rs

dO. B. Bo

Orthotropic plates

thetesndnditramwidtorimathat

are usropic mnical,uch astress an

concen

holes in a plate structures and pressure vessels. Kotousov andWang (2002) presented analytical solutions for (3-D) stress

plates. Shivakumar et al. (2007), and Bhargava and Shivakumar(2007, 2008) presented two detailed equations for the stress andstrain concentration factors in countersunk holes of isotropic platessubjected to tensile loading. Darwish et al. (2012) introduceda modied parametric equation for the SCF in isotropic plates.Raghavan and Raju (2009) discussed the stress concentration dueto countersunk holes in adhesively bonded layered Aluminumthrough a 3-D FEmodel to estimate the location and themagnitudeof the maximum SCF.

* Corresponding author. Tel.: 962 2 720 1000x22735, 962 79 995 6787(mobile); fax: 962 2 720 1000.

E-mail addresses: fhdarwish@just.edu.jo (F. Darwish), gtash@just.edu.jo(G. Tashtoush), mohd_gh@hotmail.com (M. Gharaibeh).

1 Tel: 962 2 720 1000x22570, 962 79 562 0645 (mobile).2

Contents lists available at

European Journal of M

journal homepage: www.els

European Journal of Mechanics A/Solids 37 (2013) 69e78Tel.: 962 79 951 4353 (mobile).sional (2-D) plates with circular holes subjected to several loadingtypes are reported in the literature and summarized by Pilkey andPilkey (2008). Three dimensional (3-D) FE results of the SCF werepresented by Shivakumar and Newman (1992) for plates withcircular straight-shank holes subjected to remote tension. Theresults showed that the maximum SCF lies at the mid-thickness ofthe isotropic plate and drops near the free surface. Wu and Mu(2003) performed FEA on uniaxial and biaxial loaded isotropicand orthotropic plates with circular holes and examined the SCF of

thickness. They studied the effects of plate thickness and notchconguration on the stress distributions, the out-of-plane stressconstraint factor, and the strain energy density. She and Guo(2007) used FEM to analyze the through-the-thickness variationof the SCF along the wall of elliptic holes in isotropic plates sub-jected to a remote tensile stress.

Few studies considered the SCF of countersunk holes in isotropicplates. Wharely (1965) determined the local stresses experimen-tally by the birefringent-plastic-coating method on Aluminum1. Introduction

In industrial applications, rivetsthat are made of isotropic or anisotthe isotropic materials, the mechaproperties of anisotropic materials sdirectional, which complicates the shood of holes.

Numerous studies on the stress0997-7538/$ e see front matter 2012 Elsevier Masdoi:10.1016/j.euromechsol.2012.04.006ed to join componentsaterials. In contrast tophysical and thermalbrous composites arealysis in the neighbor-

tration of two dimen-

distribution around typical stress concentrators in an isotropicplate of arbitrary thickness. Few years later, Kotousov et al. (2010)showed that ignoring the plate thickness by the classical planesolutions could lead to an error in the assessment of the stressstate especially in the neighborhood of notches. Li et al. (2000)investigated through FEA the elastic notch-root elds in plateswith different thicknesses and notch congurations subjected touniaxial tension. Berto et al. (2004) presented an analytical solu-tion for the stress eld at a notch root in a plate of arbitraryless than 14%. 2012 Elsevier Masson SAS. All rights reserved.within 7% of the nite element (FE) results for 96% of the runs and that the maximum overall error isStress concentration analysis for counte

Feras Darwish a,*, Ghassan Tashtoush b,1, MohammaaAeronautical Engineering Department, Jordan University of Science and Technology, P.bMechanical Engineering Department, Jordan University of Science and Technology, P.O

a r t i c l e i n f o

Article history:Received 4 March 2012Accepted 29 April 2012Available online 15 May 2012

Keywords:Countersunk holeStress concentration factorFinite element analysis

a b s t r a c t

This research investigatesorthotropic laminated plasoftware is used to build aboundary and loading cogeometric and material pacountersunk depth, platemulti parameter t and facthe various geometric andon the results, it is foundson SAS. All rights reserved.unk rivet holes in orthotropic plates

Gharaibeh b,2

ox 3030, Irbid 22110, Jordanx 3030, Irbid 22110, Jordan

in-plane stress concentration factor (SCF) in countersunk rivet holes inunder uniaxial tension load. Finite element analysis (FEA) using ANSYSmesh the geometry of a plate containing a countersunk hole, dene theions, run the analysis and obtain the SCF results. The effect of severaleters such as plate thickness, straight-shank radius, countersunk angle,th, and the laminate ply angles on the SCF is also investigated. Finally,al analyses are applied to establish the relationships between the SCF andterial parameters, and to formulate a general equation for the SCF. Basedthe values of the SCF obtained by means of the formulated equation are

