CC BY
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Математика. Физика
УДК 539.3
LAMINATED COMPOSITE SHELL IN CYLINDRICAL BENDING G.M. Kulikov1, A.A. Mamontov2, S.V. Plotnikova1, A.V. Erofeev2
Departments: “Applied Mathematics and Mechanics” (1),
“Construction of Buildings and Structures” (2), TSTU; kulikov@apmath. tstu.ru
Key words and phrases: cross-ply composite; cylindrical shell; plane strain elasticity; sampling surfaces method.
Abstract: This paper presents an efficient method of solving the plane strain problem of elasticity for laminated composite cylindrical shells. The method is based on the new concept of sampling surfaces (SaS) proposed recently by the authors. According to this concept, we introduce inside the nth layer In not equally spaced SaS parallel to the middle surface of the shell and choose displacements of these surfaces as shell unknowns. Such choice of displacements allows the derivation of strain-displacement relationships, which exactly represent all rigid-body shell motions in any curvilinear surface coordinate system. The latter gives in turn the opportunity to find the solutions of plane strain elasticity for thick and thin laminated cylindrical shells with a prescribed accuracy by using a sufficiently large number of SaS, which are located at layer interfaces and Chebyshev polynomial nodes.
1. Introduction
A conventional way for developing the higher order shell formulation consists in the expansion of displacements into power series with respect to the transverse coordinate, which referred to the direction normal to the middle surface of the shell. For the approximate representation of the displacement field, it is possible to use finite segments of power series because the principal purpose of the shell theory consists in the derivation of approximate solutions of elasticity. Such a way has been extensively utilized for development of the higher order layer-wise shell models accounting for thickness stretching [1-3]. However, the implementation of this approach for thick laminated shells is not easy to realize, since it is necessary to retain a sufficiently large number of terms in corresponding expansions to obtain the comprehensive results.
More efficient approach for the analysis of laminated shells can be achieved through the use of a SaS method. This method has been recently proposed by the authors for homogeneous shells [4, 5] and laminated plates [6]. In the present paper, we extend the SaS method to laminated composite shells to solve the plane strain problem
of elasticity. As SaS denoted by Q(n)1, Q(n)2,..., Q(n)In, we choose outer surfaces and any inner surfaces inside the nth layer and introduce displacement vectors u(n)1, u(n)2,..., u(n)In of these surfaces as fundamental shell unknowns, where In is the total number of SaS selected for each layer (In > 3). The index n identifies the belonging of any quantity to the nth layer and runs from 1 to N, where N is the number of layers. Such choice of displacements with the consequent use of Lagrange
polynomials of degree In -1 in the thickness direction for each layer allows the derivation of strain-displacement equations, which exactly represent all rigid-body motions of the shell in any convected curvilinear coordinate system.
Unfortunately, the above polynomial interpolation in the thickness direction implemented for equally spaced SaS [4-6] does not work properly with Lagrange polynomials of high degree because of Runge's phenomenon [7]. However, the use of Chebyshev polynomial nodes [8] can help to improve significantly the behaviour of Lagrange polynomials of high degree for which the error will go to zero as In ^ ro . This fact gives an opportunity to derive the elasticity solutions for thick laminated shells with a prescribed accuracy employing a sufficient number of SaS located at layer interfaces and Chebyshev polynomial nodes.
Consider a thick laminated shell of the thickness h. Let the middle surface Q be described by orthogonal curvilinear coordinates 91 and 02, which are referred to the lines of principal curvatures of its surface. The coordinate 93 is oriented along the unit vector a3 = e3 normal to the middle surface. Introduce the following notations: r = r(91,02) is the position vector of any point of the middle surface; aa are the base vectors of the middle surface given by
aa = r,a = Aaea , (1)
where ea are the orthonormal base vectors; Aa are the coefficients of the first
fundamental form; 93n)in are the transverse coordinates of SaS of the nth layer expressed as
03n)1 =03n-1], 03n )In =93n],
03n)mn = _2(03n-1] +03n]) -1 hn cos
. (2)
. 2(In - 2) V '
where 93n-1] and 93n] are the transverse coordinates of the bottom and top surfaces
Q[n-1] and Q[n] of the nth layer (Fig. 1) such that 93°] = -h /2 and 03N] = h /2;
hn = 93n] - 93n-1] is the thickness of the nth layer; R = r + 93e3 is the position vector of
any point in the shell body; R(n)in = r + 93n )in e3 are the position vectors of SaS of the nth layer; gi are the base vectors in the shell body defined as
ga = R,a = Aacaea, g3 = R,3 = e3 , (3)
where ca = 1 + ka93 are the components of the shifter tensor; ka are the principal
curvatures of the middle surface; gi(n)in are the base vectors of SaS of the nth layer given by
g(an)in = R(a)in = Aacan)in ea, g<n)in = e3, (4)
where can)in = 1 + ka93n)in are the components of the shifter tensor at SaS. Here and in the following developments, (...); stands for the partial derivatives with respect to coordinates 9i ; the index mn identifies the belonging of any quantity to the inner SaS of the nth layer and runs from 2 to In -1, whereas the indices in, jn, kn describe all SaS of the nth layer and run from 1 to In; Greek tensorial indices a, p range from 1 to 2; Latin tensorial indices i, j, k, l range from 1 to 3.
