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European Journal of Mechanics A/Solids 41 (2013) 111e122

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European Journal of Mechanics A/Solids

journal homepage: www.elsevier .com/locate/ejmsol

A new laminated composite beam element based on eigenvalueproblem

Nam-Il Kim a,1, Chan-Ki Jeon b,2, Jaehong Lee a,*

aDepartment of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, South KoreabDepartment of Urban Construction Engineering, University of Incheon, 12-1 Songdo-dong, Yeonsu-gu, Incheon 406-772, South Korea

a r t i c l e i n f o

Article history:Received 10 October 2012Accepted 11 February 2013Available online 14 March 2013

Keywords:Laminate beamStiffness matrixEigenvalue

* Corresponding author. Tel.: þ82 2 3408 3287.E-mail addresses: kni8501@gmail.com (N.-I. K

(C.-K. Jeon), jhlee@sejong.ac.kr, jaehong.lee00@gmail.1 Tel.: þ82 10 6668 7656.2 Tel.: þ82 10 3210 6766.

0997-7538/$ e see front matter � 2013 Elsevier Mashttp://dx.doi.org/10.1016/j.euromechsol.2013.02.004

a b s t r a c t

A new laminated composite beam element is developed based on the eigenvalue problem for theflexural and torsional analyses of beams with I- and channel-sections. Analytical technique is used topresent the laminated composite beam theory considering the transverse shear and the restrainedwarping induced shear deformation based on an orthogonal Cartesian coordinate system. From theenergy principle, the equilibrium equations are derived and the member stiffness matrix is determinedusing the forceedisplacement relations of the beam. In order to demonstrate the validity of this study,the deflections and the twist angles of composite beams with I- and channel-sections are presented andcompared with the results from other researchers and the finite beam element using the Lagrangeinterpolation functions, and the detailed finite element analysis results using ABAQUS. Especially theflexural and torsional behavior and the effect of shear deformation are investigated with respect to themodulus ratio and the fiber angle change.

� 2013 Elsevier Masson SAS. All rights reserved.

1. Introduction

The beams made with composite materials are becomingincreasingly popular for use as structural members in the architec-ture, aviation, space flight, and automotive industries. Owing to thehigh strength-to-weight ratio offered by composites, the structuralweight is much less of an issue than it is for isotropic materials.Theses composite beams might be subjected to the flexural andtwist moments when used in above applications. Therefore, theaccurate prediction of the flexural and torsional behavior of com-posite beam is of fundamental importance in the design.

During the past couple of decades, considerable attention hasbeen paid to understanding the flexural and torsional behavior ofthe laminated composite beams due to the many advantages theyoffer over isotropic beams. Oin and Librescu (2002) and Librescuand Song (1991) incorporated a number of non-classical effectsand investigated the cross-ply, circumferentially uniform stiffness,and circumferentially asymmetric stiffness for the thin-walledbeam based on the extended Galerkin’s method. They showedthat the non-classical effects such as transverse shear and non-

im), johnland@incheon.ac.krcom (J. Lee).

son SAS. All rights reserved.

uniformity of membrane shear stiffness could significantly influ-ence the accuracy of predictions. Shadmehri et al. (2007) consid-ered a single-cell composite thin-walled beam under the tipbending load and the twist moment and solved the governingequations using the extended Galerkin’s method.

On the other hand, a lot of research effort has been devoted toimprove the finite element because of its versatility and usability.Lee (2005) developed the displacement-based one-dimensionalfinite element method based on the Lagrange interpolation func-tion and used a reduced integration technique for the evaluation ofthe stiffness matrix in order to avoid shear locking. Back and Will(2008) developed the two-, three-, and four-node isoparametricbeam elements using the reduced integration for the flexural andbuckling analyses of thin-walled bisymmetric and monosymmetricI-beams. However, this reduced integration technique is suscepti-ble to display inherent numerical disturbances like the occurrenceof spurious modes. Sheikh and Thomsen (2008) developed a threenode beam element, where the nodes at the two ends contained 7DOFs, while the internal node contained 5 DOFs. This efficient finiteelement formulation did not require the reduced integrationtechnique and was adopted for the transverse shear deformation,which helped to conveniently implement the C1 continuousformulation required by torsional deformation due to incorporationof warping deformation. But it has a shortcoming in that about 6beam elements should be used to obtain accurate results for thetorsional analysis of beamwith channel-section subjected to a twist

Fig. 1. Displacement parameters and stress resultants.

N.-I. Kim et al. / European Journal of Mechanics A/Solids 41 (2013) 111e122112

moment. Feo and Mancusi (2010) suggested a general approach tomodel the transverse shear deformability in thin-walled compositebeams based on the use of suitable monomials. The one-dimensional finite element approach was proposed to overcomethe difficulties related to a tri-dimensional analysis of the defor-mation associated with shear forces. Wu et al. (2004) presented theinitial-value solutions of static equilibrium equations for the single-cell thin-walled laminated composite beams under bending loads.In order to verify their theory, Wu et al. (2004) used the finite beamelement with 4 DOFs for every node. Piovan and Cortínez (2007)introduced a non-shear-locking 7 degrees-of-freedom (DOFs) pera node to validate the model. Cardoso and Valido (2011) performedthe optimal design of cross-section properties of thin-walledlaminated composite beams. The cross-section geometry was dis-cretized by eight-node isoparametric plate elements in order todetermine its bending-torsion properties. McCarthy andChattopadhyay (1997) used a four-node plate element to dis-cretize the individual plates of the beam. This element was C1

continuous in the zeroth-order mid-surface displacements of eachelement and was C0 continuous for the higher-order terms in therotations of the normals to the mid-surface. As a result, the elementcontained 11 DOFs per a node. Maddur and Chaturvedi (2000)proposed a four-node finite plate element procedure for the lami-nated graphite epoxy cantilever I-beam subjected to a twist load.The finite element modeling was accomplished by using theLagrange interpolation function for the geometric coordinate var-iables and Hermitian interpolation function for the unknownfunctions. And the sub parametric formulation was used forachieving continuity requirement of the first order derivatives ofthe dependent variables. However, the main drawback of the abovementioned finite element analysis techniques is that a considerablenumber of finite elements are required to obtain the satisfactoryresults due to the use of the approximate shape functions.

