Refined First-Order Shear

Fe

e

e

9

Downloaded Ftheories: the equivalent single-layer theories ~ESLTs! and the lay-erwise theories ~LWTs!.

The ESLTs represent the direct extension of plate theories tolaminates, so that the laminate is reduced to a single-layer platewith equivalent anisotropic material properties. In fact, the classi-cal laminate theory ~CLT!, @3,4#, is an extension of the classicalplate theory based on Kirchhoff-Love assumptions, i.e., it neglectsthe shear deformation in the thickness of the laminate. The first-order shear deformation theory ~FSDT! is an extension of theReissner, @5#, and Mindlin, @6#, plate models to the case of lami-nated anisotropic plates. The FSDT presented in Refs. @7,8# allowsthe determination of satisfactory solutions for a wide class oflaminate problems. In particular, accurate results are obtained ifproper values of the shear correction factors are adopted. Unfor-tunately, the exact values of the shear correction factors are knowna priori only for very simple cases, @9#. To overcome this diffi-

higher-order terms of the thickness coordinate in the representa-tion form of the displacements.

The LWTs are obtained assuming independent shear deforma-tion within each laminate layer, @2125#, so that the displacementfield is continuous in the thickness, while the transverse shearstrain can be discontinuous along the out-of-plane coordinate. Theunknown functions for the LWTs depend on the number of layersin the laminate. A layerwise finite element formulation, which canbe implemented in FEA commercial codes, was proposed in Ref.@26#; therein, Barbero discretized each layer in displacement-based three-dimensional elements with two-dimensional kine-matic constraints.

The zig-zag theories are deduced from the LWT enforcing thecontinuity of the out-of-plane shear stresses; thus, the number ofunknowns in the zig-zag theories do not depend on the number oflayers, @2730#. In particular, Carrera @30# developed multilayerand zig-zag theories in the framework of the Reissner mixedvariational theorem.

Among the several laminate theories, the FSDT appears simpleand efficient for many structural problems. In fact, as emphasizedabove, FSDT is able to predict the response of laminates withsatisfactory approximations for most structural problems. On theother hand, finite element commercial codes, mainly based ondisplacement formulations, requires as input data the values of theshear correction factors. Moreover, the displacement formulations

Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OFMECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED ME-CHANICS. Manuscript received by the ASME Applied Mechanics Division, March 3,2002; final revision, October 4, 2002. Associate Editor: M.-J. Pindera.Discussion on the paper should be addressed to the Editor, Prof. Robert M. McMeek-ing, Department of Mechanical and Environmental Engineering University ofCaliforniaSanta Barbara, Santa Barbara, CA 93106-5070, and will be accepted untilfour months after final publication of the paper itself in the ASME JOURNAL OFAPPLIED MECHANICS.

Copyright 2003 by ASMEJournal of Applied Mechanics MAY 2003, Vol. 70 381F. AuricchioDipartimento di Meccanica Strutturale,

Universita` di Pavia,Via Ferrata 1,

27100 Pavia, Italye-mail: [email protected]

E. SaccoDipartimento di Meccanica, Stuutture, A. & T.,

Universita` di Cassino,Via Di Biasio 43,

03043 Cassino, Italye-mail: [email protected]

RefinedDeformaComposiIn the present work, ntion laminate theory avariables of the problindependent piecewisdimensional equilibriuThe developed formulathe use of shear corrderived without post-pin order to verify thesolutions obtained usdimensional solution,element results. Finallbase for the developm@DOI: 10.1115/1.1572

1 IntroductionThe modeling of composite laminated structures is one of the

most active research fields of the last decades, since accuratestress analyses are required to design structural parts of mechani-cal, naval, aeronautical, and aerospace, as well as civil construc-tions.

In fact, composite laminates present an anisotropic response,with extension-bending coupling and non-negligible shear defor-mations in the thickness. Furthermore, to prevent the developmentof the delamination, which strongly limit the performances ofcomposites, @1#, an accurate evaluation of the interlaminar out-of-plane stresses ~i.e., the shear stress and the normal stress in thethickness direction at the interface between two adjacent laminae!is required.

Actually, several laminate models are available in the literature,@2#. In particular, two different approaches may be distinguishedin the laminate modeling, which lead to two classes of laminaterom: http://appliedmechanics.asmedigitalcollection.asme.org/ on 02/06/20irst-Order Sheartion Theory Models forte Laminatesw mixed variational formulations for a first-order shear deforma-re proposed. The out-of-plane stresses are considered as primaryem. In particular, the shear stress profile is represented either by

quadratic functions in the thickness or by satisfying the three-m equations written in terms of midplane strains and curvatures.tions are characterized by several advantages: They do not requireection factors as well as the out-of-plane shear stresses can berocessing procedures. Some numerical applications are presentedffectiveness of the proposed formulations. In particular, analyticaling the developed models are compared with the exact three-with other classical laminate analytical solutions and with finitey, we note that the proposed formulations may represent a rationalent of effective finite elements for composite laminates.01#

culty, two different approaches can be found in literature. The firstapproach consists in the development of iterative predictor-corrector techniques, as proposed by Noor and co-workers, @1012#. Numerical procedures, developed within the finite elementmethod, were proposed in Refs. @13,14#, where new effectivelaminate elements were presented. The second approach is basedon the refinement of the FSDT model, e.g., see Refs. @15,16#.Within this context, Rolfes and Rohwer @17# and Rolfes @18# pro-posed an improved composite finite element based on FSDT,which does not need the computation of the shear correction fac-tors. They computed the transverse shear stresses from equilib-rium equations and they were able to determine a priori the shearprofile introducing suitable simplifications, i.e., neglecting thepresence of membrane forces and assuming two simultaneous cy-lindrical bending modes.

