremental)</th><th>Relative error</th></tr><tr><th><i>p</i><sub>p</sub><sup>ellip</sup><br>(MPa)</th><th><i>p</i><sub>p-num</sub><sup>ellip</sup><br>(MPa)</th><th><sup><i>p</i><sub>p</sub><sup>ellip</sup> - <i>p</i><sub>p-num</sub><sup>ellip</sup></sup>/<i>p</i><sub>p</sub><sup>ellip</sup><br>(%)</th></tr></thead><tbody><tr><td>4000</td><td>10.228</td><td>9.8747</td><td>3.90</td></tr><tr><td>2000</td><td>7.2822</td><td>7.0364</td><td>3.37</td></tr><tr><td>1000</td><td>5.1672</td><td>4.9959</td><td>3.32</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>Figure 5: Non-linear buckling analysis of an ellipsoidal shell under external pressure (a = 500 mm, b = 400 mm, a/t = 50, H = 2000 MPa)</figcaption></figure>
### 3. Buckling analysis of a complete pressure vessel

The buckling analysis of a complete pressure vessel, made up of a cylindrical shell and two ellipsoidal (or hemispherical) heads, can be seen as the joint analysis of the buckling behavior of each part independently. However, the buckling response of each component may be slightly modified by the presence of the other one. Therefore, in this section, the buckling formulae are possibly revisited so as to take into account the more realistic configuration of a complete pressure vessel. First, the influence of ellipsoidal heads on the buckling behavior of the cylindrical shell is investigated. Then, the possible influence of the cylindrical shell on the buckling response of the closure ends will be discussed.

#### 3.1. Buckling of cylindrical shells in the presence of ellipsoidal heads

Let us first consider the case of a complete pressure vessel where buckling arises in the cylindrical shell, which is generally the case with sufficiently long equipments. The buckling formulae derived in [32] and recalled in Equations (2,3) were obtained by considering idealized simply-supported boundary conditions. In practice, both circular ends of the cylindrical shell are connected to a semi-ellipsoidal closure head, which also deforms during buckling, even if the buckling phenomenon mainly concerns the cylindrical shell. Figure 6 displays these local deformations (in blue line) which induce non-conventional boundary conditions for the buckling analysis of the cylindrical shell. This more practical configuration may thus give rise to different buckling modes and modified critical values with respect to the idealized simply-supported cylindrical shell by itself.

Based on numerical computations, it can be shown that the critical pressure of the complete equipment is slightly lower than the one of the cylindrical shell alone. As the new boundary conditions at the interface are far too difficult to take into account in an exact derivation of the buckling pressure, an idea would be to use the previous formulae, but with an equivalent cylindrical shell of modified length $L_{eq} = L + \Delta L$. L is the actual length of the cylindrical shell, and $\Delta L$ the added length to take into account the influence of the closure ends.

Several ideas have been proposed so as to determine the most appropriate value for $\Delta L$. Galletly and Aylward [34] suggested to define the extra length in the following way:

$$ \Delta L = \frac{D}{n} \qquad (10) $$
<figure><img src="image_7.png" /><figcaption>Figure 6: Buckling deformations at the interfaces between the cylindrical shell and the closure ends</figcaption></figure>

where *D* is the diameter of the equipment and *n* still represents the circumferential wave number of the first buckling mode of the initial cylindrical shell. Besides, the French CODAP [2] recommends to penalize the admissible pressure by using an increased length depending on the dome height *h* (see Figure 7):

$$ \Delta L = \frac{2h}{3} \qquad (11) $$

In this study, several series of finite element calculations have been carried out in order to estimate the optimal equivalent length to consider in the sequel (linearized buckling analyses have been performed in elasticity with *E* = 200000 MPa and *ν* = 0.3). The particular case of a cylindrical shell of length *L* = 2000 mm and radius *R* = 500 mm has been examined. For three different thickness ratios *t*<sub>h</sub> (head thickness) / *t*<sub>c</sub> (cylindrical shell thickness), several dome heights have been considered, from the flat case (*h* = 0) to the hemispherical case (*h* = 500 mm). In each particular configuration, the critical pressure of the equipment (related to the buckling of the cylindrical shell) has been first numerically evaluated. Then, an optimal length has been analytically identified (using Equation (2)), which corresponds to the length of the cylindrical shell alone whose critical pressure is the sought numerical value. From these calculations whose results are listed in Table 3, one can plot the optimal added length Δ*L* (between the modified length and the real length of the cylindrical shell) as a function of the dome height. Figure 8 shows that a quasi-linear relationship is observed for each thickness ratio, but with various slopes. However, as the head thickness is classically less than or equal to the cylindrical shell thickness in practice, and in view of
the very close slopes observed in these cases, an added length of $\Delta L = 2h/3$ (as recommended by the CODAP) is finally retained for future applications. This choice is represented schematically in Figure 7. It will be used for both elastic and plastic buckling cases by extrapolation.

<figure><img src="image_8.png" /><figcaption>Figure 7: Definition of the modified length of the equivalent cylindrical shell in the presence of ellipsoidal ends</figcaption></figure>

<table><caption>Table 3: Added length $\Delta L$ as a function of the dome height $h$ for various thicknesses</caption><thead><tr><th>Dome height h<br>(mm)</th><th>t<sub>c</sub> = t<sub>h</sub> = 5 mm<br>ΔL<br>(mm)</th><th>t<sub>c</sub> = 5 mm, t<sub>h</sub> = 2.5 mm<br>ΔL<br>(mm)</th><th>t<sub>c</sub> = 5 mm, t<sub>h</sub> = 10 mm<br>ΔL<br>(mm)</th></tr></thead><tbody><tr><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>100</td><td>71.0</td><td>89.9</td><td>40.2</td></tr><tr><td>200</td><td>130.5</td><td>148.0</td><td>91.2</td></tr><tr><td>300</td><td>190.5</td><td>208.1</td><td>141.2</td></tr><tr><td>400</td><td>250.5</td><td>271.0</td><td>187.7</td></tr><tr><td>500</td><td>312.0</td><td>337.1</td><td>228.7</td></tr></tbody></table>

