Sairam and Sinha

8/4/2019 Sairam and Sinha

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Journal of ound and Vibration (1992) 158(l), 133-148

HYGROTHERMAL EFFECTS ON THE FREE VIBRATIONOF LAMINATED COMPOSITE PLATES

K. S. SAI RAM AND P. K. SINHADepartment of Aerospace Engineering, Indian Institute of Technology, Kharagpur-721 302, India

(Received 4 October 1990,and in jnal form 20 June 1991)Effects of moisture and temperature on the free vibration of laminated composite platesare investigated. The analysis is carried out by the finite element method with the qu adratic

isoparametric element, which takes transverse shear deformation into account. The analysisalso accounts for lamina material properties at elevated moisture concentration and tem-perature. Results are presented showing the reduction in the natural frequency with theincrease in uniform moisture concentration and temperature for symmetric and anti-symmetric laminates with simply supported and clamped boundary conditions.

1. INTRODUCTIONFibre reinforced plastics are being increasingly used in aerospace applications. They maybe subjected to moisture and temperature environments during service life. Moisture andtemperature have a sign& ant effect on the free vibration of fibre reinforced plastic lamin-ated plates.

The effect of environment on the free vibration of laminated plates has been consideredearlier by Whitney and Ashton [I]. They used the Ritz method to analyze sym metriclaminates and equilibrium equations of motion in the case of antisymmetric angle-plylaminates, based upon the classical laminated plate theory. A few results were presentedfor only symm etric angle-ply lamina tes. Very recently, Dhanaraj and Palaninath an [2]used the semi-loof shell element to study the free vibrational characteristics of compositelaminates under initial stress, which may also arise due to temperature. Resu lts werepresented showing how temperature affects the fundam ental frequencies of antisymmetriclaminates. Except for the above-mentioned publications, no literature has been found onhygrothermal effects on free vibration of laminated composite plates. Yang and Shieh [3]considered vibrations of initially stressed antisymmetric cross-ply laminates. Initial stressesincluded both force and moment resultants. Rotary inertia and transverse shear effectswere taken into account. On the other han d, vibrations of isotropic plates under arbitraryinitial stress have received much attention [4-71.

In the authors previous paper [8], the quadratic isoparametric element wa s applied tostudy hygrothermal effects on the bending characteristics of laminated composite plates.In the present work, attention is focused prim arily on investigating the effects of moistureand temperature on the free vibration of laminated composite plates. The conventionalfinite element formulation is modified to include the strain energy of all types of initialstresses, viz. compressive, bending and shear stresses, developed due to moisture andtemperature. Transverse shear deformation and rotary inertia are accounted for accordingto the Yang-N orris-Stavsky theory [ll], which extends the Mindlin theory [12] to13 3002240X/92/190133 16 $08.00/O 0 1992AcademicPressLimited


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13 4 K. S. SAI RAM AND P. K . SINHAlaminated composite p lates. Furthermore, lamina material properties at elevated moistureconcentration and temperature are used in the present an alysis. The reduction in naturalfrequency with the increase in uniform moisture concentration and temperature is studiedin the cases of symmetric and antisymm etric laminates with simply supported and clampedboundary conditions.

2. GOVERNING EQUATIONSConsider a laminated plate of uniform thickness t consisting of a number of thin laminae,

each of which may be arbitrarily oriented at an angle 8 with reference to the x-axis of theco-ordinate system (see Figures 1 and 2). The constitutive equations [ 11, 13, 141 for theplate, when it is subjected to moisture and tem perature, are given by (a list of notation isgiven in the Appendix)

where

Figure 1. Arbitrarily oriented lamina in a laminated plate. Axis 3 coincides with z axis.

Figure 2. Layer details.


