Buckling of circular cylindrical shells under in-plane and/or pressure loads

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<div id="_heading=h.gjdgxs" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<big>'''Buckling of circular cylindrical shells under in-plane and/or pressure loads'''</big></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
J. Enrique Herencia </div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Senior Stress Engineer, Airbus Operations GmbH, Kreetslag 10, 21129, Hamburg, Germany  </div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
[mailto:enrique.herencia@airbus.com enrique.herencia@airbus.com]</div>

'''RESUMEN:''' En este trabajo se presenta un método semi analítico para calcular de manera rápida pero precisa el pandeo de cilindros circulares con distintas condiciones de contorno y sometidos a cargas en el plano y/o de presión usando el método de Rayleigh-Ritz. Los resultados se han comparado con soluciones analíticas y modelos de elementos finitos de detalle de otras publicaciones. El método propuesto permite realizar un cálculo rápido del pandeo de cilindros circulares lo que lo hace ser un candidato ideal para ser utilizado como parte de un proceso de optimización y/o para reducir potencialmente el número de elementos finitos de detalle en las primeras fases de diseño.
-->

'''ABSTRACT:''' This work presents a semi-analytical approach to calculate rapidly but accurately the buckling onset of metallic and composite circular cylindrical shells with various boundary conditions under in-plane and/or pressure loads by the Rayleigh-Ritz method. Results are compared with analytical solutions and detailed finite element models reported in the literature. The proposed approach allows a quick buckling analysis of circular cylindrical shells, which makes it an ideal candidate to be used as part of an optimization scheme and/or to reduce potentially the number of detailed finite element models employed in the early design phases.

'''Keywords: '''Analytical method, Rayleigh-Ritz, buckling, metallic, composite, cylinders, FEM

==1. Introduction==

Circular cylindrical shells are extensively used in the aerospace industry. They can be made of metallic or composite materials and designed with or without stiffeners as well as step thickness variations. A typical example of a cylindrical shell could be an aircraft fuselage or rocket pressure fuel tank.

Although the nature of shell buckling is a highly non-linear phenomenon, mostly due to its sensitivity to imperfections, it has been addressed by using a combination of empirical knockdown factors and linear buckling analysis [1]. Several shell theories [1-7] have been developed throughout the years to perform shell buckling analysis. A good overview can be found in [2]. One of the most extensively used is the classical or thin shell theory attributed to Love [3]. Examples of thin shell theories that build upon Love’s postulates are those attributed to Timoshenko [4], Flügge [5], Vlasov [6], and Donnell [7] among others.

In the early stages of shell design, when a large number of optimization studies and trends need to be performed, the use of non-linear or even linear detailed finite elements is not feasible due to their complexity and computational cost [1]. An alternative to the use of finite elements is the use of variational or energy methods such as the Rayleigh-Ritz method (e.g., see Ref. [8]). Energy-based methods have been used by many authors (e.g., see Refs. [1,9-10]) and have demonstrated to provide very accurate results. Furthermore, they are proven to be computationally more efficient than traditional finite element models used for structural calculations [11].

This paper focuses on the linear buckling analysis and presents a semi-analytical approach to calculate rapidly but accurately the buckling onset of metallic and composite cylindrical shells with various boundary conditions under in-plane and/or pressure loads by the Rayleigh-Ritz method. The results of the linear buckling analysis are compared with analytical solutions and detailed finite element models reported in the literature.

==2. Geometry, loads and boundary conditions==

<span id='_Hlk133836156'></span>The cylindrical shell is circular, of radius (''R'') and length (''L'') and has a thickness (''t'') reinforced with longitudinal stiffeners or stringers and transverse stiffeners or rings. The pitch between the longitudinal and transverse stiffeners is <math display="inline">{d}_{s}</math> and <math display="inline">{d}_{r}</math>, respectively. An orthogonal curvilinear  coordinate system is defined such that ''x ''is the longitudinal direction, ''y ''is the transverse direction following the shell contour and ''z'' is the vertical direction being positive pointing towards the shell exterior. A sketch of the shell under study is shown in <span id='cite-_Ref134263930'></span>[[#_Ref134263930|Figure 1]] below.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Herencia_516249341-image1.png|306px]] </div>

<div id="_Ref134263930" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Figure 1.''' '''Circular cylindrical shell geometry</div>

The longitudinal and transverse stiffeners are idealised as beam elements and defined by their area ( <math display="inline">{A}_{i}</math>), inertia at the centroid ( <math display="inline">{I}_{i}</math>), torsional constant ( <math display="inline">{J}_{i}</math>), and offset ( <math display="inline">{e}_{i}</math>) with respect to the shell midplane where the subscript ( <math display="inline">i</math>) is equal to <math display="inline">s</math> or <math display="inline">r</math>, if the stiffener is longitudinal or transverse, respectively. See <span id='cite-_Ref134263990'></span>[[#_Ref134263990|Figure 2]] below.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Herencia_516249341-image2.png|252px]] </div>

<div id="_Ref134263990" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Figure 2.''' '''Longitudinal and transverse stiffeners idealisation</div>

The shell is assumed to be under in-plane and/or external pressure loads. For pure external pressure loads <math display="inline">{N}_{y}=</math><math>-\frac{{q}_{ext}}{R}</math> , similarly, for uniform hydrostatic pressure, <math display="inline">{N}_{x}=</math><math>-\frac{{q}_{ext}}{2R}</math>  and   <math display="inline">{N}_{y}=-\frac{{q}_{ext}}{R}</math>, where <math display="inline">{q}_{ext}</math> is the pressure value. <span id='cite-_Ref134264039'></span>[[#_Ref134264039|Figure 3]] below shows the applied loads.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Herencia_516249341-image3.png|282px]] </div>

<div id="_Ref134264039" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Figure 3.''' '''Applied loads</div>

The shell is assumed to be simply supported or clamped at the circular loading edges.

==3. Fundamental equations of shell theory==

The following assumptions following Love’s first approximation [3] are considered:

:* The shell is thin. That is the thickness of the shell is small when compared to other dimensions.

