Vietnam Journal of Mechanics, VAST, Vol.36, No. 3 (2014), pp. 201-214
A N A N A L Y T I C A L A P P R O A C H T O ANALYZE N O N L I N E A R D Y N A M I C R E S P O N S E O F E C C E N T R I C A L L Y S T I F F E N E D F U N C T I O N A L L Y
G R A D E D C I R C U L A R C Y L I N D R I C A L SHELLS S U B J E C T E D T O T I M E D E P E N D E N T AXIAL C O M P R E S S I O N A N D E X T E R N A L P R E S S U R E . P A R T 1: G O V E R N I N G E Q U A T I O N S E S T A B L I S H M E N T
D a o Van D u n g ^ Vu Hoai Nsun^*
^ Hanoi University of Science, VNU, Vietnam University of Transport Technology, Hanoi, Vietnam
*E^mail: [email protected] Received December 25, 2013
A b s t r a c t . Based on the classical thin shell theory with the geometrical nonlinearity in von Karman-Donnell sense, the smeared stiffener technique and Galerkin method, this paper deals with the nonlinear dynamic problem of eccentrically stiffened functionally graded circular cylindrical shells subjected to time dependent axial compression and external pressure by analytical approach. The present novelty is that an approximate three-term solution of deflection taking into account the nonhnear buckUng shape is cho- sen, the nonlinear dynamic second-order differential three equations system is established and the frequency-amplitude relation of nonlinear vibration is obtained in explicit form.
Keywords: Functionally graded material, discontinuous reinforcement, bucklmg, elastic- ity, analytical modelling.
1. I N T R O D U C T I O N
Many authors studied the static buckling and postbuckling of FGM cylindrical shells subjected to the mechanic and thermal loading. Shen [1,2] investigated the non- linear postbuckling of thin FGM cyhndrical shells and FGM hybrid cyhndrical shells in thermal environments under lateral pressure and axial loading, respectively. Bahtui and Eslami [3] investigated the coupled thermo-elasticity of FGM cylindrical shells. Batra and laccarino [4] presented the exact solutions for radial deformations of a functionally graded isotropic and incompressible second-order elastic cylinder. Huang and Han [5-7] studied the buckling and postbuckling of un-stiffened FGM cyUndrical shells under axial compres- sion, radial pressure and combined axial compression and radial pressure based on the Donnell shell theory and the nonlinear strain-displacement relations and the nonlinear
SOS Dao Van Dung, Vu Hoai Nam
three term solution form is used. Sun et al. [8] proposed the accurate symplectic space solutions for thermal buckling of functionally graded cylindrical shells. The postbuckhng of shear deformable FGM cyUndrical shells surrounded by an elastic medium was studied by Shen [9]. Dung and Hoa [10] investigated the nonlinear torsional buckling and post- buckling of eccentrically stiffened functionally graded thin circular cylindrical shells. Liew et al. [11] studied postbuckling responses of functionaUy graded cylindrical sheUs under axial compression and thermal loads. Sofiyev [12] analyzed the buckling of FGM circular shells under combined loads and resting on the Pasternak type elastic foundation. The non-linear static buckling of FGM conical shells which is more general than cyUndrical shells, were studied by Sofiyev [13,14]. Torabi et al. [15] studied the linear thermal buck- ling analysis of truncated hybrid FGM conical shells.
For dynamic analysis of FGM cylindrical shells, Singh et al. [16] investigated tor- sional vibrations of functionally graded finite cylinders. Darabi et al. [17] presented respec- tively Unear and nonlinear parametric resonance analyses for un-stiffened FGM cylindri- cal shells. Three-dimensional vibration analysis of fluid-filled orthotropic FGM cylindrical shells was investigated by Chen et al. [18]. Sofiyev and Schnack [19] and Sofiyev [20]
obtained critical parameters for un-stiffened cylindrical thin shells under linearly increas- ing dynamic torsional loading and under a periodic axial impulsive loading by using the Galerkin technique together with Ritz type variation method. Sofiyev [21-24] and Deniz and Sofiyev [25] investigated the vibration and dynamic instability of FGM conical shells.
