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Applied Mathematics and Mechanics (English Edition), 2006, 27(12):17091718cEditorial Committee of Appl. Math. Mech., ISSN 0253-4827

ANALYSIS OF COMPOSITE LAMINATE BEAMS USINGCOUPLING CROSS-SECTION FINITE ELEMENT METHOD

JIANG Wen-guang ()1,2, John L. Henshall3

(1. School of Mechanical Engineering, Yanshan University,

Qinhuangdao 066004, Hebei Province, P. R. China;

2. Department of Aerospace Engineering, University of Bristol, BS8 1TR, UK;

3. Department of Mechanical Engineering and Manufacturing Engineering,

Nottingham Trent University, Nottingham, NG1 4BU, UK)

(Communicated by FU Ming-fu)

Abstract: Beams and plates manufactured from laminates of composite materials havedistinct advantages in a signicant number of applications. However, the anisotropy aris-ing from these materials adds a signicant degree of complexity, and thus time, to thestress and deformation analyses of such components, even using numerical approachessuch as nite elements. The analysis of composite laminate beams subjected to uniformextension, bending, and/or twisting loads was performed by a novel implementation ofthe usual nite element method. Due to the symmetric features of the deformations,only a thin slice of the beam to be analysed needs to be modelled. Conventional three-dimensional solid nite elements were used for the structural discretization. The accuratedeformation relationships were formulated and implemented through the coupling of nodaltranslational degrees of freedom in the numerical analysis. A sample solution for a rect-angular composite laminate beam is presented to show the validity and accuracy of theproposed method.

Key words: composite material; beam; coupling equation; nite element method; ex-tension; bending; torsion

Chinese Library Classification: O343.682000 Mathematics Subject Classification: 65Y10; 74S05; 74K10; 74G15Digital Object Identifier(DOI): 10.1007/s 10483-006-1213-z

Introduction

Laminated composite materials have been developed over the last three decades to havemechanical and other properties that make them attractive alternatives to the conventionalstructural materials such as metallic alloys. These materials were initially used in structuressuch as high-performance aircraft which exploited their high strength-to-weight ratio, but theyare increasingly being used in structural applications across all engineering industries. Thestiness mismatch between the contiguous plies, which present very steep stress gradients inthe free edge region, may cause delamination with a consequent lose in the laminate mechanicalproperties and possible failure at loads that are lower than those expected according to classicalfailure theories. An accurate prediction of the interlaminar stresses is therefore crucial for designpurposes. Due to the complex nature of the problem, numerical techniques have been usuallyemployed, such as the nite dierence method[1,2], boundary element method[3], nite element

Received Jul.1, 2005; Revised Aug.8, 2006Corresponding author JIANG Wen-guang, Professor, Doctor, E-mail: [email protected];wenguang [email protected]

1710 JIANG Wen-guang and John L. Henshall

method[4,5] requiring domain discretization, series expansion in conjunction with boundarycollocation[6], or variational methods[79].

In this paper, accurate and rapid analyses of stress and strain are achieved by using thecoupling cross-section nite element method presented herein. The coupling cross-sectionmethod has been successfully used to model the extension-torsion coupling of helically sym-metric components[1016]. In this paper, this methodology is extended to analyse the generaluniform deformation of composite laminate beams.

Fig.1 A model cross-sectional slice takenfrom a prismatic beam subjectedto uniform deformations

Consider a long prismatic beam with arbitrarycross-section, referred to a coordinate system (x, y, z),which is such that the z-direction is parallel to thelongitudinal direction of the beam as shown in Fig.1.The material behaviour is only a function of x- andy-coordinates, and not dependent on z. The appliedstress distribution on the ends of the beam is staticallyequivalent to an applied extensional force Fz that actsalong the axial z-direction and a couple with compo-nents Mx,My and Mz about the x-, y-, and z-axes,respectively. Assuming that the body forces are negli-gible in comparison to the applied stress distribution,suciently far from the beam ends, end eects canalso be neglected according to the St Venant princi-ple and only the small deformation case is considered,therefore a uniform deformation of the beam will oc-cur. Under such a deformation mode, the warpingand distortion of all cross-sections, and the stress dis-tributions over all cross-sections of the beam will beidentical. In other words, the problem may be reducedto an analysis of a typical small slice of the beam asshown in Fig.1. Precise boundary conditions are en-forced by using the constraint equations which relatethe degrees of freedom of the corresponding nodes onthe opposite articial beam cross-sections of the niteelement model, as will be detailed in the following sec-

tions. The advantages of the method are not only its conceptual conciseness, but also its highaccuracy and easy of implementation.