Fig. 1. Geometry of the thick laminated shell
It should be mentioned that transverse coordinates of inner SaS (2) coincide with the nodes of Chebyshev polynomials [7]. This fact has a great meaning for a convergence of the SaS method.
2. Three-dimensional kinematic description of laminated shell
A position vector of the deformed shell is written as
R = R + u,
(5)
where u is the displacement vector, which is always measured in accordance with the total Lagrangian formulation from the initial configuration to the current configuration directly. In particular, the position vectors of SaS of the nth layer are
R(n)in = R(n)in + u(n)in , (6)
u(n)in = u(03n)in ), (7)
where u(n )in (9j, 02) are the displacement vectors of SaS of the nth layer (Fig. 2). Due to continuity conditions, we have
u(1)1 = u[0], u(N) JN = u[N
u(m)Im = u(m+1)1 = uIm]
(8)
where u[m] (9j, 02) are the displacement vectors of layer interfaces Q[m]
(m = 1,2,...,N -1).
The base vectors in the current shell configuration are defined as
& = R,i = gi + u,i.
In particular, the base vectors of deformed SaS of the nth layer are
8T n = g3(03n)in) = e3 +P
(n)in = R(n)in = g(n)in + u(n)in g(n)in = g (e(n)in ) = + R(n)in
: R">ln = gv AV,a но
P(n)in = U3(e3n)!n),
(9)
(10) (11)
Fig. 2. Initial and current configurations of the shell
where p(n)in (0j, 02) are the values of the derivative of the 3D displacement vector with respect to coordinate 93 at SaS.
The Green-Lagrange strain tensor in an orthogonal curvilinear coordinate system [4] can be written as
^ ( ( - gi ‘gj), (12)
where A3 = 1 and c3 = 1. In particular, the Green-Lagrange strains at SaS are
9P(n)in = 2p (9(n)in ) =_________1_(g(n)in g(n)in - g(n)in g(n)in )
iJ iJ( 3 ) . . (n)in (n)in Vgi gJ gi g J r
(13)
)in ' J J
ij ^ j
A-A с(n)inc(n)in
Substituting (4) and (10) into the strain-displacement relationships (13) and discarding the non-linear terms, one derives
2c(n)in = 1 u(n)in . e„ I 1 u(n)in . e
ap A c(n)in “a eP + A c(n)in ,P ea,
2e(an3)in =p(n )in . ea +^— u(a)i'n • e3, 83^ = p(n >n . e3. (14)
A sX "n aa
Next, we represent the displacement vectors u(n)in and p(n)in in the reference
surface frame ei as follows:
u(n )in = £ u(n)'n ei, (15)
i
p(n)in =^p(n)i'n ei. (16)
i
Using (15) and well-known formulas [9] for derivatives of unit vectors ei with respect to orthogonal curvilinear coordinates 9a, we have
A- uain =zx(a)in ei, (17)
Aa i
where
^ = A-uaa + Baupn)in + kau3n)in for
Aa
A-(pndin = A~ upa- Ba«an)in for
Aa
^ = Ar “3Sn - ku^. (18)
Aa
Inserting (16) and (17) into the strain-displacement relationships (14), we arrive at the following equations:
2p(n)in = 1 1(n)in + 1 1(n)i«
ap c(n)in ap c(n)in pa ,
cp ca
2p(n)in =R(n )in + 1 i(n )in P(n )in = R(n)in (19)
2ba3 _ Pa + (n)in A3a , b33 _ P3 . (19)
3. Displacement and strain distributions in thickness direction
Up to this moment, no assumptions concerning displacement and strain fields have been made. We start now with the first fundamental assumption of the proposed higher order layer-wise shell theory. Let us assume that the displacements are distributed through the thickness of the nth layer as follows:
u(n) = £ L(n)inu(n)in, 03”-1] <03 < 03n], (20)
where L(n)in (93) are the Lagrange polynomials of degree In -1 expressed
as
л(п)jn
L(n)in = n 03 93 (21)
M 9(n)in -9(n)Jn • 1 )
Jn *}n 3 3
The use of relations (11), (16) and (20) yields