Several higher-order shear deformation theories were devel-oped in the literature, @19,20#, within the ESLTs; they consider14 Terms of Use: http://asme.org/terms

of plates and laminate theories are able to recover satisfactoryvalues for the in-plane stresses, while out-of-plane shear and nor-

where the plane z50 identifies the midplane A of the undeformedplate. The laminate is made of n layers and the typical kth layer

Downloaded Fmal stresses are obtained after manipulations of the in-plane re-sults by post-processing the solution, @13,14,31,32#. Post-processing techniques are generally simple and often efficient.Within finite element formulation the use of these techniques isnot always straightforward. In fact, the determination of the shearstresses from equilibrium equations requires the computation ofthe in-plane stress ~resultant axial forces and bending moments!derivatives. This can be accomplished developing mixed formula-tions, or displacement-based laminate finite elements character-ized by high-degree polynomial interpolation functions, @13#. Thestress derivatives can be also computed performing regularizationof the extensional and flexural strain, @14#. The so-called extendedtwo-dimensional method presented by Rolfes et al. @32# does notrequires the computation of the stress derivatives; in fact, the as-sumption of neglecting the membrane forces and of consideringthe presence of two simultaneous cylindrical bending simplify thelaminate equations, so that the bending moment derivatives areequal to the resultant shear stresses.

Furthermore, it can be emphasized that the FSDT allows todetermine satisfactory values for the in-plane and out-of-planestresses. Thus, once the most stressed zones of the laminate aredetermined using the FSDT, layerwise or zig-zig theories can beadopted in these zone to investigate on the possible delaminationand failure.

Aim of the present paper is the development of suitable andviable laminate models based on the equivalent single layertheory. In particular, refined FSDT models, based on new partialmixed formulations, are developed, without introducing any sim-plification on the laminate problem.

The following features characterize the proposed approach:

It does not need shear correction factors. It does not need to post-process the in-plane solution to get

out-of-plane shear stresses. It may represent the basis for the development of new and

efficient laminate finite elements.

The proposed approach is based on a variational formulationthat considers the out-of-plane shear stresses as primarily vari-ables of the problem. A new approach is proposed; in fact, theshear stress profile introduced in the partial mixed functional isobtained considering new independent variables or it is deducedfrom the three-dimensional local equilibrium equations. In fact,the explicit expression of the shear stresses is obtained by inte-grating the first two equilibrium equations with respect to thethickness direction. Thus, the shear stresses are expressed as func-tions of the in-plane stresses, which can be written as functionseither of the in-plane strains or of the displacement and rotationfields. Hence, several formulations are obtained. In order to assessthe performances of the proposed models, analytical solutions aredetermined for the proposed models. It can be emphasized thatanalytical solutions are available only for special cases; in fact,simply supported rectangular cross-ply and angle-ply laminatesare considered within the paper. The solutions computed for theproposed models are compared with the exact three-dimensionalsolution, @33#, with other classical laminate analytical solutions,@2#, and with finite element results, @13#.

In the following the subscript comma indicates the partial de-rivative f

,a5] f /]xa and f ,z5] f /]z .

2 First-Order Shear Deformation Theory FSDTLaminate Model

A laminate plate V refers to a flat body, with constant thicknessh:

V5H ~x1 ,x2 ,z !PR3/zPS 2 h2 , h2 D ,~x1 ,x2!PA,R2J (1)382 Vol. 70, MAY 2003rom: http://appliedmechanics.asmedigitalcollection.asme.org/ on 02/06/20lies between the thickness coordinates z5zk and z5zk11 , suchthat z152h/2 and zn115h/2.

The first-order shear deformation theory can be obtained intro-ducing suitable assumptions on both the strain and the stress fieldsdefined in the three-dimensional continuous body V, as empha-sized in @34# for the case of homogeneous plates. In fact, theFSDT for laminated plates is based on the following well-knownassumptions, @35#:

1. The through-the-thickness transverse normal stress is nil,i.e., szz50.

2. Straight lines perpendicular to the midplane cannot bestretched, i.e., zz50.

3. Straight lines perpendicular to the midplane remain straightafter deformation, i.e., 1z ,z52z ,z50.

Displacement Field. The kinematics is restrained to satisfythe following conditions:

zz50 1z ,z52z ,z50 (2)which leads to the classical representation form for the displace-ment field:

s~x1 ,x2 ,z !5u~x1 ,x2!1zw~x1 ,x2!

sz~x1 ,x2 ,z !5w~x1 ,x2! (3)where

s5 H s1s2J u5 H u1u2J w5 Hw1w2J (4)with u and w the vectors of the midplane membrane displace-ments and rotations, respectively.

