Buckling Cylinder Bueno

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Asia-Pacific Conference on FRP in Structures (APFIS 2007)

S.T. Smith (ed)

© 2007 International Institute for FRP in Construction

465

BUCKLING AND REDUCED STIFFNESS CRITERIA FOR FRP CYLINDRICAL

SHELLS UNDER COMPRESSION

K. Matsumoto1*, S. Yamada1, H.T. Wang2 and J.G.A. Croll 3

1 Toyohashi University of Technology, Toyohashi, J apan.Email: [email protected] University College London, U.K.3 University College London, U.K.

ABSTRACT

This paper deals with buckling of thin-walled cylindrical shells under compression. It is well-known that axially

compressed cylindrical shells have buckling behaviour which is very sensitive to initial geometric imperfections.However, current approaches using mathematical algorithms to optimise the linearised classical critical loadswith respect to many design variables, generally ignore the potential reductions in elastic load carrying capacitiesthat result from the initial imperfections. They present major problems for incorporation into design processesand often involve excessive computational effort. Adopting 6-ply symmetric, glass-epoxy cylindrical shells, thispaper carries out classical buckling analysis for perfect shells and nonlinear analysis for imperfect shells. Inaddition the present paper applies an alternative lower bound design concept called the reduced stiffness method.By confirming the correspondence between the lower bounds of nonlinear buckling analysis and the reducedstiffness analysis, it suggested that the reduced stiffness method could provide an important basis for design.

KEYWORDS

Cylindrical Shells, Buckling, Imperfection Sensitivity, Reduced Stiffness Method, Angle of Fibre Orientation.

INTRODUCTION

In many situations FRP composites provide opportunities for enhanced efficiency, primarily because of theirhigh strength-to-weight ratios and corrosion resistance. These advantages are significant for the bridges, storagetanks, pressure vessels, and especially for the reductions in weight for aircraft and other aerospace structures. The present paper undertakes buckling of orthotropic cylindrical shells under compression. In this area one of themost important aspects is that real cylindrical shells always have initial geometric imperfections. Theseimperfections play a defining role as to the reductions of elastic buckling capacities from the upper boundprovided by the classical critical loads (Yamaki 1984). Moreover different design parameters and differentbuckling modes will exhibit very different levels of imperfection related knock-down to the lower bound,buckling loads. One possible way to tackle this dilemma is to predict the worst effects of initial imperfections onthe buckling capacities of composite shells. The reduced stiffness method is such a lower bound designphilosophy that is able to predict the worst possible effects of initial imperfections. It is based on the physical

argument that the reductions in the buckling loads of shells result from the loss of initial stabilizing membraneenergy. By eliminating these membrane energies from the classical critical analysis allows a lower bound to theimperfection sensitive buckling loads of shells to be obtained (e.g. Yamada and Croll 1993, 1999; Y amadaet al.2006). Using nonlinear analysis this paper will demonstrate that for increasing amplitudes of initialimperfections the elastic buckling loads exhibit well defined lower bounds. The present study concentrates theeffects of angle of fibre orientation (Wang and Croll 2007), and confirms the coexistence between the theoreticallower bound predicted by the reduced stiffness analysis and the lowest buckling loads exhibited by nonlinearnumerical experiments. For this reason it is suggested that the reduced stiffness method could provide asignificant contribution to more effective optimization design.

METHOD OF ANALYSIS

The geometry and coordinate system of a laminated shell are shown in Figure 1 with thickness, t, length, L, and

radius,R. The shell is simply supported as described in Eq. 1 and subject to uniform axial stress σ .



APFIS 2007 466

Figure 1 Convention for shell geometry and coordinate

σ

u

w

v

x

zy

t

2R

L

2

20, 0 , 0 , 0 at 0,x L

xx

∂ ∂= = = = =

∂∂

w u w v

Classical Critical Analysis

Displacement functionu , v , w can be taken as harmonic functions that satisfy the boundary conditions of Eq. 1.

( ) ( )( ) ( )

( ) ( )

,

,

,

cos cossin sin

cos sin

i j

i j

i j

iy R j x Liy R j x L

iy R j x L

π π

π

==

=

u u

v v

w w

wherei and j are the circumferential full-wave and longitudinal half wave number.

