Bending response of thin-walled laminated composite cylinders

Composite Structures 2 2 (1992) 87-107

!~!,l

Bending response of thin-walled laminated composite cylinders H. P. Fuchs & M. W. Hyer Department of Engineering Science & Mechanics, Virginia Polytechnic Institute & State University, Blacksburg, Virginia 24061-0219, USA A theory for computing the linear response of thin, symmetrically laminated circular cylinders subject to bending by end rotations is presented. The Donnell theory is used to compute the displacements and intralaminar stresses. Integration of the three-dimensional equilibrium equations of elasticity is used to compute the interlaminar stresses. Three laminates with stacking sequences [ _+30/0z]2s , [ + 45/0212s and [ _ 45/0/9012~, and an aluminum cylinder are used to illustrate the displacement response, intralaminar, and interlaminar stresses for two length-to-radius ratios, L/R, of 1 and 5. The radius-to-wall-thickness ratio, R/H, is taken to be 50. Also studied is the boundary layer near the ends of the cylinder, which is assumed to be clamped. By using an analogy to a cylinder subjected to a compressive loading, the boundary layer is studied in detail for the [ + 0/0212sfamily of laminates, where 0 ranges from 0 ° to 90 °. The results of the study show that if only fiber breakage causes ultimate failure, material failure will occur on the compression side of the cylinders, at the ends, due to fiber compression for the three laminates under consideration. The three interlaminar stress components are small compared to the intralaminar stress components, yet it is seen that delamination may be initiated before ultimate failure due to low interlaminar shear strength. The interlaminar normal stress, 03, is an order of magnitude smaller than the interlaminar shear stresses and it is not likely to initiate failure. Also, the study concludes that laminate material properties influence the length of the boundary layer for the [_+ 0/0212~ family of laminates, but the material properties do not influence boundary layer strength. The amplitudes of the boundary layer on the compression side of the cylinder, taken here as a measure of the strength of the boundary layer, are only 4% greater than the radial displacement away from the ends of the cylinder, independent of the ply orientation, 0.

1

INTRODUCTION

experimental study of the stability of the more complicated case of bending of stiffened and unstiffened isotropic cylinders (see for example Refs 2-11) while relatively little work has been done in regard to the stability of sandwich and composite cylinders in bending) 2-2° Even fewer studies have examined the stable bending prebuckling state. The ones that have generally consider a three-dimensional elasticity approach (see for example Refs 21 & 22) or a thin-walled beam type of solution,23 both of which neglect any boundary effects. With bending, in addition to the stability issues associated with the compression side of the cylinder, there is also the possibility of material failure on the compression and tension sides of the cylinder. As with axisymmetric loadings, large displacement gradients can occur near the

Composite cylindrical structures are known for their efficiency. The curvature effectively increases the stiffness of the structure, resulting in a more economical use of material. In addition, cylindrical structures are well suited for fabrication by automated fiber-placement techniques such as filament winding and tow placement. Since the advent of advanced fiber-reinforced composite materials, researchers have studied, experimentally, numerically, and analytically, the response of cylinders to axial loads, torsion, external pressure, internal pressure, and combinations of these. The vast majority of these efforts have focused on the stability issues associated with axisymmetric loadings, see e.g. Ref. 1. Much attention has been focused on the analytical and 87

Composite Structures 0263-8223/92/S05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain

88

H. P. Fuchs, M. W. Hyer

boundaries of a cylinder, be they clamped, simplysupported, or subjected to some other support condition. In bending, there are also gradients in the circumferential direction in addition to those gradients in the axial direction. The intralaminar stresses associated with these displacement gradients, coupled with the brittle nature of most composite material systems, can be severe enough to cause material failure within these boundary regions, ff the intralaminar stresses do not cause failure directly, they may cause it indirectly. The gradients in the intralaminar stresses induce interlaminar stresses. Since laminated materials have low interlaminar strength, the interlaminar stresses could cause failure. These issues may be a function of the geometry and wall construction of the cylinder. The primary goal of this paper is to evaluate the influence of geometry and ply orientation on the overall prebuckling response of cylinders, paying particular attention to the response in the boundary layer region. Here, response is measured in terms of displacements and stresses. The study considers only the geometrically linear case. The paper begins by briefly outlining the specific problem by presenting the geometry, governing differential equations, and boundary conditions that describe the cylinder response. The solution of the governing equations is obtained and expressions for displacements and intralaminar stresses are derived. The interlaminar stresses are then computed using the three-dimensional equations of elasticity in polar form. The procedure for obtaining the solution to these three-dimensional equations is described. Numerical results for the displacement response, for the traditionally computed intralaminar stresses, and for the not-so-traditionally computed interlaminar stresses are presented and discussed. Some of the similarities between the bending problem and the problem of axial compression are discussed. The axial compression problem is then used to provide valuable insight into the bending problem. Concluding comments complete the paper.

mechanical system is zero. Mathematically, this is stated as:

av=au+awo=o (1) where 6U and 5W e are the first variation of the internal energy and the first variation of the potential energy of external forces, respectively. For the cylinder of Fig. 1 these are defined as:

6U=jx=-L/2 J0=0

Sz=-HI2

(

+ oo6 o

(2)

+ Zxo6Yxo) dzR dO dx and

6W~= -I,s (Tx6u°+ T°bv°+ Tz6w°) dS

(3)

where Tx, To, and Tz are known applied tractions acting on the bounding surfaces of the cylinder, S. For example, if the only external traction force considered is internal pressure, then Tz- p . Body forces are neglected. In the above, use has been made of both the Kirchhoff hypothesis and the assumption that the cylinder wall is thin. The equilibrium equations and boundary conditions are derived using Hooke's law and the polar strain-displacement relationship in the above equations. 2.1

Kinematics and Hooke's law

For thin shells, the displacement field is assumed to be given by:

u(x,O,z)= u°(x,O) + Z x(x,O) o v(x,O,z)=v°(x,O) + Z o(x,O) w(x,O,z)= w°(x,O)

Z

(4a) (4b) (4c)

Z

z,W°

y ,V*

--y

2 DERIVATION OF GOVERNING EQUATIONS

The governing equations for laminated thinwalled cylinders are derived from the principle of virtual work. The principle states that the first variation of the total potential energy of a

Fig. 1. Cylinder geometry and nomenclature.

Bending responseof thin laminatedcomposite cylinders

The boundary conditions at the ends of the cylinder, x = + L/2 are:

where the rotations are defined as: 0w °

fl;--- - - -

Ox

; fl~-

0w °

(4d)

RO0

The total axial, circumferential, and radial displacements are given by u, v, and w. The superscript '°' signifies midsurface quantities. Note that z/R has been neglected compared to unity in the volume integral of eqn (2) and in the expression for circumferential displacement, eqn (4b), due to the thin shell assumption. Substituting the assumed displacement field into the linear polar strain-displacement relationship results in the following definitions of the midsurface strains and curvatures:

Ou°

~ = ax

"c~=

Ov ° + - -w ° e°=RO0 R o YxO =

au °

RO0

+

o _O,ao ROO

(5)

,Co =

a,., °

o

Ox

"cxO =

Nx = specified or u ° = specified

(8a)

Nxo= specified or v ° = specified

(8b)

Qx-

OOgx

gxo0 = specified or w o specified +2O R0

a/~

RO0

a~o

Mx = specified or fix = specified

0,~

022

016 026 ;,(k)

(8d)

For symmetric laminates:

jAil z,2

(9)

and

o61

Ox

Mxo

=

(8c)

026| (~66J

o

(10)

Mo = D12 D22 D26]

+ --

which are the midsurface kinematics of the Donnell theory. 24 Hooke's law for laminated materials in a state of plane stress is given by:

o0f rxOJ

=

No -- A,2 An A261$eo~ Nxo A16 A26 A66J [ • x O J

OOflx

o

89

(6) /,o+z'cS~ [Txo+Z'cxOJ

where the ~dij (i,j = 1,2,6) are the reduced stiffnesses of the kth ply.

