Flexural failure analysis of anisotropic elliptic plates

Computers & Strwturrs, Printed in Great Britain.

Vol. 14. No. 5-f~. pp. 463-468, 1981

@345-7949/81/i 1046MW.00/0 0 19% Pergamon Press Ltd.

FLEXURAL FAILURE ANALYSIS OF ANISOTROPIC ELLIPTIC PLATES G. J. TURVEY Department of Engineering, University of Lancaster, Lancaster, England (Receiued 8 May 1980; in revised form 15 January 1981)

Abstract-An initial flexural failure analysis of laminated anisotropic elliptic clamped plates subjected to a uniform lateral pressure is presented. Two failure modes are distinguished: general lamina failure and interlaminar shear failure. For each of these modes dimensionless initial failure loads have been computed for a range of aspect ratios and tibre orientations. These results are complemented by a set of critical thickness ratios which serve to distinguish which mode of failure governs for a particular case and hence which failure loads apply. It is concluded that general lamina failure is the dominant failure mode for the practical range of thickness ratios, though the likelihood of interlaminar shear failure occurring is greater in CARPplates than in GFRP plates. NOMENCLATURE length of elliptic plate semi-major axis f length of elliptic plate semi-minor axis c.,XV C”.Y cosines of the angles between the normal (directed outwards) and the x and y axes plate rigidities Dij(i=1,2;j=l,2,6) e,, e,, e,, plate strain components et, es, elz lamina strain components lamina longitudinal and transverse elastic EL.ET moduli GLT lamina shear modulus h plate thickness 6*(= ha-‘) critical thickness ratio lateral pressure dimensionless initial failure load (general lamina failure mode) q*(= q&ah-‘) dimensionless initial failure load (interlaminar shear failure mode) lamina compliances Q(i,j=1,2,6) Qi~(i,j = 1,2,61 transformed compliances plate aspect ratio R(= ba-‘) L $9 sxy plate in-plane stress components L> syz plate transverse shear stress components 3, plate normal stress component $13S2r 312 lamina in-plane stress components shear stress comSIZ,$22 lamina transverse ponents SL, sTt,<,SLT single lamina longitudinal, transverse (tensile, compressive) and shear strengths Cartesian plate co-ordinates Cartesian lamina co-ordinates Poisson’s ratios vLT, vTL dimensionless failure function (general 4 lamina failure mode) dimensionless failure functions (interlaminar shear failure mode) partial differentiation with respect to x 0 partial differentiation with respect to y ( )- partial differentiation with respect to z ( 1.” partial differentiation with respect to n (the outward normal co-ordinate direction)

1.INTRODUCTION

Recently, the author has been conducting analytical studies of the onset of flexural failure in laminated plates [ 1,2]. These studies were primarily concerned with examining the effects of lay-up antisymmetry on the

initial failure loads. Furthermore, only one type of failure mode was considered, viz. general lamina failure as predicted by the Tsai-Hill failure criterion. However, in laminated plates made of high modulus fibre reinforced materials it is known that the through-thickness shear stresses may be of the same order of magnitude as the lamina shear strength even when the plate thickness: plate breadth ratio is small (
463

464

G.

Z.PLATE GEOMETRY AND MATEMAL

J.

PROPERTIES

TURVEY

in which

A sketch of the clamped elliptic plate is shown in Fig. 1, which also shows the positive plate and lamina coordinate directions and the fibre orientation angle, 8. The plate consists of an aggregation of equal thickness unidirectional laminae. Typical lamina stiffness and strength properties have been used for computational purposes and these are given in Table 1. As only uni-directional lay-ups are considered in the present study, the precise number of laminae in the laminate/plate is unimportant, i.e. the plate may be considered as homogeneous and anisotropic.

A = &(I - VLTYTJ’ (Y= ELET-’ /I = G,,A -‘. These lamina compliances may be referred to any other set of orthogonal axes by substituting eqn (1) into the transformation relationship given in Ref. [3]. Hence, the transformed compliances may be expressed as: Q1, = h{OC4t 2(VLTt 2p)cY t x4} 012= A&Y t 1- 4/.I)C2S2 t VLT(C4 t $4))

3.LAMNACOMPLIANCESAND

&=A{(a-

PLATElUGIDlTIES

&

Each lamina is assumed to be specially orthotropic with respect to the lamina principal axes, 1 and 2, so that the lamina compliances may be expressed as:

[Ql,, (212, Q22,&I = A[%VLT,LPI

(1)

VLT-

2/3)C'S+ (vLT- lt2/.3)cs'}

= A{c4+ 2(vL~ t 2/3)c2s2t as4}

1 •t 2/?)c3s + (a - vr_T-2j?)cs’} & = A{(a + 1 - 2vLT - 2/3)c2s2+ ,8(c4t s4)} 026

=

A{(VL’LT

-

in which c = cos 0 and s = sin 8. LAMINA PRINCIPAL AXIS

PLATE AXIS_ 2 Y'

Fig. 1. Elliptic plate geometry and co-ordinate systems.

