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a

A refined model for eigenfrequency analysis of

thin and thick composite beam structures

M. A. Ramos Loja*, J. Infante Barbosa* & C. M. Mota Scares'"

ENIDH-Escola Ndutica Infante D Henrique, Dpt Mdq.

Maritimas Av. Bonneville Franco, Pago de Arcos, 2780 Oeiras,

Portugal.

IDMEC-Instituto de Engenharia Mecdnica, Instituto Superior

Tecnico, Av. Rovisco Pais, 1096 Lis boa Codex, Portugal.

Abstract

A higher order shear deformation theory, assuming a non-linear variation for the displacementfield, is used to develop a finite element model to predict static and free vibration behaviour ofanisotropic multilaminated thick and thin beams, which are currently used in ship design. Themodel is based on a single-layer Lagrangean four-node straight beam element with fourteendegrees of freedom per node. It considers bending into two orthogonal planes, stretching andtwisting. The most common cross section and symmetric and non symmetric layups arestudied. The behaviour of the model is tested on thin and thick isotropic and composite beams.Comparisons show that the model is accurate and versatile. The good performance of thepresent model is evident on the prediction of the fundamental and higher modes naturalfrequencies of thin and thick highly anisotropic beam structures. Several applications areshown and discussed.

1 Introduction

Laminated beams are presently used as structural elements in general highperformance mechanical applications, namely naval applications, where highstrength and high stiffness to weight ratios are desired. The beams made ofcomposite materials have the ability of being tailored according to specifiedresponse constrained requirements, thus achieving optimum structuralobjectives. As part of the design process it is required to predict accurately thefree vibrations behaviour, to establish the performance capabilities of this typeof structural elements.

Research publications which study the dynamic response in free vibrationsof multilaminated beams are very rare. A closed form solution for freevibrations of simply supported beam based on the Euler-Bemoulli theory ispresented on Vinson & Sierakowski^ and Whitney among other text books.Free vibrations of orthotropic cantilever beams have been studied by Teoh and

Transactions on the Built Environment vol 24, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509

96 Marine Technology II

Huang^ using an energy approach with shear deformation and rotary inertiaeffects included. Closed form solutions for free vibrations of composite beamswith symmetric layups based on first order shear deformation theory andconsidering rotary inertia effects has been proposed by Chandrashekhara et al\Teh and Huang* presented two finite element models for the prediction of thenatural frequencies of fixed-free beams of general orthotropy, based ontransverse shear deformation theory and rotary inertia effects. Kapania andRaciti* have developed a simple one-dimensional finite element for the non-linear analysis of symmetrically and unsymmetrically laminated beamsincluding shear deformation. An overview of developed dynamic studies onvibration analysis of composite beams up to 1988, is given in Kapania andRaciti*. Recently Maiti and Sinha presented a model based on a higher ordertheory where all the displacement functions are assumed to contain the cubicpower of z, and alternatively the conventional first order theory is used. Theydeveloped a nine-noded isoparametric plate type element, which can havetwelve (HSDT) or five (FSDT) degrees of freedom per node. Abramovich et al*have presented a closed form solution which includes the effects of sheardeformation and rotary inertia, but omits their joint contribution from thedifferential equations of motion. Their solution can be applied to laminatedbeams of rectangular cross section, with symmetric layups and differentboundary conditions.

Presently the use of composite beam structural elements increases thedemand for the development and implementation of numerical tools which canpredict the response of these type of structures with accuracy. Among otherfactors, failure analysis needs the accurate prediction of eigenfrequencies forhighly anisotropic composite beams. Although three-dimensional finiteelements can be used, with refined meshes, to contemplate reasonable aspectratios, to avoid locking, they are very computational expensive. Acompromising less expensive situation can be reached by the use of higherorder discrete models using a single layer theory. From the literature it isevident the lack of studies on higher order anisotropic multilaminated beamelements, contemplating general layups and considering the most used crosssections.

In the present study, the development of a higher order discrete model forfree vibration analysis is presented. The model is based on a straight beamfinite element with four nodes and fourteen degrees of freedom per node,considering bi-axial bending, stretching and twisting effects, but warping isneglected. The development takes into consideration non-symmetric layups andthe most common cross-section.

The present discrete model is part of a package of finite element programsfor the optimisation of composite beam structures. This package also includesEuler-Bemoulli (EBT) and Timoshenko (FSDT) beam elements. Theperformance of the model developed is discussed with reference to severalapplications.