D16

D26

D66_]

K xOJ

For the particular case of bending by end rotations for a cylinder with clamped supports, the boundary conditions at x -- +_L/2 are specified to be: u °= T-RQ cos 0

(lla)

v°=0

(llb)

w°=0

(llc)

fix = -T-f2 cos 0

(lld)

2.2 Equilibriumequations and boundary conditions

where Q is the applied end rotation angle in radians, as seen in Fig. 2.

Substituting the relations (5) and (6) into eqn (2) along with eqn (3), and finally into eqn (1), integrating through the thickness of the laminate, applying the standard definitions of the stress resultants, and integrating by parts, results in the linear equilibrium equations for thin-walled cylinders. These equations are given by:

2.3 Governingequations for bending

-ONx -+

0x

In this study, it will be assumed that variables are separable and that the primary response variables

Z

ONxo R00 =0

ONxo ONo --+ =0 Ox RO0 OQx.~ 02go No= 0 Ox R2002 R

(7)

I i i

i

i

I

X,x,u °

Fig. 2. Applicationof end rotationf~.

90

H. P. Fuchs, M. W. Hyer

of the problem, u °, v o, w o, to, Nx ' Nxo, Qx, and Mx, are given by products of functions of x and 0. Due to the 0-dependence of the bending boundary conditions, eqn ( 11 ), and the physical requirement that the response be periodic in 0, it will be assumed that the primary response variables vary with sin 0 and cos 0. With this assumption it will prove convenient to write the governing equations as a set of coupled first-order partial differential equations, writing the derivatives with respect to x on the left side of the equations. This form is achieved by algebraic manipulation of the equilibrium equations (7), the constitutive relations (9) and (10), the definitions of fl_~and Qx, (4d) and (8c), and the strain-displacement relations (5). For symmetric laminates, the first-order form becomes: 0 tit'~ -Cl

Ox

0U -~

Ox

-

-

OV ° ~

RO0 +

w° C2--+C~N,

R

" "

+C4N~o

(12a)

0//° OU ° W° C5 - - ~-C6 -~-o+ C7 - ~ + CsN,, + CgN, o

RO0

(12b) OW °

ax

OtiS.

o

C,,,flx

(12c)

a2w °

afl~

Ox - C ~ R200----------5+C,2RO----~+C,3M~

ONr

--= Ox

aNxo

aNxo

C14 - Ri)O

(12d) (12e)

a2v °

aw °

aNx

O---~ = C,5 R200-----~+ c,~ R2a---~ + C,7 RO---O 0N.

12f)

+ C~,~ RO0 aQx av ~' 0-7 = c , . R a---0+

w°

a4w ° + c2, R 4a o 4

a3/3;

q- C22 R300-------~+ C23Nx + C24Nro 02 Mx

(12g)

+ C25 R 2 0 0 2

OM~

a-~w °

a~/3]

0~--~--- C26 -'----'~ R3 ()O + C27 R2i)O 2

0Mx

+ C29 RBO

~ C~Q~

(12h)

where C~ through C29 are constants which

depend on laminate stiffnesses and cylinder radius (see Appendix 1). 2.4 Comments on the validity of the Donnell equations The Donnell equations are the most basic of many available circular cylindrical shell equilibrium equations and also possibly the most controversial. The question of accuracy of these equations arises naturally due to their simplicity and approximate nature. There have been a number of comparisons between various geometrically linear shell theories in the literature for isotropic cylinders (e.g. Ref. 25). Theories incorporating Donnell, 24 Sanders, 26 and Fliigge~-7 kinematics appear to be the most popular ones, with Fliigge's equations being perceived as the most accurate. Reference 25 assessed the range of validity of the Donnell equations versus the Fliigge equations, among others, by comparing the characteristic roots of the governing equations for isotropic cylinders. The characteristic roots were computed for a wide range of geometric parameters for both Donnell and Fliigge for the case of line loads applied along the circumference (e.g. bending of a cylinder). It was seen that the Donnell theory gives good results compared to Fliigge for an extremely large range of thickness ratios with no restriction on the cylinder length. It was concluded that the shell theories under consideration are in good agreement for the case of axisymmetric loadings. However, it was suggested that some inaccuracy may occur for the present loading case of bending for which the circumferential wave number is unity. Additionally, for the present study, it might be expected that any computed strains and stresses using the Donnell theory may differ slightly from those computed by using Fliigge strain-displacement relationships. It should be kept in mind that these conclusions apply to isotropic cylinders. The range of validity of the Donnell equations has yet to be explored for composite cylinders. However, it is felt by the authors that any error due to the discrepancies between the Donnell and Fliigge theories will not significantly alter the cylinder bending responses presented or any conclusions made in this paper. 3 SOLUTION OF GOVERNING EQUATIONS As indicated in the Section 2.3, the separable solution is then assumed to be of the form:

Bending response of thin laminated composite cylinders {y(x,O)}={y¢(x)} cos O+{y,(x)} sin 0

(13)

where

{y(x, 0)} r= {u°(x, O),v°(x, 0), w°(x, O),fl°~(x,0), Nx(X,O),Nxo(X,O),Qx(X,O),Mx(X,O)} (14) The sin 0 and cos 0 components of {y(x,O)}, namely {y¢(x)} and {ys(x)}, are functions of x to be determined. Substituting this assumed solution form into first-order form of the governing equations (12) results in a system of 16 ordinary differential equations in the sin 0 and cos 0 components of the variables. This system of 16 equa ~ tions can be written in matrix notation as:

91

3.1 Symmetrically laminated cylinder bending response Having now solved for the displacement responses, u ,o v o, w o, flox, and the stress resultant variables, Nx, Nxo, Qx, and Mx, in the functional form of eqns (13) and (14), the complete response of the cylinder, in the sense of classical lamination theory, is known. The intralaminar strains, curvatures, and stresses can be written in a common functional form. Writing the midsurface strains and curvatures of eqns (5) in vector notation results in: 16

{e°(x,O)} = Z ({E~.}cos 0+{ E js} sin 0) e~x

(20)

j=l

where [F] depends on the laminate stiffnesses and the cylinder radius, and

and 16

{r°(x,0)} = Z ({K{~}cos 0+{K~} sin O)e six

{33(x)}T= {u~(x),vs(x), w¢(x),flc(x),Nx~(X),N~os(X),

(21)

]=1

Q~,.(x),Mx~(X),G(x),vc(x), w~(x),fl~(x), Nx~(x),Nxo~(X), Qx~(x),Mx~(X)}

(16)

The solution of this coupled set of first-order ordinary differential equations is of the form: 16

{)3(x)} = E cj{ Uj} e ~'X

(17)

j=l

{O(x,O,z)}IkI={oM(x,O)}(kI+ z{oB(x,O)} lkl

where the {UJ} and sj are the complex-valued eigenvectors and eigenvalues of the problem, respectively. The Cyare complex-valued constants to be determined from the application of the boundary conditions. For the special case of axisymmetric loadings, such as compression, tension, torsion, and internal or external pressure, the assumed solution is independent of 0, namely:

{y(x,O)} = {y0(x)}

The strain components {E j,.s} and {K j,.~} depend only on the known values of cj, {UJ}, and sj (see Appendix 2). The intralaminar stresses in the kth ply may be written in terms of a membrane component and a bending component, denoted by a superscript 'M' and 'B', respectively, as: (22)

where the membrane and bending stress components are: 16

{oM(x,O)}Ikl= ~, ({M{.}/k/cos 0+{M~} tk/sin 0)e ~jx j=l

(23) and

(18) 16

This case requires the solution of eight simultaneous first-order ordinary differential equations. These have a solution which can be written as:

{oB(x,O)} Ik)= Z ({B{}/k/cos 0+{B~}/k/sin 0)e s'x j=l

(24)

4

{y0(x)}= ~, cj{Uq eS'X+ c5{U5}+ c6{U6}x j=l

+ c7{UT}x z + c,{US}x 3

(19)

Solutions to problems with combined loading can be obtained by superposing solutions (13) and

(18).