The Donnell-Mushtari-Vlasov approximations of the linear incremental strain-displacement relations at thecritical state are given as

2

2

2

2

2

,

,

1,

2

d d

x x

d dy y

d dxy xy

x x

y R y

y x x y

ε κ

ε κ

ε κ

∂ ∂= = −

∂ ∂

∂ ∂= − = −

∂ ∂

⎛ ⎞∂ ∂ ∂= + = −⎜ ⎟

∂ ∂ ∂ ∂⎝ ⎠

u w

v w w

u v w

and the non-linear incremental components22

1 1 1, ,

2 2 2

dd dd dd

x y xyx y x y

ε ε ε ⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞

= = =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

w w w w

Here (ε x, ε y, ε xy) are the incremental membrane strains; (κ x, κ y, κ xy) the incremental bending strains. Superscriptof “d” denotes the linear components, and “dd” the quadratic components.

Based on the classical lamination theory the constitutive relations corresponding to the linear incremental straincomponents are

11 12 16 11 12 16

12 22 26 12 22 26

16 26 66 16 26 66

11 12 16

12 22 26

16 26

2 2

d d dx x x

d d dy y y

d d dxy xy xy

dx

dy

dxy

n A A A B B B

n A A A B B B

A A A B B Bn

m B B B

m B B B

B B Bm

ε κ

ε κ

ε κ

⎧ ⎫ ⎧ ⎫ ⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥= +⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ ⎩ ⎭

⎧ ⎫⎪ ⎪⎪ ⎪

=⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

11 12 16

12 22 26

66 16 26 662 2

d dx x

d dy y

d dxy xy

D D D

D D D

D D D

ε κ

ε κ

ε κ

⎧ ⎫ ⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥+⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

and those corresponding to the nonlinear incremental strain components are

11 12 16

12 22 26

16 26 66 2

dd ddx x

dd ddy y

dd ddxy xy

n A A A

n A A A

A A An

ε

ε

ε

⎧ ⎫ ⎧ ⎫⎡ ⎤⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪

⎢ ⎥⎣ ⎦⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

(2)

(3)

(4)

(5)

(1)

(6)



APFIS 2007 467

Following the usual convention (nx, ny, nxy) are the stress resultants; (mx, my, mxy) the incremental momentresultants. Aij, Bij andDij (i, j =1, 2, 6) are respectively the membrane, bending-membrane coupling and bendingstiffness of a laminate for which the components adopt the Halpin-Tsai equation (Jones 1975 and Wang 2007).

For a circular cylindrical shell subject to uniform axial compression the fundamental state prior to bifurcationpoint can be taken as axi-symmetric membrane in the classical critical analysis. Hence the fundamental stress

resultants are( ) ( ), , , 0, 0E E E

x y xyn n n tσ = −

Here we assume that the effect of membrane-bending coupling stiffnessBij (i, j=1,2,6) can be neglected in thefundamental state, and that the extension-shear coupling stiffnessA16 and A26 equal zero.

Using constitutive relation Eq. 5 the corresponding fundamental strain components are

( ) 22 12

2 211 22 12 11 22 12

, , , , 0E E Ex y xy

A t A t

A A A A A A

σ σ ε ε ε

⎛ ⎞= −⎜ ⎟

− −⎝ ⎠

In the classical critical analysis the principle of stationary total potential energy gives a compact and systematicframework for interpreting buckling behaviour. Of present interest is the quadratic term of the total potentialenergy from which the condition of the stationarity results in the eigenvalue problem that yields the classicalcritical load spectra. The quadratic term of the total potential energy can be represented as

2 2 2 2 2x y

M B M MU U U V V= + + +

where U2B is the linear bending energy, U2M the linear membrane energy, and 2xMV the linearised membrane

component associated with axial direction, while 2yMV is associated with circumferential direction, that is

( )

( )

( )

( )

2

20 0

2

20 0

2

20 0

2

2 0 0

12

2

12

2

1

2

1

2

R Ld d d d d d

B x x y y xy xy

R Ld d d d d d

M x x y y xy xy

R Lx dd E E ddM x x x x

R Ly dd E

M y y

U m m m dxdy

U n n n dxdy

V n n dxdy

V n dxdy

π

π

π

π

κ κ κ

ε ε ε

ε ε

ε

= + +

= + +

= +

=

∫ ∫

∫ ∫

∫ ∫

∫ ∫

Hence the condition of the stationarity is expressed as follows, and solution of the eigenvalue problem gives

critical loadσ c as eigenvalue and buckling mode (u c, v c, w c) as eigenvector.

