2008 Strengthening Metallic Cylindrical Shells Against Elephant's Foot Buckling With FRP

ARTICLE IN PRESS

Thin-Walled Structures 47 (2009) 1078–1091

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Thin-Walled Structures

0263-82

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/tws

Strengthening metallic cylindrical shells against elephant’s foot bucklingwith FRP

M. Batikha a, J.F. Chen a,�, J.M. Rotter a, J.G. Teng b

a Institute for Infrastructure & Environment, The University of Edinburgh, Edinburgh EH9 3JL, UKb Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, China

a r t i c l e i n f o

Available online 21 December 2008

Keywords:

Metallic shells

Cylindrical shells

Silos

Elephant’s foot buckling

FRP

Strengthening

31/$ - see front matter & 2008 Elsevier Ltd. A

016/j.tws.2008.10.012

esponding author. Tel.: +44131650 6768; fax

ail address: [email protected] (J.F. Chen).

a b s t r a c t

Thin metal cylindrical shell structures such as silos and tanks are susceptible to an elastic–plastic

instability failure at the base boundary known as elephant’s foot buckling, due to its characteristic

deformed shape. This form of buckling occurs under high internal pressure accompanied by axial

compression in the shell structure. This is a common failure mode for tanks under earthquake loading.

Another common situation is in a silo where the silo wall is subjected to both normal pressures from the

stored granular solid and vertical compressive forces developed from the friction between the stored

solid and the silo wall. This paper presents a novel method of strengthening cylindrical shells against

elephant’s foot buckling in which a small amount of fibre-reinforced polymer (FRP) composite, used at a

critical location, can effectively eliminate the problem and increase the buckling strength. The

strengthened shell is analysed using linear elastic bending theory in this preliminary study. Within the

scope of this research, the strengthening effect is shown to be sensitive to the thickness, height and

location of the FRP sheet. The issue of optimal FRP strengthening to allow the shell to attain pure

membrane-state deformation is examined in detail as strengthening with too much and too little FRP

are both undesirable. Both pinned-based and fixed-based shells are examined and their responses are

compared.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Thin metal cylindrical shells are widely used as containers,such as silos and tanks. These shells are sensitive to themagnitude of the imperfections, which can cause elastic buckling,but under high internal pressure, this sensitivity is much reduced[1–5]. Existing research [4,5] has demonstrated that cylindricalshells fail near local imperfections by elastic buckling if theinternal pressure is small. By increasing the internal pressure,yielding of the wall near the base boundary leads to localreductions in flexural stiffness and local amplifications ofdisplacements. The circumferential membrane stress resultantsare raised [6] and elastic–plastic buckling occurs [1]. Thiselastic–plastic instability failure near the base boundary is knownas elephant’s foot buckling. Further, Rotter [1] showed that, forthin cylinders, a clamped base is considerably stronger than apinned base, whilst in thicker cylinders, clamped and pinnedbases have similar strengths.

Chen et al. [7,8] proposed to strengthen the shell againstelephant’s foot buckling by using a small ring stiffener. The

ll rights reserved.

: +44131650 6781.

optimal dimensions and location for this stiffener were derived. Inthis paper, an alternative method of strengthening cylindricalshells against elephant’s foot buckling using fibre-reinforcedpolymer (FRP) composites is proposed. FRP composites havesuperior properties, including a high strength-to-density ratio andsuperior durability to many traditional materials. Applying FRPcomposites to the external surface of the shell is particularlyadvantageous in many cases because it avoids the need to accessthe interior of the storage structure and even to take the structureout of service, thus minimising health and safety concerns andoperational disruptions. Bonding FRP to the shell is also easierthan welding reinforcement to it, since the latter would not onlyresult in residual stresses and imperfections in the shell but mightalso have significant safety implications when sensitive orflammable materials such as petroleum products have beenstored in the structure.

Extensive research on the use of FRP for strengthening concretestructures has been undertaken since the 1990s (e.g. [9–11]). FRPstrengthening research has been extended to metallic beams andcolumns (e.g. [9,12–15]), masonry (e.g. [16,17]) and timberstructures (e.g. [18]). In the vast majority of these cases, it isthe strength rather than the stability of the structure that isthe main concern. Teng and Hu [19,20] explored the behaviourof circular steel tubes subjected to axial load and confined by

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Nomenclature

[A] coefficient matrix (Eq. (48)){B} coefficient vector (Eq. (48)){C} constant vector (Eq. (48))Br extensional stiffness of the composite FRP–metal shell

in the circumferential directionD shell flexural rigidityDb flexural rigidity of the composite FRP–metal shell in

the meridional directionDfz flexural rigidity of the FRP shell in the meridional

directionDs flexural rigidity of the metal shellEb Young’s modulus of the composite FRP–metal shellEfz Young’s modulus of the FRP shell in the meridional

directionEfy Young’s modulus of the FRP shell in the circumfer-

ential directionEs Young’s modulus of the metal shellh height of cylindrical shellhf height of FRP sheetMz meridional bending momentMy circumferential bending momentNz vertical load per unit circumferenceNzf axial force in the FRP shellNzs axial force in the metal shell

Ny circumferential stress resultantp uniform internal pressurepf confining pressure induced by the FRP shellQz meridional shear stress resultantr radius of cylindrical shelltb effective thickness of the composite FRP–metal shellts thickness of the metal cylinderw radial displacementwm membrane theory normal deflectionwmb membrane theory normal deflection of the composite

FRP–metal shellxf starting position of FRP sheet above the basea extensional stiffness ratio (Eq. (8))b rotation around circumferencel bending half-wavelengthlb meridional bending half-wavelength of the composite

FRP–metal shellnb Poisson’s ratio of the composite FRP–metal shellnfzy Poisson’s ratio of the FRP shell in the meridional

directionnfyz Poisson’s ratio of the FRP shell in the circumferential

directionns Poison’s ratio of the metal cylinder$b normalised FRP height (Eq. (25e))$c normalised FRP starting position (Eq. (29e))

M. Batikha et al. / Thin-Walled Structures 47 (2009) 1078–1091 1079

FRP jackets, and also studied elephant’s foot buckling ofcylindrical shells under combined axial compression and internalpressure when the whole shell is wrapped with FRP. Bothexperiments and finite element (FE) analysis demonstrated thatthe ductility of the tubes was greatly enhanced by the FRP. Theirnumerical investigation also showed that the elephant’s footbuckling strength of a cylindrical shell under combined axialcompression and internal pressure can be significantly enhancedby the FRP wraps.

