Grillages of maximum strength and maximum stiffness

Int. J. mech. 8~i. Pergamon Press. 1972. Vol. 14, pp. 651-666. Printed in Great Britain

GRILLAGES OF MAXIMUM STRENGTH A N D MAXIMUM STIFFNESS G. I. N. ROZVANY Department of Engineering, Monash University, Clayton, Victoria, Australia

(Received 29 February 1972) Summary--Assuming a preassigned beam depth, m i n i m u m weight solutions are derived for both perfectly plastic and elastic grillages of given strength as well as elastic grillages of given stiffness. The solutions presented also give a m i n i m u m reinforcement volume for perfectly plastic fibre-reinforced plates. Common kinematic optimality conditions are stated for the foregoing classes of problems and a method is outlined for finding the optimal solution for a n y clamped boundary. Morley's claim regarding the non-existence of certain kinematically admissible optimal solutions is shown to be erroneous. The proposed technique is illustrated with a n u m b e r of examples. 1. I N T R O D U C T I O N

IT w~.T. be seen from the following comparison that a close relationship exists between the class of optimal structures to be considered and the class of frameworks commonly known as Mitchell structures. 1 Optimal flex~ral systems considered

Mitchell structures Forces are parallel to the plane of the structure. Weight per unit area is proportional to the sum of the moduli of principal forces. Principal strains take on a given constant value in directions of non-zero principal forces.

Forces are normal to the plane of the structure. Weight per unit area is proportional to the sum of the moduli of principal moments. Principal curvatures take on a given constant value in directions of non-zero principal moments.

The class of structures considered have the following common properties: the structural components to be optimized (beams in grillages or fibres in plates) are placed in the directions of principal moments; the lever arm between the compressive and tensile stress resultant in any cross-section is assumed to be both preassigned and constant over the domain of the structure; the average value of both compressive and tensile stresses along any line normal to the middle plane takes on a constant value throughout the structure; membrane action is neglected; and the spacing of the structural components is much smaller than the smallest span of the structure. Examples of structures that fall in the foregoing category are grillages consisting of beams having a rectangular cross-section of given depth b u t variable width or sandwich crosssection of constant core depth and small cover plate thickness; and fibrereinforced plates in which the plate material has an infinite compressive 651

652

G.I.N.

RozvA~r

strength but a zero tensile strength) A more refined formulation that allows for finite compressive strength was developed by Mroz) First the optimality criteria for maximum strength will be given for p e r f e c t l y p l a s t i o g r i l l a g e s a n d f i b r e - r e i n f o r c e d p l a t e s a n d t h e n i t is s h o w n t h a t the same solutions also satisfy elastic compatibility and the requirement of m a x i m u m o v e r a l l stiffness.

2. M A T H E M A T I C A L

FORMULATION

F o r the class of structures considered, the material consumption per unit area is ~b = k(lM1 I ÷ l M s I),

(1)

where M~(x, y) and M2(x, y) are principal moments such t h a t M~>~M~, k is a known constant and x and y are Cartesian co-ordinates. I n Prager's terminology, a ~b is called the "specific cost" and equation (1) the "specific cost function". Considering perfectly plastic systems designed on the basis of the lower bound theorem of plastic limit analysis, ~ the only constraint on the optimization problem is the equilibrium equation. Hence the problem considered can be stated as min #9 = k f f D ([ M~ [+[M2 l) d x d y

(2)

~ M~/~x s ÷ ~ M~/~y ~÷ 2~~ M ~ / ~ x ~y = p(x, y),

(a)

subject to

M~+M~+

[[/M~-M~\ ~

~ ÷ML],

(4)

where D is the domain of the structure to be optimized, (I) is the total "cost" to be minimised and M~, M~ and M ~ are bending twisting moments in Cartesian coordinates and p(x,y) is the load. Other symbols were defined earlier.

3. K I N E M A T I C

CONDITIONS

FOR

OPTIMALITY

According to the general theory of optimal plastic design b y Prager and Shield, ~ a n y optimal solution for the problem in equations (2)-(4) is associated with a statically admissible generalized stress field {~(x, y) and a kinematically admissible displacement field u(x, y) such t h a t a t a n y point of the domain considered the generalized strains el(x, y) are given b y the gradient of the specific cost function with respect to the generalized stresses. I n this problem, the generalized stresses are the principal moments M 1 and M s and the generalized strains are the principal curvatures K1 and Ks. The specific cost function in equation (1) and its gradient vectors are shown graphically in Fig. 1. I n an analytical form, the Prager and Shield theory gives the following strain-stress relations: ~ = k for M~>O ~¢=--k forMi<0 -k<~<~k

forM~= 0

( i = 1,2),

(5) (6) (7)

where the principal curvatures ~1, and Ks must have the same directions as the principal moments M1 and M s. The same optimality conditions can be obtained b y using variational methods, s, 7 the principle of uniform energy dissipation s or Morley's derivation, s

Grillages of m a x i m u m strength and m a x i m u m stiffness

653

An upper bound on the m i n i m u m cost ¢opt can be obtained b y adopting a n y statically admissible moment field, and then evaluating the right-hand side of equation (2). A lower bound on ¢opt can be established ~, 9 b y finding a kinematically admissible displacement field u(x, y) satisfying -/c~
(9)

i~ = f f D u p d x d y .

%

\

/ "o ~./

FIG. 1.

Cost function and kinematic optimality conditions.

4. E X T E N S I O N S

OF THE OPTIMALITY ELASTIC GRILLAGES

CONDITIONS

TO

The main difference between the optimization of perfectly plastic and elastic grillages is the introduction of a further constraint, viz. the elastic compatibility requirement in addition to the equilibrium constraint. However, it can be shown readily that all optimal solutions obtained for perfectly plastic grillages satisfy elastic compatibility and hence they are also valid for elastic grfllages. Considering a n y linearly or non-linearly elastic strain-stress relation and assuming t h a t all cross-sections are designed to develop a given value of permissible stress, both stress and strain values in the extreme fibres of a beam cross-section will take on a constant value throughout structure. Since all beams are assumed to have the same depth and the beams are placed in the directions of the principal moments, the absolute value of the principal curvatures will be the elastic strain value corresponding to the permissible stress, divided b y the (constant) distance of the extreme fibres from the middle plane of the grillage, l~aturally, the sign of the curvature will be the same as the sign of the corresponding moment and hence the elastic momenV-curvature relation take on the same form as equations (5) and (6). I f a beam has a zero cross-section over a finite length, then a n y curvature value satisfies elastic compatibility including the ones given b y equation (7). Since the optimal displacement field u(x, y) based on the Prager-Shield theory 4 also satisfies the kinematic boundary conditions, the optimal solution obtained for perfectly plastic grfllages is valid for elastic grillages. I t follows from some recent work b y Prager ix that the foregoing solutions also give a m i n i m u m volume for a n elastic grillage of given stiffness (i.e. a given value of the integral SSDup dx dy). The same conclusion can be obtained b y standard variational methods.* Further, the same kinematic optimality conditions can be obtained for both given strength and given stiffness from a general optimality criterion by Masur. x° * A complete proof based on the calculus of variations is available from the author.

654

G.I.N. 5. G E N E R A L

PROPERTIES

ROZVANY OF

OPTIMAL

SOLUTIONS

E q u a t i o n s (5)-(7) a d m i t t h e following t y p e s of " r e g i o n s " i n a n o p t i m a l s o l u t i o n {see Fig. l, M~>.~Ms): (i)

R+

-Y/I~> 0,

M s = 0,

K~ = 1,

--l~
(ii)

R-

M~ = 0,

M s ~< 0,

-1,.
(iii)

R +-

M~>

0,

M s ~< 0,

K I = 1,

(iv)

R ++

M~ ~> 0,

M~ ~> 0,

(V)

/~----

M 1 ~ 0,

M S ~ 0,

Ks = - 1

Ks = - 1

(AinFig.

1),

(BinFig.

1),

(ABinFig.

1),

K~ = Ks = 1

(ACinFig.

1),

K1 -- K2 = -- 1

( B D i n Fig. 1).

I n all figures s h o w i n g o p t i m a l solutions, R + a n d R - a r e m a r k e d w i t h a r r o w s in o n e d i r e c t i o n (Fig. 2), R + - w i t h a r r o w s i n t w o d i r e c t i o n s a n d R ++ a n d R - - w i t h s m a l I circles. T h e a r r o w s s h o w t h e d i r e c t i o n s of p r i n c i p a l m o m e n t s a n d t h e a p p r o p r i a t e signs a r e also i n d i c a t e d . R e g i o n s R +-, R ++ a n d R - - give a n infinite n u m b e r o f s t a t i c a l l y a d m i s s i b l e m o m e n t fields. T h e d i r e c t i o n s o f t h e p r i n c i p a l m o m e n t s axe fixed for R + - b u t a r e a r b i t r a r y for R ++ a n d R - - .

R÷

R-

®

5)

R+ ÷

R--

R~~iG. ~.

Syl~bols representing various types of optimal fields.

U s i n g s o m e earlier r e s u l t s b y Shield, is it c a n b e s h o w n t h a t o p t i m a l lines o f p r i n c i p a l m o m e n t s a r e a l w a y s s t r a i g h t i n t h e d i r e c t i o n of n o n - z e r o m o m e n t s . Shield o b t a i n e d t h e following r e l a t i o n 1~ ~,¢~/~ss = (1/p~) (Ks-- K~), (10) w h e r e s S is a c u r v f l i n e a r c o - o r d i n a t e m e a s u r e d in t h e d i r e c t i o n o f t h e line o f p r i n c i p a l c u r v a t u r e Ks a n d P1 is t h e i n - p l a n e c u r v a t u r e of t h e line of p r i n c i p a l c u r v a t u r e K1. I f M~ ¢ 0, t h e n b y e q u a t i o n s (5) a n d (6), K1 = c o n s t = (k or - k). H e n c e t h e d e r i v a t i v e i n e q u a t i o n (10) t a k e s o n a zero v a l u e a n d p~ = co. T h i s m e a n s t h a t i n R + a n d R - regions t h e lines of p r i n c i p a l m o m e n t s M~ a n d M s r e s p e c t i v e l y a r e s t r a i g h t . T h e g e n e r a l e q u a t i o n s for d i s p l a c e m e n t fields in t h e o t h e r t h r e e regions a r e : R + - : ½k(yS-xS)+a+bx+cy, } R++: ½k(-yS-x~)+a+bx+cy, R - - : ½k(yS+ xS) + a + bx + cy,

(11)

w h e r e a, b a n d c a r e c o n s t a n t s , x a n d y a r e i n t h e d i r e c t i o n s o f M~ a n d M S, r e s p e c t i v e l y a n d K~ a n d Ks a r e defined as

K l _ ~ - ~ u / ~ x ~, K s = - ~ s u / ~ y s. 6. R U L E S

FOR

(12)

DERIVING OPTIMAL SOLUTIONS CLAMPED BOUNDARY

FOR

ANY

Along clamped boundaries the kinematic boundary conditions are

u(x,y) = O, ~u/~x = O, egu/~y = 0. T h e rules t h a t follow are v a l i d for a n y n o n - n e g a t i v e l o a d i n g t h e r e a d e r is r e f e r r e d t o t h e A p p e n d i x .

(13)

q(x, y)>~O. F o r proofs,

Grillages of m a x i m u m strength and m a x i m u m stiffness

655

Ru~e 1. I f par~ of the domain D lies in between two clamped edges, then the optimal regions can be determined b y the construction shown in Fig. 3.

2< ~

FIG. 3.

~

Graphical representation of Rule 1.

Line segments A-0-and 0 ~ are of equal length and t h e y are normal to the adjacent boundaries. Points B a n d D divide these segments into two equal parts a n d are contained b y the region boundaries between R+ a n d / ~ - t y p e regions. Line segments ~ and DE give the direction of principal moments in R - regions adjacent to the boundaries and line segment BD gives the direction of principal moment in the central/~+ region. The curve FF" in Fig. 3, will be referred to as the eenterline a t the R + region, where ~-~ = ~-~ and C is not necessarily contained b y the eenter-line.

Example 1. Clamped elliptic baundary Let the b o u n d a r y of a clamped grillage be given b y (Fig. 4(a)) =

,

(T4)

~Y

i,

a

.I.

a

,I (a)

b) FIG. 4. E x a m p l e based on Rule 1. Optimal regions for a clamped elliptical boundary.

656

G.I.N.

RozvA~x

which gives dy xb d--~ = -+h ~~/[1 - (x~/a~)] '

dy xb ~ y "~ = "~-"

Then from Fig. 4, the co-ordinates of the region boundary are ~=½b

j(x;) 1--

,

~=x

1-

(15)

which reduces to ~ = ~J(1

a~[l_~b~/2a~)].~).

(16)

The optimal regions are shown in Fig. 4(b). Rule 2. The intersection of the centerline of a n R + region and a boundary between R + and R - (point G in Fig. 5) always occurs at a distance r/2 from a point of the boundary where the curvature of the boundary takes on a local maximum, r is the radius of curvature of the boundary, and the curvature is taken as positive if r is contained by the domain D.

(~) Fro. 5.

(b) Graphical representation of Rule 2.

This rule is important, because it gives the correct position of the centerline (see Appendix). In E x a m p l e

I, the m a x i m u m

curvatures

occur at y = O, x = + a.

The radius of

curvature at that point is 1 y=o b2 d 2 x/dy ~ = -~.

(17)

~ = +a~b2/2a.

(18)

For ~) = 0, equation (16) gives

~ is the horizontal co-ordinate of point G which is the intersection of the centerline and the region boundary (cf. Rule 2). Ru/e 3. A n R ++ type region occurs at the intersection of three or more centerlines. The appropriate region boundaries can be consSructed in accordance with Fig. 6. Example 2. Boundary consisting o f / o u r circular arcs (Fig. 7(a))

I n polar co-ordinates, the region boundary is given b y (Fig. 7(b)) R

(1

1

(19)

Grillages of m a x i m u m strength and m a x i m u m stiffness

/ F~G. 6.

Graphical representation of Rule 3.

.

~>

e)

(b)

F~G. 7.

Example illustrating Rules 1, 2 and 3.

657

658

G. I. N. RozvA~Y

which gives ~ = (R]2) ( 1 - c o s 0 ) ,

• = (R/2) (sin O+ t a n 0),

y. = 2 ( R - 9 ) 4 ( R 9 - 9 ~)

(20) (21)

R - 29 ~ e o p t ~ a l r e # o n s are s h o ~ ~ Fig. 7(c). ~ o j ~ c t i o n p o ~ t (H ~ Fig. (7c)) can be located b y m a ~ g ~ = R - 5 ~ equation (21): R - - X H = y , = R(~2--1)/2 ~2.

(22)

Note t h a t local maxima of the c ~ v a t ~ e o c c ~ at p o ~ t s H where r = 0 ~ d 1/r = ~ . S~ce r/2 = 0, R ~ o 2 ~ p l i e s that the ~temection of the centerl~es and the region b o ~ d a ~ ~ c o n t a ~ e d by the b o ~ d a r y of the d o m a ~ D (see p o ~ t s J). I t can be checked readily that u(x, y) satisfies the slope c o n t r a r y c o n ~ t i o ~ along all region b o ~ d a r i e s . For example, the displacement f i e l ~ ~ ~ho n m b e r e d r e g i o ~ ~ Fig. 7(c) ~ e * u~ = a~/2 = (~[x ~+ (y - R) ~] - R}~/2, (23) (24)

u~ = ( ~ - ye)/2 + u~(x, ~),

where ~(x) ~ ~he vertical co-ord~ate of the region b o ~ d a ~ for a ~ v e n value of x. The c o ~ e s p o n d ~ g ~ s t derivatives with respect to y ~re ~u~/~y

=

-

(25)

y,

~u~/~y = y - R - R ( y - R)/~[x ~+ ( y - R)~].

(26)

S u b s t i t u t ~ g ~ for y and the righ~ hand side of equation (21) for x, e q u a t i o ~ (25) and

(26) ~ve ~u~ = ~u , = ~Y y=~ ~Y y=~

-~.

(~)

S~ce equation (24) also e n s u e s continuity and slope c o n t r a r y along and ~ the d~ection of the r e , o n b o ~ d a r y y ~ ~, e q u a t i o ~ (24) and (27) ~ p l y slope c o n t r a r y along the b o ~ d a ~ ~ M1 directions. R u ~ 4. ~ R - - ¢ ~ e r e , o n o c c ~ s ~ r o ~ d reentrant c o ~ e r s and an R +- t ~ e region o c c ~ ~ between two such R - - t ~ e regions (Fig. 8).

Fro. 8.

Graphical representation of Rule 4.

* k = 1 is adopted in all exaznples.

Grillages of m a x i m u m strength and m a x i m u m stiffness

_ L/4__ l-I,

Fxo. 9.

~

-I-

659

_I_L~ .~ L

-l-

-i d

Example illustrating Rule 4.

Example 3. Intersection of three one-way clamped grillages (plates) The optimal regions for this problem are given in Fig. 9. 7. F U R T H E R EXAMPLES The first solution of equations (2)-(4) for a non-axisymmetrlc boundary was reported in 1966 b y the author 14 who used a statical method and considered a simply supported square boundary. The same solution was soon confirmed b y a static-kinematic method, s Solutions for clamped rectangular and triangular domains were obtained recently b y Lowe and Melchers 15 who employed a rather ingenious statical method. Using the rules described in Section 6, the optimal solution can be determined readily for a n y clamped boundary. The author has derived solutions for a very large number of boundary shapes, out of which only a few typical cases are discussed herein. ~5, 31 Example 4. Clamped quadrilateral domain I t will be seen from Fig. 10{a) that, in general, the optimal solution for a quadrilateral domain contains two R ++ regions. The only exceptions are kite-shaped domains (Fig. 10(b)) including rhombic and square domains (Fig. 10(c)). For the latter, the displacement fields are the following: ul = y~/2, ~ us = ( 4 x y - x 2 - Y S ) / 6 ' u a = a(x ÷ y)/2 -- (x ~+ y*)/2 - a*/8.

I

(28)

I t can be checked easily that the displacement field and its slopes are continuous along region boundaries. For rectangular boundaries, the optimal displacement field is showa in Fig. 10(d). Exam/ple 5. Clamped support along a line and a point For a grillage supported along the line EE" and the point A (Fig. I1), Rule 1 yields the region boundaries shown in Fig. 11. The co-ordinates of the boundaries corresponding to points C and B in Cartesian and polar co-ordinates, respectively, are* Yc = a / 2 ( l + c o s 0 ) , rs --- a/2(1 ÷ cos ~).

xc = a s i n O / ( l ÷ c o s 8 ) ,

(29) (30)

* The author has shown~a that the line segment C B in Fig. 11 gives the direction of principal curvatives in the central R + type region.

660

G. I. N. R o z v ~ v

(a

(b)

))X

(c) (after Melchers)

~Y

0 Fro. 10.

Optimal regions for quadrilateral domains.

O,l

!

~ X~

(d)

.-.-/ /

~[~

1~

E

~

xc

.I

E,

Fro. 11. Optimal regions for a grillage with clamped supporbs along a line and at a point,

Grillages of maximum strength and maximum stiffness

661

Example 6. T.shaped domain

In the optimal solution given in Fig. 12, the curved boundaries are defined by equations (30) and (31). Example 7. Square grillage supported on four square columns

This solution (Fig. 13) is given as an example of a more complicated region configuration.

8

~,-~-' -~

8

~

~:_~

~: ~=.~ - -

i I£{#

Fm. 12. Optimal regions for a T-shaped domain.

e

8

e

/

@

:b rl~

®

8 -i-

x~ I

8

/<

{

e 8 @~ _\,~ ~,

Fm. 13. Optimal regions for a grillage supported on four square columns. 8. C O N C L U D I N G R E M A R K S Grillages o f m a x i m u m s t r e n g t h a n d m a x i m u m stiffness were considered a n d sufficient conditions for o p t i m a l i t y were presented. A m e t h o d giving the o p t i m a l solution for a n y clamped b o u n d a r y was outlined. An extension o f this m e t h o d t o a n y simply s u p p o r t e d b o u n d a r y a n d combined b o u n d a r y conditions will be given elsewhere. Clearly, Morley's d o u b t s ~ regarding t h e

662

Rozv~r

G.I.N.

non-existence of a kinematically admissible solution for c l a m p e d corners are unfounded. Whilst research workers seem to e n c o u n t e r difficulties in fmding the e x a c t solution in problems involving Michell structures or plastic limit analysis of plates, ~8 it is r a t h e r r e m a r k a b l e t h a t the m e t h o d presented gives the optimal solution directly a n d s y s t e m a t i c a l l y for even the m o s t complicated b o u n d a r y conditions. The close relationship between Michell structures a n d the class of structures considered was p o i n t e d o u t in this p a p e r a n d a dual relationship between plastic analysis a n d optimal plastic design was outlined in a n o t h e r paper. ~v The problems of o p t i m a l design of grillages a n d fibre-reinforced plates for multiple load conditions, serviceability conditions a n d partially preassigned g e o m e t r y were discussed elsewhere!, ~9, 20 REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

A. G. M. MICm~LL, Phil Mag. 8, 589 (1904). C. T. M o ~ r , Int. J. mech. Sci. 8, 305 (1966). Z. M~oz and F. G. Smc_~a~v, Archiwum Inzynierii Ladowej 16, 575 {1970). W. PRA(~.R and R. T. Sm-~.D, J. appl. Mech. 184, (1967). W. PRAO~.~, A n Introduction to Plasticity. Addison-Wesley, Reading, Mass. (1959). W . S. I-IE~e, Opti~num Structures. Oxford University, Dept. of Engng Sci., Report (November 1968). D. E. C ~ R E ~ r and G. I. N. R o z v ~ r , Int. J. non-linear Mech., 7, 51 (1972). D. C. DRUC~:E~and R. T. SHr~L]), Prec. 9th Int. Congress appl. Mech. Brussels, 1956, Book 5. G. I. N. RozvA~'Y and S. R. A])IDA~, Prec. Int. Conf. Design Automation (A.S.~I.E.) Toronto, 1971; also J. Engng Industry, Trans A . S . M . E . 94 (series B), 409 (1972). E. F. M~sv~, J. engng Mech. Div., Prec. A.S.C.E. 96, 621 (1970). W. PRAG~.~,J. Optimiz. Theory Appl. (J.O.T.A.) 6, 1 (1970). R. T. SHmLD, Q. appl. Math. 18, 131 (1960). G. I. N. ROZVAN~, Optimal Load Transmission by Flexure. Monash University, Dept. of Civil Engng, Report 1/1972. G. I. ~q. Rozv~a~Y, Civil Engrg Trans, Inst. Engrs. Aust. {~ES, 158 (1966); also Prec. Conf. Inst. Engrs Aust. (March 1966). P. G. Low~ and R. E. I~LC'~RS, Int. J. mech. Sci. 14, 311 (1972). G. SACCHIand M. SAvE, Int. Assoc. Bridge Struct. Engng 29-II, 157 (1969). G. I. N. R o z v ~ Y , J. engng Mech. Div., Proc. A.S.C.E. 97 1703 (1971). R. H. Woo]), Mag. Conc. Research 21, 79 (1969). D. E. CH~I~I~E~Y2and G. I. N. R o z v ~ v , Archives appl. Mech. 24, 89 (1972). G. I. N. R o z v ~ , J. Optimiz. Theory Appl. (J.O.T.A.) (in press). G. I. N. R o z v ~ Y and S. R. A])~DA~, Computer Methods in Applied Mechanics and Engineering (in press).

APPENDIX--JUSTIFICATION

OF

THE

"RULES"

IN

SECTION

6

Rule 1. It can be shown that the displacement u(x, y) and its slopes do not change their value along the polygons A B D E (Figs. 3 and 14) if the curved boundaries shown are replaced with straight boundaries (FA and _FE in Fig. 14) which are tangent to the original ones at points A and E. Adopting Cartesian co-ordinates (x, y) such that the axis x bisects the angle A F E (Fig. 14), le~ the new external boundaries be defined by y = +_rex = _+xtan8

(31)

663

Grillages of m a x i m u m strength and m a x i m u m stiffness and the yet u n k n o w n region boundaries b y y

=

+nx

=

(32)

+xtem[L

The value of n (and fl) can be determined from the slope continuity requirement for u(x, y). The displacement u(x, y) in regions R1 and R 8 has a zero value a n d zero slope along the clamped external boundaries and have a curvature K - - - / ¢ normal to the boundary. Hence at a n y point P of these regions (Fig. 14), the displacement is (33)

u = kt~/2,

y

A

~

R~=R-

~

R~:R + /

~y---nx __

.I ~m, 14, ~ns~i~o~$io~ o£ ~

L

where t is the distance from the clamped boundary. After co-ordinate transformation, equation (33} yields ul = l~(xsinO-ycosO)~/2,

]

u a = k ( - x sin 0 - y cos 0)~]2"

I

(34)

The boundary and curvature conditions for the central region are ( f o r x = O, y = O)

u~ = Ou~/Ox = Ou~/Oy = O,

O~u~/Oy ~ = I¢,

(onRe)

]

(35)

/

O2u/OxOy --- O,

which gives

(36)

ul ---- -- Icy~/2 + dx ~,

where d is a n u n k n o w n constant. Then the equalities ux = u~

a~d

Ou~[Ox = OUa/OX

along y = n~ yield /¢ [sin2 0-- 2n sin 0 cos 0 + n 2 cos 20] ------ kn ~+ 2d, ~ ~ [ s ~ 8(s~ ~-- n cos ~)] = 2d.

)}

(37)

Equation (37) reduces to tariff=n= ~

s~cos~ l+cos~,

t~(~-fl)= ~

t~-tanfl l+t~8+t~fl--

t~ 2

(38)

~

which ~ p l i ~ A B = B G ~ Fig. 14. ~ s h o ~ that ~ g the c o , t r a c t i o n ~ Fig. 3, the e n t ~ d o ~ c~ ~ 0nfl~te) set of p o l y g o ~ ( A B D E ) such that the ~ a t ~ of u(~, y) ~ of the p o l y g o ~ ~ 2 k ~ d the slope ~ c o n t ~ u o ~ along e ~ h polygon. I t ~ to e ~ t~t (i) the m o d ~ of the c ~ a t ~ does not e x c ~ d ~ ~ other ~ c t i o ~ ; (ii) each ~ m a l p o ~ t of D ~ c o n t a ~ e d ~ o ~ y one such polygon.

be covered b y the ~ e c t i o ~ stffi n e c ~ ~d

664

G.I.N.

Rozv~¥

Rule 2. Rule 1 does not define the position of the centerline of an R + type region and can result in non-uniqueness of the solution. However, condition (ii) above can be used to find the correct position of the centerline. To illustrate the importance of this condition, consider again the solution in Fig. 4. I n that problem, the centerline of R + is the axis x and it intersects the boundary at the m a x i m u m curvature of the latter giving the correct optimal solution.

