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Analysis of piezolaminated plates by the spline finite strip method
Computers and Structures 79 (2001) 2321±2333
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Analysis of piezolaminated plates by the spline ®nite strip method M.A.R. Loja a, J. Infante Barbosa a, C.M. Mota Soares b,*, C.A. Mota Soares b b
a ENIDH ± Escola N autica Infante D. Henrique, Av. Eng.° Bonneville Franco, 2780-572 Pacßo de Arcos, Portugal IDMEC ± Instituto de Engenharia Mec^ anica, Instituto Superior T ecnico, Av. Rovisco Pais, 1049-001 Lisboa Codex, Portugal
Received 18 October 1999; accepted 22 February 2001
Abstract This paper deals with the development of a family of higher order B-spline ®nite strip models applied to the static and free vibration analysis of laminated plates, with arbitrary shape and lay-ups, loading and boundary conditions. The lamination scheme can be such that the embedded and/or surface bonded piezoelectric actuating and sensing layers are included. The structure is discretised in a speci®ed number of strips, and the geometry and displacement components of each strip are represented by interpolating functions that are products of linear or cubic B-spline, and linear or quadratic Lagrange functions along the y and x orthonormal directions. The accuracy and relative performance of the proposed discrete models are compared and discussed among the developed and alternative models. Ó 2001 CivilComp Ltd. and Elsevier Science Ltd. All rights reserved. Keywords: Piezolaminated plates; Composite materials; Piezoelectric materials; B-spline ®nite strip method
1. Introduction Higher order laminated plate theories have been proposed to overcome the limitations of the classical plate theory to accurately predict the structural response of highly anisotropic composite laminates, which exhibit more pronounced transverse shear eects in laminated plates than in the isotropic ones under similar loading conditions. The use of piezolaminated composite materials and structures with sensory and activation capabilities, require the combination of tailor-made mechanical properties of the laminates and the associated smart materials, to change the structural behaviour of advanced structures. The ease of integrating piezoelectric laminae or patches, by means of embedding or simply bonding them to laminated structures, has contributed to a wide range of applications. All these factors
*
Corresponding author. Fax: +351-218-417-915. E-mail address: [email protected] (C.M. Mota Soares).
have led to an increasing demand for the development of ecient computational tools to represent the electromechanical behaviour of these new adaptive structures. During recent years, several researchers have carried out the development of mathematical models and associated computational models to predict the response of laminated structures with actuators/sensors capabilities. However, only a few consider higher order shear deformations theories. No published works have been found in the analysis of laminated structures with piezoelectric laminae or patches, using the B-spline ®nite strip method approach. The use of this type of model can reduce the computational eort, maintaining at the same time, the versatility of the traditional equivalent higher order displacement ®nite element models, for the analysis of piezolaminated plates with arbitrary geometry and loading. Another advantage is the ease of implementation of general shape, boundary conditions and loading, when compared with the better known Fourier series based ®nite strip models [1]. These reasons have provided a strong motivation for the present study.
0045-7949/01/$ - see front matter Ó 2001 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 1 ) 0 0 0 6 5 - 7
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B-spline ®nite strip models without piezoelectric laminae have been developed by some researchers. Fan and Luah [2], presented a nine-node spline ®nite element model for the bending analysis of isotropic thin-plate structures, using the classical Kirchho plate theory. Li et al. [3], also carried out a similar study for arbitrarily shaped plates. Vermeulen and Heppler [4] presented an application of the B-spline approximation method to the Timoshenko isotropic beam, in order to study its locking behaviour. Wang [5] also analysed the shear locking phenomenon on beams and plates, under the ®rst order shear deformation theory (FSDT) assumptions, having extended its study to laminates. In order to ease the use of these functions, Leung and Au [6], carried out boundary transformations associated to the ®rst and last two spline parameters. Gagnon et al. [7] proposed a C0 linear family of ®nite strip elements for the static analysis of isotropic rectangular thick plate of variable thickness using the FSDT. The possibility of varying the thickness of isotropic structures, was also considered by the C1 strip model of Uko and Cusens [8]. The last two papers use cubic B-splines along the strip length. The static analysis of thin axisymmetric isotropic shells using isoparametric cubic B-spline functions was considered by Gupta et al. [9]. Isoparametric strips for the analysis of Mindlin plates were also proposed by Au and Cheung [10] and Ng and Chen [11]. Tan and Dawe [12,13] presented a general spline ®nite strip based on the ®rst order shear deformation plate theory for the buckling and free vibration analysis of prismatic laminated structures. A higher order B-spline ®nite strip approach based on the Reddy [14] third order shear deformation theory, was used for the static analysis of isotropic and cross-ply laminated plates by Kong and Cheung [15]. Wang and Dawe [16] presented buckling stresses for composite laminated prismatic shell structures using the curved ®nite strips models. In the ®eld of the piezoelectric materials and structures, one can refer to several published research works using the traditional ®nite element displacement formulation, some of them being summarised in this paper. A pioneering ®nite element formulation, which includes the piezoelectric or electroelastic eect has been developed by Allik and Hughes [17] for a tetrahedral ®nite element model. A plate ®nite element based on the ®rst order shear deformation plate theory, in which the electrical potential degree of freedom does not present a nodal dependency by remaining constant within each element, was proposed by Suleman and Venkayya [18]. An analogous study for the de¯ection control of plates with piezoelectric actuators was carried out by Lin et al. [19] which used an eight-node isoparametric plate ®nite element.
Wang et al. [20] developed a plate bending element for static shape control, which can have distributed piezoelectric sensors and actuators, and nodal electrical potential degrees of freedom. A self-sensing piezolaminated actuator model for shells, based on the FSDT, was proposed by Miller and Abramovich [21]. Varadarajan et al. [22], discussed the adaptive shape control of laminated composite plates with integrated piezoelectric actuators and position sensors, under quasi-static loads. Detwiler et al. [23] developed a similar study but considering only static loads. Heyliger [24], Heyliger and Brooks [25], Ray et al. [26] and Ray et al. [27] presented exact solutions for the static behaviour of simply supported laminated piezoelectric plates. Closed form solutions based on a FSDT were proposed by Abramovich [28] for the de¯ection control of laminated beams with piezoceramic layers. Oguamanam et al. [29] presented a ®nite element formulation including Von K arm an non-linear strain± displacement relations for the static and dynamic analysis of laminated Timoshenko beams with surface bonded sensors and actuators. The stress stiening effects, induced piezoelectrically are also studied. Batra and Liang [30] used the three-dimensional linear theory of elasticity to analyse the steady-state vibrations of a simply supported rectangular linear elastic laminated plate with embedded PZT layers. Lam and Ng [31], presented theoretical formulations based on the classical plate theory and Navier solutions, for the active control of piezolaminated plates which are submitted to both mechanical and electrical loadings. Chandrashekhara and Agarwal [32] formulated a ®nite element model based on FSDT, for the active vibration control of piezolaminated composites. A similar study was carried out by Chen et al. [33]. Ha et al. [34], presented a ®nite element formulation for modelling the dynamic as well as the static response of laminates with piezoceramics, using a eight-node three-dimensional composite brick element. Crawley and Lazarus [35], developed and veri®ed experimentally the induced strain actuation of plate components of a smart structure. Crawley and Luis [36], proposed an analytical and experimental set-up of piezoelectric actuators as elements of intelligent structures. Hwang and Park [37], Hwang et al. [38], proposed a ®nite element formulation based on classical laminate theory, for the vibration control of piezolaminated plates. The element developed was a four-node quadrilateral plate bending element with 12 mechanical elastic generalised displacements degrees of freedom and one electrical degree of freedom. The development of a piezolaminated shell ®nite element, for the vibration control of composite shells containing bonded piezoelectric layers, was carried out by Lammering [39]. Another approach to shell-type structures was carried out by Tzou and Tseng [40], using
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a piezoelectric solid element with internal degrees of freedom for dynamic measurement and active vibration control. Batra et al. [41] developed exact solutions for the forced vibration analysis of a simply supported rectangular elastic plate having piezoelectric actuators bonded to its top and bottom surfaces. A similar study for the dynamic analysis of piezolaminated plates, was also carried out by Bhattacharya and Samantha [42], and for beams by Abramovich and Meyer-Piening [43], through the method of Fourier series. Huang and Sun [44] developed a model, based on the third order shear deformation theory of Reddy [14], for the vibration analysis of composite beams with piezoelectric layers. A higher order theory for modelling composite laminates with induced strain actuators was developed by Chattopadhyay and Seeley [45], in order to analyse surface bonded or laminate embedded smart materials. Another higher order theory based study is presented by Franco Correia et al. [46], where static and free vibrations analysis of piezolaminated plates is carried out, using a family of nine-node higher order plate ®nite element models. Mota Soares et al. [47] applied these HSDT models to sensitivity analysis and structural optimisation to maximise the piezoelectric actuators eciency, and to improve the structural performance and/ or minimise the weight of the structure. Robbins and Reddy [48] presented a study about the static and dynamic interaction between a bonded actuator and the underlying beam structure, by using a layerwise displacement theory. Finite element models were developed. Saravanos et al. [49], and Heyliger et al. [50] developed ®nite element formulations for the analysis of piezolaminated plates and thin shells respectively. Saravanos and Heyliger [51] also developed discrete layer formulations for the study of the coupled electromechanical response of composite beams with embedded sensory and active piezoelectric layers. An hybrid theory, in which the mechanical displacement ®eld is modelled according to an equivalent single-layer theory and the potential function is represented through a layerwise discretisation in the thickness direction, was carried out by Mitchell and Reddy [52]. In Reddy [53] a detailed theoretical formulation for Navier solution and ®nite element models based on the classical and shear deformation plate theories are presented for the analysis of composite plates with integrated sensors and actuators. A negative velocity feedback control algorithm coupling the direct and converse piezoelectric eects is discussed to control the dynamic response of an integrated structure. A discrete layer model and the associated ®nite element formulation, was developed by Douthireddy and Chandrashekhara [54], in order to achieve the shape
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control of laminated composite beams with piezoelectric actuators. Han and Lee [55] presented a layerwise formulation for the vibration control of composite plates with piezoelectric actuators, in which the in-plane displacements through the thickness are modelled using a discrete layer approach. An adaptive sandwich-beam ®nite element capable of dealing with either extension or shear actuated mechanisms, was developed by Benjeddou et al. [56]. Very recently, comprehensive surveys by Benjeddou [57] and Gopinathan et al. [58], discuss the advances, trends and applications of the ®nite element modelling of adaptive structural elements. The objective of the present work is to present, a family of B-spline higher order ®nite strip models for the static and free vibrations analysis of laminated plates with embedded or surface bonded piezoelectric laminae or patches. In order to achieve this objective, a package of six strip models were developed, being four of them based in the ®rst order theory (FSDT) and the remaining two, based on Lo et al. [59] higher order displacement ®eld (HSDT). The dierences among the same theory based models, lies on the number of nodal lines of the strip and also on the degree of the longitudinal interpolating functions. The FSDT models have the option of choosing between two or three nodal lines. For the HSDT models only three nodal lines are assumed. In all the cases piecewise linear or cubic B-spline functions are used along the y-direction. The performance of some of the models that constitute this package will be shown through the illustrative cases presented.
