Comput Mech (2009) 43:493–514DOI 10.1007/s00466-008-0324-9

ORIGINAL PAPER

Coupled deflection analysis of thin-walled Timoshenko laminatedcomposite beams

Nam-Il Kim · Dong Ku Shin

Received: 6 July 2008 / Accepted: 20 July 2008 / Published online: 9 September 2008© Springer-Verlag 2008

Abstract For the deflection analyses of thin-walledTimoshenko laminated composite beams with the mono-symmetric I-, channel-, and L-shaped sections, the stiffnessmatrices are derived based on the solutions of the simulta-neous ordinary differential equations. A general thin-walledcomposite beam theory considering shear deformation effectis developed by introducing Vlasov’s assumptions. The shearstiffnesses of thin-walled composite beams are explicitlyderived from the energy equivalence. The equilibrium equa-tions and force-deformation relations are derived from energyprinciples. By introducing 14 displacement parameters, ageneralized eigenvalue problem that has complex eigenva-lues and multiple zero eigenvalues is formulated. Polynomialexpressions are assumed as trial solutions for displacementparameters and eigenmodes containing undetermined para-meters equal to the number of zero eigenvalues are deter-mined by invoking the identity condition to the equilibriumequations. Then the displacement functions are constructedby combining eigenvectors and polynomial solutions cor-responding to nonzero and zero eigenvalues, respectively.Finally, the stiffness matrices are evaluated by applying themember force-displacement relations to the displacementfunctions. In addition, the finite beam element formulationbased on the classical Lagrangian interpolation polynomialis presented. In order to verify the validity and the accuracyof this study, the numerical solutions are presented and com-pared with the finite element results using the isoparametricbeam elements and the detailed three-dimensional analysis

N.-I. Kim · D. K. Shin (B)Department of Civil and Environmental Engineering,Myongji University, San 38-2, Nam-Dong,Yongin, Kyonggi-Do 449-728, South Koreae-mail: dkshin@mju.ac.kr

N.-I. Kime-mail: kni8501@gmail.com

results using the shell elements of ABAQUS. Particularly theeffects of shear deformations on the deflection of thin-walledcomposite beams with the mono-symmetric I-, channel-, andL-shaped sections with various lamination schemes are inves-tigated.

Keywords Composite beam · Thin-walled · Deflectionanalysis · Shear deformation · Stiffness matrix

1 Introduction

The thin-walled open-section composite beams are usedextensively in civil, marine and mechanical engineering aswell as in aerospace engineering. These structural compo-nents made of advanced composite materials are ideal forstructural applications because of the high strength-to-weightand stiffness-to-weight ratios and their ability to be tailoredto meet the design requirements of stiffness and strength.

To present, considerable attention over the past fewdecades has been paid to understanding the behavior of fiber-reinforced composite beams due to the many advantages theyoffer over isotropic beams. Especially, the research efforts[1–11] to improve the laminated beam theory have been madeby many researchers. Berkowitz [1] pioneered a theory ofsimple beams and columns for anisotropic materials. Vinsonand Sierakowski [2] applied classical lamination theory alongwith a plane strain assumption to obtain the extensional, cou-pling and bending stiffnesses for an Euler–Bernoulli typelaminated beam. Bank and Bednarczyk [3] proposed a theoryfor orthotropic thin-walled composite beams, where the in-plane material properties were obtained using classical lami-nation theory and coupon tests. A theory for thin-walledsymmetrically laminated beams of open profile was propo-sed by Skudra et al. [4] and they illustrated the distribution

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of forces in a flat homogeneous anisotropic strip. Tsai [5]defined the engineering constants from the laminate com-pliances, and employed them to obtain the deflections forlaminated beams. Dharmarajan and McCutchen [6] exten-ded the formulation of Cowper [7] who derived a shear coef-ficient for isotropic materials from the elasticity solution,for orthotropic beams without addressing the case of thin-walled sections. Bert [8] presented a derivation of the staticshear factor for rectangular beams of nonhomogeneous crosssection. Barbero et al. [9] derived an explicit expression forthe static shear correction factor of the thin-walled compositebeams with doubly-symmetric I-section from energy equiva-lence. Salim and Davalos [10] extended the classical Vlasovtheory of isotropic thin-walled section [11] to sections madeof composite laminates including the shear deformation ofthe cross section.

Analytical studies for the static analysis of compositebeam have been performed by several authors [12–20]. Kollárand Pluzsik [12] derived a closed-form solutions for the cal-culation of the 4 × 4 stiffness matrix of thin-walled opensection composite beam. In their study, there was no restric-tion on the layup of the wall segments and the local bendingstiffnesses were taken into account. But they neglected theeffects of shear deformation and restrained torsional war-ping. Then Pluzsik and Kollár [13] analyzed the effect ofshear deformation and restrained torsional warping for thin-walled beam with layups that are symmetrical or unsymme-trical, orthotropic or unorthotropic. A closed-form solutionfor the response of both symmetric and anti-symmetric layupI-beam with transverse shear coupling was derived by Jungand Lee [14]. The analysis included the effects of elasticcoupling, wall thickness, and transverse shear deformation.Song et al. [15] presented an analytical solution for the res-ponse of anisotropic composite I-beams loaded at their freeend and highlighted the importance of a number of essentialnonclassical effects in composite I-beams. Dufort et al. [16]proposed a simple analytical approach based on the higherorder theory that account for the cross section warping inbeams under three-point bending. Khdeir and Reddy [17]developed an analytical solution based on the state-spaceconcept in conjunction with the Jordan canonical form tosolve the governing equations for the bending of symmetricand anti-symmetric cross-ply beams with arbitrary boundaryconditions. The Capurso’s theory to the static analysis of FRPpultruded profile was generalized by Lorenzis and Tegola[18]. Kabir and Sherbourne [19] presented a theoretical studyon the structural response of thin-walled open section beamsmade from mid-plane symmetric fiber-reinforced laminatesbased on a Vlasov-type linear hypothesis. Chandra andChopra [20] performed a theoretical-experimental study onthe structural response of thin-walled composite I-beamswith elastic coupling based on Vlasov’s theory.

The finite element method, as an effective approachsolving the problem of composite beams, has been widelyused because of its versatility and a large amount of works[21–26] was devoted to the improvement of composite finiteelements in order to obtain the acceptable results. Subra-manian [21] developed a two-noded C1 finite element of8 DOF per node, based on the higher order shear defor-mation theory for flexural analysis of symmetric laminatedcomposite beams assuming a parabolic variation of trans-verse shear stress through the thickness of beams. A gene-ral finite element with 10 DOF per node was derived byWu and Sun [22] for the thin-walled laminated compositebeams by modifying the assumptions of the Vlasov theory.Maddur and Chaturvedi [23] presented a Vlasov type modi-fied first order shear deformation theory which can accountfor shear deformations of open profile sections and includesthe effect of interlaminar shear stresses in order to obtainwarping displacements. Also they simplified their theory forI-beams [24] as a special case and analyzed the deformationresponse of I-sections made of orthotropic laminated com-posites based on finite element procedure by using Lagrangeinterpolation function for the geometric coordinate variablesand Hermitian interpolation function for the unknown func-tions. Shi et al. [25] investigated the influence of the inter-polation order of bending strains on the solution accuracyof composite beam element based on HSDT and presenteda simple but accurate third-order composite beam elementusing the assumed strain finite element method. Lee and Lee[26] developed a general model applicable to the flexural-torsional behavior of I-section composite beam with arbitrarylaminate stacking sequence using the displacement-basedfinite element method. They expressed the generalized dis-placements as a linear combination of the one-dimensionalLagrangian interpolation function for axial displacement andthe Hermite-cubic interpolation function for lateral displace-ments and twist angle. In their formulation, the effect of shearwas not considered.

As an alternative numerical method, the boundary elementmethod (BEM) [27–32] was developed to solve the static pro-blems of the homogeneous or composite beams. It is knownthat the laminate theories do not satisfy the continuity condi-tions of transverse shear stress at layer interfaces and assumethat the transverse shear stress along the thickness coordi-nate remains constant, leading to the fact that kinematic orstatic assumptions cannot be always valid. To overcome thisdrawback, Mokos and Sapountzakis [27] developed a BEMfor the solution of the general transverse shear loading pro-blem of composite beams of arbitrary cross section. In theirstudy, a stress function was introduced, which fulfilled theequilibrium and compatibility equations and from which thetransverse shear stresses at any interior point were obtai-ned by direct differentiation. The boundary condition of the

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Comput Mech (2009) 43:493–514 495

stress function was derived following physical consideration:the traction vectors in the direction of the normal vectoron the interfaces separating the two different materials areequal in magnitude and opposite in direction. Sapountza-kis and Mokos [28] developed a BEM for the construc-tion of the stiffness matrix and the nodal load vector of amember of arbitrary homogeneous or composite cross sec-tions taking into account both warping and shear deforma-tion effects. The concept of shear deformation coefficientswas used and the evaluation of its coefficients was accompli-shed from stress functions using only boundary integration.Also Sapountzakis and Mokos [29] presented a BEM for thesolutions of the general transverse shear loading problemsof beams with homogeneous cross section and Sapountzakisand Mokos [30] developed a BEM for the nonuniform torsionof composite bars of arbitrary constant cross section, subjec-ted to a concentrated or distributed twisting moment. Twoboundary-value problems with respect to variable along thebeam angle of twist and to the warping function with respectto the shear center were formulated. Then Sapountzakis andMokos [31] improved their previous study [30] by accom-plishing the evaluation of the secondary warping functionleading to the computation of the secondary shear stressesdue to warping. Friedman and Kosmatka [32] presented aboundary element solution procedure for studying the cou-pled torsion and flexure problems of an isotropic beam havingan arbitrary cross section using the St. Venant semi-inversemethod.

