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Large deflection of adaptive multilayered Timoshenko beams
Composite Structures 31(1995) 49-60 Q 1995 Elsevier Science Limited
Printedin Great Britain. AU rights reserved 0263~8223/95/S9.50 0263-8223(95)00001-l
ELSEVIER
Large deflection of adaptive multilayered Timoshenko beams Marco Di Sciuva & Ugo Icardi Department of Aerospace Engineering, Politecnio di Torino, Corso Duca degli Abruzzi, 24-10129 Turin, Italy
The paper deals with the formulation of the nonlinear equations governing the mechanical behavior of anisotropic, laminated Timoshenko beams having any number of arbitrarily positioned and orientated actuator and/or sensor layers. Use is made of the von KarmBn nonlinear strain-displacement relations. Subsequently, the static stability equations for the initial (bifurcation) buckling under transverse and compressive loads are formulated via the Euler method of the adjacent equilibrium configurations. The present analysis is quite general in .that no assumptions are made on the placements of the active layers, their symmetry, and their constitutive relations. The only assumptions pertain to the behavior of the adaptive multilayered beam as one equivalent, linear elastic, anisotropic beam (smeared laminate model). Numerical results deal with the nonlinear flexural response of unsymmetrically laminated beams under transverse and compressive axial loads. It is concluded that the effectiveness of the control depends on the boundary conditions, mechanisms of activation and lay-ups.
1 INTRODUCTION Adaptive structures are those which can adjust their geometry, stiffness and damping on demand to meet a high degree of adaptability to changes in their external and internal loading environment. External conditions m.ay consist of environment, loads, or the desire to change the scope, purpose, or geometry of the structure after it has been built and is in service. Internal conditions may be damage or failure to isolated portion of the material or structure. Adaptive composite structures built by using fiber reinforced composites where some layers behave as induced strain actuators and/or sensors have been already vogue in aerospace industry. The success of these structures depends largely on the effective control of the structure through the actuators and the sensors embedded in their load carrying members. To do this, further theoretical research work for a better understanding of the elastodynamical behavior of adaptive structures is needed. Theoretical analysis of adaptive multilayered composite structures is currently accomplished using the so-called Classical Lamination Theory (CLT) (see Tzou & Gadre,’ Rogers et a1.,2 Crawley & Lazarus3) coupled with some numerical schemes (for example, Ritz’s method).
Other numerical approaches include finite element formulations (Tzou & Tseng,4 Ha et aL5). The addressed topic in these papers is mainly the vibration control, while it appears that static control has been investigated only with respect to Euler’s beams (Crawley & de Luis,6 Im & Atl~ri,~ Song et al*). The range of applicability of the CLT in evaluating the global behavior (deflections, lower natural frequencies and buckling loads) is well understood to be that of slender beams and thin multilayered plates and shells. Thus, the inclusion of the transverse shear deformability effect in modeling moderately thick, adaptive multilayered plates and shells appears to be of great interest. Furthermore, the study of the active control of the geometrically nonlinear response and of the buckling behavior of adaptive beams appears the best of authors’ knowledge to be rather scarse if not absent in the open literature. Another topic which as received less attention is the effect of changes in the boundary conditions. The aim of the present study is to fill some of the previous gaps. To this end, we use the general formulation given by Di Sciuva & Icardi9 to derive the variationally consistent nonlinear equations governing the mechanical behavior of anisotropic, laminated Timoshenko beams having any
50
M. Di Sciuva, U. Icardi
number of arbitrarily positioned and oriented actuator and/or sensor layers. Use is made of the von K&-man nonlinear strain-displacement relations. Subsequently, the static stability equations for the initial (bifurcation) buckling under transverse and compressive loads are formulated via the Euler method of the adjacent equilibrium configurations. The present analysis is quite general in that no assumptions are made on the placements of the active layers, their symmetry, and their constitutive relations. The only assumptions pertain to the behavior of the adaptive multilayered beam as one equivalent, linear elastic, anisotropic beam (smeared laminate model). This assumption generally results in incompatible transverse shearing stresses between two adjacent layers, Di Sciuva,r” and can produce poor predictions of thickness distributions of displacements and stresses across the thickness for thick laminates. Because in this paper we are interested only in the global response, the smeared laminate model in conjunction with the von K5rn&n straindisplacement relations appears to be adequate. Numerical results deal with the nonlinear flexural response of unsymmetrically laminated beams under transverse and compressive axial loads. They show how effective may be the control of the response by induced strain actuator (in this case, some piezoelectric material layers). Various boundary conditions and control laws are investigated. 2 FORMULATION OF THE EQUATIONS OF MOTION OF ADAPTIVE BEAMS 2.1 Kinematics Let us consider a multilayered beam of total thickness h, length 2L and width b consisting of a finite number IV of linearly elastic orthotropic layers of uniform thickness perfectly bonded together. Each layer can exhibit different individual physico-mechanical properties. The points of the beam are referred to a Cartesian orthogonal co-ordinates system Xi, i= 1,2,3 (see Fig. 1) with the origin at midlength of the beam, x1 being the longitudinal axis of the beam (reference axis), (x1, x3) the plane of flexure, and (x2, x3) the plane of the beam cross section. The beam model developed in this study is based on the following representation of the displacement field across the beam thickness,
=3 Midsurface 1
Fig. 1.
Actuator lavers -
t
Geometrical
’
n
beam configuration.
where 6, and z& are the displacements of a generic point of the beam in the x1-, and x3-directions, respectively; u1 and ug are reference axis displacements in the same coordinate directions; #, denotes the total rotation in the (x,, x3) plane of a line normal to the reference axis. To formulate a geometrically nonlinear theory of a shear-deformable (Timoshenko), adaptive straight beam, the strain-displacement relationships are assumed to be of the following form (small strain and moderately small rotation approximation; von K&man strain tensor) El1
=&11+X3kl;
&3=q3
where El1 = u1.1
+ b3,J2/2;
k1=
$4,1;
El3 = $4 + U3,l
(3) are the strains and the curvature of the reference axis. The notation ( .),1 denotes partial derivative (d( * ml ). 2.2 Variational formulation of nonlinear equations of motion Let us consider a piezoelectric continuum filling a volume T/ of boundary 9. Under the assumptions that and pryoelectric (i) thermal, thermoelastic phenomena do not take place; (ii) mechanical and electrical forces during the vibration are balanced at any given tune, thus allowing for a quasi-static approximation in the analysis; the following relations hold: - Linear piezoelectric constitutive equations 9,=
e,kl.6kl+ xi,&,,, direct effect
(4)
uii = C &gkl - emiiRminverse effect in terms of strain tensor components, 3, = dnkrGk,+x&J’, F,.= S%,ii,_,+ d 2..
direct effect inverse
effect
or
(5)
Large deflection of Timoshenko beams
in terms of stress tensor components, where 6’ii and Eliidenote the mechanical stress elastic tensor and the nonlinear strain tensor components, respectively; C$, denotes the sixth-order, symmetric positive definite elastic stiffness matrix elements of the pieozoelectric material (here assumed to be an elastic piezoelectric monoclinic material), under constant electric field conditions and at prescribed temperature; S$, is the symmetric, positive definite elastic compliance matrix of the piezoelectric malterial, under constant electric field and at prescribed temperature; emii denotes the dielectric permissivity matrix elements, also known as the piezoelectric stress coefficients matrix; it couples the mechanical and electrical equations; xi,,, is the symmetric matrix of dielectric constants or dielectric susceptibility under conditions of constant strain; 8, is the electric field vector in the material, maintained constant by potentials. applied to suitable electrodes; gn is the electric displacement or dielectric induction vector; d,,= e,,S& is the piezoelectric strain coefficient matrix; x& is the permissivity (dielectric susceptibility) coefficients matrix under constant elastic stress. The superscripts, such as ( )O, on material properties typically denote the mechanical or electrical boundary conditions under which the constants are valid. Obviously, eqn (4J with emu= 0 holds for any monoclinic material. In the expressions for e,$ and dmii the first subscript refers to the electrical axis, while the subsequent two refer to the mechanical strain (for example, e1i3 refers to the shear strain developed in the plane (x1,x1) in response to an electric field in the x,-direction). - Field equations In the given Cartesian system of coordinates, the field equations (under the hypothesis that the material is insulated so that no free electric charges exist) read [( 6,+ L$i) ~ki],k= pi equations Of motion (i,j,k,n= %,n=O
1,2,3)
Maxwell-Gauss
(6) equation
where pj=fi”+fp+fi”
(7)
lists the three components of the body forces f!, of the inertial forces f:, and of dissipative forces
f$
the piezoelectric material, are: (i) the surface tractions pi applied on the surfaceYe, and the displacement tii orPI,; (ii) the electrical charge density 4 applied on the surface ~7~ and the electric potential @ onYQ. The previous equations of motion and the associated natural boundary conditions follow from the Euler’s equations of the following variational statement pi?:isliidV+ V
with t?Fm=sDm
The boundary conditions (mechanical and electrical) on Sp, the boundary of the volume V filled by
(9)
In eqn (8) the overdots indicate differentiation with respect to time, Substitution of eqns (4) into eqn (8) yields
Jv
PUidUi d V+ =
-
(C$kk,-
e,,B,)6iq
dV
Jv
(10) Here, the variational statement ( 10) is used to obtain the equations of motion and the variationally consistent natural boundary conditions for the adaptive multilayered Timoshenko beam under consideration. This is accomplished by substituting into eqn ( 10) the approximate displacement field ( 1 ), the approximate strain field (2) and (3), and the constitutive relations for the individual layers. These relations for the beam under consideration and under the assumption that the x,-direction is associated with the direction of poling (i.e., the direction of the applied electric field vector) read .,
- Boundary conditions
51
011
=Q
1141-44e&
(11)
M. Di Sciuva, U. Icardi
52
where Q 11 and Qa4 are the transformed reduced elastic stiffnesses, depending on the elastic coefficients of the piezoelectric material, and e31 is the reduced dielectric permissivity coefficient. As usual, we have adopted the contracted notation for the reduced stiffnesses and dielectric permissivity coefficients. Applying the well-known techniques of variational calculus, the vanishing of the virtual variations in the surface and line integrals yields the equations of motion and the variationally consistent natural (mechanical) boundary conditions: - Nonlinear equations of motion N 11.1
-
T~=m,zi,+m,$l
(12)
Boundary conditions At the ends x1 = f L, one element of the pair of conditions
(NsG);(T,+N,,~,
(15) where the quantities affected by an overhat (^)and overcheck (‘) identify the pure mechanical and induced strain actuator contributions to the indicated quantities, respectively. The resulting expressions for the force and moment elastic stress resultants in terms of the generalized displacements follow from the constitutive equations 1 NH =A,,+ +B,+, (16)
where A,l, B,, and D,, are extensional, coupling and bending stiffenesses, and A,, is transverse shear stiffness given by [A,, B,, OH]=@
(13)
’
is assigned. In eqn ( 12) p is the transverse load per unit length of the beam. The following expressions for the force and moment resultants of the stress and of inertial forces hold e11>6.13Px36.11));
~3
x%211(4>;
4, = xu/Q&h
,-M,-m,&;~,);
(M,,; $1)
A,,%,,,
BIA,,,
+ %#1,,1 +Jy1+ &4&J
=RHS, (18)
+A44~3,1l+JG=RfG +DI,~1,11+~3+6,~,,,,=RHS,
where pi given by
where
(17)
x44 being the shear correction factor. When expressed in terms of generalized displacements, the geometrically nonlinear differential equations of motion become
A,,#,,,
mi=(xip(s)>
(14)
N
M,, = d,J&,
8 = A,,&,,
-~+m,ii,,,+m,ii,+m,~,,,
(NH, T,JG)=((
Tl = PI;
N,, = A,, + &fill;
fi,, = &El1 + D,, x1
=VZ,Z&+t?I,J,
T,,,+(N,,u,,,),,=
M 11.1
can write
are nonlinear
differential
operators
XJS (...)
