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12Magazine of Civil Engineering. 2019. 89(5). Pp. 61-78 Инженерно-строительный журнал. 2019. № 5(89). С. 61-78
Magazine of Civil Engineering
journal homepage: http://engstroy.spbstu.ru/
ISSN
2071-0305
DOI: 10.18720/MCE.89.6
Finite element model of Reisner's plates in stresses
Yu.Ya. Tyukalov
Vyatka State University, Kirov, Russia
Keywords: Reisner's plates, possible displacements, finite elements, bending plate
Abstract. A method for calculating bending plates by the finite element method based on Reisner's theory is proposed. The method is based on the fundamental principles of minimum of additional energy and possible displacements. For discretization of the subject area, arbitrary quadrangular finite elements are used. Over the area of the finite element, the moment fields and shear forces are approximated by constant functions that satisfy the differential equilibrium equations in the area of the finite element in the absence of a distributed load. Using the principle of possible displacements, algebraic equilibrium equations of the nodes of the finite element grid are compiled. In accordance with Reisner's theory, vertical displacements and angles of rotation of the middle surface of the plate are taken as nodal possible displacements as independent. The proposed method of calculation allows you to calculate both thick and thin plates. There is no effect of «locking» of the solution for thin plates, which is confirmed by calculations of rectangular plates with different support conditions of side and different ratios of thickness to plate sizes. The solutions obtained by the proposed method for plates of various shapes are compared with analytical solutions. Sufficiently fast convergence and accuracy of the proposed calculation method for both thick and thin plates is shown.
Bending plates are one of the main elements of the supporting structures for various construction objects. Often in construction, for example as a foundation, thick reinforced concrete slabs are used. When calculating thick plates, it is necessary to consider, in addition to bending deformations, transverse shear deformations of sections, which can significantly affect the stress-strain state of the plate. The theory of bending plates, based on the hypothesis of direct normals, does not allow to consider shear deformations. Finite elements, developed based on the Kirchhoff theory, can be used only for the calculation of thin plates
The Reisner's bending plate theory is widely used for calculating thick plates [3-4]. In contrast to the classical Kirchhoff theory, in Reisner's theory, the angles of rotation of the middle surface and vertical displacements are considered as independent variables. Such way makes it possible to lower the maximum order of derivatives in the strain energy functional and makes it possible to use first-order functions for approximating the displacement functions. As known, the direct use of Reisner's theory for constructing finite elements in displacements leads to the «locking» effect which consists in the impossibility of using these finite elements for calculating thin plates, which limits their applicability only to the area of thick plates. In order to overcome the "blocking" effect, various additional coordination of the finite element's unknowns is often used. For example, the hypothesis of direct normals is enforced gone at discrete points or high order shift theory is used [5-6]. The finite elements based on satisfying the hypothesis of direct normals in the middle of the sides of finite elements are widely used in software and demonstrate good accuracy.
Various variational principles are successfully used to solve the stability and dynamics problems of various structures, including plates of variable stiffness [7-8]. A detailed analysis of the applicability of variational principles to the solution of a wide range of problems of the theory of elasticity by the finite element method is performed in [9-10]. In these works, the formulations of the variational principles of Lagrange,
Tyukalov, Yu.Ya. Finite element model of Reisner's plates in stresses. Magazine of Civil Engineering. 2019. 89(5). Pp. 61-78. DOI: 10.18720/MCE.89.6
Тюкалов Ю.Я. Конечно-элементная модель в напряжениях для пластин Рейснера // Инженерно-строительный журнал. 2019. № 5(89). С. 61-78. DOI: 10.18720/MCE.89.6
1. Introduction
[1-2].
(°0
This open access article is licensed under CC BY 4.0 (https://creativecommons.org/licenses/bY/4.0/)
Castilian, Reisner are considered in detail and the relevance of obtaining a two-sided estimate of the obtained approximate solutions is noted.
When constructing analytical solutions of the problems of bending rectangular plates, high-order shear theories are applied. In this case, along the cross section the deformations change according to law different from linear principle [11-12]. To construct finite elements that consider shear deformations, the third-order shear theory is applied successfully [13-14]. In [14], quadrangular finite element is presented that has seven degrees of freedom at each node: three displacements of the middle surface along the axes of coordinates, two shear angles and two angles of rotation of the normals. Such an approach allows one to more accurately consider shear deformations, when the properties of the material are changing in different directions. In [15], an additional procedure is used to account for shear deformations in the finite elements already construct, which are used to calculate thin plates. Transverse shear deformations are calculated of based on the direct application of the equations of the three-dimensional theory of elasticity. To construct quadrangular finite element the Galerkin's method is used in a weak form [16]. Also, the Galerkin's method is used to construct triangular and quadrangular finite elements according to Reisner-Mindlin theory in [17]. In [18, 19], analytical solutions for rectangular plates, obtained based on Reisner's theory, are proposed.
Mixed and hybrid variational formulations are also used to construct finite elements of plates with allowance for shear deformations [20-23]. On the one hand, such formulations simplify the inclusion of shear deformations by using both displacements and shear forces and moments as unknowns, but, on the other hand, matching of displacements and forces is required to ensure convergence.
Thus, most of the developed methods for calculating plates with considering shearing deformations, in one form or another, are based either on approximations of displacement functions or on solving differential equations expressed in terms of displacement functions. In this case, stresses are determined as derivatives of displacements expressed by approximate functions, which leads to an inevitable loss of accuracy in their determination, although stresses are more important for assessing the strength of a structure. In addition, the accuracy of the solution depends on the selected finite element mesh and we cannot, in the general case for an arbitrary construction, determine the accuracy of the solution obtained. Therefore, it remains actual to build alternative models, to the finite element method in displacements, for considering shear deformations when calculating flexible plates based on Reisner's theory. If we have solutions obtained on an alternative basis, then by comparing two (or more) solutions we will be able to obtain a more reliable estimate of the accuracy of the solutions obtained.