Strain Field. Denoting by i j the typical component of thestrain tensor, the in-plane strain vector 5$11 22 212%T, asso-ciated to the displacement representation ~3!, is written as

5e1zk (5)where the membrane strain vector e and the curvature strain vec-tor k are given by

e5Lu k5Lw L5F ]]x1 00 ]]x2]

]x2

]

]x1

G . (6)The in-plane strain vector is a linear function of the thicknesscoordinate z.

The out-of-plane strain field vector g5$21z 22z%T is obtainedas

g5w1w (7)where the symbol indicates the gradient operator.

Stress Field. The in-plane stresses within each lamina of thecomposite laminate are computed using the constitutive relation-ships. In particular, it is assumed that the body V is obtainedassembling in a staking sequence orthotropic layers, with z50representing a plane of material symmetry. Thus, denoting by s i jthe typical component of the stress tensor, the in-plane stress vec-tor sk5$s11

k s22k s12

k %T for the kth lamina is given by

sk5Ck5Ck~e1zk! (8)where Ck is the so-called reduced in-plane constitutive elasticmatrix associated to the kth lamina. Note that Ck is derived from

Transactions of the ASME14 Terms of Use: http://asme.org/terms

the three-dimensional constitutive matrix, enforcing the condition1, i.e., szz50, @34#. Since each lamina of the laminate presents

3 Variational Formulation

Downloaded Fdifferent elastic properties, the in-plane stress vector s is a dis-continuous piecewise linear function of the coordinate z.

The out-of-plane shear stress vector uk5$u1zk u2z

k %T can beevaluated through constitutive equation as

uk5Qkg (9)where Qabk 5xabQ abk with a ,b51,2. Note that Q abk are the com-ponents of the shear elastic matrix of the kth lamina and xab arethe shear correction factors. As it is well known, the transverseshear stress vector computed by formula ~9! is absolutely unsatis-factory; in fact, formula ~9! leads to a transverse shear stress fieldwhich is not equilibrated at the interfaces of adjacent laminae andit does not satisfy the boundary conditions on the top and on thebottom of the laminate.

A satisfactory field for the out-of-plane shear stress vector t5$t1z

k t2zk %T can be recovered using the equilibrium equations; in

the following no body forces and no tangential surface forces onthe top and bottom of the laminate are considered; thus, the equi-librium equations give

t52E2h/2

z

LTsd , (10)

i.e., in components

taz52E2h/2

z

~sa1,11sa2,2!d with a51,2. (11)

Note that, according to formula ~11!, it implicitly resultst(2h/2)50; the further boundary condition t(h/2)50 has alsoto be satisfied.

Once the shear stresses in the laminate thickness are deter-mined, the transverse normal stress, which is very important forthe delamination failure, can be determined adopting a post-processing method, i.e., by integrating with respect to z the thirdequilibrium equation:

szz52E2h/2

z

~t1z ,11t2z ,2!d . (12)

It can be emphasized that the in-plane strain components ~5! arelinear functions of the z thickness coordinate, so that the in-planestresses ~8! are piecewise linear functions of z. As a consequence,the transverse shear stresses computed by the equilibrium Eq. ~10!are piecewise quadratic functions. Since the piecewise quadraticshear stress profiles computed by Eq. ~10! are widely recognizedas the best transverse shear stresses evaluation, within the FSDTmodel, it can be assumed as basis for the model construction thecondition that the through-the-thickness shear stresses saz arecontinuous piecewise quadratic functions of the z-coordinate.

A classical problem arising in conjunction with the use of theFSDT is the determination of the shear factors x11 , x22 , and x12appearing in the matrix Qk of Eq. ~9!. Denoting by Et and Eu thecomplementary shear energies in the thickness obtained consider-ing the t and u shear profiles, respectively, characterized by thesame resultant shear stress, the shear correction factors are deter-mined enforcing Et5Eu. The exact values of x11 , x22 , and x12can be evaluated analytically only for special cases. In particular,Whitney @9# derived an analytical formula of the shear correctionfactor for the case of cross-ply laminates in cylindrical bending.For a more general case, it is possible to evaluate the shear cor-rection factors developing an iterative procedure. This can bebased on the determination of the displacement solution, of in-plane stresses computation via constitutive equations, of theequilibrated shear stresses, of the complementary shear energiesand, finally, of new shear correction factors which are adopted tocompute a new solution, and so on, @13#.

Journal of Applied Mechanicsrom: http://appliedmechanics.asmedigitalcollection.asme.org/ on 02/06/20A mixed functional for the three-dimensional laminate-likebody V is now considered:

H~u,w ,w,e,k,g,s,t!5Hmb~u,w,e,k,s!1Hs~w ,w,g,t!2Pext(13)

where Hmb is a Hu-Washizu functional accounting for themembrane-bending terms, Hs is a Prange-Hellinger-Reissnerfunctional accounting for the transverse shear terms and Pext ac-counts for the boundary conditions and loading forces.

