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NITINOL-reinforced plates: Part II. Static and buckling characteristics
Composites Engineering, Vol. 5, No. 1, pp. 77-90, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0961-9526/95 $9.50 + .00
Pergamon
0961-9526(94) E00085-9
N I T I N O L - R E I N F O R C E D P L A T E S : P A R T II. STATIC AND BUCKLING CHARACTERISTICS* J. Ro and A. Baz Mechanical Engineering Department, The Catholic University of America, Washington, DC 20064, U.S.A.
(Received 8 July 1994; final version accepted 27 September 1994) Abstract--The static and buckling characteristics of flexible fiberglass NITINOL-reinforced composite plates are controlled by activating optimal sets of NITINOL fibers embedded along the midplane of these plates. The NITINOL fibers are pre-tensioned and activated to generate significant phase recovery forces in order to increase the membrane strain energy which in turn increases the critical buckling load of the NITINOL-reinforced plates. With such control capabilites, the plates can be manufactured from light weight sections without compromising their elastic stability. This feature is invaluable in building light weight structures that have high resistance to failure due to buckling. The NITINOL fibers are trained to memorize the shape of the unbuckled plate and when the plate is deflected under the action of external compressive loads, the controller activates the NITINOL fibers by heating them above their transformation temperature. The generated phase recovery forces bring the plate back to its memorized undeflected position. A finite element model of NITINOL-reinforced plates is developed to describe the interaction between the external loads, operating conditions and the geometrical and physical parameters of the composite plate and the NITINOL fibers. This model predicts the critical buckling loads of NITINOL-reinforced plates. The predicted loads are compared with results available in the literature for symmetrically isotropic, orthotropic and anisotropic laminates. The mathematical model described in this paper provides an invaluable means of predicting realistic performance of NITINOL-reinforced composites.
NOMENCLATURE A a
b E
E~ e~ E, h ho [Re] [Ka,]
[KoA Nx,y, xy rl t
P,, TK t U W
WI
w~ w, X, y
surface area of the unit element length of the rectangular element width of the rectangular element Young's modulus of the isotropic material longitudinal Young's modulus transverse Young's modulus longitudinal shear modulus recovery force generated by NITINOL fiber in x-direction nodal forces vector thickness of the plate thickness of single ply element stiffness matrix flexural stiffness matrix geometric stiffness matrix in-plane force in x, y and xy direction per unit length total number of plies external forces in i direction thermal loads in i direction kinetic energy time strain energy transverse deflection work done due to the nodal forces work done due to the in-plane forces total work done Cartesian coordinates along the plate neutral axis, respectively
*This paper was submitted for presentation at the International Conference on Composite Engineering (ICCE/1), New Orleans, 28-31 August 1994. 77
78
P 0 Y
v~ vy [~] [v] [x]
J. Ro and A. Baz
mass per unit volume ply orientationangle Poisson's ratio of isotropic material longitudinal Poisson's ratio transverse Poisson's ratio 12 x 1 nodal displacementvectorof the four nodes of the plate element 12 × 12 matrix dependingon nodal coordinates curvature
1. INTRODUCTION The static characteristics of NITINOL-reinforced composite plates are primarily governed by their stiffness. The plate stiffness is made up of different components which include: the flexural rigidity of the plate, the geometric stiffness that accounts for the axial and thermal loading, as well as the stiffness imparted by the elasticity of the NITINOL fibers. However, when the compressive membrane forces, acting on the plate, are large enough to reduce its flexural stiffness to zero then the plate becomes elastically unstable and buckling occurs. It is worth noting that if the membrane forces become tensile rather than compressive, the flexural stiffness can be increased with the decrease of the geometric stiffness. Such a characteristic is utilized by Baz and Tampe (1989) to enhance the critical buckling loads of long slender beams by using shape memory actuators which are mounted externally on the beams. In 1991, Baz et al., increased the critical buckling loads of beams by embedding NITINOL fibers inside the composite fabric of these beams. The shape memory effect of the NITINOL fibers is used to increase the strain energy of the beams which in turn is utilized to delay the occurrence of buckling. Extension of their work to NITINOL-reinforced plates is the subject of the present study. It is important to note that piezo-electric actuators have also been considered to control the buckling of beams (Meressi and Paden, 1992), plates (Chandrashekhara and Bhatia, 1993) and trusses (Berlin, 1994). Such actuators have, however, low control authority that make them only suitable for controlling soft structures as compared to the shape memory (NITINOL) actuators. This paper is organized in five sections. In Section 1, a brief introduction is given. Section 2, a finite element model describing the static characteristics of the NITINOLreinforced plates is developed. The model quantifies the role that the NITINOL fibers play in enhancing the stiffness and the critical buckling loads of the composite plates. In Section 3, comparisons are presented between the critical buckling loads calculated by the present finite element method with data available in the literature in order to verify the accuracy of the model. The buckling control characteristics of NITINOL-reinforced plates are presented in Section 4 and Section 5 summarizes the conclusions of the present study. 2. FINITEELEMENTFORMULATIONOF NITINOL-REINFORCEDPLATE In the present study, NITINOL fibers are embedded inside vulcanized rubber sleeves placed along the neutral plane of composite plates made of randomly oriented glass fibers as shown in Fig. 1. In this arrangement, the fibers are free to move during the phase transformation process in order to avoid degradation and/or destruction of the shape memory effect which may result when the fibers are completely bonded inside the composite matrix. The static characteristics of this class of NITINOL-reinforced plates are obtained by applying the mechanical finite element model developed in the following section. The model is integrated with the thermal finite element model described in Part I in order to compute the physical properties of the composite and thermal loading as a function of the activation strategy of the NITINOL fibers. The model utilizes quadrilateral plate elements which are bounded by four nodal points each of which has three degrees of freedom (w, Ow/Ox and Ow/Oy).
79
NITINOL-reinforcedplates: Part II. Static and buckling characteristics b ~
SLEEVE - /
COMPOSITES ~ /
L
NITINOL FIBER
Fig. 1. A schematic drawing of a NITINOL-reinforcedplate.
2.1. Theoretical formulation (a) Strain energy. According to the Classical Plate Theory, the normal stress a z , and shear strains ~'yz and Yz: are negligible. Thus, the overall strain energy U may be written as (Young, 1986): U-24(]---
l t[
v 2) A
\-ff-~,] + \OY2,] + 2 ( 1 - v)~o--~y ~ + 2v Ox2 0 y 2 j d x d y ,
(1)
where E, v, A and h denote modulus of elasticity, Poisson's ratio, area and thickness of the plate. Also, w defines the transverse deflection of the plate at any location (x, y). (b) External work done. The external work done We on the unit NITINOL-plate element consists of two parts. The first part W1 is due to the nodal forces [f~]. The second part WE is due to the in-plane forces which include the recovery forces of the N I T I N O L fibers, thermal load induced by heating the N I T I N O L fibers and the external in-plane loads. The work W~ is given by: W 1 = l[fn]T[~e] =
½[oe]rtKe][~]
(2)
where [Ke] is the element stiffness matrix and [ae] is the element displacement vector. The work done W2 by the in-plane loads (N x, Ny, and N,:y) is given by: W2 =
-~- g y ~ - ~ y )
+
dxdy
(3)
where
Nx=Pmx+P,x-Fw~, Ny = P m y + Pt~,, Nxy = Pmxy + Pt~,
(4)
with Pmx,y.~r and Pt,.y,~ denoting the compressive in-plane mechanical and thermal loads in the x, y and xy directions. Also, Fwx denotes the total tension developed in the N I T I N O L fibers which are aligned along the x-axis. Such a tension is due to initial tightening and phase recovery forces resulting from activating the shape memory effect. 2.2. Finite element displacement formulation The spatial distribution of the transverse deflection w, over a unit element of the NITINOL-reinforced plate can be expressed by the following 12-term polynomial: W = Ot~1 -'[- O/2X "~ o/3y "1- 0/4 x2 q- Ol5xY.-t- ot6Y 2 q- OL7x3
+ OtaX2y + a 9 x Y 2 + Otl0Y 3 + OtllX3y + Otl2XY 3.