Table 1. CFRP and GFRP stiffness and strength ratios (a) Stiffness

:Iatios

CF;RP*

40.0

0.5

0.25

3.0

0.5

0.25

GFRP*

(b) Strenpth

Ratios

:,:ateria1

*

Data

ys’I 2

-1

s,p

c

-1 SC, ;t

-1

sLTs’i’t

CF.?$*

11.85

7.64

1.43

GFIIP+

37.50

5.00

1.50

taken

from Ref.3.

** Datr. provided

by D. ?urslovr, R.A.E.,

Farnborough,

ENGLAND.

(2)

Flexural failure analysis of anisotropic elliptic plates

The plate rigidities may then be simply evaluated as: ID, 1,&, Da =

D22146,

W12)lQH.

465

Similarly, the in-plane stress field referred to plate principal axes is given by:

&tl

&

0,6r0z2, 0X1 &I.

(3)

(MB. in eqn (3) the factor, (P/12) is a simplification of the expression, (l/3) ,f, (h’- hil_r), in which hi and hi-, are the distances of the ith and (i - I)th lamina interfaces from the plate mid-piane.)

The through-thickness shear stresses are obtainedapproximately at least-by integrating the three-dimensional elasticity equations. These equations may be written in terms of stress derivatives as follows: s; t s:, t s, = 0

4.ELASTIC SOLUTIONOF THE PLATEEQUILIBRIUM EQUATION

For the Cartesian co-ordinate system of Fig. 1 the anisotropic plate equilibrium equation may be expressed as: D,,w

s;,ts;ts,=o

(11)

s;, -I-s;* t s,-= 0. Neglecting the through-thickness normal stress, s,, and

.“‘+4D,6W“‘,

integrating the first two of eqn (11) with respect to z, the t2(D,2t2Dw)w*"' throu~-thickness or transverse shear stresses at the -f,tDz2$$P= @ t4Dxw (4) upper face of the kth Iamina are:

The solution of eqn (4)[4] for a clamped elliptic plate subjected to a uniform normal pressure is:

s,, = -

w = 0.125 w&l - x*u-* - yZb-2)z

syz= -

in which w. = qa’D_’ and

N.B. It should be appreciated that eqn (5) represents the anisotropic plate deflection function despite the absence of the characteristic flexural-twisting stiffness, D16 and D26.These terms are present in the expressions for other relevant quantities (see, e.g. eqn (13)). Equation (5) also satisfies the clamped edge boundary conditions:

(s;t s:,) dz

%-I

(5)

D = 3D,, t 2(D,, t 2Dw)aZb-’ t 3DZ2a4b-4.

I2i

*i (s;t I ?-I

s;,)dz.

(12)

Now substituting eqn (7) into eqn (lo), differentiating with respect to x and y and then substituting the stress derivatives into eqn (12) and summing for the k laminae, the transverse shear stresses may be expressed as:

fxz=

ck{&w--.

i3d,aw'*lt(~,2t2~6)W.lt zi t Q2)26w"'} x zdz I+1

w=o W.” = W%“,. t W’C”,y.

(6)

and eqn (5) may now be used to determine the complete stress and strain fields within the plate.

Finally these transverse shear stresses may be transformed to lamina principal axes as follows:

4.1 Strain field The strain field relative to the plate principal axes may be derived as follows: [e_e,,e,,]=-z[w”,

w”,~w”].

