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orthotropic plates and cylinders with a circular hole
Composites: Part B 34 (2003) 127–134 www.elsevier.com/locate/compositesb
On stress concentrations for isotropic/orthotropic plates and 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 finite width isotropic plate with a circular hole and under uniaxial loading. In this paper, these SFs were also found to be valid in an isotropic plate with biaxial loading and an isotropic cylinder with uniaxial loading or internal pressure, if a suitable hole to structure dimension ratio was chosen. The study 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 influence of the structural dimension on the SCF was also studied. An empirical calculation method for the stress concentrations for isotropic/orthotropic plates and cylinders with a circular hole was proposed and the results agreed well with the FEM simulations. This research work may provide structure engineers a simple and efficient way to estimate the 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
1.1. Isotropic plate
Stress concentrations around cutouts have great practical importance because they are normally the cause of failure. For most materials, the failure strengths of the materials are strongly notch (or hole) sensitive. The net failure stress, taking into account the reduction in cross-sectional area, is typically much less than the ultimate tensile strength of the same materials without the notch or hole. For example, strength reductions of 40 – 60% have been reported for a glass fiber reinforced plastic plate [1]. Hence a better approach dealing with strength reductions due to stress concentration around a geometric discontinuity (such as a hole) is the use of the stress concentration model or point stress model [2]. Failure is predicted by the use of elastic stress concentration factor (SCF), KT, without considering sharp edge cracks around the hole. KT is defined as the ratio of the maximum stress in the presence of a geometric irregularity or discontinuity to the stress that would exist at the same point if the irregularity was not present.
A stress concentration is typically introduced in plates and cylinders in the form of circular holes. This form of cutout has many practical applications and is familiar to most engineers. Most of the strength analyses involving SCFs are based on the conditions of infinite-width/diameter plate/cylinder because closed form stress distributions are available. Therefore, there is a need to account for the effect of finite width/diameter on SCFs. Such effect can be suitably considered through the use of a scale factor (SF) defined as the ratio of the SCF of finite-size structure to that of infinitesize structure. The SFs for isotropic plates are not a function of the material properties. Therefore, the SFs for an isotropic plate with a center hole can be determined accurately using a curve fitting technique [3,4]. For instance, the following relationship has been obtained for a finite-width plate [3]:
* Corresponding author. Tel.: þ 1-313-577-0745; fax: þ1-313-577-3881. E-mail address: [email protected] (H.-C. Wu).
1;1 KT;i;p;u 1 KT;i;p;u
¼
3ð1 2 d=WÞ 2 þ ð1 2 d=WÞ3
ð1Þ
1;1 1 where KT;i;p;u and KT;i;p;u represent the SCF at point-1 in an isotropic plate under uniaxial tension with infinite and finite
1359-8368/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 8 3 6 8 ( 0 2 ) 0 0 0 9 7 - 5
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1.3. Isotropic pipe In actual pipes, a hole is located on the circular surface of a cylinder rather than on a flat surface. Therefore, the effect of such plate curvature on the SCF should pffiffiffiffiffiffibe investigated. For an isotropic cylinder with d=D p 2t=D and under axial loading (along cylinder direction, Fig. 2), the SCFs at point1 and point-2 are [7] sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ðm2 2 1Þ p d2 1;1 ð3aÞ ¼3þ KT;i;c;u 8 Dt m2 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3ðm 2 1Þ p d 1;2 A KT;i;c;u ¼ 2@1 þ ð3bÞ 8 Dt m2 Fig. 1. A plate with a central hole and under uniaxial tension.
1;1 width, respectively (Fig. 1). For an infinite plate, KT;i;p;u ¼ 1;2 3:0 and KT;i;p;u ¼ 21:0 (point-2 in Fig. 1); d is the diameter of the hole and W is the width of the plate. Subscripts ‘i’, ‘p’, and ‘u’ represent isotropic material, plate and uniaxial loading, respectively. Superscript ‘1’ represents infinite size structure, such as infinite-width plate and infinitediameter cylinder.
1.2. Orthotropic plate As for finite-width orthotropic plates, the stress analyses have been produced mainly by the finite element method. By assuming an approximate stress distribution for a finite orthotropic plate containing a circular hole, Tan [5,6] derived a closed form solution for SCFs of finite orthotropic plates containing a central circular opening and under uniaxial loading. The SCFs depend on the material parameters. 6 1;1 KT;o;p;u 3ð1 2 d=WÞ 1 d 1;1 M ðKT;o;p;u ¼ þ 2 3Þ 1 2 W 2 þ ð1 2 d=WÞ3 KT;o;p;u " 2 # d M ð2aÞ 12 W 1;1 KT;o;p;u
1 KT;o;p;u
where and represent the SCF at point-1 for an orthotropic plate under uniaxial tension with infinite and finite width, respectively. M is a magnification factor and is only a function of d=W: Detailed expression of M can be found in Ref. [6]. Subscript ‘o’ represents orthotropic material. The SCF for an infinite orthotropic plate can be written as [2] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u sffiffiffiffiffi u Ey Ey 1;1 t KT;o;p;u ¼ 1 þ 2 ð2bÞ 2 nyx þ Ex 2Gyx where Ex and Ey are Young’s modulus in x and y direction (see Fig. 1), respectively. Gyx is shear modulus in x –y plane and nyx is Poisson’s ratio.
