Vietnam Joirnia] of Medianics, VAST, VoL 38, No. 2 (2016), pp. 103 -122 DOI:10.15625/0866-7136/38/2/6730
F R E E V I B R A T I O N O F F U N C T I O N A L L Y G R A D E D S A N D W I C H P L A T E S W I T H S T I F F E N E R S B A S E D O N T H E
T H I R D - O R D E R S H E A R D E F O R M A T I O N T H E O R Y Pham Tien Dal, Do Van Thom*, Doan Trac Luat
le Quy Don Technical University, Hanoi, Vietnam
*E-mail: promotion6699@gmail.com Received August 11,2015
Abstiatt In this paper, tlie ft» vibration o( functionally sandwidi grades plates willi stifieners is mvestigated by using llie fimte dement metliod. -Ilie material pioperties are usumed to 1» graded in the Ihiclcness di«fion by a p„wer-Uw distribution B a S i n the , 1 ^ " ! - ^ . ' ' " 7 " " ° " * ' ° ' ^ - * ' S ' " " ™ " S equahons of modon am derived from aie Hamilton s pimcipie. A panimetnc study is carried out to highlight the effect of ma- tenal distribution, stiffener parameters on the fn« vibration d i a n S e S i i s of te p l . t e T Keywords: Stiffened plate, sandwich, vibration, fiinctionally graded materials
1. INTRODUCTION
FmictionaUy graded malerials (FGMs) made of hvo constitiients, mainly metal and ceramic are widely used. T^e composition is varied continuously along c e S c t e c tons accordmg lo volume h-ation from a ceramic-rid, surface to a meW-ridlTuxfa^e S d r i ^ S r d d ^ v f l T ' " d " ° * ' ^ " P - '•> - ' i - ' - d behaviorTf tocjf^y graded plates. Reddy [1] can-ied oul nonliear finite element static and dynamic analvses of hmctionaUy graded plates ustag Navier's sotation, Vel and Batia [2] ^ s ^ l e T t e e e ttanendional exact solutions for free and forced vibrations of smiply supported taSSn' aUy graded rectangular plates, Tmh Quoc Bui et al [3] used finite e l e M m e t h l t H T static bendtag and vibration of heated FG plates. Dao fluy K A el S M T l J H i Due etal [6,7] used analytical metiiod lo stady vibration norfTnel '' ^ 7 " " ^ ^
caUy stiffened frnictionaUy graded cylindrUi.lJi^THS^^Z^''"'''''"' T"*":
FGM plate and eccentiicaUy stiffened Bun F G M S S on eTaic ^ . r ' ^ " ' ^ ° environmentit [6,7]. Several stadies have been p e A ^ l d to S v t T h e l H™ " * " 7 S " ' sandwidi Stiuctiires, Zenkour [8[, Ahpour and a a n ^ a l [9^ ^ ' °' ^'^
ta this paper, the free vibartion of stiffened FG sanHwi,-!, „i 1 j . j , tagtiniteeleme„tmetiiod.HerewepresentFGs»'dtf^^rv^raterifp:^^^^^^^^
© 2016 Vietnam Academy of Science and Tedmology
104 Pham Turn Dal, Do VanThom,DcHmTniclHal
symmetric aboul the mid-plane. The faces of the plate consist of a FGM witii propt varytag only ta the thickness tlirection. Such faces can be made by mixing two di£E material phases, for example, a metal and a ceramic. The core material may be h geneous and can be made by one of these materials, for example, a ceramic or a r.
Eigenfrequendes and mode shapes of stiffened FG sandwitii pMtfes ate piesit^ed u the higtier-order shear deformation theory. ' / / 3
2. GOVERNING EQUATIONS AND FINITE ELEMENT ELEMENT FORMULATION
Consider a stiffened FG sandwich plate with thickness, long, wide of the plate ale I 2A, a, b, and depflv width of the stiffener are h„ b„ respectively (Fig. 1). The ;cy-plane is 1 ttie mid-plane of tiie plate, and the positive z-axis is upward from ttie mid-plane The ; power law distribution is used for describmg ttie volume fraction of the ceramic (vP) ' and ttie metal (vi'') ta i-tti layer (i = 1,2,3) as follow
v i i ' - U ' W - l .