The purpose of this work is to develop a method for calculating plates with considering to the shear deformations of Reisner's plates, which based on the functional of additional energy and the principle of possible displacements, as well as comparing the solutions obtained for plates of various shapes and with different supporting conditions with solutions obtained by other methods. Such an approach for solving the plane problem of the theory of elasticity is used in [24], for rod systems in [25-26] and for bendable plates in [27-28]. In [28], a method for calculating thick bent plates with allowance for shear deformations based on the extended Castiliano functional is proposed. As additional terms, using the Lagrange multipliers method, algebraic equilibrium equations of nodes, obtained using the principle of possible displacements, are added to the functional. Possible vertical displacements of nodes causing bending and possible vertical displacements of nodes causing shear are used separately. As a result of the calculation, the rotation angles for the nodes are not directly determined. In this work, as possible displacements, both possible displacements in the form of vertical displacements and rotation angles will be used. In this case, as a result of the calculation, we can obtain both the magnitudes of the vertical displacements and the magnitudes of the rotation angles, which is more convenient.
2. Methods
Solving the problems of plate bending we obtain based on the functional of additional energy for an isotropic plate (for simplicity, we assume that there are no specified displacements) considering the shear deformations:
n = ^El^iK +M2 - 2yMMy + 2 (1+v) M%) a2 + ) d^ min. (1)
E is the modulus of material elasticity;
t is the plate thickness;
vis Poisson's ratio;
k = 6/5 is coefficient considering the parabolic law of change of tangential stress across the plate thickness. The functional (1) is can written in matrix form that is more convenience for solving by the finite element method:
Magazine of Civil Engineering, 89(5), 2019
n = 2\{M}T [E]-1 {M}dn + 2\{Q}T [Eh]-1 {Q}dn^ min. (2)
23 <■ > l J <■ > 2 In expression (2) the following notation is entered:
{M } =
Mx
M,.
M
xy
, {Q} =
Qy
[ E ]-1 =
12
E • t '
1 -v 0 -v 1 0 0 0 2 (1 + v)
]-1 = 12 (1 + v)
[Eh ] = 5E • t
1 0 0 1
(3)
In functional (2), the first addend is associated with the bending deformations of the plate, the second is due to shear deformations by transverse forces.
Figure 1. An arbitrary quadrangular finite element.
For discretization of the subject area, we use arbitrary quadrangular finite elements (Figure 1). Over the area of the finite element the moments and shear forces will be approximated by the constant functions. It is obvious that such functions satisfy differential equilibrium equations in the area of a finite element in the absence of distributed loads.
For the finite element k in the global coordinate system, we combine the unknown internal forces into the vector {Fk}, and shall denote the flexible matrix as [Dk].
F } =
M M M
x,k
y ,k
xy,k Qx,k
Qy.k
[ Dk ] = A
12 -12•v
E • t3 E • t3 -12•v 12
E • t3 E • t3 0 0
0 0
24 (1 + v)
0 0
0 0
E • t3 0
0
0 0
0
12 (1 + v)
5 • E • t
0
0 0
0
0
12 (1 + v)
5 • E • t
(4)
Ak is the area of the finite element;
n is number of finite elements. Then the expression of the functional (1) can be written in the following matrix form:
nc = 1 x;=1(Fk}T D]{Fk} = 2{F}T [D]{F} ^ min;
1 A] " {F}
[ D] = , {F}=-
. [ Dn ]. {Fni
(5)
(6)
In accordance with the principle of minimum of the additional energy, the functions for moments Mx, My, Mxy and shear forces Qx, Qy must satisfy the corresponding differential equilibrium equations (7) and static boundary conditions.
M dM
dx
xy
dy
- Qx = o,
dMv dM dQ Qv
y ■ xy - Qy = 0, + q = 0.
dy
dx
dx dy
(7)
Dividing the region of problem into finite elements and using approximations for internal efforts, then we shall obtain a finite-dimensional analog of the differential equations (7). For this we use the method of weighted residuals. Then for the first differential equation, we can write the following equations for the method of weighted residuals:
A
'dM dM.
dx
Л
xy
dy
- Qx
WxldA = 0, i = 1,2,...N.
(8)
N is the number of nodes of the finite element grid; Sdx i is weight functions;
Ai is area of the region where the weight function is nonzero. We assume that the weight functions 50xi
are nonzero only in the region of finite elements adjacent to the node i under consideration. Such weight functions can be called "local" or "finite". Using for (8) the integration procedure in parts, we obtain the following expression:
f
Ai
M,
) + M d(S0x,i)
dx xy dy
\
- QxS9x
f
d(Mxô6Xi ) d(Mxyôev )
Л
dx
cy
dA=0, i=1,2,..., N. (9)
i
The second integral in expression (9) can be transformed using the Gauss theorem [1] (integration by parts in the plane case) into the integral over the boundary of the region, then we get:
дj d(ô&xi) d(seXii)
M^ „ ' + Mm „ ' - QXSGX
\
dx
xy
A V
dy
dA -\( Mxlx +Mxyly )sexJdr = 0, i = 1,2,..., N. (10)
г
r is the boundary of the area in which the weight function S9xi is nonzero;
lx,l are direction cosines of normal to the boundary of the region r . If we shall give the weight
functions Sdx { a physical meaning and take them in the form of possible angles of rotation along the Xaxis,
then expression (10) coincides with the expression of the principle of possible displacements. The first integral in (10) is the work of internal forces on possible displacements, the second integral is the potential of external, given moments on the boundary. Bending and torsional moments perform work on the corresponding curvatures, and shear forces perform work on shear angles.