## 3.2. Buckling of ellipsoidal heads in the presence of cylindrical shells

The buckling phenomenon may also concern the closure ends of the equipment at first. This is mainly the case when the cylindrical shell and/or the
<figure><img src="image_9.png" /><figcaption>Figure 8: Relationship between the optimal added length and the dome height for various thickness ratios</figcaption></figure>

dome height are particularly short, or when the thickness of the ellipsoidal heads is sufficiently lower than the one of the cylindrical shell. In the literature, the influence of the boundary conditions on the buckling behavior of a spherical cap was investigated, as in Rotter and Schmidt [3] who proposed several boundary conditions that may be equivalent to the presence of a cylindrical shell. In the particular case of an oblate ellipsoidal head, the modal deformation is mostly located around the pole, namely far from the interface with the cylindrical shell (as illustrated in Figure 9). As a consequence, the presence of the cylindrical shell is unlikely to have any influence on the buckling response of the ellipsoidal shell.

For verification purposes, several configurations of pressure equipments were considered, with either hemispherical or ellipsoidal ends, for which buckling is likely to occur at the poles of the heads. The critical pressures of the equipments are then compared to the ones of the corresponding complete spheres or ellipsoids, using respectively numerical linearized buckling analyses (still in elasticity) and the appropriate elastic analytical solutions. The
<figure><img src="image_10.png" /><figcaption>Figure 9: Buckling of a pressure equipment around the pole of an ellipsoidal end</figcaption></figure>

results are summarized in Tables 4 and 5 for the sphere and the ellipsoid, respectively. The critical pressures are almost similar in each configuration, whatever the length of the equipment, the dome height and the respective thicknesses, with a relative error never higher than 1%. These results encourage us to refrain from any modification relating to such a phenomenon, and the previous formulae (Equations (4,5) for the spheres and Equations (8,9) for the ellipsoids) will be directly used for the buckling design of the pressure equipment as a whole.

<table><caption>Table 4: Comparison between critical pressures of hemispherical ends and corresponding complete spheres ($R = 500$ mm)</caption><thead><tr><th colspan="5">Pressure equipment</th><th>Complete sphere</th></tr><tr><th colspan="2">Cylinder</th><th>Spherical head</th><th>Critical pressure</th><th>Critical pressure</th><th></th></tr><tr><th>L<br>(mm)</th><th>t<sub>c</sub><br>(mm)</th><th>t<sub>h</sub><br>(mm)</th><th>p<sup>pe-head</sup><br>(MPa)</th><th>p<sup>sph</sup><br>(MPa)</th><th></th></tr></thead><tbody><tr><td rowspan="3">1000</td><td>10</td><td>1</td><td>0.9680</td><td>0.9675</td><td></td></tr><tr><td>12.5</td><td>2.5</td><td>6.0294</td><td>6.0329</td><td></td></tr><tr><td>25</td><td>5</td><td>24.033</td><td>23.960</td><td></td></tr><tr><td rowspan="3">2000</td><td>10</td><td>1</td><td>0.9681</td><td>0.9675</td><td></td></tr><tr><td>15</td><td>2.5</td><td>6.0272</td><td>6.0329</td><td></td></tr><tr><td>25</td><td>5</td><td>24.033</td><td>23.960</td><td></td></tr></tbody></table>
<table><caption>Table 5: Comparison between critical pressures of ellipsoidal ends and corresponding complete ellipsoids ($R = 500$ mm, $h = 200$ mm)</caption><thead><tr><th colspan="3">Pressure equipment</th><th>Complete ellipsoid</th></tr><tr><th colspan="2">Cylinder</th><th>Ellipsoidal head</th><th>Critical pressure</th></tr><tr><th>L<br>(mm)</th><th>t<sub>c</sub><br>(mm)</th><th>t<sub>h</sub><br>(mm)</th><th>p<sup>pe-head</sup><br>(MPa)</th><th>p<sup>ellip</sup><br>(MPa)</th></tr></thead><tbody><tr><td rowspan="5">1000</td><td>4.5</td><td>1</td><td>0.16338</td><td>0.16338</td></tr><tr><td>10</td><td>2</td><td>0.6661</td><td>0.6661</td></tr><tr><td>10</td><td>2.5</td><td>1.0492</td><td>1.0492</td></tr><tr><td>10</td><td>5</td><td>4.333</td><td>4.333</td></tr><tr><td>19</td><td>10</td><td>18.076</td><td>18.100</td></tr><tr><td rowspan="5">2000</td><td>5</td><td>1</td><td>0.16338</td><td>0.16338</td></tr><tr><td>10</td><td>2</td><td>0.66607</td><td>0.66608</td></tr><tr><td>10</td><td>2.5</td><td>1.0492</td><td>1.0492</td></tr><tr><td>15</td><td>5</td><td>4.333</td><td>4.333</td></tr><tr><td>25</td><td>10</td><td>18.084</td><td>18.100</td></tr></tbody></table>

# 4. Development of a VBA Excel analytical tool

All the previous analytical solutions, including the necessary adjustments discussed above, will now serve as a basis for the implementation of a VBA Excel program devoted to the buckling design of pressure vessels. At this time, the pressure equipments considered in the analytical tool are solely made up of a cylindrical shell and two ellipsoidal (possibly hemispherical) heads, and they are only submitted to external pressure. In the sequel, the different buckling types pertaining to elastoplastic structures are identified and the different buckling scenarii which are likely to occur for such a problem are then deduced. A few details about the design tool are finally provided.

