CC BY
4doi: 10.18720/MCE.76.20
Stress-strain state of clamped rectangular Reissner plates
Напряженно-деформированное состояние защемленной прямоугольной пластины Рейсснера
Abstract. The paper focuses on obtaining numerical results for a rectangular Reissner plate with clamped contour under the influence of a uniform load using the iteration superposition method of four types of trigonometric series (correcting functions). The initial function of bendings is selected as a quartic polynomial which turns into zero on the contour and is a specific solution to the main bending equation. Discrepancies in rotation angles from the initial polynomial are eliminated in turn on parallel edges by pairs of correcting functions of bendings and stresses which cause angular discrepancies themselves. During an infinite process of the superposition of these pairs, all discrepancies tend to zero, which gives a precise solution at the limit. The paper presents results of bending computations, bending moments, and shearing forces for square plates different thickness. The obtained results are compared with the results of other authors, as well as with Kirchhoff theory. It is shown that with the relative thicknesses less than 1/20, the results gained with both theories are almost the same.
Аннотация. В статье получены численные результаты для защемленной по контуру прямоугольной пластины Рейсснера под действием равномерной нагрузки итерационным методом суперпозиции четырех видов тригонометрических рядов (исправляющих функций). Начальная функция прогибов выбирается в виде многочлена четвертой степени, который обращается в нуль на контуре и является частным решением основного уравнения изгиба. Невязки по углам поворота от начального многочлена поочередно устраняются на параллельных краях парами исправляющих функций прогибов и напряжений, которые сами порождают угловые невязки. В ходе бесконечного процесса суперпозиции этих пар все невязки стремятся к нулю, что в пределе дает точное решение. Приведены результаты расчетов прогибов, изгибающих моментов и перерезывающих сил для квадратных пластин различной толщины. Дается сравнение с результатами других авторов, а также с теорией Кирхгофа. Показано, что при относительных толщинах, меньших 1/20 результаты по обеим теориям практически совпадают.
Modern structures widely use metallic and non-metallic materials (composite, synthetic, etc.) which have increased pliability to an interlaminar shear. Such materials are often used for making plates (panels, slabs) which are main elements in ship, aero-, and other structures as well as in nanoengineering.
Solution to 3D problems of the elasticity theory, which include problems of the plates' elastic behavior, is connected with solving a complex system of differential equations and boundary conditions. It caused the necessity of shifting from 3D problems to more simple 2D ones. Historically, a simplified theory of thin plates, based on the hypotheses of Kirchhoff-Love, was the first to put forward; it is called the classical theory. Many engineering problems were successfully solved using this theory. However, it provides poor accuracy near the plate's contour, around the points of sharp change in boundary conditions and the points of applied concentrated forces, as well as when making computations for plates
M.V. Sukhoterin, S.O. Baryshnikov, T.P. Knysh,
Admiral Makarov State University of Maritime and Inland Shipping, St. Petersburg, Russia
Д-р техн. наук, заведующий кафедрой М.В. Сухотерин,
д-р техн. наук, ректор С.О. Барышников, канд. физ.-мат. наук, заместитель директора института водного транспорта Т.П. Кныш,
Государственный университет морского и
речного флота имени адмирала
С.О. Макарова, г. Санкт-Петербург, Россия
Key words: Plate Reissner; clamped contour; bending; Fourier series; computations
Ключевые слова: Пластина Рейсснера; защемленный контур; ряды Фурье; компьютерные вычисления
Introduction
of average thickness. Therefore, there emerged a problem of shifting to more precise two-dimensional theories with using the altered hypotheses by Kirchhoff-Love. These theories were called refined theories (intermediate between the classical two-dimensional theory and the three-dimensional one).
Today, a variety of refined theories are developed and used, including theories that take into account the influence of transverse shear strain on the bending. Timoshenko [1] was the first to note the necessity of considering this influence when solving rod vibration problems. The number of refined theories today is quite large because there is no universal theory providing acceptable results for all types of problems.
The linear plate bending theory, which qualitatively refined the classical theory, was firstly put forward by Reissner [2]. Author rejected the hypothesis of the rectilinear element normality to the median surface and suggested replacing it with a hypothesis of rectilinearity of this element and introducing a law of stress variation based on thickness of the plate. Reissner, using a balance equation of the three-dimensional elasticity theory, compatibility conditions, and Castiglian's principle of minimum strain-energy, obtained new differential equations of the plate bending and the corresponding boundary conditions allowing for the transverse shear effect. The fundamental system consists of two equations. The first equation of the fourth order characterizes the plate bending. The second equation of the second order describes the stress state which is of local character and disappears quickly when moving away from the plate's edge. It increased the system's order to the sixth which allowed satisfying three boundary conditions (instead of two in the classical theory). The given and similar shear theories are often called Reissner - Mindlin [3] - Timoshenko [4] theories due to their similarity. Particularly, the difference between the theories by Reissner and Mindlin is basically values of the transverse shear coefficient: Reissner has it equal to 5/6 (= 0.833) and Mindlin to n2/12 (= 0.822), which is very close.
Variant of shear theories presented in the work Ambartsumyan [5].
Applicability limits of the theories by Kirchhoff-Love, Poisson, and Reissner, as well as revision of refined theories are discussed in the works Goldenveizer et al. [6, 7], Vasiliev [8, 9], Zhilin [10, 11] an others.
Goldenveizer et al. [6, 7] in the refined theory divide the stress state into internal and edge. Researchers use the asymptotic method in combination with the variational principle. The authors state that the Reissner system of basic equations is incorrect because it does not result from the asymptotic method. Vasiliev [8, 9] notes that there are problems which cannot be solved with the Kirchhoff theory. The author reckons that the asymptotic method of the Goldenveizer refined theory is ambiguous and approximate. In the work Vasiliev [9], the author makes an attempt to show that with the help of certain transformations the sixth order refined theories can be presented as the modern form of the classical plate theory. Zhilin [10] points out that the Reissner theory is in line with the three-dimensional elasticity theory and the Kirchhoff theory should be considered as an asymptotic consequence of the Reissner theory. In the work Zhilin [11] author warns about possible negative consequences of the formal use of the classical plate theory in the Finite Difference Method (FDM), the Finite Element Method (FEM) and other computing systems if spatial structures have rectangular plates with free support on the framework. Actually, in the shear plate theory support reactions coincide with contour transverse forces Qx and Qy, which balance the pressure on the plate. It excludes any angular forces in the case of free contour support which takes place in the Kirchhoff theory.
Revision, refinement and generalization of the theory of Reissner-Mindlin-Timoshenko and dedicated work in recent years [12-19].
There is little information about numerical results of the bending problem of a rectangular plate with a clamped edge using shear theories due to the problem's complexity. Let us note the works [20-29].
A rigidly clamped uniformly-loaded plate was examined in the work of Rudiger [20]. The author used hyperbolic-trigonometric series. The series' ratios were found by the principle of virtual displacements. Numerical computations for two kinds of rectangular plates show that allowing for the transverse shear deformation significantly affects the plate bending (no computations were done for a square plate). The works [21, 22] is based on the Ambartsumyan [5] shear theory. To solve the problem, the author used trigonometric series with hyperbolic functions in a different coordinate. Indefinite coefficients are found from the problem's boundary conditions. The problem reduces to solving an infinite system of linear algebraic equations.