The aim of this paper is to investigate the strengthening ofthin metallic cylindrical shells by local application of FRP toincrease the elephant’s foot buckling strength. The elephant’s footbuckling strength falls below the von Mises failure criterionapplied to the membrane stress resultants in the shell wallbecause of the effect of local bending near the boundarycondition, which is exaggerated by geometrically nonlinear effects[6]. In this paper, a linear shell bending analysis is used to obtain afirst estimate of the effect of local FRP strengthening on increasingthe elephant’s foot buckling strength. Equations are developed forthe behaviour of the cylindrical shell with FRP strengthening, forboth simply supported and fixed boundary conditions. Theoptimal dimensions and location of the FRP strengthening areexamined.

2. Stress resultants in cylindrical shells strengthened with anFRP sheet

2.1. Geometry, boundary conditions, loading and material properties

An isotropic elastic cylindrical shell of height h, radius r,and thickness ts is strengthened by bonding FRP sheets toits exterior. The wall thickness of the cylindrical shell isassumed to be constant over the whole structure. The Young’smodulus and the Poisson’s ratio of the shell are Es and ns,respectively.

In the following analysis, the boundary condition at the base ofthe shell is considered to be either simply supported or fixed. Thesimply supported case (radial, axial and circumferential displace-ments restrained, but not the meridional rotation) is consideredfirst, but a fixed support is also later included. The shell isassumed to be relatively long, so that the top boundary conditiondoes not influence the local behaviour at the location of theelephant’s foot buckle.

An FRP sheet with a height of hf is bonded to the externalsurface of the cylindrical shell starting at a distance xf above thebase. It is assumed that the FRP sheet consists of uniaxial fibres inthe circumferential direction. The FRP sheet is therefore treated asorthotropic with Young’s modulus Efy in the circumferentialdirection and Efz in the meridional direction, with Poisson’s ratiosnfyz and nfzy (nfyz/Efy ¼ nfzy/Ez).

It is assumed that the elephant’s foot buckling phenomenonoccurs over such a short length of shell that the internal pressureand vertical load can both be treated as invariant with height z.Thus a uniform internal pressure p and a vertical load per unitcircumference Nz at a distant top boundary are applied to the shellas shown in Fig. 1.

2.2. Governing equations

For the purposes of the analysis, the shell is divided into threesections. In Sections A and C, above and beneath the FRP sheet, theshell is isotropic and unstrengthened. In Section B, FRP jacketing isapplied, and the equations must capture the interaction betweenthe base shell and the FRP.

In Sections A and C, where the cylindrical isotropic shell hasuniform thickness t, the radial displacement w under a uniforminternal pressure p and vertical load per unit circumference Nz isgoverned [21] by

Dd4w

dz4þ

Est

r2w ¼ pþ

vsNz

r(1)

ARTICLE IN PRESS

p

p

p

Nz

Shell section A

Shell section Cx fh f Shell section B

QaMa

Za

Zb

Zc

Qb2

Qb1

Qc

Mb2

Mb1

Mc

Fig. 1. Coordinate systems and stress resultants.

Steel shell

p pf pf

FRP shell

Fig. 2. Interaction pressure between the FRP and the metal shells within Section B.

M. Batikha et al. / Thin-Walled Structures 47 (2009) 1078–10911080

in which D is the shell flexural rigidity given by

D ¼Est

3

12ð1� n2s Þ

(2)

In Section B, the metal shell expands as a result of both theapplied internal pressure p and (through Poisson effects) thevertical loading Nz. The FRP sheet then induces a confiningpressure pf on the metal shell as shown in Fig. 2. The governingequation for the metal shell in Section B is thus given by

Dsd4w

dz4þ

Ests

r2w ¼ p� pf þ

vsNzs

r(3)

and that for the FRP sheet is

Dfzd4w

dz4þ

Efytf

r2w ¼ pf þ

vfzyNzf

r(4)

where Nzs is the axial force in the cylindrical metal shell and Nzf isthe value in the FRP shell in Section B.

In Eqs. (3) and (4), Ds and Dfz are the flexural rigidity of themetal and FRP shells, respectively. They can be obtained bysubstituting the thickness and respective properties for the z

direction into Eq. (2).Eliminating pf from Eqs. (3) and (4) gives

ðDs þ DfzÞd4w

dz4þ

Ests þ Efytf

r2w ¼ pþ ns

Nzs

Nzþ nfzy

Nzf

Nz

� �Nz

r(5)

By comparing Eq. (5) with Eq. (1), the differential equation forSection B can be written as follows:

Dbd4w

dz4þ

Ebtb

r2w ¼ pþ

vbNz

r(6)

in which

Db ¼ Ds þ Dfz (7a)

Ebtb ¼ Ests þ Efytf ¼ Estsð1þ aÞ (7b)

nb ¼ nsNzs

Nzþ nfzy

Nzf

Nz(7c)

in which the extensional rigidity ratio of the FRP to the metal shellis characterised by

a ¼Efytf

Ests(8)

Section B may be treated as an equivalent metal shell.The effective thickness tb of the transformed Section B may bederived by equating the ratio of the flexural to the extensionalrigidity of the equivalent shell to that of the metal–FRP compositeshell:

Db

Ebtb¼

Ebt3b=12ð1� n2

bÞ

Ebtb(9)

Substituting Eqs. (7a) and (7b) into the left hand side of Eq. (9)and rearranging the resulting equation gives

tb ¼

ffiffiffiffiffiffiffiffiffiffiffiffi12Db

Br

s(10)

where Br is the extensional rigidity

Br ¼Ebtb

1� n2b

¼Estsð1þ aÞ

1� n2b

(11)

The same expression for tb can be found in Baker et al. [22] andZienkiewicz [23] for an orthotropic material.