The {M{;s}and {B{.s}stress components depend on the ply properties, ()(k) ~ i j , and the strain comp o n e n t s , { E j,.s}, and {KJ,.~} (see Appendix 2). Stresses in the principal material coordinate system are simply obtained by applying the stress transformation equations. The force and moment resultants are computed via eqns (9) and (10).

H. P. Fuchs, M. W. Hyer

92

4 INTERLAMINAR STRESS COMPUTATIONS

(k)

•r r x ( X , O , r )

=-

1

r

C(lk)(x,O).-~ - F(lk)(x,O)--[--~ r ,,

F~kl(x,O)

2

Although the interlammar stresses are neglected compared to the intralaminar stresses in the framework of classical lamination theory, the interlaminar stresses may be calculated once the intralaminar stresses are known by integrating the three-dimensional polar stress equilibrium equations. The equilibrium equations are:

(25a)

~-O'l?x'O-[-O'l[rx"l- "[r'~= O Ox rO0 Or r

OOx

r F(k)Ix O) "[--3\,

(27a)

3

r~'(x,O,r)= 4 c(2k)(x,O)+ 2 F{4k)(x,O)+r F(~'(x,0) r 3 " r

2

+-- F(rkl(x,O)

(27b)

4

o~l(x,O,r)

1 CI~)(x,O)+4 r

r

F(7k)(x,O)+F(~)(x,O) 2

OrxO+ 0o0+ 0rr0+ 2 rr---°=0 Ox rO0 Or r

+

(25b)

r F~k)(x,O)+ -g r Ft~(x,O ) r

3

+ - - F((,I(x,O) 0rr~ ~-0rr0 + 0O% Or-- O0= 0

OX

rO0

Or

r

The intralaminar stresses are known explicitly as functions of x, 0, and r. Rearranging eqns (25) and writing the intralaminar stresses in terms of r, where z = r - R , for purposes of integration with respect to r, we obtain:

r ~ r + rrx= - r

12

(25C)

~O(rxM +(r-R)r~o)

+O~x(cr~+(r-R)o~)

(26a)

(27c)

where the Flik)(x,O), i=1,...,11, are known functions of the intralaminar stresses and their gradients in both the x and 0 directions (see Appendix 3). The c~k)(x,O), j = 1 , 2 , 3 , are functions of integration to be determined by the application of interlaminar stress boundary conditions. A similar procedure is followed for the axisymmetric loading of a cylinder. The results in the axisymmetric case are considerably simplified due to the absence of terms which involve partial derivatives with respect to 0. Interlaminar stresses may be referred to the principal material coordinate system by applying a simple stress transformation.

r

Orr__yo [ 0 M a r Or+2rro=-r[-~-o(Oo+(r-R)oo)

0 M

B]

+--(rxo+(r-R)vxO) Ox

4.1

Evaluation of functions of integration, C~k)(x, O),j = 1,2,3

(26b)

The boundary conditions for the interlaminar stresses in the case of bending are:

(i) Stress-free surface at inner radius, r=r i = r

+or = - r

R-(H/2),k=I:

Vro+(r-R)l:rO)

r0x)= 0 m

O

+ Ox

M

r00~= 0

B

(28)

(Vrx+(r_R).t.rx)

_lr (°~ + ( r - R)a~)]

OIr~) = 0 (26c)

(ii) Continuity of interlaminar stresses at kth interface, r = rk, k # 1: T(krx) = Zrx-(k- 1)

Equations (26) may now be integrated to obtain the following expressions for the interlaminar stresses in the kth ply of the laminate:

(k-l) T(rk)o = TrO O(k) r ~ Or(k- 1)

(29)

Bending response of thin laminated composite cylinders 3

Substitution of eqns (27) into boundary condi" for Cj(1)(X , 0): tions (28) results in expressions

93

(F~%-I)(x,O)-- ~lk)o(X,O))

4

2

C?)(x,O)= - riF?)(x,O)+ 2 Ft])(x,O)

+rk(F~-I)(x,O)--Pll,--(k){x,O))

]

4

+ 3 Fl~)(x' O)

(30a)

(31C)

Equations (30) and (31) simplify greatly for axisymmetric problems.

c~'~(x,O>=- ¢2(x,O1+3 ¢,'(x,O)

4

5

]

+ r, F<6,)(x,O)

(30b)

4

C(31)(x,O)= -[~ F(71)(x,O)+r,~l)(x,O) 1

2

3

ri F<91)(x,o)+ri F]l)o(x,O)

4

r i

]

+--~ F(ll((x,O)

(30c)

while substitution of eqns (27) into continuity conditions (29) results in the expressions for c~kI(x, 0), k # 1, namely:

C?)(x,O) = C]k-')(x,O) - rk(F(kl-l)(x,O) - F?)(x,O) 2

+ 2 (F<2k-ll(X'O)-- F~2kl(x'O))

+r~ (F~- ~)(x,O>- e~%, 0)> 3

c~%,o)= c?-~>(x,O) -

(31a)

(1~4~-'~(x,o)

5.1

Material parameters and cylinder geometry

Table 1 gives the material properties which were used to study bending responses. The subscripts '1' and '2' refer to the principal material directions of a layer. A layer thickness is taken to be h -- 0"006 in (0.152 ram). Two cylinder geometries are considered here. The radius-to-thickness ratio is taken as R/H= 50, while length-to-radius ratios of L/R = 1 and 5 are used to represent 'short' and 'long' cylinders, respectively.

5.2

Displacement response

Bending responses are computed for a [_+30/ 0212s , a [ _+45/0212s , and a [ _+45/0/90]2 s laminated cylinder, as well as for an aluminum cylinder. For purposes of discussion, the end rotation Q (see Fig. 2) is chosen such that the overall applied strain at the top of the cylinder is 100/~s compression. That is: AL 2Rf2 . . . . L L

ea-

Q = 50 × 10 -6 rad for L/R = 1

- F?)(x,O))+ 3 (F~k-'l(x,O) - FIk)(x,O))

]

+ 4 (F<6~-l)(x'O)- F~6k>(x'O))

and

(31b)

(/L"(:- i)(x' 0)-- ~k)(x' 0))

(33b)

Since the analysis is linear, the results may simply be scaled to obtain the response for any end rotation, Q, as long as the deformations remain in the

Table 1. Material properties

+ rk(F~-'(x,O) - F~%,O)) +2

(33a)

if] = 250 × 10 -6 rad for L/R = 5

c~%,o>= c~-~(x,O)- ; W?-'(x,O)- F?>(x,O)>

2

(32)

1 0 0 X 10 -6

The end rotations are then:

3

4

RESULTS

Material

E 1 (Msi)

