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Materials Science and Engineering A 498 (2008) 115–124

Contents lists available at ScienceDirect

Materials Science and Engineering A

journa l homepage: www.e lsev ier .com/ locate /msea

inite element analysis for stress concentration and deflection in isotropic,rthotropic and laminated composite plates with central circular hole underransverse static loading

.K. Jain ∗, N.D. Mittalepartment of Applied Mechanics, Maulana Azad National Institute of Technology, Bhopal, India

r t i c l e i n f o

rticle history:eceived 22 June 2007eceived in revised form5 September 2007ccepted 14 April 2008

eywords:inite element method

a b s t r a c t

The distributions of stresses and deflection in rectangular isotropic, orthotropic and laminated compositeplates with central circular hole under transverse static loading have been studied using finite elementmethod. The aim of author is to analyze the effect of D/A ratio (where D is hole diameter and A is platewidth) upon stress concentration factor (SCF) and deflection in isotropic, orthotropic and laminatedcomposite plates under different transverse static loading condition. Analysis has been done for sym-metric and anti-symmetric composite laminates. The results are obtained for three different boundaryconditions. The variations of SCF and deflection with respect to D/A ratio are presented in graphical

tress concentration factoreflection

form and discussed. The finite element formulation is carried out in the analysis section of the ANSYSpackage.

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aminatesoundary conditions

. Introduction

An isotropic, orthotropic and laminated composite plateith central circular hole under transverse loading, have foundidespread applications in various fields of engineering such as

erospace, marine, automobile and mechanical. For design of suchlates with hole, accurate knowledge of deflection, stresses andtress concentration are required. Stress concentration arises fromny abrupt change in geometry of plate under loading. As a result,tress distribution is not uniform throughout the cross section. Fail-res such as fatigue cracking and plastic deformation frequentlyccur at points of stress concentration.

Paul and Rao [1,2] presented a theory for evaluation of stressoncentration factor (SCF) of thick and fibre reinforced polymerFRP) laminated plate with the help of Lo-Christensen-Wu higherrder bending theory under transverse loading. Shastry and Raj [3]ave analysed the effect of fibre orientation for a unidirectionalomposite laminate with finite element method by assuming a

lane stress problem under in plane static loading. Xiwu et al. [4,5]valuated stress concentration of finite composite laminates withlliptical hole and multiple elliptical holes based on classical lam-nated plate theory. Iwaki [6] worked on stress concentrations in

∗ Corresponding author.E-mail address: nitinkumarjain2001@yahoo.co.in (N.K. Jain).

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921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2008.04.078

© 2008 Elsevier B.V. All rights reserved.

plate with two unequal circular holes. Ukadgaonker and Rao [7]roposed a general solution for stresses around holes in symmetric

aminates by introducing a general form of mapping function andn arbitrary biaxial loading condition in to the boundary conditions.ing et al. [8] presented a theory for stress analysis by using rhombicrray of alternating method for multiple circular holes. Chaudhuri9] worked on stress concentration around a part through holeeakening a laminated plate by finite element method. Mahiou

nd Bekaou [10] studied for local stress concentration and for therediction of tensile failure in unidirectional composites. Toubal etl. [11] studied experimentally for stress concentration in a circu-ar hole in composite plate. Younis [12] investigated by reflectedhoto elasticity method that the assembly stresses contributes toeducing stresses around the holes in a plate. Peterson [13] haseveloped good theory and charts on the basis of mathematicalnalysis and presented excellent methodology in graphical form forvaluation of stress concentration factors in isotropic plates withifferent types of abrupt change, but no techniques are presentedor evaluation of the stress concentration factor in orthotropic andaminated plate. In literature [14–21], works for various cases oftress concentration in plates with hole can be found.

In this article, a study of rectangular isotropic, orthotropic andaminated composite plates with central circular hole for the effectf D/A ratio where D is hole diameter and A is plate width on SCFnder different transverse static loading condition is made. Theurpose of this research work is to investigate the effect of D/A ratio

116 N.K. Jain, N.D. Mittal / Materials Science and Engineering A 498 (2008) 115–124

F ted plate with central hole. (b) Uniformly distributed loading of P Newton. (c) Loading atb

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ig. 1. Basic model of the problem under different loading conditions. (a) A laminaoundary of hole.

n SCF for normal stress in X, Y directions (�x, �y), shear stress in XYlane (�xy) and von Mises (equivalent) stress (�eqv), and on deflec-ion in transverse direction (Uz). The Uz for different ratio of D/A isompared with deflection in transverse direction in plate withoutole (U∗

z ). Results are obtained for three different boundary con-itions. The analytical treatment for such type of problem is veryifficult and hence the finite element method is adopted for wholenalysis.

. Description of problem

The influence of D/A ratio upon deflection and SCF for differenttresses, isotropic, orthotropic and four ply laminated compositelates of dimension 200 mm × 100 mm × 1 mm with a circular holef diameter D at centre under two different loading conditions asniformly distributed loading P Newton and four point loads F New-on at edges of hole in transverse direction for all cases is analysedy finite element method. Fig. 1 shows the details and dimensionsf the problem.