(5)
80
J. R o a n d A . B a z
If [J] is the vector of the nodal displacements and [od is the vector of the coefficients ai, i = 1, 2 . . . . . 12, then the nodal displacement can be written as [Jl =
[
w
Ox
Ow
= [FI[~I
(6)
\
where [FI is a known 3 x 12 matrix. The unknown constants c~i can be calculated in terms of the nodal displacements if the nodal coordinates are successively introduced into (6). The resulting matrix equation may be solved for the constants ~ yielding [Og] =
[ ~ ] - 1 [E~'e]
(7)
where [~] is 12 × 12 matrix whose entries depend on nodal coordinates, [if] is the 12 × 1 nodal displacement vector of the four nodes. Based on eqns (6) and (7), the curvature [r] and angular displacement [0] vectors are written as follows (Chajes, 1974): [K] =
_02w 0x 2
[01=
_02w _202wit Oy2
OxOy I
Ox
= [o1[~
= [C][od = [B][~I, = [HI[~I,
(8)
(9)
where [CI is a 3 x 12 matrix, [BI = [C][~] -1, and [el is a 2 x 12 matrix and [/-/] = [a][~1-1. 2.3. Stiffness matrix Equilibrium of the element is attained when the external work done becomes equal to the strain energy, We = Wl + W2. Thus from eqns (1)-(3), and eqns (8) and (9), we have:
[BIT[DI[BI d x d y ['~1 - [ j q r
[0~lr[Kel[b~l = [~1 r 0 0
[/-/lr[Nl d x d y [~1,
(10)
dO dO
i.e. [Ke]
(11)
= [K,e.I - [K~.I
where
[BIr[DI[BI dxdy,
[/(Be] = 0
and
[K~A =
fol b[_rflr[NJ[H] 0x@
(12)
0 0
0
where [DI is the flexural rigidity matrix and
[NI=[ Nx N~ In eqn (11), [Kse] denotes the conventional flexural stiffness matrix associated with the bending vibration o f the thin rectangular plate element. For a NITINOL-reinforced plate, the dependence of the modulus of the elasticity of the matrix and flexural rigidity matrix [D] on the temperature is particularly emphasized. Also, [Kc,] defines the geometric stiffness matrix which results from the in-plane loads. Based on eqn (11), it is easy to understand the role that the NITINOL fibers play in controlling the overall stiffness matrix. For example, heating the NITINOL fibers in a clamped plate reduces the modulus of the elasticity of the matrix and generates in-plane compressive thermal loads. These two effects tend to reduce the overall stiffness of the plate. If the heating is high enough, such that the determinant of the overall stiffness becomes equal to zero, the plate will buckle. However, it is important to note that the tensile phase recovery forces which are developed by NITINOL shape memory effect, tend to increase the overall stiffness matrix to resist the plate buckling. Therefore, the NITINOL fibers can be used to compensate for the softening of the composite matrix and counterbalance the effect of the compressive
NITINOL-reinforced plates: Part II. Static and buckling characteristics
81
thermal loads. Accordingly, the N I T I N O L fibers can be utilized to alter the structural frequency and increase the critical buckling loads of the plate in response to the external loads acting on it. 2.4. Flexural rigidity matrix Emphasis is placed in here on flexural rigidity matrix [D] for isotropic and symmetric anisotropic laminates. (a) F o r isotropic plate element. For plane stresses in an isotropic material, the flexural rigidity matrix can be expressed by (Zienkiewicz, 1989)
[D1-
Eh 3
1
v
0
v
l
0
12(1 - v)
(13)
1-v
o o--~
(b) F o r s y m m e t r i c anisotropic laminate plate element. Consider a laminated NITINOL-reinforced composite plate which is constructed from a number of laminae with the principal axis of any single lamina making an angle 0 with the global coordinate system of the plate. The elasticity relationships for the single lamina which has orthotropic properties, due to orientation of the N I T I N O L fibers, and has its principal axis not parallel to the global coordinate system has been defined by:
oil
011 Ol2 Q16
el
012 022 026
~2
O"6
Q16 026 Q66
C6
0"2
:.
(14)
where ai, ei and Qo denote the stress, strain and elastic stiffness, respectively. With Qll °
C04
S04
2C02 s02
4c02 s02
Q22
s04
c04
2c02 s02
4 c 0 2 s0 2
Q12
CO2 SO2
CO2 SO2
CO4 + SO4
-- 4c02 sO2
Q~6
cO2 sO2
cO2 sO2
2C0 2 sO2
(cO2 - s02) 2
Q16
cO3 S0
--CO S03 COSO3 -- C03 S0
2(C0 S03 -- CO3 S0)
. Q26.
CO SO3
-- CO3 SO CO3 SO -- CO SO3
2(C0 3 SO -- CO SO3
(15) Qxy
Qs_l
where cO = cos0, s0 = sin0, Qx = rvEx, Qy = rvEy, Q~y = rvvxEy, Qs = Es, and rv = (1 - VxVy) -1. For a symmetric anisotropic laminate, [D] is given by (Mottram, 1987):
[D] =
Dll
DIE
DI6 ] !
D12
022
D26 / .
D26
! D66_]
DI6
(16)
where D o are given by Tsai (1980): Qo(r) [r 3 - (r - 1)3]
Do = ~
n t is even,
(17)
r= 1 - n t / 2
D 0 = -hg ~ ~ Q}])[1 + 12q 2]
n t is odd, q = (n t - 1)/2,
-q
where ho is the thickness of single lamina and n t is the total number of laminae.
(18)
82
J. Ro and A. Baz
2.5. Buckling analysis of NITINOL-reinforced plates For the buckling analysis of the NITINOL-reinforced plates, the in-plane loads (N x, Ny, N,:y), which include the external compressive loads Pm,.y,~, the thermal loads Pt,.y.~ and the recovery forces of fibers F~,, need to be considered separately. First, the external compressive loads are either uniaxial loads as Pmx and P% or shear loads Pro. Second, the thermal loads generated by heating the NITINOL fibers can also be either uni- or hi-axial compressive loads. These loads increase the geometric stiffness and decrease the flexural stiffness of the plate because the associated high temperatures tend to reduce the effective modulus of elasticity of its composite matrix. Such a dual effect makes the plate buckle under small thermal loads rather than under pure mechanical loading. Third, the phase recovery forces which are created by activating the NITINOL fibers can be utilized to generate appropriate axial tensile loads according to their orientation angles inside the composite. The combined three types of in-plane loads affect the buckling analysis of the NITINOL-reinforced plates. When these combined loads are large enough to make the plate buckle, then this load is denoted by Nor. Since the combined thermal loads and recovery forces in direction y can be normalized with respect to those in the x-direction ( P t x - Fwx) such that rl = Pty/(Ptx - Fw~). Also, the external loads in direction y are normalized with respect to Pm~ such that r2 = P m ~,~Prox and r3 = P m x y ~Prox • Note that Pt~, = 0 and F,,y = 0. In eqn (10), the effect of the in-plane loads on the geometric matrix [K~] can be divided to separate the contribution of the thermal loads Pti and recovery forces F~i from that of the external loads Pm~ as follows: [K~] = [KB.] - (Pt.
-
Fw.)[KGle]
Pm.[K~2e]
-
(19)
where [Kol~] and [KG2~] are defined by eqn (12), except
[NI =
0
rl
and
,
r3 r2
respectively. In order to determine the critical buckling load, it is necessary to obtain the nontrivial solution of the following equation: ([KB.] - (Pt~ - Fw~)[Ko,o] - Pm~[Kc2.])[,~e] = O.
(20)
The elements of the matrix ([/(Be] - (Ptx - Fw~)[KGle]) are completely known for a given plate configuration with a given activation strategy of its NITINOL reinforcing fibers. The plate configuration determines the flexural stiffness matrix [/(Be] and the activation strategy determines the thermal loads Pti and the phase recovery forces Fwx. In eqn (20), [tV] denotes the nodal deflection vector. If the order of the system of eqn (20) is relatively small, the critical buckling force can be obtained by setting the determinant of the equations equal to zero, solving for the smallest root of the resulting polynomial equation. However, if the order of the system is considerably large, it is more convenient to determine the critical buckling force iteratively. Prior to carrying out the iterations, it is desirable to put eqn (20) into the more commonly encountered form where the eigenvalues appear only along the main diagonal. This form is obtained by premultiplying eqn (20) by [KG2e] -1 a n d inverting the entire equation. Thus eqn (20) becomes: ([KB.] -- (Pt x - Fwx)[gGle])-l[KG2 e] -- ~
/"mx
[I]'~[~]
/
---- O,
(21)
where [/] = Identity matrix. Therefore, if the eigenvalues of the matrix in (21) are obtained by iteration such that the highest eigenvalue is obtained first, then eqn (21) will give the highest value of 1/Pmx or the lowest value of Pmx"
NITINOL-reinforced plates: Part II. Static and buckling characteristics
83
It should be noted that the recovery forces Fwx of the NITINOL fibers are obtained by monitoring their temperature-recovery force characteristics for a specific pre-strain e. The variation of the modulus of elasticity with temperature is obtained experimentally using the Dynamic Mechanical Thermal Analyzer (DMTA). Thus, the thermal loads Pt~ and the flexural stiffness matrix [/(Be] can be computed based on the temperature distribution inside the composites which results from the activation of the NITINOL fibers. 3. NUMERICAL ANALYSIS OF NITINOL-REINFORCED PLATE BUCKLING
3.1. Single layer isotropic plate Figure 2 shows a rectangular plate whose opposite sides are subjected to uniform compressive force Nx. The boundary conditions used are denoted, for example, by SSCC to imply that sides 1 and 2 are simply supported and sides 3 and 4 are clamped. The plate is divided into 36 equal square plate elements and its buckling characteristics are determined in terms of the nondimensional buckling coefficient C k which is defined as: N/Zb2~fl - V2~
for uniaxial loads.
