Research Article Investigation of Through-Thickness...

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Hindawi Publishing Corporation International Journal of Engineering Mathematics Volume 2013, Article ID 676743, 11 pages http://dx.doi.org/10.1155/2013/676743 Research Article Investigation of Through-Thickness Stresses in Composite Laminates Using Layerwise Theory Hamidreza Yazdani Sarvestani and Ali Naghashpour Concordia Centre for Composites (CONCOM), Department of Mechanical and Industrial Engineering, Concordia University, 1455 De Maisonneuve Boulevard West, Montreal, QC, Canada H3G1M8 Correspondence should be addressed to Hamidreza Yazdani Sarvestani; h [email protected] Received 13 August 2013; Revised 10 October 2013; Accepted 11 October 2013 Academic Editor: George S. Dulikravich Copyright © 2013 H. Yazdani Sarvestani and A. Naghashpour. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this study, an analytical method is developed to exactly obtain the interlaminar stresses near the free edges of laminated composite plates under the bending moment based on the reduced form of elasticity displacement field for a long laminate. e analytical and numerical studies were performed based on the Reddy’s layerwise theory for the boundary layer stresses within cross-ply, symmetric, angle-ply, and general composite laminates. Finally, a variety of numerical results are presented for the interlaminar normal and shear stresses along the interfaces and through thickness of laminates near the free edges. e results showed high stress gradient of interlaminar normal and shear stresses near the edges of laminates. 1. Introduction Due to high specific strength and stiffness of fiber reinforced polymer composite, laminated composites have found aug- mented use in many industrial applications. In the boundary- layer regions, due to geometry and material discontinuities, the interlaminar stresses exhibit much higher values than those predicted by the classical lamination theory (CLT). ese highly concentrated stresses cause delaminate fail- ure in the laminates. However, no exact solution is found for elasticity equations because of inherent complexities involved in the problem of finding exact stress values in the edges. Hence, different analytical and numerical methods for finding the interlaminar stresses are used to describe the interlaminar stresses at the free edges of composite laminates. Complete literature surveys on this subject are available in review articles of Kant and Swaminathan [1] which obviously show the detailed path of development of methods. e first approximate analysis of interlaminar stresses was presented by Puppo and Evensen [2]. ey studied interlaminar shear stresses in an idealized lami- nate consisting of orthotropic layers separated by isotropic shear layers with interlaminar normal stress being neglected through the laminate. Other approximate analytical methods utilized to examine the problem consist of the use of the higher-order plate theory by Pagano [3], the perturbation technique by Hsu and Herakovich [4], the boundary layer theory by Tang and Levy [5], and the approximate elasticity solutions by Pipes and Pagano [6]. An approximate theory is also employed by Pagano [7] based on the assumed inplane stresses and the use of Reissner’s variational principle. Wang and Choi [8] utilized Lekhnistskii’s stress potential and the theory of anisotropic elasticity for examining the free edge singularities. A variational approach concerning Lekhnitskii’s stress functions is used by Yin [9] for the evaluation of free-edge stresses in laminates under uniaxial tension, bending, and torsion. Wang and Crossman [10] developed a quasi-three-dimensional finite element solution to determine the free-edge stresses in a symmetric balanced composite laminate under uniaxial tension and uniform thermal loading. Whitcomb et al. [11] studied the differences in numerical results for interlaminar stresses obtained by various methods (finite difference methods, finite element methods, and perturbation techniques). Boundary element method and the integral equation theory were used by Dav` ı[12] to study the stresses in a general laminate under uniform axial strain. Carrera and Demasi [13, 14] studied the accuracy of the finite-element mixed layerwise solutions

Transcript of Research Article Investigation of Through-Thickness...

Page 1: Research Article Investigation of Through-Thickness ...downloads.hindawi.com/archive/2013/676743.pdf · (,, ) is located at the middle plane of the laminate, that is of thickness

Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2013 Article ID 676743 11 pageshttpdxdoiorg1011552013676743

Research ArticleInvestigation of Through-Thickness Stresses inComposite Laminates Using Layerwise Theory

Hamidreza Yazdani Sarvestani and Ali Naghashpour

Concordia Centre for Composites (CONCOM) Department of Mechanical and Industrial EngineeringConcordia University 1455 De Maisonneuve Boulevard West Montreal QC Canada H3G1M8

Correspondence should be addressed to Hamidreza Yazdani Sarvestani h yazdencsconcordiaca

Received 13 August 2013 Revised 10 October 2013 Accepted 11 October 2013

Academic Editor George S Dulikravich

Copyright copy 2013 H Yazdani Sarvestani and A Naghashpour This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

In this study an analyticalmethod is developed to exactly obtain the interlaminar stresses near the free edges of laminated compositeplates under the bending moment based on the reduced form of elasticity displacement field for a long laminate The analyticaland numerical studies were performed based on the Reddyrsquos layerwise theory for the boundary layer stresses within cross-plysymmetric angle-ply and general composite laminates Finally a variety of numerical results are presented for the interlaminarnormal and shear stresses along the interfaces and through thickness of laminates near the free edges The results showed highstress gradient of interlaminar normal and shear stresses near the edges of laminates

1 Introduction

Due to high specific strength and stiffness of fiber reinforcedpolymer composite laminated composites have found aug-mented use inmany industrial applications In the boundary-layer regions due to geometry and material discontinuitiesthe interlaminar stresses exhibit much higher values thanthose predicted by the classical lamination theory (CLT)These highly concentrated stresses cause delaminate fail-ure in the laminates However no exact solution is foundfor elasticity equations because of inherent complexitiesinvolved in the problem of finding exact stress values in theedges Hence different analytical and numerical methodsfor finding the interlaminar stresses are used to describethe interlaminar stresses at the free edges of compositelaminates Complete literature surveys on this subject areavailable in review articles of Kant and Swaminathan [1]which obviously show the detailed path of developmentof methods The first approximate analysis of interlaminarstresses was presented by Puppo and Evensen [2] Theystudied interlaminar shear stresses in an idealized lami-nate consisting of orthotropic layers separated by isotropicshear layers with interlaminar normal stress being neglectedthrough the laminate Other approximate analytical methods

utilized to examine the problem consist of the use of thehigher-order plate theory by Pagano [3] the perturbationtechnique by Hsu and Herakovich [4] the boundary layertheory by Tang and Levy [5] and the approximate elasticitysolutions by Pipes and Pagano [6] An approximate theoryis also employed by Pagano [7] based on the assumedinplane stresses and the use of Reissnerrsquos variational principleWang and Choi [8] utilized Lekhnistskiirsquos stress potentialand the theory of anisotropic elasticity for examining thefree edge singularities A variational approach concerningLekhnitskiirsquos stress functions is used by Yin [9] for theevaluation of free-edge stresses in laminates under uniaxialtension bending and torsion Wang and Crossman [10]developed a quasi-three-dimensional finite element solutionto determine the free-edge stresses in a symmetric balancedcomposite laminate under uniaxial tension and uniformthermal loading Whitcomb et al [11] studied the differencesin numerical results for interlaminar stresses obtained byvarious methods (finite difference methods finite elementmethods and perturbation techniques) Boundary elementmethod and the integral equation theory were used byDavı [12] to study the stresses in a general laminate underuniform axial strain Carrera and Demasi [13 14] studiedthe accuracy of the finite-element mixed layerwise solutions

2 International Journal of Engineering Mathematics

y

z

x

a

a

2b

MO

MOh

Figure 1 Laminate geometry and coordinate system

by using the Reissner mixed variational theorem (RMVT)They compared the numerical results for interlaminar stressesin several finite-element models and elasticity theory withincomposite laminates and sandwich plates Nguyen andCaron[15] employed a multiparticle finite element method to studythe interlaminar stresses near the free edges of generalcomposite laminates under mechanical and thermal loadingRobbins and Reddy [16] used a displacement-based variablekinematic global-local finite elementmethodMittelstedt andBecker [17] utilizedReddyrsquos layerwise laminate plate theory tofind the closed-form analysis of free-edge effects in layeredplates of arbitrary nonorthotropic layups The approachconsists of the subdivision of the physical laminate layers intoan arbitrary number of mathematical layers through the platethickness Na [18] used a finite element model based on thelayerwise theory He employed the von Karman type nonlin-ear strains to analyze damage in laminated composite beamsIn his formulation the Heaviside step function is employedto express the discontinuous interlaminar displacement fieldat the delaminated interfaces Plagianakos and Saravanos [19]presented a higher-order layerwise theoretical frameworkwhich enables prediction of the static response of thick com-posite and sandwich composite plates Ullah et al [20] carriedout some experimental tests to characterize the behavior of awoven CFRP material under large-deflection bending Two-dimensional finite element (FE) models were implementedin the commercial code Abaqus They performed series ofsimulations to study the deformation behavior and damage inCFRP for cases of high-deflection bendingHelenon et al [21]presented an experimental and numerical investigation intofailure of T-shaped laminated composite structures Threeout-of-plane bending cases are studied They found thatvery high free-edge maximum principal transverse tensilestresses perpendicular to the fiber direction occur at thefailure locations Thai et al [22] indicated an isogeometricfinite element formulation for static free vibration and buck-ling analysis of laminated composite and sandwich platesTheir method allows removing shear correction factors andimproves the accuracy of transverse shear stresses Thai et al[23] investigated the behavior of laminated composites usingseveral high order or layerwise finite element calculationsA layerwise model and its dedicated 119862

∘ eight-node finite

element were specifically developed for interlaminar stressesanalysis in free edge problem Malekzadeh [24] developed ahigh accuracy and rapid convergence hybrid approach fortwo-dimensional static analyses of circular arches with dif-ferent boundary conditions The method essentially consistsof a layerwise theory used for the thickness direction inconjunction with differential quadrature method in the axialdirection

There have been very limited works to study the interlam-inar stresses subjected to the bendingmoment In the presentpaper by the use of Reddyrsquos LWT an analytical solutionis presented to evaluate interlaminar stresses in cross-plysymmetric angle-ply and general composite laminates underthe bending moment To commence with based on physicalarguments regarding the deformations of a long generally andother laminated composite plates an appropriate reducedelasticity displacement field is established The boundary-layer stresses within the laminate are obtained analyticallybased on Reddyrsquos LWT

2 Problem Formulation

21 Elasticity Displacement Field An Nth-layered compositeplate (with arbitrary lamination) under the bending momentis considered as shown in Figure 1 The coordinate system(119909 119910 119911) is located at the middle plane of the laminate thatis of thickness ℎ width 2119887 and length 2119886 and is assumed tobe long in the 119909 direction so that the strains away from theends (119909 = plusmn119886) of the laminate are functions of only 119910 and 119911

The integrations of the three-dimensional elasticitystrain-displacement relations [25] inside the kth layer of thelaminate will generate the most general form of displacementfield which can be shown to be [26]

119906(119896)

1

(119909 119910 119911) = 119861(119896)

4

119909119910 + 119861(119896)

6

119909119911 + 119861(119896)

2

119909 + 119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = minus119861(119896)

1

119909119911 + 119861(119896)

3

119909 minus1

2119861(119896)

4

1199092

+ V(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 119861(119896)

1

119909119910 + 119861(119896)

5

119909 minus1

2119861(119896)

6

1199092

+ 119908(119896)

(119910 119911)

(1)

where 119906(119896)

1

119906(119896)2

and 119906(119896)

3

represent the displacement com-ponents of the material point (119909 119910 119911) in the 119909 119910 and 119911

International Journal of Engineering Mathematics 3

directions respectively To satisfy the displacement continu-ity conditions at the interfaces of the adjoining layers theintegration constants in (1) must be the same for all layerswithin the laminate Therefore the most general displace-ment field in the kth layer is shown

119906(119896)

1

(119909 119910 119911) = 1198614119909119910 + 119861

6119909119911 + 119861

2119909 + 119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = minus1198611119909119911 + 119861

3119909 minus

1

211986141199092

+ V(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 1198611119909119910 + 119861

5119909 minus

1

211986161199092

+ 119908(119896)

(119910 119911)

(2)

If loading conditions at 119909 = minus119886 and 119886 are similar based onthe physical grounds the following conditions hold

119906(119896)

1

(119909 119910 119911) = minus119906(119896)

1

(minus119909 minus119910 119911)

119906(119896)

2

(119909 119910 119911) = minus119906(119896)

2

(minus119909 minus119910 119911)

119906(119896)

3

(119909 119910 119911) = 119906(119896)

3

(minus119909 minus119910 119911)

(3)

Upon imposing these restrictions on (1) it is readily seen thatthe constants 119861

4and 119861

5must vanish and the displacement

field in (1) is reduced to the following equations

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 1198616119909119911 + 119906

(119896)

(119910 119911) (4a)

119906(119896)

2

(119909 119910 119911) = minus1198611119909119911 + 119861

3119909 + V(119896)

(119910 119911) (4b)

119906(119896)

3

(119909 119910 119911) = 1198611119909119910 minus

1

211986161199092

+ 119908(119896)

(119910 119911) (4c)

Also by replacing 119906(119896)

(119910 119911) by minus1198613119910 + 119906

(119896)

(119910 119911) in (4a) itbecomes apparent that terms involving 119861

3in (4a)ndash(4c) can

be neglected since no strains are generated by such terms(these terms will correspond to an infinitesimal rigid-bodyrotation of the laminate about the z-axis in Figure 1) Thusthemost general formof the displacement field of an arbitrarylaminate is given by

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 1198616119909119911 + 119906

(119896)

(119910 119911) (5a)

119906(119896)

2

(119909 119910 119911) = minus1198611119909119911 + V

(119896)

(119910 119911) (5b)

119906(119896)

3

(119909 119910 119911) = 1198611119909119910 minus

1

211986161199092

+ 119908(119896)

(119910 119911) (5c)

The displacement field in (5a) (5b) and (5c) may be used inprinciple for computing the stress field in any composite lam-inate subjected to arbitrary combinations of self-equilibratingmechanical and uniform hygrothermal loads The termsinvolving 119861

1 1198612 and 119861

6in (5a) (5b) and (5c) demonstrate

certain global deformations that occurred in the laminate Onthe other hand the unknown functions appearing in (5a)(5b) and (5c) illustrate the local deformations They will beevaluated by LWT of Reddy that happened in a laminate

For cross-ply laminates subjected tomechanical loadingsbased on physical grounds the following restrictions hold(see Figure 1)

119906(119896)

1

(119909 119910 119911) = minus119906(119896)

1

(minus119909 119910 119911) (6a)

119906(119896)

2

(119909 119910 119911) = 119906(119896)

2

(minus119909 119910 119911) (6b)

Upon imposing these conditions on (5a) it is readily seen that119906(119896)

(119910 119911) = 0 Also it is concluded from (6b) and (5b) that1198611= 0 Thus for cross-ply laminates the most general form

of the displacement field is given

119906(119896)

1

(119909 119910 119911) = 1198616119909119911 + 119861

2119909

119906(119896)

2

(119909 119910 119911) = V(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = minus1

211986161199092

+ 119908(119896)

(119910 119911)

(7)

Also for symmetric laminates based on physical groundsthe following conditions must hold (see Figure 1)

119906(119896)

1

(119909 119910 119911) = 119906(119873+1minus119896)

1

(119909 119910 minus119911) (8a)

119906(119896)

2

(119909 119910 119911) = 119906(119873+1minus119896)

2

(119909 119910 minus119911) (8b)

119906(119896)

3

(119909 119910 119911) = minus 119906(119873+1minus119896)

3

(119909 119910 minus119911)

(119896 = 1 2 3 119873

2)

(8c)

Upon imposing these conditions on (5a) (5b) and (5c) it isfound that 119861

1= 1198616= 0 Thus for symmetric laminates the

most general form of displacement field is expressed by

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = V(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 119908(119896)

(119910 119911)

(9)

Finally for antisymmetric angle-ply laminateswith119873 lay-ers subjected tomechanical loadings the following conditionmust hold

119906(119896)

1

(119909 119910 119911) = 119906(119873+1minus119896)

1

(119909 minus119910 minus119911) (119896 = 1 2 3 119873

2)

(10)

Concluding from (10) and (5a) 1198616= 0 and therefore the

most general displacement field for such laminates arewrittenas

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = minus1198611119909119911 + V

(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 1198611119909119910 + 119908

(119896)

(119910 119911)

(11)

4 International Journal of Engineering Mathematics

The displacement fields in (7) (9) and (11) can be repre-sented in one place as

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 1205751198621198616119909119911 + 120575

119878119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = minus1205751198601198611119909119911 + V

(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 1205751198601198611119909119910 minus

1

212057511986211986161199092

+ 119908(119896)

(119910 119911)

(12)

where 120575119860= 120575119878= 0 and 120575

119862= 1 for general cross-ply laminates

120575119860= 120575119862= 0 and 120575

119878= 1 for symmetric laminates 120575

119860= 120575119878= 1

and 120575119862= 0 for angle-ply laminates and 120575

119860= 120575119878= 120575119888= 1 for

general laminates

3 Layerwise Laminated Plate Theory of Reddy

In this section LWT is used for calculation of interlaminarstresses in arbitrary laminated composite plates with freeedges In LWT it is possible to replace the actual physicallayers to many mathematical layers [27] This can be doneby subdividing each physical layer through the thicknessto get the proper accuracy The theory assumes that thedisplacement components of a generic point in the laminateare given by [27]

1199061(119909 119910 119911) = 119906

119896(119909 119910) 120601

119896(119911)

1199062(119909 119910 119911) = V

119896(119909 119910) 120601

119896(119911)

1199063(119909 119910 119911) = 119908

119896(119909 119910) 120601

119896(119911)

119896 = 1 2 119873 + 1

(13)

It should be emphasized that a repeated index indicates sum-mation from 1 to119873 + 1 The functions 119906

119896(119909 119910) V

119896(119909 119910) and

119908119896(119909 119910) represent the displacements of the points initially

located at the kth plane of the laminate in the 119909 119910 and 119911

directions respectively The variable 119873 in (13) is the totalnumber of numerical layers introduced in any laminate Alsothe functions 120601

119896rsquos are the global approximation functions of

the thickness coordinate which are assumed here to be linear[27] This function expressed as

120601119896(119911) =

0 119911 le 119911119896minus1

1205952

119896minus1

(119911) 119911119896minus1

le 119911 le 119911119896

1205951

119896

(119911) 119911119896le 119911 le 119911

119896+1

0 119911 ge 119911119896+1

(119896 = 1 2 119873 + 1)

(14a)

where the local Lagrangian interpolation functions120595119895119896

(119911) (119895 =

1 2) related with the kth surface in the laminate are definedas

1205951

119896

(119911) =1

ℎ119896

(119911119896+1

minus 119911) 1205952

119896

(119911) =1

ℎ119896

(119911 minus 119911119896)

(14b)

where ℎ119896is the thickness of the kthmathematical layer Based

on the displacement field in (12) the displacement field ofLWT in (13) takes the following form

1199061(119909 119910 119911) = 120575

1198621198616119909119911 + 119861

2119909 + 120575119878119880119896(119910) 120601119896(119911)

1199062(119909 119910 119911) = minus120575

1198601198611119909119911 + 119881

119896(119910) 120601119896(119911)

1199063(119909 119910 119911) = 120575

1198601198611119909119910 minus

1

212057511986211986161199092

+119882119896(119910) 120601119896(119911)

119896 = 1 2 119873 + 1

(15)

It is noted that by the means of through-the-thicknesslinear interpolation functions the continuity of displacementcomponents through the thickness of laminate is identicallysatisfied On the other hand the transverse strain com-ponents remain discontinuous at the interfaces They willultimately emphasize the feasibility of having continuousinterlaminar stresses at the interfaces of adjoining layersby increasing the number of numerical layers through thephysical laminate Substituting (15) into the linear strain-displacement relations of elasticity [25] yields

120576119909= 1205751198621198616119911 + 1198612 120576

119910= 1198811015840

119896

120601119896 120576

119911= 1198821198961206011015840

119896

120574119910119911

= 1198811198961206011015840

119896

+1198821015840

119896

120601119896

120574119909119911

= 1205751198781198801198961206011015840

119896

+ 1205751198601198611119910 120574

119909119910= 1205751198781198801015840

119896

120601119896minus 1205751198601198611119911

(16)

where a prime in (16) denotes ordinary differentiation withrespect to a suitable independent variable (ie either 119910 or 119911)The three-dimensional constitutive law within the kth layer(with arbitrary fiber orientation) of the laminate may also beshown as [28]

(

(

120590119909

120590119910

120590119911

120590119910119911

120590119909119911

120590119909119910

)

)

(119896)

=

[[[[[[[[[

[

1198621111986212

11986213

0 0 11986216

11986212

11986222

11986223

0 0 11986226

11986213

11986223

11986233

0 0 11986236

0 0 0 11986244

11986245

0

0 0 0 11986245

11986255

0

11986216

11986226

11986236

0 0 11986266

]]]]]]]]]

]

(119896)

times(

(

120576119909

120576119910

120576119911

120574119910119911

120574119909119911

120574119909119910

)

)

(119896)

(17)

where [119862](119896) is the transformed (ie off axis) stiffness matrixof the kth layer By using the strain-displacement relations

International Journal of Engineering Mathematics 5

(16) in the principle of the minimum total potential energy[25] the equilibrium equations within LWT are found as

120575119880119896 120575119878(119876119896

119909

minus

d119872119896119909119910

d119910) = 0 119896 = 1 2 119873 + 1 (18a)

120575119881119896 119876119896119910

minus

d119872119896119910

d119910= 0 119896 = 1 2 119873 + 1 (18b)

120575119882119896 119873119896119911

minus

d119877119896119910

d119910= 0 119896 = 1 2 119873 + 1 (18c)

1205751198611 120575119860int

119887

minus119887

(119876119909119910 minus119872

119909119910) d119910 = 0 (19a)

1205751198612 int119887

minus119887

119873119909d119910 = 0 (19b)

1205751198616 120575119862int

119887

minus119887

119872119909d119910 = 119872

0 (19c)

Also the traction-free boundary conditions at the free edgesof the laminate are given as

119872119896

119910

= 119877119896

119910

= 120575119878119872119896

119909119910

= 0 at 119910 = plusmn119887 (119896 = 1 2 119873 + 1)

(20)

Equations (18a) (18b) and (18c) present 3(119873+1) local equilib-rium equations associated with119873+ 1 surface in the laminatewithin LWT Also (19a) (19b) and (19c) represent the globalequilibrium equation of a laminate The generalized stressandmoment resultants in (18a) (18b) (18c) (19a) (19b) (19c)and (20) are defined as

(119872119896

119910

119872119896

119909119910

119873119896

119911

) = int

ℎ2

minusℎ2

(120590119910120601119896 120590119909119910120601119896 1205901199111206011015840

) d119911 (21a)

(119876119896

119909

119876119896

119910

119877119896

119910

) = int

ℎ2

minusℎ2

(1205901199091199111206011015840

119896

1205901199101199111206011015840

119896

120590119910119911120601119896) d119911 (21b)

(119872119909 119873119909119872119909119910 119876119909) = int

ℎ2

minusℎ2

(120590119909119911 120590119909 120590119909119910119911 120590119909119911) d119911 (21c)

Substitution (16) into (17) and subsequently into (21a) (21b)and (21c) generates the following relations

(119872119896

119910

119872119896

119909119910

119873119896

119911

)

= 120575119878(119863119896119895

26

119863119896119895

66

119861119896119895

36

)1198801015840

119895

+ (119863119896119895

22

119863119896119895

26

119861119895119896

23

)1198811015840

119895

+ (119861119896119895

23

119861119896119895

36

119860119896119895

33

)119882119895minus 120575119860(119863119896

26

119863119896

66

119861119896

36

) 1198611

+ 120575119862(119863119896

12

119863119896

16

119861119896

13

) 1198616+ (119861119896

12

119861119896

16

119860119896

13

) 1198612

(22a)

(119876119896

119909

119876119896

119910

119877119896

119910

)

= 120575119878(119860119896119895

55

119860119896119895

45

119861119896119895

45

)119880119895+ (119860119896119895

45

119860119896119895

44

119861119896119895

44

)119881119895

+ (119861119895119896

45

119861119895119896

44

119863119896119895

44

)1198821015840

119895

minus 120575119860(119860119896

55

119860119896

45

119861119896

45

) 1198611119910

(22b)

(119872119910119872119909119910 119873119909)

= 120575119878(119863119896

16

119863119896

66

119861119896

16

)1198801015840

119895

+ (119863119896

12

119863119896

26

119861119896

12

)1198811015840

119895

+ (119861119896

13

119861119896

36

119860119896

13

)119882119895minus 120575119860(11986316 11986366 11986116) 1198611

+ 120575119862(11986111 11986116 11986011) 1198616+ (11986311 11986316 11986111) 1198612

(22c)

119876119909= 120575119878119860119896

55

119880119895+ 119860119896

45

119881119895+ 119861119896

45

1198821015840

119895

+ 120575119860119860551198611119910 (22d)

The explicit expressions for the laminate rigidities appearingin (22a) (22b) (22c) and (22d) in terms of 119862

119894rsquos are presented

in the Appendix Direct substitution of (22a) (22b) (22c)and (22d) into (18a) (18b) and (18c) provides the localdisplacement equilibrium equations within LWTThe resultsare expressed as

120575119880119896 120575119878(119863119896119895

66

11988010158401015840

119895

minus 119860119896119895

55

119880119895+ 119863119896119895

26

11988110158401015840

119895

minus 119860119896119895

45

119881119895+ (119861119896119895

36

minus 119861119895119896

45

)1198821015840

119895

) = 120575119860119860119896

55

1198611119910

119896 = 1 2 119873 + 1

(23a)

120575119881119896 120575119878119863119896119895

26

11988010158401015840

119895

minus 120575119878119860119896119895

45

119880119895+ 119863119896119895

22

11988110158401015840

119895

minus 119860119896119895

44

119881119895+ (119861119896119895

23

minus 119861119895119896

44

)1198821015840

119895

= 120575119860119860119896

45

1198611119910

119896 = 1 2 119873 + 1

(23b)

120575119882119896 120575119878(119861119896119895

45

minus 119861119895119896

36

)1198801015840

119895

minus (119861119896119895

44

minus 119861119895119896

23

)1198811015840

119895

+ 119863119896119895

44

11988210158401015840

119895

minus 119860119896119895

33

119882119895

= minus120575119860(119861119896

45

+ 119861119896

36

) 1198611+ 119860119896

13

1198612+ 120575119862119861119896

13

1198616

119896 = 1 2 119873 + 1

(23c)

In the same way upon substitution of (16) into (17) andthe subsequent results into (19a) (19b) and (19c) the globalequations (19a) (19b) and (19c) are found in terms ofdisplacement functions as follows

1205751198611 120575119860int

119887

minus119887

((120575119878119860119896

55

119880119895+ 119860119896

45

119881119895+ 119861119896

45

1198821015840

119895

) 119910

minus (120575119878119863119896

66

1198801015840

119895

+ 119863119896

26

1198811015840

119895

+ 119861119896

36

119882119895)

+ 120575119860(1198605511986111199102

+ 11986366) 1198611minus 119861161198612

minus120575119862119863161198616) d119910 = 0

(24a)

6 International Journal of Engineering Mathematics

1205751198612 int119887

minus119887

(120575119878119861119896

16

1198801015840

119895

+ 119861119896

12

1198811015840

119895

+ 119860119896

13

119882119895

minus 120575119860119861161198611+ 119860111198612+ 120575119862119861111198616) d119910 = 0

(24b)

1205751198616 120575119862int

119887

minus119887

(120575119878119863119896

16

1198801015840

119895

+ 119863119896

12

1198811015840

119895

+ 119861119896

13

119882119895

minus 120575119860119863161198611+ 119861111198612+ 120575119862119863111198616) d119910 = 119872

0

(24c)

where the extra laminate rigidities appearing in (24a) (24b)and (24c) are defined in the Appendix The system ofequations in (23a) (23b) and (23c) shows 3(119873 + 1) coupledordinary differential equations with constant coefficientswhich may be indicated in a matrix form as

[119872] 12057810158401015840

+ [119870] 120578 = [119871] 119861 sdot 119910 (25)

where

120578 = 119880119879

119881119879

119882119879

119879

119880 = 1198801 1198802 119880

119873+1119879

119881 = 1198811 1198812 119881

119873+1119879

119882 = 11988211198822 119882

119873+1119879

(26)

119861 = 1198611 1198612 1198616119879

119882119895= int

119910

119882119895d119910

(27)

The coefficient matrices [119872] [119870] and [119871] in (25) are dis-played in the Appendix Also from the conditions in (3) andthe displacement field in (23a) (23b) and (23c) it is observedthat the functions 119880

119895 119881119895 and119882

119895are all odd functions of the

independent variable 119910 It can be confirmed that the generalsolution of (25) may be presented as

120578 = [120595] [sinh (120582119910)] 119867 + [119870]minus1

[119871] 119861 sdot 119910 (28)

where [sinh(120582119910)] is a 3(119873 + 1) times 3(119873 + 1) diagonal matrixand

[sinh (120582119910)]

= diag (sinh (1205821119910) sinh (120582

2119910) sinh (120582

3(119873+1)119910))

(29)

Also [120595] and (1205822

1

1205822

2

1205822

3(119873+1)

) are the model matrixand eigenvalues of (minus[119872]

minus1

[119870]) respectively 119870 is anunknown vector representing 3(119873+1) integration constantsThe constants 119861

119895(119895 = 1 2 6)must be calculated within LWT

analysis So the boundary conditions in (20) are first imposedto found the vector 119870 in terms of the unknown parameters119861119895(119895 = 1 2 6) These constants are next calculated in terms

of the specific bending moment 1198720by satisfaction of the

global equilibrium conditions in (19a) (19b) and (19c)

0 01 02 03 04 05 06 07 08 09 1

0

10

20

30

minus40

minus30

minus20

minus10

120590

120590z120590xz

Goodsell and Pipes (2013)Goodsell and Pipes (2013)

yb

mdashPresent solutionmdashPresent solution

Figure 2 Interlaminar stresses along the 45∘

minus45∘ interface of

[45∘

minus45∘

]119904

laminate

Table 1 Engineering properties of unidirectional graphiteopoxy[28]

1198641

(GPa)1198642

= 1198643

(GPa)11986612

= 11986613

(GPa)11986623

(GPa) 12059212

= 12059213

12059223

132 108 565 338 024 059

4 Numerical Results and Discussions

In this part several numerical examples are shown for thedistribution of interlaminar stress All physical layers areassumed to be of equal thickness (=05mm)Themechanicalproperties of unidirectional graphiteepoxy T3005208 usedin this study are shown in Table 1 [28]

Also each physical ply is modeled as being made up of 12numerical layers within LWT (ie 119901 = 12) In addition thewidth thickness ratio (ie 2119887ℎ) is assumed to be equal to 10Furthermore the stress components are normalized as

120590119894119895=120590119894119895

1205900

(30)

where 1205900= 1198720119887ℎ2

Figure 2 shows the distribution of interlaminar normaland shear stresses along the width of [45∘minus45∘]

119904laminate

Good correspondence was found between the present resultsand the results obtained by Goodsell and Pipes [29] whichconfirms the accuracy of the present solution Figure 3 showsthe distribution of interlaminar stresses 120590

119911and 120590

119910119911along the

minus45∘

0∘ and 120590

119909119911along the 0

∘

90∘ interfaces of the general

[45∘

minus 45∘

0∘

90∘

30∘

minus 30∘

] laminate It is seen that theinterlaminar stresses show high stress gradient near the freeedge It is observed that the interlaminar stresses 120590

119911and 120590

119909119911

grow quickly near the free edge while being zero in the innerregion of the laminate Also the magnitude of the transverseshear stress 120590

119909119911is greater than that of transverse normal stress

120590119911 On the other hand 120590

119910119911rises toward the free edge and

decreases suddenly to zero at free edgeThe interlaminar normal stress 120590

119911along the 90∘0∘ inter-

face of the unsymmetric cross-ply [90∘

0∘

90∘

0∘

90∘

0∘

]