If the FRP fibres run in the circumferential direction, theYoung’s modulus of FRP in the meridional direction is very smallcompared with that in the circumferential direction and that ofthe metal shell. In addition, it is anticipated that only thin FRPsheets would be required to achieve the strengthening purpose.Therefore, in the meridional direction, both the flexural andextensional rigidities of the FRP are much smaller than those ofthe metal shell and they can be neglected. In such a case, Eq. (5)may be rewritten ignoring the terms relating to the meridionalproperties of FRP

Dsd4w

dz4þ

Estsð1þ aÞr2

w ¼ pþ nsNz

r(12)

Similarly, the properties for an equivalent metal shell forSection B can be obtained from Eqs. (7a)–(7c) and (10) as

Db ¼ Ds ¼Est

3s

12ð1� n2s Þ

(13a)

nb ¼ ns (13b)

and

tb ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ð1� n2

s ÞDs

Estsð1þ aÞ

s(13c)

2.3. General solution

The general solution of Eq. (1) may be written as [24]

w ¼ C1 cospz

lþ C2 sin

pz

l

h ie�pz=l

nþ C3 cos

pz

lþ C4 sin

pz

l

h iepz=l þ 1

owm (14)

ARTICLE IN PRESS


in which l is the meridional bending half-wavelengthgiven by

l ¼p

ð3ð1� v2s ÞÞ

1=4

ffiffiffiffiffiffirts

p(15)

and wm is the membrane theory normal deflection

wm ¼ pþvsNz

r

� �r2

Ests(16)

For Section B, the meridional bending half-wavelength, lb,and the membrane theory normal deflection, wmb, can be obtainedas

lb ¼p

½3ð1� v2bÞ�

1=4

ffiffiffiffiffiffirtb

p(17a)

wmb ¼ pþ nsNz

r

� �r2

Estsð1þ aÞ(17b)

The stress resultants in the shell may be expressed in terms ofdisplacements as

Ny ¼Et

Rwþ vNz (18a)

Mz ¼ �Dd2w

dz2(18b)

My ¼ vMz (18c)

Qz ¼ �Dd3w

dz3(18d)

2.3.1. Shell Section A

It is assumed that Section A of the metal cylindrical shell is solong that the upper boundary does not influence the deforma-tions. Writing

xa ¼pza

l¼pðz� xf � hf Þ

l(19)

From Eq. (14):

w ¼ ½ðCa1 cos xa þ Ca2 sin xaÞe�xa þ 1�wm (20a)

b ¼dw

dza¼ wm

pl½�Ca1ðsin xa þ cos xaÞ þ Ca2ðcos xa � sin xaÞ�e

�xa

(20b)

d2w

dz2a

¼ 2wmpl

� �2½Ca1 sin xa � Ca2 cos xa�e

�xa (20c)

d3w

dz3a

¼ 2wmpl

� �3½Ca1ð� sin xa þ cos xaÞ þ Ca2ðcos xa þ sin xaÞ�e

�xa

(20d)

The deformation and internal forces at the junction betweenshell Sections A and B (za ¼ 0 in Fig. 1) can be found fromEqs. (18a)–(18d) and (20a)–(20d):

wa ¼ ðCa1 þ 1Þwm (21a)

ba ¼ wmpl½�Ca1 þ Ca2� (21b)

Ma ¼ 2Dwmpl

� �2Ca2 (21c)

Qa ¼ �2Dwmpl

� �3½Ca1 þ Ca2� (21d)

2.3.2. Shell section B

Section B is a composite shell with different properties asdescribed earlier. It also needs to be treated as a short shell.Writing

xb ¼pzb

lb¼ðz� xf Þ

lb(22)

From Eq. (14), the deformations for Section B are:

w ¼ ½ðCb1 cos xb þ Cb2 sin xbÞe�xb þ ðCb3 cos xb þ Cb4 sin xbÞe

xb þ 1�wmb

(23a)

b ¼dw

dzb¼ wmb

plb½�Cb1ðsin xb þ cos xbÞ þ Cb2ðcos xb � sin xbÞ�e

�xb

nþ ½Cb3ð� sin xb þ cos xbÞ þ Cb4ðcos xb þ sin xbÞ�e

xb

o(23b)

d2w

dz2b

¼ 2wmbplb

� �2

½Cb1 sin xb � Cb2 cos xb�e�xb

n

þ ½�Cb3 sin xb þ Cb4 cos xb�exb

o(23c)

d3w

dz3b

¼ 2wmbplb

� �3

½Cb1ð� sin xb þ cos xbÞ þ Cb2ðcos xb þ sin xbÞ�e�xb

n

þ½�Cb3ðsin xb þ cos xbÞ þ Cb4ðcos xb � sin xbÞ�exb

o(23d)

The deformation and internal forces at zb ¼ 0 can be found fromEqs. (18a)–(18d) and (23a)–(23d):

wb1 ¼ ½Cb1 þ Cb3 þ 1�wmb (24a)

bb1 ¼ wmbplb½�Cb1 þ Cb2 þ Cb3 þ Cb4� (24b)

Mb1 ¼ �2Db wmbplb

� �2

½�Cb2 þ Cb4� (24c)

Qb1 ¼ �2Db wmbplb

� �3

½Cb1 þ Cb2 � Cb3 þ Cb4� (24d)

Similarly, those at zb ¼ hf are:

wb2 ¼ f½Cb1 cos $b þ Cb2 sin $b�e�$b

þ ½Cb3 cos $b þ Cb4 sin $b�e$b þ 1gwmb (25a)

bb2 ¼ wmbplb½�Cb1ðsin $b þ cos $bÞ þ Cb2ðcos $b � sin $bÞ�e

�$b�

þ½Cb3ð� sin $b þ cos $bÞ þ Cb4ðcos $b þ sin $bÞ�e$b�

(25b)

Mb2 ¼ � 2Dbwmbplb

� �2

½Cb1 sin $b � Cb2cos $b�e

�$b�

þ½�Cb3 sin $b þ Cb4 cos $b�e$b�

(25c)

Qb2 ¼ � 2Dbwmbplb

� �3

½Cb1ð� sin $b þ cos $bÞ þ Cb2ðcos $b þ sin $bÞ�e�$b

�þ½�Cb3ðsin $b þ cos $bÞ þ Cb4ðcos $b � sin $bÞ�e

$b�

(25d)

where

$b ¼phf

lb(25e)

ARTICLE IN PRESS


2.3.3. Shell Section C

Section C, between the base and the reinforced section, is alsotaken to be a short shell. Writing

xc ¼pzc

l¼pz

l(26)

The deformations in Section C can be obtained from Eq. (14):

w ¼ f½Cc1 cos xc þ Cc2 sin xc�e�xc

þ ½Cc3 cos xc þ Cc4 sin xc�exc þ 1gwm (27a)

b ¼dw

dzc¼ wm

pl½�Cc1ðsin xc þ cos xcÞ þ Cc2ðcos xc � sin xcÞ�e

�xc

nþ½Cc3ð� sin xc þ cos xcÞ þ Cc4ðcos xc þ sin xcÞ�e

xc

o(27b)

d2w

dz2c

¼ 2wmpl

� �2½Cc1 sin xc � Cc2 cos xc�e

�xc

nþ½�Cc3 sin xc þ Cc4 cos xc�e

xc

o(27c)

d3w

dz3c

¼ 2wmpl

� �3½Cc1ð� sin xc þ cos xcÞ þ Cc2ðcos xc þ sin xcÞ�e

�xc

nþ½�Cc3ðsin xc þ cos xcÞ þ Cc4ðcos xc � sin xcÞ�e

xc

o(27d)

