Composites: Part B 60 (2014) 297–305

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Composites: Part B

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Concentrated loading of a fibre-reinforced composite plate:Experimental and computational modeling of boundary fixity

1359-8368/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compositesb.2013.12.034

⇑ Corresponding author. Tel.: +1 514 398 2424; fax: +1 514 398 7361.E-mail address: hasan.nikopourdeilami@mail.mcgill.ca (H. Nikopour).

H. Nikopour ⇑, A.P.S. SelvaduraiDepartment of Civil Engineering and Applied Mechanics, McGill University, Montreal, QC H3A 2K6, Canada

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 January 2013Received in revised form 30 October 2013Accepted 22 December 2013Available online 2 January 2014

Keywords:A. Layered structuresB. ElasticityC. Finite element analysis (FEA)

This paper examines the flexural behavior of a locally loaded Carbon Fibre-Reinforced Polymer (CFRP)composite rectangular plate with different edge support conditions. The transversely isotropic elasticityproperties of the unidirectionally reinforced laminae were determined experimentally and verifiedthrough computational modeling. The assembly of the laminae is used to construct a model of the plate.The layered composite CFRP plate used in the experimental investigation consisted of 11 layers of a poly-ester matrix reinforced with carbon fibres. The bulk fibre volume fraction in the plate was approximately61%. The experimental results for the deflected shape of the plate are compared with computationalresults that take into account large deflections of the plate within the small strain range. It was found thatthe Representative Area Element (RAE) method developed for estimating the elastic properties of uni-directional fibre reinforced plates provides reliable estimates for the finite element modeling of the com-posite plate at the macro-scale.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Fibre-reinforced plates are used quite extensively in variousengineering applications, ranging from infrastructure engineering,aerospace engineering, automotive engineering, marine structuresand wind energy production [1–4]. In many of these applications,the high strength to weight ratio requirements are satisfied bythe fibre reinforced composite.

This paper examines the flexural mechanics problem for a Car-bon Fibre Reinforced Polymer (CFRP) laminated rectangular platethat is constrained at the edges. In particular, the edge supportconditions were varied to include either (i) complete fixity thatcan be achieved by clamping the plate between rigid surfaces or(ii) partial fixity that can be achieved by incorporating a deform-able rubber-like layer at clamped locations. The elasticity ofboundary supports is a research area that has received limitedattention, but it is a critical aspect in advanced applications of fi-bre-reinforced composites, involving connections to metallic andnon-metallic components [5,6]. The research also involves compu-tational modeling of the experiments, which takes into consider-ation large deflection but small strain flexural behavior of the plate.

The accurate theoretical and computational modeling of themechanical behavior of laminated plates requires the correct spec-ification of their mechanical properties. The composite action of alaminated plate is largely governed by the mechanical properties of

the individual layers and these properties are in turn governed bythe fibre–matrix constituent properties at the micro-scale (i.e. fibrescale, where the fibre diameter could be approximately 5–10 lm incomparison to the thickness of a plate which could be several mil-limeters). The simplest theoretical relationships used to estimatethe effective properties of a composite are the Voigt and Reussbounds that assume either the existence of uniform strain (upperbound-Voigt) or uniform stress (lower bound – Reuss) in the indi-vidual phases [7]. Improvements to these bounds were provided bya number of investigators; the work of Hashin and Rosen [8] isbased on energy principles and the studies by Hill [9,10] are basedon the self-consistent scheme. References to recent developmentscan be found in [11–13]. In a companion study [14], the effectivetransversely isotropic properties of a unidirectionally fibre-rein-forced CFRP composite were investigated using a micro-mechani-cal evaluation of the irregular fibre arrangements combined withcomputational simulations. It was shown that the effective trans-versely isotropic elastic properties of the unidirectionally rein-forced composite determined from experimental results togetherwith computational simulations closely matched the results basedon the theoretical relationships proposed by Hashin and Rosen [8].This paper extends the research investigations to the modeling ofrectangular laminated plates, consisting of eleven layers of unidi-rectionally reinforced elements with orthogonal fibre orientations.The plate can experience large deflection behavior during bending.The research also presents results of experimental investigations ofthe flexural behavior of a multi-laminate composite plate witheither fixed or partially fixed boundary conditions, and illustrates

298 H. Nikopour, A.P.S. Selvadurai / Composites: Part B 60 (2014) 297–305

the need for incorporation of the large flexural deflections of theplate to examine experimental results. The requirement for consid-ering the influence of large deflection behavior is demonstratedthrough a comparison of computational and experimental resultsat specified locations of the plate.