:* Displacement, strains and rotations are small.

:* Kirchhoff’s hypotheses. That is, the normals to the middle surface remain straight before and after deformation and suffer no extension. In other words, transverse shear and through thickness strains are neglected ( <math display="inline">{\gamma }_{xz}=</math><math>{\gamma }_{yz}={\epsilon }_{z}=0)</math>

:* The through thickness stress is small and can be neglected ( <math display="inline">{\sigma }_{z}=</math><math>0</math>).

:* Material behaviour is linearly elastic. Materials can be metallic or composites.

The relations between the in-plane strains, curvatures and displacements are linear with ''z'' and are obtained using Donnell’s shell theory [7]. Thus,

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="margin:auto;width: 100%;"
|-
| <math>\left\{ \mathit{\boldsymbol{\epsilon }}\right\} =\left\{ {\mathit{\boldsymbol{\epsilon }}}^{\mathit{\boldsymbol{0}}}\right\} +</math><math>z\mathit{\boldsymbol{\, }}\left\{ \mathit{\boldsymbol{\kappa }}\right\}</math> 
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref134350767'></span>(1)
|}


where, <math display="inline">\mathit{\boldsymbol{\epsilon }}=</math><math>{\left\{ {\epsilon }_{x},{\epsilon }_{y},{\epsilon }_{xy}\right\} }^{\mathit{\boldsymbol{T}}}</math>is the vector of strains''', ''' <math display="inline">{\mathit{\boldsymbol{\epsilon }}}^{\mathit{\boldsymbol{0}}}=</math><math>{\left\{ {\epsilon }_{x}^{0},{\epsilon }_{x}^{0},{\epsilon }_{xy}^{0}\right\} }^{\mathit{\boldsymbol{T}}}</math>''' '''is the vector of the in-plane strains, <math display="inline">\mathit{\boldsymbol{\kappa =}}{\left\{ {\kappa }_{x},{\kappa }_{y},{\kappa }_{xy}\right\} }^{\mathit{\boldsymbol{T}}}</math>''' '''is the vector of the curvatures, with

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="margin:auto;width: 100%;"
|-
| <math>{\epsilon }_{x}^{0}=\frac{\partial {u}_{0}}{\partial x}\quad \quad {\epsilon }_{y}^{0}=</math><math>\frac{\partial {v}_{0}}{\partial y}+\frac{{w}_{0}}{R}\quad \, {\epsilon }_{xy}^{0}=</math><math>\frac{\partial {u}_{0}}{\partial y}+\frac{\partial {v}_{0}}{\partial x}</math>

<math>{\kappa }_{x}=-\frac{{\partial }^{2}{w}_{0}}{{\partial x}^{2}}\quad {\kappa }_{y}=</math><math>-\frac{{\partial }^{2}{w}_{0}}{{\partial y}^{2}}\quad {\kappa }_{xy}=-2\frac{{\partial }^{2}{w}_{0}}{\partial x\partial y}</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|(2)
|}


where <math display="inline">{u}_{0}</math>, <math display="inline">{v}_{0}</math> and <math display="inline">{w}_{0}</math> are the displacements in ''x'', ''y'' and ''z'' directions, respectively.

The material constitutive equations of the shell are obtained by (e.g. [8]),

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="margin:auto;width: 100%;"
|-
| <math>\left\{ \begin{matrix}\mathit{\boldsymbol{N}}\\\mathit{\boldsymbol{M}}\end{matrix}\right\} =</math><math>\left[ \begin{matrix}\mathit{\boldsymbol{A}}&\mathit{\boldsymbol{B}}\\\mathit{\boldsymbol{B}}&\mathit{\boldsymbol{D}}\end{matrix}\right] \left\{ \begin{matrix}{\mathit{\boldsymbol{\epsilon }}}^{0}\\\mathit{\boldsymbol{\kappa }}\end{matrix}\right\}</math> 
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|(3)
|}


where <math display="inline">\mathit{\boldsymbol{N=}}{\left\{ {N}_{x},{N}_{y},{N}_{xy}\right\} }^{\mathit{\boldsymbol{T}}}</math> is the vector of running loads, <math display="inline">\mathit{\boldsymbol{M=}}{\left\{ {M}_{x},{M}_{y},{M}_{xy}\right\} }^{\mathit{\boldsymbol{T}}}</math>''' '''is the vector of running moments, <math display="inline">\mathit{\boldsymbol{A}}</math>''' '''is the membrane stiffness matrix, <math display="inline">\mathit{\boldsymbol{B}}</math>''' '''is the membrane-bending coupling stiffness matrix and <math display="inline">\mathit{\boldsymbol{D}}</math>''' '''is the bending stiffness matrix.

The membrane, membrane-bending and bending stiffness matrix terms of the shell considering the longitudinal and transverse stiffener properties are as follows [1];

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="margin:auto;width: 100%;"
|-
| <math>\mathit{\boldsymbol{~}}\left[ \begin{matrix}{A}_{11}&{A}_{12}&{A}_{13}\\{A}_{12}&{A}_{22}&{A}_{23}\\{A}_{13}&{A}_{23}&{A}_{33}\end{matrix}\right] =</math><math>\left[ \begin{matrix}{A}_{11}^{shell}&{A}_{12}^{shell}&0\\{A}_{12}^{shell}&{A}_{22}^{shell}&0\\0&0&{A}_{33}^{shell}\end{matrix}\right] +</math><math>\left[ \begin{matrix}\frac{{E}_{s}{A}_{s}}{{d}_{s}}&0&0\\0&\frac{{E}_{r}{A}_{r}}{{d}_{r}}&0\\0&0&0\end{matrix}\right] \mathit{\boldsymbol{~}}</math>