The deformation and internal forces at zc ¼ 0 can be found fromEqs. (18a)–(18d) and (27a)–(27d):

wc0 ¼ ½Cc1 þ Cc3 þ 1�wm (28a)

bc0 ¼ wmpl½�Cc1 þ Cc2 þ Cc3 þ Cc4� (28b)

Mc0 ¼ �2Dwmpl

� �2½�Cc2 þ Cc4� (28c)

Qc0 ¼ �2Dwmpl

� �3½Cc1 þ Cc2 � Cc3 þ Cc4� (28d)

Those at z ¼ xf are:

wc ¼ f½Cc1 cos $c þ Cc2 sin $c�e�$c

þ ½Cc3 cos $c þ Cc4 sin $c�e$c þ 1gwm (29a)

bc ¼ wmpl½�Cc1ðsin $c þ cos $cÞ þ Cc2ðcos $c � sin $cÞ�e

�$c�

þ½Cc3ð� sin $c þ cos $cÞ þ Cc4ðcos $c þ sin $cÞ�e$c�

(29b)

Mc ¼ � 2Dwmpl

� �2½Cc1 sin $c � Cc2 cos $c�e

�$c�

þ½�Cc3 sin $c þ Cc4 cos $c�e$c�

(29c)

Qc ¼ � 2Dwmpl

� �3½Cc1ð� sin $c þ cos $cÞ þ Cc2ðcos $c þ sin $cÞ�e

�$c�

þ½�Cc3ðsin $c þ cos $cÞ þ Cc4ðcos $c � sin $cÞ�e$c�

(29d)

where

$c ¼pxf

l(29e)

2.4. Solution for a pinned-based shell

2.4.1. Boundary conditions at the base

The first boundary condition examined here is a simplysupported base. The boundary conditions can be expressed as

wc0 ¼ 0 (30a)

Mc0 ¼ 0 (30b)

Substituting Eqs. (28a)–(28d) into Eqs. (30a) and (30b) gives:

Cc3 ¼ �ðCc1 þ 1Þ (31)

Cc4 ¼ Cc2 (32)

2.4.2. Continuity at Joint 1

The deformation continuity at Joint 1 requires that

wc ¼ wb1 (33a)

bc ¼ bb1 (33b)

Substituting Eqs. (24a), (24b), (29a) and (29b) into Eqs. (33a)and (33b) gives

Cc1ðcos $ce�$c Þ þ Cc2ðsin $ce�$c Þ

þ Cc3ðcos $ce$c Þ þ Cc4ðsin $ce$c Þ

� Cb1wmb

wm� Cb3

wmb

wmþ 1�

wmb

wm¼ 0 (34)

Cc1½�ðsin $c þ cos $cÞe�$c � þ Cc2½ðcos $c � sin $cÞe

�$c �

þ Cc3½ð� sin $c þ cos $cÞe$c � þ Cc4½ðcos $c þ sin $cÞe

$c �

þ Cb1 F1 � Cb2 F1 � Cb3 F1 � Cb4 F1 ¼ 0 (35)

where

F1 ¼wmb

wm

llb

(36)

2.4.3. Equilibrium of Joint 1

Equilibrium at Joint 1 requires

Mc ¼ Mb1 (37a)

Qc ¼ Qb1 (37b)

Substituting Eqs. (24c), (24d), (29c) and (29d) into Eqs. (37a)and (37b) gives

Cc1 sin $ce�$c � Cc2 cos $ce�$c � Cc3 sin $ce$c

þ Cc4 cos $ce$c þ Cb2F2 � Cb4F2 ¼ 0 (38)

Cc1½ð� sin $c þ cos $cÞe�$c � þ Cc2½ðcos $c þ sin $cÞe

�$c �

þ Cc3½�ðsin $c þ cos $cÞe$c � þ Cc4½ðcos $c � sin $cÞe

$c �

� F3Cb1 � Cb2F3 þ Cb3F3 � Cb4F3 ¼ 0 (39)

where

F2 ¼wmb

wm

l2

l2b

Db

D(40)

and

F3 ¼wmb

wm

l3

l3b

Db

D(41)

2.4.4. Continuity at Joint 2

Continuity at Joint 2 requires

wa ¼ wb2 (42a)

ba ¼ bb2 (42b)

Substituting Eqs. (21a), (21b), (25a) and (25b) into Eqs. (42a)and (42b) gives

Cb1ðcos $be�$b Þ þ Cb2ðsin $be�$b Þ þ Cb3ðcos $be$b Þ

þ Cb4ðsin $be$b Þ þ Ca1�

wm

wmb

� �þ 1�

wm

wmb¼ 0 (43)

ARTICLE IN PRESS


Cb1½�ðsin $b þ cos $bÞe�$b � þ Cb2½ðcos $b � sin $bÞe

�$b �

þ Cb3½ð� sin $b þ cos $bÞe$b � þ Cb4½ðcos $b þ sin $bÞe

$b �

þ Ca11

F1

� �þ Ca2 �

1

F1

� �¼ 0 (44)

2.4.5. Equilibrium of Joint 2

The equilibrium of Joint 2 requires that

Ma ¼ Mb2 (45a)

Qa ¼ Qb2 (45b)

Substituting Eqs. (21c), (21d), (25c) and (25d) into Eqs. (45a)and (45b) gives

Cb1ð� sin $be�$b Þ þ Cb2ðcos $be�$b Þ þ Cb3ðsin $be$b Þ

þ Cb4ð� cos $be$b Þ þ Ca2 �1

F2

� �¼ 0 (46)

Cb1½ð� sin $b þ cos $bÞe�$b � þ Cb2½ðcos $b þ sin $bÞe

�$b �

þ Cb3½�ðsin $b þ cos $bÞe$b � þ Cb4½ðcos $b � sin $bÞe

$b �

þ Ca1 �1

F3

� �þ Ca2 �

1

F3

� �¼ 0 (47)

Substituting Eqs. (31) and (32) into Eqs. (34)–(47) andrearranging the resulting equations in matrix form gives

A11 0 A13 A14 A15 A16 0 0

A21 A22 A23 A24 A25 A26 0 0

0 A32 A33 A34 A35 A36 0 0

A41 A42 A43 A44 A45 A46 0 0

0 0 A53 0 A55 0 A57 A58

0 0 0 A64 0 A66 A67 A68

0 0 A73 A74 A75 A76 A77 A78

0 0 A83 A84 A85 A86 A87 A88

2666666666666664

3777777777777775

�

Ca1

Ca2

Cb1

Cb2

Cb3

Cb4

Cc1

Cc2

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

¼

B1

0

0

0

B5

B6

B7

B8

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

(48)

which may be expressed as

½A� � fCg ¼ fBg (49)

The coefficients Aij and Bi in Eq. (48) are presented in AppendixA for a pinned base. The solution {C} is

fCg ¼ ½A��1 � fBg (50)

Note that coefficients Cc3 and Cc4 are not included inEqs. (48)–(50) but are given in Eqs. (31) and (32).

