Theory of structural design

CoA Report No

THE COLLEGE OF AERONAUTICS

CRANFIELD

THEORY OF STRUCTURAL DESIGN

by

Corrigenda to C. of A. Report Wo. 115

'Theory of Structural Design' "by ?v'.5. Hem-p

Page 55 'Equation' (L|..2)

Delete the two equations defining e-, and Sp in

terms of e-. and e^, together with the preceedlng word 'if'

and the following word 'then'. Change e, and e^ to e^ and

e- , respectively, In the rest of the page,

Page 56 'Equation' (U.5)

Change e-, to e^ and e^ to e„. Delete the preceedlng

clause 'using the definitions of e^ and e in (lj..2)'.

Pap:e 36 Last line

Replace 'Equation' "by 'Equating',

Page 37 Second line after Equation

(h.b)

Delete 'and use has heen made of the values of e^,

s given In this same equation',

Page 38 'Equation' (li.9)

Delete this equation. Replace the preceedlng line

'e- given hy' by 'and e^'. Delete the following line and

replace hy 'We see hy

(k-.^)

that these strains are .,.,., '

' REPORT M3. 115 AUGUST. 1958 T H E C O L L E G E O P A E R O N A U T I C S G R A N F I E L D

t

(Professor of Aircraft Structures and Aero-Elasticity and Head of the Department of Aircraft Design.)

To be presented as a paper at the Conference of the Struct-ures and Materials Panel of A.G.A.R.D., Copenhagen, October 1958,

2

-Table of Contents

Abstract

1» Introduction

§1.1 Statement of the General Problem

§1.2 History

2 . C l a s s i c a l Theory for Frajne Structures

§2.1 Maxwell's Theorem

§2,2 Michell's Theorem

§2.3 Cox's Design Applications

Figures 2 , 1 , 2 , 2 , 2.3

3« Developments i n Two-Dimensions

§3.1 l a n e s of P r i n c i p a l S t r a i n

§3.2 Conditions of Equilibrium

§3,3 Formulation of the General Problem

§ 3 . 4 Special Solutions

Figures 3 . 1 , 3.2

4. Plates

§4.1 Michell Type Theorem for a Plate §4.2 Development of the Theory fear a Plate §4.3 Constant Stress Solutions

§4.4 Alternative Approach

5. Suggestions for Lines of Research 6* Conclusions

References

Appendix A, Curvilinear Coordinates Figiire A.I

Appendix B, Deformation and Equilibrium Conditions in Orthogonal Curvilinear Coordinates Figures B,1, B,2, B.3

3

-Abstract

The Theory of Structures is for the most part concerned with the calculation of stresses in a given structxire \inder given

external conditions of loading and temperature. The real problem of structural design, however, in aeronautics at any rate, is to find that structure, which will equilibrate the external loads, without failure or undue deformation, under such conditions of temperat-ure, as may be appropriate, and which at the same time will have the least possible weight. The solution of tliis general design problem is obviously very difficult and cannot be resolved at the present time. However, on the basis of certain classical theorems due to Maxwell and Michell and using methods and suggestions derived from these theorems by H.L. Cox, one cein make certain progress, and in addition point the way to profitable lines of research. The present paper reviews the classical results and their cijrrent application, develops the mathematical theory for the

two-dimensional case and derives a nuniber of special solutions, It is hoped that its publication will encourage research in this very important field,

•T 4 •T

-1. Introduction

§1.1 Statement of the General Problem

The real problem of structiiral design is the disposal of material in such a way, that it will safely equilibrate given systems of applied force, unier the appropriate physical conditions of, for exanrple, temperature, without exceeding permissible amounts of deflection and at the same time using the minimum of material. This last requirement is crucial in aircraft construction.

Practical considerations relating to manufacture, maintenance or function will of course force a departure from this ideal solution, but a knov/ledge of the optimum is clearly desirable as a control,

The Theory of Structures is for the most part concerned with stress or deflection analysis of given structures. This means that in practice, it can only be used in design by a process of trial and error, in vrhich the structural layout and sizes are first guessed or very roughly calculated, and are then subjected to as complete analysis as the theory will permit. The results of these oalcijlations are then used to modify the design to perhaps achieve a more uniform

distribution of stress, and the thoroioghgoing analysis repeated as a check. The theory ought to be in a position to tackle the design problem directly, that is, to begin with the given forces and to

produce by calculation the best struotvire that will safely carry them, The present report is concerned with reviewing the present position in this little developed branch of the theory and with suggesting lines of research, which may lead to developments of knowledge, such as to make direct structural design a practical part of the normal techniques of engineering.

