MATEC Web of Conferences 117, 00005 (2017)
DOI: 10.1051/ matecconf/201711700005
XXVI R-S-P Seminar 2017, Theoretical Foundation of Civil Engineering
Operational and Variational Formulations of Boundary Problems of Anisotropic Plate Analysis, Adapted for Application Within Discrete-Continual Methods Pavel Akimov1,2,*, Vladimir Sidorov3,4,5, Zbigniew Wójcicki6 1
Russian Academy of Architecture and Construction Sciences; 24-1, ul. Bolshaya Dmitrovka, Moscow, 107031, Russia 2 Scientific Research Center “StaDyO”; office 810, ul. 3ya Yamskogo Polya, Moscow, 125040, Russia 3 Kielce University of Technology; Al. 1000-lecia Państwa Polskiego 5, 25-541, Kielce, Poland 4 Moscow State University of Railway Engineering; 9b9 Obrazcova Street, Moscow, 127994, Russia 5 Perm National Research Polytechnic University; 29 Komsomolsky prospekt, Perm, Perm Krai, 614990, Russia 6 Civil Engineering Faculty, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
Abstract. This paper is devoted to operational and variational formulations [1] of boundary problems of anisotropic plate analysis (with the use of so-called method of extended domain [2]), adapted for applications within discrete-continual methods (discrete-continual finite element method (DCFEM), discrete-continual variation-difference method (DCVDM)) [1]. Generally, the field of application of these methods, which are now becoming available for computer realization, comprises structures with regular (in particular, constant or piecewise constant) physical and geometrical parameters in some dimension. Considering problems remain continual along “basic” direction while along other directions discretecontinual methods presuppose mesh approximation.
1 Introduction As is known, structures from anisotropic materials are widely used in modern construction. Anisotropic materials normally have sharp difference in the elastic properties for different directions. A classic vital sample here is natural wood. Its elastic modulus in tension parallel to the grain is well above corresponding modulus in tension across the grain. Besides, elastic constants of natural wood depend on the direction with respect to the wood grains. Anisotropic (and inhomogeneous) materials are also include composite materials (particularly rather popular in aircraft engineering), crystals, some rocks, concrete, etc. Here we indicated materials with “natural” anisotropy. However, materials with so-called “artificial” anisotropy are applied in construction as well. We should mention here as a samples corrugated plates and shells from isotropic material or rib*
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© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).
MATEC Web of Conferences 117, 00005 (2017)
DOI: 10.1051/ matecconf/201711700005
XXVI R-S-P Seminar 2017, Theoretical Foundation of Civil Engineering reinforced plates and shell. Thus we have to consider boundary problems of anisotropic plate analysis [3-6].
2 Standard operational and variational formulations of boundary problem within method of extended domain Operational formulation of the problem in the domain has the form: 2
2
B 2 j
i 1
i, j
i2 w 41 2 B3,31 2 w
j 1
2
2 ( Bi , 31 2 wk 1 2 Bi ,3 wk ) c w q M Q , 2 i
(1)
2 i
i 1
