A Timoshenko Piezoelectric Beam Finite Element with Consistent Performance Irrespective of Geometric and Material Configurations

Transcription

1 99 A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve of Geometrc and Materal Confguratons Abstract The conventonal Tmoshenko pezoelectrc beam fnte elements based on Frst-order Shear Deformaton Theory (FSDT) do not mantan the accuracy and convergence consstently over the applcable range of materal and geometrc propertes. In these elements, the naccuracy arses due to the nduced potental effects n the transverse drecton and neffcency arses due to the use of ndependently assumed lnear polynomal nterpolaton of the feld varables n the longtudnal drecton. In ths work, a novel FSDT-based pezoelectrc beam fnte element s proposed whch s devod of these defcences. A varatonal formulaton wth consstent through-thckness potental s developed. The governng equlbrum equatons are used to derve the coupled feld relatons. These relatons are used to develop a polynomal nterpolaton scheme whch properly accommodates the bendng-extenson, bendng-shear and nduced potental couplngs to produce accurate results n an effcent manner. It s noteworthy that ths consstently accurate and effcent beam fnte element uses the same nodal varables as of conventonal FSDT formulatons avalable n the lterature. Comparson of numercal results proves the consstent accuracy and effcency of the proposed formulaton rrespectve of geometrc and materal confguratons, unlke the conventonal formulatons. Ltesh N. Sulbhewar a P. Raveendranath b Department of Aerospace Engneerng, Indan Insttute of Space Scence and Technology, Thruvananthapuram, Inda. a (Correspondng author) Research Scholar Emal: lteshsulbhewar@gmal.com b Adjunct Professor Emal: raveendranath@st.ac.n Receved In revsed form Accepted Avalable onlne Keywords Coupled feld, fnte elements, nduced potental, pezoelectrc, smart structures. 1 INTRODUCTION Beam formulatons are wdely used for the numercal analyss of one dmensonal pezoelectrc structure (Marnkovc and Marnkovc, 01). Analytcal formulatons (Crawley and de Lus, 1987; Abramovch and Pletner, 1997; Crawley and Anderson, 1990) and fnte elements (Bendary et al.,

2 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve ; Kumar and Narayanan, 008; Sulbhewar and Raveendranath, 014a) based on the Euler- Bernoull beam theory can be effectvely used for the analyss of thn and slender pezoelectrc smart beams. However, Euler-Bernoull theory neglects the deformaton due to shear and hence not sutable for thck and short beams. Sandwch Beam Theory (SBT) based analytcal formulaton (Zhang and Sun, 1996) and fnte element formulatons (Benjeddou et al., 1997, 000; Raja et al., 00) consdered the thcker core as a Tmoshenko beam and the relatvely thnner faces as Euler- Bernoull beams. However, SBT s not sutable for short beams wth thck pezoelectrc layers. Tmoshenko beam formulatons based on the Frst-order Shear Deformaton Theory (FSDT) consder constant shear stran across the beam cross-secton. Analytcal formulatons (Abramovch, 1998; Aldrahem and Khder, 000; Khder and Aldrahem, 001) and fnte elements (Robbns and Reddy, 1991; Shen, 1995; Narayanan and Balamurugan, 003; Ray and Mallk, 004; Neto et al., 009) based on FSDT are wdely used n the lterature for the analyss of pezoelectrc smart structures. Accuracy of the conventonal FSDT-based pezoelectrc beam fnte elements (Shen, 1995; Narayanan and Balamurugan, 003; Ray and Malk, 004; Neto et al., 009) s adversely affected by the nduced potental effects. These elements consder lnear through-thckness dstrbuton of electrc potental whch s actually nonlnear by vrtue of the nduced potental. The accuracy can be mproved usng assumed hgher-order approxmaton of through-thckness electrc potental (Jang and L, 007; Kapura and Hagedorn, 007; Wang et al., 007; Behesht-Aval and Lezgy-Nazargah, 01, 013). However, assumed hgher-order potental dstrbuton n the formulaton ntroduces addtonal nodal electrcal degrees of freedom n the transverse drecton and hence ncreases the computatonal cost. An alternate effcent way to nclude the hgher-order nduced potental n FSDT-based formulaton s to use the consstent through-thckness potental dstrbuton derved from the electrostatc equlbrum equaton (Sulbhewar and Raveendranath, 014b). Also, convergence of the conventonal two-noded soparametrc FSDT-based pezoelectrc beam element (Narayanan and Balamurugan, 003) depends on the extent of the extenson-bendng and bendng-shear couplngs. Recently, Sulbhewar and Raveendranath (015) proposed a novel FSDT pezoelectrc beam fnte element based on coupled polynomals for feld varables whch showed mproved convergence. However, ths element s not consstently accurate as the governng equatons used to defne coupled shape functons n ths formulaton are based on the assumed lnear through-thckness potental. The assocate errors are promnent for beams wth pezoelectrcally domnant cross-sectons and/or wth hgher pezoelectrc coeffcents. An deal FSDT-based formulaton whch s accurate and effcent over all geometrc and materal confguratons of the pezoelectrc beam should ncorporate the nduced potental couplng along wth other mechancal couplngs at the feld nterpolaton level tself. In the present work, an attempt s made to develop a novel FSDT pezoelectrc beam formulaton whch s consstently accurate and effcent throughout the applcable range of geometrc and materal propertes. The governng equatons are derved usng the varatonal formulaton based on FSDT n conjuncton wth the consstent through-thckness potental. The relatons establshed tween feld varables are used to defne coupled quadratc polynomals for axal dsplacement ( u 0 ) and secton rotaton ( ), havng contrbutons from the assumed cubc polynomal for transverse dsplacement ( w 0 ) and assumed lnear polynomals for layerwse electrc potental varables ( ). The shape functons based on these polynomals effcently handle change n stffness due to the