Torsional vibration and stability of functionally graded orthotropic cylindrical shells on elastic foundations is presented by Najafov et al. [26]. Sofiyev and Kuruoglu [27] investi- gated the torsional vibration and buckling of the cylindrical shell with functionally graded coatings surrounded by an elastic medium. Tornabene and Viola [28] studied free vibra- tion analysis of functionally graded panels and shells of revolution. Huang and Han [29]
presented the nonlinear dynamic buckling problems of un-stiffened functionally graded cylindrical shells subjected to time-dependent axial load by using the Budiansky-Roth dynamic buckling criterion [30]. Various effects of the inhomogeneous parameter, loading speed, dimension parameters; environmental temperature rise and initial geometrical im- perfection on nonlinear dynamic buckling were discussed. Dynamic analysis of thick short length FGM cyUnders was investigated by Asemi et al. [31].
In engineering design, plates and shells are usually reinforced by stiffeners for the benefit of added load carrying capability with a relatively small additional weight. How- ever, the investigation on this field has received comparatively little attention. Najafizadeh et al. [32] have studied linear static buckling of FGM cylindrical sheU under axial compres- sion reinforced by FGM stiffeners. Bich et al. [33-36] investigated the nonlinear static and dynamic analysis of FGM plates, cylindrical panels, shallow shells and cyUndrical shells with eccentrically homogeneous stiffener system. Dung and Hoa [37] presented an analyt- ical study of nonlinear static buckling and post-buckling analysis of eccentrically stiffened functionally graded circular cylindrical shells under external pressure with FGM stiffen- ers and approximate three-term solution of deflection taking into account the nonlinear buckhng shape.
To best of authors' knowledge, there is no analytical approach on the nonlinear dy- namic analysis of stiffened FGM shells subjected to time dependent external pressure and
An analyttcal approach to analyze nonlinear dynamic response of eccentmcaily stiffened . . 203 axial compression by analytical approach. In addition, the nonlinear three term solution of deflection is popular used to investigate the nonlinear static analysis of sheU [5-8 and 37], but there are a mathematical difficulty on the nonlinear dynamic analysis. This paper studies the dynamic behavior of stiffened FGM cylindrical circular shells under mechanic loads. The nonlinear dynamic equations are derived by using the classical sheU theory with the nonlinear strain-displacement relation of large deflection, the smeared stiffeners technique and Galerkin method. The three-term solution of deflection is used and the firequency-amplitude relation of nonlinear vibration is obtained in explicit form.
2. E C C E N T R I C A L L Y S T I F F E N E D F G M CYLINDRICAL SHELLS (ES-FGM C Y L I N D R I C A L SHELLS)
An ES-FGM cylindrical shell as shown in Fig. 1 is assumed to be thin with length L, mean radius R, reinforced by homogeneous ring and stringer stiffener systems. Stiffener material is similarly designed with Refs. [33-36] is full ceramic if it is located at ceramic-rich surface and is pure-metal if is located at metal-rich surface. The origin of the coordinate 0 locates on the middle plane of the shell, x,y = R9, z axes are in the axial, circumferential, and inward radial directions, respectively.
'W^"^^^''
Fig. I. Geometric and the coordinate system of an eccentrically stiffened FGM cylindrical shell Functionally graded material is assumed to be made from a mixture of ceramic and metal with the simple power law exponent of volume fraction distribution
K = K W = ( ^ ) ' v„. = v„W = i-K.W.
where h is the thickness of shell; *: > 0 is the volume fraction index; z is the thickness coordinate and varies from - f t / 2 to ft/2; the subscripts m and c refer to the metal and ceramic constituents, respectively.
20.i Dao Van Dung, Vu Hoai Nam
Effective properties Preff of fimctionally graded material are determined by linea rule of mixture
Pr.ff = Prm{z)V^iz) + Prc{z)Vc(z).
