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DOI: 10.1177/002199838802201105 1988 22: 1080Journal of Composite Materials

Seng C. TanOpening

Finite-Width Correction Factors for Anisotropic Plate Containing a Central

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Finite-Width Correction Factorsfor Anisotropic Plate Containing

a Central Opening

SENG C. TAN

US Air Force Materials LaboratoryWright Patterson AFB

AFWAL/MLBM

Dayton, OH 45433-6533

(Received October 2, 1987)(Revised March 21, 1988)

ABSTRACT

The finite-width correction factors for anisotropic and orthotropic plates containing anelliptical opening are presented in a tractable and closed-form solution. Examining withexperimental data and finite element solutions shows that the present theory is very ac-curate for a broad

rangeof

opening-to-width ratio, 2a/W,and

opening aspect ratio,a/b.

The application of isotropic finite-width correction factors to estimate the anisotropic or

orthotropic finite-width correction factors can cause significant error in many cases. Thesensitivity of the finite-width correction factors as a function of the opening aspect ratiois also discussed.

1. INTRODUCTION

B YDEFINITION, A FWC (finite-width correction) factor is a scale factor whichis applied to multiply the notched infinite-plate solution to obtain the notched

finite-plate result. The FWC factor can be discussed in the following aspects: (1)the stress or strain concentration such as the maximum tangential stress at theopening edge; (2) the stress intensity factor of a plate with a crack and; (3) the

interpolation of finite plate testing data (such as strength) to infinite plate result;this concept has been utilized for the strength analysis of isotropic and anisotro-pic plates. In this aspect, most ofthe strength analyses are based on infinite platesolution because a compact form stress distribution is available. Therefore, thereis a need for the data reduction between finite and infinite-width plates. The firsttwo aspects have been discussed a great deal for isotropic material [1-9] and or-

thotropic material [10-12]. The first and the third aspects are the inspirations forthe present paper. The third aspect has seldom been discussed in the literature.

The FWC factors for isotropic plate are not a function of the material proper-ties. Therefore, the isotropic FWC factors for a plate with a hole or a crack can

Journal of COMPOSITE MATERIALS, Vol. 22 -November 1988

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be determined accurately using a curve fitting technique [5]. Whereas for finite-width anisotropic or orthotropic plates, the stress analyses have been proceededmainly by using finite element methods. Due to the lack of a closed-form solutionfor an

anisotropicfinite-width

plate containinga

cutout,the

isotropicFWC fac-

tors have been applied for the data reduction of notched orthotropic finite-widthlaminates to notched infinite plates. In some cases, the isotropic FWC factors canbe applied for anisotropic plates, for instance, the stress intensity of an infinitelength composite laminate with a crack [10]. This is because the stress distribu-tion around the crack is independent of the material properties. However the ap-plication of an anisotropic FWC factor to transform the maximum stress concen-tration and the notched strength data between a finite and an infinite-width platewith an opening other than a crack has seldom been discussed in the literature.

In the present study, a compact form solution is derived for the FWC factors

of anisotropic and orthotropic plates containing a central elliptical opening. Pub-lished numerical and experimental results for respective orthotropic and isotropicplates are applied to examine the present theory. The present approach can be ap-plied to anisotropic laminate with an opening of some other shapes such as rec-

tangular, oval, triangular, etc.

2. FUNDAMENTAL THEORY

An anisotropic FWC factor can be derived by proper utilization of the stressdistribution around the opening of an infinite anisotropic plate. Two derivationsare

presentedin the

following:one is

by usingthe exact two-dimensional aniso-

tropic normal stress distribution; another one is by considering an approximateorthotropic stress distribution for an infinite plate.

2.1. Anisotropic FWC Factor

The stress distribution of an infinite anisotropic laminate containing a central

elliptical opening, Figure 1, can be derived using a complex variable method [13-15]. The normal stress, a,, along the x-axis of the laminate, under the appliedstress, ory, is

where y = xla, X = bla and ~,1 and JJ-2 are the solutions of the characteristic

equation:

where a,&dquo;i~j = 1,2,6 are the compliances of the laminate with 1 and 2 parallel




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Figure 1. A finite-width anisotropic laminate containing a central elliptical opening.

and transverse to the loading direction respectively. Only the principal roots ofEquation (2) should be chosen, i.e., two of the four roots that a have positiveimaginary part. According to the definition of the FWC factor stated in Section 1; and an

assumption that the normal stress profile for a finite plate is identical to that for

an infinite plate except for a FWC factor, the following relation is obtained:

where KTlKT is the FWC factor and K, and KT denote the stress concentrationat the opening edge on the axis normal to the applied load for a finite plate andan infinite plate, respectively. The parameter ay is the y-component of normalstress for a finite-width plate. Justification of the above assumption is made inSection 4. The

comparisonof the

presentsolution with the

experimentalresult

[2] in Figure 2; the finite element result in Figure 4 and those in Reference [16]show that this assumption is acceptable for alb >_ 1 with 2alW < 0.5 and for asmaller 2alW ratio with alb < 1. Outside these regions, the stress distributiontends to deviate from the assumption. Therefore, an improved theory is also de-

veloped in Section 3. The solution of the improved theory agrees very well withthe experimental and numerical results.From the consideration ofequilibrium condition (force resultant) in the loading

direction, the following equation is obtained from the integration of Equation (3):




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Figure 2. The maximum tangential stress concentration of an isotropic finite-width platecontaining an elliptical opening.




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Figure 3. Comparison for the predictions ofFWC factors ofmaximum tangential stress of or-

thotropic laminates with a circular hole using the improved solutions: (a) Equation (21); (b)Equation (24) and the finite element solution [16].




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Figure 4. The normal stress distribution of a GrIEp T30015208 [O/90J4s laminate containinga 1 in. diameter hole, 2aAV = 1/3.

Substituting Equation (1) into Equation (4) yields

Equation (5) is the reciprocal of the anisotropic FWC factor. The imaginary partof the solution for the term with a square root should keep the same sign as thatfor the parameter under the square root. This also holds for Equation (10). Another way of approaching the stress concentration factor and FWC factor is

the

applicationof the

averagestress across the net section instead of the whole

width of the laminate. This approach will show well the influence of the geome=try of the elliptical opening upon the FWC factors when the diameter, 2a, is closeto the plate width. This stress concentration, KTn, can be related to the previous




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stress concentration factor, KT by

Substituting Equation (6) into Equation (5) we obtain

For orthotropic laminate, the characteristic equation, Equation (2) reduces to

which has the solution:

Solving Equation (9) yields

where

Again, only the two principal roots of Equation (10) should be used for comput-

ingthe FWC factors. Note that if

0, Equation (llb),lies in the second or third

quadrant, it has to be subtracted from 7r before substituting into Equation (10).The anisotropic FWC factors can also be derived using the stress component

a7(0,y) under the applied normal stress o., (given in [13-14]). The solution is




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obtained below following the same procedure described previously:

where it, and ~,2 are obtained from the characteristic equation:

where 1 and 2 are parallel and transverse to the loading direction respectively.For an orthotropic plate, ILl and ~,2 can be computed using Equations (8-11) by in-terchanging 1 and 2 for the a,, . The solution of Equation (12) is identical to thatof Equation (5) although the solutions of the complex parameters under thesquare root sign of these two equations are different. This is a justification for thesolution technique of the complex roots discussed in Equations (9-11).

2.2. Approximate Orthotropic FWC Factor

In an earlier work [15], an approximate stress distribution for an infinite ortho-tropic laminate containing an elliptical opening has been derived. The solution,which was found very accurate within the range 0 ~ bla s 1, is

where the parameters have all been defined and

where A.niJ = 1,2,6 denote the effective laminate in-plane stiffnesses with 1and 2 parallel and transverse to the loading direction respectively.




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Substituting Equation (14) into (4) yields

When X = bla = 1 (a circular hole), the first part of Equation (16) is undeter-mined. After applying L’Hospital’s rule twice w.r.t. ~ and substituting X = 1 weobtain

When X = bla = 0(a crack), Equation (16)

reduces to

which is the same as the Dixon formula [7] and is independent of the materialproperties. Apparently, the approximate Arn and K:; I KTn for an orthotropic laminate can

easily be obtained by substituting Equation (6) into Equations (16-18).

2.3. Isotropic FWC Factor

The FWC factors for an isotropic plate with an elliptical opening can be ob-tained by substituting KT- = 1 + 2/X, isotropic stress concentration factor, into

Equation (16), i.e.

Equation (19) can also be obtained by substituting ~,1 = /A-2 =i into Equation(5). When X = bla = 1 (a circular hole), the solution can easily be seen from




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Equation (17) as

When X = bla = 0 (a crack), the FWC factor is given in Equation (18). The KTnand K Tool KTn for an isotropic plate can be obtained by substituting Equation (6)into Equations (19-20).

3. IMPROVED THEORY (FOR alb < 4)

A parametric study in Section 4, Figure 2, shows that the maximum finite-width stress concentration predicted by Equations (19) and (20) are more accuratefor alb ? 4 than for alb < 4. The

predictions usingthe fundamental

theoryfor

alb < 4 are too low compared to the experimental data. Therefore, a magnifica-tion factorM is developed to improve the previous basic theory. Knowing the factthat the accuracy of the Heywood formula [8] is very good for an isotropic platewith a circular opening, we derive an anisotropic solution that will recover to theHeywood formula under the isotropic and same opening condition. This can beaccomplished by multiplying the 2a/W ratio of Equation (20) by a magnificationfactor M ; equating it to the Heywood formula, Equation (26), and solving for M.Finally, multiplying the opening-to-width ratio, 2alW, of Equation (5) by themagnification factor M, we obtain the improved anisotropic FWC factor as:

where

The approximate orthotropic FWC factor, Equation (16), becomes




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When X = bla = 1, Equation (23) reduces to

For isotropic plate the FWC factor, Equation (19), becomes

When X = 1, Equation (25) is simplified to

Equation (26) recovers the familiar form of the Heywood formula [8], which hasbeen widely used for the isotropic plate.

The second stress concentration factor, KTn, and the second FWC factor, KTn~KT’, for the improved theory can be obtained by substituting Equation (6) intoEquations (21-26).

4. EXPERIMENTAL AND NUMERICAL COMPARISONS

Using the experimental data presented in Reference [2], the prediction of themaximum tangential stress concentration of a finite-width isotropic plate contain-ing an elliptical opening under uniaxial loading is examined in Figures 2(a) and

(b). Comparisonshows that the

presentbasic

approach, Equations(19) and (20),

is highly accurate for alb > 1 with 2alW < 0.6 and the improved theory Equa-tions (25) and (26), has excellent agreement with the data for alb < 4. TheNeuber solution [9] is too high for alb > 1 while the Isida solution [6] is quite




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Table 1. Elastic properties of typical graphitelepoxy laminae.

accurate for the cases considered. When alb = 1, the solutions by Heywood,Wahl-Beeuwkes and Isida are practically the same.

The predictions for the FWC factors of orthotropic laminates containing a cir-cular hole are compared to the finite element solution [16] in Figures 3(a) and (b).The material properties, Example 1, are listed in Table 1. It is shown that both theimproved anisotropic theory, Equation (21), and the improved orthotropic solu-tion, Equation (24), agree very well with the finite element solution. Even whenthe 2alW ratio is over 90 percent, the predictions are still in excellent agreementwith the finite element solution, Table 2.The present approach assumes that the normal stress, 0~,0) (Figure 1), across

the net section of a finite-width plate is proportional to that of an infinite-width

plate bya factor. The solution of this

assumptioncan be examined

usinga

typicalfinite element solution [17]. In Figure 4, comparisons are made for: (1) the resultof FWC factor obtained from Equation (5) multiplied by the infinite plate solu-tion ; (2) the FWC factor obtained from Equation (17) multiplied by the approx-imate orthotropic solution, Equation (14), and; (3) the isotropic FWC factor mul-tiplied by the infinite orthotropic plate solution. The material properties for thegraphite/epoxy (Gr/Ep) T300/5208 [0/90]4S laminate being used is listed in Table1, example 2. The result appears that the normal stress distribution for the finiteelement solution correlates better with the solutions obtained by using the ortho-

tropic FWC factor, Equations (5) and (17), than by using the isotropic FWC fac-

tor. In this comparison, the Heywood formula, Equation (26), was used for theisotropic FWC factor.

Table 2. Comparison of the improved theory, Equations (21) and (24), andthe finite element solution [16] for orthotropic laminates, Example 1 in

Table 1, with a circular hole and 2alW = 0.91.




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Figure 5. The FWC factors for some GrIEp T30015208 laminates containing an ellipticalopening with (a) a/b = 1/2 and; (b) alb = 2 using the improved solution (21).




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Table 3. The FWC factors for GrIEp T30015208 laminates with an ellipticalopening, alb = 112, and opening-to-width ratio, 2alW = 113, using the

improved solutions.

5. ANISOTROPIC PLATE WITH AN ELLIPTICAL OPENING

Using the material properties shown in Table 1, some graphite-epoxy T300/5208 laminates [~75]5> [O6/90]S, [0/45]S and an isotropic plate, containing an el-

liptical opening with aspect ratio a/b, equal to 1/2 and 2 were studied in Figure5(a) and 5(b) respectively. These results show that the FWC factors are in de-

scending order for the [±75]., isotropic, [0/45]S and [06/90]S laminates. Thisorder suggests that the FWC factor is more significant for matrix dominated lami-nates than for fiber dominated laminates. A comparison of the curves, alb =

1/2, 1, 2 and 5, show the difference in FWC factors ofthese laminates in descend-

ing order with increasing ratio of alb. When alb >_ 5, the effect of anisotropy

Figure 6. The FWC factors of a GrIEp T300/5208 [0/90Js laminate containing an ellipticalopening, using Equation (21).




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upon the FWC factors is practically negligible. If alb is less than unity, aniso-tropy can be very significant for the FWC factors even when the opening-to-widthratio is small. An example for alb = 1/2 and 2a/w = 1/3 for these laminates isshown in Table 3.

To discuss the FWC factors as a function of the opening aspect ratio, a/b, forfiber dominated and matrix dominated laminates, two results are illustrated in

Figures 6 and 7 which comprise an elliptical opening with alb = 1/5, 1/2, 1, 2,5 and a crack. The improved solutions, Equations (21-26) are applied for alb< 4 whereas the basic theories, Equations (5), (16), (18) and (19) are applied foralb >_ 4. A Gr/Ep [04/90]s is used to represent a fiber dominated laminate,Figure 6, and [ =E 60] is demonstrated for a matrix dominated laminate, Figure 7.For [04/90]s laminate with a 2alWratio smaller than 1/3, the FWC factors changeslightly for alb >_ 1. When alb >_ 5 the values of the FWC factor are the same

as that with a crack. When alb=

1/5, the difference ofthe FWC factors betweenan isotropic and a [04/90]s laminate is about 10 percent, Table 4. In Figure 7, theFWC factor of a [ =)= 60], laminate is shown to be very sensitive to the openingaspect ratio. For instance, when 2alW ratio is equal to 1/3, the variation of theFWC factor as a function of the opening aspect ratio is already quite significant,Table 5. The difference of the FWC factors for a [ t 60], and an isotropic plate atalb = 1/5 is 13 percent.

Figure 7. The FWC factors of a T30015208 (t 60]s laminate with an elliptical opening, us-ing Equation (21).




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Table 4. The FWC factors for a: (1) GrIEp T30015208 (OQ/90JS and;(11) isotropic laminates with an elliptical opening and opening-to-width ratio,

2a/W = 1/3.

6. DISCUSSIONS AND CONCLUSION

The finite-width correction factors of anisotropic, orthotropic and isotropicplates containing an elliptical opening are presented in a closed-form solution.

The fundamental solution is derived from the consideration of resultant forceequilibrium in the loading direction. The basic theory can be improved sig-nificantly by multiplying the opening-to-width ratio, 2a/W, by a magnificationfactor M. Comparisons with experimental data and some finite element solutionsshow that the basic solution has good accuracy for alb >_ 4 while the improvedtheory has excellent accuracy for the domain alb < 4.

Illustration with figures and tables reveals that within the region of alb > 1with 2alW < 1/3, FWC factors of most orthotropic laminates can be estimatedby using the isotropic FWC factors with less than 6 percent error. Beyond thisrange, however, significant error can be caused by estimating the anisotropic and

orthotropic FWC factors using the isotropic FWC factors, Figures 3 and 5.The result of the present study also concludes that:

1. Matrix dominated laminates, e.g. [:i:: 75]s, have higher FWC factors thanthose of the isotropic plate (for a same opening length) while the fiberdominated laminates, e.g. [06/90]s, are just the opposite, Figures 3 and 5.

2. For each ratio of 2a/W, the FWC factors are increasingly sensitive and thevalues are higher for the decreasing ratio of a/b, Figures 6 and 7.

3. The FWC factors of matrix dominated laminates are extremely sensitive tothe opening aspect ratio, a/b, whereas the fiber dominated laminates are

somewhat less sensitive than that of the matrix-dominated laminates.4. For alb _> 5 the influence of the anisotropy upon the FWC factors vanishes

(for infinite length).

Table 5. The FWC factors for a: (1) GrIEp T30015208 (t 601s and;(11) isotropic laminates with an elliptical opening and opening-to-width ratio,

2alW = 1/3.




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It is important to point out that the FWC factors can be used to transform thelaminate notched strength (finite-width) to infinite plate notched strength only ifthe stress distribution at the neighboring points around the maximum stress con-

centration agrees with the actualsolution. This is

because,in the failure

analysis,the maximum stress concentration cannot explain the notched strength due to thewidth effect. The component of normal stress at a very small distance (usuallywithin 2.54 mm) away from the maximum stress point, however, has been ap-plied successfully to explain the width and the hole size effects [14,15,17,18] for thenotched strength of composite laminates. For a finite-width plate, if the stressprofile at a small distance around the maximum tangential stress point (usually atthe opening edge) agrees with the stress profile of an infinite plate multiplied bythe FWC factor, then the notched strength can be correlated to that of an infinite

plate. In Figure 4, the normal stress distribution of a finite-width plate with a cen-

tral hole predicted using the present theory agrees very well with a typical finiteelement solution. This result supports the application of the FWC factors to inter-

polate a finite-width-specimen data to an infinite plate result, especially the solu-tions for the maximum tangential stress and the notched strength. The widtheffect (hold 2a and change W) and the hole size effect (hold W and change 2a )both change the magnitude of the stress concentration. The correction factors forthese effects can all be computed utilizing the present analysis if their opening-to-width ratio, 2a/W, is known.

