Calculation of the Three-layer Shell Taking into Account Creep

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ScienceDirect Procedia Engineering 165 (2016) 990 – 994

15th International scientific conference “Underground Urbanisation as a Prerequisite for Sustainable Development”

Calculation of the three-layer shell taking into account creep Anton Chepurnenko a,*, Levon Mailyana, Batyr Jazyeva a

Don State Technical University, Sotsialisticheskaya, 162, Rostov-on-Don, 344022, Russia

Abstract We obtained resolving equations of finite element method to calculate the three-layer shells taking into account the creep of the middle layer. The outer layers of the shell are considered elastic and isotropic. We investigated the influence of curvature of a shell on the growth of deflection due to creep. The example of calculation is given for a spherical shell, hinged along the contour and loaded by uniformly distributed load. It was found that with increasing of curvature of the shell influence of creep on deflection has reduced. 2016Published The Authors. Published by Elsevier Ltd. © 2016 by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 15th International scientific conference “Underground Peer-review under scientific committee of the 15th International scientific conference “Underground Urbanisation as a Urbanisation as aresponsibility Prerequisite of forthe Sustainable Development. Prerequisite for Sustainable Development Keywords: three-layer shallow shell; creep; strain energy; finite element method.

1. Introduction There are lot of publications devoted to calculation of sandwich plates and shells, including [1-11]. But most of these calculations are usually conducted exclusively in the elastic formulation. However, the foams used in the sandwich structures as the middle layer, in addition to the elastic properties inherent viscosity [11, 12]. Therefore, to adequately describe the stress-strain state of sandwich shells it is necessary to involve the creep theory [13,14]. Approximate methods of calculations of the sandwich plate taking into account creep of the middle layer are provided in [12]. In this paper the finite element analysis of three-layer shells is considered. The shell is represented as a set of flat triangular elements. Finite element used is shown in Fig. 1. Each node in the element has 5 degrees of freedom: movements of the upper plating ui , v i ; movements of the lower face ui ,

* Corresponding author. Tel.:+7-918-571-87-38. E-mail address: [email protected]

1877-7058 © 2016 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 15th International scientific conference “Underground Urbanisation as a Prerequisite for Sustainable Development

doi:10.1016/j.proeng.2016.11.810

Anton Chepurnenko et al. / Procedia Engineering 165 (2016) 990 – 994

vi and deflection w . The thickness of faces t  и t  we consider small compared to the thickness of the shell. Furthermore, we accept that the outer layers transmit the normal and tangential forces in their plane, and the middle layer works only on shear.

Fig. 1. The three-layer triangular finite element.

2. Theory of calculation Movements within the element can be expressed in terms of nodal displacements as follows:

u ( )

N1u1( )  N 2 u2( )  N 3u3( ) ; v ( )

N1v1( )  N 2 v2( )  N 3v3( ) ;

N1w1  N 2 w2  N 3 w3 , 1 where N i (ai  bi x  ci y ) – form functions; A – area of the element. 2A a1 x2 y3  x3 y2 ; a2 x3 y1  x1 y3 ; a3 x1 y 2  x2 y1; b1 y 2  y3 ; w

b2

y3  y1; b3

y1  y 2 ; c1

x 3  x 2 ; c2

x1  x3 ; c3

x2  x1 ,

where xi , yi – coordinates of the nodes. Deformations of elements are defined as follows [15]: H x (  )

wu  (  ) wx

1 (b1u1(  )  b2 u 2(  )  b3u 3(  ) ); 2A

H y (  )

wv (  ) wy

1 (c1v1(  )  c 2 v 2(  )  c 3 v 3(  ) ); 2A

J xy(  )

wu (  ) wv (  )  wy wx

1 (c1u1(  )  c 2 u 2(  )  c 3u 3(  )  b1v1(  )  b2 v 2(  )  b3 v 3(  ) ); 2A

Jm zx

u   u  ww  wx h

b1 w1  b2 w2  b3 w3 N 1 (u1  u1 )  N 2 (u 2  u 2 )  N 3 (u 3  u 3 )  ; h 2A

Jm zy

v   v  ww  h wy

c1 w1  c 2 w2  c 3 w3 N 1 (v1  v1 )  N 2 (v 2  v 2 )  N 3 (v 3  v 3 )  . h 2A

Or in matrix form:

^H` [B] ˜ ^U `, where {H} {H x H y J xy H x H y J xy J mzx ^U ` ^{U1} {U2 } {U3}`T ; {Ui } {ui vi ui vi wi } .