International Journal of Engineering Mathematics 7

0

0

01 02

02

03 04

04

05 06

06

07 08 09 1

minus02

minus04

minus06

minus08

120590

120590xz at z = 0

120590z at z = h6120590yz at z = h6

yb

Figure 3 Distribution of interlaminar stresses 120590119911

and 120590119910119911

alongthe minus45∘0∘ and 120590

119909119911

along the 0∘90∘ interfaces of the [45∘ minus 45∘0∘

90∘

30∘

minus30∘

] laminate

0 02 04 06 08 1yb

minus4

0

minus8

minus12

minus16

120590z

2bh = 502bh = 20

2bh = 102bh = 5

Figure 4 Distribution of the interlaminar normal stress 120590119911

alongthe 90∘0∘ interface of the laminate for various width-to-thicknessratios

laminate for various width-to-thickness ratios is demon-strated in Figure 4 It is observed that by decreasingthe width-to-thickness ratio the boundary-layer region canbe expanded towards the internal region of the laminate withits width being almost equal to the thickness of the laminateIt is seen that the magnitude of the interlaminar stress at thefree edge is not varied as the width-to-thickness ratio of thelaminate is changed In addition the highly localized natureof interlaminar stresses near and exactly at the free edges ofthe laminate is apparent

The distribution of interlaminar stresses along the0∘

90∘ and 90

∘

0∘ interfaces of the unsymmetric cross-ply

[0∘90∘0∘90∘] and [90∘0∘90∘0∘] laminates respectivelyare compared in Figure 5 It is seen that the maximuminterfacial value 120590

119911at the 0

∘

90∘ and 90

∘

0∘ interfaces take

place at the free edge Figure 6 shows the variation ofinterlaminar normal stress 120590

119911on through-the-thickness and

near the free edge of the angle-ply [45∘minus 45∘45∘minus 45∘]laminate as the free edge is approached It is observed that

120590z at z =h4120590z at z

120590yz

120590yz at z120590yz at z = h4

minus14minus12minus1

minus08minus06minus04minus02

0020406

0 02 04 06 08 1

= h4= h4

in [0∘90∘0∘90∘]

in [0∘90∘0∘90∘]

in [90∘0∘90∘0∘]in [90∘0∘90∘0∘]

yb

Figure 5 Distribution of interlaminar stresses along the 0∘

90∘

and 90∘

0∘ interfaces of the [0∘90∘0∘90∘] and [90∘0∘90∘0∘]

laminates respectively

minus1

minus05

0

05

1

minus200 minus150 minus100 minus50 0 50 100 150 200

zh

y = 094by = 096b

y = 099by = b

120590z

Figure 6 Variations of interlaminar normal stress 120590119911

through thethickness of the [45∘minus45∘45∘minus45∘] laminate for various width-to-thickness ratios

the maximum negative and positive values of 120590119911occur within

the top minus80∘ layer and the bottom 80

∘ layer at the free edge(ie 119910 = 119887) respectively It is also noticed that 120590

119911diminishes

away from the free edge as the interior region of the laminateis approached Moreover 120590

119911reduces by moving slightly away

from the free edge and its distribution becomes smootherThe variations of the interlaminar shear stress 120590

119910119911at 119910 =

119887 through the thickness of the [0∘

60∘

minus60∘

]119904laminate are

displayed in Figure 7 By increasing the number of numericallayers in each lamina 120590

119910119911becomes slightly closer to zero but

8 International Journal of Engineering Mathematics

minus15

minus1

minus05

0

05

1

15

minus20 minus10 0 10 20 30

zh

p = 20

p = 12

p = 8

120590yz

Figure 7 Variations of interlaminar shear stress 120590119910119911

on through-thickness of the [0∘60∘minus60∘]119904

laminate as a function of layer subdivisionnumber 119901

012

01

008

006

004

002

0

minus002

120590z

p = 30p = 20

p = 12p = 8

0 02 04 06 08 1

yb

Figure 8 Variations of interlaminar normal stress 120590119911

on through-thickness of the [minus15∘

90∘

90∘

minus15∘

] laminate as a function of layersubdivision number 119901

it never becomes zero It should be noted that increasing thenumber of subdivisions results in no convergence for 120590

119911and

120590119909119911

at interface-edge junction of two dissimilar layers Thenumerical values of these components (120590

119911and 120590

119909119911) continue

to grow as the number of sublayers increased Since thegeneralized stress resultants 119877119896

119910

instead of 120590119910119911

are forced todisappear at the free edge in LWT the numerical value of120590119910119911

may never become zero at the interface-edge intersection

(even by increasing the number of sublayers in each physicallayer)

The distribution of interlaminar normal stress 120590119911on

through thickness of the general [minus15∘90∘90∘minus15∘] lam-inate is displayed in Figure 8 It is noted that increasingthe number of numerical layers (119901) in each lamina has nosignificant effect on the values of the interlaminar stress 120590

119911

within the boundary-layer region of the laminate expect

International Journal of Engineering Mathematics 9

exactly at the free edge (ie 119910 = 119887) Furthermore theinterlaminar normal stress 120590

119911grows rapidly in the vicinity of

the free edge while being zero in the interior region of thelaminate

5 Conclusions

In the present paper analytical solutions are developed toanalyze the interlaminar stresses for different lay-up con-figurations in laminated composite plates subjected to thebending moment The solutions were obtained based onthe reduced elasticity displacement field for long laminatedplates It is found that the components of the displacementsfield are composed of two distinct parts signifying theglobal and local deformations within laminates The localdisplacement functions as well as the interlaminar stressesthrough the layers of the laminate were found by Reddyrsquoslayerwise theory (LWT)Thenumerical results based onLWTwere shown for the free-edge interlaminar stresses throughthe thickness and across the interfaces of various cross-plysymmetric angle-ply and general composite laminates

Appendix

The laminate rigidities in (22a) (22b) (22c) (22d) (24a)(24b) and (24c) are defined as

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

1206011015840

119895

1206011198961206011015840

119895

120601119896120601119895) d119911

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

120601119896 1206011015840

119896

119911 120601119896119911) d119911

(119896 119895 = 1 2 119873 + 1)

(119860119901119902 119861119901119902 119863119901119902) =

119873

sum

119894=1

int

119911119894+1

119911119894

119862(119894)

119901119902

(1 119911 1199112

) d119911

(A1)

which upon integration are presented in the following form

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

)

=

(minus

119862(119896minus1)

119901119902

ℎ119896minus1

minus

119862(119896minus1)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

6) if 119895 = 119896 minus 1

(

119862(119896minus1)

119901119902

ℎ119896minus1

+

119862(119896)

119901119902

ℎ119896

119862(119896minus1)

119901119902

2minus

119862(119896)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

3+

ℎ119896119862(119896)

119901119902

3) if 119895 = 119896

(minus

119862(119896)

119901119902

ℎ119896

119862(119896)

119901119902

2

ℎ119896119862(119896)

119901119902

6) if 119895 = 119896 + 1

(0 0 0) if 119895 lt 119896 minus 1 or 119895 gt 119896 + 1

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

)

=

(minus119862(1)

119901119902

ℎ1119862(1)

119901119902

2 119862(1)

119901119902

1199112

1

minus 1199112

2

2ℎ1

119862(1)

119901119902

ℎ1

[1199113

1

minus 1199113

2

3minus 1199112

1199112

1

minus 1199112

2

2]) if 119896 = 1

(minus119862(119896minus1)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2]) if 119896 = 119873 + 1

(119862(119896minus1)

119901119902

minus 119862(119896)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2+

ℎ119896119862(119896)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

+ 119862(119896)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2] +

119862(119896)

119901119902

ℎ119896

[1199113

119896

minus 1199113

119896+1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896+1

2]) if 1 lt 119896 lt 119873 + 1

(A2)

10 International Journal of Engineering Mathematics

Also

(119860119901119902 119861119901119902 119863119901119902)

=

119873

sum

119894=1

119862(119894)

119901119902

([119911119894+1

minus 119911119894] [

1199112

119894+1

minus 1199112

119894

2] [

1199113

119894+1

minus 1199113

119894

3])

(A3)

The coefficient matrices [119872] [119870] and [119871] appearing in(25) are given as

[119872] =[[

[

120575119878[11986366] 120575119878[11986326] 120575119878([11986136] minus [119861

45]119879

)

120575119878[11986326] [119863

22] [119861

23] minus [119861

44]119879

[0] [0] [11986344]

]]

]

[119870]=[

[

minus120575119878([11986055] + [120572]) minus120575

119878[11986045] [0]

minus120575119878[11986045] minus ([119860

44] + [120572]) [0]

120575119878([11986145]minus[11986136]119879

) [11986144]minus[11986123]119879

minus ([11986033]+[120572])

]

]

[119871] = [

[

12057511986011986055 0 0

12057511986011986045 0 0

minus120575119860(11986145 + 119861

36) 119860

13 12057511986211986113

]

]

(A4)

where [119860119901119902] [119861119901119902] and [119863

119901119902] are (119873 + 1) times (119873 + 1) square

matrices containing 119860119896119895119901119902 119861119896119895119901119902 and 119863

119896119895

119901119902 respectively and the

vectors 119860119901119902 119861119901119902 and 119861

119901119902 are (119873+1)times1 columnmatrices

containing 119860119896119901119902

119861119896119901119902

and 119861119896

119901119902

respectively Also [0] is (119873 +

1) times (119873 + 1) square zero and 0 is a zero vector with 119873 + 1

rows The artificial matrix [120572] is also an (119873 + 1) times (119873 + 1)

square matrix whose elements are given by

120572119896119895

= 120572int

ℎ2

minusℎ2

120601119896120601119895d119911 (A5)

with 120572 being a relatively small parameter in comparison withthe rigidity constants 119860119896119895

119901119902(119901119902 = 33 44 55) It is noted that

the insertion of [120572] in the matrix [119870] makes the eigenvaluesof the matrix (minus[119872]

minus1

[119870]) distinct

References

[1] T Kant and K Swaminathan ldquoEstimation of transverseinter-laminar stresses in laminated compositesmdasha selective reviewand survey of current developmentsrdquo Composite Structures vol49 no 1 pp 65ndash75 2000

[2] A H Puppo and H A Evensen ldquoInterlaminar shear ınlamınated composıtes under generalızed plane stressrdquo Journalof Composite Materials vol 4 pp 204ndash220 1970

[3] N J Pagano ldquoOn the calculatıon of ınterlamınar normal stressın composıte lamınaterdquo Journal of Composite Materials vol 8pp 65ndash81 1974

[4] P W Hsu and C T Herakovich ldquoEdge effects in angle-plycomposite laminatesrdquo Journal of CompositeMaterials vol 11 pp422ndash428 1977

[5] S Tang and A Levy ldquoBoundary layer theory 2 extension oflaminated finite striprdquo Journal of CompositeMaterials vol 9 no1 pp 42ndash52 1975

[6] R B Pipes andN J Pagano ldquoInterlaminar stresses in compositelaminates an approximate elasticity solutionrdquo Journal of AppliedMechanics vol 41 no 3 pp 668ndash672 1974

[7] N J Pagano ldquoStress fields in composite laminatesrdquo InternationalJournal of Solids and Structures vol 14 no 5 pp 385ndash400 1978

[8] S S Wang and I Choi ldquoBoundary-layer effects in composıtelaminates 1 free-edge stress singulaiıtiesrdquo Journal of AppliedMechanics vol V 49 no 3 pp 541ndash548 1982

[9] W L Yin ldquoFree-edge effects in anisotropic laminates underextension bending and twisting part I a stress-function-basedvariational approachrdquo Journal of Applied Mechanics vol 61 no2 pp 410ndash415 1994

[10] A S D Wang and F W Crossman ldquoSome new results on edgeeffect in symmetric composite laminatesrdquo Journal of CompositeMaterials vol 11 no 1 pp 92ndash106 1977

[11] J D Whitcomb I S Raju and J G Goree ldquoReliability ofthe finite element method for calculating free edge stresses incomposite laminatesrdquo Computers and Structures vol 15 no 1pp 23ndash37 1982

[12] G Davı ldquoStress fields in general composite laminatesrdquo AIAAJournal vol 34 no 12 pp 2604ndash2608 1996

[13] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT Part 1 derivationof finite element matricesrdquo International Journal for NumericalMethods in Engineering vol 55 no 2 pp 191ndash231 2002

[14] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT part 2 numericalimplementationsrdquo International Journal for Numerical Methodsin Engineering vol 55 no 3 pp 253ndash291 2002

[15] V-T Nguyen and J-F Caron ldquoFinite element analysis of free-edge stresses in composite laminates under mechanical anthermal loadingrdquo Composites Science and Technology vol 69no 1 pp 40ndash49 2009

[16] D H Robbins Jr and J N Reddy ldquoModelling of thick compos-ites using a layerwise laminate theoryrdquo International Journal forNumerical Methods in Engineering vol 36 no 4 pp 655ndash6771993

[17] C Mittelstedt and W Becker ldquoReddyrsquos layerwise laminate platetheory for the computation of elastic fields in the vicinity ofstraight free laminate edgesrdquoMaterials Science and EngineeringA vol 498 no 1-2 pp 76ndash80 2008

[18] W J Na Damage analysis of laminated composite beams underbending loads using the layer-wise theory [Dissertation thesis]Texas AampM University Texas Tex USA 2008

[19] T S Plagianakos and D A Saravanos ldquoHigher-order layerwiselaminate theory for the prediction of interlaminar shear stressesin thick composite and sandwich composite platesrdquo CompositeStructures vol 87 no 1 pp 23ndash35 2009

[20] H Ullah A R Harland T Lucas D Price and V V Sil-berschmidt ldquoFinite-element modelling of bending of CFRPlaminates multiple delaminationsrdquo Computational MaterialsScience vol 52 no 1 pp 147ndash156 2012

[21] F Helenon M R Wisnom S R Hallett and R S TraskldquoInvestigation into failure of laminated composite T-piecespecimens under bending loadingrdquo Composites A vol 54 pp182ndash189 2013

[22] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013

International Journal of Engineering Mathematics 11

[23] N D Thai M DrsquoOttavio and J-F Caron ldquoBending analysis oflaminated and sandwich plates using a layer-wise stress modelrdquoComposite Structures vol 96 pp 135ndash142 2013

[24] PMalekzadeh ldquoA two-dimensional layerwise-differential quad-rature static analysis of thick laminated composite circulararchesrdquoAppliedMathematicalModelling vol 33 no 4 pp 1850ndash1861 2009

[25] Y C Fung and P Tong Classical and Computational SolidMechanics World Scientific New Jersey NJ USA 2001

[26] S G LekhnitskiiTheory of Elasticity of anAnisotropic BodyMirPublishers Moscow Russia 1981

[27] A Nosier R K Kapania and J N Reddy ldquoFree vibrationanalysis of laminated plates using a layerwise theoryrdquo AIAAJournal vol 31 no 12 pp 2335ndash2346 1993

[28] C T HerakovichMechanics of Fibrous Composites Wiley NewYork NY USA 1998

[29] O Goodsell and R B Pipes ldquoInterlaminar stresses in compositelaminates subjected to anticlastic bending deformationrdquo Journalof Applied Mechanics vol 80 no 4 pp 1ndash7 2013

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Investigation of Through-Thickness ...downloads.hindawi.com/archive/2013/676743.pdf · (,, ) is located at the middle plane of the laminate, that is of thickness

2 International Journal of Engineering Mathematics

y

z

x

a

a

2b

MO

MOh

Figure 1 Laminate geometry and coordinate system

by using the Reissner mixed variational theorem (RMVT)They compared the numerical results for interlaminar stressesin several finite-element models and elasticity theory withincomposite laminates and sandwich plates Nguyen andCaron[15] employed a multiparticle finite element method to studythe interlaminar stresses near the free edges of generalcomposite laminates under mechanical and thermal loadingRobbins and Reddy [16] used a displacement-based variablekinematic global-local finite elementmethodMittelstedt andBecker [17] utilizedReddyrsquos layerwise laminate plate theory tofind the closed-form analysis of free-edge effects in layeredplates of arbitrary nonorthotropic layups The approachconsists of the subdivision of the physical laminate layers intoan arbitrary number of mathematical layers through the platethickness Na [18] used a finite element model based on thelayerwise theory He employed the von Karman type nonlin-ear strains to analyze damage in laminated composite beamsIn his formulation the Heaviside step function is employedto express the discontinuous interlaminar displacement fieldat the delaminated interfaces Plagianakos and Saravanos [19]presented a higher-order layerwise theoretical frameworkwhich enables prediction of the static response of thick com-posite and sandwich composite plates Ullah et al [20] carriedout some experimental tests to characterize the behavior of awoven CFRP material under large-deflection bending Two-dimensional finite element (FE) models were implementedin the commercial code Abaqus They performed series ofsimulations to study the deformation behavior and damage inCFRP for cases of high-deflection bendingHelenon et al [21]presented an experimental and numerical investigation intofailure of T-shaped laminated composite structures Threeout-of-plane bending cases are studied They found thatvery high free-edge maximum principal transverse tensilestresses perpendicular to the fiber direction occur at thefailure locations Thai et al [22] indicated an isogeometricfinite element formulation for static free vibration and buck-ling analysis of laminated composite and sandwich platesTheir method allows removing shear correction factors andimproves the accuracy of transverse shear stresses Thai et al[23] investigated the behavior of laminated composites usingseveral high order or layerwise finite element calculationsA layerwise model and its dedicated 119862

∘ eight-node finite

element were specifically developed for interlaminar stressesanalysis in free edge problem Malekzadeh [24] developed ahigh accuracy and rapid convergence hybrid approach fortwo-dimensional static analyses of circular arches with dif-ferent boundary conditions The method essentially consistsof a layerwise theory used for the thickness direction inconjunction with differential quadrature method in the axialdirection

There have been very limited works to study the interlam-inar stresses subjected to the bendingmoment In the presentpaper by the use of Reddyrsquos LWT an analytical solutionis presented to evaluate interlaminar stresses in cross-plysymmetric angle-ply and general composite laminates underthe bending moment To commence with based on physicalarguments regarding the deformations of a long generally andother laminated composite plates an appropriate reducedelasticity displacement field is established The boundary-layer stresses within the laminate are obtained analyticallybased on Reddyrsquos LWT

2 Problem Formulation

21 Elasticity Displacement Field An Nth-layered compositeplate (with arbitrary lamination) under the bending momentis considered as shown in Figure 1 The coordinate system(119909 119910 119911) is located at the middle plane of the laminate thatis of thickness ℎ width 2119887 and length 2119886 and is assumed tobe long in the 119909 direction so that the strains away from theends (119909 = plusmn119886) of the laminate are functions of only 119910 and 119911

The integrations of the three-dimensional elasticitystrain-displacement relations [25] inside the kth layer of thelaminate will generate the most general form of displacementfield which can be shown to be [26]

119906(119896)

1

(119909 119910 119911) = 119861(119896)

4

119909119910 + 119861(119896)

6

119909119911 + 119861(119896)

2

119909 + 119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = minus119861(119896)

1

119909119911 + 119861(119896)

3

119909 minus1

2119861(119896)

4

1199092

+ V(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 119861(119896)

1

119909119910 + 119861(119896)

5

119909 minus1

2119861(119896)

6

1199092

+ 119908(119896)

(119910 119911)

(1)

where 119906(119896)

1

119906(119896)2

and 119906(119896)

3

represent the displacement com-ponents of the material point (119909 119910 119911) in the 119909 119910 and 119911

International Journal of Engineering Mathematics 3

directions respectively To satisfy the displacement continu-ity conditions at the interfaces of the adjoining layers theintegration constants in (1) must be the same for all layerswithin the laminate Therefore the most general displace-ment field in the kth layer is shown

119906(119896)

1

(119909 119910 119911) = 1198614119909119910 + 119861

6119909119911 + 119861

2119909 + 119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = minus1198611119909119911 + 119861

3119909 minus

1

211986141199092

+ V(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 1198611119909119910 + 119861

5119909 minus

1

211986161199092

+ 119908(119896)

(119910 119911)

(2)

If loading conditions at 119909 = minus119886 and 119886 are similar based onthe physical grounds the following conditions hold

119906(119896)

1

(119909 119910 119911) = minus119906(119896)

1

(minus119909 minus119910 119911)

119906(119896)

2

(119909 119910 119911) = minus119906(119896)

2

(minus119909 minus119910 119911)

119906(119896)

3

(119909 119910 119911) = 119906(119896)

3

(minus119909 minus119910 119911)

(3)

Upon imposing these restrictions on (1) it is readily seen thatthe constants 119861

4and 119861

5must vanish and the displacement

field in (1) is reduced to the following equations

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 1198616119909119911 + 119906

(119896)

(119910 119911) (4a)

119906(119896)

2

(119909 119910 119911) = minus1198611119909119911 + 119861

3119909 + V(119896)

(119910 119911) (4b)

119906(119896)

3

(119909 119910 119911) = 1198611119909119910 minus

1

211986161199092

+ 119908(119896)

(119910 119911) (4c)

Also by replacing 119906(119896)

(119910 119911) by minus1198613119910 + 119906

(119896)

(119910 119911) in (4a) itbecomes apparent that terms involving 119861

3in (4a)ndash(4c) can

be neglected since no strains are generated by such terms(these terms will correspond to an infinitesimal rigid-bodyrotation of the laminate about the z-axis in Figure 1) Thusthemost general formof the displacement field of an arbitrarylaminate is given by

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 1198616119909119911 + 119906

(119896)

(119910 119911) (5a)

119906(119896)

2

(119909 119910 119911) = minus1198611119909119911 + V

(119896)

(119910 119911) (5b)

119906(119896)

3

(119909 119910 119911) = 1198611119909119910 minus

1

211986161199092

+ 119908(119896)

(119910 119911) (5c)

The displacement field in (5a) (5b) and (5c) may be used inprinciple for computing the stress field in any composite lam-inate subjected to arbitrary combinations of self-equilibratingmechanical and uniform hygrothermal loads The termsinvolving 119861

1 1198612 and 119861

6in (5a) (5b) and (5c) demonstrate

certain global deformations that occurred in the laminate Onthe other hand the unknown functions appearing in (5a)(5b) and (5c) illustrate the local deformations They will beevaluated by LWT of Reddy that happened in a laminate

For cross-ply laminates subjected tomechanical loadingsbased on physical grounds the following restrictions hold(see Figure 1)

119906(119896)

1

(119909 119910 119911) = minus119906(119896)

1

(minus119909 119910 119911) (6a)

119906(119896)

2

(119909 119910 119911) = 119906(119896)

2

(minus119909 119910 119911) (6b)

Upon imposing these conditions on (5a) it is readily seen that119906(119896)

(119910 119911) = 0 Also it is concluded from (6b) and (5b) that1198611= 0 Thus for cross-ply laminates the most general form

of the displacement field is given

119906(119896)

1

(119909 119910 119911) = 1198616119909119911 + 119861

2119909

119906(119896)

2

(119909 119910 119911) = V(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = minus1

211986161199092

+ 119908(119896)

(119910 119911)

(7)

Also for symmetric laminates based on physical groundsthe following conditions must hold (see Figure 1)

119906(119896)

1

(119909 119910 119911) = 119906(119873+1minus119896)

1

(119909 119910 minus119911) (8a)

119906(119896)

2

(119909 119910 119911) = 119906(119873+1minus119896)

2

(119909 119910 minus119911) (8b)

119906(119896)

3

(119909 119910 119911) = minus 119906(119873+1minus119896)

3

(119909 119910 minus119911)

(119896 = 1 2 3 119873

2)

(8c)

Upon imposing these conditions on (5a) (5b) and (5c) it isfound that 119861

1= 1198616= 0 Thus for symmetric laminates the

most general form of displacement field is expressed by

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = V(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 119908(119896)

(119910 119911)

(9)

Finally for antisymmetric angle-ply laminateswith119873 lay-ers subjected tomechanical loadings the following conditionmust hold

119906(119896)

1

(119909 119910 119911) = 119906(119873+1minus119896)

1

(119909 minus119910 minus119911) (119896 = 1 2 3 119873

2)

(10)

Concluding from (10) and (5a) 1198616= 0 and therefore the

most general displacement field for such laminates arewrittenas

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = minus1198611119909119911 + V

(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 1198611119909119910 + 119908

(119896)

(119910 119911)

(11)

4 International Journal of Engineering Mathematics

The displacement fields in (7) (9) and (11) can be repre-sented in one place as

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 1205751198621198616119909119911 + 120575

119878119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = minus1205751198601198611119909119911 + V

(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 1205751198601198611119909119910 minus

1

212057511986211986161199092

+ 119908(119896)

(119910 119911)

(12)

where 120575119860= 120575119878= 0 and 120575

119862= 1 for general cross-ply laminates

120575119860= 120575119862= 0 and 120575

119878= 1 for symmetric laminates 120575

119860= 120575119878= 1

and 120575119862= 0 for angle-ply laminates and 120575

119860= 120575119878= 120575119888= 1 for

general laminates

3 Layerwise Laminated Plate Theory of Reddy

In this section LWT is used for calculation of interlaminarstresses in arbitrary laminated composite plates with freeedges In LWT it is possible to replace the actual physicallayers to many mathematical layers [27] This can be doneby subdividing each physical layer through the thicknessto get the proper accuracy The theory assumes that thedisplacement components of a generic point in the laminateare given by [27]

1199061(119909 119910 119911) = 119906

119896(119909 119910) 120601

119896(119911)

1199062(119909 119910 119911) = V

119896(119909 119910) 120601

119896(119911)

1199063(119909 119910 119911) = 119908

119896(119909 119910) 120601

119896(119911)

119896 = 1 2 119873 + 1

(13)

It should be emphasized that a repeated index indicates sum-mation from 1 to119873 + 1 The functions 119906

119896(119909 119910) V

119896(119909 119910) and

119908119896(119909 119910) represent the displacements of the points initially

located at the kth plane of the laminate in the 119909 119910 and 119911

directions respectively The variable 119873 in (13) is the totalnumber of numerical layers introduced in any laminate Alsothe functions 120601

119896rsquos are the global approximation functions of

the thickness coordinate which are assumed here to be linear[27] This function expressed as

120601119896(119911) =

0 119911 le 119911119896minus1

1205952

119896minus1

(119911) 119911119896minus1

le 119911 le 119911119896

1205951

119896

(119911) 119911119896le 119911 le 119911

119896+1

0 119911 ge 119911119896+1

(119896 = 1 2 119873 + 1)

(14a)

where the local Lagrangian interpolation functions120595119895119896

(119911) (119895 =

1 2) related with the kth surface in the laminate are definedas

1205951

119896

(119911) =1

ℎ119896

(119911119896+1

minus 119911) 1205952

119896

(119911) =1

ℎ119896

(119911 minus 119911119896)

(14b)

where ℎ119896is the thickness of the kthmathematical layer Based

on the displacement field in (12) the displacement field ofLWT in (13) takes the following form

1199061(119909 119910 119911) = 120575

1198621198616119909119911 + 119861

2119909 + 120575119878119880119896(119910) 120601119896(119911)

1199062(119909 119910 119911) = minus120575

1198601198611119909119911 + 119881

119896(119910) 120601119896(119911)

1199063(119909 119910 119911) = 120575

1198601198611119909119910 minus

1

212057511986211986161199092

+119882119896(119910) 120601119896(119911)

119896 = 1 2 119873 + 1

(15)

It is noted that by the means of through-the-thicknesslinear interpolation functions the continuity of displacementcomponents through the thickness of laminate is identicallysatisfied On the other hand the transverse strain com-ponents remain discontinuous at the interfaces They willultimately emphasize the feasibility of having continuousinterlaminar stresses at the interfaces of adjoining layersby increasing the number of numerical layers through thephysical laminate Substituting (15) into the linear strain-displacement relations of elasticity [25] yields

120576119909= 1205751198621198616119911 + 1198612 120576

119910= 1198811015840

119896

120601119896 120576

119911= 1198821198961206011015840

119896

120574119910119911

= 1198811198961206011015840

119896

+1198821015840

119896

120601119896

120574119909119911

= 1205751198781198801198961206011015840

119896

+ 1205751198601198611119910 120574

119909119910= 1205751198781198801015840

119896

120601119896minus 1205751198601198611119911

(16)

where a prime in (16) denotes ordinary differentiation withrespect to a suitable independent variable (ie either 119910 or 119911)The three-dimensional constitutive law within the kth layer(with arbitrary fiber orientation) of the laminate may also beshown as [28]

(

(

120590119909

120590119910

120590119911

120590119910119911

120590119909119911

120590119909119910

)

)

(119896)

=

[[[[[[[[[

[

1198621111986212

11986213

0 0 11986216

11986212

11986222

11986223

0 0 11986226

11986213

11986223

11986233

0 0 11986236

0 0 0 11986244

11986245

0

0 0 0 11986245

11986255

0

11986216

11986226

11986236

0 0 11986266

]]]]]]]]]

]

(119896)

times(

(

120576119909

120576119910

120576119911

120574119910119911

120574119909119911

120574119909119910

)

)

(119896)

(17)

where [119862](119896) is the transformed (ie off axis) stiffness matrixof the kth layer By using the strain-displacement relations

International Journal of Engineering Mathematics 5

(16) in the principle of the minimum total potential energy[25] the equilibrium equations within LWT are found as

120575119880119896 120575119878(119876119896

119909

minus

d119872119896119909119910

d119910) = 0 119896 = 1 2 119873 + 1 (18a)

120575119881119896 119876119896119910

minus

d119872119896119910

d119910= 0 119896 = 1 2 119873 + 1 (18b)

120575119882119896 119873119896119911

minus

d119877119896119910

d119910= 0 119896 = 1 2 119873 + 1 (18c)

1205751198611 120575119860int

119887

minus119887

(119876119909119910 minus119872

119909119910) d119910 = 0 (19a)

1205751198612 int119887

minus119887

119873119909d119910 = 0 (19b)

1205751198616 120575119862int

119887

minus119887

119872119909d119910 = 119872

0 (19c)

Also the traction-free boundary conditions at the free edgesof the laminate are given as

119872119896

119910

= 119877119896

119910

= 120575119878119872119896

119909119910

= 0 at 119910 = plusmn119887 (119896 = 1 2 119873 + 1)

(20)

Equations (18a) (18b) and (18c) present 3(119873+1) local equilib-rium equations associated with119873+ 1 surface in the laminatewithin LWT Also (19a) (19b) and (19c) represent the globalequilibrium equation of a laminate The generalized stressandmoment resultants in (18a) (18b) (18c) (19a) (19b) (19c)and (20) are defined as

(119872119896

119910

119872119896

119909119910

119873119896

119911

) = int

ℎ2

minusℎ2

(120590119910120601119896 120590119909119910120601119896 1205901199111206011015840

) d119911 (21a)

(119876119896

119909

119876119896

119910

119877119896

119910

) = int

ℎ2

minusℎ2

(1205901199091199111206011015840

119896

1205901199101199111206011015840

119896

120590119910119911120601119896) d119911 (21b)

(119872119909 119873119909119872119909119910 119876119909) = int

ℎ2

minusℎ2

(120590119909119911 120590119909 120590119909119910119911 120590119909119911) d119911 (21c)

Substitution (16) into (17) and subsequently into (21a) (21b)and (21c) generates the following relations

(119872119896

119910

119872119896

119909119910

119873119896

119911

)

= 120575119878(119863119896119895

26

119863119896119895

66

119861119896119895

36

)1198801015840

119895

+ (119863119896119895

22

119863119896119895

26

119861119895119896

23

)1198811015840

119895

+ (119861119896119895

23

119861119896119895

36

119860119896119895

33

)119882119895minus 120575119860(119863119896

26

119863119896

66

119861119896

36

) 1198611

+ 120575119862(119863119896

12

119863119896

16

119861119896

13

) 1198616+ (119861119896

12

119861119896

16

119860119896

13

) 1198612

(22a)

(119876119896

119909

119876119896

119910

119877119896

119910

)

