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On stress concentrations for isotropic/orthotropic platesand cylinders with a circular hole

Hwai-Chung Wu *, Bin Mu Infrastructure Materials/Systems Laboratory, Department of Civil and Environmental Engineering, Wayne State University,

5050 Anthony Wayne Drive, Detroit, MI 48202, USA

Received 17 April 2002; accepted 27 August 2002

Abstract

Scale factors (SFs) are widely used in engineering applications to describe the stress concentration factor (SCF) of a nite width isotropicplate with a circular hole and under uniaxial loading. In this paper, these SFs were also found to be valid in an isotropic plate with biaxialloading and an isotropic cylinder with uniaxial loading or internal pressure, if a suitable hole to structure dimension ratio was chosen. Thestudy was further expanded to consider orthotropic plates and cylinders with a center hole and under uniaxial loading. The applicable range of the SFs was given based on the orthotropic material parameters. The inuence of the structural dimension on the SCF was also studied. Anempirical calculation method for the stress concentrations for isotropic/orthotropic plates and cylinders with a circular hole was proposed andthe results agreed well with the FEM simulations. This research work may provide structure engineers a simple and efcient way to estimatethe hole effect on plate structures or pressure vessels made of isotropic or orthotropic materials.q 2003 Elsevier Science Ltd. All rights reserved.

Keywords: A. Plate; Stress concentration factor; Finite element method; Stress analysis; Cylinder

1. Introduction

Stress concentrations around cutouts have great practicalimportance because they are normally the cause of failure.For most materials, the failure strengths of the materials arestrongly notch (or hole) sensitive. The net failure stress,taking into account the reduction in cross-sectional area, istypically much less than the ultimate tensile strength of thesame materials without the notch or hole. For example,strength reductions of 4060% have been reported for aglass ber reinforced plastic plate [1]. Hence a betterapproach dealing with strength reductions due to stressconcentration around a geometric discontinuity (such as ahole) is the use of the stress concentration model or pointstress model [2]. Failure is predicted by the use of elasticstress concentration factor (SCF), K T , without consideringsharp edge cracks around the hole. K T is dened as the ratioof the maximum stress in the presence of a geometricirregularity or discontinuity to the stress that would exist atthe same point if the irregularity was not present.

1.1. Isotropic plate

A stress concentration is typically introduced in platesand cylinders in the form of circular holes. This form of cutout has many practical applications and is familiar tomost engineers. Most of the strength analyses involvingSCFs are based on the conditions of innite-width/diameterplate/cylinder because closed form stress distributions areavailable. Therefore, there is a need to account for the effectof nite width/diameter on SCFs. Such effect can be suitablyconsidered through the use of a scale factor (SF) dened as

the ratio of the SCF of nite-size structure to that of innite-size structure. The SFs for isotropic plates are not a functionof the material properties. Therefore, the SFs for anisotropic plate with a center hole can be determinedaccurately using a curve tting technique [3,4]. Forinstance, the following relationship has been obtained fora nite-width plate [3]:

K 1 ;1T;i;p;u

K 1T;i;p;u 31 2 d = W

2 1 2 d = W 3 1

where K 1 ;1T;i;p;u and K

1T;i;p;u represent the SCF at point-1 in an

isotropic plate under uniaxial tension with innite and nite

1359-8368/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.PII: S1359- 8368(02 )00097- 5

Composites: Part B 34 (2003) 127134www.elsevier.com/locate/compositesb

* Corresponding author. Tel.: 1-313-577-0745; fax: 1-313-577-3881.E-mail address: [email protected] (H.-C. Wu).
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width, respectively ( Fig. 1). For an innite plate, K 1 ;1T ;i;p;u 3:0 and K 1 ;2T;i;p;u

2 1:0 (point-2 in Fig. 1); d is the diameterof the hole and W is the width of the plate. Subscripts i, p,and u represent isotropic material, plate and uniaxialloading, respectively. Superscript 1 represents innitesize structure, such as innite-width plate and innite-diameter cylinder.