Strain energy is written in the form of:

T Jm yx } ;

991

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Anton Chepurnenko et al. / Procedia Engineering 165 (2016) 990 – 994

П

1               ³ t (V x H x  V y H y  W xy J xy )  t (V x H x  V y H y  W xy J xy )  2A

m m m m m  h [W m zx ( J zx  J zx )  W zy ( J zy  J zy )]dA

where ^N `T

{N x

N y

 N xy

{H } {0 0 0 0 0 0

Jm zx

N x

N y

T Jm zy }

1 T

³ ^N ` ({H}  {H })dA, 2A  N xy Qzx

Qzy } - vector of the internal forces in the element;

- vector of creep deformations.

The relationship between the internal forces and deformations has the form:

^N `

[ D]({H}  {H }),

(1)

ª[ D  ] º « »  [D ] where [ D ] « » — matrix of elastic constants; « [ D m ]» ¬ ¼ 0 º ª1 Q ª1 0º E (  ) t (  ) « ». m (  ) [ D ] Gm h « ] 0 Q 1 » ; [D » 1  Q 2 «« ¬0 1¼ ¬0 0 (1  Q) / 2»¼ Substituting (1) into the expression for the strain energy and then applying the principle of minimum total energy, we obtain a system of linear algebraic equations: [ K ]{U } {Fq }  {F *},

where [ K ]

T T

³ [ B] [ D][ B]dA – stiffness matrix; {F } ³ >B@ dA ˜[ D] ˜ {H } – contribution of creep deformations in A

A

the vector of nodal loads; {Fq } – vector of external nodal loads. The expressions for the stiffness matrix and load vector are written in the local-coordinate system. To compile the FEM system of equations it is necessary for each element to convert to global coordinates. 3. Results and discussion The calculation was made for a spherical shell, shown in Fig. 2. The shell is loaded with a uniformly distributed load, with the following initial data: the faces elasticity modulus E = 2 ∙ 105 MPa, the Poisson's ratio of faces ν = 0.3, thickness of faces t = 1.5 mm, shell thickness h = 80 mm, the shear modulus of the middle layer Gm 2.5 MPa , the shell radius in the plan r = 8 m.

Fig. 2. The spherical shell hinged along the contour.

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Anton Chepurnenko et al. / Procedia Engineering 165 (2016) 990 – 994

Creep law was taken in the form of:

Gm J i

t

W i  ³ W i K (t  W)dW, i

( xz , yz ),

(2)

f

where Wi – the shear stress in the middle layer; J i – shear deformation of the core. The creep kernel was taken in the form of:

K (t  W ) C m e D m (t W ) , C m

Dm

0.077

1 . hour

With this core creep law (2) appears in the differential form: wJ i wt

cm W i  D m J i , Gm

where J i – shear creep deformation of the middle layer. The differential form of the law of creep allows calculation by step method, determining the creep deformation in the next time by using a linear approximation [16-20]. The calculation was performed for different values of the radius of curvature of the shell. Fig. 3 shows graphs of growth of deflection in relation to the bending deflection at the initial time w(t ) / w(0) at the point A with the following values of R: a – R = 112 m; b – R = 224 m; c – R = 336 m; d– R = 672 m; e – R = ∞. The figure shows that an increase in the curvature of the shell reduces the creep effect. For the plate (at R = ∞) ratio was 1.34, and for R = 112 m deflection due to creep increased only in 3%.

Fig. 3. Deflection growth charts.