SciVerse ScienceDirect

echanics A/Solids

evier .com/locate/ejmsol

of Mechanics A/Solids 37 (2013) 69e78Lekhnitskii (1963) presented an equation to approximate theSCF in thin innite orthotropic plates with circular hole. Thisequation was used to verify the experimental results of Pipes et al.(1979) and Toubal et al. (2005). Whitney and Nuismer (1974)introduced two fracture criteria known as the point stress crite-rion (PSC) and the average stress criterion (ASC) to predict theuniaxial tensile strength of a composite laminate with a circularhole. Potti et al. (1999) made an improvement on the accuracy of

Nomenclature

b straight shank depthCs countersink depthe element size at the holeE Youngs modulusG shear modulusKt theoretical stress concentration factorKh,o a parameter that accounts for the effect of the

plates width and the ply angle on KtKss,o a parameter that accounts for the effect of the

plates thickness and the ply angle on KtKCs,o a parameter that accounts for the effect of the

countersink depth and the ply angle on KtKqc,o a parameter that accounts for the effect of the

countersink angle and the ply angle on Ktr straight shank radiust plate thicknessw plates half-widthx,y,z global coordinate system1,2,3 principle coordinate systemn Poisons ratioqc countersink angleqp ply angleso remote stress

F. Darwish et al. / European Journal70failure stress predictions by the (PSC) and (ASC). In (2009), Jenet al. used a modied PSC to predict the notch strength ofcomposite laminates at elevated temperatures. The predictionsagreed well with the experimental results. Hong and Crews (1979)presented a 2-D FEA for nite width orthotropic laminate witha circular hole subjected to uniaxial loading. They also discussedthe effect of width and length to radius ratios on the stressconcentration. Konish and Whitney (1975) developed threeequations to calculate the SCF in orthotropic plates with a circularhole. Tan (1987, 1994) modied the PSC and derived severalformulas for the SCF for orthotropic plate with an elliptical hole.Jain and Mittal (2008) presented FE study on both isotropic andorthotropic plates with central circular hole subjected to trans-verse static loading to study the effect of the hole diameter to platewidth ratio on the SCF.

The main objectives of the present research are to performstress analysis around countersunk holes in orthotropic plates andto develop a parametric equation for the SCF as function of thematerial properties and the geometric parameters.

2. Geometry and materials

The conguration of the plate containing a countersunk holeand subjected to a tensile load is shown in Fig. 1.

An orthotropic material system of eight plies of carbon/epoxy(AS4/3501-6) with stacking sequence [qp]2s is considered for theanalysis of the present study as shown in Fig. 2. The mechanicalproperties of a single lamina of unidirectional (AS4/3501-6) in theprincipal directions are: E1 149 GPa, E2 10.3 GPa, E3 10.3 GPa,G12 6.9 GPa,G23 3.7 GPa,G13 6.9 GPa, n12 0.27, n23 0.54 andn13 0.27, Daniel and Ishai (2006).

3. Approach and verication of the analysis

Extensive 3-D FEA is conducted to obtain the SCF at the coun-tersunk hole at different geometric and material parameters. TheSCF is dened as the ratio of the maximum in-plane stress in theload direction (x-direction) at the notch edge to the nominalapplied stress.

Kt smaxso (1)

Based on the obtained FE results, factorial and multi parametert analyses are performed to develop an equation for the SCF.Factorial analysis, which is amethod used to describe the variabilityamong uncorrelated variables, is implemented to determine therelative signicance of the geometric and material parameters that

Fig. 1. Conguration of countersunk hole.will be used in formulating the dimensionless parameters of theSCF equation.

3.1. Finite element modeling

ANSYS Release 13.0 was used to construct the FE model. Ageneral exible FE code was written using the ANSYS Parametric

Fig. 2. The stacking sequence of the orthotropic laminate.

3.2.2. Thin nite orthotropic plate with a circular holeHere, the values of the SCF (Kh,o) become function of the plates

3.84

3.86

3.88

3.90

Kt

F. Darwish et al. / European Journal of Mechanics A/Solids 37 (2013) 69e78 71Design Language (APDL). This code was used to dene the materialsystem, to generate the geometric model, the FE model and to runthe analysis of the problem. Only 3-D hexahedron elements,specied as solid45 in ANSYS package, were chosen to generate themesh in the volumes through the strategic selection of the iso-parametric mapping concept. In order to achieve accurate resultsfor the SCF in the neighborhood of the hole, mesh gradation wasengineered to produce a very ne mesh near the hole and coursemesh elsewhere as shown in Fig. 3.