Performing similar transformations to the second and third equations (7), we obtain two more equilibrium equations:
f
M,
ôK),„ dK)
Л
"i V i
i
dy +Mxy dx
sh\ Л
- QvS0.
y y, 1
dA-i(Myly + Mxylx ]Sdyidr = 0, i = 1,2,..., N. (11)
г
d(ôwsh ) 5 (ôwsh )
dx
dy
dA - i ( Qxlx + Qyly )Swfdr - i qSwfdA = 0, i = 1,2, ., N. (12)
J
r A
S9vi is possible displacement (weight function) in the form of a rotation angle along the Yaxis;
dwf is possible displacement (weight function) in the form of a vertical displacement associated with a shear.
Thus, from differential equilibrium equations (7), for the Reisner plate, we obtained algebraic equilibrium equations (10)-(12) for nodes of the finite elements grid, which, when crushing the grid, will tend to differential equations. Note that expressions for derivatives of internal forces are not included in expressions (2), (10)-(12), so they can be accepted to be constant over the finite element region, and possible displacements can be taken as linear functions.
To build a solution, instead of the three differential equilibrium equations (7), having completed the transformations, we can get only two:
d2 Mx
dx x
„ d 2Mxy d 2M„ + 2-^ +
+ q = 0, dQx + Ql + q = o.
dxdy dy21 dx dy
(13)
Obviously, the first equation is associated with bending deformations, and the second with shear deformations. For the second differential equation, algebraic equilibrium equations (12) are obtained above.
For the first equation, we shall take the function Swb as a weight function. Then we get
5 2Mx d2Mxy d2My ^
x + 2 , , 7 +--T- + q
dx2
dxdy dy2
SwbdA = 0, i = 1,2,..., N.
(14)
For the convenience of recording transformations, we introduce intermediate notation:
S(PX. i =
dwb
5(Pv, i =
dwb
x,' dx ' ^ dy Using (15) we get
■ V =
' x dx
dM dMv dM dM
x v =_y u = xy t t = xy
y~ cy ' x" dx ' y~'
dy
'dV dU dUy dVy )
+ + + q
dx dy dx dy
SwbdA = 0, i = 1,2,..., N.
(15)
(16)
Let us demonstrate the further transformations scheme by the example of the transformation of the first addend in (16). Using procedure of integration in parts we get:
A = jd^wfdA = -j v
diâwb ) rd(vxâwb )
V ' ) A A , f V x ' )
dx
-dA + j'Ai
dx
dA.
(17)
Next, using the Gauss theorem, we replace the second integral over the area by the integral along the boundary contour:
f d(5wb ) f b
A =-j V.^^dA + j VjMdT.
Ai r
Using (15), we replace the variables again:
/1=sm w+\sMr'Mdr
Ai r
To the first integral from (19), we also use integration by parts:
d(s<t>*.<L, i^MM,,.,)
'i = j M*^rldA
dM„
dx
dA + j^MxLSwbdr.
J dx x 1
A A r
Applying the Gauss theorem to the second integral, we finally get
A = j M
d2 (âwb )
d(âwb )
dA - j MxÇ^rdr + j ^^wbdF.
dx2
r
r
(18)
(19)
(20)
(21)
Using similar transformations for all terms and using the expression for the transverse forces from (7), we obtain the following expression for (16):
f
i
d2 (Swb) d2 (Swb) d2 (Swb
—V-L + 2M —Ь-+M —Ь— '
dx1 xy dxdy y By2
M,
dA -J(Mxlx +M yly,)
г
ly
\(Myly + Mxylx+ \ (Qxlx + Qyly HdT + \ qJw-dA = 0, i = 1,2,..., N.
(22)
г
г
Equations (22) are equilibrium equations for nodes on possible vertical displacements, causing flexural deformations. In [27, 28], equilibrium equations like (22) were used in constructing a solution for Kirchhoff plates, which based on the functional of additional energy. Linear functions were taken as possible displacements in the finite element region. Therefore, in the triangular finite elements, the first integral was zero. For the rectangular finite elements in the first integral, only the term associated with the torsional curvature was nonzero. The fourth integral is zero, both at the boundaries between the finite elements and at the outer boundaries. The fourth integral is the work of transverse forces normal to the boundary. Therefore, if the external boundary is free, then the transverse forces are zero. If there are no displacements at the border, then there is also zero the possible displacement. The second and third integrals in (22) are the work of moments at the angles of rotation along the normal to the sides of the finite elements. With linear possible displacements, these angles get fractures (breaks) and therefore the third and fourth integrals will not be equal to zero. In papers [27, 28] for triangular finite elements, these integrals were calculated using geometric constructions. Note that when using this approach to the solution, it is impossible to use the approximations of forces constant over the finite element region, since in this case the second and third integrals will be equal to zero.
Comparing the two options have considered above, we note that in the second version, the states of bending and shear are completely separated, and in the first version they are connected. But at the same time, it should be noted that the original differential equations equilibrium of are the same.