Fig-1.
typel
type 2
A FG plate stiffened by an x-direction and an y-direction stifieners
vf = 1
' \h~h) •'
\ ^« =0 ; V(3)_/'2-'!lV I ' {h-hj
'
-h<z< -h-l -hi<z<h hi<z<h
-h<z< -hi -hi<z<h hi<z<h
Free vHmition offuncllonalhf gnuled samlwieh plales with stigeners bastai on the Ihird-onler shear dtfiirmntlon theory 105
Where, 21i-ttiickness of the plate, 2Jti-ttiidmess of second layer; n-ttie gradient ta- dex (n > 0); z-lhe thickness coordtaate variable, and subscripts c and m represent the ceramic and metid constitiients, respectively
ta this shidy, the material properties, i.e.. Young's modulus E, Poisson's ratio v and ttie mass density p, can be expressed by the rule of mixtiire as [1]
P'-'H'^) = Pm+{Po-Pm)Vi'K (4) where P<') is ttie material property of i-tti layer.
ta [10[ ttie three-dunensional displacement field (u, v, w) can be expressed ta lerms ot nme unknown variables as follows
u (x,y,z) = ,t»(x,y) -\ z.if^{x,y) -(-z'.5,(i,y) -|- 2 ^ * , ( I , y ) ,
" ("'V'^) = ""(^-y) + z-ifj,(r,y) -I- z\^x,y) -I- z\0,{x,y), (5) 'o{x,y,z) = w„{x,y),
where u", n", w<> represent ttie displacements at flie mid-pbne of ttie plate ta flte z, y and z durechons, respectively; tf^tf^ denote ttie tiansverse normal rotations of tile y aod j : axes; & , ? , denote ttie higher order displacements and * „ * „ denote ttie higher order tiansverse rotations. ' 6 ci 2.1. Plate element
Fig 2 ) ^ ^ '""^^'" " ' ' ' ' " ' ' ' ' ™ ' ^ ' " ' ' * * ""'"''"' '''"* "'^ ^ ' ' ^ ' ' 8 " * ' " ' f«='=<'°" (see
n-i.-i) 2(1,-1) (b) Natural coordmate system (a) Plate element in Decaster system
Fig. 2. Rectangular isoparametric plate element For ttie plate element, displacement vector of i-tti node has Uie form
{l,} = {ll,V.,lO,i<l',.,iP,„li,i,i,l,0,„,i^,Y i ^ I J J (gj o X a ^ C 7 * ' ' * " ' ' ' P ° ' " * ° ' * ' ' " = * " ' ™ ' = " ' f " > ' - ° - P " ' ^ d u s m g s h a p e
function N,- as foUows
{"} = E H . {?,-}, (7)
Pham Tfen Dat, Do Van Thom, Doan TruEEtBtfj'
Nl = 0,25(1 - r)(l - s ) , N2 = 0,25(1 + r ) ( l -s), N3 = 0,25(1+r)(l + s), N k - 0 , 2 5 ( 1 ^ r ) ( l + s ) . The strain vector can be expressed in the form
{e} = {s°}-^z{x}-^^^{;t}-^z'W- M = {7°]+^W)+z^{x'}, i {,;''} = L,{t,} = l,ENi{,i}=[B,]{,}.,
/=l
M=L2{u}=L2'£Ni{q,MB2]{t,}„{x}=L3{u}=L3£N,{q,)=[B^]{,f}^, ( l |
« 4
{.;}=L4{u}=L4EM{?,}=[B4l{?K,{7°}=L;{«}=L;£Ni{,,.}=[B'j{,} {*
{x'}=L;{„}=L;£N,{,i}=[B;i{,}„{^'}=4{,t}=L;ENi{,,,-}=[B;]{,,}„ ( ^
(M)f
[B/i = t,-EH, [B;] = L;EN,-, (i4|
1 = 1 f = l .-'^
"''*. ' ' ] . ' = ( (?> J {ll}}'^- displacement vector of plate element. L,, L,'- stant*e stiam-displacement matrix of plate element.
The shKis vector at any potal ta i-tt, layer of tt,e plate is expressed as / {"}'" 1 _ f [Dmi] [0] 1 / {£}(') 1 where
W'" = HV(')4)}^ {r}(') = {x«r«},
^_E«(z)_ ' l-(v(.)) 2.2. A:-Sti£fener element
The displacement field of i-Stiffened can be expressed as (see Fig. 3)
""fey-'zl^o,"'''^' ^"^"''^'^' +^'S»(^,y) +^<s>„{x,y),
''«{x,y,z) = ^^{x,y).