In particular, the membrane-bending functional Hmb and thetransverse shear functional Hs are written as

Hmb~u,w,e,k,s!512 EV~e1zk!TC~e1zk!dv1E

V@~Lu2e!1z~Lw2k!#Tsdv (14)

Hs~w ,w,t!5EV

~w1w!Ttdv212 EVtTTtdv (15)

where Tk5(Qk)21 is the shear compliance matrix of the kth layer.Performing the integration along the thickness coordinate, the

membrane-bending mixed functional ~14! takes the form

H mb~u,w,e,k,N,M!512 EA~eTAe12eTBk1kTDk!dA1E

A$@~Lu!2e#TN1@~Lw!2k#TM%dA .

(16)The matrices A, B, and D represent the membrane, the membrane-bending coupling and the bending elastic stiffness matrices of a nlayer laminate, respectively, defined by equations

A5(k51

n

Ck~zk112zk! (17)

B512 (k51

n

Ck~zk112 2zk2! (18)

D513 (k51

n

Ck~zk113 2zk3!. (19)

Moreover, the resultant membrane force and bending momentvectors, N and M, are defined as

N5E2h/2

h/2sdz M5E

2h/2

h/2zsdz . (20)

Because of the constitutive Eqs. ~8!, taking into account the defi-nitions ~17!, ~18!, and ~19!, it results:

N5Ae1Bk M5Be1Dk. (21)Hence, the mixed functional ~13! for the laminate can be writ-

ten in terms of the introduced resultant forces and bending mo-ments as

H ~u,w ,w,e,k,g,N,M,t!5H mb~u,w,e,k,N,M!

1Hs~w ,w,g,t!2Pext . (22)A full displacement formulation of the membrane-bending

functional H mb is recovered, implicitly satisfying the resultantconstitutive laws ~21! and the compatibility Eqs. ~6!:

MAY 2003, Vol. 70 38314 Terms of Use: http://asme.org/terms

Emb~u,w!5 1 E @~Lu!TALu12~Lu!TBLw1~Lw!TDLw#dA . 05]tHs~w ,w,t!5E ~w1w!Tdtdv2 1 E tTTdtdv

Downloaded F2 A(23)

It can be emphasized that different laminate models can berecovered depending on the expression considered for thethrough-the-thickness shear stress. In particular, two classes ofmodels are herein considered:

The shear stress profile are approximated introducing inde-pendent variables.

The shear stress profile are written as function of the mid-plane strains and curvatures using the equilibrium Eqs. ~10!.

In the following, four variational formulations of the laminateproblem are derived, considering different representation forms ofthe out-of-plane shear stress vector.

4 Independent Approximation of the Shear StressesThe first refined model, denoted in the following as RM1, is

derived considering independent approximations of the shearstresses. In fact, the shear stress profile is represented as a con-tinuous piecewise quadratic function in the thickness, satisfyingthe boundary conditions. Hence, within the kth layer, it is assumed

tk5tok zk112z

zk112zk1to

k11 z2zkzk112zk

2tk~zk112z !~z2zk! (24)

with to15to

n1150. In the formula ~24! tok represents the shear stressvector at the interface between the layers k21 and k, while tkgives the curvature profile of the shear stress in kth layer.

Introducing the representation formula ~24! in the transverseshear energy ~15!, it applies

H s5212 E A(k51

n

@ ~Rktok !Ttok1~Rktok11!Ttok2~R ktk!Ttok#dA

1212 EA(k51

n

@ ~Rktok11!Ttok112~R ktk!Ttok11#dA

1212 EA(k51

n

@~R ktk!Ttk#dA1EA~w

1w!T(k51

nzk112zk

2 S tok1tok112tk ~zk112zk!2

3 D dA (25)where

Rk5zk112zk

3 Tk R k5

~zk112zk!3

6 Tk R k5

~zk112zk!5

30 Tk.

(26)Finally, the mixed functional ~22! takes the form

H ~u,w ,w,e,k,N,M,to2,. . ,to

n,t1,. . ,tn!

5H mb~u,w,e,k,N,M!1H s~w ,w,to2,. . ,to

n,t1,. . ,tn!2Pext .

(27)The number of the unknowns in the FSDT refined model RM1

depends on the number of layers. Since the transverse shear stressprofile does not depend on the in-plane stresses, the membraneand bending terms can be written adopting a full displacementapproach functional, substituting H mb with Emb in Eq. ~27!.

