# Numerical analysis of piping elbows for in-plane bending and internal pressure

**ABSTRACT** This work presents the development of two different finite piping elbow elements with two nodal tubular sections for mechanical analysis. The formulation is based on thin shell displacement theory, where the displacement is based on high-order polynomial or trigonometric functions for rigid-beam displacement, and uses Fourier series to model warping and ovalization phenomena of cross-tubular section. To model the internal pressure effect an additional formulation is used in the elementary stiffness matrix definition. Elbows attached to nozzle or straight pipes produce a stiffening effect due to the restraint of ovalization provided by the adjacent components. When submitted to any efforts, the excessive oval shape may reduce the structural resistance and can lead to structural collapse. For design tubular systems it is also important to consider the internal-pressure effect, given its effect on the reduction of the pipe flexibility. Some conclusions and examples are compared with results produced by other authors.

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- Finite Elements in Analysis and Design - FINITE ELEM ANAL DESIGN. 01/2007; 20(1):26-26.
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**ABSTRACT:**An alternative formulation to current meshes dealing with finite shell elements is presented to solve the problem of stress analysis of curved pipes subjected to in-plane bending forces. The solution is based on finite curved elements, where displacements are defined from a total set of trigonometric functions or a fifth-order polynomial, combined with Fourier series. Global shell displacements are achieved through the one associated with curved arch bending and the other referred to the toroidal thin-walled shell distortion. Beam-type displacement and in-plane rotation are uncoupled and separately formulated, using trigonometric shape functions, as in Timoshenko or Mindlin beam theory. To build up the solution, a simple deformation model was adopted, based on the semi-membrane concept of the doubly curved shells behaviour. Several studies are presented and compared with experimental and numerical analyses reported by other authors.Thin-Walled Structures 02/2010; 48(2):103–109. · 1.23 Impact Factor

Page 1

Thin-Walled Structures 44 (2006) 393–398

Numerical analysis of piping elbows for in-plane bending

and internal pressure

E.M.M. Fonsecaa,?, F.J.M.Q. de Melob, C.A.M. Oliveirac

aDepartment of Applied Mechanics, Polytechnic Institute of Braganc -a, 5301-857, Portugal

bDepartment of Mechanical Engineering, University of Aveiro, 3810-193, Portugal

cDepartment of Mechanical Engineering, Faculty of Engineering of University of Porto, 4050-345, Portugal

Received 27 October 2005; accepted 12 April 2006

Available online 19 June 2006

Abstract

This work presents the development of two different finite piping elbow elements with two nodal tubular sections for

mechanical analysis. The formulation is based on thin shell displacement theory, where the displacement is based on

high-order polynomial or trigonometric functions for rigid-beam displacement, and uses Fourier series to model warping

and ovalization phenomena of cross-tubular section. To model the internal pressure effect an additional formulation is used

in the elementary stiffness matrix definition. Elbows attached to nozzle or straight pipes produce a stiffening effect due to the restraint

of ovalization provided by the adjacent components. When submitted to any efforts, the excessive oval shape may reduce the

structural resistance and can lead to structural collapse. For design tubular systems it is also important to consider the internal-pressure

effect, given its effect on the reduction of the pipe flexibility. Some conclusions and examples are compared with results produced by

other authors.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Finite piping elbow element; Flexibility; Tubular section; In-plane bending; Internal pressure

0. Introduction

The main objective of this investigation has focused on

the development of a computational model for the analysis

of the stiffness piping elbows under any type of mechanical

loading. Piping systems are constituted by straight and

curved elements and exhibit complex deformations given

their toroidal geometry and the multiplicity of the loading

conditions. Several authors have studied the bending

stiffness of curved thin-walled pipes: von Ka ´ rma ´ n [1]

solved the problem using the Rayleigh–Ritz method, O¨ry

and Wilczek [2] presented a economical method which uses

a transfer matrix to able stress and deformation calcula-

tion, Thomson [3] worked with an analytical formulation

using a development with trigonometric series functions

and realized many experimental studies, Thomas [4]