where is the domain, occupied by structure with boundary ; is extended domain, embordering ; x1 , x2 are coordinates in plan; x 3 is coordinate of plate thickness; ( x1 , x 2 ) is the characteristic function of domain ; ( x1 , x 2 ) is the delta-function of border ; n [ n1 n 2 ]T is unit normal vector of domain boundary ; ( x1 , x 2 ) is corresponding normal derivative; w is plate deflection in domain ; q is the load density in domain ; Q , M are shear force and torque moment at the border ; c is modulus of foundation; k / xk , k / xk , k 1, 2 ;
c c ;
( x1 , x 2 ) ( x1 , x 2 )/n ;
(2) (3)
B1,1 , B2, 2 , B3,3 , B1, 2 B2,1 , B2,3 B3, 2 , B1,3 B3,1 are constants (parameters) that characterize the elastic properties of structure;
B1,1 B1,1 , B1,3 B1,3 , B2,3 B2,3 , B1, 2 B1, 2 , B3,3 B3,3 ; B1, 2 B1, 2 , B3,3 B3,3 , B2, 2 B2, 2 ; (4) and in the particular case of an isotropic plate, we have
B1,1 B2, 2 D ; B1, 2 B2,1 D ; B3, 3 0.5 D(1 ) ; B1,3 B2,3 B3,1 B3, 2 0 . (5) Internal moments and plate forces are determined by formulas
M 1 B1,1 1 B1, 2 2 B1,3 12 ; M 2 B1, 2 1 B2, 2 2 B2,3 12 ;
(6)
M 12 M 21 B1,3 1 B2,3 2 B3,3 12 ;
(7)
N1 B1,11 1 3B1,3 2 1 ( B1, 2 2B3,3 )1 2 B2,3 2 2 ;
(8)
N 2 B1,31 1 ( B1, 2 2B3,3 ) 2 1 3B2,31 2 B2,2 2 2 ;
(9)
where 1 , 2 , 12 21 are strains for x3 1 ,
1 12 w ; 2 22 w ; 12 21 21 2 w 2
(10)
MATEC Web of Conferences 117, 00005 (2017)
DOI: 10.1051/ matecconf/201711700005
XXVI R-S-P Seminar 2017, Theoretical Foundation of Civil Engineering Variational formulation of the problem is formulated in the form of the corresponding energy functional (this formulation is convenient in the algorithmic sense): 1 2
Ф( w)
[ (M 1
1
M 2 2 2M 1, 2 1, 2 ) c w2 ]dx1dx2
qwdx dx 1
2
.
(11)
3 Operational and variational displacement formulations of boundary problem within method of extended domain If we combine (6)-(10) we can get formulation (1) in the following form
Lw F ,
(12)
where L is so-called stiffness operator,
L 12 B1,112 22 B1, 212 12 B2,1 22 22 B2, 2 22 41 2 B3,31 2 212 B1,31 2 2 22 B2,31 2 21 2 B1,312 21 2 B2,3 22 c ; F q Г M Г Г Q.
(13) (14)
We can rewrite (11) in matrix form: Ф( w)
1 2
[( N , ) c w ]dx dx qwdx dx 2
1
2
1
2
,
(15)
where
N [ M1 M 2 M1, 2 M 2,1 ]T ; [ 1 2 1, 2 2,1 ]T .
(16)
Using (6)-(9), we get
N A ,
Bw ,
(17)
where
B1,1 B A 1, 2 B1, 3 B1, 3
B1, 2 B2, 2 B2,3 B2,3
0.5 B1,3 0.5 B2, 3 0.5 B3,3 0.5 B3,3
0.5 B1,3 12 2 0.5 B2, 3 ; B 2 ; 0.5 B3,3 2 1 2 0.5 B3,3 21 2
(18)
Thus we have: Ф( w)
1 2
[( B A Bw, w) c w ]dx dx qwdx dx
2
1
2
1
2
,
(19)
where
B1,1 B A A 1, 2 B1,3 B 1,3
B1, 2 B2, 2 B2,3 B2,3
0.5 B1,3 0.5 B2, 3 0.5 B3,3 0.5 B3,3
0.5 B1,3 0.5 B2,3 ; B [ 12 22 2 21 2 21 ] . 0.5 B3, 3 0.5 B3, 3 3
(20)
MATEC Web of Conferences 117, 00005 (2017)
DOI: 10.1051/ matecconf/201711700005
XXVI R-S-P Seminar 2017, Theoretical Foundation of Civil Engineering
4 Operational and variational formulations of boundary problem in displacements with extraction of basic direction Without loss of generality we suppose constancy of physical and geometrical parameters of plate along coordinate x 2 (it is so-called “basic direction”). We can rewrite (12) in the following form:
L L4 42 L332 L2 22 L1 2 L0 с ,
(21)
where
L4 B2, 2 ; L3 2B2,3 1 2 1 B2,3 ; L2 B1, 212 12 B2,1 41B3,31 ; L1 212 B1,31 21B1,312 ; L0 12 B1,1 12 . (22) We can rewrite (21):
L4 42 w L332 w L2 22 w L1 2 w ( L0 с )w F .