In this paper, we will use the first version of the equilibrium equations construction, based on three independent possible displacements. In accordance with the minimum of additional energy principle the functions of moments and shear forces must satisfy the corresponding differential equilibrium equations and static boundary conditions. Since, in the general case, it is almost impossible to select such functions, we shall act as follows. Using the possible displacements principle, we shall compose algebraic equilibrium equations for the nodes of the finite element grid. In this case, in accordance with Reisner's theory we shall take, as the nodal possible displacements, the vertical displacements and rotation angles of middle surface independently. Further, the resulting algebraic equilibrium equations will be added to the functional (5) using the Lagrange multipliers method. This ensures the equilibrium of the selected stress fields in a discrete sense (in nodes) and the solution can be obtained by minimizing the resulting extended functional. It should be noted that the number of equilibrium equations must be less than the total number of unknown nodal forces. Since the forces in the domain of finite elements are approximated by constant functions, then small grids are necessary for obtaining sufficiently accurate solutions, therefore the above requirement will be fulfilled. For example, in a 4x4 square grid, the number of unknown nodal forces will already be greater than the number of equilibrium equations, even if the superposed supports are not taken into account.
Figure 2. Possible displacements of the node 1 of finite element.
To get the solution, we use the well-known transformation of an arbitrary quadrilateral (Figure 1a) into the square element (Figure 1b). Such transformation can be written in the following form:
x =
Z4=1 Ni(Ы-x^ y = Ti=№,n)-N(Ы=4(1 + Ш1+ Ы
(23)
xi k, yik are coordinates nodes of finite element k in the global coordinate system.
Displacements in the finite element area, caused by possible displacements of the node i, we shall express using the linear basis functions introduced above - Ni (4,n):
Sw = Ni (^,n)5Wi, Sdx = Ni (Z,rj)>Ox,i, S^ = Ni (4,nM,i. (24)
With the possible displacement dwt = 1 only shear deformations occur in the plate:
= d(Sw) = 6Nj (4,n) Sy = d(Sw) = dN (4,n)
/ xz ~ ~ ' / yz ~ ~ • (25)
dx dx dy dy
Then, the work of the internal forces of finite element k by possible displacement is expressed in the following form:
SUWik = Qx,k JJ SYxzdA + Qyk JJ SyyZdA.
(26)
Ak Ak
The work of external forces
SVWi,k = Pzi +JJN qzkdA. (27)
P2i is force which concentrated in a node;
qz k is load which is distributed over area of the finite element;
Ak is area of the finite element. In accordance with the principle of possible displacements, we shall obtain the equilibrium equation for the node i
I SUWi, k + I SVWi, k = (28)
kGSi kGSi
s, is the set of finite elements adjacent to the node i. Equations (28) for all nodes can be written in the following general matrix form:
t fk,i}+ P = °, i (29)
{Cwi} is vector containing the coefficients before unknown forces of finite elements in the equilibrium equation of the node i;
{Fwi} is vector of unknown forces of finite elements adjacent to node i.
P is generalized force equal to the work of external forces;
Sw is set of nodes that have degree of freedom along the Z axis. The expressions for the elements of the vectors and the generalized forces will be given below.
With the possible displacement in the form of rotation angle along the Xaxis Sdx i = 1, both bending and torsional deformations and shear deformations take place:
s d(S6 ) dN, _ d(S6 ) dN, (%,rj) _ __ ^
SXxx = ' , Xxy = = T , SYxz =-S^x =-Nt (30)
dx dx dy dy
In this case, the work of the internal forces for the finite element has the following form:
SU0xi,k = Qx,k JJSYxzdA + Mx,k JJ SXxxdA + Mxy,k JJ SXxydA (31)
Ak Ak Ak
The work of external forces in the general case
SVex,,k = Mexi + JJNi (4, n\mex,kdA
Инженерно-строительный журнал, N° 5(89), 2019
Me i is the moment of external forces concentrated in the node, which acting along the Xaxis; - acting along the X axis moment of external forces, which distributed over the finite element area;
Ak is the area of the finite element. Equilibrium equation, like equation (29), will have the following form:
C ,i }T {Fdx ,i}+ Mxi = 0, i . (33)
Ce i} is vector containing coefficients before unknown forces of finite elements in the equilibrium equation of the node i;
F i} is unknown forces vector for finite elements adjacent to node i.
Mx i is generalized moment equal to the work of external forces;
Se is nodes set, which have degrees of freedom in the form turn angles Ox.
For possible displacement, in the form a rotation angle along the Y axis 5Oy i = 1, such expressions are written similarly:
Xyy dy dy ' Xxy dx dx ' Yyz y m ( )
5Udyl ,k = Qy,k U 5 YyzdA + My,k U 5XyydA + Mxy,k jj 5XxydA (35)
Ak Ak Ak
5\,k = M0y,i + jjNi n\mey,kdA. (36)
)% }+ Myi = 0 i ^y • (37)
Using the Lagrange multipliers, equations (29), (33) and (37) we shall add to the functional (5). Then we get
ПС = 2{F}T [D]{F}+ 2 wi ({Cw,f {Fwi} + Л) + С - \ ( T - Л (38)
2 O (К,i} {Fox,i}+Mx,i 1+ 2 O (K,i} К,i}+Mysh min-
We introduce the notation for the nodal displacements global vector - { W}. The vector combines all unknown vertical displacement and rotation angles for the whole area:
{W } = ( W 6xa 6y,з ••• wm-2 6Xm-! 6ym f .
Then, equating to zero the derivatives of the functional (38) with respect to the vectors { W) and (F), we shall obtain the following linear algebraic equations system:
ЙШ -
M W
_L] [0]
The matrix [L] and vector {P} is formed from the coefficients of the equilibrium equations (29), (33) and (37).
The matrix [D] is block diagonal form and easily analytically invertible. Therefore, we can express the vector {F} from first matrix equation:
{F}= [d]-1 {P}- [D]-1 [L]t (W} (40)
Then we express the vector { W} from the second matrix equation
[k]{W }=[D]-1{P} [K]=[L][D]-1[L]T. (41)
Note that the matrix [K] has tape structure of nonzero elements.