The^stiata veclor for tae «tiffener elemeni has ttie form
{^}„ - {£"}.„ -t-z {x}„ -^z^ (^}^^ +,3 j ^ j ^ ^ _ j ^ j ^ ^ ^ ^^„j^^ ^ ^ j_^j^^ ^ ^ ^ ^ j
"(19) [D„,
1 fC) 0 v") 1 0
0 0 1 ^
' i°")=i3)*"^'''''- ''*!
(18)
^'^^^'^offm:cti»»uaiygradeds«ndwidtplateswithst^cneni^
x-Stiffener . _
X~y-Stiffener
Fig. 3. Plate element and stiffener element where
{ ' = ° } „ = i l » { " } „ = [ B i „ ] { ? } „ ,
M„ = 1.2,, {<t}„ = [Bj^J {,}^^^, Ix^^ = 1.3^^ {„}^^ ^ [B^^j {^j^^
{!?}„ = 1-4,, {.<}„ = [B^,] { , } „ , , {/}^^ =!,;__ {„j^^ ^ [j|J {^j^^^
{"•}„ = C {"}„ = 1B;,J {,}„., {x }^^ = 4 „ {u}„ = (B;„J {,,}„
4
(20) (21) (22) (23) where [B;„] = L,„ E N,„; [B;„] = L',„ t N „ . ; N,„ - a,e shape tanctions of ;r-stt(fener which can be obtained by substitating 7= s, tato Eq. (8); { , } „ - displacement vector s l ^ i f d e m ^ r ^ ' ' ' ' ^ " " ^ ' " " ' - • '*^^-' * ^ - d ^ P l - ™ e n t matiices of x-
The stiess veclor at any pomt ta tae stiffener is expressed as
where
• W „ U f [D„„l [OJ 1 / {c}^. -1
W}r. = W>l^y'Cr,}l, , { T } „ = { T „ T , . } J ^ ,
[D„] =
l - v j ,0 0
'°""'°2(iTir)*"^<''''-
(24)
(25) (26) 2.3. y-Stiffener element
The displacement field of y-stiffened has tile form
",.(:c,y,z) = 0,
Vy. {x,y,z) = 1,0 (i:,y) + z,f„(x,y) -f z^f,.(,,y) + z'*,.(x,y), w„ (x,y,2) = a,o^(i,y).
The stiata vector for die y-stiflener elemem is expressed ta ttie form
^^^y-^^X^^^'^)y.^^Mys^^Mys,{'r}y.H7X+z{^}„W{,-)^^,(2S)
(27)
Pham Tten Dat, Do Van Thom, Doan "DveLioa
{^}ys=Ltys{u}^=lBi^]{q)^, My, = Lzys {"}ys = [Bl,,] M,,, • {l)y. s i " } j , = [B4,J{?}™
f'},.= CW!»=[B2,JW„
{X}y. = i3,S {U]y. = [B^] {l,}^, b'')y, = ^iA^}y.'=[BJ{ci}^, 01),
(32J!
where [B^] = hiy, E N|,s; [Bj^[ = tj^, j N,,,; Niyj - shape functions of y-stiffener wHcbi can be obtataed by substitiiting r = ro mto Eq. (8); {^}^, - displacement veclor of y^
stiffener coordinate system; Ljy,, LJ^^ - standard sham-displacement matrix of y-stiffener'.
elemeni.
The stress vector at any potat ta the stiffener is expressed as
/ Wy4=f P"»'] m 1/w,» 1
(33|W „ = K "-y i i , } j ; , {T},, = {T;„ r , , } j ; .
• 1 V 0 V 1 0
2 2.4. Transformation matinees
- B>.
1-vJ.
0 0 2 ( l - ^ v „ ) rft<ig(2,2).
We consider taat tae i:-Stiffener is attached to tae lower side of flie plate, tae con-, ditions of displacement compatibihty along tae Une of connection can be written as *
'"Ji!"-0.5I. = {"}«[!.0.5ll„
Ustag Eqs. (5) and (18), tae conditions ta Eq. (36) ftus lead to ML- = [""], + 4 [fJ,- -i-<^5 [ a , +4 [*,],-, l w , = [if»li ,
[ « , = I ? » 1 , , [*.L-=[*«1,-,
(37>;
-Q< + h„)n, el=.{h^-hl,)/i, 4 = - (it'-l-Al,)/8.
^"*^^<^ioni^fmuiimaaygradedsandtmdtpbdeswi&st^^mTsbasedmthea^
Expressing in matrix form
"2
0
Wx
f,-:.