Also, based on the extended Galerkin’s method, Oinand Librescu [33] investigated the effects of three types oflay-ups, namely, the cross-ply, circumferentially uniformstiffness, and circumferentially asymmetric stiffness for ananisotropic thin-walled beam. A stiffness matrix methodbased on the solution of the differential equation of beamwas developed by Murthy et al. [34]. They presented anaccurate beam element with 4 DOF per node subjected touniformly distributed load based on an exact solution of sta-tic governing equations. This beam theory was based on thehigher order shear deformation theory and used the axialand transverse displacement fields proposed by Heyliger andReddy [35].

The existing literatures reveal that, even though a signifi-cant amount of research has been intensively conducted onthe development of improved theories for the static analysisof composite beam, to the best of authors’ knowledge, therewas no study reported on the stiffness matrix of thin-walledcomposite beam with open cross sections considering theeffect of shear deformation in the literature. It is well knownthat evaluating the stiffness matrix of the thin-walled com-posite beam with shear deformation is very difficult due tothe complexities arising from coupling effects of extensio-nal, flexural, and torsional deformation as can be seen in 7simultaneous second order ordinary differential equations.

Accordingly, the primary aim of this study is to presentexplicit expressions for the stiffness matrices of thin-walledTimoshenko laminated composite beams with the mono-symmetric I-, channel-, and L-shaped sections. The presentapproach employs direct evaluation schemes using symbolicmanipulation rather than using a discrete integration schemeor a method based on energy principles. The important pointsof this study are summarized as follows:

1. Equilibrium equations and force-deformation relationsof thin-walled composite beam considering the effectsof shear deformation and restrained torsional warpingare derived by introducing Vlasov’s assumptions.

2. The shear stiffnesses for the mono-symmetric I- andchannel-sections, and the L-shaped section are expressedexplicitly from the energy equivalence.

3. The stiffness matrices of thin-walled Timoshenkolaminated composite beam are evaluated using the dis-placement state vector consisting of 14 displacementparameter.

4. The finite element model is presented based on theLagrangian interpolation polynomials.

5. To demonstrate the accuracy of this study, the coupleddeflection of composite beams with various types ofcross-sections subjected to a bending are evaluated andcompared with the finite element solutions using theisoparametric beam elements and the shell elements ofABAQUS [36]. Additionally, the effect of laminationschemes on the flexural behavior of beams is investi-gated.

2 Equilibrium equations of composite beam

2.1 Geometrical relationships and laminate constitutiverelations

The theoretical developments presented in this study requirethree sets of coordinate systems which are mutually interre-lated as shown in Fig. 1. The first coordinate system is theorthogonal Cartesian coordinate system (x1, x2, x3) and thesecond coordinate system is the local plate coordinate system(n, s, x1), wherein the n axis is normal to the middle surfaceof a plate element, the s axis is tangent to the middle surfaceand is directed along the contour line of the cross section.The third one is the contour coordinate s along the profileof the section with its origin at any point O on the profilesection. The point P is called the pole axis. For the deflec-tion analysis of thin-walled composite beam, the followingassumptions are adopted.

1. The strains are assumed to be small.2. The beam is linearly elastic and prismatic.

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Fig. 1 Pictorial definitions of coordinates in thin-walled section

3. The cross section is assumed to maintain its shape duringdeformation, so that there is no distortion.

4. Each laminate is thin and perfectly bonded.5. The hoop stress σs in the contour direction s is small

compared to the axial stress σx .6. Each plate element of a cross section is a thin

fiber-reinforced laminate that obeys the usual laminateconstitutive relations.

According to the assumption 3, the mid-plane displace-ment components u and v of an arbitrary point in the contourcoordinate system can be expressed as follows:

u(s, x1) = Uy(x1) sinψ(s)−Uz(x1) cosψ(s)−ω1(x1)q(s)

(1a)

v(s, x1) = Uy(x1) cosψ(s)+Uz(x1) sinψ(s)+ ω1(x1)r(s)

(1b)

where Uy , Uz , and ω1 are the rigid body translations and therotations of the cross section, respectively. Also the angleψ defines the relative orientation of the (x1, x2, x3) and(n, s, x1) coordinate systems, and is equal to the angle bet-ween x1 and s directions at G. The transverse shear strainγxs of the mid-plane of a plate element is as follows:

γxs = ∂w

∂s+ ∂v

∂x1= γy cosψ(s)+ γz sinψ(s) (2)

where γy and γz are the transverse shear strains in the x − yand x − z planes, respectively, and are expressed as follows:

γy = U ′y − ω3 (3a)

γz = U ′z + ω2 (3b)

where ω2 and ω3 are the rigid body rotations of the crosssection about x2 and x3 axes, respectively, and the super-script ‘prime’ indicates the derivative with respect to x1.Thus, Eqs. (1a), (1b), and (2) lead to the expression:

∂w

∂s= ω2(x1) sinψ(s)− ω3(x1) cosψ(s)− ω′

1(x1)r (4)

In the case of restrained warping of the cross section, whenthe shear deformation effect is considered, there is an additio-nal rate of twist of the cross section θs . Thus, in the presenceof shear deformation, the rate of twist of the beam is as fol-lows [22]:

θs = ω′1 + f (5)

where θs is the rate of twist due to the shear deformation whenwarping is zero and f is the rate of twist due to warping ofthe cross section when the shear strain is zero.

After integration of Eq. (4) with respect to s consideringEq. (5), yields

w(s, x1) = Ux (x1)+ ω2(x1)x3 − ω3(x1)x2 + f (x1)φ (6)

where φ is the prescribed warping function or the sectionproperty called the sectorial area and is defined by

φ =∫

C

rds (7)

We note that x2 and x3 are the coordinates of a point on thecontour C .

The constitutive relations between the membrane forces,the bending and torsional moments, as shown in Fig. 2 andtheir strains and curvatures for a mid-plane symmetric lami-nate [37] are as follows:⎧⎨⎩

Nx

Ns

Nxs

⎫⎬⎭ =

⎡⎣ A11 A12 A16

A12 A22 A26

A16 A26 A66

⎤⎦

⎧⎨⎩εx

εs

γxs

⎫⎬⎭ (8)

⎧⎨⎩

Mx

Ms

Mxs

⎫⎬⎭ =

⎡⎣ D11 D12 D16

D12 D22 D26

D16 D26 D66

⎤⎦

⎧⎨⎩κx

κs

κxs

⎫⎬⎭ (9)

where

Ai j =N∑

k=1

Qki j (tk − tk−1) = tai j (10a)

Di j = 1

3

N∑k=1

Qki j

(t3k − t3

k−1

)= t3di j (10b)

in which Ai j and Di j are the extensional and bending stiff-nesses, respectively. Also Qi j denotes the lamina stiffnesscoefficient, tk is the thickness of each lamina, and t is theoverall thickness of a plate element. The membrane strains

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Comput Mech (2009) 43:493–514 497

Fig. 2 Plate stress resultantsand their sign convention

x

s n

NsxNs

Nx Nxs

x

s n

Ms

Msx

Mxs

Mx

Qs

Qx

εx and εs are related to displacements as

εx = ∂w

∂x1(11a)

εs = ∂v

∂s(11b)

The axial, tangential, and twisting curvatures of the middlesurface, κx , κs , and κxs are respectively, expressed as follows.