. Finite element analysis

An eight nodded structural three-dimensional (3-D) Shell Ele-ent (specified as Shell93 in ANSYS package) with element length

f 2 mm for isotropic and orthotropic plates and another eightodded Linear Layered Structural 3-D Shell Element (specified ashell99 in ANSYS package) with element length of 2 mm for com-osite laminated plates, were selected based on convergence testnd used through out the study. For both elements, each node has

ix degrees of freedom, making a total 48 degrees of freedom perlement. In order to construct the graphical image of the geome-ries of models for different D/A ratios, a plate examined usinghe ANSYS (Advanced Engineering Simulation). Mapped meshings used for all models so that more elements are employed near the

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Fig. 2. Typical example of finite element model for D/A = 0.2.

ole boundary. Due to the symmetric nature of different modelsnvestigated, it was necessary to discretize the quadrant plate fornite element analysis. Main task in finite element analysis is selec-ion of suitable elements. Numbers of checks and convergence testre made for selection of suitable elements from different avail-ble elements and to decide the element length. Results were thenisplayed by using post processor of ANSYS programme. Fig. 2 pro-ides the example of the discretized models for D/A = 0.2, used intudy.

. Results and discussion

Numerical results are presented for isotropic, orthotropic andaminated composite plates with a central circular hole. Laminatesre composite plates consisting of four plies of same material andqual thickness of 0.25 mm. Symmetric cross ply laminate has sym-

etric lay up [0◦/90◦/90◦/0◦] and anti-symmetric cross ply has

nti-symmetric lay up [90◦/0◦/90◦/0◦]. The material properties ofsotropic plate are as: [E, �]: [39 GPa, 0.3] and, of orthotropic andaminated composite plate are as: [(E1, E2, E3, G12, G23, G31) �12,

23, �31]: [(39, 8.6, 8.6, 3.8, 3.8, 3.8) GPa, 0.28, 0.28, 0.28]. Here, E,

N.K. Jain, N.D. Mittal / Materials Science and Engineering A 498 (2008) 115–124 117

t all e

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Fig. 3. Boundary conditions a

and � represent modulus of elasticity, modulus of rigidity andoisson’s ratio.

Plates with three different boundary conditions, as plate (a), (b)nd (c) are analysed. In plate (a), all edges are simply supported, inlate (b); all edges are fixed and in plate (c); two edges are fixednd other two are simply supported. Fig. 3 provides the details ofoundary conditions at all edges of plates (a), (b) and (c). Stressesnd deflections are obtained for two different loading conditions asniformly distributed loading and loading at boundary of hole forll D/A ratios.

Fig. 4 shows the effect of D/A on SCF (for �x, �y, �xy and �eqv)nd Uz/U∗

z in isotropic plates (a), (b) and (c) under uniformly dis-ributed loading. Following observations can be made from theseesults. Variation of SCF for �x with respect to D/A is maximum inase of plate (a) and almost negligible in plates (b) and (c). Maxi-um stress concentration for �x always occurred in plate (a) for all/A ratios. In the case of plate (a), SCF for �x continuously decreased

rom 2.09 to 1.12, when D/A increased from 0.1 to 0.5. In the case oflate (b), SCF for �x continuously increased from 1.00 to 1.07, when/A increased from 0.1 to 0.5. In the case of plate (c), SCF for �x

ontinuously increased from 1.01 to 1.11, when D/A increased from

.1 to 0.5. Maximum SCF for �x is obtained as 2.09 in plate (a) at/A = 0.1. Variation of SCF for �y with respect to D/A is significant

n case of plates (a) and (c) and negligible in plate (b). Maximumtress concentration for �y occurred for plate (a) for almost all D/A

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Fig. 4. Effect of D/A ratio on SCF (for �x , �y , �xy and (�eqv) and Uz/U∗z in

dges of plates (a), (b) and (c).

atios. In the case of plate (a), SCF for �y continuously decreasedrom 1.81 to 1.57, when D/A increased from 0.1 to 0.5. In the casef plate (b), SCF for �y continuously decreased from 1.00 to 0.90,hen D/A increased from 0.1 to 0.5. In the case of plate (c), SCF fory continuously decreased from 1.82 to 1.41, when D/A increased

rom 0.1 to 0.5. Maximum SCF for �y is obtained as 1.82 in platec) at D/A = 0.1. Variation of SCF for �xy with respect to D/A is max-mum in plate (b), significant in plate (c) and minimum in platea). Maximum stress concentration for �xy occurred for plate (b)or all D/A ratios. For plate (a), SCF for �xy continuously decreasedrom 1.22 to 1.04, when D/A increased from 0.1 to 0.5. In the casef plate (b), SCF for �xy continuously decreased from 2.36 to 1.35,hen D/A increased from 0.1 to 0.5. For plate (c), SCF for �xy con-

inuously decreased from 1.81 to 1.38, when D/A increased from.1 to 0.5. Maximum SCF for �xy is obtained as 2.36 in plate (b) at/A = 0.1. Variation of SCF for �eqv with respect to D/A is maximum

n the case of plate (c), significant in plate (a) and almost negligiblen plate (b). Maximum stress concentration for �eqv occurred forlate (a) for all D/A ratios. In the case of plate (a), SCF for �eqv con-inuously decreased from 1.87 to 1.63, when D/A increased from 0.1o 0.5. For plate (b), SCF for � continuously decreased from 1.00

eqv

o 0.90, when D/A increased from 0.1 to 0.5. In the case of plate (c),CF for �eqv continuously decreased from 1.48 to 1.15, when D/Ancreased from 0.1 to 0.5. Maximum SCF for �eqv is obtained as 1.87n plate (a) at D/A = 0.1. Variation of Uz/U∗

z with respect to D/A is

isotropic plates (a), (b) and (c) under uniformly distributed load.