A mesh renement study was also conducted to optimize the FEsolution. The geometric parameters associated with the meshrenement study were as follows: t/r 2, Cs/t 0.4, l/r 15 andw/r 15 at qc 100. Five FE models with different mesh sizes wereconsidered for the mesh renement as shown in Table 1.

For each mesh level the maximum SCF (Kt) was obtained andcompared with the next ner mesh. The results of the study areshown in Fig. 4. Apparently, the Kt value of model 3 with the radius-to-element size ratio at the hole (r/e 25.3) has reacheda convergence value with an error less than 0.01% from theasymptote. This error is very small; accordingly model 3 wasconsidered throughout the analysis of the present FE model.

The boundary conditions of the quarter model were imposed by

Fig. 3. The meshed conguration of the quarter model with the boundary and loadingconditions.constraining the x-displacement at x 0 and the y-displacement aty 0 to account for the planes of symmetry of the full model. Auniform remote tensile stress (so) was applied at the plane x l.Fig. 3 shows the details of the boundary and loading conditions ofthe quarter model.

In the FE model, the properties of the 8-plies of the orthotropicangle ply laminate were homogenized. This type of homogeniza-tion only alters the normal-shear coupling terms and not the othercomponents. Therefore, the plate is treated as one thick ply withhomogenized orthotropic properties that depends on the ply angle.The homogenization technique was found used in the literature bymany researchers including Whitney and Nuismer (1974), Pipes

Table 1The mesh size specications of the ve FE models of the mesh renement study.

Model # Numberof elements

Number ofnodes

Radius-to-elementsize ratio at the hole (r/e)

1 1380 1848 12.72 5100 6160 19.03 25200 28158 25.34 48960 53475 38.05 144000 152971 44.3et al. (1979), Tan (1987), Soutis and Hu (2000) and Darwish andShivakumar (2011).

3.2. Verication analysis

The verication analysis was performed by independentlygenerating FE models by means of the prescribed element type andmeshing procedures for certain cases in the literature and byreproducing their results.

3.2.1. Thin innite orthotropic plate with a circular holeThe SCF values (KN) in an innite orthotropic plate with

a circular hole were obtained from the present FE model fordifferent material systems and different lay-ups and comparedwith the results of Lekhnitskiis equation (1963) in Table 2.

It is shown in Table 2 that the FE results are in good agreementwith the results of Lekhnistkiis equation.

3.82

10 15 20 25 30 35 40 45r/e

Fig. 4. Mesh renement results.width in addition to the materials orthotropy. The FE results of theSCF versus the radius-to-width ratio are compared with the resultsfrom Tans equation (1994). Fig. 5 presents the results of thecomparison.

It is shown in Fig. 5 that the FE results are in good agreementwith the results of Tans equation. In the next sections, FE andformulation analyses will be conducted to obtain themaximum SCFaround countersunk holes in orthotropic plates.

Table 2Comparison between KN from the FE results and Lekhnistkiis equation for thininnite orthotropic plate with a central circular hole.

Material Angle (qp) FE Results Lekhnistkii Equation %Error

Aluminum Isotropic 3.00 3.00 0.0Carbon Epoxy 0 6.06 6.32 4.1

10 5.24 5.34 2.020 4.04 3.94 2.660 1.99 1.92 3.770 2.01 2.00 0.7

E-glass Epoxy 0 3.67 3.65 0.530 3.10 3.01 2.9

Carbon Polyimide 0 6.37 6.57 3.130 4.61 4.54 1.5

of M4. Formulation analysis of the stress concentration factor

As introduced earlier, the SCF is function of the plates geometricparameters and the material orthotropy which by itself carries 9different material properties. In this study, factorial analysis(Montgomery (2000), Montgomery and Runger (2006), Dar et al.(2002)) was performed to eliminate any insignicant parameterfrom the formulation analysis. Nonlinear regression and multiparameter t analyses were also conducted on different sets of theFE results to put them on tted forms of equations.

The formulation approach of the present study is, in general,similar to that presented by Shivakumar et al. (2007) except for theinvolvement of the material properties in the nal SCF equation ofthe present study due to the orthotropic nature of the material ofthe plate.

The nal Kt equation is suggested to be expressed in thefollowing form:

3.6

3.8

4.0

4.2

4.4

4.6

4.8

5.0

5.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6

FETan Eq.

Kho

r/w

Fig. 5. Comparison between the FE results and Tans equation.

F. Darwish et al. / European Journal72Kt Kh;o Kss;o KCs;o Kqc;o (2)

where Kh,o, Kss,o, KCs,o and Kqc,o are dimensionless parameters thataccount for the effect of the width of the plate, the thickness, thecountersink depth and the countersink angle respectivelycombined with the effect of the material orthotropy on the value ofKt. The subscript o indicates for an orthotropic plate.