(...)=bC
dx,
s=l i xjs-,
It is concluded that the equations of equilibrium and the boundary conditions for multilayered Timoshenko beams having induced strain actuator layers are formally the same as those for the non-adaptive multilayered Timoshenko beams, provide that they are expressed in terms of force and moment stress resultants. - Equations of motion in terms of displacements The above force and moment stress resultants take into account pure mechanical and actuation strain contributions. The use of the tracer 6,) which identifies the contribution brought by the actuation strain (it takes the values 1 or 0), makes it possible to recast these in a form in which the actuator effect appears in explicit form. Thus, we
N3=ifMb43,A21,,
(1%
and RHS,(i= 1,2,3) stands for the right-hand side of eqn(l2,). 2.3 Static stability equations The linearized equations governing the bifurcation buckling of the multilayered adaptive beam subjected to compressive axial load N,, are formulated by using the Euler method of the adjacent equilibrium configurations, Refs 9 and 11. Let us assumed that: (i) the external forces vary neither in magnitude nor in direction during
53
Large dejlection of Timoshenko beams
the buckling; (ii) before the onset of the buckling, the deformation state is characterized by absence of rotations of the normals to the reference axis, then the beam will be in a state of uniform axial stress corresponding to the applied load NI1 and the stability problern reduces to the classical (bifurcation) buckling problem. Under these assumptions, the principle of virtual work for the adjacent equilibrium reads
and the vanishing of one element of the pairs (T,;
-
T,)W,l
dx-
1 W,h
(20)
+(~~+N,,u,,,)tSu,+M,,~~,14L=O
Note that according to the adjacent equilibrium criterion, terms of third or higher order have been ignored in the above equation; in addition, all of the force and moment stress resultants which appear in eqn (20) (with the exception of the external forces), as well as the generalized displacements ul, u3, and iI are measured from the original equilibrium configuration, i.e., they represent the buckled mode. The incremental displacement field is of the form given by eqn (l), while for the incremental strain components we have the following linearized expressions
(25)
h).
3 METHOD OF SOLUTION The expressions for the in-plane force RI, and moment resultants a,, induced by the strain actuator piezoelectric layers are r;i,,
+M,,,
UK,;
~3);
Ml,
=&d,,g,;
=&d,,~,
(26)
where A,, and 6),, refer to the activated layers only, for an applied electric field vector with the only non-vanishing component in the thickness direction. Following the procedure presented by Chen and Shu16 (see Appendix), the set of eqns ( 12) is transformed into the following governing equation for the deflection u3
u3,1111-
(till -~ll),ll
fill -riT1
u3,11-
DC
D‘
l-j =----D‘
(27)
where Eli =
q,;
k=h,,;
(21)
&13=h+U3,1.
Taking into accotmt that the virtual variations are independent varia.tions, the principle of virtual work yields the following system of differential equations and boundary conditions governing the bifurcation buckling: - Field equations N 11,1=
0
T,,l+%,~3,ll=o
M 11,1-
(22)
T,=o
or, in terms of generalized displacements, A,,%,,,
(23)
BC2: simply supported u3( &L)=O;
=0
- Boundary conditions The appropriate bou:ndary conditions are given by N,,=O
(29)
M,,( f L)=O;
A44~1,1+A44U3,11-+~~1U3,1*=0
+ 6,AIIJ
and x44 is taken to be equal to ~t*/12. Equation (27) is a linear ordinary differential equation with constant coefficients, whose general solution depends on the coefficients, on the equivalent loading term (resulting from the control and external loading), on the boundary conditions and on the x,-distribution of 8, (see eqn (26)). The following set of boundary conditions has been considered: BCl:pinnedu,(+L)=O;u,(+L)=O;
+~,,9,,,,+&4~1,,1=0
BII~I,,, + LW,,,,
03)
M,,( &L)=O;
BC3: loosely-clamped
at x1 = f L
(24)
NI r( + L ) = IV1 1; (30)
NI 1(+ L ) = A, 1;
u,(fL)=O;u,,,(fL)=O;
together with the following x,-distributions electric field (whose only non-vanishing
(31)
of the com-
54
M. Di Sciuva, U. Icardi
where
ponent is assumed to be 8,): D 1: B, = const; D2: B, proportional
to ug max,
As we said, the effects of the pizeoelectric layers which appear both in the second and in the third terms of eqn (27) depend on the xr-distributions of the g3 (see eqn (26)), i.e. by the strategy of the activation. Specifically, for &,-distribution depending on the x,-distribution of u3 (adaptive case D2 by feed-back activation), the membrane force Nl, and the equivalent transverse loading (M,, -(B,,IA,,)BPUSI,),~,/D, generate additional coefficients for z+,11 and u3 11, 1. For a uniform 8,distribution (case Dl ), the contribution of the piezoelectric layers to membrane force Njl1 and to the equivalent transverse loading (A?, 1 - (B, , / A,,)l\s, 1), 1JD, constitute an additional term for the right-hand side of eqn (27). - BCl +Dl For a constant 6, (Dl ) and pinned ends BCl, the solution of eqn (27) is: u3 =
C,F(Kx,)
+
c,
Pd
2(Nl
-NJ
where the three unknowns C,, C,, firI are determined by solving numerically the following nonlinear algebric system, which results from the satifaction of the boundary conditions
F(c)=cosh(c)and
G(t)=sinh(e)
whenK2=(fi,,-N,r)/D,>O F(e)=cos(t)and
G(~)=sin(~)
when K2=(l)ujll-fi11)/Dc>0 -BC2+Dl For a constant &‘,(Dl ) and simply-supported ends BC2, the previous solution to eqn (27) and the expressions for C, and C, still apply (note than N,, is now imposed). The deflection curve is
I
+!$l11
p(L2-x:) 2K2 I
(36)
-BC2+D2 Whether B, is assumed to be proportional to the deflection u3 (adaptive case D2) and to the feedback gain G of the active control apparatus, the magnitude of which is varied in such a way to maintain the saturation level in the PZT G1195 layers, the expression for the moment resultant becomes ~,,=r3,&3@3 (37) Corresponding to the activation control laws CL 1 and CL2, the expression for deflection is
(33)
I
P
(34)
1
PW2-XT)
K4
2K2
(38)
-BC3 + D2 Corresponding to the activation control laws CL1 and CL2 and loosely-clamped ends, the deflection is
(39) Here, as in the BC2 + D2 case, F(c)=cosh(E)and
P2L2 +6(&1-~d2
(35)
when K2=
G(~)=sinh(~)
&I -h&, G DC
Large deflection of Timoshenko beams
F(c)=cos(c)and whenK2=
(Y(t)=sin(t) -
flu-&d& D,
4 NUMERICAL TESTS
The aim of the present numerical tests is two-fold: (i) to investigate the effect of the induced strains (i.e. of the strategy of activation) and (ii) to assess the effectiveness of the active control of the large deflection and of the buckling behavior of multilayered Timoshenko beams coupled with piezoelectric actuator/sensor layers. As an illustrative example, the possibility of controlling the nonlinear static response and the buckling behavior of unsymmetrically laminated beams in bending under combined uniformly distributed transverse and compressive end loads, @ and Nil respectively, is investigated. Here the positive direction for loading and deflections is assumed to be the upward direction. Four-layered beam.s with various lay-ups and boundary conditions are considered. The basic structure is a two-layered unsymmetric laminate; the top and bottom surface layers are piezoceramic actuator layers. Table 1 gives the values of the constants characterizing the PZT G1195 N piezoceramic actuator layers and the unidirectional lamina of the basic structure. The length to thickness ratio of the beam is here assumed to range from 2.04 to 4.63, depending upon the thickness of the actuator layers (see Table 2). Due to the exploratory nature of the present investigation, it appears interesting to test two Table 1. Electromechanical
55
extreme cases: the first one with very thin piezoceramic layers, the second one with very thick piezoceramic layers. For this purpose, the length to thickness ratio of the single actuator layers is chosen to be 126.58 for the lay-ups named (a) and (c), and 6.90 for the lay-ups named (b) and (d), whereas the length to thickness ratio of the two laminae constituting the basic structure is arbitrarily set equal to 5. To give some insight on the perspective of the piezoelectric effect to change the mechanical response, six activation strategies are investigated: CLl, CL2, which are both antisymmetric, the former with a positive activation in the upper layer and a negative one in the lower layer; vice versa for CL2. The strategies of CL3 and Cl4 are characterized by the activation (positive for CL3 and negative for CL4) of the upper layer only. The strategies CL5 and CL6 are the same as CL3 and CL4, with the activated layer being the lower one. Tables 3 and 4 list the values of the extensional and coupling stitiesses of the activated layers resulting from the previous strategies of activation. Figures 2-13 summarize the results. The behavior of the (90”/0’) basic structure under uniform transverse load ~7and pinned end boundary conditions has been investigated by Icardi,lS and Chen & Shu.16 From these studies it results that the induced in-plane stress resultant can be tensile (positive), compressive (negative) or zero depending upon the sign of the coupling stiffness B, l, the sign and the magnitude of the transverse load and the slenderness ratio. For B,, and p opposite in sign, a transition point can be found, corresponding to which the in-plane resultant turns from tension to compression. In conseproperties of the layers
Material
F’ZT G1195 N Basic
9.135 E+06 20.00 E + 06
3.513 E+06 0.800 E + 06
;:;
9.843 E - 09
1524E+04
Table 2. Lay-ups investigated Lay-ups
(a) (b) (c) (d)
PZT PZT PZT PZT
G1195 G1195 G1195 G1195
N/90“/0”/PZT G1195 N N/9V/O”/l?ZT G1195 N N/45”/ - 30”/F’ZT G1195 N N/45”/-30”/F’ZT G1195 N
Thicknesses (in.)