Marine Technology II 97

2 Displacement and strain fields

The present finite element model considers a displacement field which isexpanded by Taylor's series to the cubic power of the thickness and width inthe axial direction, and to the quadratic power in the transverse directions. Thedisplacement field can be represented on matrix form as follows :

, f )

1 0 0 0 z -y

0 1 0 0 0 0

0 0 1 0 0 0

0

0

0

0

0

0

0 0 z'

.y 0 0

0 z 0

0

0

= u 8? 6? %°

where q is the vector of generalised displacements. The first six terms are

related with displacements and rotations as defined on Figure 1. The remainingparameters are higher-order terms which represent higher-order transversecross-sectional deformation modes.

Figure 1 : Typical laminated beam geometry. Coordinate system xyz.

All generalised displacement components are functions of time t. By assumingthat plane sections remain plane after deformation, but not perpendicular to thegeometrical axis one obtains the first-order shear deformation displacementfield for the Timoshenko's model (FSDT). Further, considering that normals tothe reference surface remain normal after deformation, i.e. neglecting



transverse shear strains yields 0° = dw*/d x and 0° = - 9 v°/9 x, then the

corresponding Euler-Bernoulli model (EBT) can be formulated.Considering the kinematic relations for linear elasticity and the HSDT

displacement field (eq.l), one obtains for the strain field :

"l0

0

0

0

0

0

0

0

2-y0

0

0

0

s°"

0

1

0

0

0

0

0

1

0

0

z*

z

0

0

0

0

0

0

0

0

k*

0

z

0

0

0

-y0

0

0

0

6,

-/0

0

0

0

0

0

y0

0

L

0

0

0

1

0

*«

0

0

0

z

0

4C

0

0

0

0

4C

0

0

0

0

1

<L

0

0

0

0

y

o"

0

0

0

/.

4C 4

where each component of the generalised strain vector is given as :(2)

9j/

dx dx dx0 _ t > 0Z "Pi

9% '

dx '

(3)

3 Constitutive relations

Considering the orthogonal referential xyz, the constitutive relation for anorthotropic layer, which can have an arbitrary fibre orientation, are related tothe strains through the relations :

Qn

Qu

2n

0

06

Qn

4

&,

0

0

£5 0

o

(4)


Neglecting


, one has for the elastic coefficient Q^ the expression

855 = Qss ~ Q*s I Qu , where the coefficients of matrix Q , for the kth layer aregiven in Vinson & Sierakowski*. Integrating the stresses through the crosssection area one obtains the resultant forces and moments acting on thelaminate.

AT

\]dydz\

(5)

where N is the number of layers. The constitutive relations can then be writtenas :

(6)

" TV "

M*

Qxy

=

A B* B** 0 C?~

B^ D** D** 0 C?jx*y J)*y* D* n (~*xy

30 0 0 5" 0

Cf Cf Cf 0 S**

;:

Matrices A, E^ , Z)» , 5* , D** and C* with (ij=xz, xy • ijk=xyz • n=i..3), can befound in Loja'.

4 Finite element model

A four-node straight beam element is developed for free vibrations analysis.The generalised displacements (eq.l), can then be represented as :



Hence, the displacement field can be represented by :

fz'=V,2,J,4) (9)

where N is the shape function matrix and q^ the element nodal displacementvector. By differentiating (eq.9), in accordance with the generalised strain field:

q, q,

(10)

where matrices BM, BF*, Bpxy, 5« and B&cy relate the degrees of freedomwith the generalised strains, for membrane, flexure and shear.

The total Lagrangean for the eth element is :

LJ[TV dx

(11)

where Q^ is the element load vector, p* the mass per unit volume of each

layer, u = dujdt and L is the element length. Using Hamilton's variationalprinciple, and specialising for free vibrations analysis, one obtains thefollowing equilibrium equation :

(12)

where q^ is the element acceleration vector. The element stiffness and massmatrix are respectively :

T« -= J( BM+ B F«



C? BS^ + ~Bs C? B

(13)

The Jacobian operator, relating the natural coordinate derivatives to the localcoordinate derivative, is J=L/2 for equal spaced nodes. The element mass

matrix M, and terms relating with stretching, bending and bending-stretching

of element stiffness matrix K^ are evaluated analytically in the £ directionusing symbolic manipulator Maple'°. The last two sub-matrices of matrix (13)related with transverse shear elastic strain energy are evaluated numericallyusing three Gauss points. The degrees of freedom 8, (i=L..4) are related with

angles of twist on a plane normal to the x axis of the element. Then assumingthat they do not affect displacements other then their own, the stiffness andmass matrices for a four-node Lagrangean bar element in free torsion are super-imposed to matrices (13) and (14), in the usual assembly way.