1.2. Orthotropic plate

As for nite-width orthotropic plates, the stress analyseshave been produced mainly by the nite element method.By assuming an approximate stress distribution for a niteorthotropic plate containing a circular hole, Tan [5,6]

derived a closed form solution for SCFs of nite orthotropicplates containing a central circular opening and underuniaxial loading. The SCFs depend on the materialparameters.

K 1 ;1T;o;p;u

K 1T;o;p;u 31 2 d = W

2 1 2 d = W 3 12

d W

M 6

K 1 ;1T;o;p;u

2 3

1 2 d

W M

2" # 2awhere K

1 ;1T;o;p;u and K

1T;o;p;u represent the SCF at point-1 for an

orthotropic plate under uniaxial tension with innite andnite width, respectively. M is a magnication factor and isonly a function of d = W : Detailed expression of M can befound in Ref. [6]. Subscript o represents orthotropicmaterial. The SCF for an innite orthotropic plate can bewritten as [2]

K 1 ;1T;o;p;u 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ffiffiffiffiffiE yE xs 2 n yx E y2G yx !vuut 2b

where E x and E y are Youngs modulus in x and y direction(see Fig. 1), respectively. G yx is shear modulus in x y planeand n yx is Poissons ratio.

1.3. Isotropic pipe

In actual pipes, a hole is located on the circular surface of a cylinder rather than on a at surface. Therefore, the effectof such plate curvature on the SCF should be investigated.For an isotropic cylinder with d = D p

ffiffiffiffi2t = Dp and under axial

loading (along cylinder direction, Fig. 2), the SCFs at point-1 and point-2 are [7]

K 1 ;1T;i;c;u 3 ffiffiffiffiffiffiffiffi3m2 2 1m2s p8 d

2

Dt 3a

K 1 ;2T;i;c;u 2 1 ffiffiffiffiffiffiffiffi3m2 2 1m2s p8 d

2

Dt 0@ 1A 3b

where K 1 ;1T;i;c;u and K

1 ;2T;i;c;u represent the SCFs at point-1 and

point-2, respectively; m 1 = n ; n is Poissons ratio; d is thediameter of the hole and D is the diameter of the cylinder; t is wall thickness.

In the case of an isotropic cylinder under two wayloading (e.g. under internal pressure, P 0 ), with d = D p

ffiffiffiffiffiffi2t = Dp ; Savin [7] also gave the SCFs at point-1 and point-2K

1 ;1T;i;c;b

s u 0P 0 D = 4t 1

2 ffiffiffiffiffiffi3m2 2 1m2s p8 d 2

Dt 4a

K 1 ;2T;i;c;b

s u p = 2P 0 D = 2t

52

1 ffiffiffiffiffiffi3m2 2 1m2s 9p40 d 2

Dt 0@ 1A 4b

where P0 D = 2t and P0 D = 4t are the stresses in the hoop andlongitudinal directions, respectively.

Neglecting the curvature terms in Eqs. (3a)(4b), it canbe found that the Eqs. (3a) and (3b) recover K 1 ;1T;i;p;u ; K

1 ;2T;i;p;u

and the Eqs. (4a) and (4b) are identical with the resultsobtained by superposition of K

1 ;1T;i;p;u; K

1 ;2T;i;p;u: This reminds us

whether we can make use of the SFs for isotropic plateswhich have been already deduced to estimate the SCFs andSFs for isotropic cylinders through some simple calculationmethod and whether this calculation method is also valid fororthotropic structures (plates and cylinders)?

1.4. Proposed approach

The objective of this paper is to provide structuralengineers a simple and reliable estimation method for SCFsin common structures. For this purpose, a systematic studyof SCFs and SFs for isotropic/orthotropic plates/curved

Fig. 1. A plate with a central hole and under uniaxial tension.

Fig. 2. A pipe with a hole under uniaxial tension.