1 Kinematic Deformation Relationships

Consider the kinematic relationship between general pairs of corresponding nodes n(x, y, 0)and n(x, y,h), and an arbitrarily pre-selected specic pair of reference corresponding nodesn0(x0, y0, 0) and n0(x0, y0,h), see Fig.2.

In the undeformed conguration, the relative position vectors L and L are

L n0n = (x x0, y y0, 0)T = L n0n

. (1)

In the deformed conguration, the corresponding relative position vectors are

L = L + u u0 (2)

andL = L + u u0, (3)

Coupling Finite Element Method of Composite Laminate Beams 1711

Fig.2 Diagram showing displace-ment vectors between pairsof corresponding nodes onthe corresponding cross-sections of the slice model

in which,

u = (ux, uy, uz)T, u = (ux, uy, u

z)

T,

u0 = (u0x, u0y, u0z)T, u0 = (u0x, u

0y, u

0z)

T (4ad)

are the displacement vectors of nodes n, n, n0 and n0,respectively.

Owing to the symmetrical nature of the beam, theclass of innitesimal deformations, for which the stressesare independent of the z-coordinate, is related to a dis-placement eld having the following general form:

L = RL, (5)

where

R =

1 z yz 1 xy x 1

(6)

is the linear rotational matrix, in which, x and y arethe bending angles between the cross-sections about thex- and y-axes, respectively, and z is the twisting anglebetween the cross-sections about the z-axis. These canbe expressed in terms of x and y, the bending curvatures about x- and y-axes, respectively,and z, the twisting curvature about the z-axis (i.e., twist angle in radians per unit length):

x = xh, y = yh, z = zh, (7a,b,c)

in which h is the slice thickness (see Fig.1).

2 Finite Element Implementation

To make use of the formulated kinematic deformation relationships in the nite elementanalysis, only a representative small slice of the beam needs to be analysed. The nite elementmesh must be established in such a way that there are matching pairs of corresponding nodesn(x, y, 0) and n(x, y,h) on the two corresponding articial cross-sections. With an arbitrarilypre-selected pair of corresponding nodes n0(x0, y0, 0) and n0(x0, y0,h) specied as referencecorresponding nodes, the kinematic deformation relationships as discussed in the previous sec-tion can be established. Conventional three-dimensional solid brick elements are used for thestructural discretization. The nodes of the elements each have three degrees of freedom, i.e.translations in the x-, y-, and z-directions. Due to the nature of uniform deformation, onlyone element division is required in the axial direction of the beam. By substituting Eq.(6) intoEq.(5), the deformation relationships can be explicitly expressed as

ux u0x + (x x0)uy u0y + (y y0)

uz u0z

=

1 z yz 1 xy x 1

ux u0x + (x x0)uy u0y + (y y0)

uz u0z

. (8)

This equation adequately describes the required deformation eld patterns including arbitraryrigid body movements. To avoid singularity of the system of equations, rigid body movementmust be restrained. In 3-D space, six degrees of freedom need to be constrained in order to keepa deformed body completely xed in space. When introducing constraints, a basic requirement


is that the constraints must not come into conict with the deformation eld (Eq.(8)). Apractical way of realizing this is illustrated as follows:

(a) All of the three translational degrees of freedom of the reference node n0 are completelyconstrained, i.e.,

u0 = (u0x, u0y, u0z)T = (0, 0, 0)T. (9)

With these constraints, the model can still rotate freely around this pivoting node n0.(b) The x and y degrees of freedom of the corresponding reference node n0 are also con-

strained, i.e.,u0x = 0, u

0y = 0. (10a,b)

With Eqs.(9) and (10a,b), Eq.(8) can be further simplied to

uxuyuz

=

1 z yz 1 xy x 1

ux + (x x0)uy + (y y0)

uz

x x0y y0u0z

. (11)

Under these constraints the model can still rotate about the xed line n0n0.(c) The rigid body rotation about line n0n0 may be removed by constraining an arbitrarily

selected node n1(x1, y1, 0), which diers from n0, from rotating relative to n0.This can be realized by the constraint equation:

u1y = u1x tan 01 = u1x(y1 y0)/(x1 x0) (12)

and the constraints on its corresponding node, n1(x1, y1,h), can be derived by substitutingEq.(12) into Eq.(11), i.e.,

u1xu1yu1z

=

1 z yz 1 xy x 1

u1x + (x1 x0)u1x tan 01 + (y1 y0)

u1z

x1 x0y1 y0u0z

. (13)

With these constraints, the nite element model is completely xed in the space, and the systemof equations has a unique solution.