2.5. Radial displacement of the strengthened shell

Having obtained the coefficients from Eq. (50), the radialdisplacement of the strengthened shell can be calculated from

w ¼

f½Cc1 cos xc þ Cc2 sin xc�e�xc þ ½Cc3 cos xc þ Cc4 sin xc�exc þ 1gwm for 0pzoxf

f½Cb1 cos xb þ Cb2 sin xb�e�xb þ ½Cb3 cos xb þ Cb4 sin xb�e

xb þ 1gwmb for xfpzoxf þ hf

f½Ca1 cos x̄þ Ca2 sin x̄�e�x̄ þ 1gwm for hf þ xfpz

8>><>>: (51)

3. Patterns of deformation in the shell

The radial displacements and internal forces of the cylindricalshell, with an axisymmetric FRP sheet of any size bonded at anylocation, can be obtained from the equations above. An examplecylindrical shell with a height h ¼ 5000 mm, radius r ¼ 5000 mm

and thickness ts ¼ 5 mm was studied here. This shell has a radius-to-thickness ratio of 1000, which is a medium length cylinderaccording to Eurocode 3 Part 1.6 [25] for the purposes of bucklingstrength evaluation. The Young’s modulus Es was taken as 200 GPaand the Poisson’s ratio ns as 0.3. An FRP sheet was bonded onto theshell at a distance xf above the base. The FRP sheet had a height hf

and thickness tf. The elastic moduli Efy, Efz in circumferential andvertical directions and the Poisson’s ratio nfyz were taken as 230 GPa,3 GPa and 0.35, respectively. An internal pressure of 0.25 N/mm2

and a vertical load of 500 N/mm were applied. The effects of the FRPsheet rigidity, height and position were explored.

Fig. 3 shows the deformed shape of the shell near the base, fordifferent quantities of FRP. The maximum deflection is closelyrelated to the peak circumferential stress, so a peak in thisdeformed shape is strongly coupled with the formation of theelephant’s foot buckle. The effect of changing the quantity of FRPcan be seen in Fig. 3 for the condition where hf/l ¼ 1 and the FRPstarts from the base of the shell (xf ¼ 0). The amount of FRP isrepresented by the normalised FRP rigidity parameter a (Eq. (8)).When no FRP sheet is used, the maximum radial displacementoccurs at a height of 0.75l above the base with the maximumdeflection equal to 1.067wm.

A small amount of FRP (e.g. a ¼ 0.05) can reduce the maximumdeflection but it cannot eliminate the peak. Whilst a larger amountof FRP (e.g. aX0.2) can eliminate the peak within the region withFRP, it produces a peak above the FRP zone and the maximumdeflection above the FRP zone increases as the amount of FRPincreases. Using a very large FRP thickness (a ¼ 50) reduces thedeflection in the strengthened zone close to zero and the maximumdeflection occurs instead at another position above the FRP sheet.Clearly, using a large amount of FRP does not help in preventing theelephant’s foot buckling from occurring, but simply moves thelocation of the buckle to another location. There is an optimalamount of FRP, with a value of a around 0.1, which can reduce themaximum deflection near the base to the membrane theorydeflection, without introducing a new peak elsewhere.

Fig. 4 shows the effect of the normalised FRP height, hf/l, whenthe thickness and the starting position of FRP remain constantwith a ¼ 0.1 and xf/l ¼ 0. When the FRP is only attached to a smallsection at the bottom of the shell (e.g. hf/l ¼ 0.5), the strengthen-ing is not very effective because a large peak value remains. If theFRP height is increased, the peak deflection reduces. However, ifthe FRP height is too large the cylinder starts to behave as if itwere two separate sections of shell with the lower sectionbehaving as a metal–FRP composite shell and the upper as theoriginal metal shell. The lower composite shell results in a peak ata distance of 0.75lb from the base. A peak also appears above theposition where the FRP ends due to bending arising fromdiscontinuity at the joint. This repeats and reinforces theobservation made above that both too little and too much FRPwill reduce the elephant’s buckling strength compared with thecase with an optimal amount of FRP.

Fig. 5 demonstrates the effect of the starting position of theFRP, xf/l, when both the thickness and the height of the FRP arekept constant (a ¼ 0.1 and hf/l ¼ 1.0). It appears that themaximum radial displacement is minimised when xf is about0.25, in which case the FRP sheet is centred at a height of 0.75labove the base.

ARTICLE IN PRESS

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

0

Nor

mal

ized

dis

tanc

e ab

ove

base

z/λ Without FRP

Alpha = 0.05Alpha = 0.1Alpha = 0.2Alpha = 0.5Alpha = 1Alpha = 50

Normalized radial displacement w/wm

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Fig. 3. Effect of FRP rigidity on shell deformation (hf/l ¼ 1 and xf/l ¼ 0).

0.00

0.75

1.50

2.25

3.00

3.75

4.50

5.25

6.00

0.9Normalized radial displacement w/wm

Nor

mal

ized

dis

tanc

e ab

ove

base

z/λ Without FRP

Normalized FRP height = 0.5Normalized FRP height = 0.75Normalized FRP height = 1Normalized FRP height = 1.25Normalized FRP height = 4

0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

Fig. 4. Effect of normalised FRP height on shell deformation (a ¼ 0.1 and xf/l ¼ 0).

0.000.250.500.751.001.251.501.752.002.252.502.753.00


Nor

mal

ized

dis

tanc

e ab

ove

base

z/λ Without FRP

xf/lamda = 0

xf/lamda = 0.25

xf/lamda = 0.5

xf/lamda = 0.75

0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1

Fig. 5. Effect of FRP start position on shell deformation (a ¼ 0.1 and hf/l ¼ 1).


