Abstract
A onedimensional finite element model for buckling analysis of hybrid piezoelectric beams under electromechanical load is presented in this work. The coupled zigzag theory is used for making the model. The inplane displacement is approximated as a combination of a global third order variation across the thickness with an additional layer wise linear variation. The longitudinal electric field is also taken into account. The deflection field is approximated to account for the transverse normal strain induced by electric fields. Two nodded elements with four mechanical and a variable number of electric degrees of freedom at each node are considered. To meet the convergence requirements for weak integral formulation, cubic Hermite interpolation function is used for deflection and electric potential at the sublayers and linear interpolation function is used for axial displacement and shear rotation. The expressions for the variationally consistent stiffness matrix and load vector are derived and evaluated in closed form using exact integration. The present 1DFE formulation of zigzag theory is validated by comparing the results with the analytical solution for simplysupported beam and 2DFE results obtained using ABAQUS. The finite element model is free of shear locking. The critical buckling parameters are obtained for clampedfree and clampedclamped hybrid beams. The obtained results are compared with the 2DFE results to establish the accuracy of the zigzag theory for above boundary conditions. The effect of lamination angle on critical buckling load is also studied.
Piezoelectric beam; FEM; Buckling; Zigzag theory; ABAQUS
Finite element modeling for buckling analysis of hybrid piezoelectric beam under electromechanical loads
Najeeb ur Rahman^{I,}^{*} * Author email: najeebalig@rediffmail.com ; M.Naushad Alam^{II}
^{I}Department of Mechanical Engineering, Aligarh Muslim University, Aligarh,U.P202002, India
^{II}Department of Mechanical Engineering, Aligarh Muslim University, Aligarh,U.P202002, India
ABSTRACT
A onedimensional finite element model for buckling analysis of hybrid piezoelectric beams under electromechanical load is presented in this work. The coupled zigzag theory is used for making the model. The inplane displacement is approximated as a combination of a global third order variation across the thickness with an additional layer wise linear variation. The longitudinal electric field is also taken into account. The deflection field is approximated to account for the transverse normal strain induced by electric fields. Two nodded elements with four mechanical and a variable number of electric degrees of freedom at each node are considered. To meet the convergence requirements for weak integral formulation, cubic Hermite interpolation function is used for deflection and electric potential at the sublayers and linear interpolation function is used for axial displacement and shear rotation. The expressions for the variationally consistent stiffness matrix and load vector are derived and evaluated in closed form using exact integration. The present 1DFE formulation of zigzag theory is validated by comparing the results with the analytical solution for simplysupported beam and 2DFE results obtained using ABAQUS. The finite element model is free of shear locking. The critical buckling parameters are obtained for clampedfree and clampedclamped hybrid beams. The obtained results are compared with the 2DFE results to establish the accuracy of the zigzag theory for above boundary conditions. The effect of lamination angle on critical buckling load is also studied.
Keywords: Piezoelectric beam, FEM, Buckling, Zigzag theory, ABAQUS.
1 INTRODUCTION
Structural components made with hybrid composite laminates and sandwich structures with surface bonded or embedded piezoelectric layers are increasingly being used in various engineering applications in aerospace, naval, civil, and mechanical industries. This is due to their attractive properties in strength, stiffness, and lightness. In such applications buckling phenomenon is often observed which is critically dangerous to structural components as it usually occurs at a lower applied stress for such structures, and it generates large deformation. Understanding their dynamic and buckling behaviour is of increasing importance.
