ARTICLE IN PRESS

International Journal of Mechanical Sciences 52 (2010) 836–846

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences

0020-74

doi:10.1

� Corr

Univers

E-m

journal homepage: www.elsevier.com/locate/ijmecsci

The stress and strain concentrations of out-of-plane bending platecontaining a circular hole

Zheng Yang a,b,�, Chang-Boo Kim b, Hyeon Gyu Beom b, Chongdu Cho b

a Department of Civil Engineering, Xi’an Jiaotong University, Xi’an 710049, Chinab Department of Mechanical Engineering, Inha University, Incheon 402-751, Korea

a r t i c l e i n f o

Article history:

Received 25 March 2008

Received in revised form

25 October 2008

Accepted 1 February 2010Available online 6 February 2010

Keywords:

Circular hole

Stress concentration factor

Strain concentration factor

Out-of-plane bending

03/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ijmecsci.2010.02.001

esponding author at: Department of Civil

ity, Xi’an 710049, China.Tel.: +86 29 8266587

ail address: zyang1964@hotmail.com (Z. Yang

a b s t r a c t

The elastic stress and strain fields of a finite thickness plate containing a circular hole subjected to out-

of-plane bending are systematically investigated using the finite element method. It is found that the

stress and strain concentration factors of the finite thickness plate are different except at the notch root

of the free surface even if the plate is in elasticity state. The through-thickness distributions of strain

components are not linear with the distance from the mid-plane in the stress concentration region. The

nonlinearity of these distributions is very severe near the free surface especially in thick plate.

The Euler–Bernoulli hypothesis and Kane–Mindlin’s plate theory may not be reasonable to be used in

the stress concentration region especially near the free surface. The maximum stress and strain do not

always occur on the free surface and their locations depend on the moment ratio and the plate

thickness. The maximum stress and strain concentration factors occur on the free surface only in thin

plates of small moment ratio. The differences between the maximum value and surface value of stress

concentration factor increase with the plate thickness and the moment ratio. This relation of strain

concentration factor is similar to the one of stress concentration factor. But the difference magnitude of

stress concentration factor is larger than that of strain concentration factor in same plate.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Stress and strain concentration locations, such as holes andnotches are the critical structural details to determine crackinitiation and growth in engineering structures. Holes existwidely in mechanical systems for either design or manufacturereasons and their stress and strain concentrating effects are suchthat they are common regions of fatigue crack initiation. On theother hand, although the stress concentration regions are notdesired in engineering design of any structure, the concentrationregion can be very well used to amplify the stress in some types ofsensor. Knowledge of stress and strain concentrations near thehole is frequently required for an accurate design of structuralcomponents. Particularly, in modern structures designed to carryhigh loads, stress concentration problem becomes the mostserious issue in the safe design and maintenance of machinesand structures.

Exhaustive studies on stress concentrations for hole in two-dimensional bodies subjected to a wide variety of loading havebeen published by Pilkey [1] and Young [2]. However, many of

ll rights reserved.

Engineering, Xi’an Jiaotong

6; fax: +86 29 83395100.

).

these readily available sources for determining stress concentra-tion factor consider only two-dimensional theory of elasticitysolutions. It is well recognized that the three-dimensional stressfield near the hole in a plate of finite thickness is very complex.There are only few analytical three-dimensional solutions avail-able in the literatures. For finite thickness plates subjected toremote tension loads, Sternberg et al. obtained an approximatesolution for 3D stress distributions near a circular hole in aninfinite plate of arbitrary thickness by using series expansion andtaking finite terms into account [3]. Folias and Wang developed a3D solution using Navier’s equation for plates of uniformthickness and with plate faces free of stress [4]. Recently,Kotousov and Wang presented analytical solutions for the three-dimensional stress distribution around typical stress concentra-tors in an isotropic plate of arbitrary thickness basing on thegeneralized plane strain theory assumption [5], which assumesthat the out-of-plane strain is a constant in the thickness direction[6]. The influence of the plate thickness on three-dimensionalstress field near notch root was examined by Li et al. using 3Dfinite element analyses [7,8]. The influence of Poisson’s ratio onthe thickness-dependent stress concentration factor (SCF) alongthe root of elliptic holes in elastic plates subjected to tension wasinvestigated by use of three-dimensional finite element method.Some empirical formulae had been obtained by fitting thenumerical results [9,10].

ARTICLE IN PRESS

y

z

x

2H

t

p1

2r

2W

p2

p2p1

p1

z

x

t

y

z

x

t

�zz �xx

�yy

Fig. 1. Illustration of the problem. (a) Geometric configuration of plate; (b) the

coordinates; and (c) the 3D finite element model near the hole.

Z. Yang et al. / International Journal of Mechanical Sciences 52 (2010) 836–846 837

The problem of stress concentration around a circular hole inthe Kirchhoff–Love theory for flexural deformations of plate wasinvestigated by Goodier [11,12]. This bending theory does notconsider the effect of transverse shear deformation. Reissner [13]addressed the stress concentration problem within the context ofa theory that accounted for transverse shear deformation. BothReissner’s shear deformation plate theory and Goodier’s platetheory assumed that the stress distribution was linear throughthe thickness and the maximum stress concentration factoralways occurs on the free surface. A finite-element study on theplate subjected to remote bending and remote tension had beenpresented by Shivakumar and Newman [14]. Their finite-elementsolutions showed that the location of the maximum stressconcentration factor is at free surface for thin plate, but themaximum stress concentration factor is slightly interior from thefree surface for thick plate. It is obvious that both Reissner’stheory and Goodier’s are inadequate within the stress concentra-tion region especially for thick plate.

