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7Ученый XXI века • 2022 • № 5-2 (86)
Технические науки
OPTIMAL LINEARIZATION OF YIELD CONDITIONS
OF RIGID-PLASTIC SHELLS
Yu. H. Akhmedov1, R. Kh. Shamsiev2
Optimization problem of the developed solution: the minimum number of faces of an inscribed or circumscribed polyhedron is found with a specified accuracy of solution of the limiting-equilibrium problem. Since this requires consideration of different shapes and numbers of faces, the basis of the work is the procedure for the automatic construction of convex polyhedra inscribed and circumscribed around some surface, f(Q^) = f0, as well as the automatic formation of the matrix of the linear programming problem.
Key words: shells, construction, nonlinear problem, surfaces, fluidity surface, linear programming, optimization, problem, polyhedron, limit load, edges, vertex, matrix, thickness, load, minimization.
Searching for two-sided bounds on the carrying capacity of ideal rigid - plastic structures by static and kinematic methods of the theory of limiting equilibrium in classical form [2,5-7,9,10 ] entails using the plasticity condition f(Q^) = f0. in explicit form. For shells, this condition is usually formulated in the six-dimensional space of internal forces Qtj,; in the general case, f represents a closed convex hypersurface. In view of the convexity of f, the problem of calculating the shell carrying capacity is significantly non-linear.
One of the most effective means of accelerating the solution is to linearize the initial nonlinear problem, so that linear-programming methods may then be used for its solution. [ 5-8, 10 ]. Such linearization requires the replacement of the nonlinear convex surface f(Q^) = f0 by an inscribed or circumscribed polyhedron.
It is known [2,8], that the estimate obtained on the basis of the accurate yield surface lies between the estimates found using the external and internal approximations. The problem is to choose the polyhedron guaranteeing specified accuracy of solution. It is understood that increasing the number of faces improves the approximation of the accurate surface, but significantly increases the magnitude of the linear-programming problem.
In the present work, the optimization problem solved is as follows: to find the minimum number of faces of the inscribed and circumscribed polyhedron with a specified accuracy of solution of the limiting-equilibrium problem. Since different forms and numbers of faces must be considered here, the basis of the work is a procedure for automatic construction of convex polyhedra inscribed in and circumscribed around some surface f(Q^) = f0, and also automatic formation of the matrix of the linear-programming problem.
P. G. Hodge [11] considered a two-dimensional case, when the yield condition is specified with some degree of indeterminacy and hence represented by two congruent lines: an internal and an external line.
Since it has been established [1,4], that the form of the yield surface has a limited influence on the estimate of the limiting load, it is assumed, in choosing the method of constructing the polyhedra, that the carrying capacity of the shell depends weakly on tne form of the polyhedral faces. Attention has been focussed mainly on the
1Akhmedov Yunus Hamidovich - Bukhara Engineering-Technological Institute, Bukhara, Uzbekistan.
2 Shamsiev Rustam Khalilovich --Bukhara Engineering-Technological Institute, Bukhara, Uzbekistan.
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Ученый XXI века • 2022 • № 5-2 (86)
development of a segmentation procedure that is identical for the inscribed and circumscribed polyhedra. It turned out that a sufficiently convenient method is to choose points on the initial surface which are at the same time vertices of the inscribed polyhedron and points at which the faces touch the circumscribed polyhedron. This method may be used not only for smooth but also for piecewise-smooth yield surfaces with edges and vertices.