118 N.K. Jain, N.D. Mittal / Materials Science and Engineering A 498 (2008) 115–124

∗z in i

mmpc010U0tD

UoVaftt�ff2�oao�fidpD2

t(a5fDDfSSafSi2�f(iUUfftf(

at

Fig. 5. Effect of D/A ratio on SCF (for �x , �y , �xy and �eqv) and Uz/U

aximum in the case of plate (a), significant in plate (c) and mini-um in plate (b). Maximum ratio of Uz/U∗

z occurred in the case oflate (a) for all D/A ratios. In the case of plate (a), the ratio of Uz/U∗

zontinuously increased from 1.06 to 1.32 with increase of D/A from.1 to 0.5. For plate (b), the ratio of Uz/U∗

z increased from 1.06 to.10 with increase of D/A from 0.1 to 0.2 and decreased from 1.10 to.98 with increase of D/A from 0.2 to 0.5. For plate (c), the ratio ofz/U∗

z increased from 1.06 to 1.24 with increase of D/A from 0.1 to.4 and decreased from 1.24 to 1.22 with increase of D/A from 0.4o 0.5. Maximum ratio of Uz/U∗

z is obtained as 1.32 in plate (a) at/A = 0.5.

Fig. 5 shows the effect of D/A on SCF (for �x, �y, �xy and �eqv) andz/U∗

z in isotropic plates (a), (b) and (c) under loading at boundaryf hole. Following observations can be made from these results.ariation of SCF for �x with respect to D/A is maximum for plates (a)nd (c) and minimum in plate (b). Maximum stress concentrationor �x always occurred for plates (a) and (c) for all D/A ratios. Inhe case of plate (a), SCF for �x continuously increased from 2.21o 2.54, when D/A increased from 0.1 to 0.5. For plate (b), SCF forx continuously increased from 2.23 to 2.44 with increase of D/A

rom 0.1 to 0.3 and decreased from 2.44 to 2.28 with increase of D/Arom 0.3 to 0.5. For plate (c), SCF for �x continuously increased from.20 to 2.54, when D/A increased from 0.1 to 0.5. Maximum SCF forx is obtained as 2.54 in plates (a) and (c) at D/A = 0.5. Variationf SCF for �y with respect to D/A is significant in plates (a) and (c)nd negligible in plate (b). Maximum stress concentration for �y

ccurred in plate (a) for almost all D/A ratios. In plate (a), SCF fory continuously increased from 1.92 to 2.26, when D/A increased

rom 0.1 to 0.5. In the case of plate (b), SCF for �y continuously

ncreased from 1.93 to 2.10 with increase of D/A from 0.1 to 0.3 andecreased from 2.10 to 2.06 with increase of D/A from 0.3 to 0.5. Inlate (c), SCF for �y continuously increased from 1.92 to 2.23, when/A increased from 0.1 to 0.5. Maximum SCF for �y is obtained as.26 in plate (a) at D/A = 0.5. Variation of SCF for �xy with respect

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sotropic plates (a), (b) and (c) under loading at boundary of hole.

o D/A is maximum in plate (a) and significant in plates (b) andc). Maximum stress concentration for �xy occurred in plate (a) forll D/A ratios. In plate (a), SCF for �xy continuously decreased from.07 to 2.44, when D/A increased from 0.1 to 0.5. In plate (b), SCFor �xy continuously decreased from 5.66 to 3.06 with increase of/A from 0.1 to 0.3 and increased from 3.06 to 3.46 with increase of/A from 0.3 to 0.5. In plate (c), SCF for �xy continuously decreased

rom 4.66 to 2.11, when D/A increased from 0.1 to 0.5. MaximumCF for �xy is obtained as 5.66 in plate (b) at D/A = 0.1. Variation ofCF for �eqv with respect to D/A is significant in plates (a) and (c)nd almost negligible in plate (b). Maximum stress concentrationor �eqv occurred in plate (a) for almost all D/A ratios. In plate (a),CF for �eqv continuously increased from 2.19 to 2.65, when D/Ancreased from 0.1 to 0.5. For plate (b), SCF for �eqv changed from.35 to 2.53, when D/A increased from 0.1 to 0.5. In plate (c), SCF foreqv continuously increased from 2.24 to 2.59, when D/A increased

rom 0.1 to 0.5. Maximum SCF for �eqv is obtained as 2.65 in platea) at D/A = 0.5. Variation of Uz/U∗

z with respect to D/A is maximumn plates (a) and (c) and significant in plate (b). Maximum ratio ofz/U∗

z occurred in plate (a) for all D/A ratios. In plate (a), the ratio ofz/U∗

z continuously increased from 1.14 to 1.70 with increase of D/Arom 0.1 to 0.5. In the case of plate (b), the ratio of Uz/U∗

z increasedrom 1.18 to 1.40 with increase of D/A from 0.1 to 0.5. For plate (c),he ratio of Uz/U∗

z increased from 1.14 to 1.64 with increase of D/Arom 0.1 to 0.5. Maximum ratio of Uz/U∗

z is obtained as 1.70 in platea) at D/A = 0.5.