In the works [23-29] various modifications of FEM were used.
Xu [23, 24] used a triangular finite element. Values of bendings and bending moments in the center of a square plate with the relative thickness of 0.1 were obtained.
In the work by Zienkiewicz et al. [25], FEM with linear quadrilateral elements is used. Numerical results are obtained for a square plate with a clamped edge with under a uniform load for relative thicknesses 0.001, 0.01, and 0.1. The number of the elements increased from 4 to 1024.
In the work Weiming and Guangsong [26] "Rational FEM" is used for Reissner plates with various boundary conditions under a uniform load and a central force. The accuracy of computations with the number of elements up to 64 is studied.
Ayad et al. [27] used the hybrid-mixed variational FEM with triangular and quadrilateral elements which is based on the Hellinger-Reissner variational principle. There are numerical results, particularly the graphs of bendings in the center of a square plate for different relative plate thicknesses when dividing the plate into 144 elements.
The work Dhananjaya [28] presents a closed form solution for equilibrium and flexibility matrices of the Mindlin-Reissner plates using the Integrated Force Method (IFM) based on 4 node rectangular elements. The author obtained the numerical results for square clamped plates with the relative thicknesses of 0.01 and 0.2 as the graphs of bendings and moments in the center for different numbers of finite elements, but, unfortunately, the scale of images is small.
In the work Aghdam et al. [29], an approximate solution is obtained for the bending of a rectangular Reissner plate with clamped edges. Resolving equations are a system of three differential equations of the second order. The solution procedure is based on using the extended Kantorovich method (EKM) to transform resolving systems of equations into ordinary differential equations.
In [30] uses the method of Bergan-Wang for moderately thick plates (modified finite integral transform method - FIT method). The results of a clamped square plate are compared with the results of the classical plate theory, Reissner-Mindlin theory and the three dimensional theory of elasticity for different relative thickness of the plate.
The goal of this work is to obtain reliable numerical results on the stress-strain state of a rectangular plate with a clamped edge allowing for the transverse shear deformation within the Reissner theory, to compare with the classical theory and with works of other authors, to determine applicability limits of the classical theory.
Methods
The fundamental system of differential equations of the Reissner elastic plate (see [2, 4]) has the
form:
, , H2 2 -v , DV2V2W = q---V2q ,
10 1-V (1)
10
V2T--- T = 0.
H2
where D = EH3/ [12(1-v2)] - cylindrical stiffness; E - Young's modulus; H - plate's thickness; v - Poisson's ratio; v2 - Laplace two-dimesional operator; W(X, Y) - function of bending of the middle surface of the plate; X, Y - coordinates; q(X, Y) - transverse load; W(X, Y) - stress function (edge potential).
For the uniform transverse load, directed at the negative side of oz axis, the system (1) in its dimensionless form will look as follows:
V2 V2 w (x, y) = - 1,
(2)
W(x,y) -a V x,y) = 0.
where w(x, y) = W/(qb4/D) - dimensionless bending function; b - the width of the plate; x =X/b, y = Y/b-dimensionless coordinates; y(x, y) = W(X, Y) /qb2 - dimensionless stress function; a = h2/10 - shear factor; h = H/b - dimensionless plate's thickness.
Boundary conditions of the rectangle plate with a clamped edge x = ± y/2, y = ± 1/2 have the form:
w = 0, <Px= 0, cpy= 0 (3)
where y = a/b - ratio of the plate's sides; a - length of the plate; yx, fy - the angles of rotation of
sections:
d
° i w2 \ W
à ( V72 \ ■
Vyw + aiV w)+ai"T" ; y 3yv 7 Sx
2a
The task is set to find bending functions w and stress functions satisfying fundamental Eq. (2) and the given conditions (3) on each edge.
To solve the problem, we use the system of functions:
Wo(x, y ) = - ^
„ 2 7
2 A
2 l y " 4
Wi
k* Akn
in
(x,y) = 2 (-l) u,
t=i,3>... cosh/t^ v
B.„
Y
xsinh/Lx - ^tanh/L cosh/Lx
cos À ky
w
2 n
(x,y) = 2 (-l)
5=i,3,...
ysinh jusy - - tanh jus cosh jusy
cosh/us v
œ
Win (xy) = 2 (-i)k*Ckn sinhPkx sinÀky
cos jusx
k=i,3,.
œ
E*
(-i)s Dsn sinh£y sin jUs
X
(4)
(5)
(6)
(7)
(8)
s=i,3,.
where wo(x, y) is the initial bending function (null approximation); win, W2n, yin, y2n are the correcting bending functions w and stress functions y; n is the number of the iteration; Akn , Bsn , Ckn , Dsn are indefinite coefficients;
, , sn * k +1 Àk = k n , u s = —, k =-
7 2
s = ■
s + i
À , Us, =Us, A=Jà'+ i, IJP;-+1.
2 2 V ol \ a
The correcting bending functions are biharmonic and turn into zero on the plate's contour; the stress functions satisfy the second basic Eq. (2).
The initial function wo(x, y) is an isolated solution to the first Eq. (2). It equals to zero on the plate's contour but causes the contour to turn, i.e. creates the main discrepancies px , (y in boundary conditions (4) which should be expanded into the Fourier series (on the edges x = - y/2 and y = - 1/2 they differ in signs):
V ol 7
x =— 2
i 1 œ
y2-i+2ai 1= 2 (-i)k aocosÀky,
4 J k=i,3....
Vy o1
7 x =— 2
aiy
Vxo1 i =-aax = 2 (-i ) Us0sinUsx,
y=ô 2
œ k* = 2 (-i ) bkosiHy.
k = i,3,
(9)
V ol=I =-^
s = i,3.
,2
x2 -— + 2a 4 i
= 2 (-i )s gsocosUsx >
s = i,3,
where ako, bk0, uso, gso - the coefficients of the decomposition.
The correcting functions during the infinite iteration process of their superposition must reduce these discrepancies to zero.
œ
s
œ
The idea of an infinite superposition of functions to elimination of the main deviations (residuals) from a private solution belongs to V.Z. Vasiliev [31].
The second discrepancy (9) (the first discrepancy will be allowed for in the next iteration to improve the series convergence) is eliminated by the first pair of correcting functions wii and yii with satisfying the conditions on the edges x = ± y/2 at the expense of coefficients Aki and Cki.
However, the functions themselves cause angular discrepancies on the edges y = ± 1/2:
bn\ 1 = Z
! =1 _
y"2 к=Тз,.
ÂfcAfci ( Y ~ I
cosh /L V 2
(10)
They should be expanded into the Fourier series in cos^x, we should invert the summation, plug expressions for the coefficients Aki, Cki, in them and put them together with the corresponding discrepancies yyo\y=i/2 from the initial polynomial (the fourth function (9)), i.e. transform into
re *
1 =Py1ll 1 +Py0| 1 = Z (_1 )S gs*iCOS Vsx . (11)
y y=2 y=2 y=2 s =u,...
where gsi* = gso +gsi are the series' ratios.
Discrepancy (11) and the third discrepancy (9) are compensated by the second pair of correcting functions W21, y 2i at the expense of coefficients Bsi and Dsi.