0 iie-x 0 0
=
1
1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 0 4
0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
4 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
4"
0 0 0 0 0 0 0 0
• u' V Wo Ipx ' ify
?I
* y
i = 1,4 (39)
W . i = T „ W , . The nodal displacement veclor can be written ta tae form
W„=T,W.,
where
T,= T„.diag{4,4,). (42) tte^tiansformation matibc for tae y-stiffener element can be derived ta a sunilar way ta
(40)
(41)
My.= Ty{l),- 2.5. Weak form of the stiffened plate problem
The elastic stiata energy of tae plate is written as [11]
or in the matrix form
'''ILU}UK].M.^ILMIIK].,U}14EMI
(43)
(44)
%ll}l,- (45)
•[B,f[A][B,J -t- [B^j'-miB,] + [Bj'-IDKBjl + [Bj'-[£1[BJ + \B,flB]m 1
+ Bzl^lDllBJ + [B,];[£,[B3l + [B/IFHBJ + [BJ^Dl i , | I U S
• "" »*,T'P' f'i + l^^'mi".] + lB,f lA'llB/l -MB ' I V [B/H
+ B, Q D ' ] [ B 3 ' 1 -,- [ B , f [B'1[B/J + [ B , f I D ' H B , - ; B V E'l B '1+
L + l B s f [D'1[B,1 -f [B3f [E'J[B,'] + [B,'!'' F'J B3l ' ^ ^ '"^ " " ' ' + dSe =
Pham Tien Dal, Do Van Thorn, Doan TyacLmt
=EE
. . i , - . i
[Kl»=f-=/
=6»E
"[B,r[AJ[B,K[B,r[Bl[BJ-MB,r[DJ[B3l-HB,]'-(El[B4H-[BJ^[B)[Bil -t-[Bj''[D][BJ-l-[B,l''[E][B3H-[B,f[Fl[BJ-KB3]''lDJ(fl,]-f[B3]''(E][Bj -H[B3r[F][B3l -1 [B3r[Gl|B,l -H [BJ^EHBJ -F'[B4r[fl[BJ.f +[B4flGl[B3l -I- (Bif [HJ[B,I + [B,f[A'l[B,') + [ B i f l S ' l l B / H +lB,'r(D'][B3'l -I- [B^f [B'llBi'l -F [B^'flD'KB/l -1- [Bj^lE'llB,'!-!- L-KBjf [Dl[B,'l + [ B , f |E'1IB,'1 + [Bj'f [F'][B3']
[ B , j l ^ J | B , J - F [ B , „ r [ D J [ B 3 j - K B , j ' - p j l B ^ ] - K B ^ ] ' - [ F J [ f l -KB3„f(DJ[B,.H-[B3j'-|FJB3j+[B,j'-[FJ[Bg-KB^]'-[HJ[B, + ( B , „ f [A,'][B,„'l -f [B,„'nD„'][B3„'] + [ B , „ f [D„'1[B^'J-F .+[B3»f P„'l[Bi„'l -F lB3„'nf„'l[B3„'J
«
.-<-[B3™r[D.'][B,„'J + [B3„f[F„'l[B3„'J
"•ff
ntx
I A, B, D, E, F, H, G) = I [D„] ( l A Z I Z ^ , 2 V , z ' ) dz, -»/2
(/l',B',D',£',F')= I [DJ (l,z,22) tiz.
-ft/2 I is det Jacobian malrix
Xl yi X2 yi X3 ys [xi yt
(^
' iNi BN2 BN, 3Nt [;| ^ 3r dr Br dr ..j. „
BUt_ BNi BN3 BNi X, u, ' (°W 9s 9s 3s "ai" .
and tti, = Hij = 1 are Gauss weights for two Gauss potats (4 node plate elemeni) and
Gauss potats = ± — iw v'S
17m I is det Jacobian matibt
l/xj = 8N, BN2 . Sr Br
fc].[«^^-
Xl . * 2
yi . y z .
for ar-stifiiener
for y-stiffener
Fi»i«Sm<tai^/iiTOlJm«IS,g,,jMamiioicftpIa„rtliajji:^ , j j
and Wi = 1 is Gauss weight for two Gauss points (2 node beam element) and Gauss potats = ±—=.