= 120575119878(119860119896119895

55

119860119896119895

45

119861119896119895

45

)119880119895+ (119860119896119895

45

119860119896119895

44

119861119896119895

44

)119881119895

+ (119861119895119896

45

119861119895119896

44

119863119896119895

44

)1198821015840

119895

minus 120575119860(119860119896

55

119860119896

45

119861119896

45

) 1198611119910

(22b)

(119872119910119872119909119910 119873119909)

= 120575119878(119863119896

16

119863119896

66

119861119896

16

)1198801015840

119895

+ (119863119896

12

119863119896

26

119861119896

12

)1198811015840

119895

+ (119861119896

13

119861119896

36

119860119896

13

)119882119895minus 120575119860(11986316 11986366 11986116) 1198611

+ 120575119862(11986111 11986116 11986011) 1198616+ (11986311 11986316 11986111) 1198612

(22c)

119876119909= 120575119878119860119896

55

119880119895+ 119860119896

45

119881119895+ 119861119896

45

1198821015840

119895

+ 120575119860119860551198611119910 (22d)

The explicit expressions for the laminate rigidities appearingin (22a) (22b) (22c) and (22d) in terms of 119862

119894rsquos are presented

in the Appendix Direct substitution of (22a) (22b) (22c)and (22d) into (18a) (18b) and (18c) provides the localdisplacement equilibrium equations within LWTThe resultsare expressed as

120575119880119896 120575119878(119863119896119895

66

11988010158401015840

119895

minus 119860119896119895

55

119880119895+ 119863119896119895

26

11988110158401015840

119895

minus 119860119896119895

45

119881119895+ (119861119896119895

36

minus 119861119895119896

45

)1198821015840

119895

) = 120575119860119860119896

55

1198611119910

119896 = 1 2 119873 + 1

(23a)

120575119881119896 120575119878119863119896119895

26

11988010158401015840

119895

minus 120575119878119860119896119895

45

119880119895+ 119863119896119895

22

11988110158401015840

119895

minus 119860119896119895

44

119881119895+ (119861119896119895

23

minus 119861119895119896

44

)1198821015840

119895

= 120575119860119860119896

45

1198611119910

119896 = 1 2 119873 + 1

(23b)

120575119882119896 120575119878(119861119896119895

45

minus 119861119895119896

36

)1198801015840

119895

minus (119861119896119895

44

minus 119861119895119896

23

)1198811015840

119895

+ 119863119896119895

44

11988210158401015840

119895

minus 119860119896119895

33

119882119895

= minus120575119860(119861119896

45

+ 119861119896

36

) 1198611+ 119860119896

13

1198612+ 120575119862119861119896

13

1198616

119896 = 1 2 119873 + 1

(23c)

In the same way upon substitution of (16) into (17) andthe subsequent results into (19a) (19b) and (19c) the globalequations (19a) (19b) and (19c) are found in terms ofdisplacement functions as follows

1205751198611 120575119860int

119887

minus119887

((120575119878119860119896

55

119880119895+ 119860119896

45

119881119895+ 119861119896

45

1198821015840

119895

) 119910

minus (120575119878119863119896

66

1198801015840

119895

+ 119863119896

26

1198811015840

119895

+ 119861119896

36

119882119895)

+ 120575119860(1198605511986111199102

+ 11986366) 1198611minus 119861161198612

minus120575119862119863161198616) d119910 = 0

(24a)

6 International Journal of Engineering Mathematics

1205751198612 int119887

minus119887

(120575119878119861119896

16

1198801015840

119895

+ 119861119896

12

1198811015840

119895

+ 119860119896

13

119882119895

minus 120575119860119861161198611+ 119860111198612+ 120575119862119861111198616) d119910 = 0

(24b)

1205751198616 120575119862int

119887

minus119887

(120575119878119863119896

16

1198801015840

119895

+ 119863119896

12

1198811015840

119895

+ 119861119896

13

119882119895

minus 120575119860119863161198611+ 119861111198612+ 120575119862119863111198616) d119910 = 119872

0

(24c)

where the extra laminate rigidities appearing in (24a) (24b)and (24c) are defined in the Appendix The system ofequations in (23a) (23b) and (23c) shows 3(119873 + 1) coupledordinary differential equations with constant coefficientswhich may be indicated in a matrix form as

[119872] 12057810158401015840

+ [119870] 120578 = [119871] 119861 sdot 119910 (25)

where

120578 = 119880119879

119881119879

119882119879

119879

119880 = 1198801 1198802 119880

119873+1119879

119881 = 1198811 1198812 119881

119873+1119879

119882 = 11988211198822 119882

119873+1119879

(26)

119861 = 1198611 1198612 1198616119879

119882119895= int

119910

119882119895d119910

(27)

The coefficient matrices [119872] [119870] and [119871] in (25) are dis-played in the Appendix Also from the conditions in (3) andthe displacement field in (23a) (23b) and (23c) it is observedthat the functions 119880

119895 119881119895 and119882

119895are all odd functions of the

independent variable 119910 It can be confirmed that the generalsolution of (25) may be presented as

120578 = [120595] [sinh (120582119910)] 119867 + [119870]minus1

[119871] 119861 sdot 119910 (28)

where [sinh(120582119910)] is a 3(119873 + 1) times 3(119873 + 1) diagonal matrixand

[sinh (120582119910)]

= diag (sinh (1205821119910) sinh (120582

2119910) sinh (120582

3(119873+1)119910))

(29)

Also [120595] and (1205822

1

1205822

2

1205822

3(119873+1)

) are the model matrixand eigenvalues of (minus[119872]

minus1

[119870]) respectively 119870 is anunknown vector representing 3(119873+1) integration constantsThe constants 119861

119895(119895 = 1 2 6)must be calculated within LWT

analysis So the boundary conditions in (20) are first imposedto found the vector 119870 in terms of the unknown parameters119861119895(119895 = 1 2 6) These constants are next calculated in terms

of the specific bending moment 1198720by satisfaction of the

global equilibrium conditions in (19a) (19b) and (19c)

0 01 02 03 04 05 06 07 08 09 1

0

10

20

30

minus40

minus30

minus20

minus10

120590

120590z120590xz

Goodsell and Pipes (2013)Goodsell and Pipes (2013)

yb

mdashPresent solutionmdashPresent solution

Figure 2 Interlaminar stresses along the 45∘

minus45∘ interface of

[45∘

minus45∘

]119904

laminate

Table 1 Engineering properties of unidirectional graphiteopoxy[28]

1198641

(GPa)1198642

= 1198643

(GPa)11986612

= 11986613

(GPa)11986623

(GPa) 12059212

= 12059213

12059223

132 108 565 338 024 059

4 Numerical Results and Discussions

In this part several numerical examples are shown for thedistribution of interlaminar stress All physical layers areassumed to be of equal thickness (=05mm)Themechanicalproperties of unidirectional graphiteepoxy T3005208 usedin this study are shown in Table 1 [28]

Also each physical ply is modeled as being made up of 12numerical layers within LWT (ie 119901 = 12) In addition thewidth thickness ratio (ie 2119887ℎ) is assumed to be equal to 10Furthermore the stress components are normalized as

120590119894119895=120590119894119895

1205900

(30)

where 1205900= 1198720119887ℎ2

Figure 2 shows the distribution of interlaminar normaland shear stresses along the width of [45∘minus45∘]

119904laminate

Good correspondence was found between the present resultsand the results obtained by Goodsell and Pipes [29] whichconfirms the accuracy of the present solution Figure 3 showsthe distribution of interlaminar stresses 120590

119911and 120590

119910119911along the

minus45∘

0∘ and 120590

119909119911along the 0

∘

90∘ interfaces of the general

[45∘

minus 45∘

0∘

90∘

30∘

minus 30∘

] laminate It is seen that theinterlaminar stresses show high stress gradient near the freeedge It is observed that the interlaminar stresses 120590

119911and 120590

119909119911

grow quickly near the free edge while being zero in the innerregion of the laminate Also the magnitude of the transverseshear stress 120590

119909119911is greater than that of transverse normal stress

120590119911 On the other hand 120590

119910119911rises toward the free edge and

decreases suddenly to zero at free edgeThe interlaminar normal stress 120590

119911along the 90∘0∘ inter-

face of the unsymmetric cross-ply [90∘

0∘

90∘

0∘

90∘

0∘

]

International Journal of Engineering Mathematics 7

0

0

01 02

02

03 04

04

05 06

06

07 08 09 1

minus02

minus04

minus06

minus08

120590

120590xz at z = 0

120590z at z = h6120590yz at z = h6

yb

Figure 3 Distribution of interlaminar stresses 120590119911

and 120590119910119911

alongthe minus45∘0∘ and 120590

119909119911

along the 0∘90∘ interfaces of the [45∘ minus 45∘0∘

90∘

30∘

minus30∘

] laminate

0 02 04 06 08 1yb

minus4

0

minus8

minus12

minus16

120590z

2bh = 502bh = 20

2bh = 102bh = 5

Figure 4 Distribution of the interlaminar normal stress 120590119911

alongthe 90∘0∘ interface of the laminate for various width-to-thicknessratios

laminate for various width-to-thickness ratios is demon-strated in Figure 4 It is observed that by decreasingthe width-to-thickness ratio the boundary-layer region canbe expanded towards the internal region of the laminate withits width being almost equal to the thickness of the laminateIt is seen that the magnitude of the interlaminar stress at thefree edge is not varied as the width-to-thickness ratio of thelaminate is changed In addition the highly localized natureof interlaminar stresses near and exactly at the free edges ofthe laminate is apparent

The distribution of interlaminar stresses along the0∘

90∘ and 90

∘

0∘ interfaces of the unsymmetric cross-ply

[0∘90∘0∘90∘] and [90∘0∘90∘0∘] laminates respectivelyare compared in Figure 5 It is seen that the maximuminterfacial value 120590

119911at the 0

∘

90∘ and 90

∘

0∘ interfaces take

place at the free edge Figure 6 shows the variation ofinterlaminar normal stress 120590

119911on through-the-thickness and

near the free edge of the angle-ply [45∘minus 45∘45∘minus 45∘]laminate as the free edge is approached It is observed that

120590z at z =h4120590z at z

120590yz

120590yz at z120590yz at z = h4

minus14minus12minus1

minus08minus06minus04minus02

0020406

0 02 04 06 08 1

= h4= h4

in [0∘90∘0∘90∘]

in [0∘90∘0∘90∘]

in [90∘0∘90∘0∘]in [90∘0∘90∘0∘]

yb

Figure 5 Distribution of interlaminar stresses along the 0∘

90∘

and 90∘

0∘ interfaces of the [0∘90∘0∘90∘] and [90∘0∘90∘0∘]

laminates respectively

minus1

minus05

0

05

1

minus200 minus150 minus100 minus50 0 50 100 150 200

zh

y = 094by = 096b

y = 099by = b

120590z

Figure 6 Variations of interlaminar normal stress 120590119911

through thethickness of the [45∘minus45∘45∘minus45∘] laminate for various width-to-thickness ratios

the maximum negative and positive values of 120590119911occur within

the top minus80∘ layer and the bottom 80

∘ layer at the free edge(ie 119910 = 119887) respectively It is also noticed that 120590

119911diminishes

away from the free edge as the interior region of the laminateis approached Moreover 120590

119911reduces by moving slightly away

from the free edge and its distribution becomes smootherThe variations of the interlaminar shear stress 120590

119910119911at 119910 =

119887 through the thickness of the [0∘

60∘

minus60∘

]119904laminate are

displayed in Figure 7 By increasing the number of numericallayers in each lamina 120590

119910119911becomes slightly closer to zero but

8 International Journal of Engineering Mathematics

minus15

minus1

minus05

0

05

1

15

minus20 minus10 0 10 20 30

zh

p = 20

p = 12

p = 8

120590yz

Figure 7 Variations of interlaminar shear stress 120590119910119911

on through-thickness of the [0∘60∘minus60∘]119904

laminate as a function of layer subdivisionnumber 119901

012

01

008

006

004

002

0

minus002

120590z

p = 30p = 20

p = 12p = 8

0 02 04 06 08 1

yb

Figure 8 Variations of interlaminar normal stress 120590119911

on through-thickness of the [minus15∘

90∘

90∘

minus15∘

] laminate as a function of layersubdivision number 119901

it never becomes zero It should be noted that increasing thenumber of subdivisions results in no convergence for 120590

119911and

120590119909119911

at interface-edge junction of two dissimilar layers Thenumerical values of these components (120590

119911and 120590

119909119911) continue

to grow as the number of sublayers increased Since thegeneralized stress resultants 119877119896

119910

instead of 120590119910119911

are forced todisappear at the free edge in LWT the numerical value of120590119910119911

may never become zero at the interface-edge intersection

(even by increasing the number of sublayers in each physicallayer)

The distribution of interlaminar normal stress 120590119911on

through thickness of the general [minus15∘90∘90∘minus15∘] lam-inate is displayed in Figure 8 It is noted that increasingthe number of numerical layers (119901) in each lamina has nosignificant effect on the values of the interlaminar stress 120590

119911

within the boundary-layer region of the laminate expect

International Journal of Engineering Mathematics 9

exactly at the free edge (ie 119910 = 119887) Furthermore theinterlaminar normal stress 120590

119911grows rapidly in the vicinity of

the free edge while being zero in the interior region of thelaminate

5 Conclusions

In the present paper analytical solutions are developed toanalyze the interlaminar stresses for different lay-up con-figurations in laminated composite plates subjected to thebending moment The solutions were obtained based onthe reduced elasticity displacement field for long laminatedplates It is found that the components of the displacementsfield are composed of two distinct parts signifying theglobal and local deformations within laminates The localdisplacement functions as well as the interlaminar stressesthrough the layers of the laminate were found by Reddyrsquoslayerwise theory (LWT)Thenumerical results based onLWTwere shown for the free-edge interlaminar stresses throughthe thickness and across the interfaces of various cross-plysymmetric angle-ply and general composite laminates

Appendix

The laminate rigidities in (22a) (22b) (22c) (22d) (24a)(24b) and (24c) are defined as

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

1206011015840

119895

1206011198961206011015840

119895

120601119896120601119895) d119911

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

120601119896 1206011015840

119896

119911 120601119896119911) d119911

(119896 119895 = 1 2 119873 + 1)

(119860119901119902 119861119901119902 119863119901119902) =

119873

sum

119894=1

int

119911119894+1

119911119894

119862(119894)

119901119902

(1 119911 1199112

) d119911

(A1)

which upon integration are presented in the following form

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

)

=

(minus

119862(119896minus1)

119901119902

ℎ119896minus1

minus

119862(119896minus1)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

6) if 119895 = 119896 minus 1

(

119862(119896minus1)

119901119902

ℎ119896minus1

+

119862(119896)

119901119902

ℎ119896

119862(119896minus1)

119901119902

2minus

119862(119896)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

3+

ℎ119896119862(119896)

119901119902

3) if 119895 = 119896

(minus

119862(119896)

119901119902

ℎ119896

119862(119896)

119901119902

2

ℎ119896119862(119896)

119901119902

6) if 119895 = 119896 + 1

(0 0 0) if 119895 lt 119896 minus 1 or 119895 gt 119896 + 1

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

)

=

(minus119862(1)

119901119902

ℎ1119862(1)

119901119902

2 119862(1)

119901119902

1199112

1

minus 1199112

2

2ℎ1

119862(1)

119901119902

ℎ1

[1199113

1

minus 1199113

2

3minus 1199112

1199112

1

minus 1199112

2

2]) if 119896 = 1

(minus119862(119896minus1)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2]) if 119896 = 119873 + 1

(119862(119896minus1)

119901119902

minus 119862(119896)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2+

ℎ119896119862(119896)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

+ 119862(119896)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2] +

119862(119896)

119901119902

ℎ119896

[1199113

119896

minus 1199113

119896+1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896+1

2]) if 1 lt 119896 lt 119873 + 1

(A2)

10 International Journal of Engineering Mathematics

Also

(119860119901119902 119861119901119902 119863119901119902)

=

119873

sum

119894=1

119862(119894)

119901119902

([119911119894+1

minus 119911119894] [

1199112

119894+1

minus 1199112

119894

2] [

1199113

119894+1

minus 1199113

119894

3])

(A3)

The coefficient matrices [119872] [119870] and [119871] appearing in(25) are given as

[119872] =[[

[

120575119878[11986366] 120575119878[11986326] 120575119878([11986136] minus [119861

45]119879

)

120575119878[11986326] [119863

22] [119861

23] minus [119861

44]119879

[0] [0] [11986344]

]]

]

[119870]=[

[

minus120575119878([11986055] + [120572]) minus120575

119878[11986045] [0]

minus120575119878[11986045] minus ([119860

44] + [120572]) [0]

120575119878([11986145]minus[11986136]119879

) [11986144]minus[11986123]119879

minus ([11986033]+[120572])

]

]

[119871] = [

[

12057511986011986055 0 0

12057511986011986045 0 0

minus120575119860(11986145 + 119861

36) 119860

13 12057511986211986113

]

]

(A4)

where [119860119901119902] [119861119901119902] and [119863

119901119902] are (119873 + 1) times (119873 + 1) square

matrices containing 119860119896119895119901119902 119861119896119895119901119902 and 119863

119896119895

119901119902 respectively and the

vectors 119860119901119902 119861119901119902 and 119861

119901119902 are (119873+1)times1 columnmatrices

containing 119860119896119901119902

119861119896119901119902

and 119861119896

119901119902

respectively Also [0] is (119873 +

1) times (119873 + 1) square zero and 0 is a zero vector with 119873 + 1

rows The artificial matrix [120572] is also an (119873 + 1) times (119873 + 1)

square matrix whose elements are given by

120572119896119895

= 120572int

ℎ2

minusℎ2

120601119896120601119895d119911 (A5)

with 120572 being a relatively small parameter in comparison withthe rigidity constants 119860119896119895

119901119902(119901119902 = 33 44 55) It is noted that

the insertion of [120572] in the matrix [119870] makes the eigenvaluesof the matrix (minus[119872]

minus1

[119870]) distinct

References

[1] T Kant and K Swaminathan ldquoEstimation of transverseinter-laminar stresses in laminated compositesmdasha selective reviewand survey of current developmentsrdquo Composite Structures vol49 no 1 pp 65ndash75 2000

[2] A H Puppo and H A Evensen ldquoInterlaminar shear ınlamınated composıtes under generalızed plane stressrdquo Journalof Composite Materials vol 4 pp 204ndash220 1970

[3] N J Pagano ldquoOn the calculatıon of ınterlamınar normal stressın composıte lamınaterdquo Journal of Composite Materials vol 8pp 65ndash81 1974

[4] P W Hsu and C T Herakovich ldquoEdge effects in angle-plycomposite laminatesrdquo Journal of CompositeMaterials vol 11 pp422ndash428 1977

[5] S Tang and A Levy ldquoBoundary layer theory 2 extension oflaminated finite striprdquo Journal of CompositeMaterials vol 9 no1 pp 42ndash52 1975

[6] R B Pipes andN J Pagano ldquoInterlaminar stresses in compositelaminates an approximate elasticity solutionrdquo Journal of AppliedMechanics vol 41 no 3 pp 668ndash672 1974

[7] N J Pagano ldquoStress fields in composite laminatesrdquo InternationalJournal of Solids and Structures vol 14 no 5 pp 385ndash400 1978

[8] S S Wang and I Choi ldquoBoundary-layer effects in composıtelaminates 1 free-edge stress singulaiıtiesrdquo Journal of AppliedMechanics vol V 49 no 3 pp 541ndash548 1982

[9] W L Yin ldquoFree-edge effects in anisotropic laminates underextension bending and twisting part I a stress-function-basedvariational approachrdquo Journal of Applied Mechanics vol 61 no2 pp 410ndash415 1994

[10] A S D Wang and F W Crossman ldquoSome new results on edgeeffect in symmetric composite laminatesrdquo Journal of CompositeMaterials vol 11 no 1 pp 92ndash106 1977

[11] J D Whitcomb I S Raju and J G Goree ldquoReliability ofthe finite element method for calculating free edge stresses incomposite laminatesrdquo Computers and Structures vol 15 no 1pp 23ndash37 1982

[12] G Davı ldquoStress fields in general composite laminatesrdquo AIAAJournal vol 34 no 12 pp 2604ndash2608 1996

[13] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT Part 1 derivationof finite element matricesrdquo International Journal for NumericalMethods in Engineering vol 55 no 2 pp 191ndash231 2002

[14] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT part 2 numericalimplementationsrdquo International Journal for Numerical Methodsin Engineering vol 55 no 3 pp 253ndash291 2002

[15] V-T Nguyen and J-F Caron ldquoFinite element analysis of free-edge stresses in composite laminates under mechanical anthermal loadingrdquo Composites Science and Technology vol 69no 1 pp 40ndash49 2009

[16] D H Robbins Jr and J N Reddy ldquoModelling of thick compos-ites using a layerwise laminate theoryrdquo International Journal forNumerical Methods in Engineering vol 36 no 4 pp 655ndash6771993

[17] C Mittelstedt and W Becker ldquoReddyrsquos layerwise laminate platetheory for the computation of elastic fields in the vicinity ofstraight free laminate edgesrdquoMaterials Science and EngineeringA vol 498 no 1-2 pp 76ndash80 2008

[18] W J Na Damage analysis of laminated composite beams underbending loads using the layer-wise theory [Dissertation thesis]Texas AampM University Texas Tex USA 2008

[19] T S Plagianakos and D A Saravanos ldquoHigher-order layerwiselaminate theory for the prediction of interlaminar shear stressesin thick composite and sandwich composite platesrdquo CompositeStructures vol 87 no 1 pp 23ndash35 2009

[20] H Ullah A R Harland T Lucas D Price and V V Sil-berschmidt ldquoFinite-element modelling of bending of CFRPlaminates multiple delaminationsrdquo Computational MaterialsScience vol 52 no 1 pp 147ndash156 2012

[21] F Helenon M R Wisnom S R Hallett and R S TraskldquoInvestigation into failure of laminated composite T-piecespecimens under bending loadingrdquo Composites A vol 54 pp182ndash189 2013

[22] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013

International Journal of Engineering Mathematics 11

[23] N D Thai M DrsquoOttavio and J-F Caron ldquoBending analysis oflaminated and sandwich plates using a layer-wise stress modelrdquoComposite Structures vol 96 pp 135ndash142 2013

[24] PMalekzadeh ldquoA two-dimensional layerwise-differential quad-rature static analysis of thick laminated composite circulararchesrdquoAppliedMathematicalModelling vol 33 no 4 pp 1850ndash1861 2009

[25] Y C Fung and P Tong Classical and Computational SolidMechanics World Scientific New Jersey NJ USA 2001

[26] S G LekhnitskiiTheory of Elasticity of anAnisotropic BodyMirPublishers Moscow Russia 1981

[27] A Nosier R K Kapania and J N Reddy ldquoFree vibrationanalysis of laminated plates using a layerwise theoryrdquo AIAAJournal vol 31 no 12 pp 2335ndash2346 1993

[28] C T HerakovichMechanics of Fibrous Composites Wiley NewYork NY USA 1998

[29] O Goodsell and R B Pipes ldquoInterlaminar stresses in compositelaminates subjected to anticlastic bending deformationrdquo Journalof Applied Mechanics vol 80 no 4 pp 1ndash7 2013

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Investigation of Through-Thickness ...downloads.hindawi.com/archive/2013/676743.pdf · (,, ) is located at the middle plane of the laminate, that is of thickness

International Journal of Engineering Mathematics 3

directions respectively To satisfy the displacement continu-ity conditions at the interfaces of the adjoining layers theintegration constants in (1) must be the same for all layerswithin the laminate Therefore the most general displace-ment field in the kth layer is shown

119906(119896)

1

(119909 119910 119911) = 1198614119909119910 + 119861

6119909119911 + 119861

2119909 + 119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = minus1198611119909119911 + 119861

3119909 minus

1

211986141199092

+ V(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 1198611119909119910 + 119861

5119909 minus

1

211986161199092

+ 119908(119896)

(119910 119911)

(2)

If loading conditions at 119909 = minus119886 and 119886 are similar based onthe physical grounds the following conditions hold

119906(119896)

1

(119909 119910 119911) = minus119906(119896)

1

(minus119909 minus119910 119911)

119906(119896)

2

(119909 119910 119911) = minus119906(119896)

2

(minus119909 minus119910 119911)

119906(119896)

3

(119909 119910 119911) = 119906(119896)

3

(minus119909 minus119910 119911)

(3)

Upon imposing these restrictions on (1) it is readily seen thatthe constants 119861

4and 119861

5must vanish and the displacement

field in (1) is reduced to the following equations

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 1198616119909119911 + 119906

(119896)

(119910 119911) (4a)

119906(119896)

2

(119909 119910 119911) = minus1198611119909119911 + 119861

3119909 + V(119896)

(119910 119911) (4b)

119906(119896)

3

(119909 119910 119911) = 1198611119909119910 minus

1

211986161199092

+ 119908(119896)

(119910 119911) (4c)

Also by replacing 119906(119896)

(119910 119911) by minus1198613119910 + 119906

(119896)

(119910 119911) in (4a) itbecomes apparent that terms involving 119861

3in (4a)ndash(4c) can

be neglected since no strains are generated by such terms(these terms will correspond to an infinitesimal rigid-bodyrotation of the laminate about the z-axis in Figure 1) Thusthemost general formof the displacement field of an arbitrarylaminate is given by

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 1198616119909119911 + 119906

(119896)

(119910 119911) (5a)

119906(119896)

2

(119909 119910 119911) = minus1198611119909119911 + V

(119896)

(119910 119911) (5b)

119906(119896)

3

(119909 119910 119911) = 1198611119909119910 minus

1

211986161199092

+ 119908(119896)

(119910 119911) (5c)

The displacement field in (5a) (5b) and (5c) may be used inprinciple for computing the stress field in any composite lam-inate subjected to arbitrary combinations of self-equilibratingmechanical and uniform hygrothermal loads The termsinvolving 119861

1 1198612 and 119861

6in (5a) (5b) and (5c) demonstrate

certain global deformations that occurred in the laminate Onthe other hand the unknown functions appearing in (5a)(5b) and (5c) illustrate the local deformations They will beevaluated by LWT of Reddy that happened in a laminate

For cross-ply laminates subjected tomechanical loadingsbased on physical grounds the following restrictions hold(see Figure 1)

119906(119896)

1

(119909 119910 119911) = minus119906(119896)

1

(minus119909 119910 119911) (6a)

119906(119896)

2

(119909 119910 119911) = 119906(119896)

2

(minus119909 119910 119911) (6b)

Upon imposing these conditions on (5a) it is readily seen that119906(119896)

(119910 119911) = 0 Also it is concluded from (6b) and (5b) that1198611= 0 Thus for cross-ply laminates the most general form

of the displacement field is given

119906(119896)

1

(119909 119910 119911) = 1198616119909119911 + 119861

2119909

119906(119896)

2

(119909 119910 119911) = V(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = minus1

211986161199092

+ 119908(119896)

(119910 119911)

(7)

Also for symmetric laminates based on physical groundsthe following conditions must hold (see Figure 1)

119906(119896)

1

(119909 119910 119911) = 119906(119873+1minus119896)

1

(119909 119910 minus119911) (8a)

119906(119896)

2

(119909 119910 119911) = 119906(119873+1minus119896)

2

(119909 119910 minus119911) (8b)

119906(119896)

3

(119909 119910 119911) = minus 119906(119873+1minus119896)

3

(119909 119910 minus119911)

(119896 = 1 2 3 119873

2)

(8c)

Upon imposing these conditions on (5a) (5b) and (5c) it isfound that 119861

1= 1198616= 0 Thus for symmetric laminates the

most general form of displacement field is expressed by

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = V(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 119908(119896)

(119910 119911)

(9)

Finally for antisymmetric angle-ply laminateswith119873 lay-ers subjected tomechanical loadings the following conditionmust hold

119906(119896)

1

(119909 119910 119911) = 119906(119873+1minus119896)

1

(119909 minus119910 minus119911) (119896 = 1 2 3 119873

2)

(10)

Concluding from (10) and (5a) 1198616= 0 and therefore the

most general displacement field for such laminates arewrittenas

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = minus1198611119909119911 + V

(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 1198611119909119910 + 119908

(119896)

(119910 119911)

(11)

4 International Journal of Engineering Mathematics

The displacement fields in (7) (9) and (11) can be repre-sented in one place as

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 1205751198621198616119909119911 + 120575

119878119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = minus1205751198601198611119909119911 + V

(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 1205751198601198611119909119910 minus

1

212057511986211986161199092

+ 119908(119896)

(119910 119911)

(12)

where 120575119860= 120575119878= 0 and 120575

119862= 1 for general cross-ply laminates

120575119860= 120575119862= 0 and 120575

119878= 1 for symmetric laminates 120575

119860= 120575119878= 1

and 120575119862= 0 for angle-ply laminates and 120575

119860= 120575119878= 120575119888= 1 for

general laminates

3 Layerwise Laminated Plate Theory of Reddy

In this section LWT is used for calculation of interlaminarstresses in arbitrary laminated composite plates with freeedges In LWT it is possible to replace the actual physicallayers to many mathematical layers [27] This can be doneby subdividing each physical layer through the thicknessto get the proper accuracy The theory assumes that thedisplacement components of a generic point in the laminateare given by [27]

1199061(119909 119910 119911) = 119906

119896(119909 119910) 120601

119896(119911)

1199062(119909 119910 119911) = V

119896(119909 119910) 120601

119896(119911)

1199063(119909 119910 119911) = 119908

119896(119909 119910) 120601

119896(119911)

119896 = 1 2 119873 + 1

(13)

It should be emphasized that a repeated index indicates sum-mation from 1 to119873 + 1 The functions 119906

119896(119909 119910) V

119896(119909 119910) and

119908119896(119909 119910) represent the displacements of the points initially

located at the kth plane of the laminate in the 119909 119910 and 119911

directions respectively The variable 119873 in (13) is the totalnumber of numerical layers introduced in any laminate Alsothe functions 120601

119896rsquos are the global approximation functions of

the thickness coordinate which are assumed here to be linear[27] This function expressed as

120601119896(119911) =

0 119911 le 119911119896minus1

1205952

119896minus1

(119911) 119911119896minus1

le 119911 le 119911119896

1205951

119896

(119911) 119911119896le 119911 le 119911

119896+1

0 119911 ge 119911119896+1

(119896 = 1 2 119873 + 1)

(14a)

where the local Lagrangian interpolation functions120595119895119896

(119911) (119895 =

1 2) related with the kth surface in the laminate are definedas

1205951

119896

(119911) =1

ℎ119896

(119911119896+1

minus 119911) 1205952

119896

(119911) =1

ℎ119896

(119911 minus 119911119896)

(14b)

where ℎ119896is the thickness of the kthmathematical layer Based

on the displacement field in (12) the displacement field ofLWT in (13) takes the following form

1199061(119909 119910 119911) = 120575

1198621198616119909119911 + 119861

2119909 + 120575119878119880119896(119910) 120601119896(119911)

1199062(119909 119910 119911) = minus120575

1198601198611119909119911 + 119881

119896(119910) 120601119896(119911)

1199063(119909 119910 119911) = 120575

1198601198611119909119910 minus

1

212057511986211986161199092

+119882119896(119910) 120601119896(119911)

119896 = 1 2 119873 + 1

(15)

It is noted that by the means of through-the-thicknesslinear interpolation functions the continuity of displacementcomponents through the thickness of laminate is identicallysatisfied On the other hand the transverse strain com-ponents remain discontinuous at the interfaces They willultimately emphasize the feasibility of having continuousinterlaminar stresses at the interfaces of adjoining layersby increasing the number of numerical layers through thephysical laminate Substituting (15) into the linear strain-displacement relations of elasticity [25] yields