Due to theoretical difficulties in dealing with complicated3D problems, the finite element method is often used to studythree-dimensional stress concentration problems. Despite itsimportance, to the best of authors’ knowledge, there is no reporton the coupled influence of moment ratio and plate thickness onthe stress or strain concentration factor, the maximum stress andstrain locations for the plate with a hole subjected to differentout-of-plane bending without any precondition. In this paper, thefinite thickness plates containing a circular hole subjected todifferent out-of-plane bending are investigated using the finiteelement method. The three-dimensional distributions of stressand strain near the hole are analyzed. The locations of themaximum stress and strain concentration factors are discussed.The sensitivity of the three-dimensional stress and strainconcentration factor to plate thickness as well as the momentratio is examined.

2. Computational procedure and modeling

2.1. Definition

This study considers large plate of finite thickness with acircular hole, subjected to a pair of normal tension varyinglinearly in the thickness direction on the remote boundary plane,shown in Fig. 1(a). So a combination loading of out-of-planebending and tension can be applied in the plate. The momentratio, m=p1/p2, is defined as the ratio of tension values on two freesurface. The loading type of the plate changes from the out-of-plane bending to tension when the moment ration varies for –1 to+1. The radius of hole, the thickness, height and width of plate arer, t, 2H and 2W, respectively. The plate material is homogeneous,isotropic and elastic. The x–y plane (plane z=0) is on the mid-plane of the plate and the two free surfaces are z=t/2 and �t/2,respectively. For convenience, the z direction line defined by x=0and y=0 is named as notch root in this paper, shown in Fig. 1(b).The stress concentration factor (Ks) is

Ks ¼ syy=snominal ð1Þ

The strain concentration factor (Ke) is

Ke ¼ eyy=enominal ð2Þ

snominal ¼p1W

ðW�rÞand enominal ¼

p1W

EðW�rÞð3Þ

Here syy and eyy are the stress and strain components of plate,snominal and enominal are the nominal stress and strain of the platesurface in the critical net section. E is Young’s modulus. The Ksand Ke along the notch root (x=y=0) are denoted as Ks0 and Ke0,

respectively. The maximum values of Ks0 and Ke0 along the notchroot are denoted as Ksmax and Kemax, and their values on freesurface are denoted as Kssur and Ke sur, respectively.

For the stress fields in the finite thickness plate, two constraintparameters are introduced to describe the 3D characteristics ofthe stress fields near the notch root. The out-of-plane stressconstraint factor, which is the ratio of through-thickness stress toin-plane mean stress, is

Tz ¼szz

sxxþsyyð4Þ

and the in-plane stress factor is

Tx ¼sxx

syyð5Þ

Tz=0 for the plane stress state and Tz=n for the plane strain state.For the finite thickness plate of elastic material, 0rTzrn, n isPoisson’s ratio of material.

2.2. Finite element model

In our computations, the width and height of the plate are100 times the diameter of hole due to the effect of plate, widthand height are not considered in this paper. The radius of hole is1 mm, both the half-height and half-width of plate are set to be100 mm. In view of the symmetries of geometry and loading, onlyone quarter of each plate is modeled. Appropriate boundaryconstraints are placed on all planes of symmetry. A pair of normaltension varying linearly in the thickness direction is applied onthe remote boundary plane that is parallel to z–x plane. Themagnitude of p1 is 100 MPa and the magnitude of p2 variesbetween �100 and +100 MPa, thus the moment ratio, m, variesbetween �1 and +1 and the loading state of plate can be purebending, combination of bending and tension or pure tension. The

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Z. Yang et al. / International Journal of Mechanical Sciences 52 (2010) 836–846838

3D finite element mesh is constructed with 8-node solid linearelements. A series of planar element layers are divided throughthe plate thickness. To accommodate the variations of the stressand strain field quantities through the plate thickness, thethickness of each successive element layer is gradually reducedtoward the free surface (z=t/2 and z=�t/2). The size of elementdecreases gradually with decreasing distance from the notch rootwithin each element layer, shown in Fig. 1(c). These 3D modelsrepresent a compromise between the required level of meshrefinement to solve the in-plane and through-thickness gradientsof the stress, or strain fields and the extensive computation timesrequired for each analysis. In our 3D FE computations, Poisson’sratio, n, is 0.3 and Young’s modulus, E, is 200 GPa.

3. The three-dimensional stress distributions near thenotch root

The stress distributions near the hole of finite thickness platediffer from the stress distributions of plate in-plane stress or planestrain state. The stress and strain fields near the hole are three-dimensional. They are very complex and depend on the momentratio and the plate thickness.

3.1. The in-plane stress sxx distribution near the notch root

The distributions of in-plane stress sxx/snominal in front ofnotch root of m= +1 and �1 are shown in Fig. 2. x is the distancefrom the notch root and it is normalized by the radius of hole. Thevalue of sxx is zero at the notch root and then increases with thedistance from the notch root. There is a maximum sxx in eachsxx/snominal–x/r curve and then the sxx value decreases graduallywith the distance from the notch root after the maximum value. Itcan be easily observed that the sxx/snominal–x/r curves of different2z/t are different, but they are similar to each other. The locationsof maximum sxx are insensitive to the location of the sxx/snominal–x/r curve. The sxx distributions are different for different m. Form=�1, the sxx is zero on the mid-plane and the location of thehighest sxx/snominal–x/r curve depends on the plate thickness. Thehighest sxx/snominal–x/r curves of the thin plate are on the freesurface. But the increasing rate of sxx/snominal–x/r curve withinthe plate near the free surface is larger than the one of the freesurface with increasing thickness. When the plate thickness islarger than a specific value, the highest sxx/snominal–x/r curve will

Eq. (8) FE results

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

m = -1

m = +1

/ �no

min

al

x/r

t / r = 8 2z/t = 0 2z/t = 0.2 2z/t = 0.4 2z/t = 0.6 2z/t = 0.8 2z/t = 1.0

� xx

Fig. 2. The distributions of the in-plane stress sxx in front of notch root.

be within the plate near the free surface and not on the freesurface. The distance between the highest sxx/snominal–x/r curveand the free surface increases with increasing thickness andmoment ratio. The specific thickness of the plate where thehighest sxx/snominal–x/r curve moves from the free surfacedecreases with increasing m. For m= +1, the lowest sxx/snominal–x/r curve is on the free surface. The highest sxx/snominal–x/r curvesof the thin plate are on the mid-surface. The highest sxx/snominal–x/r curve moves from the mid-plane with increasing thickness.The thicker the plate, the farther the highest sxx/snominal–x/r curveis from the free surface, shown in Fig. 3.