E 2 (Msi)

G12 (Msi)

'k'12

Graphite-epoxy Aluminum

20.0 10.0

1.30 10.0

1.03 3.846

0.30 0.30

94

H. P. Fuchs, M. W. Hyer

realm of linear elasticity. For example, if it were desired to compute the radial displacements for an applied maximum compressive strain of e, = 4000 ~s, one would only need to multiply the given results by a factor of 40. The normalized radial displacement response w°/H, is plotted as a function of the normalized axial coordinate, x/L, for the three laminates and aluminum in Fig. 3 for L/R = 5 and Fig. 4 for L/R = 1. The circumferential location is taken as

0 0 °. This is the top of the cylinder, the location where the compressive strains are maximum. Two entirely different responses are observed for the two cylinder lengths. The long cylinders of Fig. 3 show the expected overall parabolic deflection shape with downward deflection as a function of the axial coordinate. The aluminum cylinder and quasi-isotropic laminate show nearly the same response, each exhibiting only small changes in curvature in the boundary layer region. The =

0.005

I ^=umiuum l~.4.5.~.~..]=

0.0025 i 0

o ooo,,

,

/.-"-..

-0.0025

-.

I

/,'/

/" ~- I o=2A x'-'\ I .oooo,l V ", \1 oo~I \ ', ~I

\',,".., \ "~ ~.."oooo.I .*o.5

~

-o,~

'.

.o,6

-o,~

" ~ "., "~..~.

-0.01

..'." 7

,","/ ," , ' / ,-,,/ / ,'/

°[

\ ' , ",,. \ ' , ",,. X","..,

-0.005 -0.0075

.(.,4.5.~j.= "(-*"~-'~']-=I

.o.,2

"-d .o4

./"• , , ' // " "

/

~...~ ,

'

~

-0.0125 I I I I -0.015 * i i * -0.5 -0.4 -0.3 -0.2 -0.1

I

*

I

*

I

i

I

*

I

I

0 0.1 0.2 0.3 0.4 x/L Radial displacement response at top of cylinder (0 = 0 °) for L/R = 5.

Fig. 3.

I

0.5

0.005 0.0025 0 -0.0025 -0.005 -0.000,

-0.0075

-00006

-o.o~

-0.01 -0.0125

l

Aluminum

-0.015 -0.5

I

I

-0.4

-0.3

Fig. 4.

-o~ .o.~ .o~ -o.~ -o.,

[.-I-.4..5.~.../~.!.= I

-0.2

,

I

-0.1

,

!::E45/~]~ I

0 x/L

,

I

0.1

.[=E3..O_~.]= ,

I

0.2

,

] I

0.3

,

I

0.4

Radial displacement response at top of cylinder ( 0 = 0 °) for L/R = 1.

0.5

Bending response of thin laminated composite cylinders [+ 30/0212s laminate, in contrast, responds with much larger changes in curvature in the boundary region, and it deflects significantly less at midspan. The response of the [+ 45/0212s lies in between these two responses. Recall that the boundary conditions being applied are kinematic, and they are the same for all cases. Although the moment required to produce a given level of end rotation varies from one laminate to the next, one would not necessarily expect the midspan deflection to be so different for each laminate. One might expect the same kinematic conditions at the ends to result in the same deflections, independent of the laminate. As seen in Fig. 3 this is not the case. The short cylinders of Fig. 4 exhibit an outward barreling along the entire length, similar to what one would expect for the case of axial compression. There is little evidence of the overall parabolic shape associated with bending. The boundary layer region near the ends does not appear to be nearly as severe as those in Fig. 3. Of interest here is that the [+30/0212~ laminate barrels out significantly more than the other laminates and the aluminum cylinder. The insets in Figs 3 and 4 show the details of the boundary layer region. By the periodic nature of the solution, the deflection at the bottom of the cylinder (0 = 180 °) is identical to the deflection at the top (0 = 0 °) except for the sign. Where the short cylinder barrels outward on the top or compression side, it deflects inward on the bottom or tension side. On the sides of the cylinder (0-- + 90°), the circumferential displacement, v °, exhibits the parabolic response of a strength of materials solution. The radial displacements are minimal and there is no boundary layer. The axial displacement response, u °, is linear along the length for these particular cylinders. For the special case of orthotropic laminates, i.e. A ~6= A 26 = D~6 = D 26 = Bij = 0 (i,j= 1,2,6), not discussed here, it is observed that u ° and w ° vary with cos 0, while v ° varies with sin 0.

5.3

Intralaminar and interlaminar stresses

All six stress components are plotted as functions of the thickness location in Figs 5-8 for L/R = 5 for each cylinder construction. The stresses on the top of the cylinder (compression side) are illustrated. Stresses at the bottom of the cylinder (tension side) are obtained by changing the sign. The stresses are plotted at cylinder midspan

95

(x/L = 0) and at the end (x/L =½) in each case. Since the stresses are virtually identical at these two axial locations for the cases of L/R = 5 and L/R = 1, the results for L/R = 1 are not presented. The stresses in the global cylindrical coordinate system are shown for the aluminum cylinder in Figs 5(a-c). It is seen in Fig. 5(a) that the axial stress component, ox, is compressive through the thickness both at midspan and at the end. A sizable gradient is present at the end, while the stresses are nearly constant with thickness at midspan. Figure 5(b) shows the circumferential stress, o0. This stress component is all compressive at the end due to the presence of o x and Poisson's effect, and nearly zero at midspan. Due to the absence of the reduced stiffnesses 016 and 026 in aluminum, there is no intralaminar shearing stress, rx0, as indicated in Fig. 5(c). The out-of-plane stresses in the aluminum cylinder are induced by the presence of gradients in the intralaminar stresses, as can be seen by eqns (27). These stress components are shown in Figs 5(d-f). The transverse shear stress, r,~, exhibits the classical parabolic shape at the cylinder boundary, as seen also in the bending of isotropic slender beams and thin plates. Its magnitude is much smaller than that of the intralaminar stresses, as expected, and it is zero at midspan. The transverse shearing stress fro is identically zero everywhere due again to the absence of the 016 and 026 reduced stiffness terms. The 'interlaminar' normal stress, or, is an order of magnitude smaller than the transverse shear stress and displays the cubic r-dependency. All 'interlaminar' stress components are zero at z/H= + ½, as required by the boundary conditions, eqns (28). Figures 6-8 show all six stress components referred to the principal material system for each of the laminated cylinders. Figure 6(a) shows that tr~ is piecewise linear and compressive through the thickness in the [ + 30/0212s laminate. A rather significant gradient is present at the end, while only a slight gradient is present at midcylinder. A o2 stress is generated by the axial stress component and Poisson's ratio. It is tensile at midspan, while entirely compressive at the end. Recall that 02 would be entirely tensile at the end on the bottom of the cylinder. Once again it is seen that significant through-the-thickness gradients are induced at the boundaries. The intralaminar sheafing stress, r12, is generated by the presence of 016 and 026 and is of the same magnitude as o2, its sign being a function of the ply angles. Gradients are again present at the boundary.