0~00158/0~02/0~02/@00158 @029/0.02/0*02/0.029 0~00158/0~02/0~02/@00158 0~029/0~02/0~02/0~029
Length/ thickness 4.63 2.04 4.63 2.04
M. Di Sciuva, U. Icardi
56
Table 3. Strategies of activation for lay-ups (a) and (c): extensional and coupling stiffnesses of activated layers
A,, (lb/in.) B,, (lb in&)
CL1
CL2
CL3
CL4
CL5
CL6
0.0 3,894
-,9”s94
128,171 1,947
- 128,171 - 1,947
128,171 - 1,947
- 128,171 1,947
Table 4. Strategies of activation for lay-ups (b) and (d): extensional and coupling stiffnesses of activated layers
A,, (lb/in.) 8,, (lb h/in.)
0.7
0.5
CL2
CL3
CL4
CL5
CL6
0.0 657
0.0 -657
15,809 329
- 15,809 - 329
15,809 - 329
- 15,809 329
2500
A
0.6
CL1
-
r/ 4
= 462174 II
B,,
= -3742
D,, = 71 A 44 = 32129
/
-
strategy of activation .
A,, = 462174 6,
2000
1= -3742
D,, = 71 A44 = 32129
~
1500
<2-
Strategy of activation
. 0 CL1 l CL2
G
4
s
0.1
z t 9 r-”
I
/
I
A
CL5
0
CL3
.
CL4
A CL5
0 -0.1
-0.3 -0.4
/ r
Transverse
load B -500
Pinned edges Pinned edges
Fig. 2. Non-dimensional transverse deflection UJ/I vs transverse load p (psi): lay-up (a); boundary conditions BCl.
Fig. 3. In-plane stress resultant A,, (lb/in.) vs transverse load p (psi): lay-up (a); boundary conditions BCl.
quence of compression at positive p, deflections became larger than those for negative @. Thus, it appears to be of some interest to investigate the possibility of changing this behavior by induced strains. The results of this investigation are summarized in Figs 2-7. Specifically, Fig. 2 gives the load-deflection curves for the beam with lay-up (a) under transverse load p and pinned end boundary conditions, with unactivated and activated piezoelectric layers. For the same beam, Fig. 3 gives the behavior of the in-plane stress resultant ZVI1 vs transverse load. The effect of activation appears to be negligible for this sample case. Numerical results, not reproduced here, show that the same conclusions still apply to the behavior of beams with lay-up (c).
Figures 4 and 5 refer to the beam with lay-up (b); Figs 6 and 7 refer to the beam with lay-up (d). For both beams, the same transverse load, boundary conditions and strategies of activation as Figs 2 and 3 hold. Here the effect of activation appears to be substantial. In detail, the effect of inducing strains in the activated layers appears significant for deflection at higher values of the transverse load, while the effect on the in-plane stress resultant appears only at low values of the load, i.e., in the field of inversion for the in-plane stress resultant. Note that the lay-ups (b) and (d) differ from the corresponding lay-ups (a) and (c) only in the thickness of the activated layers. Based on the Bernoulli-Euler beam theory, the linearized buckling behavior of the (90”/00) basic
Large deflection of Timoshenko beams 0.20
0.20
A,,= 445921
r
57
A,,
= 419764
B,, = -223.75 0.15
D,,
= 66.86
-
0
. 0 CL1 * CL2 0
CL3
.
CL4
A CL5
/
0
+ %
/d l
A,, = 84296 strategy of activation
I
I
I
8 s
I
I
I
$
E
g
1 o
1 8
Strategy of activation . 0
CL1
l CL2 OCL3 ;;;;
0 CL6 :, 9
-0.05
-
-0.20
L
Transverse load j!
0
CL6
i:
-0.20
L
Pinned
edges
Fig. 4. Non-dimensional transverse deflection uJh vs transverse load p (psi): lay-up (b); boundary conditions BC 1.
1800 c
1600
A,,
= 445922
%I
= -725.5
edges
Fig. 6. Non-dimensional transverse deflection Q//I vs transverse load @(psi): lay-up (d); boundary conditions BCl.
1600
1400
D11= 67.53 ‘444 =
Pinned
83933
A,,
= 419764
B,,
= -223.75
D,,
= 66.86
A,,
= 84296
1200 strategy of activation .
activation
0 CL1 * CL2 \ 1
0
CL3
.
CL4
\
0
Transverse load B Pinned
edges
Transverse load 8 Pinned
edges
Fig. 5. In-plane stress resultant A,, (lb/in.) vs transverse load p (psi): lay-up (ID);boundary conditions BC 1.
Fig. 7. In-plane stress resultant fi,, (lb/in.) vs transverse load i, (psi): lay-up (d); boundary conditions BCl.
structure under combined uniform transverse load p and axial compression, and simply supported and loosely-clamped end boundary conditions has been investigated by Sun & Chin17 For simply supported ends, contrary to looselyclamped ends, it results in the possibility that the
buckling deflection is opposite in sign to the applied transverse load, provided that the coupling stiffnessB, 1 and the applied transverse load p are opposite in sign, and that the transverse loads to not exceed a certain level. Concerning the linearized buckling behavior, unbounded deflec-
M. Di Sciuva, U. Zcardi
58
A
;
8 .I ii
0.20
-
0.15
-
0.10
-
, , = 462174
B,,
= -3742
D,,
= 71
A 44
= 32129
A
strategy of activation
0 CL1
D,,
0.04
l CL2
3” 8 ._ t; 2
0 A CL2
c 4
1= -725.5
B,
.
* 2 5 zl 9
4 5 2 2 G
strategy of activation
, , = 445922
.
= 67.53
0 CL1
0.03 0.02 0.01 0
c -0.01 -0.02 -0.15
In-plane t
-0.20
Simply
stress
resultant
supported
edges
p=
Simply
200
Fig. 8. Transverse deflection uj (in.) vs in-plane load I?,, (lb/in.): lay-up (a); BC2 +Dl; lower curve (p=200 psi), upper curve (p = 2000 psi).
A
3”
II = 462174
B1,
= -3742
D,,
= 71
A
A 44 = 32129
CL1
0
._ 2
0.15
4 CL1
G 4
0.10
zP
0.05
G
p=
2000
P
&A--d--AN
0 P ---_
I
I
= -725.5
Dl,
= 67.53
A
P
A CL2
I
,, = 445922
B,,
I
I
I
I
1
E ._
0.03
z c
0.02
4 k% 5 z2 c
-0.05
I -0.20
In-plane stress resultant A,, Simply
supported
p=
supported
50
edges
Control .