2. Displacement and strain ®elds The displacement ®eld considered is based on the Lo et al. [59] higher order shear deformation theory (Eq. (1)), in which the in-plane and the transverse displacements are respectively cubic and quadratic functions of the thickness co-ordinate (Fig. 1), as follows: u Zq; T
u x; y; z; t u x; y; z; t v x; y; z; t w x; y; z; t ; h q uo vo wo hox hoy boz uo vo wo hox 2 3 1 0 0 z 0 0 z2 0 0 z3 0 6 7 Z 4 0 1 0 0 z 0 0 z2 0 0 z3 5 0 0 1 0 0 z 0 0 z2 0 0
hoy
iT
;
1
In Eq. (1), uo , vo , wo represent the plate mid-plane displacements along the co-ordinate lines x; y; z, and
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eT E
r Qe
4
D ee pE
where r is the stress vector, e is the strain vector, D is the electrical displacements vector, E is the applied electric ®eld vector, Q is the transformed elastic stiness coecients matrix (which can be subdivided in MB S the submatrices Q and Q according to Eq. (6)), whose coecients are explicitly given in Reddy [60] for the system reference co-ordinates, e is the transformed piezoelectric stress coecients matrix and p is the permittivity matrix.
Fig. 1. Generic B-spline strip element.
hox , hoy are the rotations of the mid-plane perpendiculars about the x and y axes respectively (Fig. 1). The remaining parameters constitute higher order deformation modes, obtained through the expansion of the displacement ®eld in Taylor series referring to the mid-plane. Through the adequate simpli®cations of this displacement ®eld, one can successively obtain the ®eld corresponding to the FSDT. Considering the displacement ®eld given in Eq. (1), and the kinematic relations from the elasticity for small deformations, one has for the extension and curvature vectors and for the transverse shear strain vector, the following expressions:
e ZMB eo ; eo eox eoy 2 1 0 60 1 6 ZMB 6 40 0
eoy
0 z2 0 0
0 z2
0 0
0 0
eoz
1 0 0 0
eox
0 0
cxy
kx
ky
kz
kx
ky
0 0
z 0 0 z
0 0
z3 0
0 z3
0 0
0 0 1 z2
0 0 0 0
2z 0
0 0
0 0
0 z
cxy
/ ZS /o ; /o coyz cozx coyz cozx coyz 1 0 z 0 z2 0 ZS 0 1 0 z 0 z2
o
czx
T
5
" Q
MB
Q 0
0 S Q
# 6
Assuming the possibility of the piezoelectric plies to be polarised only in the thickness direction, one can write: E
ru 0
0 Ez T
7
and if one considers a kth ply, we will have Ezk uk =tk where Ezk is the electrical ®eld in the thickness direction for a given laminae k, uk is the electric potential and tk is
kxy 3 0 07 7 7 05
kxy ; 2
z3
the kth layer thickness. The transformed piezoelectric stress coecient matrix for the same generic layer, in the system co-ordinates, is given as:
; 3
3. Constitutive relations By considering the existence in a generic structure, of both orthotropic and piezoelectric laminae or patches, one has for the kth layer, the following constitutive relation, accounting for the piezoelectric eect:
2 0 0 6 0 6 eTk 6 0 0 0 4 e31 e32 e33 h i e^TkMB e^TkS
where
0 0 e36
.. . .. . .. .
3 e14 e24 0
e15 7 7 e25 7 5 0 8
M.A.R. Loja et al. / Computers and Structures 79 (2001) 2321±2333
e31 e31 cos2 h e32 sin2 h; 2
given as the product of B-spline functions and Lagrangean polynomials according to:
2
e32 e31 sin h e32 cos h; e33 e33 ; e36 e31 e14 e15
Nij x; y Li x/j y:
e32 sin h cos h; e24 sin h cos h;
9
e15 e15 cos2 h e24 sin2 h; e24 e24 cos2 h e15 sin2 h; e25 e15 e24 sin h cos h in which, the eij are the piezoelectric stress coecients in the layer material axes, and h (positive counterclockwise) is the angle between the x1 -direction and the xdirection of the reference system for a generic layer (the x1 -direction coincides with the ®bre direction in the x1 x2 z material co-ordinate system).
A generic function U, say a generalised displacement, can be represented as a linear combination of local Bspline functions [61] /j and parameters aj for the jth node, according to the expression: ju X jjl
/j aj
10
The summation limits of Eq. (10), vary according to the B-spline function degree. In the case of linear functions one has jl 0, ju NS, and for the cubic case, jl 1, ju NS 1 (NS ± number of sections). Depending on the degree of the x-direction Co Lagrange polynomials, a strip will have two or more nodal lines. In the present study, the strip models are linear and quadratic, having an arbitrary number of longitudinal nodes. According to this representation, the plate structure is divided into S strips along the y-direction, each of which is subdivided into NS sections. The strip elements developed here can model plate type structures with other shapes than the rectangular one. The generalised displacements within a strip can then be represented as follows: uo
N X M X i1
j1
Li /j uoij . . . hy
N X M X i1
j1
o
Li /j hyij
11
yielding ue ZN qe
12
where qe is the nodal generalised displacement parameter vector and N is the shape function matrix, which is
13
The local B-spline linear and cubic functions /j , which allow for uneven spacing between the nodes, are explicitly given in Ref. [61]. According to the strain ®eld (Eqs. (2) and (3)), the strain matrices concerning to the membrane/bending and shear deformations are then given by: ee ZMB BMB qe ;
/e ZS BS qe
14
where BMB and BS are the shape functions derivative matrices which relate the element degrees of freedom to the extensions, curvatures and shear strains. The electric ®eld vector, can be represented for an arbitrary section within a strip, by: E
4. B-spline ®nite strips
U y
2325
Bu uSec
15
where uSec is the electric potential vector for a section and Bu is the electric potential matrix, which relates the electrical ®eld vector to the voltages, as follows: 2 3 1=t1 0 6 .. .. 7 .. Bu 4 . 16 . 5 . 0
1=tNP
where NP is the total number of piezoelectric section plies. The principle of virtual work, applied to an element of volume Ve , leads to [17]: Z Z Z n deT Qe deT eE dET eT e dET pE Ve o u dVe dWu dW u 0 17 qduT where dWu and dWu are the virtual work done by the external mechanical loading and the applied surface electrical charge. By making the appropriate substitutions and simpli®cations in Eqs. (1)±(3), (6)±(8) and (17) for the whole discretised piezolaminated plate domain, yields: Fu t Kuu Kuu Muu 0 u u Kuu Kuu Fu t u 0 0 u 18 where Muu and Kuu are the mass and elastic stiness matrices, Kuu is the dielectric `stiness' matrix, and T Kuu Kuu are the elasto-electric coupling eects matrix. The vectors u and u are the generalised electromechanical response of the structure. Fu t and Fu t are the generalised electromechanical load vectors. For free harmonic vibrations, Eq. (18) is rewritten as:
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x2n Muu u Kuu u 0;
Kuu
Kuu u Kuu u 0 19
where xn is the natural frequency associated to the mode n. After condensing the electrical degrees of freedom (Eq. (19)), yields: x2n Muu u 0
K
K Kuu
21
Eq. (18) can be simpli®ed for static analysis. When a known voltage is applied to the actuators, an additional load vector due to the converse piezoelectric eect, is given by: Fa Kuu u
22
yielding: Fa
23
The element matrices derived from Eq. (17), which after assembled constitute the whole system matrices are explicitly given by: "
#
Z Z (
BMB 0 T 0 Bu A " # NL Z hk X ZMB 0 T QMB e^MB k 0 1 hk 1 e^TMB p k1 !) BMB 0 ZMB 0 dx dy dz 0 Bu 0 1 ! ) Z Z ( NL Z hk X S BTS ZST Qk ZS dz BS dx dy
e Kuu e Kuu
e Kuu e Kuu
A
e Muu
MB
k1
Z Z (
N
A
T
NL X k1
S
Section initial node
Section ®nal node
/12 /12 2 /1 /21 /30 /30 4 / 10
/12 0 /21 /21 4/12 3 /0 /30 /12 /4 1 /4 1
4
/ 1 4/4 1
20
Kuu Kuu1 Kuu :
Kuu u Fu
Table 1 Modi®ed local cubic B-spline function
hk
qk
5. Numerical applications 24
1
Z
hk hk
! ) T
Z Z dz N dx dy
for a parabolic transverse shear stress distribution across the laminate, adequate reduced transformed elastic coecients are used, which are explicitly given by Vinson and Sierakowski [62]. Simpli®cations are also extended to the piezoelectric constants eij , only being required the e31 , e32 and e36 coecients, under the same assumption of polarisation through the thickness. The constraints associated with existing boundary conditions, are represented by carrying out, for all the models, the adequate modi®cations to the local B-spline functions. These constraints which can be, associated to a generic degree of freedom are imposed by making the modi®cations presented in Table 1, for the cubic Bspline case. For simplicity, the integration of the elementary matrices is carried out numerically in both in-plane directions, and analytically in the thickness direction. To ease the whole procedure, the integration is carried out section by section, after the transformation of the initial functions given in system co-ordinates into a natural coordinates system, to use the Gauss quadrature methodology.
