Composite Structures 24 (1993) 139-147

1 INTRODUCTION

Pultruted composite beams and columns arebeing' extensively used for civil engineering struc­tural applications. They have many advantagesover conventional materials (steel,concrete,wood, etc.), such as light weight and high corro­Sion resistance. Mass production of compositestructural members (e.g. by pultrusion) makescomposite materials cost competitive with con­ventional ones.

In .the pultrusion process, fibers are pulledthrough a heated die that provides the shape ofthe cross-section to the final product. Pultrusion isa continuous process of prismatic sections ofvirtually any shape.1 Other mass production tech­niques like automatic tape layout can also be usedto produce prismatic sections. .

Pultruted structural members have open orclosed thin-walled cross-sections. For long com­posite columns, overal~ (Euler) buckling is morelikely to occur before any other instability failureand the buckling equation has to account for theanisotropic nature of the material. For shortcolumns, local buckling occurs first leading eitherto large deflections and finally to overall buckling,or to material degradation due to large deflections(crippling). Because of the large elongation tofailure allowed by both the fibers and the resin,the composite material remains linearly elastic forlarge deflections and strains unlike conventionalmaterials that yield (steel) or crack (concrete) formoderate strains. Therefore, buckling is thegoverning failure for this type of cross-sectionstnd the critical buckling load is directly relatedwith the carrying capacity of the member. TheEuler buckling problem is considered herein.

Ever J. Barbero & Ioannis G. RaftoyiannisMechanical and Aero. Engineering, West Virginia University, West Virginia 26506-6101, USA

Pultruted composite structural members with open or closed thin-walled sec­tions are being used as columns for structural applications where buckling isthe main consideration in the design. An analytical model for Euler buckling isdeveloped herein. Failure envelopes for some commercially available structuralshapes are presented. The analytical model presented can be used to predictthe behavior of any new material.

Buckling is an instability phenomenon that'causes failure on a structure and is accompaniedby large deflections and non-linear behavior.Buckling failure has been observed in longcolumns under axial loading where failure occursfor axial stress much lower than the yield stress ofthe material. Euler, based upon this fact, con­cluded that this instability was due to the geo­metry of the column (i.e. length I and bendingstiffness EI) and he solved the problem mathe­matically.2 Many researchers followed the sameprocedure in order to solve more specific cases,where the boundary conditions or eccentricities inapplication of the load are important.3,4 Consider­ing a long simply supported column under axialloading there is, from a mathematical viewpoint,an infinite number of buckling loads, each oneassociated with a specific deflection shape (buck­ling mode shape). The minimum buckling load isthe critical buckling load of interest. It is assumedthat once this load is r~ached, the structure hasfailed. Approximate methods of analysis, such asenergy method, finite differences method, andfinite element method, have been employed tosolve buckling problems, whose mathematicalsolution may be very difficult to obtain in closedform.5

In the case of short columns, Euler's theorycannot be applied because short column buck­ling failure is associated more likely with localbuckling (i.e. buckling of a part of the cross­section of the column) or material failure that canbe encountered before any instability failure. Avery short column of solid cross-section, forinstance, is expected to fail due to material failure.On the other hand, hollow cross-section may faildue to local buckling of part of the cross-section.

139Composite Structures 0263-8223/93/$06.00 © 1993 Elsevier Science Publishers Ltd, England. Printed in Great Britain

140 E. J. Barbero, 1. G. Rajtoyiannis

The case of local buckling of a cross-section withthin parts, such as wide flange I-beams and thin~

walled box-beams, is considered by Barbero andRaftoyiannis,6 Vakiener,7 Yuan,8 Raftoyiannis9and Barbero. tO The problem of local buckling hasbeen already considered for steel cross-sectionsand considerable research has been done in thisarea in order to increase the carrying capacity of ast~el member against local buckling by intro­ducing stiffeners (Desmond,11,12 Dwight,13Sharp,14 Winter,15 Bleichl6 ).

For a long composite column, the ClassicalBuckling Theory4 in combination with basicconcepts of the Classical Lamination Theory(CLT)l7,18 are applied in order to determine thebending stiffness of the column and the criticalbuckling load. The beam bending stiffness can beobtained experimentally from full size tests19 orby considering each part of the cross-section(flange and web) as an orthotropic plate withproperties determined by coupon testing.20

For the design of new composite beams opti­mized for buckling strength, it is advantageous tobe able to predict member properties from theconstituent -(fiber, resin) properties and theirarrangement in the cross-section (stacking se­quence). Berkowitz developed expressions21 forthe bending, axial, and shear stiffness of laminatedrectangular beams with laminae stacked perpen­dicularly to the plane of bending (horizontally).