κx = ∂2u

∂x21

(12a)

κs = ∂2u

∂s2 (12b)

κxs = −2∂2u

∂x1∂s(12c)

Substituting Eqs. (1a,b) and Eq. (6) which include the trans-verse shear and the restrained warping induced shear defor-mation, into Eqs. (11a), (12a), and (12c) yields

εx = U ′x + ω′

2x3 − ω′3x2 + f ′φ (13a)

κx = ω′2 cosψ + ω′

3 sinψ + f ′q (13b)

κxs = −2ω′1 (13c)

Since the pile in the laminated composites behave in a highlytwo-dimensional manner due to the Poisson’s effect [38], theappropriate assumptions for constitutive relations are essen-tial for a refined composite beam theory. In this regard, thezero hoop stress assumption is employed in this study. Assu-ming zero hoop stress leads to Ns = Ms = 0, then εs and κs

can be expressed from Eqs. (8) and (9) as

εs = − A12

A22εx − A26

A22γxs (14a)

κs = − D12

D22κx − D26

D22κxs (14b)

Substituting Eqs. (14a) and (14b) into Eqs. (8) and (9) givesthe reduced constitutive relation as follows:⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Nx

Nxs

Mx

Mxs

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎣

A∗11 A∗

16 − −A∗

16 A∗66 − −

− − D∗11 D∗

16

− − D∗16 D∗

66

⎤⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

εx

γxs

κx

κxs

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(15)

where

A∗11 = A11 − A2

12

A22(16a)

A∗16 = A16 − A12 A26

A22(16b)

A∗66 = A66 − A2

26

A22(16c)

D∗11 = D11 − D2

12

D22(16d)

D∗16 = D16 − D12 D26

D22(16e)

D∗66 = D66 − D2

26

D22(16f)

2.2 Force-deformation relationships and equilibriumequations

Based on the principle of virtual work by Gjelsvik [11], thebeam stress resultants that are equivalent to the distributionsof the plate stress resultants acting on a cross section of abeam are listed as

F1 =∫

C

Nx ds (17a)

M2 =∫

C

(Nx x3 + Mx cosψ) ds (17b)

M3 = −∫

C

(Nx x2 − Mx sinψ) ds (17c)

Mφ = −∫

C

(Nxφ + Mx q) ds (17d)

Ts = −∫

C

(Msx + Mxs) ds (17e)

where F1 is an axial force; M2 and M3 are bending momentsabout x2 and x3 axes, respectively; Mφ is the bimoment about

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498 Comput Mech (2009) 43:493–514

x1 axis; Ts is the St. Venant torsional moment with respectto x1 axis.

Substituting the plate stress resultants in Eq. (15) intoEqs. (17a–e), the axial force, the bending moments, the bimo-ment, and the St. Venant torsional moment of compositebeam are obtained by⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

F1

M2

M3

−Mφ

Ts

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

A S2 −S3 Sw −S2 I2 −I23 Iφ2 −Hc

−S3 −I23 I3 −Iφ3 −Hs

Sw Iφ2 −Iφ3 Iφ −Hq

· −Hc −Hs −Hq J G

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

U ′x

ω′2

ω′3

f ′

ω′1

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭(18)

where the detailed expressions of the sectional quantities inEq. (18) are presented in Appendix A.

Also we consider formalistically that the shear deforma-tions are related to the shear forces and the warping inducedtorque. For beam with arbitrary cross section, the shear defor-mations due to the coupling effect of shear forces and therestrained warping torsion are considered and consistentlypresented as⎧⎪⎪⎨⎪⎪⎩

F2

F3

MR

⎫⎪⎪⎬⎪⎪⎭

=

⎡⎢⎢⎣

k11 k12 k13

k21 k22 k23

k31 k32 k33

⎤⎥⎥⎦

⎧⎪⎪⎨⎪⎪⎩

U ′y − ω3

U ′z + ω2

ω′1 + f

⎫⎪⎪⎬⎪⎪⎭

(19)

where F2 and F3 are the horizontal and vertical compo-nents of cross-sectional shear forces, respectively; MR is therestrained (nonuniform) torsional moment about x1 axis.

In the linear regime, the strain energy density expressionfor a composite beam undergoing extension, bending, twis-ting, and warping deformations is given by

∏= 1

2

l∫

o

{F1U ′

x+F2

(U ′

y−ω3

)+F3

(U ′

z+ω2)+M2ω

′2

+ M3ω′3 + Tsω

′1 + Mφ f ′+MR

(ω′

1+ f)}

dx1 (20)

where l denotes the length of beam and substituting Eqs. (18)and (19) into Eq. (20), we can obtain the following strainenergy of the thin-walled composite beam.

∏= 1

2

l∫

o

{AU ′2

x + I2ω′22 + I3ω

′23 − 2I23ω

′2ω

′3 + J Gω′2

1

+ Iφ f ′2 + 2S2U ′xω

′2 − 2S3U ′

xω′3 + 2SwU ′

x f ′

− 2Hcω′1ω

′2 − 2Hsω

′1ω

′3 − 2Hqω

′1 f ′ + 2Iφ2ω

′2 f ′

− 2Iφ3ω′3 f ′ + k11

(U ′

y − ω3

)2 + k22(U ′

z + ω2)2

+ k33(ω′

1 + f)2 + 2k12

(U ′

y − ω3

) (U ′

z + ω2)

+ 2k13

(U ′

y − ω3

) (ω′

1 + f)

+ 2k23(U ′

z + ω2) (ω′

1 + f)}

dx1 (21)

The equilibrium equations can be obtained by variationof the strain energy in Eq. (21) with respect to the sevendisplacements measures Ux , Uy , Uz , ω1, ω2, ω3, and f .

F ′1 = 0 (22a)

F ′2 = 0 (22b)

F ′3 = 0 (22c)

M ′1 = T ′

s + M ′R = 0 (22d)

M ′2 − F3 = 0 (22e)

M ′3 + F2 = 0 (22f)

M ′φ − MR = 0 (22g)

Lastly, substituting the force-deformation relationships inEqs. (18) and (19) into Eqs. (22a–g), the equilibrium equa-tions can be expressed in terms of seven displacement para-meters.

AU ′′x + S2ω

′′2 − S3ω

′′3 + Sw f ′′ = 0 (23a)

k11

(U ′′

y −ω′3

)+ k12

(U ′′

z +ω′2

)+ k13(ω′′

1+ f ′) = 0 (23b)

k12

(U ′′

y −ω′3

)+ k22

(U ′′

z +ω′2

) + k23(ω′′

1+ f ′) = 0 (23c)

−Hcω′′2 − Hsω

′′3 − Hq f ′′ + J Gω′′

1 + k13

(U ′′

y − ω′3

)

+ k23(U ′′

z + ω′2

) + k33(ω′′

1 + f ′) = 0 (23d)

S2U ′′x + I2ω

′′2−I23ω

′′3+Iφ2 f ′′−Hcω

′′1−k12

(U ′

y−ω3

)

−k22(U ′

z+ω2)−k23

(ω′

1 + f) = 0 (23e)

−S3U ′′x −I23ω

′′2+I3ω

′′3−Iφ3 f ′′−Hsω

′′1 + k11

(U ′

y −ω3

)

+ k12(U ′

z +ω2) + k13

(ω′

1+ f) = 0 (23f)

SwU ′′x + Iφ2ω

′′2−Iφ3ω

′′3 + Iφ f ′′−Hqω

′′1−k13

(U ′

y−ω3

)

− k23(U ′

z +ω2) − k33

(ω′

1 + f) = 0 (23g)

3 Calculation of shear stiffnesses

In this section, based on the study by Barbero et al. [9] whoderived an explicit expression for the shear correction fac-tor of thin-walled composite beams from the energy equi-valence, we determine the elements of the shear stiffnessmatrix in Eq. (19) for the three cross-sections considered.The shear flow N∗

xs evaluated in the eth wall constitutes arefinement over the laminate shear stress resultant calculatedfrom constitutive Equation (15) and the detailed derivation

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Comput Mech (2009) 43:493–514 499

procedure is presented in [9] as follows:

N∗xs = − F2

I2

{Se + 1

2

(A∗

11

)e sinψe

(s2

e − b2e

4

)

+ (A∗

11

)e

(se + be

2

)}for − be

2≤ se ≤ be

2(24)

where

Se =e−1∑i=1

(A∗

11

)i

(yi − yn

)bi (25)

in which Se is the weighted static moment of the portion ofthe cross section corresponding to the first e−1 walls; be andyi are the wall width and the position of the wall centroid,respectively; yn is the position of the neutral axis of bendingof the cross section, which yields

yn =∑n

i=1 yi(

A∗11

)i bi∑n

i=1

(A∗

11

)i bi

(26)

The shear stiffness is introduced by equating the shearstrain energy obtained from the shear flow in the cross sec-tion, and the shear strain energy predicted by the Timoshenkobeam theory.

1

2

n∑i=1

bi /2∫

−bi /2

N∗xsγ

∗xs sin2 ψi dsi = 1

2F2 γz (27)

where γ ∗xs is the shear strain distribution in the i th wall and

is expressed as

γ ∗xs = N∗

xs

(A66)i(28)

Also the shear force F2 of beam in Eq. (27) can be expressedas

F2 = Kz Fz γz = k22 γz (29)

where Kz is the shear correction factor and

Fz =n∑

i=1

(A66)i bi sin2 ψi (30)

After substitution of the expression for γz from Eq. (29), andfor γ ∗

xs obtained form the equilibrium in Eq. (28), we obtain

1

2

n∑i=1

bi /2∫

−bi /2

1

(A66)i

(N∗

xs sinψi)2

dsi = F22

2k22(31)

where the beam shear force F2 can be expressed as

F2 =n∑

i=1

bi /2∫

−bi /2

N∗xs sinψi dsi (32)

Substituting the expression in Eq. (32) into Eq. (31), we arriveat

k22 =[∑n

i=1

(sinψi

∫ bi /2−bi /2

N∗xsdsi

)]2

∑ni=1

1(A66)i

sin2 ψi∫ bi /2

−bi /2

(N∗

xs

)2dsi

(33)

The explicit expression for the shear stiffness k22 is obtainedby replacing the expression for the shear flow Eq. (24) in Eq.(33) and performing the integrals, yielding

k22 =[∑n

i=1 bi sinψi (Si + ci )]2

∑ni=1

bi(A66)i

sin2 ψi(S2

i + 2ci Si + di) (34)

where the stiffness parameters of the i th wall, ci and di , aredefined as follows:

ci = 1

2bi

(A∗

11

)i

(yi − yn − 1

6bi sinψi

)(35a)

di = 1

3b2

i

(A∗

11

)2i

×{

b2i

40sin2 ψi − bi

4

(yi − yn

)sinψi + (

yi − yn)2

}

(35b)

The explicit expressions of shear stiffnesses for the mono-symmetric I-, channel-, and L-shaped sections are presentedin Appendix B.