Fig. 6 shows the effect of D/A ratio on SCF (for �x, �y, �xy and �eqv)nd Uz/U∗

z in orthotropic plates (a), (b) and (c) under uniformly dis-ributed loading. Following observations can be made from these

esults. Variation of SCF for �x with respect to D/A is maximum inlate (a) and almost negligible in plates (b) and (c). Maximum stressoncentration for �x always occurred in plate (a) for all D/A ratios. Inlate (a), SCF for �x decreased from 2.00 to 1.13 with increase of D/Arom 0.1 to 0.3 and increased from 1.13 to 1.23 with increase of D/A

N.K. Jain, N.D. Mittal / Materials Science and Engineering A 498 (2008) 115–124 119

∗z in o

f0�fai(a2�f2��(�fitf0Diirt�0D3ipa1

U0iia

�iti((dpf11�w((22�f2ora

Fig. 6. Effect of D/A ratio on SCF (for �x, �y, �xy and �eqv) and Uz/U

rom 0.3 to 0.5. In plate (b), SCF for �x continuously increased from.72 to 0.80, when D/A increased from 0.1 to 0.5. In plate (c), SCF forx continuously increased from 0.83 to 0.91, when D/A increased

rom 0.1 to 0.5. Maximum SCF for �x is obtained as 2.00 in plate (a)t D/A = 0.1. Variation of SCF for �y with respect to D/A is maximumn case of plate (c), significant in plate (a) and minimum in plateb). Maximum stress concentration for �y occurred in plate (c) forll D/A ratios. In plate (a), SCF for �y continuously decreased from.75 to 1.64, when D/A increased from 0.1 to 0.5. In plate (b), SCF fory continuously decreased from 1.29 to 0.87, when D/A increased

rom 0.1 to 0.5. In plate (c), SCF for �y continuously decreased from.99 to 1.21, when D/A increased from 0.1 to 0.5. Maximum SCF fory is obtained as 2.99 in plate (c) at D/A = 0.1. Variation of SCF forxy with respect to D/A is maximum in plate (a), significant in plateb) and minimum in plate (c). Maximum stress concentration forxy occurred in plate (a) for almost all D/A ratios. In plate (a), SCFor �xy continuously increased from 2.94 to 3.58, when D/A rationcreased from 0.1 to 0.5. In plate (b), SCF for �xy changed from 3.16o 2.92, when D/A ratio increased from 0.1 to 0.5. In plate (c), SCFor �xy changed from 2.82 to 1.94, when D/A ratio increased from.1 to 0.5. Maximum SCF for �xy is obtained as 3.58 in plate (a) at/A = 0.5. Variation of SCF for �eqv with respect to D/A is maximum

n case of plate (a) and almost negligible in plates (b) and (c). Max-mum stress concentration for �eqv occurred in plate (a) for all D/Aatios. In plate (a), SCF for �eqv continuously increased from 2.61o 3.19, when D/A increased from 0.1 to 0.5. In plate (b), SCF foreqv fluctuated between 0.75 and 0.80, when D/A increased from.1 to 0.5. In plate (c), SCF for �eqv increased from 0.83 to 0.91, when/A increased from 0.1 to 0.5. Maximum SCF for �eqv is obtained as

.19 in plate (a) at D/A = 0.5. Variation of Uz/U∗

z with respect to D/As maximum of plate (a), significant in plate (c) and minimum inlate (b). Maximum ratio of Uz/U∗

z occurred in plate (a) for almostll D/A ratios. In plate (a), the ratio of Uz/U∗

z changed from 1.07 to.25 with increase of D/A from 0.1 to 0.5. In plate (b), the ratio of

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rthotropic plates (a), (b) and (c) under uniformly distributed load.

z/U∗z increased from 0.57 to 0.65 with increase of D/A from 0.1 to

.5. In plate (c), the ratio of Uz/U∗z increased from 1.11 to 1.17 with

ncrease of D/A from 0.1 to 0.2 and decreased from 1.17 to 1.02 withncrease of D/A from 0.2 to 0.5. Maximum ratio of Uz/U∗

z is obtaineds 1.26 in plate (a) at D/A = 0.4.

Fig. 7 shows the effect of D/A ratio on SCF (for �x, �y, �xy andeqv) and Uz/U∗

z in orthotropic plates (a), (b) and (c) under load-ng at boundary of hole. Following observations can be made fromhese results. Variation of SCF for �x with respect to D/A observed,s significant in plate (a) and almost negligible in plates (b) andc). Maximum stress concentration for �x always occurred in platec) for almost all D/A ratios. In plate (a), SCF for �x continuouslyecreased from 1.36 to 1.13, when D/A increased from 0.1 to 0.5. Inlate (b), SCF for �x changed from 0.83 to 0.79, when D/A increasedrom 0.1 to 0.5. In plate (c), SCF for �x continuously decreased from.24 to 1.11 with increase of D/A from 0.1 to 0.3 and increased from.11 to 1.17 with increase of D/A from 0.3 to 0.5. Maximum SCF forx is obtained as 1.36 in plate (a) at D/A = 0.1. Variation of SCF for �y

ith respect to D/A is significant in plate (b) and negligible in platesa) and (c). Maximum stress concentration for �y occurred in platea) for almost all D/A ratios. In plate (a), SCF for �y decreased from.42 to 2.29 with increase of D/A from 0.1 to 0.3 and increased from.29 to 2.44 with increase of D/A from 0.3 to 0.5. In plate (b), SCF fory continuously increased from 1.74 to 2.21, when D/A increased

rom 0.1 to 0.5. In plate (c), SCF for �y fluctuated between 2.26 and.40, when D/A increased from 0.1 to 0.5. Maximum SCF for �y isbtained as 2.44 in plate (a) at D/A = 0.5. Variation of SCF for �xy withespect to D/A is maximum in plate (a) and significant in plates (b)nd (c). Maximum stress concentration for �xy occurred in plate (c)

or almost all D/A ratios. In plate (a), SCF for �xy decreased from 4.65o 2.89 with increase of D/A from 0.1 to 0.2 and increased from 2.89o 3.60 with increase of D/A from 0.2 to 0.5. In plate (b), SCF for �xy

ecreased from 3.72 to 2.88 with increase of D/A from 0.1 to 0.2 andncreased from 2.88 to 3.40 with increase of D/A from 0.2 to 0.5. In