(7)

These strains may then be transformed into lamina principal strains with the help of the following transformation relationship: c2 s2 es el e2 = sz c2 -cs -2cs 2cs (CZ- s2) I[ H[e12

e, ey . eXyI

(8)

4.2 Stress field The in-plane stress field may now be determined via the lamina constitutive relationship:

(14)

5.FAJ.LURECRlTERIA AND INITIALFAILURELOADS

Since two failure modes have been postulatedgeneral lamina failure and interlaminar shear failurethen two failure criteria are required in order to establish the onset of failure in each of these modes. The Tsai-Hill failure criterion (in its two-dimensional form) has been selected for establishing the onset of general lamina failure. This criterion may be expressed as: S2S,(Sl- $2)+ 7j2s2*

,y*s:*

=

S$#

in which S = sT,sL-’

rl= (9)

i-

1 when sz is tensile. sTts~i when s2 is compressive.

(15)

G. J. TURVEY

466

A much simpler criterion is used to establish the onset of interlaminar shear failure, viz.: $1, =

SLT

or $27. = SLT.

loads for the high modulus anisotropic elliptic plates, i.e. the CFRP plates, are presented in Fig. 2 and those for low moduius, or GFRP, plates are shown in Fig. 3. Both sets of results are plotted, for convenience, against the plate aspect ratio, R, for the complete range of fibreorientation angles, 8.

(16)

On substituting eqn (9) into eqn (15) a dimensionless expression for the lateral pressure associated with general lamina failure may be obtained as: 4 = t ,--1/z

(17)

in which I$ is a coniplex function of the displacement derivatives and the strength ratios. The initial f;liIure load for general lamina failure corresponds to C$= (bmax. Likewise, on substituti~ eqn (14) into eqn (IS) the dimensionless expression for the lateral pressure associated with interlaminar shear failure may be obtained as: G”=*#,

2

(18)

so that when & is a maximum, cl* assumes its minimum value which is equal to the interlaminar shear failure load. Since,the nondimensional forms of 4 and 4* differ, it is not immediately obvious from their numerical values which of eqns (17) and (18) corresponds to the smaller pressure and hence in which mode failure initiates. This dificulry is readily surmounted by combining eqns (17) and (18) to establish a critical thickness ratio below which the interlaminar shear failure load always exceeds the general lamina failure load. This thickness ratio is:

0% R

06

Fig. 2(a).

3

6.COMPUTATIONOFTHEINITIALFAILURELOADS d, and &,.z are both rather complex functions and hence analytical expressions for their maximum values are not readily obtainable. Recourse, therefore, has been made to a numerical evaluation of these &values at the nodes of a non-uniform rectangular grid extending over one half of the plate in order to_determjne their maxima. These computations show the I$ and I#J,,~maxima often occur at the same point on the plate boundary, but that the precise location of this point on the boundary varies with the fibre-orientation angle, 0 and the plate aspect ratio, R. Once the maximum #-values have been computed, it is a straightforward matter to evaluate the two initial failure loads from eqns (17)andW)and hencedetermine whichof these governs the failure from eqn (19). 7.NUMERlCAL RESULTSANDDISCUSSION

From the standpoint of design, a study of the kind described herein may be expected to provide two important pieces of information, namely the initial failure load and the associated plate centre deflection. However, as the latter quantity is readily obtained on scaling eqn (5) by the magnitude of the initial failure load, no results for the associated plate centre deflection are given here. Instead, attention is facussed solely on the initial failure results obtained. In Section 1 of the paper it was pointed out that initial failure loads have only been computed for typical high and low modulus fibre-reinforced materials. The failure

E

7

6

5 --* 9 4

3

2

I

I 9

0.2

I

I

04 R

0.6

Fig. 2(b).

I 04

I

1.0

Flexural failure analysis of anisotropic elliptic plates I.00

F

467

9-

a-

6-

5-. 1 A-

3-

0.1

02

I

I

0.1

0%

I

I

I

I

I

1

05

@6

0.7

0.8

0.9

I.0

2-

R

l-

Fig. 2. CFRP plate results plotted against plate aspect ratio. (a) General lamina failure loads. (b) Interlaminar shear failure loads. (c) Critical thickness ratios.

I

0’ 04l

04

0.6

of CFRP plates are shown in Fig. 2(a). From this figure it is evident that, in general, the failure loads decrease as the value of R increases. However, for plates with 0 = 0” the initial failure loadplate aspect ratio dependency is somewhat different, i.e. the curve exhibits a local minimum at R--O.4 and a local maximum at kO.7 and therefore does not decrease continuously with increasing R. It is further evident from Fig. 2(a) that the initial failure loads increase as 0 increases for all aspect ratios

04

I

I

I

0.2

0%

0.6

R

Fig. 3(a).

