Open access peer-reviewed chapter - ONLINE FIRST

# Static and Dynamic Analysis of Piezoelectric Laminated Composite Beams and Plates

By Chung Nguyen Thai, Thinh Tran Ich and Thuy Le Xuan

Submitted: April 11th 2019Reviewed: August 22nd 2019Published: April 29th 2020

DOI: 10.5772/intechopen.89303

## Abstract

In this chapter, the mechanical behavior analysis of piezoelectric laminated composite beams and plates is influenced subjected to static, dynamic, and aerodynamic loads. Algorithm for dynamic, stability problem analysis and vibration control of laminated composite beams and plates with piezoelectric layers is presented. In addition, numerical calculations, considering the effect of factors on static, dynamic, and stability response of piezoelectric laminated composite beams and plates are also clearly presented. The content of this chapter can equip readers with the knowledge used to calculate the static, dynamic, and vibration control of composite beams, panels made of piezoelectric layers applied in the field different techniques.

### Keywords

• beams
• plates
• static
• dynamic
• piezoelectric
• composite
• stiffened

## 1. Introduction

The content of this chapter is the inheritance and development of the research results of the authors and other authors by published scientific works on composite materials, piezoelectric and structural calculation by piezoelectric composite materials.

## 2. Electromechanical interaction of piezoelectric materials

### 2.1 Mechanical-electrical behavior relations

Let us consider a block of elastic material in an environment with an electric field of zero, the relationship between stress and strain is followed Hooke’s law, and written as follows [1, 2]:

σ=cε,E1

where {σ} is the mechanical stress vector, {ε} is the mechanical strain vector, and [c] is the material stiffness matrix of beam.

Mechanical-electrical relations in piezoelectric materials have an interactive relationship, strain {ε} will produce eε - polarization, where e is the voltage stress factor when there is no mechanical strain. The imposed electric field E produces the -eE stress in the piezoelectric material according to the reverse voltage effect. Therefore, we have a mathematical model that describes the mechanical-electrical interaction relationship in piezoelectric materials as follows [3, 4, 5, 6]:

σ=cεeE,E2
D=eTε+pE,E3
orD=dTσ+pE,E4

where [e] is the piezoelectric stress coefficient matrix, [p] is the dielectric constant matrix, {E} is the vector of applied electric field (V/m), and {D} is the vector of electric displacement (C/m2).

For the linear problem and small strain, strain vector in the piezoelectric structures can be defined as follows:

ε=sσ+dE,E5

in which [s] is the matrix of compliance coefficients (m2/N), [d] is the matrix of piezoelectric strain constants (m/V).

In the field of engineering, piezoelectric materials are used by two types. The first type, the piezoelectric layers or the piezoelectric patches act as actuators, called the piezoelectric actuators. In this case, the piezoelectric layers are strained when imposing an electric field on it. The second type, the piezoelectric layers or piezoelectric patches act as sensors, called piezoelectric sensors. In this case, the voltage is generated in piezoelectric layers when there is mechanical strain.

### 2.2 Piezoelectric actuators and sensors

#### 2.2.1 Piezoelectric actuators

Eq. (5) can be written in the matrix form as follows [4, 6]:

ε=ε11ε22ε33γ23γ13γ12=s11s12s13s14s15s16s21s22s23s24s25s26s31s32s33s34s35s36s41s42s43s44s45s46s51s52s53s54s55s56s61s62s63s64s65s66σ11σ22σ33τ23τ13τ12+d11d21d31d12d22d32d13d23d33d14d24d34d15d25d35d16d26d36E1E2E3,E6

Assuming that the device is pulled along the axis 3, and viewing the piezoelectric material as a transversely isotropic material, which is true for piezoelectric ceramics, many of the parameters in the above matrices will be either zero, or can be expressed through each other. In particular, the non-zero compliance coefficients are s11, s12, s13, s21, s22, s23, s31, s32, s33, s44, s55, s66, in which s12 = s21, s13 = s31, s23 = s32, s44 = s55, s66 = 2(s11 − s12).