* CL2
44 = a3933
tions are found to occur (whether or not the transverse load is present) as I?,, approaches the buckling load. The question whether the active control is effective in shifting this load level and in altering the mechanical behavior of the beam after and before this load level, appears to be of some importance. Concerning the possibility of shifting the value of p at which the inversion in sign of NII takes place, numerical results (not reproduced here) indicate that none of the activation strategies is able to meet this goal, at least for the tested sample problems. The numerical results plotted in Figs 8-13 shows the effectiveness of the piezoelectric effect to change the mechanical response of the adaptive beam. Specifically Figs 8 and 9 give
1000
A CL1 A CL2
0 -0.01 -0.02
/
-A
---
0.01
l s g-i-
I
I
“:
7
8 \ ”
In-plane slrcss resultant A,, Simply
edges
Fig. 9. Transverse deflection u3 (in.) vs in-plane load N,, (lb/in.): lay-up (a); BC2 +D2; lower curve (p= 200 psi), upper curve (p= 2000 psi).
p=
0
-0.10 -0.15
R,,
0 CL1
0.04
+ CL2
0.20
1L
resultant
law
. 0
stress
Fig. 10. Transverse deflection uj (in.) vs in-plane load N,, (lb/in.): lay-up (b); BC2 + D 1; lower curve (@= 50 psi), upper curve (p= 1000 psi).
Control law
5
9zl
In-plane
i?,,
supported
I 8 s
F=
50
edges
Fig. 11. Transverse deflection uj (in.) vs in-plane load A,, (lb/in.): lay-up (b); BC2 + D2; lower curve (p = 50 psi), upper curve (p= 1000 psi).
Loosely-clamped 0.09 0.08
= 462174 E
A,,
= -3742
0.07
B,,
0.06
D,,
0.05
A dl = 32129
edges
Control law
F= 2000
. = 71
0
CL1
+ CL2
I
In-plane stress resultants f?,, Fjg. 12. Transverse deflection ujm (in.) vs in-plane load N, I (lb/in.): lay-up (a); BC3 + D2; lower curve (p= 200 psi), upper curve (j= 2000 psi).
Large deflection of Timoshenko beams Loosely---clamped edges
been studied. The following observations drawn from the numerical tests performed.
0.12 3; z ._
0.10
:
0.08
5 u s 5 $
0.04
p9
0.02
0.06
200
0
In-plane stress resultants A,, Transverse deflection u3 (in.) vs in-plane load A,, ~$i~.~’ lay-up (b); BC3 +D2; lower curve (p= 200 psi), upper curve (p= 2000 psi).
results on the transverse deflection of the beam with lay-up (a) and simply-supported boundary conditions under transverse load p and in-plane compressive load N, I. Figure 8 shows the effect of the strategy of activation; Fig. 9 shows the effect of the active control (D2). Figures 10 and 11 give the same results, but for the beam with lay-up (b). Here the effect of activation appears to be substantial for deflections at higher values of the transverse load. It is observed that, for downward bending, CL1 reduces deflections, whereas CL2 increases them; the opposite holds for upward bending. The response of the adaptive beam under boundary conditions BC3 and D2 x,-distribution of the electric field is plotted in Figs 12 and 13. For this case only the effect of adaptive control is investigated for lay-ups (a) and (b). As for BC2 boundary conditions, CL1 reduces deflections, CL2 increa.ses them, specially when the in-plane stress resultant approaches the buckling load. Note that, as CL1 reduces deflections, the response of the unsymmetric plate (which behaves like an imperfect symmetrically laminated beam as a result of the coupling) becomes close to that of a symmetric one subjected to bifurcation buckling. This suggests the possibility of modifying the buckling behavior of shape imperfection sensitive structures.
5 CONCLUSIONS The nonlinear deflection behavior of multilayered Timoshenko beams including fibre-reinforced and piezoelectric actuator and sensor layers have
59
are
- Boundary conditions BC 1: (1) The effect of activation appears to be influenced by the lay-up considered. Precisely, it is shown to depend on the coupling stiffness of the basic strucure and on the thickness of actuator layers. Whereas it appears negligible for lay-ups (a) and (c), it appears substantial for the lay-ups (b) and (d), for which the response becomes activation law dependent. The effect of inducing strains in the acti(2) vated layers appears significant for deflection at higher values of the transverse load while, on the countrary, the effect on the inplane stress resultant appears only at low values of the load. (3) The effect of activation on deflections is reduced for the lay-ups with a lower coupling; the opposite occurs for the induced inplane resultant, a greater influence being shown at low values of the transverse load p for all the examined cases. - Boundary conditions BC2: (1) For this case, the response of the lay-ups (a) and (b) is investigated along with the activation and the adaptive control laws. (2) As for the unactivated structure, the bending direction appears to be altered both by the direction and by the magnitude of the transverse load, a load level being found at which the deflection ceases to follow the sign of B,,. The numerical analysis shows that the effect of activation appears negligible for such inversion phenomenon. (3) For the investigated cases (two transverse loads above and below the inversion load), the influence of the constant electric field Dl on the response appears to be rather scarse for lay-up (a). The opposite holds for the lay-up (b), where the deflections appear to be increased or reduced depending on the activation strategy. (4 For downward bending, the activation law CL1 reduces deflections, whereas CL2 increases them; the opposite holds for upward bending. - Boundary conditions BC3: (1) The effect of adaptive control is investigated for lay-ups (a) and (b). As for BC2 boundary conditions, CL1 reduces deflec-
M. Di Sciuva, U. Icardi
60
tions and Cl2 increases them, above all when approaching the buckling load. As a consequence, the behavior of the unsymmetric beam, which behaves like an imperfect symmetrically laminated beam as a result of the coupling, becomes close to that of a symmetric one subjected to bifurcative buckling. This suggests the possibility of modifying the behavior of shape imperfection sensitive structures. The influence of the transverse load is (2) smaller than that for BC2; furthermore, transverse loads cannot cause inversion of the deflection. The influence of the transverse load is considerable when the compressive load approaches the buckling load. In this same range, the effect of the control law is substantial. Preliminary numerical tests show that the effect of induced strains is enhanced at local level, for example, when analysing the variation of the thickness-wise distributions of the strains and stresses consequent to the activation of the piezoelectric layers. As a consequence, modeling of adaptive multilayered beams and plates claims for discrete-layer theories. In this respect, work is in progress. REFERENCES 1. Tzou, H. S. & Gadre, M., Theoretical analysis of a multilayered thin shell coupled with piezoelectric shell actuators for distributed vibration controls. J. Sound and I%., 132 (3) (1989) 433-50. 2. Rogers, C. A., Liang, C. & Jia, J., Structural modification of simply-supported laminated plates using embedded shape memory alloy fibers. Computers & Structures, 38 (1991) 569-80. 3. Crawley, E. F. & Lazarus, K. B., Induced strain actuation of isotropic and anisotropic plates. AZAA J., 29 (1991) 944-51. 4. Tzou, H. S. & Tseng, C. I., Distributed piezoelectric sensor/actuator design for dynamic measurement/ control of distributed parameter systems: a piezolectric finite element approach. J. Sound and I%., 138 (1990) 17-34. 5. Ha, S. K., Keilers, C. & Chang, F. K., Finite element analysis of composite structures containing distributed piezoceramic sensors and actuators. AZAA J., 30 (1992) 772-80. 6. Crawley, E. F. & de Luis, J., Use of piezoelectric actuators as elements of intelligent structures. AZAA J., 25 (1987) 1373-84.
7. Im, S. & Atluri, S. N., Effects of a piezo-actuator on a finitely deformed beam subjected to general loading. AZAAJ., 27 (1989) 1801-7. 8. Song, O., Librescu, L. & Rogers, C. A., Application of adaptive technology to static aeroelastic control of wing structures. AZAA J., 30 (1992) 2882-9. 9. Di Sciuva, M. & Icardi, U., Stability behavior of adaptive multilayered plates and shells under combined loads. In MEET’W93, First Joint ASCE-EMD, ASME-AMD, SES Meeting, Charlottesville, Virginia, 6-9 June, 1993. 10. Di Sciuva, M., An improved shear-deformation theory for moderately thick multilayered anisotropic shells and plates. J. Appl. Mech., 54 (1987) 589-96. 11. Brush, D. 0. & Ahnroth, B. O., Buckling of Bars, Plates and Shells. McGraw-Hill Book Co., N.Y., 1975. 12. Kicher, T. P. & Mandell, J. F., A study of the buckling of laminated composite plates. AZAA J., 9 ( 197 1) 605- 13. 13. Leissa, A. W., Conditions for laminated plates to remain flat under inplane loading. Comp. Structures, 6 (1986) 261-70. 14. Zhang, Y. & Matthews, F. L., Large deflection behavior of simply supported laminated panels under in-plane loading. J. Appl. Mech., 52 (1985) 553-8. 15. Icardi, U., The nonlinear response of unsymmetric multilayered plates using smeared laminate and layerwise models. Comp. Structures, 29 (1994) 349-64. 16. Chen, H. P. & Shu, J. C., Cylindrical bending of unsymmetric composite laminates. AZAA J., 30 (1991) 1438-40. 17. Sun, C. T. & Chin, H., On large deflection effects in unsymmetric cross-ply composite laminates. J. Comp. Materials,22 (1988) 1045-59.
APPENDIX The set of eqns (12) is transformed into the governing eqn (27) for the deflection u3 as follows. First, let us consider the equilibrium equations N 11,1= 0 T,,,
+u%l~3,1),1
A4 11,1-
=
-P
(A4
T,=o
Then, computer s1 1 from the constitutive equation ( 16 ) and substitute it in ( 16,); the result is I
By substituting eqn (A.2) and eqn (A. 13) into eqn (A.l,), eqn (27) is obtained.