p(n)in = ^M(n)Jn (93n)in )u(n)]n, (22)
jn
where M(n)Jn = L(n1 n are the derivatives of Lagrange polynomials. The values of these derivatives at SaS are calculated as
9(n)in 9(n)k
9( ) n - 9( ) n
M(n)Jn (93n)in )= ( 1 ( ). n 9(3). 3 )k for Jn * in,
V3 ! 0(n)Jn -0(n)in ; 1 1 . 0(n)Jn -0(n)kn n
3 3 kn *in,1 n 3 3
M(n)in (93n)in )=- ^M(n)]n (93n)in ). (23)
J n *in
Thus, the key functions p(n)in of the proposed layer-wise shell theory are represented according to (22) as a linear combination of displacements of SaS of the nth layer
u(n) n . ui .
n
The following step consists in a choice of the correct approximation of strains through the thickness of the nth layer. It is apparent that the optimal solution of the problem is to choose the strain distribution, which is similar to the displacement distribution (20), that is,
Strain-displacement relationships (14) and (24) are invariant under rigid-body motions of a laminated shell in any curvilinear surface coordinate system. The idea of a proof can be found in [10, 11]. The ability of proposed strain-displacement relationships exactly represent all rigid-body shell motions admits the development of exact geometry solid-shell elements [12, 13]. The term “exact geometry” reflects the fact that the parametrization of the reference surface is known and, therefore, coefficients of the first and second fundamental forms of the surface can be taken exactly at each element node.
Inserting strains (24) in the total potential energy of a laminated shell and introducing stress resultants [4]
at bottom and top surfaces Q[0] and Q[N]; p[0] and piN] are the loads acting on bottom and top surfaces of a shell; WE is the work done by external loads applied to the boundary surface E.
For simplicity, we restrict ourselves to the case of linear elastic materials. The natural choice for constitutive equations is the generalized Hook’s law:
Substituting stresses (27) in (25) and accounting for the strain distribution (24), we have
(24)
4. Total potential energy of laminated shell
(25)
one derives
Q L n 1n h j 1
w
J
where 4° = 1 + ^a®3°] and caN] = 1 + ^a03N] are the components of the shifter tensor
(27)
k ,i
where
(28)
(29)
5. Exact solution for laminated cylindrical shell
Let us consider a cylindrical bending of a simply supported laminated cylindrical shell of the radius R subjected to the sinusoidally distributed load acting on the top surface
p3N] = P0sinm&2, 02 e [0,0*], (30)
where 02 is the circumferential coordinate of the middle surface.
The analytical solution of the problem satisfying the boundary conditions can be written as
M|n)'n = 0, «2n)!n = 4n>n cosm02, u^^'n = «30^ sinm02. (31)
Substituting (30) and (31) in the total potential energy (26) with WE = 0 and
allowing for relations (18), (19) and (28), one finds
n = n(w2n)!n, «30^ ). (32)
Invoking the principle of the minimum total potential energy, we arrive at the system of linear algebraic equations
dr) . = 0, . = 0 (33)
s«2Sn ^n
of order 2l ^In - N +1 . The linear system (33) can be solved by using a method of
V n )
Gaussian elimination.
The described algorithm was performed with the Symbolic Math Toolbox, which incorporates symbolic computations into the numeric environment of MATLAB. This gave the possibility to derive the exact solutions of plane strain elasticity for laminated composite cylindrical shells with a prescribed accuracy.