.\h3]\ E ]
(22)
Comparisons between the exact buckling coefficients by Roark and Young (1975) and those obtained by the present finite element method are shown in Fig. 3 for different boundary conditions. side
3
- - t i m -
~
---tD, - -
side
1
side
--DD--
2
-
-,~1-----~
side
4
-
T
--
,i
Fig. 2. Rectangular plate under uniform compression N x on two opposite edges b. 10
/ f
/Z
¢./3
i
CCSS
//~/SSSS
lJ 0~, 0
sss , , , . 2
i
i
i
4
6
8
10
CK - Roark and Young (1975) Fig. 3. Comparison between buckling coefficient C x obtained by present FEM and by Roark and Young (1975) for various boundary conditions.
J. RO and A. Baz
84
The figure indicates a close agreement between the exact results and the predictions obtained by the present finite element method. Additional comparisons are displayed in Fig. 4(a) and (b) to demonstrate the effect of the aspect ratio on the buckling coefficients for simply-supported and clamped plates, respectively. Again, close agreement is evident between predictions of the present FEM and the exact solutions, particularly for small aspect ratios. From Fig. 4(a) and (b) for small a/b ratios, the differences between the exact results reported by Roark and Young (1975) and the results of the present FEM are reasonable. For example, when a/b = 1, the differences are 2.83070 for SSSS and 2.47°70 for CCCC. But when a/b = 3, the differences are 11.43070 for SSSS and 26.4070 for CCCC. Such large errors are attributed to the fact that the number of elements is not large enough to converge to the accurate solution for such large aspect ratio. Figure 5(a) and (b) shows the effect of the number of elements in the present FEM on the accuracy of the solution in comparison with the exact solutions for simply-supported and clamped plates, respectively. For the case of a simply supported plate, as the number of elements increases to 49 (7 x 7), the difference decreases to 9.68070. For the case of clamped plate, the difference decreases to 22.43°70.
3.2. Multi-layers symmetric anisotropic laminated plates During bending deformations, symmetrically laminated composite plates display anisotropy in the form of material-induced coupling between pure bending and twisting due to the D16 and D26 constitutive terms. These terms appear whenever a ply is stacked with a fiber orientation other than 0 or 90 ° to the reference axes of the plate. Hence, their values and importance depend on ply orientation, numbers of plies and stacking 8.0
(a)ALLEDGESSIMPLY-SUPPORTED - - O-- PRESENTFEM 6.0 ]
4.0A
2"0I 0.0t, 0.0
., 0.4
' "
I '
0.8
'
'
I
1.2
'
'
-
-
1.6
,
'
'
i
'
2.0
,
,
ASPECTRATIO(a/b)
+
' '
2.4
'
1
'
'
2.8
'
3.2
15.0 (b)ALLEDGESCLAMPED - - Q-- PRESENTFEM 10.0
Q "
5.0-
RoarkandYoung(1975)
-o- -o- -o- -o-
-o
0.0 0.0' " 014
' '028
' '112 ' 1A ' 2'.0' ' '214" ' 2'.8 ASPECT RATIO (a/b)
' 3.2
Fig. 4. Comparison between critical buckling coefficient C k obtained by present FEM and by Roark and Young (1975) for different aspect ratio when all edges are (a) simply-supported and (b) clamped.
N I T I N O L - r e i n f o r c e d plates: P a r t II. Static and buckling characteristics
85
6
(a) ALLEDGESSIMPLY-SUPPORTED 5
.....
PRESENT FEM EXACT SOLUTION
4
3
o
. . . . . . . . . . 0
10
20 30 NUMBER OF ELEMENTS
, ....
~
40
50
40
50
12(b) ALL EDGESCLAMPED
PRESENTFEM
10A
.....
EXACT SOLUTION
g. 6. 4 [
10
o
20 30 NUMBER OF ELEMENTS
Fig. 5. Effect of number of finite elements on the critical buckling coefficient C K under uniaxial loading N x. All edges are (a) simply-supported and (b) c l a m p e d .
sequence. Often in a preliminary analysis, anisotropy is neglected in the problem formulation. This simplifying assumption allows the use of existing solutions for specially orthotropic plates and the exploitation of symmetry in the plate modeling to reduce computational effort and cost. Therefore, the effect of the bending-twisting terms, D16 and D 2 6 , will be investigated in this section. For orthotropic plates, the bending-twisting terms D]6 and D26 are set equal to zero and D1]/D22 = 1 0 , (O12 + 2 0 6 6 ) / 0 2 2 = 1. Also, the plate is assumed to be simply-supported with uniaxial in-plane load N x. The critical buckling coefficient Ck = (Nxb2/It2022) is obtained for different aspect ratios. The results are compared with those of Nemeth (1986) as shown in Fig. 6. When a/b = 1, the
2o!
ALL EDGES SIMPLY-SUPPORTED
Nemeth (1986) . . . . . . . PRESENT FEM
J
10j Z
7 r~
0!--,
,
A S P E C T R A T I O (a/b) Fig. 6. Comparison between critical buckling coefficient obtained by present FEM and by Nemeth (1986) of specially orthotropic plates under uniform compressive loads N x.
86
J. Ro and A. Baz 10 (a) ALL EDGES SIMPLY-SUPlK)RTED ----o---- LINE 1 ,, LINE 3 __£>._ LINE2 --o-- LINE4
8.
0
15 30 45 60 75 FIBERORENTATIONANGLE(0)
90
2O Co) ALL EDGES CLAMPED - - . i f - - LINE I , LINE 3 __~__ LINE2 -_¢.-_ LINE4
~
I0
500~
' 1'5 ' 3 0 4'5 6*0 ' 7~5 90 FIBERORENTATIONANGLE(0) Fig. 7. Comparison between the effect of two stacking sequence laminates [(+/-0)6] ` (line 1: Nemeth (1986), line 2: present FEM) and [(+ 06/-06)], (line 3: Nemeth (1986), line 4: present FEM) on buckling coefficient Cx = Nc, b2/(n2D~HD~2) for different fiber orientation angle 0.
difference is 4.07070. However if a/b = 3.9 the difference increases to 9.86°70. Such differences are due to the fact that the mesh size is not fine enough to converge to the exact solution. The effect of anisotropy on the buckling of symmetrically laminated composite plates loaded in compression, is investigated by considering the bending-twisting coupling terms, D~6 and/)26, comparing them to the generally orthotropic plates. The material properties of the ply are such that El~E2 = 11.6, GI2/E 2 = 0.5, vl2 = 0.35 and thickness = 0.127 ram. The results shown in Fig. 7(a) and (b) indicate close agreement between the predictions of the present FEM and those of Nemeth for 24-ply square laminates which are either clamped or simply-supported, respectively. The solid lines shown in Fig. 7(a) and (b) denote the value obtained by Nemeth (1986), and the dotted lines correspond to the present FEM results. The stacking sequences of the laminates are [( + 0)6L and ( + 06/- 06)s. Such comparison shows the FEM can be utilized to predict the critical buckling force of thin anisotropic laminated plates. 4. BUCKLING CONTROL OF NITINOL-REINFORCED COMPOSITE PLATES
In this section, the mechanical finite element models and thermal distribution inside the composite will be utilized to analyze the buckling of NITINOL-reinforced plates under different boundary conditions. A NITINOL-reinforced plate is made up of randomly oriented glass fibers and a low cure temperature polyester resin (Fibre GlassEvercoat Co. Inc. Ohio 45242) matrix. The plate is 0.1905 m long, 0.1905 m wide and 2.54mm thick mounted with all its edges in a clamped arrangement. A total of 58 NITINOL fibers (Innovative Technology International Inc. MD 20705), 0.55 mm in diameter, are embedded in the mid-plane and along the x-direction inside the plate
NlTINOL-reinforcedplates: Part I1. Static and buckling characteristics
87
through vulcanized rubber sleeves that have outer diameter of 0.95 mm. The N I T I N O L fibers have a 50°C phase transformation temperature. The NITINOL fibers are activated by passing electric current through the fibers in order to heat the fibers above their transformation temperature. Since the side-to-thickness ratio of the NITINOL-reinforced plate is relatively large, the present finite element method should be accurate enough to predict the critical buckling force for different boundary conditions. Also, as the aspect ratio is equal to unity then a relatively small number of elements (49 elements) can be used to accurately predict the critical buckling load of the NITINOL-reinforced plates. There are four different models, plain unreinforced plate (model A), plate with unactivated N I T I N O L fibers (model B), plate reinforced with heated but untrained NITINOL fibers (model C) and plate with activated and trained N I T I N O L fibers (model D). The initial tension of the fibers is selected to be 31.14 N/fiber. A value of 6.8 W/fiber is utilized to activate the N I T I N O L fibers. When fully activated the fibers generate a total phase recovery force of 116.0 N/fiber. The temperature distribution of the NITINOLreinforced plate is as shown in Fig. 8. Such distributions are obtained using the thermal finite element model described in Part I of this paper. The buckling characteristics of the plain plate (model A) will be used to serve as a datum for measuring the effect of reinforcing the plate with plain pre-tensioned fibers (model B), heated untrained N I T I N O L fibers (model C) and activated trained N I T I N O L fibers (model D). In this manner, the individual contribution of the tension, heating and shape memory to the buckling characteristics of the NITINOL-reinforced plate can be isolated and studied separately. In other words, model B will quantify the effect of the initial tension, model C will determine the effect of softening of the matrix due to heating
.... ii i
ili i :! ¸
",m(~) o.~ o o.~
X- , ~ '
Fig. 8. The FEM predictions of the temperature distribution on the NITINOL-reinforcedplate: when (a) all the NITINOL fibers are activated, and (b) half of the NITINOL fibers (alongx-axis) are activated.