Finally, Eq. (6) becomes:

ε=ε11ε22ε33γ23γ13γ12=s11s12s13000s12s22s23000s13s23s33000000s44000000s55000000s66σ11σ22σ33τ23τ13τ12+00d3100d3200d330d240d1500000E1E2E3,E7

where E1, E2, and E3 are electric fields in the 1, 2, and 3 directions, respectively.

#### 2.2.2 Piezoelectric sensors

The induction charge equation of piezoelectric sensor layers is derived from Eq. (4) can be written in the matrix form as [4, 6, 7]:

D=D1D2D3=d11d12d13d14d15d16d21d22d23d24d25d26d31d32d33d34d35d36σ11σ22σ33τ23τ13τ12+p11p12p13p21p22p23p31p32p33E1E2E3,E8

The non-zero piezoelectric strain constants are d31, d32, d15, d24, and d33, in which d31 = d32, d15 = d24. And the non-zero dielectric coefficients are p11, p22, and p33, where p11 = p22. Eq. (8) becomes:

D=D1D2D3=0000d150000d2400d31d32d33000σ11σ22σ33τ23τ13τ12+p11000p22000p33E1E2E3,E9

where D1, D2, D3, p11, p22, and p33 are the displacement charge, dielectric constant in the 1, 2, and 3 directions, respectively.

Normally, the voltage is transmitted through the thickness of the actuator layers.

## 3. Static and dynamic analysis of laminated composite beams with piezoelectric layers

### 3.1 Displacement and strain

Based on the first-order shear deformation theory (FSDT), the displacement field at any point of the beam is defined as [1, 2]:

uxz=u0x+zθyx,wxz=w0x,E10

where u, w denotes the displacements of a point (x, z) in the beam; u0, w0 are the displacements of a point at the beam neutral axis, and θy is the rotation of the transverse normal about the y axis. The bending and shear strains associated with the displacement field in Eq. (10) are defined as:

ε=εxγxz=dudxdudz+dwdx=du0dx+zzdxθy+dw0dx=ddx0zddx0ddx1u0w0θz,E11

in which εx, γxz are the normal strain, and shear strain, respectively.

Using finite element method, we consider 2-node bending elements with 3 degrees of freedom per node (Figure 1).

The displacements of the beam neutral axis are expressed in local coordinate system in the form:

d0=u0v0θz=NuquNvqvNθzqθz=NMqbe,E12

where {qb}e is the vector of vector of nodal displacements of element, [NM] is the matrix mechanical shape functions:

qbe=q1q2q3q4q5q6T,E13
NM3×6=Nu000Nv000Nθz,E14

in which [Nu], [Nv], [Nθz] are, in this order, the row vectors of longitudinal, transverse along y, and rotation about z shape functions.

Substituting Eq. (12) into Eq. (11), we obtain:

ε2×1=Bb2×6qbe6×1,E15
whereBb=ddx0zddx0ddx1NM.E16

The electric potential is constant over the element surface:

ϕk=i=1nNiϕi,E17

where n is the element node number.

A voltage ϕ is applied across an actuator of layer thickness tp generates an electric field vector {E}, such that [4, 8, 9, 10]:

Ek=ϕk=00Ekz,E18

in which

Ekz=ϕktpk=Bϕϕ=001tp1000000001tp2Tϕ1ϕ2,E19

where tpk is the thickness of the kth piezoelectric layer.

Substituting Eq. (19) into Eq. (18), the electric field vector {E} can also be defined in terms of nodal variables as:

E=Bϕϕe,E20

Using Eqs. (15), and (20), the linear piezoelectric constitutive equations coupling the elastic and electric fields will be completely determined by Eqs. (2) and (3).

### 3.2 Finite element equations

Using Hamilton’s principle, we have [11, 12, 13]:

t1t2TeUeWedt=0,E21

where Te, Ue are the kinetic and potential energy, respectively and We is the work done by external forces. They are determined by:

Te=12Veρq̇eTqedV,E22
Ue=12VeεeTσedV,E23
We=VeqeTfbedV+SeqeTfsedS+qeTfce,E24

in which fbe,fse,fceare the body, surface, and concentrated forces acting on the element, respectively. Ve and Se are elemental volume and area.