The geometrical and mechanical parameters of a three-ply cylindrical shell are taken to be R = 10, m = 3, 0* = n /3 and El = 25Et, ^lt = 0,5Et, ^tt = 0,2Et,
Et = 106, v LT = v TT = 0,25, where subscripts L and T refer to the fiber and transverse directions of the ply, and L-direction coincides with 01-direction. Here, we study a shell
with the lamination scheme hn = h /3 and [90o /0° /90° ]. To compare the results derived with Ren’s exact solution of the plane strain elasticity [14], the following dimensionless variables are introduced:
U3 = 100ETh3w3(n/6, z)/p0R4, Sn = 100h2CT11(n/6, z)/p0R2,
S22 = h2CT22(n/6, Z)/p0R2, S23 = 10hCT23(0, z)/p0R,
S33 = CT33(n/6, z)/p0, z = 03/h. (34)
The data listed in Tables show that the SaS technique allows the derivation of exact solutions for thick and thin shells with a prescribed accuracy using a sufficient number of SaS. Figs. 3 present the distribution of stresses through the thickness of the shell for different values of the slenderness ratio R / h by choosing 9 SaS for each layer. These results demonstrate convincingly the high potential of the proposed layer-wise shell formulation. This is due to the fact that boundary conditions on the bottom and top surfaces and continuity conditions at layer interfaces for transverse stresses are satisfied precisely without integration of the equilibrium equations of elasticity, i.e. only constitutive equations (27) are applied. It is important that the enhanced SaS method provides the uniform convergence that is impossible with equally spaced SaS [4-6].
Table 1
Results for a thick three-ply cylindrical shell
In U3(0) £„(- 0,5) Sn(0,5) S22(- 0,5) £22(0,5) £23(0) £33(0,5)
R/h = 2
3 13,853 - 5,2469 7,7902 - 2,9025 2,3088 3,7576 0,8770
5 14,360 - 3,8054 8,6232 - 3,4761 2,4618 3,353 0,9858
7 14,357 - 3,4951 8,7140 - 3,4672 2,4631 3,9352 1,0001
9 14,357 - 3,4709 8,7140 - 3,4670 2,4631 3,9352 1,0001
11 14,357 - 3,4678 8,7133 - 3,4669 2,4631 3,9352 1,0000
13 14,357 - 3,4672 8,7132 - 3,4669 2,4631 3,9352 1,0000
Ren [14] 14,36 - 3,47 8,71 - 3,467 2,463 3,94 1,0000
R/h = 4
3 4,5060 - 2,3348 2,7340 - 1,7382 1,3483 4,7085 0,8868
5 4,5816 - 1,7830 2,9208 - 1,7717 1,3670 4,7574 0,9945
7 4,5815 - 1,7730 2,9301 - 1,7714 1,3670 4,7573 1,0004
9 4,5815 - 1,7717 2,9296 - 1,7714 1,3670 4,7573 1,0001
11 4,5815 - 1,7715 2,9296 - 1,7714 1,3670 4,7573 1,0000
Ren [14] 4,57 - 1,77 2,93 - 1,772 1,367 4,76 1,0000
R/h = 100
3 0,78578 - 0,79062 0,77808 - 0,78661 0,77907 5,2258 0,3973
5 0,78578 - 0,78656 0,78162 - 0,78656 0,77912 5,2341 0,9986
7 0,78578 - 0,78656 0,78162 - 0,78656 0,77912 5,2341 1,0002
9 0,78578 - 0,78656 0,78162 - 0,78656 0,77912 5,2341 1,0000
Ren [14] 0,787 - 0,79 0,78 - 0,786 0,781 5,23 1,0000
0,5
z
0,25
-0,25
-0,5
100 l/l 0/4 ^^2 0,5 z 0,25 moo
10|p4 /2 2 A #■-10
100 0 _ (
т=Шлъ ti ll 100 l -0,25 -0.5 [ О ( 4 R/h=2
-5
5 Su 10 -4 -2 0 2 S22 4
a)
Fig. 3. Distribution of stresses through the thickness of the three-ply shell:
--------present analysis; O - Ren’s solution [14];
a - S22
0,5
z
0,25 (р Ф
Г 44-
R/h=Z\ 0 *100 10Ф<-
А Ф с
-0,25 R/h=2
, ~—"1 -0.5 1 1
О 2 4 6 S23 8 -20 -15 -10 - 5 0 S33 5
b)
Fig. 3. (continue):
b - S23, ^33
6. Conclusion
An efficient method of solving the plane strain problem of elasticity for laminated composite shells has been proposed. It is based on the new technique of SaS located at the Chebyshev polynomial nodes inside the shell body and layer interfaces as well. The stress analysis of composite shells is based on the complete constitutive equations and gives an opportunity to obtain the exact solutions of plane strain elasticity for thick and thin laminated cylindrical shells with a prescribed accuracy.