3 994 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve nduced potental along wth bendng-extenson and bendng-shear couplngs, n an effcent manner. Comparson of results from the present and the conventonal formulatons aganst ANSYS D benchmark smulaton results proves the mproved accuracy of the present formulaton over the conventonal formulatons. Convergence studes are carred out to prove the mproved convergence characterstcs of the present FSDT element over the conventonal soparametrc FSDT beam elements. It s noteworthy that owng to the fully coupled polynomal representaton for secton rotaton and coupled quadratc term n the nterpolaton polynomal for axal dsplacement and transverse electrc potental, the mproved performance has been acheved wth the same number of nodal degrees of freedom as of conventonal two-noded soparametrc FSDT-based pezoelectrc beam element. THEORETICAL FORMULATION An equvalent sngle layer (ESL) FSDT model for mechancal felds and a layerwse model for electrc potental ( ) are employed for the proposed formulaton. Consder a general multlayered extenson mode pezoelectrc smart beam wth total number of layers n, as shown n Fgure 1. The layers can be host layer(s) of conventonal materal or bonded/embedded layers of pezoelectrc materal. The beam layers are assumed to be made up of sotropc or specally orthotropc materals wth perfect bondng among them. Top and bottom faces of pezoelectrc layers are fully covered wth electrodes. Mechancal and electrcal quanttes are assumed to be small enough to apply lnear theores of elastcty and pezoelectrcty and assumptons of beam theory apply. Fgure 1: Geometry of a general multlayered pezoelectrc smart beam.

4 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve Reduced Consttutve Relatons For a general pezoelectrc smart structure, the elastc Cj (, j 1...6), pezoelectrc e ( k 1,,3; j 1...6) and delectrc ( k 1,,3) constants relate the mechancal and electrcal var- kj k ables through the three-dmensonal consttutve relatons. An extenson mode pezoelectrc smart beam wth axes of materal symmetry parallel to the beam axes s consdered here. For extenson mode, the transversely poled pezoelectrc materal s subjected to the transverse electrc feld. For such a beam, the general consttutve relatons are reduced accordng to beam theory, whch are gven as (Sulbhewar and Raveendranath 014 b): k k x Q11 0 e 31 x k k xz 0 Q 55 0 xz Dz e Ez (1) where ( =1.number of pezoelectrc layers), ( k =1..n).,,,,D and E denote the axal stress ( N m ), shear stress ( N m ), normal stran, shear stran, electrc dsplacement ( C m ) and electrc feld (V m), respectvely. The constants Q ( 1,5), e and denote reduced elastc ( N m ), pezoelectrc ( C m ) and delectrc ( F m ) propertes, respectvely.. Mechancal Dsplacements and Strans The mechancal dsplacement felds n the longtudnal and transverse drectons for FSDT are gven as (Narayanan and Balamurugan, 003): uxz (,) u() x z () x () 0 wxz (, ) w( x) 0 (3) u 0 ( x ) and w 0 ( x ) are the centrodal axal and transverse dsplacements, respectvely. s the secton rotaton of the beam. Dmensons Lbh,, denote the length, wdth and the total thckness of the beam, respectvely. Axal and shear stran felds are derved usng usual stran-dsplacement relatons as: (, ) ' ' x (, ) uxz x z u0 ( x ) z ( x x ) uxz (, ) wxz (, ) ' xz ( x, z) w0 ( x) ( x) z x (4) (5) where ()' denotes dervatve wth respect to x.

5 996 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve.3 Electrc Potental and Electrc Feld The layerwse two dmensonal electrc potental ( x, z) takes the values of 1 ( x ) and () x at the top and bottom faces of th pezolayer, respectvely as shown n Fgure 1. The through-thckness dstrbuton of electrc potental ( x, z) consstent wth FSDT s used (Sulbhewar and Raveendranath, 014b): z z e31 h 4( z z) ' ( x, z) ( x) ( x) 1 ( x) h 3 8 h (6) where /; ; z ( z z )/ The frst two terms of expresson (6) descrbe the conventonal lnear part n whch and are the mean and dfference, respectvely, of the top and bottom surface potentals of the th pezoelectrc layer. The quadratc term represents the couplng between curvature stran and electrc potental whch consttutes nduced potental. z The layerwse electrc feld ( E ) s obtaned from equaton (6) as (Sulbhewar and Raveendranath, 014b): 31 3 ( xz, ) ( x) e ' Ez( x, z) zz ( x) z h (7) 3 VARIATIONAL FORMULATION The formulaton s based on Hamlton s prncple whch mplctly takes care of natural boundary condtons. It s expressed as (Chee et al., 1999): t t (8) ( K H W) dt ( K H W) dt 0 t1 t1 where, K =knetc energy, H =electrc enthalpy densty functon for pezoelectrc materal and mechancal stran energy for the lnear elastc materal and W =external work done. 3.1 Varaton of Electromechancal/Stran Energy The electromechancal/stran energy varaton of the pezoelectrc smart beam s gven as (Chee et al., 1999): V k k x x xz xz z z H ED dv (9)

6 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve 997 Substtutng values of axal stran ( x ), shear stran ( xz ), electrc feld ( z E ) from equatons (4), (5), (7) and usng them along wth consttutve relatons gven by equaton (1) n expresson (9); the total varaton on the potental energy of the smart beam s gven as: t t ' k k ' k k ( e31) ' Hdt u0 Q 11I0 u0 Q 11I1 I1 I0z e 31I0 h t1 t1 x 3 ' k k ( e31) ' k k ( e31) ' Q 11I1 I 1 I0z u0 Q 11I I I0z e 31I0z h 3 3 k k ' ' k k ' Q 55I 0 ( w0 ) w 0 Q 55I 0 ( w0 ) ' ' e I h u e I z h I h dxdt (10) where k q1 q1 q k1 k I b( z z ) ( q1). 3. Varaton of Knetc Energy Total knetc energy of the beam s gven as (Chee et al., 1999): zk1 k (11) x zk 1 K b u w dzdx where k s the mass densty of k th 3 layer n kgm and ( k =1 n). Substtutng values of u and w from equatons () and (3) and applyng varaton, to derve at: where t t k t1 t1 x. () denotes t. k k k k k K dt u I u I I u I w I w dx dt (1) 3.3 Varaton of Work of External Forces Total vrtual work of the structure can be defned as the product of vrtual dsplacements wth forces for the mechancal work and the product of the vrtual electrc potental wth the charges for the electrcal work. The varaton of total work done by external mechancal and electrcal loadng s gven by (Chee et al., 1999): Wdt t t V V V S S u w u w S C C ufu wfw q ds uf wf dv uf wf ds dt t1 t1 0 S V S C n whch f, f, f are volume, surface and pont forces, respectvely. q 0 and S are the charge densty and area on whch charge s appled. (13)