<math>\mathit{\boldsymbol{~}}\left[ \begin{matrix}{B}_{11}&{B}_{12}&{B}_{13}\\{B}_{12}&{B}_{22}&{B}_{23}\\{B}_{13}&{B}_{23}&{B}_{33}\end{matrix}\right] =</math><math>\left[ \begin{matrix}{B}_{11}^{shell}&{B}_{12}^{shell}&0\\{B}_{12}^{shell}&{B}_{22}^{shell}&0\\0&0&{B}_{33}^{shell}\end{matrix}\right] +</math><math>\left[ \begin{matrix}\frac{{E}_{s}{A}_{s}}{{d}_{s}}{e}_{s}&0&0\\0&\frac{{E}_{r}{A}_{r}}{{d}_{r}}{e}_{r}&0\\0&0&0\end{matrix}\right] \mathit{\boldsymbol{~}}</math>

<math>\mathit{\boldsymbol{~}}\left[ \begin{matrix}{D}_{11}&{D}_{12}&{D}_{13}\\{D}_{12}&{D}_{22}&{D}_{23}\\{D}_{13}&{D}_{23}&{D}_{33}\end{matrix}\right] =</math><math>\left[ \begin{matrix}{D}_{11}^{shell}&{D}_{12}^{shell}&0\\{D}_{12}^{shell}&{D}_{22}^{shell}&0\\0&0&{D}_{33}^{shell}\end{matrix}\right] +</math><math>\left[ \begin{matrix}\frac{{E}_{s}}{{d}_{s}}\left( {I}_{s}+{A}_{s}{e}_{s}^{2}\right) &0&0\\0&\frac{{E}_{r}}{{d}_{r}}\left( {I}_{r}+{A}_{r}{e}_{r}^{2}\right) &0\\0&0&\frac{1}{4}\left( \frac{{G}_{s}{J}_{s}}{{d}_{s}}+\frac{{G}_{r}{J}_{r}}{{d}_{r}}\right) \end{matrix}\right] \mathit{\boldsymbol{~}}</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref134350775'></span>(4)
|}


where <math display="inline">{E}_{i}</math> and <math display="inline">{G}_{i}</math> ( <math display="inline">i=</math><math>s</math> or <math display="inline">r</math>) are the Young’s and Shear moduli for the longitudinal and transverse stiffeners, respectively. For this work, shell anisotropy is not considered.

==4. Problem formulation==

The Rayleigh-Ritz method is used to perform the buckling analysis of the circular cylindrical shell. This is an approximate method that uses energy principles that are equivalent to the governing differential equations (i.e., Euler-Lagrange) and natural boundary conditions [12]. The energy principle used in this work is the principle of minimum potential energy (e.g., see Ref. [8]).

In the Rayleigh-Ritz method the displacements are approximated in the form of a finite double series, such that

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="margin:auto;width: 100%;"
|-
| <math>{u}_{0}\left( x,y\right) =\sum _{i}^{I}\sum _{j}^{J}{u}_{ij}{X}_{i'x}\left( x\right) {Y}_{j}\left( y\right)</math>

<math>{v}_{0}\left( x,y\right) =\sum _{i}^{I}\sum _{j}^{J}{v}_{ij}{X}_{i}\left( x\right) {Y}_{j'y}\left( y\right)</math>

<math>{w}_{0}\left( x,y\right) =\sum _{i}^{I}\sum _{j}^{J}{w}_{ij}{X}_{i}\left( x\right) {Y}_{j}\left( y\right)</math>


|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref134263262'></span>(5)
|}


where the functions <math display="inline">{X}_{i}\left( x\right)</math>  and <math display="inline">{Y}_{j}(y)</math> are the ''i''th and ''j''th known functions in the ''x'' and ''y'' directions, respectively that must satisfy the geometrical boundary conditions in the whole domain [13]. <math display="inline">{u}_{ij}</math> , <math display="inline">{v}_{ij}</math> and <math display="inline">{w}_{ij}\,</math> are unknown constants for the displacements in ''x'', ''y'' and ''z'', respectively and <math display="inline">I</math> and <math display="inline">J</math> are the number of terms in the series in the ''x'' and ''y'' direction, respectively. Note that <math display="inline">{(\, )}_{'\alpha }</math> denotes <math display="inline">\frac{\partial }{\partial \alpha }(\, )</math>.

The geometrical boundary conditions considered for the shell at the circular edges (e.g., see Ref.  [14]) and are shown in <span id='cite-_Ref74509411'></span>[[#_Ref74509411|Table 1]] below.

{| style="width: 53%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Type
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Geometrical boundary condition
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|S2(SS)
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>v(0,y)=v(L,y)=0\,</math>

<math>w(0,y)=w(L,y)=0\,</math>

<math>{w}_{''x}(0,y)={w}_{''x}(L,y)=0</math>
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|C1(CC)
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>u(0,y)=u(L,y)=0\,</math>

<math>v(0,y)=v(L,y)=0\,</math>

<math>w(0,y)=w(L,y)=0\,</math>

<math>{w}_{'x}(0,y)={w}_{'x}(L,y)=0\,</math> 
|}


<span id='_Ref74509411'></span><span id='_Ref74509404'></span><span id='_Toc75884591'></span><div id="_Toc77291704" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Table 1. Geometrical boundary conditions at the circular edges</div>

The trigonometric functions for in the ''x'' direction that satisfy the simply supported and clamped conditions (e.g., see Ref. [15]) in <span id='cite-_Ref74509411'></span>[[#_Ref74509411|Table 1]] are given by,

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="margin:auto;width: 100%;"
|-
| <math>{X}_{i}\left( x\right) =sin\left( \frac{i\pi x}{L}\right) \quad \, \, i=</math><math>1,\ldots ,~I\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \rightarrow SS</math>

<math>{X}_{i}\left( x\right) =cos\left( (i-1)\frac{\pi x}{L}\right) -cos\left( (i+1)\frac{\pi x}{L}\right) \quad i=</math><math>1,\ldots ,~I\, \, \rightarrow CC</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|(6)
|}


The trigonometric function in the ''y'' direction is given by

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="margin:auto;width: 100%;"
|-
| <math>{Y}_{j}\left( y\right) =sin\left( \frac{jy}{R}\right) \quad \, \, j=1,\ldots ,J</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|(7)
|}


For the buckling problem, the principle of minimum potential energy takes the form,

