Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 153 (2016) 444 – 449
XXV Polish – Russian – Slovak Seminar “Theoretical Foundation of Civil Engineering”
Dissection Method Applications for Complex Shaped Membranes and Plates R.F. Gabbasova,V.V. Filotova,N.B. Ovarovaa, A.M. Mansoura* a
Moscow State University of Civil Engineering (National Research University), 26 Yaroslavskoye Shosse, Moscow, 129337, Russia
Abstract Dissection (disintegration) for Poisson’s two-dimensional problems differential equation takes us to 2nd-order ordinary differential equations, the thing that simplifies a solution algorithm and makes it programmable. This algorithm illustrated by examples of membranes and calculations of bend plates. New obtained results are compared with those known and stated in the references listed at the last part of the article. The new algorithm obtained by dissecting ordinary differential equations of the 1storder, based on applying the method of successive approximations and the generalized equations of finite difference method. Solution convergence are illustrated in both methods, also; the out coming algorithm shows the great combination between both techniques. The super advantage of the new developed algorithm is the capability of calculating plates sophistically shaped, which stands as a great benefit for engineers and designers. © by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license © 2016 2016The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility ofthe organizing committee of the XXV Polish – Russian – Slovak Seminar “Theoretical Peer-review responsibility of the organizing committee of the XXV Polish – Russian – Slovak Seminar “Theoretical Foundation Foundation under of Civil Engineering”. of Civil Engineering”. Keywords: Plates; membranes; differential equations; numerical solutions; successive approximation; finite difference
Nomenclature w ȟ,Ș p h i j
Dimensionless Deflection Cartesian Coordinates Distributed Load Mesh Spacing (Step) Measuring Along The Axis (ȟ) Measuring Along The Axis (Ș)
* Corresponding author. Tel.: +7-985-366-3123. E-mail address:
[email protected]
1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the XXV Polish – Russian – Slovak Seminar “Theoretical Foundation of Civil Engineering”.
doi:10.1016/j.proeng.2016.08.150
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n v a q0
Number of Domain Divisions (Meshes) Poisson’s Coefficient Plate Side Length Load Intensity at a Certain Node
1. Main text In engineering practice, plates of different shapes are widely used. The far known numerical methods and techniques allow engineers to calculate such complicated plates, but they are quite complex. This article presents the dissection method (disintegration), and numerical implementation represented by the combination of successive approximations method and generalized equations of finite difference method. It also targets a development of highly accurate and simple algorithm for calculating plates and membranes in complex shapes. Considering dissection method to calculate a strained membrane, the following differential equation in partial derivatives describes the strain-load relation: பమ ୵ பஞమ
பమ ୵ பమ
ൌ െሺɌǡ Ʉሻ
(1)
It’s well known that the dissection method is applicable only for differential equations systems, in which the differential operator is as the sum of one-dimensional differential operators. (Eq.1) satisfies this condition. Suppose
பమ ୵ பȟమ
ൌ െ ȟȟ
(2)
Thus, பమ ୵ பమ
ൌ െ ȟȟ Ǥ
(3)
Consequently, instead of (Eq.1); we have a system of two homogeneous algebraic equations, equations for such difference operators are formulated much easier. Now, the difference equations (successive approximations), approximating (Eq.2) and (Eq.3) considering uniform mesh with no discontinuities, this produces:
୧ିଵǡ୨ െ ʹ୧ǡ୨ ୧ାଵǡ୨ ൌ െ ୧ǡ୨ିଵ െ ʹ୧ǡ୨ ୧ǡ୨ାଵ ൌ െ
୦మ ଵଶ
୦మ ଵଶ
ஞஞ
ஞஞ
ஞஞ
ቀ୧ିଵǡ୨ ͳͲ୧ǡ୨ ୧ାଵǡ୨ ቁ
(4)
ஞஞ
ஞஞ
ஞஞ
ቀ୧ǡ୨ିଵ ͳͲ୧ǡ୨ ୧ǡ୨ାଵ െ ୧ǡ୨ିଵ െ ͳͲ୧ǡ୨ െ ୧ǡ୨ାଵ ቁ
(5)
Then, problem is solved using Seidel’s iterative method, initially conditioned to zero, that results in converting (Eq.4) and (Eq.5) to the form: ȟȟ
ଵ
ଵ
ସ
ଶସ
୧ǡ୨ ൌ െ ሺ୧ିଵǡ୨ െ ʹ୧ǡ୨ ୧ାଵǡ୨ ሻ െ ଵ
ଵ
ଶ
ଵଶ
୧ǡ୨ ൌ ሺ୧ǡ୨ିଵ ୧ǡ୨ାଵ ሻ
ȟȟ
ȟȟ
ȟȟ
ሺ୧ିଵǡ୨ െ ͳͶ୧ǡ୨ ୧ାଵǡ୨ ሻ ஞஞ
(6) ஞஞ
ஞ
ሺ୧ǡ୨ିଵ ͳͲ୧ǡ୨ ୧ǡ୨ାଵ െ ୧ǡ୨ିଵ െ ͳͲ୧ǡ୨ െ ୧ǡ୨ାଵଵ
(7)
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Where ൌ
ଶ୵ ୦మ
(7.1)
As a test problem, considered a square membrane under a uniformly distributed load P = 1 with length-less side equals to the unity. Since the boundary conditions w = 0, (Eq.2), (Eq.3) lead toൌ Ͳ,Ɍ ൌ ͳ , ஞஞ =1; whenɄ ൌ Ͳǡ Ʉ ൌ ͳǡ ஞஞ =0. Origin of the vertical axisሺɌሻlocates in the upper left corner of the membrane, In the first and second rows of (Table.1) the values of dimensionless deflection in the center of the membrane are given, calculated by successive approximations method and generalized equations of finite difference method for different values of n. Proved evidently that the solution using successive approximations method quickly converges to that obtained through (Eq.4) which equals 0.07367. (Eq.5) outlines a matrix form of dissection method numerical implementation referring-less to the predecessors, in particular (Eq.3). According to the new developed algorithm, results for a hinged resting square plate with a side-length of unity, loaded with a uniform distribution are obtained, and for the sake of that; a system of two 2nd-order differential equations had been solved as shown: பమ ୫ பȟ
మ
பమ ୫ பȘమ
ൌ െ
Where ൌ பమ ୵ பȟమ
பమ ୵ பȘమ
୯ ୯బ
(8)
(8.1)
ൌ െ
(9)
ାಏ
ಖ Where ൌ ሺଵାሻ୯
మ బୟ
(9.1)
Deflections in the center of the plate, calculated for different values of n, listed in the third row of (Table.1). Solution within (Eq.6) gives a result of 0.00406. At the position where the contour lines do not coincide with the direction axes ȟ and Ș, (Eq.4) and (Eq.5) are preserved. There is a need to identify ܑǡ ܒat nodes in a chamfered edge (Fig.1). If we used the equation that calculates the deformation of boundary node by successive approximations method; instead of (Eq.2) and (Eq.3), we will consider the difference approximation at (i,j):
Figure (1). Rotated 45º membrane on ȟ, Ș Cartesian axes On which shown node on chamfered edge
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ȟ
െ୧ǡ୨ ୧ାଵǡ୨ =െ
୦య ଵଶ
୦య
Ș
୦ଶ
ȟȟȟ
୧ǡ୨ െ Ș
ଵଶ
ȟȟ
ȟȟȘ
െ୧ǡ୨ ୧ǡ୨ାଵ =െ (୧ǡ୨ -୧ǡ୨ ሻ െ ଵଶ
Where ஞ = =
ப୵ பஞ
୦మ ଵଶ
(10) ȟȟ
ȟȟ
(5୧ǡ୨ +୧ǡ୨ାଵ െ ͷ୧ǡ୨ െ ୧ǡ୨ାଵ ሻǡ
(11)
(11.1)
ப୵
(11.2)
பȘ
பమ ୵
ஞஞ ൌ
ȟȟ
(5୧ǡ୨ +୧ାଵǡ୨ );
பஞమ
பయ ୵
ஞஞஞ ൌ
பஞయ
ሺͳͳǤ͵ሻ
ሺͳͳǤͶሻ
ሺͳͳǤͷሻ
ሺͳͳǤሻ
ப୵ಖಖ
ஞஞ ൌ
ப
ப୮
ൌ ப
ஞ
ஞஞஞ
ப୵
ஞஞ
In (Eq.10) and (Eq.11) the unknowns are୧ǡ୨ ,୧ǡ୨ ǡ ୧ǡ୨ ,୧ǡ୨ , as per contour line where= 0 , the derivative பஓ ப୵ ப୵ ஞ =
Ƚ Ƚ ൌ Ͳ; therefore: ୧ǡ୨ =െ୧ǡ୨ . ப
பஞ
ஞஞஞ ୧ǡ୨
ஞஞ
, ୧ǡ୨ , ୧ǡ୨ approximate square parabola and express it in terms of the values of Derivatives corresponding functions, for example: ஞஞ
୧ǡ୨ =
ଵ ଶ୦
ஞஞ
ஞஞ
ஞஞ
(െ͵୧ǡ୨ + 4୧ǡ୨ାଵ െ ୧ǡ୨ାଶ ).
the
(12)
Then from (Eq.10), (Eq.11) with the help of such terms given by (Eq.12), the following could be generated: ஞஞ ଵ ଶ
୧ǡ୨ = ൫୧ǡ୨ାଵ െ ୧ାଵǡ୨ ൯+
ଵ ଶସ
ஞஞ
ஞஞ
ஞஞ
ஞஞ
ஞஞ
(10୧ǡ୨ െ ୧ାଵǡ୨ െ ୧ǡ୨ାଵ +୧ǡ୨ାଶ+୧ାଶǡ୨ +9୧ǡ୨ +4୧ǡ୨ାଵ െ ୧ǡ୨ାଶ ሻ
(13)
(Fig.1) shows a specially rotated 45° square shaped membrane by means of the above derived procedure as an illustrative example; the deflection values at its center are given in the fourth row of (Table.1). Table (1)
n 1 2 3 4
Membrane by successive approximations method generalized equations of finite difference method Plate by method of successive approximations Membrane Checked by method of successive approximations
2 0.07292 0.07031 ---------
4
6
0.07363 0.07366 0.07278 0.07327 0.004041 0.004057 0.07800 -----
8
10
12
0.07377 --------0.07357 --------0.004059 0.004060 ----0.07400 0.07381 0.07372
As noticed, the above procedure uses a uniform grid size (uniform meshing). When it comes to solve problems with much more complexity in shapes of its plates or membranes, we have to deal with a non-uniform grid (nonuniform meshing). Consequently; difference formulas are getting more complicated and defining ஞஞ . That’s why a combination between successive approximations method and the generalized equations of finite difference method for algorithm simplification aims is proposed. This illustrates an example of calculating membrane in the shape
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rectangular trapezoid under a uniformly distributed load (Fig.2). Except for computational grid (mesh) and designated nodes, Figure shows boundary nodes of (1,4) ; (2,3) ; (3,2) use the generalized equations of finite difference method equation taking into consideration that ଵ =
ሺȽሻ ஞஞ
୧ିଵǡ୨ െ ʹ୧ǡ୨ ୧ାଵǡ୨ ൌ െଵଶ ୧ǡ୨
(ͳͶሻ ஞஞ
୧ିଵǡ୨ െ ʹ୧ǡ୨ ୧ାଵǡ୨ ൌ െଵଶ ሺ୧ǡ୨ െ ୧ǡ୨ ).
ሺͳͷሻ
For other nodes, successive approximations method in the general case for non-uniform grid (mesh) is functioned. As in (Eq.14) and (Eq.15) the value of ஞஞ is counted only in the border node (i,j) , there is no need to determine ஞஞ for chamfered edge nodes in a membrane or a plate. Solution with successive approximations allows to obtain high accuracy and also at other nodes. Results obtained using different number of divisions and ˔ ןǤwhere of the membranes vertical side = 4, deflections for node (2,6) are provided in (Table.2) as shown in (Fig.2) .Comparing the results to another solution, the membrane discussed above was calculated atןൌ Ͷͷ° , ൌ Ͷusing successive approximation method; ܟǡ =0.0904; ଵǡସ = 0,0542 ;ଵǡହ ൌ ͲǤͲͶ͵Ǥ
Figure (2). Deflections for different nodes on an understudy membrane...
Table (2)
ࢉ࢚ ࢇ ࢝ǡ
0.8 0.0872
0.9 0.0887
1 0.0903
1.1 0.0919
1.2 0.0904
That way, the new proposed algorithm that based on a combination of successive approximations method and the generalized equations of finite difference method; proved for complex shaped bent plates calculations. For simply supported (hinged-edged) plates, and according to (Eq.8) and (Eq.9) is determined at the beginning then w, for other boundary conditions; m and w are determined by solving the equations simultaneously. Proceeding method of Dissection for Poisson’s two-dimensional problems differential equation leads us to a pair of single-dimensional-equations. It’s a fact that the number of equations increases, but they get significantly simplified compared to difference (differential) equations. It’s not hard to solve problems using numerical techniques especially the above used successive approximations method, just a matter of getting the continuous right hand side of the original differential equation and its derivatives, discredited. A small number of divisions; also, high accuracy speaks up about the possibility of using the new developed method together besides the method of finite elements…
R.F. Gabbasov et al. / Procedia Engineering 153 (2016) 444 – 449
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