The presented approach leads to serious drawbacks. In fact, thestationary condition of the mixed functional ~22! with respect tothe shear stress t gives

384 Vol. 70, MAY 2003rom: http://appliedmechanics.asmedigitalcollection.asme.org/ on 02/06/20V 2 V(28)

i.e., Tktk5w1w; thus, Eq. ~28! represents the complementaryconstitutive equation written in variational form. Because of thedisplacement representation form ~3! for the FSDT, the secondterm of Eq. ~28! is constant in the thickness, so that the shearstress vector t is enforced to be piecewise constant in the thick-ness. When the stress t is represented by formula ~24!, the varia-tional Eq. ~28! enforces the constitutive law in approximatedform; enlarging the space of the shear parameters to

2,. . ,to

n,t1,. . ,tn,

i.e., increasing the number of independent parameters defining thestress t given by formula ~24!, the constitutive Eq. ~28! tends tobe enforced in a stronger manner, so that the shear stress profiletends to become piecewise constant in the laminate thickness.

5 Equilibrated Shear Stress5.1 Shear Stress Computation. The out-of-plane shear

stress t, computed using the equilibrium Eqs. ~10!, is a continuouspiecewise quadratic function of the thickness coordinate; thus, thetransverse shear stress tk at the kth lamina is given by

tk~z !52Ezk

z

LTskd1tok (29)

where tok is the stress evaluated at z5zk :

tok52E

2h/2

zkLTsd . (30)

Substituting the expression ~8! into the formula ~30! gives

tok52E

2h/2

zkLTC~e1k!d52LT~A ke1B kk! (31)

with

A k5(i51

k21

Ci~zi112zi! B k512 (i51

k21

Ci~zi112 2zi2!. (32)

Then, taking into account expressions ~8! and ~31! and perform-ing the integration in the thickness, the out-of-plane shear stress tk~29! becomes

tk~z !52LTF ~z2zk!Cke1 12 ~z22zk2!CkkG2LT~A ke1B kk!52LT~Ak~z !e1Bk~z !k! (33)

where

Ak~z !5~z2zk!Ck1A k

Bk~z !512 ~z

22zk2!Ck1B k. (34)

Moreover, in order to satisfy the boundary condition t(h/2)50 exactly, the formula ~33! is enhanced by adding a linear termwhich is zero at z52h/2:

tk~z !52LT~Ak~z !e1Bk~z !k!1aS z1 12 h D . (35)The vector a is evaluated enforcing the boundary condition:

05tn~h/2!52LT~Ae1Bk!1ah (36)where An(h/2)5A and Bn(h/2)5B are the membrane andmembrane-bending coupling elastic matrices of the laminate de-fined by Eqs. ~17! and ~18!, respectively. Solving Eq. ~36! withrespect to the vector a, we obtain


a51

LT~Ae1Bk!. (37)tk~z !52LT@~Ak~0 !1zAk~1 !!e1~Bk~0 !1zBk~1 !1z2Bk~2 !!k#

(39)

A

L

F k11 k k11 k G

k~2 !

Downloaded F1 4 ~zk112zk !B GYk~2 !52F13 ~zk113 2zk3!Bk~0 !1 14 ~zk114 2zk4!Bk~1 !

115 ~zk11

5 2zk5!Bk~2 !G .

The mixed functional ~22! for the model RM2 takes the form

H ~u,w ,w,e,k,N,M!5H mb~u,w,e,k,N,M!1H s~w ,w,e,k!2Pext .(45)

The number of the unknowns in the FSDT refined RM2 modeldoes not depend on the number of layers; this fact represents anadvantageous feature of the proposed RM2 formulation with re-spect to the RM1 model.

Journal of Applied Mechanicsrom: http://appliedmechanics.asmedigitalcollection.asme.org/ on 02/06/20E~u,w ,w,e,k!5Emb~u,w!1H s~w ,w,e,k!1L~u,w,e,k!2Pext(46)

where the penalty term L is defined by

L51

2h EA@~Lu2e!T~Lu2e!1~Lw2k!T~Lw2k!#dA (47)

with h the penalty parameter.

MAY 2003, Vol. 70 3852

Xk~1 !52F12 ~zk112 2zk2!Ak~0 !1 13 ~zk113 2zk3!Ak~1 !GYk~0 !52F ~zk112zk!Bk~0 !1 12 ~zk112 2zk2!Bk~1 !

113 ~zk11

3 2zk3!Bk~2 !G (44)

Yk~1 !52F12 ~zk112 2zk2!Bk~0 !1 13 ~zk113 2zk3!Bk~1 !1 4 4

formulation. In fact, functional ~45! depends on five displacementparameters (u1 ,u2 ,w ,w1 ,w2), on six midplane strains and curva-tures (e11 ,e22 ,e12 ,k11 ,k22 ,k12) on six axial and bending result-ants (N11 ,N22 ,N12 ,M 11 ,M 22 ,M 12).