-1^3
The ktaetk energy of Uie stiffened plate is computed by
= IL{«J iMp], {^}.+i E «}J [MJ, U}A E Ml [My]. {?},,
wiwre
/• " ^ / 2 2 \
[Mp],=y[N]'-[p][NJdl/,= /• E E [ N l ' ' [ f l [ N l l / l = ' , - » ' , - | & ,
1'. -ft/2 v = i / . i 7 [M,l, = T j J y [ W J ^ [ , J [ N J ,i,„J r , = T j L "/"(t [ N , r [pJ [NJ | / „ | u i A t f z j r , ,
K ] . = I - j | v / [ N y ] ^ W [Ny] J T y = T j ^ 6 , 2 " ( i : [ ~ . ] ' W K ] | / , > , ) 4 ^ 1,„ ril!" T^ ^^^' r , ^P'^''' * ^ " ™ 5 " ° n ' s principle to find tae weak form of tae prob- lem (ttie dampmg of plate is neglected). The principle is started ttmt
where W is ttie work done by external forces on tae stiffened olate So ttiat Eq. (56) leads to '^
j (S'm + 5T-SU)dt = 0, (55,
ll
xlemal forces on tae sliffene
([Ml + [MJ) {?} + ([K] + (KJ) {,,} = {F}. (57, b e s o l ^ d " ' ^ ™ ' " ' " " ' " " " ' ' ' " ' ' * = { " " f t ^ M o w m g eigenvalue equation must
{([Kl + [KJ) -uf^ ([Ml -I- IMJ)} {,,„} = {0}, (58) where Cl, {^o} - nahiral frequency and modal shapes
tal^ ta Eqs. (46)-{47) and (55) will be computed by Gauss quadiatare 1111 Th»
program for solving Eq. (58) was coded ta MaUab. quadiahire [11]. The
igW
Pham Tien Dal, Do Van Thom, Doan Trac Luat
.,. NUMERICAL RESULTS 3.1. Comparison study
3.1.1. Free uibration ofa simply supported homogeneous rectangular stiffemd plate with tied \ \ stiffeners ''\ \ \ A fully simply supported square plate havtag two cenfrally placed stiffener has been analyzed by L. X. Peng [12]. The plate and stiffener were made of flie same materiaL j wita Young's modulus 3.10' Pa, densily 2820 kg/m^", and Poisson's ratio 0.3 (Fig. 4). ThM first five nahiral frequencies of this stiffened plate were calculated by ustag tae present ] taeory. Tab. 1 shows calculation results compared witti fliose by mesh-fiee method of
Peng. A good agreement can be seen ta this table. j
Fi^. 4. The stiffened rectangular plate wita two sliffeneis Table I. Comparisons of frequencies for tae simply supported homogeneous
stiffened plate wita two stiffeners
3.1.2. A free vibration of square FG sandwich plate
and 2*\ti^^r;;(P^nr^"m':rr'*'''^-'^^'^^'^'^^
Em = 70 GPa, tiii. L 03 J ^ - 2 7 n ? ! T l T I'^P^'"^' ^'P^^" ^L. Hadji [11] are ke/m3forAl,0 Th- v '^" "; 2707 k g / m ' for Al; Ec = 380 GPa, Vc = 0.3,p, = 3800.
nfs'stt::LttS'tt^w:if;r^!r"'*^r^p'^^-«^^^^
(where PO=1 k„/^ ;;i-"p •,'!^?'^!r°"'''''"''"'^°^^
first-oX shear drfo™°.« ^ ' u™'='' "^^ *^^ P'"^* P^P^ ^ compared witti fte s h e a r d e L t T o n 5 : ™ X ' S ^ * - ^ f f S D T ) (analytical metaod), ^ e fttai-order
piare meory (rsDT) (analytical metaod), and ttie four-variable refined
I'reeolbnllonoffunctionMlygniiledsnntbinch plales uiithiiHffineis based an the third-order iiheardetorm^^ theory 113
plate taeory [11] (analytical mefcod) ta Tab. 2. This comparison once agam shows clearly that good agreements are obtataed.