7 998 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve 4 DERIVATION OF COUPLED FIELD RELATIONS The relatonshp between the feld varables s establshed here usng statc governng equatons. For statc condtons wthout any external loadng, the varatonal prncple gven n equaton (8) reduces to (Sulbhewar and Raveendranath, 015): H 0 (14) Applyng varaton to the basc varables n equaton (10), the statc governng equatons are obtaned as: where u '' u '' u ' u : A u A A 0 (15) '' '' ' ' : Au A A( w) A 0 (16) w ' '' w : A ( w ) 0 (17) u k k u k k ( e 31) u A1 Q 11I0; AQ 11I1 I1I0z ; A3 e31i0 h; 3 k k ( e31) k k ( 31) k k A1 Q 11I1 I 1 I0z A Q 11I I I0z A3 Q 55I0 A4 e 31I0z h 3 3 A w Q k I k e ; ; ; Assumng that the hgher order contnuous dervatves of varables appearng n the governng equaton (17) exst, we can wrte: '' ''' w 0 (18) Usng equatons (15) and (18), we can wrte the relatonshp of axal dsplacement ( u 0 ) wth transverse dsplacement ( w 0 ) and electrc potental varable ( ) as: u u u u u '' ''' ' 0 1w0 (19) where 1 A / A1 and A3 / A1. From equatons (16)-(19), we can wrte the relatonshp of secton rotaton ( ) wth transverse dsplacement ( w 0 ) and electrc potental varable ( ) as: u where A A1 A 3 u A A A and A1 A 3 A4 4 u A A A ' ''' ' 0 3w0 4 w (0) u 3 1 3

8 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve 999 These equatons for u 0 '' and are used n the next Secton to derve coupled polynomal expressons for the feld varables. It s clear that the couplng coeffcents j ( j 1,,3,4) whch depend on geometrc and materal propertes of the beam, relate all the feld varables by properly accommodatng bendng-extenson, bendng-shear and nduced potental couplngs. It s noteworthy that the constants Am n ( m1,,3,4) ( n u,, w) appearng n equatons (15)-(17), whch are used to defne the couplng coeffcents j ( j 1,,3,4) are dfferent from those gven n the Sulbhewar and n Raveendranath (015). The constants A m n the present formulaton contan addtonal stffness terms (shown n curly braces) due to the nduced potental effects. It may be noted that ths nduced stffness s proportonal to ( e ) whch bears the same unt (N/m ) as of elastc modulus Q 11. Hence, the quantty FINITE ELEMENT FORMULATION ( e ) may be termed as nduced modulus. Usng the varatonal formulaton descrbed above, a fnte element model s developed here. The two-noded beam element consdered here s based on FSDT wth layerwse electrc potental n the transverse drecton. There are three mechancal varables n the formulaton namely, u 0, w 0 and and layerwse electrc potental varables where ( =1...number of pezoelectrc layers n the beam). The equatons (19) and (0), derved usng the governng equlbrum equatons, demand contnuous thrd order dervatve of w 0 and frst order dervatve of. Hence, n terms of the natural coordnate, a cubc polynomal for transverse dsplacement w 0 and lnear polynomals for layerwse electrc potental varable are assumed as gven n equatons (1 a) and (1 b), respectvely. The transformaton between the local coordnate and the global coordnate x along the length of the beam s gven as ( xx ) / ( x x ) 1 and ( x x1) l, length of the beam element w b b b b (1a) 0 1 c c (1b) Usng equatons (1 a) and (1 b) n equaton (19) and ntegratng wth respect to, we get the coupled polynomal for axal dsplacement u 0 as: 6 / /4 u0 1 l b3 l c1 a1 a 0 () It s noted that the coupled quadratc term n equaton () contans contrbutons from w 0 and felds and does not brng n any addtonal generalzed degree of freedom. Substtutng equatons (1 a) and (1 b) n equaton (0), the coupled polynomal expresson for the secton rotaton s derved as: 3 b1(/ l) b (/ l) b 3 3 (/ l) 6 3(/ l) 4(/ l) c1 (3)

9 1000 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve Equaton (3) nterpolates by purely coupled terms wth contrbutons from w 0 and felds. It s noteworthy that equatons () and (3) take care of extenson-bendng, bendng-shear and nduced potental couplngs n a varatonally consstent manner wth the help of coupled terms present n the descrpton of axal dsplacement and secton rotaton. u w Usng equatons (1)-(3), the coupled shape functons [ Nm ] ( m 1..8), [ Nm ] ( m 1..6), [ Nm] ( m 1..6) and [ Nm ]( m 1,) whch relate the feld varables to ther nodal values as gven n equaton (4) are derved by usual method. The expressons for shape functons are gven n Appendx. 1 u 0 1 w0 1 u u u u u u u u u0 N 1 N N3 N4 N5 N6 N7 N 8 w w w w w w 1 w0 0 N1 N N3 0 N4 N5 N6 0 N1 N N3 0 N4 N5 N6 u N N w0 (4) As noted from the equaton (4), whle employng quadratc polynomals for axal dsplacement u0 and secton rotaton n the present FSDT formulaton, the number and type of nodal varables are mantaned the same as of the conventonal soparametrc FSDT formulaton. The varaton on basc mechancal and electrcal varables can now be transferred to nodal degrees of freedom. Substtutng equaton (4) n equatons (10), (1), (13) and usng them n equaton (8), the followng dscretzed form of the model s obtaned: M Kuu K u Q U U F Ku K where M s mass matrx, Kuu, Ku, Ku, K are global stffness sub-matrces. U, are the global nodal mechancal dsplacement and electrc potental degrees of freedom vectors, respectvely. F and Q are global nodal mechancal and electrcal force vectors, respectvely. The matrx equatons are now solved accordng to electrcal condtons (open/closed crcut), confguraton (actuator/sensor) and type of analyss (statc/dynamc). 6 NUMERICAL EXAMPLES AND DISCUSSIONS The software mplementaton of the present formulaton has been carred out n MATLAB envronment. The accuracy and effcency of the proposed FSDT fnte element are tested here for statc (actuaton/sensng) and modal (open/closed crcut) analyses and ts performance s compared (5)