Besides, functions W2i and y 2i on the edges x = ± y/2 also create angular discrepancies:
^2ll L = Z
X= 2 s=1,3,...
MsBsl . , 1 ^
cosh jus
ysmhjUsy--tmhjüscoshjUsy +2ax/us cosh jusy +axDslÇsœs\\Çsy
j
(12)
which should be expanded into the Fourier series in coshy we should invert the summation, plug expressions for the coefficients Bsi, Dsi in them and put them together with the corresponding discrepancies $xo\x = y/2 from the initial polynomial (the first function (9)), i.e. transform into
re
(p*o+p*2i ) r = - S H )k*cos 4y ; ■ (13)
-L
X " 2 к=1,3.
where a k1 = am + ak1.
Discrepancies (13) are compensated by the correcting pair of w12 and y 12 of the second iteration when satisfying the boundary conditions on the edges x = ± y/2. This gives the system of two equations to determine the coefficients Ak2 , Ck2.
The discrepancies of this pair yyi2\y=i/2 will have the form similar to (10, 11):
re *
Py 12 1 1 = " Z (_1 )Sgs2COS MsX . (14)
y = 2 s = 1,3,...
Then series W22 and y22 are used to eliminate the discrepancies of this pair.
And then the process is repeated.
The convergence of the method
During the iteration process, discrepancies in boundary conditions should tend to zero, i.e. the iteration process should be convergent. Due to linearity of the problem, it is sufficient to prove, for example, that
limAn = 0 (k = 1,3,...; n = 1,2,...). (15)
n ^re v '
It is established that the coefficients Ak of two adjacent iterations are linked linearly. The dependence of Akn+i on Akn is a homogeneous infinite system of linear algebraic equations. This system should be regular [32]. Then, successive approximations will lead to a trivial solution from whatever initial values of the coefficients Ak, limited in total, we would start.
Since the discrepancy coefficients linearly depend on coefficients Ak, during the iteration process they will also tend to zero.
Analysis of the series convergence for bending moments and shearing forces
Moments M, and shearing forces Q according to [4] will have the following form:
Mv = -
^ d2 w d2 w
+ v—r- + a —t V2 w
J* m 2 ôx2
M> =-
v dx2 ay2 o w d w
a „2
+ v—T + a—tV 2 w
dy2 ' ' dx2 ' 2 dy2 d2 w
d W i i a,
dxdy
d 2w - a2--+ a3,
2 dxdy 3
M = (i -v)-+ a2-V2 w -a
dxdy dxdy
22 d w dw
~dyT "dx2"
„ d 2 d^ d 2 dw
Qx =--V2 w + , Qy =--V2 w--— .
dx dy dy dx
Here, the moments are referred to value qb2, shearing forces - to value qb; a 2 = 2a; a3 = va/(i- v).
Let us show final expressions for bending moments Mx, torsion moments Mxy and shearing forces Qx which were used for computing:
^ ai i Mx = — + -x 2 4
2 i
y — + v 4
2
x 2 - —
4 j
- V (-1)* Ah \ \2coshIx + 4aA? [ coshIx - cos^^
1 I coshA
cosh J3kx
+(l - v)(AkxsinhAkx - Ak tanh Ak cosh^x)] + Los^y
v } coshAk Ak coshj3k 1
(16)
+
2 (-1)" jus \ [~2vcosh jusy + 4aju2s cosh ¡usy - ^^-^cosh Çsy
s=i,3,.
cosh<^
+(1 - v)(fisy sinhy - p,s tanh jus coshy)]
BJZ 4a
cosh ^ I
cosh jus VjLis cosh J
COS JUSX ,
i-v
M, =--xy
2 k=i,3.
xy
00 ( / r~ —\
- 2 (~1)* |( ^j + 4aAk sinh^x + (l-v^xcosh/t^x
-2
A cosh/?/
A A2 + B2 I
+2a—r^— ~ sinh/^xUin/t^y
AA
I cosh Ak Ak Pk cosh (3k
CO * / /
- z {([('-v/)0 "Â tanh//v) + 4«//;]sinh//j' + (l - y cosh//
s=i,3,...
,y
-2 «AcoshA
£cosh£
UsBs! , 2a Us2 + £
■ +
cosh /7v 7 ¿/v£,cosh£
■sinh^ j Uin//Sx,
Qx = x - 2 (-i)
k=i,3,.
A
sinh COS|1 ^ sinh (3kx
Pk cosh (3k
Ak z + i sinh Ax
+2 2 (-i)
s=i,3,.
Us
cosh //j' - CQS|1/} cosh £j cosh .i
5.
cosh A, cosh/?,
i i cosh^j
cos\y
-+-
cosh//,. coshçs
Sin JUSX,
(17)
(18)
y
k
where Akz = AH + Ak2 +... + Akn;...; B^ = B^ + Bs2 +... + Bsn are overall coefficients in all iterations;
Pk=Pky! 2, ¿=£/2.
Let us study the convergence of functional series that occur in formulae (16), (18). The fastest to converge are the series of bending moments (16) in the center of the plate, where general terms have the order <9(l/cosh\) or 0(l/cosh/iJ, and the slowest - in the middle of clamped edges, where expressions for bending moments will take the form:
Mv
M.
* 2
^ - ^ - 2 Ê H)' hAz ,
k=1,3,.
= V
a
Z_ 16
2 Л
(19)
- 2v Ê (-1) mAs* ■
s=1,3,.
The coefficients Akn (5) and Bsn (6) have similar estimations Akn = 0(1 / k2), Bsn = O(1 / s2), and
the corresponding series that occurs in (19), starting from some number, converges not worse than
« *
alternating series ^ (-1)m / m ■ Although such a series converges slowly, it is good for computations
m=1,3,...
because pursuing the Leibniz theory, it is possible to estimate the inaccuracy of computing its sum (from the moment when the series terms start to decay).
Let us note that in angular points of the plate
a
Mv
'±y; * i4
2 2,
h2
2 10 (1 -v)' while for the Kirchhoff plate these moments equal to zero.
The most slowly the series of shearing forces converges on the side x = ± y/2:
(20)
Qx
'y Л
2; У,
V2 У
= y - 2 ÊÊ (-1)
2 Ê
s=1,3,...
k=1,3,,
r
Л:
tan h Як -
^tanhÄ
Pk У
tanh ßk
h Pk
COSÀky
Ms
cosh jUsy cosh ¿;sy cosh jus cosh|^
^ y
1 cosh Çsy m) cosh|v
(21)
It is proved that the series for the shear forces converge no worse than numerical
series
ê 1/
m
m=1,3,.
Similar conclusions are also valid for bending moments My, torsion moments Mxy and shearing forces Qy.
Thus, a series of moments and shear forces are quite suitable for computer calculations.
Results and Discussion
Numerical results were obtained for square plates with relative thicknesses h = 0.05, 0.1, 0.2, 0.3 and Poisson's ratio v = 0.3. Up to 150 terms were held in the series depending on the speed of convergence of a particular series. The process converged in a geometrical progression with the ratio < 1/3 for all considered examples. The discrepancy coefficients were printed out in every iteration. The calculation stopped after ten iterations, when all discrepancies were nearly equal to zero; in the process, the overall coefficients Akz and Bsz were calculated (due to linearity of the problem), using which the bendings, bending moments MX, and shearing forces Qx in different points of plates were obtained. Near the contour, computational points clustered in order to refine the influence of ends.