4. Conclusions From obtained results it follows that for the sandwich shells of greater curvature creep of filler has no influence on the deflection. However, in solution we considered the linear theory of creep, which is valid only at low stresses. Therefore it is necessary to conduct a similar study using a non-linear creep law, such as Maxwell-Gurevich equation. References [1] V.A. Kovalenko, A.V. Kondratyev, The use of polymeric composite materials in rocket and space technology as a reserve to increase its mass and functional efficiency, Aerospace technics and technology. 5 (2011) 14-20. [2] D.V. Leonenko, Radial natural vibrations of elastic three-layer cylindrical shells, Mechanics of machines, tools and materials. 3 (2010) 12-15.

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[3] V.N. Bakulin, Non-classical refined models in the mechanics of sandwich shells, Journal of the Nizhny Novgorod University of N.I. Lobachevsky. 4 (2011) 1989–1991. [4] A.V. Zemskov, V.A. Pukhliy, A.K. Pomeranskaya, D.V. Tarlakovskiy, Calculation of stress-strained state of sandwich shells of variable rigidity, Bulletin of the Moscow Aviation Institute. 1 (2011) 45–48. [5] V.F. Kirichenko, Existence of solutions to a problem related to thermoelastic-away of sandwich shells, Proceedings of the universities. Mathematics. 9 (2012) 66–71. [6] S.N. Sukhinin, Matematicheskoe i fizicheskoe modelirovanie v zadachakh ustoychivosti trekhsloynykh kompozitnykh obolochek [Mathematical and physical modeling in problems of stability of three-layer composite shells], Vestnik Nizhegorodskogo universiteta im. N.I. Lobachevskogo [Bulletin of the Nizhny Novgorod university of N.I. Lobachevsky]. 4 (2011) 2521–2522. [7] Ya.M. Grigorenko, A.T. Vasilenko, On some approaches to the construction of the specified models of the theory of anisotropic shells of variable thickness, Mathematical methods and physical-mechanical fields. 7 (2014) 21 – 25. [8] V.N. Bakulin, Effective models for proximate analysis of deformed state-balanced, three-layered non-axisymmetric cylindrical shells, Reports of the Russian Academy of Sciences. 5 (2007) 613 – 617. [9] A.A. Smerdov, Fan Tkhe Sean, Design analysis and optimization composite bearing shells, News of higher educational institutions. Engineering industry. 11 (2014) 90-98. [10] V.N. Bakulin, Construction of ap-proximations and models for investigation of stressed-stained state of layered not- axisymmetric shells, Mathematical modeling. 19 (2007) 118−128. [11] M. Garrido, J. Correia, F. Branco, Creep behavior of sandwich panels with rigid polyurethane foam core and glass-fibre reinforced polymer faces: Experimental tests and analytical modeling, Journal of Composite Materials. (2013) 21–28. [12] B.M. Jazyev, A.S. Chepurnenko, S.V. Litvinov, S.B. Jazyev, Calculation of three-layer plates using finite element method taking into account the creep of the middle layer, Bulletin of the Dagestan State Technical University. 33 (2014) 47–55. [13] Yu.N. Rabotnov, Creep of structural element, Moscow, Nauka, Moscow 1966. [14] L.M. Kachanov, Creep theory, Fizmatgiz, Moscow, 1960. [15] A.S. Vol'mir, Flexible plates and shells, Publishing House of Technical and theoretical literature, Moscow, 1956. [16] V. I. Andreev, B. M. Yazyev, A. S Chepurnenko, On the Bending of a Thin Plate at Nonlinear Creep, Advanced Materials Research. 900 (2014) 707-710. [17] V.I. Andreev, The stability of the polymer rods at creep, Mechanics of composite materials. 1 (1968) 22. [18] A.S. Chepurnenko, V.I. Andreev, B.M. Jazyev, Energy method for calculation of the stability of compressed polimer bars with creep, Vestnik MGSU. 1 (2013) 101–108. [19] V.I. Andreev, B.M. Jazyev, A.S. Chepurnenko, Axisymmetric bending of a flexible circular plate under creep, Vestnik MGSU. 5 (2014) 16– 24. [20] M.Ju. Kozel'skaja, A.S. Chepurnenko, S.V. Litvinov, Calculation of the stability of the compressed polymer rods the effects of temperature and vysokoela-plastic deformation, Scientific and Technical Gazette of Volga region. 4 (2013) 190–194.