Substituting Eqs. (15), (2), (20), (22), (23), and (24) into Eq. (21), one obtains:

Mbbeq¨e+Kbbeqe+Keϕe=fe,E25
KϕbeqeKϕϕeϕe=Qe,E26

where

Element mass matrix:Mbbe=VeρNMTNMdV,E27
Element mechanical stiffness matrix:Kbbe=SeBbTHBbdS,E28

Element mechanical-electrical coupling stiffness matrix:

Ke=SeBbTe¯BϕdS,E29

Element electrical-mechanical coupling stiffness matrix:

Kϕbe=KeT,E30

Element piezoelectric permittivity matrix:

Kϕϕe=SeBϕTp¯BϕdS,E31
whereH=c1100c22,e¯=e11e12e21e22,p¯=tp1p1100tp2p22,E32

{f}e, {Q}e are the applied external load and charge, respectively.

#### 3.2.1 Static analysis

In the case of beams subjected to static loads, zero acceleration, from Eqs. (25) and (26), we obtain the static equations of the beam as follows:

Kbbeqe+Keϕe=fe,E33
KϕbeqeKϕϕeϕe=Qe,E34

Assembling the element equations yields general static equation:

Kbbq+Kϕ=f,E35
KϕbqKϕϕϕ=Q.E36

where [Kbb], [Kϕϕ] are the overall mechanical stiffness and piezoelectric permittivity matrices respectively; [K] and [Kϕb] are the overall mechanical - electrical and electrical - mechanical coupling stiffness matrices, respectively, and {q}, {ϕ} are respectively the overall mechanical displacement, and electric potential vector.

Substituting Eq. (36) into Eq. (35) yields:

Kbb+KKϕϕ1Kϕbq=f+KKϕϕ1Q,E37

Substituting {q} from Eq. (37) into Eq. (36), we obtain the vector {ϕ}.

#### 3.2.2 Dynamic analysis

From Eqs. (25) and (26), assembling the element equations yields general dynamic equation of motion:

Mbbq¨+Kbbq+Kϕ=f,E38
KϕbqKϕϕϕ=Q,E39

Substituting Eq. (39) into Eq. (38), we obtain:

Mbbq¨+Kbb+KKϕϕ1Kϕbq=f+KKϕϕ1Q,E40

#### 3.2.3 Free vibration analysis

Mbbq¨+Kbb+KKϕϕ1Kϕbq=0.E41

The beam vibrations induce charges and electric potentials in sensor layers. Therefore, the control system allows current to flow and feeds back to the actuators. In this case, if we apply no external charge Q to a sensor, from Eq. (39), we will have:

Kϕϕs1Kϕbsqs=ϕs.E42

and Qs=Kϕbsqsis the induced charge due to strain.

The operation of the amplified control loop implies, the actuating voltage is determined by the following relationship [1, 10, 14]:

ϕa=Gdϕs+Gvϕ̇s,E43

where Gd and Gv are the feedback control gains for displacement and velocity.

Substituting Eq. (43) into Eq. (39), the charge in the actuator due to actuator strain in response to the beam vibration modified by control system feedback is:

KϕbaqaKϕϕaGdϕs+Gvϕ̇s=Qa.E44

Substituting (42) into (44) leads to:

Qa=KϕbaqaGdKϕϕaKϕϕs1KϕbsqsGvKϕϕaKϕϕs1Kϕbsq̇s.E45

Substituting Eq. (45) into (40), we obtain:

Mbbq¨+Kbb+KKϕϕ1Kϕbq=f++KKϕϕ1KϕbaqaGvKϕϕaKϕϕs1Kϕbsq̇sGdKϕϕaKϕϕs1Kϕbsqs,E46

in which {q}s ≡ {q}a ≡ {q} is the beam displacement vector, [Kϕϕ]a = [Kϕϕ]s = [Kϕϕ] is the piezoelectric permittivity matrix, and [Kϕb]a = [Kϕb]s = [Kϕb] is the mechanical-electrical coupling stiffness matrix.