120 N.K. Jain, N.D. Mittal / Materials Science and Engineering A 498 (2008) 115–124

∗z in or

pDop

Fig. 7. Effect of D/A ratio on SCF (for �x , �y , �xy and �eqv) and Uz/U

late (c), SCF for �xy decreased from 5.13 to 3.00 with increase of/A from 0.1 to 0.2 and increased from 3.00 to 3.10 with increasef D/A from 0.2 to 0.5. Maximum SCF for �xy is obtained as 5.13 inlate (c) at D/A = 0.1. Variation of SCF for �eqv with respect to D/A

iaot

Fig. 8. Effect of D/A ratio on SCF (for �x , �y , �xy and �eqv) and Uz/U∗z in symme

thotropic plates (a), (b) and (c) under loading at boundary of hole.

s maximum in case of plate (b) and almost negligible in plates (a)nd (c). Maximum stress concentration for �eqv occurred in casef plate (a) for almost all D/A ratios. In plate (a), SCF for �eqv con-inuously increased from 1.71 to 1.85, when D/A increased from 0.1

tric cross ply laminates (a), (b) and (c) under uniformly distributed load.

ce an

tt�0Dm(I1U00iia

a(vrn�fff0�fDimDtc

0tor(oipD2��atSi0�f(miDftf10i

N.K. Jain, N.D. Mittal / Materials Scien

o 0.5. In plate (b), SCF for �eqv continuously increased from 1.28o 1.88, when D/A increased from 0.1 to 0.5. In plate (c), SCF foreqv fluctuated between 1.47 and 1.59, when D/A increased from.1 to 0.5. Maximum SCF for �eqv is obtained as 1.88 in plate (b) at/A = 0.5. Variation of Uz/U∗

z with respect to D/A ratio observed isaximum in plate (a), significant in plate (c) and minimum in plate

b). Maximum ratio of Uz/U∗z occurred in plate (a) for all D/A ratios.

n plate (a), the ratio of Uz/U∗z continuously increased from 1.23 to

.53 with increase of D/A from 0.1 to 0.5. In plate (b), the ratio ofz/U∗

z decreased from 0.74 to 0.73 with increase of D/A from 0.1 to.2 and increased from 0.73 to 0.87 with increase of D/A from 0.2 to.5. In plate (c), the ratio of Uz/U∗

z increased from 1.29 to 1.37 withncrease of D/A from 0.1 to 0.3 and decreased from 1.37 to 1.33 withncrease of D/A from 0.3 to 0.5. Maximum ratio of Uz/U∗

z is obtaineds 1.53 in plate (a) at D/A = 0.5.

Fig. 8 shows the effect of D/A on SCF (for �x, �y, �xy and �eqv)nd Uz/U∗

z in symmetric cross ply laminated composite plates (a),b) and (c) under uniformly distributed loading. Following obser-ations can be made from these results. Variation of SCF for �x withespect to D/A ratio observed is maximum in plate (a) and almostegligible in plates (b) and (c). Maximum stress concentration forx always occurred in plate (a) for all D/A ratios. In plate (a), SCF

or �x suddenly decreased from 1.55 to 1.08 with increase of D/Arom 0.1 to 0.2 and increased from 1.08 to 1.22 with increase of D/Arom 0.2 to 0.5. In plate (b), SCF for �x continuously increased from.84 to 0.92, when D/A increased from 0.1 to 0.5. In plate (c), SCF forx continuously increased from 0.84 to 0.94, when D/A increased

rom 0.1 to 0.5. Maximum SCF for �x is obtained as 1.55 in plate (a) at/A = 0.1. Variation of SCF for �y with respect to D/A ratio observed

s maximum in plates (a) and (c) and significant in plate (b). Maxi-

um stress concentration for �y occurred in plate (a) for almost all/A ratios. In plate (a), SCF for �y continuously decreased from 2.43

o 1.44, when D/A increased from 0.1 to 0.5. In plate (b), SCF for �y

ontinuously decreased from 1.81 to 1.29, when D/A increased from

a(tr

Fig. 9. Effect of D/A ratio on SCF (for �x , �y , �xy and �eqy) and Uz/U∗z in symmet

d Engineering A 498 (2008) 115–124 121

.1 to 0.5. In plate (c), SCF for �y continuously decreased from 2.30o 1.18, when D/A increased from 0.1 to 0.5. Maximum SCF for �y isbtained as 2.43 in plate (a) at D/A = 0.1. Variation of SCF for �xy withespect to D/A observed is maximum in plate (a), significant in platec) and minimum in plate (b). Maximum stress concentration for �xy