4 Stiffness matrices of Timoshenko laminated compositebeams

4.1 Evaluation of displacement function

For the calculation of the stiffness matrix of the thin-walledTimoshenko laminated composite beam, the displacementfunction for the beam is first evaluated. We transform thesecond order simultaneous ordinary differential equations(SODEs) in Eqs. (23a–g) into the first order SODEs by intro-ducing a displacement state vector consisting of 14 displace-ment parameters.

d =⟨Ux ,U

′x ,Uy ,U

′y ,Uz,U

′z, ω1, ω

′1, ω2, ω

′2, ω3, ω

′3, f, f ′⟩T

= 〈d1, d2, d3, d4, d5, d6, d7, d8, d9, d10, d11, d12, d13, d14〉T

(36)

Using Eqs. (23a–g), (36) are transformed into the follo-wing SODEs of the first order with constant coefficients.

d ′1 = d2 (37a)

Ad ′2 + S2d ′

10 − S3d ′12 + Swd ′

14 = 0 (37b)

d ′3 = d4 (37c)

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500 Comput Mech (2009) 43:493–514

k11d ′4 + k12d ′

6 + k13d ′8 = −k12d10 + k11d12 − k13d14

(37d)

d ′5 = d6 (37e)

k12d ′4 + k22d ′

6 + k23d ′8 = −k22d10 + k12d12−k23d14 (37f)

d ′7 = d8 (37g)

k13d ′4 + k23d ′

6+(k33+ J G) d ′8−Hcd ′

10−Hsd ′12−Hqd ′

14

= −k23d10 + k13d12 − k33d14 (37h)

d ′9 = d10 (37i)

S2d ′2 − Hcd ′

8 + I2d ′10 − I23d ′

12 + Iφ2d ′14

= k12d4 + k22d6 + k23d8 + k22d9 − k12d11 + k23d13

(37j)

d ′11 = d12 (37k)

−S3d ′2 − Hsd ′

8 − I23d ′10 + I3d ′

12 − Iφ3d ′14

= −k11d4−k12d6−k13d8−k12d9+k11d11−k13d13 (37l)

d ′13 = d14 (37m)

Swd ′2 − Hqd ′

8 + Iφ2d ′10 − Iφ3d ′

12 + Iφd ′14

= k13d4 + k23d6 + k33d8+k23d9−k13d11+k33d13 (37n)

which can be compactly expressed as

A d′ = B d (38)

where the detailed expressions for matrices A and B are givenin Appendix C. To find the homogeneous solution of theSODEs in Eq. (38), the following eigenvalue problem withnonsymmetric matrix is considered.

λ A Z = B Z (39)

The general eigenvalue problem of Eq. (39) has the com-plex eigenvalue λ and the associated eigenvector because thematrix B is nonsymmetric. In this study, an IMSL subroutineDGVCRG (IMSL Library [39]) is used to obtain the complexeigen solutions of Eq. (39).

4.1.1 Beams with mono-symmetric I- and channel-sections

In case of evaluating the displacement function of thin-walledcomposite beams with the mono-symmetric I- and channel-sections, first we can obtain 2 nonzero eigenvalues and 12zero eigenvalues from the eigen problem in Eq. (39). Thenonzero eigenvalues and eigenvectors corresponding to 2

pairs may be e xpressed as follows:

(λi , Zi) , i = 1, 2 (40)

where

Zi = ⟨z1,i , z2,i , z3,i , z4,i , z5,i , z6,i , z7,i , z8,i , z9,i , z10,i ,

z11,i , z12,i , z13,i , z14,i⟩T (41)

Based on the above eigen solutions, it is possible that thegeneral solution of Eq. (38) corresponding to nonzeroeigenvalues is represented by the linear combination of eigen-vectors with complex exponential functions as follows:

dI =2∑

i=1

Zi eλi x ai = Z1eλ1x a1 + Z2eλ2x a2 = XIaI (42)

where

XI = ⟨Z1eλ1x ; Z2eλ2x ⟩ (43a)

aI = 〈a1, a2〉T (43b)

in which XI and aI denote the 14 × 2 matrix function madeup of 2 eigen solutions and the integration constant vector,respectively.

In the next step, seven displacement parameters, to deter-mine the displacement modes corresponding to 12 zero eigen-values, are assumed to be

Ux = µ1 + µ2x (44a)

Uy = α1 + α2x + α3x2 + α4x3

3! (44b)

Uz = β1 + β2x + β3x2 + β4x3

3! (44c)

ω1 = γ1 + γ2x + γ3x2 + γ4x3

3! (44d)

ω2 = δ1 + δ2x + δ3x2 + δ4x3

3! (44e)

ω3 = ζ1 + ζ2x + ζ3x2 + ζ4x3

3! (44f)

f = τ1 + τ2x + τ3x2 + τ4x3

3! (44g)

where µ1, µ2 and αi , βi , γi , δi , ζi , τi (i = 1−4) are theundetermined coefficients for each displacement parameters.

Substituting Eqs. (44a–g) into the equilibrium equationsin Eqs. (23a–g) and applying the undetermined parameter

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Comput Mech (2009) 43:493–514 501

method, the following relations are obtained as

I3ζ3 − I23δ3 + k11 (α2 − ζ1)+ k12 (β2 + δ1)

+ k13 (γ2 + τ1) = 0 (45a)

I2δ3 − I23ζ3 − k12 (α2 − ζ1)− k22 (β2 + δ1)

− k23 (γ2 + τ1) = 0 (45b)

−k13 (α2 − ζ1)− k23 (β2 + δ1)− k33 (γ2 + τ1) = 0 (45c)

ζ3 = α4 (45d)

ζ2 = α3 (45e)

δ3 = −β4 (45f)

δ2 = −β3 (45g)

Eqs. (45a–c) are can be expressed as

⎧⎪⎪⎨⎪⎪⎩

α2 − ζ1

β2 + δ1

γ2 + τ1

⎫⎪⎪⎬⎪⎪⎭

= −

⎡⎢⎢⎣

k11 k12 k13

k12 k22 k23

k13 k23 k33

⎤⎥⎥⎦

−1 ⎡⎢⎢⎣

I3 I23

I23 I2

− −

⎤⎥⎥⎦

⎧⎪⎪⎨⎪⎪⎩

α4

β4

−

⎫⎪⎪⎬⎪⎪⎭

=

⎡⎢⎢⎣

W11 W12

W21 W22

W31 W32

⎤⎥⎥⎦

⎧⎪⎪⎨⎪⎪⎩

α4

β4

−

⎫⎪⎪⎬⎪⎪⎭

(46)

From Eq. (46), ζ1, δ1, and η1 are expressed as

ζ1 = α2 − W11α4 − W12β4 (47a)

δ1 = −β2 + W21α4 + W22β4 (47b)

τ1 = −γ2 + W31α4 + W32β4 (47c)

Substituting Eqs. (45d–g) and (47a–c) into Eqs. (44–g), sevendisplacement parameters with 12 nonzero undeterminedcoefficients are determined as follows:

Ux = µ1 + µ2x (48a)

Uy = α1 + α2x + α3x2

2+ α4

x3

3 ! (48b)

Uz = β1 + β2x + β3x2

2+ β4

x3

3 ! (48c)

ω1 = γ1 + γ2x (48d)

ω2 = −β2 + W21α4 − β3x − β4

(x2

2− W22

)(48e)

ω3 = α2 − W12β4 + α3x + α4

(x2

2− W11

)(48f)

f = −γ2 + W31α4 + W32β4 (48g)

From which the homogeneous solution corresponding to 12zero eigenvalues may be written as

dII =14∑

i=3

Ziai =XIIaII (49)

where

XII = [Z3; Z4; Z5; Z6; Z7; Z8; Z9; Z10; Z11;Z12; Z13; Z14] (50a)

aII =〈a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14〉T

= 〈µ1, µ2, α1, α2, α3, α4, β1, β2, β3, β4, γ1, γ2〉T (50b)

Consequently, from Eqs. (42) and (49), the solution of thedisplacement state vector corresponding to the nonzero andzero eigenvalues is obtained as follows:

d = XIaI + XIIaII = Xa (51)

where the detailed components for matrix X are presented inAppendix D.