1;1 1;2 and KT;i;c;u represent the SCFs at point-1 and where KT;i;c;u point-2, respectively; m ¼ 1=n; n is Poisson’s ratio; d is the diameter of the hole and D is the diameter of the cylinder; t is wall thickness. In the case of an isotropic cylinder under two way loading pffiffiffiffiffiffi (e.g. under internal pressure, P0), with d=D p 2t=D; Savin [7] also gave the SCFs at point-1 and point-2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi su¼0 3ðm2 2 1Þ p d 2 1;1 KT;i;c;b ¼ ¼12 ð4aÞ 8 Dt P0 D=4t m2 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 s 5 3ðm 2 1Þ 9p d 1;2 A KT;i;c;b ¼ u¼p=2 ¼ @1 þ ð4bÞ 2 40 Dt P0 D=2t m2
where P0 D=2t and P0 D=4t are the stresses in the hoop and longitudinal directions, respectively. Neglecting the curvature terms in Eqs. (3a)– (4b), it can 1;1 1;2 be found that the Eqs. (3a) and (3b) recover KT;i;p;u ; KT;i;p;u and the Eqs. (4a) and (4b) are identical with the results 1;1 1;2 obtained by superposition of KT;i;p;u ; KT;i;p;u : This reminds us whether we can make use of the SFs for isotropic plates which have been already deduced to estimate the SCFs and SFs for isotropic cylinders through some simple calculation method and whether this calculation method is also valid for orthotropic structures (plates and cylinders)? 1.4. Proposed approach The objective of this paper is to provide structural engineers a simple and reliable estimation method for SCFs in common structures. For this purpose, a systematic study of SCFs and SFs for isotropic/orthotropic plates/curved
Fig. 2. A pipe with a hole under uniaxial tension.
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129
Fig. 3. FEM mesh for a plate with a central hole.
plates (cylinders) containing a circular hole was carried out in this paper. Based on the SCFs and SFs of an isotropic plate and superposition principle, a simple computational method was proposed to estimate SFs and SCFs of an isotropic cylinder under one way or two way (internal pressure) loading. Then this method is extended to orthotropic structures. The proposed method can be described as follows: † For an isotropic plate under biaxial tension, the SCFs are calculated from the individual SCFs of the same plate under uniaxial tension then by summing up the SCFs together at the same location according to the superposition principle. It should be noted that the SFs obtained from the FEM results (see Fig. 4) for point-2 location would be used in the calculation, since there is no analytical equation available at present. † For a finite-width orthotropic plate under uniaxial tension, the SCFs at point-1 are estimated by the SCFs of a corresponding finite-width isotropic plate (Fig. 4) 1;1 1;1 multiplied by the ratio of KT;o;p;u to KT;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 finite-width isotropic plates (Fig. 4) multiplied by the 1;1 1;1 1;2 1;2 to KT;i;p;u or KT;i;c;u to KT;i;p;u : ratios of KT;i;c;u † The superposition principle will be employed if an isotropic cylinder is under biaxial loading (e.g. internal pressure). † For an orthotropic cylinder under uniaxial tension, two calculation steps should be adopted. The SCFs at point-1 are calculated by the SCFs of the finite-width isotropic 1;1 to plates (Fig. 4) multiplied first by the ratio of KT;i;c;u 1;1 KT;i;p;u to account for the cylindrical effect and then by Table 1 Geometrical size of plates and cylinders
P1
P2
C
W (mm)
d (mm)
t (mm)
D (mm)
L (mm)
d=W or d=D
101.6 101.6 101.6 101.6 152.4 152.4 152.4 – – – –
15.24 30.48 50.8 76.2 22.86 76.2 114.3 7.62 30.48 76.2 127
10.16 10.16 10.16 10.16 10.16 10.16 10.16 10.16 10.16 10.16 10.16
– – – – – – – 203.2 203.2 203.2 203.2
254 254 254 254 254 254 254 508 508 508 508
0.15 0.3 0.5 0.75 0.15 0.5 0.75 0.0375 0.15 0.375 0.625
Fig. 4. The SCFs for isotropic plates under uniaxial tension. 1;1 1;1 to KT;i;p;u to account for the orthotropic the ratio of KT;o;p;u property.
1.5. FEM simulation In the FEM simulations, the first order shear deformation shell theory is adopted [9]. The displacement field is given by uðx; y; zÞ ¼ u0 ðx; yÞ þ zcx ðx; yÞ vðx; y; zÞ ¼ v0 ðx; yÞ 2 zcy ðx; yÞ
ð5Þ
wðx; y; zÞ ¼ wðx; yÞ where u, v, w are displacement components in the x, y, z directions, respectively; cx and cy are rotations of the crosssection about the x and y axis; and u0 and v0 are displacement components at the mid-plane of the plate. In ABAQUS, the plate/cylinder was modeled with 4-noded quadrilateral and 3-noded triangular shell elements with six degrees of freedom: three displacement components and three rotation components ðu; v; w; ux ; uy ; uz Þ at each node (S4R and S3R). In the following sections, the calculation results are verified by the FEM simulations (using ABAQUS). The application range of the proposed approach and difficulties are discussed. The research findings are found useful to structural design.
2. SCFs of an isotropic/orthotropic plate under uniaxial or biaxial tension The SCFs of an isotropic plate under uniaxial or biaxial tension have been studied extensively and good results have been reported [3,6 – 8]. There are also formulae for SCFs of an orthotropic plate under uniaxial tension [5,6]. For completeness, this paper begins with FEM formulation and compares the FEM results with the closed form solutions. The analysis emphasizes cutout size effect on
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Fig. 7. The SCFs for orthotropic plates under uniaxial tension (in E22 direction). Fig. 5. The SCFs for isotropic plates under biaxial tension.