The equilibrium equation for the whole and discretised beam for freevibrations is then written as :

(15)

where K and M are the system stiffness and mass matrices and q. is the

eigenvector associated to natural frequency CD.. The eigenvalue problem(eq.15) can easily be solved once the boundary conditions are introduced.

5 Numerical applications

Some illustrative numerical results are obtained using the present higher ordershear deformation model HSDT, for free vibrations, to show its potentiality todifferent situations.

5.1 Simply supported sandwich beam

A simply supported sandwich beam discretised in six elements is considered toanalyse the behaviour of the present model for various core to face Young'smodulus ratios (E/E). The geometrical and material properties are :L=l .Om (Length) ; h=24.71x!0'* m (thickness of the core)y =0.2875x10"* m (thickness of the bottom face layer)^ =0.5750x10'* m (thickness of the top face layer)Ef=142 GPa (Young's modulus of the outer layers)



Gf=0 GPa (Transverse elasticity modulus of the outer layers)Gc=E/2 GPa (Transverse elasticity modulus of the core layer)pf=1400 kg/nf ; p^=0 kg/nr* (mass density of outer and core layers)Boundary conditions :

Simply supported : u« = v° = w° = ej = u* = u°" = v°* = w°* = pj = p° = o

Table I shows the natural frequencies for the three first flexure modes in the xzplane, and compares them with the available results of the FSDT one plane-bending finite element model of Wennerstrom & Backlund".

Table I: Natural frequencies (Hz) for simply supported sandwich beam.

Mode1

HSDT 231

Ref." 23

10-2198.773778.8971702.495198.489788.0741788.52

E/Ef10-"

133.625339.579541.112134.882346.337561.095

10-*18.94638.04557.11619.29338.87759.102

From table I, one can conclude from the good agreement between the HSDTmodel and the reference results. This is visible even for lower EJG ratios, andfor higher modes, where the present model underpredicts the naturalfrequencies.

5.2 Non-symmetric multilaminated orthotropic beam

An orthotropic multilaminated beam discretised in six finite elements isconsidered, to study the effects of the L/h ratio and the stacking sequence onthe fundamental frequencies. The material and geometric properties used, arethe following :E, = 129.207 GPa ; E^ = E, = 9.42512 GPa ; G^ = 5.15658 GPaG,3 = 4.3053 GPa ; G% = 2.5414 GPa ;v^= v^ = 0.3 ;v%, =0.218837p=l 550.0666 kg.m ; b = 0.0127m ; L = 0.1905m

Tables II and III illustrate the non-dimensionalised fundamental frequencies forlaminated composite beams with different L/h ratios and different boundary

conditions. The non-dimensional multiplier is co = o L (pA/E l) where A=b.h

and I=b.h*/12. The present results are compared with Maiti and Sinha HSDTnine node isoparametric plate type element where all the displacementfunctions are assumed to contain the cubic power of z. A good correlation isfound with this alternative model.



Table H : Effect of fibre orientation on the fundamental frequency wof simply supported multilaminated beams

Stackingsequence

0/30/00/45/00/60/00/90/0

0/90/0/900/45/-45/00/60/-60/0

CPU ratios ref

Maiti & Sinha

36.0535.8535.7835.7726.3334.3934.26

erred to EBT model

EBT36.3436.1436.0235.9726.4235.0134.58

L/h=60FSDT (k=5/6)

362636.0635.9335.8826.3734.9334.501.3

HSDT36.1735.9735.8435.78263234.8434.4110.5

As one can observe from table II, there is a good agreement for all cases, eithersymmetric and non-symmetric, between all the models and the referenceresults. However, it can be observed that the HSDT model is more demandingin computational terms. Table ffl also illustrates non-dimensional fundamentalfrequencies, using the same multiplier. In this table one can observe theinfluence of the L/h ratio on the fundamental frequency of multilayered non-symmetric clamped-clamped beams.

Table III: Influence of L/h ratio on the fundamental frequency wof clamped-clamped multilaminated beams.

StackingSequence0/90/0/900/45/-45/00/60/-60/0

R(171818

5f.'.45.61.04

L/hEBT

20.2622.4420.88

i=5FSDT"

18.7121.8320.28

HS

17.20.19.

DT

89,32,53

Ref/

55.8777.0576.62

L/hEBT

56.5079.3678.39

=60FSDT"

56.0678.5277.53

HS

55,77.76.

DT

,67,64,60

*(k=5/6)

As one would expect, in the lower L/h ratio case the shear effect becomesvisible, being the HSDT model, the one that shows a better agreement with theMaiti & Sinha results by giving lower natural frequency values. The presentmodel has the advantage of being less computationally expensive.

