Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf

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Mechanics of Advanced Materials and Structures, 9:359 373, 2002

Copyright C2002 Taylor & Francis

1537-6494/02 $12.00 + .00

DOI: 10.1080/1537649029009699 1

Free Vibration Analysis of Fiber-Reinforced Plastic

Composite Cantilever I-Beams

PIZHONGQIAO

GUIPINGZOU

Department of Civil Engineering, The University of Akron,

Akron, Ohio, USA

ABSTRACT

An analytical study for dynamic behavior of pultruded ber-reinforced plastic(FRP) composite cantilever I-beams is presented. Based on a Vlasov-type linear hypoth-esis, dynamic beam mass and stiffness coefcients, which account for both cross-sectiongeometry and material anisotropy of the beam, are obtained. The eigenfrequency prob-lem is solved by a Ritz energy method, and both exact transcendental and polynomialshape functions satisfying the boundary conditions of cantilever beams are used to de-scribe the modal shapes. Good agreement between the proposed analytical method andnite-element analysis is obtained. The effect of beam span length, ber orientation,

and ber volume fraction on natural frequencies is investigated. The proposed analyti-cal solution can be used to effectively predict the vibration behavior of FRP cantileverI-beams.

Pultruded ber-reinforced plastic(FRP) composite structural shapes have been used for

civil engineering construction when corrosion resistance is important. Pultruded FRP struc-

tures resemble cold-rolled steel structural shapes and are used as replacements for steel.

The FRP shapes(e.g., beams and columns) are usually thin-walled structures manufactured

by a pultrusion process, and materials used are high-strength E-glass bers embedded in

either vinyl ester or polyester polymer resins [1, 2]. Due to the complexity of compositesand the thin-walled conguration of FRP shapes, consistent theories and pertinent analyti-

cal tools are needed to determine their static and dynamic response. A number of theories

for isotropic thin-walled beams have been developed. Conventional theories of thin-walled

isotropic structures were given by Vlasov [3], Gjelsvik [4], and Murray [5]. A Vlasov-type

theory for composite thin-walled beams with open cross sections was established by Bauld

and Tzeng [6], and the thin-walled beams considered were composed of a number of sym-

metric laminated plates. Based on the assumption that the cross section of the beam does

not deform in its own plane and the laminate has a symmetric layup, Kobelev and Larichev[7] studied thin-walled beams with close cross sections. Libove [8] established a simple

theory for anisotropic thin walled beams with a single cell closed cross section Manseld


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360 P. Qiao and G. Zou

and Sobey [9] and Manseld [10] developed theories for one- or two-cell beams of ber

composites. Bauchau et al. [11] conducted a combined theoretical and experimental study

of a thin-walled box beam theory with orthotropic material properties. Chandra and Chopra

[12] presented a theory along with experimental results for laminated composite I-beams

subjected to bending and torsional loads. Including both torsional warping and beam sheardeformation, Wu and Sun [13] developed a simplied theory for composite thin-walled

beams. Zvarick and Cruse [14] used a strength of materials approach to derive a general-

ized beam theory for statically determinate conditions for open section laminated composite

beams and compared the results with nite-element studies for typical graphite/epoxy lam-

inated composite beams. Badir et al. [15] proposed a variational and asymptotic analysisof two-dimensional shell theory for thin-walled open-cross-section beams. Based on kine-

matic assumptions consistent with Timoshenko beam theory, Barbero et al. [16] presented

a formal engineering approach for analysis of thin-walled laminated beams under bending,

and the accuracy of the theory was later validated by experimental and nite-element study[1]. Massa and Barbero [17] used a strength-of-materials approach to formulate a simple

methodology for the analysis of thin-walled composite beams subjected to bending, tor-

sion, shear, and axial forces. Maddur and Chaturvedi [18] proposed a simplied theory for

laminated composite I-sections under nonuniform torsion.