the SCFs under a fixed ratio of the cutout size (such as hole) to the structural dimension (such as a plate width or cylinder diameter). It also emphasizes the relationship between the SFs of an isotropic plate and an orthotropic plate. A typical FEM mesh of a plate with a center hole is given in Fig. 3. The physical sizes of the plates are given in Table 1. There are two different plate widths and various hole-diameters to give the same d=W ratios (P1 and P2 in Table 1). Fig. 4 presents the SCFs of an isotropic plate with different d=W and under uniaxial tension. At point-1 location, the FEM simulations for SCFs agree well with Eq. (1) (Heywood’s curve [3]). For the same d=W; the actual hole size or plate width has little influence on the SCFs. Fig. 4 also gives the SCFs at the point-2 location (compression). For an isotropic infinite width plate, this SCF equals 2 1. Based on the SCFs (at point-1 and point-2) given in Fig. 4 and by the superposition principle, the SCFs at the point-1 location of an isotropic plate under biaxial tension (in x and y directions) can be calculated following the approach described in this paper. For the two groups of plates (P1 and P2), the SCFs are shown in Fig. 5. It should be noted that for rectangular plates, one should use two hole-diameter to plate-width ratios (d=W and d=L; see
Table 1) to account for the biaxial effect. There are two such predictions based on the geometry of the two rectangular plates (P1 and P2 in Table 1). Again, the influence of the hole size and plate width is insignificant when the d=W ratio is fixed. SCFs at point-2 under biaxial loading can be obtained in a similar manner based on the point-2 curves in Fig. 4. Next, the SFs of isotropic plates under uniaxial tension are extended to orthotropic plates with a central hole. The SCF of an infinite orthotropic plate is given by Eq. (2a). For finite width orthotropic plates, the SCFs can be predicted by the proposed approach, as shown in Figs. 6 and 7 for uniaxial tension in the longitudinal (E11 is in y direction, Fig. 1) and transverse (E22) directions, respectively. The relevant orthotropic material parameters are listed in Table 2 [10]. In these two figures, the proposed computation approach results in good agreements with the FEM simulations, as well as Eq. (2a). Comparing Eqs. (1) and (2a), we can find that the first part of Eq. (2a) is identical to Eq. (1). It suggests that if the second part of Eq. (2a) is much smaller than the first part, i.e. Eq. (1), the differences in the SFs for isotropic plates and orthotropic plates are very small. This comparison is given in Fig. 8. In Fig. 8, A2 represents the second part of Eq. (2a) and A1 represents the 1;1 are used in first part of Eq. (2a), i.e. Eq. (1). Different KT;o;p;u the figure. We can see that when the dimension ratio, d=W; is less than 0.5, A2 can be always neglected and hence the SFs for isotropic plates can be also used for orthotropic plates. When d=W is larger than 0.5, the applicable range would 1;1 : For ordinary fiber depend on the actual values of KT;o;p;u 1;1 reinforced composites, KT;o;p;u is usually less than 10. Table 2 The orthotropic material parameters E11 (GPa) E22 (GPa) u12 20
Fig. 6. The SCFs for orthotropic plates under uniaxial tension (in E11 direction).
9.2
u21
G12 (GPa) G13 (GPa) G23 (GPa)
0.341 0.157 5.0
5.0
2.6
E11, modulus along the fiber direction; E22, modulus in the perpendicular direction; u12 and u21, Poisson’s ratios; G12, G13, and G23, shear moduli.
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131
Fig. 10. FEM mesh for a cylinder with a hole.
1;1 Fig. 8. Orthotropic influence (KT;o;p;u ¼ 3:5; 10, 30, 100) on SCFs of plates under uniaxial tension.
1;1 For instance as shown in Figs. 6 and 7, KT;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 applicable to orthotropic plates up to almost d=W ¼ 0:75: From Figs. 4 –7, when the dimension ratios are fixed, the SCFs are determined and are independent of the sizes of the plates or the holes. However, at point-2, the negative SCFs can be very different for the orthotropic plates with different material properties or under different uniaxial loading directions (comparing Figs. 6 and 7). The SCFs at point-2 depend strongly on the composite modulus. As reported in the literature [11,12], for some infinite-width orthotropic plates under uniaxial tension, the SCF at point-2 can reach 2 4, much different from that of corresponding isotropic plates which is always 2 1. Another difficulty is that currently there is no theoretical equation to predict the SCF at point-2 location of an infinite-width orthotropic plate under uniaxial tension, unlike the SCF at point-1 (Eq. (2b)). Due to these two reasons, it is not always reliable to predict the SCFs and SFs both at point-1 and point-2 of finite-width orthotropic plates under biaxial tension based on the proposed calculation method. Prior to applying the proposed method
Fig. 9. The SCFs for orthotropic plates under biaxial tension.
to biaxial tension, the applicable range (i.e. the values of material properties and loading directions) should be investigated. Such applicable range should be a function of the material parameters. In this preliminary study, the SCF at point-1 location of an orthotropic plate (with the materials properties given in Table 2) under biaxial loading is estimated by the proposed calculation method. The SCFs are calculated from the SCFs of the same plate under separate uniaxial tension in E11 and E22 directions, then by the superposition principle to add the SCFs together at the same location. The predictions match the FEM results quite well (Fig. 9). Further study will be carried out to examine the applicable range of this calculation method.
3. SCFs of an isotropic/orthotropic cylinder under uniaxial tension or internal pressure In Section 2, the SFs of isotropic plates under uniaxial tension are used to estimate the SCFs of isotropic plates under biaxial tension and SCFs of orthotropic plates under uniaxial tension. All of the plates studied have a center circular hole. In this section, the proposed calculation method will be applied to hollow cylinders with a circular cutout 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-defined dimension ratio, defined as the ratio of the hole diameter to the plate width ðd=WÞ: However, this ratio may not be valid in the cylinder case. Finding a valid dimension ratio is a prerequisite for the proposed calculation method. For cylinders (Fig. 2), there are four possible dimension ratios which can be defined as: d=D; d=pD; d2 =2Dt and arcsinðd=DÞ= 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 the cylinder, pD: Actually, the second and the fourth are quite close because in real engineering applications, the circular cutout in a cylinder cannot be very large. Therefore we can consider and compare only the first 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 axial tension. The SCFs at point-1 or point-2 are calculated by
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Fig. 11. The SCFs for isotropic cylinders under axial tension.
the SCFs of the finite-width isotropic plates (Fig. 4) 1;1 1;1 1;2 multiplied by the ratios of KT;i;c;u to KT;i;p;u or KT;i;c;u to 1;2 : The relationships of the SCFs versus the first three KT;i;p;u dimension ratios are calculated based on the proposed calculation method. The results (Fig. 11) show that the applicable range of the second ratio ðd=pÞ is very small (only to 0.2) because the cutout diameter is seldom larger than 60% of the cylinder’s diameter. Using d 2 =2Dt; Savin’s results [7] overestimates the hole effect of the cylinder for larger dimension ratios. In fact, as discussed by pSavin ffiffiffiffiffiffi [7], his prediction is only valid in the case of d=D p 2t=D; i.e. for very small hole sizes. If the diameter ratio, d=D; is selected, the pre-edited SCFs agree well with FEM results
both at point-1 and point-2. It means that by using this dimension ratio, the proposed calculation based on the SFs of an isotropic plate under one way loading can describe the SCFs well around a circular cutout in a cylinder under axial tension. It has been noticed that under axial tension, an unsymmetric effect resulting from the cutoff leads to some axial bending moment around the hole. Nevertheless, by checking the FEM results, this axial bending stress (or additional axial stress due to bending) is relatively small (less than 15%) compared with the stress caused by the stress concentration of the hole. After identifying the suitable dimension ratio for cylinders, the proposed calculation method is applied to
Fig. 12. The SCFs for isotropic cylinders under internal pressure.