2 BEAM STIFFNESS FRO CLT

Bert developed a simple expression22 ' for theshear correction factor, needed when usingTimoshenko beam theory, that complements thework of Berkowitz. Bank developed expressions20for the shear correction 'factor of beams com­posed of thin orthotropic walls.

Angle-ply layers in both web and flanges aredesirable to increase the local buckling, strengthand the efficiency of connections. Arbitrary orien­tation of the fibers introduces shear-extensioncoupling that must be accounted for in the com­putation of the ben.ding stiffness. In this paper, thebending stiffness is computed directly from thedescription of the cross-section used by the manu­facturer. In this way, an efficient optimization ofthe section for maximum buckling ,strength can beattempted.

Although pultruted beams are not manufac­tured by lamination, they do contain differentmaterial combinations through the thickness, thusjustifying the use of lamination theory. Under thistheory, each layer is modeled as a homogeneousequivalent material that macroscopically behavessimilarly to the fibrous 'composite. Next, CLT isused" to' model all entire flmge"()r web as ayetequivalent homogeneous material. Flanges andweb can be dealt with separately for ease of inter­pretation. The beam can be derived from the platestiffnesses of CLT or by reformulating CLT fOIbeam bending.

Using CLT, the stiffness component of an anisotropic laminated plate (e.g. the flange) can be computed.Hence, the constitutive equation for ach laminate is

[{N}) = [[Al [Bl]' [{E,}){M} [B] [D] {1(}

(1)

(2)

where·{N}, {M}, {E} and {1(} are stress resultants, moment resultants, strains and curvatures respectively.The coefficients Aij' Bij and Dij are defined for the flanges of thickness h = t f and width b (Fig. 1) as

1

fb/2 fh/2 fh/2- 2 - 2

(Aij,Bij,Dij)=-b Qq{l,z,z ) dydz= Qii 1,z,z ) dz- b/2 - h /2 - h/2

and for the webs of thickness a= t W and depth d as

1 fa/2

fd/2 - 2 (d d3) f

a/2

- .(Aij,Bij,Dij)=- Qii1,0,z ) dydz= -,0, -12 Qijdy

a - a /2 -d/2 a a - a /2

where Qij are the transformed reduced stiffness.17

(3)

Euler buckling ofpultruted composite columns 141

e

d

Fig. 1. Typical structural shape and coordinate system.

The units for Aij' Bij and Dija~e (kNfm), (kN) and (kNm), respectively. In using this approach, the cross­sectio~ is transformed to an equivalent laminate of width b and height h. For that reason, the webs arespread to ~hewidth of the flanges by introducing a width correction factorajb.

.Applying the parallel axis theorem18 with respect to the middle surface of the cross-section, a constitu­tive matrix for the whole cross-section can be determined. For a symmetric I-beam of width b and depthd+ 2h the total stiffness components are

aA··=- A~+2Af.

IJ b lJ lj

D .. =!!. D~+ 2Df.+ 2 (_.e)2 AI.IJ b I) lJ 2 I)

For a symmetric box-beam.of width b and depth d + 2h the stiffness components are

aA··= 2 - A~+ 2Af.

IJ b IJ I)

D .. = 2 !!. D~+ 2Df.+ 2 (!!-)2 AI.IJ b IJ I) 2 I)

(4)

(5)

where e= d+ h. Note that symmetry involves not only geometry symmetry but also material symmetrywith respect to the middle surface.

So far, the analysis is similar to that of Vinson and Sierakowski23 or Tsai24 and is valid for a plane strainsituation in the y-direction (Fig. 1). The reduced stiffnesses (Qij) are obtained using CLT17 that assumesplane stress through the thickness of the flanges or web (Fig. 1). However, a plane stress assumption mustbe used through the width of the beam,21 i.e.

Ny = Nxy = My = Mxy = 0

Enforcing eqn (6) on the consitutive eqn (1), the following can be written:

M~ot= Mxb= D'Kx

where

Chen and Yang presented a similar expression25 but limited to rectangular crQss-sections.