25
1
Qk and Qk are the membrane and bending, and transverse shear terms of the constitutive matrix for the kth laminae. qk and hk are respectively the density and the vector distance from the laminate mid-plane and the outer surface of the kth ply. NL is the number of plies in the laminate. Matrix I is an identity matrix, which dimension depends on the number of piezoelectric plies within each section in a strip. When considering the FSDT strip models, some simpli®cations are carried out. One is related to the displacement ®eld. Another is associated to the elastic constants Qij . In fact, in such conditions, reduced transformed elastic coecients [60] are used. To account
In this section, several test cases are analysed in order to show the performance of the ®nite strip models developed. The following type of abbreviated designations are used for referring to the dierent models: FSDT-XY: Strip using Xth degree Lagrange functions on the x-direction, and Yth degree B-spline functions along the y-direction. These ®rst order strip models have ®ve degrees of freedom per node. HSDT-XY: Strip using Xth degree Lagrange functions on the x-direction, and Yth degree B-spline functions along the y-direction. These higher order strip models have 11 degrees of freedom per node. The elastic and piezoelectric materials properties are given in Tables 2 and 3 for all examples but one (example 5.1), where units are not given in the comparing reference.
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Table 2 Elastic properties Examples E1 (GPa) E2 (GPa) E3 (GPa) G23 (GPa) G13 (GPa) G12 (GPa) m23 m13 m12
5.2
5.3
5.4
5.5 Gr-Ep
5.5 PZT
5.6 Gr-Ep
5.6 PZT-4
145.0 10.0 10.0 4.8 4.8 4.8 0.25 0.25 0.25
40.0 1.0 1.0 0.5 0.5 0.6 0.25 0.25 0.25
2.0 2.0 2.0 1.0 1.0 1.0 0.0 0.0 0.0
150.0 9.0 9.0 2.5 7.1 7.1 0.3 0.3 0.3
63.0 63.0 63.0 24.2 24.2 24.2 0.3 0.3 0.3
132.38 10.76 10.76 3.61 5.65 5.65 0.49 0.24 0.24
81.3 81.3 64.5 25.6 25.6 30.6 0.43 0.43 0.33
Upper curve
Table 3 Piezoelectric properties
y 0:225b3 0:225b2 2:975b 2:975
Examples 5.4 12
d31 (10 m/V) d32 (10 12 m/V) d33 (10 12 m/V) e31 (C/m2 ) e32 (C/m2 ) e33 (C/m2 ) e33 (10 10 F/m)
± ± ± 0.046 0.046 0.0 1.062
5.5 PZT
± ± ± ±
166.0 166.0 360.0
5.6 PZT-4
± ± ±
122.0 122.0 285.0
x 0:2475b3
5.1. S-shaped isotropic plate The ®rst case considered is a s-shaped two-span isotropic slab bridge, which can be observed in Fig. 2. According to Au and Cheung [10], the upper and lower curves which limit the slab, are expressed by the following equations:
0:89b 2:6319
Lower curve y 3b 3 x 0:9b3
115.05
0:8494b2
0:9b2
1:3b 0:5
where b is a parameter which varies within [ 1. . .1]. The slab is supported along three straight lines de®ned in the same ®gure. According to Ref. [10], the values corresponding to the material properties are: E 104 (Young's modulus), m 0:15 (Poisson's ratio), h 0:2 (thickness) ± (no units were given). A unitary concentrated load is applied at the point, with co-ordinates 4:523; 0:872, located in the central line of the structure.
Fig. 2. S-shaped two-span continuous slab bridge.
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Fig. 3. Central line de¯ections of the s-shaped slab.
Au and Cheung [10], used an isotropic B-spline strip FSDT model, and considered a discretisation in two quadratic ®nite strips with 12 sections each. From Fig. 3 one can observe the ®tted curves, referred to the de¯ection pro®le of the centre line of the sshaped plate, from both the present higher order models, and the comparing solution. It is visible a good agreement between the present strip models, and the reference results. 5.2. Cantilever skew plate A cantilever skew orthotropic plate, with four unidirectional 0°=0°=0°=0° layers of graphite-epoxy, with a thickness of 7.5 mm each, is analysed. The plate geometric properties can be observed in Fig. 4, where dimensions are given in meters. The plate is submitted to a uniform transverse pressure loading of 5000 Pa.
Fig. 4. Skew cantilever orthotropic plate (dimensions in meters).
Table 4 Nodal transverse displacements (mm) x (m)
HSDTQL
HSDTQC
Q9FSDT5
Q9HSDT11
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
0.217 0.230 0.244 0.258 0.272 0.286 0.299 0.313 0.326
0.217 0.230 0.244 0.258 0.272 0.285 0.299 0.312 0.325
0.217 0.231 0.245 0.259 0.272 0.286 0.300 0.313 0.326
0.216 0.229 0.243 0.257 0.270 0.284 0.297 0.311 0.323
In order to compare with an available numerical higher order solution, a discretisation of four strips with eight sections each, parallel to the inclined edges of the plate, is considered. The nodal transverse displacements are given in Table 4. These results are compared with two available numerical solutions using a 4 4 mesh of nine-node quadratic elements. Full details of the models are given in Ref. [46]. These alternative plate ®nite element models, have ®ve (Q9-FSDT5), and 11 degrees of freedom per node (Q9-HSDT11). The Q9-HSDT11 uses the same displacement ®eld as Eq. (1). From Table 4, one can conclude from the good agreement between the present higher order models and the comparing alternative models. It is also possible to conclude from the computational less expensive behaviour of the present higher order strip models when compared to the alternative higher order plate ®nite element model. For the present case, the relation between the number of degrees of freedom required by the higher order strip models and the alternative higher order model is 0.4 (792/1936).
M.A.R. Loja et al. / Computers and Structures 79 (2001) 2321±2333
5.3. Anti-symmetric angle-ply laminated plate
Table 6 Non-dimensional fundamental frequency x a=h 10
A simply supported square plate (SS2 conditions: vo vo 0, wo wo 0, hox hox 0 ± for y 0 and y L; uo uo 0, wo wo boz 0, hoy hoy 0 ± for x 0 and x b) with an anti-symmetric lamination scheme, a variable number of layers (n) and various ®bre orientations h are considered for two aspect ratios, a=h 4 and a=h 10. A discretisation in eight strips and eight sections is used. The fundamental frequencies shown in Tables 5±7, are put in a p non-dimensional form, with the multiplier x x a2 =h q=E2 . The results are compared with alternative solutions [60], based on the third order shear deformation theory of Reddy, and FSDT. For both the a=h 4 and a=h 10 ratios one can observe a global good agreement for all the strip models developed. Some discrepancies in the thicker plate can be explained by the fact that the present models do not assume zero shear stresses at the outermost surfaces of the laminate. Also full transformed elastic coecients are used instead of the reduced ones used by the comparing reference. Another reason for the discrepancies lies on the fact that the present HSDT models do not consider the normal transverse displacement w constant through the thickness.
Table 5 Non-dimensional fundamental frequency x a=h 4 HSDT [60]
HSDT-QL
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h
n
HSDT-QC
5°
2 6
8.715 8.859
8.666 8.814
8.672 8.819
30°
2 6
9.446 10.577
8.673 10.238
8.670 10.237
45°
2 6
9.759 10.895
8.846 10.524
8.837 10.518
h
n
HSDT [60]
HSDT-QL
HSDT-QC
5°
2 6
14.230 14.848
14.211 14.828
14.231 14.847
30°
2 6
12.873 18.170
12.551 17.979
12.556 17.977
45°
2 6
13.263 19.025
12.883 18.813
12.878 18.798
Table 7 Non-dimensional fundamental frequency x a=h 10 h
n
FSDT [60]
FSDT-LL
FSDT-LC
5°
2 6
14.179 14.840
13.889 14.486
13.899 14.504
30°
2 6
12.681 18.226
12.856 18.264
12.798 18.215
45°
2 6
13.044 19.025
13.205 19.283
13.118 19.193
5.4. Piezoelectric bimorph beam A cantilever bimorph beam constituted of two polyvinyldene¯uoride layers bonded together and polarised in dierent directions, is analysed as an actuator. The geometric characteristics of the bimorph beam can be observed in Fig. 5 (all dimensions given in meters). In order to compare with the reference results, a one-strip discretisation with ®ve sections is considered. The upper and lower surfaces of the bimorph beam are subjected to a unitary electrical potential (0.5 V= 0.5 V). The de¯ections obtained with the dierent models through the imposition of this potential, are presented in Table 8.
Fig. 5. Bimorph beam.