120576119909= 1205751198621198616119911 + 1198612 120576

119910= 1198811015840

119896

120601119896 120576

119911= 1198821198961206011015840

119896

120574119910119911

= 1198811198961206011015840

119896

+1198821015840

119896

120601119896

120574119909119911

= 1205751198781198801198961206011015840

119896

+ 1205751198601198611119910 120574

119909119910= 1205751198781198801015840

119896

120601119896minus 1205751198601198611119911

(16)

where a prime in (16) denotes ordinary differentiation withrespect to a suitable independent variable (ie either 119910 or 119911)The three-dimensional constitutive law within the kth layer(with arbitrary fiber orientation) of the laminate may also beshown as [28]

(

(

120590119909

120590119910

120590119911

120590119910119911

120590119909119911

120590119909119910

)

)

(119896)

=

[[[[[[[[[

[

1198621111986212

11986213

0 0 11986216

11986212

11986222

11986223

0 0 11986226

11986213

11986223

11986233

0 0 11986236

0 0 0 11986244

11986245

0

0 0 0 11986245

11986255

0

11986216

11986226

11986236

0 0 11986266

]]]]]]]]]

]

(119896)

times(

(

120576119909

120576119910

120576119911

120574119910119911

120574119909119911

120574119909119910

)

)

(119896)

(17)

where [119862](119896) is the transformed (ie off axis) stiffness matrixof the kth layer By using the strain-displacement relations

International Journal of Engineering Mathematics 5

(16) in the principle of the minimum total potential energy[25] the equilibrium equations within LWT are found as

120575119880119896 120575119878(119876119896

119909

minus

d119872119896119909119910

d119910) = 0 119896 = 1 2 119873 + 1 (18a)

120575119881119896 119876119896119910

minus

d119872119896119910

d119910= 0 119896 = 1 2 119873 + 1 (18b)

120575119882119896 119873119896119911

minus

d119877119896119910

d119910= 0 119896 = 1 2 119873 + 1 (18c)

1205751198611 120575119860int

119887

minus119887

(119876119909119910 minus119872

119909119910) d119910 = 0 (19a)

1205751198612 int119887

minus119887

119873119909d119910 = 0 (19b)

1205751198616 120575119862int

119887

minus119887

119872119909d119910 = 119872

0 (19c)

Also the traction-free boundary conditions at the free edgesof the laminate are given as

119872119896

119910

= 119877119896

119910

= 120575119878119872119896

119909119910

= 0 at 119910 = plusmn119887 (119896 = 1 2 119873 + 1)

(20)

Equations (18a) (18b) and (18c) present 3(119873+1) local equilib-rium equations associated with119873+ 1 surface in the laminatewithin LWT Also (19a) (19b) and (19c) represent the globalequilibrium equation of a laminate The generalized stressandmoment resultants in (18a) (18b) (18c) (19a) (19b) (19c)and (20) are defined as

(119872119896

119910

119872119896

119909119910

119873119896

119911

) = int

ℎ2

minusℎ2

(120590119910120601119896 120590119909119910120601119896 1205901199111206011015840

) d119911 (21a)

(119876119896

119909

119876119896

119910

119877119896

119910

) = int

ℎ2

minusℎ2

(1205901199091199111206011015840

119896

1205901199101199111206011015840

119896

120590119910119911120601119896) d119911 (21b)

(119872119909 119873119909119872119909119910 119876119909) = int

ℎ2

minusℎ2

(120590119909119911 120590119909 120590119909119910119911 120590119909119911) d119911 (21c)

Substitution (16) into (17) and subsequently into (21a) (21b)and (21c) generates the following relations

(119872119896

119910

119872119896

119909119910

119873119896

119911

)

= 120575119878(119863119896119895

26

119863119896119895

66

119861119896119895

36

)1198801015840

119895

+ (119863119896119895

22

119863119896119895

26

119861119895119896

23

)1198811015840

119895

+ (119861119896119895

23

119861119896119895

36

119860119896119895

33

)119882119895minus 120575119860(119863119896

26

119863119896

66

119861119896

36

) 1198611

+ 120575119862(119863119896

12

119863119896

16

119861119896

13

) 1198616+ (119861119896

12

119861119896

16

119860119896

13

) 1198612

(22a)

(119876119896

119909

119876119896

119910

119877119896

119910

)

= 120575119878(119860119896119895

55

119860119896119895

45

119861119896119895

45

)119880119895+ (119860119896119895

45

119860119896119895

44

119861119896119895

44

)119881119895

+ (119861119895119896

45

119861119895119896

44

119863119896119895

44

)1198821015840

119895

minus 120575119860(119860119896

55

119860119896

45

119861119896

45

) 1198611119910

(22b)

(119872119910119872119909119910 119873119909)

= 120575119878(119863119896

16

119863119896

66

119861119896

16

)1198801015840

119895

+ (119863119896

12

119863119896

26

119861119896

12

)1198811015840

119895

+ (119861119896

13

119861119896

36

119860119896

13

)119882119895minus 120575119860(11986316 11986366 11986116) 1198611

+ 120575119862(11986111 11986116 11986011) 1198616+ (11986311 11986316 11986111) 1198612

(22c)

119876119909= 120575119878119860119896

55

119880119895+ 119860119896

45

119881119895+ 119861119896

45

1198821015840

119895

+ 120575119860119860551198611119910 (22d)

The explicit expressions for the laminate rigidities appearingin (22a) (22b) (22c) and (22d) in terms of 119862

119894rsquos are presented

in the Appendix Direct substitution of (22a) (22b) (22c)and (22d) into (18a) (18b) and (18c) provides the localdisplacement equilibrium equations within LWTThe resultsare expressed as

120575119880119896 120575119878(119863119896119895

66

11988010158401015840

119895

minus 119860119896119895

55

119880119895+ 119863119896119895

26

11988110158401015840

119895

minus 119860119896119895

45

119881119895+ (119861119896119895

36

minus 119861119895119896

45

)1198821015840

119895

) = 120575119860119860119896

55

1198611119910

119896 = 1 2 119873 + 1

(23a)

120575119881119896 120575119878119863119896119895

26

11988010158401015840

119895

minus 120575119878119860119896119895

45

119880119895+ 119863119896119895

22

11988110158401015840

119895

minus 119860119896119895

44

119881119895+ (119861119896119895

23

minus 119861119895119896

44

)1198821015840

119895

= 120575119860119860119896

45

1198611119910

119896 = 1 2 119873 + 1

(23b)

120575119882119896 120575119878(119861119896119895

45

minus 119861119895119896

36

)1198801015840

119895

minus (119861119896119895

44

minus 119861119895119896

23

)1198811015840

119895

+ 119863119896119895

44

11988210158401015840

119895

minus 119860119896119895

33

119882119895

= minus120575119860(119861119896

45

+ 119861119896

36

) 1198611+ 119860119896

13

1198612+ 120575119862119861119896

13

1198616

119896 = 1 2 119873 + 1

(23c)

In the same way upon substitution of (16) into (17) andthe subsequent results into (19a) (19b) and (19c) the globalequations (19a) (19b) and (19c) are found in terms ofdisplacement functions as follows

1205751198611 120575119860int

119887

minus119887

((120575119878119860119896

55

119880119895+ 119860119896

45

119881119895+ 119861119896

45

1198821015840

119895

) 119910

minus (120575119878119863119896

66

1198801015840

119895

+ 119863119896

26

1198811015840

119895

+ 119861119896

36

119882119895)

+ 120575119860(1198605511986111199102

+ 11986366) 1198611minus 119861161198612

minus120575119862119863161198616) d119910 = 0

(24a)

6 International Journal of Engineering Mathematics

1205751198612 int119887

minus119887

(120575119878119861119896

16

1198801015840

119895

+ 119861119896

12

1198811015840

119895

+ 119860119896

13

119882119895

minus 120575119860119861161198611+ 119860111198612+ 120575119862119861111198616) d119910 = 0

(24b)

1205751198616 120575119862int

119887

minus119887

(120575119878119863119896

16

1198801015840

119895

+ 119863119896

12

1198811015840

119895

+ 119861119896

13

119882119895

minus 120575119860119863161198611+ 119861111198612+ 120575119862119863111198616) d119910 = 119872

0

(24c)

where the extra laminate rigidities appearing in (24a) (24b)and (24c) are defined in the Appendix The system ofequations in (23a) (23b) and (23c) shows 3(119873 + 1) coupledordinary differential equations with constant coefficientswhich may be indicated in a matrix form as

[119872] 12057810158401015840

+ [119870] 120578 = [119871] 119861 sdot 119910 (25)

where

120578 = 119880119879

119881119879

119882119879

119879

119880 = 1198801 1198802 119880

119873+1119879

119881 = 1198811 1198812 119881

119873+1119879

119882 = 11988211198822 119882

119873+1119879

(26)

119861 = 1198611 1198612 1198616119879

119882119895= int

119910

119882119895d119910

(27)

The coefficient matrices [119872] [119870] and [119871] in (25) are dis-played in the Appendix Also from the conditions in (3) andthe displacement field in (23a) (23b) and (23c) it is observedthat the functions 119880

119895 119881119895 and119882

119895are all odd functions of the

independent variable 119910 It can be confirmed that the generalsolution of (25) may be presented as

120578 = [120595] [sinh (120582119910)] 119867 + [119870]minus1

[119871] 119861 sdot 119910 (28)

where [sinh(120582119910)] is a 3(119873 + 1) times 3(119873 + 1) diagonal matrixand

[sinh (120582119910)]

= diag (sinh (1205821119910) sinh (120582

2119910) sinh (120582

3(119873+1)119910))

(29)

Also [120595] and (1205822

1

1205822

2

1205822

3(119873+1)

) are the model matrixand eigenvalues of (minus[119872]

minus1

[119870]) respectively 119870 is anunknown vector representing 3(119873+1) integration constantsThe constants 119861

119895(119895 = 1 2 6)must be calculated within LWT

analysis So the boundary conditions in (20) are first imposedto found the vector 119870 in terms of the unknown parameters119861119895(119895 = 1 2 6) These constants are next calculated in terms

of the specific bending moment 1198720by satisfaction of the

global equilibrium conditions in (19a) (19b) and (19c)

0 01 02 03 04 05 06 07 08 09 1

0

10

20

30

minus40

minus30

minus20

minus10

120590

120590z120590xz

Goodsell and Pipes (2013)Goodsell and Pipes (2013)

yb

mdashPresent solutionmdashPresent solution

Figure 2 Interlaminar stresses along the 45∘

minus45∘ interface of

[45∘

minus45∘

]119904

laminate

Table 1 Engineering properties of unidirectional graphiteopoxy[28]

1198641

(GPa)1198642

= 1198643

(GPa)11986612

= 11986613

(GPa)11986623

(GPa) 12059212

= 12059213

12059223

132 108 565 338 024 059

4 Numerical Results and Discussions

In this part several numerical examples are shown for thedistribution of interlaminar stress All physical layers areassumed to be of equal thickness (=05mm)Themechanicalproperties of unidirectional graphiteepoxy T3005208 usedin this study are shown in Table 1 [28]

Also each physical ply is modeled as being made up of 12numerical layers within LWT (ie 119901 = 12) In addition thewidth thickness ratio (ie 2119887ℎ) is assumed to be equal to 10Furthermore the stress components are normalized as

120590119894119895=120590119894119895

1205900

(30)

where 1205900= 1198720119887ℎ2

Figure 2 shows the distribution of interlaminar normaland shear stresses along the width of [45∘minus45∘]

119904laminate

Good correspondence was found between the present resultsand the results obtained by Goodsell and Pipes [29] whichconfirms the accuracy of the present solution Figure 3 showsthe distribution of interlaminar stresses 120590

119911and 120590

119910119911along the

minus45∘

0∘ and 120590

119909119911along the 0

∘

90∘ interfaces of the general

[45∘

minus 45∘

0∘

90∘

30∘

minus 30∘

] laminate It is seen that theinterlaminar stresses show high stress gradient near the freeedge It is observed that the interlaminar stresses 120590

119911and 120590

119909119911

grow quickly near the free edge while being zero in the innerregion of the laminate Also the magnitude of the transverseshear stress 120590

119909119911is greater than that of transverse normal stress

120590119911 On the other hand 120590

119910119911rises toward the free edge and

decreases suddenly to zero at free edgeThe interlaminar normal stress 120590

119911along the 90∘0∘ inter-

face of the unsymmetric cross-ply [90∘

0∘

90∘

0∘

90∘

0∘

]

International Journal of Engineering Mathematics 7

0

0

01 02

02

03 04

04

05 06

06

07 08 09 1

minus02

minus04

minus06

minus08

120590

120590xz at z = 0

120590z at z = h6120590yz at z = h6

yb

Figure 3 Distribution of interlaminar stresses 120590119911

and 120590119910119911

alongthe minus45∘0∘ and 120590

119909119911

along the 0∘90∘ interfaces of the [45∘ minus 45∘0∘

90∘

30∘

minus30∘

] laminate

0 02 04 06 08 1yb

minus4

0

minus8

minus12

minus16

120590z

2bh = 502bh = 20

2bh = 102bh = 5

Figure 4 Distribution of the interlaminar normal stress 120590119911

alongthe 90∘0∘ interface of the laminate for various width-to-thicknessratios

laminate for various width-to-thickness ratios is demon-strated in Figure 4 It is observed that by decreasingthe width-to-thickness ratio the boundary-layer region canbe expanded towards the internal region of the laminate withits width being almost equal to the thickness of the laminateIt is seen that the magnitude of the interlaminar stress at thefree edge is not varied as the width-to-thickness ratio of thelaminate is changed In addition the highly localized natureof interlaminar stresses near and exactly at the free edges ofthe laminate is apparent

The distribution of interlaminar stresses along the0∘

90∘ and 90

∘

0∘ interfaces of the unsymmetric cross-ply

[0∘90∘0∘90∘] and [90∘0∘90∘0∘] laminates respectivelyare compared in Figure 5 It is seen that the maximuminterfacial value 120590

119911at the 0

∘

90∘ and 90

∘

0∘ interfaces take

place at the free edge Figure 6 shows the variation ofinterlaminar normal stress 120590

119911on through-the-thickness and

near the free edge of the angle-ply [45∘minus 45∘45∘minus 45∘]laminate as the free edge is approached It is observed that

120590z at z =h4120590z at z

120590yz

120590yz at z120590yz at z = h4

minus14minus12minus1

minus08minus06minus04minus02

0020406

0 02 04 06 08 1

= h4= h4

in [0∘90∘0∘90∘]

in [0∘90∘0∘90∘]

in [90∘0∘90∘0∘]in [90∘0∘90∘0∘]

yb

Figure 5 Distribution of interlaminar stresses along the 0∘

90∘

and 90∘

0∘ interfaces of the [0∘90∘0∘90∘] and [90∘0∘90∘0∘]

laminates respectively

minus1

minus05

0

05

1

minus200 minus150 minus100 minus50 0 50 100 150 200

zh

y = 094by = 096b

y = 099by = b

120590z

Figure 6 Variations of interlaminar normal stress 120590119911

through thethickness of the [45∘minus45∘45∘minus45∘] laminate for various width-to-thickness ratios

the maximum negative and positive values of 120590119911occur within

the top minus80∘ layer and the bottom 80

∘ layer at the free edge(ie 119910 = 119887) respectively It is also noticed that 120590

119911diminishes

away from the free edge as the interior region of the laminateis approached Moreover 120590

119911reduces by moving slightly away

from the free edge and its distribution becomes smootherThe variations of the interlaminar shear stress 120590

119910119911at 119910 =

119887 through the thickness of the [0∘

60∘

minus60∘

]119904laminate are

displayed in Figure 7 By increasing the number of numericallayers in each lamina 120590

119910119911becomes slightly closer to zero but

8 International Journal of Engineering Mathematics

minus15

minus1

minus05

0

05

1

15

minus20 minus10 0 10 20 30

zh

p = 20

p = 12

p = 8

120590yz

Figure 7 Variations of interlaminar shear stress 120590119910119911

on through-thickness of the [0∘60∘minus60∘]119904

laminate as a function of layer subdivisionnumber 119901

012

01

008

006

004

002

0

minus002

120590z

p = 30p = 20

p = 12p = 8

0 02 04 06 08 1

yb

Figure 8 Variations of interlaminar normal stress 120590119911

on through-thickness of the [minus15∘

90∘

90∘

minus15∘

] laminate as a function of layersubdivision number 119901

it never becomes zero It should be noted that increasing thenumber of subdivisions results in no convergence for 120590

119911and

120590119909119911

at interface-edge junction of two dissimilar layers Thenumerical values of these components (120590

119911and 120590

119909119911) continue

to grow as the number of sublayers increased Since thegeneralized stress resultants 119877119896

119910

instead of 120590119910119911

are forced todisappear at the free edge in LWT the numerical value of120590119910119911

may never become zero at the interface-edge intersection

(even by increasing the number of sublayers in each physicallayer)

The distribution of interlaminar normal stress 120590119911on

through thickness of the general [minus15∘90∘90∘minus15∘] lam-inate is displayed in Figure 8 It is noted that increasingthe number of numerical layers (119901) in each lamina has nosignificant effect on the values of the interlaminar stress 120590

119911

within the boundary-layer region of the laminate expect

International Journal of Engineering Mathematics 9

exactly at the free edge (ie 119910 = 119887) Furthermore theinterlaminar normal stress 120590

119911grows rapidly in the vicinity of

the free edge while being zero in the interior region of thelaminate

5 Conclusions

In the present paper analytical solutions are developed toanalyze the interlaminar stresses for different lay-up con-figurations in laminated composite plates subjected to thebending moment The solutions were obtained based onthe reduced elasticity displacement field for long laminatedplates It is found that the components of the displacementsfield are composed of two distinct parts signifying theglobal and local deformations within laminates The localdisplacement functions as well as the interlaminar stressesthrough the layers of the laminate were found by Reddyrsquoslayerwise theory (LWT)Thenumerical results based onLWTwere shown for the free-edge interlaminar stresses throughthe thickness and across the interfaces of various cross-plysymmetric angle-ply and general composite laminates

Appendix

The laminate rigidities in (22a) (22b) (22c) (22d) (24a)(24b) and (24c) are defined as

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

1206011015840

119895

1206011198961206011015840

119895

120601119896120601119895) d119911

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

120601119896 1206011015840

119896

119911 120601119896119911) d119911

(119896 119895 = 1 2 119873 + 1)

(119860119901119902 119861119901119902 119863119901119902) =

119873

sum

119894=1

int

119911119894+1

119911119894

119862(119894)

119901119902

(1 119911 1199112

) d119911

(A1)

which upon integration are presented in the following form

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

)

=

(minus

119862(119896minus1)

119901119902

ℎ119896minus1

minus

119862(119896minus1)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

6) if 119895 = 119896 minus 1

(

119862(119896minus1)

119901119902

ℎ119896minus1

+

119862(119896)

119901119902

ℎ119896

119862(119896minus1)

119901119902

2minus

119862(119896)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

3+

ℎ119896119862(119896)

119901119902

3) if 119895 = 119896

(minus

119862(119896)

119901119902

ℎ119896

119862(119896)

119901119902

2

ℎ119896119862(119896)

119901119902

6) if 119895 = 119896 + 1

(0 0 0) if 119895 lt 119896 minus 1 or 119895 gt 119896 + 1

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

)

=

(minus119862(1)

119901119902

ℎ1119862(1)

119901119902

2 119862(1)

119901119902

1199112

1

minus 1199112

2

2ℎ1

119862(1)

119901119902

ℎ1

[1199113

1

minus 1199113

2

3minus 1199112

1199112

1

minus 1199112

2

2]) if 119896 = 1

(minus119862(119896minus1)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2]) if 119896 = 119873 + 1

(119862(119896minus1)

119901119902

minus 119862(119896)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2+

ℎ119896119862(119896)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

+ 119862(119896)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2] +

119862(119896)

119901119902

ℎ119896

[1199113

119896

minus 1199113

119896+1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896+1

2]) if 1 lt 119896 lt 119873 + 1

(A2)

10 International Journal of Engineering Mathematics

Also

(119860119901119902 119861119901119902 119863119901119902)

=

119873

sum

119894=1

119862(119894)

119901119902

([119911119894+1

minus 119911119894] [

1199112

119894+1

minus 1199112

119894

2] [

1199113

119894+1

minus 1199113

119894

3])

(A3)

The coefficient matrices [119872] [119870] and [119871] appearing in(25) are given as

[119872] =[[

[

120575119878[11986366] 120575119878[11986326] 120575119878([11986136] minus [119861

45]119879

)

120575119878[11986326] [119863

22] [119861

23] minus [119861

44]119879

[0] [0] [11986344]

]]

]

[119870]=[

[

minus120575119878([11986055] + [120572]) minus120575

119878[11986045] [0]

minus120575119878[11986045] minus ([119860

44] + [120572]) [0]

120575119878([11986145]minus[11986136]119879

) [11986144]minus[11986123]119879

minus ([11986033]+[120572])

]

]

[119871] = [

[

12057511986011986055 0 0

12057511986011986045 0 0

minus120575119860(11986145 + 119861

36) 119860

13 12057511986211986113

]

]

(A4)

where [119860119901119902] [119861119901119902] and [119863

119901119902] are (119873 + 1) times (119873 + 1) square

matrices containing 119860119896119895119901119902 119861119896119895119901119902 and 119863

119896119895

119901119902 respectively and the

vectors 119860119901119902 119861119901119902 and 119861

119901119902 are (119873+1)times1 columnmatrices

containing 119860119896119901119902

119861119896119901119902

and 119861119896

119901119902

respectively Also [0] is (119873 +

1) times (119873 + 1) square zero and 0 is a zero vector with 119873 + 1

rows The artificial matrix [120572] is also an (119873 + 1) times (119873 + 1)

square matrix whose elements are given by

120572119896119895

= 120572int

ℎ2

minusℎ2

120601119896120601119895d119911 (A5)

with 120572 being a relatively small parameter in comparison withthe rigidity constants 119860119896119895

119901119902(119901119902 = 33 44 55) It is noted that

the insertion of [120572] in the matrix [119870] makes the eigenvaluesof the matrix (minus[119872]

minus1

[119870]) distinct

References

[1] T Kant and K Swaminathan ldquoEstimation of transverseinter-laminar stresses in laminated compositesmdasha selective reviewand survey of current developmentsrdquo Composite Structures vol49 no 1 pp 65ndash75 2000

[2] A H Puppo and H A Evensen ldquoInterlaminar shear ınlamınated composıtes under generalızed plane stressrdquo Journalof Composite Materials vol 4 pp 204ndash220 1970

[3] N J Pagano ldquoOn the calculatıon of ınterlamınar normal stressın composıte lamınaterdquo Journal of Composite Materials vol 8pp 65ndash81 1974

[4] P W Hsu and C T Herakovich ldquoEdge effects in angle-plycomposite laminatesrdquo Journal of CompositeMaterials vol 11 pp422ndash428 1977

[5] S Tang and A Levy ldquoBoundary layer theory 2 extension oflaminated finite striprdquo Journal of CompositeMaterials vol 9 no1 pp 42ndash52 1975

[6] R B Pipes andN J Pagano ldquoInterlaminar stresses in compositelaminates an approximate elasticity solutionrdquo Journal of AppliedMechanics vol 41 no 3 pp 668ndash672 1974

[7] N J Pagano ldquoStress fields in composite laminatesrdquo InternationalJournal of Solids and Structures vol 14 no 5 pp 385ndash400 1978

[8] S S Wang and I Choi ldquoBoundary-layer effects in composıtelaminates 1 free-edge stress singulaiıtiesrdquo Journal of AppliedMechanics vol V 49 no 3 pp 541ndash548 1982

[9] W L Yin ldquoFree-edge effects in anisotropic laminates underextension bending and twisting part I a stress-function-basedvariational approachrdquo Journal of Applied Mechanics vol 61 no2 pp 410ndash415 1994

[10] A S D Wang and F W Crossman ldquoSome new results on edgeeffect in symmetric composite laminatesrdquo Journal of CompositeMaterials vol 11 no 1 pp 92ndash106 1977

[11] J D Whitcomb I S Raju and J G Goree ldquoReliability ofthe finite element method for calculating free edge stresses incomposite laminatesrdquo Computers and Structures vol 15 no 1pp 23ndash37 1982

[12] G Davı ldquoStress fields in general composite laminatesrdquo AIAAJournal vol 34 no 12 pp 2604ndash2608 1996

[13] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT Part 1 derivationof finite element matricesrdquo International Journal for NumericalMethods in Engineering vol 55 no 2 pp 191ndash231 2002

[14] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT part 2 numericalimplementationsrdquo International Journal for Numerical Methodsin Engineering vol 55 no 3 pp 253ndash291 2002

[15] V-T Nguyen and J-F Caron ldquoFinite element analysis of free-edge stresses in composite laminates under mechanical anthermal loadingrdquo Composites Science and Technology vol 69no 1 pp 40ndash49 2009

[16] D H Robbins Jr and J N Reddy ldquoModelling of thick compos-ites using a layerwise laminate theoryrdquo International Journal forNumerical Methods in Engineering vol 36 no 4 pp 655ndash6771993

[17] C Mittelstedt and W Becker ldquoReddyrsquos layerwise laminate platetheory for the computation of elastic fields in the vicinity ofstraight free laminate edgesrdquoMaterials Science and EngineeringA vol 498 no 1-2 pp 76ndash80 2008

[18] W J Na Damage analysis of laminated composite beams underbending loads using the layer-wise theory [Dissertation thesis]Texas AampM University Texas Tex USA 2008

[19] T S Plagianakos and D A Saravanos ldquoHigher-order layerwiselaminate theory for the prediction of interlaminar shear stressesin thick composite and sandwich composite platesrdquo CompositeStructures vol 87 no 1 pp 23ndash35 2009

[20] H Ullah A R Harland T Lucas D Price and V V Sil-berschmidt ldquoFinite-element modelling of bending of CFRPlaminates multiple delaminationsrdquo Computational MaterialsScience vol 52 no 1 pp 147ndash156 2012

[21] F Helenon M R Wisnom S R Hallett and R S TraskldquoInvestigation into failure of laminated composite T-piecespecimens under bending loadingrdquo Composites A vol 54 pp182ndash189 2013

[22] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013

International Journal of Engineering Mathematics 11

[23] N D Thai M DrsquoOttavio and J-F Caron ldquoBending analysis oflaminated and sandwich plates using a layer-wise stress modelrdquoComposite Structures vol 96 pp 135ndash142 2013

[24] PMalekzadeh ldquoA two-dimensional layerwise-differential quad-rature static analysis of thick laminated composite circulararchesrdquoAppliedMathematicalModelling vol 33 no 4 pp 1850ndash1861 2009

[25] Y C Fung and P Tong Classical and Computational SolidMechanics World Scientific New Jersey NJ USA 2001

[26] S G LekhnitskiiTheory of Elasticity of anAnisotropic BodyMirPublishers Moscow Russia 1981

[27] A Nosier R K Kapania and J N Reddy ldquoFree vibrationanalysis of laminated plates using a layerwise theoryrdquo AIAAJournal vol 31 no 12 pp 2335ndash2346 1993

[28] C T HerakovichMechanics of Fibrous Composites Wiley NewYork NY USA 1998

[29] O Goodsell and R B Pipes ldquoInterlaminar stresses in compositelaminates subjected to anticlastic bending deformationrdquo Journalof Applied Mechanics vol 80 no 4 pp 1ndash7 2013

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Investigation of Through-Thickness ...downloads.hindawi.com/archive/2013/676743.pdf · (,, ) is located at the middle plane of the laminate, that is of thickness

4 International Journal of Engineering Mathematics

The displacement fields in (7) (9) and (11) can be repre-sented in one place as

119906(119896)

1

(119909 119910 119911) = 1198612119909 + 1205751198621198616119909119911 + 120575

119878119906(119896)

(119910 119911)

119906(119896)

2

(119909 119910 119911) = minus1205751198601198611119909119911 + V

(119896)

(119910 119911)

119906(119896)

3

(119909 119910 119911) = 1205751198601198611119909119910 minus

1

212057511986211986161199092

+ 119908(119896)

(119910 119911)

(12)

where 120575119860= 120575119878= 0 and 120575

119862= 1 for general cross-ply laminates

120575119860= 120575119862= 0 and 120575

119878= 1 for symmetric laminates 120575

119860= 120575119878= 1

and 120575119862= 0 for angle-ply laminates and 120575

119860= 120575119878= 120575119888= 1 for

general laminates

3 Layerwise Laminated Plate Theory of Reddy

In this section LWT is used for calculation of interlaminarstresses in arbitrary laminated composite plates with freeedges In LWT it is possible to replace the actual physicallayers to many mathematical layers [27] This can be doneby subdividing each physical layer through the thicknessto get the proper accuracy The theory assumes that thedisplacement components of a generic point in the laminateare given by [27]

1199061(119909 119910 119911) = 119906

119896(119909 119910) 120601

119896(119911)

1199062(119909 119910 119911) = V

119896(119909 119910) 120601

119896(119911)

1199063(119909 119910 119911) = 119908

119896(119909 119910) 120601

119896(119911)

119896 = 1 2 119873 + 1

(13)

It should be emphasized that a repeated index indicates sum-mation from 1 to119873 + 1 The functions 119906

119896(119909 119910) V

119896(119909 119910) and

119908119896(119909 119910) represent the displacements of the points initially

located at the kth plane of the laminate in the 119909 119910 and 119911

directions respectively The variable 119873 in (13) is the totalnumber of numerical layers introduced in any laminate Alsothe functions 120601

119896rsquos are the global approximation functions of

the thickness coordinate which are assumed here to be linear[27] This function expressed as

120601119896(119911) =

0 119911 le 119911119896minus1

1205952

119896minus1

(119911) 119911119896minus1

le 119911 le 119911119896

1205951

119896

(119911) 119911119896le 119911 le 119911

119896+1

0 119911 ge 119911119896+1

(119896 = 1 2 119873 + 1)

(14a)

where the local Lagrangian interpolation functions120595119895119896

(119911) (119895 =

1 2) related with the kth surface in the laminate are definedas

1205951

119896

(119911) =1

ℎ119896

(119911119896+1

minus 119911) 1205952

119896

(119911) =1

ℎ119896

(119911 minus 119911119896)

(14b)

where ℎ119896is the thickness of the kthmathematical layer Based

on the displacement field in (12) the displacement field ofLWT in (13) takes the following form

1199061(119909 119910 119911) = 120575

1198621198616119909119911 + 119861

2119909 + 120575119878119880119896(119910) 120601119896(119911)

1199062(119909 119910 119911) = minus120575

1198601198611119909119911 + 119881

119896(119910) 120601119896(119911)

1199063(119909 119910 119911) = 120575

1198601198611119909119910 minus

1

212057511986211986161199092

+119882119896(119910) 120601119896(119911)

119896 = 1 2 119873 + 1

(15)

It is noted that by the means of through-the-thicknesslinear interpolation functions the continuity of displacementcomponents through the thickness of laminate is identicallysatisfied On the other hand the transverse strain com-ponents remain discontinuous at the interfaces They willultimately emphasize the feasibility of having continuousinterlaminar stresses at the interfaces of adjoining layersby increasing the number of numerical layers through thephysical laminate Substituting (15) into the linear strain-displacement relations of elasticity [25] yields

120576119909= 1205751198621198616119911 + 1198612 120576

119910= 1198811015840

119896

120601119896 120576

119911= 1198821198961206011015840

119896

120574119910119911

= 1198811198961206011015840

119896

+1198821015840

119896

120601119896

120574119909119911

= 1205751198781198801198961206011015840

119896

+ 1205751198601198611119910 120574

119909119910= 1205751198781198801015840

119896

120601119896minus 1205751198601198611119911

(16)

where a prime in (16) denotes ordinary differentiation withrespect to a suitable independent variable (ie either 119910 or 119911)The three-dimensional constitutive law within the kth layer(with arbitrary fiber orientation) of the laminate may also beshown as [28]

(

(

120590119909

120590119910

120590119911

120590119910119911

120590119909119911

120590119909119910

)

)

(119896)