88
J. Ro and A. Baz
of the reinforcing fibers whereas model D will demonstrate the contribution of the shape memory effect. In the case of model B, the initial tension of the fibers is found to increase the stiffness of the plate which, in turn, increases the critical buckling coefficient of the plate as shown in Fig. 9(a) and (b) for simply-supported and clamped plates, respectively. Passing electric current through memoryless NITINOL fibers which are pre-tensioned, as in case of model C, results in softening the matrix and including thermal loads which tend to reduce the critical buckling coefficient as shown in Fig. 9(a) and (b). Therefore, the stiffening effect resulting from the passive pre-tensioning of the fibers is offset by the softening of the matrix and by the induced thermal loads. If the pre-tensioning is not large enough to counterbalance the thermal effects, the plate can thermally buckle. When the shape memory effect is introduced in the fibers above their phase transformation temperature, large phase recovery forces are generated, as in the case of model D. These forces will increase the strain energy stored in the plate to enhance its critical buckling coefficient. Figure 9(a) and (b) presents comparisons between the dimensionless buckling coefficients for the four models as functions of the aspect ratio for simply-supported and clamped plates, respectively. For clamped plates with a/b = l, the critical buckling coefficients Ck normalized with respect to that of the plain plate are 105.58070, 60.46070 and 271.74% for models B, C and D, respectively. The corresponding ratios become 260070 (model B), 182.7% (model C) and 703.87070 (model D) when the plate becomes simply-supported. From these figures, it is evident that reinforcing the plates with NITINOL fibers results in significant increase in the critical buckling loads of these plates. Basically, the use of the plain pre-tensioned fibers to reinforce composite plates can enhance their capability of resisting buckling as in the case of model B. However, such 8O
6o! =
(a) A L L E D G E S S I M P L Y - S U P P O R T E D P L A I N P L A T E (A) ..... P L A T E W I T H U N A C T I V A T E D N I T I N O L F I B E R S (B) .... P L A T E W I T H M E M O R Y L E S S F I B E R S (C) P L A T E W I T H A L L N I T I N O L F I B E R S A C T I V A T E D (D)
40-
20-
0
.
0.0 ' '0~.4' '0'.8
.
.
.
.
~ , • , , , , , i , , '
1'.2 ~-116 2.0 2.4 ASPECT RATIO (a/b)
2.8
3.2
SO (b) A L L _ _ ..... ....
EDGES CLAMPED p L A I N P L A T E (A) p L A T E W I T H U N A C T I V A T E D N I T I N O L F I B E R S (B) p L A T E W I T H M E M O R Y L E S S F I B E R S (C) p L A T E W I T H A L L N I T I N O L F I B E R S A C T I V A T E D (D)
0.4
0.8
¢~ 40~ i
0.0
1.2 1.6 2.0 2.4 ASPECT RATIO (a/b)
2.8
3.2
Fig. 9. The critical buckling coefficient as function of the aspect ratio for plates which are (a) simply supported and (b) clamped. All the fibers are activated.
NITINOL-reinforced plates: Part II. Static and buckling characteristics
~0 ~ 60 ~
89
(a) A L L EDGES SIMPLY-SUPPORTED _ _ PLAIN PLATE (A) .... PLATE WITH UNACTIVATED NITINOL FIBERS (B) __ ~ _ PLATE WITH MEMORYLESS FIBERS (C) PLATE WITH A L L NITINOL FIBERS ACTIVATED (D)
J rOz 40 t
20 "
0.0
014
015
1.2 1.6 210 2.4 ASPECT RATIO (a/b)
2.8
3.2
80 i i
60
]
~4o i
(b) A L L EDGES CLAMPED _ _ PLAIN PLATE (A) _ _ _ PLATE WITH UNACTIVATED NITINOL FIBERS (B) .... PLATE WITH MEMORYLESS FIBERS (C) PLATE WITH A L L NITINOL FIBERS ACTIVATED (D)
~',',\,
i, ] i
t 0.0
............................. 014
0.8
1.2 1.6 2.0 2.4 ASPECT RATIO 0~b)
2.8
/
3.2
Fig. 10. The critical buckling coefficient as function of the aspect ratio for plates which are (a) simply supported and (b) clamped. Half the fibers are activated.
an effect is limited in the sense that the enhancement is not significant. But, if the plain fibers are replaced by the shape memory NITINOL fibers, the enhancement is dramatic. By controlling the initial tension of the fibers and the power passing through them, the developed recovery forces can be tuned to counterbalance excessively large external loads. Figure 10(a) and (b) shows the effect of activating half the NITINOL fibers on the critical buckling coefficients of simply-supported and clamped plates, respectively. The Cx values increase by 260.57% (model B), 246.59°70 (model C) and 435.80070 (model D) when the plate is simply-supported. The corresponding increase becomes 105.68% (model B), 98.67070 (model C) and 178.21% (model D) when the plate is clamped. It is important to note that since the temperature distribution shown in Fig. 8(b) is not uniform throughout the plate, the fibers nearby the activated fibers will be activated unintentionally by the heat propagation effect. As a result, additional recovery forces are generated which contribute significantly to the strain energy of the plate. These effects are considered in the present analysis. 5. CONCLUSIONS
This paper has presented an analysis of the buckling of NITINOL-reinforced plates. A finite element model is described and utilized to predict the critical buckling loads of plates. The predictions of the model are validated against results available in the literature for plain unreinforced isotropic, orthotropic and anisotropic plates. The model is then used to study the buckling characteristics of NITINOL-reinforced composites. The individual contributions of the initial tension, the softening of the composite matrix and the shape memory effect to the critical buckling loads are determined. The effect that the boundary conditions, aspect ratio and activation strategy have on the buckling loads of plates are
90
J. Ro and A. Baz
analyzed in great detail. It is found that reinforcing composite plates with NITINOL fibers can dramatically enhance their critical buckling loads even when these plates are clamped from all their edges. Acknowledgements--This work is funded by a grant from Army Research Office (grant number DAAL03-89-G0084). Special thanks are due to Dr Gary Anderson, the technical monitor and chief of the Structures and Dynamics Branch of ARO, for his invaluable and continuous technical inputs. REFERENCES Baz, A., Poh, S., Ro, J., Mutua, M. and Gilbeany, J. (1992). Active control of NITINOL-reinforced composite beams. In Intelligent Structural Systems (Edited by H. S. Tzou and G. Anderson). Kluwer, The Netherlands. Baz, A., Ro, J., Mutua, M. and Gilheany, J. (1991b). Active control of buckling of NITINOL-reinforced composite beams. In Active Materials and Adaptive Structures Conference, Arlington, VA, pp. 167-176. Berlin, A. (1994). Active control of buckling. Ph.D. Dissertation, M.I.T., Cambridge, MA. Chajes, A. (1974). Principles of Structural Stability Theory. Prentice-Hall, NJ. Chandrashekhara, K. and Bhatia, K. (1993). Active buckling control of smart composite plates--finite element analysis. Smart Mater. Struct. 2, 31-39. Fenner, R. T. 0975). Finite Element Methods for Engineers. Macmillan, London. Meressi, T. and Paden, B. (1992). Buckling control of a flexible beam using piezoelectric actuators. J. Guidance, Control Dynam. 16(5), 977-980. Mottram, J. T. and Selby, A. R. (1987). Bending of thin laminated plates. Comp. & Struct. 25(2), 271-280. Nemeth, M. P. (1986). Importance of anisotropy on buckling of compression-loaded symmetric composite plates. AIAA. J. 24(11), 1831-1835. Polymer Laboratories, Ltd (1990). PL-DMTA, MKII Dynamic Mechanical Thermal Analyzer. Loughborough, U.K. Roark, R. J. and Young, W. C. (1975). Formulas for Stress and Strain, 5th edn. McGraw-Hill, New York. Tsai, S. W. and Hahn, H. T. (1980). Introduction to Composite Material. Technomic Publishing Co., Pennsylvania. Young, T. Y. 0986). Finite Element Structural Analysis. Prentice-Hall, NJ. Zienkiewicz, O. C. and Taylor, R. L. (1989). The Finite Element Method, Vol. 1, 4th edn. McGraw-Hill, New York.
Pergamon
0961-9526(94) E00085-9
N I T I N O L - R E I N F O R C E D P L A T E S : P A R T II. STATIC AND BUCKLING CHARACTERISTICS* J. Ro and A. Baz Mechanical Engineering Department, The Catholic University of America, Washington, DC 20064, U.S.A.
(Received 8 July 1994; final version accepted 27 September 1994) Abstract--The static and buckling characteristics of flexible fiberglass NITINOL-reinforced composite plates are controlled by activating optimal sets of NITINOL fibers embedded along the midplane of these plates. The NITINOL fibers are pre-tensioned and activated to generate significant phase recovery forces in order to increase the membrane strain energy which in turn increases the critical buckling load of the NITINOL-reinforced plates. With such control capabilites, the plates can be manufactured from light weight sections without compromising their elastic stability. This feature is invaluable in building light weight structures that have high resistance to failure due to buckling. The NITINOL fibers are trained to memorize the shape of the unbuckled plate and when the plate is deflected under the action of external compressive loads, the controller activates the NITINOL fibers by heating them above their transformation temperature. The generated phase recovery forces bring the plate back to its memorized undeflected position. A finite element model of NITINOL-reinforced plates is developed to describe the interaction between the external loads, operating conditions and the geometrical and physical parameters of the composite plate and the NITINOL fibers. This model predicts the critical buckling loads of NITINOL-reinforced plates. The predicted loads are compared with results available in the literature for symmetrically isotropic, orthotropic and anisotropic laminates. The mathematical model described in this paper provides an invaluable means of predicting realistic performance of NITINOL-reinforced composites.
NOMENCLATURE A a
b E
E~ e~ E, h ho [Re] [Ka,]
[KoA Nx,y, xy rl t
P,, TK t U W
WI
w~ w, X, y
surface area of the unit element length of the rectangular element width of the rectangular element Young's modulus of the isotropic material longitudinal Young's modulus transverse Young's modulus longitudinal shear modulus recovery force generated by NITINOL fiber in x-direction nodal forces vector thickness of the plate thickness of single ply element stiffness matrix flexural stiffness matrix geometric stiffness matrix in-plane force in x, y and xy direction per unit length total number of plies external forces in i direction thermal loads in i direction kinetic energy time strain energy transverse deflection work done due to the nodal forces work done due to the in-plane forces total work done Cartesian coordinates along the plate neutral axis, respectively
*This paper was submitted for presentation at the International Conference on Composite Engineering (ICCE/1), New Orleans, 28-31 August 1994. 77
78
P 0 Y
v~ vy [~] [v] [x]
J. Ro and A. Baz
mass per unit volume ply orientationangle Poisson's ratio of isotropic material longitudinal Poisson's ratio transverse Poisson's ratio 12 x 1 nodal displacementvectorof the four nodes of the plate element 12 × 12 matrix dependingon nodal coordinates curvature
1. INTRODUCTION The static characteristics of NITINOL-reinforced composite plates are primarily governed by their stiffness. The plate stiffness is made up of different components which include: the flexural rigidity of the plate, the geometric stiffness that accounts for the axial and thermal loading, as well as the stiffness imparted by the elasticity of the NITINOL fibers. However, when the compressive membrane forces, acting on the plate, are large enough to reduce its flexural stiffness to zero then the plate becomes elastically unstable and buckling occurs. It is worth noting that if the membrane forces become tensile rather than compressive, the flexural stiffness can be increased with the decrease of the geometric stiffness. Such a characteristic is utilized by Baz and Tampe (1989) to enhance the critical buckling loads of long slender beams by using shape memory actuators which are mounted externally on the beams. In 1991, Baz et al., increased the critical buckling loads of beams by embedding NITINOL fibers inside the composite fabric of these beams. The shape memory effect of the NITINOL fibers is used to increase the strain energy of the beams which in turn is utilized to delay the occurrence of buckling. Extension of their work to NITINOL-reinforced plates is the subject of the present study. It is important to note that piezo-electric actuators have also been considered to control the buckling of beams (Meressi and Paden, 1992), plates (Chandrashekhara and Bhatia, 1993) and trusses (Berlin, 1994). Such actuators have, however, low control authority that make them only suitable for controlling soft structures as compared to the shape memory (NITINOL) actuators. This paper is organized in five sections. In Section 1, a brief introduction is given. Section 2, a finite element model describing the static characteristics of the NITINOLreinforced plates is developed. The model quantifies the role that the NITINOL fibers play in enhancing the stiffness and the critical buckling loads of the composite plates. In Section 3, comparisons are presented between the critical buckling loads calculated by the present finite element method with data available in the literature in order to verify the accuracy of the model. The buckling control characteristics of NITINOL-reinforced plates are presented in Section 4 and Section 5 summarizes the conclusions of the present study. 2. FINITEELEMENTFORMULATIONOF NITINOL-REINFORCEDPLATE In the present study, NITINOL fibers are embedded inside vulcanized rubber sleeves placed along the neutral plane of composite plates made of randomly oriented glass fibers as shown in Fig. 1. In this arrangement, the fibers are free to move during the phase transformation process in order to avoid degradation and/or destruction of the shape memory effect which may result when the fibers are completely bonded inside the composite matrix. The static characteristics of this class of NITINOL-reinforced plates are obtained by applying the mechanical finite element model developed in the following section. The model is integrated with the thermal finite element model described in Part I in order to compute the physical properties of the composite and thermal loading as a function of the activation strategy of the NITINOL fibers. The model utilizes quadrilateral plate elements which are bounded by four nodal points each of which has three degrees of freedom (w, Ow/Ox and Ow/Oy).
79
NITINOL-reinforcedplates: Part II. Static and buckling characteristics b ~
SLEEVE - /
COMPOSITES ~ /
L
NITINOL FIBER
Fig. 1. A schematic drawing of a NITINOL-reinforcedplate.
2.1. Theoretical formulation (a) Strain energy. According to the Classical Plate Theory, the normal stress a z , and shear strains ~'yz and Yz: are negligible. Thus, the overall strain energy U may be written as (Young, 1986): U-24(]---
l t[
v 2) A
\-ff-~,] + \OY2,] + 2 ( 1 - v)~o--~y ~ + 2v Ox2 0 y 2 j d x d y ,
(1)
where E, v, A and h denote modulus of elasticity, Poisson's ratio, area and thickness of the plate. Also, w defines the transverse deflection of the plate at any location (x, y). (b) External work done. The external work done We on the unit NITINOL-plate element consists of two parts. The first part W1 is due to the nodal forces [f~]. The second part WE is due to the in-plane forces which include the recovery forces of the N I T I N O L fibers, thermal load induced by heating the N I T I N O L fibers and the external in-plane loads. The work W~ is given by: W 1 = l[fn]T[~e] =
½[oe]rtKe][~]
(2)
where [Ke] is the element stiffness matrix and [ae] is the element displacement vector. The work done W2 by the in-plane loads (N x, Ny, and N,:y) is given by: W2 =
-~- g y ~ - ~ y )
+
dxdy
(3)
where
Nx=Pmx+P,x-Fw~, Ny = P m y + Pt~,, Nxy = Pmxy + Pt~,
(4)
with Pmx,y.~r and Pt,.y,~ denoting the compressive in-plane mechanical and thermal loads in the x, y and xy directions. Also, Fwx denotes the total tension developed in the N I T I N O L fibers which are aligned along the x-axis. Such a tension is due to initial tightening and phase recovery forces resulting from activating the shape memory effect. 2.2. Finite element displacement formulation The spatial distribution of the transverse deflection w, over a unit element of the NITINOL-reinforced plate can be expressed by the following 12-term polynomial: W = Ot~1 -'[- O/2X "~ o/3y "1- 0/4 x2 q- Ol5xY.-t- ot6Y 2 q- OL7x3
+ OtaX2y + a 9 x Y 2 + Otl0Y 3 + OtllX3y + Otl2XY 3.
(5)
80
J. R o a n d A . B a z
If [J] is the vector of the nodal displacements and [od is the vector of the coefficients ai, i = 1, 2 . . . . . 12, then the nodal displacement can be written as [Jl =
[
w
Ox
Ow
= [FI[~I
(6)
\
where [FI is a known 3 x 12 matrix. The unknown constants c~i can be calculated in terms of the nodal displacements if the nodal coordinates are successively introduced into (6). The resulting matrix equation may be solved for the constants ~ yielding [Og] =
[ ~ ] - 1 [E~'e]
(7)
where [~] is 12 × 12 matrix whose entries depend on nodal coordinates, [if] is the 12 × 1 nodal displacement vector of the four nodes. Based on eqns (6) and (7), the curvature [r] and angular displacement [0] vectors are written as follows (Chajes, 1974): [K] =
_02w 0x 2
[01=
_02w _202wit Oy2
OxOy I
Ox
= [o1[~
= [C][od = [B][~I, = [HI[~I,
(8)
(9)
where [CI is a 3 x 12 matrix, [BI = [C][~] -1, and [el is a 2 x 12 matrix and [/-/] = [a][~1-1. 2.3. Stiffness matrix Equilibrium of the element is attained when the external work done becomes equal to the strain energy, We = Wl + W2. Thus from eqns (1)-(3), and eqns (8) and (9), we have:
[BIT[DI[BI d x d y ['~1 - [ j q r
[0~lr[Kel[b~l = [~1 r 0 0
[/-/lr[Nl d x d y [~1,
(10)
dO dO
i.e. [Ke]
(11)
= [K,e.I - [K~.I
where
[BIr[DI[BI dxdy,
[/(Be] = 0
and
[K~A =
fol b[_rflr[NJ[H] 0x@
(12)
0 0
0
where [DI is the flexural rigidity matrix and
[NI=[ Nx N~ In eqn (11), [Kse] denotes the conventional flexural stiffness matrix associated with the bending vibration o f the thin rectangular plate element. For a NITINOL-reinforced plate, the dependence of the modulus of the elasticity of the matrix and flexural rigidity matrix [D] on the temperature is particularly emphasized. Also, [Kc,] defines the geometric stiffness matrix which results from the in-plane loads. Based on eqn (11), it is easy to understand the role that the NITINOL fibers play in controlling the overall stiffness matrix. For example, heating the NITINOL fibers in a clamped plate reduces the modulus of the elasticity of the matrix and generates in-plane compressive thermal loads. These two effects tend to reduce the overall stiffness of the plate. If the heating is high enough, such that the determinant of the overall stiffness becomes equal to zero, the plate will buckle. However, it is important to note that the tensile phase recovery forces which are developed by NITINOL shape memory effect, tend to increase the overall stiffness matrix to resist the plate buckling. Therefore, the NITINOL fibers can be used to compensate for the softening of the composite matrix and counterbalance the effect of the compressive
NITINOL-reinforced plates: Part II. Static and buckling characteristics
81
thermal loads. Accordingly, the N I T I N O L fibers can be utilized to alter the structural frequency and increase the critical buckling loads of the plate in response to the external loads acting on it. 2.4. Flexural rigidity matrix Emphasis is placed in here on flexural rigidity matrix [D] for isotropic and symmetric anisotropic laminates. (a) F o r isotropic plate element. For plane stresses in an isotropic material, the flexural rigidity matrix can be expressed by (Zienkiewicz, 1989)
[D1-
Eh 3
1
v
0
v
l
0
12(1 - v)
(13)
1-v
o o--~
(b) F o r s y m m e t r i c anisotropic laminate plate element. Consider a laminated NITINOL-reinforced composite plate which is constructed from a number of laminae with the principal axis of any single lamina making an angle 0 with the global coordinate system of the plate. The elasticity relationships for the single lamina which has orthotropic properties, due to orientation of the N I T I N O L fibers, and has its principal axis not parallel to the global coordinate system has been defined by:
oil
011 Ol2 Q16
el
012 022 026
~2
O"6
Q16 026 Q66
C6
0"2
:.
(14)
where ai, ei and Qo denote the stress, strain and elastic stiffness, respectively. With Qll °
C04
S04
2C02 s02
4c02 s02
Q22
s04
c04
2c02 s02
4 c 0 2 s0 2
Q12
CO2 SO2
CO2 SO2
CO4 + SO4
-- 4c02 sO2
Q~6
cO2 sO2
cO2 sO2
2C0 2 sO2
(cO2 - s02) 2
Q16
cO3 S0
--CO S03 COSO3 -- C03 S0
2(C0 S03 -- CO3 S0)
. Q26.
CO SO3
-- CO3 SO CO3 SO -- CO SO3
2(C0 3 SO -- CO SO3
(15) Qxy
Qs_l
where cO = cos0, s0 = sin0, Qx = rvEx, Qy = rvEy, Q~y = rvvxEy, Qs = Es, and rv = (1 - VxVy) -1. For a symmetric anisotropic laminate, [D] is given by (Mottram, 1987):
[D] =
Dll
DIE
DI6 ] !
D12
022
D26 / .
D26
! D66_]
DI6
(16)
where D o are given by Tsai (1980): Qo(r) [r 3 - (r - 1)3]
Do = ~
n t is even,
(17)
r= 1 - n t / 2
D 0 = -hg ~ ~ Q}])[1 + 12q 2]
n t is odd, q = (n t - 1)/2,
-q
where ho is the thickness of single lamina and n t is the total number of laminae.
(18)
82
J. Ro and A. Baz
2.5. Buckling analysis of NITINOL-reinforced plates For the buckling analysis of the NITINOL-reinforced plates, the in-plane loads (N x, Ny, N,:y), which include the external compressive loads Pm,.y,~, the thermal loads Pt,.y.~ and the recovery forces of fibers F~,, need to be considered separately. First, the external compressive loads are either uniaxial loads as Pmx and P% or shear loads Pro. Second, the thermal loads generated by heating the NITINOL fibers can also be either uni- or hi-axial compressive loads. These loads increase the geometric stiffness and decrease the flexural stiffness of the plate because the associated high temperatures tend to reduce the effective modulus of elasticity of its composite matrix. Such a dual effect makes the plate buckle under small thermal loads rather than under pure mechanical loading. Third, the phase recovery forces which are created by activating the NITINOL fibers can be utilized to generate appropriate axial tensile loads according to their orientation angles inside the composite. The combined three types of in-plane loads affect the buckling analysis of the NITINOL-reinforced plates. When these combined loads are large enough to make the plate buckle, then this load is denoted by Nor. Since the combined thermal loads and recovery forces in direction y can be normalized with respect to those in the x-direction ( P t x - Fwx) such that rl = Pty/(Ptx - Fw~). Also, the external loads in direction y are normalized with respect to Pm~ such that r2 = P m ~,~Prox and r3 = P m x y ~Prox • Note that Pt~, = 0 and F,,y = 0. In eqn (10), the effect of the in-plane loads on the geometric matrix [K~] can be divided to separate the contribution of the thermal loads Pti and recovery forces F~i from that of the external loads Pm~ as follows: [K~] = [KB.] - (Pt.
-
Fw.)[KGle]
Pm.[K~2e]
-
(19)
where [Kol~] and [KG2~] are defined by eqn (12), except
[NI =
0
rl
and
,
r3 r2
respectively. In order to determine the critical buckling load, it is necessary to obtain the nontrivial solution of the following equation: ([KB.] - (Pt~ - Fw~)[Ko,o] - Pm~[Kc2.])[,~e] = O.
(20)
The elements of the matrix ([/(Be] - (Ptx - Fw~)[KGle]) are completely known for a given plate configuration with a given activation strategy of its NITINOL reinforcing fibers. The plate configuration determines the flexural stiffness matrix [/(Be] and the activation strategy determines the thermal loads Pti and the phase recovery forces Fwx. In eqn (20), [tV] denotes the nodal deflection vector. If the order of the system of eqn (20) is relatively small, the critical buckling force can be obtained by setting the determinant of the equations equal to zero, solving for the smallest root of the resulting polynomial equation. However, if the order of the system is considerably large, it is more convenient to determine the critical buckling force iteratively. Prior to carrying out the iterations, it is desirable to put eqn (20) into the more commonly encountered form where the eigenvalues appear only along the main diagonal. This form is obtained by premultiplying eqn (20) by [KG2e] -1 a n d inverting the entire equation. Thus eqn (20) becomes: ([KB.] -- (Pt x - Fwx)[gGle])-l[KG2 e] -- ~
/"mx
[I]'~[~]
/
---- O,
(21)
where [/] = Identity matrix. Therefore, if the eigenvalues of the matrix in (21) are obtained by iteration such that the highest eigenvalue is obtained first, then eqn (21) will give the highest value of 1/Pmx or the lowest value of Pmx"
NITINOL-reinforced plates: Part II. Static and buckling characteristics
83
It should be noted that the recovery forces Fwx of the NITINOL fibers are obtained by monitoring their temperature-recovery force characteristics for a specific pre-strain e. The variation of the modulus of elasticity with temperature is obtained experimentally using the Dynamic Mechanical Thermal Analyzer (DMTA). Thus, the thermal loads Pt~ and the flexural stiffness matrix [/(Be] can be computed based on the temperature distribution inside the composites which results from the activation of the NITINOL fibers. 3. NUMERICAL ANALYSIS OF NITINOL-REINFORCED PLATE BUCKLING
3.1. Single layer isotropic plate Figure 2 shows a rectangular plate whose opposite sides are subjected to uniform compressive force Nx. The boundary conditions used are denoted, for example, by SSCC to imply that sides 1 and 2 are simply supported and sides 3 and 4 are clamped. The plate is divided into 36 equal square plate elements and its buckling characteristics are determined in terms of the nondimensional buckling coefficient C k which is defined as: N/Zb2~fl - V2~
for uniaxial loads.