Next, the complex coefficients a can be represented withrespect to 14 nodal displacement parameters. For this, thenodal displacement vector at p and q which mean the twoends of the member (x = 0, l) is defined by

Ue = ⟨Up, Uq ⟩T (52)

where

Uχ =⟨Uχ

x ,Uχy ,U

χz , ω

χ1 , ω

χ2 , ω

χ3 , f χ

⟩T, χ = p, q (53)

Also Eq. (53) can be rewritten as

Up = ⟨Ux (o),Uy(o),Uz(o), ω1(o), ω2(o), ω3(o), f (o)

⟩T(54a)

Uq = ⟨Ux (l),Uy(l),Uz(l), ω1(l), ω2(l), ω3(l), f (l)

⟩T(54b)

Substituting coordinates of the element end into Eq. (51)and accounting for Eq. (54), the nodal displacement vectorUe can be obtained as follows:

Ue = E a (55)

where E is evaluated from X in Eq. (51) and is presented inAppendix D.

Elimination of the complex coefficients a from Eq. (51)using Eq. (55) yields the displacement state vector consistingof 14 displacement components.

d = X E−1Ue (56)

where the inverse of E is calculated using an IMSL subroutineDLINCG (IMSL Library [39]). It should be noted that X E−1

in Eq. (56) denotes the exact interpolation matrix and thisdisplacement state vector satisfies the homogenous form ofthe SODEs in Eqs. (23a–g).

4.1.2 Beam with L-shaped section

It is well known that for beam with L-shaped section, the sec-tional properties, Iφ , Iφ2, Iφ3, and the shear stiffnesses, k33,k13, k23 associated with the restrained warping are negligible

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502 Comput Mech (2009) 43:493–514

since the warping function of cross section at the shear centerbecomes zero. In this case, the nonzero eigenvalues are notgenerated from the eigenvalue problem in Eq. (39). There-fore, applying the identity condition used in previous section,the eigenmodes for six displacement parameters except thewarping displacement, f are given by

Ux = µ1 + µ2x (57a)

Uy = α1 + α2x + α3x2

2+ α4

x3

3 ! (57b)

Uz = β1 + β2x + β3x2

2+ β4

x3

3 ! (57c)

ω1 = γ1 + γ2x (57d)

ω2 = −β2 + W ∗21α4 − β3x − β4

(x2

2− W ∗

22

)(57e)

ω3 = α2 − W ∗12β4 + α3x + α4

(x2

2− W ∗

11

)(57f)

where

W ∗11 = −k22 I3 + k12 I23

k11k22 − k212

, W ∗12 = −k22 I23 + k12 I2

k11k22 − k212

,

W ∗21 = k12 I3 − k11 I23

k11k22 − k212

, W ∗22 = k12 I23 − k11 I2

k11k22 − k212

(58-d)

The matrix X is given in Eqs. (57a–f). Evaluating the nodaldisplacement vector related to the complex coefficients a, thematrix E can be obtained. The detailed matrices X and E arepresented in Appendix E.

4.2 Calculation of element stiffness matrices

The procedure calculating element stiffness matrices of com-posite beam are presented based on the displacement functionderived in the previous section. We consider the nodal forcevector at two ends p and q of the beam defined by

Fe = ⟨Fp, Fq ⟩T (59)

where

Fχ =⟨Fχ1 , Fχ2 , Fχ3 ,Mχ

1 ,Mχ2 ,Mχ

3 ,Mχφ

⟩T, χ= p, q (60)

Next, the force-deformation relations in Eqs. (18) and (19)can be expressed in a matrix form.

f = S d (61)

in which each component of 7 × 14 matrix S is given inAppendix C. Here this matrix is reduced to 6×12 matrix forthe beam with L-shaped section.

Now, substitution of the displacement function in Eq. (56)into Eq. (61) leads to

f = S X E−1 Ue (62)

And nodal forces at ends of element are evaluated as

Fp = −f(0) = −S X(0) E−1 Ue (63a)

Fq = f(l) = S X(l) E−1 Ue (63b)

Consequently the element stiffness matrix of thin-walledcomposite beam is evaluated as

Fe = K Ue (64)

where

K =[−SX(0)E−1

SX(l)E−1

](65)

It should be noted that the stiffness matrix in Eq. (65) is for-med by the shape functions which are exact solutions of theequilibrium equations. Therefore, the accurate beam elementbased on the stiffness matrix developed by this study elimi-nates discretization errors and is free from the shear locking.

5 Finite element formulation

For comparison, the finite element formulation based on theisoparametric beam elements having the thin-walled crosssections and the shear deformations is presented. In thisstudy, the 2-noded isoparametric beam element with 7 DOFper node is introduced to interpolate displacement parame-ters that are defined at the centroid-shear center axes. Thecoordinate and all the displacement parameters of the beamelement can be interpolated with respect to the nodal coor-dinates and displacements, respectively, as follows:

x1 = l

2

(1 + r∗) (66a)

Ui =2∑α=1

Rα(r∗) Uα

i i = x, y, z (66b)

ωi =2∑α=1

Rα(r∗) ωαi i = 1, 2, 3 (66c)

f =2∑α=1

Rα(r∗) f α (66d)

where Uαi ,ωαi and f α are the translational and rotational dis-

placements in the xi direction and warping parameter at nodeα, respectively; Rα is the Lagrangian interpolation functionwhose the detailed expression is presented in Bathe [40]; r∗is a natural coordinate that varies from −1 to +1.

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Comput Mech (2009) 43:493–514 503

The element displacement vector Ue and force vector Fe

for the isoparametric beam element are defined as

Ue =[U 1,U 2

](67a)

Uα =[Uα

x ,Uαy ,U

αz , ω

α1 , ω

α2 , ω

α3 , f α

]T, α = 1, 2 (67b)

Fe =[

F1, F2]

(67c)

Fα =[

Fα1 , Fα2 , Fα3 ,Mα1 ,Mα

2 ,Mα3 ,Mα

φ

]T(67d)

where Uα and Fα are the nodal point displacement and forcevectors, respectively.

Substituting the shape functions and cross-sectional pro-perties into Eq. (21) and integrating along the element length,the potential energy of thin-walled beam element is obtainedin a matrix form as

∏T

= 1

2UT

e KeUe − UTe Fe (68)

where Ke is the element elastic stiffness in local coordi-nate. The elastic stiffness matrix is evaluated using a redu-ced Gauss numerical integration scheme and the assemblyof element stiffness matrix for the entire structure based onthe coordinate transformation leads to the equilibrium matrixequation in a global coordinate system.

6 Numerical examples

The coupled deflection analyses of thin-walled Timoshenkolaminated composite beams with the mono- symmetric I- andchannel-sections, and the L-shaped section are performed inorder to demonstrate the accuracy of the proposed procedure.The solutions obtained from this study are compared with thefinite element solutions using the isoparametric beam ele-ments and the nine-node laminated shell elements (S9R5)by ABAQUS [36] using the reduced integration with hour-glass control. For ABAQUS calculation, a total of 60 S9R5shell elements (6 along the beam span and 10 through thecross section) for beam with the mono-symmetric I-sectionare used to obtain the results. Also the 160 S9R5 shell ele-ments (8 and 20 along the beam span and through the crosssection, respectively) for beams with the channel-section andL-shaped section are used.

The influence of the shear deformation, the fiber orienta-tion, and boundary conditions on the deflection of beams isalso investigated. The material of beams used in followingexamples is glass-epoxy and its material properties are asfollows:

Fig. 3 Configuration of a mono-symmetric I-section

E1 = 53.78 GPa, E2 = E3 = 17.93 GPa,

G12 = G13 = 8.96 GPa G23 = 3.45 GPa, (69)

ν12 = ν13 = 0.25, ν23 = 0.34

where subscripts ‘1’ and ‘2’ correspond to directions paralleland perpendicular to fibers, respectively.

6.1 Mono-symmetric I-beam

The mono-symmetric I-beam as shown in Fig. 3 is conside-red. The length of beam is 25 cm, and the top and bottomflange widths are 3 and 4 cm, respectively. The height ofbeam is 5 cm and the total thicknesses of top flange, bot-tom flange, and web are assumed to be 0.208, 0.312, and0.104 cm, respectively. The detailed stacking sequences ofmono-symmetric I-beam considered in this example are sum-marized in Table 1. The vertical tip deflection of a clamped-free (CF) beam subjected to a tip force 10 kN by the presentstiffness matrix method (SMM) is given in Table 2 for thelamination schemes considered. The results are comparedwith those obtained from the various numbers of isopara-metric beam elements and from ABAQUS analysis. Alsopresented in Table 2 are the results from the classical beamtheory (CBT) which neglects the shear deformation effect. Itcan be found from Table 2 that the solutions by SMM usingonly a single element are in an excellent agreement with thoseusing 40 isoparametric beam elements. Also the correlation

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504 Comput Mech (2009) 43:493–514

Table 1 Stacking sequence of mono-symmetric I-beams

Stacking sequence Top flange Bottom flange Web Shear stiffness k22 (N)

ANG 0 [0]16 [0]24 [0]8 4.599E+05

ANG15 [15/−15]4S [15/−15]6S [15/−15]2S 5.504E + 05

ANG30 [30/−30]4S [30/−30]6S [30/−30]2S 7.316E + 05

ANG45 [45/−45]4S [45/−45]6S [45/−45]2S 8.222E + 05

ANG60 [60/−60]4S [60/−60]6S [60/−60]2S 7.316E + 05

ANG75 [75/−75]4S [75/−75]6S [75/−75]2S 5.504E + 05

QSISO [0/45/90/−45]2S [0/45/90/−45]3S [0/45/90/−45]S 4.599E + 05

Table 2 Vertical tip deflection of mono-symmetric CF I-beams (cm)

Stacking sequence CBT Isoparametric beam elements SMM ABAQUS

2 4 10 40

ANG 0 0.82223 1.3145 1.3530 1.3638 1.3657 1.3659 1.426

ANG15 0.91007 1.3074 1.3500 1.3620 1.3641 1.3642 1.419

ANG30 1.2256 1.4907 1.5482 1.5643 1.5671 1.5673 1.608

ANG45 1.7705 1.9639 2.0469 2.0702 2.0743 2.0746 2.101

ANG60 2.2376 2.4395 2.5444 2.5738 2.5790 2.5793 2.606

ANG75 2.4289 2.7313 2.8451 2.8770 2.8827 2.8831 2.919

QSISO 1.4216 1.7228 1.7894 1.8081 1.8114 1.8116 1.850

between the SMM and the ABAQUS analysis is seen to beexcellent for the whole range of fiber angle considered.

It should note that both of SMM and isoparametric beamformulation use the same elastic strain energy. The diffe-rence of two methods is that the SMM is based on the shapefunctions which satisfy the homogeneous forms of the equi-librium equations exactly. Therefore it is possible to obtainthe solutions though only a minimum number of beam ele-ments are used. However, the solutions obtained from theisoparametric beam elements using the Lagrangian inter-polation function which satisfies only displacement conti-nuity at nodal point are approximate. Resultantly, as anumber of isoparametric beam element used in the beamstructure increases, its solutions converge to those from theSMM.

Figure 4 shows the variation of the tip deflection withrespect to the fiber angle change for CF beams with andwithout shear deformation. Also the deflection at the mid-span for both sides simply supported (SS) and clamped (CC)beams subjected to a vertical force 10 kN acting at the mid-span are plotted in Figs. 5 and 6, respectively. From Figs. 4to 6, it is interesting to observe that the deflection by CBTincreases as the fiber angle increases for the three boundaryconditions considered. On the other hand, Fig. 4 shows thatthe deflection by SMM decreases and is minimum near 9◦of fiber angle and then increases with the increase of fiberangle for CF beam. The deflections for SS and CC beams

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

0.5

1.0

1.5

2.0

2.5

3.0

0.8

1.3

1.8

2.3

2.8

Dis

pla

cem

ent

(cm

)

SMM

CBT

Fig. 4 Variation of the tip deflection with respect to the fiber anglechange for a mono-symmetric CF I-beam

are minimum near 28◦ and 38◦ of fiber angle, respectively,as can be seen in Figs. 5 and 6.

To investigate the influence of the shear deformation onthe deflection behavior of beams with respect to the fiber

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Comput Mech (2009) 43:493–514 505

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

0.04

0.08

0.12

0.16

0.20

0.24

0.28

0.06

0.10

0.14

0.18

0.22

0.26

0.30D

isp

lace

men

t (c

m)

SMM

CBT

Fig. 5 Variation of the deflection at the mid-span with respect to thefiber angle change for a mono-symmetric SS I-beam

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

0.00

0.04

0.08

0.12

0.16

0.02

0.06

0.10

0.14

0.18

Dis

pla

cem

ent

(cm

)

SMM

CBT

Fig. 6 Variation of the deflection at the mid-span with respect to thefiber angle change for a mono-symmetric CC I-beam

angle change, the relative difference of deflection betweenthe CBT and the SMM is depicted in Fig. 7. It is observedfrom Fig. 7 that the shear effect is most significant at 0◦and least significant at 57◦ for all boundary conditions. Thediscrepancy of the deflections due to the ignorance of sheardeformation effect is over 90% at 0◦ for CC beam since theratio of height to length (h/ l = 5) is very small.

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

0

20

40

60

80

100

10

30

50

70

90

(SM

M-C

BT

)/S

MM

×100

(%

)

CF

SS

CC

Fig. 7 Relative deflection ratio between SMM and CBT for the mono-symmetric I-beams

Fig. 8 Configuration of a channel-section

6.2 Beam with channel-section

We consider the beam with channel section as shown in Fig. 8.The beam length is 25 cm, the widths of top and bottomflanges are 2.5 cm, and the height is 5 cm. The two flangeshave 16 layers with the thickness of 0.208 cm and the webhas 8 layers with its thickness of 0.104 cm. The stacking

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506 Comput Mech (2009) 43:493–514

Table 3 Stacking sequence of beams with channel section

Stacking sequence Top and bottom flanges Web Shear stiffness k22 (N)

ANG 0 [0]16 [0]8 4.640E + 05

ANG15 [15/−15]4S [15/−15]2S 5.554E + 05

ANG30 [30/−30]4S [30/−30]2S 7.382E + 05

ANG45 [45/−45]4S [45/−45]2S 8.296E + 05

ANG60 [60/−60]4S [60/−60]2S 7.382E + 05

ANG75 [75/−75]4S [75/−75]2S 5.554E + 05

QSISO [0/45/90/−45]2S [0/45/90/−45]S 4.640E + 05

Table 4 Vertical deflection at the mid-span of CC beams with channel section (cm)

Stacking sequence CBT Isoparametric beam elements SMM ABAQUS

4 10 40 60

ANG 0 0.19945 1.4965 1.5384 1.5459 1.5461 1.5464 1.605

ANG15 0.22075 1.2908 1.3372 1.3455 1.3458 1.3460 1.402

ANG30 0.29729 1.0696 1.1320 1.1432 1.1436 1.1439 1.195

ANG45 0.42947 1.0754 1.1656 1.1817 1.1823 1.1828 1.229

ANG60 0.54279 1.2537 1.3677 1.3880 1.3888 1.3894 1.433

ANG75 0.58916 1.5671 1.6908 1.7129 1.7138 1.7144 1.761

QSISO 0.34486 1.2249 1.2973 1.3102 1.3107 1.3111 1.358

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

1.0

2.0

3.0

4.0

5.0

1.5

2.5

3.5

4.5

Dis

pla

cem

ent

(cm

)

SMM

CBT

Fig. 9 Variation of the deflection at the free end with respect to thefiber angle change for a CF beam with channel section

sequences of the channel-section are presented in Table 3.Table 4 shows the vertical deflections at the mid-span of theCC beam subjected to a force 100 kN acting at the mid-spanfor various lamination schemes. As can be seen in Table 4,a very good agreement between the results from the present

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

0.00

0.10

0.20

0.30

0.40

0.05

0.15

0.25

0.35

Dis

pla

cem

ent

(cm

)

SMM

CBT

Fig. 10 Variation of the deflection at the mid-span with respect to thefiber angle change for a SS beam with channel section

SMM and ABAQUS analysis is achieved. When the resultsby the isoparametric beam elements are compared with thoseby SMM, 60 beam elements are required to achieve the accu-rate results. It can also be observed that the coupled twistangle, which is not presented in Table 4, is the largest with

123

Comput Mech (2009) 43:493–514 507

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

0.00

0.04

0.08

0.12

0.16

0.20

0.02

0.06

0.10

0.14

0.18

Dis

pla

cem

ent

(cm

)

SMM

CBT

Fig. 11 Variation of the deflection at the mid-span with respect to thefiber angle change for a CC beam with channel section

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

0

20

40

60

80

100

10

30

50

70

90

(SM

M-C

BT

)/S

MM

×100

(%

)

CF

SS

CC

Fig. 12 Relative deflection ratio between SMM and CBT for beamswith channel sections

its value of 0.00184 rad at 30◦ of ply angle. This is due to thefact that the maximum bending -twisting coupling stiffnessD16 occurs near 30◦.

The variation of the tip deflection for CF beam under theforce 10 kN acting at free end is plotted in Fig. 9 and the mid-span deflections for SS and CC beams under 10 kN acting at

the mid-span are presented in Figs. 10 and 11, respectively.It is indicated that the deflection behavior of channel-sectionbeam appears quite similar to the previous deflection res-ponse of mono-symmetric I-beam. The response from CBTincreases with the increase of fiber angle for all boundaryconditions. While it is seen to have a minimum value at 5◦ply angle for CF beam, and at 20◦ and 34◦ for SS and CCbeams, respectively. Also the inclusion of shear deformationincreases the deflection by 87% at 0◦ for CC beam as shown inFig. 12. The trend for the variation of deflection with respectto the fiber angle is similar to that of the mono-symmetricI-beam.