ccurred in plate (b) for all D/A. In plate (a), SCF for �xy continuouslyncreased from 2.20 to 2.57, when D/A increased from 0.1 to 0.5. Inlate (b), SCF for �xy continuously decreased from 4.18 to 3.88, when/A increased from 0.1 to 0.5. In plate (c), SCF for �xy decreased from.26 to 1.94, when D/A increased from 0.1 to 0.5. Maximum SCF forxy is obtained as 4.18 in plate (b) at D/A = 0.1. Variation of SCF foreqv with respect to D/A observed is maximum in case of plate (a)nd almost negligible in plates (b) and (c). Maximum stress concen-ration for �eqv occurred in plate (a) for all D/A ratios. In plate (a),CF for �eqv continuously increased from 2.35 to 2.76, when D/Ancreased from 0.1 to 0.5. In plate (b), SCF for �eqv changed from.96 to 0.92, when D/A increased from 0.1 to 0.5. In plate (c), SCF foreqv continuously increased from 0.84 to 0.94, when D/A increased

rom 0.1 to 0.5. Maximum SCF for �eqv is obtained as 2.76 in platea) at D/A = 0.5. Variation of Uz/U∗

z with respect to D/A observed isaximum in case of plate (a), significant in plate (c) and minimum

n plate (b). Maximum ratio of Uz/U∗z occurred in plate (a) for all

/A ratios. In plate (a), the ratio of Uz/U∗z continuously increased

rom 0.97 to 1.15 with increase of D/A from 0.1 to 0.5. In plate (b),he ratio of Uz/U∗

z increased from 0.68 to 0.75 with increase of D/Arom 0.1 to 0.5. In plate (c), the ratio of Uz/U∗

z increased from 0.99 to.06 with increase of D/A from 0.1 to 0.3 and decreased from 1.06 to.97 with increase of D/A from 0.3 to 0.5. Maximum ratio of Uz/U∗

zs obtained as 1.15 in plate (a) at D/A = 0.5.

Fig. 9 shows the effect of D/A ratio on SCF (for �x, �y, �xy and �eqv)

nd Uz/U∗

z in symmetric cross ply laminated composite plates (a),b) and (c) under loading at boundary of hole. Following observa-ions can be made from these results. Variation of SCF for �x withespect to D/A is almost negligible in all plates. Maximum stress

ric cross ply laminates (a), (b) and (c) under loading at boundary of hole.

1 ce an

cIi0f0DisIi1f0Di(ffSDoffffsIDfSf(i

(Dftr1r

�pifcIrfS0DiMDtc0tiwcS

22 N.K. Jain, N.D. Mittal / Materials Scien

oncentration for �x always occurred in plate (a) for all D/A ratios.n plate (a), SCF for �x fluctuated between 0.90 and 1.02, when D/Ancreased from 0.1 to 0.5. In plate (b), SCF for �x fluctuated between.76 and 0.85, when D/A increased from 0.1 to 0.5. In plate (c), SCFor �x fluctuated between 0.83 and 0.99, when D/A increased from.1 to 0.5. Maximum SCF for �x is obtained as 1.02 in plate (a) at/A = 0.5. Variation of SCF for �y with respect to D/A is significant

n case of plate (b) and negligible in plates (a) and (c). Maximumtress concentration for �y occurred in plate (c) for all D/A ratios.n plate (a), SCF for �y fluctuated between 2.02 and 2.42, when D/Ancreased from 0.1 to 0.5. In plate (b), SCF for �y fluctuated between.86 and 2.59, when D/A increased from 0.1 to 0.5. In plate (c), SCFor �y fluctuated between 2.07 and 2.43, when D/A increased from.1 to 0.5. Maximum SCF for �y is obtained as 2.59 in plate (b) at/A = 0.5. Variation of SCF for �xy with respect to D/A is significant

n all plates. Maximum stress concentration for �xy occurred plateb) for all D/A ratios. In plate (a), SCF for �xy suddenly decreasedrom 4.13 to 2.57 with increase of D/A from 0.1 to 0.2 and increasedrom 2.57 to 3.20 with increase of D/A from 0.2 to 0.5. In plate (b),CF for �xy suddenly decreased from 4.21 to 3.35 with increase of/A from 0.1 to 0.2 and increased from 3.35 to 4.43 with increasef D/A from 0.2 to 0.5. In plate (c), SCF for �xy suddenly decreasedrom 4.35 to 2.58 with increase of D/A from 0.1 to 0.2 and increasedrom 2.58 to 2.74 with increase of D/A from 0.2 to 0.5. Maximum SCFor �xy is obtained as 4.43 in plate (b) at D/A = 0.5. Variation of SCFor �eqv with respect to D/A is significant in all plates. Maximumtress concentration for �eqv occurred in plate (a) for all D/A ratios.n plate (a), SCF for �eqv fluctuated between 1.37 and 1.83, when/A increased from 0.1 to 0.5. In plate (b), SCF for �eqv increased

rom 1.27 to 2.00, when D/A increased from 0.1 to 0.5. In plate (c),

CF for �eqv fluctuated between 1.26 and 1.59, when D/A increasedrom 0.1 to 0.5. Maximum SCF for �eqv is obtained as 2.00 in plateb) at D/A = 0.5. Variation of Uz/U∗

z with respect to D/A is maximumn case of plate (a), significant in plate (c) and minimum in plate