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HYGROTH ERMAL EFFECTS ON FREE VIBRATION 13 5

PI =

All AIZ -416 &I &2 &6 0 0

Al2 A22 A26 &2 I322 &6 0 0

Al6 A26 -466 B16 826 866 0 0

Bll Bl2 &6 Dll 012 016 0 0

B12 B22 B26 012 022 026 0 0

Bl6 B26 B66 016 026 066 0 0

0 0 0 0 0 0 Az,z, A45.o 0 0 0 0 0 A45 A55

Non-mechanical force and mom ent resultants are expressed as

i, j= 1, 2, 6,. (2)

The stiffness coefficients are defined as(A,, B,, Dii)= f s Q [&M L z, ~1 dz, i, j= 1, 2, 6,.k*I Zkk-I

&)=a 2 r &]k dz, i, j=4, 5.[ &]k in equations (2) and (3) is defined as

@ii lk= ~W[Qii lU,l -T@,lk= [~2I- [Q,M~21

where

(3)

for i, j=l, 2, 6,for i, j=4, 5 (4)

[

cos2e sin20[T,] = sin20 cos2e

-sin 8 cos 8 sin 8 cos 8-~ $t e~ z$]. LT21=[;;; -;:;I,

QII Q12 0[Q ~lk= [ Q12 Q22 0 1i, j= 1, 2, 4- tQijlk= [ ; 1 , j=4, 5,0 0 QM 55

in whichQII =&A 1 - v12v21), Q1 2 = v12452/(1- ~12v21)r

Q22=EzlU - VI~V ZI), Q4=G 3, Qss = G23.


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13 6 K. S. SAI RAM AND P. K. SINHAThe linear strains are defined as

6, = ii,, , Ey= fi,y Yxy= 6.y+ 0,x Kx = e,, 1 Ky= -&,y,~~~ e,, e,, b= ey+w,x, #y=-ex+w,y. (5 )

Upon assum ing that w does not vary with z, the non-linear strains of the plate [3, lo] canbe expressed as

Since u = U+ ze, and o = 6 - ze,, equations (6) may be written as

Eynr= [ii, + 0; + w;+ z(u,yey,y - tyo,,~)+ zW y + e;,ywTyxynl= [ti,,iiy + ~,,ii,~ + w,x~,y 4u,yey.x c.xey,y)

- z(~,y e,,, + u,~,,y) +zVy,xey,yex,xexg)i9yxrnl= kxey - a,d, + z(eyey,x e,e,,d,yyznl= [iiyey - v,yex+ 4eyey,y+ wb,)i. (7)

3 . F I N I T E E L E M E N T F O R M U L A T IO NThe eight-noded isoparametric element (see Figure 3) which is used for static analysis[8-lo] is applied to the present free vibration problem. Five degrees of freedom are consid-ered at each node. The stiffness matrix, the initial stress stiffness matrix, the ma ss matrix

and the nodal load vectors of the element are derived by using the principle of minimumpotential energy. The element d isplacements are expressed in terms of their nodal values

4* ,,?.7 ??

0 a1 5 2Figure 3. Eight-noded isoparame tric element.


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HYGROTHERMAL EFF ECTS ON FRE E VIBRATION

by using the element shape functions and are given byii= f Niiii, IY= f NiCi, IV= f NiW i,

i = I i=l i=l

3.1. ELEMENT STIFFNESS MATRIXThe linear strain matrix {E} is obtained by

expressed as

13 7

substituting equations (8) into (5), and is

Figure 4. Discretizational details.

TABLE 1Eiastic modu ii of graphite/epoxy lamin a at different moistu re concentrations; G13 = G12,

Gzj=O .SG,z, VIZ= 0.3, /?, =0 and p2=044Moisture concentration, C (%)Elastic moduli@P a) 0.00 0.25 0.50 0.75 I.00 1.25 I.50

EI 130 130 130 130 130 130 130EZ 9.5 9.25 9.0 8.75 8.5 8.5 8.5Cl2 6.0 6.0 6.0 6.0 6.0 6-O 6.0

TABLE 2E l a s t i c m o d u l i o f g ra p h i t e / ep o x y l a m i n a at d@ erent temperatures; Gls = G12, G23= 0. 5G,2,

V12~0.3, al =-O-3 X 10-6/K and az=28.1 x 10-6/KT e m p e r a t u r e T (K )Elastic moduliWa) 300 325 350 375 400 425

El 130 130 130 130 130 130E2 9.5 8.5 8.0 7.5 7.0 6.75(712 6.0 6.0 5.5 5.0 4*75 4.5