5

-§1.2 Hi.-.tory

The f i r s t important c o n t r i b u t i o n to the Theory of Structijiral

t

Design wa^ made by Maxwell . He proved a theorem about the

equilibrium of a series of attracting and repelling centres of force and applied it to a fi-ame structure in which the bars replaced the

"actions at a distance" except in the case of the exteriial forces, and by this mear^s effectively obtained the result of equation (2,1), He commented upon the englnsering significance of his theorem in these words:

"The importance of the theorem to the engineer arises from the circijmstance that the strength of a piece is in general proportional to its section, so that if the strength of each piece is proportional to the stress which it has to bear, its weight will be proportional to the product of the stress mixLtiplied by the length of the piece,

Hence these sums of products give an estimate of the total quantity of material which must be used in sustaining tension and pressure

respectively".

Nothing that Maxwell uses the word "stress" for what we sho\ild term "load" we see that in effect he has obtained equation (2.3) and has

draivn the practical conclusion about the required weight of the structure.

tt

Michell made the second important contribution to our subject, He recognised the importance of Maxvrell's result and enunciated

eqioation (2.3) in its present form, applying it to the calculation of optimiira structural weights. However he went much further than that by generalising one of Maxwell's proofs of his theorem by the method of virtual work, tising, instead of Maxwell's uniform dilatation, the more general deformation of (2.6). This led him to sufficient conditions for a structure to be an optimum. He proved the geometric restriction of equation (3.3)* which determines the classes of orthogonal sets of

' Ref. 1 pp.1 75-7

tt

curves along which the members of an optimum structure must lie and gave figures illvistrating all the results of section §3.1 with the exception of the general integral of equations (3.8) and (3.II), He also gave an example of a three-dimensional structure for

transmission of torque, the members of which lie on the surface of +

a sphere .

These important contributions to our subject passed unnoticed for some forty years until Foxolkes of the Department of Engineering at Cambridge University read Michell's paper and realised its theoretical importance. He drew the attention of H.L. Cox to the paper and by so doing created a champion for the ca\ise of direct structural design,

14-Cox has done much by l e c t \ i r e s and papers to draw the a t t e n t i o n of engineers to Maxwell's and Michell's resxilts ajid to convince them of the important gains tliat may w e l l be made by further development of t h i s s u b j e c t . His own important c o n t r i b u t i o n s to the p r a c t i c a l a p p l i c a t i o n of, i n p a r t i c u l a r . Maxwell's Theorem are o u t l i n e d i n s e c t i o n §2.3

t

Reproduced i n Ref. 3 , P i g . 1 2

I j l

1

-2. Classical Theory for Frame Structures §2.1 Maxtvell's Theorem

Consider any frame structure which equilibrates a set of forces P. acting at points with position vectors r.(i=1,2..,n). Let T, be the

X X x load carried in a typical tension member with length L, and section area A..

Let (-T ) be the load carried in a typical compression member with length c

L and section area A .

c c

Impose a virtiial displacement on the structure which consists of a uniform dilatation of space of magnitude 3e, chosen so that the origin

for the vectors r. is at rest. Every linear element of space is extended by a strain e and so the virtual displacements at the points of application of the forces are er..

We can thus write,

n

Virtiial work of the external forces = e \ P. . r.