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133
Fig. 13. The SCFs for orthotropic cylinders under axial tension.
an isotropic cylinder with a hole under internal pressure (two way loading). This case can also be considered as a pressure vessel containing a circular cutout that is commonly used in many engineering applications. The superposition principle is employed based on the results of the same cylinder under uniaxial tension. The current predictions and the FEM results are given in Fig. 12. At point-1, it can be found that the two results are very close. However, at point-2, a large difference (more than 20%) is found when the dimension ratio increases to more than 0.5. A possible explanation comes from a twoway bending caused by the unsymmetrical effect of the cylinder under the internal pressure. The influence of the bending in axial direction is small as just explained in Fig. 11 and also verified here. However, the bending in hoop direction changes the local stress at point-2 and adds extra tension along the hoop direction. This influence may be large, and may be responsible for the larger differences between the predictions and the FEM results at large d=D ratios. This local stress increment due to the unsymmetrical effect is very complicated and is not considered in the proposed calculation method. Nevertheless, this local tensile stress increment at point-2 has little influence on the SCF predictions at point-1. Further study on the unsymmetrical effect is needed. From Fig. 12, when the dimension ratio is less than 0.5 which is the usual case in the pressure vessel industry, the proposed method is effective. The predictions of the SCFs of an orthotropic cylinder under axial tension is shown in Fig. 13. The SCFs at point-1 are calculated by the SCFs of the finite-width isotropic plates (Fig. 4) multiplied first by the ratio of 1;1 1;1 KT;i;c;u to KT;i;p;u to account for the cylindrical effect and
1;1 1;1 to KT;i;p;u to account for the then by the ratio of KT;o;p;u orthotropic property. Two cases are investigated: axial tension in E11 direction and in E22 direction. The material parameters are given in Table 2. The proposed approach gives predictions which is in very good agreements with the FEM results.
4. Conclusion For designing engineering structures with a circular cutout, a reliable estimation of SCFs is a must. The paper proposed a simple computation method to estimate the SCFs of finite-width isotropic/orthotropic plates/cylinders with a circular cutout and under uniaxial or biaxial tension. This method is based on the SFs of finite-width isotropic plates (Fig. 4) and SCFs of an infinite-width isotropic/orthotropic pffiffiffiffiffiffi plate or an isotropic cylinder with d=D p 2t=D in one way 1;1 1;2 1;1 1;1 1;2 loading. KT;i;p;u ; KT;i;p;u ; KT;o;p;u ; KT;i;c;u and KT;i;c;u can be conveniently calculated by Eqs. (2b) –(3b). For an isotropic plate under biaxial tension, the SCFs are calculated by the superposition principle according to Fig. 4. For an orthotropic plate under uniaxial tension, the SCFs are estimated by the SCFs of the finite-width isotropic plates 1;1 1;1 (Fig. 4) multiplied by the ratio of KT;o;p;u to KT;i;p;u : For an isotropic cylinder under uniaxial tension, the SCFs at point1 or point-2 are calculated by the SCFs of the finite-width 1;1 isotropic plates (Fig. 4) multiplied by the ratios of KT;i;c;u to 1;1 1;2 1;2 KT;i;p;u or KT;i;c;u to KT;i;p;u : The superposition principle will be employed if the cylinder is under biaxial loading (internal pressure). For an orthotropic cylinder under uniaxial tension, the SCFs are calculated by the SCFs of the finitewidth isotropic plates (Fig. 4) multiplied first by the ratio of
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1;1 1;1 KT;i;c;u to KT;i;p;u to account for the cylindrical shape and then 1;1 1;1 by the ratio of KT;o;p;u to KT;i;p;u to account for the orthotropic property. From the investigation, it can concluded that:
† The proposed computation method is simple and efficient. It is well verified by the FEM simulations. † The SCFs only depend on the dimension ratio defined as the hole diameter to plate width for plates, or the hole diameter to cylinder diameter for cylinders. It is true both for isotropic and orthotropic structures. Variations in the actual structural dimensions are quite small. † For common orthotropic materials, the SFs of finitewidth isotropic plates and orthotropic plates are quite close. If the dimension ratio of an orthotropic plate is smaller than 0.5, the SFs can be replaced by those of an isotropic plate. † At present, the SCFs of orthotropic plates and cylinders under bi-axial loading cannot be reliably predicted by the proposed method because of the difficulties in determining analytically the negative SCFs at the point-2 location of an infinite-width orthotropic plate. † In the case of isotropic cylinders under two-way loading (internal pressure), the influence of bending due to unsymmetrical cutoff in the hoop direction on the SCF at the point-2 location (Fig. 2) is quite large if the dimension ratio is larger than 0.5. Nevertheless, the influence is very small at the point-1 location.
References [1] Adams K. The effect of cutouts on strength of GRP for naval ship hulls. 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 high strength 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 Mater 1987;21:925–48. [6] Tan SC. Stress concentrations in laminated composites. Lancaster: Technomic; 1994. [7] Savin GN. Stress concentration around holes. New York: Pergamon Press; 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 12–16 bars. Internal report. Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor; 1997. [11] Hong CS, Crews JH. Stress-concentration factors for finite orthotropic laminates with a circular hole and uniaxial loading. NASA TP-1469; 1979. [12] Crew JH, Hong CS, Raju IS. Stress-concentration factors for finite orthotropic laminates with a pin-loaded hole. NASA TP-1862; 1981.