In the present method, the loading process is realized by applying bending angles x andy, and twisting angle z between the cross-sections. When these values are given, theformulated constraint equations become linear relationships between nodal translational degreesof freedom, and can be conveniently incorporated into the global system equations. The degreesof freedom on the left-hand side in Eqs.(11), (12) and (13) can be expressed in terms of thedegrees of freedom on the right-hand side, and therefore eliminated during the solution of thesystem equations. After solution, the eliminated degrees of freedom can be recovered fromthese constraint equations. The resultant moments of bending and torsion can be derived bythe summation of the reaction nodal forces on one of the cross-sections of the nite elementmodel of the beam. The total axial force can be applied as a nodal force to the degree offreedom u0z of the node n

0. By means of the coupling equations, the axial force will actually

be distributed to all the nodes over the cross-section.

3 Numerical Example

A (45/ 45)s composite laminate beam, as shown in Fig.3, is analyzed here using theproposed algorithm. The plies are perfectly bonded along the interfaces of the laminate. Eachlayer is assumed to be a homogeneous and linear elastic medium obeying the generalized or-thotropy constitutive law[17] with one of the orthotropic directions parallel to its thickness


direction. The laminates consist of plies with rectangular cross-section having thickness h andwidth b = 16h. The material properties of individual layer in the principal material coordinatesare taken to be E1 = 137.9 GPa, E2 = E3 = 14.5 GPa, G12 = G31 = G23 = 5.99 GPa, and12 = 23 = 31 = 0.21. Conventional 20-noded quadratic iso-parametric brick elements wereused. The nite element meshes used are shown in Fig.4.

Fig.3 Geometry and lamination sche-me of the (45/ 45)s compo-site laminate analysed

Fig.4 Finite element meshes used for the compositelaminate beam analyses

Being an elastic body, the constitutive relationship relating the axial extension strain (z),bending curvatures (x,y), and twisting curvature (z) to the resultant axial force (Fz),bending moments (Mx,My), and twisting moment (Mz) may be expressed in the general form:

k11 k12 k13 k14k21 k22 k23 k24k31 k32 k33 k34k41 k42 k43 k44

zxyz

=

FzMxMyMz

, (14)

where k is symmetric based upon reciprocity.

To use the present method, the loading can be realised by explicitly specifying set of valuesof (z,x,y,z). Any four sets of linearly independent solutions, for example, pure extensiondeformation (Fz = 0,x = y = z = 0), pure bending deformations (x = 0, Fz = y = z =0) and (y = 0, Fz = x = z = 0), and pure torsion deformation (z = 0, Fz = x = y = 0)may be performed to derive all of the stiness constants kij , i, j = 1, , 4. In practice, thesefour fundamental deformation modes themselves are of primary interest. After the derivationof all of the stiness constants, in the case that Mx,My,Mz and z are primarily known values,the values of Fz,x,y,z can be derived from Eq.(14).

The stiness matrix predicted by the present analysis for the composite beam with dimen-


sions of b = 16h = 16 mm is given as

k =

1287 kN 0.000 Nm 0.000 Nm 0.000 Nm0.000 Nm 1.701 Nm2 0.000 Nm2 0.723 Nm20.000 Nm 0.000 Nm2 26.49 Nm2 0.000 Nm20.000 Nm 0.723 Nm2 0.000 Nm2 5.596 Nm2

. (15)

It is worth noting that the relative discrepancy of the stiness coecients between the twosets of results predicted using a ne mesh and coarse mesh is well below 0.1 percent, and thesymmetric relationship kij = kji is very precisely satised. The o-diagonal non-zero valuesk24 = k42 = 0.723 Nm2 indicate that coupling exists between bending about the x axis andtorsion about the z-axis.

The stress elds over the beam cross-section predicted using the ne mesh are depicted inFigs.58, respectively, for the pure extension, pure bending about the x- and y-axes, and thepure twisting loading cases.

Fig.5 Stress eld for the (45/45)s laminate beam under pure extension (Fz = 0,x =y = z = 0)

Fig.6 Stress eld for the (45/ 45)s laminate beam under pure bending deformation aboutthe x-axis (x = 0, Fz = y = z = 0)


Fig.7 Stress eld for the (45/ 45)s laminate beam under pure bending deformation aboutthe y-axis (y = 0, Fz = x = z = 0)

Fig.8 Stress eld for the (45/ 45)s laminate beam under pure torsion about the z-axis(z = 0,z = x = y = 0)

The interlaminar stresses along the interface y = h are shown in Figs.912, and the resultsare compared with the solutions of Wang and Choi[6], Ye[4] and Dav` and Milazzo[3]. Excellentagreement has been found with the corresponding results from the above authors. It is alsoworth noting that the analysis performed using the coarse mesh yielded reasonably accurateresults. Stress singularities can be observed between the contiguous plies in the free edge regionfor all of the loading modes addressed above.