In Table 1 the first five coefficients Akz ( = Bs£) are given, as well as their values with k = 299 (150 terms of the series) for different relative thicknesses of a square plate.
The table shows that the highest are the first coefficients; the second are lower in an absolute value approximately by two orders, then the coefficients decay, keeping the negative sign.
Tables 2-5 show values of relative bendings; Tables 6-9 show values of bending moments Mx, Tables 10-13 show values of shearing forces Qx for square Reissner plates with the relative thicknesses of 0.05, 0.1, 0.2, 0.3.
Table 1. Values of coefficients Aki for the bending functions of a square plate (Reissner -CCCC, q = const)
h k
1 3 5 7 9 299
0.05 1.774X10-2 -1.063X10-4 -3.966X10"5 -1.173x10-5 -3.971X10-6 -1.013X10-9
0.1 1.726X10"2 -5.582X10-5 -1.923X10"5 -4.715X10-6 -1.564X10-6 -4.941X10-9
0.2 1.543*10-2 -5.191X10-5 -3.400x10-5 -2.000x10-5 -1.356x10-5 -2.229X10-8
0.3 1.248*10-2 -2.531X10-4 -1.231X10-4 -6.974x10-5 -4.492X10-5 -5.282X10-8
Table 2. Values of bendings referred to value qb4 / D x10-5 of a square plate h = 0.05 (Reissner - CCCC, q = const)
y x
0 0.1 0.2 0.3 0.4 0.42 0.44 0.46 0.48 0.5
0 -132.70 -123.80 -98.33 -60.94 -21.70 -15.11 -9.40 -4.79 -1.56 0
0.1 -123.80 -115.50 -91.84 -57.01 -20.35 -14.19 -8.83 -4.51 -1.47 0
0.2 -98.33 -91.84 -73.23 -45.68 -16.43 -11.48 -7.17 -3.67 -1.21 0
0.3 -60.94 -57.01 -45.67 -28.70 -10.42 -7.30 -4.57 -2.35 -0.78 0
0.4 -21.70 -20.35 -16.43 -10.42 -3.77 -2.62 -1.62 -0.82 -0.26 0
0.42 -15.11 -14.19 -11.48 -7.30 -2.62 -1.82 -1.11 -0.55 -0.17 0
0.44 -9.40 -8.83 -7.17 -4.57 -1.62 -1.11 -0.67 -0.32 -0.09 0
0.46 -4.79 -4.51 -3.67 -2.35 -0.82 -0.55 -0.32 -0.14 -0.03 0
0.48 -1.56 -1.47 -1.21 -0.78 -0.26 -0.17 -0.09 -0.03 -0.001 0
0.5 0 0 0 0 0 0 0 0 0 0
Table 3. Values of bendings referred to value qb4 / D *10~5 of a square plate h = 0.1 (Reissner - CCCC, q = const)
y x
0 0.1 0.2 0.3 0.4 0.42 0.44 0.46 0.48 0.5
0 -150.50 -140.90 -113.60 -72.98 -28.88 -21.07 -14.04 -8.01 -3.23 0
0.1 -140.90 -132.00 -106.50 -68.56 -27.21 -19.88 -13.26 -7.57 -3.06 0
0.2 -113.60 -106.50 -86.19 -55.75 -22.32 -16.35 -10.94 -6.28 -2.55 0
0.3 -72.98 -68.56 -55.75 -36.36 -14.73 -10.83 -7.28 -4.20 -1.72 0
0.4 -28.88 -27.21 -22.32 -14.74 -6.03 -4.44 -2.98 -1.72 -0.70 0
0.42 -21.08 -19.88 -16.35 -10.83 -4.44 -3.26 -2.19 -1.26 -0.51 0
0.44 -14.04 -13.26 -10.94 -7.28 -2.98 -2.19 -1.46 -0.83 -0.34 0
0.46 -8.01 -7.57 -6.28 -4.20 -1.72 -1.26 -0.83 -0.47 -0.19 0
0.48 -3.23 -3.06 -2.55 -1.72 -0.70 -0.51 -0.34 -0.19 -0.07 0
0.5 0 0 0 0 0 0 0 0 0 0
Table 4. Values of bendings referred to value qb4 / D *10~5 of a square plate h = 0.2 (Reissner - CCCC, q = const)
y x
0 0.1 0.2 0.3 0.4 0.42 0.44 0.46 0.48 0.5
0 -217.20 -205.40 -171.00 -118.20 -55.60 -43.20 -31.22 -19.87 -9.38 0
0.1 -205.40 -194.30 -161.90 -112.10 -52.91 -41.14 -29.77 -18.97 -8.96 0
0.2 -171.00 -161.90 -135.40 -94.29 -44.94 -35.04 -25.43 -16.26 -7.72 0
0.3 -118.20 -112.10 -94.29 -66.38 -32.24 -25.27 -18.45 -11.88 -5.68 0
0.4 -55.60 -52.91 -44.94 -32.24 -16.22 -12.85 -9.50 -6.21 -3.02 0
0.42 -43.20 -41.14 -35.04 -25.27 -12.85 -10.22 -7.59 -4.99 -2.45 0
0.44 -31.22 -29.77 -25.43 -18.45 -9.50 -7.59 -5.68 -3.76 -1.87 0
0.46 -19.87 -18.97 -16.26 -11.88 -6.21 -4.99 -3.76 -2.53 -1.28 0
0.48 -9.38 -8.96 -0.77 -5.68 -3.02 -2.45 -1.87 -1.28 -0.67 0
0.5 0 0 0 0 0 0 0 0 0 0
Table 5. Values of bendings referred to value qb4 / D *10~5 of a square plate h = 0.3 (Reissner - CCCC, q = const)
y x
0 0.1 0.2 0.3 0.4 0.42 0.44 0.46 0.48 0.5
0 -324.60 -309.10 -263.40 -190.70 -98.19 -78.38 -58.47 -38.64 -19.07 0
0.1 -309.10 -294.40 -251.20 -182.20 -94.11 -75.19 -56.14 -37.13 -18.34 0
0.2 -263.40 -251.20 -215.00 -157.00 -81.92 -65.62 -49.13 -32.59 -16.16 0
0.3 -190.70 -182.20 -157.00 -116.00 -61.80 -49.78 -37.50 -25.04 -12.51 0
0.4 -98.19 -94.11 -81.92 -61.80 -34.29 -27.96 -21.36 -14.50 -7.38 0
0.42 -78.38 -75.19 -65.62 -49.78 -27.96 -22.90 -17.59 -12.02 -6.16 0
0.44 -58.47 -56.14 -49.13 -37.50 -21.36 -17.59 -13.60 -9.38 -4.87 0
0.46 -38.63 -37.13 -32.59 -25.04 -14.50 -12.02 -9.38 -6.55 -3.46 0
0.48 -19.07 -18.34 -16.16 -12.51 -7.38 -6.16 -4.87 -3.46 -1.90 0
0.5 0 0 0 0 0 0 0 0 0 0
Table 6. Values of bending moments Mx referred to value qb2*10~6 of a square plate h = 0.05 (Reissner -CCCC, q = const)
y x
0 0.1 0.2 0.3 0.4 0.42 0.44 0.46 0.48 0.5
0 -23110 -21450 -15860 -4413 16220 21860 28090 34980 42520 50670
0.1 -21460 -19950 -14830 -4244 15040 20340 26220 32720 39870 47610
0.2 -16570 -15480 -11690 -3555 11790 16090 20890 26230 32160 38640
0.3 -8553 -8087 -6281 -1850 7450 10150 13200 16630 20470 24730
0.4 2189 1948 1463 1554 3772 4614 5593 6693 7905 9298
0.42 4603 4222 3282 2498 3311 3802 4400 5080 5821 6691
0.44 7088 6570 5183 3542 2984 3142 3389 3696 4033 4465
0.46 9643 8991 7167 4688 2800 2645 2574 2563 2589 2717
0.48 12290 11510 9251 5945 2745 2290 1927 1646 1445 1466
0.5 15120 14210 11520 7342 2713 1944 1270 720 355 357
Table 7. Values of bending moments Mx referred to value qb2*10~6 of a square plate h = 0.1
(Reissner -CCCC, q = const)
y x
0 0.1 0.2 0.3 0.4 0.42 0.44 0.46 0.48 0.5
0 -23630 -21920 -16210 -4677 15810 21340 27420 34060 41260 48940