The objective of this study is to develop the refined and accuratelaminated composite beam element based on the eigenvalueproblem for the flexural and torsional analyses. A significant pointof departure of the present numerical approach from other avail-able methods is in the solutions of coupled ordinary differentialequations that arise in the solution process. The important points ofthis study are summarized as follows:

1. The thin-walled laminated composite beam theory is devel-oped considering the effects of transverse shear and warpingbased on the orthogonal Cartesian coordinate system.

2. The element stiffness of the laminated composite beam isevaluated rigorously by substituting the displacement statevector into the equilibrium equations and forceedisplacementrelations.

3. In order to show accuracy and superiority of the beam devel-oped by this study, the deflections and the twist angles of thelaminated composite beams with I- and channel-sections areevaluated and compared with those from other researchersand the finite beam element using Lagrange interpolationfunction, and the results from two-dimensional finite elementanalysis using ABAQUS (2003). In addition, some characteris-tics of flexural and torsional behavior are reported.

2. General theory

2.1. Kinematics

Fig. 1 shows the displacement parameters and the stress re-sultants defined at the coordinate system (x, y, z) where the x axiscoincides with the centroid; y, z are the principal inertia axes; u, v,

w are the rigid body translations of the cross-section in the x di-rection at the centroid and in the y and z directions at the shearcenter, respectively; u1, u2, u3 are the rigid body rotations about theshear center and in the y and z axes, respectively; f is the parameterdefining warping of the cross-section; e2 and e3 are coordinates ofthe pole S in the y and z axes, respectively.

According to the assumption of in-plane rigid cross-sections ofbeam, the beam is deformed in a way such that the shape of thecross-section remains unchanged. As a consequence of thin-walledbeam assumptions, the longitudinal and transverse displacements(Chang et al. 1996) of the arbitrary mid-surface point can bewritten as

uðx; y; zÞ ¼ uðxÞ � u3ðxÞyþ u2ðxÞz� f ðxÞf (1a)

vðx; y; zÞ ¼ vðxÞ � ðz� e3Þu1ðxÞ (1b)

wðx; y; zÞ ¼ wðxÞ þ ðy� e2Þu1ðxÞ (1c)

where f is the warping function. The displacement fields of anygeneric point in the flanges can be expressed with respect to themid-surface displacements as

Ua�x; y; z� ¼ ua�x; y�þ n

vwa

vx

�x; y�

(2a)

Va�x; y; z� ¼ va�x; y�þ n

vwa

vy

�x; y�

(2b)

Wa�x; y; z� ¼ wa�x; y� (2c)

where ua, va, andwa are the plate displacements at the mid-surfaceof each flange and n means the coordinate normal to the mid-surface of a plate element. The displacement fields for the webmay also be expressed as

Uw�x; y; z� ¼ uw�x; z�� n

vvw

vx

�x; z�

(3a)

Vw�x; y; z� ¼ vw�x; z�

(3b)

Ww�x; y; z� ¼ ww�x; z�� nvvw

vz

�x; z�

(3c)

where uw, vw, and ww are the plate displacements at the mid-surface of the web. The definitions of the beam axial and shearstrain fields are written by

3x ¼ vUvx

; gxy ¼ vUvy

þ vVvx

; gxz ¼ vUvz

þ vWvx

(4)

N.-I. Kim et al. / European Journal of Mechanics A/Solids 41 (2013) 111e122 113

2.2. Equilibrium equations and forceedisplacement relations

The strain energy of the laminated composite beam is expressedfrom the basic assumptions for the thin-walled beams (Barbero,1999).

PL ¼ 12

ZV

�sx 3x þ sxygxy þ sxzgxz

�dV (5)

where sx, sxy, and sxz are the normal and shear stresses, respectively.Substituting Eq. (4) into Eq. (5), the strain energy expression isfollows:

P ¼ 12

Z lo

ZA

hsxn

3ox þ

�z� n cos q

�ky þ

�yþ n sin q

�kz

þ �fþ nq�kf

oþ sxy

ngoxy � gofðz� e3Þ þ kxsn cos q

oþ sxz

ngoxz þ gofðy� e2Þ þ kxsn sin q

oidAdx

(6)

where l and A are the length and the area of cross-section of beam,respectively; q is the angle tangent to the contour makes with the yaxis; q is the coordinate of the point x, y on the contour measured inthe tangential direction; go

xy, goxz and gof are the flexural shear

strains in the xey and xez planes and warping shear strain in thebeam, respectively; 3ox , ky, kz, kf and kxs are the axial strain, thebiaxial curvatures in the y and z directions, the warping curvaturewith respect to the shear center, and the twist curvature in thebeam, respectively, defined by