The sxx/snominal–x/r curve on the free surface depends onthe moment ratio and the plate thickness. The values andlocations of maximum sxx increase with the moment ratios for agiven thickness. The larger the moment ratio, the higher thesxx/snominal–x/r curve on the free surface, shown in Fig. 4(a). Thevariations of sxx/snominal–x/r curve on the free surface withthe plate thickness are different for different moment ratio. Theyincrease with the plate thickness for moment ratio m=�1 anddecrease with the plate thickness for moment ratio m= +1. Thedistances between the location of the maximum sxx and the notchroot increase with the plate thickness for moment ratio m=�1and decrease with the plate thickness for moment ratio m= +1,respectively. The values and locations of the maximum sxx areinsensitive to the moment ratio for the thick plates, but they aresensitive to the moment ratio for the thin plate, shown in Fig. 4(a)and (b).

3.2. The in-plane stress syy distribution near the notch root

The distributions of in-plane stress syy /snominal in front of notchroot for the plate of t/r=8 and m=�1 are shown in Fig. 5. Themaximum syy is at the notch root for each 2z/t and then decreaseswith the distance from the notch root rapidly except for the mid-plane of m=�1. The syy/snominal–x/r curves of different 2z/t aredifferent, but they are similar to each other. The closer to the mid-plane, the flatter the syy/snominal–x/r curve. All syy values on the mid-plane are zero for m=�1. The distributions of the in-plane stress infront of the notch root for m= +1 and �1 are shown in Fig. 6 for theplate of t/r=8. The through-thickness distributions of syy are notlinear with the distance from the mid-plane. The closer to the notchroot, the severer the nonlinearity. The syy distributions near the notchroot are different for different moment ratio. For the plate of m= +1,the minimum syy through thickness is always on the free surface. The

x/r =0.4

xx /

xx s

ur

z

t / r = 2t / r = 4t / r = 6t / r = 8t / r = 10

m = -1, m = 0, m = +1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Fig. 3. Through-thickness distributions of the in-plane stress sxx in front of notch

root.

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

xx /

nom

inal

x/r

m = -1m = -0.5m = 0m = +0.5m = +1

free surface, t / r = 2, t / r = 8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

free surface, m = +1, m = -1 t / r = 2t / r = 4t / r = 6t / r = 8t / r = 10

xx /

nom

inal

x/r

Fig. 4. The distributions of the in-plane stress sxx on the free surface for

(a) different moment ratios and (b) different thickness.

Eq. (9) FE results

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0-0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

t / r = 8, m = -1 2z/t = 0 mid plane 2z/t = 0.2 2z/t = 0.4 2z/t = 0.6 2z/t = 0.8 2z/t = 1.0 free surface

yy /

nom

inal

x/r

Fig. 5. The distributions of the in-plane stress syy in front of notch root.

t / r = 8, m = -1, m = +1

yy /

nom

inal

2 z / t

x / r = 0x / r = 0.2x / r = 0.4x / r = 0.6x / r = 0.8x / r = 1.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Fig. 6. The distributions of the in-plane stress syy in front of notch root for t/r=8.

z

yy /

yy s

ur

t / r = 1t / r = 2t / r = 4t / r = 6t / r = 8t / r = 10

m = -1, m = 0, m = +1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Fig. 7. The distributions of stress syy along the notch root.

Z. Yang et al. / International Journal of Mechanical Sciences 52 (2010) 836–846 839

location of the maximum syy through thickness moves from mid-plane to the vicinity of free surface when it closes to the notch root.But for the plate of m=�1, the syy through thickness increases withthe distance from the mid-plane till free surface and the maximum

syy through the thickness is on the free surface beyond the stressconcentration region. The syy value through thickness increases withthe distance from the mid-plane but it may decreases near the freesurface rapidly within the stress concentration region. The location ofthe maximum syy through thickness moves from the free surface tothe vicinity of free surface when it closes to the notch root. The closerit is to the notch root, the larger the maximum syy value through thethickness.

The syy distributions along the notch root are different fordifferent moment ratio or thickness. For m=�1, the syy value iszero at the notch root of the mid-plane and the location of themaximum syy depends on the plate thickness. The location of themaximum syy of thin plate is on free surface. But the increasingrate of syy within the plate near the free surface is larger than theone of the free surface with increasing thickness. When the platethickness is larger than a specific value, the maximum syy willbe slightly interior from the free surface and not on the freesurface. The distance between the location of the maximum syy

and the free surface increases with the plate thickness. Thisspecific thickness of the plate that the maximum syy begins tomove from the free surface decreases with increasing m. Form= +1, the maximum syy of all plate are not on free surface. Themaximum syy of a thin plate is on the mid-plane. The location ofmaximum syy moves from the mid-plane with increasing

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Z. Yang et al. / International Journal of Mechanical Sciences 52 (2010) 836–846840

thickness. The thicker the plate, the farther the maximum syy

from the mid-plane, shown in Fig. 7. The location of the maximumsyy for a thin plate may move from the free surface to the mid-plane with increasing moment ratio. But for a thick plate, thelocation of maximum syy is always between the mid-plane andfree surface for different moment ratio. The distance of themaximum syy location and the free surface increases with theincreasing moment ratio.

