Increasing the load-carrying capacity of three-layer plates and shallow shells
This article examines three-layer plates rectangular in plan and shallow shells of symmetrical construction subjected to bending by a transverse distributed load. The loadcarrying capacity of the plates and shells is evaluated by means of a generalized strength criterion. We solve the problem of fin...
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| Datum: | 1985 |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
1985
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| Zitieren: | Increasing the load-carrying capacity of three-layer plates and shallow shells / A.D. Panteleev // Проблемы прочности. — 1985. — № 8. — С. 1158-1162. — Бібліогр.: 6 назв. — англ. |
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Panteleev, A.D. 2022-01-22T19:42:30Z 2022-01-22T19:42:30Z 1985 Increasing the load-carrying capacity of three-layer plates and shallow shells / A.D. Panteleev // Проблемы прочности. — 1985. — № 8. — С. 1158-1162. — Бібліогр.: 6 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/182893 539.384:624.073 This article examines three-layer plates rectangular in plan and shallow shells of symmetrical construction subjected to bending by a transverse distributed load. The loadcarrying capacity of the plates and shells is evaluated by means of a generalized strength criterion. We solve the problem of finding the optimum law of variation in the thickness of the external layers of these structure. The resulting thickness distribution keeps the structures as far as possible from the limiting stress state. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Scientific-technical section Increasing the load-carrying capacity of three-layer plates and shallow shells О повышении несущей способности трехслойных пластин и пологих оболочек Article published earlier |
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Increasing the load-carrying capacity of three-layer plates and shallow shells |
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Increasing the load-carrying capacity of three-layer plates and shallow shells Panteleev, A.D. Scientific-technical section |
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Increasing the load-carrying capacity of three-layer plates and shallow shells |
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Increasing the load-carrying capacity of three-layer plates and shallow shells |
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Increasing the load-carrying capacity of three-layer plates and shallow shells |
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Increasing the load-carrying capacity of three-layer plates and shallow shells |
| title_sort |
increasing the load-carrying capacity of three-layer plates and shallow shells |
| author |
Panteleev, A.D. |
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Panteleev, A.D. |
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Scientific-technical section |
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Scientific-technical section |
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1985 |
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English |
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Проблемы прочности |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Article |
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О повышении несущей способности трехслойных пластин и пологих оболочек |
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This article examines three-layer plates rectangular in plan and shallow shells of symmetrical construction subjected to bending by a transverse distributed load. The loadcarrying capacity of the plates and shells is evaluated by means of a generalized strength criterion. We solve the problem of finding the optimum law of variation in the thickness of the external layers of these structure. The resulting thickness distribution keeps the structures as far as possible from the limiting stress state.
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0556-171X |
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https://nasplib.isofts.kiev.ua/handle/123456789/182893 |
| citation_txt |
Increasing the load-carrying capacity of three-layer plates and shallow shells / A.D. Panteleev // Проблемы прочности. — 1985. — № 8. — С. 1158-1162. — Бібліогр.: 6 назв. — англ. |
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AT panteleevad increasingtheloadcarryingcapacityofthreelayerplatesandshallowshells AT panteleevad opovyšeniinesuŝeisposobnostitrehsloinyhplastinipologihoboloček |
| first_indexed |
2025-11-25T00:15:16Z |
| last_indexed |
2025-11-25T00:15:16Z |
| _version_ |
1850501659250130944 |
| fulltext |
INCREASING THE LOAD-CARRYING CAPACITY OF THREE-
LAYER PLATES AND SHALLOW SHELLS
A. D. Panteleev UDC 539.384:624.073
This article examines three-layer plates rectangular in plan and shallow shells of sym-
metrical construction subjected to bending by a transverse distributed load. The load-
carrying capacity of the plates and shells is evaluated by means of a generalized strength
criterion. We solve the problem of finding the optimum law of variation in the thickness
of the external layers of these structure. The resulting thickness distribution keeps the
structures as far as possible from the limiting stress state.
Formulation of the Problem. As the theoretical model we take a three-layer packet of
symmetrical construction with a lightweight filler. We assume that the broken line hy-
pothesis is valid for the entire packet.