As matter of fact, in the membrane-bending Hu-Washizu func-tional H mb, defined by Eq. ~16!, the resultant stress vectors N andM can be regarded as the Lagrange multipliers of the constraintscorresponding to the compatibility Eqs. ~6!. The FSDT refinedmodel RM3 is obtained implicitly satisfying the compatibilityEqs. ~6! in the membrane-bending Hu-Washizu functional ~16!;thus, the full displacement membrane-bending functional ~23! isobtained. Moreover, the penalty approach is adopted to enforcethe constraint ~6! in the mixed functional ~42!; in fact, a penaltyterm is added into the governing functional, which results inh

Substituting expression ~37! into formula ~35!, the out-of-planeshear stress tk takes the form

tk~z !52LTH FAk~z !2 1h S z1 12 h DAGe1FBk~z !2 1h S z1 12 h DBGkJ . (38)

A suitable form for the expression of the out-of-plane shearstress tk, useful for the next developments, is proposed. In fact,the formula ~38! can be rewritten in the following equivalentform:

H s5EA(k51

n H 12 TkLTXk~0 !e1TkLTYk~0 !kJT

LTAk~0 !edA1E1

1

2 EA(k51n

@TkLTYk~0 !k#TLTBk~0 !kdA11

2 EA(k51n

@Tk

1EA~w1w!T(

k51

n

~LTXk~0 !e1LTYk~0 !k!dA

where

Xk~0 !52 ~z 2z !Ak~0 !11

~z2 2z2!Ak~1 ! (43)where

Ak~0 !5A k2zkCk212 A A

k~1 !5Ck21h A (40)

Bk~0 !5B k212 zk

2Ck212 B B

k~1 !521h B B

k~2 !512 C

k.

(41)Next, several refined FSDT laminate formulations, based on the

use of equilibrated shear stresses, are derived.

5.2 Refined Model RM2. The FSDT refined model RM2 isdeduced substituting the expression ~39! of the transverse shearstress vector t, obtained from the three-dimensional equilibriumequations, into the mixed shear functional H s defined in ~15!.Thus,

(k51

n H 12 TkLTXk~1 !e1TkLTYk~1 !kJT

LTAk~1 !edA

TYk~1 !k#TLTBk~1 !kdA11

2 EA(k51n

@TkLTYk~2 !k#TLTBk~2 !kdA

(42)

5.3 Refined Model RM3. A possible disadvantage of therefined model RM2 is represented by the large number of un-known functions with respect to the classical full displacement14 Terms of Use: http://asme.org/terms

According the penalty method, the resultant stresses are de-duced as

5.4 Refined Model RM4. The refined model RM4 is basedon the full displacement-based variational formulation. In fact, it

L

(k

RM3 0.7347060 2.9722354 0.9443031

Downloaded Fwmax5100ETwCp0hr4

(52)

386 Vol. 70, MAY 2003rom: http://appliedmechanics.asmedigitalcollection.asme.org/ on 02/06/20RM4 0.7347033 2.9721543 0.94430313D

AS 0.7316710 2.9502480 0.9306170

Transactions of the ASMErial. The subscripts L and T indicate the longitudinal and transver-sal principal material directions.

6.1 Cylindrical Bending. Initially, homogeneous and cross-ply laminated plates in cylindrical bending, subjected to the sinu-soidal load p5p0 sin(ax) with a5p/a , are studied. In particular,antisymmetric @0/90# and symmetric @0/90/0# laminates are con-sidered. In Table 1, the results obtained using the refined modelsare put in comparison with the exact three-dimensional analyticalsolution (3D2AS) obtained by Pagano @33# and with those recov-ered through the classical Mindlin-Reissner theory ~FSDT!. Inparticular, FSDT solutions are obtained considering the shear cor-rector factor x equal to x055/6 and the exact value of shearcorrector factor proposed by Whitney xW , @8#. Results are re-ported in terms of the dimensionless maximum displacement wmaxdefined as

approximating the shear stresses, the constitutive Eq. ~9! tends tobe enforced. Thus, enlarging the space of the shear parametersto2,. . ,to

n,t1,. . ,tn the shear stress profile tends to become constant

for the homogeneous plate.

Table 1 Dimensionless maximum displacement wmax definedby formula 52 for homogeneous plate and for 090 and 0900 composite laminates in cylindrical bendingwmax Homogeneous @0/90# @0/90/0#

FSDTx0 0.7347060 2.9662221 0.8136198FSDTxW 0.7347060 2.9713422 0.9443031RM1 0.7347060 2.9482925 0.7902380RM2 0.7347060 2.9722354 0.9443031N51h

~Lu2e! M51h

~Lw2k! (48)

H5 s5EA(k51

n H 12 @TkLTXk~0 !Lu#T1@TkLTYk~0 !Lw#TJ LTAk~0 !Lud1

12 EA(k51

n

@TkLTYk~0 !Lw#TLTBk~0 !LwdA112 EA(k51

n

@Tk

112 EA(k51

n

@TkLTYk~2 !Lw#TLTBk~2 !LwdA1EA~w1w!T

Hence, the full displacement functional governing the laminateproblem is

E5~u,w ,w!5Emb~u,w!1H5 s~w ,u,w!2Pext . (50)The recovered potential energy functional ~50! appears very

appealing since it presents only five unknown functions, i.e., u1 ,u2 , w, w1 , and w2 . On the other hand, second-order derivativesof the in-plane displacement and the rotation vectors, i.e., u and w,appear in the functional E5 . From a numerical point of view, thepresence of the second-order derivatives of the unknown functionsin the governing functional could represent a drawback. In fact, inthe perspective of developing suitable finite laminate elementsbased on refined FSDT theories, a greater continuity of the inter-polation functions is required.