L

^ ^

The change in energy of a tension member is

[

(T^A^ + Ee) (V^t^'

IE"

^t^t =

W

correct to the first order in e, where E is Young's Modulus. The change in energy of a compression member is -T L e and so applying the

c o

P r i n c i p l e of V i r t u a l Yfork, we find, cancelling e , t h a t

n

t c i=1 •

8

-where \ , \ are sums over the tension and compression members

l—i L-i

respectively,

t o

If f. and f are the permissible stresses in tension and

Hi C

compression and if we assi;mie that all members are stressed to the limit,

we can \'vrite,

_{T. = A.f. , T = A f . . . (2.2)}

t

t

t

'

o

o

o ^ '

and s\ibstituting in (2,1) obtain,

^ t ^ - ^c^c = ^ ^ i • ^ i ••• (2.3)

X

where V, is the volume of all the tension members and V that of the

t c

compression members,

The total volume of the framework V is given by,

V = V. + V ... (2.4)

axid so using (2.3) we can write,

i i

We see then that of all the possible frameworks that eqiJilibrate the

forces P. and satisfy the strength requirement (2,2), the lightest

structure is the one which has the least volume of compression members or

t

alternatively, the least volume of tension members , In particular the

framework, if it exists, all of whose members carry tension or

alternatively compression loads only, is the lightest framework

possible among all those which carry the given loads. The vol-ume of

this optimum structure is given by (2,5) v/ith V. = 0 or V = 0 .

t

9

-S?,2 Michell'.? T'neorem

Consider once more as in §2.1 a series of external forces P. acting at r.. Let D be a domain of space containing the points r.; in particular D can be the whole of space. Consider then all possible frameworks S , contained in D which equilibrate the forces P. and which satisfy the limiting conditions of stress (2.2). Let us assume that there is a framewoi'k S which satisfies the following condition:

"There exists a virtioal deformation of the domain D such that

the strain a3.ong all the members of S is equal to ± e, where e is a small positive niurber, and where the sign agrees 'with the sign of the end load carried by the particular member, and further that no linear element of D has a strain numerically greater than e". ... (2.6)

* *

Michell's Theorem states that the voliime V of S is less than or equal to the volume V of any of the frameworks S.

First of all we notice from (2.2),(2.3) and (2.4) that,

t c i=1 and so the frame with the least voli:ime is that v/hich has the least value of \ L,T + \ L T . Secondly we apply the virtual deformation of (2.

₆₎

10

-to any of the frameworks S. The virtual work of the external forces

will be the same for all the frameworks and so the change in strain energy

will be the same too. If e. , e are the mean values of virtual direct

t c

strain taken along the lengths of typical tension and compression members,

the change in strain energy calciolated as in the proof of (2.1) is given by

Change i n S t r a i n Energy _ NT^TLg -

for any S / t t t / ^ c o c

t c

For the special case of S we find from (2,6) that,

for S*

\

^ ^

l«}

Equating these r e s u l t s we find,

since by (2,6) we have le. I < e , Ie I < e. Dividing by the p o s i t i v e

number e we see by ( 2 . 7 ) t h a t ,

V < V . . . (2.8)

The actual value of V follows from the Principle of Virtiaal

Work, If the virtual displacements corresponding to (2,6) at the points

of application of the forces, i.e. at r. , are ev, , we have, dividing

out the e,

n

l^.h

^

l^.h

-

E

t o i=1

and so by (2.7) >

11

-n /^ ^ \ -n * (ft + ^c^ f^ (^t ^c^ V'

-t c Z_' -t c £

i=1 i=1 The character of the deformation of (2.6) imposes certain

*

restrictions upon the layout of members in S , At a node of this frame-work the directions of the strains ± e, which are along the lines of

members of S , are principal directions of strain and must thus satisfy certain conditions of orthogonality. In a three dimensional frajnework, at a node with three members, there vv'ill be no restriction, if the loads in the members have the same sign, since in that case the virtual

deformation is a pure dilatation and therefore isotropic; however, if one load is of opposite sign to the others, it must be at right angles to them. At a node with four members, there is again no restriction if all loads have the same sign; if one member has an opposite load to the other three, then it must be orthogonal to them all and so forces them to lie in a plane; finally if the members fall into pairs with opposite signed loads then one of these pairs must be in line and normal to the other two. The general natiire of the restrictions is clear from these examples. Similar requirements follow for the layout of optimism two-dimensional frames, When the loads at a node have the same sign, there is no restriction, A node with tvro members oax*rying loads of opposite sign must have these members at right ajigles. A node with three members, one of vihose loads is opposite to the two others, must have the two in line and the member with the opposite load at right angles, while one with fo\xc members, with two pairs having opposite signed loads, miost have the pairs in line and orthogonal to one another,