There have been few exact threedimensional (3D) solutions of buckling of elastic composite and sandwich plates and exact 2D solutions of buckling of composite and sandwich beams. These solutions serve as useful benchmarks for assessment of various 1D beam theories and approximate 2D numerical solutions such as the solution using the finite element method. Song and Waas (1987) presented a higher order theory for the buckling and vibration analysis of composite beams and the accuracy of HSDT was demonstrated compared to 1D EulerBernoulli, 2D classical elasticity theory and Timoshenko beam theory. Chandrashekhra and Bhatia (1993) presented a finite element model for active buckling control of composite plates, with surface bonded or embedded, continuous or segmented, piezoelectric sensors and actuators. Khdeir and Reddy (1997) developed analytical solutions for free vibration and buckling of crossply composite beams with arbitrary boundary conditions in conjunction with the state space approach. Wang (2002) and Wang and Quek (2002) have presented coupled 1D classical beam theory for buckling and further analysis of a column with a pair of piezoelectric layers partially or fully covering it. Finite difference method is used for solution and it is shown that with proper placement of the actuators, the buckling load for the statically actuated beams can be significantly increased. Thompson and Loughlan (1995) demonstrated experimentally that the buckling capacity of a column is increased by applying controlled voltage to the piezoelectric actuators. Kapuria and Alam (2004) presented a twodimensional exact piezoelasticity solution for buckling of simply supported symmetrically laminated hybrid beam and crossply panel with elastic substrate and piezoelectric layers. They considered buckling under axial strain and actuation potentials for movable inplane end conditions and under actuation potential alone for immovable inplane end conditions. Kapuria and Alam (2004) developed a new efficient coupled onedimensional (1D) geometrically nonlinear zigzag theory for buckling analysis of hybrid piezoelectric beams, under electromechanical loads. They approximated potential field layerwise as piecewise linear. Kamruzzaman et al. (2006) performed parametric studies to identify better configuration of given composite to achieve higher buckling strength for laminated antisymmetric cross and angleply simply supported rectangular orthotropic plates subjected to uniaxial compressive loads. Kapuria and Alam (2005) developed an efficient electromechanically coupled geometrically nonlinear zigzag theory for buckling analysis of hybrid piezoelectric beams, under electrothermomechanical loads. They obtained analytical solutions for buckling of symmetrically laminated simply supported beams under electrothermal loads and compared the results with the available exact twodimensional (2D) piezothermoelasticity solution. Anas et al. (2011) developed a one dimensional finite element model for the buckling analysis of laminated composite beam, using the efficient layer wise zigzag theory. They obtained the 1DFE results for cantilever beam and compared with the 2DFE results, obtained using ABAQUS. Anas and Husain (2012) have used an efficient one dimensional finite element model for the vibrational analysis of composite laminated symmetric beam, using the efficient layerwise zigzag theory.
Vo and Inam (2012) presented vibration and buckling analysis of crossply composite beams using refined shear deformation theory. The theory accounts for the parabolical variation of shear strains through the depth of beam. They obtained numerical results for composite beams to investigate modulus ratio on the natural frequencies, critical buckling loads and loadfrequency interaction curves.
Qiao et al. (2010) derived local delamination buckling formulas for laminated composite beams based on the rigid, semi rigid, and flexible joint models with respect to three bilayer beam theories. They analyzed two local delamination buckling modes and obtained their critical buckling loads based on the three joint models. Chakrabarti et al. (2012) studied stability analysis of laminated soft core sandwich beam by a FE model developed by the authors based on higher order zigzag theory (HOZT).The proposed model satisfies the condition of stress continuity at the layer interfaces and the zero stress condition at the top and bottom of the beam for transverse shear. Alam and Anas (2009) have used an efficient one dimensional finite element model developed for the buckling analysis of composite laminated beams, using the efficient layer wise zigzag theory. They compared 1DFE results for cantilever beam with the 2DFE results, obtained using ABAQUS. Sherwani and Alam (2009) have used an efficient one dimensional finite element model for the buckling analysis of smart beam, using the efficient layer wise zigzag theory. The employed finite element model is free of shear locking and obtained results of buckling parameters for cantilever smart beam. Kapuria and Alam (2005) developed a new efficient electromechanically coupled geometrically nonlinear zigzag theory for buckling analysis of hybrid piezoelectric beams, under electrothermomechanical loads. The thermal and potential fields are approximated as piecewise linear in sublayers. Analytical solutions for buckling of symmetrically laminated simply supported beams under electrothermal loads are obtained for comparing the results with the available exact twodimensional piezothermoelasticity solution. Pandit et.al (2008) have proposed a higher order zigzag theory for the static and buckling analysis of sandwich plates with soft compressible core. They employed a nine node isoparametric element with 11 field variables per node. To overcome the problem of C1continuity the authors have used separate shape functions to define the derivatives of transverse displacements. Matsunaga (1996, 2001) developed a one dimensional global higher order theory, in which the fundamental equations were derived based on the power series expansions of continuous displacement components to analyze the vibration and buckling problems. Cetkovic and Vuksanovic (2009) and many others assume unique displacement field in each layer and displacement continuity across the layers. In these theories, the number of unknowns increases directly with the increase in the number of layers due to which it required huge computational involvement. Aydogdu (2006) carried out the vibration and buckling analysis of crossply and angleply with different sets of boundary conditions by using Ritz method. Iqbal et al. (2011) have studied free vibration response of laminated sandwich beams having a soft core by using a C0 finite element beam model. The model has been developed based on higher order zigzag theory where the inplane displacement variation is considered to be cubic for both the face sheets and the core.