Printedin Great Britain. AU rights reserved 0263~8223/95/S9.50 0263-8223(95)00001-l
ELSEVIER
Large deflection of adaptive multilayered Timoshenko beams Marco Di Sciuva & Ugo Icardi Department of Aerospace Engineering, Politecnio di Torino, Corso Duca degli Abruzzi, 24-10129 Turin, Italy
The paper deals with the formulation of the nonlinear equations governing the mechanical behavior of anisotropic, laminated Timoshenko beams having any number of arbitrarily positioned and orientated actuator and/or sensor layers. Use is made of the von KarmBn nonlinear strain-displacement relations. Subsequently, the static stability equations for the initial (bifurcation) buckling under transverse and compressive loads are formulated via the Euler method of the adjacent equilibrium configurations. The present analysis is quite general in .that no assumptions are made on the placements of the active layers, their symmetry, and their constitutive relations. The only assumptions pertain to the behavior of the adaptive multilayered beam as one equivalent, linear elastic, anisotropic beam (smeared laminate model). Numerical results deal with the nonlinear flexural response of unsymmetrically laminated beams under transverse and compressive axial loads. It is concluded that the effectiveness of the control depends on the boundary conditions, mechanisms of activation and lay-ups.
1 INTRODUCTION Adaptive structures are those which can adjust their geometry, stiffness and damping on demand to meet a high degree of adaptability to changes in their external and internal loading environment. External conditions m.ay consist of environment, loads, or the desire to change the scope, purpose, or geometry of the structure after it has been built and is in service. Internal conditions may be damage or failure to isolated portion of the material or structure. Adaptive composite structures built by using fiber reinforced composites where some layers behave as induced strain actuators and/or sensors have been already vogue in aerospace industry. The success of these structures depends largely on the effective control of the structure through the actuators and the sensors embedded in their load carrying members. To do this, further theoretical research work for a better understanding of the elastodynamical behavior of adaptive structures is needed. Theoretical analysis of adaptive multilayered composite structures is currently accomplished using the so-called Classical Lamination Theory (CLT) (see Tzou & Gadre,’ Rogers et a1.,2 Crawley & Lazarus3) coupled with some numerical schemes (for example, Ritz’s method).
Other numerical approaches include finite element formulations (Tzou & Tseng,4 Ha et aL5). The addressed topic in these papers is mainly the vibration control, while it appears that static control has been investigated only with respect to Euler’s beams (Crawley & de Luis,6 Im & Atl~ri,~ Song et al*). The range of applicability of the CLT in evaluating the global behavior (deflections, lower natural frequencies and buckling loads) is well understood to be that of slender beams and thin multilayered plates and shells. Thus, the inclusion of the transverse shear deformability effect in modeling moderately thick, adaptive multilayered plates and shells appears to be of great interest. Furthermore, the study of the active control of the geometrically nonlinear response and of the buckling behavior of adaptive beams appears the best of authors’ knowledge to be rather scarse if not absent in the open literature. Another topic which as received less attention is the effect of changes in the boundary conditions. The aim of the present study is to fill some of the previous gaps. To this end, we use the general formulation given by Di Sciuva & Icardi9 to derive the variationally consistent nonlinear equations governing the mechanical behavior of anisotropic, laminated Timoshenko beams having any
50
M. Di Sciuva, U. Icardi
number of arbitrarily positioned and oriented actuator and/or sensor layers. Use is made of the von K&-man nonlinear strain-displacement relations. Subsequently, the static stability equations for the initial (bifurcation) buckling under transverse and compressive loads are formulated via the Euler method of the adjacent equilibrium configurations. The present analysis is quite general in that no assumptions are made on the placements of the active layers, their symmetry, and their constitutive relations. The only assumptions pertain to the behavior of the adaptive multilayered beam as one equivalent, linear elastic, anisotropic beam (smeared laminate model). This assumption generally results in incompatible transverse shearing stresses between two adjacent layers, Di Sciuva,r” and can produce poor predictions of thickness distributions of displacements and stresses across the thickness for thick laminates. Because in this paper we are interested only in the global response, the smeared laminate model in conjunction with the von K5rn&n straindisplacement relations appears to be adequate. Numerical results deal with the nonlinear flexural response of unsymmetrically laminated beams under transverse and compressive axial loads. They show how effective may be the control of the response by induced strain actuator (in this case, some piezoelectric material layers). Various boundary conditions and control laws are investigated. 2 FORMULATION OF THE EQUATIONS OF MOTION OF ADAPTIVE BEAMS 2.1 Kinematics Let us consider a multilayered beam of total thickness h, length 2L and width b consisting of a finite number IV of linearly elastic orthotropic layers of uniform thickness perfectly bonded together. Each layer can exhibit different individual physico-mechanical properties. The points of the beam are referred to a Cartesian orthogonal co-ordinates system Xi, i= 1,2,3 (see Fig. 1) with the origin at midlength of the beam, x1 being the longitudinal axis of the beam (reference axis), (x1, x3) the plane of flexure, and (x2, x3) the plane of the beam cross section. The beam model developed in this study is based on the following representation of the displacement field across the beam thickness,
=3 Midsurface 1
Fig. 1.
Actuator lavers -
t
Geometrical
’
n
beam configuration.
where 6, and z& are the displacements of a generic point of the beam in the x1-, and x3-directions, respectively; u1 and ug are reference axis displacements in the same coordinate directions; #, denotes the total rotation in the (x,, x3) plane of a line normal to the reference axis. To formulate a geometrically nonlinear theory of a shear-deformable (Timoshenko), adaptive straight beam, the strain-displacement relationships are assumed to be of the following form (small strain and moderately small rotation approximation; von K&man strain tensor) El1
=&11+X3kl;
&3=q3
where El1 = u1.1
+ b3,J2/2;
k1=
$4,1;
El3 = $4 + U3,l
(3) are the strains and the curvature of the reference axis. The notation ( .),1 denotes partial derivative (d( * ml ). 2.2 Variational formulation of nonlinear equations of motion Let us consider a piezoelectric continuum filling a volume T/ of boundary 9. Under the assumptions that and pryoelectric (i) thermal, thermoelastic phenomena do not take place; (ii) mechanical and electrical forces during the vibration are balanced at any given tune, thus allowing for a quasi-static approximation in the analysis; the following relations hold: - Linear piezoelectric constitutive equations 9,=
e,kl.6kl+ xi,&,,, direct effect
(4)
uii = C &gkl - emiiRminverse effect in terms of strain tensor components, 3, = dnkrGk,+x&J’, F,.= S%,ii,_,+ d 2..
direct effect inverse
effect
or
(5)
Large deflection of Timoshenko beams
in terms of stress tensor components, where 6’ii and Eliidenote the mechanical stress elastic tensor and the nonlinear strain tensor components, respectively; C$, denotes the sixth-order, symmetric positive definite elastic stiffness matrix elements of the pieozoelectric material (here assumed to be an elastic piezoelectric monoclinic material), under constant electric field conditions and at prescribed temperature; S$, is the symmetric, positive definite elastic compliance matrix of the piezoelectric malterial, under constant electric field and at prescribed temperature; emii denotes the dielectric permissivity matrix elements, also known as the piezoelectric stress coefficients matrix; it couples the mechanical and electrical equations; xi,,, is the symmetric matrix of dielectric constants or dielectric susceptibility under conditions of constant strain; 8, is the electric field vector in the material, maintained constant by potentials. applied to suitable electrodes; gn is the electric displacement or dielectric induction vector; d,,= e,,S& is the piezoelectric strain coefficient matrix; x& is the permissivity (dielectric susceptibility) coefficients matrix under constant elastic stress. The superscripts, such as ( )O, on material properties typically denote the mechanical or electrical boundary conditions under which the constants are valid. Obviously, eqn (4J with emu= 0 holds for any monoclinic material. In the expressions for e,$ and dmii the first subscript refers to the electrical axis, while the subsequent two refer to the mechanical strain (for example, e1i3 refers to the shear strain developed in the plane (x1,x1) in response to an electric field in the x,-direction). - Field equations In the given Cartesian system of coordinates, the field equations (under the hypothesis that the material is insulated so that no free electric charges exist) read [( 6,+ L$i) ~ki],k= pi equations Of motion (i,j,k,n= %,n=O
1,2,3)
Maxwell-Gauss
(6) equation
where pj=fi”+fp+fi”
(7)
lists the three components of the body forces f!, of the inertial forces f:, and of dissipative forces
f$
the piezoelectric material, are: (i) the surface tractions pi applied on the surfaceYe, and the displacement tii orPI,; (ii) the electrical charge density 4 applied on the surface ~7~ and the electric potential @ onYQ. The previous equations of motion and the associated natural boundary conditions follow from the Euler’s equations of the following variational statement pi?:isliidV+ V
with t?Fm=sDm
The boundary conditions (mechanical and electrical) on Sp, the boundary of the volume V filled by
(9)
In eqn (8) the overdots indicate differentiation with respect to time, Substitution of eqns (4) into eqn (8) yields
Jv
PUidUi d V+ =
-
(C$kk,-
e,,B,)6iq
dV
Jv
(10) Here, the variational statement ( 10) is used to obtain the equations of motion and the variationally consistent natural boundary conditions for the adaptive multilayered Timoshenko beam under consideration. This is accomplished by substituting into eqn ( 10) the approximate displacement field ( 1 ), the approximate strain field (2) and (3), and the constitutive relations for the individual layers. These relations for the beam under consideration and under the assumption that the x,-direction is associated with the direction of poling (i.e., the direction of the applied electric field vector) read .,
- Boundary conditions
51
011
=Q
1141-44e&
(11)
M. Di Sciuva, U. Icardi
52
where Q 11 and Qa4 are the transformed reduced elastic stiffnesses, depending on the elastic coefficients of the piezoelectric material, and e31 is the reduced dielectric permissivity coefficient. As usual, we have adopted the contracted notation for the reduced stiffnesses and dielectric permissivity coefficients. Applying the well-known techniques of variational calculus, the vanishing of the virtual variations in the surface and line integrals yields the equations of motion and the variationally consistent natural (mechanical) boundary conditions: - Nonlinear equations of motion N 11.1
-
T~=m,zi,+m,$l
(12)
Boundary conditions At the ends x1 = f L, one element of the pair of conditions
(NsG);(T,+N,,~,
(15) where the quantities affected by an overhat (^)and overcheck (‘) identify the pure mechanical and induced strain actuator contributions to the indicated quantities, respectively. The resulting expressions for the force and moment elastic stress resultants in terms of the generalized displacements follow from the constitutive equations 1 NH =A,,+ +B,+, (16)
where A,l, B,, and D,, are extensional, coupling and bending stiffenesses, and A,, is transverse shear stiffness given by [A,, B,, OH]=@
(13)
’
is assigned. In eqn ( 12) p is the transverse load per unit length of the beam. The following expressions for the force and moment resultants of the stress and of inertial forces hold e11>6.13Px36.11));
~3
x%211(4>;
4, = xu/Q&h
,-M,-m,&;~,);
(M,,; $1)
A,,%,,,
BIA,,,
+ %#1,,1 +Jy1+ &4&J
=RHS, (18)
+A44~3,1l+JG=RfG +DI,~1,11+~3+6,~,,,,=RHS,
where pi given by
where
(17)
x44 being the shear correction factor. When expressed in terms of generalized displacements, the geometrically nonlinear differential equations of motion become
A,,#,,,
mi=(xip(s)>
(14)
N
M,, = d,J&,
8 = A,,&,,
-~+m,ii,,,+m,ii,+m,~,,,
(NH, T,JG)=((
Tl = PI;
N,, = A,, + &fill;
fi,, = &El1 + D,, x1
=VZ,Z&+t?I,J,
T,,,+(N,,u,,,),,=
M 11.1
can write
are nonlinear
differential
operators
XJS (...)
(...)=bC
dx,
s=l i xjs-,
It is concluded that the equations of equilibrium and the boundary conditions for multilayered Timoshenko beams having induced strain actuator layers are formally the same as those for the non-adaptive multilayered Timoshenko beams, provide that they are expressed in terms of force and moment stress resultants. - Equations of motion in terms of displacements The above force and moment stress resultants take into account pure mechanical and actuation strain contributions. The use of the tracer 6,) which identifies the contribution brought by the actuation strain (it takes the values 1 or 0), makes it possible to recast these in a form in which the actuator effect appears in explicit form. Thus, we
N3=ifMb43,A21,,
(1%
and RHS,(i= 1,2,3) stands for the right-hand side of eqn(l2,). 2.3 Static stability equations The linearized equations governing the bifurcation buckling of the multilayered adaptive beam subjected to compressive axial load N,, are formulated by using the Euler method of the adjacent equilibrium configurations, Refs 9 and 11. Let us assumed that: (i) the external forces vary neither in magnitude nor in direction during
53
Large dejlection of Timoshenko beams
the buckling; (ii) before the onset of the buckling, the deformation state is characterized by absence of rotations of the normals to the reference axis, then the beam will be in a state of uniform axial stress corresponding to the applied load NI1 and the stability problern reduces to the classical (bifurcation) buckling problem. Under these assumptions, the principle of virtual work for the adjacent equilibrium reads
and the vanishing of one element of the pairs (T,;
-
T,)W,l
dx-
1 W,h
(20)
+(~~+N,,u,,,)tSu,+M,,~~,14L=O
Note that according to the adjacent equilibrium criterion, terms of third or higher order have been ignored in the above equation; in addition, all of the force and moment stress resultants which appear in eqn (20) (with the exception of the external forces), as well as the generalized displacements ul, u3, and iI are measured from the original equilibrium configuration, i.e., they represent the buckled mode. The incremental displacement field is of the form given by eqn (l), while for the incremental strain components we have the following linearized expressions
(25)
h).
3 METHOD OF SOLUTION The expressions for the in-plane force RI, and moment resultants a,, induced by the strain actuator piezoelectric layers are r;i,,
+M,,,
UK,;
~3);
Ml,
=&d,,g,;
=&d,,~,
(26)
where A,, and 6),, refer to the activated layers only, for an applied electric field vector with the only non-vanishing component in the thickness direction. Following the procedure presented by Chen and Shu16 (see Appendix), the set of eqns ( 12) is transformed into the following governing equation for the deflection u3
u3,1111-
(till -~ll),ll
fill -riT1
u3,11-
DC
D‘
l-j =----D‘
(27)
where Eli =
q,;
k=h,,;
(21)
&13=h+U3,1.
Taking into accotmt that the virtual variations are independent varia.tions, the principle of virtual work yields the following system of differential equations and boundary conditions governing the bifurcation buckling: - Field equations N 11,1=
0
T,,l+%,~3,ll=o
M 11,1-
(22)
T,=o
or, in terms of generalized displacements, A,,%,,,
(23)
BC2: simply supported u3( &L)=O;
=0
- Boundary conditions The appropriate bou:ndary conditions are given by N,,=O
(29)
M,,( f L)=O;
A44~1,1+A44U3,11-+~~1U3,1*=0
+ 6,AIIJ
and x44 is taken to be equal to ~t*/12. Equation (27) is a linear ordinary differential equation with constant coefficients, whose general solution depends on the coefficients, on the equivalent loading term (resulting from the control and external loading), on the boundary conditions and on the x,-distribution of 8, (see eqn (26)). The following set of boundary conditions has been considered: BCl:pinnedu,(+L)=O;u,(+L)=O;
+~,,9,,,,+&4~1,,1=0
BII~I,,, + LW,,,,
03)
M,,( &L)=O;
BC3: loosely-clamped
at x1 = f L
(24)
NI r( + L ) = IV1 1; (30)
NI 1(+ L ) = A, 1;
u,(fL)=O;u,,,(fL)=O;
together with the following x,-distributions electric field (whose only non-vanishing
(31)
of the com-
54
M. Di Sciuva, U. Icardi
where
ponent is assumed to be 8,): D 1: B, = const; D2: B, proportional
to ug max,
As we said, the effects of the pizeoelectric layers which appear both in the second and in the third terms of eqn (27) depend on the xr-distributions of the g3 (see eqn (26)), i.e. by the strategy of the activation. Specifically, for &,-distribution depending on the x,-distribution of u3 (adaptive case D2 by feed-back activation), the membrane force Nl, and the equivalent transverse loading (M,, -(B,,IA,,)BPUSI,),~,/D, generate additional coefficients for z+,11 and u3 11, 1. For a uniform 8,distribution (case Dl ), the contribution of the piezoelectric layers to membrane force Njl1 and to the equivalent transverse loading (A?, 1 - (B, , / A,,)l\s, 1), 1JD, constitute an additional term for the right-hand side of eqn (27). - BCl +Dl For a constant 6, (Dl ) and pinned ends BCl, the solution of eqn (27) is: u3 =
C,F(Kx,)
+
c,
Pd
2(Nl
-NJ
where the three unknowns C,, C,, firI are determined by solving numerically the following nonlinear algebric system, which results from the satifaction of the boundary conditions
F(c)=cosh(c)and
G(t)=sinh(e)
whenK2=(fi,,-N,r)/D,>O F(e)=cos(t)and
G(~)=sin(~)
when K2=(l)ujll-fi11)/Dc>0 -BC2+Dl For a constant &‘,(Dl ) and simply-supported ends BC2, the previous solution to eqn (27) and the expressions for C, and C, still apply (note than N,, is now imposed). The deflection curve is
I
+!$l11
p(L2-x:) 2K2 I
(36)
-BC2+D2 Whether B, is assumed to be proportional to the deflection u3 (adaptive case D2) and to the feedback gain G of the active control apparatus, the magnitude of which is varied in such a way to maintain the saturation level in the PZT G1195 layers, the expression for the moment resultant becomes ~,,=r3,&3@3 (37) Corresponding to the activation control laws CL 1 and CL2, the expression for deflection is
(33)
I
P
(34)
1
PW2-XT)
K4
2K2
(38)
-BC3 + D2 Corresponding to the activation control laws CL1 and CL2 and loosely-clamped ends, the deflection is
(39) Here, as in the BC2 + D2 case, F(c)=cosh(E)and
P2L2 +6(&1-~d2
(35)
when K2=
G(~)=sinh(~)
&I -h&, G DC
Large deflection of Timoshenko beams
F(c)=cos(c)and whenK2=
(Y(t)=sin(t) -
flu-&d& D,
4 NUMERICAL TESTS
The aim of the present numerical tests is two-fold: (i) to investigate the effect of the induced strains (i.e. of the strategy of activation) and (ii) to assess the effectiveness of the active control of the large deflection and of the buckling behavior of multilayered Timoshenko beams coupled with piezoelectric actuator/sensor layers. As an illustrative example, the possibility of controlling the nonlinear static response and the buckling behavior of unsymmetrically laminated beams in bending under combined uniformly distributed transverse and compressive end loads, @ and Nil respectively, is investigated. Here the positive direction for loading and deflections is assumed to be the upward direction. Four-layered beam.s with various lay-ups and boundary conditions are considered. The basic structure is a two-layered unsymmetric laminate; the top and bottom surface layers are piezoceramic actuator layers. Table 1 gives the values of the constants characterizing the PZT G1195 N piezoceramic actuator layers and the unidirectional lamina of the basic structure. The length to thickness ratio of the beam is here assumed to range from 2.04 to 4.63, depending upon the thickness of the actuator layers (see Table 2). Due to the exploratory nature of the present investigation, it appears interesting to test two Table 1. Electromechanical
55
extreme cases: the first one with very thin piezoceramic layers, the second one with very thick piezoceramic layers. For this purpose, the length to thickness ratio of the single actuator layers is chosen to be 126.58 for the lay-ups named (a) and (c), and 6.90 for the lay-ups named (b) and (d), whereas the length to thickness ratio of the two laminae constituting the basic structure is arbitrarily set equal to 5. To give some insight on the perspective of the piezoelectric effect to change the mechanical response, six activation strategies are investigated: CLl, CL2, which are both antisymmetric, the former with a positive activation in the upper layer and a negative one in the lower layer; vice versa for CL2. The strategies of CL3 and Cl4 are characterized by the activation (positive for CL3 and negative for CL4) of the upper layer only. The strategies CL5 and CL6 are the same as CL3 and CL4, with the activated layer being the lower one. Tables 3 and 4 list the values of the extensional and coupling stitiesses of the activated layers resulting from the previous strategies of activation. Figures 2-13 summarize the results. The behavior of the (90”/0’) basic structure under uniform transverse load ~7and pinned end boundary conditions has been investigated by Icardi,lS and Chen & Shu.16 From these studies it results that the induced in-plane stress resultant can be tensile (positive), compressive (negative) or zero depending upon the sign of the coupling stiffness B, l, the sign and the magnitude of the transverse load and the slenderness ratio. For B,, and p opposite in sign, a transition point can be found, corresponding to which the in-plane resultant turns from tension to compression. In conseproperties of the layers
Material
F’ZT G1195 N Basic
9.135 E+06 20.00 E + 06
3.513 E+06 0.800 E + 06
;:;
9.843 E - 09
1524E+04
Table 2. Lay-ups investigated Lay-ups
(a) (b) (c) (d)
PZT PZT PZT PZT
G1195 G1195 G1195 G1195
N/90“/0”/PZT G1195 N N/9V/O”/l?ZT G1195 N N/45”/ - 30”/F’ZT G1195 N N/45”/-30”/F’ZT G1195 N
Thicknesses (in.)