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\partial V=\delta (U+\lambda W)=0</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref134352567'></span>(8)
|}


with

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>U=\frac{1}{2}\int\!\int\!\int _{}^{}({\epsilon }_{x}{\sigma }_{x}+{\epsilon }_{y}{\sigma }_{y}+{\gamma }_{xy}{\tau }_{xy})dV</math>

<math>W=-\frac{1}{2}\int\!\int _{}^{}\left[ {N}_{x}{\left( \frac{\partial {w}_{0}}{\partial x}\right) }^{2}+{N}_{y}{\left( \frac{\partial {w}_{0}}{\partial y}\right) }^{2}+2{N}_{xy}\frac{\partial {w}_{0}}{\partial x}\frac{\partial {w}_{0}}{\partial y}\right] dA</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref134350905'></span>(9)
|}


where <math display="inline">V</math> is the potential energy of the shell, <math display="inline">U</math> is the strain energy of the shell including the longitudinal and transverse stiffeners, <math display="inline">W</math> is work of external forces and <math display="inline">\lambda</math>  is the critical buckling load factor.

The expression of the potential energy of the shell is found by substituting Equations <span id='cite-_Ref134350767'></span>[[#_Ref134350767|(1)]]-<span id='cite-_Ref134350775'></span>[[#_Ref134350775|(4)]] in Equation <span id='cite-_Ref134350905'></span>[[#_Ref134350905|(9)]] and rearranging,

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>V=\frac{1}{2}\int\!\int _{}^{}\left[ {A}_{11}{\left( \frac{\partial {u}_{0}}{\partial x}\right) }^{2}+2{A}_{12}\frac{\partial {u}_{0}}{\partial x}\left( \frac{\partial {v}_{0}}{\partial y}+\frac{{w}_{0}}{R}\right) +{A}_{22}{\left( \frac{\partial {v}_{0}}{\partial y}+\frac{{w}_{0}}{R}\right) }^{2}+{A}_{33}{\left( \frac{\partial {u}_{0}}{\partial y}+\frac{\partial {v}_{0}}{\partial x}\right) }^{2}\right] dA-</math><math>\int\!\int _{}^{}\left[ {B}_{11}\frac{\partial {u}_{0}}{\partial x}\frac{{\partial }^{2}{w}_{0}}{\partial {x}^{2}}+2{B}_{12}\left( \left( \frac{\partial {v}_{0}}{\partial y}+\frac{{w}_{0}}{R}\right) \frac{{\partial }^{2}{w}_{0}}{\partial {x}^{2}}+\frac{\partial {u}_{0}}{\partial x}\frac{{\partial }^{2}{w}_{0}}{\partial {y}^{2}}\right) +{B}_{22}\left( \frac{\partial {v}_{0}}{\partial y}+\frac{{w}_{0}}{R}\right) \frac{{\partial }^{2}{w}_{0}}{\partial {xy}^{2}}+4{B}_{33}\left( \frac{\partial {u}_{0}}{\partial y}+\frac{\partial {v}_{0}}{\partial x}\right) \frac{{\partial }^{2}{w}_{0}}{\partial x\partial y}\right] dA+</math><math>\frac{1}{2}\int\!\int _{}^{}\left[ {D}_{11}{\left( \frac{{\partial }^{2}{w}_{0}}{\partial {x}^{2}}\right) }^{2}+2{D}_{12}\frac{{\partial }^{2}{w}_{0}}{\partial {x}^{2}}\frac{{\partial }^{2}{w}_{0}}{\partial {y}^{2}}+{D}_{22}{\left( \frac{{\partial }^{2}{w}_{0}}{\partial {y}^{2}}\right) }^{2}+4{D}_{33}{\left( \frac{{\partial }^{2}{w}_{0}}{\partial x\partial y}\right) }^{2}\right] dA-</math><math>\lambda \frac{1}{2}\int\!\int _{}^{}\left[ {N}_{x}{\left( \frac{\partial {w}_{0}}{\partial x}\right) }^{2}+{N}_{y}{\left( \frac{\partial {w}_{0}}{\partial y}\right) }^{2}+2{N}_{xy}\frac{\partial {w}_{0}}{\partial x}\frac{\partial {w}_{0}}{\partial y}\right] dA</math>


|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref134352549'></span>(10)
|}


The solution of the problem is found by substituting Equation <span id='cite-_Ref134263262'></span>[[#_Ref134263262|(5)]] in Equation <span id='cite-_Ref134352549'></span>[[#_Ref134352549|(10)]] and minimising with respect to each of the displacements’ unknown coefficients. Hence,

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\frac{\partial V}{\partial {u}_{ij}}=0\quad \, \frac{\partial V}{\partial {v}_{ij}}=</math><math>0\quad \frac{\partial V}{\partial {w}_{ij}}=0</math>

<math display="inline">i=1,\ldots ,~I\,</math>   and <math display="inline">j=</math><math>1,\ldots ,~J</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|(11)
|}


This generates a system of <math display="inline">3\cdot \, I\, \cdot \, J</math> linear equations with <math display="inline">3\cdot \, I\, \cdot \, J</math> unknowns. Thus,