.\h3]\ E ]
(22)
Comparisons between the exact buckling coefficients by Roark and Young (1975) and those obtained by the present finite element method are shown in Fig. 3 for different boundary conditions. side
3
- - t i m -
~
---tD, - -
side
1
side
--DD--
2
-
-,~1-----~
side
4
-
T
--
,i
Fig. 2. Rectangular plate under uniform compression N x on two opposite edges b. 10
/ f
/Z
¢./3
i
CCSS
//~/SSSS
lJ 0~, 0
sss , , , . 2
i
i
i
4
6
8
10
CK - Roark and Young (1975) Fig. 3. Comparison between buckling coefficient C x obtained by present FEM and by Roark and Young (1975) for various boundary conditions.
J. RO and A. Baz
84
The figure indicates a close agreement between the exact results and the predictions obtained by the present finite element method. Additional comparisons are displayed in Fig. 4(a) and (b) to demonstrate the effect of the aspect ratio on the buckling coefficients for simply-supported and clamped plates, respectively. Again, close agreement is evident between predictions of the present FEM and the exact solutions, particularly for small aspect ratios. From Fig. 4(a) and (b) for small a/b ratios, the differences between the exact results reported by Roark and Young (1975) and the results of the present FEM are reasonable. For example, when a/b = 1, the differences are 2.83070 for SSSS and 2.47°70 for CCCC. But when a/b = 3, the differences are 11.43070 for SSSS and 26.4070 for CCCC. Such large errors are attributed to the fact that the number of elements is not large enough to converge to the accurate solution for such large aspect ratio. Figure 5(a) and (b) shows the effect of the number of elements in the present FEM on the accuracy of the solution in comparison with the exact solutions for simply-supported and clamped plates, respectively. For the case of a simply supported plate, as the number of elements increases to 49 (7 x 7), the difference decreases to 9.68070. For the case of clamped plate, the difference decreases to 22.43°70.
3.2. Multi-layers symmetric anisotropic laminated plates During bending deformations, symmetrically laminated composite plates display anisotropy in the form of material-induced coupling between pure bending and twisting due to the D16 and D26 constitutive terms. These terms appear whenever a ply is stacked with a fiber orientation other than 0 or 90 ° to the reference axes of the plate. Hence, their values and importance depend on ply orientation, numbers of plies and stacking 8.0
(a)ALLEDGESSIMPLY-SUPPORTED - - O-- PRESENTFEM 6.0 ]
4.0A
2"0I 0.0t, 0.0
., 0.4
' "
I '
0.8
'
'
I
1.2
'
'
-
-
1.6
,
'
'
i
'
2.0
,
,
ASPECTRATIO(a/b)
+
' '
2.4
'
1
'
'
2.8
'
3.2
15.0 (b)ALLEDGESCLAMPED - - Q-- PRESENTFEM 10.0
Q "
5.0-
RoarkandYoung(1975)
-o- -o- -o- -o-
-o
0.0 0.0' " 014
' '028
' '112 ' 1A ' 2'.0' ' '214" ' 2'.8 ASPECT RATIO (a/b)
' 3.2
Fig. 4. Comparison between critical buckling coefficient C k obtained by present FEM and by Roark and Young (1975) for different aspect ratio when all edges are (a) simply-supported and (b) clamped.
N I T I N O L - r e i n f o r c e d plates: P a r t II. Static and buckling characteristics
85
6
(a) ALLEDGESSIMPLY-SUPPORTED 5
.....
PRESENT FEM EXACT SOLUTION
4
3
o
. . . . . . . . . . 0
10
20 30 NUMBER OF ELEMENTS
, ....
~
40
50
40
50
12(b) ALL EDGESCLAMPED
PRESENTFEM
10A
.....
EXACT SOLUTION
g. 6. 4 [
10
o
20 30 NUMBER OF ELEMENTS
Fig. 5. Effect of number of finite elements on the critical buckling coefficient C K under uniaxial loading N x. All edges are (a) simply-supported and (b) c l a m p e d .
sequence. Often in a preliminary analysis, anisotropy is neglected in the problem formulation. This simplifying assumption allows the use of existing solutions for specially orthotropic plates and the exploitation of symmetry in the plate modeling to reduce computational effort and cost. Therefore, the effect of the bending-twisting terms, D16 and D 2 6 , will be investigated in this section. For orthotropic plates, the bending-twisting terms D]6 and D26 are set equal to zero and D1]/D22 = 1 0 , (O12 + 2 0 6 6 ) / 0 2 2 = 1. Also, the plate is assumed to be simply-supported with uniaxial in-plane load N x. The critical buckling coefficient Ck = (Nxb2/It2022) is obtained for different aspect ratios. The results are compared with those of Nemeth (1986) as shown in Fig. 6. When a/b = 1, the
2o!
ALL EDGES SIMPLY-SUPPORTED
Nemeth (1986) . . . . . . . PRESENT FEM
J
10j Z
7 r~
0!--,
,
A S P E C T R A T I O (a/b) Fig. 6. Comparison between critical buckling coefficient obtained by present FEM and by Nemeth (1986) of specially orthotropic plates under uniform compressive loads N x.
86
J. Ro and A. Baz 10 (a) ALL EDGES SIMPLY-SUPlK)RTED ----o---- LINE 1 ,, LINE 3 __£>._ LINE2 --o-- LINE4
8.
0
15 30 45 60 75 FIBERORENTATIONANGLE(0)
90
2O Co) ALL EDGES CLAMPED - - . i f - - LINE I , LINE 3 __~__ LINE2 -_¢.-_ LINE4
~
I0
500~
' 1'5 ' 3 0 4'5 6*0 ' 7~5 90 FIBERORENTATIONANGLE(0) Fig. 7. Comparison between the effect of two stacking sequence laminates [(+/-0)6] ` (line 1: Nemeth (1986), line 2: present FEM) and [(+ 06/-06)], (line 3: Nemeth (1986), line 4: present FEM) on buckling coefficient Cx = Nc, b2/(n2D~HD~2) for different fiber orientation angle 0.
difference is 4.07070. However if a/b = 3.9 the difference increases to 9.86°70. Such differences are due to the fact that the mesh size is not fine enough to converge to the exact solution. The effect of anisotropy on the buckling of symmetrically laminated composite plates loaded in compression, is investigated by considering the bending-twisting coupling terms, D~6 and/)26, comparing them to the generally orthotropic plates. The material properties of the ply are such that El~E2 = 11.6, GI2/E 2 = 0.5, vl2 = 0.35 and thickness = 0.127 ram. The results shown in Fig. 7(a) and (b) indicate close agreement between the predictions of the present FEM and those of Nemeth for 24-ply square laminates which are either clamped or simply-supported, respectively. The solid lines shown in Fig. 7(a) and (b) denote the value obtained by Nemeth (1986), and the dotted lines correspond to the present FEM results. The stacking sequences of the laminates are [( + 0)6L and ( + 06/- 06)s. Such comparison shows the FEM can be utilized to predict the critical buckling force of thin anisotropic laminated plates. 4. BUCKLING CONTROL OF NITINOL-REINFORCED COMPOSITE PLATES
In this section, the mechanical finite element models and thermal distribution inside the composite will be utilized to analyze the buckling of NITINOL-reinforced plates under different boundary conditions. A NITINOL-reinforced plate is made up of randomly oriented glass fibers and a low cure temperature polyester resin (Fibre GlassEvercoat Co. Inc. Ohio 45242) matrix. The plate is 0.1905 m long, 0.1905 m wide and 2.54mm thick mounted with all its edges in a clamped arrangement. A total of 58 NITINOL fibers (Innovative Technology International Inc. MD 20705), 0.55 mm in diameter, are embedded in the mid-plane and along the x-direction inside the plate
NlTINOL-reinforcedplates: Part I1. Static and buckling characteristics
87
through vulcanized rubber sleeves that have outer diameter of 0.95 mm. The N I T I N O L fibers have a 50°C phase transformation temperature. The NITINOL fibers are activated by passing electric current through the fibers in order to heat the fibers above their transformation temperature. Since the side-to-thickness ratio of the NITINOL-reinforced plate is relatively large, the present finite element method should be accurate enough to predict the critical buckling force for different boundary conditions. Also, as the aspect ratio is equal to unity then a relatively small number of elements (49 elements) can be used to accurately predict the critical buckling load of the NITINOL-reinforced plates. There are four different models, plain unreinforced plate (model A), plate with unactivated N I T I N O L fibers (model B), plate reinforced with heated but untrained NITINOL fibers (model C) and plate with activated and trained N I T I N O L fibers (model D). The initial tension of the fibers is selected to be 31.14 N/fiber. A value of 6.8 W/fiber is utilized to activate the N I T I N O L fibers. When fully activated the fibers generate a total phase recovery force of 116.0 N/fiber. The temperature distribution of the NITINOLreinforced plate is as shown in Fig. 8. Such distributions are obtained using the thermal finite element model described in Part I of this paper. The buckling characteristics of the plain plate (model A) will be used to serve as a datum for measuring the effect of reinforcing the plate with plain pre-tensioned fibers (model B), heated untrained N I T I N O L fibers (model C) and activated trained N I T I N O L fibers (model D). In this manner, the individual contribution of the tension, heating and shape memory to the buckling characteristics of the NITINOL-reinforced plate can be isolated and studied separately. In other words, model B will quantify the effect of the initial tension, model C will determine the effect of softening of the matrix due to heating
.... ii i
ili i :! ¸
",m(~) o.~ o o.~
X- , ~ '
Fig. 8. The FEM predictions of the temperature distribution on the NITINOL-reinforcedplate: when (a) all the NITINOL fibers are activated, and (b) half of the NITINOL fibers (alongx-axis) are activated.