Moy et al. (2005) developed a shear deformable plate bending element based on a third order shear deformable theory (i.e. HSDT). Dawe and Yuan (2001) used a Bspline finite strip method (FSM) for predicting the buckling stresses of rectangular sandwich plates. They represented core as a threedimensional solid in which the inplane displacements vary quadratically through the thickness whilst the outofplane displacement varies linearly. Herbert et al. (2012) used a consistently linearized Eigenproblem to derive mathematical conditions in the frame of the Finite Element Method (FEM) for loss of static stability of elastic structures at prebuckling states. Cai etal. (2011) presented the buckling behaviours of composite long cylinders subjected to external hydrostatic pressure by using deterministic and probabilistic finite element analyses. They studied the effects of uncertainties of material properties and physical dimensions on the critical buckling pressure. Kheirikhah et al. (2012) presented an accurate 3D finite element model for bucking analysis of softcore rectangular sandwich plates. They studied the effect of geometrical parameters of the sandwich plate.
A detailed review of literature shows that although considerable research work has been done on buckling of composite beams, but the finite element modelling for buckling analysis of hybrid beams for various boundary conditions using zigzag theory is missing. Keeping this point in view a onedimensional finite element model is presented in this work for buckling analysis of hybrid piezoelectric beams under electromechanical load. The coupled zigzag theory by Kapuria and Alam (2004) is used for making the model. The inplane displacement is approximated as a combination of a global third order variation across the thickness with an additional layer wise linear variation. The longitudinal electric field is also taken into account. The deflection field is approximated to account for the transverse normal strain induced by electric fields. Two nodded elements with four mechanical and a variable number of electric degrees of freedom at each node are considered. The critical buckling parameters are obtained for clampedfree and clampedclamped smart beams. The 1DFE results are compared with 2D FE results obtained using ABAQUS.
2 ONE DIMENSIONAL COUPLED ZIGZAG THEORY FOR HYBRID PIEZOELECTRIC BEAM
Consider a hybrid beam having any layup, whose thickness h and the number of layers L may vary segmentwise [Kapuria and Alam (2004)] due to the presence of piezoelectric patches. The longitudinal and thickness axes are along x  and z directions. The xy  plane is chosen to be the plane which is the midplane for most of the length of the beam. Let the planes z=z_{0} and z=z_{L} be the bottom and top surfaces of the beam, which may vary segmentwise. The z  coordinate of the bottom surface of the k^{th} layer (numbered from the bottom) is denoted as z_{k}_{1} and its material symmetry direction 1 is at an angle θ_{k} to the x axis. The reference plane z = 0 either passes through or is the bottom surface of the layer. All the elastic and piezoelectric layers are perfectly bonded. It is loaded transversely on the bottom and top with no variation along the width b . The piezoelectric layers have poling direction along z  axis.
The approximations of the coupled zigzag theory presented by Kapuria and Alam (2004) are as follows. For a beam with a small width, a state of plane stress is assumed i.e. σ_{y} = τ_{yz} = τ_{xy} = 0. For infinite panels, a plane strain state (ε_{y} = γ_{yz} = γ_{xy} = 0) is considered. The transverse normal stress is neglected (i.e.σ_{z} = 0). The axial and transverse displacements u,w and electric potential ϕ are assumed to be independent of y . With these assumptions, the general 3D constitutive equations of a piezoelectric medium for stresses σ_{x},τ_{zx} and electric displacements D_{x}, D_{z} reduce to
Where are the reduced stiffness coefficients, piezoelectric stress constants and electric permittivities respectively.