Fig. 2. Tensile test setup.

Table 1Mechanical properties of resin matrix and fibre.

Property Specificgravity

Tensilestrength(MPa)

Young’smodulus(GPa)

Ultimatetensile strain(%)

Poisson’sratio

Resin 1.20 78.6 3.10 3.4 0.35Fibre 1.81 2450.4 224.40 1.6 0.20

Table 2Transverse isotropic elasticity properties obtained from Hashin and Rosen [8] and RAE[14] methods.

Hashin and Rosen RAE Percentage difference

E11 (GPa) 149.17 146.26 2.0m12 0.23 0.23 0.0E22 (GPa) 12.72 12.11 5.0

2. Plate lay-up and material properties identification

The fibre-reinforced plates were supplied by Aerospace Com-posite Products (ACP), California, USA. The tested plate measured460.0 mm � 360.0 mm � 2.4 mm. The composite lay-up was iden-tified using a scanning electron microscopy. The fibre area fractionin each layer was determined using the image processing softwareavailable in the MATLAB™ software. The longitudinal elastic prop-erties were measured in the laboratory by conducting tensile testsaccording to the ASTM D3039 standard, while the transverse prop-erties were estimated using a representative area element-basedcomputational approach.

Scanning Electron Microscopy (SEM): The details of the SEMtechnique together with MATLAB™ procedure used to image thescans for development of computational representations of thetransverse section are described in Selvadurai and Nikopour [14].Fig. 1 shows the scanned results for the physical arrangement of fi-bres in the plate. The plate consisted of 11 orthogonally orientedlayers with a lay-up of [(90�/0�)2, 90�, 0�, 90�, (90�/0�)2] relative tothe longitudinal direction of the plate. The fibre volume fractionwas approximately 61%.

Longitudinal Elastic Properties: The longitudinal elastic proper-ties of the CFRP specimens were determined from a series of ten-sion tests (ASTM D3039) on fibre-reinforced specimensmeasuring 200.0 mm � 25.0 mm � 1.4 mm. Two strain gauges, ori-ented at 0� and 90� to the fibre direction, were installed at the cen-ter of the specimens to monitor the longitudinal (1) and transverse(2) strains. These values were then used to calculate one value ofPoisson’s ratio, (m12), applicable to the composite. Fig. 2 showsthe instrumentation and loading grips used in the tension test.The experimentally determined longitudinal Young’s modulus fora single lamina, (E11), was estimated at (138.26 ± 5.26) GPa andPoisson’s ratio, m12, was estimated to be 0.23 ± 0.01. The propertiesof both the fibre and matrix materials used in the fabrication of thecomposite plate were provided by the manufacturer and these arelisted in Table 1.

Transverse Elastic Properties: The transverse Young’s modulus,E22, Poisson’s ratio, m23, and the plane strain shear modulus, G23,were identified using a computational simulation of a two-dimen-sional plane strain Representative Area Element (RAE). Region B, asshown in Fig. 1, was selected as a representative area, and the finiteelement model of that area was developed using the ABAQUS™software. The fibres and the matrix were modeled as isotropic lin-early elastic materials with the elastic constants shown in Table 1.It was also assumed that there was perfect bonding between thefibres and matrix. The RAE was subjected to suitable homogeneous

Region A

100 µm

10 µm

218

µm

Fig. 1. Fibre arrangement as observed from a scanning electron microscope.

m23 0.27 0.29 7.4G23 (GPa) 5.01 4.85 3.3

strains to estimate, through an energy equivalence, the effectivetransverse elasticity properties of the composite. Details of theRAE method used to determine each of the transverse elasticityproperties (E22, m23, and G23) of the composite region are given bySelvadurai and Nikopour [14] and Nikopour [15]. Table 2 showspredictions for the transverse elastic constants using the Hashinand Rosen model [8], which does not take into account any irreg-ularity in the fibre arrangement, and the RAE method [14] thatconsiders irregular fibre arrangements.