3ox ¼ u0; ky ¼ u0

2; kz ¼ �u03; kf ¼ f 0; kxs ¼ 2u0

1 (7)

where the superscript ‘prime’ indicates the derivative with respectto x. The variation form of the strain energy in Eq. (6) can also bewritten as

dP ¼Z lo

�F1d 3

ox þ F2dg

oyx þ F3dg

ozx þM2dky þM3dkz

þMfdkf þ Tdgof þMtdksx�dx

(8)

P ¼ 12

Z lo

hE11u

02 þ E22u022 þ E33u

023 þ E44f

02 þ 4E55u021 þ E66ðv0 � u3Þ2 þ E77ðw0 þ u2Þ2 þ E88

�u01 þ f

�2 þ 2E12u0u0

2 � 2E13u0u0

3

þ 2E14u0f 0 þ 4E15u

0u01 þ 2E16u

0ðv0 � u3Þ þ 2E17u0ðw0 þ u2Þ þ 2E18u

0�u01 þ f

�� 2E23u02u

03 þ 2E24u

02f

0 þ 4E25u01u

02

þ 2E26u02ðv0 � u3Þ þ 2E27u

02ðw0 þ u2Þ þ 2E28u

02�u01 þ f

�� 2E34u03f

0 � 4E35u01u

03 � 2E36u

03ðv0 � u3Þ � 2E37u

03ðw0 þ u2Þ

� 2E38u03�u01 þ f

�þ 4E45u01f

0 þ 2E46f0ðv0 � u3Þ þ 2E47f

0ðw0 þ u2Þ þ 2E48f0�u0

1 þ f�þ 4E56u

01ðv0 � u3Þ þ 4E57u

01ðw0 þ u2Þ

þ 4E58u01�u01 þ f

�þ 2E67ðv0 � u3Þðw0 þ u2Þ þ 2E68ðv0 � u3Þ�u01 þ f

�þ 2E78ðw0 þ u2Þ�u01 þ f

��dx

(11)

where d is the variation symbol; F1 is the axial force; F2 and F3 arethe shear forces in the y and z directions, respectively; M2 and M3are the bendingmoments about the y and z axes, respectively;Mɸ isthe bimoment; T andMt are the two contributions to the total twistmoment. The expressions of the generalized forces and momentsacting over the cross-section are presented in Appendix A.

For a laminated composite material, adopting the assumption ofthe free stress in contour direction, the stressestrain relationshipsof the kth lamina of the flanges and the web are written in theCartesian coordinate system as follows:

skx ¼ Q11 �

Q212

Q22

!k

3kx þ

Q16 �

Q12Q26

Q22

!k

gkxy (9a)

skxy ¼ Q16 �

Q12Q26

Q22

!k

3kx þ

Q66 �

Q226

Q22

!k

gkxy (9b)

where Qij are the transformed reduced stiffness coefficients (Wang,1997) including the material properties of each lamina.

Substituting Eq. (9) into the equations of forces and moments,the constitutive equations for the laminated composite beam canbe expressed as

8>>>>>>>>>><>>>>>>>>>>:

F1M2�M3Mf

MtF2F3T

9>>>>>>>>>>=>>>>>>>>>>;

¼

266666666664

E11 E12 E13 E14 E15 E16 E17 E18E22 E23 E24 E25 E26 E27 E28

E33 E34 E35 E36 E37 E38E44 E45 E46 E47 E48

E55 E56 E57 E58E66 E67 E68

Symm: E77 E78E88

377777777775

�

8>>>>>>>>>><>>>>>>>>>>:

u0u02�u03

f 02u0

1v0 �u3w0 þu2u01þ f

9>>>>>>>>>>=>>>>>>>>>>;

(10)

where Eij are the laminate stiffnesses which depend on the cross-section of the composite and detailed expressions are given inAppendix B. Substitution of Eq. (10) into Eq. (8) leads to thefollowing strain energy of the shear deformable laminated com-posite beam.

Finally, the equilibrium equations can be derived by inte-grating the derivative of the varied quantities by part and bycollecting the coefficients of du, dv, dw, du1, du2, du3, and df in Eq.(11) as follows:

F 01 ¼ 0 (12a)

N.-I. Kim et al. / European Journal of Mechanics A/Solids 41 (2013) 111e122114

F 02 ¼ 0 (12b)

F 03 ¼ 0 (12c)

2M0t þ T 0 ¼ 0 (12d)

M02 � F3 ¼ 0 (12e)

M03 þ F2 ¼ 0 (12f)

M0f þ T ¼ 0 (12g)

The forceedisplacement relations can also be obtained as thenatural boundary conditions as follows:

du: F1 (13a)

dv: F2 (13b)

dw: F3 (13c)

du1: 2Mt þ T (13d)

du2: M2 (13e)

du3: M3 (13f)

df : Mf (13g)

3. Stiffness of laminated composite beam

3.1. Displacement function

The displacement function for the flexural and torsional ana-lyses of the laminated composite beam is rigorously derived. Totransform the 2nd order simultaneous ordinary differential equa-tions (SODEs) in Eq. (12) into the first order ones, the followingdisplacement state vector is used.

d ¼DUx;U0

x;Uy;U0y;Uz;U0

z;u1;u01;u2;u

02;u3;u

03; f ; f

0ET

¼ hd1; d2;.; d14iT (14)

Based on Eq. (14), the equilibrium equations in Eq. (12) can betransformed into the SODEs of the 1st order with constant co-efficients and expressed in matrix form as follows:

Ad0 ¼ Bd (15)

where A and B are the 14 � 14 matrices. To find the homogenoussolutions in Eq. (15), the following eigenvalue problem with non-symmetric matrix is considered.

lAZ ¼ BZ (16)

The eigenvalue problem of Eq. (16) has the complex eigenvalue l

and the associated eigenvector Z because the matrix B is non-symmetric matrix. The IMSL subroutine DGVCRG (IMSL, 1984) isused to solve the above complex eigenvalue problem.

From Eq. (16), the 2 non-zero eigenvalues and the 12 zero ei-genvalues are generated and the non-zero eigenvectors corre-sponding to the 2 pairs may be expressed as follows:

Zi ¼ z1;i; z2;i; z3;i; z4;i; z5;i; z6;i; z7;i; z8;i; z9;i; z10;i;�T (17)

� z11;i; z12;i; z13;i; z14;i ; i ¼ 1;2

The general solutions of Eq. (15) corresponding to non-zero ei-genvalues are easily represented by the linear combination of ei-genvectors with complex exponential functions as follows:

dI ¼X2i¼1

aiZielix ¼ XIaI (18)

where XI and aI denote the 14� 2matrix function made up of the 2eigen solutions and the integration constant vector, respectively.

Next, the undetermined coefficient method is applied to theequilibrium equations to determine the displacement modes cor-responding to the 12 zero eigenvalues. For this, the 7 displacementparameters may be assumed as follows:

Ux ¼ x1 þ x2x (19a)

Uy ¼ a1 þ a2xþ a3x2

2þ a4

x3

6(19b)

Uz ¼ b1 þ b2xþ b3x2

2þ b4

x3

6(19c)

u1 ¼ g1 þ g2xþ g3x2

2þ g4

x3

6(19d)

u2 ¼ d1 þ d2xþ d3x2

2þ d4

x3

6(19e)

u3 ¼ z1 þ z2xþ z3x2

2þ z4

x3

6(19f)

f ¼ h1 þ h2xþ h3x2

2þ h4

x3

6(19g)

where x1, x2 and ai, bi, gi, zi, hi, (i ¼ 1e4) are the undeterminedcoefficients. By substituting Eq. (19) into the equilibrium equationsin Eq. (12) and by applying the identity condition, we can obtain thefollowing relations:

g3 ¼ g4 ¼ d4 ¼ 24 ¼ h2 ¼ h3 ¼ h4 ¼ 0 (20a)

z2 ¼ a3 (20b)

z3 ¼ a4 (20c)

d2 ¼ �b3 (20d)

d3 ¼ �b4 (20e)

E66ða2 � z1Þ þ E67ðb2 þ d1Þ þ E68h1 þ E16x2 � E36a3þ E33a4 � E26b3 þ E23b4 þ ð2E56 þ E68Þg2 ¼ 0

(20f)

E67ða2 � z1Þ þ E77ðb2 þ d1Þ þ E78h1 þ E17x2 � E37a3þ E23a4 � E27b3 þ E22b4 þ ð2E57 þ E78Þg2 ¼ 0

(20g)

E68ða2 � z1Þ þ E78ðb2 þ d1Þ þ E88h1 þ E18x2 � E38a3� E34a4 � E28b3 � E24b4 þ ð2E58 þ E88Þg2 ¼ 0

(20h)

N.-I. Kim et al. / European Journal of Mechanics A/Solids 41 (2013) 111e122 115

From Eqs. (20f)e(20h), z1, d1, and h1 are expressed as

z1 ¼a2 þ ðE16U11 þ E17U12 þ E18U13Þx2 � ðE36U11 þ E37U12 þ E38U13Þa3 þ ðE33U11 þ E23U12 � E34U13Þa4� ðE26U11 þ E27U12 þ E28U13Þb3 þ ðE23U11 þ E22U12 � E24U13Þb4 þ fð2E26 þ E68ÞU11 þ ð2E57 þ E78ÞU12

þ ð2E58 þ E88ÞU13Þgg2(21a)

d1 ¼ � b2 � ðE16U12 þ E17U22 þ E18U23Þx2 þ ðE36U12 þ E37U22 þ E38U23Þa3 � ðE33U12 þ E23U22 � E34U23Þa4þ ðE26U12 þ E27U22 þ E28U23Þb3 � ðE23U12 þ E22U22 � E24U23Þb4 � fð2E26 þ E68ÞU12 þ ð2E57 þ E78ÞU22

þ ð2E58 þ E88ÞU23Þgg2(21b)

h1 ¼ � ðE16U13 þ E17U23 þ E18U33Þx2 þ ðE36U13 þ E37U22 þ E38U33Þa3 � ðE33U13 þ E23U23 � E34U33Þa4þ ðE26U13 þ E27U23 þ E28U33Þb3 � ðE23U13 þ E22U23 � E24U33Þb4 � fð2E26 þ E68ÞU13 þ ð2E57 þ E78ÞU23

þ ð2E58 þ E88ÞU33Þgg2(21c)

where coefficients Uij, (i ¼ 1e3, j ¼ 1e3) are given by

24U11 U12 U13U12 U22 U23U13 U23 U33

35 ¼

24 E66 E67 E68E67 E77 E78E68 E78 E88

35�1

(22)

From which the homogenous solutions corresponding to the 12zero eigenvalues may be written as

dII ¼X14i¼3

aiZi ¼ XIIaII (23)

in which XII and aII are the 14 � 12 matrix function and the 12integration constants, respectively.

By summing up the 2 eigen solution of Eq. (18) and the 12modesof Eq. (23), the solutions of the displacement state vector withrespect to the integration constants are obtained as follows:

d ¼ XIaI þ XIIaII ¼ Xa (24)

where X denotes the 14 � 14 matrix function with the coefficientsof the displacement parameters.

Next, let Ue be the nodal generalized displacement of 14 DOFs atthe two ends of the beam element:

Ue ¼ �uð0Þ; vð0Þ;wð0Þ;u1ð0Þ;u2ð0Þ;u3ð0Þ; f ð0Þ;uðlÞ; vðlÞ;wðlÞ;u1ðlÞ;u2ðlÞ;u3ðlÞ; f ðlÞ

�T (25)

By substituting coordinates of the member end (x ¼ 0, l) into Eq.(24) and accounting for Eq. (25), complex coefficients a can beobtained as follows:

a ¼ E�1Ue (26)

where E is evaluated from X and the inverse of E is calculated usingIMSL subroutine DLINCG (IMSL, 1984).

Finally, the displacement function consisting of the displace-ment components is evaluated by substituting Eq. (26) into Eq. (24)as follows:

d ¼ XE�1Ue (27)

It should be noted that the interpolation function matrix XE�1 inEq. (27) is exact not approximate since this displacement statevector satisfies the homogenous form of the SODEs in Eq. (12).

3.2. Calculation of member stiffness

The member stiffness matrix of the laminated composite beamis calculated based on the displacement function derived in Section3.1. The forceedisplacement relations in Eq. (13) can be compactlyrepresented in matrix form with respect to the displacement pa-rameters in Eq. (14) as follows:

F ¼ Sd (28)

F ¼DF1; F2; F3;M1;M2;M3;Mf

ET(29)

where S is the 7� 14matrix andM1 is the total twist moment. Thensubstitution of the displacement function in Eq. (27) into Eq. (28)leads to

F ¼ SXE�1Ue (30)

Finally, the member stiffness matrix of the beam subjected toexternal forces is evaluated by substituting coordinates of themember ends into Eq. (30) as follows:

Fe ¼ KUe (31)

It is noted that the member stiffness matrix in Eq. (31) is formed bydisplacement functions which are exact solutions of the equilib-rium equations. Therefore the laminated composite beam based onthe stiffness matrix developed by this study eliminates discretiza-tion errors and is free from the shear locking.

4. Numerical examples

Numerical investigation has been carried out to validate thecurrent analysis and also to identify the effect of shear deformationon the flexural and torsional behavior of the laminated compositebeams. The results obtained from this study are compared withthose from other researchers and the finite beam element solu-tions, and the shell elements by ABAQUS.

For the finite beam element analysis, the beam element with thetwo-nodes and 7 DOFs per a node is used. In order to accuratelyexpress the deformations of the element, the pertinent shapefunctions are necessary. In this study, the Lagrange interpolationpolynomials are applied to interpolate the displacement parame-ters that are defined at the centroid-shear center axes. Bysubstituting shape functions and laminate stiffnesses into the strain

Fig. 2. Shape and warping profile of bisymmetric I-section.

Table 1Tip deflection of CF bisymmetric I-beams under a vertical tip force (mm).

Lay-up This study Park et al. (2000) ABAQUS

With shear Without shear

[15/�15]4S 46.11 45.19 45.21 46.18[30/�30]4S 61.56 60.84 60.89 61.49[45/�45]4S 88.55 87.85 87.95 88.33[60/�60]4S 111.8 111.0 111.2 111.7[75/�75]4S 121.6 120.5 120.7 121.6[0/90]4S 62.01 60.89 60.93 62.13

N.-I. Kim et al. / European Journal of Mechanics A/Solids 41 (2013) 111e122116

energy in Eq. (11) and integrating along the element length, theelement stiffness matrix can be evaluated in local coordinate. Toavoid locking, the shear strains are integrated using one-pointGaussian quadrature. Then the assembly of element stiffness ma-trix for the entire structure, based on the coordinate trans-formation, leads to the equilibrium matrix equation in globalcoordinate system.

The material of beams used in following examples is the glasseepoxy where the properties of each layer are as follows:E1 ¼ 53.78 GPa, E2 ¼ E3 ¼ 17.93 GPa, G12 ¼ G13 ¼ 8.96 GPa,G23¼ 3.45 GPa, n12¼ n13¼ 0.25, n23¼ 0.34. The subscripts ‘1’ and ‘2’,‘3’ correspond to directions parallel and perpendicular to fibers,respectively.

Fig. 3. Vertical deflections at mid-span of bisymmetric I-beams under a vertical forceacting at mid-span.

4.1. Beams with bisymmetric I-section

Fig. 2 shows the shape and warping profile of the bisymmetric I-section under consideration. The length l and the height h of beamare 1 m and 50 mm, respectively. The total thicknesses of top andbottom flanges and web are assumed to be 2.08 mm and 16 layerswith equal thickness are used in two flanges and web. The lami-nated stacking sequence is assumed to be symmetric with respectto its mid-surface.

Table 2Convergence of the finite beam element for tip deflection of CF bisymmetic I-beams (mm).

Lay-up No. of the finite beam element This study

4 8 12 16 20 30 40 60

[15/�15]4S 45.400 45.930 46.028 46.062 46.078 46.094 46.099 46.103 46.107[30/�30]4S 60.608 61.322 61.454 61.500 61.522 61.543 61.550 61.555 61.560[45/�45]4S 87.179 88.210 88.401 88.467 88.498 88.529 88.540 88.547 88.553[60/�60]4S 110.10 111.40 111.64 111.73 111.77 111.81 111.82 111.83 111.84[75/�75]4S 119.67 121.08 121.35 121.44 121.48 121.52 121.54 121.55 121.56[0/90]4S 61.061 61.775 61.907 61.953 61.975 61.996 62.003 62.008 62.013

Fig. 4. Twist angles at mid-span of bisymmetric I-beams under a twist moment actingat mid-span.

Table 3Maximum twist angle of SS bisymmetric I-beams under a twist moment acting atright end (rad.).

Lay-up This study Lee and Lee (2004) ABAQUS

[15/�15]4S 0.2073 0.2073 0.2086[30/�30]4S 0.1563 0.1563 0.1575[45/�45]4S 0.1396 0.1396 0.1408[60/�60]4S 0.1571 0.1571 0.1583[75/�75]4S 0.2080 0.2080 0.2092[0/90]4S 0.2480 0.2481 0.2498

N.-I. Kim et al. / European Journal of Mechanics A/Solids 41 (2013) 111e122 117

For beams with clamped-free (CF) boundary condition, the tipdeflections of beams under a vertical tip force 1 kN are presented inTable 1 for several lay-ups and compared with the analytical so-lutions (Park et al., 2000) neglecting shear deformation effects andthe finite element results from ABAQUS. A total of 600 nine-nodeshell elements (50 along the beam span and 12 through thecross-section), for ABAQUS calculation, are used to obtain the

Fig. 5. Effects of shear on the deflection and the twist angle of CC bisymmetricI-beams.

Fig. 6. Shape and warping profile of monosymmetric I-section.

a

Fig. 7. Shape and warping profile of channel-sections.Table 4Lay-ups of monosymmetric I-beams.

Lay-up Top flange Bottom flange Web

ANG0 [0]16 [0]24 [0]8ANG15 [15/�15]4S [15/�15]6S [15/�15]2SANG30 [30/�30]4S [30/�30]6S [30/�30]2SANG45 [45/�45]4S [45/�45]6S [45/�45]2SANG60 [60/�60]4S [60/�60]6S [60/�60]2SANG75 [75/�75]4S [75/�75]6S [75/�75]2SQSISO [0/45/90/�45]2S [0/45/90/�45]3S [0/45/90/�45]S

Table 5Tip deflection of CF monosymmetric I-beams under a vertical tip force (mm).

Lay-up No. of the finite beam element

4 8 20 30 40

ANG0 53.947 54.563 54.736 54.754 54ANG15 59.128 59.810 60.001 60.021 60ANG30 78.562 79.481 79.739 79.766 79ANG45 112.74 114.07 114.44 114.48 114ANG60 142.32 144.00 144.47 144.52 144ANG75 154.82 156.64 157.15 157.20 157QSISO 91.103 92.170 92.468 92.500 92

Table 6Tip deflection of CF channel-beams under a vertical tip force (mm).

Lay-up No. of the finite beam element

4 8 20 30 40

[0]16 71.509 72.347 72.582 72.607 7[15/�15]4S 78.838 79.765 80.025 80.053 8[30/�30]4S 105.61 106.86 107.21 107.24 10[45/�45]4S 152.16 153.97 154.47 154.52 15[60/�60]4S 192.22 194.50 195.14 195.21 19[75/�75]4S 208.82 211.30 211.99 212.06 21[0/90]4S 106.13 107.38 107.73 107.76 10

Table 7Maximum twist angle of CF channel-beams under a twist moment acting at right end (r

Lay-up No. of the finite beam element

1 2 4 6

[0]16 0.25219 0.24413 0.24456 0.24466[15/�15]4S 0.22879 0.21794 0.21808 0.21812[30/�30]4S 0.19520 0.18198 0.18199 0.18199[45/�45]4S 0.18622 0.17260 0.17254 0.17254[60/�60]4S 0.21211 0.19660 0.19647 0.19647[75/�75]4S 0.27486 0.25483 0.25473 0.25477[0/90]4S 0.28209 0.26717 0.26727 0.26731

N.-I. Kim et al. / European Journal of Mechanics A/Solids 41 (2013) 111e122118

results. It can be found from Table 1 that the values of deflectionwith shear effects from this study are in good agreement withABAQUS solutions and exhibits better results than that from theclassical beam theory (Park et al., 2000) which does not considershear effects. The convergence study of the finite beam elementbased on the Lagrange interpolation polynomials is performed inTable 2. The results from at least 60 finite beam elements are inexcellent agreement with the present solutions obtained from only

This study ABAQUS

60 With shear Without shear

.761 54.765 54.769 52.621 54.89

.029 60.034 60.038 58.273 60.12

.776 79.782 79.788 78.388 79.74

.50 114.51 114.51 113.25 114.3

.54 144.55 144.56 142.92 144.4

.22 157.23 157.24 155.55 157.3

.511 92.519 92.525 90.979 92.55

This study ABAQUS

60 With shear Without shear

2.615 72.621 72.627 71.399 72.750.062 80.069 80.075 79.211 80.157.26 107.27 107.27 106.15 107.24.54 154.56 154.57 154.22 154.35.23 195.25 195.26 193.41 195.12.09 212.11 212.12 211.16 212.17.78 107.79 107.79 106.53 107.9

ad.).

This study ABAQUS

8 With shear Without shear

0.24470 0.24475 0.24413 0.24070.21814 0.21816 0.21782 0.21680.18200 0.18200 0.18181 0.18380.17254 0.17254 0.17236 0.17540.19647 0.19647 0.19629 0.19870.25477 0.25478 0.25445 0.25490.26733 0.26735 0.26686 0.2638

Fig. 8. Vertical deflections at mid-span of channel-beams under a vertical force actingat mid-span.

N.-I. Kim et al. / European Journal of Mechanics A/Solids 41 (2013) 111e122 119

one element. It is noted that the solutions from the finite elementusing the Lagrange interpolation functions which satisfy onlydisplacement continuity at nodal point are approximate. Resul-tantly, as a number of beam element used in the beam model in-creases, its solutions converge to those from the element developedby this study. On the other hand, it is possible for this study toobtain good results though a minimum number of elements is usedsince the current element is based on the shape functions satisfyingthe homogenous forms of the equilibrium equations exactly.

Fig. 3 shows the variation of the deflection at mid-span withrespect to the fiber angle change for CF and CC (clampedeclamped)beams under a vertical force 1 kN acting at mid-span. For conve-nience, the dimensionless deflection w* ¼ wPl2/(E2h3t) is used. It isseen that the flexural stiffness is highest at q ¼ 45� for the isotropicbeam (E1/E2 ¼ 1), while the maximum stiffnesses occur in low fiberangles for the unisotropic beams. Moreover, as E1/E2 increases, theflexural stiffness increases. It can also be seen in unisotropic beamsthat the fiber angle which the minimum deflection takes placeincreases as the end boundary condition of beam is restrained.

The maximum twist angles of the simply supported (SS) beamunder a twist moment 1 N m acting at right end are presented inTable 3. The degree of freedom corresponding to twist angle at rightend is released. In this case, the shear effects due to the warping ofcross-section become to be negligible. From Table 3, the correlationbetween this study and a displacement based finite element solu-tions by Lee and Lee (2004), ABAQUS results is seen to be excellent.The variation of the twist angles at mid-span for CF and CC beamsunder a twist moment 100 N m at mid-span is plotted in Fig. 4. Thedimensionless twist angle is defined as u*

1 ¼ u1M1=ðG12ht2Þ. It isobserved that as the fiber angle increases, the twist angle increasesand the maximum twist angle is exhibited at q ¼ 45� for CF beamwith E1/E2 ¼ 1. This is due to the fact that the stiffness componentD66 in flanges and web plays an important role in torsional stiffnessE55 and the fiber angle around 45� gives theminimum of E55. On theother hand, for CC beam with E1/E2 ¼ 1, the minimum twist angletakes places at q ¼ 45� because of the negative twist moment atright place of the applied moment. Similar to the flexural behavior,the torsional stiffness also increases with increase of E1/E2.

Fig. 5 shows the effect of shear on the flexural and torsionalbehavior of CC beam with respect to the fiber angle change. It isinteresting to observe that the largest shear effect occurs at q ¼ 45�

for E1/E2 ¼ 1, while it takes place at q ¼ 0� for E1/E2 > 1. It is alsoobserved that the shear effect increases as E1/E2 increases in fiberangles q � 22� and it decreases with increase of E1/E2 in q > 22�.

4.2. Beams with monosymmetric I-section

The beam with monosymmetric I-section, as seen in Fig. 6, isconsidered. The length of beam is 1m and the detailed lay-ups usedin this example are summarized in Table 4. The tip deflections of CFbeams under the vertical tip force 1 kN by this study are given inTable 5 and compared with those from the finite beam element andABAQUS analysis for various lay-ups. It is observed that the presentresults considering shear effects are in excellent agreement withthose of the ABAQUS analysis as well as the results with at least 60finite beam elements.

4.3. Beams with channel-section

In our final example, the flexural and torsional analyses ofbeams with channel-section are studied. The configuration ofcross-section is shown in Fig. 7 and the widths of flanges and theheight are 25 mm and 50 mm, respectively. The two flanges andweb have 16 layers with the thickness of 2.08 mm. For CF beamswith l ¼ 1 m, the tip deflection under a tip force 1 kN and the

maximum twist moment under a twist moment 1 N m acting atright end are shown in Tables 6 and 7, respectively. For comparison,the results from various numbers of the finite beam elements andABAQUS analysis are presented. For all lay-ups, a very good corre-lation between this study and ABAQUS analysis is attained. It canalso be found that the present results are in close agreement withthose obtained with at least 60 and 8 finite beam elements forflexural and torsional analyses, respectively. Moreover, it is seen

Fig. 9. Twist angles at mid-span of channel-beams under a twist moment acting atmid-span.

Fig. 10. Effects of shear on the deflection and the twist angle of CC channel-beams.

N.-I. Kim et al. / European Journal of Mechanics A/Solids 41 (2013) 111e122120

that the shear effect due to flexural shear is larger than that due towarping stress.

Figs. 8 and 9 show the variation of the vertical deflection andtwist angle at mid-span under the vertical force 1 kN and the twistmoment 100 N m acting at mid-span, respectively, for CF and CCbeams. In addition, the shear effects on the flexural and torsionalbehavior are seen in Fig. 10. It can be observed from Figs. 8e10, thetrends of the flexural and torsional behavior and the shear effects

are very similar to those of the beamswith bisymmetric I-section inSection 4.1.