The Yoimg's modulus and mass density can be written by according to the men-
tioned law , E{z) = E^V^ + EeVc = -E^ + {E, - £m) ( ^ ^ )
p{z) = p ^ y ^ -H p , K = p^ + (Pc - Pm) ( ^ ^ )
and the Poisson's ratio u is assumed to be constant for simplicity. According to the von Karman nonlinear strain-displacement relations of cylindrical shell [38], the mid-surface strain components are
du 1 f dw\ f, dv w \ {dw
^' dx'^2[dx) ' '" dy R'^2\dy/ ^ j Q du dv dw dw d^w d^w d^w
where e^ and e^ are normal strains, 7^^ is the shear strain at the middle surface of shell, Xx, Xy^ Xxy are the change of curvatures and twist of sheU, and u — u{x,y),v = v{x,y), It; ^ ty (x, y) are displacements along x, y and z axes, respectively. The strains components can be written as the form
£^ = EI- ZXX, £y = £y- ZXy, 7xy = Jxy - '^ZXxy (3) The deformation compatibility equation is deduced firom Eq. (2)
d^
^ _ ? ^ „_ 1 ^
( d^w\^ _d^d^
dy"^ dx^ dxdy Rdx"^ \dxdy) dx^ dy^' Hook's stress-strain relation for the shell is presented
. f = ^ f e + . . , ) , < ^ ^ ( s „ + . e j . r - = j f ^ 7 , „ , (5) and for stiffeners
o-f = E^e^, af = ErSy, (6) where Es,Er are Yoimg's modulus of stringer and ring stiffeners, respectively.
Taking into account the contribution of stiffeners by the smeared stiffeners technique and omitting the twist of stiffeners [38] and integrating the stress-strain relations and their moments through the thickness of shell, the force and moment of an ES-FGM cylindrical sheU are
JVx = [A,^ + : ^ ~ ) 4 + A,^4 - {Bn + C.)x.- B,iXy,
Ny = Auel + (AII + ^ ^ \ 4 - Biix, - (Bn + C.) Xy, ' ' ' Hiy = Ami" - 2BmXxt
An analyttcal approach to analyze nonlinear dynamic response of eccentrically stiffened ... 205
M. = (iJl, + C) 4 + B,24 -(^Dn + ^'jx.- D,ixy,
My = BIIEI + (B22 + a ) 4 - DiiX. - (D22 + ^ \ Xy, ' ^ ^ Afa,j, = B667S„ - 2De,iXxy,
wheieA„,B,j,Dij(i,j = 1,2,6) are extensional, coupling and bending stif&iess of the un- stiffened FGM cylindrical shell. They are defined as
l-v' 1-1/2 "" 2{l + i/)
A l = .22 = J ^ , . t . ^ ^ , . . = ^ ,
in which
\ " k + 1 ) ' ' 2(k + l){k + 2)-
^3=[^ + ( ^ . - ^ . ) ( ^ - ^ ^ ^ ) ] . 3 ,
Is='-§^A.zl Ir^'-§^A.zl (10)
_ hs + h _ hr + h 2-KR L Ss = 1 Sr — ,
ris nj.
where
Ca and Cr are negative for outside stiffeners and positive for inside ones.
Ss and Sr are the spacing of the stringer and ring stiffeners, respectively.
dg, hs and dr, hr are the width and thickness of the stringer and ring stiffeners, respectively.
As, Ar are the cross-section areas of stiffeners.
h,!r,Zs, Zr are the second moments of cross section areas and the eccentricities of stiffeners with respect to the middle surface of shell, respectively.
Es,Er are Young's modulus of stringer and ring stiffeners, respectively.
From the Eq. (7), one can obtain inversely
tl = A\.^N:, - A\^Ny + B\^Xx + B\2Xy^
£% = AliNy - Al2N^ -i- B^.Xx + B'22Xy^ (11) lly = AlQN:,y + 2Bl^Xxy,
Dao Van Dung, Vu HtMi Nam
where
If. B.A.\ „ I f . , ErAr']
. Ai2
- ^ ) (
^22 + - ^ — ^ ) B'n = A'ii{Bn+C.)-AliBi2, Bii = A'n (Bi2 + Cr) - AI1B12, Bt2 = A'22Bl2 - AI2 {B22 + Cr) , Bii = ^ ! l B l 2 - ^12 (Bn + C,), R. _ BaaSubstituting Eq. (11) into Eq. (8) leads to
where
D'n = Dn + - ^ - ( S n + C,) Bl^ - B^B^^, -^22 = -^22 H B12BI2 — {B22 + Cr) B22, -^12 ^ -^12 — (-Sll + Cs) BI2 — B12B22, -^21 — -^12 ~ B12B11 — {B22 + Cr) B21, DKR = Dm — BeeBcf:.