Magazine of Civil Engineering, 89(5), 2019
Consider the matrix [L] formation algorithm for a plate represented by arbitrary quadrangular finite elements. For partial derivatives of the function N t (t,n) of the finite element with the number k, the following expressions can be written:
ôN, ôN, dx ôN, dy ôN, ôN, dx ôN, dy
, _ , + , i' , _ , + , i'
dt dx dt dy dt dn dx dn dy dn
(42)
Index i denotes the local number of a finite element node (Figure 1a). Equations (42) are written in the matrix form:
dN, dx dy f dN, dx dy
dt dN, . _ dt dx dt dy dx dN t , [J ]_ dt dx dt dy
dn dn dn dn dn dn
Using relations (7), we obtain the expressions of the Jacobi [J] matrix's elements:
J11 II 4_ t(1 + nn) 1 4 x,,k _ \[(1 -n)(x2,k - x1,k ) + (1 + n)( x3,k- x4,k,
J12 : dy dt' t(1 + nn), 1 4 y,,k _ 4 [(1 -n)( y2,k - y1,k ) + (1 + n)( y3,k - y4,k
J21 _ dx_ dn n (1+ttt) =1 4 x,,k _ 4 [(1 -t)( x4,k- - x1,k ) + (1 + t)( x3,k" " x2,k,
J22 11 4_ n (1+ttt) y 1 4 y,,k _ 4[(1 -t)(y4,k - y1,k ) + (1 + t)( y3,k - y2,k
From relation (43), we can obtain the expressions for the necessary derivatives:
(43)
(44)
№1 fdN,]
dx dN, _[J j"1- dt dN, -, [Jj"
dn dn
b11 b12 b21 b22
The elements of matrix [j j-1 have the following expressions:
J.
det J J\\J 22 J12 J21, b11
" J " J J
b _ 12 b _ 21 b _ 11
22
det J 12 det J' 21 det J
Using the expression (45) and (7), we shall get
dN, t (1 + nn) + n(1 + ttl)
— fj! . Tfi-
dx 4 ' ~12 4
dN_b t^+nn) + n(1+ttl)
dy 21 4 22 4 '
det J
(45)
(46)
(47)
We shall get integrals on the area of arbitrary quadrilateral finite element, for function n) and it
dN. dN,
derivatives —-, —L, using the integrals on area of square element.
dx dy
,k _ WitdA _ 4L L (but' (1 + nn ) + M/ (1 + tt )) det J dÇdn,
cn > _
C2i,k _
Wl^dA _ 4 i-1 i-1 (b21& (1 + nn ) + bn (1 + tt )) det J dtdn,
,k _ jj N ,dA _ 4 j-1 £ (1 + tt,) (1 + nn) det J dtdn.
Ak
C3, > _
Integrals (48) are calculated numerically using the four-point Gauss formula.
(48)
k
Using (48), we write the expressions (26), (31) and (35) for the possible displacements of the node i for the finite element with the number k in the following form:
SUw„ к = Qx, kcu, k + Qy , kc
SU,
в, к
SU,
e,,i, к
iy, k^ 2i, k~>
= Mx,kC1i,k + Mxy,kC2i,k - Qx,kC3i,k , = Mxy,kc1i,k + My,kC2i,k — Qy,kC3i,k ■
(49)
We can introduce the vector - {sUi k }=
{su,k }=Kkfc } [L,k ]
su SU su
exi ,k
в„ ,k
. Then
0 00 C1i ,k C2i ,k
1i ,k 0 C2i,k — C3i,k 0
0 C2i ,k C1i,k 0 — C3i ,k
(50)
From the matrices [_Lik ] formed for each finite elements nodes, in accordance with the numbering of the nodes and finite elements, the global matrix [L] is formed for whole system. If the boundary of the area is
not parallel to one of the global axes, but lies at an angle ato the Xaxis, then the matrix should be multiplied by the matrix of direction cosines:
Lu ]=№,k 1 M=
1 0 0 0 cos a sin a 0 — sina cosa
(51)
For the case of a load uniformly distributed on the finite element, the work of external forces on possible displacement is calculated by the following formula:
SVWi,k = J] N = qz,kc3i,k ■
A
3. Results and Discussion
(52)
It is well known that when using the theory of bending of Reisner plates for building finite elements in the form of the displacement's method, the so-called problem of "locking" the solution arises. "Locking" is that such finite elements are unsuitable for the calculation of thin plates. Therefore, as first example, we shall obtain solutions for rectangular plates with different ratios of the plate thickness to its dimensions and for different variants of supporting the sides (Figure 3).
CF SF FF
Figure 3. Conditions of support and the sizes of the sides of bending Levi's plates.
k
c
In Figure 3 dashed line and the letter S denotes a simply supported side, the oblique hatching and the letter C denotes a clamped side, and the letter F denotes the free side. Table 1 presents the results of calculations of the plate denoted SS, given in [11] for various theories, and the results obtained by the proposed method — SFEM. The lengths of the plate sides were taking equal to 3 m or 1.5 m. The side 3 m long was divided into 30 finite elements, and the side 1.5 m — into 16 elements. Poisson's ratio is v = 0.3. The load was taken evenly distributed over the plate area. The results of calculations in Table 1 are presented in dimensionless form:
*=100DwI- -1
qa
(53)
Table 1. Displacement of the center of the plate under the action of a uniformly distributed load.