11. P. Fuchs, M. W. Hyer

96

0.500

0.500

0.375

0.375 f

~250

0.250 f h

0.125 "I"~N 0.000

oooot

\

-0.125 -0.250 -0.375 -0.500 -2000

11

"t- 0.1 25 t

I

-1500

-I 000

'LI

-0.1 25 f -0.250 t

\

-0.375 I -0.500 ;00

-500

- 400

-500

(b)

(a) O'x, psi

-1N

0.500

0.500

0.375

0.375

0.250

0.250

0.125

0.1 25 -r" "~N

0.000

--0.125

-0.250

-~250

-0.575

-0.375

o15

0.0

-80

0.500

0.375

0.375

0.250

0.250

0.1 25

0.125 -r N

"I~'N 0.000

--0.250

-0.250

-0.375

-0.575 -

(e) "rre,psi Fig. 5.

oil, psi

-50

-40

-30

-20

-1 0

0

1o

i ,

L

;

f

0.000

-0.125

--0.12.5

o15

~00

(d) "rn~,psi

0.500

0.0

'i

-0.500

1.0

(c) "r,e, psi

-0.500

0

0.000

-0.125

-0.500

I -I 00

- 200

-0.500

-0.5

=/k-ao =/k--e.s 1

i

I

/

111II 0.0

0.5

(f) O'r,psi

Stresses for aluminum cylinder, L/R = 5, 0 = 0 ° (top of cylinder).

1.0

1.5

Bending response of thin laminated composite cylinders

O.5OO

O.5OO

0.375

0.375

0.250

0.25O

0.1 25

0.125

"I~N O.000

-r" "~N 0.000

97

I I I

I

I

r _

-0.125

I

I i

I

-0.125 i I I

-0.250

-0.250

I I

-0.375

-0.375 -0.500 -3000

iI

i

I

-2000

-1500

-1000

i

-2500

I . . . . . . I i

-0.500 -100

-500

I

I

-50

0.500

0.500

"-23

0.375

0.250

0.250

o12,

0.125

-r "~N 0.000 --0.125

100

(b) o2, psi

( a ) 0"1, p s i

0.5,5

i

50

(

0.000

~:?:---_?--~

-0.125

--0.250

-0.250

-o3,5

-O.375

-0.500

-200 -150 -I00

-50

0

50

100

150

-0.500 -80

200

I

I

-50

-40

(c) "rl 2, psi

-30

-20

-1 0

(d) "rl 3, ps[

0.500

0.500 ;

.

,

,

i

I 1.0

0.375 0.250

0.250

0.125

0.125

"IN

o.ooo

N

O.000

-0.1 25

-O.1 25

-0.250

-0.250

-0.375

-0.375

-0.500 -30

~/

:Z

-20

-1 0

I0

(e) "r23, psi

Fig. 6.

20

-0.500 -0.5

30

[

,/L_--~O ./,--o.s

Stresses for [ _+30//02] 2s cylinder,

L/R = 5,

i / \ \

0.0

05 (f) 0"3, psi

0 = 00 (top of cylinder).

1.5

H. P. Fuchs, M. W. Hyer

98

0.500

0.500 I

+

0.375

0.250

_

\

_1

0,375

I

\

\

0.250

0,1 25

I N

\

0.I 25

t

. . . . . . . . . q

J

+

L. . . . . . . . .

"r" 0.000

0.000

-0.125

-0.125

-0.250

-0.250

-0.375

-0.375

-0.500 - 3000

i

i

I

I

-2500

-2000

-1500

-1000

,\

\

-0.500 - I 50

-500

i

•

r,

--.4

. . . . . .

1

-50

50

0

(b) o'2, psi 0.500

0.500

0.375

I i

I

-1 O0

(a) o'1, psi

;5

0.575

0.250

T

J

0.250

,I

_

0.125

0.1 25

_

-e-

"1N

.....

0.000

-0.1 25

0.000

-0.1 25

;\\\\\\T---

-0.250

-0.250

-0..]75

-0.375

-0.500 -200

-0.500 -150

-I00

50

-50

(C)

7"1 2,

I O0

1 50

200

-60

i

i

i

i

-50

-40

-50

-20

0

(d) T13, psi

pSi 0.500

0.500

7-

0..]75

-1 0

•

i

i

i

0.5

1,0

1.5

0..]75

0.250

0.250

0.125

0.1 25

iI

"t-

"r-

/ Ill

0.000

0.000

-0.1 25

-0.1 25

-0.250

-0.250

x

)

-0.375

-0.375 P -0.500 -40

-30

-20

-10

10

(e) "r2.~, psi Fig. 7.

20

30

40

-0.500 -0.5

,/L_=_~o ,/t--o.n

0.0

(f) O'Z, psi

Stresses for [ + 45/0212+ cylinder, L/R = 5, 0 = 0 ° (top of cylinder).

2.0

Bending response of thin laminated composite cylinders

99

0.5O0

0.500

I I

\.

0.375

0.375

t

. . . .

I

. . . . .

i

I

0.250

0.250

!

I I __1

0.1 25

0.1 25

-IN

2: '~N 0.000

0.000 I_

-0.1 25

-0.1 25

-0.250

-0.250

.....

-0.375

-0.375 -0.500 -3000 -2500 -2000 -1500 -1000

( O ) 0"1,

2::

~2-V-

\ I

I

-500

0

-0.500 -200

5OO

- I 50

- I O0

-50

0

(b) 0.2, psi

psi

0.500

0.500

0.375

0.375

0.250

0.250

0.1 25

0.125 Z

0.000

N

J

/ ,/

I

/

25

h

0.000 -0.1 25

__/:\:\::

-0.250

\

-0.250

-0.375

4

1 1,

-0.375

-0.500 -150

-1 00

-50

0

50

100

-0.50

150

-40

0.500 " I ' r ' I ~ I 0.375

1

L

I

-30

-20

(c) "r| 2, psi

-I 0

0

(d) "rl 3, psi

~

0.500

/tI

0.375

0.250 0.1 25

I , 4

hl

-0.1

50

0.250 [

0.1 23

"I"

'I-

0.000

0.000

N

N r-

-0.1 25

I

-0.1 25

-0.250

I

-0.250

-0.375

-0.375 V

-0.500 ~ -30

-20

-10

I0

(e) Fig. 8.

"r2a, psi

20

30

-0.500 -1 d

40

./L~O ./L---O5

Stresses for [ _+45/0/9012s cylinder,

L/R -- 5,

i

-0.5

ols

0.0

(f)

~ , psi

0 = 0 ° (top of cylinder).

1.0

115

2.0

1O0

H.P. Fuchs, M. W. Hyer

Figure 6(d) shows that r13 somewhat resembles the classical parabolic shape and is significantly smaller than the intralaminar stresses at the end, while its magnitude is identically zero at midcylinder. Since the (~16 and (~26 reduced stiffnesses are non-zero for this laminate, we have an interlaminar shear stress, r23, at the ends. Its magnitude is of the same order as r13. The peak values occur at the interfaces between the innermost + 30 ° and - 3 0 ° plies and it is zero at the center of the cylinder. Note that although r23 appears to be piecewise linear, it is actually piecewise parabolic. The interlaminar normal stress, 03, is seen to be an order of magnitude smaller than the interlaminar shearing stresses. It is piecewise cubic and shows peak values at the interfaces between the - 30 ° and 0 ° plies. Note also that 03 is tensile over much of the thickness. It should be kept in mind that although the interlaminar stresses are small, interlaminar failure strengths are considerably smaller than their intralaminar counterparts. The interlaminar stresses should not in general be ruled out as a source of a delamination failure. This issue will be addressed in the following section. In addition, note that the interlaminar shear stress components are piecewise continuous in the material coordinate system due to a transformation from the global reference system, in which they are continuous through the thickness. Similar comments can be made about the [ + 45/0:]:~ laminated cylinder (Fig. 7). The variation of each stress component with the radial coordinate is strikingly similar to that of the [_+ 30/0212s laminate. The only significant differences are in the magnitudes of the stresses. The above remarks may be again applied to the [+45/0/90]2 s laminated cylinder (Fig. 8). The intralaminar stress distributions for ol and 02 are somewhat different for this laminate due to the different ply angles. For example, it is observed that tensile at stresses are developed in the 90 ° plies at midspan. The interlaminar stresses are similar to the [ _+30/0212s laminate except that the shearing stress, rl3, exhibits a minimum in the 90 °

plies while T23 exhibits a maximum in the innermost 90 ° plies. It is important to note that the signs of the stresses in Figs 5-8 are reversed when considering the corresponding locations at the bottom of the cylinder (0 = 180°). If the stress component 02, for example, is compressive at the top of the cylinder, it will be tensile at the same axial location on the bottom of the cylinder.