Most of the studies introduced above deal mainly with the general theory and static

response of thin-walled structures, and limited studies have been devoted to the dynamic

behavior of anisotropic thin-walled structures. Combining theory of thin-walled beams

and mechanics of anisotropic composites, Song and Librescu [19] considered the dynamic

problem of laminated composite single-cell thin-walled beams with arbitrary cross section

and incorporated nonuniform torsion. Later, Song et al. [20] employed this method [19]

for dynamic simulation of anisotropic thin-walled beams with blast and harmonically time-

dependent loads. Song et al. [21] further used this theory [19] for the dynamic analysis

of pretwist spinning thin-walled composite beams. Of the research conducted in [19 21],

only the dynamic problems of closed-cross-section composite beams are considered, and

there is little information available in the literature for the dynamic behavior of anisotropic

composite beams with open cross-sectional proles. On the other hand, as smart materials

(e.g., piezoelectric pads and shape memory alloys)are increasingly used in active controland damping of advanced composite materials and structures, a better understanding of free

vibration behavior of anisotropic thin-walled structures seems more important in facilitating

the design of smart composite structures.

In this study, an analytical study for dynamic behavior of pultruded FRP cantilever

I-section beams is presented, and a simplied formulation based on the Vlasvo-type linear

hypothesis and Ritz energy method is derived. Two types of shape functions, i.e., exact

transcendental function and polynomial function, which satisfy the cantilever boundary

conditions, are studied. Comparison of analytical solutions with numerical results of nite-

element analyses is performed. Parametric studies of beam span length, ber orientation,and volume fraction on the inuence of natural vibration frequencies are also presented.


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Vibration Analysis of Composite Cantilever Beams 361

Figure 1. Coordinate systems of thin-walled beam.

geometric considerations [4, 6], the displacement transformation can be stated as

u(z; s ) D Usin q Vcos q qw

(1)v(z; s ) D Ucos q C Vsin q C rw

whereU,V,and ware functions of the axial coordinate z only; andq, q , and rare functions

of the contour coordinatesn ,s alone.

The axial displacement component of an arbitrary point on the contour can be obtained

using the denition of the shear strain, czs D w=s C v=z, and from Eq.(1)the fol-

lowing expression is obtained:

ws

D (U0 cos q C V0 sin q C rw 0) (2)


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in Eq.(3)is dened as

x D

ZC

r ds (4)

Note that x and y are the coordinates of a point on the contour Cand are, therefore,

functions of the contour coordinate s. All other quantities in Eq. (3), except x (s), are

functions of the axial coordinate zonly;x as dened in Eq.(4)is a section property and is

called the sectional area [22].

2. GOVERNING EQUATIONS

Toward the goal of deriving the equations of motion for anisotropic thin-walled struc-

tures, Hamiltons variational principle is used. This variational principle can be stated as

d

} DZ

t1

t0 fd

d

Kg dtD 0 (

5)

where

D1

2

Z (NW0 C My U

00 Mx V00 C Mxw

00 C TSw0) dz (6)

and

KD1

2

Z\

q < < d\ (7)

denote the strain energy and kinetic energy, respectively, in which(0) denotes differentiationwith respect to z; Nis the axial force; Mx ; My are the bending moments acting about the

x and y axes, respectively; Mx is the warping moment, so called by Timoshenko and

Gere [22], or the biomoment as designated by Vlasov [3];TSis St. Venants torsion or free

warping; qis the mass per unit volume; and < D [u(z; s); v(z; s ); w(z; s )]T.

The constitutive relation for the anisotropic thin-walled beams can be stated as [6]

8>>>>>>>>>>>:

N

Mx

My

Mx

TS

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

2666664

A 0 0 0 0

0 Ix x 0 0 HC

0 0 Iyy 0 HS

0 0 0 Ixx Hq

0 HC HS Hq JG

3777775

8>>>>>>>>>>>:

W0

V00

U00

w 00

w 0

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

(8)

where

A D

Z A11ds

Ix x DZ

(A11y2

C D11cos2

q) dsZ


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in which Ai j andDi j are the extensional and bending stiffness of a laminated panel, respec-

tively, and are expressed as

Ai j D

N

XkD1

( Qi j )(zk zk1 )

(10)

Di j D1

3

NXkD1

(Q i j )z

3k z

3k1

where Q i j is known as the transformed reduced elastic constant [23].