(12)
M . = Bl^N^ + Bi,Ny - Dlix^ - D'i2Xy,
My = BJ2JV, + B^iNy - DJiXi - D'22Xy, (13) Mxy = B^gN^y - 2D'^Xiy,
(14)
The nonlinear equations of motion of a thin circular cylindrical shell based on the d u d V
assumption u -tiw and w -C w, Pi-^-j —* Oi P^~a~2 ~^ ^ ^^'^' 19,39] are given by
dx dy ' dx dy '
8 ^ 9 ^ a ^
s^,,^ +
A f ^ +dx^ dxdy ^ dy^ ^''dx^^'^dxdy^ 'dy^^
1 S^w dw +-Ny+gc=Pi-g^ + 2p,„-^,
(15)
An a n a l y t i c a l a p p r o a c h to analyze nojUtnear dynamic response of eccentrically stiffened . 207
where /i is the linear damping coefficient, and
/" , X , As Ar f Pc- Pm\ A , Ar Pl = j p{z)dz + P s — + P r ~ = [ p m + ^ ^ ) h - V P s ~ + P r — , (16)
with pr = Pm-, Ps — Pm. if stiffoners are full metal and pr — Pc, ps = Pc if stiffeners are fuU ceramic. The first two of Eq. (15) are identically satisfied by introducing a stress function
AT = ^ i V = ^ AT - - ^ (17)
' dy^' ' dx^' " ' » " dxdy- ^ ' Substituting Eq. (11) into Eqs. (4) and (13) and the third of Eq. (15), taking into account Eqs. (2) and (17), we have
-*" a ? + K e - 2^.2) g^^jg^i + ^22 aji + B21 a ? +
d'w e*w 1 a^M / a^m y o - m o - m | + (iJ,-, + Bii - 2B'„) g - 5 ^ + Bii-g-j + -i^-g^-\ [g^) - gj2 aJT = «.
(18) dw d^h.
-2pl^l.-^+D{^^r-i + {Dl2 + Dil+ Wit) -,
* ar"" "" &T ^ ^"'2 ^ "" " ""'='" fcW
1 aV a^i^a^m a y a^'m aya^m
~ S a ? " ap'a?"'" axaya!:a!/ ai^ ai/2 * ^ '
Eqs. (18) and (19) are a nonlinear governing equation system. They are used to investigate the dynamic characteristics of ES-F<3M circular cylindrical shells.
3. N O N L I N E A R D Y N A M I C B U C K L I N G ANALYSIS Suppose that an ES-FGM cylindrical shell is simply supported and subjected to uniformly distributed pressure of intensity go and axial compression of intensity ro respec- tively at its cross-section (in N/m^). The boundary conditions of this study are
TO = 0, M I = 0, N^ = -rah, N^y = 0, at i = 0, L. (20) Assume the buckling mode shape is represented by the popular form |6-8,39|, the simply supported boundary condition Eq. (20) is fulfilled on the average sense
viTTX . ny . . 2 "^^^ / o n to = /o + / i s i n - ^ s m - ^ - h / 2 S i n ' ' - j - , (21) where /o = fo{t) is time dependent unknown uniform deflection of pre-buckling state,
/ l = / i C ) is time dependent unknown linear buckling deflection, /2 = /2(() is time dependent unknown nonlinear deflection, m is numbers of half waves and n is numbers of full wave in axial and circumferential directions, respectively.
Dao Van Dung. Va Hoai Nar,
Substituting Eq. (21) into Eq. (18) and solving obtained equation, leads to 2nnTX 2ny . mirx . ny ip = (pi cos — h 'P2 cos — fs sm —^ sin -—+
Sm-Kx . ny + ip4 sm —-:— s m —— troyft
L H 2 '
Vl --
n^X^ . (4AL - lUBiimV) ill - ^ , o . . ^,..1 ^ A . '^2 = i2A',,m'iT^ 32Ar,m%2
Vi = -jfi H 2 •''•''' '^^ " G A = AlimV + {Aia - 2A'i2) m^n^jr^A^ -I- .4^2"''-^^.