Table 2. Comparisons of the dimensionless fiequency CD for simply support FG sandwich plate
FSDT [11]
1.82442
CD (thickness relation = 1-1-1) TSOT [11]
1.82445
Refined plate taeory [11]
TabU 3. Dimensionless frequency of stiffened FG sandwich plate tvilh one stiffener (a/b = l,b,= a/50, h, = lO/i, thickness relation =1:8:1)
0/(21,)
40
20
10
5
Dimensionless frequencv (cn) Boundanry
contlition n 0.5
1 2 10 0.5 1 2 10 0.5 1 2 10 0.5 1 2 10
SSSS First type
3.8447 3.6780 3.5991 3.5229 3.4174 3.3175 3.1793 3.1140 3.0512 2.9645 2.5497 2.4421 2.3914 2.3428 2.2763 0.8770 0.8726 0.8704 0.8683 0.8654
Second lype 5.3852 2.7504 2.6715 2.5920 2.4728 48090 2.7206 2.6427 2.5641 2.4464 3.7773 2.1815 2.1400 2.0985 2.0369 1.9684 1.5580 1.5268 1.4956 1.4492
CCCC First lype
6.6365 6.3388 6.1952 6.0551 5.8590 5.0946 4.8855 4.7859 4.6898 4.5567 3.6246 3.4710 3.3984 3.3288 3.2337 2.5530 2.4382 2.3841 2.3323 2.2621
Second lype 8.2091 4.1175 3.9998 3.8811 3.7032 7.4121 4.0213 3.9075 3.7929 3.6209 5.3731 3.0847 3.0244 2.9639 2.8739 3.6931 2.0938 2.0525 2.0110 1.9489
IU Pham Tien Dal, Do Van Thom, Doan Tiaclual
3.2. Free vibration of square sandwich FG plate ,^
3.2.1. Free vibration ofFG plate with one central stiffener - Effect of boundary condition and side-to-thickness ratio
A stiffened FG sandwich plate Si3N4/SUS304 wita a long, b -wide andThOag The material pioperties, as given ta Reddy and Chta [13], are £„, = 322 7 GPa v 0.2S,p„ = 2370 kg/m3 for SisNi; Ee = 207.79 GPa, v. = OSS.pe = 8166 i^/'rj'i SUS304, thickness relation =1:8:1. Tabs. 3 and 4 show the nondimensional nahiral;|
quendes of different side-hi-ttuckness ratio and volume fraction exponents wifll k boundary conditions, viz., all edges staiply supported (SSSS), all edges damped (CtSS two edges opposite simply supported and two edges opposite clamped (CSCS), and tu adjacent edges damped while tae otaer two edges simply supported (CCSS).
» f e 4. Dimensionless ftequency of stiffened FG sandwich plate wita one stifiener (a/S =l,b, = a/50,h, = lOft, thickness relation =1:8:1)
a/(2h)
40
20
10
Dimensionless frequency (a) Boundary
condition n
0.5 1 10 0.5 1 2 10
10 0.5
10
CSCS First lype
4.2496 4.0924 4.0177 3.9456 3.8457 3.6826 3.5461 3.4189 2.8807 2.7681 2.7146 2.6632 2.5928 0.8773 0.8728 0.8707 0.8685 0.8656
Second type 6.3919 3.7179 3.6466 3.5750 3.4682 5.6357 3.3235 3.2009 4.4068 2.6098 2.5619 2.5140 1.9696 1.8426 1.8068 1.7709
CCSS i First type
5.0234 4.7525 4.6195 4.4887 4.3042 4.0892 3.9068 3.7345 2.9509 2.8336 2.7'/71 2.7224 1.5559 1.5500 1.5467 1.5433
Second type 6.0410 3.0867 3.0023 2.9171 2.7892 5.5741 2.9433 2.8680 2.7917 2.6768 4.3743 2.4699 2.4171 i 2.3638 <
3.0524 1.7762 J 1.7400 1.7036
'''^'^raOenoflunetlomdlygradedsandundiplaesuMslgeneniliimlonthelhird-ordershaird.firmalionlheo^ 115 Remark: The result shows tt^t tae nahiral fimdamentid fiequendes decrease when tae power volume index mcreases. That as n decreases, tae ceramic ta plate decreases and so ttiat the ngidity of tae plate decreases, ta tae same conditions (power volume mdex st^d^to-thtdmiss ration), tae frequendes are highest for CCCC Stiffened FG sandwiA plales followed by CSCS, CCSS, and SSSS stiffened sandwidi plate while SSSS stiffened FG sandwidi plate has tae lowest value of frequendes. This is due to tae Wgher number of consti-ainls mtioduced ta CCCC stiffened plate compared lo fliat of S s £ CCSS and CbCb plales mcreasmg tae stiffness of tae plate. The frequendes are to be decreastae wita tae mcreasmg of ttie plate's fliidoiess, whidi because flie mass of plates is tacreased much more flian tae stiffness.
First four mode shape of stiffened FG sandwidi plate (type 1) shows ta Fig. 5.