a
b
t/a
Theory
Boundary conditions
CC CS SS CF SF FF
3 1.5 0.001 CRT 0.0163 0.0305 0.0633 0.1450 0.3810 1.3714
S-FSDT 0.0163 0.0305 0.0633 0.1450 0.3809 1.3713
[11] 0.0163 0.0305 0.0633 0.1450 0.3810 1.3714
SFEM 0.0161 0.0303 0.0632 0.1449 0.3808 1.3688
0.04 CRT 0.0178 0.0322 0.0647 0.1505 0.3880 1.3797
S-FSDT 0.0175 0.0318 0.0646 0.1476 0.3835 1.3770
[11] 0.0178 0.0322 0.0646 0.1504 0.3879 1.3795
SFEM 0.0176 0.0320 0.0646 0.1503 0.3890 1.3769
0.1 CRT 0.0256 0.0421 0.0725 0.1746 0.4111 1.4168
S-FSDT 0.0245 0.0386 0.0714 0.1614 0.3972 1.4070
[11] 0.0256 0.0407 0.0714 0.1721 0.4084 1.4130
SFEM 0.0254 0.0405 0.0715 0.1721 0.4100 1.4103
0.2 CRT 0.0524 0.0694 0.0958 0.2391 0.4688 1.5247
S-FSDT 0.0489 0.0630 0.0958 0.2105 0.4464 1.5141
[11] 0.0525 0.0695 0.0958 0.2395 0.4692 1.5248
SFEM 0.0524 0.0695 0.0960 0.2396 0.4712 1.5220
3 3 0.001 CRT 0.1917 0.2785 0.4062 0.5667 0.7931 1.3094
S-FSDT 0.1917 0.2786 0.4062 0.5667 0.7931 1.3094
[11] 0.1917 0.2786 0.4062 0.5667 0.7931 1.3094
SFEM 0.1916 0.2786 0.4063 0.5663 0.7924 1.3068
0.04 CRT 0.1965 0.2830 0.4096 0.5737 0.7981 1.3154
S-FSDT 0.1955 0.2819 0.4096 0.5712 0.7975 1.3151
[11] 0.1965 0.2830 0.4096 0.5737 0.7981 1.3154
SFEM 0.1964 0.2834 0.4107 0.5732 0.7983 1.3128
0.1 CRT 0.2209 0.3059 0.4273 0.6065 0.8224 1.3459
S-FSDT 0.2128 0.2996 0.4273 0.5945 0.8208 1.3451
[11] 0.2209 0.3059 0.4273 0.6065 0.8224 1.3459
SFEM 0.2209 0.3065 0.4289 0.6061 0.8230 1.3432
0.2 CRT 0.3021 0.3827 0.4904 0.7139 0.9072 1.4539
S-FSDT 0.2759 0.3827 0.4904 0.6777 0.9041 1.4523
[11] 0.3021 0.3827 0.4904 0.7139 0.9072 1.4539
SFEM 0.3023 0.3837 0.4924 0.7135 0.9081 1.4512
1.5 3 0.001 CRT 0.8445 0.9270 1.0129 1.0605 1.1496 1.2887
S-FSDT 0.8445 0.9270 1.0129 1.0605 1.1496 1.2887
[11] 0.8445 0.9270 1.0129 1.0605 1.1496 1.2887
SFEM 0.8440 0.9256 1.0106 1.0564 1.1444 1.2804
0.04 CRT 0.8511 0.9330 1.0181 1.0664 1.1547 1.2938
S-FSDT 0.8497 0.9322 1.0181 1.0660 1.1551 1.2944
[11] 0.8511 0.9330 1.0181 1.0664 1.1547 1.2938
SFEM 0.8506 0.9323 1.0172 1.0623 1.1502 1.2855
0.1 CRT 0.8850 0.9637 1.0454 1.0981 1.1829 1.3228
S-FSDT 0.8770 0.9596 1.0454 1.0946 1.1837 1.3244
[11] 0.8850 0.9637 1.0454 1.0981 1.1829 1.3228
SFEM 0.8845 0.9638 1.0420 1.0940 1.1791 1.3145
0.2 CRT 1.0000 1.0704 1.1430 1.2090 1.2844 1.4283
S-FSDT 0.9746 1.0572 1.1430 1.1970 1.2861 1.4316
[11] 1.0000 1.0704 1.1430 1.2090 1.2844 1.4283
SFEM 0.9998 1.0712 1.1446 1.2050 1.2812 1.4199
The results presented in Table 1 demonstrate good accuracy of the solution according to the proposed method, both for thin and thick plates. Thus, the proposed solution method is free from the so-called "locking" effect and can be used to calculate plates of any thickness.
To assess the accuracy of determining the bending moments, we consider the results of calculations of the SS plate for different aspect ratios. It is known that when the sides of plate are simply supported, the values of the bending moments don't depend on the plate thickness. Table 2 provides comparison of the values of moments in the center of the plate, obtained by the proposed method, with the results of analytical calculations according to the Kirchhoff theory [29]. The finite element grid was taken 21 *21.
Table 2. Moments in the center of the SS plate which loaded with a uniformly distributed load.
b/a - 100MX / qa2 100My / qa2
t/a = 0.001 t/a = 0.2 Exact [29] t/a = 0.001 t/a = 0.2 Exact [29]
1.0 4.79 4.84 4.79 4.79 4.84 4.79
1.1 5.55 5.60 5.54 4.93 4.98 4.93
1.2 6.27 6.33 6.27 5.01 5.06 5.01
1.3 6.94 7.00 6.94 5.03 5.08 5.03
1.4 7.55 7.62 7.55 5.02 5.07 5.02
1.5 8.11 8.19 8.12 4.98 5.03 4.98
1.6 8.62 8.69 8.62 4.93 4.98 4.92
1.7 9.08 9.15 9.08 4.86 4.90 4.86
1.8 9.48 9.56 9.48 4.78 4.83 4.79
1.9 9.85 9.92 9.85 4.71 4.75 4.71
2.0 10.17 10.24 10.17 4.63 4.67 4.64
Note that for thin plates, the proposed solution practically coincides with the analytical one, and for thick plates one, the resulting moments are more than analytical ones by about 1 %.
Also, the rate of the solution convergence by the proposed method was tested when was crushing the finite element grid. The test showed fast convergence of displacements for all considered variants of plates. In Figure 4 shows graphs of solutions convergence for two variants of square plates, depending on the number side's divisions.
0.35)
0.3
0.25
0.2
W
0.15
t/a-0.2
t/a=0.001
0.45r
W
0.4
0.35
0.3
10
a)
20
30
0.251
t/a=0.2
t/a-0.001 1-4
20
30
b)
Figure 4. Displacement the center of a square plate depending on the number of side divisions: a) SS plate; b) CS plate.
In Figure 5 shows the graphs of the change in bending moments in the SS plate (Figure 3) for various sides ratios. The plate was crushed into 21 finite elements along each side. Therefore, the lengths of the finite element's sides were in the same ratio as the lengths of the plate sides. Bending moments are constant values in the finite element region and so more accurately model the value in the finite element centers. Therefore, to more accurately determine the value of the moment in the clamped side middle My,2, linear extrapolation was used according to the moment values in the two finite elements nearest to the boundary. Obviously, with an increase in the ratio of the finite element sides, the error in determining the moment on the clamped side also increases.
Figure 5. Bending moments for a uniformly loaded square plate CC with different ratios of sides. Mx j = Mx,i/qa2 and My t = My,i/qa2 are bending moments in the center of the plate;
My 2 = My,2/qa2 is bending moment in the middle of the clamped side. Designations:
Red line is t/a = 0.001; Blue line is t/a = 0.2; Green line is the solutions for the Kirchhoff plate [29].
Graphs in Figure 5 shows the concurrence of the obtained moment values for thin plates with the analytical solution for Kirchhoff plates (the Green lines coincide with the Red lines). For thick plates (Blue line), only the moments My 1 coincide with the solution for Kirchhoff's plates (Figure 5b). The values of the moments
Mx1 and My 2 for thick plates differ by 10-20 % from the corresponding moments obtained for thin plates.
For an additional estimation of the results accuracy obtained, square SS plate was calculated on the LIRA-SAPR program using volume finite elements. A quarter of the plate was divided into 12 elements in height and into 20 elements along the X and Y axes. A plate with thickness t = 0.6m was calculated. The dimensions of the plate quarter in plan are 1.5 m by 1.5 m. Poisson's ratio v = 0.3. Load is q = 10 kN/m2. Modulus of elasticity is E = 10000 kN/m2.
Table 3. Stresses in a square SS plate - (Figure 2).
Solution Fiber of section Center of the plate \ax1\, kN / m2 \ay1, kN / m2 Middle of the clamped side 2,kN /m2 1 1
LIRA-SAPR top 41.85 43.39 88.11
bottom 47.50 46.29 89.13
mean 44.68 44.84 88.62
value
SFEM 43.91 49.95 93.15
Kirchhoff's plate [29] 36.54 49.86 103.46
Comparison of the results presented in Table 3 shows that the stress values obtained for thick plate based on Reisner's theory agree well with the stress values obtained by solving three-dimensional elasticity theory problem. Stresses in the clamped side ay, 2 were determined using linear extrapolation from the values in the two finite elements nearest to the boundary. Also note that the displacements of the slab center, obtained by the LIRA-SAPR program, are 8.5 % more than the displacements obtained by the proposed method.
As next example the semiring was calculated on the acting of uniformly distributed load (Figure 6). The semiring has hinge supports along the lines DE and DB, and clamped support along the line EA. Line AB is the axis of symmetry, therefore, in the nodes lying on this line, the angles of rotation along the horizontal axis were excluded. In the calculations were taken the following data: E = 10000 kN/m2, / = 0.3, t = 0.1 m, q = 10 kN/m2, R = 6 m, r = 3 m. In Figure 7 shows graphs of changes along the line AB: bending radial moments - M ; moments directed perpendicular to the line AB - M and vertical displacements w
a) b) c)
Figure 7. Bending moments and deflections of the semiring along the AB line: Red line is scheme in Figure 6a; Blue line is scheme in Figure 6b.
Table 4. Comparison of calculation results for the semiring with an analytical solution.
Solution Wc MxcC MyC MxA My,A
SFEM (Figure 66) 0.357 0.0118 0.0387 -0.00436 -0.0167
Analytical [30] 0.358 0.0118 0.0393 -0.00439 -0.0168
The values of the moments at point C were determined as the middle value of moments in two finite elements adjacent to the point on both sides. The values of the moments at point A were determined by linear extrapolation from the values of moments in the two nearest finite elements. Graphs in Figure 7 illustrate the linear variation of moments near clamped side (y = 0). Thus, the graphs in Figure 7 and the data in Table 4 show good convergence and accuracy of the proposed method and for this example.
In the following example we use the finite element in the form of a parallelogram which have maximum inner angle of 120 degrees. A skewed plate was calculated (Figure 8). For such plate the finite difference solution is given in [29].
6.06m
Figure 8. Skewed plate. Grid - 10x20 finite elements.
The following plate parameters were taken: E = 107 kN/m2, j = 0.21, t = 0.1 m, q = 10 kN/m2. Sizes of the plate are shown in Figure 8. Calculations were performed for two finite element grids: 10 by 20 elements (Figure 8) and 20 by 40 elements. In addition, for comparison, calculations of this plate were performed using the LIRA-SAPR program. All results are given in Table 5.
Table 5. Displacements and bending moments in the center of the plate (Figure 8).