On stress concentrations for isotropic/orthotropic plates and 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 finite width isotropic plate with a circular hole and under uniaxial loading. In this paper, these SFs were also found to be valid in an isotropic plate with biaxial loading and an isotropic cylinder with uniaxial loading or internal pressure, if a suitable hole to structure dimension ratio was chosen. The study 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 influence of the structural dimension on the SCF was also studied. An empirical calculation method for the stress concentrations for isotropic/orthotropic plates and cylinders with a circular hole was proposed and the results agreed well with the FEM simulations. This research work may provide structure engineers a simple and efficient way to estimate the 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
1.1. Isotropic plate
Stress concentrations around cutouts have great practical importance because they are normally the cause of failure. For most materials, the failure strengths of the materials are strongly notch (or hole) sensitive. The net failure stress, taking into account the reduction in cross-sectional area, is typically much less than the ultimate tensile strength of the same materials without the notch or hole. For example, strength reductions of 40 – 60% have been reported for a glass fiber reinforced plastic plate [1]. Hence a better approach dealing with strength reductions due to stress concentration around a geometric discontinuity (such as a hole) is the use of the stress concentration model or point stress model [2]. Failure is predicted by the use of elastic stress concentration factor (SCF), KT, without considering sharp edge cracks around the hole. KT is defined as the ratio of the maximum stress in the presence of a geometric irregularity or discontinuity to the stress that would exist at the same point if the irregularity was not present.
A stress concentration is typically introduced in plates and cylinders in the form of circular holes. This form of cutout has many practical applications and is familiar to most engineers. Most of the strength analyses involving SCFs are based on the conditions of infinite-width/diameter plate/cylinder because closed form stress distributions are available. Therefore, there is a need to account for the effect of finite width/diameter on SCFs. Such effect can be suitably considered through the use of a scale factor (SF) defined as the ratio of the SCF of finite-size structure to that of infinitesize structure. The SFs for isotropic plates are not a function of the material properties. Therefore, the SFs for an isotropic plate with a center hole can be determined accurately using a curve fitting technique [3,4]. For instance, the following relationship has been obtained for a finite-width plate [3]:
* Corresponding author. Tel.: þ 1-313-577-0745; fax: þ1-313-577-3881. E-mail address: [email protected] (H.-C. Wu).
1;1 KT;i;p;u 1 KT;i;p;u
¼
3ð1 2 d=WÞ 2 þ ð1 2 d=WÞ3
ð1Þ
1;1 1 where KT;i;p;u and KT;i;p;u represent the SCF at point-1 in an isotropic plate under uniaxial tension with infinite and finite
1359-8368/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 8 3 6 8 ( 0 2 ) 0 0 0 9 7 - 5
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1.3. Isotropic pipe In actual pipes, a hole is located on the circular surface of a cylinder rather than on a flat surface. Therefore, the effect of such plate curvature on the SCF should pffiffiffiffiffiffibe investigated. For an isotropic cylinder with d=D p 2t=D and under axial loading (along cylinder direction, Fig. 2), the SCFs at point1 and point-2 are [7] sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ðm2 2 1Þ p d2 1;1 ð3aÞ ¼3þ KT;i;c;u 8 Dt m2 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3ðm 2 1Þ p d 1;2 A KT;i;c;u ¼ 2@1 þ ð3bÞ 8 Dt m2 Fig. 1. A plate with a central hole and under uniaxial tension.
1;1 width, respectively (Fig. 1). For an infinite plate, KT;i;p;u ¼ 1;2 3:0 and KT;i;p;u ¼ 21:0 (point-2 in Fig. 1); d is the diameter of the hole and W is the width of the plate. Subscripts ‘i’, ‘p’, and ‘u’ represent isotropic material, plate and uniaxial loading, respectively. Superscript ‘1’ represents infinite size structure, such as infinite-width plate and infinitediameter cylinder.
1.2. Orthotropic plate As for finite-width orthotropic plates, the stress analyses have been produced mainly by the finite element method. By assuming an approximate stress distribution for a finite orthotropic plate containing a circular hole, Tan [5,6] derived a closed form solution for SCFs of finite orthotropic plates containing a central circular opening and under uniaxial loading. The SCFs depend on the material parameters. 6 1;1 KT;o;p;u 3ð1 2 d=WÞ 1 d 1;1 M ðKT;o;p;u ¼ þ 2 3Þ 1 2 W 2 þ ð1 2 d=WÞ3 KT;o;p;u " 2 # d M ð2aÞ 12 W 1;1 KT;o;p;u
1 KT;o;p;u
where and represent the SCF at point-1 for an orthotropic plate under uniaxial tension with infinite and finite width, respectively. M is a magnification factor and is only a function of d=W: Detailed expression of M can be found in Ref. [6]. Subscript ‘o’ represents orthotropic material. The SCF for an infinite orthotropic plate can be written as [2] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u sffiffiffiffiffi u Ey Ey 1;1 t KT;o;p;u ¼ 1 þ 2 ð2bÞ 2 nyx þ Ex 2Gyx where Ex and Ey are Young’s modulus in x and y direction (see Fig. 1), respectively. Gyx is shear modulus in x –y plane and nyx is Poisson’s ratio.