## 4.1. Elastic, plastic and threshold buckling

In the present study, the material is supposed to be elastoplastic with a bi-linear constitutive law, as represented in Figure 10. It means that the response of the material is first linearly elastic with Young's modulus $E$ until an initial yield stress $\sigma_0$, and then elastoplastic with a linear isotropic
hardening (namely a constant hardening modulus $H$ or tangent modulus $E_T$, as already assumed in the analytical developments).

<figure><img src="image_11.png" /><figcaption>Figure 10: Bi-linear elastoplastic constitutive law</figcaption></figure>

As seen before, the idealized equipments considered in this study are likely to buckle either in the cylindrical part or more locally in one head (or both heads since the equipment is supposed to be symmetric), depending on the respective critical pressures. Until now, two critical pressures were derived, one elastic and the other one elastoplastic, for each component of the equipment. Let us define now the following critical Von Mises stress levels at the critical time, which are represented schematically in Figure 11.

In the cylindrical shell, the pre-critical stress state is uniform and biaxial, so that the critical Von Mises stress writes:

$$ \sigma_{cr}^{cyl} = \frac{\sqrt{3}p_{cr}^{cyl-head}R}{2t_c} \qquad (12) $$

In Equation (12), the index ●$_{cr}$ should be replaced by ●$_e$ and ●$_p$ in the elastic and plastic buckling cases, respectively. The exponent ●$_{cyl-head}$ refers to the case of a cylindrical shell under external pressure and axial compression (due to the pressure end thrust), namely Equations (2) or (3), but with the modified length, as discussed in the previous section.

In the ellipsoidal heads, the Von Mises stress is only evaluated at the poles (where the stress state is equi-biaxial) and it is related to the external
<figure><img src="image_12.png" /><figcaption>Figure 11: Critical Von Mises stresses in the cylindrical shell and at the pole of the ellipsoidal heads</figcaption></figure>

pressure as follows:

$$ \sigma_{cr}^{head} = \frac{p_{cr}^{head} R'}{2t_h} \qquad (13) $$

where $R'$ stands for the radius of curvature of the ellipsoidal shell at the poles (which is equal to the radius of the sphere, if so). The exponent ●<sup>head</sup> should be replaced by ●<sup>sph</sup> (using Equations (4) or (5)) in the particular case of a hemispherical head and ●<sup>ellip</sup> (using Equations (8) or (9)) in the general case of an ellipsoidal head.

For any arbitrary component, the effective buckling scenario will depend on the order of appearance of the two possible buckling phenomena (elastic and plastic) and the plasticity threshold. Considering a given buckling mode, conversely to the elastic case displaying a single critical value, the elastoplastic buckling behavior can be characterized by the existence of a continuum of critical points. This continuum enables the structure to bifurcate within an interval of critical loads, by continuously modifying the unloaded zone and the structural stiffness at the critical time. Each continuous range of bifurcation points (one per mode) observed in elastoplasticity spreads from the tangent modulus critical stress $\sigma_p$ to the elastic one $\sigma_e$. Such continua of critical points were highlighted first by Cimetière [39] when dealing with the case of compressed rectangular plates.

Owing to the definition of the constitutive law (which is quite faithful to those proposed in the CODAP [2]), the buckling phenomenon may occur in
three different ways, depending on the yield stress $\sigma_0$ of the material (see Figure 12):

• Case 1:

When the yield stress precedes the continuous range ($\sigma_0 \le \sigma_p$), plastic buckling occurs at the tangent modulus critical value $\sigma_p$, which corresponds to the minimum critical value of the continuum.

• Case 2:

If $\sigma_0 \ge \sigma_e$, the buckling phenomenon is purely elastic.

• Case 3:

In between, in the intermediate case when $\sigma_p \le \sigma_0 \le \sigma_e$, the buckling response, referred to as a plastic breakdown phenomenon by Akel [40], occurs at the yield stress $\sigma_0$ precisely, due to the abrupt change of the tangent modulus at this point. Indeed, the structure behaves elastically at lower stresses, which remain smaller than the elastic buckling value. Moreover, the yield stress lies in the continuum of bifurcation points and thus, as soon as plasticity appears, buckling may arise at this particular value. This phenomenon will be further referred to as “threshold buckling”. The corresponding critical pressures can be easily derived from Equations (12) and (13), respectively:

$$p_0^{cyl-head} = \frac{2\sigma_0 t_c}{\sqrt{3R}} \qquad (14)$$

$$p_0^{head} = \frac{2\sigma_0 t_h}{R'} \qquad (15)$$

## 4.2. Buckling scenarios

A procedure has been defined so as to identify all the possible buckling scenarios and choose the effective one. First, the two components of the pressure vessel are considered separately. The previous analysis allows one to find the actual buckling type (elastic, plastic, threshold) for each of them, by comparing the appropriate critical stresses with the yield stress of the material. Then, the two retained critical pressures $p_{cr}^{cyl-head}$ and $p_{cr}^{head}$ (which may be elastic, plastic or correspond to the threshold stress, depending on the
<figure><img src="image_13.png" /><figcaption>Figure 12: Three buckling types encountered in elastoplasticity</figcaption></figure>