4.1. Innite thin orthotropic plate with a hole

In the mechanics of isotropic materials, the SCF of an innitethin plate with a circular hole is 3. This baseline value is not validwhen the plate is made of an orthotropic material due to itsdirectional properties. In the present study, the plate is made ofCarbon/Epoxy laminate with an orthotropic layup of [qp]2s. Here,the ply angle, qp, is as a key parameter in determining the ortho-tropic mechanical properties of the plate and accordingly thebaseline value of the SCF at the hole (KN). To nd an expression forKN in terms of the 9 orthotropic properties, 7 nondimensionalmaterial parameters (Ex/Ey, Ex/Ez, Ex/Gxy, Ex/Gyz, Ex/Gxz, nxy, nyz/nxz)were rst tested through factorial analysis to examine their inu-ence on the value of KN. Two levels, low and high, for eachnondimensional material parameter were chosen for this study aslisted in Table 3.The values of the low and high levels were selected carefully tocover a wide range of each material parameter. One hundred andtwenty eight ANSYS runswere performed at different combinationsof the nondimensional parameters for the geometry of w/r 15, t/r 1, Cs/t 0.5 and qc 100. The FE results from these runs wereanalyzed through Minitab software. The factorial analysis wasperformed, and the results are shown in Fig. 6.

It can be seen in Fig. 6 that among the 7 dimensionless materialparameters, only three parameters (Ex/Ey, Ex/Gxy and nxy) havemajor effect on KN. The parameter Ex/Gxz is considered to havea minor effect on KN while the remaining parameters nearlyhave no effect on KN. Based on these results, it can be stated thatKN can be expressed in terms the planar 2-D properties (Ex, Ey,Gxy and nxy) even though the analysis and geometry are threedimensional.

A 2-D analytical equation for KN was derived by Lekhnitskii(1963) for an innite thin orthotropic plate with a central hole,this equation is shown below:

KN 12

" ExEy

s nxy Ex2Gxy

#vuut (3)Since the outcome of the factorial analysis is in good agreement

with Lekhnitskiis equation, Equation (3) will be used to calculateKN which will be representing the material orthotropy in theformulation of Kh,o, Kss,o, KCs,o, Kqc,o and eventually Kt. A comparisonbetween the 3-D FE results of KN and the results of Equation (3)was previously presented in Table 2.

4.2. Formulation of Kh,o

The rst step in the formulation procedure is to account for thewidth effect on the SCF by making the plate nite and the holecentered. This change in the geometry will be represented by thegeometric parameter (r/w). The combined effect of (r/w) and (KN)on the SCFwill be introduced through the dimensionless parameter(Kh,o). Fig. 7 shows through FE results the effect of r/w and qp on thevalue of Kh,o.

By examining the best t of the curves of Fig. 7, it was found thatKh,o is in a polynomial relationship with each of r/w and KN. Many

Table 3Material parameter levels for factorial analysis.

Material parameter Low High

Ex/Ey 0.5 5.0Ex/Ez 0.5 5.0Ex/Gxy 1.67 10Ex/Gyz 1.67 10Ex/Gxz 1.67 10nxy 0.1 0.7nyz/nxz 0.6 3.0

echanics A/Solids 37 (2013) 69e78mathematical expressions were suggested and evaluated. It wasfound that the form which holds the least error is expressed as:

Kh;o X3j0

X3i0

Aij rw

iKjN (4)

where:

Aij

2664a00 a10 a20 a30a01 a11 a21 a31a02 a12 a22 a32a03 a13 a23 a33

3775

the e

F. Darwish et al. / European Journal of Mechanics A/Solids 37 (2013) 69e78 73The coefcients of the matrix Aij, except a0j, were computedthrough multi-parametric tting of the FE results by using theMinitab software. The coefcients a0j were forced to be 0,1,0,0 inorder to have Kh,o equals to KN when the plate becomes innite (r/w 0). The coefcients, as extracted from Minitab, are:

Aij

26640:00 0:11 14:04 15:751:00 1:00 12:41 20:390:00 0:09 1:84 3:100:00 0:01 0:02 0:06

3775

A comparison between the results of Equation (4) and the FEresults was conducted at non-tting values of r/w and KN. The

Fig. 6. Factorial analysis results forcomparison is shown in Fig. 8.It can be seen that the results of the equation of Kh,o are in good

agreement with the FE results with an overall maximum errorof 3.1%.