4. Optimal FRP strengthening

4.1. Failure criterion

In elephant foot failures, the membrane stress resultantsdominate the failure of the shell when it is isotropic andunstiffened [12]. The axial membrane stress resultant Nz interacts

with the local value of the circumferential stress resultant Ny toproduce failure according to von Mises criterion

N2eq ¼ ðN

2z þ NzNy þ N2

yÞ (52)

Since Nz is unchanged by deformation, the highest von Misesstress resultant arises when Ny is at its peak value. Further,

ARTICLE IN PRESS


because Ny is linearly related to the radial displacement w

according to Eq. (18a), the peak value Ny is located at the pointwhere w has its maximum value. From Eq. (14) for a longcylindrical shell, it can be shown that the maximum w is locatedat z ¼ 0.75l when the cylindrical shell is not strengthened. Thepeak value of deflection is then 1.067wm as shown in Fig. 3.

4.2. Estimation of optimal FRP strengthening

It is clear that the extensional rigidity of the FRP (related to a),the normalised height of the FRP sheet (hf/l) and its normalisedlower edge position (xf/l) affect the deformation of the strength-ened shell. Fig. 6 shows the effect of the normalised FRP height, hf/l, and the normalised start position, xf/l, on the maximumdisplacement when a ¼ 0.1. It is seen that each curve has aminimum, at which the position of the FRP is optimal. A closeanalysis shows that the minimum of the maximum displacementis achieved if the FRP is centred at a distance of 0.75l above thebase, i.e.

xf ¼ 0:75l�hf

2(53)

Further analysis shows that the same relationship applies fordifferent values of a.

0.98

1

1.02

1.04

1.06

1.08

0

Nor

mal

ized

max

imum

radi

aldi

spla

cem

ent w

max

/wm

Minimize the maximum displace

Normalized F0.25 0.5 0.75

Fig. 6. Effects of hf/l and xf/l on the max

0.000.250.500.751.001.251.501.752.002.252.502.753.00

0Normalized radia

Nor

mal

ized

dis

tanc

e ab

ove

base

z/λ

0.1 0.2 0.3 0.4 0.5

Fig. 7. The ‘ideal’ deformation of a she

Based on the failure criterion of Eq. (52), the deformation of thecylindrical shell strengthened with optimal FRP may be expectedto look like Fig. 7, which satisfies the boundary condition at thebase whilst the radial deformation is not greater than themembrane theory solution anywhere in the shell.

For the deformed shape of the shell as shown in Fig. 7, theradial displacement and the rotation at the joint between shellSections A and B (Fig. 1) should be wm and 0, respectively. Thisleads to the condition that Ca1 ¼ Ca2 ¼ 0 according to Eqs. (21a)and (21b). Substituting this condition into Eqs. (43)–(47) gives

Cb1 ¼cos $b

2� e$b �

wm

wmb� 1

� �(54a)

Cb2 ¼sin $b

2� e$b �

wm

wmb� 1

� �(54b)

Cb3 ¼cos $b

2� e�$b �

wm

wmb� 1

� �(54c)

Cb4 ¼sin $b

2� e�$b �

wm

wmb� 1

� �(54d)

where $b is given in Eq. (25e).Further, if it is assumed that the deformations match the me-

mbrane theory values (w ¼ wm, b ¼ 0) at z ¼ 0.75l above the base

hf/lamda = 0.1

hf/lamda = 0.25

hf/lamda = 0.35

hf/lamda = 0.5

ment: xf = 0.75λ-hf/2

RP start position xf / λ1 1.25 1.5 1.75 2

imum radial displacement (a ¼ 0.1).

l displacement w/wm

0.6 0.7 0.8 0.9 1 1.1

ll with optimal FRP strengthening.

ARTICLE IN PRESS

0.000.250.500.751.001.251.501.752.002.252.502.753.00

0

Nor

mal

ized

dis

tanc

e ab

ove

base

z/λ

alpha = 0.1, xf/Lamda = 0.53, hf/Lamda = 0.45


alpha = 0.2, Xf/Lamda = 0.64, hf/Lamda = 0.217


Without FRP


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Fig. 8. Deformation of the example shell strengthened with different ‘optimal’ FRP sheets.

0.000.250.500.751.001.251.501.752.002.252.502.753.00

0.97

Nor

mal

ized

dis

tanc

e ab

ove

base

z/λ alpha = 0.1, xf/Lamda = 0.53, hf/Lamda = 0.45




Without FRP


0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07

Fig. 9. Detailed deformation of the example shell strengthened with different ‘optimal’ FRP sheets.


where the displacement reaches its peak value for the unstrengthenedshell, then from Eqs. (27a) and (27b) together with the boundarycondition at the base (Eqs. (31) and (32)), the following are found

Cc1 ¼ ½sin 2$� 1� e2$�=½e2$ � 2 sin 2$� e�2$Þ� (55a)

Cc2 ¼Cc1ðe

$ � e�$Þ þ e$

tan $ðe�$ þ e$Þ(55b)

Cc3 ¼ �ðCc1 þ 1Þ (55c)

Cc4 ¼ Cc2 (55d)

where $ ¼ 0U75p.Substituting Eqs. (54a)–(54d) and (55a)–(55d) into Eq. (38)

results in:

Cc1 sin $ce�$c � Cc2 cos $ce�$c � Cc3 sin $ce$c þ Cc4 cos $ce$c

þ F2 �sin $b

2

wm

wmb� 1

� �� ðewb � e�wb Þ ¼ 0 (56)

where F2 is given in Eq. (40) and

$c ¼pxf

l¼ 0:75p� 0:5$b

lb

l(57)

Substituting Eq. (57) into Eq. (56), the resulting equationcontains only a single unknown $b. The equation can be solved

for $b. From Eqs. (25e) and (57), xf and hf can then be obtaineddirectly.

4.3. A worked example

The cylindrical shell described in Section 3 with the sameproperties for both the cylindrical shell and the FRP sheet wasstudied here to verify the above procedure in estimating anoptimal FRP strengthening scheme. The shell was assumed to beunder an internal pressure of 0.25 N/mm2 and vertical forceper unit circumference of 500 N/mm. For each value of thecircumferential extensional rigidity for the FRP, a, the aboveprocedure can be used to estimate optimal values for the FRP baseheight xf and the height of the FRP strip hf. The results are shownin Figs. 8 and 9.

5. Cylindrical shells with a fixed base

5.1. Solution

To study the effect of boundary conditions, a fixed base wasalso investigated. Compared with the shell with a pinned base, theonly difference in this case is that the rotational boundary

ARTICLE IN PRESS

0.00

0.50

1.00

1.50

2.00

2.50

0

Nor

mal

ized

dis

tanc

e ab

ove

base

z/λ

Alpha = 0Alpha = 0.05Alpha = 0.1Alpha = 0.2Alpha = 0.5Alpha = 1Alpha = 50


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Fig. 10. Effect of FRP rigidity on the deformation of a shell with fixed base (hf/l ¼ 1 and xf/l ¼ 0).