Let m = (mi)( i=l, 5) be a vector function, the components of which represent displace-
ments of points of the middle surface of the bearing (external and internal) layers of the
shell (Fig. i). Here ~, = u,, m= = v,, m, = u=, m~ = v=. We assume that the deflection in
the direction of the axis is the same for all layers and m5 = w. In the case of the plates,
considering the symmetrical construction of the packet, we take m, =--m~ and ~= =-~ [i].
We will examine a symmetrical bilinear form:
�9 . ~, ~ + -~, (~;~ + ~, (o', o-) = t' / r~--~7:[ ~'~' ~ ' r , . + "~'~, ~;~ + ~#;~-t-~;~;) + ~ +
o
!
Here, e i and e~ are strain components generated by the vectors o'and ~"[i]; t(x,, x=),
h(xt, x=) are functions of point (x,, x=)6Q; ~ = [a,., bt] • [a=, b=] is a projection of the
middle surface of the shell; Et and Es are Young's moduli; G,= is the shear modulus; ~,, and
~2 are Poisson's ratios of the orthotroplc bearing layers (E,v= = E=~,) Ga, and Ga= are
!
the shear moduli of the filler. Here -~ t(x,, x=) + h(xt, x=) = const.
!
We form the energy space H~ [2], closing in the norm lloIIH(s)=at((o, o) 2 a set of smooth
functions o satisfying conditions of shell support such that ~ = 0 follows from the equation
at(o, ~) = 0.
As in [2], we will call the generalized solution of the problem of the stress--strain
state of a three-layer shell of variable thickness the function (o(xt, x=) 6 H(Q), for which
aa(o, of)= (go'dQ, Vo'6H(f,~), (I)
n
where g is the intensity of the external load.
Considering the function t(x,, xa) as a control, we determine the permissible set of
controls:
v o --= {tit 6 ~'~ (~), Iltllw~,,n~c~, p~2, f l< t< t= ,
(t) = I t~ < c2, o, (t, o (t)) ~< i, i = i .... , m},
O
(2)
Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated
from Problemy Prochnosti, No. 8, pp. 103-106, August, 1985. Original article submitted
May 12, 1983.
1158 0039-2316/85/1708-1158509.50 �9 1986 Plenum Publishing Corporation
.t ~ f / ' ' ' ' ~ ul / /
A,
Fig. i. Element of a three-layer shell.
where Ct, C2, tt, and ta are positive constants; W~(fl) is a Sobolev space; the value of the
functional ~(t) is equal to the volume of the material of the bearing part of the shell;
~.(t, m(t)) is a functional representing the chosen strength criterion and characte[izing
t~e stress state of the structure in the neighborhood of the point (xj(i)) (j = i, 3). It
is determined through the solution m(t) of problem (i). Here, we assume that the set U d is
not empty.
We introduce the object function
(t) = sup ~ (t, ~ (t)).
i
The problem of finding a structure of maximum strength is formulated in accordance with the
formulation of an infinite problem of optimum control: find the function t*(x,, Xa) for
which
t * 6 U d, ~ ( t * ) = i n f ~ ( t ) . ( 3 )
t~u d
The physical meaning of problem (3) consists of selecting the optimum law of change in the
thickness of the bearing layers of the structure in question. With such a law, the stress
state at the point liable to the greatest stress (accounted for in the form of the safety
factor) will be as mild as possible. In other words, the resulting thickness distribution
will correspond to a shell (plate) which is maximally unloaded in accordance with the chosen
strength criteria. This in turn makes it possible to additionally load the structure and
thereby increase its load-carrying capacity.
As was done in [3], the existence of a solution to problem (3) can be proven.
Numerical Realization of the Direct and Optimum Problems. Let the following subdivi-
sions be prescribed on the segments [ak, b k] (k = i, 2)
A h = a k = x k < x h < , . . . , < x ~=bh . l
We will designate h~ = x~ x t - l o - k , where I <i ~< N~; hA = max hl; =~ = E hL Assuming =k = 0,
for 0 .~< i ~ N k we note the following obvious relation: i t=l
x~=ah+~. (4)
We i n t r o d u c e i n t o t h e r e g i o n fl = [ a t , b t ] x [ a 2 , b 2 ] a g r i d ~ = &t x A2 d i v i d i n g ~ i n t o
rectangular cells, from among which we pick out
fltJ = {(xa, x~)lxxE[al , bl] N [x~ -~, x]+q ,
i = 0 . . . . . Nx;
x, E [a~, bd N [x~-', x~+q, ] = o . . . . . N,}.