Using Gauss integration for the kinetic energy, the following equation can be obtained:

Z t1

t0

dK dtD

Z t1

t0

Z\ q

<

d< d\

D

Z t1t0

Z\

f(Usin q Vcos q qw )(dUsin q dVcos q qdw )

C (Ucos q C Vsin q C rw )(dUcos q C dVsin q C rdw )

C ( W U0x V

0y w

0x)(dW dU0x dV0y dw 0x)g d\ (11)

in which()denotes the differentiation with respect to timet.

By substituting Eq.(8)into Eq.(6), the expression for the strain energy,, becomes

D1

2

Z fA W02 C Ix x V

002 C Iyy U002 C Ixx w

002 C JGw02 C 2HC V

00 w 0

2Hs U00 w 0 C 2Hq w

0 w 00g dz (12)

By substituting Eqs.(11)and(12)into Eq.(5), the differential equilibrium equations

are obtained as

8>>>>>>>>>>>>>>>>>:

A W00 I1 WC I10U0 C I11 V0 C I12w 0 D 0

Iyy U00 00 Hsw

000 C I1 UC I2w C I3 U00 I4w

00 I13 V

00D 0

Ix x V00 00 C HCw

000 C I1VC I5w C I6 V00 I13U

00 I9w

00D 0

Ixx w00 00 HCV

000 C Hs U000 JGw 00 C I2 UC I5 VC I7w I12 W

0 I4 U

00 I9 V

00

I8w00D 0

(13)

where Z Z


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I9 D

ZA

qxy dA I 10 D

ZA

qx dA

I11 D

ZA

qy dA I12 D

ZA

qx dA

I13 D

ZA

qx y dA

3. RIGIDITIES OFFRP I-SECTION BEAM

For a typical I-beam(see Figure 2), the centroid, principal pole, and principal origin

coincide. The contour has ve branches, numbered as in Figure 2, for which the principal

coordinate functions are given in Table 1.

The warping, torsional, and exural stiffness components in Eq. (9) and dynamiccoefcients in Eq.(14)are simplied for I-section beams as follows:

Ixx D1

24A11f h

2wb

3fC

1

6D11f b

3fC

1

12D11w h

3w

JG D8D33f bfC 4D33w hw

Ix x D1

2

A11f bfh2w C 2D11f b fC

1

12

A11w h3w

(15)Iyy D

1

6A11f b

3fC D11whw

Hs D2D13whw; HCD 4D13wbf

Hq D1

2

D13f bft

2fC D13w hwt

2w


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Table 1

Principal coordinate functions for an I-section beam

Branch Range ofsi q x y q(s) r(s) x

1 0 bf=2 0 s1 bf =2 hw =2 s1 bf =2 hw =2 (hw =2)(bf=2 s1 )2 0 bf=2 p bf =2 s2 hw =2 bf=2 s2 hw =2 (hw =2)(s2 bf =2)3 hw =2 hw =2 p=2 0 s3 s3 0 04 0 bf=2 p s4 hw =2 s4 hw =2 (hw =2)s45 0 bf=2 0 s5 hw =2 s5 hw =2 (hw=2)s5

I1 D q(2bftfC hw tw ) I2 D qhw

b ftf 0:5t2w

I3 D qbftf3b2

f

6bftfC 8t2

f12 I4 D qhwb3

f

tf 2b2

f

t2

f8

I5 D qbftf(tf bf)

2 I6 D qhw

3bfh wtfC 2t3w

6

(16)I7 D q

3b3ftfC 6bfh

2w tf 6b

2ft

2fC 8bft

3fC 4hwt

3w

12

I8 D qhw3b3fhwtfC 6b

2fhw t

2f 8b

2fhwt

3f

48

I9 D I10 D 0 I11 D qhwt2w

2 I12 D I13D 0

For simplicity, the coupling between the bending and exural-torsion vibration is not

considered here, and then the simplied equilibrium formulation can be stated as

8>>>>>:

A W00 I1 WD 0

Iyy U0000 C I1UC I3 U

00D 0

Ix x V0000 C I1VC I6 V

00D 0

Ixxw0000 J Gw 00 C I7w I8w

00D 0

(17)