0

1

2

3

4

5

6

0.9

Nor

mal

ized

dis

tanc

e ab

ove

base

z/λ Normalized FRP height = 0

Normalized FRP height = 0.5


Normalized FRP height = 1


Normalized FRP height = 4


0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1

Fig. 11. Effect of normalised FRP height on the deformation of a shell with fixed base (a ¼ 0.1 and xf/l ¼ 0).


condition at the base becomes bc0 ¼ 0 at z ¼ 0 (instead of Mc0 ¼ 0as in Eq. (30b)). Applying this condition to Eqs. (28a)–(28d) gives

Cc4 ¼ 2Cc1 � Cc2 þ 1 (58)

From Eqs. (31), (33a), (33b)–(47) and (58), the solution can beobtained in the same form as Eq. (48), but with differentcoefficients Aij and Bi. These coefficients for a fixed base are givenin Appendix B.

The same example as in Section 3 but with a fixed base wasstudied here. Fig. 10 shows the effect of the governing rigidityparameter a (Eq. (8)) when the normalised FRP height is hf/l ¼ 1and the FRP starts at the base (xf/l ¼ 0). For the unstrengthenedshell, the maximum radial displacement occurs at a height of 1.0labove the base (instead of 0.75l for a pinned base) and has a valueof 1.043wm (c.f. 1.067wm for a pinned base). The addition of a smallamount of FRP can significantly reduce the peak value. Theoptimal value of a for a fixed base is about 0.2 (c.f. affi0.1 for apinned base). Again if a very large thickness of FRP is used (e.g.a ¼ 50), the deflection within the strengthened zone is mostlyeliminated and a new peak develops above the FRP.

Fig. 11 shows the effect of the normalised FRP height hf/l whena ¼ 0.1 and xf/l ¼ 0. It shows that an increase of normalised FRPheight reduces the peak radial displacement initially but a furtherincrease introduces a peak above the FRP sheet which is less

effective than the optimal case. Fig. 12 demonstrates that the FRPlower edge location (normalised starting position xf/l) can alsosignificantly affect the shell deformation.

5.2. Optimal FRP strengthening

As was the case for the pinned-base cylinder, the parameters a,hf/l and xf/l all influence the deformation pattern in thestrengthened shell. Figs. 10–12 show the effects of changing thethickness (Fig. 10) and the height of the FRP sheet (Fig. 11), and itsstart position (Fig. 12). All three figures use the unstrengthenedfixed base cylinder as a reference case, where it can be seen thatthe maximum radial displacement w, located at z ¼ l, attains thepeak value of 1.043wm.

The critical design information is contained in Fig. 13, wherethe peak deflection is plotted against the normalised start positionxf/l. As the height of the FRP sheet increases, the peak deflectioncan be steadily reduced towards wmax ¼ wm, but there is clearly anarrowly defined critical height that should be used: the optimalposition for the FRP centres it on the level z ¼ l, which indicatesthat

xf ;opt ¼ l�hf

2(59)

ARTICLE IN PRESS

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.9

Nor

mal

ized

dis

tanc

e ab

ove

base

z/λ Without FRP

xf/Lamda = 0.5xf/Lamda = 0.75xf/Lamda = 1


0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1

Fig. 12. Effect of FRP start position on the deformation of a shell with fixed base (a ¼ 0.1 and hf/l ¼ 1.25).

1

1.02

1.04

1.06

0

Nor

mal

ized

max

imum

radi

aldi

spla

cem

ent w

max

/wm

hf/lamda = 0.1

hf/lamda = 0.15

hf/lamda = 0.2

hf/lamda = 0.25

Minimize the maximum displacement: xf = λ-hf/2

Normalized FRP start position xf / λ0.25 0.5 0.75 1 1.25 1.5 1.75 2

Fig. 13. Effects of hf/l and xf/l on the maximum radial displacement of a fixed-base shell.

0.000.250.500.751.001.251.501.752.002.252.502.753.00

0

Nor

mal

ized

dis

tanc

e ab

ove

base

z/λ

Normalized radial displacement w/wm0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Fig. 14. The ‘ideal’ deformation of a fixed-base shell with optimal FRP strengthening.


A similar procedure to that for a shell with a pinned base isdeveloped here for estimating the optimal FRP parameters.Assuming that the deformation of the strengthened shell with

optimal FRP strengthening is as shown in Fig. 14. Applying w ¼ wm

and b ¼ 0 at the joint between shell sections A and B leads to thesame outcome as for a pinned base (Eqs. (54a)–(54d)). Further

ARTICLE IN PRESS

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00


Nor

mal

ized

dis

tanc

e ab

ove

base

Z/λ alpha = 0.1, Xf/Lamda = 0.86, hf/Lamda = 0.28




Without FRP

0.99 1 1.01 1.02 1.03 1.04 1.05

Fig. 15. Deformation of a fixed-base shell strengthened with different ‘optimal’ FRP sheets.


applying w ¼ wm and b ¼ 0 to Eqs. (27a) and (27b) respectively, atz ¼ l together with the boundary conditions Eqs. (31) and (58)gives

Cc1 ¼ �ep

ðep � e�pÞ¼ Cc2 (60)

Cc3 ¼ �ð1þ Cc1Þ (61)

Cc4 ¼ Cc1 þ 1 (62)

Substituting Eqs. (60–62) into Eq. (56) and considering

$c ¼pxf

l¼ p� 0:5$b

lb

l(63)

the optimal FRP sheet can be derived as for the pinned base fromEq. (56).

To verify the practical procedure, the worked exampleof Section 4.3 for the pinned base were studied again forthe fixed base. Fig. 15 shows the radial displacement afteradding different optimal FRP sheets by using the equationsderived here.

Using the practical procedure, the radial displacement of theshell can be seen to have decreased significantly (Fig. 15). Thisradial displacement is very close to the membrane theory uniformdisplacement wm at the mid-point of the FRP sheet, located at labove the base, and thus it satisfies the assumptions, which weremade for this design procedure in Section 5.2.

6. Conclusions

Cylindrical shells under internal pressure accompanied byaxial loads are susceptible to elephant’s foot buckling near thebase. This paper has presented a preliminary study of thepossibility of strengthening these shells against elephant’s footbuckling by using externally bonded FRP. The linear elasticbending equations for the strengthened cylindrical shell havebeen derived and illustrated through exploratory examples. Bothpinned and fixed base boundary conditions for the cylindricalshell have been considered. It has been shown that a smallamount of FRP sheet, placed in the critical location, is optimal in

reducing the radial deformation of the shell, which wouldincrease the elephant’s foot buckling strength.