0.1 -22040 -20470 -15200 -4482 14730 19930 25670 31960 38790 46090
0.2 -17300 -16130 -12130 -3732 11760 16030 20760 25970 31670 37810
0.3 -9528 -8962 -6858 -1970 7824 10610 13740 17230 21100 25370
0.4 952 822 704 1454 4523 5566 6788 8212 9870 11830
0.42 3343 3072 2500 2400 4104 4800 5650 6681 7942 9514
0.44 5840 5428 4398 3444 3787 4140 4622 5274 6145 7307
0.46 8469 7913 6422 4600 3567 3573 3687 3956 4498 5432
0.48 11280 10580 8611 5876 3417 3067 2806 2635 2729 3348
0.5 14360 13540 11020 7338 3196 2620 1806 1456 472 1429
Table 8. Values of bending moments Mx referred to value qb2*10~6 of a square plate h = 0.2
(Reissner -CCCC, q = const)
y x
0 0.1 0.2 0.3 0.4 0.42 0.44 0.46 0.48 0.5
0 -25290 -23450 -17440 -5692 14220 19410 25050 31120 37630 44580
0.1 -23830 -22120 -16500 -5461 13380 18320 23680 29480 35700 42330
0.2 -19470 -18120 -13630 -4635 11080 15260 19840 24810 30190 35910
0.3 -12250 -11440 -8666 -2823 8012 11010 14330 18000 22030 26380
0.4 -2112 -1969 -1306 598 5157 6620 8348 10380 12750 15520
0.42 321 318 516 1554 4679 5780 7138 8809 10840 13400
0.44 2925 2770 2487 2627 4242 4946 5887 7155 8833 10880
0.46 5732 5418 4630 3834 3867 4127 4573 5333 6581 8066
0.48 8787 8305 6982 5196 3587 3366 3240 3296 3800 5341
0.5 12220 11510 9532 6654 3494 2992 2037 878 322 5714
Table 9. Values of bending moments Mx referred to value qb2*10~6 of a square plate h = 0.3 (Reissner -CCCC, q = const)
y x
0 0.1 0.2 0.3 0.4 0.42 0.44 0.46 0.48 0.5
0 -27650 -25710 -19460 -7598 11770 16720 22050 27780 33890 40490
0.1 -26300 -24480 -18580 -7348 11120 15860 20980 26490 32380 38680
0.2 -22230 -20750 -15890 -6489 9337 13460 17950 22800 28020 33500
0.3 -15340 -14370 -11160 -4692 6847 9979 13430 17230 21360 25720
0.4 -5251 -4951 -3860 -1340 4120 5845 7875 10250 12980 16170
0.42 -2763 -2613 -2000 -380 3566 4915 6572 8595 11040 14270
0.44 -82 -88 31 715 3028 3942 5148 6751 8832 11260
0.46 2818 2649 2252 1970 2552 2958 3594 4616 6205 7570
0.48 5965 5628 4696 3416 2215 2050 2007 2187 2871 4716
0.5 9552 8928 7253 4879 2245 1964 613 -1191 -1472 12860
Table 10. Values of shearing forces Qx referred to value qb of a square plate h =0.05 (Reissner -CCCC, q = const)
y x
0 0.1 0.2 0.3 0.4 0.42 0.44 0.46 0.48 0.5
0 0 0.052 0.112 0.191 0.297 0.322 0.349 0.376 0.404 0.429
0.1 0 0.047 0.103 0.177 0.279 0.303 0.329 0.356 0.384 0.408
0.2 0 0.033 0.076 0.136 0.224 0.246 0.269 0.294 0.320 0.343
0.3 0 0.011 0.030 0.068 0.132 0.149 0.166 0.186 0.206 0.227
0.4 0 -0.021 -0.034 -0.027 0.010 0.020 0.031 0.041 0.052 0.067
0.42 0 -0.028 -0.048 -0.049 -0.016 -0.007 0.003 0.013 0.022 0.036
0.44 0 -0.035 -0.062 -0.070 -0.042 -0.032 -0.022 -0.012 -0.003 0.010
0.46 0 -0.039 -0.072 -0.087 -0.063 -0.053 -0.042 -0.031 -0.021 -0.008
0.48 0 -0.035 -0.066 -0.083 -0.066 -0.057 -0.047 -0.036 -0.027 -0.013
0.5 0 0 0 0 0 0 0 0 0 0
Table 11. Values of shearing forces Qx referred to value qb of a square plate h = 0.1 (Reissner - CCCC, q = const)
y x
0 0.1 0.2 0.3 0.4 0.42 0.44 0.46 0.48 0.5
0 0 0.051 0.112 0.189 0.292 0.316 0.340 0.366 0.390 0.412
0.1 0 0.047 0.103 0.176 0.274 0.298 0.322 0.346 0.370 0.392
0.2 0 0.034 0.077 0.137 0.222 0.243 0.265 0.287 0.309 0.331
0.3 0 0.013 0.034 0.072 0.136 0.152 0.169 0.187 0.207 0.226
0.4 0 -0.016 -0.023 -0.012 0.024 0.035 0.046 0.059 0.074 0.092
0.42 0 -0.021 -0.033 -0.028 0.002 0.012 0.022 0.035 0.049 0.066
0.44 0 -0.024 -0.041 -0.041 -0.016 -0.008 0.001 0.012 0.025 0.043
0.46 0 -0.024 -0.043 -0.047 -0.029 -0.022 -0.014 -0.005 0.006 0.023
0.48 0 -0.018 -0.033 -0.038 -0.027 -0.023 -0.018 -0.013 -0.005 0.009
0.5 0 0 0 0 0 0 0 0 0 0
Table 12. Values of shearing forces Qx referred to value qb of a square plate h = 0.2 (Reissner - CCCC, q = const)
y x
0 0.1 0.2 0.3 0.4 0.42 0.44 0.46 0.4B 0.5
0 0 0.051 0.110 0.1B4 0.277 0.29B 0.319 0.340 0.362 0.3B2
0.1 0 0.047 0.103 0.172 0.262 0.2B2 0.303 0.324 0.345 0.366
0.2 0 0.036 0.0B0 0.139 0.21B 0.237 0.256 0.276 0.296 0.316
0.3 0 0.01B 0.044 0.0B4 0.146 0.162 0.179 0.197 0.215 0.235
0.4 0 -0.002 0.003 0.019 0.055 0.065 0.07B 0.092 0.109 0.12B
0.42 0 -0.004 -0.004 0.00B 0.037 0.046 0.057 0.070 0.0B6 0.105
0.44 0 -0.006 -0.00B -0.001 0.021 0.02B 0.037 0.04B 0.063 0.0B1
0.46 0 -0.007 -0.010 -0.007 0.00B 0.013 0.019 0.02B 0.040 0.057
0.4B 0 -0.005 -0.00B -0.007 0.000 0.003 0.006 0.010 0.01B 0.032
0.5 0 0 0 0 0 0 0 0 0 0
Table 13. Values of shearing forces Qx referred to value qb of a square plate h = 0.3 (Reissner - CCCC, q = const)
y x
0 0.1 0.2 0.3 0.4 0.42 0.44 0.46 0.4B 0.5
0 0 0.051 0.109 0.179 0.266 0.2B5 0.304 0.324 0.344 0.365
0.1 0 0.04B 0.102 0.169 0.253 0.272 0.291 0.311 0.331 0.351
0.2 0 0.03B 0.0B2 0.140 0.216 0.234 0.252 0.271 0.290 0.310
0.3 0 0.023 0.052 0.093 0.154 0.170 0.1B6 0.204 0.222 0.242
0.4 0 0.006 0.017 0.036 0.073 0.0B3 0.096 0.110 0.127 0.146
0.42 0 0.004 0.011 0.026 0.055 0.064 0.075 0.0B9 0.104 0.123
0.44 0 0.002 0.006 0.016 0.03B 0.046 0.055 0.066 0.0B0 0.099
0.46 0 0.000 0.002 0.00B 0.023 0.02B 0.034 0.043 0.055 0.072
0.4B 0 0.000 0.000 0.003 0.010 0.012 0.015 0.020 0.027 0.042
0.5 0 0 0 0 0 0 0 0 0 0
Hereinafter CCCC -piate is ciamped on ai! four edges.