The selection of parameters used in the models such as slice thickness h (varying from 0.1%to 10% of the maximum dimension of the cross-section), arbitrary locations of the referencenode and rotational constraint node n1, were all extensively tested. The inuences on theinduced torque and stress eld were found to be negligible. This indicates that, as desired, themodel is not sensitive to these parameters.


Fig.9 Interlaminar stress distributions along the y = h interface of the laminate under pure extension(Fz = 0,x = y = z = 0)

Fig.10 Interlaminar stress distributions along the y = h interface of the laminate under pure bendingdeformation about the x-axis (x = 0, Fz = y = z = 0)

Fig.11 Interlaminar stress distributions along the y = h interface of the laminate under pure bendingdeformation about the y-axis (y = 0, Fz = x = z = 0)


Fig.12 Interlaminar stress distributions along the y = h interface of the laminate under pure torsionabout the z-axis (z = 0, Fz = x = y = 0)

4 Concluding Remarks

The analysis of composite beams subjected to uniform extension, bending and torsion canbe solved by the analysis of only a representative slice sector of the beam. Precise boundaryconditions have been formulated and implemented. The proposed method is conceptually simpleand mathematically rigorous. No extraneous assumptions have been introduced. The methodis straightforwardly applicable to prismatic beams with arbitrary cross-sectional shapes andmaterial stacking sequences.

References

[1] Pipes R B, Pagano N J. Interlaminar stresses in composite laminates under uniform axial exten-sion[J]. Journal of Composite Materials, 1970, 4:538548.

[2] Altus E, Rotem A, Shmueli M. Free edge eect in angle ply laminatesa new three dimensionalnite dierence solution[J]. Journal of Composite Materials, 1980, 14(1):2130.

[3] DavG, Milazzo A. Boundary integral formulation for composite laminates in torsion[J]. AIAAJournal, 1997, 35(10):16601666.

[4] Ye L. Some characteristics of distributions of free-edge interlaminar stresses in composite lami-nates[J]. International Journal of Solids and Structures, 1990, 26(3):331351.

[5] Mitchell J A, Reddy J N. Study of interlaminar stresses in composite laminates subjected totorsional loading [J]. AIAA Journal, 2001, 39(7):13741382.

[6] Wang S S, Choi I. Boundary-layer eects in composite laminates: Part 2, free-edge stress solutionsand basic characteristics[J]. ASME Journal of Applied Mechanics, 1982, 49(3):549560.

[7] Pagano N J. Stress elds in composite laminates[J]. International Journal of Solids and Structures,1978, 14(5):385400.

[8] Pagano N J. Free edge stress elds in composite laminates[J]. International Journal of Solids andStructures, 1978, 14(5):401406.

[9] Yin W L. Free-edge eects in anisotropic laminates under extension, bending and twisting, PartI: a stress-function-based variational approach[J]. ASME Journal of Applied Mechanics, 1994,61(2):410415.

[10] Jiang W G. The Development of the Helically Symmetric Boundary Condition in Finite ElementAnalysis and Its Applications to Spiral Strands[D]. PhD Dissertation, Brunel University, Uxbridge,1999.


[11] Jiang W G, Henshall J L. The development and applications of the helically symmetric boundaryconditions in nite element analysis[J]. Communications in Numerical Methods in Engineering,1999, 15(6):435443.

[12] Jiang W G, Henshall J L. A novel nite element model for helical springs[J]. Finite Elements inAnalysis and Design, 2000, 35(4):363377.

[13] Jiang W G, Henshall J L. Torsion-extension coupling in initially twisted beams by nite ele-ments[J]. European Journal of Mechanics A-Solids, 2001, 20(3):501508.

[14] Jiang W G, Henshall J L, Walton J M. A concise nite element model for 3-layered straight wirerope strand[J]. International Journal of Mechanical Sciences, 2000, 42(1):6386.

[15] Jiang W G, Yao M S, Walton J M. A concise nite element model for simple wire rope strand[J].International Journal of Mechanical Sciences, 1999, 41(2):143161.

[16] Jiang W G, Henshall J L. A coupling cross-section nite element model for torsion analysis ofprismatic bars[J]. European Journal of Mechanics A-Solids, 2002, 21(3):513522.

[17] Lekhnitskii S G. Theory of Elasticity of an Anisotropic Elastic Body[M]. Holden-Day, Inc,Uxbridge, 1963.

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