1;1 1;2 and KT;i;c;u represent the SCFs at point-1 and where KT;i;c;u point-2, respectively; m ¼ 1=n; n is Poisson’s ratio; d is the diameter of the hole and D is the diameter of the cylinder; t is wall thickness. In the case of an isotropic cylinder under two way loading pffiffiffiffiffiffi (e.g. under internal pressure, P0), with d=D p 2t=D; Savin [7] also gave the SCFs at point-1 and point-2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi su¼0 3ðm2 2 1Þ p d 2 1;1 KT;i;c;b ¼ ¼12 ð4aÞ 8 Dt P0 D=4t m2 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 s 5 3ðm 2 1Þ 9p d 1;2 A KT;i;c;b ¼ u¼p=2 ¼ @1 þ ð4bÞ 2 40 Dt P0 D=2t m2
where P0 D=2t and P0 D=4t are the stresses in the hoop and longitudinal directions, respectively. Neglecting the curvature terms in Eqs. (3a)– (4b), it can 1;1 1;2 be found that the Eqs. (3a) and (3b) recover KT;i;p;u ; KT;i;p;u and the Eqs. (4a) and (4b) are identical with the results 1;1 1;2 obtained by superposition of KT;i;p;u ; KT;i;p;u : This reminds us whether we can make use of the SFs for isotropic plates which have been already deduced to estimate the SCFs and SFs for isotropic cylinders through some simple calculation method and whether this calculation method is also valid for orthotropic structures (plates and cylinders)? 1.4. Proposed approach The objective of this paper is to provide structural engineers a simple and reliable estimation method for SCFs in common structures. For this purpose, a systematic study of SCFs and SFs for isotropic/orthotropic plates/curved
Fig. 2. A pipe with a hole under uniaxial tension.
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129
Fig. 3. FEM mesh for a plate with a central hole.
plates (cylinders) containing a circular hole was carried out in this paper. Based on the SCFs and SFs of an isotropic plate and superposition principle, a simple computational method was proposed to estimate SFs and SCFs of an isotropic cylinder under one way or two way (internal pressure) loading. Then this method is extended to orthotropic structures. The proposed method can be described as follows: † For an isotropic plate under biaxial tension, the SCFs are calculated from the individual SCFs of the same plate under uniaxial tension then by summing up the SCFs together at the same location according to the superposition principle. It should be noted that the SFs obtained from the FEM results (see Fig. 4) for point-2 location would be used in the calculation, since there is no analytical equation available at present. † For a finite-width orthotropic plate under uniaxial tension, the SCFs at point-1 are estimated by the SCFs of a corresponding finite-width isotropic plate (Fig. 4) 1;1 1;1 multiplied by the ratio of KT;o;p;u to KT;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 finite-width isotropic plates (Fig. 4) multiplied by the 1;1 1;1 1;2 1;2 to KT;i;p;u or KT;i;c;u to KT;i;p;u : ratios of KT;i;c;u † The superposition principle will be employed if an isotropic cylinder is under biaxial loading (e.g. internal pressure). † For an orthotropic cylinder under uniaxial tension, two calculation steps should be adopted. The SCFs at point-1 are calculated by the SCFs of the finite-width isotropic 1;1 to plates (Fig. 4) multiplied first by the ratio of KT;i;c;u 1;1 KT;i;p;u to account for the cylindrical effect and then by Table 1 Geometrical size of plates and cylinders
P1
P2
C
W (mm)
d (mm)
t (mm)
D (mm)
L (mm)
d=W or d=D
101.6 101.6 101.6 101.6 152.4 152.4 152.4 – – – –
15.24 30.48 50.8 76.2 22.86 76.2 114.3 7.62 30.48 76.2 127
10.16 10.16 10.16 10.16 10.16 10.16 10.16 10.16 10.16 10.16 10.16
– – – – – – – 203.2 203.2 203.2 203.2
254 254 254 254 254 254 254 508 508 508 508
0.15 0.3 0.5 0.75 0.15 0.5 0.75 0.0375 0.15 0.375 0.625
Fig. 4. The SCFs for isotropic plates under uniaxial tension. 1;1 1;1 to KT;i;p;u to account for the orthotropic the ratio of KT;o;p;u property.
1.5. FEM simulation In the FEM simulations, the first order shear deformation shell theory is adopted [9]. The displacement field is given by uðx; y; zÞ ¼ u0 ðx; yÞ þ zcx ðx; yÞ vðx; y; zÞ ¼ v0 ðx; yÞ 2 zcy ðx; yÞ
ð5Þ
wðx; y; zÞ ¼ wðx; yÞ where u, v, w are displacement components in the x, y, z directions, respectively; cx and cy are rotations of the crosssection about the x and y axis; and u0 and v0 are displacement components at the mid-plane of the plate. In ABAQUS, the plate/cylinder was modeled with 4-noded quadrilateral and 3-noded triangular shell elements with six degrees of freedom: three displacement components and three rotation components ðu; v; w; ux ; uy ; uz Þ at each node (S4R and S3R). In the following sections, the calculation results are verified by the FEM simulations (using ABAQUS). The application range of the proposed approach and difficulties are discussed. The research findings are found useful to structural design.
2. SCFs of an isotropic/orthotropic plate under uniaxial or biaxial tension The SCFs of an isotropic plate under uniaxial or biaxial tension have been studied extensively and good results have been reported [3,6 – 8]. There are also formulae for SCFs of an orthotropic plate under uniaxial tension [5,6]. For completeness, this paper begins with FEM formulation and compares the FEM results with the closed form solutions. The analysis emphasizes cutout size effect on
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Fig. 7. The SCFs for orthotropic plates under uniaxial tension (in E22 direction). Fig. 5. The SCFs for isotropic plates under biaxial tension.
