Table of Contents
ISRN Mechanical Engineering
Volume 2011 (2011), Article ID 362030, 11 pages
http://dx.doi.org/10.5402/2011/362030
Research Article

Thermal Buckling of Piezoelectric Composite Beam

1ME Department, Islamic Azad Universit, South Tehran Branch, Tehran, Iran
2ME Department, Amirkabir University of Technology, Tehran, Iran
3Academy of Sciences, ME Department, Amirkabir University of Technology, Tehran, Iran

Received 12 January 2011; Accepted 6 February 2011

Academic Editors: A. Combescure, A. Postelnicu, A. Z. Sahin, K. Yasuda, and D. Zhou

Copyright © 2011 S. Yazdani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Buckling analysis of laminated composite beams with piezoelectric layers subjected to thermal loading and constant voltage is studied. The material properties are assumed to be homogeneous in any layer through the beam thickness. The first-order beam theory and nonlinear strain-displacement relation are used to obtain the governing equations of the composite beam. The beam is assumed under uniform type of thermal loading and various types of boundary conditions. For each case of boundary conditions, closed-form solutions are obtained. The effects of the applied actuator voltage, beam geometry, and boundary conditions on the buckling temperature are investigated.

1. Introduction

Static and dynamic analysis for multilayer composite structures have been well established for various engineering applications during the last decades. Brush and Almroth [1] have a general treatment on the subject of structural stability, including beams, plates, and shells. Wang et al. [2] presented the closed-form solutions for buckling of beams, plates, and shells based on the classical, first-order, and higher-order displacement theories under compressive loads. Eslami and Shariyat [3, 4] used the improved equations to obtain the elastic, plastic, and creep buckling of thin cylindrical shells under different mechanical loading conditions. Analytical solutions of refined beam theories are developed to study the buckling behavior of cross-ply rectangular beams with arbitrary boundary conditions [5]. Kolakowski et al. [6] presented a modal interactive buckling of thin-walled composite beam columns regarding distortional deformations. Buckling analysis of cross-ply laminated beams with general boundary conditions by Ritz method is studied by Aydogdu [7].

If the membrane stresses due to a temperature distribution in a composite laminate are compressive and sufficiently large, equilibrium may become unstable, and thermal buckling may occur. In recent years, many studies have focused on the analysis of the thermal buckling and postbuckling responses of composite laminates. Eslami et al. [8] obtained the thermoelastic buckling of thin cylindrical shells under a number of practical thermal loadings. Shear deformation effects on thermal buckling of cross-ply composite laminates have been studied by Mannini [9]. In this paper, thermal buckling of symmetric and antisymmetric cross-ply composite laminates is investigated. The first-order shear deformation theory in conjunction with the Rayleigh-Ritz method is used for the evaluation of the thermal buckling parameters of structures.

Jordan canonical form solution for thermally induced deformation of cross-ply laminated composite beams has been presented by Khdeir and Reddy [10]. Also, Khdeir [11] studied the thermal buckling of thick, moderately thick, and thin cross-ply laminated beams subjected to uniform temperature distribution. He presented the exact analytical solutions of refined beam theories to obtain the critical buckling temperature of cross-ply beams with various boundary conditions. Li and Song [12] studied the large thermal deflections of Timoshenko beams under transversely nonuniform temperature rise. Thermal buckling analysis of cross-ply laminated composite beams with general boundary conditions is presented by Aydogdu [13]. The study is concerned with the thermal buckling analysis of cross-ply laminated beams subjected to different sets of boundary conditions. The analysis is based on a three-degrees-of-freedom shear deformable beam theory. The governing equations are obtained by means of the minimum energy principle. Thermal buckling load optimization of angle-ply symmetrically laminated composite beams is studied by Topal [14]. The objective of the optimization problem is to maximize the critical thermal buckling load of the laminated beams, and the fibre orientation is considered as the design variable.