10 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve 1001 aganst the conventonal two-noded soparametrc FSDT pezoelectrc beam fnte element avalable n the lterature. The followng desgnatons are used: FSDT-Coupled: The present formulaton whch uses the coupled polynomals (cubc for w 0 gven by equaton (1 a), coupled quadratc for u 0 gven by equaton (), coupled quadratc for gven by equaton (3) and lnear for gven by equaton (1 b)) for nterpolaton of feld varables and layerwse consstent through-thckness potental ( coupled quadratc approxmaton n z drecton gven by equaton (6)). FSDT: The conventonal FSDT formulaton of Narayanan and Balamurugan (003) whch uses ndependent polynomals for feld nterpolaton (lnear for u0, w0, and ) and layerwse assumed lnear through-thckness potental. ANSYS D: For a comparatve evaluaton of the above FSDT formulatons, benchmark solutons have been obtaned from a refned two-dmensonal analyss usng ANSYS fnte element software, for whch PLANE 183 elements are used to mesh conventonal materal layers, whle PLANE 3 elements are used to mesh pezoelectrc materal layers. 6.1 Example 1: A Symmetrc Bmorph Beam A bmorph cantlever beam wth oppostely poled pezoelectrc layers as shown n Fgure s consdered here. In order to study the effect of materal propertes on the performance of the FSDT elements, the followng materals are used whle the geometry s fxed ( h10 mm, L 100mm). PVDF (Sun and Huang, 000): E Gpa, 0.9, e0.046 Cm, Fm, 1800kgm PZT (EFunda.com, 014) C, C, C, C, C, C, C, C, C ,67.89,68.09,134.87,68.09,113.30,.,.,33.44 GPa e, e, e , , Cm,, ,8.7655, Fm kgm PZT 4 (EFunda.com, 014) C, C, C, C, C, C, C, C, C 139,77.84,74.8,139,74.8,115.41,5.64,5.64,30.58 GPa e, e, e 5.08, 5.08, Cm,, ,1.3060, Fm kgm PZT-5H (Kapura and Hagedorn, 007): C, C, C, C, C, C, C, C, C 16,79.5,84.1,16,84.1,117,3,3,3.5 GPa , 3, , 6.5,3.3 1,, ,1.503, e e e Cm Fm kgm PZT 5A (EFunda.com, 014) C, C, C, C, C, C, C, C, C 10.35,75.18,75.09,10.35,75.09,110.87,1.05,1.05,.57 GPa e, e, e 5.351, 5.351, Cm,, ,1.5317, Fm kgm.

11 100 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve PZT 8 (EFunda.com, 014) C, C, C, C, C, C, C, C, C ,81.09,81.05,146.88,81.05,131.71,31.35,31.35,3.89 GPa e, e, e , , Cm,, 1.14,1.14, Fm kgm 3 G1195N (Peng et al., 1998): E 63 GPa, 0.3, d 5410 mv, e E d Cm, Fm, 7600kgm. Fgure : Example 1: Geometry of a bmorph cantlever beam. For a comparatve evaluaton of accuracy of varous FSDT-based formulatons, converged results from a refned mesh of 40 elements have been used. Converged results from an ANSYS D smulaton wth a mesh of elements are used as a benchmark. Statc Analyss-Actuator Confguraton: In ths confguraton, the nterface of the bmorph s grounded and the voltages of 10volts are appled on the free surfaces. Table 1 and show the results for the tp deflecton and the maxmum axal stress developed n the bmorphs of dfferent materals, respectvely. Also, the assocated absolute errors calculated wth respect to ANSYS D benchmark solutons are presented n brackets. As seen from the tables, the conventonal FSDT formulaton (Narayanan and Balamurugan, 003) fals to produce consstently accurate results. The percentage errors ncrease wth the modulus rato e31 3 / Q 11 (the rato of nduced modulus to elastc modulus) of the materal. The present FSDT-Coupled element predcts accurate results for all the bmorphs, rrespectve of the modulus rato. Ths consstent performance of the present formulaton can be attrbuted to the accommodaton of nduced potental effects through the coupled nterpolaton polynomals. Statc Analyss-Sensor Confguraton: Here, the beam shown n Fgure s subjected to a tp load of 1000 N. The results for the tp deflecton, potental developed at the md-span and the maxmum axal stress developed at the root of the bmorphs of dfferent materals are tabulated n the Tables 3, 4 and 5, respectvely. The assocated absolute errors (n percentage) wth respect to AN- SYS D benchmark solutons are presented n brackets. As seen from the tables, the present FSDT- Coupled consstently reproduces the ANSYS D smulaton results for all the bmorphs, unlke the conventonal formulaton. The accuracy of the present FSDT-Coupled formulaton s practcally nsenstve to the materal propertes of the beam.

