free vibration analysis of fiber-reinforced plastic composite cantilever i-beams.pdf
TRANSCRIPT
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
1/16
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
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
2/16
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.
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
3/16
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)
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
4/16
362 P. Qiao and G. Zou
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
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
5/16
Vibration Analysis of Composite Cantilever Beams 363
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
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
6/16
364 P. Qiao and G. Zou
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
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
7/16
Vibration Analysis of Composite Cantilever Beams 365
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
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
8/16
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
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
9/16
Vibration Analysis of Composite Cantilever Beams 367
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),
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
10/16
368 P. Qiao and G. Zou
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.
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
11/16
Vibration Analysis of Composite Cantilever Beams 369
(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
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
12/16
370 P. Qiao and G. Zou
Figure 5. Inuence of ber ply angles on natural vibration frequencies.
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
13/16
Vibration Analysis of Composite Cantilever Beams 371
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
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
14/16
372 P. Qiao and G. Zou
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
Open Cross Sections,Int. J. Solids Struct., vol. 20, no. 3, pp. 277 297, 1984.
[7] V. V. Kobelev and A. D. Larichev, Model of Thin-Walled Anisotropic Rods, Mechanika
Kompozitnykh Materialov, vol. 24, no. 2, pp. 102 109, 1988.
[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,
pp. 338 353, 1981.
[11] O. A. Bauchau, B. S. Coffenbery, and L. W. Reheld, Composite Box Beam Analysis: Theory
and Experiments,J. Reinforced Plastics Composites, vol. 6, no. 1, pp. 25 35, 1987.
[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-
posite Beams Subjected to Generalized Beam Loading, The 34th AIAA /ASME/ASCE/AHS/ASC
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
Cross Section Beams,The 35th AIAA/ASME/ASCE/AHS/ASC Structures: Structural Dynamics
and Material Conf., La Jolla, CA, 1993, pp. 2761 2770.
[16] E. J. Barbero, R. Lopez-Anido, and J. F. Davalos, On the Mechanics of Thin-Walled Laminated
Composite Beams,J. Composite Mater., vol. 27, no. 8, pp. 806 829, 1993.
[17] J. C. Massa and E. J. Barbero, A Strength of Materials Formulation for Thin Walled Composite
Beams with Torsion,J. Composite Mater., vol. 32, no. 17, pp. 1560 1594, 1998.
[18] S. S. Maddur and S. K. Chaturvedi, Laminated Composite Open Prole Sections: Non-Uniform
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
15/16
Vibration Analysis of Composite Cantilever Beams 373
[23] E. J. Barbero,Introduction to Composite Materials Design, Taylor & Francis, New York, 1999.
[24] G. P. Zou, An Exact Symplectic Solution for the Dynamic Analysis of Reissner Plates,Comput.
Meth. Appl. Mech. Eng.,vol. 156, pp. 171 178, 1998.
[25] I. Elishakoff and Z. Guede, Novel Closed-Form Solutions in Buckling of Inhomogeneous
Columns under Distributed Variable Loading, Chaos, Solitons and Fractals, vol. 12, pp. 1075
1089, 2001.
[26] L. Meirovitch and L. Kwak, Convergence of the Classical Rayleigh-Ritz Method and the Finite
Element Method,AIAA J., vol. 8, pp. 15091516, 1990.
-
8/10/2019 Free Vibration Analysis of Fiber-Reinforced Plastic Composite Cantilever I-Beams.pdf
16/16