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plates (cylinders) containing a circular hole was carried outin this paper. Based on the SCFs and SFs of an isotropicplate and superposition principle, a simple computationalmethod was proposed to estimate SFs and SCFs of anisotropic cylinder under one way or two way (internalpressure) loading. Then this method is extended toorthotropic structures. The proposed method can bedescribed as follows:

For an isotropic plate under biaxial tension, the SCFs arecalculated from the individual SCFs of the same plate

under uniaxial tension then by summing up the SCFstogether at the same location according to the super-position principle. It should be noted that the SFsobtained from the FEM results (see Fig. 4) for point-2location would be used in the calculation, since there isno analytical equation available at present.

For a nite-width orthotropic plate under uniaxialtension, the SCFs at point-1 are estimated by the SCFsof a corresponding nite-width isotropic plate ( Fig. 4)multiplied by the ratio of K

1 ;1T;o;p;u to K

1 ;1T;i;p;u :

For an isotropic cylinder under uniaxial tension, theSCFs at point-1 or point-2 are calculated by the SCFs of

the nite-width isotropic plates ( Fig. 4) multiplied by theratios of K 1 ;1T;i;c;u to K

1 ;1T;i;p;u or K

1 ;2T;i;c;u to K

1 ;2T;i;p;u :

The superposition principle will be employed if anisotropic cylinder is under biaxial loading (e.g. internalpressure).

For an orthotropic cylinder under uniaxial tension, twocalculation steps should be adopted. The SCFs at point-1are calculated by the SCFs of the nite-width isotropicplates (Fig. 4) multiplied rst by the ratio of K

1 ;1T;i;c;u to

K 1 ;1T;i;p;u to account for the cylindrical effect and then by

the ratio of K 1 ;1T;o;p;u to K

1 ;1T;i;p;u to account for the orthotropic

property.

1.5. FEM simulation

In the FEM simulations, the rst order shear deformationshell theory is adopted [9]. The displacement eld is givenby

u x; y; z u0 x; y zc x x; yv x; y; z v0 x; y2 zc y x; y 5w x; y; z w x; ywhere u, v, w are displacement components in the x, y, z

directions, respectively; c x and c y are rotations of the cross-section about the x and y axis; and u0 and v0 aredisplacement components at the mid-plane of the plate. InABAQUS, the plate/cylinder was modeled with 4-nodedquadrilateral and 3-noded triangular shell elements with sixdegrees of freedom: three displacement components andthree rotation components u; v; w; u x; u y; u z at each node(S4R and S3R).

In the following sections, the calculation results areveried by the FEM simulations (using ABAQUS). Theapplication range of the proposed approach and difcultiesare discussed. The research ndings are found useful tostructural design.

2. SCFs of an isotropic/orthotropic plate under uniaxialor biaxial tension

The SCFs of an isotropic plate under uniaxial or biaxialtension have been studied extensively and good results havebeen reported [3,68] . There are also formulae for SCFs of an orthotropic plate under uniaxial tension [5,6]. Forcompleteness, this paper begins with FEM formulationand compares the FEM results with the closed formsolutions. The analysis emphasizes cutout size effect on

Table 1Geometrical size of plates and cylinders

W (mm) d (mm) t (mm) D (mm) L (mm) d = W or d = D

P1 101.6 15.24 10.16 254 0.15101.6 30.48 10.16 254 0.3101.6 50.8 10.16 254 0.5101.6 76.2 10.16 254 0.75

P2 152.4 22.86 10.16 254 0.15152.4 76.2 10.16 254 0.5152.4 114.3 10.16 254 0.75

C 7.62 10.16 203.2 508 0.0375 30.48 10.16 203.2 508 0.15 76.2 10.16 203.2 508 0.375 127 10.16 203.2 508 0.625

Fig. 4. The SCFs for isotropic plates under uniaxial tension.

Fig. 3. FEM mesh for a plate with a central hole.