In the region ~, considering (4), we assign the fundamental splines:
Zi~ =
I
at
0 at
(x . x2) E fll~;
(5)
1159
]
where ~i -~- [ ~ j ; 62 = 6, + i; [a] is the integral part of the number a.
We use functions (5) to form an N-dimensional space of controls HN(~)~ Wo*(~):
Hu (f~) = [ tu (x. x~) I
Nt N2
tN (xx, X2)= ~-a X~" t i jTi j t .v,, .v,2_),
/ = 0 1=0
(6)
1
N = ( N , + I ) ( N , + I ) I o
The geometric sense of the coefficients tij is given by the relation
li I = IN (~, x~). (7)
Regarding HN(~) as a space of controls, we can approximate ~-dimensional optimization
problem (3) by a sequence of flnite-dimensional problems: for each fixed number N we find
the function
(8) u �9 (t~v) = inf O) (tu). g, C u d ,
tNEU d
Here, the allowable set is represented in the form
tx ~ ttj <~ t2, ~h (tu, N (tN)) ~ I,
N, N: ] , . . . , m ,
~=o j=o o ( 9 )
where ~N(t N) is the approximate solution of problem (i) found by the Ritz method, and it can
in turn be approximated by means of bicubic polynomial splines [4].
It should be noted that the functions tN(X, , x~) from (6) are ambiguously determined by
specification of the coefficients tij , which can be considered components of an N-dimensional
vector. This fact allowed us to examine problem (8) as a nonlinear programming problem,
which was solved by the methods of penalty functions and coordinate descent [5].
In the problems examined below, the load-carrying capacity of the plates was evaluated
by means of a generalized strength criterion [6] which can be represented in the form of a
functional
l
~h (tN) = A l a n + A2~2 + "~ [A3o~t + A4o~2 + Asa11~ , + ~ m , ( 1 0 )
where A i (i = i, 6) are constants expressed through the physical constants of the material
and characterizing its strength in compression, tension, and shear.
Considering the nonlinearity of the stresses ~pq in (i0) with respect to t, it can be
shown that the greatest value of ~k(t) is reached on one of the surfaces bounding the bear-
ing layers of the plate: xs = --t-h, xs =--h, xs = h, xs = h + t.
In approximating the thickness of the plate t(x,, xa) by means of the functions
TiJ(x,, x2) from (5) the segments [ak, bk] (k = i, 2) were subdivided into an odd number of
intervals N k. Here, the nodes of the subdivision (4) were chosen so that the lengths of
the corresponding intervals satisfied the relations
h~ ~ = h~,o, h~ i-z >>hk,o, i == 1 . . . . . nh; ( 1 1 )
N h + 1
nh ~ 2 - -
In the calculations we assumed that the constant hk, o is equal to
t,2i--I 1 rain ,,k �9
h#,o --~ ~ I ~<i.<nk
1160
qg"
0.75
5 59
Z.2~
I
2:25 3.250 3.375 It
Fig. 2. Comparison of the load-carrying
capacity of an optimum plate (i) and a
plate of constant thickness (2).
-...~
Fig. 3. Form of the optimum plate.
Also, the coefficients tij (7) were subject to the following conditions:
tH:-= ti+li = tii+l = ti+li+l = OH,
(12)
i = 0, 2 . . . . . N l - - 1 ; ] ~ 0 , 2 . . . . . N 2 - - 1 ,
where Oij are positive constants.
Equations (II) and (12) correspond to the fact that controls tN(xt , x2) from the allow-
able set U N (9) are continuous functions which are close to being piecewise-constant. The d
proposed approximation for the thickness of the plate makes it possible to diminish the
dimensionality of flnite-dimensional problem (8) (to reduce the number of optimization
parameters tij) and to thereby reduce computer operating time.