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="margin:auto;width: 100%;"
|-
| <math>\left( \left[ \begin{matrix}{\mathit{\boldsymbol{K}}}_{\mathit{\boldsymbol{uu}}}&{\mathit{\boldsymbol{K}}}_{\mathit{\boldsymbol{uv}}}&{\mathit{\boldsymbol{K}}}_{\mathit{\boldsymbol{uw}}}\\{\mathit{\boldsymbol{K}}}_{\mathit{\boldsymbol{uv}}}^{\mathit{\boldsymbol{T}}}&{\mathit{\boldsymbol{K}}}_{\mathit{\boldsymbol{ww}}}&{\mathit{\boldsymbol{K}}}_{\mathit{\boldsymbol{vw}}}\\{\mathit{\boldsymbol{K}}}_{\mathit{\boldsymbol{uw}}}^{\mathit{\boldsymbol{T}}}&{\mathit{\boldsymbol{K}}}_{\mathit{\boldsymbol{vw}}}^{\mathit{\boldsymbol{T}}}&{\mathit{\boldsymbol{K}}}_{\mathit{\boldsymbol{ww}}}\end{matrix}\right] +\right. </math><math>\left. \lambda \left[ \begin{matrix}\mathit{\boldsymbol{0}}&\mathit{\boldsymbol{0}}&\mathit{\boldsymbol{0}}\\\mathit{\boldsymbol{0}}&\mathit{\boldsymbol{0}}&\mathit{\boldsymbol{0}}\\\mathit{\boldsymbol{0}}&\mathit{\boldsymbol{0}}&{\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{ww}}}\end{matrix}\right] \right) \cdot \left\{ \begin{matrix}\mathit{\boldsymbol{u}}\\\mathit{\boldsymbol{v}}\\\mathit{\boldsymbol{w}}\end{matrix}\right\} =</math><math>\mathit{\boldsymbol{0}}</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref134358283'></span>(12)
|}


where <math display="inline">\mathit{\boldsymbol{u}}</math>,''' ''' <math display="inline">\mathit{\boldsymbol{v}}</math> and <math display="inline">\mathit{\boldsymbol{w}}</math> are the unknown coefficients of the displacement vectors, each of <math display="inline">I\, \cdot \, J</math> dimension, <math display="inline">{\mathit{\boldsymbol{K}}}_{\mathit{\boldsymbol{mn}}}\, (m,n=</math><math>u,v,w)</math> are squared matrices each of <math display="inline">I\cdot J\times I\cdot J</math> dimensions related to the shell’s stiffnesses and <math display="inline">{\mathit{\boldsymbol{L}}}_{\mathit{\boldsymbol{ww}}}</math> is a squared matrix of <math display="inline">I\cdot J\times I\cdot J</math> dimensions, related to the applied loads. This linear system is homogeneous and can be solved as an eigenvalue problem. The first eigenvalue will provide the critical buckling load such that,

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="margin:auto;width: 100%;"
|-
| <math>\left\{ \begin{matrix}{N}_{x}^{cr}\\{N}_{y}^{cr}\\{N}_{xy}^{cr}\end{matrix}\right\} =</math><math>\lambda \left\{ \begin{matrix}{N}_{x}\\{N}_{y}\\{N}_{xy}\end{matrix}\right\}</math> 
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|(13)
|}


==5. Numerical results==

The semi-analytical method presented in this paper was developed in Fortran [16]. Integrals to obtain the matrices in Equation <span id='cite-_Ref134358283'></span>[[#_Ref134358283|(12)]] were computed numerically. Results from the semi-analytical approach were compared against analytical solutions and detailed finite element models reported in the literature. Sensitivity studies were performed to determine the number of terms in the ''x'' and ''y'' directions in the series that guarantee accurate predictions for the eigenvalue obtained with Equation <span id='cite-_Ref134358283'></span>[[#_Ref134358283|(12)]]. Even for a large number of terms in the series the method was very fast providing result within seconds. Unstiffened and stiffened circular cylindrical shells made of metallic or composite materials subjected to different states of loading were considered.

First, results and comparisons are presented for unstiffened shells. ]]<span id='cite-_Ref134359961'></span>[[#_Ref134359961|Table 2]] shows the non-dimensional buckling pressure <math display="inline">\frac{{q}_{ext}^{cr}R{L}^{2}}{{\pi }^{2}D}</math> for isotropic unstiffened circular cylinders with simply supported edges and subjected to uniform external pressure for several Batdorf parameters <math display="inline">\left( Z\right)</math> . The shell is made of a metallic material with <math display="inline">\nu =</math><math>0.3</math> (Poisson’s ratio). From <span id='cite-_Ref134359961'></span>[[#_Ref134359961|Table 2]], it can clearly be seen that the present method provides identical and nearly identical results to those reported in [1] and [17], respectively. The maximum difference is approximately 0.6% for <math display="inline">Z=</math><math>100000</math>.

{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Unstiffened shell<sup>a</sup>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>Z=\frac{{L}^{2}}{Rt}\sqrt{1-{\nu }^{2}}</math>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Ref. [17]

Donnell]
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Ref. [1]

Donnell
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Present
|-
|  rowspan='5' style="border: 1pt solid black;text-align: center;width: 29%;"|[[Image:Draft_Herencia_516249341-image4.png|162px]] 
|  style="border: 1pt solid black;text-align: center;"|10
|  style="border: 1pt solid black;text-align: center;"|5.1836
|  style="border: 1pt solid black;text-align: center;"|5.1840

(1,44)<sup>b</sup>
|  style="border: 1pt solid black;text-align: center;"|5.1840

(1,44)[10x45]<sup>c</sup>
|-
|  style="border: 1pt solid black;text-align: center;"|100
|  style="border: 1pt solid black;text-align: center;"|11.883
|  style="border: 1pt solid black;text-align: center;"|11.884

(1,26)
|  style="border: 1pt solid black;text-align: center;"|11.884

(1,26)[10x30]
|-
|  style="border: 1pt solid black;text-align: center;"|1000
|  style="border: 1pt solid black;text-align: center;"|34.254
|  style="border: 1pt solid black;text-align: center;"|34.255

(1,15)
|  style="border: 1pt solid black;text-align: center;"|34.255

(1,15)[10x20]
|-
|  style="border: 1pt solid black;text-align: center;"|10000
|  style="border: 1pt solid black;text-align: center;"|106.95
|  style="border: 1pt solid black;text-align: center;"|106.95

(1,9)
|  style="border: 1pt solid black;text-align: center;"|106.95

(1,9)[10x10]
|-
|  style="border: 1pt solid black;text-align: center;"|100000
|  style="border: 1pt solid black;text-align: center;"|330.09
|  style="border: 1pt solid black;text-align: center;"|332.84

(1,5)
|  style="border: 1pt solid black;text-align: center;"|332.84

(1,5)[10x10]
|-
|  colspan='5'  style="border: 1pt solid black;vertical-align: top;"|<math>=\frac{E{t}^{3}}{12\left( 1-{\nu }^{2}\right) }</math>