0~00158/0~02/0~02/@00158 @029/0.02/0*02/0.029 0~00158/0~02/0~02/@00158 0~029/0~02/0~02/0~029
Length/ thickness 4.63 2.04 4.63 2.04
M. Di Sciuva, U. Icardi
56
Table 3. Strategies of activation for lay-ups (a) and (c): extensional and coupling stiffnesses of activated layers
A,, (lb/in.) B,, (lb in&)
CL1
CL2
CL3
CL4
CL5
CL6
0.0 3,894
-,9”s94
128,171 1,947
- 128,171 - 1,947
128,171 - 1,947
- 128,171 1,947
Table 4. Strategies of activation for lay-ups (b) and (d): extensional and coupling stiffnesses of activated layers
A,, (lb/in.) 8,, (lb h/in.)
0.7
0.5
CL2
CL3
CL4
CL5
CL6
0.0 657
0.0 -657
15,809 329
- 15,809 - 329
15,809 - 329
- 15,809 329
2500
A
0.6
CL1
-
r/ 4
= 462174 II
B,,
= -3742
D,, = 71 A 44 = 32129
/
-
strategy of activation .
A,, = 462174 6,
2000
1= -3742
D,, = 71 A44 = 32129
~
1500
<2-
Strategy of activation
. 0 CL1 l CL2
G
4
s
0.1
z t 9 r-”
I
/
I
A
CL5
0
CL3
.
CL4
A CL5
0 -0.1
-0.3 -0.4
/ r
Transverse
load B -500
Pinned edges Pinned edges
Fig. 2. Non-dimensional transverse deflection UJ/I vs transverse load p (psi): lay-up (a); boundary conditions BCl.
Fig. 3. In-plane stress resultant A,, (lb/in.) vs transverse load p (psi): lay-up (a); boundary conditions BCl.
quence of compression at positive p, deflections became larger than those for negative @. Thus, it appears to be of some interest to investigate the possibility of changing this behavior by induced strains. The results of this investigation are summarized in Figs 2-7. Specifically, Fig. 2 gives the load-deflection curves for the beam with lay-up (a) under transverse load p and pinned end boundary conditions, with unactivated and activated piezoelectric layers. For the same beam, Fig. 3 gives the behavior of the in-plane stress resultant ZVI1 vs transverse load. The effect of activation appears to be negligible for this sample case. Numerical results, not reproduced here, show that the same conclusions still apply to the behavior of beams with lay-up (c).
Figures 4 and 5 refer to the beam with lay-up (b); Figs 6 and 7 refer to the beam with lay-up (d). For both beams, the same transverse load, boundary conditions and strategies of activation as Figs 2 and 3 hold. Here the effect of activation appears to be substantial. In detail, the effect of inducing strains in the activated layers appears significant for deflection at higher values of the transverse load, while the effect on the in-plane stress resultant appears only at low values of the load, i.e., in the field of inversion for the in-plane stress resultant. Note that the lay-ups (b) and (d) differ from the corresponding lay-ups (a) and (c) only in the thickness of the activated layers. Based on the Bernoulli-Euler beam theory, the linearized buckling behavior of the (90”/00) basic
Large deflection of Timoshenko beams 0.20
0.20
A,,= 445921
r
57
A,,
= 419764
B,, = -223.75 0.15
D,,
= 66.86
-
0
. 0 CL1 * CL2 0
CL3
.
CL4
A CL5
/
0
+ %
/d l
A,, = 84296 strategy of activation
I
I
I
8 s
I
I
I
$
E
g
1 o
1 8
Strategy of activation . 0
CL1
l CL2 OCL3 ;;;;
0 CL6 :, 9
-0.05
-
-0.20
L
Transverse load j!
0
CL6
i:
-0.20
L
Pinned
edges
Fig. 4. Non-dimensional transverse deflection uJh vs transverse load p (psi): lay-up (b); boundary conditions BC 1.
1800 c
1600
A,,
= 445922
%I
= -725.5
edges
Fig. 6. Non-dimensional transverse deflection Q//I vs transverse load @(psi): lay-up (d); boundary conditions BCl.
1600
1400
D11= 67.53 ‘444 =
Pinned
83933
A,,
= 419764
B,,
= -223.75
D,,
= 66.86
A,,
= 84296
1200 strategy of activation .
activation
0 CL1 * CL2 \ 1
0
CL3
.
CL4
\
0
Transverse load B Pinned
edges
Transverse load 8 Pinned
edges
Fig. 5. In-plane stress resultant A,, (lb/in.) vs transverse load p (psi): lay-up (ID);boundary conditions BC 1.
Fig. 7. In-plane stress resultant fi,, (lb/in.) vs transverse load i, (psi): lay-up (d); boundary conditions BCl.
structure under combined uniform transverse load p and axial compression, and simply supported and loosely-clamped end boundary conditions has been investigated by Sun & Chin17 For simply supported ends, contrary to looselyclamped ends, it results in the possibility that the
buckling deflection is opposite in sign to the applied transverse load, provided that the coupling stiffnessB, 1 and the applied transverse load p are opposite in sign, and that the transverse loads to not exceed a certain level. Concerning the linearized buckling behavior, unbounded deflec-
M. Di Sciuva, U. Zcardi
58
A
;
8 .I ii
0.20
-
0.15
-
0.10
-
, , = 462174
B,,
= -3742
D,,
= 71
A 44
= 32129
A
strategy of activation
0 CL1
D,,
0.04
l CL2
3” 8 ._ t; 2
0 A CL2
c 4
1= -725.5
B,
.
* 2 5 zl 9
4 5 2 2 G
strategy of activation
, , = 445922
.
= 67.53
0 CL1
0.03 0.02 0.01 0
c -0.01 -0.02 -0.15
In-plane t
-0.20
Simply
stress
resultant
supported
edges
p=
Simply
200
Fig. 8. Transverse deflection uj (in.) vs in-plane load I?,, (lb/in.): lay-up (a); BC2 +Dl; lower curve (p=200 psi), upper curve (p = 2000 psi).
A
3”
II = 462174
B1,
= -3742
D,,
= 71
A
A 44 = 32129
CL1
0
._ 2
0.15
4 CL1
G 4
0.10
zP
0.05
G
p=
2000
P
&A--d--AN
0 P ---_
I
I
= -725.5
Dl,
= 67.53
A
P
A CL2
I
,, = 445922
B,,
I
I
I
I
1
E ._
0.03
z c
0.02
4 k% 5 z2 c
-0.05
I -0.20
In-plane stress resultant A,, Simply
supported
p=
supported
50
edges
Control .