(6)

(7)

(8)

142

3 LAMINATED BEAM THEORY

E. J. Barbero, I. G. Raftoyiannis

The computation of bending stiffness presented in- the previous section is based on a plate analysis byCLT and subsequent reduction to beam theory. Therefore, the analysis of beams becomes cOlIl:plicated,requiring understanding of plate theory to develop beam theory. This is contrary to the usual approacl.where beam theory is developed from fundamental concepts21 ,22 evolving in a theory much simpler thanthe theory of plates. The objective of this section is to present such a theory for thin-walled beams withunsymmetric material and/or geometry and vertically as well as horizontally laminated parts. The result­ing bending-extension coupling is replaced by indentifying the location of the neutral axis and formulat-ing the stiffness with respect to this point instead of the middle surface. -

The constitutive equations of a composite beam with thin, laminated flanges are developed under theassumption of plane stress21 through the thickness and the width of the beam (Fig. 1), which is only anapproximation for the case of laminates.25,31

U z~ Uy ~ Uyz

:: Uxy ~ 0

Therefore, the material constitutive equations are

(9)

(10)ux=ExEx

uxz = Gxzyxz

where Ex is the equivalent axial stiffness and Gxz is the equivalent shear stiffness of the material. For anisotropic material, Ex = E, the modulus of elasticity and Gxz = G, the shear modulus. The constitutiveequations of a laminate (flange or w~b), so called laminate constitutive equations, are derived by integrat­ing the expression of the stress resultants, to get

Nx=AE~+ B'Kx

Mx =BE~+ D'Kx (11)

Qx=Fyxz

where A is the extensional stiffness, D is the bending stiffness, F is the shear stiffness, and B is thebending-extension coupling- for unsymmetrical laminates (note that A, B, D and F are not matrices). Forthe flange, i.e. section of the beam with layers on a plane perpendicular to the plane of bending, we have

d = b I: J~k E~ d~= bI E~tkk=l ~k-) k=l

(12)

Nf J~k NJ

pi=b k~l ~k= 1 Gxz d~ = b k~l G~ztk

where ~ is a local coordinate attached to the flange (Fig. 1), ~ is the position of the middle surface of thekth layer in the local coordinate system, b is the width of the flange, Nt is the number of layers of theflange, and ~z is the out-of-plane shear modulus of the kth layer in the flange. For the web, i.e. section ofthe beam with layers on a plane parallel to the plane of bending, we have

A W = d k~I J:~l E~ dy

Euler buckling ofpultruted composite columns 143

d3 Nil' fYk

DW=- L E;dy

12 k= 1 Yk-l

(13)

'where d is the depth of the web, NW is the number of layers of the web, and ~z is the in-plane shearmodulus of the kth layer in the web. Equation (12) was previously used by Bert,22 and Bert and Gorda-ninejad.26

To obtain the stiffnesses of the whole section, the contribution of the flanges (eqn (12)) and the webs(eqn (13)) are combined using the parallel axis theoremI8 with respect to the axis of symmetry of thecross-section. For example consider an I-beam as in Fig. 1. The flange has widthb and thickness t~ theweb has width tWand depth d. Therefore, the total depth of the I-beam is h= d+ 2tf and e= d-t- tf(Fig. 1).The equivalent transformed stiffnesses are

A=2Af+Aw; p=pw (14)

B= e(KtOP- KbOt)+ Btop+ Bbot=O

Note that the contribution of the flanges to the shear stiffness is omitted in eqn (14). The assumption ofconstant shear strain through the thickness used in Timoshenko beam theory is unrealistic for an I-beam,where the shear strain approaches zero in the flanges.27 '

Th~·stIuctl.J.ral equivalent.pTQ_perties ..E*andGx-z_fQreac_hJayt~rap_ply~inthe ..structllratcoordinatesy-stem(Fig. 1). They are obtained by rotation from the material coordinate system to the structural coordinatesystem (Fig. 2). The constitutive equation for each layer, assuming a state of plane stress through thethickness of the beam is given by Jones.17 Introducing .the plane stress assumption through the width of"be beam,21 ay=axy = 0, and replacing the constitutive equation for ax, we obtain

ax= Exex

(15)

As an example if () = 0 then Ex = E1 (the. modulus of elasticity along the fiber direction). Similarly, if0= n/2 then Ex = E 2 (the modulus of elasticity in the direction perpendicular to the fibers). To compute

y

2z

x

13

Fig. 2. Material (1-2-3) and structural (x,y,z) coordinatesystem for the flange.

Fig. 3. Material (1-2-3) and structural (x, y, z) coordinatesystem for the web.