=

[[[[[[[[[

[

1198621111986212

11986213

0 0 11986216

11986212

11986222

11986223

0 0 11986226

11986213

11986223

11986233

0 0 11986236

0 0 0 11986244

11986245

0

0 0 0 11986245

11986255

0

11986216

11986226

11986236

0 0 11986266

]]]]]]]]]

]

(119896)

times(

(

120576119909

120576119910

120576119911

120574119910119911

120574119909119911

120574119909119910

)

)

(119896)

(17)

where [119862](119896) is the transformed (ie off axis) stiffness matrixof the kth layer By using the strain-displacement relations

International Journal of Engineering Mathematics 5

(16) in the principle of the minimum total potential energy[25] the equilibrium equations within LWT are found as

120575119880119896 120575119878(119876119896

119909

minus

d119872119896119909119910

d119910) = 0 119896 = 1 2 119873 + 1 (18a)

120575119881119896 119876119896119910

minus

d119872119896119910

d119910= 0 119896 = 1 2 119873 + 1 (18b)

120575119882119896 119873119896119911

minus

d119877119896119910

d119910= 0 119896 = 1 2 119873 + 1 (18c)

1205751198611 120575119860int

119887

minus119887

(119876119909119910 minus119872

119909119910) d119910 = 0 (19a)

1205751198612 int119887

minus119887

119873119909d119910 = 0 (19b)

1205751198616 120575119862int

119887

minus119887

119872119909d119910 = 119872

0 (19c)

Also the traction-free boundary conditions at the free edgesof the laminate are given as

119872119896

119910

= 119877119896

119910

= 120575119878119872119896

119909119910

= 0 at 119910 = plusmn119887 (119896 = 1 2 119873 + 1)

(20)

Equations (18a) (18b) and (18c) present 3(119873+1) local equilib-rium equations associated with119873+ 1 surface in the laminatewithin LWT Also (19a) (19b) and (19c) represent the globalequilibrium equation of a laminate The generalized stressandmoment resultants in (18a) (18b) (18c) (19a) (19b) (19c)and (20) are defined as

(119872119896

119910

119872119896

119909119910

119873119896

119911

) = int

ℎ2

minusℎ2

(120590119910120601119896 120590119909119910120601119896 1205901199111206011015840

) d119911 (21a)

(119876119896

119909

119876119896

119910

119877119896

119910

) = int

ℎ2

minusℎ2

(1205901199091199111206011015840

119896

1205901199101199111206011015840

119896

120590119910119911120601119896) d119911 (21b)

(119872119909 119873119909119872119909119910 119876119909) = int

ℎ2

minusℎ2

(120590119909119911 120590119909 120590119909119910119911 120590119909119911) d119911 (21c)

Substitution (16) into (17) and subsequently into (21a) (21b)and (21c) generates the following relations

(119872119896

119910

119872119896

119909119910

119873119896

119911

)

= 120575119878(119863119896119895

26

119863119896119895

66

119861119896119895

36

)1198801015840

119895

+ (119863119896119895

22

119863119896119895

26

119861119895119896

23

)1198811015840

119895

+ (119861119896119895

23

119861119896119895

36

119860119896119895

33

)119882119895minus 120575119860(119863119896

26

119863119896

66

119861119896

36

) 1198611

+ 120575119862(119863119896

12

119863119896

16

119861119896

13

) 1198616+ (119861119896

12

119861119896

16

119860119896

13

) 1198612

(22a)

(119876119896

119909

119876119896

119910

119877119896

119910

)

= 120575119878(119860119896119895

55

119860119896119895

45

119861119896119895

45

)119880119895+ (119860119896119895

45

119860119896119895

44

119861119896119895

44

)119881119895

+ (119861119895119896

45

119861119895119896

44

119863119896119895

44

)1198821015840

119895

minus 120575119860(119860119896

55

119860119896

45

119861119896

45

) 1198611119910

(22b)

(119872119910119872119909119910 119873119909)

= 120575119878(119863119896

16

119863119896

66

119861119896

16

)1198801015840

119895

+ (119863119896

12

119863119896

26

119861119896

12

)1198811015840

119895

+ (119861119896

13

119861119896

36

119860119896

13

)119882119895minus 120575119860(11986316 11986366 11986116) 1198611

+ 120575119862(11986111 11986116 11986011) 1198616+ (11986311 11986316 11986111) 1198612

(22c)

119876119909= 120575119878119860119896

55

119880119895+ 119860119896

45

119881119895+ 119861119896

45

1198821015840

119895

+ 120575119860119860551198611119910 (22d)

The explicit expressions for the laminate rigidities appearingin (22a) (22b) (22c) and (22d) in terms of 119862

119894rsquos are presented

in the Appendix Direct substitution of (22a) (22b) (22c)and (22d) into (18a) (18b) and (18c) provides the localdisplacement equilibrium equations within LWTThe resultsare expressed as

120575119880119896 120575119878(119863119896119895

66

11988010158401015840

119895

minus 119860119896119895

55

119880119895+ 119863119896119895

26

11988110158401015840

119895

minus 119860119896119895

45

119881119895+ (119861119896119895

36

minus 119861119895119896

45

)1198821015840

119895

) = 120575119860119860119896

55

1198611119910

119896 = 1 2 119873 + 1

(23a)

120575119881119896 120575119878119863119896119895

26

11988010158401015840

119895

minus 120575119878119860119896119895

45

119880119895+ 119863119896119895

22

11988110158401015840

119895

minus 119860119896119895

44

119881119895+ (119861119896119895

23

minus 119861119895119896

44

)1198821015840

119895

= 120575119860119860119896

45

1198611119910

119896 = 1 2 119873 + 1

(23b)

120575119882119896 120575119878(119861119896119895

45

minus 119861119895119896

36

)1198801015840

119895

minus (119861119896119895

44

minus 119861119895119896

23

)1198811015840

119895

+ 119863119896119895

44

11988210158401015840

119895

minus 119860119896119895

33

119882119895

= minus120575119860(119861119896

45

+ 119861119896

36

) 1198611+ 119860119896

13

1198612+ 120575119862119861119896

13

1198616

119896 = 1 2 119873 + 1

(23c)

In the same way upon substitution of (16) into (17) andthe subsequent results into (19a) (19b) and (19c) the globalequations (19a) (19b) and (19c) are found in terms ofdisplacement functions as follows

1205751198611 120575119860int

119887

minus119887

((120575119878119860119896

55

119880119895+ 119860119896

45

119881119895+ 119861119896

45

1198821015840

119895

) 119910

minus (120575119878119863119896

66

1198801015840

119895

+ 119863119896

26

1198811015840

119895

+ 119861119896

36

119882119895)

+ 120575119860(1198605511986111199102

+ 11986366) 1198611minus 119861161198612

minus120575119862119863161198616) d119910 = 0

(24a)

6 International Journal of Engineering Mathematics

1205751198612 int119887

minus119887

(120575119878119861119896

16

1198801015840

119895

+ 119861119896

12

1198811015840

119895

+ 119860119896

13

119882119895

minus 120575119860119861161198611+ 119860111198612+ 120575119862119861111198616) d119910 = 0

(24b)

1205751198616 120575119862int

119887

minus119887

(120575119878119863119896

16

1198801015840

119895

+ 119863119896

12

1198811015840

119895

+ 119861119896

13

119882119895

minus 120575119860119863161198611+ 119861111198612+ 120575119862119863111198616) d119910 = 119872

0

(24c)

where the extra laminate rigidities appearing in (24a) (24b)and (24c) are defined in the Appendix The system ofequations in (23a) (23b) and (23c) shows 3(119873 + 1) coupledordinary differential equations with constant coefficientswhich may be indicated in a matrix form as

[119872] 12057810158401015840

+ [119870] 120578 = [119871] 119861 sdot 119910 (25)

where

120578 = 119880119879

119881119879

119882119879

119879

119880 = 1198801 1198802 119880

119873+1119879

119881 = 1198811 1198812 119881

119873+1119879

119882 = 11988211198822 119882

119873+1119879

(26)

119861 = 1198611 1198612 1198616119879

119882119895= int

119910

119882119895d119910

(27)

The coefficient matrices [119872] [119870] and [119871] in (25) are dis-played in the Appendix Also from the conditions in (3) andthe displacement field in (23a) (23b) and (23c) it is observedthat the functions 119880

119895 119881119895 and119882

119895are all odd functions of the

independent variable 119910 It can be confirmed that the generalsolution of (25) may be presented as

120578 = [120595] [sinh (120582119910)] 119867 + [119870]minus1

[119871] 119861 sdot 119910 (28)

where [sinh(120582119910)] is a 3(119873 + 1) times 3(119873 + 1) diagonal matrixand

[sinh (120582119910)]

= diag (sinh (1205821119910) sinh (120582

2119910) sinh (120582

3(119873+1)119910))

(29)

Also [120595] and (1205822

1

1205822

2

1205822

3(119873+1)

) are the model matrixand eigenvalues of (minus[119872]

minus1

[119870]) respectively 119870 is anunknown vector representing 3(119873+1) integration constantsThe constants 119861

119895(119895 = 1 2 6)must be calculated within LWT

analysis So the boundary conditions in (20) are first imposedto found the vector 119870 in terms of the unknown parameters119861119895(119895 = 1 2 6) These constants are next calculated in terms

of the specific bending moment 1198720by satisfaction of the

global equilibrium conditions in (19a) (19b) and (19c)

0 01 02 03 04 05 06 07 08 09 1

0

10

20

30

minus40

minus30

minus20

minus10

120590

120590z120590xz

Goodsell and Pipes (2013)Goodsell and Pipes (2013)

yb

mdashPresent solutionmdashPresent solution

Figure 2 Interlaminar stresses along the 45∘

minus45∘ interface of

[45∘

minus45∘

]119904

laminate

Table 1 Engineering properties of unidirectional graphiteopoxy[28]

1198641

(GPa)1198642

= 1198643

(GPa)11986612

= 11986613

(GPa)11986623

(GPa) 12059212

= 12059213

12059223

132 108 565 338 024 059

4 Numerical Results and Discussions

In this part several numerical examples are shown for thedistribution of interlaminar stress All physical layers areassumed to be of equal thickness (=05mm)Themechanicalproperties of unidirectional graphiteepoxy T3005208 usedin this study are shown in Table 1 [28]

Also each physical ply is modeled as being made up of 12numerical layers within LWT (ie 119901 = 12) In addition thewidth thickness ratio (ie 2119887ℎ) is assumed to be equal to 10Furthermore the stress components are normalized as

120590119894119895=120590119894119895

1205900

(30)

where 1205900= 1198720119887ℎ2

Figure 2 shows the distribution of interlaminar normaland shear stresses along the width of [45∘minus45∘]

119904laminate

Good correspondence was found between the present resultsand the results obtained by Goodsell and Pipes [29] whichconfirms the accuracy of the present solution Figure 3 showsthe distribution of interlaminar stresses 120590

119911and 120590

119910119911along the

minus45∘

0∘ and 120590

119909119911along the 0

∘

90∘ interfaces of the general

[45∘

minus 45∘

0∘

90∘

30∘

minus 30∘

] laminate It is seen that theinterlaminar stresses show high stress gradient near the freeedge It is observed that the interlaminar stresses 120590

119911and 120590

119909119911

grow quickly near the free edge while being zero in the innerregion of the laminate Also the magnitude of the transverseshear stress 120590

119909119911is greater than that of transverse normal stress

120590119911 On the other hand 120590

119910119911rises toward the free edge and

decreases suddenly to zero at free edgeThe interlaminar normal stress 120590

119911along the 90∘0∘ inter-

face of the unsymmetric cross-ply [90∘

0∘

90∘

0∘

90∘

0∘

]

International Journal of Engineering Mathematics 7

0

0

01 02

02

03 04

04

05 06

06

07 08 09 1

minus02

minus04

minus06

minus08

120590

120590xz at z = 0

120590z at z = h6120590yz at z = h6

yb

Figure 3 Distribution of interlaminar stresses 120590119911

and 120590119910119911

alongthe minus45∘0∘ and 120590

119909119911

along the 0∘90∘ interfaces of the [45∘ minus 45∘0∘

90∘

30∘

minus30∘

] laminate

0 02 04 06 08 1yb

minus4

0

minus8

minus12

minus16

120590z

2bh = 502bh = 20

2bh = 102bh = 5

Figure 4 Distribution of the interlaminar normal stress 120590119911

alongthe 90∘0∘ interface of the laminate for various width-to-thicknessratios

laminate for various width-to-thickness ratios is demon-strated in Figure 4 It is observed that by decreasingthe width-to-thickness ratio the boundary-layer region canbe expanded towards the internal region of the laminate withits width being almost equal to the thickness of the laminateIt is seen that the magnitude of the interlaminar stress at thefree edge is not varied as the width-to-thickness ratio of thelaminate is changed In addition the highly localized natureof interlaminar stresses near and exactly at the free edges ofthe laminate is apparent

The distribution of interlaminar stresses along the0∘

90∘ and 90

∘

0∘ interfaces of the unsymmetric cross-ply

[0∘90∘0∘90∘] and [90∘0∘90∘0∘] laminates respectivelyare compared in Figure 5 It is seen that the maximuminterfacial value 120590

119911at the 0

∘

90∘ and 90

∘

0∘ interfaces take

place at the free edge Figure 6 shows the variation ofinterlaminar normal stress 120590

119911on through-the-thickness and

near the free edge of the angle-ply [45∘minus 45∘45∘minus 45∘]laminate as the free edge is approached It is observed that

120590z at z =h4120590z at z

120590yz

120590yz at z120590yz at z = h4

minus14minus12minus1

minus08minus06minus04minus02

0020406

0 02 04 06 08 1

= h4= h4

in [0∘90∘0∘90∘]

in [0∘90∘0∘90∘]

in [90∘0∘90∘0∘]in [90∘0∘90∘0∘]

yb

Figure 5 Distribution of interlaminar stresses along the 0∘

90∘

and 90∘

0∘ interfaces of the [0∘90∘0∘90∘] and [90∘0∘90∘0∘]

laminates respectively

minus1

minus05

0

05

1

minus200 minus150 minus100 minus50 0 50 100 150 200

zh

y = 094by = 096b

y = 099by = b

120590z

Figure 6 Variations of interlaminar normal stress 120590119911

through thethickness of the [45∘minus45∘45∘minus45∘] laminate for various width-to-thickness ratios

the maximum negative and positive values of 120590119911occur within

the top minus80∘ layer and the bottom 80

∘ layer at the free edge(ie 119910 = 119887) respectively It is also noticed that 120590

119911diminishes

away from the free edge as the interior region of the laminateis approached Moreover 120590

119911reduces by moving slightly away

from the free edge and its distribution becomes smootherThe variations of the interlaminar shear stress 120590

119910119911at 119910 =

119887 through the thickness of the [0∘

60∘

minus60∘

]119904laminate are

displayed in Figure 7 By increasing the number of numericallayers in each lamina 120590

119910119911becomes slightly closer to zero but

8 International Journal of Engineering Mathematics

minus15

minus1

minus05

0

05

1

15

minus20 minus10 0 10 20 30

zh

p = 20

p = 12

p = 8

120590yz

Figure 7 Variations of interlaminar shear stress 120590119910119911

on through-thickness of the [0∘60∘minus60∘]119904

laminate as a function of layer subdivisionnumber 119901

012

01

008

006

004

002

0

minus002

120590z

p = 30p = 20

p = 12p = 8

0 02 04 06 08 1

yb

Figure 8 Variations of interlaminar normal stress 120590119911

on through-thickness of the [minus15∘

90∘

90∘

minus15∘

] laminate as a function of layersubdivision number 119901

it never becomes zero It should be noted that increasing thenumber of subdivisions results in no convergence for 120590

119911and

120590119909119911

at interface-edge junction of two dissimilar layers Thenumerical values of these components (120590

119911and 120590

119909119911) continue

to grow as the number of sublayers increased Since thegeneralized stress resultants 119877119896

119910

instead of 120590119910119911

are forced todisappear at the free edge in LWT the numerical value of120590119910119911

may never become zero at the interface-edge intersection

(even by increasing the number of sublayers in each physicallayer)

The distribution of interlaminar normal stress 120590119911on

through thickness of the general [minus15∘90∘90∘minus15∘] lam-inate is displayed in Figure 8 It is noted that increasingthe number of numerical layers (119901) in each lamina has nosignificant effect on the values of the interlaminar stress 120590

119911

within the boundary-layer region of the laminate expect

International Journal of Engineering Mathematics 9

exactly at the free edge (ie 119910 = 119887) Furthermore theinterlaminar normal stress 120590

119911grows rapidly in the vicinity of

the free edge while being zero in the interior region of thelaminate

5 Conclusions

In the present paper analytical solutions are developed toanalyze the interlaminar stresses for different lay-up con-figurations in laminated composite plates subjected to thebending moment The solutions were obtained based onthe reduced elasticity displacement field for long laminatedplates It is found that the components of the displacementsfield are composed of two distinct parts signifying theglobal and local deformations within laminates The localdisplacement functions as well as the interlaminar stressesthrough the layers of the laminate were found by Reddyrsquoslayerwise theory (LWT)Thenumerical results based onLWTwere shown for the free-edge interlaminar stresses throughthe thickness and across the interfaces of various cross-plysymmetric angle-ply and general composite laminates

Appendix

The laminate rigidities in (22a) (22b) (22c) (22d) (24a)(24b) and (24c) are defined as

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

1206011015840

119895

1206011198961206011015840

119895

120601119896120601119895) d119911

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

120601119896 1206011015840

119896

119911 120601119896119911) d119911

(119896 119895 = 1 2 119873 + 1)

(119860119901119902 119861119901119902 119863119901119902) =

119873

sum

119894=1

int

119911119894+1

119911119894

119862(119894)

119901119902

(1 119911 1199112

) d119911

(A1)

which upon integration are presented in the following form

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

)

=

(minus

119862(119896minus1)

119901119902

ℎ119896minus1

minus

119862(119896minus1)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

6) if 119895 = 119896 minus 1

(

119862(119896minus1)

119901119902

ℎ119896minus1

+

119862(119896)

119901119902

ℎ119896

119862(119896minus1)

119901119902

2minus

119862(119896)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

3+

ℎ119896119862(119896)

119901119902

3) if 119895 = 119896

(minus

119862(119896)

119901119902

ℎ119896

119862(119896)

119901119902

2

ℎ119896119862(119896)

119901119902

6) if 119895 = 119896 + 1

(0 0 0) if 119895 lt 119896 minus 1 or 119895 gt 119896 + 1

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

)

=

(minus119862(1)

119901119902

ℎ1119862(1)

119901119902

2 119862(1)

119901119902

1199112

1

minus 1199112

2

2ℎ1

119862(1)

119901119902

ℎ1

[1199113

1

minus 1199113

2

3minus 1199112

1199112

1

minus 1199112

2

2]) if 119896 = 1

(minus119862(119896minus1)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2]) if 119896 = 119873 + 1

(119862(119896minus1)

119901119902

minus 119862(119896)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2+

ℎ119896119862(119896)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

+ 119862(119896)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2] +

119862(119896)

119901119902

ℎ119896

[1199113

119896

minus 1199113

119896+1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896+1

2]) if 1 lt 119896 lt 119873 + 1

(A2)

10 International Journal of Engineering Mathematics

Also

(119860119901119902 119861119901119902 119863119901119902)

=

119873

sum

119894=1

119862(119894)

119901119902

([119911119894+1

minus 119911119894] [

1199112

119894+1

minus 1199112

119894

2] [

1199113

119894+1

minus 1199113

119894

3])

(A3)

The coefficient matrices [119872] [119870] and [119871] appearing in(25) are given as

[119872] =[[

[

120575119878[11986366] 120575119878[11986326] 120575119878([11986136] minus [119861

45]119879

)

120575119878[11986326] [119863

22] [119861

23] minus [119861

44]119879

[0] [0] [11986344]

]]

]

[119870]=[

[

minus120575119878([11986055] + [120572]) minus120575

119878[11986045] [0]

minus120575119878[11986045] minus ([119860

44] + [120572]) [0]

120575119878([11986145]minus[11986136]119879

) [11986144]minus[11986123]119879

minus ([11986033]+[120572])

]

]

[119871] = [

[

12057511986011986055 0 0

12057511986011986045 0 0

minus120575119860(11986145 + 119861

36) 119860

13 12057511986211986113

]

]

(A4)

where [119860119901119902] [119861119901119902] and [119863

119901119902] are (119873 + 1) times (119873 + 1) square

matrices containing 119860119896119895119901119902 119861119896119895119901119902 and 119863

119896119895

119901119902 respectively and the

vectors 119860119901119902 119861119901119902 and 119861

119901119902 are (119873+1)times1 columnmatrices

containing 119860119896119901119902

119861119896119901119902

and 119861119896

119901119902

respectively Also [0] is (119873 +

1) times (119873 + 1) square zero and 0 is a zero vector with 119873 + 1

rows The artificial matrix [120572] is also an (119873 + 1) times (119873 + 1)

square matrix whose elements are given by

120572119896119895

= 120572int

ℎ2

minusℎ2

120601119896120601119895d119911 (A5)

with 120572 being a relatively small parameter in comparison withthe rigidity constants 119860119896119895

119901119902(119901119902 = 33 44 55) It is noted that

the insertion of [120572] in the matrix [119870] makes the eigenvaluesof the matrix (minus[119872]

minus1

[119870]) distinct

References

[1] T Kant and K Swaminathan ldquoEstimation of transverseinter-laminar stresses in laminated compositesmdasha selective reviewand survey of current developmentsrdquo Composite Structures vol49 no 1 pp 65ndash75 2000

[2] A H Puppo and H A Evensen ldquoInterlaminar shear ınlamınated composıtes under generalızed plane stressrdquo Journalof Composite Materials vol 4 pp 204ndash220 1970

[3] N J Pagano ldquoOn the calculatıon of ınterlamınar normal stressın composıte lamınaterdquo Journal of Composite Materials vol 8pp 65ndash81 1974

[4] P W Hsu and C T Herakovich ldquoEdge effects in angle-plycomposite laminatesrdquo Journal of CompositeMaterials vol 11 pp422ndash428 1977

[5] S Tang and A Levy ldquoBoundary layer theory 2 extension oflaminated finite striprdquo Journal of CompositeMaterials vol 9 no1 pp 42ndash52 1975

[6] R B Pipes andN J Pagano ldquoInterlaminar stresses in compositelaminates an approximate elasticity solutionrdquo Journal of AppliedMechanics vol 41 no 3 pp 668ndash672 1974

[7] N J Pagano ldquoStress fields in composite laminatesrdquo InternationalJournal of Solids and Structures vol 14 no 5 pp 385ndash400 1978

[8] S S Wang and I Choi ldquoBoundary-layer effects in composıtelaminates 1 free-edge stress singulaiıtiesrdquo Journal of AppliedMechanics vol V 49 no 3 pp 541ndash548 1982

[9] W L Yin ldquoFree-edge effects in anisotropic laminates underextension bending and twisting part I a stress-function-basedvariational approachrdquo Journal of Applied Mechanics vol 61 no2 pp 410ndash415 1994

[10] A S D Wang and F W Crossman ldquoSome new results on edgeeffect in symmetric composite laminatesrdquo Journal of CompositeMaterials vol 11 no 1 pp 92ndash106 1977

[11] J D Whitcomb I S Raju and J G Goree ldquoReliability ofthe finite element method for calculating free edge stresses incomposite laminatesrdquo Computers and Structures vol 15 no 1pp 23ndash37 1982

[12] G Davı ldquoStress fields in general composite laminatesrdquo AIAAJournal vol 34 no 12 pp 2604ndash2608 1996

[13] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT Part 1 derivationof finite element matricesrdquo International Journal for NumericalMethods in Engineering vol 55 no 2 pp 191ndash231 2002

[14] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT part 2 numericalimplementationsrdquo International Journal for Numerical Methodsin Engineering vol 55 no 3 pp 253ndash291 2002

[15] V-T Nguyen and J-F Caron ldquoFinite element analysis of free-edge stresses in composite laminates under mechanical anthermal loadingrdquo Composites Science and Technology vol 69no 1 pp 40ndash49 2009

[16] D H Robbins Jr and J N Reddy ldquoModelling of thick compos-ites using a layerwise laminate theoryrdquo International Journal forNumerical Methods in Engineering vol 36 no 4 pp 655ndash6771993

[17] C Mittelstedt and W Becker ldquoReddyrsquos layerwise laminate platetheory for the computation of elastic fields in the vicinity ofstraight free laminate edgesrdquoMaterials Science and EngineeringA vol 498 no 1-2 pp 76ndash80 2008

[18] W J Na Damage analysis of laminated composite beams underbending loads using the layer-wise theory [Dissertation thesis]Texas AampM University Texas Tex USA 2008

[19] T S Plagianakos and D A Saravanos ldquoHigher-order layerwiselaminate theory for the prediction of interlaminar shear stressesin thick composite and sandwich composite platesrdquo CompositeStructures vol 87 no 1 pp 23ndash35 2009

[20] H Ullah A R Harland T Lucas D Price and V V Sil-berschmidt ldquoFinite-element modelling of bending of CFRPlaminates multiple delaminationsrdquo Computational MaterialsScience vol 52 no 1 pp 147ndash156 2012

[21] F Helenon M R Wisnom S R Hallett and R S TraskldquoInvestigation into failure of laminated composite T-piecespecimens under bending loadingrdquo Composites A vol 54 pp182ndash189 2013

[22] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013

International Journal of Engineering Mathematics 11

[23] N D Thai M DrsquoOttavio and J-F Caron ldquoBending analysis oflaminated and sandwich plates using a layer-wise stress modelrdquoComposite Structures vol 96 pp 135ndash142 2013

[24] PMalekzadeh ldquoA two-dimensional layerwise-differential quad-rature static analysis of thick laminated composite circulararchesrdquoAppliedMathematicalModelling vol 33 no 4 pp 1850ndash1861 2009

[25] Y C Fung and P Tong Classical and Computational SolidMechanics World Scientific New Jersey NJ USA 2001

[26] S G LekhnitskiiTheory of Elasticity of anAnisotropic BodyMirPublishers Moscow Russia 1981

[27] A Nosier R K Kapania and J N Reddy ldquoFree vibrationanalysis of laminated plates using a layerwise theoryrdquo AIAAJournal vol 31 no 12 pp 2335ndash2346 1993

[28] C T HerakovichMechanics of Fibrous Composites Wiley NewYork NY USA 1998

[29] O Goodsell and R B Pipes ldquoInterlaminar stresses in compositelaminates subjected to anticlastic bending deformationrdquo Journalof Applied Mechanics vol 80 no 4 pp 1ndash7 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Investigation of Through-Thickness ...downloads.hindawi.com/archive/2013/676743.pdf · (,, ) is located at the middle plane of the laminate, that is of thickness

International Journal of Engineering Mathematics 5

(16) in the principle of the minimum total potential energy[25] the equilibrium equations within LWT are found as

120575119880119896 120575119878(119876119896

119909

minus

d119872119896119909119910

d119910) = 0 119896 = 1 2 119873 + 1 (18a)

120575119881119896 119876119896119910

minus

d119872119896119910

d119910= 0 119896 = 1 2 119873 + 1 (18b)

120575119882119896 119873119896119911

minus

d119877119896119910

d119910= 0 119896 = 1 2 119873 + 1 (18c)

1205751198611 120575119860int

119887

minus119887

(119876119909119910 minus119872

119909119910) d119910 = 0 (19a)

1205751198612 int119887

minus119887

119873119909d119910 = 0 (19b)

1205751198616 120575119862int

119887

minus119887

119872119909d119910 = 119872

0 (19c)

Also the traction-free boundary conditions at the free edgesof the laminate are given as

119872119896

119910

= 119877119896

119910

= 120575119878119872119896

119909119910

= 0 at 119910 = plusmn119887 (119896 = 1 2 119873 + 1)

(20)

Equations (18a) (18b) and (18c) present 3(119873+1) local equilib-rium equations associated with119873+ 1 surface in the laminatewithin LWT Also (19a) (19b) and (19c) represent the globalequilibrium equation of a laminate The generalized stressandmoment resultants in (18a) (18b) (18c) (19a) (19b) (19c)and (20) are defined as

(119872119896

119910

119872119896

119909119910

119873119896

119911

) = int

ℎ2

minusℎ2

(120590119910120601119896 120590119909119910120601119896 1205901199111206011015840

) d119911 (21a)

(119876119896

119909

119876119896

119910

119877119896

119910

) = int

ℎ2

minusℎ2

(1205901199091199111206011015840

119896

1205901199101199111206011015840

119896

120590119910119911120601119896) d119911 (21b)

(119872119909 119873119909119872119909119910 119876119909) = int

ℎ2

minusℎ2

(120590119909119911 120590119909 120590119909119910119911 120590119909119911) d119911 (21c)

Substitution (16) into (17) and subsequently into (21a) (21b)and (21c) generates the following relations

(119872119896

119910

119872119896

119909119910

119873119896

119911

)

= 120575119878(119863119896119895

26

119863119896119895

66

119861119896119895

36

)1198801015840

119895

+ (119863119896119895

22

119863119896119895

26

119861119895119896

23

)1198811015840

119895

+ (119861119896119895

23

119861119896119895

36

119860119896119895

33

)119882119895minus 120575119860(119863119896

26

119863119896

66

119861119896

36

) 1198611

+ 120575119862(119863119896

12

119863119896

16

119861119896

13

) 1198616+ (119861119896

12

119861119896

16

119860119896

13

) 1198612

(22a)

(119876119896

119909

119876119896

119910

119877119896

119910

)

= 120575119878(119860119896119895

55

119860119896119895

45

119861119896119895

45

)119880119895+ (119860119896119895

45

119860119896119895

44

119861119896119895

44

)119881119895

+ (119861119895119896

45

119861119895119896

44

119863119896119895

44

)1198821015840

119895

minus 120575119860(119860119896

55

119860119896

45

119861119896

45

) 1198611119910

(22b)

(119872119910119872119909119910 119873119909)

= 120575119878(119863119896

16

119863119896

66

119861119896

16

)1198801015840

119895

+ (119863119896

12

119863119896

26

119861119896

12

)1198811015840

119895

+ (119861119896

13

119861119896

36

119860119896

13

)119882119895minus 120575119860(11986316 11986366 11986116) 1198611

+ 120575119862(11986111 11986116 11986011) 1198616+ (11986311 11986316 11986111) 1198612

(22c)

119876119909= 120575119878119860119896

55

119880119895+ 119860119896

45

119881119895+ 119861119896

45

1198821015840

119895

+ 120575119860119860551198611119910 (22d)

The explicit expressions for the laminate rigidities appearingin (22a) (22b) (22c) and (22d) in terms of 119862

119894rsquos are presented

in the Appendix Direct substitution of (22a) (22b) (22c)and (22d) into (18a) (18b) and (18c) provides the localdisplacement equilibrium equations within LWTThe resultsare expressed as

120575119880119896 120575119878(119863119896119895

66

11988010158401015840

119895

minus 119860119896119895

55

119880119895+ 119863119896119895

26

11988110158401015840

119895

minus 119860119896119895

45

119881119895+ (119861119896119895

36

minus 119861119895119896

45

)1198821015840

119895

) = 120575119860119860119896

55

1198611119910

119896 = 1 2 119873 + 1

(23a)

120575119881119896 120575119878119863119896119895

26

11988010158401015840

119895

minus 120575119878119860119896119895

45

119880119895+ 119863119896119895

22

11988110158401015840

119895

minus 119860119896119895

44

119881119895+ (119861119896119895

23

minus 119861119895119896

44

)1198821015840

119895

= 120575119860119860119896

45

1198611119910

119896 = 1 2 119873 + 1

(23b)