Advanced structures with integrated self-monitoring and control capabilities are increasingly becoming important due to the rapid development of smart structure and mechanical systems. Bailey and Hubbard [15] reported vibration control of a piezoelectric beam with a simplified beam model. Recently, discrete layer theories are utilized for the analysis of composite structures with piezoelectrics in order to fully consider the effects of the transverse shear and variable in-plane displacements [16]. Tzou et al. [17, 18] proposed the mathematical modelling of nonlinear thermopiezoelastic laminates and investigated the static and dynamic control of beams and plates. Abramovich [19] presented the closed-form solutions for deflection control of laminated composite beams with piezoceramic layers. In his study, the three coupled equations of motion of a general nonsymmetric piezolaminated composite beam subjected to axial and lateral traction, and its corresponding boundary conditions are derived using a variational approach. The static shape control is performed using either continuous piezoceramic layers or patches embedded or bonded to the surface of the beam structure. Closed-form solutions for the bending angle and the axial lateral displacements along the beam are presented for various configurations of layup, boundary conditions, and mechanical loading. Waisman and Abramovich [20] studied the active stiffening of laminated composite beams using piezoelectric actuators. The present study deals with the stiffening effects of a smart piezolaminated composite beam. The structure consists of piezoceramic layers or patches bonded on the surface of the beam. The analysis considers the linear piezoelectric constitutive relations and the first-order shear deformation theory. Aldraihem and Khdeir [21, 22] presented the exact deflection solutions of beams with shear piezoelectric patches and actuators. Jerome and Ganesan [23] developed a generalized plane strain finite element formulation to predict the critical buckling voltage and temperature of a piezo composite beam. Akhras and Li [24] proposed the three-dimensional thermal buckling analysis of piezoelectric antisymmetric angle-ply laminates using finite layer method.

In this paper, the thermal buckling of piezoelectric laminated composite beams is studied. The first-order shear deformation beam theory is employed, and the closed-form solutions are presented for different types of boundary conditions.

2. Formulation of Problem

Consider a laminated composite beam with length 𝐿, width 𝑐, and total thickness . The rectangular Cartesian coordinates is used such that the 𝑥 axis is along the length of the beam on its middle surface and 𝑧 is measured from the middle surface and is positive upward, as shown in Figure 1. The analysis is based on the first-order beam theory. The displacement field for the beam is 𝑢 and 𝑤, which is based on Timoshenko beam theory, can be written as 𝑢(𝑥,𝑧)=𝑢(𝑥)+𝑧𝜙(𝑥),𝑤(𝑥,𝑧)=𝑤(𝑥),(1) where 𝑢 and 𝑤 are the axial and lateral displacements of a point on the midplane and 𝜙 is the bending rotation of the normal to the mid plane. The normal strain 𝜀𝑥 and the transverse shear strain 𝛾𝑥𝑧 at any point in the laminate are 𝜀𝑥=𝜕𝑢+1𝜕𝑥2𝜕𝑤𝜕𝑥2=𝑢+12𝑤2+𝑧𝜙,𝛾𝑥𝑧=𝜕𝑢+𝜕𝜕𝑧𝑤𝜕𝑥=𝜙+𝑤,(2) where a    stands for a derivation respect to 𝑥. When piezo composite beam is subjected to thermal load, the force and moment equations are written as [11, 19]𝑁𝑥𝑀𝑥𝑄𝑥𝑧=𝐴11𝐵110𝐵11𝐷11000𝐴55𝑢+12𝑤2𝜙𝜙+𝑤𝑁𝑇𝑀𝑇0𝑁𝐸𝑀𝐸0,(3) where in this equation 𝑁𝑥=/2/2𝑐𝜎𝑥𝑑𝑧,𝑀𝑥=/2/2𝑐𝜎𝑥𝑄𝑧𝑑𝑧,𝑥𝑧=/2/2𝑐𝜏𝑥𝑧𝑑𝑧,(4)𝜎𝑥 and 𝜏𝑥𝑧 being the normal and shear stresses, respectively. Thermal force and thermal moment are 𝑁𝑇=𝑐𝑁𝑛=1𝑧𝑛𝑧𝑛1𝑄𝑛11𝛼𝑛𝑥𝑀Δ𝑇𝑑𝑧,𝑇=𝑐𝑁𝑛=1𝑧𝑛𝑧𝑛1𝑄𝑛11𝛼𝑛𝑥Δ𝑇𝑧𝑑𝑧.(5) Here, 𝑁 is the number of layers and 𝛼𝑥 is the axial coefficient of thermal expansion. Terms 𝑁𝐸and 𝑀𝐸 are the piezoelectric force and moment and are 𝑁𝐸=𝑐𝑁𝑎𝑛=1(𝑄11)𝑛𝑎𝑉𝑛𝑑𝑛31,𝑀𝐸=𝑐2𝑁𝑎𝑛=1𝑄11𝑛𝑎𝑉𝑛𝑑𝑛312𝑧𝑛𝑎+𝑛𝑎.(6) A subscript 𝑎 stands for quantities associated with piezoelectric layers. Here, 𝑉𝑛 is the applied actuator voltage to the surface of 𝑛𝑡 piezoelectric layer nsd 𝑑31 is the piezoelectric constant. Also, 𝐴11, 𝐵11, 𝐷11, and 𝐴55 are the usual extensional, bending-extension, bending, and transverse shear stiffness coefficients defined as 𝐴11=𝑐/2/2𝑄11𝑑𝑧=𝑐𝑁𝑛=1𝑄11𝑛𝑧𝑛+1𝑧𝑛,𝐵11=𝑐/2/2𝑄11𝑐𝑧𝑑𝑧=2𝑁𝑛=1𝑄11𝑛𝑧2𝑛+1𝑧2𝑛,𝐷11=𝑐/2/2𝑄11𝑧2𝑐𝑑𝑧=3𝑁𝑛=1𝑄11𝑛𝑧3𝑛+1𝑧3𝑛,𝐴55=𝑐𝑘/2/2𝑄55𝑧𝑑𝑧=𝑐𝑘𝑁𝑛=1𝑄55𝑛𝑧𝑛+1𝑧𝑛,(7) where 𝑁 is the number of layers, 𝑘 is a shear correction factor, and 𝑄11 and 𝑄55 are the transformed material constants given by 𝑄11=𝑄11cos4𝜃+𝑄22sin4𝑄𝜃+212+2𝑄66sin2𝜃cos2𝜃,𝑄55=𝐺13cos2𝜃+𝐺23sin2𝜃.(8) The angle 𝜃 is the angle between the fibre direction and longitudinal axis (𝑥 axis) of the beam, and the constants 𝑄11, 𝑄12, 𝑄22, and 𝑄66 are 𝑄11=𝐸111𝜈12𝜈21,𝑄22=𝐸221𝜈12𝜈21,𝑄12=𝐸11𝜈121𝜈12𝜈21,𝑄66=𝐺12.(9) Using the principle of minimum total potential energy, the governing equations for the displacement field of (1) are derived in [1, 11, 19] as𝑑𝑁𝑥𝑑𝑥=0,𝑑𝑀𝑥𝑑𝑥𝑄𝑥𝑧=0,𝑑𝑄𝑥𝑧𝑑𝑥+𝑁𝑥𝑑2𝑤𝑑𝑥2=0.(10)

362030.fig.001
Figure 1: Geometry of cross-ply composite beam.

In this paper, it is assumed that the thermal load is uniform. The equilibrium equations in terms of the displacement components are obtained by substituting (3) into  (10)𝐴11𝑢+𝑤𝑤+𝐵11𝜙𝐵=0,11𝑢+𝑤𝑤+𝐷11𝜙𝐴55𝜙+𝑤𝐴=0,55𝜙+𝑤+𝑁𝑥𝑤=0.(11)

3. Prebuckling Deformation

The flat prebuckling configurations are assumed. For this purpose, the prebuckling deformation of laminated composite beam should be studied to assure that the beam remains flat under uniform thermal loading. The deformation of a beam prior to buckling may be obtained by solving the equilibrium equations (11) with the nonlinear terms set equal to zero [11]𝐴11𝑢+𝐵11𝜙𝐵=0,11𝑢+𝐷11𝜙𝐴55𝜙+𝑤𝐴=0,55𝜙+𝑤=0.(12) Solving these equations, we obtain 𝐵𝑢=11𝐴11𝐴55𝐵211/𝐴11𝐷11𝑏1𝑥22+𝑏5𝑥+𝑏6,𝐴𝑤=55𝐵211/𝐴11𝐷11𝑏1𝑥36+𝑏2𝑥22+𝑏3𝑥+𝑏4,𝐴𝜙=55𝐵211/𝐴11𝐷11𝑏1𝑥22𝑏2𝑥+𝑏3+𝑏1,𝑁𝑥=𝐴11𝑏5𝐵11𝑏2𝑁𝑇𝑁𝐸,𝑀𝑥=𝐴55𝑏1𝑥+𝐵11𝑏5𝐷11𝑏2𝑀𝑇𝑀𝐸,𝑄𝑥𝑧=𝐴55𝑏1,(13) where 𝑏1 to 𝑏6 are constants which have to be determined using the associated boundary conditions. The prebuckling boundary conditions are listed in Table 1. For each case of boundary conditions, constants 𝑏1 to 𝑏6 have been evaluated and listed in Table 2,𝐴𝐼=55𝐵211/𝐴11𝐷11,𝐺=𝐴55𝐵𝐿211𝐴11𝐼𝐿2+𝐷11𝐿1+𝐼2/6.𝐿/2(14) From this table, one may obtain that except the Clamped-Clamped and Clamped-Roller laminated composite beams, the other types of boundary conditions under thermal loading initially start to deflect rather than buckling. But the 𝐶-𝐶 and 𝐶-𝑅 boundary condition follow the bifurcation type buckling for uniform temperature rise loading.

tab1
Table 1: Prebuckling boundary conditions for various edge supports. (𝐶 indicates clamped, 𝑆 shows simply supported and 𝑅 is used for roller edge).
tab2
Table 2: Constants 𝑏𝑖 for various edge conditions.

4. Stability Equations

To derive the stability equations, the adjacent-equilibrium criterion is used. Assume that the equilibrium state of a laminated composite beam is defined in terms of the displacement components 𝑢0, 𝑤0, and 𝜙0 and the displacement components of a neighboring stable state differ by 𝑢1, 𝑤1, and 𝜙1 with respect to the equilibrium position. Thus, the total displacements of a neighboring state are [1] 𝑢=𝑢0+𝑢1,𝑤=𝑤0+𝑤1,𝜙=𝜙0+𝜙1.(15) Similar to the displacements, the force and moment of a neighboring state may be related to the state of equilibrium as 𝑁𝑥=𝑁𝑥0+𝑁𝑥1,𝑀𝑥=𝑀𝑥0+𝑀𝑥1,𝑄𝑥𝑧=𝑄𝑥𝑧0+𝑄𝑥𝑧1.(16) Here, 𝑁𝑥1, 𝑀𝑥1, and 𝑄𝑥𝑧1 represent the linear parts of the force and moment increments corresponding to 𝑢1, 𝑤1, and 𝜙1. The stability equations may be obtained by substituting (15) and (16) in (3). Upon substitution, the terms in the resulting equations with subscript 0 satisfy the equilibrium conditions and, therefore, drop out of the equations. The remaining terms form the stability equations as𝑑𝑁𝑥1𝑑𝑥=0,𝑑𝑀𝑥1𝑑𝑥𝑄𝑥𝑧1=0,𝑑𝑄𝑥𝑧1𝑑𝑥+𝑁𝑥0𝑑2𝑤1𝑑𝑥2=0.(17) Using (3) and (15), the force and moment with subscript 1 may be defined by𝑁𝑥1=𝐴11𝑢1+𝐵11𝜙1,𝑀𝑥1=𝐵11𝑢1+𝐷11𝜙1,𝑄𝑥𝑧1=𝐴55𝜙1+𝑤1.(18) For 𝐶-𝐶 and 𝐶-𝑅 composite beam subjected to uniform temperature rise, one may obtain𝑁𝑥0=𝑁𝑇𝑁𝐸,𝑀𝑥0=𝑀𝑇𝑀𝐸.(19) Combining (17) and (18) by eliminating 𝑢1 and 𝜙1 provides an ordinary differential equation in terms of 𝑤1, which is the stability equation of composite beam under thermal loading𝑑4𝑤1𝑑𝑥4+𝜇2𝑑2𝑤1𝑑𝑥2=0,(20) with𝜇2=𝑁𝑥0𝐷11𝐵211/𝐴11𝑁1+𝑥0/𝐴55.(21) When the temperature distribution in composite beam is uniform, the parameter 𝜇 is constant, and then the exact solution of (20) is𝑤1(𝑥)=𝐶1sin(𝜇𝑥)+𝐶2cos(𝜇𝑥)+𝐶3𝑥+𝐶4.(22) Using (17), (18), and (22), the expressions for 𝑢1, 𝜙1, and 𝑁𝑥1, 𝑀𝑥1, 𝑄𝑥𝑧1 become𝑢1𝐵(𝑥)=11𝐴11𝜇12𝐷11𝐵211/𝐴11𝐴55+𝜇2𝐷11𝐵211/𝐴11𝐶×𝜇1cos(𝜇𝑥)𝐶2sin(𝜇𝑥)+𝐶5𝑥+𝐶6,𝜙(23)1𝜇(𝑥)=12𝐷11𝐵211/𝐴11𝐴55+𝜇2𝐷11𝐵211/𝐴11×𝜇𝐶1cos(𝜇𝑥)+𝐶2sin(𝜇𝑥)𝐶3,𝑁(24)𝑥1(𝑥)=𝐴11𝐶5,𝑀𝑥1𝐷(𝑥)=11𝐵211𝐴11𝜇12𝐷11𝐵211/𝐴11𝐴55+𝜇2𝐷11𝐵211/𝐴11×𝜇2𝐶1cos(𝜇𝑥)+𝐶2sin(𝜇𝑥)+𝐵11𝐶5,𝑄𝑥𝑧1𝜇(𝑥)=3𝐷11𝐵211/𝐴11𝜇1+2/𝐴55𝐷11𝐵211/𝐴11×𝐶1cos(𝜇𝑥)𝐶2.sin(𝜇𝑥)(25)

Constants of these equations (𝐶1 to 𝐶6) are obtained using the boundary conditions of the composite beam. To find the minimum value of 𝑁𝑥0 associated with the thermal buckling load, the parameter 𝜇 must be minimized. Five types of boundary conditions are assumed for the composite beam. Consider a beam with both edges clamped. The edge conditions of the clamped-clamped composite beam are𝑢1(0)=𝑤1(0)=𝜙1(0)=𝑢1(𝐿)=𝑤1(𝐿)=𝜙1(𝐿)=0.(26)

Using (22)–(24) and (26), the constants 𝐶1 to 𝐶6 must satisfy the system of equations𝐵010100sin(𝜇𝐿)cos(𝜇𝐿)𝐿10011𝐴11𝐵𝑃𝜇0000111𝐴11𝐵𝑃𝜇cos(𝜇𝐿)11𝐴11𝐶𝑃𝜇sin(𝜇𝐿)00𝐿1𝑃𝜇01000𝑃𝜇cos(𝜇𝐿)𝑃𝜇sin(𝜇𝐿)10001𝐶2𝐶3𝐶4𝐶5𝐶6=000000,(27) where𝜇𝑃=12𝐷11𝐵211/𝐴11𝐴55+𝜇2𝐷11𝐵211/𝐴11.(28) To have a nontrivial solution, the determinant of coefficient matrix must be zero, which yields𝑃𝜇𝐿(22cos(𝜇𝐿)+𝑃𝜇𝐿sin(𝜇𝐿))=0.(29) The smallest positive value of 𝜇 which satisfies (29) is 𝜇min=2𝜋/𝐿. Table 3 shows different types of boundary conditions and the minimum values of 𝜇 associated with the thermal buckling loads. Now, the critical force for buckling from (21) (except for 𝐶-𝑆 beam, where the approximate solution from [2] is considered) is𝑁𝑥0𝜇=2𝐷11𝐵211/𝐴11𝜇1+2/𝐴55𝐷11𝐵211/𝐴11.(30) Then with this equation and (19), the buckling force of the beam for all cases of boundary conditions can be written in the form𝑁𝑇+𝑁𝐸=𝜇2𝐷11𝐵211/𝐴11𝜇1+2/𝐴55𝐷11𝐵211/𝐴11.(31)

tab3
Table 3: Boundary conditions and minimum value of 𝜇 for various edge supports. (𝐶 indicates clamped, 𝑆 shows simply supported, and 𝑅 is used for roller edge).

5. Thermal Loading

Consider a beam under uniform temperature rise. That is, consider a beam at reference temperature 𝑇0. The uniform temperature may be raised to 𝑇0+Δ𝑇 such that the beam buckles. Substituting (5) and (6) into (31) gives𝑐Δ𝑇𝑁𝑛=1𝑧𝑛𝑧𝑛1𝑄𝑛11𝛼𝑛𝑥𝑑𝑧+𝑐𝑁𝑎𝑛=1𝑄11𝑛𝑎𝑉𝑛𝑑𝑛31=𝜇2𝐷11𝐵211/𝐴11𝜇1+2/𝐴55𝐷11𝐵211/𝐴11.(32)

6. Numerical Result and Discussions

In this section, various combinations of composite beams comprising piezoelectric layers are assumed. General boundary conditions are considered on both sides to determine the critical buckling temperatures.

6.1. Aluminium Beam

Consider an aluminium beam with surface-bonded piezoelectric layers. we consider PZT-5A for piezoelectric layers. The beam thickness and length are =0.01 m and 𝐿=0.25 m, and the actuator layer thickness is 𝑎=0.001 m. The shear correction factor is 𝑘=5/6. Young's modules, coefficient of thermal expansion, Poisson's ratio, and the shear modules for aluminum are 𝐸=72.4 GPa, 𝛼=22.5×106/C, 𝜈=0.3, and 𝐺=27.8 GPa, respectively [25]. The PZT-5A properties are 𝐸𝑎=63 GPa, 𝛼𝑎=0.9×106/C, 𝜈12𝑎=0.3, 𝐺𝑎=24.2 GPa, and 𝑑31=2.54×1010 m/V [26]. Five electric loading cases are considered 𝑉0=0, ±200 V, ±500 V. Here, 𝑉0=0V denotes a grounding condition. Figure 2 and Table 4 depict the critical buckling temperature for various types of boundary conditions, and various voltages subjected to the uniform temperature rise. Also, the critical buckling temperature for the 𝑆-𝑆 and 𝐶-𝑅 types of boundary conditions are equal and larger than the value related to the 𝑆-𝑅 beams but lower than 𝐶-𝐶 and 𝐶-𝑆 beams.

tab4
Table 4: Effect of applied voltage on Δ𝑇𝑐𝑟, (piezo aluminium beam).
fig2
Figure 2: Critical buckling temperatures for piezoelectric aluminium beams with various boundary conditions and various voltages.
6.2. Glass-Epoxy Symmetric Beam

Consider a three-layered cross-ply composite beam (0/90/0), with surface-bonded piezoelectric layers. Also, similar to the previous example, consider PZT-5A for piezoelectric layers. The beam thickness and length are =0.0045 m, and 𝐿=0.25 m and the actuator layer thickness is 𝑎=0.001 m. The shear correction factor is 𝑘=5/6. It is assumed that the thickness and the material for all laminae are the same, (glass-epoxy) with the following characteristics [25]:𝐸11=50GPa,𝐸22𝐺=15.2GPa,12=𝐺13=4.7GPa,𝐺23𝛼=3.28GPa,1=6×106/C,𝛼2=𝛼3=23.3×106/𝜈C,12=𝜈13=0.254,𝜈23=0.428.(33)

Figure 3 and Table 5 depict the critical buckling temperature for various types of boundary conditions and various voltages subjected to the uniform temperature rise. The critical buckling temperature for the 𝑆-𝑆 and 𝐶-𝑅 types of boundary conditions are equal and larger than the value related to the 𝑆-𝑅 beams, but lower than 𝐶-𝐶 and 𝐶-𝑆 beams.

tab5
Table 5: Effect of applied voltage on Δ𝑇𝑐𝑟 (three-layered cross-ply composite beam).
fig3
Figure 3: Critical buckling temperature for three layered cross-ply beams with various boundary conditions and various voltages.
6.3. Glass-Epoxy Antisymmetric Beam

Consider an antisymmetric four-layered composite beam (0/90/0/90), with surface-bonded piezoelectric layers. Similar to the previous examples, consider PZT-5A for piezoelectric layers. The beam thickness and length are =0.004 m and 𝐿=0.25 m, and the actuator layer thickness is 𝑎=0.001 m. The shear correction factor is 𝑘=5/6. The thickness and the material for all laminae are the same, (glass-epoxy), with material properties given in the previous example. In this example, we first consider one piezoelectric layer on the top surface of the beam, and then with two piezoelectric layers on the top and bottom surfaces of the beam.

Figures 4 and 5 and Tables 6 and 7 depict the critical buckling temperature for various types of boundary conditions and various voltages subjected to the uniform temperature rise. The critical buckling temperature for the 𝑆-𝑆 and 𝐶-𝑅 types of boundary conditions are equal and larger than the values related to the 𝑆-𝑅 beams but lower than the 𝐶-𝐶 and 𝐶-𝑆 beams.

tab6
Table 6: Effect of applied voltage on Δ𝑇𝑐𝑟. (four-layered antisymmetric composite beam with one piezoelectric layer on the top surface of the beam).
tab7
Table 7: Effect of applied voltage on Δ𝑇𝑐𝑟. (four-layered antisymmetric composite beam with two piezoelectric layers on top and bottom surfaces of the beam).
fig4
Figure 4: Critical buckling temperature for antisymmetric four layered beam (0/90/0/90) with one piezoelectric layer on the top surface of the beam with various boundary conditions and various voltages.
fig5
Figure 5: Critical buckling temperature for antisymmetric four layered beam (0/90/0/90) with two piezoelectric layers on the top and bottom surfaces of the beam with various boundary conditions.

Figure 6 depicts the difference between the buckling temperature for the four-layered antisymmetric beam (0/90/0/90) with one and two piezoelectric layers with various boundary conditions.

362030.fig.006
Figure 6: Difference between the buckling temperature for four layered antisymmetric beam (0/90/0/90) with one and two piezoelectric layers with various boundary conditions.

The results show that for this type of piezoelectric layer, the buckling temperature decreases with the increase of the applied voltage and increases with the increase of applied voltage in opposite phase. The changes are, however, small. It should be mentioned that increasing or decreasing the buckling temperature by applying voltage in comparison with the grounding condition depends upon both the sign of applied voltage and the sign of the piezoelectric constant.

6.4. Influence of Geometry on Critical Buckling Temperature

Consider three cross-ply composite beams with three layers (0/90/0) that are bonded with two piezoelectric layers on the top and bottom surfaces of the beams. The thickness of the beams are =0.006 m, =0.0045 m, and =0.003 m. The lengths of the beams are equal and is 𝐿=0.25 m. The thickness and the material properties for all laminae are the same, (glass-epoxy), and the actuator layer is PZT-5A with thickness 𝑎=0.001 m. The influence of beam geometry on the buckling temperature Δ𝑇cr for various types of boundary conditions under applied voltages is shown in Figure 7. As shown, when the thickness increases, the critical buckling temperature increases for various types of boundary conditions, as expected.

362030.fig.007
Figure 7: Influence of thickness on the buckling temperature.

7. Conclusion

In this paper, the buckling analysis of composite beams with piezoelectric layers under various types of boundary conditions is investigated. Exact analytical solutions for the critical buckling temperature differences of beams are presented. The following are concluded.(1) The buckling temperature difference for homogeneous, symmetric composite, and antisymmetric composite beams can be controlled by applying suitable voltage on the actuator layers, but the effect of this control voltage is small.(2)For composite beams under uniform temperature rise, by increasing the beam thickness, the critical buckling temperature increases for any type of boundary conditions.

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