<table><caption>Table 6: Description of the six buckling scenarios of a pressure equipment</caption><thead><tr><th>Designation</th><th>Buckling scenario</th><th>Conditions</th></tr></thead><tbody><tr><td>CE</td><td>Elastic buckling<br>(cylindrical shell)</td><td>σ<sub>p</sub><sup>cyl-head</sup> &lt; σ<sub>e</sub><sup>cyl-head</sup> &lt; σ<sub>0</sub><br>p<sub>cr</sub><sup>cyl-head</sup> &lt; p<sub>cr</sub><sup>head</sup></td></tr><tr><td>CT</td><td>Threshold buckling<br>(cylindrical shell)</td><td>σ<sub>p</sub><sup>cyl-head</sup> &lt; σ<sub>0</sub> &lt; σ<sub>e</sub><sup>cyl-head</sup><br>p<sub>cr</sub><sup>cyl-head</sup> &lt; p<sub>cr</sub><sup>head</sup></td></tr><tr><td>CP</td><td>Plastic buckling<br>(cylindrical shell)</td><td>σ<sub>0</sub> &lt; σ<sub>p</sub><sup>cyl-head</sup> &lt; σ<sub>e</sub><sup>cyl-head</sup><br>p<sub>cr</sub><sup>cyl-head</sup> &lt; p<sub>cr</sub><sup>head</sup></td></tr><tr><td>HE</td><td>Elastic buckling<br>(ellipsoidal heads)</td><td>σ<sub>p</sub><sup>head</sup> &lt; σ<sub>e</sub><sup>head</sup> &lt; σ<sub>0</sub><br>p<sub>cr</sub><sup>head</sup> &lt; p<sub>cr</sub><sup>cyl-head</sup></td></tr><tr><td>HT</td><td>Threshold buckling<br>(ellipsoidal heads)</td><td>σ<sub>p</sub><sup>head</sup> &lt; σ<sub>0</sub> &lt; σ<sub>e</sub><sup>head</sup><br>p<sub>cr</sub><sup>head</sup> &lt; p<sub>cr</sub><sup>cyl-head</sup></td></tr><tr><td>HP</td><td>Plastic buckling<br>(ellipsoidal heads)</td><td>σ<sub>0</sub> &lt; σ<sub>p</sub><sup>head</sup> &lt; σ<sub>e</sub><sup>head</sup><br>p<sub>cr</sub><sup>head</sup> &lt; p<sub>cr</sub><sup>cyl-head</sup></td></tr></tbody></table>

component) are compared to each other so as to find the effective buckling phenomenon and corresponding critical pressure. All in all, six scenarios may be encountered in theory, which are listed in Table 6.

This procedure has been implemented in a VBA Excel analytical tool, whose interface is displayed in Figure 13. The program is intended to provide in a systematic and very efficient way the buckling scenario of a pressure
vessel (where and how it buckles) and the corresponding critical pressure, for any set of geometric and material parameters, using all the previous analytical expressions. The dashboard allows one to deal with a particular component (cylindrical shell or ellipsoidal head) or the entire equipment. Several data windows enable one to enter the geometric and material information by filling the appropriate fields, and the results are finally summarized in a specific table.

# 5. Numerical results and validation of the analytical tool

## 5.1. Mapping of the buckling types and critical pressures

By varying some geometric and/or material parameters in certain ranges, it is possible to plot some maps displaying the distribution of the buckling types and critical pressures. For illustration purposes, a particular set of material properties has been retained, corresponding to a specific elastoplastic constitutive law originating from the CODAP abacuses. The material parameters are listed in Table 7.

<table><caption>Table 7: Material properties</caption><thead><tr><td>Young's modulus E</td><td>Poisson's ratio ν</td><td>Yield stress σ₀</td><td>Tangent modulus E<sub>T</sub></td></tr></thead><tbody><tr><td>200000 MPa</td><td>0.3</td><td>324 MPa</td><td>1000 MPa</td></tr></tbody></table>

Concerning the geometric parameters, the radius *R* of the cylindrical part is always taken to be equal to 500 mm. First, ellipsoidal heads are considered with a dome height *h* of 250 mm and a uniform thickness all over the equipment ($t_h = t_c$). The length and thickness of the cylinder vary from 50 mm to 5000 mm in increments of 2 mm, and from 2 mm to 80 mm in increments of 0.05 mm, respectively. These extreme values correspond to the limit values authorized in the application of the CODAP rules. From all these calculations, using the analytical tool, it is possible to draw a mapping of the different buckling types encountered according to the geometric characteristics of the equipment (see Figure 14).

Several comments can be made about this figure. First, in this specific (but realistic) material configuration, it can be shown that, for any length
<figure><img src="image_14.png" /><figcaption>(a) Dashboard</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Figure 13: Interface of the analytical tool</figcaption></figure>

and thickness considered over these very wide ranges, the elastic buckling of the ellipsoidal heads (HE) never appears in the first instance. Besides, all the
<figure><img src="image_16.png" /><figcaption>Figure 14: Mapping of the different buckling types according to the length and thickness of the cylindrical shell - $h = 250$ mm and $t_c = t_h$</figcaption></figure>

other five buckling types occur more or less frequently in the ranges of length and thickness considered. It can be observed that very thin equipments tend to buckle elastically in the cylindrical part (CE). As the thickness increases, they may buckle first in the heads at the critical pressure corresponding to the initiation of plasticity at the poles (HT), once the cylindrical shell is sufficiently short. Then, there is a limit value for the thickness (regardless of the length) beyond which buckling first occurs in the heads at the minimum tangent modulus critical pressure (HP), and then in the cylindrical part of the equipment (essentially at the pressure corresponding to the onset of plasticity (CT) but possibly at the minimum tangent modulus critical value too (CP)). Only particularly short equipments buckle invariably in the heads, strictly after the occurrence of plasticity (HP) when considering somewhat thick structures. A second mapping is represented in Figure 15 in the same conditions, with $t_c = 2t_h$, namely with a head thickness half than in the previous case. In this new configuration, the pressure equipment buckles virtually all the time in the heads. A small zone corresponding to very short and very thin equipments even reveals the elastic buckling of the heads
in the first instance (HE). Otherwise, most geometric configurations give
rise to an expected threshold buckling phenomenon (HT). Conversely, when
considering hemispherical heads (again with a uniform thickness all over the
structure), the heads are much less likely to buckle, and the cylindrical shell
almost systematically buckles first (see Figure 16). Again, plastic buckling
(CP) is preferably observed than elastic buckling (CE) when considering a
short and/or thick equipment. However, in most geometric configurations
between these two extreme cases, buckling occurs again at the yield pressure
(CT).