2 22 2 0

x yM M

M B c

V VU Uδ σ

σ σ

⎛ ⎞⎛ ⎞∂ ∂+ + + =⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠

Substitutingσ c and (u c, v c, w c) into Eq. 10, we can confirm that the total potential energy equal zero

2 22 2 0

x yM M

M B c

V VU U σ

σ σ

⎛ ⎞∂ ∂+ + + =⎜ ⎟

∂ ∂⎝ ⎠

Nonlinear Buckling Analysis

In the present nonlinear buckling analysis displacement functions are taken to be linear combination of harmonicexpressions

( ) ( )

( ) ( )

( ) ( )

,

,

,

cos cos

sin sin

cos sin

i ji j

i ji j

i ji j

iy R j x L

iy R j x L

iy R j x L

π

π

π

=

=

=

∑ ∑

∑ ∑

∑ ∑

u u

v v

w w

For convergence of the nonlinear post-buckling response, a total of 64 modes in Eq. 12 in which each degrees-of-freedom was 28, 17 and 19 for u , v and w , respectively, was adopted as completely the same in Eq. 30 of Yamada and Croll (1999). The initial geometric imperfection is taken to consist of a single harmonic

( ) ( )0 0, cos sinb f by R f x Lπ =w w

In which b and f represent the circumferential full-wave number and longitudinal half-wave number for theimperfection.

(7)

(8)

(9)

(10)

(11)

(12)

(13)



APFIS 2007 468

Strain-displacement relations are20 2

2

20 2

2

0 0 2

1,

2

1,

2

1,

2

x x

y y

xy xy

x x x x x

y R y y y y

y x y y x x y x y

ε κ

ε κ

ε κ

∂ ∂ ∂ ∂ ∂⎛ ⎞= + + = −⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂ ∂= − + + = −⎜ ⎟

∂ ∂ ∂ ∂ ∂⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + + + = −⎜ ⎟

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

u w w w w

v w w w w w

u v w w w w w w w

x

The sets of nonlinear algebraic equations are obtained through the stationarity of the total potential energy withrespect to each of the displacement degrees of freedom included in Eq. 14. Solution of these sets of nonlinearequations is achieved using a step-by-step process in which either load or suitable displacements is used as thecontrol parameter. At each step a Newton-Raphson iteration is used to provide convergence to an acceptablelevel of precision. A more complete description of the theoretical model is for the isotropic cylinders (Yamadaand Croll 1993), which lists the integration coefficients for all terms up to and including the quartic (fourth-power) energy terms.

Reduced Stiffness Analysis

As evidenced in Yamada and Croll (1999), the linearised axial energy component 2xMV is negative in compressed

cylindrical shell buckling problems; all the other energy components in Eq. 8 are positive definite, implying that

all but 2xMV contribute to the shell’s initial resistance to buckling. The reduced stiffness method is based on the

physical argument that mode coupling, catalyzed by geometric imperfections, results in the loss of initial

stabilising membrane energy. By eliminating U2M and 2yMV from Eq. 8 and applying the condition of stationarity

to the reduced quadratic form of the total potential energy we obtain the reduced stiffness critical load *

cσ .

* 22 0

xM

B c

VUδ σ

σ

⎛ ⎞⎛ ⎞∂+ =⎜ ⎟⎜ ⎟⎜ ⎟∂⎝ ⎠⎝ ⎠

As will be shown in the following example, each choice of axial half-wave j will result in a classical critical load

spectrum from Eq. 10 that exhibits a minimum critical load σ c at some value of i. This load is defined as σ cm,j.

The lowest of σ cm,j, occurring in a mode( ) ( ), ,cm cmi j i j= , is what is usually referred to as the classical criticalload, here denoted byσ cm. For each choice of j the reduced stiffness critical spectra of Eq. 15 will predict a value

of *cσ associated with the circumferential wave number i corresponding with the lowest classical critical load.

The lowest of these reduced stiffness critical loads *cmσ will occur in a mode ( ) ( )* *, ,cm cmi j i j= that could be

different to the classical critical mode ( ),cm cmi j . It is this least value *cmσ that has been shown to represent a

lower bound to imperfection sensitivity (Yamadaet al. 2006).