Therefore, Eq. (46) becomes:

Mbbq¨+Kbbq+GvKKϕϕ1KϕϕKϕϕ1Kϕbq̇++GdKKϕϕ1KϕϕKϕϕ1Kϕbq=f.E47

In the case of considering the structural damping, the equation of motion of the beam is:

Mbbq¨+CA+CRq̇+Kq=f,E48

where CA=GvKKϕϕ1KϕϕKϕϕ1Kϕbis the active damping matrix, K=Kbb+GdKKϕϕ1KϕϕKϕϕ1Kϕbis the total of mechanical stiffness matrix and piezoelectric, CR=αRMbb+βRKbbis the overall structural damping matrix, αR, and βR are respectively the Rayleigh damping coefficients, which are generally determined by the first and second natural frequencies (ω1, ω2) and ratio of damping ξ, {f} is the overall mechanical force vector.

Eq. (48) can be solved by the direct integration Newmark’s method.

### 3.3 Numerical analysis

An example for free vibration of laminated beam affected by piezoelectric layers is presented here. The beam is made of four layers symmetrically (0°/90°/90°/0°) of epoxy-T300/976 graphite material with 2.5 mm thickness per layer, and with one layer piezo ceramic materials bonded to the top and bottom surfaces, 2.0 mm thickness per layer as shown in Figure 2 is considered (a = 0.254 m, b = 0.0254 m). The material properties of the piezo ceramic layers and graphite-epoxy are shown in Table 1.

PropertiesPZT G1195 NT300/976
E11 [N/cm2]0.63 × 1061.50 × 106
E22 = E33 [N/cm2]0.63 × 1060.09 × 106
ν12 = ν13 = ν230.30.3
G12 = G13 [N/cm2]0.242 × 1060.071 × 106
G23 [N/cm2]0.242 × 1060.025 × 106
ρ [kg/m3]76001600
d31 = d32 (m/V)254 × 10−12
p11 = p22 (F/m)15.3 × 10−9
p33 (F/m)15.0 × 10−9

### Table 1.

Relevant mechanical properties of respective materials.

The direct integration Newmark’s method is used with parameters αR = 0.5, βR = 0.25; integral time step Δt = 0.005 s with total time calculated t = 15 s.

Figures 3 and 4 illustrate the vertical displacement w at the free end of the beam for two cases:

1. Case 1: With structural damping, and without piezoelectric damping (Gv = 0, Gd = 0).

2. Case 2: With structural damping, with piezoelectric damping (Gv = 0.5, Gd = 30).

## 4. Dynamic analysis of laminated piezoelectric composite plates

### 4.1 The electromechanical behavioral relations in the plate

Consider laminated composite plates with general coordinate system (x, y, z), in which the x, y plane coincides with the neutral plane of the plate. The top and bottom surfaces of the plate are bonded to the piezoelectric patches or piezoelectric layers (actuator and sensor). The plate under the load acting on its neutral plane has any temporal variation rule (Figure 5).

Hypothesis: The piezoelectric composite plate corresponds with Reissner-Mindlin theory. The material layers are arranged symmetrically through the neutral plane of the plate, ideally adhesive with each other.

#### 4.1.1 Strain - displacement relations

Based on the first-order shear deformation theory, the displacement fields at any point in the plate are [7, 8]:

uxyzt=u0xyt+zθyxyt,vxyzt=v0xytzθxxyt,wxyzt=w0xyt,E49

where u, v and w are the displacements of a general point (x, y, z) in the laminate along x, y and z directions, respectively. u0, v0, w0, θx and θy are the displacements and rotations of a midplane transverse normal about the y-and x-axes respectively.