144 E. J. Barbero, I. G. Raftoyiannis

the out-of-plane shear of the flange, a typical layer laminated at an angle (J is considered. The constitutiveequations for out-of-plane shear of this layer, treated as a plate under the assumption of plane stress are

[Oyz] = [~44 ~45] [Yyz]'Oxz C45 C55 Yxz

Introducing Oyz = 0 into eqn (16) we obtain

(16\

- 2C45 -Gxz = ----+ C55C44

(17)

where overline indicates a rotated quantity.28

The lamina constitutive equations in global coordinates for a typical lamina in the web are used tocompute the out-of-plane shear of the web (Fig. 3). To be consistent with beam theory, the assumptionGz = 0 must be introduced to obtain

(18)

For th~ particular case of fibers oriented along the length of the beam.( ()= 0), Q16 = Q26 = 0 and egn (18)reduces to the usual relationship used in beam theory:

(IS

Uxz = Q66Yxz= Gxzyxz

If the material is isotropic Ex = E and Gxz = G in eqn (19). It is customary for unsmmetrically laminatedbeams to compute the location of the neutral axis rather than to use a bending-extension coupling coeffi­cient as in CLT. The structural behavior of a composite beam depends entirely on the bending stiffness D,axial stiffness A, and shear stiffness F. The position of the neutral axis can be found by repeating theintegration of eqns (12)-( 13) with respect to a coordinate system located at a distance ~0 from the middlesurface, in such a way that the bending-extension coupling vanishes,

Bf= b I E~tk(~k- ~o)=O

therefore

(20)

(21 )

While ; 0 in eqn (21 ) is the location of the neutral axis for laminate, eqn (22) describes the position of theneutral axis for a general section of a thiil-walled beam or column,

(22)

where eqn (14) gives B and A . The axial stiffness A and shear stiffness F remain unchanged. An expresion similar to eqn (22) was derived by Bert22 for laminated beams of constant cross-section and by Berland Gordaninejad26 for bimodular materials based on the assumption that the in-plane normal strain

Euler buckling ofpultruted composite columns 145

vanishes at the neutral axis. The bending stiffness D with respect to the neutral axis is computed using theparallel axis theorem

D= Dweb + Dtop + Dbot + A tOP(e- ZO)2 - A bot(e+ ZO)2 + A webzo (23)

Considering transverse isotropy on each layer,17 four material properties per layer are needed. Usingmicromechanics, the material properties for each lamina (E I , E 2, V12' G12 ) are determined from thematerial properties of fiber (Eft vf) and matrix (Em' Vm).

The fiber volume fraction J.f is obtained from the pultrusion m.anufacturing information. ~or unidirec-·tional fibers aligned with the axis of the beam J.f= n/(yptc )' where n is the number of roving per unitwidth, y is the yield (number of meters of roving weighing 1 kg), p is the density in kg/m3 , and tc is thethickness of the layer. For the continuous strand mat (DC) J.f= W/(ptc ), where W is the weight per unitarea of the DC mat.

For the material under study, Ef =72·393 GPa, Em =3·445 GPa, vf =0·22, and vm =0·35. Due to thehigh tension that pultrusion e~erts on the fibers, the fiber misalingment factor used is K = 1. As anexample; consider an E-glass-vinylester pultruted material with J.f= 0·25 for which the following areobtained £1 = 20·632 GPa, E2 =4·433 GPa, V I 2 = 0·318. These predictions correlate well with experi­mental data.29

The elasticity solution with contiguity is used for the determination of the shear modulus. Using eqn(3·68) in Ref. 17, GI2 = 1·985 GPa is obtained. The predicted value does not correlate well with experi­mental data. Therefore, the stress partition parameter18 is obtained using experimental d~ta for currentlyproduced pultruted material and it is assumed to remain constant while varymg' the fiber volume andresin properties during material optimization studies. Predictions using this approach correlate well withexperimental data.29

In unidirectional composites all the fibers have a specifi~ orient~tion iIl tl1e m~tJix for each particularlayer. A special case is when the fibers are randomly oriented in the matrix. The composite acts as aplane-isotropic material and the properties were obtained using the following formulas30

3 5E=8 E 1 +8 E2,

1 1G=-E +-E8 1 4 2,

Ev=--1

2G(24)

4 BUCKLING EQUATION AND CRITICALLOAD

The governing equation for buckling is obtainedby introducing eqn (9) into the Euler bucklingequation for a pinned-.pinned perfect columnunder axial compressive load (eqn (2.1) in Ref. 4).Then, the critical buckling load is found to be

(25)

Equations (8) and (25) are used to draw the longcolumn failure envelopes presented in Figs 4-8,for structural shapes produced by Creative Pul­trusions Inc. The failure envelopes are producedby using the analysis presented herein for Eulerbuckling and the local buckling predictions ofRaftoyiannis.9

Five different wide-flange I-beam sections anda thin-walled box-beam section are considered. Itmust be noted that for the 10 x 10 x 0·432 cm

(Fig. 4) and 15 x 15 x 0·635 cm (Fig. 5), the trans­ition from local to Euler buckling occurs for relat­ively short lengths (1 - 1·5 m). Therefore, accurateprediction of Euler buckling becomes quiteimportant for structural applications. For the

700 -------------.,

600

~:\\=.~~:, , ,"'--' ....-'...-.....------

200

100

0.1 0.2 2

Length (m)

Fig. 4. Buckling failure envelope for a 10 x 10 x 0·432 emwide flange I-beam from Creative Pultrusions Inc.

146 E. J. Barbero, 1. G. Raftoyiannis

600 -------------, 200

0.1 0.2 2

Length (m)

Fig. 5. Buckling failure envelope for a 15 x 15 x 0·635 cmwide',flange I-beam from Creative Pultrusions Inc.

HingedElasticClampedEuler

0.1

\

,-/,_ ........ _------

,\\ "

\ I \

\ '-~ ' ... ",',_ ... _... _---------

50

150 \.J/\".J'\,.-",.-.. _

"...-....

Z.:::l100

.........."

Length (m)

Fig. 8. Buckling failure envelope for a 10 x 10 x 0·432 emthin-walled box-beam from Creative Pultrusions Inc.

- Euler---- Local

Weak axis

.Strong .axis

,,,,,,,,,\\\\\\\

"" "' ...._..' ,-.---.------------5

c..200

400,-...Z.:::l...........

5 CONCLUSIONS

ACKNOWLEDGEMENTS

The analytical solution developed herein has beenus.ed to predict critical buckling loads as well a.the buckling mode failure for commercially avail­able pultruted columns. Instability failure is veryimportant. be~ause ·it leads to the collapse of themember. Prediction of the buckling load is veryimportant for the prediction of ultimate axialstrength of pultruted columns. The analyticalsolutions presented are based on the descriptionof the cross-section used by the manufacturer. Acomprehensive analysis, from the properties offiber and matrix to the determination of the buck­ling load has been presented, which can be usedby the manufacturer to tailor the properties of thematerial for specific applications.

The financial support of the National ScienceFoundation, West Virginia Department of High­ways, and the Federal Highway Administration isacknowledged. The cooperation of CreativePultrusions, Inc. which provided detailed infor­mation about its product is appreciated. ThanJto John Tomblin for his help with the computa­tions.

behavior that defines the intermediate length forstructural shapes is currently under investigation~

The buckling envelopes for a 10 x 10 x 0·432 cmbox-beam is shown in Fig. 8.

2

- Euler---- Local

,,,,,,,,,\,\\" /\\ , \

,..._, ' ..._,'......-.".------

0.1 0.2

600

1200Weak axis

Strong axis900 ,,,,

,-... ,\

Z ,\

.:::l \

........... 600\" '\

" '"~

' ...' ,--,'.......-----0a..

300 - Euler---- Local

00.1 0.2 2

200

,-...Z.:::l

........... 400

5n.

800 ,....---------.-----.----,

Length (m)

Fig. 6. Buckling failure envelope for a 15 x 15 x 0·952 cmwide flange I-beam from Creative Pultrusions Inc.

Length (m)

Fig. 7. Buckling failure envelope for a 20 x 20 x 0·952 emwide flange I-beam from Creative Pultrusions Inc.

15 X 15 X 0·952 cm (Fig. 6) and 20 X 20 x 0·952cm (Fig. 7), the transition occurs for 2-3 m longcolumns. Therefore, the behavior of columns ofintermediate length becomes critical. The interac­tion between local and Euler column buckling

Euler buckling ofpultruted composite columns 147

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2. Euler, L., Sur la force de colonnes. Memoires de l'Acad-mie de Berlin, (1759). ' <

3. 'Brush, D. O. & Almroth, B. 0., Buckling ofBars, Platesand Shells. McGraw-Hill, New York, USA, 1975.