Test frame

Upper view of the test plate and steel grips

Bottom view of the test plate and the actuator

Fixed edge

Free edge

CFRP plate

Spherical indentor

Load cell

Actuator

CFRP plate

4mm screw

2m

eter

s

365 mm

Fig. 3. Experimental setup.

B

Potentiometer

CFRP plate

Spherical indentor

Load cell

Electro-mechanical linear actuator

Motor

Emergency shut-off breaker

Steel grip

To computer control system

Control & power supply

Data acquisition system

Data processing computer

Fig. 4. Schematic view of test setup.

CFRP plate

Fixed steel support

h2

Rubber sheet

4 mm Steel bolt

Fixed steel support

h1h1

4 mm Steel boltFixed steel support

25.4

mm

Fixed steel support

38.1 mm

25.4

mm

CFRP plate

(a) Regular fixed boundary condition (b) Partially fixed boundary condition

Fig. 5. Large schematic view of area B in Fig. 4 (h1 = 2.4 mm, h2 = 3 mm before and 2.7 mm after tightening the steel screws).

b/4b/

2

a/4a/2

(a) Central loading (b) Eccentric loading

Y

X

Y

X

Fig. 6. Point loading positions (a = 365 mm; b = 332 mm).

H. Nikopour, A.P.S. Selvadurai / Composites: Part B 60 (2014) 297–305 299

3. Experimental localized loading of a plate

The test facility used in the research is shown in Fig. 3. It was anadaptation and modification of an arrangement used by Selvaduraiand Yu [16]. The facility was modified to accommodate testing offibre-reinforced plates with variable boundary conditions applica-ble to a rectangular region. The boundary conditions includedeither (i) complete fixity or (ii) partial fixity on a combination ofedges. The effective dimensions of the plate measured from theboundary of the plate were 365.0 mm � 332.0 mm � 2.4 mm. Theplate was subjected to a monotonically increasing quasi-static load

Table 3Transversely isotropic stiffness coefficients for a composite layer obtained fromHashin and Rosen [8] and RAE [14] methods.

.

Elastic coefficient D1111 D2222 D1122 D2233 D1212 D2323

RAE method (GPa) 143.77 13.35 4.33 3.97 55.20 4.85Hashin and Rosen (GPa) 142.08 13.86 4.42 3.84 55.20 5.01

(b) Fixed, central loading

(d) Fixed, eccentric loading

Displacement (mm)

Displacement (mm)

Forc

e (N

)Fo

rce

(N)

(a) Mesh configuration fo(Total number of elem

FEM-RAE

Experiment

FEM-Hashin and Rosen

Linear Plate Theory

FEM-RAE

Experiment

FEM-Hashin and Rosen

Linear Plate Theory

Fig. 7. Mesh configuration and experimental and computational results for load vers

300 H. Nikopour, A.P.S. Selvadurai / Composites: Part B 60 (2014) 297–305

applied through a rigid stainless steel spherical indentor withdiameter of 15 mm, which was connected to an electro-mechanicalactuator (Parker™) with a capacity of 3750 N. Fig. 4 shows a sche-matic view of the test setup and steel grips arrangement used inthe experiments. Details of the steel grips and the fixed and par-tially fixed boundary conditions are shown in Fig. 5. Four potenti-ometers (ATech™) with an accuracy of ±0.01 mm werepositioned in the test setup to monitor the deflection of the plateat salient locations. The applied load and the resulting displace-ments were recorded using a digital data acquisition system (Mea-surements COMPUTING™) connected to a computer thatincorporated the TracerDAQ™ software. The fixed boundary condi-tion on an edge was achieved by clamping the CFRP plate directlybetween two steel plates of thickness 25.4 mm, using four 4 mmscrews along each edge. The partial fixity at a boundary was at-tained by incorporating two natural rubber strips with a thicknessof 3 mm on the upper and lower surfaces of CFRP plates as shown

(c) Partially fixed, central loading

(e) Partially fixed, eccentric loading

Displacement (mm)

Displacement (mm)

Forc

e (N

)Fo

rce

(N)

r three fixed edges ents: 13356)

FEM-RAE

Experiment

FEM-Hashin and Rosen

Linear Plate Theory

FEM-RAE

Experiment

FEM-Hashin and Rosen

Linear Plate Theory

us deflection at position of loading for the CFRP plate having three fixed edges.