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138 K. S . SAI R AM AND P. K. SINHATABLE 3

VeriJication of non-dimension al frequency /I by c omp arison with Ritz meth od results; a/b =1, a/t = 100, (O/90/90/0), simp ly supported

c= 0.1% T= 325 KMode number,m Present FEM Ritz Method Present FEM Ritz Method1 9.429 9.411 8.088 8.0682 20.679 19.911 19.196 18.3783 40.068 39.528 39.324 38.7784 46.752 45.815 45.43 1 44-778

where{~,}={~~,~~,w~,~,~,~,~,...,~~,~~,w~,~,~,~~~~~,Ni,, 0 0 0 00 Ni,y 0 0 0

Ni,y Ni,.x 0 0 0[l&, ; ; ; O yf .-Ni,y0 0 0 -Ni,x Ni.y0 0 Ni,.r 0 Ni

-0 0 Ni,y -Ni 0 sThe element stiffness matrix is given by

[%I = ss PITPWl dx dy. (10)3.2. ELEMENT INITIAL STRESS STIFFNESS M ATRIX

The non-linear strains, equations (7), are represented in matrix form as{sn,)= {&xn[,+z/, ~xyn/, ~xznr, ~y/yzn~~ ~=[Rl{d~ /2, (11)

where {d} = {&, ii,, fi,X,Us, w,~, w,~, e,, , by, ey,x, ey,y, 8, , 8 , IT, ad [RI is obkusfrom equations (7) and (11).By using equations (8), {d } may be expressed as{d) = KW~eL

where

wf,

ri,, 0 0 0 0? i ,y 0 0 0 00 Ni,, 0 0 00 Ni,y 0 0 00 0 N,, 0 00 0 Ni,y 0 00 0 0 Ni,x 00 0 0 Ni,y 00 0 0 0 Ni;0 0 0 0 Ni,0 0 0 1 00 0 0 0 1

(12)

rY


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HYGRO THERMA L EFFECXS ON FREE VIBRATION-___--___/t,=2,O/t =40--ll__ -- ---____ - - -___x55? ----__

/_aLb_=2 - cl! 7 l_O-- ---_- _-__ ___

._.

=G

4A1

10 0.25 0.5 0.75 1. 0 1.25 .5

139

Moistu re concent ra t ion . C (%)Figure 5. Effect of moisture on non-dimensional fundam ental frequency of (O/9 0/90 /0) lamina te. --, Simplysupported ; - - -, clamped.

The initial stress stiffness matrix is given by[J&,1 [YTISl[Gld-xdy,

where

PI=

SllS21000000

s91S 101

0S 121

s22000000

S 92SlO2

0s122

S 3S 4300

&3

s8300

S 1130

S4400

S 4S3400

s1140

Ss::660 0 s710 0 s87 s080 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

(13)

s99

SO9 S 10100 00

0 0 Of 0


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140 K. S . SAI R AM AND P . K . SINHA601

t

---___ ---___ --_ l- ,,/b=2,o/t=40---___ ------------

,o/b=2,o/,=lO

___-__ ____ _-___-_---____________--o_/_b_0~5,0/t=40o/b=l ,o/ t=lO

01 10 0.25 0.5 0.75 I.0 I.25 5Moistu re conc entrat ion, C (%I

Figure 6. Effect of moisture on non-dimensional fundam ental frequency of (45/-45/-45/45) laminate.

in whichS,,=&=&=N;, Szz=SM=Ss6=N;,S,, = S,, = Sas = N& , ST, = SW = N: tZ/ 12,

s 8 8 = S,o,o=N;t*/12, S87=S,09=N~yt2/12, -s,3=s$l,=lu~, -&=&=M;,- s 1 4 = - s 8 3 = &2 = &, , = d ' f ; y , - &, 3 = &2 , = : ,Q -s114 & 2 2 = Q;,

3.3. E L EM E N T M A S S M A T R I XT he element m ass matrix is obtained from the integral

[MI-IS- [NITPIINlk dx (14)