120575119882119896 120575119878(119861119896119895

45

minus 119861119895119896

36

)1198801015840

119895

minus (119861119896119895

44

minus 119861119895119896

23

)1198811015840

119895

+ 119863119896119895

44

11988210158401015840

119895

minus 119860119896119895

33

119882119895

= minus120575119860(119861119896

45

+ 119861119896

36

) 1198611+ 119860119896

13

1198612+ 120575119862119861119896

13

1198616

119896 = 1 2 119873 + 1

(23c)

In the same way upon substitution of (16) into (17) andthe subsequent results into (19a) (19b) and (19c) the globalequations (19a) (19b) and (19c) are found in terms ofdisplacement functions as follows

1205751198611 120575119860int

119887

minus119887

((120575119878119860119896

55

119880119895+ 119860119896

45

119881119895+ 119861119896

45

1198821015840

119895

) 119910

minus (120575119878119863119896

66

1198801015840

119895

+ 119863119896

26

1198811015840

119895

+ 119861119896

36

119882119895)

+ 120575119860(1198605511986111199102

+ 11986366) 1198611minus 119861161198612

minus120575119862119863161198616) d119910 = 0

(24a)

6 International Journal of Engineering Mathematics

1205751198612 int119887

minus119887

(120575119878119861119896

16

1198801015840

119895

+ 119861119896

12

1198811015840

119895

+ 119860119896

13

119882119895

minus 120575119860119861161198611+ 119860111198612+ 120575119862119861111198616) d119910 = 0

(24b)

1205751198616 120575119862int

119887

minus119887

(120575119878119863119896

16

1198801015840

119895

+ 119863119896

12

1198811015840

119895

+ 119861119896

13

119882119895

minus 120575119860119863161198611+ 119861111198612+ 120575119862119863111198616) d119910 = 119872

0

(24c)

where the extra laminate rigidities appearing in (24a) (24b)and (24c) are defined in the Appendix The system ofequations in (23a) (23b) and (23c) shows 3(119873 + 1) coupledordinary differential equations with constant coefficientswhich may be indicated in a matrix form as

[119872] 12057810158401015840

+ [119870] 120578 = [119871] 119861 sdot 119910 (25)

where

120578 = 119880119879

119881119879

119882119879

119879

119880 = 1198801 1198802 119880

119873+1119879

119881 = 1198811 1198812 119881

119873+1119879

119882 = 11988211198822 119882

119873+1119879

(26)

119861 = 1198611 1198612 1198616119879

119882119895= int

119910

119882119895d119910

(27)

The coefficient matrices [119872] [119870] and [119871] in (25) are dis-played in the Appendix Also from the conditions in (3) andthe displacement field in (23a) (23b) and (23c) it is observedthat the functions 119880

119895 119881119895 and119882

119895are all odd functions of the

independent variable 119910 It can be confirmed that the generalsolution of (25) may be presented as

120578 = [120595] [sinh (120582119910)] 119867 + [119870]minus1

[119871] 119861 sdot 119910 (28)

where [sinh(120582119910)] is a 3(119873 + 1) times 3(119873 + 1) diagonal matrixand

[sinh (120582119910)]

= diag (sinh (1205821119910) sinh (120582

2119910) sinh (120582

3(119873+1)119910))

(29)

Also [120595] and (1205822

1

1205822

2

1205822

3(119873+1)

) are the model matrixand eigenvalues of (minus[119872]

minus1

[119870]) respectively 119870 is anunknown vector representing 3(119873+1) integration constantsThe constants 119861

119895(119895 = 1 2 6)must be calculated within LWT

analysis So the boundary conditions in (20) are first imposedto found the vector 119870 in terms of the unknown parameters119861119895(119895 = 1 2 6) These constants are next calculated in terms

of the specific bending moment 1198720by satisfaction of the

global equilibrium conditions in (19a) (19b) and (19c)

0 01 02 03 04 05 06 07 08 09 1

0

10

20

30

minus40

minus30

minus20

minus10

120590

120590z120590xz

Goodsell and Pipes (2013)Goodsell and Pipes (2013)

yb

mdashPresent solutionmdashPresent solution

Figure 2 Interlaminar stresses along the 45∘

minus45∘ interface of

[45∘

minus45∘

]119904

laminate

Table 1 Engineering properties of unidirectional graphiteopoxy[28]

1198641

(GPa)1198642

= 1198643

(GPa)11986612

= 11986613

(GPa)11986623

(GPa) 12059212

= 12059213

12059223

132 108 565 338 024 059

4 Numerical Results and Discussions

In this part several numerical examples are shown for thedistribution of interlaminar stress All physical layers areassumed to be of equal thickness (=05mm)Themechanicalproperties of unidirectional graphiteepoxy T3005208 usedin this study are shown in Table 1 [28]

Also each physical ply is modeled as being made up of 12numerical layers within LWT (ie 119901 = 12) In addition thewidth thickness ratio (ie 2119887ℎ) is assumed to be equal to 10Furthermore the stress components are normalized as

120590119894119895=120590119894119895

1205900

(30)

where 1205900= 1198720119887ℎ2

Figure 2 shows the distribution of interlaminar normaland shear stresses along the width of [45∘minus45∘]

119904laminate

Good correspondence was found between the present resultsand the results obtained by Goodsell and Pipes [29] whichconfirms the accuracy of the present solution Figure 3 showsthe distribution of interlaminar stresses 120590

119911and 120590

119910119911along the

minus45∘

0∘ and 120590

119909119911along the 0

∘

90∘ interfaces of the general

[45∘

minus 45∘

0∘

90∘

30∘

minus 30∘

] laminate It is seen that theinterlaminar stresses show high stress gradient near the freeedge It is observed that the interlaminar stresses 120590

119911and 120590

119909119911

grow quickly near the free edge while being zero in the innerregion of the laminate Also the magnitude of the transverseshear stress 120590

119909119911is greater than that of transverse normal stress

120590119911 On the other hand 120590

119910119911rises toward the free edge and

decreases suddenly to zero at free edgeThe interlaminar normal stress 120590

119911along the 90∘0∘ inter-

face of the unsymmetric cross-ply [90∘

0∘

90∘

0∘

90∘

0∘

]

International Journal of Engineering Mathematics 7

0

0

01 02

02

03 04

04

05 06

06

07 08 09 1

minus02

minus04

minus06

minus08

120590

120590xz at z = 0

120590z at z = h6120590yz at z = h6

yb

Figure 3 Distribution of interlaminar stresses 120590119911

and 120590119910119911

alongthe minus45∘0∘ and 120590

119909119911

along the 0∘90∘ interfaces of the [45∘ minus 45∘0∘

90∘

30∘

minus30∘

] laminate

0 02 04 06 08 1yb

minus4

0

minus8

minus12

minus16

120590z

2bh = 502bh = 20

2bh = 102bh = 5

Figure 4 Distribution of the interlaminar normal stress 120590119911

alongthe 90∘0∘ interface of the laminate for various width-to-thicknessratios

laminate for various width-to-thickness ratios is demon-strated in Figure 4 It is observed that by decreasingthe width-to-thickness ratio the boundary-layer region canbe expanded towards the internal region of the laminate withits width being almost equal to the thickness of the laminateIt is seen that the magnitude of the interlaminar stress at thefree edge is not varied as the width-to-thickness ratio of thelaminate is changed In addition the highly localized natureof interlaminar stresses near and exactly at the free edges ofthe laminate is apparent

The distribution of interlaminar stresses along the0∘

90∘ and 90

∘

0∘ interfaces of the unsymmetric cross-ply

[0∘90∘0∘90∘] and [90∘0∘90∘0∘] laminates respectivelyare compared in Figure 5 It is seen that the maximuminterfacial value 120590

119911at the 0

∘

90∘ and 90

∘

0∘ interfaces take

place at the free edge Figure 6 shows the variation ofinterlaminar normal stress 120590

119911on through-the-thickness and

near the free edge of the angle-ply [45∘minus 45∘45∘minus 45∘]laminate as the free edge is approached It is observed that

120590z at z =h4120590z at z

120590yz

120590yz at z120590yz at z = h4

minus14minus12minus1

minus08minus06minus04minus02

0020406

0 02 04 06 08 1

= h4= h4

in [0∘90∘0∘90∘]

in [0∘90∘0∘90∘]

in [90∘0∘90∘0∘]in [90∘0∘90∘0∘]

yb

Figure 5 Distribution of interlaminar stresses along the 0∘

90∘

and 90∘

0∘ interfaces of the [0∘90∘0∘90∘] and [90∘0∘90∘0∘]

laminates respectively

minus1

minus05

0

05

1

minus200 minus150 minus100 minus50 0 50 100 150 200

zh

y = 094by = 096b

y = 099by = b

120590z

Figure 6 Variations of interlaminar normal stress 120590119911

through thethickness of the [45∘minus45∘45∘minus45∘] laminate for various width-to-thickness ratios

the maximum negative and positive values of 120590119911occur within

the top minus80∘ layer and the bottom 80

∘ layer at the free edge(ie 119910 = 119887) respectively It is also noticed that 120590

119911diminishes

away from the free edge as the interior region of the laminateis approached Moreover 120590

119911reduces by moving slightly away

from the free edge and its distribution becomes smootherThe variations of the interlaminar shear stress 120590

119910119911at 119910 =

119887 through the thickness of the [0∘

60∘

minus60∘

]119904laminate are

displayed in Figure 7 By increasing the number of numericallayers in each lamina 120590

119910119911becomes slightly closer to zero but

8 International Journal of Engineering Mathematics

minus15

minus1

minus05

0

05

1

15

minus20 minus10 0 10 20 30

zh

p = 20

p = 12

p = 8

120590yz

Figure 7 Variations of interlaminar shear stress 120590119910119911

on through-thickness of the [0∘60∘minus60∘]119904

laminate as a function of layer subdivisionnumber 119901

012

01

008

006

004

002

0

minus002

120590z

p = 30p = 20

p = 12p = 8

0 02 04 06 08 1

yb

Figure 8 Variations of interlaminar normal stress 120590119911

on through-thickness of the [minus15∘

90∘

90∘

minus15∘

] laminate as a function of layersubdivision number 119901

it never becomes zero It should be noted that increasing thenumber of subdivisions results in no convergence for 120590

119911and

120590119909119911

at interface-edge junction of two dissimilar layers Thenumerical values of these components (120590

119911and 120590

119909119911) continue

to grow as the number of sublayers increased Since thegeneralized stress resultants 119877119896

119910

instead of 120590119910119911

are forced todisappear at the free edge in LWT the numerical value of120590119910119911

may never become zero at the interface-edge intersection

(even by increasing the number of sublayers in each physicallayer)

The distribution of interlaminar normal stress 120590119911on

through thickness of the general [minus15∘90∘90∘minus15∘] lam-inate is displayed in Figure 8 It is noted that increasingthe number of numerical layers (119901) in each lamina has nosignificant effect on the values of the interlaminar stress 120590

119911

within the boundary-layer region of the laminate expect

International Journal of Engineering Mathematics 9

exactly at the free edge (ie 119910 = 119887) Furthermore theinterlaminar normal stress 120590

119911grows rapidly in the vicinity of

the free edge while being zero in the interior region of thelaminate

5 Conclusions

In the present paper analytical solutions are developed toanalyze the interlaminar stresses for different lay-up con-figurations in laminated composite plates subjected to thebending moment The solutions were obtained based onthe reduced elasticity displacement field for long laminatedplates It is found that the components of the displacementsfield are composed of two distinct parts signifying theglobal and local deformations within laminates The localdisplacement functions as well as the interlaminar stressesthrough the layers of the laminate were found by Reddyrsquoslayerwise theory (LWT)Thenumerical results based onLWTwere shown for the free-edge interlaminar stresses throughthe thickness and across the interfaces of various cross-plysymmetric angle-ply and general composite laminates

Appendix

The laminate rigidities in (22a) (22b) (22c) (22d) (24a)(24b) and (24c) are defined as

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

1206011015840

119895

1206011198961206011015840

119895

120601119896120601119895) d119911

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

120601119896 1206011015840

119896

119911 120601119896119911) d119911

(119896 119895 = 1 2 119873 + 1)

(119860119901119902 119861119901119902 119863119901119902) =

119873

sum

119894=1

int

119911119894+1

119911119894

119862(119894)

119901119902

(1 119911 1199112

) d119911

(A1)

which upon integration are presented in the following form

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

)

=

(minus

119862(119896minus1)

119901119902

ℎ119896minus1

minus

119862(119896minus1)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

6) if 119895 = 119896 minus 1

(

119862(119896minus1)

119901119902

ℎ119896minus1

+

119862(119896)

119901119902

ℎ119896

119862(119896minus1)

119901119902

2minus

119862(119896)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

3+

ℎ119896119862(119896)

119901119902

3) if 119895 = 119896

(minus

119862(119896)

119901119902

ℎ119896

119862(119896)

119901119902

2

ℎ119896119862(119896)

119901119902

6) if 119895 = 119896 + 1

(0 0 0) if 119895 lt 119896 minus 1 or 119895 gt 119896 + 1

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

)

=

(minus119862(1)

119901119902

ℎ1119862(1)

119901119902

2 119862(1)

119901119902

1199112

1

minus 1199112

2

2ℎ1

119862(1)

119901119902

ℎ1

[1199113

1

minus 1199113

2

3minus 1199112

1199112

1

minus 1199112

2

2]) if 119896 = 1

(minus119862(119896minus1)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2]) if 119896 = 119873 + 1

(119862(119896minus1)

119901119902

minus 119862(119896)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2+

ℎ119896119862(119896)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

+ 119862(119896)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2] +

119862(119896)

119901119902

ℎ119896

[1199113

119896

minus 1199113

119896+1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896+1

2]) if 1 lt 119896 lt 119873 + 1

(A2)

10 International Journal of Engineering Mathematics

Also

(119860119901119902 119861119901119902 119863119901119902)

=

119873

sum

119894=1

119862(119894)

119901119902

([119911119894+1

minus 119911119894] [

1199112

119894+1

minus 1199112

119894

2] [

1199113

119894+1

minus 1199113

119894

3])

(A3)

The coefficient matrices [119872] [119870] and [119871] appearing in(25) are given as

[119872] =[[

[

120575119878[11986366] 120575119878[11986326] 120575119878([11986136] minus [119861

45]119879

)

120575119878[11986326] [119863

22] [119861

23] minus [119861

44]119879

[0] [0] [11986344]

]]

]

[119870]=[

[

minus120575119878([11986055] + [120572]) minus120575

119878[11986045] [0]

minus120575119878[11986045] minus ([119860

44] + [120572]) [0]

120575119878([11986145]minus[11986136]119879

) [11986144]minus[11986123]119879

minus ([11986033]+[120572])

]

]

[119871] = [

[

12057511986011986055 0 0

12057511986011986045 0 0

minus120575119860(11986145 + 119861

36) 119860

13 12057511986211986113

]

]

(A4)

where [119860119901119902] [119861119901119902] and [119863

119901119902] are (119873 + 1) times (119873 + 1) square

matrices containing 119860119896119895119901119902 119861119896119895119901119902 and 119863

119896119895

119901119902 respectively and the

vectors 119860119901119902 119861119901119902 and 119861

119901119902 are (119873+1)times1 columnmatrices

containing 119860119896119901119902

119861119896119901119902

and 119861119896

119901119902

respectively Also [0] is (119873 +

1) times (119873 + 1) square zero and 0 is a zero vector with 119873 + 1

rows The artificial matrix [120572] is also an (119873 + 1) times (119873 + 1)

square matrix whose elements are given by

120572119896119895

= 120572int

ℎ2

minusℎ2

120601119896120601119895d119911 (A5)

with 120572 being a relatively small parameter in comparison withthe rigidity constants 119860119896119895

119901119902(119901119902 = 33 44 55) It is noted that

the insertion of [120572] in the matrix [119870] makes the eigenvaluesof the matrix (minus[119872]

minus1

[119870]) distinct

References

[1] T Kant and K Swaminathan ldquoEstimation of transverseinter-laminar stresses in laminated compositesmdasha selective reviewand survey of current developmentsrdquo Composite Structures vol49 no 1 pp 65ndash75 2000

[2] A H Puppo and H A Evensen ldquoInterlaminar shear ınlamınated composıtes under generalızed plane stressrdquo Journalof Composite Materials vol 4 pp 204ndash220 1970

[3] N J Pagano ldquoOn the calculatıon of ınterlamınar normal stressın composıte lamınaterdquo Journal of Composite Materials vol 8pp 65ndash81 1974

[4] P W Hsu and C T Herakovich ldquoEdge effects in angle-plycomposite laminatesrdquo Journal of CompositeMaterials vol 11 pp422ndash428 1977

[5] S Tang and A Levy ldquoBoundary layer theory 2 extension oflaminated finite striprdquo Journal of CompositeMaterials vol 9 no1 pp 42ndash52 1975

[6] R B Pipes andN J Pagano ldquoInterlaminar stresses in compositelaminates an approximate elasticity solutionrdquo Journal of AppliedMechanics vol 41 no 3 pp 668ndash672 1974

[7] N J Pagano ldquoStress fields in composite laminatesrdquo InternationalJournal of Solids and Structures vol 14 no 5 pp 385ndash400 1978

[8] S S Wang and I Choi ldquoBoundary-layer effects in composıtelaminates 1 free-edge stress singulaiıtiesrdquo Journal of AppliedMechanics vol V 49 no 3 pp 541ndash548 1982

[9] W L Yin ldquoFree-edge effects in anisotropic laminates underextension bending and twisting part I a stress-function-basedvariational approachrdquo Journal of Applied Mechanics vol 61 no2 pp 410ndash415 1994

[10] A S D Wang and F W Crossman ldquoSome new results on edgeeffect in symmetric composite laminatesrdquo Journal of CompositeMaterials vol 11 no 1 pp 92ndash106 1977

[11] J D Whitcomb I S Raju and J G Goree ldquoReliability ofthe finite element method for calculating free edge stresses incomposite laminatesrdquo Computers and Structures vol 15 no 1pp 23ndash37 1982

[12] G Davı ldquoStress fields in general composite laminatesrdquo AIAAJournal vol 34 no 12 pp 2604ndash2608 1996

[13] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT Part 1 derivationof finite element matricesrdquo International Journal for NumericalMethods in Engineering vol 55 no 2 pp 191ndash231 2002

[14] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT part 2 numericalimplementationsrdquo International Journal for Numerical Methodsin Engineering vol 55 no 3 pp 253ndash291 2002

[15] V-T Nguyen and J-F Caron ldquoFinite element analysis of free-edge stresses in composite laminates under mechanical anthermal loadingrdquo Composites Science and Technology vol 69no 1 pp 40ndash49 2009

[16] D H Robbins Jr and J N Reddy ldquoModelling of thick compos-ites using a layerwise laminate theoryrdquo International Journal forNumerical Methods in Engineering vol 36 no 4 pp 655ndash6771993

[17] C Mittelstedt and W Becker ldquoReddyrsquos layerwise laminate platetheory for the computation of elastic fields in the vicinity ofstraight free laminate edgesrdquoMaterials Science and EngineeringA vol 498 no 1-2 pp 76ndash80 2008

[18] W J Na Damage analysis of laminated composite beams underbending loads using the layer-wise theory [Dissertation thesis]Texas AampM University Texas Tex USA 2008

[19] T S Plagianakos and D A Saravanos ldquoHigher-order layerwiselaminate theory for the prediction of interlaminar shear stressesin thick composite and sandwich composite platesrdquo CompositeStructures vol 87 no 1 pp 23ndash35 2009

[20] H Ullah A R Harland T Lucas D Price and V V Sil-berschmidt ldquoFinite-element modelling of bending of CFRPlaminates multiple delaminationsrdquo Computational MaterialsScience vol 52 no 1 pp 147ndash156 2012

[21] F Helenon M R Wisnom S R Hallett and R S TraskldquoInvestigation into failure of laminated composite T-piecespecimens under bending loadingrdquo Composites A vol 54 pp182ndash189 2013

[22] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013

International Journal of Engineering Mathematics 11

[23] N D Thai M DrsquoOttavio and J-F Caron ldquoBending analysis oflaminated and sandwich plates using a layer-wise stress modelrdquoComposite Structures vol 96 pp 135ndash142 2013

[24] PMalekzadeh ldquoA two-dimensional layerwise-differential quad-rature static analysis of thick laminated composite circulararchesrdquoAppliedMathematicalModelling vol 33 no 4 pp 1850ndash1861 2009

[25] Y C Fung and P Tong Classical and Computational SolidMechanics World Scientific New Jersey NJ USA 2001

[26] S G LekhnitskiiTheory of Elasticity of anAnisotropic BodyMirPublishers Moscow Russia 1981

[27] A Nosier R K Kapania and J N Reddy ldquoFree vibrationanalysis of laminated plates using a layerwise theoryrdquo AIAAJournal vol 31 no 12 pp 2335ndash2346 1993

[28] C T HerakovichMechanics of Fibrous Composites Wiley NewYork NY USA 1998

[29] O Goodsell and R B Pipes ldquoInterlaminar stresses in compositelaminates subjected to anticlastic bending deformationrdquo Journalof Applied Mechanics vol 80 no 4 pp 1ndash7 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Investigation of Through-Thickness ...downloads.hindawi.com/archive/2013/676743.pdf · (,, ) is located at the middle plane of the laminate, that is of thickness

6 International Journal of Engineering Mathematics

1205751198612 int119887

minus119887

(120575119878119861119896

16

1198801015840

119895

+ 119861119896

12

1198811015840

119895

+ 119860119896

13

119882119895

minus 120575119860119861161198611+ 119860111198612+ 120575119862119861111198616) d119910 = 0

(24b)

1205751198616 120575119862int

119887

minus119887

(120575119878119863119896

16

1198801015840

119895

+ 119863119896

12

1198811015840

119895

+ 119861119896

13

119882119895

minus 120575119860119863161198611+ 119861111198612+ 120575119862119863111198616) d119910 = 119872

0

(24c)

where the extra laminate rigidities appearing in (24a) (24b)and (24c) are defined in the Appendix The system ofequations in (23a) (23b) and (23c) shows 3(119873 + 1) coupledordinary differential equations with constant coefficientswhich may be indicated in a matrix form as

[119872] 12057810158401015840

+ [119870] 120578 = [119871] 119861 sdot 119910 (25)

where

120578 = 119880119879

119881119879

119882119879

119879

119880 = 1198801 1198802 119880

119873+1119879

119881 = 1198811 1198812 119881

119873+1119879

119882 = 11988211198822 119882

119873+1119879

(26)

119861 = 1198611 1198612 1198616119879

119882119895= int

119910

119882119895d119910

(27)

The coefficient matrices [119872] [119870] and [119871] in (25) are dis-played in the Appendix Also from the conditions in (3) andthe displacement field in (23a) (23b) and (23c) it is observedthat the functions 119880

119895 119881119895 and119882

119895are all odd functions of the

independent variable 119910 It can be confirmed that the generalsolution of (25) may be presented as

120578 = [120595] [sinh (120582119910)] 119867 + [119870]minus1

[119871] 119861 sdot 119910 (28)

where [sinh(120582119910)] is a 3(119873 + 1) times 3(119873 + 1) diagonal matrixand

[sinh (120582119910)]

= diag (sinh (1205821119910) sinh (120582

2119910) sinh (120582

3(119873+1)119910))

(29)

Also [120595] and (1205822

1

1205822

2

1205822

3(119873+1)

) are the model matrixand eigenvalues of (minus[119872]

minus1

[119870]) respectively 119870 is anunknown vector representing 3(119873+1) integration constantsThe constants 119861

119895(119895 = 1 2 6)must be calculated within LWT

analysis So the boundary conditions in (20) are first imposedto found the vector 119870 in terms of the unknown parameters119861119895(119895 = 1 2 6) These constants are next calculated in terms

of the specific bending moment 1198720by satisfaction of the

global equilibrium conditions in (19a) (19b) and (19c)

0 01 02 03 04 05 06 07 08 09 1

0

10

20

30

minus40

minus30

minus20

minus10

120590

120590z120590xz

Goodsell and Pipes (2013)Goodsell and Pipes (2013)

yb

mdashPresent solutionmdashPresent solution

Figure 2 Interlaminar stresses along the 45∘

minus45∘ interface of

[45∘

minus45∘

]119904

laminate

Table 1 Engineering properties of unidirectional graphiteopoxy[28]

1198641

(GPa)1198642

= 1198643

(GPa)11986612

= 11986613

(GPa)11986623

(GPa) 12059212

= 12059213

12059223

132 108 565 338 024 059

4 Numerical Results and Discussions

In this part several numerical examples are shown for thedistribution of interlaminar stress All physical layers areassumed to be of equal thickness (=05mm)Themechanicalproperties of unidirectional graphiteepoxy T3005208 usedin this study are shown in Table 1 [28]

Also each physical ply is modeled as being made up of 12numerical layers within LWT (ie 119901 = 12) In addition thewidth thickness ratio (ie 2119887ℎ) is assumed to be equal to 10Furthermore the stress components are normalized as

120590119894119895=120590119894119895

1205900

(30)

where 1205900= 1198720119887ℎ2

Figure 2 shows the distribution of interlaminar normaland shear stresses along the width of [45∘minus45∘]

119904laminate

Good correspondence was found between the present resultsand the results obtained by Goodsell and Pipes [29] whichconfirms the accuracy of the present solution Figure 3 showsthe distribution of interlaminar stresses 120590

119911and 120590

119910119911along the

minus45∘

0∘ and 120590

119909119911along the 0

∘

90∘ interfaces of the general

[45∘

minus 45∘

0∘

90∘

30∘

minus 30∘

] laminate It is seen that theinterlaminar stresses show high stress gradient near the freeedge It is observed that the interlaminar stresses 120590

119911and 120590

119909119911

grow quickly near the free edge while being zero in the innerregion of the laminate Also the magnitude of the transverseshear stress 120590

119909119911is greater than that of transverse normal stress

120590119911 On the other hand 120590

119910119911rises toward the free edge and

decreases suddenly to zero at free edgeThe interlaminar normal stress 120590

119911along the 90∘0∘ inter-

face of the unsymmetric cross-ply [90∘

0∘

90∘

0∘

90∘

0∘

]

International Journal of Engineering Mathematics 7

0

0

01 02

02

03 04

04

05 06

06

07 08 09 1

minus02

minus04

minus06

minus08

120590

120590xz at z = 0

120590z at z = h6120590yz at z = h6

yb

Figure 3 Distribution of interlaminar stresses 120590119911

and 120590119910119911

alongthe minus45∘0∘ and 120590

119909119911

along the 0∘90∘ interfaces of the [45∘ minus 45∘0∘

90∘

30∘

minus30∘

] laminate

0 02 04 06 08 1yb

minus4

0

minus8

minus12

minus16

120590z

2bh = 502bh = 20

2bh = 102bh = 5

Figure 4 Distribution of the interlaminar normal stress 120590119911

alongthe 90∘0∘ interface of the laminate for various width-to-thicknessratios

laminate for various width-to-thickness ratios is demon-strated in Figure 4 It is observed that by decreasingthe width-to-thickness ratio the boundary-layer region canbe expanded towards the internal region of the laminate withits width being almost equal to the thickness of the laminateIt is seen that the magnitude of the interlaminar stress at thefree edge is not varied as the width-to-thickness ratio of thelaminate is changed In addition the highly localized natureof interlaminar stresses near and exactly at the free edges ofthe laminate is apparent

The distribution of interlaminar stresses along the0∘

90∘ and 90

∘

0∘ interfaces of the unsymmetric cross-ply

[0∘90∘0∘90∘] and [90∘0∘90∘0∘] laminates respectivelyare compared in Figure 5 It is seen that the maximuminterfacial value 120590

119911at the 0

∘

90∘ and 90

∘

0∘ interfaces take

place at the free edge Figure 6 shows the variation ofinterlaminar normal stress 120590

119911on through-the-thickness and

near the free edge of the angle-ply [45∘minus 45∘45∘minus 45∘]laminate as the free edge is approached It is observed that

120590z at z =h4120590z at z

120590yz

120590yz at z120590yz at z = h4

minus14minus12minus1

minus08minus06minus04minus02

0020406

0 02 04 06 08 1

= h4= h4

in [0∘90∘0∘90∘]

in [0∘90∘0∘90∘]

in [90∘0∘90∘0∘]in [90∘0∘90∘0∘]

yb

Figure 5 Distribution of interlaminar stresses along the 0∘

90∘

and 90∘

0∘ interfaces of the [0∘90∘0∘90∘] and [90∘0∘90∘0∘]

laminates respectively

minus1

minus05

0

05

1

minus200 minus150 minus100 minus50 0 50 100 150 200

zh

y = 094by = 096b

y = 099by = b

120590z

Figure 6 Variations of interlaminar normal stress 120590119911

through thethickness of the [45∘minus45∘45∘minus45∘] laminate for various width-to-thickness ratios

the maximum negative and positive values of 120590119911occur within

the top minus80∘ layer and the bottom 80

∘ layer at the free edge(ie 119910 = 119887) respectively It is also noticed that 120590

119911diminishes

away from the free edge as the interior region of the laminateis approached Moreover 120590

119911reduces by moving slightly away

from the free edge and its distribution becomes smootherThe variations of the interlaminar shear stress 120590

119910119911at 119910 =

119887 through the thickness of the [0∘

60∘

minus60∘

]119904laminate are

displayed in Figure 7 By increasing the number of numericallayers in each lamina 120590

119910119911becomes slightly closer to zero but

8 International Journal of Engineering Mathematics

minus15

minus1

minus05

0

05

1

15

minus20 minus10 0 10 20 30

zh

p = 20

p = 12

p = 8

120590yz

Figure 7 Variations of interlaminar shear stress 120590119910119911

on through-thickness of the [0∘60∘minus60∘]119904

laminate as a function of layer subdivisionnumber 119901

012

01

008

006

004

002

0

minus002

120590z

p = 30p = 20

p = 12p = 8

0 02 04 06 08 1

yb

Figure 8 Variations of interlaminar normal stress 120590119911

on through-thickness of the [minus15∘

90∘

90∘

minus15∘

] laminate as a function of layersubdivision number 119901

it never becomes zero It should be noted that increasing thenumber of subdivisions results in no convergence for 120590

119911and

120590119909119911

at interface-edge junction of two dissimilar layers Thenumerical values of these components (120590

119911and 120590

119909119911) continue

to grow as the number of sublayers increased Since thegeneralized stress resultants 119877119896

119910

instead of 120590119910119911

are forced todisappear at the free edge in LWT the numerical value of120590119910119911

may never become zero at the interface-edge intersection

(even by increasing the number of sublayers in each physicallayer)

The distribution of interlaminar normal stress 120590119911on

through thickness of the general [minus15∘90∘90∘minus15∘] lam-inate is displayed in Figure 8 It is noted that increasingthe number of numerical layers (119901) in each lamina has nosignificant effect on the values of the interlaminar stress 120590

119911

within the boundary-layer region of the laminate expect

International Journal of Engineering Mathematics 9

exactly at the free edge (ie 119910 = 119887) Furthermore theinterlaminar normal stress 120590

119911grows rapidly in the vicinity of

the free edge while being zero in the interior region of thelaminate

5 Conclusions

In the present paper analytical solutions are developed toanalyze the interlaminar stresses for different lay-up con-figurations in laminated composite plates subjected to thebending moment The solutions were obtained based onthe reduced elasticity displacement field for long laminatedplates It is found that the components of the displacementsfield are composed of two distinct parts signifying theglobal and local deformations within laminates The localdisplacement functions as well as the interlaminar stressesthrough the layers of the laminate were found by Reddyrsquoslayerwise theory (LWT)Thenumerical results based onLWTwere shown for the free-edge interlaminar stresses throughthe thickness and across the interfaces of various cross-plysymmetric angle-ply and general composite laminates

Appendix

The laminate rigidities in (22a) (22b) (22c) (22d) (24a)(24b) and (24c) are defined as

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

1206011015840

119895

1206011198961206011015840

119895

120601119896120601119895) d119911

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

120601119896 1206011015840

119896

119911 120601119896119911) d119911

(119896 119895 = 1 2 119873 + 1)