4. Timoshenko, S. P. & Gere, J. M., Theory of Elastic Sta­bility. McGraw-Hill, New York, 1961.

5. Chajes, A., Principles of Structural Stability Theory.Prentice Hall, Inc., Englewood Cliffs, NJ, USA, 1974.

6. Barbero, E. J. & Raftoyiannis, I. G., Buckling analysis ofpultruted composite columns. In Impact and Buckling ofStructures, ASME AD-Vol. 20; AMD-Vol. 114 Ameri­can'Soc,iety of Mechanical Engineers, New York, USA,ed. D. Hui. 1990, pp. 47-52. '

7. Vakiener, A. R., Zureick, A. & Will, K. M., Prediction oflocal flange' buckling in pultruted shapes by finite ele­ment analysis. In Advanced Composite Mate!'ials in c;ivilEngineering Structures, ed. S. L. Iyer, Amencan SOCIetyof Civil Engineers, New York, 1991, pp. 302-12.

8. Yuan, R. L., Hashen, Z., Green A. & Bisarnsin, T.,Fiber-reinforced plastic composite columns. In Ad­vanced Composite Materials in Civil Engineering Struc­tures, ed. S. L. Iyer, American Society of Civil Engineers,New York, 1991, pp. 205-11. .

9. Raftoyiannis, I. G., Buckling of pultruted compOSItecolumns. MSCE Thesis, West Virginia University, WV,lj-SA~l-991 ~

10. Ba.rb~ro, E. J., Fu, S. H. & Raftoyiannis, I. G., Ultimatebending strength of composite beams. ASCE J. Mater.Civil Eng., 3 (4) (1991) 292-306.

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12. Desmond, T. P., Pekoz, T. & Winter, G., Intermediatestiffeners for thin-walled members. ASCE J. Struct. Div.,(Proc. Pap. 16194), 107 (ST4), (1981) 627-48. .

13. Dwight, J. B. & Little, G. H., Stiffened steel compressIonflanges, a simpler approach. Struct. Eng., 52 (12) (1976)501-9.

14. Sharp, M. L., Longitudinal stiffeners for compressionmembers. ASCE J. Struct. Div." 92 (ST5) (1966)187-212.

15. Winter, G., Strength of thin steel compression flanges.Trans. Am. Soc. Civ. Engng., 112 (1947) 527.

16. Bleich, F., Buckling Strength of Metal Structures~

McGraw-Hill, New York, USA, 1952.17. Jones; R. M., Mechanics ofComposite Materials. Hemis-, phere Publishing Corporation, New York, 1975.18. Tsai, S. W. & Hahn, H. T., Introduction to Composite

Materials. Technomic Publishing Co, Lancaster, PA,USA, 1980. "'

19. Bank, L. C., Flexural and shear modulii of full-sectionfiber reinforced plastic (FRP) pultruted beams. J. TestingEval.,17(1)(1989)40-5. ,

20. Bank, L. 'C., Shear coefficients for thin-walled compositebeams. Compo Struct., 8 (1987) 47-61.

21. Berkowitz, H. M., A theory of simple -beams andcolumns for anisotropic materials. J. Compo Mater., 3(1969) 196-200.

22. Bert, C. W., Simplified analysis of static shear factors forbeams of non-homogeneous cross-section.' J. CompoMater.,7 (1973) 525-9.

23. Vinson, J. R. & Sierakowski,. R. L., The Behavior ofStructures Composed of Composite Materials. MartinusNijhoff, The Netherlands, 1987. .

24. Tsai, S. W., Composites Design (4th edn.) ThInk Compo­sites, Dayton, OH, USA, 1989.

25. Chen, A.'T. & Yang, T. Y., Static and dynamic formula­tion of a symmetrically laminated beam finite elementfor a microcomputer. J. Compo Mater., 19 (1985)'459-75.

26. Bert, C. W. & Gordaninejad, F., Transverse shear effectsin bimodular composite laminates. J. Compo Mater., 17(1983) 282-9.

27. ~Q!1jb~_ell,_R. V., All Introduction to the Theory of Elas­ticity (2nd edn.), Oxford University Press, Oxford, UK,1941.

28. Reddy, J. N., Energy and Variational Methods in AppliedMechanics. Wiley, New York, USA, 1984.

29. Barbero, E.J. & Sonti, S. S., Micro-mechanical modelingof pultruted composite beams. Paper presented at the32nd AIAA SDM Conference, Baltimore, MD, April1991.

30. Naughton, B. P., Panhuizen, F. & Venmeulen, A. C., Theelastic properties of chopped strand mat and wovenroving in g.r.laminae. J. Reinforced Plastics Comp., 4 (2)(1985) 195-204.

31. Barbero, E. J., Lopez-Anido, R. & Davalos, J. F., On themechanics of thin-walled laminated composites. J.Compo Mater., 27 (6) (1993).