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Table 8 Nodal de¯ections (10
7
m)
y (mm) HSDT-QL HSDT-QC Analytic [18] FEM [18]
20
40
60
80
100
0.139 0.139 0.138 0.14
0.553 0.553 0.552 0.55
1.24 1.24 1.24 1.24
2.21 2.21 2.21 2.21
3.45 3.45 3.45 3.45
The results are compared with an analytical solution and with an FSDT four-node ®nite element model. The present strip models show a good performance. One advantage of the present strip models is their less expensive characteristics in terms of the total number of system degrees of freedom. For instances, in this case, the strip mesh comprises 208 electromechanical degrees of freedom, while the alternative model mesh has 340. 5.5. Laminated plate with embedded actuators This case considers a cantilever graphite-epoxy laminated plate with the lamination scheme 0°=0°= 45°= 45°S , with embedded PZT actuators in the outermost plies. The thickness of the plies follows the scheme [0:05h=0:2h=0:125h=0:125h]S , where h is the total thickness of the laminate. Each pair of actuators is submitted to equal but opposite electric ®elds of magnitude 500 V/mm. There are no mechanical loads applied. The geometric characteristics of the plate can be observed in Fig. 6, where dimensions are given in meters. The actuators are embedded in the outermost plies, their locations being represented by the rectangular shaded area. The strains through the thickness are calculated at the element Gauss points, and the plots, refer to a point in Section 2, close to the central longitudinal line of the cantilever plate.
Fig. 6. Laminated plate with embedded actuators (dimensions in meters).
The plate was discretised in eight strips with 10 sections each. The comparing results [45] were obtained using a 8 10 mesh model based on Reddy's third order shear deformation theory. The axial strain distribution is shown in Figs. 7 and 8. As one can observe there are slight dierences between the HSDT-QC strip model and the comparing model [45], which are due to the in¯uence of the d33 coecient and to the use of non-reduced transformed elastic coecients, in opposition to the coecients which are used in the comparing formulation. 5.6. Piezolaminated plate A piezolaminated simply supported plate is considered in order to determine its fundamental frequency. The laminate has the lay-up: p=0°=90°=0°=p, where the
Fig. 7. Axial strain 10 6 distribution pro®le, L=h 10.
M.A.R. Loja et al. / Computers and Structures 79 (2001) 2321±2333
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Fig. 8. Axial strain 10 6 distribution pro®le, L=h 4.
Table 9 Non-dimensional fundamental frequency x Model Saravanos et al. [49] (w constant) Saravanos et al. [49] (w variable) FSDT-LL FSDT-LC HSDT-QL HSDT-QC Exact [49]
a=h 4
a=h 50
Closed
Open
Closed
Open
145.323 146.269 146.059 145.645 146.845 146.322 145.339
151.222 151.964 146.125 145.711 146.917 146.488 145.377
236.833 239.628 233.444 231.943 239.110 237.508 245.941
259.173 261.703 233.646 232.144 239.316 237.714 245.942
three internal layers are of graphite-epoxy and the faces laminae are of PZT-4 material. Two dierent electric boundary conditions are analysed: the ®rst one, corresponding to the closed-circuit condition, where the electric potential is forced to be null, and the second situation, where the electric potential remains free, corresponds to the open-circuit condition. The in¯uence of the aspect ratio is also studied, by considering dierent side to thickness relations, corresponding to a thick a=h 4 and a thin plate a=h 50, where a is the side of the plate and h its thickness. In this analysis, and in order to compare with the reference results, one considers a unit density. For both the models developed in this study a discretisation in eight ®nite strips with eight sections each was considered, according to the 8 8 mesh used by Saravanos et al. [49]. The non-dimensional fundamental frequencies obtained with the present strip models are presented in p Table 9, by using the multiplier x xa2 = h q 103 1=2 Hz(kg/m) . They are compared with the exact solutions
based on the piezoelasticity theory, and with the solutions obtained by Saravanos et al. [49], which used a layerwise four-node model with linear interpolating functions. One can conclude a good performance of the models. A comparison between the exact piezoelasticity solutions [49] and the present numerical solutions do not show a signi®cant dierence between the natural frequencies in the two electrical conditions considered.
6. Conclusions A family of B-spline ®nite strip models for the static and free vibration analysis of thin to thick laminated plate structures with surface bonded and embedded piezoelectric laminae or patches was presented. The models developed are based in the FSDT and on the higher order shear deformation theory of Lo et al. [59], having the possibility of choosing among several degree interpolating functions combinations.
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The performance of the present models is analysed by comparing them with some exact and numerical alternative solutions. From the studies carried out, one can conclude from its global good performance, and from its less computationally expensive characteristics, when compared to standard displacement ®nite element models based on the same higher order theory. As demonstrated by other researchers, the B-spline ®nite strip technique has also advantages when compared to the ®nite strip method, based on Fourier series, namely with the ease of modelling arbitrary shaped plates, boundary conditions and loading. Acknowledgements The authors wish to thank the partial ®nancial support given by FCT ± Project PRAXIS/P/EME/12028/ 1998, POCTI/FEDER and Fundacß~ao Calouste Gulbenkian. References [1] Cheung YK, Tham LG. The ®nite strip method. Boca Raton, USA: CRC Press; 1997. [2] Fan SC, Luah MH. New spline ®nite element for plate bending. J Eng Mech 1992;118(6):1065±81. [3] Li WY, Cheung YK, Asce F, Tham LG. Spline ®nite strip analysis of general plates. J Eng Mech 1985;112(1):43±55. [4] Vermeulen AH, Heppler GR. Prediction and avoiding shear locking in beam vibration problems using the Bspline ®eld approximation method. Comput Meth Appl Mech Eng 1998;158:311±27. [5] Wang S. A uni®ed Timoshenko beam B-spline Rayleigh± Ritz method for vibration and buckling analysis of thick and thin beams and plates. Int J Numer Meth Eng 1997; 40:473±91. [6] Leung AYT, Au FTK. Spline ®nite elements for beam and plate. Comput Struct 1990;37(5):717±29. [7] Gagnon P, Gosselin C, Cloutier L. A ®nite strip element for the analysis of variable thickness rectangular thick plates. Comput Struct 1997;63(2):349±62. [8] Uko CEA, Cusens AR. Application of spline ®nite strip analysis to variable depth bridges. Commun Appl Numer Meth 1988;4:273±8. [9] Gupta A, Kiusalaas J, Saraph M. Cubic B-spline for ®nite element analysis of axisymmetric shells. Comput Struct 1991;38(4):463±8. [10] Au FTK, Cheung YK. Isoparametric spline ®nite strip for plane structures. Comput Struct 1993;48(1):23±32. [11] Ng SF, Chen X. Analysis of arbitrary Mindlin plates or bridge decks by spline ®nite strip methods. Comput Struct 1995;54(1):111±8. [12] Tan D, Dawe DJ. General spline ®nite strip analysis for buckling and vibration of prismatic composite laminated plate and shell structures. Composites: Part B 1998;29B: 377±89.
[13] Tan D, Dawe DJ. Buckling and vibration analysis of composite laminated plates and shells using general spline function. Compos Struct 1998;40(1):25±42. [14] Reddy JN. A simple high order theory for laminated composite plates. J Appl Mech 1984;51:745±52. [15] Kong J, Cheung YK. Application of the spline ®nite strip to the analysis of shear-deformable plates. Comput Struct 1993;46(6):985±8. [16] Wang S, Dawe DJ. Buckling of composite shell structures using the spline ®nite strip method. Composites: Part B 1999;30:351±64. [17] Allik H, Hughes TJR. Finite element method for piezoelectric vibration. Int J Numer Meth Eng 1970;2:151±7. [18] Suleman A, Venkayya VB. A simple ®nite element formulation for a laminated composite plate with piezoelectric layers. J Intell Mater Syst Struct 1995;6:776±82. [19] Lin CC, Hsu CY, Huang HN. Finite element analysis on de¯ection control of plates with piezoelectric actuators. Compos Struct 1996;35:423±33. [20] Wang Z, Chen S, Han W. The static shape control for intelligent structures. Finite Elements Anal Des 1997; 26:303±14. [21] Miller SE, Abramovich H. A self-sensing piezolaminated actuator model for shells using a ®rst order shear deformation theory. J Intell Mater Syst Struct 1995;6: 624±38. [22] Varadarajan S, Chandrashekhara K, Agarwal S. Adaptive shape control of laminated composite plates using piezoelectric materials. AIAA J 1998;36(9):1694±8. [23] Detwiler DT, Shen MHH, Venkayya VB. Finite element analysis of laminated composite structures containing distributed actuators and sensors. Finite Elements Anal Des 1995;20:87±100. [24] Heyliger P. Exact solutions for simply supported laminated piezoelectric plates. J Appl Mech 1997;64(2):299±306. [25] Heyliger P, Brooks S. Exact solutions for laminated piezoelectric plates in cylindrical bending. J Appl Mech 1996;63(4):903±10. [26] Ray MC, Rao KM, Samanta B. Exact solution for static analysis of an intelligent structure under cylindrical bending. Comput Struct 1993;47(6):1031±42. [27] Ray MC, Bhattacharya R, Samanta B. Exact solutions for static analysis of intelligent structures. AIAA J 1993; 31(9): 1684±91. [28] Abramovich H. De¯ection control of laminated composite beams with piezoelectric layers ± closed form solution. Compos Struct 1998;43:217±31. [29] Oguamanam DCD, Almeida SFM, Hansen JS. Stress stiening eects in laminated beams with piezoelectric actuators. J Intell Mater Syst Struct 1998;9:137±45. [30] Batra RC, Liang XQ. The vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators. Comput Struct 1997;63(2):203±16. [31] Lam KY, Ng TY. Active control of composite plates with integrated piezoelectric sensors and actuators under various dynamic loading conditions. Smart Mater Struct 1999;8:223±37. [32] Chandrashekhara K, Agarwal AN. Active vibration control of laminated composite plates using piezoelectric devices: a ®nite element approach. J Intell Mater Syst Struct 1993;4:496±508.