The potential field ϕ is assumed as piecewise linear between n_{ϕ} points across the thickness:
where
(z) are linear interpolation functions and summation convention is used with the summation index j, taking values 1,2,...,n_{ϕ}. This description allows the piezoelectric layers to be divided into a number of sublayers and a series of elastic layers to be combined into one, for effective modeling of ϕ across the thickness. The variation of deflection w is obtained by integrating the constitutive equation for ε_{z} by neglecting the contribution of σ_{x} via Poisson's effect compared to that due to the electric field: w_{,z} ; d_{33}ϕ_{,z}⇒
where is a piecewise linear function. The axial displacement u for the k^{th} layer is approximated to follow a global third order variation across the thickness with a layerwise linear variation:
For the layer through which the plane z = 0 passes, denote u_{0} (x) = u_{k0} (x) = u (x,0), ψ_{k0} (x) = ψ_{k0} (x). Thus u_{0} and ψ_{0} are the axial displacement and the shear rotation at z=0, respectively. Using the (L 1) conditions each for the continuity of τ_{zx} and u at the layer interfaces and the two shear tractionfree conditions τ_{zx} = 0 at z = ±h / 2 , the functions u_{k},ψ_{k},ξ,η are expressed in terms of u_{0} and ψ_{0} to yield
where R^{k} (z),R^{k} (z) are cubic functions of z whose coefficients are dependent on the material properties and layup.
Thus, even though w,u have layerwise distributions, they are expressed in terms of only three displacement variables u_{0}, w_{0}, ψ_{0} by Eqns. (4) and (6).
Eqns. (6) and (4) for u , w can expressed as
with
where elements with index j mean a sequence of elements with j = 1 to n_{ϕ}. Using Eqn. (7) and (2), the strains and the electric fields can be expressed as
where
Let be the normal forces per unit area on the bottom and top surfaces of the beam in direction z . Let there be distributed viscous resistance force with the distributed viscous damping coefficient c_{1} per unit area per unit transverse velocity of the top surface of the beam. At the interface at z = where the potential is prescribed, the extraneous surface charge density is q_{ij}. Using the notation ... = for integration across the thickness, the extended Hamilton's principle for the beam reduces to
Substituting the expressions (7) and (2) for u,w,ϕ and (9) for ε_{x}, γ_{zx}, E_{x}, E_{z} into Eqn. (11) yields
where an overbar on the stress and electric resultants and on u_{0}, w_{0}, ψ_{0}, ϕ^{j} means values at the ends.
3 FINITE ELEMENT MODEL OF HYBRID BEAM
A finite element model using the 1D coupled zigzag theory is developed for the buckling analysis of hybrid piezoelectric beams (Fig.1) under electromechanical loads. Two noded elements are used for the electromechanical variables.
The highest derivatives of u_{0}, ψ_{0}, w_{0}, ϕ^{j} appearing in the variational Eqn. (12) are u_{0,x}, ψ_{0,xx}, w_{0,x}, . To meet the convergence requirements, the interpolation functions for u_{0}, ψ_{0}, w_{0,x}, must be continuous at the element boundaries. Hence w_{0}, ϕ^{j} are expanded using cubic Hermite interpolation in terms of the nodal values of w_{0}, w_{0,x} and ϕ^{j}, respectively, and a linear interpolation is used for u_{0}, ψ_{0}. Thus, at the element level, each node will have four degrees of freedom u_{0}, w_{0}, w_{0,x}, ψ_{0} for the displacements and 2n_{ϕ} degrees of freedom of ϕ^{j}, for the electric potential. This leads to elements with variable numbers of degrees of freedom, since n_{ϕ} can be different for different elements.