(119860119901119902 119861119901119902 119863119901119902) =

119873

sum

119894=1

int

119911119894+1

119911119894

119862(119894)

119901119902

(1 119911 1199112

) d119911

(A1)

which upon integration are presented in the following form

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

)

=

(minus

119862(119896minus1)

119901119902

ℎ119896minus1

minus

119862(119896minus1)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

6) if 119895 = 119896 minus 1

(

119862(119896minus1)

119901119902

ℎ119896minus1

+

119862(119896)

119901119902

ℎ119896

119862(119896minus1)

119901119902

2minus

119862(119896)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

3+

ℎ119896119862(119896)

119901119902

3) if 119895 = 119896

(minus

119862(119896)

119901119902

ℎ119896

119862(119896)

119901119902

2

ℎ119896119862(119896)

119901119902

6) if 119895 = 119896 + 1

(0 0 0) if 119895 lt 119896 minus 1 or 119895 gt 119896 + 1

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

)

=

(minus119862(1)

119901119902

ℎ1119862(1)

119901119902

2 119862(1)

119901119902

1199112

1

minus 1199112

2

2ℎ1

119862(1)

119901119902

ℎ1

[1199113

1

minus 1199113

2

3minus 1199112

1199112

1

minus 1199112

2

2]) if 119896 = 1

(minus119862(119896minus1)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2]) if 119896 = 119873 + 1

(119862(119896minus1)

119901119902

minus 119862(119896)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2+

ℎ119896119862(119896)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

+ 119862(119896)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2] +

119862(119896)

119901119902

ℎ119896

[1199113

119896

minus 1199113

119896+1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896+1

2]) if 1 lt 119896 lt 119873 + 1

(A2)

10 International Journal of Engineering Mathematics

Also

(119860119901119902 119861119901119902 119863119901119902)

=

119873

sum

119894=1

119862(119894)

119901119902

([119911119894+1

minus 119911119894] [

1199112

119894+1

minus 1199112

119894

2] [

1199113

119894+1

minus 1199113

119894

3])

(A3)

The coefficient matrices [119872] [119870] and [119871] appearing in(25) are given as

[119872] =[[

[

120575119878[11986366] 120575119878[11986326] 120575119878([11986136] minus [119861

45]119879

)

120575119878[11986326] [119863

22] [119861

23] minus [119861

44]119879

[0] [0] [11986344]

]]

]

[119870]=[

[

minus120575119878([11986055] + [120572]) minus120575

119878[11986045] [0]

minus120575119878[11986045] minus ([119860

44] + [120572]) [0]

120575119878([11986145]minus[11986136]119879

) [11986144]minus[11986123]119879

minus ([11986033]+[120572])

]

]

[119871] = [

[

12057511986011986055 0 0

12057511986011986045 0 0

minus120575119860(11986145 + 119861

36) 119860

13 12057511986211986113

]

]

(A4)

where [119860119901119902] [119861119901119902] and [119863

119901119902] are (119873 + 1) times (119873 + 1) square

matrices containing 119860119896119895119901119902 119861119896119895119901119902 and 119863

119896119895

119901119902 respectively and the

vectors 119860119901119902 119861119901119902 and 119861

119901119902 are (119873+1)times1 columnmatrices

containing 119860119896119901119902

119861119896119901119902

and 119861119896

119901119902

respectively Also [0] is (119873 +

1) times (119873 + 1) square zero and 0 is a zero vector with 119873 + 1

rows The artificial matrix [120572] is also an (119873 + 1) times (119873 + 1)

square matrix whose elements are given by

120572119896119895

= 120572int

ℎ2

minusℎ2

120601119896120601119895d119911 (A5)

with 120572 being a relatively small parameter in comparison withthe rigidity constants 119860119896119895

119901119902(119901119902 = 33 44 55) It is noted that

the insertion of [120572] in the matrix [119870] makes the eigenvaluesof the matrix (minus[119872]

minus1

[119870]) distinct

References

[1] T Kant and K Swaminathan ldquoEstimation of transverseinter-laminar stresses in laminated compositesmdasha selective reviewand survey of current developmentsrdquo Composite Structures vol49 no 1 pp 65ndash75 2000

[2] A H Puppo and H A Evensen ldquoInterlaminar shear ınlamınated composıtes under generalızed plane stressrdquo Journalof Composite Materials vol 4 pp 204ndash220 1970

[3] N J Pagano ldquoOn the calculatıon of ınterlamınar normal stressın composıte lamınaterdquo Journal of Composite Materials vol 8pp 65ndash81 1974

[4] P W Hsu and C T Herakovich ldquoEdge effects in angle-plycomposite laminatesrdquo Journal of CompositeMaterials vol 11 pp422ndash428 1977

[5] S Tang and A Levy ldquoBoundary layer theory 2 extension oflaminated finite striprdquo Journal of CompositeMaterials vol 9 no1 pp 42ndash52 1975

[6] R B Pipes andN J Pagano ldquoInterlaminar stresses in compositelaminates an approximate elasticity solutionrdquo Journal of AppliedMechanics vol 41 no 3 pp 668ndash672 1974

[7] N J Pagano ldquoStress fields in composite laminatesrdquo InternationalJournal of Solids and Structures vol 14 no 5 pp 385ndash400 1978

[8] S S Wang and I Choi ldquoBoundary-layer effects in composıtelaminates 1 free-edge stress singulaiıtiesrdquo Journal of AppliedMechanics vol V 49 no 3 pp 541ndash548 1982

[9] W L Yin ldquoFree-edge effects in anisotropic laminates underextension bending and twisting part I a stress-function-basedvariational approachrdquo Journal of Applied Mechanics vol 61 no2 pp 410ndash415 1994

[10] A S D Wang and F W Crossman ldquoSome new results on edgeeffect in symmetric composite laminatesrdquo Journal of CompositeMaterials vol 11 no 1 pp 92ndash106 1977

[11] J D Whitcomb I S Raju and J G Goree ldquoReliability ofthe finite element method for calculating free edge stresses incomposite laminatesrdquo Computers and Structures vol 15 no 1pp 23ndash37 1982

[12] G Davı ldquoStress fields in general composite laminatesrdquo AIAAJournal vol 34 no 12 pp 2604ndash2608 1996

[13] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT Part 1 derivationof finite element matricesrdquo International Journal for NumericalMethods in Engineering vol 55 no 2 pp 191ndash231 2002

[14] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT part 2 numericalimplementationsrdquo International Journal for Numerical Methodsin Engineering vol 55 no 3 pp 253ndash291 2002

[15] V-T Nguyen and J-F Caron ldquoFinite element analysis of free-edge stresses in composite laminates under mechanical anthermal loadingrdquo Composites Science and Technology vol 69no 1 pp 40ndash49 2009

[16] D H Robbins Jr and J N Reddy ldquoModelling of thick compos-ites using a layerwise laminate theoryrdquo International Journal forNumerical Methods in Engineering vol 36 no 4 pp 655ndash6771993

[17] C Mittelstedt and W Becker ldquoReddyrsquos layerwise laminate platetheory for the computation of elastic fields in the vicinity ofstraight free laminate edgesrdquoMaterials Science and EngineeringA vol 498 no 1-2 pp 76ndash80 2008

[18] W J Na Damage analysis of laminated composite beams underbending loads using the layer-wise theory [Dissertation thesis]Texas AampM University Texas Tex USA 2008

[19] T S Plagianakos and D A Saravanos ldquoHigher-order layerwiselaminate theory for the prediction of interlaminar shear stressesin thick composite and sandwich composite platesrdquo CompositeStructures vol 87 no 1 pp 23ndash35 2009

[20] H Ullah A R Harland T Lucas D Price and V V Sil-berschmidt ldquoFinite-element modelling of bending of CFRPlaminates multiple delaminationsrdquo Computational MaterialsScience vol 52 no 1 pp 147ndash156 2012

[21] F Helenon M R Wisnom S R Hallett and R S TraskldquoInvestigation into failure of laminated composite T-piecespecimens under bending loadingrdquo Composites A vol 54 pp182ndash189 2013

[22] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013

International Journal of Engineering Mathematics 11

[23] N D Thai M DrsquoOttavio and J-F Caron ldquoBending analysis oflaminated and sandwich plates using a layer-wise stress modelrdquoComposite Structures vol 96 pp 135ndash142 2013

[24] PMalekzadeh ldquoA two-dimensional layerwise-differential quad-rature static analysis of thick laminated composite circulararchesrdquoAppliedMathematicalModelling vol 33 no 4 pp 1850ndash1861 2009

[25] Y C Fung and P Tong Classical and Computational SolidMechanics World Scientific New Jersey NJ USA 2001

[26] S G LekhnitskiiTheory of Elasticity of anAnisotropic BodyMirPublishers Moscow Russia 1981

[27] A Nosier R K Kapania and J N Reddy ldquoFree vibrationanalysis of laminated plates using a layerwise theoryrdquo AIAAJournal vol 31 no 12 pp 2335ndash2346 1993

[28] C T HerakovichMechanics of Fibrous Composites Wiley NewYork NY USA 1998

[29] O Goodsell and R B Pipes ldquoInterlaminar stresses in compositelaminates subjected to anticlastic bending deformationrdquo Journalof Applied Mechanics vol 80 no 4 pp 1ndash7 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Investigation of Through-Thickness ...downloads.hindawi.com/archive/2013/676743.pdf · (,, ) is located at the middle plane of the laminate, that is of thickness

International Journal of Engineering Mathematics 7

0

0

01 02

02

03 04

04

05 06

06

07 08 09 1

minus02

minus04

minus06

minus08

120590

120590xz at z = 0

120590z at z = h6120590yz at z = h6

yb

Figure 3 Distribution of interlaminar stresses 120590119911

and 120590119910119911

alongthe minus45∘0∘ and 120590

119909119911

along the 0∘90∘ interfaces of the [45∘ minus 45∘0∘

90∘

30∘

minus30∘

] laminate

0 02 04 06 08 1yb

minus4

0

minus8

minus12

minus16

120590z

2bh = 502bh = 20

2bh = 102bh = 5

Figure 4 Distribution of the interlaminar normal stress 120590119911

alongthe 90∘0∘ interface of the laminate for various width-to-thicknessratios

laminate for various width-to-thickness ratios is demon-strated in Figure 4 It is observed that by decreasingthe width-to-thickness ratio the boundary-layer region canbe expanded towards the internal region of the laminate withits width being almost equal to the thickness of the laminateIt is seen that the magnitude of the interlaminar stress at thefree edge is not varied as the width-to-thickness ratio of thelaminate is changed In addition the highly localized natureof interlaminar stresses near and exactly at the free edges ofthe laminate is apparent

The distribution of interlaminar stresses along the0∘

90∘ and 90

∘

0∘ interfaces of the unsymmetric cross-ply

[0∘90∘0∘90∘] and [90∘0∘90∘0∘] laminates respectivelyare compared in Figure 5 It is seen that the maximuminterfacial value 120590

119911at the 0

∘

90∘ and 90

∘

0∘ interfaces take

place at the free edge Figure 6 shows the variation ofinterlaminar normal stress 120590

119911on through-the-thickness and

near the free edge of the angle-ply [45∘minus 45∘45∘minus 45∘]laminate as the free edge is approached It is observed that

120590z at z =h4120590z at z

120590yz

120590yz at z120590yz at z = h4

minus14minus12minus1

minus08minus06minus04minus02

0020406

0 02 04 06 08 1

= h4= h4

in [0∘90∘0∘90∘]

in [0∘90∘0∘90∘]

in [90∘0∘90∘0∘]in [90∘0∘90∘0∘]

yb

Figure 5 Distribution of interlaminar stresses along the 0∘

90∘

and 90∘

0∘ interfaces of the [0∘90∘0∘90∘] and [90∘0∘90∘0∘]

laminates respectively

minus1

minus05

0

05

1

minus200 minus150 minus100 minus50 0 50 100 150 200

zh

y = 094by = 096b

y = 099by = b

120590z

Figure 6 Variations of interlaminar normal stress 120590119911

through thethickness of the [45∘minus45∘45∘minus45∘] laminate for various width-to-thickness ratios

the maximum negative and positive values of 120590119911occur within

the top minus80∘ layer and the bottom 80

∘ layer at the free edge(ie 119910 = 119887) respectively It is also noticed that 120590

119911diminishes

away from the free edge as the interior region of the laminateis approached Moreover 120590

119911reduces by moving slightly away

from the free edge and its distribution becomes smootherThe variations of the interlaminar shear stress 120590

119910119911at 119910 =

119887 through the thickness of the [0∘

60∘

minus60∘

]119904laminate are

displayed in Figure 7 By increasing the number of numericallayers in each lamina 120590

119910119911becomes slightly closer to zero but

8 International Journal of Engineering Mathematics

minus15

minus1

minus05

0

05

1

15

minus20 minus10 0 10 20 30

zh

p = 20

p = 12

p = 8

120590yz

Figure 7 Variations of interlaminar shear stress 120590119910119911

on through-thickness of the [0∘60∘minus60∘]119904

laminate as a function of layer subdivisionnumber 119901

012

01

008

006

004

002

0

minus002

120590z

p = 30p = 20

p = 12p = 8

0 02 04 06 08 1

yb

Figure 8 Variations of interlaminar normal stress 120590119911

on through-thickness of the [minus15∘

90∘

90∘

minus15∘

] laminate as a function of layersubdivision number 119901

it never becomes zero It should be noted that increasing thenumber of subdivisions results in no convergence for 120590

119911and

120590119909119911

at interface-edge junction of two dissimilar layers Thenumerical values of these components (120590

119911and 120590

119909119911) continue

to grow as the number of sublayers increased Since thegeneralized stress resultants 119877119896

119910

instead of 120590119910119911

are forced todisappear at the free edge in LWT the numerical value of120590119910119911

may never become zero at the interface-edge intersection

(even by increasing the number of sublayers in each physicallayer)

The distribution of interlaminar normal stress 120590119911on

through thickness of the general [minus15∘90∘90∘minus15∘] lam-inate is displayed in Figure 8 It is noted that increasingthe number of numerical layers (119901) in each lamina has nosignificant effect on the values of the interlaminar stress 120590

119911

within the boundary-layer region of the laminate expect

International Journal of Engineering Mathematics 9

exactly at the free edge (ie 119910 = 119887) Furthermore theinterlaminar normal stress 120590

119911grows rapidly in the vicinity of

the free edge while being zero in the interior region of thelaminate

5 Conclusions

In the present paper analytical solutions are developed toanalyze the interlaminar stresses for different lay-up con-figurations in laminated composite plates subjected to thebending moment The solutions were obtained based onthe reduced elasticity displacement field for long laminatedplates It is found that the components of the displacementsfield are composed of two distinct parts signifying theglobal and local deformations within laminates The localdisplacement functions as well as the interlaminar stressesthrough the layers of the laminate were found by Reddyrsquoslayerwise theory (LWT)Thenumerical results based onLWTwere shown for the free-edge interlaminar stresses throughthe thickness and across the interfaces of various cross-plysymmetric angle-ply and general composite laminates

Appendix

The laminate rigidities in (22a) (22b) (22c) (22d) (24a)(24b) and (24c) are defined as

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

1206011015840

119895

1206011198961206011015840

119895

120601119896120601119895) d119911

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

120601119896 1206011015840

119896

119911 120601119896119911) d119911

(119896 119895 = 1 2 119873 + 1)

(119860119901119902 119861119901119902 119863119901119902) =

119873

sum

119894=1

int

119911119894+1

119911119894

119862(119894)

119901119902

(1 119911 1199112

) d119911

(A1)

which upon integration are presented in the following form

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

)

=

(minus

119862(119896minus1)

119901119902

ℎ119896minus1

minus

119862(119896minus1)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

6) if 119895 = 119896 minus 1

(

119862(119896minus1)

119901119902

ℎ119896minus1

+

119862(119896)

119901119902

ℎ119896

119862(119896minus1)

119901119902

2minus

119862(119896)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

3+

ℎ119896119862(119896)

119901119902

3) if 119895 = 119896

(minus

119862(119896)

119901119902

ℎ119896

119862(119896)

119901119902

2

ℎ119896119862(119896)

119901119902

6) if 119895 = 119896 + 1

(0 0 0) if 119895 lt 119896 minus 1 or 119895 gt 119896 + 1

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

)

=

(minus119862(1)

119901119902

ℎ1119862(1)

119901119902

2 119862(1)

119901119902

1199112

1

minus 1199112

2

2ℎ1

119862(1)

119901119902

ℎ1

[1199113

1

minus 1199113

2

3minus 1199112

1199112

1

minus 1199112

2

2]) if 119896 = 1

(minus119862(119896minus1)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2]) if 119896 = 119873 + 1

(119862(119896minus1)

119901119902

minus 119862(119896)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2+

ℎ119896119862(119896)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

+ 119862(119896)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2] +

119862(119896)

119901119902

ℎ119896

[1199113

119896

minus 1199113

119896+1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896+1

2]) if 1 lt 119896 lt 119873 + 1

(A2)

10 International Journal of Engineering Mathematics

Also

(119860119901119902 119861119901119902 119863119901119902)

=

119873

sum

119894=1

119862(119894)

119901119902

([119911119894+1

minus 119911119894] [

1199112

119894+1

minus 1199112

119894

2] [

1199113

119894+1

minus 1199113

119894

3])

(A3)

The coefficient matrices [119872] [119870] and [119871] appearing in(25) are given as

[119872] =[[

[

120575119878[11986366] 120575119878[11986326] 120575119878([11986136] minus [119861

45]119879

)

120575119878[11986326] [119863

22] [119861

23] minus [119861

44]119879

[0] [0] [11986344]

]]

]

[119870]=[

[

minus120575119878([11986055] + [120572]) minus120575

119878[11986045] [0]

minus120575119878[11986045] minus ([119860

44] + [120572]) [0]

120575119878([11986145]minus[11986136]119879

) [11986144]minus[11986123]119879

minus ([11986033]+[120572])

]

]

[119871] = [

[

12057511986011986055 0 0

12057511986011986045 0 0

minus120575119860(11986145 + 119861

36) 119860

13 12057511986211986113

]

]

(A4)

where [119860119901119902] [119861119901119902] and [119863

119901119902] are (119873 + 1) times (119873 + 1) square

matrices containing 119860119896119895119901119902 119861119896119895119901119902 and 119863

119896119895

119901119902 respectively and the

vectors 119860119901119902 119861119901119902 and 119861

119901119902 are (119873+1)times1 columnmatrices

containing 119860119896119901119902

119861119896119901119902

and 119861119896

119901119902

respectively Also [0] is (119873 +

1) times (119873 + 1) square zero and 0 is a zero vector with 119873 + 1

rows The artificial matrix [120572] is also an (119873 + 1) times (119873 + 1)

square matrix whose elements are given by

120572119896119895

= 120572int

ℎ2

minusℎ2

120601119896120601119895d119911 (A5)

with 120572 being a relatively small parameter in comparison withthe rigidity constants 119860119896119895

119901119902(119901119902 = 33 44 55) It is noted that

the insertion of [120572] in the matrix [119870] makes the eigenvaluesof the matrix (minus[119872]

minus1

[119870]) distinct

References

[1] T Kant and K Swaminathan ldquoEstimation of transverseinter-laminar stresses in laminated compositesmdasha selective reviewand survey of current developmentsrdquo Composite Structures vol49 no 1 pp 65ndash75 2000

[2] A H Puppo and H A Evensen ldquoInterlaminar shear ınlamınated composıtes under generalızed plane stressrdquo Journalof Composite Materials vol 4 pp 204ndash220 1970

[3] N J Pagano ldquoOn the calculatıon of ınterlamınar normal stressın composıte lamınaterdquo Journal of Composite Materials vol 8pp 65ndash81 1974

[4] P W Hsu and C T Herakovich ldquoEdge effects in angle-plycomposite laminatesrdquo Journal of CompositeMaterials vol 11 pp422ndash428 1977

[5] S Tang and A Levy ldquoBoundary layer theory 2 extension oflaminated finite striprdquo Journal of CompositeMaterials vol 9 no1 pp 42ndash52 1975

[6] R B Pipes andN J Pagano ldquoInterlaminar stresses in compositelaminates an approximate elasticity solutionrdquo Journal of AppliedMechanics vol 41 no 3 pp 668ndash672 1974

[7] N J Pagano ldquoStress fields in composite laminatesrdquo InternationalJournal of Solids and Structures vol 14 no 5 pp 385ndash400 1978

[8] S S Wang and I Choi ldquoBoundary-layer effects in composıtelaminates 1 free-edge stress singulaiıtiesrdquo Journal of AppliedMechanics vol V 49 no 3 pp 541ndash548 1982

[9] W L Yin ldquoFree-edge effects in anisotropic laminates underextension bending and twisting part I a stress-function-basedvariational approachrdquo Journal of Applied Mechanics vol 61 no2 pp 410ndash415 1994

[10] A S D Wang and F W Crossman ldquoSome new results on edgeeffect in symmetric composite laminatesrdquo Journal of CompositeMaterials vol 11 no 1 pp 92ndash106 1977

[11] J D Whitcomb I S Raju and J G Goree ldquoReliability ofthe finite element method for calculating free edge stresses incomposite laminatesrdquo Computers and Structures vol 15 no 1pp 23ndash37 1982

[12] G Davı ldquoStress fields in general composite laminatesrdquo AIAAJournal vol 34 no 12 pp 2604ndash2608 1996

[13] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT Part 1 derivationof finite element matricesrdquo International Journal for NumericalMethods in Engineering vol 55 no 2 pp 191ndash231 2002

[14] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT part 2 numericalimplementationsrdquo International Journal for Numerical Methodsin Engineering vol 55 no 3 pp 253ndash291 2002

[15] V-T Nguyen and J-F Caron ldquoFinite element analysis of free-edge stresses in composite laminates under mechanical anthermal loadingrdquo Composites Science and Technology vol 69no 1 pp 40ndash49 2009

[16] D H Robbins Jr and J N Reddy ldquoModelling of thick compos-ites using a layerwise laminate theoryrdquo International Journal forNumerical Methods in Engineering vol 36 no 4 pp 655ndash6771993

[17] C Mittelstedt and W Becker ldquoReddyrsquos layerwise laminate platetheory for the computation of elastic fields in the vicinity ofstraight free laminate edgesrdquoMaterials Science and EngineeringA vol 498 no 1-2 pp 76ndash80 2008

[18] W J Na Damage analysis of laminated composite beams underbending loads using the layer-wise theory [Dissertation thesis]Texas AampM University Texas Tex USA 2008

[19] T S Plagianakos and D A Saravanos ldquoHigher-order layerwiselaminate theory for the prediction of interlaminar shear stressesin thick composite and sandwich composite platesrdquo CompositeStructures vol 87 no 1 pp 23ndash35 2009

[20] H Ullah A R Harland T Lucas D Price and V V Sil-berschmidt ldquoFinite-element modelling of bending of CFRPlaminates multiple delaminationsrdquo Computational MaterialsScience vol 52 no 1 pp 147ndash156 2012

[21] F Helenon M R Wisnom S R Hallett and R S TraskldquoInvestigation into failure of laminated composite T-piecespecimens under bending loadingrdquo Composites A vol 54 pp182ndash189 2013

[22] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013

International Journal of Engineering Mathematics 11

[23] N D Thai M DrsquoOttavio and J-F Caron ldquoBending analysis oflaminated and sandwich plates using a layer-wise stress modelrdquoComposite Structures vol 96 pp 135ndash142 2013

[24] PMalekzadeh ldquoA two-dimensional layerwise-differential quad-rature static analysis of thick laminated composite circulararchesrdquoAppliedMathematicalModelling vol 33 no 4 pp 1850ndash1861 2009

[25] Y C Fung and P Tong Classical and Computational SolidMechanics World Scientific New Jersey NJ USA 2001

[26] S G LekhnitskiiTheory of Elasticity of anAnisotropic BodyMirPublishers Moscow Russia 1981

[27] A Nosier R K Kapania and J N Reddy ldquoFree vibrationanalysis of laminated plates using a layerwise theoryrdquo AIAAJournal vol 31 no 12 pp 2335ndash2346 1993

[28] C T HerakovichMechanics of Fibrous Composites Wiley NewYork NY USA 1998

[29] O Goodsell and R B Pipes ldquoInterlaminar stresses in compositelaminates subjected to anticlastic bending deformationrdquo Journalof Applied Mechanics vol 80 no 4 pp 1ndash7 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Investigation of Through-Thickness ...downloads.hindawi.com/archive/2013/676743.pdf · (,, ) is located at the middle plane of the laminate, that is of thickness

8 International Journal of Engineering Mathematics

minus15

minus1

minus05

0

05

1

15

minus20 minus10 0 10 20 30

zh

p = 20

p = 12

p = 8

120590yz

Figure 7 Variations of interlaminar shear stress 120590119910119911

on through-thickness of the [0∘60∘minus60∘]119904

laminate as a function of layer subdivisionnumber 119901

012

01

008

006

004

002

0

minus002

120590z

p = 30p = 20

p = 12p = 8

0 02 04 06 08 1

yb

Figure 8 Variations of interlaminar normal stress 120590119911

on through-thickness of the [minus15∘

90∘

90∘

minus15∘

] laminate as a function of layersubdivision number 119901

it never becomes zero It should be noted that increasing thenumber of subdivisions results in no convergence for 120590

119911and

120590119909119911

at interface-edge junction of two dissimilar layers Thenumerical values of these components (120590

119911and 120590

119909119911) continue

to grow as the number of sublayers increased Since thegeneralized stress resultants 119877119896

119910

instead of 120590119910119911

are forced todisappear at the free edge in LWT the numerical value of120590119910119911

may never become zero at the interface-edge intersection

(even by increasing the number of sublayers in each physicallayer)

The distribution of interlaminar normal stress 120590119911on

through thickness of the general [minus15∘90∘90∘minus15∘] lam-inate is displayed in Figure 8 It is noted that increasingthe number of numerical layers (119901) in each lamina has nosignificant effect on the values of the interlaminar stress 120590

119911

within the boundary-layer region of the laminate expect

International Journal of Engineering Mathematics 9

exactly at the free edge (ie 119910 = 119887) Furthermore theinterlaminar normal stress 120590

119911grows rapidly in the vicinity of

the free edge while being zero in the interior region of thelaminate

5 Conclusions

In the present paper analytical solutions are developed toanalyze the interlaminar stresses for different lay-up con-figurations in laminated composite plates subjected to thebending moment The solutions were obtained based onthe reduced elasticity displacement field for long laminatedplates It is found that the components of the displacementsfield are composed of two distinct parts signifying theglobal and local deformations within laminates The localdisplacement functions as well as the interlaminar stressesthrough the layers of the laminate were found by Reddyrsquoslayerwise theory (LWT)Thenumerical results based onLWTwere shown for the free-edge interlaminar stresses throughthe thickness and across the interfaces of various cross-plysymmetric angle-ply and general composite laminates

Appendix

The laminate rigidities in (22a) (22b) (22c) (22d) (24a)(24b) and (24c) are defined as

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

1206011015840

119895

1206011198961206011015840

119895

120601119896120601119895) d119911

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

120601119896 1206011015840

119896

119911 120601119896119911) d119911

(119896 119895 = 1 2 119873 + 1)

(119860119901119902 119861119901119902 119863119901119902) =

119873

sum

119894=1

int

119911119894+1

119911119894

119862(119894)

119901119902

(1 119911 1199112

) d119911

(A1)

which upon integration are presented in the following form

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

)

=

(minus

119862(119896minus1)

119901119902

ℎ119896minus1

minus

119862(119896minus1)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

6) if 119895 = 119896 minus 1

(

119862(119896minus1)

119901119902

ℎ119896minus1

+

119862(119896)

119901119902

ℎ119896

119862(119896minus1)

119901119902

2minus

119862(119896)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

3+

ℎ119896119862(119896)

119901119902

3) if 119895 = 119896

(minus

119862(119896)

119901119902

ℎ119896

119862(119896)

119901119902

2

ℎ119896119862(119896)

119901119902

6) if 119895 = 119896 + 1

(0 0 0) if 119895 lt 119896 minus 1 or 119895 gt 119896 + 1

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

)

=

(minus119862(1)

119901119902

ℎ1119862(1)

119901119902

2 119862(1)

119901119902

1199112

1

minus 1199112

2

2ℎ1

119862(1)

119901119902

ℎ1

[1199113

1

minus 1199113

2

3minus 1199112

1199112

1

minus 1199112

2

2]) if 119896 = 1

(minus119862(119896minus1)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2]) if 119896 = 119873 + 1

(119862(119896minus1)

119901119902

minus 119862(119896)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2+

ℎ119896119862(119896)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

+ 119862(119896)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2] +

119862(119896)

119901119902

ℎ119896

[1199113

119896

minus 1199113

119896+1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896+1

2]) if 1 lt 119896 lt 119873 + 1

(A2)

10 International Journal of Engineering Mathematics

Also

(119860119901119902 119861119901119902 119863119901119902)

=

119873

sum

119894=1

119862(119894)

119901119902

([119911119894+1

minus 119911119894] [

1199112

119894+1

minus 1199112

119894

2] [

1199113

119894+1

minus 1199113

119894

3])

(A3)

The coefficient matrices [119872] [119870] and [119871] appearing in(25) are given as

[119872] =[[

[

120575119878[11986366] 120575119878[11986326] 120575119878([11986136] minus [119861

45]119879

)

120575119878[11986326] [119863

22] [119861

23] minus [119861

44]119879

[0] [0] [11986344]

]]

]

[119870]=[

[

minus120575119878([11986055] + [120572]) minus120575

119878[11986045] [0]

minus120575119878[11986045] minus ([119860

44] + [120572]) [0]

120575119878([11986145]minus[11986136]119879

) [11986144]minus[11986123]119879

minus ([11986033]+[120572])

]

]

[119871] = [

[

12057511986011986055 0 0

12057511986011986045 0 0

minus120575119860(11986145 + 119861

36) 119860

13 12057511986211986113

]

]

(A4)

where [119860119901119902] [119861119901119902] and [119863

119901119902] are (119873 + 1) times (119873 + 1) square

matrices containing 119860119896119895119901119902 119861119896119895119901119902 and 119863

119896119895

119901119902 respectively and the

vectors 119860119901119902 119861119901119902 and 119861

119901119902 are (119873+1)times1 columnmatrices

containing 119860119896119901119902

119861119896119901119902

and 119861119896

119901119902

respectively Also [0] is (119873 +

1) times (119873 + 1) square zero and 0 is a zero vector with 119873 + 1

rows The artificial matrix [120572] is also an (119873 + 1) times (119873 + 1)

square matrix whose elements are given by

120572119896119895

= 120572int

ℎ2

minusℎ2

120601119896120601119895d119911 (A5)

with 120572 being a relatively small parameter in comparison withthe rigidity constants 119860119896119895

119901119902(119901119902 = 33 44 55) It is noted that

the insertion of [120572] in the matrix [119870] makes the eigenvaluesof the matrix (minus[119872]

minus1

[119870]) distinct

References

[1] T Kant and K Swaminathan ldquoEstimation of transverseinter-laminar stresses in laminated compositesmdasha selective reviewand survey of current developmentsrdquo Composite Structures vol49 no 1 pp 65ndash75 2000

[2] A H Puppo and H A Evensen ldquoInterlaminar shear ınlamınated composıtes under generalızed plane stressrdquo Journalof Composite Materials vol 4 pp 204ndash220 1970

[3] N J Pagano ldquoOn the calculatıon of ınterlamınar normal stressın composıte lamınaterdquo Journal of Composite Materials vol 8pp 65ndash81 1974

[4] P W Hsu and C T Herakovich ldquoEdge effects in angle-plycomposite laminatesrdquo Journal of CompositeMaterials vol 11 pp422ndash428 1977

[5] S Tang and A Levy ldquoBoundary layer theory 2 extension oflaminated finite striprdquo Journal of CompositeMaterials vol 9 no1 pp 42ndash52 1975