0ftt

Fig. 10. Effect of D/A ratio on SCF (for �x , �y , �xy and �eqv) and Uz/U∗z in anti-symm

d Engineering A 498 (2008) 115–124

b). Maximum ratio of Uz/U∗z occurred in case of plate (a) for all

/A ratios. In plate (a), the ratio of Uz/U∗z continuously increased

rom 1.09 to 1.40 with increase of D/A from 0.1 to 0.5. In plate (b),he ratio of Uz/U∗

z increased from 0.83 to 0.99 with increase of D/Aatio from 0.1 to 0.5. In plate (c), the ratio of Uz/U∗

z increased from.12 to 1.27 with increase of D/A ratio from 0.1 to 0.5. Maximumatio of Uz/U∗

z is obtained as 1.40 in plate (a) at D/A = 0.5.Fig. 10 shows the effect of D/A ratio on SCF (for �x, �y, �xy and

eqv) and Uz/U∗z in anti-symmetric cross ply laminated composite

lates (a), (b) and (c) under uniformly distributed loading. Follow-ng observations can be made from these results. Variation of SCFor �x with respect to D/A is negligible in all plates. Maximum stressoncentration for �x always occurred in plate (b) for all D/A ratios.n plate (a), SCF for �x fluctuated between 1.17 and 1.18, when D/Aatio increased from 0.1 to 0.5. In plate (b), SCF for �x increasedrom 1.32 to 1.38, when D/A increased from 0.1 to 0.5. In plate (c),CF for �x increased from 0.96 to 1.04, when D/A increased from.1 to 0.5. Maximum SCF for �x is obtained as 1.18 in plate (a) at/A = 0.1 and 0.5. Variation of SCF for �y with respect to D/A is max-

mum in plate (a), significant in plate (c) and negligible in plate (b).aximum stress concentration for �y occurred in plate (a) for all/A ratios. In plate (a), SCF for �y continuously decreased from 2.82

o 1.18, when D/A increased from 0.1 to 0.5. In plate (b), SCF for �y

ontinuously increased from 0.48 to 0.58, when D/A increased from.1 to 0.5. In plate (c), SCF for �y continuously decreased from 2.59o 1.12, when D/A increased from 0.1 to 0.5. Maximum SCF for �y

s obtained as 2.82 in plate (a) at D/A = 0.1. Variation of SCF for �xy

ith respect to D/A is significant in all plates. Maximum stress con-entration for �xy occurred in plate (b) for all D/A ratios. In plate (a),CF for � increased from 2.05 to 2.26 with increase of D/A from

xy

.1 to 0.2 and decreased from 2.26 to 2.02 with increase of D/A ratiorom 0.2 to 0.5. In case of plate (b), SCF for �xy increased from 4.64o 4.96 with increase of D/A from 0.1 to 0.3 and decreased from 4.96o 4.49 with increase of D/A ratio from 0.3 to 0.5. In plate (c), SCF

etric cross ply laminates (a), (b) and (c) under uniformly distributed load.

N.K. Jain, N.D. Mittal / Materials Science and Engineering A 498 (2008) 115–124 123

symm

ft0Disr1cf1�wa(iIwidM

�po�nofl0D55�

�cpDipD3tMD2tst02tiwc(Do02orn

Fig. 11. Effect of D/A ratio on SCF (for �x , �y , �xy and �eqv) and Uz/U∗z in anti-

or �xy increased from 3.20 to 3.26 with increase of D/A from 0.1o 0.2 and decreased from 3.26 to 2.80 with increase of D/A from.2 to 0.5. Maximum SCF for �xy is obtained as 4.96 in plate (b) at/A = 0.3. Variation of SCF for �eqv with respect to D/A is significant

n plate (a) and almost negligible in plates (b) and (c). Maximumtress concentration for �eqv occurred in plate (a) for almost all D/Aation. In plate (a), SCF for �eqv continuously increased from 0.91 to.22, when D/A increased from 0.1 to 0.5. In plate (b), SCF for �eqv

ontinuously decreased from 0.93 to 0.79, when D/A ratio increasedrom 0.1 to 0.5. In plate (c), SCF for �eqv continuously increased from.01 to 1.07, when D/A increased from 0.1 to 0.5. Maximum SCF foreqv is obtained as 1.22 in plate (a) at D/A = 0.5. Variation of Uz/U∗

zith respect to D/A is maximum in plate (a), significant in plate (c)

nd minimum in plate (b). Maximum ratio of Uz/U∗z occurred plate

a) for all D/A ratios. In plate (a), the ratio of Uz/U∗z continuously

ncreased from 0.92 to 1.04 with increase of D/A from 0.1 to 0.5.n plate (b), the ratio of Uz/U∗

z fluctuated between 1.03 and 1.05ith increase of D/A from 0.1 to 0.5. In plate (c), the ratio of Uz/U∗

zncreased from 0.87 to 0.94 with increase of D/A from 0.1 to 0.4 andecreased from 0.94 to 0.93 with increase of D/A from 0.4 to 0.5.aximum ratio of Uz/U∗

z is obtained as 1.04 in plate (a) at D/A = 0.5.Fig. 11 shows the effect of D/A ratio on SCF (for �x, �y, �xy and

eqv) and Uz/U∗z in anti-symmetric cross ply laminated composite

lates (a), (b) and (c) under loading at boundary of hole. Followingbservations can be made from these results. Variation of SCF forx with respect to D/A is maximum in plates (a) and (b) and sig-ificant in plate (c). Maximum stress concentration for �x alwaysccurred in plate (a) for almost all D/A ratios. In plate (a), SCF for �x

uctuated between 5.47 and 7.55, when D/A increased from 0.1 to

.5. In plate (b), SCF for �x fluctuated between 5.29 and 7.54, when/A increased from 0.1 to 0.5. In plate (c), SCF for �x increased from.37 to 5.51 with increase of D/A from 0.1 to 0.2 and decreased from.51 to 4.66 with increase of D/A from 0.2 to 0.5. Maximum SCF forx is obtained as 7.55 in plate (a) at D/A = 0.5. Variation of SCF for

putwo

etric cross ply laminates (a), (b) and (c) under loading a t boundary of hole.