As an example, we solved problem (8) on finding the optimum law of change in the thick-
ness of a plate which is square in plan (with a side c = bk -- ak, k = i, 2). The plate is
freely supported at its contour and is bent by a uniformly distributed pressure of intensity
go. Also, for comparison we solved auxiliary problem (I) on the stress state of a plate of
constant thickness for the same boundary conditions, geometric dimensions, and external
load. Here, the intensity of the pressure was chosen so that the value of functional (I0)
at the most heavily stressed point in a plate of constant thickness reached the limiting value,
i.e., SUP ~k = i. Solving problem (8), in accordance with (9) we assumed ~(t N) = C2 =
l~k~m
V, where V is the volume of the material of the bearing part of the constant-thickness
plate. This is evidence of the fact that the optimum plate was found from among a set of
plates of equal weight in the case of constant thickness, with the stress state reaching
the limiting value in the plate. Here, the weight of the filler was not considered.
X
We introduce the dimensionless coordinate li~-T!~ ([~1.2) In these coordinates, a
rectangle ~ -- the plan of the plate being examined -- is given by the inequalities 0 ~Zi~
i. As indicated above, with fixed values of Zx and Z2 the greatest value of the functional
(I0) is reached at one of four points xa =-h-- t, -~; h; h + t (Fig. I). Calculations
showed that, in accordance with the chosen strength criterion, the most heavily stressed
zone in the plate of constant thickness is the section 72 = 0.5.
1161
Figure 2 shows distribution curves of dimensionless values of the function @*(Z,) =
max @(Z,, 0.5, xs), where the functional @ is determined by Eq. (i0). Due to the symmetry
Xs
of the geometric dimensions (square plate), the boundary, conditions, and the external load,
the results are shown for one-fourth of the plate, i.e., for 0 ~<. Z,~ 0.5. It is apparent
from the graphs that the maximum value of @*(11) is reduced by 15% in the optimum plate
compared to the plate of constant thickness.
The accuracy of the solution of problem (I) on the stress state of a plate was checked
by comparing the solution for different numbers of coordinate functions. Thus, the graphs
shown in Fig. 2 were obtained from the solution ~ = (u, v, w) approximated by 104 coordinate
functions. Here, we took 20 functions for tangential displacements u, vand 64 functions for
the deflection w. The stress state found from the refined solution differs no more than 3%
from the above solution with a twofold increase in the number of dimensions.
Figure 3 shows the form of 1/8 of the resulting optimum plate. The law of thickness
change shown in the figure was obtained by varying nine parameters @iJ (see (12)).
In conclusion, we note that the solution of problem (I) is linearly dependent on the in-
tensity of the load g, and the maximum value of @*(Z,) for the optimum plate found is 15%
lower than for the constant-thickness plate. It follows from the foregoing that the optimum
plate, having the same weight as the constant-thlckness plate, has a load-carrying capacity
15% greater than the constant-thickness plate.
Taking into account the multiple extreme character of problem (8) and increasing the
number of optimization parameters, it would be possible to find an optimum plate design
with an even greater load-carrying capacity.
LITERATURE CITED
i. S.M. Tsurpal and N. G. Tamurov, Design of Multiply Connected Layered and Nonlinear
Elastic Plates and Shells [in Russian], Vishcha Shkola, Kiev (1977).
2. S.G. Mikhlin, Variational Methods in Mathematical Physics [in Russian], Nauka, Moscow
(1970).
3. V.G. Litvinov and A. D. Panteleev, "Optimization problem for plates of variable thick-
ness," Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 2, 174-181 (1980).
4. Yu. S. Zav'yalov, B. I. Kvasov, and V. L. Miroshnlchenko, Spline-Functlon Methods [in
Russian], Nauka, Moscow (1980).
5. D.M. Himmelblau, Applied Nonlinear Programmlng, McGraw-Hill (1972).
6. I.I. Gol'denblat and V. A. Kopnov, Strength and Ductility Criteria for Structrual
Materials [in Russian], Mashinostroenie, Moscow (1968).
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