4. DISPLACEMENT FIELD OF CANTILEVERI-BEAM

The free vibration displacement elds which satisfy the cantilever beam boundary

conditions can be selected as either the exact transcendental shape functions or polynomial

shape functions. These two types of shape functions are all considered in this study. The

exact transcendental functions are assumed as [24, 25]

8>>>>

U(z; t)

V(z

; t)

w(z; t)

9>>>=>> D

8>>>>

U

Vw

9>>>=>>

XmD1;2;3;:::

sinkm

z

L

sinh

km

z

L

bm

cos

km

z

L


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Table2

Pan

elstiffnesscoefcientsofWFI-section

A11

A66

D11

D12

D22

D

16

D26

D66

q

omponent

(N/m)

(N/m)

(Nm)

(Nm)

(Nm)

(N

m)

(Nm)

(Nm)

(kg/m3)

Flang

e

1.9

95

108

3.0

80107

500

110

250

7

7

126

1,8

50

Web

1.6

97108

2.7

30107

457

107

238

7

7

122

1,8

50


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and km satises the transcendental equation

cos(km ) cosh(km ) 1 D 0 (18c)

withk1 D 1:875104; k2 D 4:694091 ; k3 D 7:854757; : : : :The polynomial deformed modal functions are assumed as [25]

8>>>>>:

U(z; t)

V(z; t)

w (z; t)

W(z; t)

9>>>=>>>;D

8>>>>>:

U

V

w

W

9>>>=>>>;

XmD1;2;3;:::

1

(m C 1)(m C 2)(m C 3)(m C 4)Gm

z

L

eix t (19a)

where

Gm

z

L

D

z

L

mC4

1

6(m C 1)(m C 3)(m C 4)

z

L

3

C1

2(m C 1)(m C 3)(m C 4)

z

L

2(19b)

The Rayleigh-Ritz method [26] is employed to solve the eigenvalues of the potential

energy equilibrium equations in Eq.(17).

5. NUMERICAL RESULTS AND DISCUSSION

The example under consideration is a wide-ange I-beam [WF 10.16 10.16

0.635 cm(WF 4 4 14

in.)] with a given span length of 3.353 m. The beam is stud-

ied in a cantilever conguration. The panels of pultruded FRP shapes are not made by hand

layup, but they can be simulated as a laminated conguration [1]. The layup of pultruded

panel components consists of two combo-stitched tri-axial(C/45 and 0)layers and one

unidirectional roving(0) layer. The laminated panel properties of WF I-beam are predicted

by a micro/macromechanics approach [1] and are given in Table 2.The commercial nite-element program ANSYS is used to perform an eigenvalue

analysis, and Mindlin eight-node isoparametric layered shell elements (SHELL 99) are em-

ployed in the modeling. The analytical assumed deformation modal shapes for the rst and

second modes in each basic direction (i.e., weak, strong, and exural torsional) are shown in

Figures 3aand 3b, respectively. The nite-element deformed shapes for bending vibration

along weak-axis, strong-axis, and exural torsion vibration are given in Figures 4a, 4b,

and 4c, respectively. Analytical frequencies using the exact transcendental and polynomial

shape functions along with nite-element results at the length ofL D 3.353 m are given in

Table 3. The present solutions using exact transcendental functions and polynomial func-

tions show good agreement with the results based on the nite-element method (FEM),


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Table 3

Comparison of natural frequencies

Exact transcendental Polynomial

Mode function(HZ) function(HZ) FEM(HZ)

1 4.88 4.88 4.81

2 8.92 8.93 8.83

3 13.63 13.55 11.45

Figure 3. Analytical assumed vibration deformed modal shapes in each basic direction of

weak bending, strong bending, or exural-torsional.


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(b)

(c)

Figure 4. (Continued) (b)Mode 2: Vibration along strong axis, and(c)Mode 3: Flexural-

t i l ib ti i t diff t i


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Figure 5. Inuence of ber ply angles on natural vibration frequencies.