5.4

Failure prediction

Application of the maximum stress failure criterion provides some insight into the mode of failure, as well as the failure load. Consider the following values for failure stresses for the composite cylinders: I

a~=180"Oksi

t

a2 = 30"0 ksi

1

o3 = 30"0 ksi

o~=210.Oksi a2 = 7"5 ksi a3 = 4"0 ksi

Laminate

[ + 30/02]2, [ + 45/0:]2,, [ + 45/0/90]2 `

max 1

o~/(7

67"4 66"1 69"8

r~2=13.5ksi

f

c

r23 -- 3"6 ksi (34)

c

r~3 -- 3"6 ksi

f

f

where the superscript 't' refers to tensile failure and the superscript 'c' refers to compression failure. Shear failure is independent of sign and is designated by a superscript 'f'. The interlaminar shear strengths are frequently assumed to be equal to the inplane shear strength while the interlaminar normal strength is assumed to be equal to the inplane transverse failure strength, o~. The above interlaminar shear strengths were inferred from recent work by Noor et al., 28 which suggests that the interlaminar failure strengths may actually be much lower than they are often assumed to be. Use of the lowest values for each failure mode in eqns (34)in conjunction with the maximum stress values occurring in each cylinder, results in Table 2, which represents the ratio of the failure strength to the maximum stress at an applied strain of e a = 100 #s. The lowest ratio predicts the mode in which the cylinders will fail. The critical locations for first ply failure for the tabulated failures are as follows. Compressive fiber failures tend to occur at the cylinder ends on the compression side. Transverse

T a b l e 2. F a i l u r e r a t i o s for • a - 1 0 0 c

c

#s

t max 0"2/0"2

f . max r 1 2 [ ~" 12

t max 0"3/0` 3

82"1 58"5 47"0

75"4 76"9 93"6

3041-2 2175"3 2609"9

f t

max

T23[ r 2 3

166"4 111"4 96"4

f max T t3 / r 13

75"1 72"2 96"5

Bending response of thin laminated composite cylinders fiber failures tend to occur also at the ends but on the tension side. Inplane shear failures appear at or near the ends on both the compression and tension sides of the cylinder. The interlaminar shear failures occur at the ends of the cylinder on both the compression and tension sides. Interlaminar tensile stresses are greatest on the compression side, at the ends, but it appears that this type of failure is highly unlikely. From Table 2 it is found that this linear analysis predicts the primary mode of failure to be compressive fiber failure for the [+ 30/0212s cylinders and a transverse fiber tensile failure for the [ + 45/ 0212sand [ -- 45/0/9012~ cylinders. Although failure perpendicular to the fibers due to 02 may not be catastrophic, it would introduce matrix cracks which could lead to ultimate failure for any subsequent application of internal pressure, for example. If matrix tension failure were considered to be a minor effect, and it is assumed that only fiber failure causes ultimate failure, as suggested by Swanson et a/., 29 it is seen that fiber compression failure would also occur in the [__45/ 0212s and [+-45/0/9012s cylinders. Table 2 also indicates that the propensity for interlaminar shear failure is nearly as great as that for compressive fiber failure. This implies that a delamination failure may be initiated before ultimate failure, depending on the exact values of T[3 and

101

tion of the boundary conditions (eqn (11)). By plotting each of the 16 components of eqn (36) as a function of x, it is clear that one ak, designated here a*, is responsible for the boundary layer attenuation. Considering the class of laminates [+0/0212s 0=0°,15°,30°,45°,60°,75°,90 ° (37) the values of a * - l can be plotted as a function of 0. The inverse of a* has the units of length and it thus provides a measure of the boundary layer length. The relationship between a*-1 and 0 for the above class of laminates is illustrated in Fig. 9. From this figure it is clear that for 0 = 0 ° the boundary layer is longest, while for 0 = 90 ° the boundary layer is shortest. Another indication of the characteristics of the boundary layer is given as follows. As mentioned previously, the displacements at the side of the cylinder are almost entirely circumferential. The displacements at the top and bottom are almost entirely radial, the boundary layer being the strongest at these two circumferential locations. For the long cylinders, the expected parabolic shape is evident at the top, bottom, and sides, the parabolic shape at the top and bottom being modified with the boundary layer displacements at each end. If the circumferential displacement at

f ~23"

_-......

5.5

" '',,..,.. ,,<< .._t..,_.._

Boundarylayer length

The length of the cylinder over gradients in the radial displacements of the cylinder attenuate is related numerically determined complex (]= 1,..., 16) of eqn (17). The sj occur of roots of the form:

+_a~+iflk

which the at the ends to the 16 roots ss in four sets

k=l,...,4

0.8

l l! !I l

0.6

(35)

These roots, in turn, lead to the expression for the x-dependence of the radial displacements of the form:

\ \X \X \

/

X%

•

0.4

,..i

',..,

¢.'

:

\

\

4

w°(x) = Z [e"*X(Akcos flkX + Bk Sin flkX)

\

0.2

\

\

k=l

+ e- a~x(Ck cos flkX + Dk Sin flkX)]

+.%. o '%',~..

(36)

The real part of the root, Ctk,is responsible for the attenuation, or amplification, of the oscillating part of the solution due to ilk. The 16 constants Ak, Bk, Ck, and Dk are determined by the applica-

0 0

I

I

10

20

~

I 30

~

I 40

J

I 50

,

I 60

,

I

i

70

80

90

lamination angle,O (deg)

Fig. 9.

Parameters that govern the boundary layer for [ -+ 0/0212s laminates.

102

H. P. Fuchs, M. W. Hyer

the side, v°(x,90°), is subtracted from the radial displacement at the top, w°(x,0°), the resulting displacement will emphasize the boundary layer displacements, the overall parabolic shape being subtracted out. In Fig. 10 this difference in side and top displacements is plotted as a function of length along the cylinder for the family of laminates of eqn (37) and the case L/R = 5. The displacement difference versus axial location relationships look very much like the axisymmetric radial displacement response of cylinders in axial compression. This is an important observation because it appears that a great deal can be learned regarding boundary layer phenomena in the bending of cylinders by studying cylinders in compression. Recall that these displacements are for the situation discussed in eqn (33b). If the results of Fig. 10 are rescaled so that the midspan deflection of all cylinders is unity, then Fig. 11 results. This figure is quite a dramatic representation of the influence of lamination angle on the length of the boundary layer. What is interesting is that the boundary layer 'overshoot' is approximately 4% of the normalized radial deflection at midspan for all laminates in the family. From this figure, since the overshoot is the same, it is seen that the unidirectional 0 ° laminate

has the longest boundary layer, and thus the least severe boundary layer gradient, while the crossply [902/0212slaminate has the shortest, and hence most severe, boundary layer gradient. It should be cautioned that since the failure strength of a laminate varies with lamination sequence, severity of the gradient is not the sole measure of the potential for failure. Similar plots may be made for the short cylinders. In these cases, however, there is no distinct boundary layer at the end of the cylinder. That which is interpreted as boundary layer response at one end of a long cylinder actually becomes the response for the entire half cylinder when the cylinder is short. This leads to the barreling effect seen in Fig. 4. Thus, a comparison of boundary layers for short cylinders is not meaningful.