<figure><img src="image_17.png" /><figcaption>Figure 15: Mapping of the different buckling types according to the length and thickness of the cylindrical shell - h = 250 mm and t_c = 2t_h</figcaption></figure>

Additionally, the critical pressure can be plotted as a surface with respect to the length and thickness of the pressure vessel, as illustrated in Figure 17 for the first previous configuration (with *h* = 250 mm and *t*<sub>c</sub> = *t*<sub>h</sub>). Such a diagram may be used for optimization purposes.

The analytical tool will be now confronted to purely numerical results, so
as to validate the general analytical procedure at the heart of this program.
<figure><img src="image_18.png" /><figcaption>Figure 16: Mapping of the different buckling types according to the length and thickness of the cylindrical shell - h = 500 mm and t_c = t_h</figcaption></figure>

## 5.2. Numerical validation

In this section, the VBA Excel program based on analytical solutions is validated against numerical results obtained through incremental computations performed on complete pressure equipments using Abaqus. The same material properties are considered as before, whereas only a few geometric configurations are processed. In each case, the smallest possible initial geometric imperfection is introduced in the model, in the shape of the first elastic buckling mode (derived from a preliminary linearized buckling analysis), so as to trigger the buckling phenomenon. A maximum imperfection amplitude of about 0.001 mm is generally sufficient, except for some elastic cases, where a value around 0.01 mm is needed. Owing to symmetry considerations, only one quarter of the whole equipment (half the length and half the circumference) is modeled so as to reduce the computation times. This reduced model allows one to find all the possible buckling modes of the whole structure, both those with even and odd circumferential wave numbers. An optimal mesh is built with the same finite elements as above (S8R5 elements in Abaqus). An external pressure is applied all over the shell structure and no particular
<figure><img src="image_19.png" /><figcaption>Figure 17: Evolution of the critical pressure with respect to the length and thickness of the cylindrical shell - h = 250 mm and t<sub>c</sub> = t<sub>h</sub></figcaption></figure>

boundary condition has to be enforced, except symmetry conditions and an additional boundary condition to prevent from rigid modes. The analytical critical pressures are confronted to the limit admissible pressures arising from the numerical equilibrium curves.

Eight configurations are retained in the first case of an equipment with ellipsoidal heads and a uniform thickness (Figure 14), covering all the possible buckling types, and another one is considered in the second case (Figure 15), with heads of half the thickness, so as to illustrate the elastic buckling of the heads. The corresponding geometric parameters and results are listed in Tables 8 and 9, in the cases where the cylindrical shell or the ellipsoidal ends buckle in the first instance, respectively. In both cases, the analytical solutions are provided and compared to the numerical results. For information purposes, some results from linearized buckling analyses are also indicated, since the first linearized buckling modes are systematically used as initial imperfection shapes. The buckling types are identified in each case, together with the critical pressures and corresponding circumferential wave
numbers (provided that buckling occurs elastically or plastically in the cylindrical shell). Finally, the relative errors between the analytical and numerical (incremental) critical pressures are specified for validation purposes.

<table><caption>Table 8: Comparison between analytical and numerical pressures - buckling of the cylindrical shell - R = 500 mm and h = 250 mm</caption><thead><tr><th></th><th>Configuration</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><td rowspan="3">Geometry</td><td>Cylinder length (mm)</td><td>1000</td><td>4000</td><td>3500</td><td>1500</td><td>2000</td></tr><tr><td>Cylinder thickness (mm)</td><td>5</td><td>15</td><td>50</td><td>70</td><td>80</td></tr><tr><td>Head thickness (mm)</td><td>5</td><td>15</td><td>50</td><td>70</td><td>80</td></tr><tr><td rowspan="3">Analytical</td><td>Buckling type</td><td>CE</td><td>CE</td><td>CT</td><td>CP</td><td>CP</td></tr><tr><td>Critical pressure (MPa)</td><td><strong>0.834</strong></td><td><strong>3.82</strong></td><td><strong>37.4</strong></td><td><strong>71.2</strong></td><td><strong>64.5</strong></td></tr><tr><td>Wave number</td><td>6</td><td>2</td><td>-</td><td>4</td><td>4</td></tr><tr><td rowspan="2">Num. lin.</td><td>Buckling type</td><td>CE</td><td>CE</td><td>CE</td><td>CE</td><td>CE</td></tr><tr><td>Wave number</td><td>6</td><td>2</td><td>2</td><td>2</td><td>2</td></tr><tr><td rowspan="4">Num. inc.</td><td>Buckling type</td><td>CE</td><td>CE</td><td>CT</td><td>CP</td><td>CP</td></tr><tr><td>Critical pressure (MPa)</td><td><strong>0.867</strong></td><td><strong>3.71</strong></td><td><strong>37</strong></td><td><strong>54</strong></td><td><strong>60.3</strong></td></tr><tr><td>Wave number</td><td>6</td><td>2</td><td>-</td><td>4</td><td>4</td></tr><tr><td>Relative error (%)</td><td>-3.85</td><td>3.09</td><td>1.11</td><td>31.9</td><td>6.95</td></tr></tbody></table>