The components of the strain vector corresponding to the displacement field (49) are defined as:

For the linear strain:

εx=ux=u0x+zθyx,εy=vy=v0yzθxy,γxy=uy+vx+∂wx∂wy=u0y+v0x+zθyxθxy,γxz=∂uz+∂wx=w0x+θy,γyz=∂vz+∂wy=w0yθx,E50

or in the vector form:

εxεyγxy=εxoεyoγxyo+zκxκyκxy=x00yyxu0v0+zy00xyxθxθy==Dεu0v0+Dκθxθy=ε0+zκ=εbL,E51
γxzγyz=x01y10woθxθy=DwIsw0θxθy=εs.E52

and for the nonlinear strain:

εxεyγxy=εbL+εN=εbN,E53
γxzγyz=εs,E54

where εN=12w0x00w0yw0yw0xxyw0is the non-linear strain vector, εbLis the linear strain vector, {εs} is the shear strain vector.

#### 4.1.2 Stress-strain relations

The equation system describing the stress-strain relations and mechanical-electrical quantities is respectively written as [8, 14]:

σb=QεbNeE,τb=Qsεs,E55
D=eεbN+pE,E56

where σb=σxσyτxyTis the plane stress vector, τb=τyzτxzTis the shear stress vector, [Q] is the ply in-plane stiffness coefficient matrix in the structural coordinate system, [Qs] is the ply out-of-plane shear stiffness coefficient matrix in the structural coordinate system. Notice that {τb} is free from piezoelectric effects.

The in-plane force vector at the state pre-buckling:

N0=Nx0Ny0Nxy0T=k=1nhk1hkσx0σy0τxy0kdz.E57

#### 4.1.3 Total potential energy

The total potential energy of the system is given by:

Π=12VpεbNTσbdV+12VpεsTτbdV12VpETDdVW,E58

where W is the energy of external forces, Vp is the entire domain including composite and piezoelectric materials.

Introducing [A], [B], [D], [As], and vectors {Np}, {Mp} as [8]:

ABD=h/2h/21zz2Qdz,As=h/2h/2Qsdz,NpMp=h/2h/21zeEdz,E59

where h is the total laminated thickness and combining with (5), (6) the total potential energy equation (8) can be written

Π=12Ωε0TAε0dΩ+12ΩκTDκdΩ+12ΩεsTAsεsdΩ++ΩεNTAε0NpdΩΩε0TNpdΩΩκTMpdΩW,E60

where Ω is the plane xy domain of the plate.

### 4.2 Dynamic stability analysis of laminated composite plate with piezoelectric layers

#### 4.2.1 Finite element models

Nine-node Lagrangian finite elements are used with the displacement and strain fields represented by Eqs. (49), (53), and (54). In the developed models, there is one electric potential degree of freedom for each piezoelectric layer to represent the piezoelectric behavior and thus the vector of electrical degrees of freedom is [6, 14]:

ϕe=..ϕje..T,j=1,,NPLe,E61

in which NPLe is the number of piezoelectric layers in a given element.

The vector of degrees of freedom for the element {qe} is:

qe=q1eq2eq9eϕeT,E62

where qie=uiviwiθxiθyiis the mechanical displacement vector for node i.

#### 4.2.2 Dynamic equations

The dynamic equations of piezoelectric composite plate can be derived by using Hamilton’s principle, accordingly, the vibration equation of the membrane (without damping) with in-plane loads is:

Mssq¨ss+Kssqss=Ft.E63

The equation of bending vibrations with out-of-plane loads is:

Mbb000q¨bbϕ¨+CR000q̇bbϕ̇+Kbb+KGKKϕbKϕϕqbbϕ=RQel,E64

where [Mss], [Kss] are the overall mass, membrane elastic stiffness matrix respectively, and qss,q̇ss,q¨ssare respectively the membrane displacement, velocity, acceleration vector. [Mbb], [Kbb] and qbb,q̇bb,q¨bbare the overall mass, bending elastic stiffness matrix and the bending displacement, velocity, acceleration vector; [KG] is the overall geometric stiffness matrix; ([KG] is a function of external in-plane loads); {F(t)} is the in-plane load vector, {R} is the normal load vector, {Qel} is the vector containing the nodal charges and in-balance charges.