H. Nikopour, A.P.S. Selvadurai / Composites: Part B 60 (2014) 297–305 301

in Fig. 5(b) to form the boundary condition corresponding to par-tial fixity. At the start of each experiment, the steel screws wheretightened to achieve a uniform initial compression strain of 10%in each rubber strip. In order to identify the mechanical propertiesof the rubber, a compression test was performed following theASTM D945-06 guidelines. In the small strain range up to 30%, rub-ber is considered to be a nearly linearly elastic material with aPoisson’s ratio, mr � 0.5 and Young’s modulus, Er = 1.96 MPa. Therubber displayed no evidence of creep or other time-dependentphenomena and this behavior is consistent with experimental re-sults obtained in other investigations (Selvadurai [17] gives a valueof Gr = 0.8267 MPa where Gr is the shear modulus of the rubber;Selvadurai and Shi [18] give average of Gr = 0.738 MPa). Centraland specific eccentric loading positions were selected for perform-ing flexure testing of the plate (Fig. 6) and sixteen different sets ofboundary conditions included 8 fixed boundary types and 8 par-tially fixed boundary types of arrangements. All tests were per-formed in a displacement control mode in a laboratory where thetemperature was maintained at approximately 20 �C, with a nom-inal variation of ±1 �C. The maximum applied displacement at the

(a) Mesh configuratio(Total number of e

(b) Fixed, central loading

(d) Fixed, eccentric loading

Displacement (mm)

Displacement (mm)

Forc

e (N

)Fo

rce

(N)

FEM-RAE

Experiment

FEM-Hashin and Rosen

Linear Plate Theory

FEM-RAE

Experiment

FEM-Hashin and Rosen

Linear Plate Theory

Fig. 8. Mesh configuration and experimental and computational results for load ver

point of application of the load, Dmax, was limited to 5 mm. Thespecified displacements were applied in a quasi-static manner ata displacement rate of _D ¼ 0:025 mm=s mainly to minimize anydynamic effects. Each test was performed 3 times in both loadingand unloading sequences to establish repeatability of the non-lin-ear load–deflection results.

4. Finite element formulation of large deflection flexure oflaminated plates

It was observed that the load–displacement behavior at thepoint of application of the load displayed a non-linear responseas the deflection increased. This characteristic response is consid-ered to be an effect that results from large flexural deflections ofthe plate. The non-linear theory of plates that accounts for largedeflection effects was developed by von Karman and is well docu-mented in the texts by Timoshenko and Woinowsky–Krieger [19]and Reddy [20]. The fundamental difference between a classicalplate theory for isotropic plates [19–21], and a plate theory for

n for four fixed edges lements: 14418)

(c) Partially fixed, central loading

(e) Partially fixed, eccentric loading

Displacement (mm)

Displacement (mm)

Forc

e (N

)Fo

rce

(N)

FEM-RAE

Experiment

FEM-Hashin and Rosen

Linear Plate Theory

FEM-RAE

Experiment

FEM-Hashin and Rosen

Linear Plate Theory

sus deflection at position of loading for the CFRP plate having four fixed edges.

Table 4Load required to achieve a 5 mm deflection at the point of application of the central load.