The value of syy/syy0 decreases with the distance fromthe notch root in the region of stress concentration rapidly. Thesyy/syy0–x/r curve on the free surface depends on the momentratio and the plate thickness. The variation of syy/syy0–x/r curveon the free surface with the moment ratio is different for differentplate thickness. The larger the moment ratio, the steeper thesyy/syy0–x/r curve. The syy/syy0–x/r curve of thin plate ismore sensitive to the moment ratio than the one of thick plate.The variations of syy/syy0–x/r curve on the free surface with theplate thickness are different for different moment ratio. Thethicker the plate, the steeper the syy/syy0–x/r curve of m=�1.The syy/syy0–x/r curve of m=�1 is much more sensitive to theplate thickness than the one of m= +1. The syy/syy0–x/r curves ofm= +1 are almost same for different thickness, shown in Fig. 8.

Eq. (9)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

free surface, t / r = 2, t / r = 8

yy /

yy 0

x/r

m = -1m = -0.5m = 0m = +0.5m = +1

Eq. (9)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

yy /

yy 0

x/r

free surface, m = +1, m = -1 t / r = 2t / r = 4t / r = 6t / r = 8t / r = 10

Fig. 8. The distributions of normalized stress syy on the free surface near the notch

root for (a) different moment ratio and (b) different thickness.

3.3. The descriptions of in-plane stress distribution near the notch

root

The in-plane stress distributions of the finite thickness platesubjected out-of-plate bending have relation to the plate thick-ness and moment ratio. The in-plane stress sxx and syy in front ofthe notch root can be uniformly expressed, respectively as

sxx ¼ hx

r;

z

t;

t

r; m

� �ð6Þ

syy ¼ gx

r;

z

t;

t

r; m

� �ð7Þ

The best-known expressions for calculating stresses in thevicinity of a notch root are for circular and elliptical notches in aninfinite plate under remote loading. When the stress concentra-tion factor was introduced, the in-plane stress components ofplane stress state in front of the notch root could be expressed asfunctions of the distance from the notch root [15]. If theseexpression functions are used to describe the in-plane stresses offinite thickness plate, they must be modified by considering thethree-dimensional effect. It is obvious that the stress syy is notuniform along the notch root. The distribution of stress syy alongthe notch root is a function of the plate thickness, moment ratioand the distance from mid-plane. It may indicate the three-dimensional effects in the stress concentration region to a certainextent. When the stress syy along the notch root is introduced, thein-plane stress components, sxx and syy, in front of the notch rootcan be expressed, respectively, as

sxx ¼syy0

21þ

x

r

� ��2

1� 1þx

r

� ��2� �

ð8Þ

syy ¼syy0

31þ

1

21þ

x

r

� ��2

þ3

21þ

x

r

� ��4� �

ð9Þ

Here syy0 is the syy value of the notch root corresponding todifferent 2z/t. The in-plane stress factor can be derived fromEqs. (8) and (9) as

Tx ¼sxx

syy¼

3 1þ xr

� �21� 1þ x

r

� �2h i

2þ 1þ xr

� �2þ3 1þ x

r

� �4h i ð10Þ

Yang et al. [16] had confirmed that the 2D solutions modified likethis could be used to describe the in-plane stress distributions of afinite thickness tension plate except in a very narrow region near theplate surface. The 3D results of stress sxx for m= +1 and �1 incomparison with Eq. (8) are shown in Fig. 2 and the 3D results ofstress syy for m=�1 in comparison with Eq. (9) are shown in Fig. 5. Itis shown that Eqs. (8) and (9) can predict the in-plane stressdistributions very well for different moment ratio except in a verynarrow region near the plate surface. But on the free surface, theseequations are suitable to express the distributions of in-plane stressesonly near the notch root. The plate thickness and moment ratioinfluence their applicable extent, shown in Fig. 8. The applicableextent of Eq. (9) on the free surface, which is about 0.2r ahead thenotch root, is smaller the one of Eq. (8), which is about 0.4r aheadthe notch root. The applicable extent of Eq. (10) is about 0.2r aheadthe notch root on the free surface.

3.4. The out-of-plane stress szz distribution near the notch root

The distributions of the out-of-plane stress, szz, in front of notchroot of m= +1 and �1 are shown in Fig. 9(a) for the plate of t/r=8. Theszz is zero on the free surface. The maximum szz through thickness ison the mid-plane for the plate of m= +1 and the szz value is zero onthe mid-plane for the plate of m=�1. The szz decreases with the

ARTICLE IN PRESS

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7t / r = 8, m = +1

t / r = 8, m = -1

x / r = 0x / r = 0.2x / r = 0.4x / r = 0.6x / r = 0.8x / r = 1.0

zz/

nom

inal

2 z / t

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t / r = 8

zz/

nom

inal

2 z / t

m = -1m = -0.25m = -0.5m = 0m = +0.5m = +1

t / r = 2

Fig. 9. The distributions of the out-of-plane stress szz (a) in front of notch root and

(b) along the notch root for different moment ratio.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

t / r = 1t / r = 2t / r = 4t / r = 6t / r = 8

/

m = -1, m = 0, m = +1

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

t / r = 8, m = -1, m = +1

2 z / t

x / r = 0x / r = 0.2x / r = 0.4x / r = 0.6x / r = 0.8x / r = 1.0

� yy

� yy

sur

/� y

y� n

omin

al

Z. Yang et al. / International Journal of Mechanical Sciences 52 (2010) 836–846 841

distance from the notch root rapidly. The farther from the notch root,the flatter the szz/snominal–2z/t curves. The out-of-plane stress, szz,disappears and the plane stress state becomes the dominant state asthe distance from the notch root increases. It had been confirmed thatthe length of 3D-perturbation region for m= +1 is about a half of theplate thickness [16,17]. The 3D-perturbation region decreases withdecreasing moment ratio.