studied the stiffening effect on thin-walled piping elbows

using a thin-shell element from a finite-difference program,

more recently Bathe and Almeida [5] formulated a

pipe elbow element using a cubic displacement interpola-

tion based on finite-element method with ovalization

phenomena contribution. In this paper an alternative

formulation is presented for the characterization of the

deformation of thin-walled piping elbow element, based

on shell displacement field and on additional formulation

to obtain the elementary stiffness matrix due to internal

pressure. A finite element with two nodes was developed,

where the displacement field is based on high-order

polynomialfunctionsfor

such as the equations developed by Fonseca et al. [6]

or a new model proposed here with trigonometric functions

andthedevelopmentof

warping and ovalization of tubular section, reported

in [3]. The two different computational models presented

in this work have been tested with some numerical

examples and experimental measurements obtained by

other authors.

rigidbeamdisplacement,

Fourierseriestomodel

ARTICLE IN PRESS

www.elsevier.com/locate/tws

0263-8231/$-see front matter r 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.tws.2006.04.005

?Corresponding author. Fax: +351273313051.

E-mail address: efonseca@ipb.pt (E.M.M. Fonseca).

Page 2

UNCORRECTED PROOF

and realized many experimental studies, Thomas [4]

studied the stiffening effect on thin-walled piping elbows

using a thin-shell element from a finite-difference program,

Thin-Walled Structures ] (]]]]) ]]]–]]]

Numerical analysis of piping elbows for in-plane bending and internal

pressure

E.M.M. Fonsecaa,?, F.J.M.Q. de Melob, C.A.M. Oliveirac

aDepartment of Applied Mechanics, Polytechnic Institute of Braganc -a, Portugal

bDepartment of Mechanical Engineering, University of Aveiro, Portugal

cDepartment of Mechanical Engineering, Faculty of Engineering of University of Porto, Portugal

Received 27 October 2005; accepted 12 April 2006

Abstract

This work presents the development of two different finite piping elbow elements with two nodal tubular sections for mechanical

analysis. The formulation is based on thin shell displacement theory, where the displacement is based on high-order polynomial or

trigonometric functions for rigid-beam displacement, and uses Fourier series to model warping and ovalization phenomena of cross-

tubular section. To model the internal pressure effect an additional formulation is used in the elementary stiffness matrix definition.

Elbows attached to nozzle or straight pipes produce a stiffening effect due to the restraint of ovalization provided by the adjacent

components. When submitted to any efforts, the excessive oval shape may reduce the structural resistance and can lead to structural

collapse. For design tubular systems it is also important to consider the internal-pressure effect, given its effect on the reduction of the

pipe flexibility. Some conclusions and examples are compared with results produced by other authors.

r 2006 Published by Elsevier Ltd.

Keywords: Finite piping elbow element; Flexibility; Tubular section; In-plane bending; Internal pressure

0. Introduction

The main objective of this investigation has focused on

the development of a computational model for the analysis

of the stiffness piping elbows under any type of mechanical

loading. Piping systems are constituted by straight and

curved elements and exhibit complex deformations given

their toroidal geometry and the multiplicity of the loading

conditions. Several authors have studied the bending

stiffness of curved thin-walled pipes: von Ka ´ rma ´ n [1]

solved the problem using the Rayleigh–Ritz method, O¨ry

and Wilczek [2] presented a economical method which uses

a transfer matrix to able stress and deformation calcula-

tion, Thomson [3] worked with an analytical formulation

using a development with trigonometric series functions

more recently Bathe and Almeida [5] formulated a pipe

elbow element using a cubic displacement interpolation

based on finite-element method with ovalization phenom-

ena contribution. In this paper an alternative formulation

is presented for the characterization of the deformation of

thin-walled piping elbow element, based on shell displace-

ment field and on additional formulation to obtain the

elementary stiffness matrix due to internal pressure. A

finite element with two nodes was developed, where the

displacement field is based on high-order polynomial

functions for rigid beam displacement, such as the

equations developed by Fonseca et al. [6] or a new model

proposed here with trigonometric functions and the

development of Fourier series to model warping and

ovalization of tubular section, reported in [3]. The two

different computational models presented in this work have

been tested with some numerical examples and experi-

mental measurements obtained by other authors.