[6] R B Pipes andN J Pagano ldquoInterlaminar stresses in compositelaminates an approximate elasticity solutionrdquo Journal of AppliedMechanics vol 41 no 3 pp 668ndash672 1974

[7] N J Pagano ldquoStress fields in composite laminatesrdquo InternationalJournal of Solids and Structures vol 14 no 5 pp 385ndash400 1978

[8] S S Wang and I Choi ldquoBoundary-layer effects in composıtelaminates 1 free-edge stress singulaiıtiesrdquo Journal of AppliedMechanics vol V 49 no 3 pp 541ndash548 1982

[9] W L Yin ldquoFree-edge effects in anisotropic laminates underextension bending and twisting part I a stress-function-basedvariational approachrdquo Journal of Applied Mechanics vol 61 no2 pp 410ndash415 1994

[10] A S D Wang and F W Crossman ldquoSome new results on edgeeffect in symmetric composite laminatesrdquo Journal of CompositeMaterials vol 11 no 1 pp 92ndash106 1977

[11] J D Whitcomb I S Raju and J G Goree ldquoReliability ofthe finite element method for calculating free edge stresses incomposite laminatesrdquo Computers and Structures vol 15 no 1pp 23ndash37 1982

[12] G Davı ldquoStress fields in general composite laminatesrdquo AIAAJournal vol 34 no 12 pp 2604ndash2608 1996

[13] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT Part 1 derivationof finite element matricesrdquo International Journal for NumericalMethods in Engineering vol 55 no 2 pp 191ndash231 2002

[14] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT part 2 numericalimplementationsrdquo International Journal for Numerical Methodsin Engineering vol 55 no 3 pp 253ndash291 2002

[15] V-T Nguyen and J-F Caron ldquoFinite element analysis of free-edge stresses in composite laminates under mechanical anthermal loadingrdquo Composites Science and Technology vol 69no 1 pp 40ndash49 2009

[16] D H Robbins Jr and J N Reddy ldquoModelling of thick compos-ites using a layerwise laminate theoryrdquo International Journal forNumerical Methods in Engineering vol 36 no 4 pp 655ndash6771993

[17] C Mittelstedt and W Becker ldquoReddyrsquos layerwise laminate platetheory for the computation of elastic fields in the vicinity ofstraight free laminate edgesrdquoMaterials Science and EngineeringA vol 498 no 1-2 pp 76ndash80 2008

[18] W J Na Damage analysis of laminated composite beams underbending loads using the layer-wise theory [Dissertation thesis]Texas AampM University Texas Tex USA 2008

[19] T S Plagianakos and D A Saravanos ldquoHigher-order layerwiselaminate theory for the prediction of interlaminar shear stressesin thick composite and sandwich composite platesrdquo CompositeStructures vol 87 no 1 pp 23ndash35 2009

[20] H Ullah A R Harland T Lucas D Price and V V Sil-berschmidt ldquoFinite-element modelling of bending of CFRPlaminates multiple delaminationsrdquo Computational MaterialsScience vol 52 no 1 pp 147ndash156 2012

[21] F Helenon M R Wisnom S R Hallett and R S TraskldquoInvestigation into failure of laminated composite T-piecespecimens under bending loadingrdquo Composites A vol 54 pp182ndash189 2013

[22] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013

International Journal of Engineering Mathematics 11

[23] N D Thai M DrsquoOttavio and J-F Caron ldquoBending analysis oflaminated and sandwich plates using a layer-wise stress modelrdquoComposite Structures vol 96 pp 135ndash142 2013

[24] PMalekzadeh ldquoA two-dimensional layerwise-differential quad-rature static analysis of thick laminated composite circulararchesrdquoAppliedMathematicalModelling vol 33 no 4 pp 1850ndash1861 2009

[25] Y C Fung and P Tong Classical and Computational SolidMechanics World Scientific New Jersey NJ USA 2001

[26] S G LekhnitskiiTheory of Elasticity of anAnisotropic BodyMirPublishers Moscow Russia 1981

[27] A Nosier R K Kapania and J N Reddy ldquoFree vibrationanalysis of laminated plates using a layerwise theoryrdquo AIAAJournal vol 31 no 12 pp 2335ndash2346 1993

[28] C T HerakovichMechanics of Fibrous Composites Wiley NewYork NY USA 1998

[29] O Goodsell and R B Pipes ldquoInterlaminar stresses in compositelaminates subjected to anticlastic bending deformationrdquo Journalof Applied Mechanics vol 80 no 4 pp 1ndash7 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Investigation of Through-Thickness ...downloads.hindawi.com/archive/2013/676743.pdf · (,, ) is located at the middle plane of the laminate, that is of thickness

International Journal of Engineering Mathematics 9

exactly at the free edge (ie 119910 = 119887) Furthermore theinterlaminar normal stress 120590

119911grows rapidly in the vicinity of

the free edge while being zero in the interior region of thelaminate

5 Conclusions

In the present paper analytical solutions are developed toanalyze the interlaminar stresses for different lay-up con-figurations in laminated composite plates subjected to thebending moment The solutions were obtained based onthe reduced elasticity displacement field for long laminatedplates It is found that the components of the displacementsfield are composed of two distinct parts signifying theglobal and local deformations within laminates The localdisplacement functions as well as the interlaminar stressesthrough the layers of the laminate were found by Reddyrsquoslayerwise theory (LWT)Thenumerical results based onLWTwere shown for the free-edge interlaminar stresses throughthe thickness and across the interfaces of various cross-plysymmetric angle-ply and general composite laminates

Appendix

The laminate rigidities in (22a) (22b) (22c) (22d) (24a)(24b) and (24c) are defined as

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

1206011015840

119895

1206011198961206011015840

119895

120601119896120601119895) d119911

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

) =

119873

sum

119894=1

int

ℎ2

minusℎ2

119862(119894)

119901119902

(1206011015840

119896

120601119896 1206011015840

119896

119911 120601119896119911) d119911

(119896 119895 = 1 2 119873 + 1)

(119860119901119902 119861119901119902 119863119901119902) =

119873

sum

119894=1

int

119911119894+1

119911119894

119862(119894)

119901119902

(1 119911 1199112

) d119911

(A1)

which upon integration are presented in the following form

(119860119896119895

119901119902

119861119896119895

119901119902

119863119896119895

119901119902

)

=

(minus

119862(119896minus1)

119901119902

ℎ119896minus1

minus

119862(119896minus1)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

6) if 119895 = 119896 minus 1

(

119862(119896minus1)

119901119902

ℎ119896minus1

+

119862(119896)

119901119902

ℎ119896

119862(119896minus1)

119901119902

2minus

119862(119896)

119901119902

2

ℎ119896minus1

119862(119896minus1)

119901119902

3+

ℎ119896119862(119896)

119901119902

3) if 119895 = 119896

(minus

119862(119896)

119901119902

ℎ119896

119862(119896)

119901119902

2

ℎ119896119862(119896)

119901119902

6) if 119895 = 119896 + 1

(0 0 0) if 119895 lt 119896 minus 1 or 119895 gt 119896 + 1

(119860119896

119901119902

119861119896

119901119902

119861119896

119901119902

119863119896

119901119902

)

=

(minus119862(1)

119901119902

ℎ1119862(1)

119901119902

2 119862(1)

119901119902

1199112

1

minus 1199112

2

2ℎ1

119862(1)

119901119902

ℎ1

[1199113

1

minus 1199113

2

3minus 1199112

1199112

1

minus 1199112

2

2]) if 119896 = 1

(minus119862(119896minus1)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2]) if 119896 = 119873 + 1

(119862(119896minus1)

119901119902

minus 119862(119896)

119901119902

ℎ119896minus1

119862(119896minus1)

119901119902

2+

ℎ119896119862(119896)

119901119902

2 119862(119896minus1)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896minus1

+ 119862(119896)

119901119902

1199112

119896

minus 1199112

119896minus1

2ℎ119896

119862(119896minus1)

119901119902

ℎ119896minus1

[1199113

119896

minus 1199113

119896minus1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896minus1

2] +

119862(119896)

119901119902

ℎ119896

[1199113

119896

minus 1199113

119896+1

3minus 119911119896minus1

1199112

119896

minus 1199112

119896+1

2]) if 1 lt 119896 lt 119873 + 1

(A2)

10 International Journal of Engineering Mathematics

Also

(119860119901119902 119861119901119902 119863119901119902)

=

119873

sum

119894=1

119862(119894)

119901119902

([119911119894+1

minus 119911119894] [

1199112

119894+1

minus 1199112

119894

2] [

1199113

119894+1

minus 1199113

119894

3])

(A3)

The coefficient matrices [119872] [119870] and [119871] appearing in(25) are given as

[119872] =[[

[

120575119878[11986366] 120575119878[11986326] 120575119878([11986136] minus [119861

45]119879

)

120575119878[11986326] [119863

22] [119861

23] minus [119861

44]119879

[0] [0] [11986344]

]]

]

[119870]=[

[

minus120575119878([11986055] + [120572]) minus120575

119878[11986045] [0]

minus120575119878[11986045] minus ([119860

44] + [120572]) [0]

120575119878([11986145]minus[11986136]119879

) [11986144]minus[11986123]119879

minus ([11986033]+[120572])

]

]

[119871] = [

[

12057511986011986055 0 0

12057511986011986045 0 0

minus120575119860(11986145 + 119861

36) 119860

13 12057511986211986113

]

]

(A4)

where [119860119901119902] [119861119901119902] and [119863

119901119902] are (119873 + 1) times (119873 + 1) square

matrices containing 119860119896119895119901119902 119861119896119895119901119902 and 119863

119896119895

119901119902 respectively and the

vectors 119860119901119902 119861119901119902 and 119861

119901119902 are (119873+1)times1 columnmatrices

containing 119860119896119901119902

119861119896119901119902

and 119861119896

119901119902

respectively Also [0] is (119873 +

1) times (119873 + 1) square zero and 0 is a zero vector with 119873 + 1

rows The artificial matrix [120572] is also an (119873 + 1) times (119873 + 1)

square matrix whose elements are given by

120572119896119895

= 120572int

ℎ2

minusℎ2

120601119896120601119895d119911 (A5)

with 120572 being a relatively small parameter in comparison withthe rigidity constants 119860119896119895

119901119902(119901119902 = 33 44 55) It is noted that

the insertion of [120572] in the matrix [119870] makes the eigenvaluesof the matrix (minus[119872]

minus1

[119870]) distinct

References

[1] T Kant and K Swaminathan ldquoEstimation of transverseinter-laminar stresses in laminated compositesmdasha selective reviewand survey of current developmentsrdquo Composite Structures vol49 no 1 pp 65ndash75 2000

[2] A H Puppo and H A Evensen ldquoInterlaminar shear ınlamınated composıtes under generalızed plane stressrdquo Journalof Composite Materials vol 4 pp 204ndash220 1970

[3] N J Pagano ldquoOn the calculatıon of ınterlamınar normal stressın composıte lamınaterdquo Journal of Composite Materials vol 8pp 65ndash81 1974

[4] P W Hsu and C T Herakovich ldquoEdge effects in angle-plycomposite laminatesrdquo Journal of CompositeMaterials vol 11 pp422ndash428 1977

[5] S Tang and A Levy ldquoBoundary layer theory 2 extension oflaminated finite striprdquo Journal of CompositeMaterials vol 9 no1 pp 42ndash52 1975

[6] R B Pipes andN J Pagano ldquoInterlaminar stresses in compositelaminates an approximate elasticity solutionrdquo Journal of AppliedMechanics vol 41 no 3 pp 668ndash672 1974

[7] N J Pagano ldquoStress fields in composite laminatesrdquo InternationalJournal of Solids and Structures vol 14 no 5 pp 385ndash400 1978

[8] S S Wang and I Choi ldquoBoundary-layer effects in composıtelaminates 1 free-edge stress singulaiıtiesrdquo Journal of AppliedMechanics vol V 49 no 3 pp 541ndash548 1982

[9] W L Yin ldquoFree-edge effects in anisotropic laminates underextension bending and twisting part I a stress-function-basedvariational approachrdquo Journal of Applied Mechanics vol 61 no2 pp 410ndash415 1994

[10] A S D Wang and F W Crossman ldquoSome new results on edgeeffect in symmetric composite laminatesrdquo Journal of CompositeMaterials vol 11 no 1 pp 92ndash106 1977

[11] J D Whitcomb I S Raju and J G Goree ldquoReliability ofthe finite element method for calculating free edge stresses incomposite laminatesrdquo Computers and Structures vol 15 no 1pp 23ndash37 1982

[12] G Davı ldquoStress fields in general composite laminatesrdquo AIAAJournal vol 34 no 12 pp 2604ndash2608 1996

[13] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT Part 1 derivationof finite element matricesrdquo International Journal for NumericalMethods in Engineering vol 55 no 2 pp 191ndash231 2002

[14] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT part 2 numericalimplementationsrdquo International Journal for Numerical Methodsin Engineering vol 55 no 3 pp 253ndash291 2002

[15] V-T Nguyen and J-F Caron ldquoFinite element analysis of free-edge stresses in composite laminates under mechanical anthermal loadingrdquo Composites Science and Technology vol 69no 1 pp 40ndash49 2009

[16] D H Robbins Jr and J N Reddy ldquoModelling of thick compos-ites using a layerwise laminate theoryrdquo International Journal forNumerical Methods in Engineering vol 36 no 4 pp 655ndash6771993

[17] C Mittelstedt and W Becker ldquoReddyrsquos layerwise laminate platetheory for the computation of elastic fields in the vicinity ofstraight free laminate edgesrdquoMaterials Science and EngineeringA vol 498 no 1-2 pp 76ndash80 2008

[18] W J Na Damage analysis of laminated composite beams underbending loads using the layer-wise theory [Dissertation thesis]Texas AampM University Texas Tex USA 2008

[19] T S Plagianakos and D A Saravanos ldquoHigher-order layerwiselaminate theory for the prediction of interlaminar shear stressesin thick composite and sandwich composite platesrdquo CompositeStructures vol 87 no 1 pp 23ndash35 2009

[20] H Ullah A R Harland T Lucas D Price and V V Sil-berschmidt ldquoFinite-element modelling of bending of CFRPlaminates multiple delaminationsrdquo Computational MaterialsScience vol 52 no 1 pp 147ndash156 2012

[21] F Helenon M R Wisnom S R Hallett and R S TraskldquoInvestigation into failure of laminated composite T-piecespecimens under bending loadingrdquo Composites A vol 54 pp182ndash189 2013

[22] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013

International Journal of Engineering Mathematics 11

[23] N D Thai M DrsquoOttavio and J-F Caron ldquoBending analysis oflaminated and sandwich plates using a layer-wise stress modelrdquoComposite Structures vol 96 pp 135ndash142 2013

[24] PMalekzadeh ldquoA two-dimensional layerwise-differential quad-rature static analysis of thick laminated composite circulararchesrdquoAppliedMathematicalModelling vol 33 no 4 pp 1850ndash1861 2009

[25] Y C Fung and P Tong Classical and Computational SolidMechanics World Scientific New Jersey NJ USA 2001

[26] S G LekhnitskiiTheory of Elasticity of anAnisotropic BodyMirPublishers Moscow Russia 1981

[27] A Nosier R K Kapania and J N Reddy ldquoFree vibrationanalysis of laminated plates using a layerwise theoryrdquo AIAAJournal vol 31 no 12 pp 2335ndash2346 1993

[28] C T HerakovichMechanics of Fibrous Composites Wiley NewYork NY USA 1998

[29] O Goodsell and R B Pipes ldquoInterlaminar stresses in compositelaminates subjected to anticlastic bending deformationrdquo Journalof Applied Mechanics vol 80 no 4 pp 1ndash7 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Investigation of Through-Thickness ...downloads.hindawi.com/archive/2013/676743.pdf · (,, ) is located at the middle plane of the laminate, that is of thickness

10 International Journal of Engineering Mathematics

Also

(119860119901119902 119861119901119902 119863119901119902)

=

119873

sum

119894=1

119862(119894)

119901119902

([119911119894+1

minus 119911119894] [

1199112

119894+1

minus 1199112

119894

2] [

1199113

119894+1

minus 1199113

119894

3])

(A3)

The coefficient matrices [119872] [119870] and [119871] appearing in(25) are given as

[119872] =[[

[

120575119878[11986366] 120575119878[11986326] 120575119878([11986136] minus [119861

45]119879

)

120575119878[11986326] [119863

22] [119861

23] minus [119861

44]119879

[0] [0] [11986344]

]]

]

[119870]=[

[

minus120575119878([11986055] + [120572]) minus120575

119878[11986045] [0]

minus120575119878[11986045] minus ([119860

44] + [120572]) [0]

120575119878([11986145]minus[11986136]119879

) [11986144]minus[11986123]119879

minus ([11986033]+[120572])

]

]

[119871] = [

[

12057511986011986055 0 0

12057511986011986045 0 0

minus120575119860(11986145 + 119861

36) 119860

13 12057511986211986113

]

]

(A4)

where [119860119901119902] [119861119901119902] and [119863

119901119902] are (119873 + 1) times (119873 + 1) square

matrices containing 119860119896119895119901119902 119861119896119895119901119902 and 119863

119896119895

119901119902 respectively and the

vectors 119860119901119902 119861119901119902 and 119861

119901119902 are (119873+1)times1 columnmatrices

containing 119860119896119901119902

119861119896119901119902

and 119861119896

119901119902

respectively Also [0] is (119873 +

1) times (119873 + 1) square zero and 0 is a zero vector with 119873 + 1

rows The artificial matrix [120572] is also an (119873 + 1) times (119873 + 1)

square matrix whose elements are given by

120572119896119895

= 120572int

ℎ2

minusℎ2

120601119896120601119895d119911 (A5)

with 120572 being a relatively small parameter in comparison withthe rigidity constants 119860119896119895

119901119902(119901119902 = 33 44 55) It is noted that

the insertion of [120572] in the matrix [119870] makes the eigenvaluesof the matrix (minus[119872]

minus1

[119870]) distinct

References

[1] T Kant and K Swaminathan ldquoEstimation of transverseinter-laminar stresses in laminated compositesmdasha selective reviewand survey of current developmentsrdquo Composite Structures vol49 no 1 pp 65ndash75 2000

[2] A H Puppo and H A Evensen ldquoInterlaminar shear ınlamınated composıtes under generalızed plane stressrdquo Journalof Composite Materials vol 4 pp 204ndash220 1970

[3] N J Pagano ldquoOn the calculatıon of ınterlamınar normal stressın composıte lamınaterdquo Journal of Composite Materials vol 8pp 65ndash81 1974

[4] P W Hsu and C T Herakovich ldquoEdge effects in angle-plycomposite laminatesrdquo Journal of CompositeMaterials vol 11 pp422ndash428 1977

[5] S Tang and A Levy ldquoBoundary layer theory 2 extension oflaminated finite striprdquo Journal of CompositeMaterials vol 9 no1 pp 42ndash52 1975

[6] R B Pipes andN J Pagano ldquoInterlaminar stresses in compositelaminates an approximate elasticity solutionrdquo Journal of AppliedMechanics vol 41 no 3 pp 668ndash672 1974

[7] N J Pagano ldquoStress fields in composite laminatesrdquo InternationalJournal of Solids and Structures vol 14 no 5 pp 385ndash400 1978

[8] S S Wang and I Choi ldquoBoundary-layer effects in composıtelaminates 1 free-edge stress singulaiıtiesrdquo Journal of AppliedMechanics vol V 49 no 3 pp 541ndash548 1982

[9] W L Yin ldquoFree-edge effects in anisotropic laminates underextension bending and twisting part I a stress-function-basedvariational approachrdquo Journal of Applied Mechanics vol 61 no2 pp 410ndash415 1994

[10] A S D Wang and F W Crossman ldquoSome new results on edgeeffect in symmetric composite laminatesrdquo Journal of CompositeMaterials vol 11 no 1 pp 92ndash106 1977

[11] J D Whitcomb I S Raju and J G Goree ldquoReliability ofthe finite element method for calculating free edge stresses incomposite laminatesrdquo Computers and Structures vol 15 no 1pp 23ndash37 1982

[12] G Davı ldquoStress fields in general composite laminatesrdquo AIAAJournal vol 34 no 12 pp 2604ndash2608 1996

[13] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT Part 1 derivationof finite element matricesrdquo International Journal for NumericalMethods in Engineering vol 55 no 2 pp 191ndash231 2002

[14] E Carrera and L Demasi ldquoClassical and advancedmultilayeredplate elements based upon PVD and RMVT part 2 numericalimplementationsrdquo International Journal for Numerical Methodsin Engineering vol 55 no 3 pp 253ndash291 2002

[15] V-T Nguyen and J-F Caron ldquoFinite element analysis of free-edge stresses in composite laminates under mechanical anthermal loadingrdquo Composites Science and Technology vol 69no 1 pp 40ndash49 2009

[16] D H Robbins Jr and J N Reddy ldquoModelling of thick compos-ites using a layerwise laminate theoryrdquo International Journal forNumerical Methods in Engineering vol 36 no 4 pp 655ndash6771993

[17] C Mittelstedt and W Becker ldquoReddyrsquos layerwise laminate platetheory for the computation of elastic fields in the vicinity ofstraight free laminate edgesrdquoMaterials Science and EngineeringA vol 498 no 1-2 pp 76ndash80 2008

[18] W J Na Damage analysis of laminated composite beams underbending loads using the layer-wise theory [Dissertation thesis]Texas AampM University Texas Tex USA 2008

[19] T S Plagianakos and D A Saravanos ldquoHigher-order layerwiselaminate theory for the prediction of interlaminar shear stressesin thick composite and sandwich composite platesrdquo CompositeStructures vol 87 no 1 pp 23ndash35 2009

[20] H Ullah A R Harland T Lucas D Price and V V Sil-berschmidt ldquoFinite-element modelling of bending of CFRPlaminates multiple delaminationsrdquo Computational MaterialsScience vol 52 no 1 pp 147ndash156 2012

[21] F Helenon M R Wisnom S R Hallett and R S TraskldquoInvestigation into failure of laminated composite T-piecespecimens under bending loadingrdquo Composites A vol 54 pp182ndash189 2013

[22] C H Thai A J M Ferreira E Carrera and H Nguyen-XuanldquoIsogeometric analysis of laminated composite and sandwichplates using a layerwise deformation theoryrdquo Composite Struc-tures vol 104 pp 196ndash214 2013

International Journal of Engineering Mathematics 11

[23] N D Thai M DrsquoOttavio and J-F Caron ldquoBending analysis oflaminated and sandwich plates using a layer-wise stress modelrdquoComposite Structures vol 96 pp 135ndash142 2013

[24] PMalekzadeh ldquoA two-dimensional layerwise-differential quad-rature static analysis of thick laminated composite circulararchesrdquoAppliedMathematicalModelling vol 33 no 4 pp 1850ndash1861 2009

[25] Y C Fung and P Tong Classical and Computational SolidMechanics World Scientific New Jersey NJ USA 2001

[26] S G LekhnitskiiTheory of Elasticity of anAnisotropic BodyMirPublishers Moscow Russia 1981

[27] A Nosier R K Kapania and J N Reddy ldquoFree vibrationanalysis of laminated plates using a layerwise theoryrdquo AIAAJournal vol 31 no 12 pp 2335ndash2346 1993

[28] C T HerakovichMechanics of Fibrous Composites Wiley NewYork NY USA 1998

[29] O Goodsell and R B Pipes ldquoInterlaminar stresses in compositelaminates subjected to anticlastic bending deformationrdquo Journalof Applied Mechanics vol 80 no 4 pp 1ndash7 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Investigation of Through-Thickness ...downloads.hindawi.com/archive/2013/676743.pdf · (,, ) is located at the middle plane of the laminate, that is of thickness

International Journal of Engineering Mathematics 11

[23] N D Thai M DrsquoOttavio and J-F Caron ldquoBending analysis oflaminated and sandwich plates using a layer-wise stress modelrdquoComposite Structures vol 96 pp 135ndash142 2013

[24] PMalekzadeh ldquoA two-dimensional layerwise-differential quad-rature static analysis of thick laminated composite circulararchesrdquoAppliedMathematicalModelling vol 33 no 4 pp 1850ndash1861 2009

[25] Y C Fung and P Tong Classical and Computational SolidMechanics World Scientific New Jersey NJ USA 2001

[26] S G LekhnitskiiTheory of Elasticity of anAnisotropic BodyMirPublishers Moscow Russia 1981

[27] A Nosier R K Kapania and J N Reddy ldquoFree vibrationanalysis of laminated plates using a layerwise theoryrdquo AIAAJournal vol 31 no 12 pp 2335ndash2346 1993

[28] C T HerakovichMechanics of Fibrous Composites Wiley NewYork NY USA 1998

[29] O Goodsell and R B Pipes ldquoInterlaminar stresses in compositelaminates subjected to anticlastic bending deformationrdquo Journalof Applied Mechanics vol 80 no 4 pp 1ndash7 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article Investigation of Through-Thickness ...downloads.hindawi.com/archive/2013/676743.pdf · (,, ) is located at the middle plane of the laminate, that is of thickness

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


SLCC Laminate Installation Guidelines - SLCC Flooring Laminate... · SLCC Laminate Installation Guidelines PLEASE READ COMPLETELY BEFORE INSTALLATION SLCC Laminate Installation 2017

SLCC Laminate Installation Guidelines - SLCC Flooring Laminate... · SLCC Laminate Installation Guidelines PLEASE READ COMPLETELY BEFORE INSTALLATION SLCC Laminate Installation 2017

LAMINATE - Order Timber Flooring Onlinediyfloorboards.com.au/brochures/Clix Laminate Brochure.pdfClix Laminate 1261 x 192 x 7mm | no bevel 1 box = 2.421m2 = 10 planks LAMINATE The

LAMINATE - Order Timber Flooring Onlinediyfloorboards.com.au/brochures/Clix Laminate Brochure.pdfClix Laminate 1261 x 192 x 7mm | no bevel 1 box = 2.421m2 = 10 planks LAMINATE The

“ Laminate veneer ,,

“ Laminate veneer ,,

Module 5: Laminate Theory Lecture 17: Laminate ... · PDF fileModule 5: Laminate Theory Lecture 17: ... Laminate Theory Lecture 17: Laminate Constitutive ... The laminates for which

Module 5: Laminate Theory Lecture 17: Laminate ... · PDF fileModule 5: Laminate Theory Lecture 17: ... Laminate Theory Lecture 17: Laminate Constitutive ... The laminates for which

Formica laminate with Protec+/media/new-zealand/documents/formica-hpl... · Dimensional tolerances (ISO 4586-1:2004, Clause no.) Typical properties Clause Units Values Thickness 6.3

Formica laminate with Protec+/media/new-zealand/documents/formica-hpl... · Dimensional tolerances (ISO 4586-1:2004, Clause no.) Typical properties Clause Units Values Thickness 6.3

Shell Elements in ABAQUS/Explicit - imechanica · –ABAQUS computes an effective thickness strain based on the total thickness change and a strain obtained by enforcing the plane

Shell Elements in ABAQUS/Explicit - imechanica · –ABAQUS computes an effective thickness strain based on the total thickness change and a strain obtained by enforcing the plane

Effect of adhesive thickness area of single lap joints in ... · COMPOSITE LAMINATE USING ACOUSTIC EMISSION TECHNIQUE ... automobiles, marine, ... Bonded structure has an advantage

Effect of adhesive thickness area of single lap joints in ... · COMPOSITE LAMINATE USING ACOUSTIC EMISSION TECHNIQUE ... automobiles, marine, ... Bonded structure has an advantage

OUTDOOR PIPELINE LAMINATE · An Adhesive tape based on Polyester Film 23 MY thick and Acrylic Polymer Thermostable Adhesive. Technical Details Total Thickness 0.06mm Backing Thickness

OUTDOOR PIPELINE LAMINATE · An Adhesive tape based on Polyester Film 23 MY thick and Acrylic Polymer Thermostable Adhesive. Technical Details Total Thickness 0.06mm Backing Thickness

Discoveryof robust in-plane ferroelectricity in atomic-thick SnTe · skite ferroelectric materials, stable out-plane spon-taneous polarization has been discovered in the thinfilmsoffewunitcells(UCs)thickness(

Discoveryof robust in-plane ferroelectricity in atomic-thick SnTe · skite ferroelectric materials, stable out-plane spon-taneous polarization has been discovered in the thinfilmsoffewunitcells(UCs)thickness(

Minimal arithmetic thickness connecting discrete planes · 2020. 9. 13. · 2-connected arithmetic discrete plane P(n,µ,ω) with ω < knk1. (a) A0-connectedarithmeticdiscrete plane(ω=11)

Minimal arithmetic thickness connecting discrete planes · 2020. 9. 13. · 2-connected arithmetic discrete plane P(n,µ,ω) with ω < knk1. (a) A0-connectedarithmeticdiscrete plane(ω=11)

PROJECTIONS OF PLANES Plane surface (plane/lamina/plate) A plane is as two dimensional surface having length and breadth with negligible thickness. They.

PROJECTIONS OF PLANES Plane surface (plane/lamina/plate) A plane is as two dimensional surface having length and breadth with negligible thickness. They.

Research Article - Semantic Scholar...10.5 kOe. The disordered FePt layer with thickness of 2nm contributed both to magnetization, in-plane and out-of-plane coercivity in FePt/FePtAg

Research Article - Semantic Scholar...10.5 kOe. The disordered FePt layer with thickness of 2nm contributed both to magnetization, in-plane and out-of-plane coercivity in FePt/FePtAg

ON THE IMPORTANCE OF IN-PLANE SHRINKAGE AND THROUGH …€¦ · tionship between drying and out-of-plane deformations, the direct effect of through-thickness drying conditions on

ON THE IMPORTANCE OF IN-PLANE SHRINKAGE AND THROUGH …€¦ · tionship between drying and out-of-plane deformations, the direct effect of through-thickness drying conditions on

WATER RESISTANT LAMINATE - DIY Floorboardsdiyfloorboards.com.au/brochures/Kronoswiss Aquastop Brochure.pdf · Do not use Steam Mops. 193 mm 1380 mm 8mm 1 size - 8mm thickness KA8

WATER RESISTANT LAMINATE - DIY Floorboardsdiyfloorboards.com.au/brochures/Kronoswiss Aquastop Brochure.pdf · Do not use Steam Mops. 193 mm 1380 mm 8mm 1 size - 8mm thickness KA8

Merino Laminate

Merino Laminate

RESTAURANT INDUSTRY BROCHURE - Eliason Doors...resistant, composite wood core, 1” total door thickness. .032” decorative high-pressure laminate (both sides) with black plastic

RESTAURANT INDUSTRY BROCHURE - Eliason Doors...resistant, composite wood core, 1” total door thickness. .032” decorative high-pressure laminate (both sides) with black plastic

Puricelli Laminate - Inter systems - · PDF filePURICELLI laminati plastici e superfici decorative LAMINATE EMPLOYMENT TECHNIQUES . Puricelli Laminate

Puricelli Laminate - Inter systems - · PDF filePURICELLI laminati plastici e superfici decorative LAMINATE EMPLOYMENT TECHNIQUES . Puricelli Laminate

rambabu/EE470/class26s.pdf ·  · 2007-11-23method c are gene of papers appeared form exp ... DIELECTRIC PLANE h —SUBSTRATE THICKNESS t CONDUCTOR THICKNESS ... dielectric constant

rambabu/EE470/class26s.pdf ·  · 2007-11-23method c are gene of papers appeared form exp ... DIELECTRIC PLANE h —SUBSTRATE THICKNESS t CONDUCTOR THICKNESS ... dielectric constant

Laminate Plant Machinerylaminateplantmachinery.com/upload/Shree_Harisidee_E_Catalog.pdf · Length of machine Height Thickness of base material treated on our machine Thickness of

Laminate Plant Machinerylaminateplantmachinery.com/upload/Shree_Harisidee_E_Catalog.pdf · Length of machine Height Thickness of base material treated on our machine Thickness of

Laminate Design

Laminate Design