6 Numerical ApplicationsWith the aim of verifying the accuracy of the proposed FSDT

refined models, some numerical calculations are developed. Re-sults are carried out for homogeneous plates as well as for com-posite laminates. In particular, square plates, characterized by thein-plane dimension a and subjected to transversal sinusoidal load-ing, are considered. The laminates have the side to thickness ratior5h/a50.10. The following elastic properties are introduced inthe computations:

ELET

525, nLT50.25,GLTET

50.5,GTTET

50.2 (51)

which correspond to a strongly orthotropic graphite-epoxy mate-is obtained enforcing the strain displacement Eqs. ~6! in the shearfunctional ~42!. Thus, the functional H s-becomes

A1EA(k51

n H 12 @TkLTXk~1 !Lu#T1@TkLTYk~1 !Lw#TJ LTAk~1 !LudATYk~1 !Lw#TLTBk~1 !LwdA

51

n

~LTXk~0 !Lu1LTYk~0 !Lw!dA . (49)

where wC represents the transversal displacement occuring in thecenter of the laminate, i.e. wC5w(a/2,a/2).

It is apparent the effectiveness of the refined models RM2,RM3, and RM4. In fact, RM2, RM3, and RM4 results are inperfect agreement with the FSDTxW and 3D2AS solutions; inother words, the RM2, RM3, and RM4 approaches are able torecover the FSDTxW model without the use of the shear correc-tion factors. Moreover, the RM1 appears satisfactory for the ho-mogeneous plate and for the antisymmetric @0/90# laminate, whileit is absolutely unsatisfactory when the symmetric @0/90/0# lami-nate is considered.

In Fig. 1 the dimensionless shear stress profile t1z /p0 for thehomogeneous plate in cylindrical bending is reported. The resultsobtained by the four proposed refined models, i.e., RM1, RM2,RM3, and RM4, are compared with the shear stress derived by theanalytical three-dimensional solution. It can be noted the perfectagreement between all the computed solutions with the exactthree-dimensional solution.

Then, the homogeneous plate in cylindrical bending is studiedconsidering fictitious staking sequences of one layer @0#, threeequal layers @0/0/0# and ten equal layers @0/0/0/0/0/0/0/0/0/0#. Re-sults in terms of dimensionless shear stress profile t1z /p0 in theplate thickness are reported in Fig. 2. It can be noted that theRM2, RM3 and RM4 proposed models lead all to the same solu-tion in perfect agreement with the exact three-dimensional solu-tion. On the contrary, the RM1 model gives different solutionsdepending on the number of layers considered for the fictitiousstaking sequence. In particular, for n510 the shear stress is al-most constant in the core of the plate. In fact, increasing the num-ber of layers, i.e., increasing the number of independent functions14 Terms of Use: http://asme.org/terms

Downloaded FIn Figs. 3 and 4 the dimensionless shear stresses t1z /p0 areplotted for the @0/90# and @0/90/0# laminates, respectively. Again itcan be noted the good agreement between the solutions obtainedusing the RM2, RM3, and RM4 models and the three-dimensionalanalytical solution. On the contrary, the RM1 model, based on theassumption of independent approximation of the shear stresses,leads to unsatisfactory solution, since the profile appears abso-lutely inadequate.

6.2 Simply Supported Laminates

Cross-Ply Laminates. Cross-ply laminates subjected to thesinusoidal load p5p0 sin(ax)sin(ay) with a5p/a , are consid-ered. The following SS1 boundary conditions are adopted:

u25w5w250 N115M 1150 at x150 and x15a

u15w5w150 N225M 2250 at x250 and x25a .Results are computed for homogeneous plate and for @0/90# and@0/90/0# laminates.

Fig. 1 Dimensionless shear stress t1z p0 at x10 for homo-geneous plate in cylindrical bending; comparison between thedifferent solutions

Fig. 2 Dimensionless shear stress t1z p0 at x10 for homo-geneous plate in cylindrical bending; comparison between thedifferent solutions computed considering one, three, and tenequal layers

Journal of Applied Mechanicsrom: http://appliedmechanics.asmedigitalcollection.asme.org/ on 02/06/20In Table 2 the dimensionless maximum displacement wmax de-fined by formula ~52! is reported; in particular, results are ob-tained, considering

the classical FSDT analytical solution with x5x055/6, the finite element solution (FEMx0) with x5x0 ,

Fig. 3 Dimensionless shear stress t1z p0 at x10 for the090 laminate in cylindrical bending; comparison between thedifferent solutions

Fig. 4 Dimensionless shear stress t1z p0 at x10 for the090 laminate in cylindrical bending; comparison between thedifferent solutions

Table 2 Dimensionless maximum displacement wmax definedby formula 52 for homogeneous plate and for 090 and 0900 cross-ply laminates

wmax Homogeneous @0/90# @0/90/0#

FSDTx0 0.6382997 1.237270 0.669302FEMx0 0.63834 1.2373 0.66930FEMx 0.63834 1.2319 0.76377RM1 0.6382997 - -RM2 0.6382997 1.231817 0.763779RM3 0.6382997 1.231817 0.763779RM4 0.6382997 1.231817 0.7637793D 0.6338085 1.224799 0.751425


Downloaded F the finite element solution ~FEMx! with exact value of theshear correction factor computed by the iterative procedureproposed in @13#,

the RM1 analytical solution only for the homogeneous plate, the RM2, RM3, and RM4 analytical solutions, and the exact three-dimensional solution 3D-AS.