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Figure 7. Inuence of ber volume fraction on natural vibration frequencies.

dramatically along the strong bending axis as the beam length is reduced; whereas similar

trends are observed for weak-axis vibration and exural-torsional vibration with respect to

the change of beam span length. The modal natural frequency versus ber volume fraction

with beam span length of 3.353 m and q D C/30 is shown in Figure 7. In the analytical

modeling, the material density input as a function of ber volume fraction is dened by arule of mixture as qD (1Vf)qm CVfqf. It can be observed that the ber volume fraction

has a direct impact on the nature frequency of FRP I-beams, particularly for vibration of

exural-torsional mode.

6. CONCLUSIONS

In this article, a theoretical vibration analysis of pultruded FRP composite cantilever

I-beams is presented. Based on a Vlasvo-type linear hypothesis, the equilibrium differential

equations are formulated and solved by the Ritz energy method. Both the exact transcen-

dental and polynomial shape functions which satisfy the cantilever boundary conditions


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approach presented can be used as an efcient and versatile tool for free vibration analysis

of FRP thin-walled structures and as a basis for further study in active control and damping

of FRP structures.

REFERENCES

[1] J. F. Davalos, H. A. Salim, P. Z. Qiao, R. Lopez-Anido, and E. J. Barbero, Analysis and Design

of Pultruded FRP Shapes under Bending, Composites BEng. J., vol. 27, no. 3 4, pp. 295 305,

1996.

[2] P. Z. Qiao, J. F. Davalos, E. J. Barbero, and D. Troutman, Equations Facilitate Composite

Designs,Modern Plastics, vol. 76, no. 11, pp. 77 80, 1999.

[3] V. Z. Vlasov, Thin-Walled Elastic Beams (translated fromRussian), National Science Foundation,

U.S. Department of Commerce, Washington, DC, TT-61-11400, 1961.

[4] A. Gjelsvik,The Theory of Thin Walled Bars, Wiley, New York, 1981.[5] N. M. Murray,Introduction to the Theory of Thin-Walled Structures, Clarendon Press, Oxford,

U.K., 1984.

[6] N. R. Bauld, Jr., and L. S. Tzeng, A Vlasov Theory for Fiber-Reinforced Beams with Thin-Walled

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[7] V. V. Kobelev and A. D. Larichev, Model of Thin-Walled Anisotropic Rods, Mechanika

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[8] C. Libove, Stresses and Rate of Twist in Single-Cell Thin-Walled Beams with Anisotropic Walls,

AIAA J., vol. 26, no. 9, pp. 11071118, 1988.

[9] E. H. Manseld and A. J. Sobey, The Fiber Composite Helicoper Blade, Part 1: Stiffness Prop-

erties; Part 2: Prospects for aeroelastic tailoring,Aeronaut. Quart., vol. 30, no. 2, pp. 413 449,

1979.

[10] E. H. Manseld, The Stiffness of a Two-Cell Anisotropic Tube,Aeronaut. Quart., vol. 32, no. 4,

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[11] O. A. Bauchau, B. S. Coffenbery, and L. W. Reheld, Composite Box Beam Analysis: Theory

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[12] R. Chandra and L. Chopra, Experimental and Theoretical Analysis of Composite I-Beams with

Elastic Coupling,AIAA J., vol. 29, no. 12, pp. 2197 2206, 1991.

[13] X. X. Wu and C. T. Sun, Simplied Theory for Composite Thin-Walled Beams,AIAA J., vol. 30,

no. 12, pp. 29452951, 1992.[14] A. G. Zvarick and T. A. Cruse, Coupled Elastic Response of Open Section Laminated Com-

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Structures, Structural Dynamics and Materials Conf., Part 2, Dallas, TX, 1992, pp. 725 735.

[15] A. M. Badir, V. L. Berdichevsky, and E. A. Armanios, Theory of Composite Thin Walled Opened

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[23] E. J. Barbero,Introduction to Composite Materials Design, Taylor & Francis, New York, 1999.

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[25] I. Elishakoff and Z. Guede, Novel Closed-Form Solutions in Buckling of Inhomogeneous

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[26] L. Meirovitch and L. Kwak, Convergence of the Classical Rayleigh-Ritz Method and the Finite

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Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf