THE ADAPTABILITY OF PLATES

L.G. Lantukh and A.V. Perel'muter UDC 624.04

The problem of the adaptability of an ideal elastoplast ic sys tem occurs when the design under consid- era t ion is under the action of repeated and varying loads, the change and alternation of which remain a r - b i t ra ry . Under the action of these loads the design may operate only elast ical ly, or fail as the resu l t of low cycle fatigue or as the resu l t of p rogress ive failure of one of its e lements , or , having experienced flow in cer ta in areas and having obtained some residual s t r e s s and deformation, may adapt to the condition of the la t ter and then operate only elast ical ly [1, 2]. Consequently, if the design is subjected to the action of repeated and varying loads, it is neces sa ry to check for adaptability in making the e las t ic and plast ic ca l - culations.

Calculations for adaptability are based on the c lass ic theo rems of Blake--Melan and Koiter [1]. The basic theorem states that adaptability occurs if there may be such an independent of t ime field of fictit ious s t r e s s e s that with any changes in load in specified l imits the sum of this field with the field of the momen- t a ry values of s t r e s se s in an ideal elast ic body is within safe l imits (necessary condition). Let us consider the adaptability of a plate not subjected to s t r e s ses .

Discre teness according to the method of final differences makes it possible to set a l imit by consid- erat ion of the final number of points and to speak of an end measured concept of the field of s t r e s s e s forme( by bending and torque which occur at the points of a lattice drawn on the surface of the plate. If at any mo- ment in t ime at each point of the lattice the fo rces f rom the load sat isfy the assumed conditions for flow, then the condition for adaptability is fulfilled. In other words , the problem of checking the calculations [3] for plates for adaptability is one of checking the consis tency of a sys tem of equations descr ibing the self - s t r e s sed condition of the plate and a sys tem of l imits imposed by the conditions of flow. The problem of d i rec t project ion of a plate of constant thickness taking into account adaptability is formulated in the fol- lowing manner : find the minimum possible thickness of plate for which at each point the fo rces of the se l f - s t r e s sed condition combined with the fo rces f rom the load sat isfy the conditions for flow, that is , it is neces sa ry to find the minimum thickness for the l imits determined by the s e l f - s t r e s s e d condition and the conditions for flow. Let us study the problem of di rect project ion.

All of the s e l f - s t r e s s conditions of the plate must sat isfy a uniform equation for equil ibrium

2 0 2 0 2 0 0 M u 0 M x 2 a A4xu ax" -~ ay, ~ = 0 , (I)

~whieh is true only for points not lying on the supported surface. In the opposite case i t would be necessary to consider equil ibrium of the moments and the intersecting forces. To Eq. (1) are added the static forces at the edges of the plate [4]. Equation (1) recorded in the final differences for each internal point of the plate together with the edge conditions fo rm a group of l inear l imiting equalit ies in the sys tem of l imits where the forces of the s e l f - s t r e s s at each point of the network and each square of the plate thickness are the unknowns.

If we designate Xj (j = 1 , 2 , . T +L) as the value of the moment M ~ at the point j (T is the number " ' ' X

of points inside the contour, L is the number of points on the contour); Yj (j = 1, 2 . . . . . T + L) is the value of the moment M~y at the point j; Aij , Bij, and Cij a re the coefficients of the final difference opera tors for par t ia l der ivat ives of the second order; then in genera l fo rm the conditions of the s e l f - s t r e s s e d conditions are descr ibed in the following manner:

Central Scient i f ic-Research Institute for the Planning of Steel Construct ion, Kiev. Trans la ted f rom Prob lemy Prochnost i , No. 6, pp. 40-43, June, 1970. Original ar t ic le submitted January 4, 1970.

�9 1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced [or any purpose whatsoever without permission o[ the publisher. A copy of this article is available Iron the publisher for $15.00.

544

where Mx, constant.

Tq-L T-I-l. T-{-L In = ~. ,4uX 1 + ~. BaZj + ~.. CuYi : 0

J= J ' J=~ (2)

( i = 1,2 . . . . . T).

Contour conditions, as a rule, requi re one or another force equal to zero [4], which reduces the num- ber of unknowns, and only in the case of a f ree contour is added the group of conditions

T+L T+L t, = ~ a,,xl + ~ b,/zl = 0 (3)

j--t i =t

( k = l . 2 . . . . . =),

appearing as the resul t of the end dif ference approximation of the condition

Q , - - oM~ = 0. 0y

In Eq. (3) a is the number of points on the contour free of support.

The system of ecluattons (2) and (3) must be taken into consideration together with the system of l im- i ts imposed by the conditions of flow.

If we accept as the condition of flow the l inear ized condition of flow of T r e s c a [3, 5], then the p rob- lem becomes a p roblem of l inear p rogramming , which may be solved by a s implex method.