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the SCFs under a xed ratio of the cutout size (such as hole)to the structural dimension (such as a plate width or cylinderdiameter). It also emphasizes the relationship between theSFs of an isotropic plate and an orthotropic plate.

A typical FEM mesh of a plate with a center hole is givenin Fig. 3. The physical sizes of the plates are given inTable 1 . There are two different plate widths and varioushole-diameters to give the same d = W ratios (P1 and P2 inTable 1 ). Fig. 4 presents the SCFs of an isotropic plate withdifferent d = W and under uniaxial tension. At point-1location, the FEM simulations for SCFs agree well withEq. (1) (Heywoods curve [3]). For the same d = W ; the actualhole size or plate width has little inuence on the SCFs.Fig. 4 also gives the SCFs at the point-2 location

(compression). For an isotropic innite width plate, thisSCF equals 2 1. Based on the SCFs (at point-1 and point-2)given in Fig. 4 and by the superposition principle, the SCFsat the point-1 location of an isotropic plate under biaxialtension (in x and y directions) can be calculated followingthe approach described in this paper. For the two groups of plates (P1 and P2), the SCFs are shown in Fig. 5. It shouldbe noted that for rectangular plates, one should use twohole-diameter to plate-width ratios ( d = W and d = L ; see

Table 1 ) to account for the biaxial effect. There are two suchpredictions based on the geometry of the two rectangularplates (P1 and P2 in Table 1 ). Again, the inuence of the

hole size and plate width is insignicant when the d = W ratiois xed. SCFs at point-2 under biaxial loading can beobtained in a similar manner based on the point-2 curves inFig. 4.

Next, the SFs of isotropic plates under uniaxial tensionare extended to orthotropic plates with a central hole. TheSCF of an innite orthotropic plate is given by Eq. (2a). Fornite width orthotropic plates, the SCFs can be predicted bythe proposed approach, as shown in Figs. 6 and 7 foruniaxial tension in the longitudinal ( E 11 is in y direction,Fig. 1) and transverse ( E 22 ) directions, respectively. Therelevant orthotropic material parameters are listed in Table 2

[10] . In these two gures, the proposed computationapproach results in good agreements with the FEMsimulations, as well as Eq. (2a). Comparing Eqs. (1) and(2a), we can nd that the rst part of Eq. (2a) is identical toEq. (1). It suggests that if the second part of Eq. (2a) is muchsmaller than the rst part, i.e. Eq. (1), the differences in theSFs for isotropic plates and orthotropic plates are verysmall. This comparison is given in Fig. 8. In Fig. 8, A2represents the second part of Eq. (2a) and A1 represents therst part of Eq. (2a), i.e. Eq. (1). Different K 1 ;1T;o;p;u are used inthe gure. We can see that when the dimension ratio, d = W ; isless than 0.5, A2 can be always neglected and hence the SFsfor isotropic plates can be also used for orthotropic plates.When d = W is larger than 0.5, the applicable range woulddepend on the actual values of K

1 ;1T;o;p;u: For ordinary ber

reinforced composites, K 1 ;1T;o;p;u is usually less than 10.

Fig. 5. The SCFs for isotropic plates under biaxial tension.

Fig. 6. The SCFs for orthotropic plates under uniaxial tension (in E 11direction).

Fig. 7. The SCFs for orthotropic plates under uniaxial tension (in E 22direction).

Table 2The orthotropic material parameters

E 11 (GPa) E 22 (GPa) u12 u21 G12 (GPa) G13 (GPa) G23 (GPa)

20 9.2 0.341 0.157 5.0 5.0 2.6

E 11 , modulus along the ber direction; E 22 , modulus in theperpendicular direction; u 12 and u 21 , Poissons ratios; G 12 , G 13 , and G 23 ,shear moduli.