3.1 Interpolation and variational equation
Denote the values of an entity (.) at the nodes 1 and 2 by (.)_{1} and (.)_{2} respectively. u_{0}, ψ_{0}, w_{0}, ϕ^{j} are interpolated in an element of length a as
with
and
where,
The integrand in the variational Eqn. (12) for the case of static mechanical load can be expressed as
the contribution T^{e} of an element to the integral in Eqn. (12) is obtained as
where,
Defining generalized displacements û, generalised strains and generalised stress resultants as
The generalized beam constitutive relation may be expressed as
3.2 Element strains
Defining the element generalized displacement vector U^{e} as
and using Eqn. (13) the generalized displacements û and strains defined in Eqn. (20) can be related to U^{e} as
where
Substituting the expressions for û and from Eqn. (23) into Eqn. (18), T^{e} can be expressed as
with
N_{x}, M_{x}, P_{x}, and Q_{x} are substituted so that to obtain general equation after integration as:
, D_{z0}, D_{zL}, q_{ji}are linearly interpolated in terms of their nodal values,
Substituting (24) and (19) into equation (26)yields
and
3.3 FEM for buckling of hybrid beam under axial loading
For buckling of laminated composite beams under axial loading it is assumed that lateral load is zero and the axial forces applied are compressive in nature at the ends. From the variational equation let N_{x} = N_{x}  N_{cr}
where N_{cr} is the buckling critical load at which buckling occurs, be substituted other terms such as shape functions and primary variables are substituted from equations (13), (14),(15) and (16) to obtain buckling eigen value equation from variational equations
N _{x}, M_{x}, P_{x}, and Q_{x} are substituted to obtain eigen value problem, after integration
N_{cr} is an eigenvalue of the generalized eigenvalue problem .The lowest eigenvalue is the critical value of the axial load at which buckling occurs.
K^{G} is the element geometric stiffness matrix
where,
The critical axial strain ε_{cr} for buckling under axial load corresponding to critical load is given by ε_{cr}= N_{cr} /A_{11} and the critical load N_{cr} is nondimensionalised as
The critical axial strain ε_{cr} is nondimensionalised as =Sε
The critical potential ϕ_{cr} is nondimensionalised as_{cr} = .
The mechanical boundary conditions for a movable simplysupported end, immovable simplysupported (hinged) end, clamped end and free end are taken as follows:
simplysupported end : N_{x} = 0 (movable) or u_{0}= 0 (immovable), w_{0} = 0, M_{x} = 0, P_{x} = 0
clamped end : u_{0} = 0, w_{0} = 0, w_{0,}_{x} = 0, Ψ_{0}= 0,
free end : N_{x} = 0, V_{x} = 0, M_{x} = 0, P_{x} = 0 .
4 RESULTS AND DISCUSSIONS
4.1 Validation
The present 1DFE formulation of zigzag theory is validated by comparing the results for critical load, and critical strain of the simplysupported beam with the results of Kapuria and Alam (2004) and 2DFE Abaqus. Results are presented for two types of simply supported hybrid beams of different symmetric laminate configurations (b) and (c) (Fig.2). The beams have two piezoelectric layers of PZT5A of thickness 0.1h bonded to their elastic substrate on top and bottom surfaces.
The substrate (b) is a graphiteepoxy composite laminate with 4 layers of equal thickness 0.2h with layup [0^{0}/90^{0}/90^{0}/0^{0}]. The substrate (c) is a 3layer sandwich having graphiteepoxy composite faces and a soft core. For a beam of span 'a' and thickness 'h', the thickness parameter S=a/h.
4.2 Numerical Example
For the numerical study, two hybrid beams (b) and (c) with composite and sandwich substrates, respectively are considered (Fig. 2). Both the beams have a PZT5A layer of thickness 0.1h bonded to the top and bottom of the elastic substrate. The PZT 5A layers have polling in + z direction. The top and bottom of the substrate are grounded. The stacking order is mentioned from bottom.
The composite substrate of beam (b) is a graphiteepoxy (material 1) composite laminate with 4 layers of equal thickness 0.2/7 with layup [0º / 90º / 90º / 0º] and the sandwich substrate of beam (c) has graphiteepoxy faces of material 2 and a soft core with thicknesses 0.08h / 0.64h / 0.08h.