However, l inear izat ion of the conditions of flow of T r e s c a makes it nece s sa ry to descr ibe each unit of the network in groups of six two-s ided l inear l imits , significantly increas ing the volume of original in- format ion and sharply reducing the possibi l i ty of a p r o g r a m suitable for solution of the problem on the com- puter . For example, for adaptability calculat ions for a square plate supported on the whole contour with a 5 x 5.network by the use of modified simplex method p rog ram would requi re holding in the memory of the machine a volume of information approaching 29,000 units, which in prac t ice el iminates the possibil i ty of solving the problem. Consequently, it is n e c e s s a r y to sea rch for other means of solving the sys tem of l im- i ts which do not requi re maintaining in the m e m o r y of the machine all of the ma t r i ces of the l imits or to deviate f rom the idea of l inear izat ion of the conditions of flow, which leads to a nonlinear problem. Togeth- er with reducing the requi red volume of information, the second method el iminates the introduction of ad- ditional inaccurac ies appearing as the resu l t of l inearizat ion.

On the nonlinear conditions of flow, the mos t useful is the condition of Von Mieses, consist ing of only one l imit

M~-F M~-- M, Mv -P 3M2 < H, ('4) My, and M are the moments occur r ing in the plate at any moment in t ime, and H is the plastic

For our problem condition (4) is wri t ten in the fo rm fit" h~

- (M ~ + M ' ) 2 - - ( ~ + .M") ~ + (M ~ + M') ( ~ + M v) - - 3 (~ , + M') ~ + ~ > o,

or for the j -point of the lat t ice o r h s

qi ---- - - (Xt q- M~ ")2 - - (YJ -I- M~') 2 a t- (X{ -[- M~) (}"i -[- M~) ~ 3 (Z i + M~) 2 -f- ~ > 0 (5)

(]: 1,2 ..... T--b L).

Here and are the ,, ues of the ben ng and torsio moments occ rri at j-point

of the plate during a repea ted changing load. They a re re la ted by the conditions

= x M+ u -

(6)

qvi - - M~ q- M + , = = . = ( y ) j ;~ 0; ~,~ = M s - - (MT) j ;~ 0; q, j - - M ; + (M.~)j > 0.

where (M~)j, (M~)j, and (M~)j are the lower values of the boundary lines of the moments , constructed on

545

~2t!

24:

2 0 I9 18 17

9c

f 4 '7 16

5 8 15 "<

dll p,r tt Itz ,t3 '<

Jk t. ~" t. ~" L ~" L. I "1 "1

Fig. 1. Calculation plan of the plate.

TABLE 1

m a x tit J n n l i n 3:0 5 0,450 6 1,162

10 0,084 11 0,130

0,098 0,327 0,186 0,030 0,057

MU

m a x r a i n m a x

i,450 0,'098 0,069 O, 327 ] 0 0,157[0

0,503 0,18610 0,777 0,36010

the assumption of ideal e las t ic operat ion of the plate, and (Mx+)j, (My+)j, and (l~z)j a re the upper values of the same curves .

Present ing the conditions of flow in the fo rm of the Von Mieses conditions makes it possible for the example p resen ted above to reduce the volume of s tored information to l e s s than a fourth. The use of special methods not requir ing holding in the mem o ry all of the tables of l imi ts makes i t possible to solve the problem of the adaptabili ty of a plate with the use of a f iner la t t ice .

The re fo re , le t us approach the prob lem of nonlinear p rogramming with a composi te sys tem of l imits and l inear functions of the end.

Rewri t ing the prob lem in compact form:

min {H/qi (X, Y, Z, M x, M u, M ~, H) :~ 0

( ] = 1 , 2 . . . . . T + L ) , h t ( X , Y , Z ) = O

(i -= 1, 2 . . . . . T), t~ (X, Y, Z) = 0

(k= I, 2 . . . . . = ) , G. (M*) > o,

% (M') > o, f~ (M ~) ~ o,

% (M ~) > O, f,~ (M') > O,

qzj(M')>~0 ( j = l , 2 . . . . . T + L ) } .

(7)

Prob lem (7) may be conveniently solved by a "penalty function n method [6], which is convenient since ~n calculations on the computer it does not r equ i re m e m o r y of the whole table of the coeff icients of the sys - tem of l imits , ' but s tores in the memory of the computer e i ther only the value of the coeff icient of the table or only the ~tle for i ts construct ion.

To use the "penalty function" method we introduce the functional

T+L r T+L r+L

FI = H + ~- X 6i (qi) [q~]' + --~ ~, [h,f A- -~- .~ [tkf 4- 6j. (f~)If.,]' j=l I~l k=l J~l

T+L T+L T+L ~ts ~ P's �9 "~ Ixs 2 + y ~ 6s (qx) [Gj] + T V 6i (fy)[/ui]" + -2- ~,, 8j (qu)[qvi ]

j ~ l j ~ l j - ~ l

T + L T + L + V-s 2 its 2

T ~ ~Jq.)ILd +-~ Y, 8;(%)[q.l, (8) j = l j = l

where #s > 0;

0, ff f >i0; 6 ( f ) = 1, ~f f < o .

546

It has been proven [7] that under cer ta in conditions which are fulfilled for a given problem there ex- i s t s a s t r ic t ly monotonic ser ies {#s} which genera tes a se r ies of values {U(#s)} appearing as points of the absolute minimum of the function {II(X, Y, Z, M x, MY, M z, H, ps)}, and then at s = 1,2 . . . . ~ ~Ps -~ 0, U(#s) ~ U* and {II(X, Y, Z, M x, MY, M z, H, #s)} converge to an optimum value of the function H (U* is the vector on which a solution to problem (7) is reached}.

The minimum 1I at each #s may be obtained by a gradient method. Relational use of the method of combined gradients [8] for each has shown convergence in the genera l case of the nonlinear functional with the rate of geomet r ic p rogress ion .

Let us show in a simple example how to set up a full sys tem of l imits for a problem of optimum pro - jection of a rec tangular plate taking into account adaptability. The plate , shown in Fig. 1, is loaded by two fo rms of conflicting loads, a uniformly distr ibuted one over the whole surface of the plate with the value q and one directed f rom bottom to top concentrated at point 5 with a value of 3qX 2, but directed in the oppo- site direct ion to the f i r s t load. As a resu l t of s y m m e t r y we need consider only 1/8 of the plate. The val- ues of the envelope, const ructed on the bas is of e last ic calculations taking into account symmet ry and con- tour conditions, a re shown in Table 1.

Let us substitute the known values in (7). Let us find the minimum H at the l imits

4X3 + X6 + Y6 + Xlo + Y,o ~ O; - - 4X 5 + 2X 6 + 2Y o - - Z 3 ~- O; - - 2X 6 --2Y o + 2X~ + X 5 + YH ~ O;

(X 3 + M~) ~ - - 3 (Z a -I- M')23 + H ~ 0; - - (X 5 q- M~)~+ H ~ 0; - - (X o -I- X~) ~ -- (Yo -[- Mg) 2 + (X6 + M~) (Yo -]- Mg}

-]- H ~ 0; - - (X n -{- M~,) 2 - - (YII -[- Mfl)~ "{- (Xll -~ g~,) (Y,l -[- M~,) ~- H > 0; "-- (X,o -]- Mfo)~ -- (Y1o -~ M~o)~

+(X I -[-M x)(Y -4-Mf)+H:~0; --0.098~M~0; 0~M]~ 0,069: 0 10 10 ' 0

-- 0.327 ~ M~ ..< 0,450; ~ O. 186 ..< Mg ~. O. 162; - - O. 157 ~: M~ ..< O; - - 0.030 ~ g~o ~ 0.084;

0.186 ~ M~o ~ 0.503; - - 0.057 ~ M~, ~ 0.130; ~ 0.360 ~< M~ ~< 0.777.

Solution of the problem gives the value H = 0.453 qX 2, f rom which hminadapt = 1.386 }" xfq-~T"

L I T E R A T U R E C I T E D

1. V .T . Koiter , General Theorems of Elastoplast ic Media [in Russian], Fizmatgiz, Moscow (1961}. 2. A.R. Rzhanitsyn, Calculating Designs Taking into Account the Plas t ic P roper t i e s of Materials [in

Russian] , Stroiizdat, Moscow (1954). 3. A.A. Chiras , Methods of Linear P rog ramming in Calculating Elastoplast ic Systems [in Russian],

Stroiizdat, Leningrad (1969). 4. P .M. Varvak, The Development and Application of a Network Method for the Calculation of Plates

[in Russian] , Nos.1 and 2, Izd-vo AN UkrSSR, Kiev (1952). 5. D .C .A . Koopman and R.H. Lense, nOn l inear p r o g r a m m i n g and plast ic l imit analysis , n J. Mech.

Phys. Solids, 13, No. 2 (1965). 6. D.B. Yudin and E.G. Gol 'shtein, Linear Programming; Theory, Methods, and Application [in Rus-

sian], Nauka, Moscow (1969). 7. Antony V. Fiacco and Garth P. McCormick, "Extensions of SUMT for nonlinear programming; equa-

lity const ra in ts and extrapolation," Manag. Sci . , 12, No. 1• (1966). 8. B .T . Polyak, nA method of combined gradients in p--roblems at the ex t reme," ZhVMMF, 9, No. 4

(1969).

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