6.3 Beam with L-shaped section

The purpose of our final example is to evaluate the verticaldeflection and lateral displacement of beam with L-shapedsection as shown in Fig. 14. The length of beam is 25 cmand the widths of flange and web are 5 cm. The height ofbeam is 5 cm and the total thicknesses of both flange andweb are assumed to be 0.208 cm with 16 layers. The ver-tical tip deflection and lateral tip displacement of the CFbeam obtained from SMM using a single element are pre-sented in Table 5 and 6, respectively, and compared withthe results from the various numbers of finite beam elementsand ABAQUS analysis. It is seen from Tables 5 and 6 thatthe solutions by SMM agree very well with those by using40 isoparametric beam elements and by ABAQUS, whilethe solutions by using a small number of the isoparame-tric beam elements are not accurate enough by substantially

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

0.2

0.4

0.6

0.8

1.0

0.3

0.5

0.7

0.9

Dis

pla

cem

ent

(cm

)

SMM

CBT

Fig. 13 Variation of the vertical tip deflection with respect to the fiberangle change for a CF beam with L-shaped section

123

508 Comput Mech (2009) 43:493–514

overestimating the flexural stiffness. It can also be observedfrom Table 6 that the lateral tip displacements by CBT andSMM are almost the same. This is because the shear stiff-nesses corresponding to the strong and weak axes are thesame and consequently the lateral tip displacement due tothe shear along the two directions is counterbalanced eachother.

Figures 13, 15 and 16 show the variations of verticaldeflection for CF, SS, and CC beams, respectively, with res-pect to the fiber angle change. It is indicated that the deflectionbehavior appears quite different from the previous deflec-tion responses of beams with the mono-symmetric I- andchannel-sections in that the deflection of CF, SS, and CCbeams considering shear effects increases as the fiber angleincreases. A relative ratio of vertical tip deflection betweenSMM and CBT is plotted in Fig. 17. It is seen in Fig. 17 thatthe effect of shear deformation for the beam with L-shapedsection is much less than that for mono-symmetric I- andchannel-sectioned beams for all boundary conditions.

7 Conclusion

In the present work, an improved thin-walled composite beamtheory considering the shear deformation is developed andthe shear stiffnesses of various cross-sections are derived

Fig. 14 Configuration of an L-shaped section

explicitly from the energy equivalence. A simple but effi-cient numerical method that exactly evaluates the elementstiffness matrix is presented by introducing 14 displacementparameters. This systematic method determines eigenmodescorresponding to multiple zero and nonzero eigenvalues andderives the displacement functions for displacement

Table 5 Vertical tip deflection of CF beams with L-shaped section (cm)

Stacking sequence k11, k22 CBT Isoparametric beam elements SMM ABAQUS

4 10 40

[0]16 1.553E+06 2.7932 2.9109 2.9476 2.9542 2.9546 3.106

[15/−15]4S 1.859E+06 3.0872 3.1740 3.2145 3.2217 3.2222 3.347

[30/−30]4S 2.471E+06 4.1573 4.1941 4.2487 4.2584 4.2591 4.332

[45/−45]4S 2.777E+06 6.0056 6.0027 6.0815 6.0956 6.0965 6.134

[60/−60]4S 2.471E+06 7.5897 7.5734 7.6731 7.6909 7.6920 7.755

[75/−75]4S 1.859E+06 8.2390 8.2460 8.3541 8.3734 8.3747 8.494

[0/45/90/−45]2S 1.553E+06 4.8218 4.8627 4.9260 4.9373 4.9380 5.037

Table 6 Lateral tip displacement of CF beams with L-shaped section (cm)

Stacking sequence CBT Isoparametric beam elements SMM ABAQUS

2 4 10 40

[0]16 1.6759 1.5714 1.6500 1.6720 1.6759 1.6762 1.678

[15/−15]4S 1.8511 1.7357 1.8225 1.8468 1.8511 1.8514 1.854

[30/−30]4S 2.4926 2.3372 2.4541 2.4868 2.4926 2.4930 2.488

[45/−45]4S 3.6008 3.3763 3.5451 3.5924 3.6008 3.6014 3.587

[60/−60]4S 4.5505 4.2667 4.4801 4.5398 4.5505 4.5512 4.542

[75/−75]4S 4.9400 4.6319 4.8635 4.9284 4.9400 4.9407 4.939

[0/45/90/−45]2S 2.8908 2.7105 2.8461 2.8840 2.8908 2.8912 2.890

123

Comput Mech (2009) 43:493–514 509

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

0.10

0.20

0.30

0.40

0.50

0.60

0.15

0.25

0.35

0.45

0.55

Dis

pla

cem

ent

(cm

)

SMM

CBT

Fig. 15 Variation of the vertical deflection at the mid-span with respectto the fiber angle change for a SS beam with L-shaped section

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

0.04

0.08

0.12

0.16

0.20

0.06

0.10

0.14

0.18

Dis

pla

cem

ent

(cm

)

SMM

CBT

Fig. 16 Variation of the vertical deflection at the mid-span with respectto the fiber angle change for a CC beam with L-shaped section

parameters based on the undetermined parameter method.Then, the stiffness matrix is obtained using the member force-deformation relationships. The theory developed by thisstudy is validated by comparing the various deflection res-ponses by this method with those by the finite element beammodel using the Lagrangian interpolation polynomials for alldisplacement parameters and the detailed three-dimensional

0 20 40 60 8010 30 50 70 90

Fiber angle (degree)

0

10

20

30

40

50

5

15

25

35

45

(SM

M-C

BT

)/S

MM

×100

(%

)

CF

SS

CC

Fig. 17 Relative vertical tip deflection ratio between SMM and CBTfor the beams with L-shaped sections

analysis results using the shell elements of ABAQUS. Goodcorrelation is achieved for various beams with different cross-sections and lamination schemes considered in this study.From numerical examples, more significant effect of sheardeformation is seen for I- and channel shaped beams than forthe beam with L-shaped section.

Consequently, it is judged that the present numerical pro-cedure provides a refined method for not only the evaluationof the stiffness matrix of thin-walled composite beam butalso general solutions of simultaneous ordinary differentialequations of the higher order. Also this thin-walled compo-site beam element eliminates discretization errors and is freefrom the shear locking since the displacement state vectorsatisfies the homogenous form of the SODEs exactly.

Acknowledgments This work is a part of a research project suppor-ted by Korea Ministry of Construction & Transportation through KoreaBridge Design & Engineering Research Center at Seoul National Uni-versity. The authors express their gratitude for the financial support.

Appendix A: Detailed expressions of the sectionalquantities in Eq. (18)

A =∫

C

A∗11ds (A-1a)

S2 =∫

C

A∗11x3ds (A-1b)

123

510 Comput Mech (2009) 43:493–514

S3 =∫

C

A∗11x2ds (A-1c)

Sw =∫

C

A∗11φds (A-1d)

Hc = 2∫

C

D∗16 cosψds (A-1e)

Hs = 2∫

C

D∗16 sinψds (A-1f)

Hq = 2∫

C

D∗16qds (A-1g)

I2 =∫

C

(A∗

11x23 + D∗

11 cos2 ψ)

ds (A-1h)

I3 =∫

C

(A∗

11x22 + D∗

11 sin2 ψ)

ds (A-1i)

I23 =∫

C

(A∗

11x2x3 − D∗11 sinψ cosψ

)ds (A-1j)

Iφ =∫

C

(A∗

11φ2 + D∗

11q2)

ds (A-1k)

Iφ2 =∫

C

(A∗

11φx3 + D∗11q cosψ

)ds (A-1l)

Iφ3 =∫

C

(A∗

11φx2 − D∗11q sinψ

)ds (A-1m)

J G = 4∫

C

D∗66ds (A-1n)

Appendix B: Shear stiffnesses for the mono-symmetricI-, channel-, and L-shaped sections

Beam with mono-symmetric I-section

We consider a beam with mono-symmetric I-section as shownin Fig. 3 which has one axis of symmetry. From the explicitexpression of the shear stiffness k22 in Eq. (34) for the deflec-tion analysis of composite beam, the shear stiffness of themono-symmetric I-section is as follows:

k22 = 5(

A∗66

)3 h

[(A∗

11

)3 h (h − 3yn)− 6

(A∗

11

)1 b1 yn

]2

3[60

(A∗

11

)21 b2

1 y2n − 20

(A∗

11

)1

(A∗

11

)3 b1hyn (h − 3yn)+ (

A∗11

)23 h2

(3h2 − 15hyn + 20y2

n

)] (A-2)

where subscripts ‘1’ and ‘3’ denote the bottom flange andthe web, respectively.

Beam with channel-section

The shear stiffness of the channel-section as shown in Fig. 8is evaluated by

k22 = 5(

A∗66

)2 h

[6(

A∗11

)1 b + (

A∗11

)2 h

]2

6[30

(A∗

11

)21 b2 + 10

(A∗

11

)1

(A∗

11

)2 bh + (

A∗11

)22 h2

]

(A-3)

where subscripts ‘1’ and ‘2’ denote the flange and the web,respectively.