1

2

3

4

5

6

7

8

0 20 40 60 80 100 120

Kh,

o FE

Ply angle p

r/w=0.10

r/w=0.25

r/w=0.40

Fig. 7. The effect of r/w and qp on Kh,o.4.3. Formulation of Kss,o

By increasing the thickness of the plate, the hole changes toa straight shank hole and Kt is expected to become dependent ofKss,o in addition to Kh,o (Kt Kh,o*Kss,o). The value of Kss,o at any (t/r)and KN was obtained by dividing Kt from FE by Kh,o from theprevious section. The results of Kss,o at different t/r and qp are shownin Fig. 9.

It can be clearly seen in the above gure that Kss,o always equalsto 1.0 regardless to the values of (t/r) and qp. Accordingly, it can bestated that Kss,o does not contribute to Kt and therefore it is droppedout from the formulation analysis. Based on this result, Equation (2)will be modied to the new form written below:

ffect of material parameters on KNKt Kh;o KCs;o Kqc;o (5)In fact, the above result was not surprising and it absolutely

agrees with the outcomes of the factorial analysis presented earlierwhich has revealed that the orthotropic properties in the z-

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.1 0.2 0.3 0.4 0.5

Kh,

o

r/w

FE Equation (4)

p=30

p=50

p=80

Fig. 8. Comparison between Kh,o from Equation (4) and FE results.

of Mechanics A/Solids 37 (2013) 69e78direction (Ez, Gxz, Gyz, nxz and nyz) have a negligible effect on theSCF.

4.4. Formulation of KCs,o

In this section, the straight shank hole is modied to a coun-tersunk hole which in fact adds a degree of complexity to theanalysis. Having a sinking depth through a portion of the holeallows the stresses to ow through the thickness of the platearound the hole. Therefore, the sinking depth represented by (Cs/t)is expected to affect the value of Kt. Here comes the signicance offormulating an expression for KCs,o that integrates the role of thegeometry of the countersunk hole and the material orthotropy indetermining the Kt value. Factorial analysis was performed todetermine the geometric parameters that directly affect the valueof KCs,o which can be obtained by dividing the Kt value from the FE

0.0

0.5

1.0

1.5

2.0

0.0 1.0 2.0 3.0 4.0 5.0

p = 10o

p = 20o

p = 40o

p = 60o

Kss

,o

t/r

Fig. 9. The effect of (t/r) on Kss,o at different qp.

F. Darwish et al. / European Journal74solution by Kh,o from Equation (4). The results of the factorialanalysis are shown in Fig. 10.

Here it can be stated that a formula for KCs,omust include r/w, t/rand Cs/t in addition to KN. A general equation of KCs,o is expressed inthe form shown below:

KCs;0 1 a1tr

b1Cst

KNc1a2

tr

b2Cst

2KNc2

a3tr

b3Cst

d1KNc3

rw

d2 6This formwas chosen tomeet the limiting geometric conditions,

for example, Equation (6) goes to 1 when Cs/t 0 (straight shankhole) or t/r 0 (thin plate). The tuning parameters (as, bs, cs, andds) were computed using the Minitab software. The resultingequation of KCs,o is:

KCs;010:16tr

0:45Cst

KN1:140:03

tr

0:74Cst

2KN1:87

1:76tr

1:65Cst

1:47KN0:56

rw

1:777

A comparison between the FE results of KCs,o and Equation (7) atdifferent geometric congurations and different ply angles isshown in Table 4.The error in Table 4 was calculated as:

%Error Kt;eq Kt;FE

Kt;FE

100 (8)

Based on Table 4, it can be stated that the results of the devel-oped equation of KCs,o match very well with the FE results.

4.5. Formulation of Kqc,o

The nal part of the formulation procedure of Kt is to nd anappropriate expression of the last parameter Kqc,o. The generalexpression of Kqc,o was written in a similar form to that presentedby Shivakumar et al. (2007) with the inclusion of the materialparameter (KN) in the slope (m) of Equation (9);

Kqc;o 1mqc 100

(9)

The slope (m) in the above equation is expressed in equation(10).

m At=rl1KNl2 (10)

where A, l1 and l2 are functions of (Cs/t) and are expressed in thefollowing equations after performing multi-parametric tting bymeans of Minitab:

Cs

Cs2 Cs

Fig. 10. Results of the factorial analysis for KCs,o equation.A t

0:008t

0:015t

0:01 (11)

Table 4Comparison between KCs,o from Equation (7) and FE.

r/w t/r 1 t/r 2 t/r 4KCs,oFE

KCs,oEq. (7)

%Error KCs,oFE

KCs,oEq. (7)

%Error KCs,oFE

KCs,oEq. (7)