I 04

I I.0

1.0

Fig. 3(b).

001

00

I

04

R

Initial failure loads for the general lamina failure mode

II

I

I

I

0.2

0.2

I

I

o-3

0 4

I

I

I

I

I

I

o-5

0.6

0.7

04

o-9

I.0

R

Fig. 3. GFRP plate results plotted against plate aspect ratio. (a) General lamina failure loads. (b) Interlaminar shear failure loads. (c) Critical thickness ratios.

468

G. J. TURVEY

except R = 1. In the latter case, the plates are no longer elliptic; they are circular and therefore the initial failure loads are coincident and hence independent of 0. The CFRP plate interlaminar shear failure loads are plotted against R for the same range of 0 values in Fig. 2(h). It is obvious that the interlaminar shear failure loads decrease rapidly as R increases. However, unlike the general lamina failure loads, the interlaminar shear failure loads to not increase with o-indeed, no obvious pattern of dependence on 6 exists. But again, for the degenerate case corresponding to R = 1 the interlaminar shear failure loads become coincident and therefore are independent of 0. In general, for a given laminated anisotropic elliptic plate only one of Figs. 2(a) and (b) will yield the true initial failure load of the plate, since it is unlikely, though not impossible, that the plate will fail in both modes simultaneously. Hence, Fig. 2(c) has been compiled in order to establish which of Figs., 2(a) and (b) would give the true initial failure load for any particular plate. Fig. 2(c) depicts the dependency of the critical plate thickness ratio, 6*, on R for a range of 0 values. With the exception of the results for 0 = O”,it is evident that 6* decreases as 0 increases and increases as R increases, i.e. the dependency is opposite to that of the initial failure loads. Fig. 2(c) may be used in the following manner. For a given elliptic plate geometry, i.e. h, R and e are all defined, the value of h* may be read-off Fig. 2(c) directly. This value is then compared with the actual 6* of the plate and if it is less than the actual value then interlaminar shear failure occurs before general lamina failure and the plate failure load should be determined by reference to Fig. 2(b). For the inverse situation Fig. 2(a) applies. The thickness ratios of CFRP plates used in practice are usually quite small. It is, therefore, likely that these ratios will be smaller than the critical values shown on Fig. 2(c) and so it may be expected that general lamina failure will be the dominant failure mode in these plates, e.g. for the degenerate case of a circular plate (R = 1) the actual h” would need to exceed 0.106 for a change from the general lamina to the interlaminar shear failure mode. The initial failure results for anisotropic elliptic GFRP

plates are presented in Figs. 3(a)-(c). These results are, for the most part, similar to their counterparts shown in Fig. 2, though, of course, the actual numerical values differ. In particular, comparing Fig. 3(c) with Fig. 2(c) it is evident that the critical thickness ratios are significantly greater for GFRP than for CFRP plates. Therefore, because of the practical limitations on plate thickness, the likelihood of the GFRP plates failing in the interlaminar shear mode is much less than that of the CFRP plates. 8. CONCLUSIONS

A simple analysis of the initial failure of laminated anisotropic elliptic clamped CFRP and GFRP plates subjected to a uniform lateral pressure has been undertaken. Failure loads for two possible failure modesgeneral lamina failure and interlaminar shear failurehave been computed. These results have been complemented with a set of critical thickness ratios, which serve to distinguish which mode of failure and hence which set of failure loads apply to a given plate geometry. These thickness ratios suggest that for the practical range of plate geometries the possibility of failure in the interlaminar shear mode in CFRP elliptic plates is small and in GFRP elliptic plates is almost negligible. Acknowledgements-The author wishes to record his indebted-

ness to the Departmentof Engineeringfor providing computing facilities and to his father, Mr. George Turvey, for his skillful preparation of the figures.

REFERENCES

1. G. J. Turvey, Flexural failure of antisymmetric cross-ply plates. I Engng Mech. Div., Pm. Amer. Sot. of Civil EWS, l(YI(EMI),279-285(1981). 2. G. J. Turvey, Initial Aexural failure of square, simply sup-

ported,angle-plyplates. Fibre scienceand technology.(To be published). 3. R. M. Jones, Mechanics of Composite Materials, p. 51. McGraw-Hill, Kogakusha, Tokyo (1975). 4. S. G. Lekhnitskii, Anisotropic Plates, pp. 365-367. Gordon and Breach Science Publishers, New York (1968).