The distributions of the out-of-plane stress, szz, along thenotch root for different moment ratios are shown in Fig. 9(b).Through-thickness distributions of szz depend on the momentratio and the plate thickness. The szz value on the mid-planeincreases with increasing moment ratio. The maximum szz valuealong the notch root moves to the mid-plane and the szz

distribution along the notch root changes from antisymmetry tosymmetry about the mid-plane as the moment ratio varies from�1 to +1. The maximum szz value along the notch root increaseswith increasing plate thickness. The thinner the plate and thesmaller the moment ratio, the flatter the szz/snominal–2z/t curves.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.0

0.1 t / r = 10

z

Fig. 10. The distributions of the in-plane strain eyy (a) in front of notch root for

different moment ratio and (b) along the notch root for different thickness.

4. The strain distributions near the notch root

The actual three-dimensional stress and strain fields within thestress concentration region are very complex. The strain distribution

is different from the stress distribution, but the distribution ofeyy/eyy sur is similar to the one of syy/syy sur. It should be noticed thatthe location of the maximum eyy is different from the location of themaximum syy. The through-thickness distributions of eyy are notlinear with the distance from the mid-plane. The closer to the notchroot, the severer the nonlinearity, shown in Fig. 10(a). Thenonlinearity of the eyy distributions along the notch root increaseswith the plate thickness and the moment ratio, shown in Fig. 10(b).The nonlinearity of thick plate is very severe near the free surface. Thestrain distribution in an elastic bending plate, which originallyproposed by Euler and Bernoulli in 1744, is popularly used incalculation of the stress distribution in a plate subjected to an out-of-plane bending. The Euler–Bernoulli hypothesis assumes that planetransverse sections of the plate remain plane during bending, i.e., thestrain eyy distribution through the thickness is linear distribution in abending plate. It is obvious that the Euler–Bernoulli hypothesiscannot be used in the stress concentration region.

The distributions of the out-of-plane strain near the notch root ofm= +1 and �1 are shown in Fig. 11(a) for the plate of t/r=8. Thethrough-thickness distributions of ezz are not linear with the distancefrom the mid-plane. The closer to the notch root, the severer thenonlinearity. This nonlinearity is very severe near the free surfaceespecially in thick plate. The distributions of ezz near the notch rootare different for different moment ratio. For the plate of m= +1, themaximum ezz through thickness is not on the free surface. Thelocation of maximum ezz through thickness is near the free surfaceand moves to the free surface when it closes to the notch root. The

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

t / r = 10

8

6

4

t / r = 1

m = -1m = 0m = +1

zz /

zz s

ur

z

2

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

t / r = 8, m = -1, m = +1x / r = 0x / r = 0.2x / r = 0.4x / r = 0.6x / r = 0.8x / r = 1.0

zz/

nom

inal

2 z / t

Fig. 11. The distributions of the out-of-plane strain ezz (a) in front of notch root for

different moment ratio and (b) along the notch root for different thickness.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

m = -0.25 m = -0.5 m = 0 m = +0.5 m = +1

t / r = 4

T z t / r = 2

t / r = 8

2 z / t

Fig. 12. The out-of-plane stress constraint factor distributions along the notch

root for different moment ratio and different thickness.

Z. Yang et al. / International Journal of Mechanical Sciences 52 (2010) 836–846842

closer to the notch root, the closer to the free surface, but it cannotreach the free surface. The closer to the notch root, the larger themaximum ezz. For the plate of m=�1, the ezz values on the mid-planeare zero and increase with the distance from the mid-plane till thefree surface beyond the stress concentration region. But near thenotch root, the location of the maximum ezz is not always on the freesurface. The maximum ezz increases, but the location of the maximumezz moves from the free surface and then moves to the free surfacewhen it closes to the notch root. The closer to the notch root, thelarger the maximum ezz.

The distributions of the out-of-plane strain along the notchroot for different thickness and different moment ratio are shownin Fig. 11(b). For m=�1, the ezz is zero at the notch root of themid-plane and the location of the maximum ezz depends on theplate thickness. The location of the maximum ezz of the thin plateis on the free surface. When the plate thickness is larger than aspecific value, the maximum ezz will be within the plate near thefree surface and not on the free surface. The distance between thelocation of the maximum ezz and the free surface increases withthe plate thickness and the moment ratio. For m= +1, the locationof maximum ezz is always between the mid-plane and freesurface. The thicker the plate and the larger the moment ratio, theseverer the nonlinearity of the out-of-plane strain distributionalong the notch root.

The through-thickness distributions of out-of-plane deforma-tion and strain are very important in simplifying the 3D finitethickness plate with free surface into a quasi-2D plane problem.Kane and Mindlin [6] proposed a plate theory of generalizedplane-strain to take into account the thickness effect byassuming constant ezz through the thickness. It is evident thatKane–Mindlin’s plate theory and similar theories assumingconstant ezz through the thickness can only be applied to thepart of plate beyond the stress concentration region or smallcentral zone of thin plate. It may not be reasonable to use thiskind of assumption in the stress concentration region especiallynear the free surface.

5. The relation of stress and strain concentration factor

In three-dimensional stress region, the stress concentrationand strain concentration distributions are different. The relationof stress and strain concentration factor can be obtained from thestress, strain concentration factor definitions and Hooke’s law[15]. This relation can be expressed as

Ke ¼ Ks½1�nðTxþTzþTx � TzÞ� ð11Þ

It is indicated that the relation of Ke and Ks depends onPoisson’s ratio, the distributions of in-plane stress factor and out-of-plane stress constraint factor. The stress concentration factorand the strain concentration factor of the finite thickness plate aredifferent as long as the in-plane stress factor and out-of-planestress constraint factor are not equal to zero simultaneously evenif the plate is in elasticity state. The Ke and Ks values are differentat different position near the notch root. The stress concentrationfactor is equal to the strain concentration factor only at the notchroot of free surface or at the notch root under the plane stressstate, where the out-of-plane stress constraint factor Tz and in-plane stress factor Tx are zero simultaneously.