Boundary condition Fully fixed edge Partially fixed edge

Experimental (N) Hashin–Rosen (N) RAE (N) Experimental (N) Hashin–Rosen (N) RAE (N)

73 76 74 56 59 58

131 138 135 83 88 86

298 311 303 245 259 251

435 451 442 407 431 422

720 742 735 581 611 598

568 609 599 490 513 506

740 763 755 640 668 655

828 865 852 703 754 742

302 H. Nikopour, A.P.S. Selvadurai / Composites: Part B 60 (2014) 297–305

isotropic plates that experience large deflections and their equiva-lents for composites, is the incorporation of the higher order termsin the rotations and the inclusion of in-plane strains. The theory oflarge deflections of laminated plates has also been investigated byBathe and Bolourchi [22], Kam et al. [23], Akhras and Li [24,25],Akhras et al. [26], Tanaka et al. [27] and Selvadurai and Nikopour[28]. A brief presentation of the basic equations is given inAppendix A.

5. Computational modeling of plates

The equations governing large deflection analysis of plates iscurrently an available option in the general-purpose finite elementcode ABAQUS™. Stiffness coefficients in the plane of isotropy[29,30] were determined using a computational approach [14]and each layer was modeled as a homogenous orthotropicmaterial. Perfect bonding was assumed to exist between compos-ite-to-composite layers. Table 3 presents the elastic orthotropic

Table 5Load required to achieve a 5 mm deflection at the point of application of the eccentric loa

Boundary condition Fully fixed edge

Experimental (N) Hashin–Rosen (N) R

29 31

54 57

103 110

641 673

1078 1143 1

675 712

1079 1157 1

1405 1481 1

coefficients calculated using the effective estimates given byHashin and Rosen [8] and those derived computationally usingthe RAE approach [14]. Upper and lower surfaces of the steel gripswere considered to be completely fixed and bonded to the compos-ite plate. The electro-mechanical actuator was loaded until theprescribed plate deflection of 5 mm was attained. The load appliedduring an experiment was monitored using an Omega™ load cellwhich had a maximum load capacity of 375 kN and an accuracyof ±1%. In the case of plates with partial boundary fixity, the platewas clamped until the strain in the individual rubber strips(Fig. 5(b)) was approximately 10%. The computational modelingof the entire system including the test plate, the rubber stripsand the steel grips was performed using a standard 9-nodeLagrangian element available in the element library of ABAQUS™.A convergence study was conducted to determine the convergenceof the method in relation to the mesh size. Mesh configuration fortwo types of boundary conditions are presented in Fig. 7(a) andFig. 8(a). Since several variables are involved in the modeling,including (i) the plate boundary conditions (ii) point of application

d.

Partially fixed edge

AE (N) Experimental (N) Hashin–Rosen (N) RAE (N)

30 23 25 24

56 52 55 54

107 59 62 61

661 493 528 520

130 810 847 835

689 531 561 547

136 820 857 845

467 1204 1271 1241

5 mm

5 mm

14.27 mm

13.69 mm

F=73 N

F= 56 N

(a) Fixed condition on one boundary

(b) Partially fixed condition on one boundary

CFRP Plate

CFRP PlateRubber sheet

182.5 mm

Fig. 9. Deflection curves under central loading for: .

H. Nikopour, A.P.S. Selvadurai / Composites: Part B 60 (2014) 297–305 303

of the load (iii) the Hashin and Rosen estimates for the elastic con-stants and (iv) the RAE-based estimates of effective elastic proper-ties, attention will be focused on the presentation of results thathighlight the influence of boundary fixity on the load–displace-ment response.

6. Computational and experimental results

Table 4 summarizes the computational and experimental re-sults for the loads required to attain a maximum deflection of5 mm at the point of application of the load for a plate with fixedand partially fixed boundary conditions and subjected to centralloading. Table 5 presents similar results for plates under non-sym-metric loading. Comparison of experimental and computational re-sults for the load–deflection behavior for the eight selected loadingconfigurations with various combinations of boundary fixity, arepresented in Figs. 7 and 8.

As expected, plate configurations with partial boundary fixityconditions required a lower maximum load to obtain a pre-scribed deflection when compared to plate configurations withfixed boundary conditions. There was negligible hysteresis dur-ing each loading–unloading cycle and energy dissipation didnot change significantly when rubber supports were incorpo-rated at the boundaries. Plates with higher stiffness and no rub-ber supports showed a greater nonlinear load–deflection trendcompared to less stiff plates. This characteristic response is con-sidered to be an effect that results from large deflections of theplate.