TYPICAL CASE STUDY

Referring to Wang and Croll (2007), a 6-ply symmetric glass-epoxy circular cylindrical shell and the materialproperties are adopted

72GPa , 0.22

3.5GPa , 0.34

f f

m m

E

E

μ

μ

= =

= =

whereE andμ are the Young’s modulus and the Poisson ratio of the composite materials, and subscript f andm denote fibre and matrix respectively. The fibre volume fraction is 0.5 for each ply and the geometries as follows(Yamada and Croll 1999)

/ 0.512

/ 405

L R

R t

=

=

Figure 2 shows the classical critical load, the nonlinear buckling load and the reduced stiffness critical load

analysis for several choices of θ . For a shell [θ ]6, which expresses the laminate configurations, “θ ” denotes the

angle of fibre orientation and “6” denotes the number of ply, and thickness of each ply is equal. For the shell

[0o ]6, the minimum critical load σ cm =18.9MPa and the lowest reduced stiffness critical load *cmσ =7.50MPa.

Accordingly the reduction is almost 60%. While in case of [ 45o ]6 the reduction indicates almost 80% because of

σ cm=20.6MPa and *cmσ =4.13MPa. This shows that the reduction from minimum critical load is very dependent

upon the angle of fibre orientation.

(14)

(15)

(16)

(17)



APFIS 2007 469

0 5 10 15 20 250

5

10

15

20

25

30

35

40σ [MPa]

i orb0 5 10 15 20 250

5

10

15

20

25

30

35

40σ [MPa]

i or b

0 5 10 15 20 250

5

10

15

20

25

30

35

40σ [MPa]

i or b

σ cm=18.9

*cmσ =7.50

icm

=17*

cmi =14.2

j=1

j=2

j=3

j=2

j=3

j=1

(a) [0o ]6 (b) [20o ]6

(c) [45o ]6 (d) [90o ]6

Figure 2. The classical critical load, reduced stiffness critical load and nonlinearload analyses for each angle of fibre orientation

5

10

15

20

25

30

35

40

00 10 3020 5040

σ [MPa]

Angle of fibre orientation(

o

)

70 9060 80

Figure 3. The lower bounds versus the varying angle of fibre orientation

j=1

j=2

j=3

j=4

j=1

j=2

j=3

j=4 j=5

j=5

j=6

j=7

σ cm=26.5

*cmσ =7.71

icm=18*cmi =15

j=1 j=2

j=3

j=1

j=2

j=3

σ cm=20.6

*cmσ =4.13

icm=0

*cmi =13.7

j=1

j=2

j=3

j=4

j=1

j=2 j=3

j=4

j=5

j=5σ cm=18.9

*cmσ =7.09

icm=17.3*

cmi =11.5

j=1 j=2

j=3

j=4

j=1

j=2 j=3 j=4

j=5

j=5

( )0,1 0.00N

b tσ =w

cσ

*cσ

( )0,1 0.02N

b tσ =w

( )0,1 0.05N

b tσ =w

( )0,1 0.10N

b tσ =w

( )0,1 0.20N

b tσ =w

( )0,1 0.40N

b tσ =w

( )0,1 0.60N

b tσ =w

( )0,1 0.80N

b tσ =w

( )0,1 1.00N

b tσ =w

( )0,1 1.20N

b tσ =w

( )0,1 1.40N

b tσ =w

( )0,1 1.60N

b tσ =w

( )0,1 1.80N

b tσ =w

( )0,1 2.00N

b tσ =w

( )c cm jσ

( )* *c cm jσ

j=6 j=7

( )0,1 0.00N

m b tσ =w

( )0,1 0.02N

m b tσ =w

( )0,1 0.05N

m b tσ =w

( )0,1 0.10N

m b tσ =w

( )0,1 0.20N

m b tσ =w

( )0,1 0.40N

m b tσ =w

( )0,1 0.60N

m b tσ =w

( )0,1 0.80N

m b tσ =w

( )0,1 1.00N

m b tσ =w

( )0,1 1.20N

m b tσ =w

( )0,1 1.40N

m b tσ =w

( )0,1 1.60N

m b tσ =w

( )0,1 1.80N

m b tσ =w

( )0,1 2.00N

m b tσ =w

Figure 3 shows the effects of changing the angle of fibre orientation. Points show the buckling loads Nmσ

obtained from the nonlinear analysis for the relevant initial imperfection. From Figures 2 and 3 the lower bounds

of the nonlinear analysis and the reduced stiffness method are equivalent because the lowest nonlinear analysis

load almost corresponds to *cmσ . From that point of view the reduced stiffness analysis exhibits well defined

lower bound.