Consider a closed convex surface z = f (ja) in the form of a triaxial ellipsoid. In the simplest case, all of its axes may coincide with the coordinate axes, and then the choice of points on the surface may be made by the rule [1]
Xu’ = р-.Хг" = ptgve.Xsi111 = ptgv,X4iIV = ptgp2J(1 + tg2<piy, (1) Where 0< ^ < 90°; p > 0
The equation of each face of the inscribed polyhedron
Ax’ + Bxu + Cxm + DxIV + E = 0 (2)
may be obtained by drawing a plane through the points i = 1:4, Then
A =
X21 X31 X41 1 X22 X32 X42 1 X23 Хзз X43 1 . X24 X35 X45 1-
B =
X31 X11 X41 1 X32 X12 X42 1 X33 X13 X43 1 X34 X14 X44 1
X11 /21 X41 1 X12 X22 X42 1 X13 X23 X43 1 - Xl4 X24 X44 1-
X11 /21 /31 1 X12 X22 X321 X13 X23 X331 -X23 X23 X34 1-
(3)
X11 /21 /31 X41 1 X12 X22 X32X42 1 X13 X23 X33X43 1 - X23 X23 X23 X44 1-
The equation of the face of the circumscribed polyhedron is obtained by drawing a plane tangential to z =f(ja ) at point i i
df f ,
ТА (X'
xl) + ТТЛ (x” - x'i,) + TZiu (xI” - xlU) + TZiv (x
df
dx11
dx111
dxIV
0 = 0 (4)
x
In the general case, the axes of the surface z =f(ja) may not coincide with the coordinate axes. Then the rule (1) allows the conventional coordinates of the points Xa, to be calculated, while the true coordinates xa are found by rotation of the axes
iXo} = M • {ja},
(5)
where A is the matrix of rotation.
The coordinates (5) may now be used to obtain dependencies (3) and (4).
Suppose that a sloping shell of quadratic outline, in plan, is of constant thickness h, and its median surface is in the form of an elliptical paraboloid
z = ^2(x2 + y2); (6)
where 2b is the shell dimension, in plan; f is the rise. The shell is fixed over its whole contour and is subject to a uniform transverse load of intensity q.
To determine the lower bound on the limiting load, consider the statically permissible fields of internal forces satisfying the equilibrium and yield conditions and the boundary conditions, and permitting a maximum of q. At first, attention is confined to a momentless formulation of the problem. It is expedient to introduce the notation nx = NXN(-1; ny = NyN-1; nxy = NxyN-1;
у = \Ъ~г; e = Щ-1; p = qa-1. (7)
It is expedient to introduce the notation
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Ученый XXI века • 2022 • № 5-2 (86)
пх = NXN0 1; пу = NyN0 1;
у = f b-1; е = Щ-1;
ftху NxyN0
р = qo-1.
(6)
Here N0 = a0h; a0 is the yield point of the material, which is identical for tension and compression.
Calculating the curvature of the surface (6)
d2z
kx = —— = yb 1; kv = k- k„, = 0
x dx2 y
the equilibrium equation is written in the form:
'■xy
nv+JL=0; d-^ +
ey2
dx
dn:
dy
■xy _ Q. дпу + dn:
dy
-21 = 0.
dx
(8)
Fixing of the shell edges means that on the contour the internal forces nx, пУ: пхУ: may take any values; therefore the boundary conditions are not considered here.
n
X
To convert to a linear-programming problem, the differential (8) is written in finite-difference form,
by means of a regular grid (Fig. 1, a), single-sided differences, and symmetry conditions. Then the equilibrium equation must be written only for nodes 1-3. For
fty2 ftys + ft-xy2 ft-хуЗ 0; (9)
where e3 is the relative shell thickness at node 3.
The coefficients for the unknown node values of the internal forces ftx,fty,ftxy form the first part [B1] of the matrix [В] of the linear-programming problem. This part does not depend on the form of the yield surface; therefore, it may be formulated just once, and there is no need for its automatic determination.
In contrast to the equilibrium condition in (8), the strength conditions must be considered not only for internal points but also for contour nodes.
The number of columns in the matrix [В] is 3N + 1,, where N is the number of nodes considered in the grid region, whereas the number of rows is associated with the form of the polyhedron and depends on its number of faces. In the general case, the number of rows L = 3N1 + M • N,, where M is the number of faces; N1 is the number of points (nodes) for which the equilibrium condition must be written. For the grid adopted (points 1-6), and in the simplest case, when the yield condition is represented by a circumscribed parallelepiped
—1 <ftx ^ 1; —1 <fty < 1; -0,57 < ftxy < 0,57, (10)
The strength constraints form 18 rows; then the dimensions of matrix are [В] —
27x19.