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pp:126ðcol:fig::NILÞ

ED:AjithG:P:

PAGN:anu SCAN:

0263-8231/$-see front matter r 2006 Published by Elsevier Ltd.

doi:10.1016/j.tws.2006.04.005

?Corresponding author. Fax: +351273313051.

E-mail address: efonseca@ipb.pt (E.M.M. Fonseca).

Page 3

UNCORRECTED PROOF

1. Essential assumptions

The geometric parameters considered for the piping

elbow element definition are: the arc length s, the mean

curvature radius R, the thickness h, the mean section radius

of the pipe r and the central angle a. Fig. 1 presents the

geometric parameters defining the two nodal tubular

sections.

The deformation field of a piping elbow element refers to

membrane strains and shell curvature variations. The

following assumptions, referred to in [6–9], were considered

in the present analysis: the curvature radius is assumed

much larger than the section radius; a semi-membrane

deformation model is adopted and it neglects the bending

stiffness along the longitudinal direction of the toroidal

shell but considers the meridional bending resulting from

ovalization. The shell is considered thin and inextensible

along the meridional direction for only the mechanical

loading case.

2. Finite piping elbow element formulation

The shell finite-element displacement field resulting from

the superposition of rigid-beam displacement and the

complete Fourier expansion for ovalization and warping

terms are as shown in the following equations:

u ¼ Us ð Þ? r cos yjs ð Þþ u s;y

ðÞ, (1a)

v ¼ ?Ws ð Þsin y þ v s;y

ðÞ,(1b)

w ¼ Ws ð Þcos y þ w s;y

where w(s,y) is the surface displacement in radial direction

resulting from ovalization, v(s,y) the meridional displace-

ment due to ovalization, u(s,y) the longitudinal displace-

ment due to warping tubular section effect, U the

tangential beam displacement, W the transversal beam

displacement and j represents the beam rotation in z

direction.

The displacements u, v and w are calculated on shell

surface from the finite piping elbow element (Fig. 1), as a

function of a displacement field under mean line arc (U, W

and j) as shown Fig. 2. Those parameters are related

through simple differential equations from beam-bending

theory.

ðÞ,(1c)

Two different models will be presented for the displace-

ment field calculation in piping elbow elements. In the first

model, a high-order formulation should be used and six

parameters are necessary to define the beam displacement

field. From this, U can be approached by the following

fifth-order polynomial (5P):

UðsÞ¼ aoþ a1s þ a2s2þ a3s3þ a4s4þ a5s5.

The transverse displacement and the rotation can be

calculated as

(2a)

Ws ð Þ¼ ?RdU

ds¼ ?R a1þ 2a2s þ 3a3s2þ 4a4s3þ 5a5s4

??,

(2b)

js ð Þ¼dW

The coefficients are determined as a function of imposed

boundary conditions under the curved referential.

The second model is a formulation based on trigono-

metric functions (TF) and four parameters are necessary to

define the beam displacement field. U can be approximated

by the following function:

? ?

ds

¼ ?R 2a2þ 6a3s þ 12a4s2þ 20a5s3

??. (2c)

UðsÞ¼ b1cos

s

R

þ b2sin

s

R

? ?

þ b3cos 2s

R

??

þ b4sin 2s

R

??

.

(3a)

The transverse displacement can be calculated as

? ?

? 2b4cos 2s

R

For the rotation field a linear polynomial may be used:

WðsÞ¼ ? RdU

ds¼ b1sin

?

s

R

? b2cos

s

R

? ?

þ 2b3sin 2s

R

??

?

.

ð3bÞ

js ð Þ¼ b5þ b6S.

The coefficients are determined using the same imposed

boundary conditions under the curved referential. For

straight-pipe elements a formulation based on third-order

polynomial (3P) was used with Hermitian shape functions.

The surface displacements in radial and in meridional

directions result from ovalization, in-plane, as discussed by

Thomson [3] and are given by the following equations:

(3c)

w s;y

ð Þ ¼

X

n?2

ancos ny

!