(c) Mode shape 3 - 3936.380 Hz (d) Mode shape 4 - 4838.207 Hz
thickness relation = 1-JJ-l) ' ^ - Effect of core's thickness
•^^'-•"^shidytaeeffectofcore'sttucknessforabovestiffenednlAto Bi„ t the non-dteiensionless ft,=quendes wita differeni v a l u r o f t a e t ^ W ^ i ' ^^S-fP^^ents taidmess ratio hi/h and wita differeni values of tae voTume factit^n T n ^ ' "
ramie ridi), n = 0.5 (FGM), and „ = 10 (metal ridi) " " " ' ^ ' ' ' ' " = ° <'^'"
Remark: The results fiom Fig 6 show taat tae freouenries ar= t„ k - . . , tacreastagoffliecore'sttticknisofstiffenedplatert^ir;';—-f~
316 Pham Tien Dat, Do Van Thom, Doan'BvcmSb'...^ , „
tobe decreasing, which because the core's thickness is increa&ExJ^iat^ie^^raiBiiiHi lyi 1 is richer than (hat in ^ e 2, so that stiffened plate of type 1 bec@ines stiffener than t h ^ of type 2.
f "
1 •
1"
S 3
.
'-h—itdotcciXD -B—rpO.B(HAa^
-o-n=io(cqGq,
^ " • ^ " " — —
i
1
(bl-^pea Fty. 6. Flequendesvary to fti/li(Ss = a/50,h, = Wh.afb = l,a/(2A) = 10) - Effect ofstiffener's position
We stady tae effed of stiffener's position by varying tae ratio ;t/<i, where x i- ."•
mension fixim one edge of tae plate and showed m Fig. 7.
, , , , , , i i i i j , i , ,
Fig. 7. The rectangular plate w i * one stiffener
plate ^ s ^ T ^ ' ^ J T ^ " " * ™ ' ^ ^ -"hen tte stifcter is doser to c a i f a of tha ta^her ^ ^ '"^'"""^ "8^"='- =° that ttie corresptmding flajumaessare;
(lypeiffoiixroi^atK^ri^r''""'''^'^*'*°^"'''**^
Free Mfeiriion t^funOiondiy graced sandwidi plales with st^eners based on the ihtrd-OTder shear defomatimt theory
(^)fypel (b) Type 2
Fig. 8. Effects of stiffener's position on frequendes {fcs = B/50,fts = 10ft,«/fc^l,fl/(2ft) = 10}
(c) Mode shape 3 - 3216.887 Hz (d) Mode shape 4 - 4394.096 Hz
^SX??ri:^2^?ftf^5::^^^2^ss^sr!;^
one 1)ni!
Pham Tien Dat, De Van Thom, Doan })nc Luaf
- Effect ofstiffener's depth
Next, free vibration analysis of stiffened sandwich plate is carried out for ^ e dif- ference of stiffener's depth. The results from Fig. 10 show that when the deptii of stiSfen^.
increases, the frequency of the stiffened plate increases. ^ T !
—•—n-0{SSSSl - 1 — i i = o ( c c c q
—B—11=0.5 <CCCO - « — i p i o ( s s s s )
_-«—
" " " (
-ri
_ _ _ j.
(a) Type 1
OlypeZ Fig 10. Frequencies vaiy to )i./21i (typel,i, = a/X,hs = 10Jl,«/6 = l,„/(2),) = 10) - E^cf ofstiffener's width
ol frJU^l ^^'^l^ ^ °^ Stiffener's widtt on tae frequendes of tae plate. The resultt o^fretjuenaes depentang on value of tae widta are show on Fig. 11 We c a n ^ C 3 J.2. Free vibration of FGM plate with two stiffeners
tae frequendes are S ^ f e c C C c l i 1 ° ^ ^ , ^ ' " ^ ".""f- ^i-te-tottidaiess ration), CCSS, l i d SSSS s t i f f e n ^ n d i^, ^ ' ^ f ^"^ ^° sandwidi plates followed by CSCS, lowest f r e q S ^ e r 5 ^ i s ^ f e T * / ' ; ' \ " ' ^ ' ^ ^^^^ ='^™^^ ' ' ^ sandwidi plate has ta^
s t i f f e n e d p l t e ^ m p I l r d T o t t S ^ S ' S S t r T r * ^ ' " ' " ' ^ ^
tae plate. °'^*='=''-CSS CSCS plates ttiat mcreases tae sUftaess of i | FirstfourmodeshapeofstiffenedFGsandwidiplate(lypel)areshowstaHg.l2 I
•nefJnncllenallygnuledmidvrichphiteswithiMfmetsbasedonlhelhinl-onlersheardipw^
(a) Typel (b) Type 2
= a/50,h, = 10h,a/b = l,a/{2h) = 10) Fig. 31. Frequendes vary to a/b, (type 1, b.