M.A.R. Loja et al. / Computers and Structures 79 (2001) 2321±2333 [33] Chen CQ, Wang CM, Shen YP. Finite element approach of vibration control using self-sensing piezoelectric actuators. Comput Struct 1996;60(3):505±12. [34] Ha SK, Keilers C, Chang FK. Finite element analysis of composite structure containing distributed piezoceramics sensors and actuators. AIAA J 1992;30(3):772±80. [35] Crawley EF, Lazarus KB. Induced strain actuation of isotropic and anisotropic plates. AIAA J 1991;29(6):944±51. [36] Crawley EF, Luis J. Use of piezoelectric actuators as elements of intelligent structures. AIAA J 1987;25(10): 1373±85. [37] Hwang WS, Park HC. Finite element modeling of piezoelectric sensors and actuators. AIAA J 1993;31(5):930±7. [38] Hwang WS, Park HC, Hwang W. Vibrational control of a laminated plate with piezoelectric sensor/actuator: ®nite element formulation and modal analysis. J Intell Mater Syst Struct 1993;4:317±29. [39] Lammering R. The application of a ®nite shell element for composites containing piezo-electric polymers in vibration control. Comput Struct 1991;41(5):1101±9. [40] Tzou HS, Tseng CI. Distributed piezoelectric sensor/ actuator design for dynamic measurement/control of distributed parameter systems: a piezoelectric ®nite element approach. J Sound Vib 1990;138(1):17±34. [41] Batra RC, Liang XQ, Yang JS. The vibration of a simply suported rectangular elastic plate due to piezoelectric actuators. Int J Solids Struct 1996;33(11):1597±618. [42] Ray MC, Bhattacharya R, Samanta B. Exact solutions for dynamic analysis of composite plates with distributed piezoelectric layers. Comput Struct 1998;66(6):737±43. [43] Abramovich H, Meyer-Piening HR. Induced vibrations of piezolaminated elastic beams. Compos Struct 1998;43:47±55. [44] Huang D, Sun B. Vibration analysis of laminated composite beams with piezoelectric layers using a higher order theory. In: Adali S, Morozov EV, Verijenko VE, editors. Proceedings of the 2nd International Conference on Composite Science and Technology, Department of Mechanical Engineering, University of Natal, Durban, South Africa, 1998. p. 395±402. [45] Chattopadhyay A, Seeley CE. A higher order theory for modeling composite laminates with induced strain actuators. Composites: Part B 1997;28B:243±52. [46] Franco Correia VM, Aguiar Gomes MA, Suleman A, Mota Sores CM, Mota Sores CA. Modelling and design of adaptive composite structures. Comput Meth Appl Mech Eng 2000;185:325±46.
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[47] Mota Soares CM, Mota Soares CA, Franco Correia VM. Optimal design of piezolaminated structures. Compos Struct 1999;47:625±34. [48] Robbins DH, Reddy JN. Analysis of piezoelectrically actuated beams using a layerwise displacement theory. Comput Struct 1991;41(2):265±79. [49] Saravanos D, Heyliger PR, Hopkins DA. Layerwise mechanics and ®nite element for the dynamic analysis of piezoelectric composite plates. Int J Solids Struct 1997; 34(3):359±78. [50] Heyliger P, Pei KC, Saravanos D. Layerwise mechanics and ®nite element model for laminated piezoelectric shells. AIAA J 1996;34(11):2353±60. [51] Saravanos D, Heyliger PR. Coupled layerwise analysis of composite beams with embedded piezoelectric sensors and actuators. J Intell Mater Syst Struct 1995;6:350±63. [52] Mitchell JA, Reddy JN. A re®ned hybrid plate theory for composite laminates with piezoelectric laminae. Int J Solids Struct 1995;32(16):2345±67. [53] Reddy JN. On laminated composite plates with integrated sensors and actuators. Eng Struct 1999;21:568±93. [54] Donthireddy P, Chandrashekhara K. Modeling and shape control of composite beams with embedded piezoelectric actuators. Compos Struct 1996;35:237±44. [55] Han JH, Lee I. Analysis of composite plates with piezoelectric actuators for vibration control using layerwise displacement theory. Composites: Part B 1998;29B:621±32. [56] Benjeddou A, Trindade MA, Ohayon R. New shear actuated smart structure beam ®nite element. AIAA J 1999;37(3):378±83. [57] Benjeddou A. Advances in piezoelectric ®nite element modelling of adaptive structural elements: a survey. Comput Struct 2000;76:347±63. [58] Gopinathan SV, Varadan VV, Varadan VK. A review and critique of theories for piezoelectric laminates. Smart Mater Struct 2000;9:24±48. [59] Lo KH, Christensen RM, Wu EM. A high-order theory of plate deformation ± Part II: Laminated plates. J Appl Mech 1977;44(4):669±76. [60] Reddy JN. Mechanics of laminated composite platestheory and analysis. Boca Raton, USA: CRC Press; 1997. [61] Prenter PM. Splines and variational methods. USA: Wiley; 1975. [62] Vinson JR, Sierakowski RL. The behavior of structures composed of composite materials. Dordrecht: Martinus Nijho Publishers; 1986.
www.elsevier.com/locate/compstruc
Analysis of piezolaminated plates by the spline ®nite strip method M.A.R. Loja a, J. Infante Barbosa a, C.M. Mota Soares b,*, C.A. Mota Soares b b
a ENIDH ± Escola N autica Infante D. Henrique, Av. Eng.° Bonneville Franco, 2780-572 Pacßo de Arcos, Portugal IDMEC ± Instituto de Engenharia Mec^ anica, Instituto Superior T ecnico, Av. Rovisco Pais, 1049-001 Lisboa Codex, Portugal
Received 18 October 1999; accepted 22 February 2001
Abstract This paper deals with the development of a family of higher order B-spline ®nite strip models applied to the static and free vibration analysis of laminated plates, with arbitrary shape and lay-ups, loading and boundary conditions. The lamination scheme can be such that the embedded and/or surface bonded piezoelectric actuating and sensing layers are included. The structure is discretised in a speci®ed number of strips, and the geometry and displacement components of each strip are represented by interpolating functions that are products of linear or cubic B-spline, and linear or quadratic Lagrange functions along the y and x orthonormal directions. The accuracy and relative performance of the proposed discrete models are compared and discussed among the developed and alternative models. Ó 2001 CivilComp Ltd. and Elsevier Science Ltd. All rights reserved. Keywords: Piezolaminated plates; Composite materials; Piezoelectric materials; B-spline ®nite strip method
1. Introduction Higher order laminated plate theories have been proposed to overcome the limitations of the classical plate theory to accurately predict the structural response of highly anisotropic composite laminates, which exhibit more pronounced transverse shear eects in laminated plates than in the isotropic ones under similar loading conditions. The use of piezolaminated composite materials and structures with sensory and activation capabilities, require the combination of tailor-made mechanical properties of the laminates and the associated smart materials, to change the structural behaviour of advanced structures. The ease of integrating piezoelectric laminae or patches, by means of embedding or simply bonding them to laminated structures, has contributed to a wide range of applications. All these factors
*
Corresponding author. Fax: +351-218-417-915. E-mail address: [email protected] (C.M. Mota Soares).
have led to an increasing demand for the development of ecient computational tools to represent the electromechanical behaviour of these new adaptive structures. During recent years, several researchers have carried out the development of mathematical models and associated computational models to predict the response of laminated structures with actuators/sensors capabilities. However, only a few consider higher order shear deformations theories. No published works have been found in the analysis of laminated structures with piezoelectric laminae or patches, using the B-spline ®nite strip method approach. The use of this type of model can reduce the computational eort, maintaining at the same time, the versatility of the traditional equivalent higher order displacement ®nite element models, for the analysis of piezolaminated plates with arbitrary geometry and loading. Another advantage is the ease of implementation of general shape, boundary conditions and loading, when compared with the better known Fourier series based ®nite strip models [1]. These reasons have provided a strong motivation for the present study.
0045-7949/01/$ - see front matter Ó 2001 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 1 ) 0 0 0 6 5 - 7
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B-spline ®nite strip models without piezoelectric laminae have been developed by some researchers. Fan and Luah [2], presented a nine-node spline ®nite element model for the bending analysis of isotropic thin-plate structures, using the classical Kirchho plate theory. Li et al. [3], also carried out a similar study for arbitrarily shaped plates. Vermeulen and Heppler [4] presented an application of the B-spline approximation method to the Timoshenko isotropic beam, in order to study its locking behaviour. Wang [5] also analysed the shear locking phenomenon on beams and plates, under the ®rst order shear deformation theory (FSDT) assumptions, having extended its study to laminates. In order to ease the use of these functions, Leung and Au [6], carried out boundary transformations associated to the ®rst and last two spline parameters. Gagnon et al. [7] proposed a C0 linear family of ®nite strip elements for the static analysis of isotropic rectangular thick plate of variable thickness using the FSDT. The possibility of varying the thickness of isotropic structures, was also considered by the C1 strip model of Uko and Cusens [8]. The last two papers use cubic B-splines along the strip length. The static analysis of thin axisymmetric isotropic shells using isoparametric cubic B-spline functions was considered by Gupta et al. [9]. Isoparametric strips for the analysis of Mindlin plates were also proposed by Au and Cheung [10] and Ng and Chen [11]. Tan and Dawe [12,13] presented a general spline ®nite strip based on the ®rst order shear deformation plate theory for the buckling and free vibration analysis of prismatic laminated structures. A higher order B-spline ®nite strip approach based on the Reddy [14] third order shear deformation theory, was used for the static analysis of isotropic and cross-ply laminated plates by Kong and Cheung [15]. Wang and Dawe [16] presented buckling stresses for composite laminated prismatic shell structures using the curved ®nite strips models. In the ®eld of the piezoelectric materials and structures, one can refer to several published research works using the traditional ®nite element displacement formulation, some of them being summarised in this paper. A pioneering ®nite element formulation, which includes the piezoelectric or electroelastic eect has been developed by Allik and Hughes [17] for a tetrahedral ®nite element model. A plate ®nite element based on the ®rst order shear deformation plate theory, in which the electrical potential degree of freedom does not present a nodal dependency by remaining constant within each element, was proposed by Suleman and Venkayya [18]. An analogous study for the de¯ection control of plates with piezoelectric actuators was carried out by Lin et al. [19] which used an eight-node isoparametric plate ®nite element.