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For instance as shown in Figs. 6 and 7, K 1 ;1T;o;p;u equals 3.503

and 2.698, for P1 and P2 plates, respectively. Then the SFs

at point-1 of isotropic plates ( Fig. 4) are readily applicableto orthotropic plates up to almost d = W 0:75: From Figs.47, when the dimension ratios are xed, the SCFs aredetermined and are independent of the sizes of the plates orthe holes.

However, at point-2, the negative SCFs can be verydifferent for the orthotropic plates with different materialproperties or under different uniaxial loading directions(comparing Figs. 6 and 7). The SCFs at point-2 dependstrongly on the composite modulus. As reported in theliterature [11,12] , for some innite-width orthotropic platesunder uniaxial tension, the SCF at point-2 can reach 2 4,

much different from that of corresponding isotropic plateswhich is always 2 1. Another difculty is that currentlythere is no theoretical equation to predict the SCF at point-2location of an innite-width orthotropic plate under uniaxialtension, unlike the SCF at point-1 (Eq. (2b)). Due to thesetwo reasons, it is not always reliable to predict the SCFs andSFs both at point-1 and point-2 of nite-width orthotropicplates under biaxial tension based on the proposedcalculation method. Prior to applying the proposed method

to biaxial tension, the applicable range (i.e. the values of material properties and loading directions) should beinvestigated. Such applicable range should be a functionof the material parameters. In this preliminary study, theSCF at point-1 location of an orthotropic plate (with thematerials properties given in Table 2 ) under biaxial loadingis estimated by the proposed calculation method. The SCFsare calculated from the SCFs of the same plate underseparate uniaxial tension in E 11 and E 22 directions, then bythe superposition principle to add the SCFs together at thesame location. The predictions match the FEM results quitewell (Fig. 9). Further study will be carried out to examinethe applicable range of this calculation method.

3. SCFs of an isotropic/orthotropic cylinder underuniaxial tension or internal pressure

In Section 2, the SFs of isotropic plates under uniaxialtension are used to estimate the SCFs of isotropic plates

under biaxial tension and SCFs of orthotropic plates underuniaxial tension. All of the plates studied have a centercircular hole. In this section, the proposed calculationmethod will be applied to hollow cylinders with a circularcutout under uniaxial tension in the longitudinal direction(axial direction) or under internal pressure (biaxial tension).The dimensions of the cylinders are given in Table 1(designated C ). In the plate case, there is a well-deneddimension ratio, dened as the ratio of the hole diameter tothe plate width d = W : However, this ratio may not be validin the cylinder case. Finding a valid dimension ratio is aprerequisite for the proposed calculation method. Forcylinders ( Fig. 2), there are four possible dimension ratioswhich can be dened as: d = D; d = p D; d 2 = 2 Dt and arcsind = D = p ; where the third (see Eqs. (3a) and (4a)) comes from Ref.[7] and the fourth is the ratio of the maximum arc length of the trimmed cylinder surface to the perimeter of thecylinder, p D: Actually, the second and the fourth arequite close because in real engineering applications, thecircular cutout in a cylinder cannot be very large. Thereforewe can consider and compare only the rst three ratios.

Fig. 10 gives the FEM mesh for a cylinder with a hole.This mesh is generated by HYPERMESH automatically.Fig. 11 presents the SCFs for isotropic cylinders under axialtension. The SCFs at point-1 or point-2 are calculated by

Fig. 8. Orthotropic inuence ( K 1 ;1T;o;p;u 3:5; 10, 30, 100) on SCFs of platesunder uniaxial tension.

Fig. 9. The SCFs for orthotropic plates under biaxial tension.

Fig. 10. FEM mesh for a cylinder with a hole.