%Error

Cs/t 0.1, qp 00.1 1.122 1.122 0.0 1.168 1.164 0.3 1.213 1.222 0.80.25 1.136 1.132 0.4 1.181 1.169 1.0 1.229 1.236 0.50.4 1.137 1.134 0.3 1.187 1.176 0.9 1.239 1.241 0.2Cs/t 0.25, qp 450.1 1.120 1.084 3.2 1.175 1.148 2.3 1.253 1.192 4.90.25 1.135 1.095 3.5 1.203 1.155 4.2 1.319 1.278 3.10.4 1.136 1.112 1.9 1.237 1.206 3.8 e e eCs/t 0.50, qp 800.1 1.126 1.171 4.0 1.205 1.236 2.6 1.333 1.345 0.90.25 1.166 1.200 2.8 1.293 1.325 2.5 1.590 1.622 2.00.4 1.251 1.245 0.5 e e e e e e

l1 Cst

4:8

Cst

210:7

Cst

6

(12)

l2 Cst

0:9

Cst

20:6

Cst

1

(13)

According to Equation (9), one can notice that Kqc,o 1 atqc 100 (reference countersink angle) and thatm 0 and Kqc,o 1at Cs/t 0 (straight shank hole) or at t/r 0 (thin plate).

A comparison between Equation (9) and the FE results of Kqc,o

5.1.1. Innite plate with countersunk hole at qc 100In this special case r/w 0 (innite plate), therefore, Kh,o KN

and Kqc,o 1 (qc 100). Accordingly, the SCF equation becomes(Kt KN*KCs,o). In this equation Kt is a function of t/r, Cs/t and qp only.The results of Kt versus qp at different values of t/r and Cs/t wereobtained from Equation (5) and FE. Figs. 11 and 12 show compari-sons between Equation (5) and the FE results at (t/r 1 and Cs/t 0.5), and (t/r 4 and Cs/t 0.1) respectively.

It can be seen in Figs. 11 and 12 that the analytical and FE resultsare in a good match with an error becoming commonly at

Table 5Comparison between Kqc,o from Equation (9) and FE at r/w 0. 1, t/r 2 and qp 40 .

qc Cs/t 0.1 Cs/t 0.25 Cs/t 0.5Kqc,o FE Kqc,o Eq. (9) %Error Kqc,o FE Kqc,o Eq. (9) %Error Kqc,o FE Kqc,o Eq. (9) %Error

80 0.9815 0.9777 0.4 0.9651 0.9489 1.7 0.9465 0.9438 0.390 0.9910 0.9888 0.2 0.9829 0.9744 0.9 0.9734 0.9719 0.2100 1.0000 1.0000 0.0 1.0000 1.0000 0.0 1.0000 1.0000 0.0110 1.0106 1.0112 0.1 1.0220 1.0256 0.4 1.0356 1.0281 0.7120 1.0215 1.0223 0.1 1.0429 1.0511 0.8 1.0725 1.0562 1.5

F. Darwish et al. / European Journal of Mechanics A/Solids 37 (2013) 69e78 756

twas performed over a wide range of (Cs/t), (t/r) and KN. Thiscomparison has shown a good agreement between the FE resultsand the analytical results. Part of this comparison is shown inTable 5 in which the error was calculated using Equation (8).

After obtaining and testing the expressions of the dimensionlessparameters of Kt, Equation (5) can be used to calculate the SCF ofa countersunk hole in an orthotropic plate.

5. Results and discussion

5.1. Results

The results of three cases of an orthotropic plate with a coun-tersunk hole obtained from Equation (5) and the FE solutions areconsidered in this section. The three cases are: (1) An innite platewith a countersunk hole at qc 100; (2) An innite plate witha countersunk hole with qc ranging from 80 to 120; (3) A nitewidth plate with a countersunk hole.

5

7

8

9

10

K

FE Equation (5) 0

1

2

3

4

0 10 20 30 40 50 60 70 80 90 100Ply angle p

Fig. 11. The stress concentration factor Kt versus ply angle qp (t/r 1, Cs/t 0.5, r/w 0).maximum at qp between 40 and 50.

5.1.2. Innite plate with a countersunk holeThis case is one step more general than the previous one since

the countersink angle (qc) varies between 80 and 120. Therefore,the SCF equation becomes (Kt KN*KCs,o*Kqc,o). Here in this equa-tion Kt is function of t/r, Cs/t, qp and qc. The results of Kt fromEquation (5) and FE over a wide range of all variables were ob-tained. Table 6 presents the Kt results from Equation (5) and the FEsolutions versus the ply angle qp at qc 80, 100 and 120.

Once more, the error is at maximumwhen the ply angle rangesfrom 40 to 50.

5.1.3. Finite width plate with a countersunk holeThis case represents the most general case of the SCF since all

the geometric parameters r/w, t/r, Cs/t, and qc; and the ply angle qpcontribute to the Kt equation (Kt Kh,o*KCs,o*Kqc,o). As there are 5independent variables affecting the value of Kt, a very large numberof Kt values as obtained from Equation (5) and FE-based data arecompared. Figs. 13e17 show a synthesis of the overall results.