Wang et al. [20] developed a plate bending element for static shape control, which can have distributed piezoelectric sensors and actuators, and nodal electrical potential degrees of freedom. A self-sensing piezolaminated actuator model for shells, based on the FSDT, was proposed by Miller and Abramovich [21]. Varadarajan et al. [22], discussed the adaptive shape control of laminated composite plates with integrated piezoelectric actuators and position sensors, under quasi-static loads. Detwiler et al. [23] developed a similar study but considering only static loads. Heyliger [24], Heyliger and Brooks [25], Ray et al. [26] and Ray et al. [27] presented exact solutions for the static behaviour of simply supported laminated piezoelectric plates. Closed form solutions based on a FSDT were proposed by Abramovich [28] for the de¯ection control of laminated beams with piezoceramic layers. Oguamanam et al. [29] presented a ®nite element formulation including Von K arm an non-linear strain± displacement relations for the static and dynamic analysis of laminated Timoshenko beams with surface bonded sensors and actuators. The stress stiening effects, induced piezoelectrically are also studied. Batra and Liang [30] used the three-dimensional linear theory of elasticity to analyse the steady-state vibrations of a simply supported rectangular linear elastic laminated plate with embedded PZT layers. Lam and Ng [31], presented theoretical formulations based on the classical plate theory and Navier solutions, for the active control of piezolaminated plates which are submitted to both mechanical and electrical loadings. Chandrashekhara and Agarwal [32] formulated a ®nite element model based on FSDT, for the active vibration control of piezolaminated composites. A similar study was carried out by Chen et al. [33]. Ha et al. [34], presented a ®nite element formulation for modelling the dynamic as well as the static response of laminates with piezoceramics, using a eight-node three-dimensional composite brick element. Crawley and Lazarus [35], developed and veri®ed experimentally the induced strain actuation of plate components of a smart structure. Crawley and Luis [36], proposed an analytical and experimental set-up of piezoelectric actuators as elements of intelligent structures. Hwang and Park [37], Hwang et al. [38], proposed a ®nite element formulation based on classical laminate theory, for the vibration control of piezolaminated plates. The element developed was a four-node quadrilateral plate bending element with 12 mechanical elastic generalised displacements degrees of freedom and one electrical degree of freedom. The development of a piezolaminated shell ®nite element, for the vibration control of composite shells containing bonded piezoelectric layers, was carried out by Lammering [39]. Another approach to shell-type structures was carried out by Tzou and Tseng [40], using
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a piezoelectric solid element with internal degrees of freedom for dynamic measurement and active vibration control. Batra et al. [41] developed exact solutions for the forced vibration analysis of a simply supported rectangular elastic plate having piezoelectric actuators bonded to its top and bottom surfaces. A similar study for the dynamic analysis of piezolaminated plates, was also carried out by Bhattacharya and Samantha [42], and for beams by Abramovich and Meyer-Piening [43], through the method of Fourier series. Huang and Sun [44] developed a model, based on the third order shear deformation theory of Reddy [14], for the vibration analysis of composite beams with piezoelectric layers. A higher order theory for modelling composite laminates with induced strain actuators was developed by Chattopadhyay and Seeley [45], in order to analyse surface bonded or laminate embedded smart materials. Another higher order theory based study is presented by Franco Correia et al. [46], where static and free vibrations analysis of piezolaminated plates is carried out, using a family of nine-node higher order plate ®nite element models. Mota Soares et al. [47] applied these HSDT models to sensitivity analysis and structural optimisation to maximise the piezoelectric actuators eciency, and to improve the structural performance and/ or minimise the weight of the structure. Robbins and Reddy [48] presented a study about the static and dynamic interaction between a bonded actuator and the underlying beam structure, by using a layerwise displacement theory. Finite element models were developed. Saravanos et al. [49], and Heyliger et al. [50] developed ®nite element formulations for the analysis of piezolaminated plates and thin shells respectively. Saravanos and Heyliger [51] also developed discrete layer formulations for the study of the coupled electromechanical response of composite beams with embedded sensory and active piezoelectric layers. An hybrid theory, in which the mechanical displacement ®eld is modelled according to an equivalent single-layer theory and the potential function is represented through a layerwise discretisation in the thickness direction, was carried out by Mitchell and Reddy [52]. In Reddy [53] a detailed theoretical formulation for Navier solution and ®nite element models based on the classical and shear deformation plate theories are presented for the analysis of composite plates with integrated sensors and actuators. A negative velocity feedback control algorithm coupling the direct and converse piezoelectric eects is discussed to control the dynamic response of an integrated structure. A discrete layer model and the associated ®nite element formulation, was developed by Douthireddy and Chandrashekhara [54], in order to achieve the shape
2323
control of laminated composite beams with piezoelectric actuators. Han and Lee [55] presented a layerwise formulation for the vibration control of composite plates with piezoelectric actuators, in which the in-plane displacements through the thickness are modelled using a discrete layer approach. An adaptive sandwich-beam ®nite element capable of dealing with either extension or shear actuated mechanisms, was developed by Benjeddou et al. [56]. Very recently, comprehensive surveys by Benjeddou [57] and Gopinathan et al. [58], discuss the advances, trends and applications of the ®nite element modelling of adaptive structural elements. The objective of the present work is to present, a family of B-spline higher order ®nite strip models for the static and free vibrations analysis of laminated plates with embedded or surface bonded piezoelectric laminae or patches. In order to achieve this objective, a package of six strip models were developed, being four of them based in the ®rst order theory (FSDT) and the remaining two, based on Lo et al. [59] higher order displacement ®eld (HSDT). The dierences among the same theory based models, lies on the number of nodal lines of the strip and also on the degree of the longitudinal interpolating functions. The FSDT models have the option of choosing between two or three nodal lines. For the HSDT models only three nodal lines are assumed. In all the cases piecewise linear or cubic B-spline functions are used along the y-direction. The performance of some of the models that constitute this package will be shown through the illustrative cases presented.
2. Displacement and strain ®elds The displacement ®eld considered is based on the Lo et al. [59] higher order shear deformation theory (Eq. (1)), in which the in-plane and the transverse displacements are respectively cubic and quadratic functions of the thickness co-ordinate (Fig. 1), as follows: u Zq; T
u x; y; z; t u x; y; z; t v x; y; z; t w x; y; z; t ; h q uo vo wo hox hoy boz uo vo wo hox 2 3 1 0 0 z 0 0 z2 0 0 z3 0 6 7 Z 4 0 1 0 0 z 0 0 z2 0 0 z3 5 0 0 1 0 0 z 0 0 z2 0 0
hoy
iT
;
1
In Eq. (1), uo , vo , wo represent the plate mid-plane displacements along the co-ordinate lines x; y; z, and
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eT E
r Qe
4
D ee pE
where r is the stress vector, e is the strain vector, D is the electrical displacements vector, E is the applied electric ®eld vector, Q is the transformed elastic stiness coecients matrix (which can be subdivided in MB S the submatrices Q and Q according to Eq. (6)), whose coecients are explicitly given in Reddy [60] for the system reference co-ordinates, e is the transformed piezoelectric stress coecients matrix and p is the permittivity matrix.
Fig. 1. Generic B-spline strip element.
hox , hoy are the rotations of the mid-plane perpendiculars about the x and y axes respectively (Fig. 1). The remaining parameters constitute higher order deformation modes, obtained through the expansion of the displacement ®eld in Taylor series referring to the mid-plane. Through the adequate simpli®cations of this displacement ®eld, one can successively obtain the ®eld corresponding to the FSDT. Considering the displacement ®eld given in Eq. (1), and the kinematic relations from the elasticity for small deformations, one has for the extension and curvature vectors and for the transverse shear strain vector, the following expressions:
e ZMB eo ; eo eox eoy 2 1 0 60 1 6 ZMB 6 40 0
eoy
0 z2 0 0
0 z2
0 0
0 0
eoz
1 0 0 0
eox
0 0
cxy
kx
ky
kz
kx
ky
0 0
z 0 0 z
0 0
z3 0
0 z3
0 0
0 0 1 z2
0 0 0 0
2z 0
0 0
0 0
0 z
cxy
/ ZS /o ; /o coyz cozx coyz cozx coyz 1 0 z 0 z2 0 ZS 0 1 0 z 0 z2
o
czx
T
5
" Q
MB
Q 0
0 S Q
# 6
Assuming the possibility of the piezoelectric plies to be polarised only in the thickness direction, one can write: E
ru 0
0 Ez T
7
and if one considers a kth ply, we will have Ezk uk =tk where Ezk is the electrical ®eld in the thickness direction for a given laminae k, uk is the electric potential and tk is
kxy 3 0 07 7 7 05
kxy ; 2
z3
the kth layer thickness. The transformed piezoelectric stress coecient matrix for the same generic layer, in the system co-ordinates, is given as:
; 3
3. Constitutive relations By considering the existence in a generic structure, of both orthotropic and piezoelectric laminae or patches, one has for the kth layer, the following constitutive relation, accounting for the piezoelectric eect:
2 0 0 6 0 6 eTk 6 0 0 0 4 e31 e32 e33 h i e^TkMB e^TkS
where
0 0 e36
.. . .. . .. .