Acknowledgements

The authors are most grateful to Damascus University forproviding the first author’s studentship. Financial support fromthe Royal Society Kan Tong Po Visiting Professorship for the thirdauthor is also gratefully acknowledged.

Appendix A. Matrices [A] and [B] in Eq. (48): pinned base

For an FRP strengthened cylindrical shell with a pinned base,the coefficients in matrices [A] and [B] in Eq. (48) are given asfollows:

A11 ¼ �wm

wmb

A13 ¼ cos $be�$b

A14 ¼ sin $be�$b

A15 ¼ cos $be$b

A16 ¼ sin $be$b

9>>>>>>>>=>>>>>>>>;

(A.1)

A21 ¼1

F1

A22 ¼ �1

F1

A23 ¼ �ðsin $b þ cos $bÞe�$b

A24 ¼ ðcos $b � sin $bÞe�$b

A25 ¼ ð� sin $b þ cos $bÞe$b

A26 ¼ ðcos $b þ sin $bÞe$b

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

(A.2)

A32 ¼ �1

F2

A33 ¼ � sin $be�$b

A34 ¼ cos $be�$b

A35 ¼ sin $be$b

A36 ¼ � cos $be$b

9>>>>>>>>=>>>>>>>>;

(A.3)

ARTICLE IN PRESS


A41 ¼ �1

F3

A42 ¼ �1

F3

A43 ¼ ð� sin $b þ cos $bÞe�$b

A44 ¼ ðcos $b þ sin $bÞe�$b

A45 ¼ �ðsin $b þ cos $bÞe$b

A46 ¼ ðcos $b � sin $bÞe$b

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

(A.4)

A53 ¼ �wmb

wm

A55 ¼ �wmb

wm

A57 ¼ cos $ce�$c � cos $ce$c

A58 ¼ sin $ce�$c þ sin $ce$c

9>>>>>>>=>>>>>>>;

(A.5)

A64 ¼ F2

A66 ¼ �F2

A67 ¼ sin $ce�$c þ sin $ce$c

A68 ¼ cos $ce$c � cos $ce�$c

9>>>=>>>;

(A.6)

A73 ¼ �F3

A74 ¼ �F3

A75 ¼ F3

A76 ¼ �F3

A77 ¼ ð� sin $c þ cos $cÞe�$c

A78 ¼ ðsin $c þ cos $cÞe�$c þ ð� sin $c þ cos $cÞe$c

9>>>>>>>>>=>>>>>>>>>;

(A.7)

A83 ¼ F1

A84 ¼ �F1

A85 ¼ �F1

A86 ¼ �F1

A87 ¼ �ðsin $c þ cos $cÞe�$c þ ðsin $c � cos $cÞe$c

A88 ¼ ð� sin $c þ cos $cÞe�$c þ ðsin $c þ cos $cÞe$c

9>>>>>>>>>=>>>>>>>>>;

(A.8)

B1 ¼wm

wmb� 1

B5 ¼wmb

wm� 1þ cos $ce$c

B6 ¼ � sin $ce$c

B7 ¼ �ðcos $c þ sin $cÞe$c

B8 ¼ ðcos $c � sin $cÞe$c

9>>>>>>>>>=>>>>>>>>>;

(A.9)

Appendix B. Matrices [A] and [B] in Eq. (48): fixed base

For an FRP strengthened cylindrical shell with a fixed base,the coefficients in matrices [A] and [B] in Eq. (48) are given asfollows:

A11 ¼ �wm

wmb

A13 ¼ cos $be�$b

A14 ¼ sin $be�$b

A15 ¼ cos $be$b

A16 ¼ sin $be$b

9>>>>>>>>=>>>>>>>>;

(B.1)

A21 ¼1

F1

A22 ¼ �1

F1

A23 ¼ �ðsin $b þ cos $bÞe�$b

A24 ¼ ðcos $b � sin $bÞe�$b

A25 ¼ ð� sin $b þ cos $bÞe$b

A26 ¼ ðcos $b þ sin $bÞe$b

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

(B.2)

A32 ¼ �1

F2

A33 ¼ � sin $be�$b

A34 ¼ cos $be�$b

A35 ¼ sin $be$b

A36 ¼ � cos $be$b

9>>>>>>>>=>>>>>>>>;

(B.3)

A41 ¼ �1

F3

A42 ¼ �1

F3

A43 ¼ ð� sin $b þ cos $bÞe�$b

A44 ¼ ðcos $b þ sin $bÞe�$b

A45 ¼ �ðsin $b þ cos $bÞe$b

A46 ¼ ðcos $b � sin $bÞe$b

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

(B.4)

A53 ¼ �wmb

wm

A55 ¼ �wmb

wm

A57 ¼ cos $ce�$c � cos $ce$c þ 2 sin $ce$c

A58 ¼ sin $cðe�$c � e$c Þ

9>>>>>>>=>>>>>>>;

(B.5)

A64 ¼ F2

A66 ¼ �F2

A67 ¼ sin $ce�$c þ sin $ce$c þ 2 cos $ce$c

A68 ¼ � cos $cðe$c þ e�$c Þ

9>>>>=>>>>;

(B.6)

A73 ¼ �F3

A74 ¼ �F3

A75 ¼ F3

A76 ¼ �F3

A77 ¼ ð� sin $c þ cos $cÞe�$c þ ðsin $c þ cos $cÞe$c

þ2ð� sin $c þ cos $cÞe$c

A78 ¼ ðsin $c þ cos $cÞe�$c � ð� sin $c þ cos $cÞe$c

9>>>>>>>>>>>=>>>>>>>>>>>;

(B.7)

A83 ¼ F1

A84 ¼ �F1

A85 ¼ �F1

A86 ¼ �F1

A87 ¼ �ðsin $c þ cos $cÞe�$c þ ðsin $c � cos $cÞe$c

þ2ðsin $c þ cos $cÞe$c

A88 ¼ ð� sin $c þ cos $cÞe�$c � ðsin $c þ cos $cÞe$c

9>>>>>>>>>>>=>>>>>>>>>>>;

(B.8)

ARTICLE IN PRESS


B1 ¼wm

wmb� 1

B5 ¼wmb

wm� 1þ ðcos $c � sin $cÞe$c

B6 ¼ �ðsin $c þ cos $cÞe$c

B7 ¼ �2 cos $ce$c

B8 ¼ �2 sin $ce$c

9>>>>>>>>>=>>>>>>>>>;

(B.9)

References

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Documents

2008 Strengthening Metallic Cylindrical Shells Against Elephant's Foot Buckling With FRP