In configurations 1 and 2, the cylindrical shell buckles elastically, and the analytical and numerical results show a very good accordance with each other. The same conclusion applies to configuration 6 where the ends buckle elastically. In the cases where the cylindrical shell or the ellipsoidal heads are theoretically supposed to buckle at the threshold value (configurations 3 and 7, respectively), the results are also very satisfactory. The pressure levels at which plasticity appears at the appropriate place (the whole cylinder and the poles, respectively) are very similar between the analytical and numerical approaches. As buckling occurs immediately after the yield pressure in these cases, the analytical and numerical critical pressures are naturally consistent.
<table><caption>Table 9: Comparison between analytical and numerical pressures - buckling of the ellipsoidal heads - R = 500 mm and h = 250 mm</caption><thead><tr><th colspan="2">Configuration</th><th>6</th><th>7</th><th>8</th><th>9</th></tr></thead><tbody><tr><td rowspan="3">Geometry</td><td>Cylinder length (mm)</td><td>200</td><td>1000</td><td>500</td><td>3000</td></tr><tr><td>Cylinder thickness (mm)</td><td>2</td><td>20</td><td>50</td><td>35</td></tr><tr><td>Head thickness (mm)</td><td>1</td><td>20</td><td>50</td><td>35</td></tr><tr><td rowspan="2">Analytical</td><td>Buckling type</td><td>HE</td><td>HT</td><td>HP</td><td>HP</td></tr><tr><td>Critical pressure (MPa)</td><td><strong>0.242</strong></td><td><strong>13</strong></td><td><strong>50.6</strong></td><td><strong>24.8</strong></td></tr><tr><td rowspan="2">Num. lin.</td><td>Buckling type</td><td>HE</td><td>CE</td><td>CE</td><td>CE</td></tr><tr><td>Wave number</td><td>-</td><td>4</td><td>4</td><td>2</td></tr><tr><td rowspan="3">Num. inc.</td><td>Buckling type</td><td>HE</td><td>HT</td><td>HP</td><td>HP</td></tr><tr><td>Critical pressure (MPa)</td><td><strong>0.257</strong></td><td><strong>13.3</strong></td><td><strong>36.9</strong></td><td><strong>24.6</strong></td></tr><tr><td>Relative error (%)</td><td>-5.66</td><td>-1.96</td><td>37</td><td>0.7</td></tr></tbody></table>

Dealing now with plastic buckling (at the tangent modulus critical value), it turns out that the performance of the analytical solutions depends on the cases considered. They are all the more valid as the representative point in the mapping (Figure 14) is close to the zones of threshold buckling, as seen when comparing configurations 4 and 5 (for the cylindrical shell) and configurations 8 and 9 (for the heads). In practice, in the vicinity of such zones, the yield pressure is only slightly lower than the buckling one, such that at the critical point, plastic strains remain small. Conversely, when moving away from the threshold zones, plastic strains may become more and more important before buckling. Owing to the strong imperfection sensitivity in plasticity, it implies that the numerical solution is all the more degenerated by such pre-critical (most often heterogeneous) strains (which act like initial imperfections), leading to a significant drop of the limit admissible pressure. Besides, in any case, the analytical solutions are no more valid in the context of finite pre-critical deformations.

In Table 8, one can find out that the wave numbers provided by the incremental calculations always correspond to the analytical ones, even if
the initial geometric imperfections (namely the associated first linearized buckling modes) display other smaller wave numbers (see configurations 4 and 5).

All these results are summarized in Figure 18 for clarity purposes.

<figure><img src="image_20.png" /><figcaption>Figure 18: Comparison between analytical and numerical critical pressures for various buckling types of pressure vessels</figcaption></figure>

## 6. Comparison with current design codes

This last section aims at comparing the previous analytical solutions to results stemming from the use of standard design rules. Use will be made of the current methodology implemented in the French CODAP [2] for the buckling design of pressure equipments (and especially Section C4.2), which is mostly derived from the ASME Boiler and Pressure Vessel Code (Section VIII) [1]. Owing to the configurations retained above, the only cases of cylindrical shells (including the influence of closure ends) and ellipsoidal
heads will be considered in the sequel. First, the calculation procedures will be briefly described. Then, numerical applications will be performed for the same previous configurations, so as to confront the results obtained by the current design rules to the solutions given by the analytical tool.

## 6.1. Calculation rules

### 6.1.1. Cylindrical shells

In the CODAP, the critical external pressure of an elastic cylindrical shell, accounting for the pressure end thrust, is classically obtained through the following formula:

$$p_{e-codap}^{cyl-head} = KE \left(\frac{t}{D_o}\right)^3 \qquad (16)$$

where the coefficient K is supposed to depend only on the geometric characteristics of the shell, namely the length L, thickness t and outside diameter D_o of the cylinder. Based on this, the critical circumferential stress can be deduced:

$$\Sigma_{\theta\theta}^{cr} = p_{e-codap}^{cyl-head} \frac{R}{t} = KE \left(\frac{t}{D_o}\right)^3 \frac{R}{t} \approx \frac{KE}{2} \left(\frac{t}{D_o}\right)^2 \qquad (17)$$

and a pseudo-strain is evaluated as follows:

$$A = \frac{\Sigma_{\theta\theta}^{cr}}{E} \approx \frac{K}{2} \left(\frac{t}{D_o}\right)^2 \qquad (18)$$

This critical strain value A also depends only on the geometric parameters, and it is determined in practice by using abacuses C4.2.9.1 [2]. From this numerical value and Young's modulus, the critical pressure can be obtained in a straightforward way, as soon as elasticity is concerned.

In the more general case of elastoplastic buckling, a stress-strain uniaxial curve like the one displayed in Figure 19 is used so as to estimate a stress parameter B. This particular curve will be used in further numerical applications, since it corresponds to the material parameters retained in the previous validation section. The elastoplastic buckling pressure writes then:

$$p_{cr-codap}^{cyl-head} = \frac{4B}{D_o/t} \qquad (19)$$

It can be checked that this expression gives naturally the same value in elasticity as obtained by using directly Equations (16) and (18) (since the linear elastic part of the stress-strain curve in Figure 19 satisfies actually the equation $B = EA/2$).
<figure><img src="image_21.png" /><figcaption>Figure 19: Elastoplastic constitutive law used for the determination of the elastoplastic buckling pressures of equipments under external pressure made of carbon or low-alloy steels under a temperature $T < 150°C$ (from the CODAP, abacus C4.3 [2])</figcaption></figure>

When considering a cylindrical shell with two (ellipsoidal) closure ends, as already mentioned above, the French CODAP recommends to penalize the admissible pressure by using a length increased by $\Delta L = \frac{2h}{3}$ (where $h$ represents the dome height), so as to take into account the influence of the heads on the buckling response of the cylinder.

Finally, let us mention that, in practice, a knock-down factor of 3 is then usually applied for design purposes.

### 6.1.2. Ellipsoidal heads

The design of hemispherical shells also starts from the determination of a critical strain value $A$ which is, in this case, directly given by the following
formula:

$$A = \frac{0.125}{R_o/t} \qquad (20)$$

where $R_o$ is the outside radius of the sphere. Using the same constitutive law, the stress parameter $B$ is derived and the elastoplastic buckling pressure writes then:

$$p_{cr-codap}^{sph} = \frac{B}{R_o/t} \qquad (21)$$

which already includes a knock-down factor (through the choice of the expression of $A$).

In the case of elastic buckling, $B = EA/2$ and the elastic buckling pressure can be deduced as follows:

$$p_{e-codap}^{sph} = \frac{E}{16} \left(\frac{t}{R_o}\right)^2 \qquad (22)$$

Let us recall here the fundamental critical pressure of an elastic perfect sphere under external pressure, initially obtained by Zoelly [14] and also found in the present analytical study:

$$p_e^{sph} = \frac{2E}{\sqrt{3(1-\nu^2)}} \left(\frac{t}{R}\right)^2 \qquad (23)$$

In the end, it turns out that, in the particular case of $\nu = 0.3$, the ratio between the elastic critical value from the CODAP (22) and the corresponding theoretical solution (23) is approximately equal to 20, which thus represents the implicit safety factor considered in this case. This high value, compared to the one selected for the cylindrical shell, is certainly due to the stronger imperfection sensitivity of spherical shells under external pressure.

Next, the CODAP suggests that the design of an ellipsoidal head be performed using the same formulae as before, just replacing the radius of the sphere by the following equivalent radius:

$$R' = 0.9 \frac{a^2}{b} \qquad (24)$$

which depends on the two semi-axes of the ellipsoid but slightly differs from the choice made in the previous analytical study. Therefore, the elastoplastic buckling pressure of an ellipsoidal head writes:

$$p_{cr-codap}^{ellip} = \frac{B}{R'/t} = \frac{Bt}{0.9 \frac{a^2}{b}} \qquad (25)$$
and the associated elastic value becomes:

$$p_{e-codap}^{ellip} = \frac{E}{16} \left( \frac{t}{0.9 \frac{a^2}{b}} \right)^2 \qquad (26)$$

Again, one can deduce the ratio between the elastic critical value from the CODAP (26) and the analytical solution (8), which is equal to 15.7 here, due to the coefficient 0.9 introduced in the design rule.

## 6.2. Comparison between analytical solutions and CODAP

The nine previous configurations, which were used as validation cases, are now considered again in the CODAP context. According to whether the cylindrical shell or the ellipsoidal heads were found analytically (and numerically) to buckle in the first instance, the appropriate procedure is applied to each configuration. All the results are listed in Tables 10 and 11, respectively. The previous analytical solutions are also recalled for comparison purposes. All these results are summarized in Figure 20 for clarity purposes.

<table><caption>Table 10: Comparison between analytical solutions and results from CODAP - buckling of the cylindrical shell - R = 500 mm and h = 250 mm</caption><thead><tr><th></th><th>Configuration</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><td rowspan="3">Geometry</td><td>Cylinder length (mm)</td><td>1000</td><td>4000</td><td>3500</td><td>1500</td><td>2000</td></tr><tr><td>Cylinder thickness (mm)</td><td>5</td><td>15</td><td>50</td><td>70</td><td>80</td></tr><tr><td>Head thickness (mm)</td><td>5</td><td>15</td><td>50</td><td>70</td><td>80</td></tr><tr><td rowspan="2">Analytical</td><td>Buckling type</td><td>CE</td><td>CE</td><td>CT</td><td>CP</td><td>CP</td></tr><tr><td>Critical pressure (MPa)</td><td><strong>0.834</strong></td><td><strong>3.82</strong></td><td><strong>37.4</strong></td><td><strong>71.2</strong></td><td><strong>64.5</strong></td></tr><tr><td rowspan="2">CODAP</td><td>Wave number</td><td>6</td><td>2</td><td>-</td><td>4</td><td>4</td></tr><tr><td>Critical pressure (MPa)</td><td><strong>0.836</strong></td><td><strong>3.43</strong></td><td><strong>32.6</strong></td><td><strong>48.7</strong></td><td><strong>47.8</strong></td></tr><tr><td></td><td>Relative error (%)</td><td>0.24</td><td>-10.2</td><td>-12.9</td><td>-31.6</td><td>-25.89</td></tr></tbody></table>

When buckling occurs first in the cylindrical shell, the critical pressures obtained through Equation (19) (without knock-down factors) are compared
<table><caption>Table 11: Comparison between analytical solutions and results from CODAP - buckling of the ellipsoidal heads - R = 500 mm and h = 250 mm</caption><thead><tr><th colspan="2">Configuration</th><th>6</th><th>7</th><th>8</th><th>9</th></tr></thead><tbody><tr><td rowspan="3">Geometry</td><td>Cylinder length (mm)</td><td>200</td><td>1000</td><td>500</td><td>3000</td></tr><tr><td>Cylinder thickness (mm)</td><td>2</td><td>20</td><td>50</td><td>35</td></tr><tr><td>Head thickness (mm)</td><td>1</td><td>20</td><td>50</td><td>35</td></tr><tr><td rowspan="2">Analytical</td><td>Buckling type</td><td>HE</td><td>HT</td><td>HP</td><td>HP</td></tr><tr><td>Critical pressure (MPa)</td><td><strong>0.242</strong></td><td><strong>13</strong></td><td><strong>50.6</strong></td><td><strong>24.8</strong></td></tr><tr><td rowspan="2">CODAP</td><td>Critical pressure (MPa)</td><td><strong>0.0154</strong></td><td><strong>3.57</strong></td><td><strong>8.91</strong></td><td><strong>6.28</strong></td></tr><tr><td>Ratio analytical/CODAP</td><td>15.71</td><td>3.64</td><td>5.68</td><td>3.95</td></tr></tbody></table>

to the corresponding analytical solutions, either in elasticity or in plasticity, depending on the case.