Beam with L-shaped section

Figure 14 shows the cross section of the L-shaped sectionwhich has the flange length b and the height h. Here, ξ andη are principal axes, respectively, and the angle θp is posi-tive counterclockwise relative to the x2 axis, which is givenby

θp = 1

2tan−1

(2I23

I3 − I2

)(A-4)

From Eq. (34), the shear stiffness kη22 along the η axis isas follows:

kη22 =[b sin θpcη1+h cos θp

(Sη2 +cη2

)]2

[b

(A∗66)1

sin2 θpdη1 + h(A∗

66)2cos2 θp

{(Sη2

)2 + 2Sη2 cη2 + dη2

}]

(A-5)

In Eq. (A-3), the stiffness parameters of the wall, cη1 , cη2 ,dη1 , dη2 , and Sη2 are

cη1 = 1

2b(

A∗11

)1

{−2 cos θp yn+ sin θp (2xn−b)

2−1

6b sin θp

}

(A-6a)

cη2 = 1

2h(

A∗11

)2

{cos θp (h−2yn)+2 sin θpxn

2−1

6h cos θp

}

(A-6b)

dη1 = 1

3b2 (

A∗11

)21

[b2

40sin2 θp+b

4cos θp sin θp yn−b

8sin2 θp

× (2xn−b)+{−2 cos θp yn+ sin θp (2xn−b)

2

}2]

(A-6c)

123

Comput Mech (2009) 43:493–514 511

dη2 = 1

3h2 (

A∗11

)22

[h2

40cos2 θp−h

8cos2 θp (h−2yn)−h

4sin θp

× cos θpxn+{

cos θp (h−2yn)+2 sin θpxn

2

}2]

(A-6d)

Sη2 = b(

A∗11

)1

{−2 cos θp yn + sin θp (2xn − b)

2

}(A-6e)

where

xn =(

A∗11

)1 b2

2{(

A∗11

)1 b + (

A∗11

)2 h

} , (A-7a)

yn =(

A∗11

)2 h2

2{(

A∗11

)1 b + (

A∗11

)2 h

} (A-7b)

Also the shear stiffness kξ22 along the ξ axis is as follows:

kξ22 =[b cos θpcξ1−h sin θp

(Sξ2 + cξ2

)]2

[b

(A∗66)1

cos2 θpdξ1 + h(A∗

66)2sin2 θp

{(Sξ2

)2 +2Sξ2 cξ2+dξ2

}]

(A-8)

where

cξ1 = 1

2b(

A∗11

)1

{cos θp (2xn−b)+2 sin θp yn

2−1

6b cos θp

}

(A-9a)

cξ2 = 1

2h(

A∗11

)2

{2 cos θpxn+ sin θp (h−2yn)

2−1

6h sin θp

}

(A-9b)

dξ1 = 1

3b2 (

A∗11

)21

[b2

40cos2 θp−b

4cos θp sin θp yn

− b

8cos2 θp (2xn−b)+

{cos θp (2xn−b)+2 sin θp yn

2

}2]

(A-9c)

dξ2 = 1

3h2 (

A∗11

)22

[h2

40sin2 θp− h

8sin2 θp (h−2yn)

+ h

4sin θp cos θpxn+

{2 cos θpxn− sin θp (h−2yn)

2

}2]

(A-9d)

Sξ2 = b(

A∗11

)1

{cos θp (2xn−b)+2 sin θp yn

2

}(A-9e)

Finally, by using the transformation equations for the shearstiffnesses, the k11, k12, and k22 are obtained as

k11 = kξ22 cos2 θp + kη22 sin2 θp (A-10a)

k12 =(

kξ22 − kη22

)sin θp cos θp (A-10b)

k22 = kξ22 sin2 θp + kη22 cos2 θp (A-10c)

Appendix C: Components for matrices A and B in Eq. (38)and matrix S in Eq. (61)

1) Components of matrix A

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1.0 − − − − − − − − − − − − −− A − − − − − − − S2 − −S3 − Sw− − 1.0 − − − − − − − − − − −− − − k11 − k12 − k13 − − − − − −− − − − 1.0 − − − − − − − − −− − − k12 − k22 − k23 − − − − − −− − − − − − 1.0 − − − − − − −− − − k13 − k23 − k33 + J G − −Hc − −Hs − −Hq

− − − − − − − − 1.0 − − − − −− S2 − − − − − −Hc − I2 − −I23 − Iφ2

− − − − − − − − − − 1.0 − − −− −S3 − − − − − −Hs − −I23 − I3 − −Iφ3

− − − − − − − − − − − − 1.0 −− Sw − − − − − −Hq − Iφ2 − −Iφ3 − Iφ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A-11)

123

512 Comput Mech (2009) 43:493–514

2) Components of matrix B

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

− 1.0 − − − − − − − − − − − −− − − − − − − − − − − − − −− − − 1.0 − − − − − − − − − −− − − − − − − − − −k12 − k11 − −k13

− − − − − 1.0 − − − − − − − −− − − − − − − − − −k22 − k12 − −k23

− − − − − − − 1.0 − − − − − −− − − − − − − − − −k23 − k13 − −k33

− − − − − − − − − 1.0 − − − −− − − k12 − k22 − k23 k22 − −k12 − k23 −− − − − − − − − − − − 1.0 − −− − − −k11 − −k12 − −k13 −k12 − k11 − −k13 −− − − − − − − − − − − − − 1.0

− − − k13 − k23 − k33 k23 − −k13 − k33 −

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A-12)

3) Components of matrix S

S =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

− A − − − − − − − S2 − −S3 − Sw− − − k11 − k12 − k13 k12 − −k11 − k13 −− − − k12 − k22 − k23 k22 − −k12 − k23 −− − − k13 − k23 − k33 + J G k23 −Hc −k13 −Hs k33 −Hq

− S2 − − − − − −Hc − I2 − −I23 − Iφ2

− −S3 − − − − − −Hs − −I23 − I3 − −Iφ3

− Sw − − − − − −Hq − Iφ2 − −Iφ3 − Iφ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A-13)

Appendix D: Matrices X and E for mono-symmetric I- and channel-sections

X =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

f1,1 f1,2 1.0 x − − − − − − − − − −f2,1 f2,2 − 1.0 − − − − − − − − − −f3,1 f3,2 − − 1.0 x x2/2 x3/6 − − − − − −f4,1 f4,2 − − − 1.0 x x2/2 − − − − − −f5,1 f5,2 − − − − − − 1.0 x x2/2 x3/6 − −f6,1 f6,2 − − − − − − − 1.0 x x2/2 − −f7,1 f7,2 − − − − − − − − − − 1.0 x

f8,1 f8,2 − − − − − − − − − − − 1.0

f9,1 f9,2 − − − − − W21 − −1.0 −x −x2/2 + W22 − −f10,1 f10,2 − − − − − − − − −1.0 −x − −f11,1 f11,2 − − − 1.0 x x2/2 − W11 − − − −W12 − −f12,1 f12,2 − − − − 1.0 x − − − − − −f13,1 f13,2 − − − − − W31 − − − W32 − −1.0

f14,1 f14,2 − − − − − − − − − − − −

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A-14)

123

Comput Mech (2009) 43:493–514 513

where

fi, j = zi, j eλ j x , i = 1 ∼ 14; j = 1, 2. (A-15)

E =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

z1,1 z1,2 1.0 − − − − − − − − − − −z3,1 z3,2 − − 1.0 − − − − − − − − −z5,1 z5,2 − − − − − − 1.0 − − − − −z7,1 z7,2 − − − − − − − − − − 1.0 −z9,1 z9,2 − − − − − W21 − −1.0 − W22 − −z11,1 z11,2 − − − 1.0 − −W11 − − − −W12 − −z13,1 z13,2 − − − − − W31 − − − W32 − −1.0y1,1 y1,2 1.0 l − − − − − − − − − −y3,1 y3,2 − − 1.0 l l2/2 l3/6 − − − − − −y5,1 y5,2 − − − − − − 1.0 l l2/2 l3/6 − −y7,1 y7,2 − − − − − − − − − − 1.0 ly9,1 y9,2 − − − − − W21 − −1.0 −l −l2/2 + W22 − −y11,1 y11,2 − − − 1.0 l l2/2 − W11 − − − −W12 − −y13,1 y13,2 − − − − − W31 − − − W32 − −1.0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A-16)

where

yi, j = zi, j eλ j l , i = 1, 3, 5, 7, 9, 11, 13; j = 1, 2 (A-17)

Appendix E: Matrices X and E for L-shaped section

X =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1.0 x − − − − − − − − − −− 1.0 − − − − − − − − − −− − 1.0 x x2/2 x3/6 − − − − − −− − − 1.0 x x2/2 − − − − − −− − − − − − 1.0 x x2/2 x3/6 − −− − − − − − − 1.0 x x2/2 − −− − − − − − − − − − 1.0 x− − − − − − − − − − − 1.0− − − − − W ∗

21 − −1.0 −x −x2/2 + W ∗22 − −

− − − − − − − − −1.0 −x − −− − − 1.0 x x2/2 − W ∗

11 − − − −W ∗12 − −

− − − − 1.0 x − − − − − −

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A-18)

and

E =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1.0 − − − − − − − − − − −− − 1.0 − − − − − − − − −− − − − − − 1.0 − − − − −− − − − − − − − − − 1.0 −− − − − − W ∗

21 − −1.0 − W ∗22 − −

− − − 1.0 − −W ∗11 − − − −W ∗

12 − −1.0 l − − − − − − − − − −− − 1.0 l l2/2 l3/6 − − − − − −− − − − − − 1.0 l l2/2 l3/6 − −− − − − − − − − − − 1.0 l

− − − − − W ∗21 − −1.0 −l −l2/2 + W ∗

22 − −− − − 1.0 l l2/2 − W ∗

11 − − − −W ∗12 − −

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A-19)

123

514 Comput Mech (2009) 43:493–514

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