12

-The optimum structure S has another very important property, In a general sense, it has greater stiffness than any other structure of S which satisfies (2,2) without meeting the requirement of (2.6) . Let us

think of the structures S loaded by forces XF. acting at r.,where X is a parameter which varies from 0 to 1. The stresses developed are X f in the tension members and X f in the compression members. The strain energy stored V/ is thus,

The displacement "corresponding to the force system P." is by Castigliano's First Theorem given by

^''^ - i( V,,f^ . V f,^ 9Xy EV T T c c

X=1

Slabs t i t u t i n g from (2.5) f or V. and V we find

X c

n

Displacement corresponding to P = ^ P^+f ^ + (f - f ^ ) \ P^ . r. ,,. (2,10) i=1

The fact that S has the greatest possible stiffness then follows by (2.8),

tt

Applications of the theorems of Maxwell and Michell to simple design problems have been made by H,L, Cox. He has considered first of all the problem of three coplanar forces. In the case where their point of intersection lies within the triajigle formed by their points of

application, the optimum framework can consist of tension (or compression)

t

H.L. Cox. Ref.3

tt

13

-members only. Some of his layouts are given in Fig. 2.1. Others can be obtained by superposition of these and analogous strvctures in suitable proportions yielding a series of redundant frameworks. Equation (2.5)j with V = 0 , shows that all these structures have equal weight. One can remark, as pertinent to the general philosophy, that we have here an infinity of solutions, ranging from mechanisms to simply stiff structures to structiires of any degree of redundancy!

The case where the point of intersection of the three forces lies outside the triangle of points of application is more difficult. Cox gives solutions for a number of symmetrical cases including the case of parallel forces illiostrated in Pig. 2.2. Here the structure consists of a circular rod, conceived as the limit of infinitesimal chords pinned end to end, two straight members and a continuum of spokes all lying along radii of the circle. The radial members are all in tension and the curved member in compression. Michell's criterion (2.6) is satisfied using a constant deformation with direct strain e radially and -e

circxomferentially. The fact that this is a consistent strain system will be shown later (in§3,l); it also follows readily from B,3 in polar

coordinates. The structure of Pig. 2.2 is thus an optimum.

Cox uses this last construction to build up a structure for the transmission of a bending moment (see Pig. 2.3). He shows that for

l/d > 4 this structure is considerably lighter than a "simple tie and strut" and that for larger values of l/d multiple constructions, on the lines of Pig. 2,3, can be even lighter. He produces a competitive 14-b£jr framework and a variation on Fig, 2.3j in which the circles are replaced by spirals, which for z/d > 4 is ligher than any other construction

considered. These structures for the transmission of bending moments are not Michell optimum structures, since they fail to satisfy the orthogonality conditions for members with opposite signed loads (§2.2). They are however by Maxwell's Theorem the best of their "class".

H

FIG 21

IS

16

-3 . Developments i n Two-Dimensions

§3.1 Lines of Principt\l Strain

The deformations, associated with two-dimensional Michell optimum structures, are by the criterion (2.6) of two kinis. In the first kind the principal strains at a point are equal in magnitude and sign and so correspond to a state of viniform dilatation. The lines of principal strain are thus completely indeterminate and as remarked before there is no restriction on the layout of the corresponding structure. The situation is quite different for the second kind of deformation in which the principal strains are equal and opposite, say +e and -e ;

here the lines of principal strain are restricted to certain classes of orthogonal cvtrves. This can be seen as follows.

Let us take the lines of principal strain as coordinate curves for a system of curvilinear coordinates (a,/9). The formulae of

Appendix A will then apply with CJ = ^ QJ^ those of Appendix B as they stand» The state of strain under consideration is defined by

^aa =^ '

^^(3

=-® ' V = ° ••• (5.1)

Siobstitution from (3.1) in the compatibility equation (B.3) yields,

cancelling (-2e)

1?