3 e14 e24 0
e15 7 7 e25 7 5 0 8
M.A.R. Loja et al. / Computers and Structures 79 (2001) 2321±2333
e31 e31 cos2 h e32 sin2 h; 2
given as the product of B-spline functions and Lagrangean polynomials according to:
2
e32 e31 sin h e32 cos h; e33 e33 ; e36 e31 e14 e15
Nij x; y Li x/j y:
e32 sin h cos h; e24 sin h cos h;
9
e15 e15 cos2 h e24 sin2 h; e24 e24 cos2 h e15 sin2 h; e25 e15 e24 sin h cos h in which, the eij are the piezoelectric stress coecients in the layer material axes, and h (positive counterclockwise) is the angle between the x1 -direction and the xdirection of the reference system for a generic layer (the x1 -direction coincides with the ®bre direction in the x1 x2 z material co-ordinate system).
A generic function U, say a generalised displacement, can be represented as a linear combination of local Bspline functions [61] /j and parameters aj for the jth node, according to the expression: ju X jjl
/j aj
10
The summation limits of Eq. (10), vary according to the B-spline function degree. In the case of linear functions one has jl 0, ju NS, and for the cubic case, jl 1, ju NS 1 (NS ± number of sections). Depending on the degree of the x-direction Co Lagrange polynomials, a strip will have two or more nodal lines. In the present study, the strip models are linear and quadratic, having an arbitrary number of longitudinal nodes. According to this representation, the plate structure is divided into S strips along the y-direction, each of which is subdivided into NS sections. The strip elements developed here can model plate type structures with other shapes than the rectangular one. The generalised displacements within a strip can then be represented as follows: uo
N X M X i1
j1
Li /j uoij . . . hy
N X M X i1
j1
o
Li /j hyij
11
yielding ue ZN qe
12
where qe is the nodal generalised displacement parameter vector and N is the shape function matrix, which is
13
The local B-spline linear and cubic functions /j , which allow for uneven spacing between the nodes, are explicitly given in Ref. [61]. According to the strain ®eld (Eqs. (2) and (3)), the strain matrices concerning to the membrane/bending and shear deformations are then given by: ee ZMB BMB qe ;
/e ZS BS qe
14
where BMB and BS are the shape functions derivative matrices which relate the element degrees of freedom to the extensions, curvatures and shear strains. The electric ®eld vector, can be represented for an arbitrary section within a strip, by: E
4. B-spline ®nite strips
U y
2325
Bu uSec
15
where uSec is the electric potential vector for a section and Bu is the electric potential matrix, which relates the electrical ®eld vector to the voltages, as follows: 2 3 1=t1 0 6 .. .. 7 .. Bu 4 . 16 . 5 . 0
1=tNP
where NP is the total number of piezoelectric section plies. The principle of virtual work, applied to an element of volume Ve , leads to [17]: Z Z Z n deT Qe deT eE dET eT e dET pE Ve o u dVe dWu dW u 0 17 qduT where dWu and dWu are the virtual work done by the external mechanical loading and the applied surface electrical charge. By making the appropriate substitutions and simpli®cations in Eqs. (1)±(3), (6)±(8) and (17) for the whole discretised piezolaminated plate domain, yields: Fu t Kuu Kuu Muu 0 u u Kuu Kuu Fu t u 0 0 u 18 where Muu and Kuu are the mass and elastic stiness matrices, Kuu is the dielectric `stiness' matrix, and T Kuu Kuu are the elasto-electric coupling eects matrix. The vectors u and u are the generalised electromechanical response of the structure. Fu t and Fu t are the generalised electromechanical load vectors. For free harmonic vibrations, Eq. (18) is rewritten as:
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M.A.R. Loja et al. / Computers and Structures 79 (2001) 2321±2333
x2n Muu u Kuu u 0;
Kuu
Kuu u Kuu u 0 19
where xn is the natural frequency associated to the mode n. After condensing the electrical degrees of freedom (Eq. (19)), yields: x2n Muu u 0
K
K Kuu
21
Eq. (18) can be simpli®ed for static analysis. When a known voltage is applied to the actuators, an additional load vector due to the converse piezoelectric eect, is given by: Fa Kuu u
22
yielding: Fa
23
The element matrices derived from Eq. (17), which after assembled constitute the whole system matrices are explicitly given by: "
#
Z Z (
BMB 0 T 0 Bu A " # NL Z hk X ZMB 0 T QMB e^MB k 0 1 hk 1 e^TMB p k1 !) BMB 0 ZMB 0 dx dy dz 0 Bu 0 1 ! ) Z Z ( NL Z hk X S BTS ZST Qk ZS dz BS dx dy
e Kuu e Kuu
e Kuu e Kuu
A
e Muu
MB
k1
Z Z (
N
A
T
NL X k1
S
Section initial node
Section ®nal node
/12 /12 2 /1 /21 /30 /30 4 / 10
/12 0 /21 /21 4/12 3 /0 /30 /12 /4 1 /4 1
4
/ 1 4/4 1
20
Kuu Kuu1 Kuu :
Kuu u Fu
Table 1 Modi®ed local cubic B-spline function
hk
qk
5. Numerical applications 24
1
Z
hk hk
! ) T
Z Z dz N dx dy
for a parabolic transverse shear stress distribution across the laminate, adequate reduced transformed elastic coecients are used, which are explicitly given by Vinson and Sierakowski [62]. Simpli®cations are also extended to the piezoelectric constants eij , only being required the e31 , e32 and e36 coecients, under the same assumption of polarisation through the thickness. The constraints associated with existing boundary conditions, are represented by carrying out, for all the models, the adequate modi®cations to the local B-spline functions. These constraints which can be, associated to a generic degree of freedom are imposed by making the modi®cations presented in Table 1, for the cubic Bspline case. For simplicity, the integration of the elementary matrices is carried out numerically in both in-plane directions, and analytically in the thickness direction. To ease the whole procedure, the integration is carried out section by section, after the transformation of the initial functions given in system co-ordinates into a natural coordinates system, to use the Gauss quadrature methodology.
25
1
Qk and Qk are the membrane and bending, and transverse shear terms of the constitutive matrix for the kth laminae. qk and hk are respectively the density and the vector distance from the laminate mid-plane and the outer surface of the kth ply. NL is the number of plies in the laminate. Matrix I is an identity matrix, which dimension depends on the number of piezoelectric plies within each section in a strip. When considering the FSDT strip models, some simpli®cations are carried out. One is related to the displacement ®eld. Another is associated to the elastic constants Qij . In fact, in such conditions, reduced transformed elastic coecients [60] are used. To account
In this section, several test cases are analysed in order to show the performance of the ®nite strip models developed. The following type of abbreviated designations are used for referring to the dierent models: FSDT-XY: Strip using Xth degree Lagrange functions on the x-direction, and Yth degree B-spline functions along the y-direction. These ®rst order strip models have ®ve degrees of freedom per node. HSDT-XY: Strip using Xth degree Lagrange functions on the x-direction, and Yth degree B-spline functions along the y-direction. These higher order strip models have 11 degrees of freedom per node. The elastic and piezoelectric materials properties are given in Tables 2 and 3 for all examples but one (example 5.1), where units are not given in the comparing reference.
M.A.R. Loja et al. / Computers and Structures 79 (2001) 2321±2333
2327
Table 2 Elastic properties Examples E1 (GPa) E2 (GPa) E3 (GPa) G23 (GPa) G13 (GPa) G12 (GPa) m23 m13 m12
5.2
5.3
5.4
5.5 Gr-Ep
5.5 PZT
5.6 Gr-Ep
5.6 PZT-4
145.0 10.0 10.0 4.8 4.8 4.8 0.25 0.25 0.25
40.0 1.0 1.0 0.5 0.5 0.6 0.25 0.25 0.25
2.0 2.0 2.0 1.0 1.0 1.0 0.0 0.0 0.0
150.0 9.0 9.0 2.5 7.1 7.1 0.3 0.3 0.3
63.0 63.0 63.0 24.2 24.2 24.2 0.3 0.3 0.3
132.38 10.76 10.76 3.61 5.65 5.65 0.49 0.24 0.24
81.3 81.3 64.5 25.6 25.6 30.6 0.43 0.43 0.33
Upper curve
Table 3 Piezoelectric properties
y 0:225b3 0:225b2 2:975b 2:975
Examples 5.4 12
d31 (10 m/V) d32 (10 12 m/V) d33 (10 12 m/V) e31 (C/m2 ) e32 (C/m2 ) e33 (C/m2 ) e33 (10 10 F/m)
± ± ± 0.046 0.046 0.0 1.062
5.5 PZT
± ± ± ±
166.0 166.0 360.0
5.6 PZT-4
± ± ±
122.0 122.0 285.0
x 0:2475b3
5.1. S-shaped isotropic plate The ®rst case considered is a s-shaped two-span isotropic slab bridge, which can be observed in Fig. 2. According to Au and Cheung [10], the upper and lower curves which limit the slab, are expressed by the following equations:
0:89b 2:6319
Lower curve y 3b 3 x 0:9b3
115.05
0:8494b2
0:9b2
1:3b 0:5
where b is a parameter which varies within [ 1. . .1]. The slab is supported along three straight lines de®ned in the same ®gure. According to Ref. [10], the values corresponding to the material properties are: E 104 (Young's modulus), m 0:15 (Poisson's ratio), h 0:2 (thickness) ± (no units were given). A unitary concentrated load is applied at the point, with co-ordinates 4:523; 0:872, located in the central line of the structure.