12 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve 1003 Materals Modulus rato FSDT (Narayanan and Balamurugan, 003) ANSYS D FSDT-Coupled PVDF (0.91 %) (0.000 %) PZT (1.408 %) (0.056 %) PZT (1.677 %) (0.035 %) PZT (1.906 %) (0.08 %) PZT-5A (.090 %) (0.040 %) PZT-5H (4.38 %) (0.038 %) G1195N (6.73 %) (0.04 %) Table 1: Example 1: Absolute tp deflecton ( m ) of the bmorph cantlever beams of dfferent pezoelectrc materals actuated by ±10 volts. (The absolute errors n percentage are gven wth respect to ANSYS D smulaton.) Materals Modulus rato FSDT (Narayanan and Balamurugan, 003) ANSYS D FSDT-Coupled PVDF (0.648 %) (0.000 %) PZT (4.174 %) (0.000 %) PZT (4.810 %) (0.009 %) PZT (5.366 %) (0.019 %) PZT-5A (5.868 %) (0.000 %) PZT-5H (11.18 %) (0.000 %) G1195N (15.99 %) (0.000 %) Table : Example 1: Absolute maxmum axal stress developed (kpa) n the bmorph cantlever beams of dfferent pezoelectrc materals actuated by ±10 volts. (The absolute errors n percentage are gven wth respect to ANSYS D smulaton.) Materals Modulus rato FSDT (Narayanan and Balamurugan, 003) ANSYS D FSDT-Coupled PVDF (0.090 %) (0.090 %) PZT (1.456 %) (0.061 %) PZT (1.373 %) (0.39 %) PZT (1.661 %) (0.145 %) PZT-5A (1.75 %) (0.1 %) PZT-5H (3.757 %) (0.077 %) G1195N (5.47 %) (0.319 %) Table 3: Example 1: Absolute tp deflecton ( m ) of the bmorph cantlever beams of dfferent pezoelectrc materals subjected to a tp load of 1000 N. (The absolute errors n percentage are gven wth respect to ANSYS D smulaton.)

13 1004 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve Materals Modulus rato FSDT (Narayanan and Balamurugan, 003) ANSYS D FSDT-Coupled PVDF (0.61 %) (0.019 %) PZT (1.411 %) (0.000 %) PZT (1.64 %) (0.011 %) PZT (1.835 %) (0.007 %) PZT-5A (.001 %) (0.005 %) PZT-5H (3.871 %) (0.000 %) G1195N (5.68 %) (0.00 %) Table 4: Example 1: Potental developed (volts) at the md-span of the bmorph cantlever beams of dfferent pezoelectrc materals subjected to a tp load of 1000 N. (The absolute errors n percentage are gven wth respect to ANSYS D smulaton.) Materals Modulus rato FSDT (Narayanan and Balamurugan, 003) ANSYS D FSDT-Coupled PVDF (0.750 %) (0.117 %) PZT (.517 %) (0.745 %) PZT (.861 %) (0.860 %) PZT (3.011 %) (0.87 %) PZT-5A (3.305 %) (0.958 %) PZT-5H (5.076 %) (0.893 %) G1195N (6.81 %) (0.893 %) Table 5: Example 1: Absolute maxmum axal stress developed (MPa) at the root of the bmorph cantlever beams of dfferent pezoelectrc materals subjected to a tp load of 1000 N. (The absolute errors n percentage are gven wth respect to ANSYS D smulaton.) The convergence graphs plotted n Fgures 3 and 4 for the tp deflecton and potental developed at the root, respectvely, compare the effcency of the FSDT-based pezoelectrc beam fnte elements. The G1195N bmorph whch has the hghest modulus rato among the chosen materals s taken as a partcular example for ths study. The FSDT-Coupled shows sngle-element convergence, closely reproducng the ANSYS-D solutons for both the tp deflecton and the potental developed. The conventonal FSDT (Narayanan and Balamurugan, 003) overestmates the response as t neglects nduced potental effects.

14 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve 1005 Fgure 3: Example 1: Sensor confguraton: Convergence characterstcs of the FSDT-based pezoelectrc beam fnte elements to predct the tp deflecton of the G1195N bmorph cantlever beam subjected to a tp load of 1000 N. Fgure 4: Example 1: Sensor confguraton: Convergence characterstcs of the FSDT-based pezoelectrc beam fnte elements to predct the potental developed at the root of the G1195N bmorph cantlever beam subjected to a tp load of 1000 N. Modal Analyss: The accuracy and effcency of the FSDT elements n predctng the natural frequences of the bmorph cantlever beam shown n Fgure are compared here. The natural frequences are evaluated for closed and open crcut electrcal boundary condtons, wth dfferent materals. For open crcut, only the nterface of the bmorph s grounded whle, for closed crcut all the faces of bmorph are grounded. The results tabulated n Table 6 reveal the nablty of conventonal FSDT formulaton to mantan the accuracy over the dfferent bmorph materals. The consstent accuracy of the present FSDT-Coupled results valdates the use of coupled polynomal shape functons n generatng the element mass matrx consstent wth the element stffness matrx.

15 1006 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve Materals Modulus rato FSDT (Narayanan and Balamurugan, 003) Open crcut ANSYS D FSDT-Coupled PVDF (0.135 %) (0.018 %) PZT (0.736 %) (0.051 %) PZT (0.8 %) (0.05 %) PZT (0.85 %) (0.106 %) PZT-5A (1.01 %) (0.043 %) PZT-5H (1.814 %) (0.045 %) G1195N (.68 %) (0.040 %) Closed crcut PVDF (0.14 %) (0.018 %) PZT (0.739 %) (0.0 %) PZT (0.960 %) (0.14 %) PZT (0.991 %) (0.053 %) PZT-5A (1.189 %) (0.158 %) PZT-5H (.70 %) (0.177 %) G1195N (3.06 %) (0.017 %) Table 6: Example 1: Frst natural frequency (Hz) of the bmorph cantlever beams of dfferent pezoelectrc materals wth open and closed crcut electrcal boundary condtons. (The absolute errors n percentage are gven wth respect to ANSYS D smulaton.) Fgures 5 and 6 show the comparson of convergence characterstcs of FSDT-based pezoelectrc beam fnte element formulatons to predct the frst natural frequency of the G1195N bmorph n open and closed crcut condtons, respectvely. FSDT-Coupled shows quck convergence, closely reproducng the ANSYS-D solutons for both open and closed crcut condtons. The conventonal FSDT (Narayanan and Balamurugan, 003) model underestmates the response as t neglects nduced potental effects. Fgure 5: Example 1: Modal analyss: Convergence characterstcs of the FSDT-based pezoelectrc beam fnte elements to predct the frst natural frequency of the G1195N bmorph cantlever beam n open crcut electrcal boundary condton.