Figure 1 illustrates bending lines of square Reissner plates under a uniform load at the section y = 0. Curve 1(the dotted line) represents the Kirchhoff plate, the following numbers are given to the Reissner plates with relative thickness h = 0.05, 0.1, 0.2, 0.3. Figures 2, 3 illustrate curves of bending moments Mx for these plates at the clamped section x = ± y/2, and Figures 4, 5 - on the adjacent side y = ± 1/2. The curves numeration is similar to Figure 1.
coordinate, x
Figure 1. Lines of relative bendings of square plates (Reissner -CCCC, q = const)
at the section y = 0
coordínate, у
-а
О-----Г---1---.--го о,] о_а о.? п.-*
Figure 2. Curves of bending moments Mx of square plates (Reissner -CCCC, q = const) at the
section x = ± y/2
Figure 3. Magnified fragment of the curve of bending moments Mx of square plates (Reissner -CCCC, q = const) at the section x = ± y/2 near the plate's angle
Figure 4. Curves of bending moments Mx of square plates (Reissner -CCCC, q = const)
at the section y = ± 1/2
Figure 5. Magnified fragment of the curve of bending moments Mx of square plates (Reissner -CCCC, q = const) at the section y = ± 1/2 near the plate's angle
The computations and graphs show that with small relative thicknesses h < 1/20, the results for the Kirchhoff and Reissner plates are almost equal. With the increase in the relative thickness, relative bendings also increase. Absolute bendings, of course, decrease, because they are obtained by multiplying relative bendings with the expression qb4/D = 12(1-v2) qb/(Eh3). If the bending in the center for the square Kirchhoff plate equals to 0.00126 [4], for the Reissner plates with thickness h = 0.05, 0.1, 0.2, 0.3 it amounts to 0.001327, 0.001505, 0.002172, 0.003246 respectively.
Thus, the Kirchhoff plate can be considered as a limit behavior of the Reissner plate, when h ^ 0.
Bending moments in the middle of clamped edges decrease when h increases, but they rise when closer to angles of the plate. In angular points the bending moments different from zero and increase in a proportion to the square of relative thickness (see (17)). This is the fundamental difference from the Kirchhoff plate.
In the center of the plate, bending moments slightly increase in an absolute value when h increases; shearing forces change moderately.
The Shirakawa work [21] presents calculated correlations w/wci of the plate's bendings within the shear theory to the bendings within the classical theory for central points of the median zo/h = 0 and top zo/h = 0.5 surfaces. For a square plate with relative thicknesses h/a = 0.1, 0.2, 0.3 these values amounted to = 1.25, 1.85, 2.95 and 1.2, 1.7, 2.6 respectively. In this paper, average values of thickness amounted to 1.2, 1.7, 2.6, i.e. were equal to the corresponding values [21] on the plate's surface.
In [22] shows diagrams of shear forces on the contour of the uniformly loaded clamped square plates with the relative thickness of 0.001, 0.04, 0.1 and 0.3. These results practically coincide with those obtained in the present work.
In the works of Xu [23, 24], for a square plate with the relative thickness of 0.1 the bending in the center amounted to 0.001499 and the bending moment amounted to 0.0231. In the work [25] of Zienkiewicz et al. 1993 with the grid of 1024 elements these values amounted 0.00150442 and 0.023195 respectively, opposing to 0.0015050 and 0.023630 in our work. It indicates a good agreement of the results.
In the work [26] by Weiming and Guangsong, the bending in the center of a square plate with the relative thickness h/a = 0.3 amounted to 0.0028997 and the bending moment amounted to 0.023538, while in this work - to 0.0032460 and 0.027650 respectively. The values in the aforementioned work were obtained using FEM with the grid of 8*8 elements; however, they poorly correlate with our results.
In the work of Ayad et al. [27], the maximum bending for a square clamped plate with the relative thickness of 0.1 amounted to = 0.001575 (according to the graph).
The work of Dhananjaya [28] provides numerical results for square plates with the relative thicknesses of 0.01 and 0.2, represented as graphs of bendings and moments in the center depending on the number of finite elements. The scale of the images does not allow making a proper comparison, although the proximity of the results is obvious.
In the article [30] for a square plate with the relative thickness of 0.1 the bending in the center amounted to 0.0013636 (method FIT), 0.0015040 (FEM, theory of Reissner - Mindlin), 0.0014918 (FEM, 3D solution). The last two values are in good agreement with the value 0.0015050 obtained in the present work.
Conclusions
1. In the present work the iterative process of superposition of hyperbolic-trigonometric series to solve the problem of bending rectangular Reissner plate clamped along the contour as a result of the action of a uniform load is constructed and its convergence to the exact solution of the problem is proved.
2. Increasing the number of members in the ranks and the number of iterations, we can obtain the numerical solution with high accuracy having used a simple algorithm.