In Figs. 5 and 6 the dimensionless shear stress profiles t1z /p0and t2z /p0 for the @0/90# laminate are plotted, respectively.Analogously, in Figs. 7 and 8 the shear stress profiles are plottedfor the @0/90/0# laminate. It can be noted that the proposed modelsare able to approximate very accurately the exact three-dimensional solution for both the considered laminations.

Finally, in Fig. 9 the dimensionless displacement wmax definedby formula ~52!, computed for the @0/90/0# lamination, is plottedversus the ratio r5h/a . In particular, the RM3 solution is com-pared with three dimensions with the FSDTx0 , with the FSDTxand with the classical laminate theory ~CLT! solutions. It can beemphasized the good accordance between the RM3 and the ana-lytical three-dimensional solutions for a wide range of the ratio r.

Fig. 5 Dimensionless shear stress t1z p0 for the 090 lami-nate computed at x10, x2a2; comparison with the three-dimensional analytical solution

Fig. 6 Dimensionless shear stress t2z p0 for the 090 lami-nate computed at x1a2, x20; comparison with the three-dimensional analytical solution

388 Vol. 70, MAY 2003rom: http://appliedmechanics.asmedigitalcollection.asme.org/ on 02/06/20Angle-Ply Laminate. The antisymmetric @245/45# angle-plylaminate subjected to the sinusoidal load p5p0 sin(ax)sin(ay)with a5p/a , is considered. The following SS2 boundary condi-tions are adopted:

u15w5w250 N125M 1150 at x150 and x15a

u25w5w150 N125M 2250 at x250 and x25a .In Table 3 the dimensionless maximum displacement wmax de-

fined by formula ~52! is reported; in particular, results are ob-tained, considering

the classical FSDT analytical solution with x5x0 and the RM2, RM3, and RM4 analytical solutions.

It can be noted that the RM2, RM3, and RM4 models give allthe same results which differ from the FSDT solution obtainedadopting the shear correction factor x55/6. Finally, in Fig. 10 theshear stress profile t1z /p0 for the angle-ply laminate is plotted.

Fig. 7 Dimensionless shear stress t1z p0 for the 0900 lami-nate computed at x10, x2a2; comparison with the three-dimensional analytical solution

Fig. 8 Dimensionless shear stress t2z p0 for the 0900 lami-nate computed at x1a2, x20; comparison with the three-dimensional analytical solution


needs the use of a penalty parameter, which is not always simpleto set; on the other hand, the RM4 requires the use of smoother

Downloaded F7 ConclusionsRefined laminate models are presented. They are derived con-

sidering mixed variational formulations of the laminate problem,introducing suitable representation forms of the shear stresses inthe plate thickness. It is obtained that independent approximationsof the shear stresses can lead to unsatisfactory models, which arenot able to recover the correct profiles of the stresses.

The proposed RM2, RM3, and RM4 models, obtained repre-senting the transverse shear stress profile by using the three-dimensional equations, are very satisfactory; in fact, the solutionsobtained by these models are in very good accordance with thethree-dimensional analytical solution.

The RM2 approach appears the more suitable in view to de-velop effective laminate finite elements. In fact, the RM3 model

Fig. 9 Dimensionless displacement wmax versus the thicknessto side ratio

Table 3 Dimensionless maximum displacement wmax definedby formula 52 for the 4545 angle-ply composite laminateModel FSDTx0 RM2 RM3 RM4

wmax 0.8828107 0.8929168 0.8929168 0.8929168

Fig. 10 Dimensionless shear stress t1z p0 for the 4545laminate computed at x10, x2a2; comparison with first-order shear deformation theory x56 analytical solution

Journal of Applied Mechanicsrom: http://appliedmechanics.asmedigitalcollection.asme.org/ on 02/06/20approximation functions in the finite element formulation, sincesecond-order derivatives appear in the governing functional.

The proposed model does not suffer of any limitation about thenumber of layers defining the stacking sequence and of the plyangles; more complex situations can be investigated, includingquasi-isotropic laminates. More complex cases can be studied de-veloping suitable finite elements based on the proposed formula-tions. In fact, the presented mixed principles, in particular theRM2 model, are the bases for the development of new and per-forming finite elements. One of the major problems in developingmixed laminate finite elements, is the definition of the approxima-tion functions used for the midplane strains and curvatures inorder to verify the stability requirement, related to the LBB con-dition. Finally, full displacement finite elements can be recoveredfrom the refined model RM2 performing static condensation ofstrain and stress variables.

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