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the SCFs of the nite-width isotropic plates ( Fig. 4)multiplied by the ratios of K

1 ;1T;i;c;u to K

1 ;1T;i;p;u or K

1 ;2T;i;c;u to

K 1 ;2T;i;p;u: The relationships of the SCFs versus the rst three

dimension ratios are calculated based on the proposedcalculation method. The results ( Fig. 11) show that theapplicable range of the second ratio d = p is very small (onlyto 0.2) because the cutout diameter is seldom larger than60% of the cylinders diameter. Using d 2 = 2 Dt ; Savinsresults [7] overestimates the hole effect of the cylinder forlarger dimension ratios. In fact, as discussed by Savin [7],his prediction is only valid in the case of d = D p

ffiffiffiffiffiffi2t = Dp ; i.e.

for very small hole sizes. If the diameter ratio, d = D; isselected, the pre-edited SCFs agree well with FEM results

both at point-1 and point-2. It means that by using thisdimension ratio, the proposed calculation based on the SFsof an isotropic plate under one way loading can describe theSCFs well around a circular cutout in a cylinder under axialtension. It has been noticed that under axial tension, anunsymmetric effect resulting from the cutoff leads to someaxial bending moment around the hole. Nevertheless, bychecking the FEM results, this axial bending stress (oradditional axial stress due to bending) is relatively small(less than 15%) compared with the stress caused by thestress concentration of the hole.

After identifying the suitable dimension ratio forcylinders, the proposed calculation method is applied to

Fig. 11. The SCFs for isotropic cylinders under axial tension.

Fig. 12. The SCFs for isotropic cylinders under internal pressure.


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an isotropic cylinder with a hole under internal pressure(two way loading). This case can also be considered as apressure vessel containing a circular cutout that iscommonly used in many engineering applications. Thesuperposition principle is employed based on the resultsof the same cylinder under uniaxial tension. The currentpredictions and the FEM results are given in Fig. 12. Atpoint-1, it can be found that the two results are very

close. However, at point-2, a large difference (more than20%) is found when the dimension ratio increases tomore than 0.5. A possible explanation comes from a two-way bending caused by the unsymmetrical effect of thecylinder under the internal pressure. The inuence of thebending in axial direction is small as just explained inFig. 11 and also veried here. However, the bending inhoop direction changes the local stress at point-2 andadds extra tension along the hoop direction. Thisinuence may be large, and may be responsible for thelarger differences between the predictions and the FEMresults at large d = D ratios. This local stress increment dueto the unsymmetrical effect is very complicated and isnot considered in the proposed calculation method.Nevertheless, this local tensile stress increment atpoint-2 has little inuence on the SCF predictions atpoint-1. Further study on the unsymmetrical effect isneeded. From Fig. 12, when the dimension ratio is lessthan 0.5 which is the usual case in the pressure vesselindustry, the proposed method is effective.

The predictions of the SCFs of an orthotropic cylinderunder axial tension is shown in Fig. 13. The SCFs atpoint-1 are calculated by the SCFs of the nite-widthisotropic plates ( Fig. 4) multiplied rst by the ratio of K

1 ;1T;i;c;u to K

1 ;1T;i;p;u to account for the cylindrical effect and

then by the ratio of K 1 ;1T;o;p;u to K 1 ;1T;i;p;u to account for the

orthotropic property. Two cases are investigated: axialtension in E 11 direction and in E 22 direction. Thematerial parameters are given in Table 2 . The proposedapproach gives predictions which is in very goodagreements with the FEM results.

4. Conclusion

For designing engineering structures with a circularcutout, a reliable estimation of SCFs is a must. The paperproposed a simple computation method to estimate the SCFsof nite-width isotropic/orthotropic plates/cylinders with acircular cutout and under uniaxial or biaxial tension. Thismethod is based on the SFs of nite-width isotropic plates(Fig. 4) and SCFs of an innite-width isotropic/orthotropicplate or an isotropic cylinder with d = D p ffiffiffiffi2t = Dp in one wayloading. K 1 ;1T;i;p;u; K 1 ;2T;i;p;u; K 1 ;1T;o;p;u; K 1 ;1T;i;c;u and K 1 ;2T;i;c;u can beconveniently calculated by Eqs. (2b) (3b). For an isotropicplate under biaxial tension, the SCFs are calculated by thesuperposition principle according to Fig. 4. For anorthotropic plate under uniaxial tension, the SCFs areestimated by the SCFs of the nite-width isotropic plates(Fig. 4) multiplied by the ratio of K