(23)
Let us introduce the following notation
y1 y1 ( x1 , x2 ) w( x1 , x2 ) ;
yi yi ( x1 , x2 ) i2 1w( x1 , x2 ), i 2, 3, 4 .
(24)
If we combine (23) and (24) we get:
L4 2 y4 L3 y4 L2 y3 L1 y2 ( L0 с ) y1 F .
(25)
Uniting (24) and (25) we obtain
1 0 0 0
0 1 0 0
0 0 y1 0 0 0 y2 0 1 0 y3 0 0 L4 y4 L0 с
1 0 0 L1
0 1 0 L2
0 y1 0 0 y2 0 , 1 y3 0 L3 y4 F
(26)
where
yi( x1 , x2 ) 2 yi ( x1 , x2 ), i 2, 3, 4 .
(27)
Thus we have the following system of the first-order ordinary differential (with respect to x2 ) equations with operator coefficients
0 1 y1 y2 0 0 y 0 0 3 1 1 с L L L L1 ( ) y 4 4 4 0
0 1 0 L4 1 L2
0 y1 0 0 y2 0 1 y3 0 1 L4 L3 y4 L4 1F
(28)
or
~ ~ U LU F , where
4
(29)
MATEC Web of Conferences 117, 00005 (2017)
DOI: 10.1051/ matecconf/201711700005
XXVI R-S-P Seminar 2017, Theoretical Foundation of Civil Engineering
0 1 ~ 0 0 L 0 0 1 1 L ( L с ) L L 4 1 4 0
0 1 0 L 41L2
0 0 0 ~ 0 ; F ; 1 0 1 1 L4 L3 L 4 F
2 y1 y1 y1 y y y2 U ; U 2U 2 2 2 . y 2 y 3 y 3 3 y y y 2 4 4 4
(30)
(31)
Equations (29), of course, should be supplemented with boundary conditions given in cross-sections with coordinates x2b,k , k = 1, ...,nk This boundary conditions have form Bk U k 1 ( x2b,k 0) BkU k ( x2b,k 0) g k g k , k = 2, ...,nk 1 ;
B1U1 ( x2b,1 0) Bn kU nk 1 ( x2b,nk 0) g1 gn k ,
(32) (33)
where Bk , Bk , k = 2, ...,nk 1 , B1 , Bn k are matrices of boundary conditions of the fourth order; gk , gk , k = 2, ...,nk 1 , g1 , g n k are right-side vectors of boundary conditions. Operators (18) and (21) can be rewritten in the following form:
B 12 B2, 0 1 2 B1,1 22 B0, 2 ; B 12 B2 , 0 1 2 B1 ,1 22 B0 , 2 ,
(34)
B2, 0 [ 1 0 0 0 ]T ; B1,1 [ 0 0 2 2 ]T ; B0, 2 [ 0 1 0 0 ]T ;
(35)
B2 , 0 [ 1 0 0 0 ] ; B1 ,1 [ 0 0 2 2 ] ; B0 , 2 [ 0 1 0 0 ] .