4

5

6

7

8

Kt

FE Equation (5) 0

1

2

3

0 10 20 30 40 50 60 70 80 90 100Ply angle p

Fig. 12. The stress concentration factor Kt versus ply angle qp (t/r 4, Cs/t 0.1, r/w 0).

Fig. 13 shows the relationship between Kt and r/w at twodifferent values of Cs/t. It can be seen that the results of Equation (5)are in good agreement with the FE results. Also it is observed that asthe plate becomes narrower the Kt value increases.

The effect of plates thickness on the SCF is presented in Fig.14 atdifferent ratios of r/w.

Fig. 15 presents the effect of Cs/t on Kt at different values of r/w.

plate (the width after which the edges of the plate would have noeffect on the SCF) is discussed hereinafter. For an innite plate,

Table 6Comparison between Kt from Equation (5) and FE at Cs/t 0. 25 and t/r 1.

qp qc 80 qc 100 qc 120

Kt FE Kt Eq. (5) %Error Kt FE Kt Eq. (5) %Error Kt FE Kt Eq. (5) %Error

0 7.35 7.85 6.8 7.71 8.02 3.9 8.03 8.18 1.810 6.25 6.41 2.6 6.52 6.55 0.5 6.82 6.69 1.920 4.56 4.49 1.6 4.71 4.6 2.4 4.89 4.70 3.930 3.25 3.14 3.4 3.33 3.23 3.0 3.43 3.31 3.540 2.67 2.31 13.5 2.75 2.37 13.5 2.83 2.44 13.850 2.22 1.99 10.5 2.29 2.05 10.4 2.35 2.10 10.360 2.01 2.01 0.0 2.02 2.07 2.2 2.06 2.13 3.570 2.08 2.10 1.2 2.10 2.16 2.9 2.13 2.22 4.280 2.33 2.34 0.4 2.36 2.41 2.2 2.41 2.48 2.890 2.56 2.58 0.8 2.61 2.65 1.6 2.7 2.72 0.7

5

6

7

8

9

10

0 1 2 3 4 5 6

Kt

t/r

r/w = 0.1

r/w = 0.25

FE Equation (5)

9

10

r/w = 0.25 FE Equation (5)

F. Darwish et al. / European Journal of Mechanics A/Solids 37 (2013) 69e7876The relationships between Kt and the countersink angle qc atdifferent Cs/t values and between Kt and the ply angle qp are shownin Figs. 16 and 17 respectively.

In summary of the comparison study between the results of thedeveloped equation and the FE results, it was found that the error inKt is within 7% for more than 96% of the FE runs (1098 out of 1140runs) with an overall maximum error less than 14%.

The location of the maximum SCF was found at the side of thecountersunk hole (90 from the direction of the load application) atthe edge between the sinking depth and the straight shank portionof the hole. Figs. 18 and 19 present the SCF variations through thethickness of the hole at different Cs/t and qc values respectively.

It can be seen in Fig. 18 that the location of the maximum SCFvaries with changing the value of Cs/t (different edge locations). Onthe other hand, Fig. 19 shows a xed location for the maximum SCFbecause Cs/t is constant (xed edge location).

5.2. Discussion

By inspecting the results of Kt at the different values of r/w, t/r,Cs/t, qc and qp it was found that by increasing the value of any of the

10

11

12

Cs/t = 0.5 FE Equation (5) 5

6

7

8

9

0.0 0.1 0.2 0.3 0.4 0.5

Kt

r/w

Cs/t = 0.1

Fig. 13. The stress concentration factor Kt versus r/w (t/r 1, qc 90 and qp 0).nondimensional geometric parameters (r/w, t/r, Cs/t, and qc) thevalue of Kt would increase. First, when r/w increases, the edges ofthe plate become closer to the hole. Accordingly, the force lines willbe compressed and the SCF will increase. Similarly, increasing (t/r)or (Cs/t) while maintaining the other parameters xed, the sinkingdepth of the countersunk hole will increase. This change ingeometry tends to have more force lines changing its directiondownward towards the root of the sinking surface which turns toincreases the SCF. The direct proportionality between Kt and qc isreferred to the fact that as qc increases the countersunk hole furtherdeviates from the straight shank hole and therefore Kt increases.Finally, as shown in the results of the previous section, the trend ofthe Kt versus the ply angle qp can be described as descending,reaching a minimum value at qp 50e60, and then ascending. Toexplain this behavior, the effect of the ber orientation on the stressow around the hole and on determining the innite width of the

Fig. 14. The stress concentration factor Kt versus t/r (Cs/t 0.1, qc 110 and qp 0 ).5

6

7

8

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Kt

Cs/t

r/w = 0.1

Fig. 15. The stress concentration factor Kt versus Cs/t (t/r 1, qc 100 and qp 0).