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HYGRO THERMAL EFFECTS ON FREE VIBRATION 141-------_________ o/b=2,o/t=40---_r__ ---- ----___ __

60 -

50 -

40.___ ___ _ _-------- --- J_d_/_b~z_*_ol_=o___-__

o/b2,o/t = 10--- -------____ r o/b=l,o/l=40--__ ---- __ ---- -_ __20 . _-_ _ /~lb_=_l,a/f=_io___________ __---------_ ___ --- __ Co/b=0.5,0/1=40

_ _~~~=_O_~~:


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14 260

K. S. SAI RAM AND P. K. SINHA

----- ___ -----____ o/b=2,o/t=40-/___ ---- -__ _ -_50 -

30 - Ka/b=2,a/t10

----^_____ -----__ a/b=l,a/t=4020 - L----_____ --__ --__

_____lr__/~l10/1_=_10__________________-

0 I0 0.25 0.5 0.75 1.0 1.25 1.5Moistur e conc entrat ion, C (%I

Figure 8. Effect of moisture on non-dimensional fundam ental frequency of (45/-4 5/45 /-45) lamina te.The element load vector due to hygrothermal forces and moments is given by

[~I@? d.xdy. (16)3 5. SOLUTION PROCESS

The stiffness matrix, the initial stress stiffness matrix, the mass matrix and the loadvectors of the element, given by equations (10) and (13)-(16), are evaluated by firstexpressing the integrals in local natural co-ordinates, 5 and q of the element and thenperforming numerical integration by using Gaussian quadrature. Then the element matri-ces are assembled to obtain the respective global matrices [K], [K,], [Ml, {P} and (P }.The first part of the solution is to obtain the initial stress resultants induced by theexternal transverse static load and by moisture and temperature in static conditions. Theinitial displacemen ts {S}, are found from the equilibrium condition[K](8) = (P} + {PI. (17)

Then the initial stress resultants Ni, N; , iY , M$ , A4;, ML ,,, QL and Qi are obtained fromequations (1) and (9).


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HYGRO THERMAL EFFECTS ON FREE VIBRATION 14 3

----_ - --_ ---_ ---_ x o/b=2,o/f=40-_ -----___ -----__

------------ --- ---lf o/b=2,0/t=10--- -_____ _ _-__

---------- --______/----=-1___________ _________f_O_/bP0.5,0/t=40/b=2,0/1=4- -----------

,/b=2,o/t=K? -----------_-_____J_ o/b=l , o/t=lO_Kolbi_0.5,0/t= 10 --- -_-___--___- - -------- _-- - - -- - --- _-__-_-_

ro/b=l .o/ t=40

3 325 350 37 5 400 1Moistu re concen l ro t i on , C (o /o )

15

Figure 9. Effect of tempe rature on non-dimensional fundam ental frequency of ( O / 9 0 / 9 0 / 0 ) aminate.

The second part of the solution involves determination of natural frequencies from thecondition

Iwl+K7l-&w =a W -9This is a generalized eigenvalue problem and is solved by using the subspace iterationmethod [15].

4. RESULTS AND DISCUSSIONThe analysis described in the previous sections is applicable to the free vibration of

laminated plates subjected to an external transverse static load and non-uniform distribu-tion of moisture and temperature through the volume of the plate. In the present investiga-tion, results are presented for symm etric and antisymm etric laminates subjected to uniformdistribution of moisture and temperature in the absence of external static load. Four-layered, graphite/epoxy laminates with simply supported and clamped boundary conditionshave been analyzed. In the cases of all the laminates, both the in-plane displacemen ts ofthe mid-plane at an edge are restrained for both sim ply supported and clamped boundaryconditions. Lam ina material properties at the elevated moisture concentrations and tem-peratures [ 141 used in the present analysis are presented in Tables 1 and 2. Since theevaluation of shea r correction factor from the exact theory of elasticity i s diiku lt in thepresent case, a commonly used value of 5/6 is assum ed [ 16, 171.


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144 K. S . SAI R AM AND P . K . SINHA60 -

---____ --__ -- __ ---/ o/b=Z,a/t=40----__ ----____

50 -

r

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