(36)
where
Thus, we have: ( B A Bw, w) (( 12 B2 , 0 2 1 B1 ,1 22 B0 , 2 ) A (12 B2 , 0 1 2 B1,1 22 B0 , 2 ) w, w) (12 B2 , 0 A 12 B2 , 0 w, w) (12 B2 , 0 A 1 2 B1,1 w, w) (12 B2 , 0 A 22 B0 , 2 w, w) ( 2 1 B1 ,1 A 12 B2 , 0 w, w) ( 2 1 B1 ,1 A 1 2 B1,1 w, w) ( 2 1 B1 ,1 A 22 B0, 2 w, w) ( 22 B0 , 2 A 12 B2 , 0 w, w) ( 22 B0 , 2 A 1 2 B1,1 w, w) ( 22 B0 , 2 A 22 B0 , 2 w, w) ( B0 , 2 A B0 , 2 y3 , y3 ) ( B0 , 2 A 1 B1,1 y 2 , y3 ) (1 B1 ,1 A B0 , 2 y3 , y 2 ) ( B0 , 2 A B2 , 0 12 y1 , y3 ) (12 B2 , 0 A B0 , 2 y3 , y1 ) (1 B1 ,1 A B2 , 0 12 y1 , y2 ) (12 B2 , 0 A 1 B1,1 y 2 , y1 ) (1 B1 ,1 A B1,11 y2 , y 2 ) (12 B2 , 0 A B2 , 0 12 y1 , y1 ),
and finally
~ ~ ( B A Bw, w) ( B0 , 2 A B0, 2 y3 , y3 ) ( B0 , 2 A B1,1 y2 , y3 ) ( B1 ,1 Ak B0, 2 y3 , y2 ) ~ ~ ~ ~ ( B0 , 2 A B2, 0 y1 , y3 ) ( B2 , 0 A B0, 2 y3 , y1 ) ( B1 ,1 A B2, 0 y1 , y2 ) ~ ~ ~ ~ ~ ~ ( B2, 0 A B1,1 y2 , y1 ) ( B1,1 A B1,1 y2 , y2 ) ( B2, 0 A B2, 0 y1 , y1 ), 5
(37)
MATEC Web of Conferences 117, 00005 (2017)
DOI: 10.1051/ matecconf/201711700005
XXVI R-S-P Seminar 2017, Theoretical Foundation of Civil Engineering where
B2, 0 [ 12 0 0 0 ]T ; B1,1 [ 0 0 21 21 ]T ;
(38)
B2 , 0 [ 12 0 0 0 ] ; B1 ,1 [ 0 0 21 21 ] .
(39)
We have
( B A Bw, w) ( KU r ,U r ) ,
(40)
where
~ ~ B 2 , 0 A B2 , 0 ~ ~ K B1,1 A B2, 0 ~ B0 , 2 A B 2,0
~ ~ B2 , 0 A B1,1 ~ ~ B1 ,1 A B1,1 ~ B0 , 2 A B1,1
~ B2 , 0 A B0, 2 y1 ; U r y . ~ B1,1 Ak B0, 2 y2 3 B0 , 2 A B0, 2
(41)
Thus, functional (19) can be rewritten in the following form: Ф( w) Ф(U r )
1 2
~
(KU
r
,U r )dx1dx2
(F ,U
r
) wdx1dx2 ,
(42)
where
~ ~ B A B2, 0 c 2,0 ~ ~ ~ K B1,1 A B2, 0 ~ B0 , 2 A B 2,0
~ ~ B2 , 0 A B1,1 ~ ~ B1 ,1 A B1,1 ~ B0 , 2 A B1,1
~ B2 , 0 A B0, 2 F ~ B1 ,1 Ak B0, 2 ; F 0 . 0 B0 , 2 A B0, 2
(43)
The solution of this problem is the point (function) of the constrained extremum of this functional with the condition (27), taking into account (32), (33). These formulations are the initial ones for the realization of the discrete-continual methods for the problems of anisotropic plate analysis. The Reported study was Funded by Government Program of the Russian Federation “Development of science and technology” (2013-2020) within Program of Fundamental Researches of Ministry of Construction, Housing and Utilities of the Russian Federation and Russian Academy of Architecture and Construction Sciences, the Research Project 7.1.1”.
References 1. 2. 3. 4. 5. 6.
P.A. Akimov, Applied Mechanics and Materials, 204-208, 4502-4505 (2012). P.A. Akimov, M.L. Mozgaleva, Applied Mechanics and Materials, 580-583, 28982902 (2014). K.-J. Bathe, Finite Element Procedures (Prentice Hall Inc., 1996). O.C. Zienkiewicz, R.L. Taylor, D.D. Fox, The Finite Element Method for Solid and Structural Mechanics (Butterworth-Heinemann, 2013). O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method: Its Basis and Fundamentals (Butterworth-Heinemann, 2005). A.B. Zolotov, P.A. Akimov, Proceedings of the International Symposium LSCE 2002 organized by Polish Chapter of IASS (Warsaw, Poland, 2002).
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