For predictions of the load required to attain the 5 mm deflec-tion, the FEM-RAE method provided results slightly closer to theexperimental values compared to the results derived from theFEM-Hashin–Rosen technique. The FEM-RAE method had a maxi-mum error of 5.6% at peak deflection, whereas the FEM-Hashin–Rosen method had a maximum error of 7.3%, for the case of fourpartially fixed edge boundary conditions. Since the only variablein the modeling was the effective properties, it can be concludedthat the RAE method [14] provides a more accurate estimate ofcomposite elastic properties at the micro-scale compared to theidealized model proposed by Hashin and Rosen [8].

Fig. 9 shows deflection curves for plates where the loading iscentral and the edges have either a fixed or partially fixed condi-tion. The maximum deflection at the free edges and the load re-quired to attain a deflection of 5 mm at the center of thepartially fixed boundary configuration were, respectively, 4.2%and 30.3% lower compared to the plate with the regular fixedboundary.

7. Conclusions

The mechanical behavior of a CFRP composite plate, with vari-ous fixed and partially fixed boundary conditions, and subjectedto either localized central or eccentric transverse loading wasexamined experimentally and modeled computationally. It wasfound that the RAE method developed for estimating the elasticproperties of uni-directional fibre reinforced plates provides reli-able estimates for the finite element modeling of the compositeplate at the macro-scale. The limitation of the RAE approach isthe need to perform micro-mechanical assessment (SEM) of the fi-bre arrangements that would enable the estimation of the effectiveorthotropic elastic constants based on reliable fibre configurations.The RAE approach, however, accounts for geometric features of thefibre configuration, which is absent in the Hashin–Rosen type esti-mates for effective properties. The fact that both approaches givevery similar results suggests that the theoretical estimates can beused with confidence in instances where SEM data are unavailable.The other important observation is that loading configurationswith higher stiffness and fixed boundaries were more likely to ex-hibit a nonlinear trend in the applied load-transverse deflection.Installation of rubber strips had little effect on the energy dissipa-tion of composite plates under quasi-static cyclic loading; how-ever, it had a significant effect in reducing the maximum load inthe CFRP plates.

Acknowledgments

The work described in the paper was supported by a NSERC Dis-covery Grant awarded to A.P.S. Selvadurai. The authors are gratefulto research assistants, Brian Siciliani and Adrian Głowacki, for theirassistance in the fabrication of the experimental setup. The com-posite plates used in this research were supplied by AerospaceComposite Products, CA, USA; the interest and support of Mr. JustinSparr, Executive Vice President, of ACP is greatly acknowledged.

Appendix A

For finite element formulation of large deflections of the plate,the origin of the x- and y-coordinates used is taken as the centerof the plate (Fig. 6). The displacement field in a thin laminatedplate undergoing large deflections is assumed to be of the form:

uxðx; y; zÞ ¼ u0ðx; yÞ þ zwxðx; yÞuyðx; y; zÞ ¼ v0ðx; yÞ þ zwyðx; yÞuzðx; y; zÞ ¼ wðx; yÞ

ðA:1Þ

304 H. Nikopour, A.P.S. Selvadurai / Composites: Part B 60 (2014) 297–305

where ux, uy, uz are the deflections in the x, y, z directions, respec-tively, u0, v0, w are the associated mid-plane deflections, and wx

and wy the rotations due to shear. The strain–displacement rela-tions in the von Karman plate theory can be expressed in the form[22,27,28]:

ex ¼@u0

@xþ 1

2@w@x

� �2

þ z@wx

@x¼ e0

x þ zjx

ey ¼@v0

@yþ 1

2@w@y

� �2

þ z@wy

@y¼ e0

y þ zjy

es ¼@u0

@yþ @v0

@xþ @w@x

@w@yþ z

@wx

@yþ@wy

@x

� �¼ e0

s þ zjs

e4 ¼ wy þ@w@y

e5 ¼ wx þ@w@x

ðA:2Þ

where e0i ði ¼ x; y; sÞ are the in-plane strains; ej(j = 4, 5) are transverse

shear strains, and ji(i = x, y, s) are bending curvatures. The associ-ated symmetric second Piola-Kirchoff stress vector r is