Thus, the problem of the lower bound on the limiting uniform load on the shell (6) consists in minimizing p with the constraints described by matrix [В].
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Ученый XXI века • 2022 • № 5-2 (86)
For the automatic formation of that part [В2] of matrix [В] which is associated with the form of the yield condition, consider the equations of the faces of the inscribed or circumscribed polyhedra
Afnx + Сfnxy + Bjny = -Dj, (11)
where j is the face number; 1 <j<M,M is the number of faces.
The set of coefficients of Eq. (11) is a vector, for example, А = (a11, a21,... ам1)т; B = (a12, a22,... ам2)т;, etc. The components of these vectors are calculated by the rule:
for an inscribed polyhedron
an = [(пуУ + 1 - (ny)i][(nxy)i + 2 - (nxy)d -
-[(пхУУ + 1 - (nxy)i][(ny)i + 2 - (ny)i]; (12)
for a circumscribed polyhedron
«ii = (nx)i-1(ny)i.
(13)
Here i is the number of the node in the grid region; i = l,2,...N;N is the number of nodes.
On the basis of (11)-(13), the part [В2] of matrix [В], depending on the form of the yield condition may be written in the following form
В
2
в* B* ... О ; В* = a11a12a13 а21а22а23
.0 - в*. .^M1^M2^M3.
Similarly, the right-hand side of Eq. (11) takes the form
Dj = (di, d.2,... dM)T; dj = (bi, &2, ■ ■■ bN)T;
and the components of the vector bj are calculated by the rule
b = -[(nx)iaii + (ny)iai2 + (nxy)iai3]
(14)
(15)
for inscribed polyhedra and b = -1 for circumscribed polyhedra. Thus, the dimensions of the matrix [В2] are L1XL, where L1 = N • M.
For the simplest yield condition in (10), the solution of the problem of the shell carrying capacity may be obtained in closed form. Suppose that in a shell with fixed edges the normal and shear forces vary weakly on passing from one point of the surface to another, and it may be approximately supposed that nx(x, y) = ny(x, y) = const = F.
Then the second and third relations in (8) are satisfied, and from the first equation it follows that p = 2ey2F.
Finding the lower bound p(-) entails finding max p.
It follows from (10) that max f = max nx = 1,; therefore
p(— = 2ey2. (16)
The intensity of the limiting load q(-) = po0 = 2o0ey2,, and the total limiting load Q(-) = Qo0b2ey2.
Passing to a dimensionless total load at the shell, it is found that
к'-'=т!у = 8еуг (17)
Consider, as an example, a shell with a slope у = 0,2 and relative thickness e = 0,05. . It follows from (17) that
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Ученый XXI века • 2022 • № 5-2 (86)
К(-) = 1,6 • 10-2 (18)
This estimate is in good agreement with the lower bound obtained in [3] for a shell with freely supported edges. Fig. 1.
Now suppose that the shell material conforms to the Mises yield condition, which in momentless formulation, taking account of the notation in (7), takes the form
n2x — nxny + n2 + 3n2xy = 1. (19)
For a shell with a slope у = 0,2 and relative thickness e = 0,05, the lower bound on the uniform transverse load is calculated with a different grid discretization (Fig. 1, а, б) and a different number of faces of the inscribed and circumscribed polyhedra. The number of faces depends on I1 and I2 the number of points chosen on the octant of the yield surface in the azimuthal and meridional directions (Fig. 2, а, б). Some results of the calculations are shown in Fig. 3, (1 - inscribed polyhedron; 2 -circumscribed polyhedron). The calculation is undertaken by a program on a modern computer, with reference to the library subprogram of the SIMPLX linear-programming software.
It is established that the results of the calculations for the two grid-division schemes practically coincide. In the given examples, the two approximate estimates converge rapidly to that in (18) and already at l2> 5 the difference between the approximate estimates is no more than 5%.
Now consider the transition from a momentless formulation of the problem to one that takes account of the moments. The first relation in (8) is then replaced by
ж ''■у -1
,e('
ПУ+4(
e fd2mv
^ d2mxy
+
d2m-
9f2 9f2
my = MyM-1; where £ = xb-1; % = yb-1.
^) + 772 = 0;
mx = MXM0
ey
m
Xy = MXyM0 ;
(20)
The yield condition (19) takes the form
n2x - nxny +n2y + 3nxy + m2 — mxmy + m2y + 3m2xy = 1. (21)
Passing to a formulation of the problem that takes account of moments entails constructing inscribed and circumscribed hyperpolyhedra in the six-dimensional space of internal forces nxny_mxy. The size of the linear-programming matrix is markedly increased here, but no so much as to lead to fundamental difficulties.
The lower bounds on the shell carrying capacity obtained here, as well as the results of [1], permit the hypothesis that, on most practical calculations, acceptable accuracy may be achieved even with a small number of faces; the guarantee of reliability here is the simultaneous construction of the inscribed and circumscribed polyhedra. Nevertheless, cases of calculation in which the specified accuracy is reached with a large number of faces may also be encountered. Therefore, the given linearization method is formulated as a subprogram and may be included in any pack of programs for singleparameter or optimizational shell calculations.
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Ученый XXI века • 2022 • № 5-2 (86)
The approach here described allows not only regular but also piecewise-smooth yield conditions — for example, with restricted interaction — to be considered from the same perspective. [4].
References:
1. Дехтярь А.С., Липский А.Г. Об условиях текучести железобетонных оболочек // Изв.вузв. Стр-во и архитек.-1983, № 9.-С. 38-43.
2. Ерхов М.И. Теория идеально пластических тел и конструкций -М.: Наука, 1978.352 с.
3. Мирзабекян Б.Ю. К определению нижней граниыи несущей способности оболочек // Строит. Механика и расчет сооружений. -1968. № 3.-С. 42-47.
4. Ольшак В., Савчук А. Неупругое поведение оболочек. - М.: Мир, 1969.-143 с.
5. Проценко А.М., Власов В. В. Определение несущей способности арок и куполов с помощью ЭВМ // Расчет пространст. Конструкций. -1971.-Вып. 14-с. 14-24.
6. Рейтман М.И., Ярин Л.И. Оптимизация параметров железобетонных конструкций на ЭЦВМ.-М.: Стройиздат, 1974.-95 с.
7. Ржаницын А.Р. Расчет оболочек методом предельного равновесия при помощи линейного программирования // тр. VI Всесоюз. конф. По теории пластин и оболочек.-М.: Наука. 1967.-С. 656-665.
8. Ходж Ф. Дж. Сравнение условий текучести в теории пластических оболочек // Пробл. Механики сплошных сред.-М.: Изд-во АН СССР. 1961.-С. 74-91.
9. Ахмедов Ю.Х. Автоматическая аппроксимация односвязных гиперповехностей полиэдрами применительно к расчётам несущей способности оболочек покрытий.: Дис.канд.техн.наук.-Киев,1984,-202с. Чирас А.А., Методы линейного программирования при расчете упругопластических систем.- Л.: Стройиздат, 1969.-148 с.
10. Hodge P.G. Automatic piecewise linearization in ideal plasticity //Comp. methods in appl. eng-1977.-10, N 3.-P. 249-272/.
11. Махмудов М.Ш., Ю. Ахмедов. use ofe space en describing agrarh analytical representationof multi-factor events ano processes International journal Innovative Engineering and Management Research, (2020): volime 09, Issue 09, Pages 194-197
© Yu.H. Akhmedov, R.Kh. Shamsiev, 2022.
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