The in-plane stress factor Tx is zero along the notch root. FromEq. (11), the relation of Ke and Ks along the notch root can beexpressed as

Ke ¼ Ks½1�nTz� ð12Þ

It is indicated that the relation of Ke and Ks along the notchroot depends on Poisson’s ratio and the out-of-plane stressconstraint factor distribution along the notch root.

The out-of-plane stress constraint factor distributions alongthe notch root for different moment ratios and different thickness

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0.94

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0.96

0.97

0.98

0.99

1.00

t / r = 4

K /

K

t / r = 2

t / r = 8

2 z / t

m = -0.25 m = -0.5 m = 0 m = +0.5 m = +1

Fig. 13. The distributions of Ke/Ks along the notch root for different moment ratio

and different thickness.

Eq. (10)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.00

0.05

0.10

0.15

0.20

0.25

0.30

free surfacet / r = 2, t / r = 8

T x

x/r

m = -1 m = -0.5 m = 0 m = +0.5 m = +1

Eq. (10)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.00

0.05

0.10

0.15

0.20

0.25

0.30

free surfaceT x

x/r

m = +1, m = -1 t / r = 2 t / r = 4 t / r = 6 t / r = 8 t / r = 10

Fig. 14. The distributions of in-plane stress ratio on the free surface near the notch

root for (a) different moment ratio and (b) different thickness.

Z. Yang et al. / International Journal of Mechanical Sciences 52 (2010) 836–846 843

are shown in Fig. 12. The out-of-plane stress constraint factor iszero on the free surface of plate and it increases with the distancefrom free surface. It should be noticed that the out-of-plane stressdoes not always increase with the distance from free surface suchas in the plate of m=�0.25 and �0.5, shown in Fig. 9. But the out-of-plane stress constraint factors of different moment ratios arealmost same at the notch root of mid-plane for a given thickness.The Tz distributions along the notch root are sensitive to the platethickness and insensitive to the moment ratio. The shape of theTz�2z/t curve becomes flatter as the plate thickness decreases.The thicker the plate, the higher the Tz�2z/t curve and the largerthe out-of-plane stress constraint factor on the mid-plane. TheTz values of the mid-plane are dependent on the plate thicknessand are almost independent of the moment ratio. So the Ke/Ks is 1at the notch root of the free surface and it decreases with thedistance from the free surface. The Ke/Ks of different momentratios are almost same at the notch root of mid-plane for a giventhickness. The Ke/Ks distributions along the notch root aresensitive to the plate thickness and insensitive to the momentratio for a given material, shown in Fig. 13. Even though thelocations of the maximum szz or maximum syy are not always atthe mid-plane of the notch root for different moment ratios, thelocation of minimum Ke/Ks value along the notch root is on themid-plane. The difference between stress concentration factor(Ks) and strain concentration factor (Ke) at notch root of mid-plane increases with increasing thickness. The Ke is 6% lower thanthe Ks as t/r=8. This difference is insensitive to the moment ratioand will approach a constant with increasing thickness for a givenmaterial.

The out-of-plane stress constraint factor Tz is zero on the freesurface of plate. From Eq. (11), the relation of Ke and Ks on the freesurface can be expressed as

Ke ¼ Ks½1�nTx� ð13Þ

It is indicated that the relation of Ke and Ks on the free surfacedepends on Poisson’s ratio and the in-plane stress factordistribution on the free surface.

The in-plane stress factor distributions on the free surface infront of notch root for different moment ratio and differentthickness are shown in Fig. 14. The Tx distributions depend onthe sxx and syy distributions and the sxx and syy distributions onthe free surface depend on the plate thickness and the momentratio. So the Tx distributions on the surface depend on the plate

thickness and the moment ratio. On the free surface, the in-planestress factor is zero at the notch root and there is a maximum Tx

ahead the notch root. The values and locations of maximum Tx

increase with the moment ratios for a given thickness. But for agiven moment ratio, the variations of Tx–x/r curve on the freesurface with the plate thickness are different for different momentratio. The maximum Tx and the distance from the maximum Tx

position to the notch root increase with the plate thickness for theplate of m=�1 and decrease with the plate thickness for the plateof m= +1. So the Ke/Ks distributions on the free surface depend onthe moment ratio and plate thickness. The Ke/Ks–x/r curve issensitive to the moment ratio and this curve of thin plate is moresensitive to the moment ratio than the one of thick plate, shown inFig. 15. There is a minimum Ke/Ks ahead the notch root and itsvalue and location relate to the moment ratio and plate thickness.The Ke/Ks is about 0.919 at x/r=0.67 and is about 0.949 at x/r=0.41for the thin plate, t/r=2, of m= +1 and m=�1, respectively. But theKe/Ks is about 0.925 at x/r=0.58 and is about 0.929 at x/r=0.57 forthe thick plate, t/r=10, of m= +1 and m=�1, respectively.

If we know the distribution of in-plane stress on the freesurface, we can obtain the exact relation of stress concentrationfactor and stress concentration factor on it. But the distribution ofstress near the notch root of finite thickness plate is verycomplicated. Here we substitute Eq. (10) into Eq. (13) to get the

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Eq. (14)

Eq. (14)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1.00

t / r = 2, t / r = 8, free surface m = -1 m = -0.5 m = 0 m = +0.5 m = +1

K /

K

x/r

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1.00

K /

K

x/r

m = +1, m = -1, free surface t / r = 2 t / r = 4 t / r = 6 t / r = 8 t / r = 10

Fig. 15. The distributions of Ke/Ks on free surface near the notch root for

(a) different moment ratio and (b) different thickness.