The element coefficient matrices are:

KGe=KGxe+KGye+KGxye,E65
whereKGxe=AeNx0Nx'Nx'TdAe,KGye=AeNy0Ny'Ny'TdAe,KGxye=AeNxy0Nx'Ny'TdAe,E66
in whichNx'=xNxy,Ny'=yNxy,E67
∂w∂x=∂N∂xqbbe=Nx'qbbe,∂wy=∂Nyqbbe=Ny'qbbeE68
KG=neKGeE69

#### 4.2.3 Dynamic stability analysis

When the plate is subjected to in-plane loads only ({R} = {0}), the in-plane stresses can lead to buckling, from Eqs. (63) and (64) the governing differential equations of motion of the damped system may be written as:

Mssq¨ss+Kssqss=Ft,Mbbq¨bb+CRq̇bb+Kbb+KGqbb+Kϕ=0,KϕbqbbKϕϕϕ=Qel.E70

Eq. (70) is rewritten as:

Mssq¨ss+Kssqss=Ft,Mbbq¨bb+CA+CRq̇bb+K+KGqbb=0.E71

The overall geometric stiffness matrix [KG] is defined as follows:

• In the case of only tensile or compression plates (w = 0): Solving Eq. (71) helps us to present unknown displacement vector {qss}, and then stress vector:

σss=AsBsqss,E72

where [As] and [Bs] are the stiffness coefficient matrix and strain-displacement matrix of the plane problem.

• In the case of bending plate (w ≠ 0), the stress vector is:

σsb=σss+σbb,σbb=AbBbqbb,E73

where [Ab] and [Bs] are the stiffness coefficient matrix and strain-displacement matrix of the plane bending problem.

Stability criteria [14]:

• In the case of plate subjected to periodic in-plane loads and without damping, the elastic stability problems become simple only by solving the linear equations to determine the eigenvalues.

• In case of the plate under any in-plane dynamic load and with damping, the elastic stability problems become very complex. This iterative method can be proved effectively and the following dynamic stability criteria are used:

• Plate is considered to be stable if the maximum bending deflection is three times smaller than the plate’s thickness:Eq. (71)has the solution (wi)max satisfying the condition0wimax<3h, where wi is the deflection of the plate at node number i.

• Plate is called to be in critical status if the maximum bending deflection of the plate is three times equal to the plate’s thickness.Eq. (71)has the solution (wi)max satisfying the conditionwimax=3h.

• Plate is called to be at buckling if the maximum deflection of the plate is three times larger than the plate’s thickness:Eq. (71)has the solution (wi)max satisfying the conditionwimax>3h.

The identification of critical forces is carried out by the iterative method.

#### 4.2.4 Iterative algorithm

1. Step 1. Defining the matrices, the external load vector and errors of load iterations.

2. Step 2. Solving Eq. (71) to present unknown displacement vector, {qss} and the stress vector is defined by (72), updating the geometric stiffness matrix [KG].

3. Step 3. Solving Eq. (71) to present unknown bending displacement vector {qbb}, and then testing stability conditions.

4. − If for all wi=0: increase load, recalculate from step 2;

5. − If at least one value wi0:

6. + In case: 0<wimax<3h: Define stress vector by Eq. (73), update the geometric stiffness matrix [KG]. Increase load, recalculate from step 2;

7. + In case: 0wimax3hwimaxεD: Critical load p = pcr. End.

#### 4.2.5 Numerical analysis

Stability analysis of piezoelectric composite plate with dimensions a × b × h, where a = 0.25 m, b = 0.30 m, h = 0.002 m. Piezoelectric composite plate is composed of three layers, in which two layers of piezoelectric PZT-5A at its top and bottom are considered, each layer thickness hp = 0.00075 m; the middle layer material is Graphite/Epoxy material, with thickness h1 = 0.0005 m. The material properties for graphite/epoxy and PZT-5A are shown in Section 5.1 above. One short edge of the plate is clamped, the other three edges are free. The in-plane half-sine load is evenly distributed on the short edge of the plate: p(t) = p0sin(2πft), where p0 is the amplitude of load, f = 1/T = 1/0.01 = 100 Hz (0 ≤ t ≤ T/2 = 0.005 s) is the excitation frequency, voltage applied V = 50 V. The iterative error of the load εD = 0.02% is chosen.