Solution (grid) w, mm Mx, kN • m/m My, kN • m/m
SFEM (10x20) 9.074 3.371 8.635
SFEM (20x40) 9.187 3.437 8.780
LIRA-SAPR (10x20) 9.115 4.758 7.237
LIRA-SAPR (20x40) 9.149 4.828 7.347
Timoshenko [29] 9.719 - 8.712
The values of the bending moments in the plate center (for SFEM and for LIRA-SAPR) were determined as the middle value of moments in two elements adjacent to the center point on one half plate. Note that the moments in the two finite elements differ by about 5 percent. The bending moments obtained by the proposed method are close to the corresponding values given in [29], and the value of plate center displacement is less than the corresponding value given in [29] by about 6 %, but almost coincides with the value obtained using the LIRA-SAPR program. The results of the skew plate calculations show a good accuracy of the proposed method for calculating the Reisner's plates with using finite elements in the parallelogram form. Note that the plate under consideration is thin, but nevertheless the solution "locking" effect is absent, as well as for the plate in the semiring form from the previous example.
4. Conclusion
1. A method for calculating of bending plates based on Reisner's theory by the finite element method is proposed. The method is based on the fundamental principles of additional energy minimum and possible displacements. The necessary relations for an arbitrary quadrangular finite element are obtained. Mathematically, the transition from differential equilibrium equations of bent plates to algebraic equilibrium equations for nodes of finite element grid.
2. The bending moments, torques and shear forces are constant in the finite element area, which is necessary condition for ensuring the solution convergence, when the mesh is crushed.
3. The proposed calculation method allows both thick and thin plates to be calculated. There is no effect of solution "locking" for thin plates, which is confirmed by rectangular plates calculations with different support conditions of side and different ratios of thickness to plate sizes.
4. Comparison of solutions obtained by the proposed method for plates of various shapes with analytical solutions shows fast convergence and proposed method accuracy of calculating both thick and thin plates.
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Contacts:
Yury Tyukalov, +7(912)8218977; yutvgu@mail.ru
© Tyukalov, Yu.Ya., 2019
Инженерно-строительный журнал
сайт журнала: http://engstroy.spbstu.ru/
ISSN
2071-0305
DOI: 10.18720/MCE.89.6
Конечно-элементная модель в напряжениях для пластин
Рейснера
Ю.Я. Тюкалов
Вятский государственный университет, г. Киров, Россия
Ключевые слова: пластины Рейсснера, возможные перемещения, конечные элементы, изгибаемые пластины
Аннотация. Предложена методика расчета изгибаемых пластин методом конечных элементов на основе теории Рейснера. Метод основывается на фундаментальных принципах минимума дополнительной энергии и возможных перемещений. Для дискретизации предметной области используются произвольные четырехугольные конечные элементы. По области конечного элемента поля моментов и поперечных сил аппроксимируются постоянными функциями, которые удовлетворяют дифференциальным уравнениям равновесия в области конечного элемента при отсутствии распределенной нагрузки. Используя принцип возможных перемещений, составляются алгебраические уравнения равновесия узлов сетки конечных элементов. При этом, в соответствии с теорией Рейсснера, в качестве узловых возможных перемещений принимаются, независимо, вертикальные перемещения и углы поворота срединной поверхности пластины. Предлагаемый метод расчета позволяет рассчитывать как толстые, так и тонкие пластины. Эффект «заклинивания» решения для тонких пластин отсутствует, что подтверждено расчетами прямоугольных пластин с различными условиями опирания и различными отношениями толщины к размеру пластины. Сравниваются решения, полученные по предлагаемой методике для пластин различной формы, с аналитическими решениями. Показана достаточно быстрая сходимость и точность предлагаемой методики расчета как для толстых, так и для тонких пластин.
Литература
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2. Секулович М. Метод конечных элементов. М.: Стройиздат, 1993. 664 с.
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4. Fallah N. On the use of shape functions in the cell centered finite volume formulation for plate bending analysis based on Mindlin— Reissner plate theory // Computers & Structures. 2006. Vol. 84. Pp. 1664-1672.
5. Katili I., Batoz J.L., Maknun I.J., Hamdouni A., Millet O. The development of DKMQ plate bending element for thick to thin shell analysis based on the Naghdi/Reissner/Mindlin shell theory // Finite Elements in Analysis and Design. 2015. Vol. 100. Pp. 12-27.
6. Khezri M., Gharib M., Rasmussen K.J.R. A unified approach to meshless analysis of thin to moderately thick plates based on a shear-locking-free Mindlin theory formulation // Thin-Walled Structures, 2018. Pp. 161-179.
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11. Park M., Choi D.-H. A two-variable first-order shear deformation theory considering in-plane rotation for bending, buckling and free vibration analysis of isotropic plates // Applied Mathematical Modelling. 2018. Vol. 61. Pp. 49-71.
12. Kumara R., Lala A., Singhb B.N., Singhc J. New transverse shear deformation theory for bending analysis of FGM plate under patch load // Composite Structures. 2019. Vol. 208. Pp. 91-100.
13. Thai H.-T., Choi D.-H. Finite element formulation of various four unknown shear deformation theories for functionally graded plates // Finite Elements in Analysis and Design. 2013. Vol. 75. Pp. 50-61.
14. Doa T.V., Nguyenb D.K., Ducc N.D., Doanc D.H., Buie T.Q. Analysis of bi-directional functionally graded plates by FEM and a new third-order shear deformation plate theory // Thin-Walled Structures. 2017. Vol. 119. Pp. 687-699.
15. Mulcahy N.L., McGuckin D.G. The addition of transverse shear flexibility to triangular thin plate elements // Finite Elements in Analysis and Design. 2012. Vol. 52. Pp. 23-30.
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