L 2
(22)
(23)
B = B J i m * ! ' + (Bn -1 B ; 2 - "^B^) m'n'n^X^ + Bl2n'X* - -^ni'ir', H D = Diim'ir* + ( D ^ + J^Ji + 4^26) m'^nVx' + Diin^X*, G = 8l7i;imV^ -1 9 (Al^ - 2JCy2) m^nVx' + Aiin*X*,
Substituting the expressions (21) and (22) into Eq. (19) and then applying Galerkin method lead to
(. D D ^^fo r>Pl<^^f2 o n ^/o ID
.oyh = R,o - Rpi^ - R ~ ^ - 2Rpi^- - Rpi ri <i'^h „rA 'Vl fr, B^\, f mV n*\''\^,
[ 2 B n A i V A 2 Ti2A2(AL-4B2'imV)"
dfi
(24)
(25)
hh+
(26)( ^^ + ^ j / i / l - m^TT^L^hrofi - uoyhn^L^X^fi = 0,
/m,n\^ / T i \ 2 " ) ,
( x ) (fl) }^'+
' ' ' ^ + 2 ' ' " ' d r + ' " 4 * ^ + 2''"'dr+
n^A^
^ /m7r\4 1 /m7r\2
^ H ~ r J " ^ R I T J J leAJ^m^TrS ' 2 A
.l.V.^A^(r^)^(^)^(i-i);J;2.
.{-n(f)^-[«2-i(f)^4(^)T^^j^
-r„ft(I^)V2 + t ^ = ,o.
(27)
An analytical approach to analyze nonlinear dynamic response of eccentrically stiffened . For circular cyUndrical shell, the circiunferential closed condition must be considered 9,39]
J Jry'^^y-J J
0 0 0 0
From Eqs. (11), (17), (21) and (22), this integral becomes
-2A;i<To„ft + 2A;2roft +\(h + 2/o) " J ( ^ ) ' / ? = 0- EUminating aoy from Eqs. (25)-(27) and the condition of closed form (29),
(29)
^ + ^"f) + K l ^ + ^"f) + - ' (^^ + ^^») - "-^? - - 3 * + «u- = 0,
dt2
"f)
0 2 1 , f<ef2^^ dfl\+ ^22/1 + 023/1/2 + Q24/f 4- Oiisfifi - aieqofi - a27rofi = 0, ( ^ -1 2 ^ ^ j + au!l + a32/?/2 -f 033/2 - a34ro/2 = 0,
(30) (31)
(32)
" 2Al,R'pi' " 8 / i ; i B V i
1 |"2BirAiVA^ _ n^A" (AX - /Blim^Tr^)
1 .4*2ft 013 = — , o i 4 ^ T r " D — • Pi A'liRpi
1 / m''7r'' ' Upl \ KA'22
Rn^X^
n*X*\
• i V i '
n-'n-'Tr'A'' m V x ^ A
031
"32
^ LVi • " i V i •
=^{h(^)^-^(T)1i^-l(x)^(i)T
033 = 1 {ia.;i [^r - h (X)^ - ^ (x)l ^^ip3^}'
4 , /mTTN
(33)
(34)
(35)
Dao Van Dung, Vu Hoai Nam
SimpUfying Eqs. (30)-(32), we have
(36)
( ^ + 2 M ^ ) + 022/1 + ftl/1/0 + ft2/i/2 + 025/1/1 + & / f - A 4 / i r o = 0, (37) (38) where
1 „ 1 „ 1 o 1
fti = 2oii,/3i2 = -031 4-ai2,ft3 = rQ32,A4 = On - - 0 3 3 , f t s = 2°34, (39) fei = - f t l Q 2 l , Ihl = -/3i402i - 0 3 3 - r - -F 023,
fts = ft2021 - 0 3 1 — -i- Q24, hi = 027 + Q14O2I.
(40)
Denoting / = uJmax, from Eq. (21), the maximal deflection of the shells
/ = /o + / i + /2, (41) locates at X ^ — , y = where i,j are odd integer numbers. Note that /o = /o(0>
2m 2n / i = flit). /2 = flit) and / = fit) in Eq. (41).
From Eqs. (36)-(38) and (41), the effects of input parameters on the dynamic re- sponse of shells are investigated.