y with respect to D/A is significant in all plates. Maximum stressoncentration for �y occurred plate (b) for almost all D/A ratios. Inlate (a), SCF for �y continuously increased from 1.76 to 2.46, when/A increased from 0.1 to 0.5. In plate (b), SCF for �y continuously

ncreased from 2.11 to 3.09, when D/A increased from 0.1 to 0.5. Inlate (c), SCF for �y continuously increased from 1.78 to 2.51, when/A increased from 0.1 to 0.5. Maximum SCF for �y is obtained as.09 in plate (b) at D/A = 0.5. Variation of SCF for �xy with respecto D/A is maximum in plate (b) and significant in plates (a) and (c).

aximum stress concentration for �xy occurred in plate (b) for all/A ratios. In plate (a), SCF for �xy suddenly decreased from 3.98 to.50 with increase of D/A from 0.1 to 0.3 and increased from 2.50o 3.17 with increase of D/A from 0.3 to 0.5. In plate (b), SCF for �xy

uddenly decreased from 4.11 to 3.71 with increase of D/A from 0.1o 0.2 and increased from 3.71 to 6.28 with increase of D/A from.2 to 0.5. In plate (c), SCF for �xy suddenly decreased from 3.53 to.42 with increase of D/A from 0.1 to 0.3 and increased from 2.42o 3.00 with increase of D/A from 0.3 to 0.5. Maximum SCF for �xy

s obtained as 6.28 in plate (b) at D/A = 0.5. Variation of SCF for �eqv

ith respect to D/A is significant in all plates. Maximum stress con-entration for �eqv occurred in plate (b) for all D/A ratios. In platea), SCF for �eqv continuously increased from 1.96 to 2.70, when/A increased from 0.1 to 0.5. In plate (b), SCF for �eqv continu-usly increased from 2.37 to 2.98, when D/A increased from 0.1 to.5. In plate (c), SCF for �eqv continuously increased from 2.00 to.77, when D/A increased from 0.1 to 0.5. Maximum SCF for �eqv isbtained as 2.98 in plate (b) at D/A = 0.5. Variation of Uz/U∗

z withespect to D/A is maximum in case of plates (a) and (b) and sig-ificant in plate (c). Maximum ratio of Uz/U∗

z occurred in case of

late (b) for all D/A ratios. In plate (a), the ratio of Uz/U∗

z contin-ously increased from 0.92 to 1.26 with increase of D/A from 0.1o 0.5. In plate (b), the ratio of Uz/U∗

z increased from 0.98 to 1.37ith increase of D/A from 0.1 to 0.5. In case of plate (c), the ratio

f Uz/U∗z increased from 0.90 to 1.19 with increase of D/A from 0.1

1 ce an

tD

5

ttF�aooaertocd�lmMotU(oDi(ti

tppiwdobwo

R

[

[[[

[[[[

24 N.K. Jain, N.D. Mittal / Materials Scien

o 0.5. Maximum ratio of Uz/U∗z is obtained as 1.37 in plate (b) at

/A = 0.5.

. Conclusions

In general, for plates (a) and (c), the maximum stress concentra-ion always occurred on the hole boundary and in case of plates (b),he maximum stress concentration always occurred on supports.or uniformly distributed loading condition the SCF for �x, �y andeqv plays a significant role and SCF for �xy plays an important role inll cases of plates and boundary conditions. For loading at boundaryf hole, the SCF for all stresses plays an important role in all casesf plates and boundary conditions. The stress concentration forll stresses occurred on maximum, significant and minimum lev-ls laminate, orthotropic plate and isotropic plate respectively forespective loading and boundary condition. The stress concentra-ion for all stresses occurred more in plate with loading at boundaryf hole than uniformly distributed loading for respective boundaryondition and type of plate. It has been seen that SCF for all stressesecreases with increase of D/A for almost all cases. The SCF for �x,y and �eqv varied most strongly with D/A in plates (a) and (c) and

east in plate (b) for almost all cases, while the SCF for �xy variedost with D/A in plate (b) and significantly in plates (a) and (c).aximum and minimum deflection in transverse direction always

ccurred for plates (a) and (b) for all cases. It has been observedhat for uniformly distributed loading conditions, the variation ofz/U∗

z with D/A is most significant in plate (a), significant in platec) and negligible in plate (b) for all cases. For loading at boundaryf hole, there is always some significant variation of Uz/U∗

z with

/A in plate (b). Even in the anti-symmetric laminate under load-

ng at boundary of hole, the variation of Uz/U∗z with D/A in plate

b) is more extensive than in plate (a). It is also observed that therend of variation of Uz/U∗

z with D/A becomes almost the same forsotropic, orthotropic and laminated composite plates for respec-

[[[

[

d Engineering A 498 (2008) 115–124

ive boundary and loading conditions. In isotropic and orthotropiclates, Uz/U∗

z , continuously increased with increasing of D/A inlate (a), first increased and after some increase, decreased with

ncrease of D/A in plate (c) and fluctuated becomes a small rangeith respect to D/A in plate (b) for all types of loading. For uniformlyistributed loading condition, laminates also follow the same trendf variation of Uz/U∗

z with D/A as in isotropic and orthotropic plates,ut for loading at boundary of hole, Uz/U∗

z continuously increasedith increasing of D/A for all plates (a), (b) and (c). Maximum ratio

f Uz/U∗z is obtained in plate (a) at D/A = 0.5 for almost all cases.

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