5.6

Response of cylinder to compression load

The general form of the eight primary response variables for the axisymmetric case of compression loading is given by eqn (19). The radial displacement solution is of the form: AI 2

y,](x)= w°(x) = - R

4

Ko + Z Kje `>x A22

,~'(z,o')-v'(,,9o') /./

times lOE-3

(38)

/=1

Normalized w°~z'°°)-~°(~ 90°) H #=0"

0 = 15"

a=lS*

s

/

0.8

i

0.6

\,i

0 = 30"

0=4,5"

0=60 •

# = 75'

'i

F77fi

;2;

', li

,,li

~

o=o °

0.4

',~

0 . 9 6 ,0.94

0.2

0.92

0"8,15 0.2 0.25 0.3 0.35 0.4 0.45~"--~"~"m~~m~.5 . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

0 0.1

0.2

0.3

0.4

x/L

Fig. 10.

Radial expansion of [ _+ 0/02]2s laminates.

0.1

0.2

0.3

0.4

0.5

x/L

0.5

Fig. 11.

Effect of 0 o n b o u n d a r y layer for [ + 0/0212~ laminates.

Bending response of thin laminated composite cylinders for balanced symmetric laminates. K 0 through K 4 are constants determined by application of the boundary conditions. While the roots sj must be determined numerically for the bending problem, they may be determined in closed form for the axial compression problem. The roots are: $1,2,3,4 ---- a(

+1 + i)

(39)

where

[A11A22-A~2]TM a = L 4A11DllR 2 j

(40)

The boundary layer lengths, a-1, can be studied explicitly as a function of material properties and geometry with eqn (40). The relation between a - 1 and 0 for the laminates of eqn (37) is plotted in Fig. 9. It is seen that the relationship is practically identical to the relationship between a*-1 and 0 for the bending problem. Thus, boundary layer lengths for the bending problem can be studied by considering the simpler case of the compression problem. Taking the analogy between bending and compression a step further, if the constants of eqn (38) are evaluated by considering an overall applied axial compressive strain of Ca--100/As (see eqn (32)) and the response plotted in the manner of Fig. 10, it is not possible to distinguish between the quantity w°(x,O°)-v°(x,90 °) for the bending problem and the radial displacement for axisymmetric compression. Hence, the case of compressive loading can be used to study other aspects of the bending problem. 5.7

Overall deflection characteristics

In addition to influencing the boundary layer characteristics, the lamination angle influences the character of the overall deflection response. This was evident in Fig. 3 in the comparison of parabolic shapes for the three laminates. In Fig. 4 the [ _ 30/0212s laminate barreled outward more at the top of the cylinder than any other cylinder. In Fig. 10, the net displacement w*(x, OO)-vO(x,90*) further demonstrated the influence of lamination angle on the radial displacement characteristics of the cylinder. These cylinders all have identical kinematic boundary conditions at the ends, yet their radial displacements are far from identical. It is clear that the difference in the radial displacement responses of the laminates is related to the material properties. Considering again the

103

analogy between the bending and axisymmetric compression problems, it is seen that the first term in eqn (38) is responsible for the overall radial deformation characteristics away from the boundary, while the summation term governs the radial deformation response in the boundary layer. The quantity is Vxy, one of the Poisson's ratios for the laminate, and it is entirely a material property. The relationship between Vxy and 0 for the [+_ 0/0212s family of laminates is shown in Fig. 9. The figure indicates that at 0 = 3 0 ° the Poisson's ratio is the largest. This corresponds to the laminate that exhibits the greatest amount of barreling in Fig. 4, and to the laminate with the least overall parabolic deflection shape in Fig. 3, the tendency to barrel countering the tendency to become parabolic. Figure 10 reinforces the fact that the [+ 30/0212s laminate experiences the largest radial deflection. Thus, it can be concluded that the Poisson's ratio of the laminate has a strong influence on the overall deflection characteristics of a cylinder in bend-

A12/AE2

ing. 6

SUMMARY AND COMMENTS

Closed-form linear response solutions were presented for the bending response of thin-walled laminated composite cylinders. The displacement response varies widely with the choice of laminate construction and geometry. Boundary layer length and the overall deflection characteristics are directly related to results from the axisymmetric problem. The case of compression can be recovered from the bending problem by 'subtracting' out the parabolic bending response. Intralaminar stresses are by far larger than the interlaminar stress components, implying that intralaminar ply failure will lead to ultimate failure. Application of the maximum stress criterion leads to the conclusion that the graphiteepoxy cylinders fail due to fiber compression on the compression side of the cylinder, at the end, if it is assumed that only fiber breakage causes failure. Since inteflaminar strengths are significantly smaller than their intralaminar counterparts, it is seen that the propensity for delamination is nearly as great as that for direct fiber breakage. It should be noted in general that linear analyses are valid only for low load levels. The range of validity of the geometrically linear response presented here has not been established as yet. It is felt that the geometrically non-linear

104

H. P. Fuchs, M. W. Hyer

response could change the conclusions so that care must be taken in extrapolating the results. ACKNOWLEDGMENTS

The work reported on herein is being supported by the Aircraft Structures Branch of the NASALangley Research Center through the NASAVirginia Tech Composites Program, Grant NAG1-343. The Grant Monitor is Dr J. H. Starnes, Jr. The financial support of the grant is greatly appreciated. REFERENCES 1. Simitses, G. J., Buckling and postbuckling of imperfect cylindrical shells: A review. AppL Mech. Reviews, 39 (1986) 1517-24. 2. Axelrad, E. L. & Emmerling, F. A., Elastic tubes. AppL Mech. Reviews, 37 (1984) 891-7. 3. Donnell, L. H., A new theory for the buckling of thin cylinders under axial compression and bending. Trans. Am. Soc. Mech. Engr., 56 (1934) 795-806. 4. Dow, M. B. & Peterson, J. P., Bending and compression tests of pressurized ring-stiffened cylinders. NASA Tech. Note D-360, National Aeronautics and Space Administration, Washington, DC, April 1960. 5. Lundquist, E., Strength tests of thin-walled duralumin cylinders in pure bending. NACA Tech. Note 479, National Advisory Committee for Aeronautics, Washington, DC, Dec. 1933. 6. Guggenheim Aeronautical Laboratory, California Institute of Technology, Some investigations of the general instability of stiffened metal cylinders V -- Stiffened metal cylinders subjected to pure bending. NACA Tech. Note 909, National Advisory Committee for Aeronatics, Washington DC, Aug. 1943. 7. Hedgepeth, J. M. & Hall, D. B., Stability of stiffened cylinders. A1AA Journal, 3 (1965) 2275-86. 8. Hoff, N. J., Boley, B. A. & Nardo, S. V., The inward bulge type buckling of monocoque cylinders IV -Experimental investigation of cylinders subjected to pure bending. NACA Tech. Note 1499, National Advisory Committee for Aeronautics, Washington, DC, Sept. 1948. 9. Mossman, R. & Robinson, R., Bending tests of metal monocoque fuselage construction. NACA Tech. Note 357, National Advisory Committee for Aeronautics, Washington DC, Nov. 1930. 10. Peterson, J. P., Bending tests of ring-stiffened circular cylinders. NACA Tech. Note 3735, National Advisory Committee for Aeronautics, Washington, DC, July 1956. 11. Seide, P., Weingarten, V. I. & Morgan, E. J., Final report on the development of design criteria for elastic stability of thin shell structures. Space Technology Labs STL/ TR-60-0000-19425, Inc., Los Angeles, CA, 31 Dec. 1960.