5.3 Multi-span multilaminated beam

A multi-span multilaminated beam made of AS/3501-6 graphite-epoxysubjected to different external boundary conditions, was discretised in six finiteelements, to study the influence of the number of spans on the fundamentalfrequency. The geometrical and material properties are :E, = 21.0x10' psi ; £2 = £3 = 1.4x10'psi

= 0.5x10* psi ;v^= v^ = v^ =0.3



p= 0.13x10* IbsMn^ ; h = b=1.0in ; L = 30inLayups considered : [0°/90<Y90°/0°] and [90°/0°]

Figures 2 and 3 show the performance of the HSDT model, compared with theAbramovich et al*, FSDT finite element beam model.

3,5 -3 -L

2,5

Hinged-hinged multilaminated beam L/h=30

_ Abramovich et al -0/90/90/0. HSDT-0/90/90/0

_ Abramovich et al - 90/0

_ HSDT-90/0

Clamped-clamped multilaminated beam L/h=30

_ Abramovich et al -0/90/90/0-HSDT-0/90/90/0

_ Abramovich et al - 90/0

_ HSDT-90/0

Figures 2 / 3 : Fundamental frequency (kHz) of multi-spanmultilaminated hinged-hinged and clamped-clamped beams

From figures 2 and 3, one can conclude from the good agreement between thepresent model and the reference® results. As expected, the natural frequencyrises as the number of spans becomes higher. It can also be seen that despitehaving the same thickness, the symmetric layup beam presents a betterperformance when compared with the non-symmetric one.



6 Conclusions

A brief review of research carried out on the analysis of multilaminated beamshas shown the need for further development of efficient numerical tools for freevibrations studies. The present work takes this motivation into consideration bycarrying out extensional work on a single layer higher order shear deformationmodel (HSDT), based on a four node Lagrangean straight beam element, withfourteen degrees of freedom per node, which considers bi-axial bending,stretching, bending-stretching and twisting, for general layups and for the mostused cross section. From the extended numerical studies and comparisons withthe available solutions it can be concluded that the proposed model can beapplied to free vibrations of thin and thick multilaminated anisotropic beams,with good accuracy, especially concerning the prediction of naturalfrequencies. The present model is however computationally more expensivethat the alternative models based on Euler-Bernoulli (EBT) and first ordershear deformation theory (FSDT).

7 Acknowledgements

The authors thank the financial support received from H.C.M. Project(CHRTX-CT93-0222), "Diagnostic and Reliability of Composite Material andStructures for Advanced Transportation Applications".

9 References

1. Vinson, J.R. & Sierakowski, R.L. "The behavior of structures composed ofcomposite materials", Martinus Nijhoff Publishers, Dordrecht, TheNetherlands, 1986.

2. Whitney, J. M. "Structural analysis of laminated anisotropic plates",Technomic Publishing Company, Inc., Pennsylvania, U.S.A., 1987.

3. Teoh, L.S. & Huang, C.C., "The vibration of beams of fiber reinforcedmaterial", Journal of Sound and Vibration, 1977, Vol. 51, pp. 467-473.

4. Chandrashekhara, K.; Krishnamurthy K. & Roy, S., "Free vibration ofcomposite beams including rotary inertia and shear deformation",Composite Structures, 1990, Vol. 14, pp. 269-279.

5. Teh, K. K. & Huang, C. C., "The vibration of generally orthotropic beams- a finite element approach", Journal of Sound and Vibration, 1970, Vol.62, pp. 195-206.



6. Kapania, K.R. & Raciti, S., "Nonlinear vibrations of unsymmetricallylaminated beams", AIM Journal, 1989, Vol. 27(2), pp. 201-210.

7. Maiti, Kr D. & Sinha, P. K., "Bending and free vibration analysis of sheardeformable laminated composite beams by finite element method",Composite Structures, 1994, Vol. 29, pp. 421-431.

8. Abramovich, H., Eisenberger, M., & Shulepov, O., "Vibrations of multi-span non-symmetric composite beams", Composites Engineering, 1995,Vol. 5(4), pp. 397-404.

9. Loja, M.A.R., "Higher-order shear deformation models - development andimplementation of stiffness coefficients for T, I, channel and rectangularbox beams", Report EDMEC/IST, Project STRDA/C/TPR/592/92, Lisbon,1995.

10. Maple V Release 4, Waterloo Maple - Advancing Mathematics, WaterlooMaple Inc., 450 Phillip Street, Waterloo, Ontario, Canada N2L 5J2

11. Wennerstrom, H. & Backlund, J., "Static, free vibration and bucklinganalysis of sandwich beams", Report 86-3, Department of AeronauticalStructures and Materials, The Royal Institute of Technology, Stockholm,Sweden, 1986.


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