Fig. 2. S-shaped two-span continuous slab bridge.
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M.A.R. Loja et al. / Computers and Structures 79 (2001) 2321±2333
Fig. 3. Central line de¯ections of the s-shaped slab.
Au and Cheung [10], used an isotropic B-spline strip FSDT model, and considered a discretisation in two quadratic ®nite strips with 12 sections each. From Fig. 3 one can observe the ®tted curves, referred to the de¯ection pro®le of the centre line of the sshaped plate, from both the present higher order models, and the comparing solution. It is visible a good agreement between the present strip models, and the reference results. 5.2. Cantilever skew plate A cantilever skew orthotropic plate, with four unidirectional 0°=0°=0°=0° layers of graphite-epoxy, with a thickness of 7.5 mm each, is analysed. The plate geometric properties can be observed in Fig. 4, where dimensions are given in meters. The plate is submitted to a uniform transverse pressure loading of 5000 Pa.
Fig. 4. Skew cantilever orthotropic plate (dimensions in meters).
Table 4 Nodal transverse displacements (mm) x (m)
HSDTQL
HSDTQC
Q9FSDT5
Q9HSDT11
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
0.217 0.230 0.244 0.258 0.272 0.286 0.299 0.313 0.326
0.217 0.230 0.244 0.258 0.272 0.285 0.299 0.312 0.325
0.217 0.231 0.245 0.259 0.272 0.286 0.300 0.313 0.326
0.216 0.229 0.243 0.257 0.270 0.284 0.297 0.311 0.323
In order to compare with an available numerical higher order solution, a discretisation of four strips with eight sections each, parallel to the inclined edges of the plate, is considered. The nodal transverse displacements are given in Table 4. These results are compared with two available numerical solutions using a 4 4 mesh of nine-node quadratic elements. Full details of the models are given in Ref. [46]. These alternative plate ®nite element models, have ®ve (Q9-FSDT5), and 11 degrees of freedom per node (Q9-HSDT11). The Q9-HSDT11 uses the same displacement ®eld as Eq. (1). From Table 4, one can conclude from the good agreement between the present higher order models and the comparing alternative models. It is also possible to conclude from the computational less expensive behaviour of the present higher order strip models when compared to the alternative higher order plate ®nite element model. For the present case, the relation between the number of degrees of freedom required by the higher order strip models and the alternative higher order model is 0.4 (792/1936).
M.A.R. Loja et al. / Computers and Structures 79 (2001) 2321±2333
5.3. Anti-symmetric angle-ply laminated plate
Table 6 Non-dimensional fundamental frequency x a=h 10
A simply supported square plate (SS2 conditions: vo vo 0, wo wo 0, hox hox 0 ± for y 0 and y L; uo uo 0, wo wo boz 0, hoy hoy 0 ± for x 0 and x b) with an anti-symmetric lamination scheme, a variable number of layers (n) and various ®bre orientations h are considered for two aspect ratios, a=h 4 and a=h 10. A discretisation in eight strips and eight sections is used. The fundamental frequencies shown in Tables 5±7, are put in a p non-dimensional form, with the multiplier x x a2 =h q=E2 . The results are compared with alternative solutions [60], based on the third order shear deformation theory of Reddy, and FSDT. For both the a=h 4 and a=h 10 ratios one can observe a global good agreement for all the strip models developed. Some discrepancies in the thicker plate can be explained by the fact that the present models do not assume zero shear stresses at the outermost surfaces of the laminate. Also full transformed elastic coecients are used instead of the reduced ones used by the comparing reference. Another reason for the discrepancies lies on the fact that the present HSDT models do not consider the normal transverse displacement w constant through the thickness.
Table 5 Non-dimensional fundamental frequency x a=h 4 HSDT [60]
HSDT-QL
2329
h
n
HSDT-QC
5°
2 6
8.715 8.859
8.666 8.814
8.672 8.819
30°
2 6
9.446 10.577
8.673 10.238
8.670 10.237
45°
2 6
9.759 10.895
8.846 10.524
8.837 10.518
h
n
HSDT [60]
HSDT-QL
HSDT-QC
5°
2 6
14.230 14.848
14.211 14.828
14.231 14.847
30°
2 6
12.873 18.170
12.551 17.979
12.556 17.977
45°
2 6
13.263 19.025
12.883 18.813
12.878 18.798
Table 7 Non-dimensional fundamental frequency x a=h 10 h
n
FSDT [60]
FSDT-LL
FSDT-LC
5°
2 6
14.179 14.840
13.889 14.486
13.899 14.504
30°
2 6
12.681 18.226
12.856 18.264
12.798 18.215
45°
2 6
13.044 19.025
13.205 19.283
13.118 19.193
5.4. Piezoelectric bimorph beam A cantilever bimorph beam constituted of two polyvinyldene¯uoride layers bonded together and polarised in dierent directions, is analysed as an actuator. The geometric characteristics of the bimorph beam can be observed in Fig. 5 (all dimensions given in meters). In order to compare with the reference results, a one-strip discretisation with ®ve sections is considered. The upper and lower surfaces of the bimorph beam are subjected to a unitary electrical potential (0.5 V= 0.5 V). The de¯ections obtained with the dierent models through the imposition of this potential, are presented in Table 8.
Fig. 5. Bimorph beam.
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Table 8 Nodal de¯ections (10
7
m)
y (mm) HSDT-QL HSDT-QC Analytic [18] FEM [18]
20
40
60
80
100
0.139 0.139 0.138 0.14
0.553 0.553 0.552 0.55
1.24 1.24 1.24 1.24
2.21 2.21 2.21 2.21
3.45 3.45 3.45 3.45
The results are compared with an analytical solution and with an FSDT four-node ®nite element model. The present strip models show a good performance. One advantage of the present strip models is their less expensive characteristics in terms of the total number of system degrees of freedom. For instances, in this case, the strip mesh comprises 208 electromechanical degrees of freedom, while the alternative model mesh has 340. 5.5. Laminated plate with embedded actuators This case considers a cantilever graphite-epoxy laminated plate with the lamination scheme 0°=0°= 45°= 45°S , with embedded PZT actuators in the outermost plies. The thickness of the plies follows the scheme [0:05h=0:2h=0:125h=0:125h]S , where h is the total thickness of the laminate. Each pair of actuators is submitted to equal but opposite electric ®elds of magnitude 500 V/mm. There are no mechanical loads applied. The geometric characteristics of the plate can be observed in Fig. 6, where dimensions are given in meters. The actuators are embedded in the outermost plies, their locations being represented by the rectangular shaded area. The strains through the thickness are calculated at the element Gauss points, and the plots, refer to a point in Section 2, close to the central longitudinal line of the cantilever plate.
Fig. 6. Laminated plate with embedded actuators (dimensions in meters).
The plate was discretised in eight strips with 10 sections each. The comparing results [45] were obtained using a 8 10 mesh model based on Reddy's third order shear deformation theory. The axial strain distribution is shown in Figs. 7 and 8. As one can observe there are slight dierences between the HSDT-QC strip model and the comparing model [45], which are due to the in¯uence of the d33 coecient and to the use of non-reduced transformed elastic coecients, in opposition to the coecients which are used in the comparing formulation. 5.6. Piezolaminated plate A piezolaminated simply supported plate is considered in order to determine its fundamental frequency. The laminate has the lay-up: p=0°=90°=0°=p, where the
Fig. 7. Axial strain 10 6 distribution pro®le, L=h 10.
M.A.R. Loja et al. / Computers and Structures 79 (2001) 2321±2333
2331
Fig. 8. Axial strain 10 6 distribution pro®le, L=h 4.
Table 9 Non-dimensional fundamental frequency x Model Saravanos et al. [49] (w constant) Saravanos et al. [49] (w variable) FSDT-LL FSDT-LC HSDT-QL HSDT-QC Exact [49]
a=h 4
a=h 50
Closed
Open
Closed
Open
145.323 146.269 146.059 145.645 146.845 146.322 145.339
151.222 151.964 146.125 145.711 146.917 146.488 145.377
236.833 239.628 233.444 231.943 239.110 237.508 245.941
259.173 261.703 233.646 232.144 239.316 237.714 245.942
three internal layers are of graphite-epoxy and the faces laminae are of PZT-4 material. Two dierent electric boundary conditions are analysed: the ®rst one, corresponding to the closed-circuit condition, where the electric potential is forced to be null, and the second situation, where the electric potential remains free, corresponds to the open-circuit condition. The in¯uence of the aspect ratio is also studied, by considering dierent side to thickness relations, corresponding to a thick a=h 4 and a thin plate a=h 50, where a is the side of the plate and h its thickness. In this analysis, and in order to compare with the reference results, one considers a unit density. For both the models developed in this study a discretisation in eight ®nite strips with eight sections each was considered, according to the 8 8 mesh used by Saravanos et al. [49]. The non-dimensional fundamental frequencies obtained with the present strip models are presented in p Table 9, by using the multiplier x xa2 = h q 103 1=2 Hz(kg/m) . They are compared with the exact solutions
based on the piezoelasticity theory, and with the solutions obtained by Saravanos et al. [49], which used a layerwise four-node model with linear interpolating functions. One can conclude a good performance of the models. A comparison between the exact piezoelasticity solutions [49] and the present numerical solutions do not show a signi®cant dierence between the natural frequencies in the two electrical conditions considered.
6. Conclusions A family of B-spline ®nite strip models for the static and free vibration analysis of thin to thick laminated plate structures with surface bonded and embedded piezoelectric laminae or patches was presented. The models developed are based in the FSDT and on the higher order shear deformation theory of Lo et al. [59], having the possibility of choosing among several degree interpolating functions combinations.
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