16 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve 1007 Fgure 6: Example 1: Modal analyss: Convergence characterstcs of the FSDT-based pezoelectrc beam fnte elements to predct the frst natural frequency of the G1195N bmorph cantlever beam n closed crcut electrcal boundary condton. 6. Example : A Two-Layer Asymmetrc Pezoelectrc Beam A two-layer asymmetrc pezoelectrc cantlever beam havng a steel host layer wth a surface bonded pezoelectrc layer of G1195N at the top, as shown n Fgure 7 s consdered here. The materal propertes used are: Steel (Carrera and Brschetto, 008): E 10 GPa, 0.3, 7850kgm 1 1 G1195N (Peng et al., 1998): E 63 GPa, 0.3, d 5410 mv, e E d Cm, Fm, 7600kgm. Fgure 7: Example : Geometry of a two-layer asymmetrc pezoelectrc cantlever beam. The length and total heght of the beam are fxed ( L 100 mm, h 5mm), whle thcknesses of the pezoelectrc layer ( h p ) and the host layer ( h c ) are vared. The performance of the FSDT-based pezoelectrc beam fnte elements s evaluated over a wde range of the pezoelectrc materal proporton n the total beam thckness (thckness rato: r h / h). For a comparatve evaluaton of accuracy of varous FSDT based formulatons, converged results from a refned mesh of 40 elements have been used. The converged results from an ANSYS D smulaton wth a mesh of 00 0 elements are used as a benchmark. p

17 1008 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve Statc Analyss-Sensor Confguraton: In ths confguraton, the beam shown n Fgure 7 s subjected to a tp load of N. The results for transverse tp deflecton, axal tp deflecton and potental developed across the pezoelectrc layer at the md-span of the beam for varous thckness ratos are tabulated n Tables 7, 8 and 9, respectvely. The present FSDT-Coupled formulaton proves ts versatlty, yeldng consstently accurate predctons over the entre range of thckness rato. The conventonal FSDT formulaton (Narayanan and Balamurugan, 003) does not mantan the consstent accuracy, as t neglects the nduced potental couplng. The assocated error ncreases rapdly n the hgher thckness rato regmes. Thckness rato (r) ANSYS D FSDT-Coupled FSDT (Narayanan and Balamurugan, 003) (0.000 %) (0.000 %) (0.000 %) (0.000 %) (0.047 %) (0.047 %) (0.04 %) (0.97 %) (0.079 %) (0.793 %) (0.115 %) (1.651 %) (0.153 %) (.937 %) (0.190 %) (4.89 %) (0.148 %) (7.766 %) (0.100 %) (1.98 %) (0.075 %) (6.95 %) Table 7: Example : Transverse tp deflecton ( mm ) of the asymmetrc pezoelectrc cantlever beam (Steel/G1195N) subjected to a tp load of N. (The absolute errors n percentage are gven wth respect to ANSYS D smulaton.) Thckness rato (r) ANSYS D FSDT-Coupled FSDT (Narayanan and Balamurugan, 003) (0.053 %) (0.053 %) (0.050 %) (0.074 %) (0.034 %) (0.13 %) (0.014 %) (0.37 %) (0.091 %) (0.77 %) (0.393 %).197 (1.365 %) (0.095 %).3867 (3.008 %) (0.004 %).3656 (5.030 %) (0.065 %).1376 (8.014 %) (0.078 %) (13.0 %) (0.000 %) (0.000 %) Table 8: Example : Axal tp deflecton ( m ) of the asymmetrc pezoelectrc cantlever beam (Steel/G1195N) subjected to a tp load of N. (The absolute errors n percentage are gven wth respect to ANSYS D smulaton.)

18 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve 1009 Thckness rato (r) ANSYS D FSDT-Coupled FSDT (Narayanan and Balamurugan, 003) (0.006 %) (0.006 %) (0.03 %) (0.03 %) (0.000 %) (0.091 %) (0.008 %) (0.338 %) (0.000 %) (0.865 %) (0.000 %) (1.761 %) (0.000 %) (3.10 %) (0.005 %) (5.03 %) (0.006 %) (7.94 %) (0.008 %) (13.11 %) (0.000 %) (0.000 %) Table 9: Example : Potental developed (volts) at the md-span of the asymmetrc pezoelectrc cantlever beam (Steel/G1195N) subjected to a tp load of N. (The absolute errors n percentage are gven wth respect to ANSYS D smulaton.) The convergence graphs plotted n Fgures 8 and 9 for the transverse tp deflecton and the potental developed at the root, respectvely, prove the consstent effcency of the present FSDT- Coupled formulaton, whch exhbts sngle-element convergence to ANSYS-D solutons. FSDT (Narayanan and Balamurugan, 003) model shows very slow convergence to the naccurate results, due to nduced potental effects. Fgure 8: Example : Sensor confguraton: Convergence characterstcs of FSDT-based formulatons to predct the transverse tp deflecton of the two-layer asymmetrc pezoelectrc cantlever beam (r=0.5) subjected to a tp load of N.

19 1010 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve Fgure 9: Example : Sensor confguraton: Convergence characterstcs of FSDT-based formulatons to predct the potental developed at root of the two-layer asymmetrc pezoelectrc cantlever beam (r=0.5) subjected to a tp load of N. Statc Analyss-Actuator Confguraton: Here, the beam shown n Fgure 7 s actuated by 100 volts. The varatons of transverse and axal deflectons at the tp, wth thckness rato are tabulated n Tables 10 and 11, respectvely. The FSDT-Coupled formulaton consstently gves accurate predctons of results over the entre range of thckness rato as gven by ANSYS D smulaton. The conventonal FSDT formulaton (Narayanan and Balamurugan, 003) does not yeld consstently accurate results. Thckness rato (r) ANSYS D FSDT-Coupled FSDT (Narayanan and Balamurugan, 003) (0.009 %) (0.01 %) (0.009 %) (0.08 %) (0.000 %) (0.096 %) (0.007 %) (0.380 %) (0.014 %) (0.986 %) (0.007 %) (1.999 %) (0.000 %) 1.58 (3.504 %) (0.000 %) (5.570 %) (0.00 %) (8.516 %) (0.006 %) (13.5 %) (0.000 %) (0.000 %) Table 10: Example : Transverse tp deflecton ( m ) of the asymmetrc pezoelectrc cantlever beam (Steel/G1195N) actuated by 100 volts. (The absolute errors n percentage are gven wth respect to ANSYS D smulaton.)

20 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve 1011 Thckness rato (r) ANSYS D FSDT-Coupled FSDT (Narayanan and Balamurugan, 003) (0.000 %) (0.000 %) (0.049 %) (0.049 %) (0.039 %) (0.077 %) (0.094 %) (0.50 %) (0.07 %) (0.399 %) (0.048 %) (0.91 %) (0.8 %) (1.645 %) (0.5 %) (.069 %) (0.045 %) (1.951 %) (0.154 %) (1.411 %) (0.000 %) (0.000 %) Table 11: Example : Axal tp deflecton ( m ) of the asymmetrc pezoelectrc cantlever beam (Steel/G1195N) actuated by 100 volts. (The absolute errors n percentage are gven wth respect to ANSYS D smulaton.) Modal Analyss: The FSDT-Coupled formulaton s evaluated here for ts accuracy and effcency to predct the natural frequences of the asymmetrc pezoelectrc smart beam. The frst natural frequency of the asymmetrc Steel/G1195N beam shown n Fgure 7 s computed for both open and closed crcut electrcal boundary condtons. Table 1 shows the varaton of frst natural frequences wth thckness rato. The results of FSDT-Coupled formulaton agree very well wth the ANSYS D smulaton results. The results of the conventonal FSDT formulaton (Narayanan and Balamurugan, 003), show sgnfcant devaton n the hgher thckness rato regmes where the nduced potental effects are predomnant. The consstent effcency of the present FSDT-Coupled s revealed by the convergence graphs for frst natural frequency n both open and closed crcut electrcal boundary condtons plotted n Fgures 10 and 11, respectvely. As seen from the fgures, the FSDT-Coupled gves fast convergence, whle FSDT (Narayanan and Balamurugan, 003) model shows very slow convergence to the naccurate results, due to nduced potental effects. Fgure 10: Example : Modal Analyss: Convergence characterstcs of FSDT-based formulatons to predct the frst natural frequency of the two-layer asymmetrc pezoelectrc cantlever beam (r=0.5) n open crcut electrcal boundary condton.

21 101 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve Thckness rato (r) ANSYS D FSDT-Coupled FSDT (Narayanan and Balamurugan, 003) Open crcut (0.035 %) (0.035 %) (0.037 %) (0.039 %) (0.037 %) (0.081 %) (0.036 %) (0.01 %) (0.01 %) (0.450 %) (0.003 %) (0.863 %) (0.034 %) (1.481 %) (0.050 %) (.371 %) (0.044 %) (3.780 %) (0.06 %) (5.931 %) (0.003 %) 3.14 (11.7 %) Closed crcut (0.035 %) (0.035 %) (0.037 %) (0.04 %) (0.043 %) (0.090 %) (0.043 %) 3.06 (0.6 %) (0.035 %) (0.518 %) (0.016 %) (1.000 %) (0.010 %) (1.695 %) (0.033 %) (.637 %) (0.04 %) (3.95 %) (0.07 %) (6.094 %) (0.003 %) 3.14 (11.7 %) Table 1: Example : Natural frequences of the asymmetrc pezoelectrc cantlever beam (Steel/G1195N) n open and closed crcut electrcal boundary condtons. (The absolute errors n percentage are gven wth respect to ANSYS D smulaton.) Fgure 11: Example : Modal Analyss: Convergence characterstcs of FSDT-based formulatons to predct the frst natural frequency of the two-layer asymmetrc pezoelectrc cantlever beam (r=0.5) n closed crcut electrcal boundary condton.

22 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve CONCLUSIONS Based on coupled polynomal feld nterpolatons n conjuncton wth a consstent through-thckness electrc potental, a novel FSDT based extenson mode pezoelectrc beam fnte element has been presented. The derved set of coupled shape functons handles bendng-extenson, bendng-shear and nduced potental couplngs n a varatonally consstent manner. Numercal evaluaton has proven the merts of the present formulaton over the conventonal formulatons avalable n the lterature, n terms of accuracy and effcency. From the numercal analyss, t was found that: The performance of the conventonal FSDT pezoelectrc beam fnte elements depends on the geometrc and materal parameters of the beam. The accuracy depends on the proporton of pezoelectrc materal n the total beam thckness (thckness rato) and the rato of nduced modulus to the elastc modulus (modulus rato). The convergence rate of conventonal FSDT elements s deterorated by the presence of bendng-shear and bendngextenson couplngs. The performance of the proposed FSDT-Coupled formulaton proves to be nsenstve to the materal and geometrc confguraton of the beam cross-secton. It consstently mantans the level of accuracy and effcency for all modulus and thckness ratos. References Abramovch, H. (1998). Deflecton control of lamnated composte beams wth pezoceramc layers-closed form solutons, Composte Structures 43: Abramovch, H., Pletner, B. (1997). Actuaton and sensng of pezolamnated sandwch type structures, Composte Structures 38: Aldrahem, O.J., Khder, A.A. (000). Smart beams wth extenson and thckness-shear pezoelectrc actuators, Smart Materals and Structures, 9: 1-9. Behesht-Aval, S.B., Lezgy-Nazargah, M. (01). A coupled refned hgh-order global-local theory and fnte element model for statc electromechnacal response of smart maltlayered/sandwch beams, Archeve of Appled Mechancs 8: Behesht-Aval, S.B., Lezgy-Nazargah, M. (013). Coupled refned layerwse theory for dynamc free and forced response of pezoelectrc lamnated composte and sandwch beams, Meccanca 48: Bendary, I.M., Elshafe, M.A., Rad, A.M. (010). Fnte element model of smart beams wth dstrbuted pezoelectrc actuators, J. Intellgent Materal Systems and Structures 1: Benjeddou, A., Trndade, M.A., Ohayon, R. (1997). A unfed beam fnte element model for extenson and shear pezoelectrc actuaton mechansms, J. Intellgent Materal Systems and Structures 8: Benjeddou, A., Trndade, M.A., Ohayon, R. (000). Pezoelectrc actuaton mechansms for ntellgent sandwch structures, Smart Materals and Structures 9: Carrera, E., Brschetto, S. (008). Analyss of thckness lockng n classcal, refned and mxed multlayered plate theores, Composte Structures 8: Chee, C.Y.K., Tong, L., Steven, G.P. (1999). A mxed model for composte beams wth pezoelectrc actuators and sensors, Smart Materals and Structures 8: Crawley E.F., Anderson, E.H. (1990). Detaled models of pezoceramc actuaton of beams, J. Intellgent Materal Systems and Structures 1: 4-5.

23 1014 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve Crawley, E.F., de Lus, J. (1987). Use of pezoelectrc actuators as elements of ntellgent structures, AIAA J. 5: Efunda.com, (014). efunda:pezo Materal Data, [onlne] Avalable at: materal_data/matdata_ndex.cfm [Accessed Aug. 014]. Jang, J.P., L, D.X. (007). A new fnte element model for pezothermoelastc composte beam, J. Sound and Vbraton 306: Kapura, S., Hagedorn, P. (007). Unfed effcent layerwse theory for smart beams wth segmented extenson/shear mode, pezoelectrc actuators and sensors, J. Mechancs of Materals and Structures : Khder, A.A., Aldrahem, O.J. (001). Deflecton analyss of beams wth extenson and shear pezoelectrc patches usng dscontnuty functons, Smart Materals and Structures 10: 1-0. Kumar, K.R., Narayanan, S. (008). Actve vbraton control of beams wth optmal placement of pezoelectrc sensor/actuator pars, Smart Materals and Structures 17: Marnkovc, D., Marnkovc, Z. (01). On FEM modelng of pezoelectrc actuators and sensors for thn-walled structures, Smart Structures and Systems 9: Narayanan, S., Balamurugan, V. (003). Fnte element modellng of pezolamnated smart structures for actve vbraton control wth dstrbuted sensors and actuators, J. Sound and Vbraton 6: Neto, M.A., Yu, W., Roy, S. (009). Two fnte elements for general composte beams wth pezoelectrc actuators and sensors, Fnte Elements n Analyss and Desgn 45: Peng, X.Q., Lam, K.Y., Lu, G.R. (1998). Actve vbraton control of composte beams wth pezoelectrcs: a fnte element model wth thrd order theory, J. Sound and Vbraton 09: Raja, S., Prathap, G., Snha, P.K. (00). Actve vbraton control of composte sandwch beams wth pezoelectrc extenson-bendng and shear actuators, Smart Materals and Structures 11: Ray, M.C., Mallk, N. (004). Actve control of lamnated composte beams usng a pezoelectrc fber renforced composte layer, Smart Materals and Structures 13: Robbns, D.H., Reddy, J.N. (1991). Analyss of pezoelectrcally actuated beams usng a layer-wse dsplacement theory, Computers and Structures 41: Shen, M.H.H. (1995). A new modelng technque for pezoelectrcally actuated beams, Computers and Structures 57: Sulbhewar, L.N., Ravenndranath, P. (014a). A numercally accurate and effcent coupled polynomal feld nterpolaton for Euler-Bernoull pezoelectrc beam fnte element wth nduced potental effect, J. Intellgent Materal Systems and Structures, DOI: X , 6(1): Sulbhewar, L.N., Raveendranath, P. (014b). An accurate novel coupled feld Tmoshenko pezoelectrc beam fnte element wth nduced potental effects, Latn Amercan J. Solds and Structures 11: Sulbhewar, L.N., Raveendranath, P. (015). A lockng-free coupled polynomal Tmoshenko pezoelectrc beam fnte element, Engneerng Computatons, 3(5): Sun, B., Huang, D. (000). Analytcal vbraton suppresson analyss of composte beams wth pezoelectrc lamnae, Smart Materals and Structures 9: Wang, F., Tang, G.J., L, D.K. (007). Accuarate modelng of a pezoelectrc composte beam, Smart Materals and Structures 16: Zhang, X.D., Sun, C.T. (1996). Formulaton of an adaptve sandwch beam, Smart Materals and Structures 5:

24 L.N. Sulbhewar and P. Raveendranath / A Tmoshenko Pezoelectrc Beam Fnte Element wth Consstent Performance Irrespectve 1015 Appendx: Coupled Shape Functons 3 u u 31l u 31l u l 1 l( 14 3) l 4834l 9638l (1 ) N ; N (1 ); N ( 1); N ( 1); 3 u u 31l u 31l u l 1 l( 14 3) l 4834l 9638l (1 ) N ; N ( 1); N ( 1); N (1 ); 3 3 w 1 l (43 3 l ) w l l w 4l N1 ; N ( 1); N3 ( 1); 4834l l 4834l 3 3 w 1 l (43 3 l ) w l l w 4l N4 ; N 5 (1 ); N 6 (1 ); 4834l l 4834l 3l 4 3 l (1 3 ) 34l l 4834l 43 l N ( 1); N ; N (1 ); 3l 4 3 l (1 3 ) 34l l 4834l 43l N (1 ); N ; N ( 1); N (1 ) (1 ) ; ; 1 N