56

7

8

9

10

70 80 90 100 110 120 130

Kt

Countersink angle c

Cs/t = 0.1

Cs/t = 0.25

Emax= 3.5%

FE Equation (5)

Fig. 16. The stress concentration factor Kt versus the countersink angle (qc) (r/w 0.1,t/r 1 and qp 0).

0.0 0.2 0.4 0.6 0.8 1.0z/t

Fig. 18. Stress concentration factor variation through the thickness of the hole (r/w 0.1, t/r 1 and qc 100).

0.5

1.0

1.5

2.0

2.5

3.0

3.5

c = 80o

c = 100o

c = 120o

Kt FE

F. Darwish et al. / European Journal of Mincreasing (qp) gradually from 0 (bers parallel to the load direc-tion) to 90 (bers perpendicular to the load direction), themodulus in the load direction Ex decreases from maximum tominimum and the lateral modulus Ey increases from minimum tomaximum. This change in the mechanical properties reduces theconcentration of the axial stresses (sx) at the sides of the hole. Inaddition, the in-plane shear modulus and Poisons ratio, Gxy and nxy,reach their peak values at 45-degrees and near 25-degreesrespectively which also affect the Kt vs. the ply angle distribution.On the other hand, for xed r/w, stiffening the lateral direction ofthe plate increases the edge effect, or in other words, the width atwhich the plate becomes innite increases. In summary of thispoint, increasing qp from 0 to 90, the SCF decreases due to thechange of the mechanical properties of the plate, and on the otherhand increases due to the increasing edge effect. This counteracteffect is responsible for the descending-ascending trend of the Ktversus q curve.p

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70 80 90 100

Kt

Ply angle p

r/w = 0.1

r/w = 0.25

FE Equation (5)

Fig. 17. The stress concentration factor Kt versus the ply angle (Cs/t 0.25, t/r 1 andqc 100).0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Cs/t = 0.10Cs/t = 0.25Cs/t = 0.50

KtFE

echanics A/Solids 37 (2013) 69e78 77At the end of the discussion, it can be stated that the SCF can beminimized by selecting small values of (r/w, t/r, Cs/t and qc) and a plyangle between 50 and 60 for the orthotropic laminate. Theminimum Kt value obtained in this study was 1.983 for an inniteplate with r/w 0, t/r 1, Cs/t 0.1, qc 80 and qp 60.

6. Conclusions

A detailed 3-D FE stress analysis was conducted on countersunkholes in orthotropic plates subjected to tensile loading. The analysisincluded wide range of geometric parameters: radius-to-widthratio, thickness-to-radius ratio, countersink depth-to-thicknessratio and countersink angle. Eight plies of carbon/epoxy (AS4/3501-6) with stacking sequence of [qp]2s were considered for theorthotropic plate. The effect of varying the geometric parameters atdifferent orthotropic congurations on the SCF was examined.Based on the FE results, it was found that the maximum Kt occurredat the countersunk edge. It was also found that Kt increasesmonotonically with r/w, t/r, Cs/t and qc, and that it has a descending

0.00.0 0.2 0.4 0.6 0.8 1.0

z/t

Fig. 19. Stress concentration factor variation through the thickness of the hole (r/w 0.1, t/r 1 and Cs/t 0.25).

- ascending trend with the ply angle, with a minimum value of Kt atqp 50e60.

By means of factorial and multi parameter t analyses of the FEresults, it was possible to specify and drop out any insignicantparameters and to t the remaining ones in a general parametricequation that predicts the maximum SCF of countersunk holes inorthotropic plates.

Several comparison studies were performed to examine theaccuracy of the developed equation. It was found that the resultsobtained by Equation (5) are accurate within 7% of the FE results for96% of the runs. The maximum overall error was less than 14%.

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Stress concentration analysis for countersunk rivet holes in orthotropic plates1. Introduction2. Geometry and materials3. Approach and verification of the analysis3.1. Finite element modeling3.2. Verification analysis3.2.1. Thin infinite orthotropic plate with a circular hole3.2.2. Thin finite orthotropic plate with a circular hole

4. Formulation analysis of the stress concentration factor4.1. Infinite thin orthotropic plate with a hole4.2. Formulation of Kh,o4.3. Formulation of Kss,o4.4. Formulation of KCs,o4.5. Formulation of Kc,o

5. Results and discussion5.1. Results5.1.1. Infinite plate with countersunk hole at c = 1005.1.2. Infinite plate with a countersunk hole5.1.3. Finite width plate with a countersunk hole

5.2. Discussion

6. ConclusionsReferences