3. The convergence of the series and their suitability for computations of bending moments and shear forces an investigated.
4. Numerous examples of calculating deflections, bending moments and shear forces for square plates with different relative thickness are given.
5. It is shown that in case of small relative thicknesses theories of Reissner and Kirchhoff produced the same results.
6. We also analyzed the differences of the above theories when changing the relative thickness.
Acknowledgement
The authors express their gratitude to the government for financial support of this work.
The authors would like to gratefully acknowledge the unknown reviewers for their review and helpful comments.
References
1. Timoshenko S.P. On the Correction for Shear of the Differential Equation for Transverse Vibration of Prismatic Bars. Philosophical Magazine. 1921. No. 245(41). Series 6. Pp. 744-746.
2. Reissner E. The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics. 1945. No. 1(12). Pp. 69-77.
3. Mindlin R.D. Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. Journal of Applied Mechanics. 1951. No. 1(18). Pp. 31-38.
4. Timoshenko S., Woinowsky-Krieger S. Theory of Plates and Shells. McGraw-Hill Book Company. New York, Toronto, London.1959. 595 p.
5. Ambartsumyan S.A. Theory of Anisotropic Plates. Technomic Publishing Co. New York, 1970. 255 p.
6. Goldenveizer A.L., Kaplunov J.D., EV Nolde E.V. Asymptotic analysis and refinement of Timoshenko-Reisner-type theories of plates and shells. Mechanics of Solids. (Izvestiya AN SSSR, Mekhanika Tverdogo Tela) .1990. No. 6(25). Pp. 126-139.
7. Goldenveizer A.L, Kaplunov J.D. and Nolde E.V. On Timoshenko-Reissner type theories of plates and shells, International Journal of Solids and Structures. 1993. No. 5(30). Pp. 675-694.
8. Vasiliev V.V. An asymptotic method of justification of the plate theory. Mechanics of Solids. (Izvestiya AN SSSR, Mekhanika Tverdogo Tela). 1997. No. 3. Pp. 150-155.
9. Vasiliev V.V. Modern conceptions of plate theory. Composite Structures. 2000. No. 1(48). Pp. 39-48.
10. Zhilin P.A. The view on Poisson's and Kirchhoffs theories of plates in terms of modern theory of plates. 1992. Mechanics of Solids (Izvestia RAN, Mechanika Tverdogo Tela). No. 3. Pp. 48-64.
11. Zhilin P.A. On the classical theory of plates and the Kelvin-Teit transformation. Mechanics of Solids. (Izvestia RAN, Mechanika Tverdogo Tela). 1995. No. 4. Pp. 133-140.
12. Shimpi R.P., Patel H.G., Arya H. New first order shear deformation plate theories. Journal of Applied Mechanics. 2007. No. 74. Pp. 523-533.
13. Vijayakumar K. A relook at Reissner's theory of plates in bending. Archive of Applied Mechanics. 2011. No. 11(81). Pp. 1717-1724.
14. Tovstik P.E., Tovstik T.P. A thin-plate bending equation of second-order accuracy. Doklady Physics. 2014. No. 8(59). Pp. 389-392.
15. Martínez Valle J.M., Martínez-Jiménez P. Modified Bolle - Reissner Theory of Plates Including Transverse Shear Deformations. Latin American Journal of Solids and Structures. 2015. No. 2(12). Pp. 295-316.
16. Nareen K., Shimpi R.P. Refined hyperbolic shear deformation plate theory. Archive Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science. 2015. Vol. 203-210. Pp. 1989-1996
17. Senjanovic I., Vladimir N., Tomic M. On new first-order shear deformation plate theories. Mechanics Research Communications. 2016. Vol. 73. Pp. 31-38.
18. Huu-Tai Thai, Trung-Kien Nguyen, Thuc P.Vo, Tuan Ngo. A new simple shear deformation plate theory. Composite Structures. 2017. Vol. 171. Pp. 277-285.
Литература
1. Timoshenko S.P. On the correction for shear of the differential equation for transverse vibration of prismatic bars // Philosophical Magazine. 1921. № 245(41). Series 6. Pp. 744-746.
2. Reissner E. The effect of transverse shear deformation on the bending of elastic plates // Journal of Applied Mechanics. 1945. № 1(12). Pp. 69-77.
3. Mindlin R.D. Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates // Journal of Applied Mechanics. 1951. № 1(18). Pp. 31-38.
4. Timoshenko S., Woinowsky-Krieger S. Theory of Plates and Shells. McGraw-Hill Book Company. New York, Toronto, London. 1959. 595 p.
5. Ambartsumyan S.A. Theory of Anisotropic Plates, Technomic Publishing Co. New York, 1970. 255 p.
6. Goldenveizer A.L., Kaplunov J.D., EV Nolde E.V. Asymptotic analysis and refinement of Timoshenko-Reisner-type theories of plates and shells // Mechanics of Solids. (Izvestiya AN SSSR, Mekhanika Tverdogo Tela).1990. № 6(25). Pp. 126-139.
7. Goldenveizer A.L, Kaplunov J.D., Nolde E.V. On Timoshenko-Reissner type theories of plates and shells // International Journal of Solids and Structures. 1993. No. 5(30). Pp. 675-694.
8. Vasiliev V.V. An asymptotic method of justification of the pl ate theory // Mechanics of Solids. (Izvestiya AN SSSR, Mekhanika Tverdogo Tela). 1997. № 3. Pp. 150-155.
9. Vasiliev V.V. Modern conceptions of plate theory // Composite Structures. 2000. № 1(48). Pp. 39-48.
10. Zhilin P.A. The view on Poisson's and Kirchhoff s theories of plates in terms of modern theory of plates // Mechanics of Solids (Izvestia RAN, Mechanika Tverdogo Tela). 1992. № 3. Pp. 48-64.
11. Zhilin P.A. On the classical theory of plates and the Kelvin-Teit transformation // Mechanics of Solids. (Izvestia RAN, Mechanika Tverdogo Tela). 1995. № 4. Pp. 133-140.
12. Shimpi R.P., Patel H.G., Arya H. New first order shear deformation plate theories // Journal of Applied Mechanics. 2007. № 74. Pp. 523-533.
13. Vijayakumar K. A relook at Reissner's theory of plates in bending // Archive of Applied Mechanics. 2011. № 11(81). Pp. 1717-1724.
14. Tovstik P.E., Tovstik T.P. A thin-plate bending equation of second-order accuracy // Doklady Physics. 2014. № 8(59). Pp. 389-392.
15. Martínez Valle J.M., Martínez-Jiménez P. Modified Bolle -Reissner Theory of Plates Including Transverse Shear Deformations // Latin American Journal of Solids and Structures. 2015. № 2(12). Pp. 295-316.
16. Nareen K., Shimpi R.P. Refined hyperbolic shear deformation plate theory. Archive Proceedings of the Institution of Mechanical Engineers. Part C. // Journal of Mechanical Engineering Science. 2015. Vol. 203-210. Pp. 1989-1996
17. Senjanovic I., Vladimir N., Tomic M. On new first-order shear deformation plate theories // Mechanics Research Communications. 2016. Vol. 73. Pp. 31-38.
18. Huu-Tai Thai, Trung-Kien Nguyen, Thuc P.Vo, Tuan Ngo. A new simple shear deformation plate theory // Composite Structures. 2017. Vol. 171. Pp. 277-285.
19. Morozov N.F., Tovstik P.E., Tovstik T.P. Generalized Timoshenko-Reissner model for a multilayer plate. Mechanics of Solids. 2016. No. 5(51). Pp. 527-537.
20. Rudiger D. Eingespannte Rechteckplatten mit Schubverzerrungen. Zeitschrift für Angewandte Mathematik und Mechanik. 1963. No. 1-2(43). Pp. 71-79.
21. Shirakawa K. Bending of plates based on improved theory. Mechanics Research Communications. 1983. No. 4(10). Pp. 205-211.
22. Belubekyan M.V., Sanoyan Yu.G. Raschet izgiba zhestko zakreplennoy plastiny pri ravnomernoy nagruzke po utochnennoy teorii [Calculation of the bending of a rigidly fixed plate with uniform load according to the refined theory]. Trudy IV Vserossiyskoy nauchnoy konferentsii, ch. 1. [Proceedings of All-Russian Scientific Conference]. Samara: Izd-vo SamGTU, 2007. Pp. 42-45. (rus)
23. Xu Z. A simple and efficient triangular finite element for plate bending. Acta Mechanica Sinica.1986. No. 2(2). Pp. 185-192.
24. Xu Z. A thick-thin triangular plate element. International Journal for Numerical Methods in Engineering.1992. No. 5(33). Pp. 963-973.
25. Zienkiewicz O.C., Xu Z., Zeng L.F., Samuelsson A. and Wiberg N-E. Linked interpolation for Reissner-Mindlin plate elements: Part I - A simple quadrilateral. International Journal for Numerical Methods in Engineering. 1993. No. 18(36). Pp. 3043-3056.
26. Weiming S., Guangsong Y. Rational finite method for elastic bending of Reissner plates. Applied Mathematics and Mechanics (English Edition). 1999. No. 2(20). Pp. 193-199.
27. Ayad R., Dhatt G. and Batoz J.L. A new hybrid-mixed variational approach for Reissner - Mindlin plates. The MiSP model. International Journal for Numerical Methods in Engineering. 1998. No. 7(42). Pp. 1149-1179.
28. Dhananjaya H.R.. Closed form solutions for element matrices of 4-node rectangular plate element using IFM. Electronic Journal of Structural Engineering. 2010. No. 10. Pp. 1-10.
29. Aghdam M.M., Shakeri M., Fariborz S.J. Solution to Reissner Plate with Clamped Edges. Journal of Engineering Mechanics. 1996. No. 7(122). Pp. 679-682.
30. Hassan K., Guirguis S., El-Hamouly H. Bending of an elastic rectangular clamped plate using Bergan-Wang approach. African Journal of Engineering Research. 2017. Vol. 5(1). Pp. 7-17.
31. Vasilyev V.Z. Primeneniye metoda nalozheniya nepolnykh resheniy v sluchaye pervoy osnovnoy zadachi dlya polubeskonechnogo tsilindra [The application of the method of imposing incomplete solutions in the case of the first fundamental problem for a semi-infinite cylinder]. Sbornik "Mekhanika sterzhnevykh sistem i sploshnykh sred". Trudy Leningradskogo inzhenerno-stroitelnogo institute [Collection "Mechanics of rod systems and continuous media". Proceedings of the Leningrad Civil Engineering Institute]. 1973. No. 73. Pp. 15-22.
32. Kantorovich L.V., Krylov V.I. Approximate Methods of Higher Analysis. Translated by C. D. Benster. Groningen, 1958. 695 p.
Mikhail Sukhoterin,
+7(921)579-25-35; mv@sukhoterin.com
Sergey Baryshnikov, +7(812)251-12-21; rector@gumrf.ru
Tatiana Knysh,
+7(812)748-96-73; KnyshTP@gumrf.ru
19. Morozov N.F., Tovstik P.E., Tovstik T.P. Generalized Timoshenko-Reissner model for a multilayer plate // Mechanics of Solids. 2016. № 5(51). Pp. 527-537.
20. Rudiger D. Eingespannte Rechteckplatten mit Schubverzerrungen // Zeitschrift fur Angewandte Mathematik und Mechanik. 1963. No. 1-2(43). Pp. 71-79.
21. Shirakawa K. Bending of plates based on improved theory // Mechanics Research Communications. 1983. № 4(10). Pp. 205-211.
22. Белубекян М.В., Саноян Ю.Г. Расчет изгиба жестко закрепленной пластины при равномерной нагрузке по уточненной теории // Труды IV Всероссийской научной конференции, ч. 1. Самара: Изд-во СамГТУ. 2007. С. 42-45.
23. Xu Z. A simple and efficient triangular finite element for plate bending // Acta Mechanica Sinica.1986. № 2(2). Pp. 185-192.
24. Xu Z. A thick-thin triangular plate element // International Journal for Numerical Methods in Engineering. 1992. № 5(33). Pp. 963-973.
25. Zienkiewicz O.C., Xu Z., Zeng L.F., Samuelsson A. and Wiberg N-E. Linked interpolation for Reissner-Mindlin plate elements: Part I - A simple quadrilateral // International Journal for Numerical Methods in Engineering. 1993. № 18(36). Pp. 3043-3056.
26. Weiming S., Guangsong Y. Rational finite method for elastic bending of Reissner plates // Applied Mathematics and Mechanics (English Edition). 1999. № 2(20). Pp. 193-199.
27. Ayad R., Dhatt G. and Batoz J.L. A new hybrid-mixed variational approach for Reissner - Mindlin plates. The MiSP model // International Journal for Numerical Methods in Engineering. 1998. № 7(42). Pp. 1149-1179.
28. Dhananjaya H.R. Closed form solutions for element matrices of 4-node rectangular plate element using IFM // Electronic Journal of Structural Engineering. 2010. № 10. Pp. 1-10.
29. Aghdam M.M., Shakeri M., Fariborz S.J. Solution to Reissner Plate with Clamped Edges // Journal of Engineering Mechanics. 1996. № 7(122). Pp. 679-682.
30. Hassan K., Guirguis S., El-Hamouly H. Bending of an elastic rectangular clamped plate using Bergan-Wang approach // African Journal of Engineering Research. 2017. Vol. 5(1). Pp. 7-17.
31. Васильев В.З. Применение метода наложения неполных решений в случае первой основной задачи для полубесконечного цилиндра // Сборник "Механика стержневых систем и сплошных сред". Труды Ленинградского инженерно-строительного института. 1973. № 73. С. 15-22.
32. Kantorovich L.V., Krylov V.I. Approximate Methods of Higher Analysis. Translated by C. D. Benster. Groningen, 1958. 695 p.
Михаил Васильевич Сухотерин,
+7(921)579-25-35;
эл. почта: mv@sukhoterin.com
Сергей Олегович Барышников, +7(812)251-12-21; эл. почта: rector@gumrf.ru
Татьяна Петровна Кныш,
+7(812)748-96-73;
эл. почта: KnyshTP@gumrf.ru
© Бик^ет М.У.ЗагуэИт^ Б.О.ДпуэИ Т.Р., 2017