3.1. Nonlinear vibration anedysis
This section considers an ES-FGM cylindrical thin shell under uniformly lateral pressure QQ = QsinQt and ro ^ 0, Eqs. (36)-(38) become
^ -b 2 M ^ ) + 011 fo - 012f! - 0uf!f2 + 0uf2 - auQ sin Qt ^ 0, (42)
{IF ^ ^^f) ^ "^^^' ^ '^''^'^'+^^^-^^=°' '^^
where Q is excitation force and fl is excitation frequency. From these equations, nonlinear response of ES-FGM shell is investigated by using the fourth order Runge-Kutta iteration method.
It is difficult to determine the fundamental frequencies of natural vibration, frequency- amplitude relation of nonlinear vibration of shell. In this paper, ones can be investigated by ignoring the uniform buckling shape and nonlinear buckling shape, Eq. (31) becomes
ilF "^ ^ ^ ^ ) ^ ""'^^^ + ''''•'"' ^ " 2 6 / i Q s i n n t = 0. (45)
An anaiytical approach to analyze noriiinear dynamic response of eccentrically stiffened .
For the free and linear vibration without damping, the Eq. (45) becomes
(46) The fundamental frequency of natural vibration can be determined by
^mn = V^- (47) where Umn is fundamental frequency of natural vibration of sheU.
Using the solution / i (t) = 77 sin (fit) and applying the procedure like Galeridn method to Eq. (45), the frequency-ampUtude relation of nonlinear vibration is obtained
n^ - -pn - 022 + 7024^^ - ^ a 2 6 Q . (48) TT 4 37r
By introducing the non-dimension firequency parameter ^ = , Eq. (48) becomes e 2 _ J ^ ^ = 1 + 3 ^ , 2 8 ^
TTLOmn 4 ljJ^„ 37r LJ^in
The frequency-amplitude relation of nonlinear free vibration Q = 0 is determined by
^ 2 _ J ^ j ^ l + ! i | l , 2 . (50)
noJmn '*^mn
3.2. Buckling analysis
3.2.1. Linear static buckhng analysis of ES-FGM cylindrical shells
Omitting the linear damping, uniform buckling shape, nonlinear buckling shape and putting / i = 0, / i = 0, and taking / i / 0 Eq. (SI) becomes
^22 + 0 2 4 / 1 ~ "2690 - 027^0 = 0. (51)
By ignoring the nonlinear term of A and rg - 0 in Eq. (51), the linear upper static buckling load of ES-FGM cylindrical shells under ordy external pressure can be determined by
022 , ,
qsbu = • (52)
Similarly, the linear upper static buckhng load of ES-FGM cylindrical shells under only axial compression (50 = 0) leads to
r,i,„ = ^ . (53)
Q27
FVom Eqs. (52)-(53), the linear static critical buckling loads of shells are determined by r,„ = minr^ku V(m,n) and q,or = ming,i,„ V(m,n).
212 Dao Van Dung, Vu Hoai Nam 3.2.2. Dynamic buckling analysis of ES-FGM cylindrical shells
The nonlinear dynamic critical buckling analysis of ES-FGM circular cyUndrical shells based on Eqs. (36)-(38), is investigated for two load types as foUows.
Firstly, ES-FGM cyUndrical shell is subjected to only lateral pressure varying as linear function of time qo = Cqt in which Cq (N/m^s) is the loading speed of external pressure.
Secondly, ES-FGM cylindrical shell is subjected to only axial compression varying as linear function of time ro — Cj-i where Cr (N/m^s) is the loading speed of axial compression.
Eqs. (36)-(38) are the nonlinear second-order differential three equations system.
This equation system may be numerically solved.
4. C O N C L U S I O N S
A formulation of governing equations of eccentrically stiffened functionally graded circular cylindrical thin shells subjected to time dependent axial compression and external pressure based upon the classical shell theory and the smeared stiffeners technique with von Karman-Donnell nonlinear terms is proposed in this paper. An approximate three- term solution of deflection taking into account the nonlinear buckling shape is used. The nonlinear dynamic equations of ES-FGM circular cylindrical shells are obtained by using the Galerkin method. Fundamental frequency of natural vibration, frequency-amplitude relation of nonlinear vibration and upper static buckling loads are obtained in expUcit forms. Dynamic responses will be numerically investigated and nonlinear dynamic buckUng loads will be determined by applying Budiansky-Roth criterion in next part.
A C K N O W L E D G E M E N T S
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2013.02.
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