<sup>b</sup> Numbers in parentheses are the number of longitudinal half-waves and circumferential waves, respectively

<sup>c </sup>Numbers in brackets are the number of terms in the trigonometric series for the ''x'' and ''y'' directions, respectively
|}


<div id="_Ref134359961" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Table 2. Non-dimensional buckling factor – isotropic unstiffened circular cylinders with simply supported edges and under external pressure. Number in parentheses are the number of terms used in the trigonometric series in the x and y directions.</div>

It can also be seen that, the number of longitudinal half-waves and circumferential waves obtained with the present method coincided with those reported in [1]. <span id='cite-_Ref134953490'></span>[[#_Ref134953490|Figure 4]] displays two examples of the buckling modes obtained with the present method that are reported in <span id='cite-_Ref134359961'></span>[[#_Ref134359961|Table 2]].

{| style="width: 68%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Herencia_516249341-image5.png|192px]] 
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Herencia_516249341-image6.png|192px]] 
|}


<div id="_Ref134953490" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Figure 4.''' '''Examples of buckling modes for isotropic unstiffened circular cylinders with simply supported edges and under external pressure for Z=1000 (left) and Z =10000 (right). </div>

Next, <span id='cite-_Ref134374660'></span>[[#_Ref134374660|Table 3]] below provides the buckling pressure <math display="inline">(\times {10}^{-4}MPa)</math> for isotropic unstiffened circular cylinders with simply supported edges and subjected to external pressure. The shell is made of a metallic material with <math display="inline">E=</math><math>200GPa</math> and <math display="inline">\nu =0.3</math>. Results are provided using 10x20 and 10x30 terms in the trigonometric series for <math display="inline">\frac{R}{t}=</math><math>300</math> and <math display="inline">\frac{R}{t}=3000</math>, respectively. From  <span id='cite-_Ref134374660'></span>[[#_Ref134374660|Table 3]] , it is clearly seen that the present method provides identical results to those reported in [1] and compares well with the detailed finite element models reported in [18]. The maximum difference in this case is approximately 5.2% for <math display="inline">\frac{R}{t}=</math><math>300</math> and <math display="inline">\frac{R}{t}=0.5</math>.

{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Unstiffened shell
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>\frac{R}{t}</math>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>\frac{L}{R}</math>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Ref. [18]

FEM
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Ref. [1]

Donnell
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Present
|-
|  rowspan='10' style="border: 1pt solid black;text-align: center;width: 29%;"|[[Image:Draft_Herencia_516249341-image4.png|162px]] 
|  rowspan='5' style="border: 1pt solid black;text-align: center;"|300
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.5
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|2632.0
|  style="border: 1pt solid black;text-align: center;"|2769
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|2769
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1251.0
|  style="border: 1pt solid black;text-align: center;"|1273.5
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1273.5
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|2
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|612.0
|  style="border: 1pt solid black;text-align: center;"|611.8
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|611.8
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|3
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|414.7
|  style="border: 1pt solid black;text-align: center;"|412.62
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|412.62
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|5
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|244.8
|  style="border: 1pt solid black;text-align: center;"|239.43
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|239.43
|-
|  rowspan='5' style="border: 1pt solid black;text-align: center;"|3000
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.5
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|7.74
|  style="border: 1pt solid black;text-align: center;"|7.822
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|7.822
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|3.809
|  style="border: 1pt solid black;text-align: center;"|3.815
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|3.815
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|2
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1.895
|  style="border: 1pt solid black;text-align: center;"|1.889
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1.889
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|3
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1.261
|  style="border: 1pt solid black;text-align: center;"|1.256
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1.256
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|5
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.754
|  style="border: 1pt solid black;text-align: center;"|0.748
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.748
|}


<div id="_Ref134374660" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Table 3. Buckling pressure – isotropic unstiffened circular cylinders with simply supported edges and under external pressure.</div>

Shells with different boundary conditions were also considered. <span id='cite-_Ref134773052'></span>[[#_Ref134773052|Table 4]] below provides the buckling pressure <math display="inline">(\times {10}^{3}Pa)</math> for isotropic unstiffened circular cylinders with clamped edges and subjected to external pressure. As before, the shell is made of a metallic material with <math display="inline">E=</math><math>200GPa</math> and <math display="inline">\nu =0.3</math>. Results are provided using 35x20 and 35x35 terms in the trigonometric series for  <math display="inline">\frac{R}{t}=</math><math>300</math> and <math display="inline">\frac{R}{t}=3000</math>, respectively. The results in <span id='cite-_Ref134773052'></span>[[#_Ref134773052|Table 4]] indicate that the present method compares well with the detailed finite element models considering clamped edges reported in [18]. The maximum difference observed is approximately 4.7% for <math display="inline">\frac{R}{t}=</math><math>300</math> and <math display="inline">\frac{L}{R}=0.5</math>.

{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Unstiffened shell
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>\frac{R}{t}</math>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>\frac{L}{R}</math>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Ref. [18]

FEM
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Present
|-
|  rowspan='10' style="border: 1pt solid black;text-align: center;width: 29%;"|[[Image:Draft_Herencia_516249341-image4.png|162px]] 
|  rowspan='5' style="border: 1pt solid black;text-align: center;"|300
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.5
|  style="border: 1pt solid black;text-align: center;"|3462
|  style="border: 1pt solid black;text-align: center;"|3624
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1
|  style="border: 1pt solid black;text-align: center;"|1703
|  style="border: 1pt solid black;text-align: center;"|1731
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|2
|  style="border: 1pt solid black;text-align: center;"|878.4
|  style="border: 1pt solid black;text-align: center;"|880.1
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|3
|  style="border: 1pt solid black;text-align: center;"|585.5
|  style="border: 1pt solid black;text-align: center;"|583.6
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|5
|  style="border: 1pt solid black;text-align: center;"|358.4
|  style="border: 1pt solid black;text-align: center;"|352.6
|-
|  rowspan='5' style="border: 1pt solid black;text-align: center;"|3000
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.5
|  style="border: 1pt solid black;text-align: center;"|10.80
|  style="border: 1pt solid black;text-align: center;"|10.96
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1
|  style="border: 1pt solid black;text-align: center;"|5.516
|  style="border: 1pt solid black;text-align: center;"|5.529
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|2
|  style="border: 1pt solid black;text-align: center;"|2.799
|  style="border: 1pt solid black;text-align: center;"|2.791
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|3
|  style="border: 1pt solid black;text-align: center;"|1.874
|  style="border: 1pt solid black;text-align: center;"|1.865
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|5
|  style="border: 1pt solid black;text-align: center;"|1.128
|  style="border: 1pt solid black;text-align: center;"|1.120
|}


<div id="_Ref134773052" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Table 4. Buckling pressure – isotropic unstiffened circular cylinders with clamped edges under external pressure </div>

In addition, shells made of composite materials were also studied. <span id='cite-_Ref134798236'></span>[[#_Ref134798236|Table 5]] below provides non-dimensional buckling loads <math display="inline">\frac{{N}_{x}^{cr}{L}^{2}}{{t}^{3}{E}_{2}}</math> for unsymmetric cross-ply circular cylinders with simply supported edges and subjected to axial compression. The radius <math display="inline">\left( R\right)</math>  and thickness of the shell <math display="inline">\left( t\right) \,</math> are 10 and 0.01 ''inches'' respectively. Ply properties used are <math display="inline">{E}_{11}=</math><math>30\times {10}^{6}psi</math>, <math display="inline">{E}_{22}=0.75\times {10}^{6}psi</math>, <math display="inline">{G}_{12}=</math><math>3.75\times {10}^{6}psi</math> and <math display="inline">{\nu }_{12}=0.25</math>. Results are provided using 10x10 terms in the trigonometric series. From <span id='cite-_Ref134798236'></span>[[#_Ref134798236|Table 5]], it can be observed that the present method provides results that are slightly stiffer than those reported in [19]. The maximum difference in this case is approximately 8.6% for the (0/90) lay-up with <math display="inline">Z=</math><math>100</math>.

{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Unstiffened shell
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>Z=\frac{{L}^{2}}{Rt}\sqrt{1-{\nu }^{2}}\,</math> 
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Lay-up
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Ref. [19]

Donnell
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Present
|-
|  rowspan='12' style="border: 1pt solid black;text-align: center;width: 30%;"|[[Image:Draft_Herencia_516249341-image4.png|162px]] 
|  rowspan='3' style="border: 1pt solid black;text-align: center;"|1
|  style="border: 1pt solid black;text-align: center;"|(0/90)
|  style="border: 1pt solid black;text-align: center;"|0.35636
|  style="border: 1pt solid black;text-align: center;"|0.36045
|-
|  style="border: 1pt solid black;text-align: center;"|(0/90)<sub>2</sub>
|  style="border: 1pt solid black;text-align: center;"|0.87041
|  style="border: 1pt solid black;text-align: center;"|0.87453
|-
|  style="border: 1pt solid black;text-align: center;"|(0/90)<sub>∞</sub>
|  style="border: 1pt solid black;text-align: center;"|1.0408
|  style="border: 1pt solid black;text-align: center;"|1.0433
|-
|  rowspan='3' style="border: 1pt solid black;text-align: center;"|10
|  style="border: 1pt solid black;text-align: center;"|(0/90)
|  style="border: 1pt solid black;text-align: center;"|0.90101
|  style="border: 1pt solid black;text-align: center;"|0.9644
|-
|  style="border: 1pt solid black;text-align: center;"|(0/90)<sub>2</sub>
|  style="border: 1pt solid black;text-align: center;"|1.5857
|  style="border: 1pt solid black;text-align: center;"|1.6629
|-
|  style="border: 1pt solid black;text-align: center;"|(0/90)<sub>∞</sub>
|  style="border: 1pt solid black;text-align: center;"|1.7982
|  style="border: 1pt solid black;text-align: center;"|1.8731
|-
|  rowspan='3' style="border: 1pt solid black;text-align: center;"|100
|  style="border: 1pt solid black;text-align: center;"|(0/90)
|  style="border: 1pt solid black;text-align: center;"|8.7406
|  style="border: 1pt solid black;text-align: center;"|9.4889
|-
|  style="border: 1pt solid black;text-align: center;"|(0/90)<sub>2</sub>
|  style="border: 1pt solid black;text-align: center;"|13.31
|  style="border: 1pt solid black;text-align: center;"|14.36
|-
|  style="border: 1pt solid black;text-align: center;"|(0/90)<sub>∞</sub>
|  style="border: 1pt solid black;text-align: center;"|14.567
|  style="border: 1pt solid black;text-align: center;"|15.604
|-
|  rowspan='3' style="border: 1pt solid black;text-align: center;"|1000
|  style="border: 1pt solid black;text-align: center;"|(0/90)
|  style="border: 1pt solid black;text-align: center;"|86.153
|  style="border: 1pt solid black;text-align: center;"|92.498
|-
|  style="border: 1pt solid black;text-align: center;"|(0/90)<sub>2</sub>
|  style="border: 1pt solid black;text-align: center;"|132.84
|  style="border: 1pt solid black;text-align: center;"|143.13
|-
|  style="border: 1pt solid black;text-align: center;"|(0/90)<sub>∞</sub>
|  style="border: 1pt solid black;text-align: center;"|145.37
|  style="border: 1pt solid black;text-align: center;"|155.52
|}


<div id="_Ref134798236" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Table 5. Non-dimensional buckling loads – unsymmetric cross-ply unstiffened circular cylinders with simply supported edges and under external pressure</div>

Secondly, results for shells stiffened with longitudinal stiffeners or stringers are presented. <span id='cite-_Ref134446657'></span>[[#_Ref134446657|Table 6]] gives the non-dimensional buckling coefficient <math display="inline">\frac{{2N}_{x}^{cr}{R}^{2}}{D}\,</math> for isotropic circular cylinders with eccentric stringers with simply supported edges and subjected to axial. The shell is made of a metallic material with <math display="inline">\nu =</math><math>0.3</math>. Results are provided using 10x10 terms in the trigonometric series. <span id='cite-_Ref134446657'></span>[[#_Ref134446657|Table 6]] shows the present method provides identical results to those reported in [1] and in [20] (as cited in [1]).

{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Shell stiffened with stringers<sup>a</sup>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>\frac{R}{t}</math>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>\frac{L}{R}</math>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>\frac{{A}_{s}}{{d}_{s}t}</math>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Ref. [20] as cited in Ref. [1]

Donnell
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Ref [1]

Donnell
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Present
|-
|  rowspan='9' style="border: 1pt solid black;text-align: center;width: 31%;"|[[Image:Draft_Herencia_516249341-image7.png|174px]] 
|  rowspan='3' style="border: 1pt solid black;text-align: center;"|100
|  rowspan='3' style="border: 1pt solid black;text-align: center;"|0.5
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.1
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|8064
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|8064
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|8064
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.5
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|14830
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|14833
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|14833
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1.5
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|19750
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|19754
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|19754
|-
|  rowspan='3' style="border: 1pt solid black;text-align: center;"|500
|  rowspan='3' style="border: 1pt solid black;text-align: center;"|2
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.1
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|7689
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|7689
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|7689
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.5
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|10700
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|10704
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|10704
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1.5
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|16260
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|16258
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|16258
|-
|  rowspan='4' style="border: 1pt solid black;text-align: center;"|2000
|  rowspan='4' style="border: 1pt solid black;text-align: center;"|4
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.1
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|28210
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|28211
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|28211
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|0.5
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|34930
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|34929
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|34929
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1.5
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|46570
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|46568
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|46568
|-
|  colspan='7'  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>=\frac{E{t}^{3}}{12\left( 1-{\nu }^{2}\right) }\, ;\, \, \frac{{I}_{s}}{{d}_{s}{t}^{3}}=</math><math>5;\, {J}_{s}=0;\, \frac{{e}_{s}}{t}=+5</math>
|}


<span id='_Ref134446657'></span><div id="_Ref134446445" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Table 6. Non-dimensional buckling coefficient – isotropic circular cylinders with stringers, simply supported edges and under axial load</div>

Finally, results for shells stiffened with transverse stiffeners or rings are presented. <span id='cite-_Ref134360064'></span>[[#_Ref134360064|Table 7]] provides the non-dimensional buckling pressure <math display="inline">\frac{{q}_{ext}^{cr}{R}^{3}}{D}\,</math>  for isotropic circular cylinders stiffened with eccentric rings with simply supported edges and subjected to uniform external pressure. The shell is made of a metallic material with <math display="inline">\nu =</math><math>0.3</math>. Results are provided using 10x10 terms in the trigonometric series. From <span id='cite-_Ref134360064'></span>[[#_Ref134360064|Table 7]], it can clearly be seen that the present method provides results that are identical to those reported in [1] and nearly identical to those in [21-22] (as cited in [1]).

{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Shell stiffened with rings<sup>a</sup>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math display="inline">\frac{R}{t},{e}_{r}</math><sup>b</sup>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>\frac{L}{R}</math>
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Refs. [21-22] as cited in Ref. [1]

Donnell
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Ref. [1]

Donnell
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|Present
|-
|  rowspan='6' style="border: 1pt solid black;text-align: center;vertical-align: top;width: 30%;"|[[Image:Draft_Herencia_516249341-image8.png|168px]] 
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|50,'' '' <math display="inline">{e}_{r}</math>+
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|2643
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|2643
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|2643
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|750,'' '' <math display="inline">{e}_{r}</math>+
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|10370
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|10370
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|10370
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|2000,'' '' <math display="inline">{e}_{r}</math>+
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|2
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|8517
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|8517
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|8517
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|50,'' '' <math display="inline">{e}_{r}</math>-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1883
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1883
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1883
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|750,'' '' <math display="inline">{e}_{r}</math>-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|11120
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|11123
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|11123
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|2000,'' '' <math display="inline">{e}_{r}</math>-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|1
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|10210
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|10208
|  style="border: 1pt solid black;text-align: center;vertical-align: top;"|10208
|-
|  colspan='6'  style="border: 1pt solid black;text-align: center;vertical-align: top;"|<math>=\frac{E{t}^{3}}{12\left( 1-{\nu }^{2}\right) }\, ;\, \frac{{A}_{r}}{{d}_{r}{t}^{3}}=</math><math>0.5;\, \frac{{I}_{r}}{{d}_{r}{t}^{3}}=5;\, {J}_{rs}=0;\, \frac{{e}_{r}}{t}=</math><math>\pm 5</math>

<sup>b</sup> <math display="inline">{e}_{r}</math>+ and <math display="inline">{e}_{r}</math>- indicate positive and negative ring eccentricity, respectively
|}


<div id="_Ref134360064" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Table 7. Non-dimensional buckling pressure – isotropic circular cylinders with rings, simply supported edges and under external pressure</div>

==6. Concluding remarks==

A semi-analytical approach using the Rayleigh-Ritz method was developed to calculate the linear buckling onset of metallic and composite circular cylindrical shells with various boundary conditions and subjected to in-plane and/or pressure loads. Donnell’s [7] shell theory was used due to its simplicity. Results for unstiffened and stiffened shells were compared with analytical solutions and detailed finite elements reported in the literature. Overall, the present method provided accurate results. For the metallic shells considered the present method provided results identical to the analytical results reported in [1] and close to the detailed finite element models results found in [18]. The maximum differences with detailed finite elements were 5.2%. For the composite shells considered, the present method provided slightly stiffer results than the analytical ones reported in [19]. For this case, the maximum differences were 8.6%. The present method was computationally fast even when large number of terms were used in the trigonometric series. Both its accuracy and computational speed makes the present method an ideal candidate to be used as part of an optimization scheme and/or to reduce potentially the number of detailed finite element models employed in the early design phases.

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