Table 5. Dmiensionless frequency of stifUmed FGM sandwid, plate wilh two siiffenel (a/b = 1, i, = a/50,h, = 10ft, ttiickness lelation =1:8:1
Pham Tien Dat. Do Van Thorn, Doan Thic Liat
Tabie 6. Dimensionless frequency of stiffened FGM sandwich plate witii two stiJ (a/b ^l.h= a/50,bs = lOh, thickness relation =1:8:1)
a/(2h)
40
20
10
5
Dimensionless fiequency (is)) . [ Boundary
condition n 0.5 1 2 10 0.5 1 2 10 0.5 1 2 10 0.5 1 2
CSCS First type Second type
5.7643 5.6155 5.5433 5.4725 5.3723 4.4584 43419 4.2857 4.2310 41545 3.1250 30371 2.9948 2.9537 2.8969 0.8773 0.8728 0.8707 0.8685 10 1 0.8656
9.4821 5.2859 5.1347 4.9823 4.7538 7.5528 4.6961 4.6160 4.5353 4.4147 5.2927 3.3491 3.2956 3.2422 3.1631 1.9696 1.8619 1.8550 1.8477 1.8358
CCSS First type
5.6178 5.4666 5.3929 5.3204 5.2173 4.3655 4.2496 41936 41390 4.0626 2.9684 2.8925 2.8555 2.8194 2.7690 1.5422 1.5360 1.5327 1.5293 1.5242
Second type 8.9041 ; ' 4.7606 f 4.6312 4.5002 4.3026 7.3151 4.3954 4.3028 4.2071 4.0584 5.1027 3.2354 3.1828 3.1301 3.0517 3.1982 2.1050 2.0702 2.0352 1.9830
f'"'^'lionoffunetlomillygralMsmiliiM,,diiUsunlh,lllenen,hasedoneielhint-ordersheard^ormMinlheory
(c) Mode shape 3 - 2621.551 Hz (d) Mode shape 4 - 4837.733 Hz
Fig 12. Fust four linear mode shape of simply support Si3N,/SUS304 sandwidi rettangular plate (type 1) wift two stiffener (a/h = l,a/h=20
n = 0.5, ii/ii, = 5Q,h, = mh, thickness relation = 1-8-1)
4. CONCLUSIONS
h,^ ™ ! r f , ! ' jf ^ ' ^ f 8 ^ ? * ' ' *•"= * ' " ' ' " ' ' ' * ' > ™ ' °f saff'^-'^d FG sandwich plates Z ^ fi ^ ° ' ^ " ' ' ^ ' I ' f o ^ a ' i ' ' " and ustag tae finite elemeni metaod Two plate configurations i.e., plate wita FG face-sheets and tae homogenous cole are meTa"
a^H ^ ' ^ ' T ' ^ ^ ^ P ' f " ^ l y - are considered. Tlie present result, are c L p a r e d t o r a l v M
^ d mesh-free metaod results given by otaer researdiers to demonstrate a g o S aSee ment. Some problems sudi as tae effects of widta depta position of stiffener taickXof layeis, p o w ^ volume tadex, boundaiy conditions on tae'L.hiral f r e q ^ n c S S e n e d FG sandwidi plate have been mvestigated. Based on taese observation., i b l T fu j be recommended for analysis of stiffened FG sandw™ p l a i e T o t r ; ft fr ™ and mode shapes wita sufficient accuracy ^ ° P"^'"'' * " frequencies
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S e c t i o n and sttesses. IntenmhomI loumai of Solids and Stmctures, 42, (18), (2005), pp. 5224- [9] M. M. Alipour and M. Shaiiyat An clastidty-equilibrium-based zigzag fteorv for axisvm-
metac bendmg and stiess analysis of tae fanctionany graded d r e u l a S d « ^ 7 p k t e r X I a^Madaunn-type series solution Euwpean Journal ofMechanics-Affolids, 3 ^ poU) pp 7^;
" ^ L ' ; ? J " f ""• ""• ? ^ " - * ™ " "^angular finite elemeni formulation based on Wdier [11] L. Hadji, H. A. Atmane, A. Tounsi I Mechah anH F A A R=^- n •-,. •