It is interesting to note that the magnification factor M is only a function of2alW and is independent ofalb ratio. Therefore, this factor, M, could be appliedas well to some other opening shapes such as rectangular, oval, etc. Finally, thesolution for finite-width anisotropic laminates is deficient in the literature. There-fore, additional work, experimental and numerical, will be conducted to examinethe present anisotropic solution, i.e., FWC factors.

REFERENCES

1.Wahl, A. M and R. Beeuwkes. "Stress Concentration Produced by Holes and Notches," Trans. American Society for Testing and Materials, 56:617-625.

2. Durelli, A. J., V. J. Parks and H. C. Feng. "Stresses Around an

EllipticalHole in a Finite Plate

Subjected to Axial Loading," Trans. American Society for Testing and Materials, J. Applied Me-chanics, 88.192-195 (1966).

3. Jones, N. and D. Hozos. "A Study of the Stresses Around Elliptical Holes in Flat Plates," Trans. American Society for Testing and Materials, J. Eng. for Industry, 93:688-695 (1971).

4. Paris, P. C. and G. C. Sih. "Fracture Toughness Testing and Its Applications," American Societyfor Testing and Materials, STP 381, pp. 84-113 (1965).

5. Brown, JW F., Jr. and J. E. Srawley. "Plane Strain Crack Toughness Testing of High StrengthMetallic Materials," American Society for Testing and Materials, STP 410 (1969).

6. Peterson, R. E. Stress Concentration Factors. New York:John Wiley & Sons (1974).7. Dixon, J. R. "Stress Distribution Around a Central Crack in a Plate Loaded in Tension: Effect

of Finite Width ofPlate,"

J.

Royal Aeronautical

Society,64:141-145

(March, 1960).8. Heywood, R. B. Designing by Photoelastacity. London:Chapman and Hall, p. 163 (1952).9. Neuber, H. Kerbspannungslehre. Berlin, Germany:Julius Springer (1937) Translation, Theory

of Notched Stresses, Office of Technical Services, Dept. of Commerce, Washington, D.C. (1961).




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10. Mar, J. "Fracture and Fatigue ofBi-Materials," Mechanics ofComposites Review, Air Force Ma-terials Laboratory and Air Force Office of Scientific Research Technical Report, pp. 117-122(Oct. 26-28, 1976).

11. Konish, H. J., Jr. "Mode I Stress Intensity Factors for Symmetrically-Cracked Orthotropic

Strips," American Society for Testing Materials, Fracture Mechanics of Composites, STP 593,pp. 99-116 (1975).

12. Atlun, S. N., A. S. Kobayashi and M. Nakagaki. "A Finite-Element Program for Fracture Me-chanics Analysis of Composite Material," American Society for Testing and Materials, FractureMechanics of Composites, STP 593, pp. 86-98 (1975).

13. Lekhnitsku, S. G. Anisotropic Plates, Translated from the second Russian edition by S. W. Tsaiand Cheron, Gordon and Breach (1968).

14. Tan, S. C. "Notched Strength Prediction and Design of Laminated Composites Under In-PlaneLoadigs," J. of Composite Materials, 21:750-780 (August, 1987).

15. Tan, S. C. "Laminated Composites Containing an Elliptical Opening. I. Approximate Stress Analyses and Fracture Models;’J. of Composite Materials, 21:925-948 (Oct., 1987).

16. Hong, C. S. and J. H. Crews, Jr. "Stress Concentration Factors for Finite Orthotropic Laminateswith a Circular Hole and Uniaxial Loading, NASA Technical Paper 1469 (1979).

17. Nuismer, R. J. and J. M. Whitney. "Uniaxial Failure of Composite Laminates Containing StressConcentrations," American Society for Testing and Materials, Fracture Mechanics of Compos-ites, STP 593, pp. 117-142 (1975).

18. Tan, S. C. "Effective Stress Fracture Models for Unnotched and Notched Multidirectional Lami-nates,"J. Composite Materials, 22:322-340 (April, 1988).

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