Niþ

X

n?2

ancos ny

!

Nj,(4)

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Fig. 1. Geometric parameters of finite piping elbow element.

Fig. 2. Degrees of freedom for in plane element.

E.M.M. Fonseca et al. / Thin-Walled Structures ] (]]]]) ]]]–]]]

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UNCORRECTED PROOF

used, while all the remaining stiffness terms were calculated

with two-point gauss integration (ovalization and warping

terms). Due to the uncoupled formulation of the displace-

ment W and the section rotation j in the TF element, it is

necessary that a different type of integration should be used

along variable s. This expedient avoids the element locking,

a numerical drawback associated with the contribution

shear terms in the stiffness matrix when exact integrations

are used. The use of selective reduced integration only for

v s;y

ð Þ ¼?

X

n?2

an

nsin ny

!

Niþ?

X

n?2

an

nsin ny

!

Nj.(5)

The longitudinal displacement due to warping tubular

section effect is calculated by the following equation,

discussed by Thomson [3]:

u s;y

ð Þ ¼

X

n?2

bncos ny

!

Niþ

X

n?2

bncos ny

!

Nj. (6)

The terms anand bnare constants to be determined as

function of developed Fourier series.

The mechanical deformation model considers that pipe

undergoes a semi-membrane strain field, discussed [8–10],

and is given by.

2

ess

gsy

wyy

8

>

>

:

where essis the longitudinal membrane strain, gsythe shear

strain and wyythe meridional curvature form ovalization.

The application of the virtual-work principle gives finally

the system of algebraic equations to be solved. The matrix

force–displacement equation for this finite-element pipe is

<

9

>

>

;

=

¼

q

qs

1

r

qy

?siny

q

qs

?1

r2

R

cosy

R

q

0

0

q

qy

1

r2

q2

qy2

66666664

3

77777775

u

v

w

8

>

>

:

<

9

>

>

;

=

, (7)

K

½ ? d f g ¼ F

d is a nodal unknown displacement vector and F the

applied nodal forces. The element stiffness matrix K is

calculated from the matrix equation

Z

where dS ¼ rdsdy, T is the transpose matrix for global

system, B results from the derivative of the shape functions

for the finite piping elbow element and the elasticity matrix

D appears with a simple algebraic definition, dependent on

the elastic modulus, the piping elbow thickness and

Poisson’s ratio.

In the piping elbow element formulation, a Gaussian

integration was carried out along variable s while an exact

one was used along the circumferential direction y. In

detail, the stiffness terms resulting from the beam shear

deformation were calculated using one-point gauss inte-

gration when a reduced integration (for TF polynomial)

and a complete integration for (5P and 3P polynomials) are

f g.(8)

Kglobal¼ T

½ ?

s

B

½ ?TD

½ ? B

½ ?dS

??

T

½ ?T, (9)

shear terms has shown a remarkable improvement in the

numerical behaviour of pipe elements, a procedure used by

Melo and Castro [8] and Prathap [11].

The total number of degrees of freedom for this element

is 2(3+2Ny), where Nyis the number of terms used in

Fourier expansions (8 terms).

3. Influence of internal pressure in stiffness matrix

determination

To consider the effect of the internal pressure in the

tubular system an infinitesimal pipe area is defined,

accounting for the work produced by the pressure load in

the pure bending of the piping elbow, as in [5]. Considering

the tubular section inextensible, the piping-elbow length is

the same but there is a volume variation produced by an

increase of pipe deformation work, due to internal

pressure. The deformation work is calculated as

ZL

ZL

where p represents the internal pressure, (R?cosy)a or L

the arc length for the mean piping-elbow surface, s the

longitudinal coordinate and dA(s,y) the modified tubular

section area.

Considering a differential element area before and after

ovalization, Fig. 3, the following expression may be used to

calculate dA(s,y).

In Fig. 3 the arc element length AB is equal to the arc

element length A0B0, considering the inextensible condition.

The infinitesimal area ABA0B0, is given by the following

equation, neglecting the second-order quantities and

dv ¼ ?wdy (pipe inextensible):

Welbow¼ pDV ¼ ?

0

Z2p

Z2p

0

pR ? r cos y

ðÞa

2

dA s;y

ðÞ, (10)

Wstraigth¼ pDV ¼ ?

00

p

2LdA s;y

ðÞ,(11)

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Fig. 3. Pipe element before and after ovalization deformation.

E.M.M. Fonseca et al. / Thin-Walled Structures ] (]]]]) ]]]–]]]

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UNCORRECTED PROOF

factor h ¼ hR= r2

Fig. 5 represents the calculated flexibility factor k ¼

4Er3tjðnode6Þ=MR 1 ? n2

using our two different models and results reported in

Ref. [5]. It was shown the internal pressure has influence on

the global stiffness of the structure. The main effect of the

internal pressure consists in decreasing of the piping-elbow

flexibility.

dA s;y

ð Þ ¼1

2ðr þ wÞðr þ wÞdy ?1

þwv

ffi w2dy þ1

þ1

If internal pressure is considered as a dominant factor into

piping elbow ovalization, w and v represents the displace-

ments by Eqs. (4) and (5), respectively.

Substituting the expressions and considering the internal

product of w with rdy equals zero, the deformation work

for tubular curved and straight pipes may be calculated:

2rrdy ?1

2ðr þ wÞdydw

2?1

2ðv þ dvÞðw ? dwÞ

2vdw þ wrdy ¼ w2dy þ wrdy

2vdw

dydy

ð12Þ

Welbow¼ ?3

Results from these equations must be added to the

diagonal terms of elementary stiffness matrix, Eq. (9),

referred to as ovalization terms.

4paRpa2

i

or

Wstraight¼ ?3

4ppLa2

i.(13)

4. Study case: Flexibility factor for in-plane bending with

internal pressure effect

Fig. 4 represents an half of a tubular structure reported

in Ref. [5]. One-dimensional mesh was used with straight

and curved elements considering the two different models

presented (5P+3P and TF+3P). It was applied a vertical

load at an extremity of the structural piping system and

different values of internal pressure were used. The elbow

ffiffiffiffiffiffiffiffiffiffiffiffiffi

?

1 ? n2

p

??

¼ 0:1084.

?

for the piping-elbow system

5. Study case: Flexibility coefficient determination in

different types of piping elbows and nozzle constraints

Fig. 6 show the geometry created for in-plane bending

and with internal-pressure effect. This figure represents also

the one-dimensional mesh used for each model. The

bending moment is equal M ¼ 73450Nm and in the

internal pressure considered is equal P ¼ 100MPa.The

pipe material has an elasticity modulus equal to 210GPa.

For simulating the stiffening effect of a nozzle, a different

length of constraint X is considered as well as a built-in-end

extremity. The numerical results obtained with our two

different formulations (5P+3P and TF+3P) for different

imposed loadings (bending moment or a bending moment

and internal pressure) are compared with solution pre-

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P

h=0.5 in

r=14.75 in

45°

R=45 in

L=59 in.

1

2

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5

67

8

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16

Y

X

Z

Fig. 4. Piping elbow geometry and loading. One-dimensional mesh used.

0

2

4

6

8

10

12

14

16

0 200 400600 80010001200

Pressão [Psi]

Flexibility factor k

ADINAP ref [7]Exper. ref [7]

(5P+3P)(TF+3P)

Fig. 5. Flexibility factor of the piping elbow due to different internal

pressure used.

E.M.M. Fonseca et al. / Thin-Walled Structures ] (]]]]) ]]]–]]]

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UNCORRECTED PROOF

M+P_(5P+3P)

M+P_(TF+3P)

Thin-shell ref [4]

sented by Thomas [4], when considering the elbow factor

h ¼ hR?r2¼ 0:1304.

The curves shown in Fig. 7 represent the elbow bending

flexibility coefficient a ¼ 2Ert2jðnode13Þ=M, using equation

k ¼ a=h, which is affected by the presence of nozzle

constraint and is obtained with different loading types.

As seen when the internal pressure is added, the structural

stiffness increases. According to the ASME Designing

Code, the calculation of the flexibility factor k in the curved

pipe under uniform bending and unflanged bends is

determined using k ¼ 1:65=h, represented in Fig. 8.

It was shown the elbow with a rigid flange at one end and

a pipe on the other is about twice as stiff as an isolated

elbow. The presence of a rigid restraint near one end of the

elbow shifts the maximum ovalization from the mid-span

of the elbow. These conclusions are in good agreement with

the observations of Thomas, when finite thin-shell elements

are used.

Fig. 8 presents the ovalization problem obtained with

our numerical results calculated at the surface tubular

section, for any different X length, at the transverse-section

elbow at mid-span for each loading situation.

6. Conclusions

Results of flexibility coefficient from mechanical actions,

including internal pressure, was presented with two

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Fig. 6. Piping elbows geometry and loading conditions used. One-dimensional mesh used.

0.0

0.5

1.0

1.5

2.0

0.00.51.01.52.0 2.53.0

Effect of nozzle constraint X/D

Flexibility coefficient α

M_(5P+3P)M_(TF+3P)

1.65 (ASME)

Fig. 7. Flexibility coefficient determination and the effect of nozzle

constraint.

E.M.M. Fonseca et al. / Thin-Walled Structures ] (]]]]) ]]]–]]]

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UNCORRECTED PROOF

different formulations. With these new finite piping elbow

elements it is possible to calculate the displacement field

due to any type of load for in-plane loading. It is a simple

finite element and easy to operate and avoids a pre-

processing expensive mesh generation for shell definition

surface. The purpose of this paper is to provide an easy and

an alternative formulation when compared with a complex

finite shell, solid or beam element analysis for the same

application. Several case studies presented were compared

with results reported by other authors and discussed.

References

[1] von Ka ´ rma ´ n Th. Die Forma ¨ nderung du ¨ nnwandiger Rohre. Insbe-

sondere federnder Ausgleichsrohre, Zeitschrift des Vereins deutscher

Ingenieure, 1911.

[2] O¨ry H, Wilczek E. Stress and stiffness calculation of thin-walled

curved pipes with realistic boundary conditions being loaded in the

plane of curvature. Int J Pressure Vessels Piping 1983;12:167–89.

[3] Thomson G. The influence of end constraints on pipe bends. PhD

Thesis, University of Strathclyde, Scotland, 1980.

[4] Thomas K. Stiffening effects on thin-walled piping elbows of adjacent

piping and nozzle constraints. Pressure Vessels Piping Div ASME

1981;50:93–108.

[5] Bathe KJ, Almeida CA. A simple and effective pipe elbow element.

Pressure stiffening effects. J Appl Mech 1982;49:914–6.

[6] Fonseca EMM, Melo FJMQ, Oliveira CAM. The thermal and

mechanical behaviour of structural steel piping systems. Int J

Pressure Vessels Piping 2005;82(2):145–53.

[7] Fonseca EMM, Melo FJMQ, Oliveira CAM. Determination of

flexibility factors on curved pipes with end restraints using a semi-

analytic formulation.IntJ

2002;79(12):829–40.

[8] Melo FJMQ, Castro PMST. A reduced integration Mindlin beam

element for linear elastic stress analysis of curved pipes under

generalized in-plane loading. Comput Struct 1992;43(4):787–94.

[9] Flu ¨ gge W. Thin elastic shells. Berlin: Springer; 1973.

[10] Kitching R. Smooth and mitred pipe bends. In: Gill SS, editor. The

stress analysis of pressure vessels and pressure vessels components.

Oxford: Pergamon Press; 1970 [Chapter 7].

[11] Prathap G. The curved beam/deep arch/finite ring element revisited.

Int J Num Meth Eng 1985;21(3):389–407.

PressureVessels andPiping

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Fig. 8. Transverse section-elbow ovalization at mid-span for in-plane bending and internal pressure.

E.M.M. Fonseca et al. / Thin-Walled Structures ] (]]]]) ]]]–]]]

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