This work was partially supported by Russian Ministry of Education and Science under Grant No 1.472.2011.
References
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Цилиндрический изгиб слоистой композитной оболочки Г.М. Куликов1, А.А. Мамонтов2, С.В. Плотникова1, А.В. Ерофеев2
Кафедры: «Прикладная математика и механика» (1),
«Конструкции зданий и сооружений» (2), ФГБОУ ВПО «ТГТУ»; kulikov@apmath. tstu.ru
Ключевые слова и фразы: метод выборочных поверхностей; перекрестно армированный композит; плоская деформация; цилиндрическая оболочка.
Аннотация: Статья представляет эффективный метод решения плоской задачи теории упругости для слоистых композитных цилиндрических оболочек. Метод основан на новой концепции выборочных поверхностей 8а8, предложенной авторами в предыдущих работах. Согласно этой концепции внутри п-го слоя вводятся ^ произвольным образом расположенных выборочных поверхностей, параллельных срединной поверхности оболочки. В качестве искомых функций выбираются перемещения этих поверхностей. Такой выбор перемещений дает возможность получать деформационные соотношения, которые представляют точно движение оболочки как жесткого тела в системе криволинейных поверхностных координат. Это в свою очередь дает возможность находить решение плоской задачи теории упругости для толстых и тонких слоистых цилиндрических оболочек с заданной точностью, используя достаточно большое число 8а8. которые размещаются на поверхностях раздела слоев и в узловых точках полинома Чебышёва.
Zylindrische Biegung der geschichteten Kompositumhillung
Zusammenfassung: Im Artikel ist die wirksame Methode der Losung der flachen Aufgabe der Theorie der Elastizitat fur die geschichteten zylindrischen Kompositumhullungen vorgelegt. Die Methode ist auf der neuen von den Autoren vor
kurzem vorgeschlagenen Konzeption der stichprobenartigen Oberflachen (SaS) gegrundet. Laut dieser Konzeption werden innerhalb der n-Schicht die in einer willkurlichen Weise angeordneten SaS parallel der Mitteloberflache der Hulle eingefuhrt und als gesuchte Funktionen werden die Umstellungen dieser Oberflachen gewahlt. Solche Auswahl der Umstellungen ermoglicht es, die Deformationsverhaltnisse zu bekommen, die genau die Bewegung der Hulle wie des harten Korpers im System der krummlinigen oberflachlichen Koordinaten vorlegen. Es ermoglicht seinerseits, die Losung der flachen Aufgabe der Theorie der Elastizitat fur die dicken und feinen geschichteten zylindrischen Hullen mit der aufgegebenen Genauigkeit zu finden, verwendend die genug grofte Zahl von SaS, die auf den Oberflachen der Abteilung der Schichten und in den Knotenpunkten des Polynoms von Tchebyschew aufgestellt werden.
Courbure cylindrique de l’enveloppe feulletee composite
Resume: L’article presente une methode efficace de la solution du probleme plane de la theorie de l’elastisite pour les enveloppes cylindriques feuiletees composites. La methode est fondee sur une nouvelle conception des surfaces selectives (SaS), proposee par les auteurs. Selon cette conception a l’interieur de la couche n sont introduites les SaS In situees librement paralleles a la surface medianne de l’enveloppe et en qualite des fonctions recherchees sont choisis les deplacements de ces surfaces. Un tel choix de deplacements donne la possibilite de recevoir les relations de deformation qui presente de la maniere precise le mouvement de l’enveloppe comme corps solide dans le systeme des coordonnees superficielles curvilignes.
Авторы: Куликов Геннадий Михайлович - доктор физико-математических наук, профессор, заведующий кафедрой «Прикладная математика и механика»; Мамонтов Александр Александрович - аспирант кафедры «Конструкции зданий и сооружений»; Плотникова Светлана Валерьевна - кандидат технических наук, докторант кафедры «Прикладная математика и механика»; Ерофеев Александр Владимирович - аспирант кафедры «Конструкции зданий и сооружений», ФГБОУ ВПО «ТГТУ».
Рецензент: Ярцев Виктор Петрович - доктор технических наук, профессор, заведующий кафедрой «Конструкции зданий и сооружений», ФГБОУ ВПО «ТГТУ».