the SCFs under a fixed ratio of the cutout size (such as hole) to the structural dimension (such as a plate width or cylinder diameter). It also emphasizes the relationship between the SFs of an isotropic plate and an orthotropic plate. A typical FEM mesh of a plate with a center hole is given in Fig. 3. The physical sizes of the plates are given in Table 1. There are two different plate widths and various hole-diameters to give the same d=W ratios (P1 and P2 in Table 1). Fig. 4 presents the SCFs of an isotropic plate with different d=W and under uniaxial tension. At point-1 location, the FEM simulations for SCFs agree well with Eq. (1) (Heywood’s curve [3]). For the same d=W; the actual hole size or plate width has little influence on the SCFs. Fig. 4 also gives the SCFs at the point-2 location (compression). For an isotropic infinite width plate, this SCF equals 2 1. Based on the SCFs (at point-1 and point-2) given in Fig. 4 and by the superposition principle, the SCFs at the point-1 location of an isotropic plate under biaxial tension (in x and y directions) can be calculated following the approach described in this paper. For the two groups of plates (P1 and P2), the SCFs are shown in Fig. 5. It should be noted that for rectangular plates, one should use two hole-diameter to plate-width ratios (d=W and d=L; see
Table 1) to account for the biaxial effect. There are two such predictions based on the geometry of the two rectangular plates (P1 and P2 in Table 1). Again, the influence of the hole size and plate width is insignificant when the d=W ratio is fixed. SCFs at point-2 under biaxial loading can be obtained in a similar manner based on the point-2 curves in Fig. 4. Next, the SFs of isotropic plates under uniaxial tension are extended to orthotropic plates with a central hole. The SCF of an infinite orthotropic plate is given by Eq. (2a). For finite width orthotropic plates, the SCFs can be predicted by the proposed approach, as shown in Figs. 6 and 7 for uniaxial tension in the longitudinal (E11 is in y direction, Fig. 1) and transverse (E22) directions, respectively. The relevant orthotropic material parameters are listed in Table 2 [10]. In these two figures, the proposed computation approach results in good agreements with the FEM simulations, as well as Eq. (2a). Comparing Eqs. (1) and (2a), we can find that the first part of Eq. (2a) is identical to Eq. (1). It suggests that if the second part of Eq. (2a) is much smaller than the first part, i.e. Eq. (1), the differences in the SFs for isotropic plates and orthotropic plates are very small. This comparison is given in Fig. 8. In Fig. 8, A2 represents the second part of Eq. (2a) and A1 represents the 1;1 are used in first part of Eq. (2a), i.e. Eq. (1). Different KT;o;p;u the figure. We can see that when the dimension ratio, d=W; is less than 0.5, A2 can be always neglected and hence the SFs for isotropic plates can be also used for orthotropic plates. When d=W is larger than 0.5, the applicable range would 1;1 : For ordinary fiber depend on the actual values of KT;o;p;u 1;1 reinforced composites, KT;o;p;u is usually less than 10. Table 2 The orthotropic material parameters E11 (GPa) E22 (GPa) u12 20
Fig. 6. The SCFs for orthotropic plates under uniaxial tension (in E11 direction).
9.2
u21
G12 (GPa) G13 (GPa) G23 (GPa)
0.341 0.157 5.0
5.0
2.6
E11, modulus along the fiber direction; E22, modulus in the perpendicular direction; u12 and u21, Poisson’s ratios; G12, G13, and G23, shear moduli.
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131
Fig. 10. FEM mesh for a cylinder with a hole.
1;1 Fig. 8. Orthotropic influence (KT;o;p;u ¼ 3:5; 10, 30, 100) on SCFs of plates under uniaxial tension.
1;1 For instance as shown in Figs. 6 and 7, KT;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 applicable to orthotropic plates up to almost d=W ¼ 0:75: From Figs. 4 –7, when the dimension ratios are fixed, the SCFs are determined and are independent of the sizes of the plates or the holes. However, at point-2, the negative SCFs can be very different for the orthotropic plates with different material properties or under different uniaxial loading directions (comparing Figs. 6 and 7). The SCFs at point-2 depend strongly on the composite modulus. As reported in the literature [11,12], for some infinite-width orthotropic plates under uniaxial tension, the SCF at point-2 can reach 2 4, much different from that of corresponding isotropic plates which is always 2 1. Another difficulty is that currently there is no theoretical equation to predict the SCF at point-2 location of an infinite-width orthotropic plate under uniaxial tension, unlike the SCF at point-1 (Eq. (2b)). Due to these two reasons, it is not always reliable to predict the SCFs and SFs both at point-1 and point-2 of finite-width orthotropic plates under biaxial tension based on the proposed calculation method. Prior to applying the proposed method
Fig. 9. The SCFs for orthotropic plates under biaxial tension.
to biaxial tension, the applicable range (i.e. the values of material properties and loading directions) should be investigated. Such applicable range should be a function of the material parameters. In this preliminary study, the SCF at point-1 location of an orthotropic plate (with the materials properties given in Table 2) under biaxial loading is estimated by the proposed calculation method. The SCFs are calculated from the SCFs of the same plate under separate uniaxial tension in E11 and E22 directions, then by the superposition principle to add the SCFs together at the same location. The predictions match the FEM results quite well (Fig. 9). Further study will be carried out to examine the applicable range of this calculation method.
3. SCFs of an isotropic/orthotropic cylinder under uniaxial tension or internal pressure In Section 2, the SFs of isotropic plates under uniaxial tension are used to estimate the SCFs of isotropic plates under biaxial tension and SCFs of orthotropic plates under uniaxial tension. All of the plates studied have a center circular hole. In this section, the proposed calculation method will be applied to hollow cylinders with a circular cutout 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-defined dimension ratio, defined as the ratio of the hole diameter to the plate width ðd=WÞ: However, this ratio may not be valid in the cylinder case. Finding a valid dimension ratio is a prerequisite for the proposed calculation method. For cylinders (Fig. 2), there are four possible dimension ratios which can be defined as: d=D; d=pD; d2 =2Dt and arcsinðd=DÞ= 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 the cylinder, pD: Actually, the second and the fourth are quite close because in real engineering applications, the circular cutout in a cylinder cannot be very large. Therefore we can consider and compare only the first 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 axial tension. The SCFs at point-1 or point-2 are calculated by
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Fig. 11. The SCFs for isotropic cylinders under axial tension.
the SCFs of the finite-width isotropic plates (Fig. 4) 1;1 1;1 1;2 multiplied by the ratios of KT;i;c;u to KT;i;p;u or KT;i;c;u to 1;2 : The relationships of the SCFs versus the first three KT;i;p;u dimension ratios are calculated based on the proposed calculation method. The results (Fig. 11) show that the applicable range of the second ratio ðd=pÞ is very small (only to 0.2) because the cutout diameter is seldom larger than 60% of the cylinder’s diameter. Using d 2 =2Dt; Savin’s results [7] overestimates the hole effect of the cylinder for larger dimension ratios. In fact, as discussed by pSavin ffiffiffiffiffiffi [7], his prediction is only valid in the case of d=D p 2t=D; i.e. for very small hole sizes. If the diameter ratio, d=D; is selected, the pre-edited SCFs agree well with FEM results
both at point-1 and point-2. It means that by using this dimension ratio, the proposed calculation based on the SFs of an isotropic plate under one way loading can describe the SCFs well around a circular cutout in a cylinder under axial tension. It has been noticed that under axial tension, an unsymmetric effect resulting from the cutoff leads to some axial bending moment around the hole. Nevertheless, by checking the FEM results, this axial bending stress (or additional axial stress due to bending) is relatively small (less than 15%) compared with the stress caused by the stress concentration of the hole. After identifying the suitable dimension ratio for cylinders, the proposed calculation method is applied to
Fig. 12. The SCFs for isotropic cylinders under internal pressure.