In the first two configurations (in elasticity), apart from whether or not a safety factor is used, the results are in good accordance. It can just be noted that the longer the cylinder, the more conservative the CODAP rules, so well that the relative difference is a little bit more pronounced in configuration 2. In the intermediate case (configuration 3) where the cylindrical shell buckles at the threshold pressure, the two approaches are still in good agreement, with a relative difference of about 10%, which is due again to the great length of the equipment. Conversely, in the plastic buckling cases (configurations 4 and 5), the average relative difference between the two solutions is about 30%. Unlike what was observed in [32] for a single cylinder, the critical values recommended by the standard in the present configurations are found to be smaller than the analytical pressures. It is important to remember that the calculation procedure in the CODAP for the plastic buckling case is somehow based on an elastic formulation and not connected at all to any exact elastoplastic formula, as is the case in the present analytical developments. The new formulae developed in this work could thus contribute positively to the standard rules in the sense that they would allow one to display a constant conservatism when associated to an appropriate safety factor (say 3, like in elasticity, or more, due to the stronger imperfection sensitivity in
<figure><img src="image_22.png" /><figcaption>Figure 20: Comparison between analytical solutions and results from CODAP for various buckling types of pressure vessels</figcaption></figure>

plasticity).

Next, when buckles occur first in the ellipsoidal heads, the critical pres-
sures obtained through Equations (26) and (25) (in elasticity and plasticity,
respectively) are compared to the corresponding analytical solutions. Here,
as the CODAP procedure inherently includes a safety factor in the expres-
sions of the critical pressure, the ratio between the analytical solutions and
the results provided by the CODAP rules is indicated in Table 11 and Figure
20 instead of the relative difference as before.

In the elastic case (configuration 6), a ratio of about 15.7 is achieved,
which corresponds to the safety factor inherently introduced in the CODAP
formula. In all other configurations (when the heads buckle at, or strictly
after, the yield pressure), a ratio of around 5 is only obtained. Owing to the
very strong imperfection sensitivity of such ellipsoidal shells (when compared
to cylindrical ones) under external pressure, and all the more so in plasticity,
it is questionable that the present standard code recommends such a small
knock-down factor, in comparison with the safety factor retained in the elastic
case which is three times greater.

7. Conclusions

The main objective of the present paper was to derive new analytical
solutions related to the buckling of pressure equipments for dimensioning
purposes. The general case of a cylindrical shell, closed with two ellipsoidal
heads, under external pressure was particularly investigated. First, analyti-
cal solutions of the elastic and plastic buckling pressures of each component
considered individually were obtained from a unified theoretical formulation
based on the elastoplastic bifurcation theory. The respective influence of
each component on the other one was then analyzed. Thereafter, analytical
solutions for the complete equipments were derived by combining the fun-
damental solutions previously obtained for the idealized geometries. Indeed,
in practice, various buckling scenarii can be envisaged. The buckling of the
pressure equipment may initially involve either the cylindrical shell or the
ends, depending on the respective critical pressures. In both cases, buckling
may be elastic, plastic or, intermediately, occur at the elastic threshold, be-
tween the critical values of the tangent and elastic moduli. An analytical tool
was created, based exclusively on all the new explicit formulae established,
and making it possible to rule on the buckling phenomenon observed, after
comparing all the possible critical pressures. A predictive tool of this kind
thus makes it possible to obtain an immediate map of the types of buck-
ling and critical pressures by varying any of the equipment's geometric or
material parameters, which can then be used to optimize the equipment's
configuration very effectively.

The analytical tool was first validated through comparison with reference
numerical results coming from finite element incremental computations on
complete equipments. Theoretically speaking, all the new solutions derive
from a rigorous bifurcation analysis of the problem and only a few approx-
imations are included in the calculations (relative to the ellipsoidal case, or
associated to the influence of the heads on the buckling response of the cylin-
drical shell), which were shown to have a limited influence on the numerical
values. It naturally follows that these analytical solutions generally agree well
with the reference finite element results computed for validation purposes,
except in some rare cases of plastic buckling. The corresponding errors (of
about 30%) may come from the emergence of finite (plastic) strains before
the critical time in the numerical computations (which are not taken into ac-
count in the analytical modeling), but they are most often due to the strong
imperfection sensitivity (in plasticity) of the problems in hand. Even in small
pre-critical strains, the greater the pre-critical deformations, the more degen-
erate the problem, giving rise to a sharp decrease of the maximum admissible
pressure, even in incremental calculations related to supposedly perfect shells
(due to unavoidable imperfect mesh and/or boundary conditions). However,
in practice, it can be shown that buckling occurs most of the time in the
elastic regime, at the yield pressure or shortly after, so well that, in these
conditions, the analytical solutions turn out to be really in accordance with
the reference numerical results.

Last, the analytical solutions were also confronted to the values provided
by the CODAP rules. While in elasticity, the relationship between the two
approaches is quite clear (the results are almost similar, provided that the
appropriate knock-down factor is applied), the elastoplastic results raise a
number of questions.

Thus, the conclusion of this paper is that the present analytical solutions
(especially in plasticity) may serve as an alternative to current empirical
design methods. Indeed, these new analytical solutions can be viewed as ref-
erence values associated to the perfect structures. In that sense, they have
proved reliable, and safety factors may then be added, according to imper-
fection sensitivity analyses. More generally, the present method is likely to
be applied to other geometries (such as conical shells) and/or loadings (such
as torsion), even combinations of loadings. Such new configurations, which
were given less consideration in the standard codes, may provide even bet-
ter results, due to less sensitivity to imperfections. This enrichment of the
standard rules is expected to enable an ever more efficient buckling dimen-
sioning of pressure equipments with a greater accuracy and a wider range of
applications.

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