r ¼ ½rx;ry;rs;r4;r5�T ðA:3Þ

The constitutive equations for the plate can be written as

Ni

Mi

� �¼

Aij Bij

Bij Dij

� �e0

j

ji

( )ði; j ¼ x; y; sÞ ðA:4aÞ

Q 2

Q 1

� �¼ A44 A45

A45 A55

" #e0

j

ji

( )ði; j ¼ x; y; sÞ ðA:4bÞ

The components Aij, Bij, Dij(i, j = x, y, s) and Aijði; j ¼ 4;5Þ are thein-plane, bending coupling, bending or twisting, and thickness-shear stiffness coefficients. Ni, Mi, Q1 and Q2 are the stress resul-tants defined by

ðNi;MiÞ ¼Z h=2

�h=2ð1; zÞridz ðA:5aÞ

ðQ1;Q 2Þ ¼Z h=2

�h=2ðr5;r4Þdz ðA:5bÞ

and

ðAij;Bij;DijÞ ¼XNL

m¼1

Z zmþ1

zm

Q ðmÞij ð1; z; z2Þdz; ði; j ¼ x; y; sÞ

Aij ¼XNL

m¼1

Z zmþ1

zm

kakbQ ðmÞij dz; ði; j ¼ 4;5; a ¼ 6; b ¼ 6� jÞ ðA:6Þ

where zm is the distance from the mid-plane to the lower surface ofthe mth layer, NL is the total number of layers, Qij are material con-stants, and ka are the shear correction coefficients which are set as

[28] k1 ¼ k2 ¼ffiffi56

q.

The basis of the formulation of the governing equations of theplate is the principle of minimum total potential energy, in whichthe total potential energy p of the system is expressed as the sumof strain energy, U, and the potential energy, P:

p ¼ U þ P ðA:7Þ

where

U ¼ 12

ZVo

eTrdV ðA:8Þ

and

P ¼ �Z

Xo

qðx; yÞwðx; yÞdX ðA:9Þ

where Vo is the plate volume, q(x, y) the distributed load intensity,and Xo the plate region.

Using the strain-deflection Eq. (A.2) and the constitutiveEq. (A.4), Eq. (A.8) can also be expressed as a function of mid-planedisplacements. Performing the through thickness integration, thestrain energy can be rewritten as [28]:

U ¼ 12

ZZe0T Ae0 þ 2e0T Bjþ jT Djþ cT Ach i

dxdy ðA:10Þ

Considering the laminated composite plate discretized into NEelements, the strain energy and potential energy of the plate canbe expressed in the form:

U ¼XNE

i¼1

Ue

¼ 12

XNE

i¼1

ZXe

e0TAe0 þ 2e0TBjþ jTDjþ cTAch i

dX� �

i

ðA:11Þ

and

P ¼XNE

i¼1

ZXe

qðx; yÞwðx; yÞdX� �

i

ðA:12Þ

where Xe, Ue are, respectively, the element area and the strain en-ergy per element.

The mid-plane displacements and rotations (u0, v0, w, wx, wy)within an element are given as a function of 5 � n discrete nodaldeflections and in matrix form they are:

u ¼Xn

i¼1

½UiI�rei ¼ U ~re ðA:13Þ

where n is the number of nodes of the element; Ui are the shapefunctions; I is a 5 � 5 unit matrix; U is the shape function matrix;~re ¼ fre1;re2; . . . ;reqgT; and the nodal displacements rei at anode are:

rei ¼ fu0i;v0i;wi;wxi;wyigT; i ¼ 1; . . . ;n

The first variation of Eq. (A.7) in terms of the nodal displace-ments can be expressed as:

dp ¼ dðU þ PÞ ¼XNE

i¼1

d ~rTe Fe

~re

h ii�XNE

i¼1

½d ~rTe Pe�i ¼ 0 ðA:14Þ

where Fe~re and, Pe are the element internal and nodal force vectors,

respectively.

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