12. Bert, C. W. & Veragen, P. M., Experiments on nonlinear transverse-curvature behavior of composite-material blades and shells. In Composites '86: Recent Advances in Japan and the United States, ed. K. Kawata, S. Umekawa & A. Kobayashi. Japan Society for Composite Materials, Tokyo, 1986, pp. 187-95. 13. Block, D. L., Buckling of eccentrically stiffened orthotropic cylinders under pure bending. NASA Tech. Note D-3351, National Aeronatics and Space Administration, Washington, DC, March 1966. 14. Holston Jr, A., Buckling of inhomogeneous anisotropic cylindrical shells by bending. AIAA Journal, 6 (1968) 1837-41. 15. Hose, D. R. & Kitching, R., Glass reinforced composites of mixed wall construction. Int. J. Pres. Vess. & Piping, 27 (1987) 305-23. 16. Lou, K. A. & Yaniv, G., Buckling of circular cylindrical composite shells under axial compression and bending loads. J. Comp. Mater., 25 (1991) 162-87. 17. Mamalis, A. G., Manolakos, D. E., Viegelhahn, G. L. & Baldoukas, A. K., Bending of fibre-reinforced composite thin-walled tubes. Composites, 21 (1990) 431-8. 18. Reese, C. D. & Bert, C. W., Buckling of orthotropic sandwich cylinders under axial compression and bending. J. Aircraft, 11 (4) (1974) 207-12. 19. Ugural, A. C. & Cheng, S., Buckling of composite cylindrical shells under pure bending. A1AA Journal, 6 (1968) 349-54. 20. Villhart, V., Bang, C. & Palazotto, A. N., Instability of short stiffened and composite cylindrical shells under bending with prebuckling displacements. Computers & Structures, 16 (1983) 773-5. 21. Pagano, N. J., Pure bending of helical wound composite cylinders. Analysis of the test methods for high modulus fibers and composites. ASTM Special Technical Publication 521. ASTM, Philadelphia, PA, 1973, pp. 255-63. 22. Zien, H. M., Bending of laminated anisotropic composite cylinders. J. Comp. Mater., 7 (1973) 394-8. 23. dos Reis, H. L. M. & Goldman, R. B., Thin-walled laminated composite cylindrical tubes: Part II -- Bending analysis. J. Composites Tech. &Res., 9 (2) (1987) 53-7. 24. Donnell, L. H., Stability of thin-walled tubes under torsion. NACA Report 479, National Advisory Committee for Aeronautics, Washington, DC, 1933. 25. Houghton, D. S. & Johns, D. J., A comparison of the characteristic equations in the theory of circular cylindrical shells. The Aeronautical Quarterly, 12 (1961) 228-36. 26. Sanders, J. L., An improved first-approximation theory for thin shells. NASA Tech. Report R-24, National Aeronautics and Space Administration, Washington DC, 1959. 27. Fliigge, W., Stresses in Shells (2nd edn). Springer Verlag, Berlin, 1973. 28. Noor, A. K., Starnes Jr, J. H. & Waters Jr, W. A., Numerical and experimental simulations of the postbuckling response of laminated anisotropic panels. A1AA Paper No. 90-0964, presented at 31st AIAA] ASME/ASCE/AHS/ACS Structures, Structural Dynamics and Materials Conf., Long Beach, CA, April 1990. 29. Swanson, S. R., Christoforou, A. E, Colvin Jr, G. E., Biaxial testing of fiber composites using tubular specimens. Experimental Mechanics, 28 (1988) 238-43.

Bendingresponseof thin laminatedcompositecylinders

105

APPENDIX 1: DEFINITION OF CONSTANTS CI-C29 A 1 6 A 2 6 -- A 1 2 A 6 6 C 1 ~--

ALIA66-A~6

6 2= C 1 A66

C3-

AI1A66-A~6 -A16

c4-

2

A 11A66 - A 16

C5~-- m I

C 6 ~--

AI2AI6-ALIA26 ALIA66-A~6

C 7= C 6 C 8 ~-- C 4 All C9 ~

2

AliA66 -A16 1

Clo ~ -

O12 C l 1 ~..~_.a

D~ 2

C12 ~..~--

016 --

Ol~

1 Cl 3 ~ m

(A1)

C14 ~-~ - 1 2

C16 ~ C15

CI7 = C1 C18 = C 6 C15 C19

~

u

m

R

C15 C2o ~ _ _ _

R

Di1D22 -D~2 C21

2

A11A~6-A22(AIIA66 -A16)+ AI2A66 - 2AI2AI6A26 ALIA66-A~6

C15

Dll

H. P. Fuchs, M. W. Hyer

106 C22 = 2

D12D16 - D l l D 2 6

Dtl

Cl C23 ~ I i

R

C6 C24 -~- _ _ _

R

C25 = - CI 1

626 ~-- 2

DllD26 -D12D16 Dll

(f27 = - 4

Dll D66 -- D~6 Dll

C28 = l C29 : CI 2

APPENDIX 2: DEFINITION OF STRAIN AND STRESS COMPONENTS

{EL}=I EEJI,,J 1

(A2)

I KKL" j 1 K;

E:,.=c~,~u; E:,= c~s~U~

0_11 012

ql6](k) [ E~c,s

012

Q__H I E j

016

K~,. = K~.,

=

-

Q66J

[xOc.,

- q s j 2 Ul~j

t n~oc.,J

o,.- R

1

KJo, = Cj--~ U[, j_

026

(A4)

cjs/ U~

1

U-CJ(u~+u])

E~,= ~ ( - U(; + U(,

IM j [(k)_ M j j [ MxO,,:'.,l

[ xOcJ

Kj

[ m~c,s](k)

2 R

(13)

011 012

'k' I K xj is1

Q26[

E;~o,,=c~(l u~+ s~U~o)

K JXO¢"m

E~o,=cj ( -~ U(+ s~U')

Kj.~--~ _ 2 cjsjVJ

qsj UlJl

o~,,

Kj 016 026 0-66J t~o~,sJ

(AS)

Bending response of thin laminated composite cylinders A P P E N D I X 3: DEFINITION OF F U N C T I O N S ~,(k~ - i (x, 0) 1~ B(k)

~

M(k)

"C xO

Vt

xO

F?)(x,O)=R -

O0

-x

O0

B(k)

~

F~k)(x, O)= R OOx Ox

M(k)

OOx Ox

-~ B(k)

OrxO O0

B(k)

ax

f~(x,O)=

Ox

F?)(x,O)=R -

O0 a(k)

F~k)(x,O)=R - "CxO Ox

O0 ;~ M(k)

,-, B(k)

V t xO

0(7 0

Ox

aO

(A6)

-x B(k)

~6k)(x, o) =

Orxo Ox

F~k)(x,O)=OC~k)(X, O) aO F~k)(x,O)= F;k)(x,O)= FIk)(x,O)=

Fl~)(x,O)=

oc~k)(x, O) 10F(4k)(x,O) M ( k ) d'tj 0 OX 2 O0 OF(,k)(x,O) 10F~k)(x,O) --t- 0 ~ (k) Ox 3 O0 OF~l'>(x,O) 10F~k)(x,O) Ox 2 O0

oe;~)(x,O) Ox

loB(k)

--lxv 0

107