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133
Fig. 13. The SCFs for orthotropic cylinders under axial tension.
an isotropic cylinder with a hole under internal pressure (two way loading). This case can also be considered as a pressure vessel containing a circular cutout that is commonly used in many engineering applications. The superposition principle is employed based on the results of the same cylinder under uniaxial tension. The current predictions and the FEM results are given in Fig. 12. At point-1, it can be found that the two results are very close. However, at point-2, a large difference (more than 20%) is found when the dimension ratio increases to more than 0.5. A possible explanation comes from a twoway bending caused by the unsymmetrical effect of the cylinder under the internal pressure. The influence of the bending in axial direction is small as just explained in Fig. 11 and also verified here. However, the bending in hoop direction changes the local stress at point-2 and adds extra tension along the hoop direction. This influence may be large, and may be responsible for the larger differences between the predictions and the FEM results at large d=D ratios. This local stress increment due to the unsymmetrical effect is very complicated and is not considered in the proposed calculation method. Nevertheless, this local tensile stress increment at point-2 has little influence on the SCF predictions at point-1. Further study on the unsymmetrical effect is needed. From Fig. 12, when the dimension ratio is less than 0.5 which is the usual case in the pressure vessel industry, the proposed method is effective. The predictions of the SCFs of an orthotropic cylinder under axial tension is shown in Fig. 13. The SCFs at point-1 are calculated by the SCFs of the finite-width isotropic plates (Fig. 4) multiplied first by the ratio of 1;1 1;1 KT;i;c;u to KT;i;p;u to account for the cylindrical effect and
1;1 1;1 to KT;i;p;u to account for the then by the ratio of KT;o;p;u orthotropic property. Two cases are investigated: axial tension in E11 direction and in E22 direction. The material parameters are given in Table 2. The proposed approach gives predictions which is in very good agreements with the FEM results.
4. Conclusion For designing engineering structures with a circular cutout, a reliable estimation of SCFs is a must. The paper proposed a simple computation method to estimate the SCFs of finite-width isotropic/orthotropic plates/cylinders with a circular cutout and under uniaxial or biaxial tension. This method is based on the SFs of finite-width isotropic plates (Fig. 4) and SCFs of an infinite-width isotropic/orthotropic pffiffiffiffiffiffi plate or an isotropic cylinder with d=D p 2t=D in one way 1;1 1;2 1;1 1;1 1;2 loading. KT;i;p;u ; KT;i;p;u ; KT;o;p;u ; KT;i;c;u and KT;i;c;u can be conveniently calculated by Eqs. (2b) –(3b). For an isotropic plate under biaxial tension, the SCFs are calculated by the superposition principle according to Fig. 4. For an orthotropic plate under uniaxial tension, the SCFs are estimated by the SCFs of the finite-width isotropic plates 1;1 1;1 (Fig. 4) multiplied by the ratio of KT;o;p;u to KT;i;p;u : For an isotropic cylinder under uniaxial tension, the SCFs at point1 or point-2 are calculated by the SCFs of the finite-width 1;1 isotropic plates (Fig. 4) multiplied by the ratios of KT;i;c;u to 1;1 1;2 1;2 KT;i;p;u or KT;i;c;u to KT;i;p;u : The superposition principle will be employed if the cylinder is under biaxial loading (internal pressure). For an orthotropic cylinder under uniaxial tension, the SCFs are calculated by the SCFs of the finitewidth isotropic plates (Fig. 4) multiplied first by the ratio of
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1;1 1;1 KT;i;c;u to KT;i;p;u to account for the cylindrical shape and then 1;1 1;1 by the ratio of KT;o;p;u to KT;i;p;u to account for the orthotropic property. From the investigation, it can concluded that:
† The proposed computation method is simple and efficient. It is well verified by the FEM simulations. † The SCFs only depend on the dimension ratio defined as the hole diameter to plate width for plates, or the hole diameter to cylinder diameter for cylinders. It is true both for isotropic and orthotropic structures. Variations in the actual structural dimensions are quite small. † For common orthotropic materials, the SFs of finitewidth isotropic plates and orthotropic plates are quite close. If the dimension ratio of an orthotropic plate is smaller than 0.5, the SFs can be replaced by those of an isotropic plate. † At present, the SCFs of orthotropic plates and cylinders under bi-axial loading cannot be reliably predicted by the proposed method because of the difficulties in determining analytically the negative SCFs at the point-2 location of an infinite-width orthotropic plate. † In the case of isotropic cylinders under two-way loading (internal pressure), the influence of bending due to unsymmetrical cutoff in the hoop direction on the SCF at the point-2 location (Fig. 2) is quite large if the dimension ratio is larger than 0.5. Nevertheless, the influence is very small at the point-1 location.
References [1] Adams K. The effect of cutouts on strength of GRP for naval ship hulls. 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 high strength 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 Mater 1987;21:925–48. [6] Tan SC. Stress concentrations in laminated composites. Lancaster: Technomic; 1994. [7] Savin GN. Stress concentration around holes. New York: Pergamon Press; 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 12–16 bars. Internal report. Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor; 1997. [11] Hong CS, Crews JH. Stress-concentration factors for finite orthotropic laminates with a circular hole and uniaxial loading. NASA TP-1469; 1979. [12] Crew JH, Hong CS, Raju IS. Stress-concentration factors for finite orthotropic laminates with a pin-loaded hole. NASA TP-1862; 1981.