5. Conclusion

In the present work, the improved laminated composite beamelementwas presented for the flexural and torsional analyses basedon the eigenvalue problem. The thin-walled laminated composite

Appendix B. Detailed expressions of the laminate stiffnesses Eij

E11 ¼ZA

Q*11 dy dz

E12 ¼ZA

Q*11ðz� n cos qÞdy dz

E13 ¼ZA

Q*11ðyþ n sin qÞdy dz

E14 ¼ZA

Q*11ðfþ nqÞdy dz

E15 ¼ZA

Q*16n dy dz

E16 ¼ZA

Q*f16 dy dz

E17 ¼ZA

Q*w16 dy dz

E18 ¼ZA

hQ

*w16 ðy� e2Þ � Q

*f16ðz� e3Þ

idy dz

E22 ¼ZA

Q*11

�z2 � 2zn cos qþ n2 cos2 q

�dy dz

E23 ¼Z

Q*11

�yz� yn cos qþ zn sin q� n2 sin q cos q

�dy dz

N.-I. Kim et al. / European Journal of Mechanics A/Solids 41 (2013) 111e122 121

beam theory considering shear effects due to the shear forces andthe restrained warping torsion was used on the basis of theorthogonal Cartesian coordinate system. The evaluation techniquefor the member stiffness of the beam was developed from theequilibrium equations and forceedisplacement relations. The the-ory was validated with the results from other researchers, the finitebeam element, and the detailed two-dimensional analysis resultsusing ABAQUS for composite beams with I- and channel-sections.Good correlation was achieved for beams with various lay-upstested in this study. Throughout numerical examples, thefollowing conclusions are made:

1) The laminated composite beam theory considering shear ef-fects and the free stress assumption in contour direction gave areliable beam model in comparison with the two-dimensionalfinite element analysis.

2) The present numerical method using a minimum number ofelements was cable of predicting the deflection and the twistangle of the beams with I- and channel-sections under theexternal forces.

3) The flexural and torsional stiffnesses increase with increases ofE1/E2. For torsional analysis of beamwith the isotropic material,the fiber angle which the minimum twist angle occurs wasdependent on the boundary conditions.

4) The shear effect was largest at q ¼ 45� and 0� for isotropic andunisotropic beams, respectively. Moreover, the shear effectincreases as E1/E2 increases in the lower range of fiber angle.However, it decreases with increase of E1/E2 in the higherranger of fiber angle.

A

E24 ¼ZA

Q*11

�fz� fn cos qþ znq� n2q cos q

�dy dz

E25 ¼ZA

Q*16

�zn� n2 cos q

�dy dz

E26 ¼ZA

Q*f16ðz� n cos qÞdy dz

E27 ¼Z

Q*w16 ðz� n cos qÞdy dz

Acknowledgments

This study was supported by Basic Science Research Programthrough the National Research Foundation of Korea (NRF) fundedby the Ministry of Education, Science and Technology (2010-0019373 & 2012R1A2A1A01007405), and the CITRC (ConvergenceInformation Technology Research Center) support program (NIPA-2013-H0401-13-1003) supervised by the NIPA (National IT IndustryPromotion Agency) of the MKE (Ministry of Knowledge Economy).

Appendix A. Expressions of the generalized forces andmoments

F1 ¼ZA

sx dy dz

F2 ¼ZA

sxy dy dz

F3 ¼ZA

sxz dy dz

M2 ¼ZA

sxðz� h cos jÞdy dz

M3 ¼ �ZA

sxðyþ h sin jÞdy dz

Mf ¼ZA

sxðfþ hqÞdy dz

T ¼ZA

sxzðy� e2Þ � sxyðz� e3Þ

�dy dz

Mt ¼ZA

�sxyh cos jþ sxzh sin j

�dy dz

(A-1)

A

E28 ¼ZA

hQ

*w16 ðy� e2Þ � Q

*f16ðz� e3Þ

iðz� n cos qÞdy dz

E33 ¼ZA

Q*11

�y2 þ 2yn sin qþ n2 sin2 q

�dy dz

E34 ¼ZA

Q*11

�fyþ fn sin qþ ynqþ n2q sin q

�dy dz

E35 ¼ZA

Q*16

�ynþ n2 sin q

�dy dz

E36 ¼ZA

Q*f16ðyþ n sin qÞdy dz

E37 ¼ZA

Q*w16 ðyþ n sin qÞdy dz

E38 ¼ZA

hQ

*w16 ðy� e2Þ � Q

*f16ðz� e3Þ

iðyþ n sin qÞdy dz

E44 ¼ZA

Q*11

�f2 þ 2qnfþ q2n2

�dy dz

E45 ¼ZA

Q*16

�fnþ qn2

�dy dz

E46 ¼ZA

Q*f16ðfþ qnÞdy dz

E47 ¼ZA

Q*w16 ðfþ qnÞdy dz

E48 ¼ZA

hQ

*w16 ðy� e2Þ � Q

*f16ðz� e3Þ

iðfþ qnÞdy dz

E55 ¼ZA

�Q

*f66n

2 cos qþ Q*w66 n

2 sin q�dy dz

E56 ¼ZA

Q*f66n cos q dy dz

E57 ¼ZA

Q*w66 n sin q dy dz

E58 ¼ZA

hQ

*w66 ðy� e2Þn sin q� Q

*f66ðz� e3Þn cos q

idy dz

E66 ¼ZA

Q*f66 dy dz

E68 ¼ �ZA

Q*f66ðz� e3Þdy dz

E77 ¼ZA

Q*w66 dy dz

E78 ¼ZA

Q*w66 ðy� e2Þdy dz

E88 ¼ZA

hQ

*f66ðz� e3Þ2 þ Q

*w66 ðy� e2Þ2

idy dz

(A-2)

N.-I. Kim et al. / European Journal of Mechanics A/Solids 41 (2013) 111e122122

where the superscripts f and w denote the flanges and the web,respectively.

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