FIO. 15.

Incorrect solution obeying Rule 1 but violating Rule 2.

Rule 1 in itself would not prevent the adoption of other centerlines. For example, if we made axis y in l~ig. 4 the centerline, then Rule 1 (Fig. 3) would give the region boundary in Fig. 15. This solution has the anomaly that any~segment (e.g. S~T in Fig. 15) of the external boundary is mapped into a segment (e.g. V W ) of the region boundary such that the two segments have a slope of opposite sign (see Fig. 15). As a result of this, at least some points (see point R) are contained b y more t h a n one polygon and hence u(x, y) would take on a non-unique value which is not permissible. I f the centerline starts at a boundary point of maxim~un curvature as required by Rule 2 (see Fig. 4), then the above anomaly cannot arise for the following reasons: I n satisfying the requirement in Rule 2, the limiting case consists of a boundary of constant curvature (Fig. 16(a)). Then point C (cf. Fig. 3) remains in a fixed position and corresponding segments (e.g. S T and V W in Fig. 16(a)) of the two boundaries have slopes of the same sign which ensures that a n y point of the domain is contained by only one poly_.gon A B C D . I f the curvature decreases (Fig. 16(b)) then point C moves to the right (C1Cz) as we move away from the centerline along the outside boundary (S~). Thus a decrease in the boundary curvature can only increase the horizontal_.compone~nt of I~W (compare Figs. 16(a) and (b)) which ensures slopes of the same sign for S T and V W for both constant and decreasing curvature of the boundary (cf. Rule 2).

C~ Cz 1~I6. 16.

Justification of Rule 2.

Grillages of m a x i m u m strength and m a x i m u m stiffness

665

Rule 3. Some polygons A B D E (Fig. 3) would also overlap in the vicinity of the intersection of three or more centerlines, if only Rule 1 were used. However, it can be shown that the introduction of an R ++ region (as shown in :Fig. 6) satisfies all continuity conditions for u(x, y). I t will be seen t h a t point K in Fig. 6 is at an equal distance from each boundary. Letting u(x, y) take on its local m a x i m u m at point K and making the same point the origin of the co-ordinate systems {x, y) (Fig. 17), the displacement field in R++, b y equations (11), becomes u = ~]¢(--y~ - x ~) + a

Fill. 17.

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Derivation of Rule 3.

and its slope along line segment QS is (Fig. 17)

O-a--ul vy] R +

= --Icy= ktsina.

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+,-~ --

The c o ~ e ~ o n d h g slope ~ R+ and along QS is (Fig. 17)

~OuR + , ~

= 1~ ~ ~-o

t~

cos az/

+ (t cos a - A t a n a~)~- t ~ - $~cos ~a]

=

( in ~ c o s ~1

m) /

~ a k ~ g ~ e of the r e l a t i o ~ (Fig. 17) tan ~ ~ ~ ~

~,

t ~ ~ = t ~ ~/(2 ~ t ~ ~ ~), s~ ~2/cos ~

~ t~

~

~

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~ c o s ~ / ( 1 + c o s ~ ~),

the r i g h t - h ~ d side of equation (41) reduces to the same value ~ the one ~ e q ~ t i o n (~0). Q.E.D.

666

G.I.N.

ROZVANY

Ru~e 4. At reentrant corners (Fig. 18(a)), Rule 1 can still be used for deriving the region boundaries, although the radius of curvature r -* 0 and 1/r -* - oo. Over the area adjacent to the reentrant corner and in between the lines ( L M and LN) normal to the domain boundary, the displacement field in polar co-ordinates (t?,r; Fig. 18(b))is u ~-r~/2 which

-

F ~ . 18.

M

Derivation of Rule 4.

gives in Cartesian co-ordirmtes u = ½(x~+y2). By equations (11), the latter defines an R - - region. Similarly, the displacements in the central region in Fig. 18(b) are given b y u = ½ ( x ~ - y2) + ½ya,

which represents an R+- region.

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