1 ;1T;o;p;u to K

1 ;1T;i;p;u: For an

isotropic cylinder under uniaxial tension, the SCFs at point-1 or point-2 are calculated by the SCFs of the nite-widthisotropic plates ( Fig. 4) multiplied by the ratios of K 1 ;1T;i;c;u toK

1 ;1T;i;p;u or K

1 ;2T;i;c;u to K

1 ;2T;i;p;u : The superposition principle will

be employed if the cylinder is under biaxial loading (internalpressure). For an orthotropic cylinder under uniaxialtension, the SCFs are calculated by the SCFs of the nite-width isotropic plates ( Fig. 4) multiplied rst by the ratio of

Fig. 13. The SCFs for orthotropic cylinders under axial tension.


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K 1 ;1T;i;c;u to K

1 ;1T;i;p;u to account for the cylindrical shape and then

by the ratio of K 1 ;1T;o;p;u to K

1 ;1T;i;p;u to account for the orthotropic

property. From the investigation, it can concluded that:

The proposed computation method is simple andefcient. It is well veried by the FEM simulations.

The SCFs only depend on the dimension ratio dened asthe hole diameter to plate width for plates, or the holediameter to cylinder diameter for cylinders. It is true bothfor isotropic and orthotropic structures. Variations in theactual structural dimensions are quite small.

For common orthotropic materials, the SFs of nite-width isotropic plates and orthotropic plates are quiteclose. If the dimension ratio of an orthotropic plate issmaller than 0.5, the SFs can be replaced by those of anisotropic plate.

At present, the SCFs of orthotropic plates and cylindersunder bi-axial loading cannot be reliably predicted by theproposed method because of the difculties in determin-ing analytically the negative SCFs at the point-2 locationof an innite-width orthotropic plate.

In the case of isotropic cylinders under two-way loading(internal pressure), the inuence of bending due tounsymmetrical cutoff in the hoop direction on the SCF atthe point-2 location ( Fig. 2) is quite large if thedimension ratio is larger than 0.5. Nevertheless, theinuence is very small at the point-1 location.

References

[1] Adams K. The effect of cutouts on strength of GRP for naval shiphulls. MS Thesis. Department of Materials Science and Engineering,MIT; 1986.

[2] Barbero EJ. Introduction to composite materials design. London:Taylor & Francis; 1998.

[3] Heywood RB. Designing by photoelasticity. London: Chapman &Hall; 1952.

[4] Brown JWF, Srawley JE. Plane strain crack toughness testing of highstrength metallic materials. STP 410, West Conshohocken, PA:American Society for Testing and Materials; 1969.

[5] Tan SC. Laminated composites containing an elliptical opening.I. Approximate stress analyses and fracture models. J Compos Mater1987;21:92548.

[6] Tan SC. Stress concentrations in laminated composites. Lancaster:Technomic; 1994.

[7] Savin GN. Stress concentration around holes. New York: PergamonPress; 1961.

[8] Timoshenko SP, Goodier JN. Theory of elasticity, 3rd ed. New York:McGraw-Hill; 1979.

[9] Zienkiewicz OC. Finite element method. New York: McGraw-Hill;

1989.[10] Wu HC. Determination of the maximum allowable circular holes

(cutouts) of the GRP pipe under working pressure of 1216 bars.Internal report. Department of Civil and Environmental Engineering,University of Michigan, Ann Arbor; 1997.

[11] Hong CS, Crews JH. Stress-concentration factors for nite orthotropiclaminates with a circular hole and uniaxial loading. NASA TP-1469;1979.

[12] Crew JH, Hong CS, Raju IS. Stress-concentration factors for niteorthotropic laminates with a pin-loaded hole. NASA TP-1862; 1981.


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