* CL2
44 = a3933
tions are found to occur (whether or not the transverse load is present) as I?,, approaches the buckling load. The question whether the active control is effective in shifting this load level and in altering the mechanical behavior of the beam after and before this load level, appears to be of some importance. Concerning the possibility of shifting the value of p at which the inversion in sign of NII takes place, numerical results (not reproduced here) indicate that none of the activation strategies is able to meet this goal, at least for the tested sample problems. The numerical results plotted in Figs 8-13 shows the effectiveness of the piezoelectric effect to change the mechanical response of the adaptive beam. Specifically Figs 8 and 9 give
1000
A CL1 A CL2
0 -0.01 -0.02
/
-A
---
0.01
l s g-i-
I
I
“:
7
8 \ ”
In-plane slrcss resultant A,, Simply
edges
Fig. 9. Transverse deflection u3 (in.) vs in-plane load N,, (lb/in.): lay-up (a); BC2 +D2; lower curve (p= 200 psi), upper curve (p= 2000 psi).
p=
0
-0.10 -0.15
R,,
0 CL1
0.04
+ CL2
0.20
1L
resultant
law
. 0
stress
Fig. 10. Transverse deflection uj (in.) vs in-plane load N,, (lb/in.): lay-up (b); BC2 + D 1; lower curve (@= 50 psi), upper curve (p= 1000 psi).
Control law
5
9zl
In-plane
i?,,
supported
I 8 s
F=
50
edges
Fig. 11. Transverse deflection uj (in.) vs in-plane load A,, (lb/in.): lay-up (b); BC2 + D2; lower curve (p = 50 psi), upper curve (p= 1000 psi).
Loosely-clamped 0.09 0.08
= 462174 E
A,,
= -3742
0.07
B,,
0.06
D,,
0.05
A dl = 32129
edges
Control law
F= 2000
. = 71
0
CL1
+ CL2
I
In-plane stress resultants f?,, Fjg. 12. Transverse deflection ujm (in.) vs in-plane load N, I (lb/in.): lay-up (a); BC3 + D2; lower curve (p= 200 psi), upper curve (j= 2000 psi).
Large deflection of Timoshenko beams Loosely---clamped edges
been studied. The following observations drawn from the numerical tests performed.
0.12 3; z ._
0.10
:
0.08
5 u s 5 $
0.04
p9
0.02
0.06
200
0
In-plane stress resultants A,, Transverse deflection u3 (in.) vs in-plane load A,, ~$i~.~’ lay-up (b); BC3 +D2; lower curve (p= 200 psi), upper curve (p= 2000 psi).
results on the transverse deflection of the beam with lay-up (a) and simply-supported boundary conditions under transverse load p and in-plane compressive load N, I. Figure 8 shows the effect of the strategy of activation; Fig. 9 shows the effect of the active control (D2). Figures 10 and 11 give the same results, but for the beam with lay-up (b). Here the effect of activation appears to be substantial for deflections at higher values of the transverse load. It is observed that, for downward bending, CL1 reduces deflections, whereas CL2 increases them; the opposite holds for upward bending. The response of the adaptive beam under boundary conditions BC3 and D2 x,-distribution of the electric field is plotted in Figs 12 and 13. For this case only the effect of adaptive control is investigated for lay-ups (a) and (b). As for BC2 boundary conditions, CL1 reduces deflections, CL2 increa.ses them, specially when the in-plane stress resultant approaches the buckling load. Note that, as CL1 reduces deflections, the response of the unsymmetric plate (which behaves like an imperfect symmetrically laminated beam as a result of the coupling) becomes close to that of a symmetric one subjected to bifurcation buckling. This suggests the possibility of modifying the buckling behavior of shape imperfection sensitive structures.
5 CONCLUSIONS The nonlinear deflection behavior of multilayered Timoshenko beams including fibre-reinforced and piezoelectric actuator and sensor layers have
59
are
- Boundary conditions BC 1: (1) The effect of activation appears to be influenced by the lay-up considered. Precisely, it is shown to depend on the coupling stiffness of the basic strucure and on the thickness of actuator layers. Whereas it appears negligible for lay-ups (a) and (c), it appears substantial for the lay-ups (b) and (d), for which the response becomes activation law dependent. The effect of inducing strains in the acti(2) vated layers appears significant for deflection at higher values of the transverse load while, on the countrary, the effect on the inplane stress resultant appears only at low values of the load. (3) The effect of activation on deflections is reduced for the lay-ups with a lower coupling; the opposite occurs for the induced inplane resultant, a greater influence being shown at low values of the transverse load p for all the examined cases. - Boundary conditions BC2: (1) For this case, the response of the lay-ups (a) and (b) is investigated along with the activation and the adaptive control laws. (2) As for the unactivated structure, the bending direction appears to be altered both by the direction and by the magnitude of the transverse load, a load level being found at which the deflection ceases to follow the sign of B,,. The numerical analysis shows that the effect of activation appears negligible for such inversion phenomenon. (3) For the investigated cases (two transverse loads above and below the inversion load), the influence of the constant electric field Dl on the response appears to be rather scarse for lay-up (a). The opposite holds for the lay-up (b), where the deflections appear to be increased or reduced depending on the activation strategy. (4 For downward bending, the activation law CL1 reduces deflections, whereas CL2 increases them; the opposite holds for upward bending. - Boundary conditions BC3: (1) The effect of adaptive control is investigated for lay-ups (a) and (b). As for BC2 boundary conditions, CL1 reduces deflec-
M. Di Sciuva, U. Icardi
60
tions and Cl2 increases them, above all when approaching the buckling load. As a consequence, the behavior of the unsymmetric beam, which behaves like an imperfect symmetrically laminated beam as a result of the coupling, becomes close to that of a symmetric one subjected to bifurcative buckling. This suggests the possibility of modifying the behavior of shape imperfection sensitive structures. The influence of the transverse load is (2) smaller than that for BC2; furthermore, transverse loads cannot cause inversion of the deflection. The influence of the transverse load is considerable when the compressive load approaches the buckling load. In this same range, the effect of the control law is substantial. Preliminary numerical tests show that the effect of induced strains is enhanced at local level, for example, when analysing the variation of the thickness-wise distributions of the strains and stresses consequent to the activation of the piezoelectric layers. As a consequence, modeling of adaptive multilayered beams and plates claims for discrete-layer theories. In this respect, work is in progress. REFERENCES 1. Tzou, H. S. & Gadre, M., Theoretical analysis of a multilayered thin shell coupled with piezoelectric shell actuators for distributed vibration controls. J. Sound and I%., 132 (3) (1989) 433-50. 2. Rogers, C. A., Liang, C. & Jia, J., Structural modification of simply-supported laminated plates using embedded shape memory alloy fibers. Computers & Structures, 38 (1991) 569-80. 3. Crawley, E. F. & Lazarus, K. B., Induced strain actuation of isotropic and anisotropic plates. AZAA J., 29 (1991) 944-51. 4. Tzou, H. S. & Tseng, C. I., Distributed piezoelectric sensor/actuator design for dynamic measurement/ control of distributed parameter systems: a piezolectric finite element approach. J. Sound and I%., 138 (1990) 17-34. 5. Ha, S. K., Keilers, C. & Chang, F. K., Finite element analysis of composite structures containing distributed piezoceramic sensors and actuators. AZAA J., 30 (1992) 772-80. 6. Crawley, E. F. & de Luis, J., Use of piezoelectric actuators as elements of intelligent structures. AZAA J., 25 (1987) 1373-84.
7. Im, S. & Atluri, S. N., Effects of a piezo-actuator on a finitely deformed beam subjected to general loading. AZAAJ., 27 (1989) 1801-7. 8. Song, O., Librescu, L. & Rogers, C. A., Application of adaptive technology to static aeroelastic control of wing structures. AZAA J., 30 (1992) 2882-9. 9. Di Sciuva, M. & Icardi, U., Stability behavior of adaptive multilayered plates and shells under combined loads. In MEET’W93, First Joint ASCE-EMD, ASME-AMD, SES Meeting, Charlottesville, Virginia, 6-9 June, 1993. 10. Di Sciuva, M., An improved shear-deformation theory for moderately thick multilayered anisotropic shells and plates. J. Appl. Mech., 54 (1987) 589-96. 11. Brush, D. 0. & Ahnroth, B. O., Buckling of Bars, Plates and Shells. McGraw-Hill Book Co., N.Y., 1975. 12. Kicher, T. P. & Mandell, J. F., A study of the buckling of laminated composite plates. AZAA J., 9 ( 197 1) 605- 13. 13. Leissa, A. W., Conditions for laminated plates to remain flat under inplane loading. Comp. Structures, 6 (1986) 261-70. 14. Zhang, Y. & Matthews, F. L., Large deflection behavior of simply supported laminated panels under in-plane loading. J. Appl. Mech., 52 (1985) 553-8. 15. Icardi, U., The nonlinear response of unsymmetric multilayered plates using smeared laminate and layerwise models. Comp. Structures, 29 (1994) 349-64. 16. Chen, H. P. & Shu, J. C., Cylindrical bending of unsymmetric composite laminates. AZAA J., 30 (1991) 1438-40. 17. Sun, C. T. & Chin, H., On large deflection effects in unsymmetric cross-ply composite laminates. J. Comp. Materials,22 (1988) 1045-59.
APPENDIX The set of eqns (12) is transformed into the governing eqn (27) for the deflection u3 as follows. First, let us consider the equilibrium equations N 11,1= 0 T,,,
+u%l~3,1),1
A4 11,1-
=
-P
(A4
T,=o
Then, computer s1 1 from the constitutive equation ( 16 ) and substitute it in ( 16,); the result is I
By substituting eqn (A.2) and eqn (A. 13) into eqn (A.l,), eqn (27) is obtained.