The Young's Moduli Y_{i}, Shear Moduli G_{ij} , Poisson's Ratio v_{ij}, Piezoelectric Strain Constants d_{ij} and Electric Permittivities η_{ij}, are given by:
[Y_{1}, Y_{2}, Y_{3}, G_{12}, G_{23}, G_{31}, v_{12}, v_{13}, v_{23}]
Material 1: [ (181, 10.3, 10.3, 7.17, 2.87, 7.17) GPa, 0.28, 0.28, 0.33]
Material 2: [ (131.1, 6.9, 6.9, 3.588, 2.3322, 3.588) GPa, 0.32, 0.32, 0.49]
Core: [(0.2208,0.2001, 2760, 16.56, 455.4, 545.1)GPa, 0.99, 3x10^{5}, 3x10^{5}]
PZT5A:[ (61, 61, 53.2, 22.6, 21.1, 21.1) GPa, 0.35, 0.38, 0.38]
[(d_{31}, d_{32}, d_{33}, d_{15}, d_{24}), (η_{11}, η_{22}, η_{33} )]
=[(171, 171, 374, 584, 584)x10^{12}m/V, (1.53, 1.53, 1.5)x10^{8}F/m].
4.3 Buckling analysis
The prebuckling load condition with uniform axial strain and zero potential i.e. [Ψ(z_{0})= Ψ(z_{L})=0] at the top and bottom surfaces is considered. The critical buckling strain and corresponding axial force are denoted as ε_{cr} and N_{cr} respectively.
N_{cr} is the eigenvalue of the generalized eigenvalue problem .The lowest eigenvalue for n=1 is the critical value of the axial compressive load for which buckling occurs. These eigenvalue problems have been solved for two end conditions of both the beams viz: ClampedClamped and ClampedFree.
4.3.1 Buckling response for clampedclamped hybrid beam
1D FE and 2DFE (ABAQUS) buckling results are obtained for the first three modes (i.e. n=1, 2 and 3) and compared for critical buckling load and strain of laminated hybrid beams, beam (b) and beam (c) for clampedclamped boundary condition. The results are listed in table5 for beam (b) and table6 for beam (c).It may be observed that for the same value of S there is little variation in values of critical load and strain for different modes. Further the 1DFE results are in good agreement with 2DFE results.
The variation of critical load for first mode (n=1)with thickness to span ratio (h/a) of both the clampedclamped hybrid beams are shown in Fig.3. The value of the critical load increases as the beams are made thicker for the same span length.
Fig.4 shows the variation of critical load with angle for clampedclamped hybrid beam, beam (b). The results are shown for span to thickness ratio, S=10.The critical load is maximum for 0 deg and exponentially reduces as the angle increases and become 90 deg. The 1DFE and 2DFE plots are in good agreement.
4.3.1 Buckling response for clampedfree hybrid beam
1D FE and 2DFE (Abaqus) buckling results are obtained and compared for critical buckling load and strain of laminated hybrid beams for clampedfree boundary condition. The results are listed in table7 and table8. It may be observed that the 1DFE results are in good agreement with 2DFE results.
Fig. 5 shows the variation of critical load for first mode (n=1) with thickness to span ratio (h/a) of fixedfree hybrid beams (b) and (c). The nature of the plot is different from clampedclamped condition as the critical load values are considerably higher for the fixed beams.
Fig.6 shows the variation of critical load with angle for clampedfree hybrid beam, beam (b) with span to thickness ratio, S=10.The critical load is maximum for 0 deg and exponentially reduces as the angle increases and becomes 90 deg. The 1DFE and 2DFE plots are in good agreement.
4 CONCLUSIONS
A new finite element model based on zigzag theory is developed for the buckling analysis of hybrid beams under electromechanical load. The accuracy of the developed 1DFE model for buckling analysis has been assessed by comparison with the analytical solution for hybrid beam and 2DFE results obtained using Abaqus for thick, moderately thick and thin hybrid beams for various boundary conditions. The 1DFE results are in good agreement with 2DFE results which shows the robustness of the model. The effect of lamination angle on critical buckling load is also substantial.
Received in 14 Feb 2013
In revised form 20 Jun 2013
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Publication Dates

Publication in this collection
03 Feb 2014 
Date of issue
Oct 2014
History

Accepted
20 June 2013 
Received
14 Feb 2013