Consider two cases: with damping (ξ = 0.05, Gv = 0.5, Gd = 15) and without damping (ξ = 0.0, Gv = 0.0, Gd = 15). The response of vertical displacement at the plate centroid over the plate thickness for the two cases is shown in Figure 6.

The results show that the critical load of the plate with damping is larger than that without damping. In the two cases above, the critical load rises by 6.8%.

Analyze the stability of the plate with damping when a voltage of −200, −150, −100, −50, 0, 50, 100, 150 and 200 V is applied to the actuator layer of the piezoelectric composite plate.

Figure 7 shows the time history of the vertical displacement at the plate centroid over the plate thickness when a voltage of 0, 50, 100, 150 and 200 V is applied. The relation between critical load and voltages is shown in Figure 8.

The results show that the voltage applied to the piezoelectric layers affects the stability of the plate. As the voltage increases, the critical load of the plate also increases.

When the amplitude of the load changes from 0.25pcr to 1.5pcr (where pcr is the amplitude of the critical load), a voltage of 50 V is applied to the actuator layer of the plate.

The results show the time history response of the vertical displacement at the plate centroid over the plate thickness as seen in Figure 9.

### 4.3 Dynamic analysis of piezoelectric stiffened composite plates subjected to airflow

Consider isoparametric piezoelectric laminated stiffened plate with the general coordinate system (x, y, z), in which the x, y plane coincides with the neutral plane of the plate. The top surface and lower surface of the plate are bonded to the piezoelectric patches (actuator and sensor). The plate subjected to the airflow load acting (Figure 10).

The dynamic equations of a finite smart composite plate are written as follows:

Meu¨e+CAeu̇e+Kbblne+Kbbnle+KAeue=fem,E74

where Kbblne=VeBblnTQBblndV,and Kbbnle=VeBbnlTQBbnldVare the element linear mechanical stiffness and nonlinear mechanical stiffness respectively, femis element external mechanical force vector.

#### 4.3.1.1 Formulation of x-Stiffener

Uxsxz=u0x+zθxsx,Wxsxz=wxsx.E75

where x-axis is taken along the stiffener centerline and the z-axis is its upward normal. The plate and stiffener element shown in Figure 11.

If we consider that the x-stiffener is attached to the lower side of the plate, conditions of displacement compatibility along their line of connection can be written as:

upz=tp/2=uxsz=txs/2,θxpz=tp/2=θxsz=txs/2,wpz=tp/2=wxsz=txs/2,E76

where tp is the plate thickness and txs is the x-stiffener depth.

The element stiffness and mass matrices are defined as follows [2, 15]:

Kxse=leBxsTDxsBxsdx,E77
Mxse=AePNu0TNu0+NwTNw+IyNθxTNθxdA,E78

with [Bxs] is the strain-displacement relations matrix, [Dxs] is the stress-strain relations matrix and le is the element length, Nu0,NwandNθxare the shape function matrices relating the primary variables u0, w, x, in terms of nodal unknowns, Iy is the area moment of inertia related to the y-axis and P=k=1nhk1hkρkdz,with ρk is density of kth layer.

#### 4.3.1.2 Formulation of y-Stiffener

The same as for x-stiffener, the element stiffness and mass matrices of the y-stiffener are defined as follows:

Kyse=leBysTDysBysdy,E79
Myse=AePNu0TNu0+NwTNw+IxNθyTNθydA,E80

#### 4.3.2 Modeling the effect of aerodynamic pressure and motion equations of the smart composite plate-stiffeners element

Based on the first order theory, the aerodynamic pressure lh and moment , can be described as [15, 16, 17]:

lw=12ρaUcosα2BkH1ẇUcosα+kH2Bθ̇Ucosα+k2H3θ+12CpρaUsinα2,mθ=12ρaUcosα2B2kA1ẇUcosα+kA2Bθ̇Ucosα+k2A3θ,E81

where k=/Uis defined as the reduced frequency, ω is the circular frequency of oscillation of the airfoil, U is the wind velocity, B is the half-chord length of the airfoil or half-width of the plate, ρa is the air density and α is the angle of attack.