0 5 10 15 20 250

5

10

15

20

25

30

35

40σ [MPa]

i orb

, , , and ,cm jσ

*

,cm jσ , , , and



APFIS 2007 470

In Figure 3 it is worth noticing that fibre orientation with high classical critical load such as [20o ]6 or [70o ]6 arerelatively low reduced stiffness load and nonlinear analysis load. This also indicates that the effects of initialimperfection are large. It is apparent that optimum design from classical critical analysis may be unfavourablefor the actual shell. In Figures 2 and 3, the fibres are in all plies in the same direction, while in Figure 4 fourtypes of lay-ups are adopted; Figure 4 shows the comparison of L/R=0.512 withL/R=2.048 adopted in Wang andCroll (2007) in which 6-ply symmetry was applied to all cases. The square bracket denotes the angle of fibre

orientation and the order of lamination from the outside to the middle plane of the shell, and “s” representssymmetric distribution about the middle plane. The results for the classical critical loads are consistent betweenthese two figures, despite the difference of L/R; this is a consequence of changes in the critical axial half-wavenumber ensuring that the wave length in axial direction remains effectively constant; The normalised axial

buckling wave length for [0 ,0 ,90o o o ]s was L/(Rj)=0.256 with j=2 for L/R=0.512, which is the same as with j=8

for L/R=2.048. Meanwhile it can be observed that the reduced stiffness loads for L/R=0.512 are somewhat higherthan those for L/R=2.048. For shorter cylindrical shells the effects of imperfection on buckling load carryingcapacities are reduced.

CONCLUSIONS

The present study has carried out three analyses that are classical critical analysis, nonlinear analysis and reducedstiffness analysis, for 6-ply symmetric, glass-epoxy cylindrical shells with parameter of angle of fibre orientation.As a result, the lower bound of the nonlinear analysis and the analytical prediction of the reduced stiffnessmethod show close correspondence with the reduced stiffness criteria providing close lower bound to theimperfection sensitive elastic buckling loads for imperfect FRP cylindrical shells. The lower bound critical loadsdue to imperfection vary remarkably with the angle of fibre orientation; the optimised angle based on thereduced stiffness analysis is shown to be very different to the optimisation by classical upper bound analysis.

REFERENCES

Jones, R. M. (1999).Mechanics of Composite Materials, 2nd Ed., Taylor & Francis.Wang, H. and Croll, J .G.A. (2007). “Buckling design optimization of fibre reinforced polymer shells using lower

bound post-buckling capacities”, 13th Int. Conf. on Exp. Mech., Alexandroupolis, Greece. Yamaki, N. (1984).Elastic Stability of Circular Cylindrical Shells, North-Holland. Yamada, S., and Croll, J .G. A. (1993). “Buckling and postbukling characteristics of pressrure loaded cylinders”,

J ournal of Applied Mechanics, ASME, 60, 290-299. Yamada, S., and Croll, J .G.A. (1999). “Contribution to understanding the behavior of axially compressed

cylinders, J ournal of Applied Mechanics, ASME, 66, 299-309. Yamada, S., Yamamoto, N., Croll, J.G.A. and Bounkhong, P. (2006). “Local buckling criteria of thin-walled

FRP circular cylinders under compression”, Int. Colloquium on Application of FRP to Bridges, JSCE,

Tokyo, Japan, 69-76.

Figure 4. The comparison of classical critical loads and reduced stiffness loads on length-radius ratio

cmσ

*cmσ

(a) [ , 0 , 0θ o o ]s

(b) [0 , ,0θ o o ]s

(c) [ 0 ,0 ,θ o o ]s

(d) [θ ]6

5

10

15

20

25

30

00 10 3020 5040

*,cm cmσ σ [MPa]

70 9060 80

5

10

15

20

25

30

00 10 3020 5040

*,cm cmσ σ [MPa]

70 9060 80

Angle of fibre orientation(o

) Angle of fibre orientation(o

)

(a) (b) (c) (d) (a) (b) (c) (d)

(a) (b) (c) (d)

(a) (b) (c) (d)

(a) L/R=0.512 (b) L/R=2.048

Buckling Cylinder Bueno

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