88
J. Ro and A. Baz
of the reinforcing fibers whereas model D will demonstrate the contribution of the shape memory effect. In the case of model B, the initial tension of the fibers is found to increase the stiffness of the plate which, in turn, increases the critical buckling coefficient of the plate as shown in Fig. 9(a) and (b) for simply-supported and clamped plates, respectively. Passing electric current through memoryless NITINOL fibers which are pre-tensioned, as in case of model C, results in softening the matrix and including thermal loads which tend to reduce the critical buckling coefficient as shown in Fig. 9(a) and (b). Therefore, the stiffening effect resulting from the passive pre-tensioning of the fibers is offset by the softening of the matrix and by the induced thermal loads. If the pre-tensioning is not large enough to counterbalance the thermal effects, the plate can thermally buckle. When the shape memory effect is introduced in the fibers above their phase transformation temperature, large phase recovery forces are generated, as in the case of model D. These forces will increase the strain energy stored in the plate to enhance its critical buckling coefficient. Figure 9(a) and (b) presents comparisons between the dimensionless buckling coefficients for the four models as functions of the aspect ratio for simply-supported and clamped plates, respectively. For clamped plates with a/b = l, the critical buckling coefficients Ck normalized with respect to that of the plain plate are 105.58070, 60.46070 and 271.74% for models B, C and D, respectively. The corresponding ratios become 260070 (model B), 182.7% (model C) and 703.87070 (model D) when the plate becomes simply-supported. From these figures, it is evident that reinforcing the plates with NITINOL fibers results in significant increase in the critical buckling loads of these plates. Basically, the use of the plain pre-tensioned fibers to reinforce composite plates can enhance their capability of resisting buckling as in the case of model B. However, such 8O
6o! =
(a) A L L E D G E S S I M P L Y - S U P P O R T E D P L A I N P L A T E (A) ..... P L A T E W I T H U N A C T I V A T E D N I T I N O L F I B E R S (B) .... P L A T E W I T H M E M O R Y L E S S F I B E R S (C) P L A T E W I T H A L L N I T I N O L F I B E R S A C T I V A T E D (D)
40-
20-
0
.
0.0 ' '0~.4' '0'.8
.
.
.
.
~ , • , , , , , i , , '
1'.2 ~-116 2.0 2.4 ASPECT RATIO (a/b)
2.8
3.2
SO (b) A L L _ _ ..... ....
EDGES CLAMPED p L A I N P L A T E (A) p L A T E W I T H U N A C T I V A T E D N I T I N O L F I B E R S (B) p L A T E W I T H M E M O R Y L E S S F I B E R S (C) p L A T E W I T H A L L N I T I N O L F I B E R S A C T I V A T E D (D)
0.4
0.8
¢~ 40~ i
0.0
1.2 1.6 2.0 2.4 ASPECT RATIO (a/b)
2.8
3.2
Fig. 9. The critical buckling coefficient as function of the aspect ratio for plates which are (a) simply supported and (b) clamped. All the fibers are activated.
NITINOL-reinforced plates: Part II. Static and buckling characteristics
~0 ~ 60 ~
89
(a) A L L EDGES SIMPLY-SUPPORTED _ _ PLAIN PLATE (A) .... PLATE WITH UNACTIVATED NITINOL FIBERS (B) __ ~ _ PLATE WITH MEMORYLESS FIBERS (C) PLATE WITH A L L NITINOL FIBERS ACTIVATED (D)
J rOz 40 t
20 "
0.0
014
015
1.2 1.6 210 2.4 ASPECT RATIO (a/b)
2.8
3.2
80 i i
60
]
~4o i
(b) A L L EDGES CLAMPED _ _ PLAIN PLATE (A) _ _ _ PLATE WITH UNACTIVATED NITINOL FIBERS (B) .... PLATE WITH MEMORYLESS FIBERS (C) PLATE WITH A L L NITINOL FIBERS ACTIVATED (D)
~',',\,
i, ] i
t 0.0
............................. 014
0.8
1.2 1.6 2.0 2.4 ASPECT RATIO 0~b)
2.8
/
3.2
Fig. 10. The critical buckling coefficient as function of the aspect ratio for plates which are (a) simply supported and (b) clamped. Half the fibers are activated.
an effect is limited in the sense that the enhancement is not significant. But, if the plain fibers are replaced by the shape memory NITINOL fibers, the enhancement is dramatic. By controlling the initial tension of the fibers and the power passing through them, the developed recovery forces can be tuned to counterbalance excessively large external loads. Figure 10(a) and (b) shows the effect of activating half the NITINOL fibers on the critical buckling coefficients of simply-supported and clamped plates, respectively. The Cx values increase by 260.57% (model B), 246.59°70 (model C) and 435.80070 (model D) when the plate is simply-supported. The corresponding increase becomes 105.68% (model B), 98.67070 (model C) and 178.21% (model D) when the plate is clamped. It is important to note that since the temperature distribution shown in Fig. 8(b) is not uniform throughout the plate, the fibers nearby the activated fibers will be activated unintentionally by the heat propagation effect. As a result, additional recovery forces are generated which contribute significantly to the strain energy of the plate. These effects are considered in the present analysis. 5. CONCLUSIONS
This paper has presented an analysis of the buckling of NITINOL-reinforced plates. A finite element model is described and utilized to predict the critical buckling loads of plates. The predictions of the model are validated against results available in the literature for plain unreinforced isotropic, orthotropic and anisotropic plates. The model is then used to study the buckling characteristics of NITINOL-reinforced composites. The individual contributions of the initial tension, the softening of the composite matrix and the shape memory effect to the critical buckling loads are determined. The effect that the boundary conditions, aspect ratio and activation strategy have on the buckling loads of plates are
90
J. Ro and A. Baz
analyzed in great detail. It is found that reinforcing composite plates with NITINOL fibers can dramatically enhance their critical buckling loads even when these plates are clamped from all their edges. Acknowledgements--This work is funded by a grant from Army Research Office (grant number DAAL03-89-G0084). Special thanks are due to Dr Gary Anderson, the technical monitor and chief of the Structures and Dynamics Branch of ARO, for his invaluable and continuous technical inputs. REFERENCES Baz, A., Poh, S., Ro, J., Mutua, M. and Gilbeany, J. (1992). Active control of NITINOL-reinforced composite beams. In Intelligent Structural Systems (Edited by H. S. Tzou and G. Anderson). Kluwer, The Netherlands. Baz, A., Ro, J., Mutua, M. and Gilheany, J. (1991b). Active control of buckling of NITINOL-reinforced composite beams. In Active Materials and Adaptive Structures Conference, Arlington, VA, pp. 167-176. Berlin, A. (1994). Active control of buckling. Ph.D. Dissertation, M.I.T., Cambridge, MA. Chajes, A. (1974). Principles of Structural Stability Theory. Prentice-Hall, NJ. Chandrashekhara, K. and Bhatia, K. (1993). Active buckling control of smart composite plates--finite element analysis. Smart Mater. Struct. 2, 31-39. Fenner, R. T. 0975). Finite Element Methods for Engineers. Macmillan, London. Meressi, T. and Paden, B. (1992). Buckling control of a flexible beam using piezoelectric actuators. J. Guidance, Control Dynam. 16(5), 977-980. Mottram, J. T. and Selby, A. R. (1987). Bending of thin laminated plates. Comp. & Struct. 25(2), 271-280. Nemeth, M. P. (1986). Importance of anisotropy on buckling of compression-loaded symmetric composite plates. AIAA. J. 24(11), 1831-1835. Polymer Laboratories, Ltd (1990). PL-DMTA, MKII Dynamic Mechanical Thermal Analyzer. Loughborough, U.K. Roark, R. J. and Young, W. C. (1975). Formulas for Stress and Strain, 5th edn. McGraw-Hill, New York. Tsai, S. W. and Hahn, H. T. (1980). Introduction to Composite Material. Technomic Publishing Co., Pennsylvania. Young, T. Y. 0986). Finite Element Structural Analysis. Prentice-Hall, NJ. Zienkiewicz, O. C. and Taylor, R. L. (1989). The Finite Element Method, Vol. 1, 4th edn. McGraw-Hill, New York.