The functions AiK,HiKare defined as follows:

H1K=πkFk,H2K=π4k1+Fk+2Gkk,H3K=π2k2FkkGk2,A1K=π4kFk,A2K=π16k1Fk2Gkk,A3K=π8k2k28+FkkGk2,E82

where F(k) and G(k) are defined as:

Fk=0.500502k3+0.512607k2+0.2104k+0.021573k3+1.035378k2+0.251293k+0.021508,Gk=0.000146k3+0.122397k2+0.327214k+0.001995k3+2.481481k2+0.93453k+0.089318.E83

Using finite element method, aerodynamic force vector can be described as:

feair=KaireueCaireu̇e+fen,E84

with Kaire,Caireand fenare the aerodynamic stiffness, damping matrices and lift force vector, respectively

Keair=ρaUcosα2Bk2AeH3kNwTNθx+BA3kNθyxTNθxdA,E85
Ceair=ρaUcosαBkAeH1kNwTNw+BH2kNwTNθxdA+AeBA1kNθyxTNw+B2A2kNθyxTNθxdA,E86
fen=CpρaUsinα2AeNwTdA,E87

where Ae is the element area, [Nw], [Nθ] are the shape functions.

From Eqs. (74) and (84), the governing equations of motion of the smart composite plate-stiffeners element subjected to an aerodynamic force without damping can be derived as:

Meu¨e+CAeu̇e+Ke+KAe+Kaireue=fem,E88

where Me=Me+Mxse+Myse, Ke=Kbblne+Kbbnle+Kxse+Kyse,fem=fem+fen.

#### 4.3.3 Governing differential equations for total system

Finally, the elemental equations of motion are assembled to obtain the open-loop global equation of motion of the overall stiffened composite plate with the PZT patches as follows:

Mu¨+CR+CAu̇+K+KA+Kairu=fm,E89

where CR=αRMbb+βRKbbln+Kbbnl.

The solution of nonlinear Eq. (89) is carried out by using Newmark direct and Newton-Raphson iteration method.

#### 4.3.4 Numerical applications

A rectangle cantilever laminated composite plate is assumed to be [0°/90°]s with total thickness 4 mm, length of 600 mm and width of 400 mm with three stiffeners along each direction x and y. The geometrical dimension of the stiffener is 5 mm of high and 10 mm of width. The plate and stiffeners are made of graphite/epoxy with mechanical properties: E11 = 181 GPa, E22 = E33 = 10.3 GPa, E12 = 7.17 GPa, ν12 = 0.35, ν23 = ν32 = 0.38, ρ = 1600 kg·m−3. Material properties for piezoelectric layer made of PZT-5A are: d31 = d32 = −171 × 10−12 m/V, d33 = 374 × 10−12 m/V, d15 = d24 = −584 × 10−12 m/V, G12 = 7.17 GPa, G23 = 2.87 GPa, G32 = 7.17 GPa, νPZT = 0.3, ρPZT = 7600 kg·m−3 and thickness tPZT = 0.15876 mm, ξ = 0.05, Gv = 0.5, Gd = 15. The effects of the excitation frequency and location of the actuators are presented through a parametric study to examine the vibration shape of the composite plate activated by the surface bonded piezoelectric actuators. The iterative error of the load εD = 0.02% is chosen. The piezoelectric stiffened composite plate is subjected to the airflow in the positive x direction as shown in Figure 10a.

Dynamic response of the piezoelectric stiffened composite plate is shown in Figure 12.

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Chung Nguyen Thai, Thinh Tran Ich and Thuy Le Xuan (April 29th 2020). Static and Dynamic Analysis of Piezoelectric Laminated Composite Beams and Plates [Online First], IntechOpen, DOI: 10.5772/intechopen.89303. Available from: