Enhanced structural behavior of flexible laminated composite beams Yulia Fridman, Haim Abramovich * Faculty of Aerospace Engineering, Technion IIT, 32000 Haifa, Israel Available online 3 June 2007 Abstract The aim of the present study is to investigate analytically and numerically the structural behavior of laminated composite beams under axial compression using piezoelectric layers. A mathematical model was developed based on a first order shear deformation theory (FSDT) which includes shear deformations, usually neglected in the classical lamination theory of composite structures. Closed form solutions for the bending angle and the axial and lateral displacements along the beam are presented for various boundary conditions. Natural frequencies and their associated mode shapes, as well as, buckling loads were computed for beams with and without piezo- electric layers influence, having various boundary conditions and lay-ups. Next, the influence of the piezoelectric layers on the axial compression load and the natural frequencies is investigated to yield an enhancement of the structural behavior of the beam. This is done using a proportional control load, in which the sensed voltage on the beam is fed back (after being amplified using a constant gain G), onto the PZT actuators which prevent the premature buckling of the flexible beam by actively increasing its stiffness. A parametric investigation was performed for beams with various lay-ups, and it was shown that for a given feedback gain value, the natural frequencies and the buckling loads can be increased by a factor of two, when using the enhancement procedure developed within the present study. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Laminated composite beam; PZT; Buckling load; Gain; Enhanced buckling load; Piezoelectric patch 1. Introduction The issue of increasing the buckling loads and natural frequencies of beams and plates using intelligent materials such as piezoelectric or shape memory alloy (SMA) mate- rials had recently become very popular. However most of the studies concentrate on basic mod- els, like Euler–Bernoulli beam theory, while most of the emphasis is made on the control part. Also most of the works had only focused on the control of structural vibra- tions. Typical references are next highlighted. Abramovich and Livshits [1] investigated the dynamical behavior of piezo-laminated composite beams with general non-symmetric lay-up. The natural frequencies of the beam with various boundary conditions were calculated and the influence of continuous piezoelectric layers bonded on the top and bottom of the beams, acting as actuators in the open loop, was investigated. Song et al. [2] studied the active vibration control of composite beams using piezoelectric ceramic patches as sensors and actua- tors. Two control algorithms were developed to achieve the active vibration damping. The results were calculated based on the theoretical Euler–Bernoulli model, and numerical simulations were obtained using the ANSYS finite element code. A good correlation was shown between the numerical predictions and the experimental results. Huang and Sun [3] developed a beam model based on the Reissner–Mindlin plate theory to demonstrate the dynamic analysis of composite beams with bonded or embedded composite sensors and actuators. Waisman and Abramovich [4] studied the stiffening effects of smart composite beams with piezo-ceramics layers or patches bonded on the surface of the beam. The analysis considers the linear piezoelectric constitutive relations and the first 0263-8223/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2007.05.007 * Corresponding author. Tel.: +972 4 8293199; fax: +972 4 8292303. E-mail addresses: [email protected](Y. Fridman), haim@ aerodyne.technion.ac.il (H. Abramovich). www.elsevier.com/locate/compstruct Composite Structures 82 (2008) 140–154
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www.elsevier.com/locate/compstruct
Composite Structures 82 (2008) 140–154
Enhanced structural behavior of flexible laminated composite beams
Yulia Fridman, Haim Abramovich *
Faculty of Aerospace Engineering, Technion IIT, 32000 Haifa, Israel
Available online 3 June 2007
Abstract
The aim of the present study is to investigate analytically and numerically the structural behavior of laminated composite beamsunder axial compression using piezoelectric layers. A mathematical model was developed based on a first order shear deformation theory(FSDT) which includes shear deformations, usually neglected in the classical lamination theory of composite structures. Closed formsolutions for the bending angle and the axial and lateral displacements along the beam are presented for various boundary conditions.
Natural frequencies and their associated mode shapes, as well as, buckling loads were computed for beams with and without piezo-electric layers influence, having various boundary conditions and lay-ups.
Next, the influence of the piezoelectric layers on the axial compression load and the natural frequencies is investigated to yield anenhancement of the structural behavior of the beam. This is done using a proportional control load, in which the sensed voltage onthe beam is fed back (after being amplified using a constant gain G), onto the PZT actuators which prevent the premature bucklingof the flexible beam by actively increasing its stiffness.
A parametric investigation was performed for beams with various lay-ups, and it was shown that for a given feedback gain value, thenatural frequencies and the buckling loads can be increased by a factor of two, when using the enhancement procedure developed withinthe present study.� 2007 Elsevier Ltd. All rights reserved.
The issue of increasing the buckling loads and naturalfrequencies of beams and plates using intelligent materialssuch as piezoelectric or shape memory alloy (SMA) mate-rials had recently become very popular.
However most of the studies concentrate on basic mod-els, like Euler–Bernoulli beam theory, while most of theemphasis is made on the control part. Also most of theworks had only focused on the control of structural vibra-tions. Typical references are next highlighted.
Abramovich and Livshits [1] investigated the dynamicalbehavior of piezo-laminated composite beams with generalnon-symmetric lay-up. The natural frequencies of thebeam with various boundary conditions were calculated
0263-8223/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.
and the influence of continuous piezoelectric layersbonded on the top and bottom of the beams, acting asactuators in the open loop, was investigated. Song et al.[2] studied the active vibration control of composite beamsusing piezoelectric ceramic patches as sensors and actua-tors. Two control algorithms were developed to achievethe active vibration damping. The results were calculatedbased on the theoretical Euler–Bernoulli model, andnumerical simulations were obtained using the ANSYSfinite element code. A good correlation was shownbetween the numerical predictions and the experimentalresults. Huang and Sun [3] developed a beam model basedon the Reissner–Mindlin plate theory to demonstrate thedynamic analysis of composite beams with bonded orembedded composite sensors and actuators. Waismanand Abramovich [4] studied the stiffening effects of smartcomposite beams with piezo-ceramics layers or patchesbonded on the surface of the beam. The analysis considersthe linear piezoelectric constitutive relations and the first
Y. Fridman, H. Abramovich / Composite Structures 82 (2008) 140–154 141
order shear deformation theory. The influence of the actu-ators is evaluated by means of the pin-force model andtheir size and location along the beam are taken intoaccount. The numerical solution of the equations ofmotion are compared with the FE results and it was foundthat piezoelectric bonded actuators are yielding significantchanges in the natural frequencies and mode shapes of thebeam. Raja et al. [5] derived a finite element formulationcapable of modeling extension–bending and shear inducedactuation in an adaptive composite sandwich beam basedon Timoshenko’s beam theory. The efficiency of bothactuators in controlling the bending vibration was studiedusing modal control analysis. The shear actuator performsbetter in controlling the first three bending modes than theextension-bending actuator. Sloss et al. [6] investigated theeffect of axial load, the piezo-sensor and the actuator feed-back control on the vibration frequencies and the modeshapes of the beam. The first three vibration frequenciesof controlled and uncontrolled beam were presented forvarious values of the feedback gain, axial load and patchsizes. Karagulle et al. [7] showed that the Ansys/Multi-physics finite element code can be used for the simulationof the active vibration control of smart structures.
Mukherjee and Chaudhuri [8] presented an exact solu-tion for the feedback vibration control of piezo-laminatedcolumns, in which the signals from the sensors attachedto the structure are fed back to the actuators with a gainmultiplier. The analytical solution has been validated withexperimental studies for tip loaded piezo-laminated cantile-ver beam with collocated PVDF layers using as sensors andas actuators.
In their previous work, Mukherjee and Chaudhuri [9]developed an imperfection approach for exact solutionsof the instability of piezo-laminated symmetrical columnsunder static and dynamics axial loads. Their solution isbased on Euler–Bernoulli beam theory. A constant gainfeedback control algorithm is derived using modified stiff-ness yielding an increased Euler’s buckling load while usingpiezoelectric sensing and actuation. A limiting actuationfeedback gain is derived. Meressi and Paden [10] observedthat the buckling of a flexible Euler–Bernoulli beam can bepostponed beyond the first and under the second criticalload by stabilization of the first bending mode by meansof a feedback control using piezoelectric actuators andstrain sensors. This is followed by the state-space modelof the reduced order system and designing of a controllerby using standard linear quadratic regulator (LQR) withconstant feedback gain. Chase et al. presented optimal sta-bilizations of column buckling [11] and plate buckling [12]using MEMS-based strain sensors and embedded piezo-electric ceramic patches. The column is fixed with pinnedends and the axial load is applied dynamically. The stabil-ity of the resulting controllers is based on multi-input,multi-output (MIMO) methodology using Lyapunov’smethods. The finite element column model is presented instate-space equations. The optimal buckling controllerswere tested on a column made of G-10 fiberglass yielding
an increase in the critical buckling load by a factor of2.9. In Ref. [12] the derivation of Ref. [11] is expanded tothe case of axially loaded composite plates clamped onall four sides. The controller is designed for Pdesired =1.5Pcr. For this case, the calculations became complicatewith a high requirement for computer time and controlpower.
Thompson and Loughlan [13] performed experimentson the active buckling control of pin-ended composite col-umn strips made of graphite-epoxy using lateral deflectiondisplacement sensor and surface bonded piezo-ceramicactuators. The test procedure is outlined and load–deflec-tion plots, obtained with and without active control, arepresented. For the lay-up configurations considered, theincrease in the load carrying capability is of the order of19.8–37.1%.
Chandrashekhara and Bhatia [14] presented a finite ele-ment analysis for active buckling control of laminatedcomposite plates using piezoelectric sensors and actuators.The finite element model is based on the first order sheardeformation plate theory in conjunction with linear piezo-electric theory. The sensor output is used to determine theinput to the actuator using proportional control algorithm.The presented finite element solutions show effectiveness ofpiezoelectric materials in enhancing the buckling loads.Berlin [15] presented results of an analysis based onEuler–Bernoulli beam theory together with an experimen-tal work and showed that buckling can be preventedtrough computer-controlled adjustment of dynamicalbehavior. He used piezo-ceramic actuators bonded on thesurface of a steel column to counteract buckling. Activecontrol of the buckling allows this column to support 5.6times more axial load then the original buckling load. Thiswas done with a complicated control law mechanism.Wang and Quek [16] showed the increase of the fluttervelocity and buckling capacity of a fixed-free column, sub-jected to a follower force using a pair of piezoelectriclayers.
In the present study, a mathematical model of a piezo-laminated composite beam was developed, formulatedand applied. It is based on a first order shear deformationtheory (FSDT) which includes shear deformations, usuallyneglected in classical lamination theory of composite struc-tures, and linear piezoelectric constitutive relations. Thethree coupled partial differential equations of motion of ageneral non-symmetric piezo-laminated composite beamsubjected to axial and lateral tractions are presented andsolved for the dynamic case – to find natural frequenciesand mode shapes for an axially compressed beam, withand without axially loads, and for the static case – to findthe buckling load, the bending angle, the axial and lateraldisplacements.
Moreover, the buckling load of a laminated compositebeam with various boundary conditions is enhanced usingpiezoelectric sensors and actuators. Various lay-ups areconsidered, including symmetric and non-symmetric ones.The enhanced buckling loads are shown to be twice the
y
z
hb
hp
hp
y
z
c
h
b c
PZT 0 Structural layer 90 Structural layer
L
zq
P xP
a
Fig. 1. A laminated composite beam with continuous piezoelectric layers.(a) The axial and lateral loads applied on the beam, (b) a symmetriclaminated beam [PZT/0/90/90/0/PZT] and (c) n non-symmetric laminatedbeam [PZT/0/90/0/90/PZT].
142 Y. Fridman, H. Abramovich / Composite Structures 82 (2008) 140–154
normal ones (without using the piezoelectric layers actua-tion (G = 0)).
2. Formulation of the problem
Fig. 1 shows a piezo-laminated type beam, referred to asystem of Cartesian coordinates with the origin of the mid-plane of the beam and the x-axis being coincident with thebeam axis. It is assumed that the piezoelectric layers havethe same length as the carrying structure (the case of con-tinuously piezoelectric layers1), are perfectly glued to it andthe thickness of the adhesive can be neglected (thus neglect-ing the effect of shear lag).
The axial displacement of the beam uðx; zÞ and the lat-eral wðx; zÞ are assumed to have the following form (implic-itly assuming incompressibility in the z direction, see Ref.[17]):
Uðx; z; tÞ ¼ U 0ðx; tÞ þ zUðx; tÞ ð1ÞW ðx; z; tÞ ¼ W 0ðx; tÞ ð2Þ
where U 0ðx; tÞ and W 0ðx; tÞ are the axial and lateral dis-placements of a point on the mid-plane and Uðx; tÞ is thebending rotation of the normal to the mid-plane.
The total strain vector is the sum of the mechanicalstrain vector and the actuator induced strain vector
feg ¼ femg þ feag ð3Þwhere the mechanical normal and transverse shear strains,em
x ; emz and cm
xz, as well as the actuator induced strains eax ; e
az
and caxz, are defined as
1 It can be shown that one can solve the issue of piezoelectric patchesbonded on a flexible laminated composite beam using the present modelwith continuous piezoelectric layers, by dividing the beam into threeregions, two without the patch and one (in the middle) with it andenforcing continuity conditions at the part with the patch. The results of alaminated composite beam with bonded piezoelectric patches will bereported in a separate paper to be issued in due time.
emx ¼
oU 0
oxþ z
oUox
ð4Þ
emz ¼ 0 ð5Þ
cmxz ¼ Uþ oW 0
oxð6Þ
eax ¼
XNa
k¼1
V kðxÞdk31
zak � za
k�1
ð7Þ
where Na is the number of actuators, V kðx; tÞ is the appliedvoltage to the kth actuator having a thickness of ðza
k � zak�1Þ
and dk31 is the piezoelectric constant. The other induced
strains vanish:
eaz ¼ ca
xz ¼ 0 ð8Þ
The beam constitutive equations can be written as
Nx
Mx
Qxz
8><>:
9>=>; ¼
A11 B11 0
B11 D11 0
0 0 A55
264
375
oU0
oxoUox
/þ oW 0
ox
8><>:
9>=>;þ
E11
F 11
0
8><>:
9>=>; ð9Þ
where
Nx ¼Z h=2
�h=2
crx dz
Mx ¼Z h=2
�h=2
crxzdz
Qxz ¼Z h=2
�h=2
csxy dz
ð10Þ
rx and sxz being the normal and shear stresses respectivelyand c is the width of the beam and h is the beams totalthickness. A11, B11, D11 and A55 are the usual extensional,bending-extensional, bending and transverse shear stiffnesscoefficients defined according to the lamination theory (seeAppendix A).
E11 and F11 are the actuator induced axial force andbending moment, respectively, defined as
E11 ¼ cXNa
k¼1
ðQ11ÞakV kðx; tÞðdk
31Þ ð11Þ
F 11 ¼c2
XNa
k¼1
ðQ11ÞakV kðx; tÞðdk
31Þðzak � za
k�1Þ ð12Þ
where zk is the distance of the kth layer from the x-axis, Na
is the number of piezoelectric layers, ðQ11Þak is calculatedaccording to Eq. (13) using the material properties of pie-zoelectric material (see Table 6).
Q11 ¼ Q11 cos4 hþ Q22 sin4 hþ 2ðQ12 þ 2Q66Þ sin2 h cos2 h
ð13ÞQ55 ¼ G13 cos2 hþ G23 sin2 h ð14Þ
The angle h is the angle between the fiber direction andthe longitudinal axis of the beam. The constants Q11,Q12, Q22 and Q66 are the usually used material constants(see Table 6).
Y. Fridman, H. Abramovich / Composite Structures 82 (2008) 140–154 143
3. Equations of motion
Using a first order shear deformation theory, the gov-erning equations of motion of a general shaped piezo-lam-inated beam (see Ref. [17])
o
oxA11
oU 0
oxþ B11
oUoxþ E11
� �¼ o
ot½ðI1
_U 0Þ þ ðI2_UÞ� ð15Þ
o
oxA55 Uþ oW
ox
� �� P
oWox
� �¼ o
ot½ðI1
_W Þ� þ q ð16Þ
o
oxB11
oU 0
oxþD11
oUoxþ F 11
� �� A55 Uþ oW
ox
� �¼ o
ot½ðI3
_UÞ þ ðI2_UÞ�
ð17Þ
where
ðI1; I2; I3Þ ¼ cZ h=2
�h=2
qð1; z; z2Þdz ð18Þ
q is the mass density of each layer, the flux denotes the par-tial derivative with respect to time, t. q denotes the trans-verse distributed loading, P is the axial compressive forceapplied at both ends of the beam (see Fig. 1). Together withthe boundary conditions for the general case:
Nx ¼ A11
oU 0
oxþ B11
oUoxþ E11 ¼ P or U 0 ¼ 0 ð19Þ
Qxz ¼ A55 Uþ oWox
� �� P
oWox¼ 0 or W ¼ 0 ð20Þ
Mx ¼ B11
oU 0
oxþ D11
oUoxþ F 11 ¼ 0 or U ¼ 0 ð21Þ
Table 1 presents the boundary conditions used in the pres-ent study.
4. Sensing effect
Due to mechanical stresses applied on the beam an elec-tric charge is generated in the piezo layer. Using the Gausslaw the charge accumulated on the piezoelectric electrodesis given by
QðtÞ ¼Z
SD31dS ð22Þ
where S is the area of the PZT layer and D31 is the electricaldisplacement. For a surface mounted flexural type PZTcontinuous layer, the electric displacement, D31 is given by
D31 ¼ Q11d31emx ¼ e31e
mx ð23Þ
Table 1Boundary conditions at beam’s ends
Boundaryconditions
x = 0 x = L
Simple–Simple W = 0 Mx = 0 U0 = 0 W = 0 Mx = 0 Nx = P
Clamped–Clamped
W = 0 U = 0 U0 = 0 W = 0 U = 0 Nx = P
Clamped–Simple
W = 0 U = 0 U0 = 0 W = 0 Mx = 0 Nx = P
Clamped–Free W = 0 U = 0 U0 = 0 Qxz = 0 Mx = 0 Nx = P
Substituting Eq. (23) into Eq. (22) and assuming that theelectrodes width is equal to the beam width, we get the totalinduced charge on the electrode, by integrating over thebeam length:
QðtÞ ¼Z
Se31e
mx dS ¼ c
ZL
e31emx dx ð24Þ
where c is the beam width.The voltage sensed per unit length of the beam is defined
as
V s ¼QðtÞ
C¼ �q
C¼ e31em
x hp
�eð25Þ
where C is the PZT capacitance given as
C ¼ �eS=hp ð26Þwhere hp is PZT layer thickness and �e is a dielectric con-stant. C and �q are the capacitance and electric charge perunit of beam’s length defined as
C ¼ CL¼ �ec=hp ð27Þ
�q ¼ ce31emx ð28Þ
5. The control law (actuation mechanism)
To enhance the axial loading capability of a laminatedcomposite beam, one has to control the deflections due toapplication of mechanical loads, by using sensing and actu-ation mechanisms. When a beam is subjected to axial com-pressive loads, it deflects laterally due to the bendingmoment. A large amount of lateral deflection is reachedwhen the axial load approaches the critical buckling load.To reduce this deflection and to make the beam nearly per-fect again we need to sense the change in the deformationof the beam and to apply a ‘‘counteractive’’ bendingmoment by using the accumulated voltage in the sensorsand fed it back into the actuators. This is given by thefollowing basic proportional control law:
V a ¼ GV s ð29Þwhere G is a constant feedback gain.
Substitution of Eqs. (4) and (25) into Eq. (29) yields thevoltage applied to the actuator as function of the displace-ments field:
V a ¼ Ge31exhp
�e¼ G
e31hp
�eoU 0
oxþ z
oUox
� �¼ G
e31hp
�eoU 0
oxþ h0
oUox
� �
ð30Þ
h0 ¼hp þ hb
2
� �ð31Þ
where hp; hb-beam dimensions (see Fig. 1).The substitution of Eq. (30) into Eq. (12) yields the
expression for the induced moment F11 as function of themechanical strain, the gain G, and the piezoelectric proper-ties of both the sensing and the actuation continuous lay-ers. In our case, we consider two sensors and two
144 Y. Fridman, H. Abramovich / Composite Structures 82 (2008) 140–154
actuators because each piezo-ceramic layer can operatesimultaneously both as sensor and as actuator:
E11 ¼ 2e2
31chp
eG
oU 0
oxþ h0
oUox
� �¼ Gf
h20
oU 0
oxþ G�f
h0
oUox
ð32Þ
F 11 ¼e2
31chp
�eGh0
oU 0
oxþ h0
oUox
� �¼ G�f
h0
oU 0
oxþ G�f
oUox
ð33Þ
where
�f ¼ e231chph2
0
�eð34Þ
The limiting condition of the system stability is that thecounteractive bending moment due to the induced bendingmoment F11 (for the case of symmetrical lay-ups) must beequal and in an opposite direction to the bending momentproduced by the externally applied axial load P.
F 11 6 Mx ð35Þwhere Mx-bending moment due to mechanical load only(see Eq. (9)).
For an axially loaded beam, the lateral displacementW 0ðx; tÞ is larger than the axial displacement U 0ðx; tÞ andtherefore we can assume that:
U 00ðx; tÞ << U0ðx; tÞ ð36ÞLeading to the following expressions:
F 11 ¼ G�f U0ðx; tÞ ð37ÞMx ¼ D11U
0ðx; tÞ ð38ÞThis yields the value of the limiting gain, Glim,
Glim ¼D11
�fð39Þ
5.1. The dynamical case – calculating the beam’s natural
frequencies and mode shapes
In the present study we apply only induced moments bytwo forces with equal magnitude and opposite direction, sothe total induced axial force E11 is zero. For the dynamicsolution we assume the lateral load to vanish, q = 0. Thegeneral solution form has the following general form:
U 0ðx; tÞ ¼ u0ðxÞ � eixt
W ðx; tÞ ¼ wðxÞ � eixt
Uðx; tÞ ¼ /ðxÞ � eixt
8><>: ð40Þ
where u0ðxÞ;wðxÞ and /(x) are the amplitudes of the axialand lateral displacements and bending rotation, respec-tively, time harmonically varying with a frequency x.
Substitution of Eq. (39) into Eqs. (15)–(17) yields:
A11u000 þ B11/00 ¼ �x2I1u0 � x2I2/ ð41Þ
A55/0 þ ðA55 � P Þw00 ¼ �x2I1w ð42Þ
B11 þG�fh0
� �u000 þ ðD11 þ G�f Þ/00 � A55ð/þ w0Þ
¼ �x2I3/� x2I2u0 ð43Þ
Together with the boundary conditions:
A11u00 þ B11/0 � P ¼ 0 or u0 ¼ 0 ð44Þ
A55ð/þ w0Þ � Pw0 ¼ 0 or w ¼ 0 ð45Þ
B11 þG�fh0
� �u00 þ ðD11 þ G�f Þ/0 ¼ 0 or / ¼ 0 ð46Þ
where the prime denotes differentiation with respect to x.To calculate the natural frequencies and mode shapes of
a beam, the solution process is divided into two cases:
(1) A symmetric case – a composite beam with a symmetriclay-up (see fig. 1b).
(2) A non-symmetric case – a composite beam with anon-symmetric lay-up (see fig. 1c).
For each case, two piezoelectric layers are bonded on thetop and bottom surfaces of the beam.
5.2. The symmetric case solution
Since for the symmetric case B11 = 0,I2 = 0 and underthe assumption from Eq. (36), the equations of motion,Eqs. (41)–(43) become:
A11u00 � P ¼ 0 or u0 ¼ 0 ð50ÞA55ð/þ w0Þ � Pw0 ¼ 0 or w ¼ 0 ð51ÞðD11 þ G�f Þ/0 ¼ 0 or / ¼ 0 ð52Þ
Eq. (46) can be solved separately assuming the followingsolution form:
u0ðxÞ ¼ B1 sinðkxÞ þ B2 cosðkxÞ; k ¼ x
ffiffiffiffiffiffiffiI1
A11
rð53Þ
where B1 and B2 are constants to be found according to thebeam’s in-plane boundary conditions. Eqs. (48) and (49)can be decoupled yielding the partial differential equationfor w only:
a2wIV þ a1w00 þ a0w ¼ 0 ð54Þ
a0 ¼ x2I1 � x4I3I1
A55
a1 ¼ � ðD11þG�f Þx2I1
A55� P þ x2I3 �1þ P
A55
� �
a2 ¼ ðD11 þ G�f Þ �1þ PA55
� �
8>>>><>>>>:
ð55Þ
The characteristic equation of Eq. (53) has the followingform:
a2s4 þ a1s2 þ a0 ¼ 0 ð56ÞThe solution for the lateral displacement w(x) and thebending rotation /(x) is
Y. Fridman, H. Abramovich / Composite Structures 82 (2008) 140–154 145
where s1 and s2 are roots of the characteristic Eq. (55).C1–C4 are constants, which will be found from the
boundary conditions of the beam. The other four con-stants, E1–E4 are depended on the first four constantsC1–C4, and can be found from the coupled equation,Eq. (48). See Appendix A for these relations.
5.3. The non-symmetric case solution
For the non-symmetric case, the three coupled equationsof motion Eqs. (41)–(43) should be first decoupled. Afterdecoupling, it is straight forward to obtain the followingform for the solutions (details see in Appendix A)
p ð63Þand s1, s2 and s3 are the roots of the characteristic uncou-pled equation for w(x).
C1–C6 are constants which are found from the beam’sboundary conditions, while A1–A6, E1–E6, are constantsdependent on C1–C6, which can be found from the cou-pled equations, Eqs. (41)–(43) (for more details seeAppendix A).
5.4. The static case – calculation of buckling loads and the
beam’s displacement field
We shall now restrict ourselves to the static case omit-ting in Eqs. (15)–(17) the inertial terms on the right-handside. The equations of motion for the static case and con-stant properties along the beam and q = const are:
A11u000 þ B11/00 ¼ 0 ð64Þ
A55/0 þ ðA55 � PÞw00 ¼ q ð65Þ
B11 þG�fh0
� �u000 þ ðD11 þ G�f Þ/00 � A55ð/þ w0Þ ¼ 0 ð66Þ
Together with the boundary conditions of the general case,Eqs. (44)–(46).
5.5. Symmetric beams
For a symmetric laminated beam B11 = 0 and it isassumed that ou0
ox << z o/ox.
The coupled equations of equilibrium have the followingform:
In this case, the longitudinal beam motion can be solvedseparately from the lateral motion yielding
u0ðxÞ ¼ g1xþ g0 ð70Þ
where g1 and g2 are constants to be found according to thebeam’s in-plane boundary conditions.
Eqs. (68) and (69) can be decoupled to yield:
KwIV þ Pw00 ¼ �q ð71ÞK/000 þ P/0 ¼ q ð72Þ
where
r ¼ 1� PA55
� �; K ¼ ðD11 þ G�f Þr ð73Þ
For a given distribution of the transverse loading, q, Eqs.(52) and (53) can be solved by looking for the homoge-neous and particular solutions. For the case of q = constthe solutions have following general form:
wðxÞ ¼ A1 cosðkxÞ þ A2 sinðkxÞ þ A3xþ A4 �qx2
2Pð74Þ
/ðxÞ ¼ B1 cosðkxÞ þ B2 sinðkxÞ þ B3 þqxP
ð75Þ
where
k ¼ffiffiffiffiPK
rð76Þ
There are seven unknowns to be determined (A1–A4, B1–B3) and only four boundary conditions. The substitutionof (74) and (75) into the uncoupled equations of motions(67) and (68) yields the additional three relations:
B1 ¼ �rkA2; B2 ¼ rkA1; B3 ¼ �A3 ð77ÞThe relations in Eq. (77) with the specific boundary condi-tions for each case give us the exact solution for w(x) and /(x). Table 2 presents the constants A1–A4 for differentboundary conditions.
For a Clamped–Simply supported beam see Appendix A(as the expression is too long).
Demanding the lateral displacement to tend to infinity(the normal definition of the buckling load) yields thebuckling load of the beam. Table 3 summarizes the closedform solutions for the buckling loads based on a FSDTapproach for different boundary conditions.
The buckling load is the minimal critical load. For Sim-ply–Simply supported and Clamped–Free symmetrical
Table 2The expressions for A1–A4-the lateral displacement solution of symmetric lay-up beams
B.C A1 A2 A3 A4
S–S � q
rk2PqðcosðkLÞ � 1Þ
Pk2r sinðkLÞqL2P
q
rk2P
C–C � qLðcosðkLÞ þ 1Þ2rkP sinðkLÞ
� qL2rkP
qL2P
qLðcosðkLÞ þ 1Þ2rkP sinðkLÞ
C–F q
P 2rk
ðLP sinðkLÞ � KkÞcosðkLÞ � qL
rPkqLP
� q
P 2rk
ðLP sinðkLÞ � KkÞcosðkLÞ
Table 3Closed form expressions for critical loads of symmetric lay-up beams
S–SPcrðnÞ ¼ p2n2ðD11 þ G�f ÞA55
p2n2ðD11 þ G�f Þ þ A55L2; n ¼ 1; 2; . . .
C–FPcrðnÞ ¼ p2n2ðD11 þ Gf ÞA55
p2n2ðD11 þ Gf Þ þ A55L2; n ¼ 1; 2; . . .
C–S F(k) � (sin(kL) � Lkrcos(kL)) = 0
C–CF ðkÞ � sinðkL
2Þ 2 sin
kL2
� �� krL cos
kL2
� �� �¼ 0
146 Y. Fridman, H. Abramovich / Composite Structures 82 (2008) 140–154
beams the analytical expression yields the minimal load forn = 1. For Clamped–Simply supported and Clamped–Clamped beams the minimal root of the characteristicequations F(k) would give the buckling load. For G = 0we get the critical loads for a symmetric laminated beamwithout the influence of the piezoelectric actuationeffect.
Table 4The expressions a1–a4; c3–c4 for lateral and axial displacements solutions of n
a1 a2
S–S �B11
A11
� q
r�k2Pa1ðcosð�kLÞ � 1Þ
sinð�kLÞ
C–C� qLðcosð�kLÞ þ 1Þ
2rkP sinðkLÞ� qL
2r�kP
C–F tanð�kLÞqL
r�kP� qK
rP 2 cosð�kLÞ� qL
Pr�k� B11
A11 cosð�kLÞ
5.6. Non-symmetric beams
For the non-symmetric beam case, the three coupledequations (64)–(66) with boundary conditions (44)–(46)have to be solved.
Eqs. (44)–(46) can be decoupled to yield:
KwIV þ Pw00 ¼ �q ð78ÞK/000 þ P/0 ¼ q ð79ÞKuIV
0 þ Pu000 ¼ 0 ð80Þ
where
B11 ¼ B11 þG�fh0
; D11 ¼ D11 þ G�f ;
r ¼ 1� PA55
; K ¼ ðD11 �B11
A11
B11Þr ð81Þ
For a given distribution of the transverse distributed load-ing q, Eqs. (78)–(80) can be solved by looking for homoge-neous and particular solutions. For the case of q = constthe solutions have the following general form:
on-symmetric lay-up beams
a3 a4 c3 c4
qL2P
�a1 � B11qA11P
þ PA11
�ar2�k
B11
A11
qL2P
�a1 � qB11 � P 2
A11P
qL2P
B11
A11
qLP
�a1 � B11qA11P
þ PA11
qLP
B11
A11
Table 5Closed form expressions for critical loads of non-symmetric lay-up beams
S–S
PcrðnÞ ¼p2n2 D11 � B11
A11B11
� �A55
p2n2 D11 � B11
A11B11
� �þ A55L2
n ¼ 1; 2; . . .
C–F
PcrðnÞ ¼p2 D11 � B11
A11B11
� �A55ð2n� 1Þ2
p2 D11 � B11
A11B11
� �ð2n� 1Þ2 þ 4A55L2
n ¼ 1:2; . . .
C–S F ð�kÞ � ðsinð�kLÞ � �kr cosð�kLÞÞ ¼ 0C–C
F ð�kÞ � sin�kL2
� �2 sin
�kL2
� �� �krL cos
�kL2
� �� �¼ 0
Y. Fridman, H. Abramovich / Composite Structures 82 (2008) 140–154 147
wðxÞ ¼ a1 cosð�kxÞ þ a2 sinð�kxÞ þ a3xþ a4 �qx2
2Pð82Þ
/ðxÞ ¼ b1 cosð�kxÞ þ b2 sinð�kxÞ þ b3 þqxP
ð83Þ
uðxÞ ¼ c1 cosð�kxÞ þ c2 sinð�kxÞ þ c3xþ c4 ð84Þ
where
�k ¼ffiffiffiffiP
K
rð85Þ
There are 11 unknowns to be determined ða1–a4; b1–b3;c1–c4Þ and only six boundary conditions. The substitutionof Eqs. (82)–(84) into the uncoupled equations of motions(64)–(66) yields the additional five relations:
b1 ¼ �r�ka2; b2 ¼ r�ka1; b3 ¼ �a3
c1 ¼ �b1
B11
A11
; c2 ¼ �b2
B11
A11
ð86Þ
The relations Eq. (86) with the specific boundary condi-tions for each case will give us the exact solution forwðxÞ;/ðxÞ and u(x). Table 4 presents the coefficients a1–a4
and c3; c4 for different boundary conditions.For a Clamped–Simply supported beam see Appendix A –
the expression is too long to be included in the table.Table 5 summarizes the closed form expressions for crit-
ical loads of a beam with a non-symmetric lay-up.The buckling load is the minimal critical load and was
found in the same way as for the symmetric lay-up case.In Table 5, the expressions for D11;B11 depend on the
gain G (see Eq. (81)). For G = 0 we get the critical load
for a non-symmetric laminated beam with structural influ-ence of piezoelectric layers but without the influence of thepiezoelectric actuation effect.
6. Results and discussion
The main aim of the present research is the investigationon the enhancement of the stability of axially loaded com-posite beams with continuous PZT layers.
Buckling loads, natural frequencies, and lateral displace-ments for beams with piezoelectric layers, having variousboundary conditions like Simply–Simply, Clamped–Clamped, Clamped–Free, Clamped–Simply, and symmetricand non-symmetric lay-ups, were computed based on aMatlab [18] written code.
Fig. 1b–c shows the cross-section of a cross-ply symmetriclaminate [PZT/0�/90�/90�/0�/PZT] and an anti-symmetric[PZT/0�/90�/0�/90�/PZT] piezo-laminated beams, respec-tively. Both structures consists of six layers where the topand bottom are PZT layers and four internal layers madeof graphite-epoxy.
The geometrical parameters of the beam and themechanical properties of the two materials (graphite-epoxyand PZT-5H) are listed in Table 6. The lateral distributedload is taken as, q = 0.05 [N/m] for all cases. The axial loadP varies between the first and second buckling load and itdepends on the feedback gain G.
The first natural frequency as function of the axial loadfor various gains for a Simply–Simply supported symmetri-cal beam is presented in Fig. 2a. For P = 0 we get the free
Fig. 2. A Simply–Simply supported symmetrical beam. (a) First natural frequency vs. axial load. (b) First squared natural frequency normalized by thefirst free frequency vs. axial load normalized by the first buckling load.
148 Y. Fridman, H. Abramovich / Composite Structures 82 (2008) 140–154
vibration frequency of a beam with a gain G and for f1 = 0the buckling load of a beam is achieved. Fig. 2b presents therelative improvement of the first natural frequency vs. therelative improvement in critical load. Fig. 3a and b showthe same results for a Clamped–Free non-symmetricalbeam. The first three natural frequencies of symmetricand anti-symmetric beams for various gains without axialload influence were calculated and presented in Tables 7and 8, respectively.
Figs. 4 and 5 show the differences of behavior between asymmetric and an anti-symmetric Clamped–Simply sup-ported beam.
As expected, one can see that the natural frequencies ofa symmetric beam are higher than the anti symmetric one,for the same boundary conditions. Increasing the gain G
Fig. 3. A Clamped–Free non-symmetrical beam. (a) First natural frequency vsfrequency vs. axial load normalized by the first buckling load.
yields an increase in the natural frequency. The relativeimprovement is in the same order for the various boundaryconditions and for a constant gain value it is higher for abeam with non-symmetric lay-up.
The first and the second critical loads and their ratio fordifferent boundary conditions for the present symmetricand anti-symmetric beams were calculating by substitutingG = 0 into the equations presented in Tables 3 and 5 andare listed in Table 9.
As expected the buckling loads of the symmetrical beamare higher then the non-symmetrical one.
The limiting gain value was found to be Glim = 24.303for a symmetric beam and Glim = 21.5531 for a non-symmetric one. The values of the buckling loads fordifferent G, and the ratios between these loads and
. axial load. (b) First squared natural frequency normalized by the first free
Fig. 4. First natural frequency vs. axial load for a Clamped–Simply supported beam. (a) Symmetric lay-up and (b) non-symmetric beam.
Fig. 5. First squared natural frequency normalized by the first free frequency vs. axial load normalized by the first buckling load A Clamped–Simplysupported beam. (a) Symmetric lay-up and (b) non-symmetric beam.
Table 9Critical loads without PZT actuation (G = 0)
Y. Fridman, H. Abramovich / Composite Structures 82 (2008) 140–154 149
the first buckling load for each case, are presented inTable 10.
The value of the feedback gain, G, depends on the flex-ural rigidity and the geometric configuration of the piezo-
electric material. Different Glim values were found forsymmetric and anti-symmetric laminates. Increasing thevalue of G, would yield a higher buckling load. The ratiobetween the enhanced buckling load and the first critical
a Note the different values of Glim for symmetric and non-symmetric laminate.
Fig. 7. A Simply–Simply supported [PZT/0/90/90/0/PZT] beam withvarious ply thickness. (a) Sensed voltage vs. normalized beam thickness.(b) Applied voltage with Glim vs. normalized beam thickness.
150 Y. Fridman, H. Abramovich / Composite Structures 82 (2008) 140–154
load depends only on the laminate structure and is higherfor non-symmetric laminates. No influence of the bound-ary condition was found. For G = Glim the enhanced buck-ling load is 2 * Pcr1 for a symmetric laminate and is2.11 * Pcr1 for a non-symmetric one.
In Fig. 6, a comparison between the enhanced bucklingload vs. feedback gain and its relation to the first bucklingload for symmetrical and non-symmetrical laminateclamped–free beam is presented. From this figure one cansee that the increase in the enhanced buckling load is linearwith the gain. The slope of anti-symmetric beam is higherthan the symmetric one as a result of in-plane out-of-planecoupling between bending moment and extension of themiddle surface (see Eq. (41)).The smallest values of Pcr
were obtained for G = 0 (the first buckling loads for bothtypes of beams).
Fig. 7 presents the sensed voltage for beams with variousgraphite-epoxy ply thicknesses where the lowest ply thick-ness is h = 1.27 * 10�4 [m] , and the other three ply thick-nesses are 2 h, 4 h and 8 h. The sensed and the appliedvoltage were calculated for different laterals loads whichwould lead to the same lateral deflection, Wmax = � 1.3 *10�4 [m] or Wmax/L = 5.23 * 10�4 for all beams loadedby axial loads of P = 0.95 * Pcr1. The applied voltage cal-culated for the limiting gain G = Glim values (see Eq.(39)) is shown in the fig. 7b. From these results one can
Fig. 6. Clamped–Free symmetrical and non-symmetrical beam. (a) Enhancedbuckling load vs. gain.
see that the real actuation power is limited by the appliedvoltage capability of the piezoelectric layer which is nor-mally for monolithic PZT under 200 V.
Fig. 8 presents the lateral displacement of a beam undervarious axial loads without piezoelectric actuation. From
buckling load vs. gain. (b) Enhanced buckling load normalized by first
Y. Fridman, H. Abramovich / Composite Structures 82 (2008) 140–154 151
these figures one can see the usual displacement form fordifferent boundary conditions.
Figs. 9 and 10 show the lateral displacement along thebeam length with PZT actuation. The PZT actuation effectis visible in these figures.
Fig. 11 presents the mid-span lateral displacement of thebeam with a symmetric lay-up, (PZT/0�/90�/90�/0�/PZT)
Fig. 8. Lateral displacement vs. beam length for various axial load for G = 0,Clamped–Free non-symmetrical beam.
Fig. 9. Lateral displacement vs. beam length for constant axial load and variousymmetrical beam.
Fig. 10. Clamped–Clamped non-symmetrical beam. (a) Lateral displacementdisplacement vs. beam length for various axial loads and gains.
vs. the axial load P, for various directions of the lateralload, q. The different lines present the displacement ofthe beam for different values of gain G. Fig. 12 presentsthe same results for a non-symmetric Clamped–Simplysupported beam (PZT/0�/90�/0�/90�/PZT). As expected, alarge lateral deflection is reached when the axial loadapproaches critical buckling load.
s gains. (a) Simply–simply supported symmetrical beam. (b) Clamped–Free
vs. beam length for constant axial load and various gains. (b) Lateral
Fig. 11. Axial load vs. beam mid-span deflection for various gains – ASimply–Simply supported symmetrical beam.
Fig. 12. Axial load vs. lateral mid-span displacement for various gains- AClamped–Simply supported beam.
152 Y. Fridman, H. Abramovich / Composite Structures 82 (2008) 140–154
Fig. 13 presents the mid-span lateral displacement of aClamped–Clamped beam vs. the axial load P normalizedby the first critical load for symmetric and non-symmetriclay-ups.
Fig. 13. Axial load normalized by first buckling load vs. beam mid-span deflectand (b) non-symmetrical lay-up.
Fig. 14. Lateral mid-span displacement vs. lateral load for a Simply–Simply sload P = 0.95 * Pcr1. (b) For various axial loads P = 0.95 * Pcr.
From Fig. 14, one can see that for a beam with an anti-symmetric lay-up, the direction of the lateral displacementdepends on the values of the lateral load, q, and itsdirection.
ion for various gains – a Clamped–Clamped beam. (a) Symmetrical lay-up
upported anti-symmetrical beam for various gains. (a) For constant axial
Y. Fridman, H. Abramovich / Composite Structures 82 (2008) 140–154 153
7. Conclusions
A mathematical model is presented for active control ofthe instability of symmetric and non-symmetric piezo-lam-inated composite beams subjected to axial and lateralloads.
The active control is obtained by using piezo-ceramiclayers as an actuators and sensors acting in closed-loop.The voltage transferred from the sensors to the actuatorshas to be amplified through a constant feedbackgain G.
An exact mathematical model, based on a first ordershear deformation theory (FSDT), was developed anddescribed. Closed form solutions for the bending angleand the axial and lateral displacements along the beamare presented for various boundary conditions.
Using the present model, the critical buckling loads andthe natural frequencies of beams and their respective modeshapes were computed for cases with and without the feed-back gain.
A parametric investigation was performed for beamswith different lay-ups, and it was found that for a givenfeedback gain value one can increase the critical bucklingload and or the natural frequency of a beam with a non-symmetric lay-up in a higher ratio then one can achievefor a symmetric lay-up. The limiting feedback gain for astable system was derived. A linear relationship was foundbetween the enhanced buckling load and the feedbackgain.
The present results are consistent with similar onespresented in the literature (see for example Refs. [8,10,13,15]), namely the buckling load of a beam or its naturalfrequencies can be actively increased above their criticalvalues, using piezoelectric sensors and actuators actingin a closed loop. The range of this increase varies froma few percents up to more than five times the criticalbuckling load, depending on the properties of the beamsand the relevant control types. The present methodologyyielded an enhancement factor of 2 for a symmetric lay-up and 2.11 for a non-symmetrical one, comparable tothe other existing results, yielding a sound and reliablegeneric model.
The sensed and the applied voltage were calculated forvarious laterals loads which would lead to the same lateraldeflection, in the order of Wmax = � 1.3 * 10�4 [m] forbeams loaded at P = 0.95 * Pcr1. It was shown that thesensed voltage is proportionally increasing, while theapplied voltage, at G = Glim, is increasing in a squaredmanner with the beam’s thickness.
One should note that the required voltage to preventbuckling is dependent on the beam’s moment of inertia.Therefore, to control laminated composite beams of prac-tical dimensions (let us say, length in meters and momentsof inertia in the range of 10+4 cm4), the applied voltageswould be a few hundred of volts, provided the piezoelectriclayers can carry this electrical load.
Appendix A
A11 ¼ cXN
k¼1
ðQ11Þkðzk � zk�1Þ ðA:1Þ
B11 ¼c2
XN
k¼1
ðQ11Þkðz2k � z2
k�1Þ ðA:2Þ
D11 ¼c3
XN
k¼1
ðQ11Þkðz3k � z3
k�1Þ ðA:3Þ
A55 ¼ cK0
XN
k¼1
ðQ55Þkðzk � zk�1Þ ðA:4Þ
where zk is the distance of the kth layer from the x-axis, N
is the number of layers (including the piezoelectric ones),K0 is a shear correction factor (chosen to be 5/6).
Relations for the dynamic solution for a symmetric case:
E1 ¼ n1C2; E2 ¼ n1C1; E3 ¼ n2C4; E4 ¼ �n2C3
ðA:5Þ
n1 ¼ �k2
1ðA55 � P Þ þ x2I1
A55k1
; n2 ¼k2
2ðA55 � P Þ � x2I1
A55k2
ðA:6ÞRelations for dynamic solution for a non-symmetric case:
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