0 1 2 3 4 5 6 7 8 9 100.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

m = -1m = -0.5m = 0m = +0.5m = +1

t / r

K /

K 0

0.80

0.85

0.90

0.95

1.00

1.05

Z. Yang et al. / International Journal of Mechanical Sciences 52 (2010) 836–846844

approximate expression of Ke/Ks�x/r curve in front of the notchroot on the free surface

Ke

Ks¼ 1�n

3 1þ xr

� �21� 1þ x

r

� �2h i

2þ 1þ xr

� �2þ3 1þ x

r

� �4

24

35 ð14Þ

It has been confirmed in Section 3 that the applicable extent ofEq. (10) is about 0.2r ahead the notch root on the free surface. Sothe applicable extent of Eq. (14) is also about 0.2r ahead the notchroot on the free surface, shown in Figs. 14 and 15.

0 1 2 3 4 5 6 7 8 9 100.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

K /

K0

t / r

m = -1m = -0.5m = 0m = +0.5m = +1

Fig. 16. The (a) stress and (b) strain concentration factors at the notch root of the

free surface and their maximum values along the notch root versus normalized

thickness for different moment ratio.

6. The stress, strain concentration factor along the notch root

6.1. The relationship of stress, strain concentration factor and the

plate thickness

In the same plate, the through-thickness distributions of stressand strain are different, so the through-thickness distributions ofstress and strain concentration factor are different. The maximumstress and strain do not always occur on the free surface. Themaximum values of Ks and Ke occur on the free surface only inthinner plate of smaller moment ratio. When the plate is thickerthan a specific thickness (t’), called transition thickness, the

location of the maximum value of Ks or Ke moves gradually fromthe free surface to the interior of plate as the plate thicknessincreases. This transition thickness decreases with increasingmoment ratio. The transition thickness of m= +1 is zero, and themaximum Ks and Ke of all different thickness are not on the freesurface. The transition thickness of m=�1 is about 3r.

For different moment ratio, the stress and strain concentrationfactors of the free surface and the maximum value along the notchroot, normalized by the stress concentration factor of m=1 in theplane stress state, versus normalized thickness are shown inFig. 16. In this figure, the K0 is the stress concentration factor ofm=1 in the plane stress state. Obviously, the stress and strainconcentration factors at the notch root of the free surface, wherethe in-plane stress factor and out-of-plane stress constraint factorare equal to zero simultaneously, are the same for all differentthickness of different moment ratio from Eq. (11). The FE resultscan also prove this conclusion. They are denoted by one term Ksur

in this paper. In the case of pure bending, m=�1, the maximumstress and strain concentration factors along the notch root are onthe free surface in thin plate. The Ksur/K0�t/r curve of m=�1 is amonotonic ascent functions of thickness. The Ksur decreases withdecreasing plate thickness. The Ksur will tend to zero when theplate thickness tends to zero. The Ksur increases with the platethickness and tends to a value which is smaller than 1. When the

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0.22

(K m

ax -

K s

ur) /

K s

ur

t / r

m = +1

m = +0.5

m = 0

m = -0.5

m = -1

0 1 2 3 4 5 6 7 8 9 10-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

m = +1

m = +0.5

m = 0

m = -0.5

m = -1

t / r

(K m

ax -

K s

ur) /

K s

ur

Fig. 17. The differences between maximum value and surface value of (a) stress

and (b) strain concentration factors versus normalized thickness for different

moment ratios.

Z. Yang et al. / International Journal of Mechanical Sciences 52 (2010) 836–846 845

plate thickness is larger than the transition thickness of m=�1,the maximum stress and strain concentration factors, denoted byKsmax and Kemax, respectively, are in the plate and the maximumstress and strain concentration factors are larger than the Ksur. Thedifference between the values of the Ksmax and the Ksur increaseswith the plate thickness. The difference between the values of theKemax and the Ksur also increases with the plate thickness, shownin Fig. 17. As we know, the stress concentration factor is differentfrom the strain concentration factor in the plate except at thenotch root of the free surface, so the Ksmax and Kemax along thenotch root are different when the plate is thicker thanthe transition thickness. The difference between the values ofthe Ksmax and Kemax increases with the plate thickness. It shouldbe noticed that the Ksmax and Kemax is not at same position.

In the case of pure tension, m= +1, the maximum stress andstrain concentration factors along the notch root are always noton the free surface except t/r=0. The Ksmax, Kemax and Ksur willtend to K0 when the plate thickness tends to zero. The Ksur/K0�t/rcurve of m= +1 is the monotonic descent functions of thicknessthat are different from the one of m=�1. The Ksur value decreasesfrom its plane stress value K0 at t/r=0 tends to a lower limit valuefor a given material when t/r becomes large enough. The Ksmax/K0�t/r curve of m= +1 is not a monotonic functions of thickness.It increases from 1 at t/r=0 to its peak point and then decreases

gradually with increasing t/r and tends to a value which is largerthan 1 when t/r becomes large enough. The Kemax/K0�t/r curve ofm= +1 is also not a monotonic functions of thickness. It increasesfrom 1 at t/r=0 to its peak point and then decreases graduallywith increasing t/r, but it tends to a value which is smaller than 1when t/r becomes large enough.

The plate of moment ratio between –1 and +1 can beconsidered as a plate, which load is a combination of purebending (m=�1) and pure tension (m= +1), i.e.

KðmÞ ¼ Km ¼ �11�m

2

� �þKm ¼ þ1

1þm

2

� �ð15Þ

When t/r-0

Km ¼ �11�m

2

� �-0 and Km ¼ þ1

1þm

2

� �-

1þm

2

So K(m)-(1+m)/2 when t/r-0. The Ksur/K0�t/r curves of m=�1,�0.5, 0, +0.5 and +1 at t/r=0 are 0, 0.25, 0.5, 0.75 and 1, respectively.When the moment ratio changes from +1 to �1, the type of Ksur/K0�t/r curve changes from a monotonic descent function of thicknessto a monotonic ascent function of thickness, the types of Ksmax/K0�t/r curve and Kemax/K0�t/r curve change from no monotonic functionto a monotonic ascent function of thickness. The Ksur/K0�t/r, Ksmax/K0�t/r and Kemax/K0�t/r curves of different moment ratio tend todifferent constants related to the moment ratio for a given materialwhen t/r becomes large enough.

6.2. The differences between surface and maximum values of stress

and strain concentration factor

The through-thickness variations of syy/syy sur and eyy/eyy sur aredependent on the plate thickness and moment ratio. The effects ofplate thickness and moment ratio on the through-thickness variationsof syy/syy sur and eyy/eyy sur are shown in Figs. 7 and 10(b), respectively.For the plate whose thickness is larger than the transition thickness,the Ksmax and Kemax are not on the free surface. It is clear that the Kson the free surface is lower than that within the plate. The differencesbetween the maximum values and surface values of stress and strainconcentration factor versus normalized thickness for differentmoment ratio are shown in Fig. 17. They all increase as thicknessincreases or moment ratio increases. The (Ksmax�Ks sur)/Ks sur�t/rcurves of different moment ratio are the monotonic ascent functionsof the plate thickness. They increase with increasing t/r when theplates are thicker than the transition thickness of each moment ratio.They will tend to the constants corresponding different moment ratio.The larger the moment ratio, the steeper the (Ksmax�Ks sur)/Ks sur�

t/r curve and the larger the (Ksmax�Ks sur)/Ks sur�t/r curve tends to.This relation of strain concentration factor is similar to the one ofstress concentration factor. But the difference magnitude of stressconcentration factor is larger than that of strain concentration factorin same plate. When t/r=10, this difference of stress concentrationfactor are 7.6%, 9.8%, 12.5%, 16.0% and 21.0% and this difference ofstrain concentration factor are 6.0%, 7.7%, 9.8%, 12.3% and 15.7% forthe moment ratio m=�1, �0.5, 0, +0.5 and +1, respectively.

The transition thickness decreases with increasing moment ratio.The larger the moment ratio, the smaller the thickness extent of theplate where the maximum stress concentration factor is on the freesurface. For the plate of large moment ratio, the Ks and Ke decreaserapidly near the free surface and are too lower to reflect the overallstress concentration as thickness increases. On the other hand, thelocations of maximum stress and strain concentration factor withinthe plate are different. But at same position, the stress concentrationfactor is larger than the strain concentration factor usually. Fig. 17 alsoshows that it is risky to use the Ke measured on the free surface oralong the notch root in the engineering design. The influence of thethickness and the moment ratio on stress concentration factor and

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Z. Yang et al. / International Journal of Mechanical Sciences 52 (2010) 836–846846

the relation of stress and strain concentration factors within the plateshould also be considered.

7. Conclusions

In the work, the three-dimensional elastic stress and strainfields of finite thickness plate with a hole subjected to out-of-plane bending are systematically investigated using the finiteelement method. The sensitivity of the three-dimensional stressand strain concentration factor to plate thickness as well as themoment ratio is examined. Some special characters of three-dimensional stress and strain distributions are revealed:

(1) The through-thickness distributions of eyy are not linearwith the distance from the mid-plane in the stress concentrationregion. The closer to the notch root, the severer the nonlinearity.The nonlinearity of the eyy distributions along the notch rootincreases with the plate thickness and the moment ratio. TheEuler–Bernoulli hypothesis, which assumes that the strain eyy

distribution through the thickness is linear distribution in abending plate, cannot be used in the stress concentration region.

(2) The through-thickness distribution of the out-of-planestrain in the stress concentration region is not linear near the freesurface. The closer to the free surface, the severer the nonlinearity.The nonlinearity of the out-of-plane strain distribution near thefree surface increases with the plate thickness and the momentratio. This nonlinearity is very severe near the free surfaceespecially in thick plate. Kane–Mindlin’s plate theory and similartheories assuming constant out-of-plane strain through thethickness may not be reasonable to be used in the stressconcentration region especially near the free surface.

(3) The stress and strain concentration factors of the finitethickness plate are different in stress concentration region exceptat the notch root of the free surface even if the plate is in elasticitystate. The maximum stress and strain do not always occur on thefree surface. The maximum values of Ks and Ke occur on the freesurface only in thinner plates of small moment ratio. Thedifferences between the maximum value and surface value ofstress concentration factor increase with the plate thickness andthe moment ratio. This relation of strain concentration factor issimilar to the one of stress concentration factor. But the differencemagnitude of stress concentration factor is larger than that ofstrain concentration factor in same plate.

Acknowledgements

This work was supported by the National Natural ScienceFoundation of China (Grant No. 50971098) and the ScientificResearch Starting Foundation for Returned Overseas ChineseScholars, Ministry of Education, China (Grant No. 09-18).

References

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[2] Young WC, Budynas RG. Roark’s formulas for stress and strain, 7th ed.McGraw-Hill: New York; 2002.

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[11] Goodier JN. The influence of circular and elliptical holeson the transverseflexure of elastic plates. Philosophical Magazine, Series VII 1936;22:69–80.

[12] Love AEH. A treatise on the mathematical theory of elasticity, 4th ed.. Dover:New York; 1944.

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[14] Shivakumar KN, Newman Jr. JC. Stress Concentrations for straight-shank andcountersunk holes in plates subjected to tension, bending, and pin loading,NASA technical paper 3192, 1992.

[15] Timoshenko S, Goodier JN. Theory of elasticity. McGraw Hill: New York;1970.

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