-Equation (A.17) which mist be s a t i s f i e d i n any coordinate system y i e l d s

with CT = r

J ^ / 1

9B\

J_/^1

M ^ n

da\^l daj "*• S^ \B dl3 J ~

We see then t h a t our l i n e s of p r i n c i p a l s t r a i n must be such as to imply,

Reference to (A. l6) shows t h a t these resiiLts can be w r i t t e n

where these l a s t are not independent since ilf2 = t i + 7'/2. This

equation can be i n t e g r a t e d a s ,

ilfi = Pi (a) +P2(/?) = t2 - | . . . (3.4)

where P^ and Pg are a r b i t r a r y functions. Equation (A. l6) then gives

the follov/ing r e s u l t s , which also follow d i r e c t l y from (3.2)

s i = -^i(") ' i f = ^5(^) -(^-s)

The form of equation (3.3) shows that our lines of principal strain

t

have the same form as the slip lines for two-dimensional perfect plastic flow . This means that we can make use of much of the known developments in this

field. Some of the integrations which follow, parallel corresponding processes in Plasticity Theory, but as the methods used are standard mathematics, it cannot be said that we are really losing the analogy,

However, it may well be that this analogy could jdeld fruitful suggestions,

18

-The integration of (3.5) is by no means a straightforward process,

However progress can be made at the expense of a Slight restriction of

generality. Let us assume that the derivatives P^ (a) and

F^(j3)

of (3.5)

maintain a constanit sign over the region of the plane in which our

structuores will lie. Reference to (A.16) shows that, since - ^ and

- ^ cannot vanish, there can be in points of inflexion on our lines of

principal strain. This is the meaning of our restriction in geometrical

terms.

Let us now apply the transformation of (A. 18) to (3.5). We find

1 9A - dPi 1 aS ^ dP,

ff^"*da'ïaa""d3

where the upper or lower signs must be taken according as ^ ^ •

•^

is positive or negative. Choosing 9^and «Pg

^°

that,

Ft l9i(a) 1 = ± a , Fa I

tf^W) ] = ±^

or since P^' and Pg do not vanish, writing,

<Pi(a) = Pi'-^(±a) , <p2(j5) =

^^(i^)

where P^ *"'•, Pg"'' are the functions inverse to P^ , P2 and the upper and lower

signs are now to be taken accordingly as P' P2 is positive or negative

»

we can write the transform of (5.5)* omitting the bars, in the form,

1 9A . 1 9B ,

(1 c\

B a:^ = -^ » A 3 ^ = ^ ••• (5-^^

Equations

in con junction with (3.6) show that with an appropriate

choice of reference direction far i|ri we can write in our new coordinates,

This is equivalent to the previous convention since E , ^ ^ = ± 1 , P^r^ = ±1

and so (P^ . P^) . (dtpi/da . dcpa/a^) = 1

19

-i|fi = a + /9 = ilrz - | . . . (3.7)

When A,B have been found equation (3.7) together with (A.5) and (A.8)

w i l l enable the determination of i n t r i n s i c eqi.iations for the l i n e s of

p r i n c i p a l s t r a i n .

Equations (3.6) jrield,

azA , ^ 92B T,

-8 ^ - ^ ^ = ° ' 3 ^ + ^ = °

the first of which can be integrated in the form, a /5 .

A = Hi (a) + H2(/9) -ƒ i?f[Hi(g) + H2(TI)1 J [(?-«)(/9-Ti)ldTi ... (3-8) o o

tt where H^, H2 are arbitrary functions and j(co) is the Bessel Function

j(co) = 1 +0)+ - S L + ... + .-«£- + ... ,,. (3.9) (2!) (n!)'

which satisfies ÜI)J"(Ü>) + J' (co) - J(Ü)) = 0 J(0) = 1

The first of (3.6) and (3.8) then give a . a j8

1 . . . (3.10)

B = -H^(/3)+ aH2(/?)+f Hi(^)d^ +ƒ d j

H2(r)) ](?-«)J'{(g-a)(/9-r]) jdn

° ° . . . (3.11)

f

Ref.6 Tome I I I §499

tt

20

-We have obtained in (3.8), (3.11) general integrals of our equations for A,B depending upon two arbitrary functions H^ and H2. These resvilts however are not very simple in form.

An important special case occurs when A = kB , where k is a

positive constant. The integrals for this are most easily obtained from (3.6) directly. We find,

A = iB =Ke^«-^/^ ... (3.12) where K is a positive constant. The intrinsic equations for the

lines of principal strain are by (A.5)» (A.8) and (3.7) given by

sa=-Ke^«re-(^^-«'"^2)/^-l]

... (3.13)

where we have taken s^ as measured from a = 0 and S2 from /5 = 0. The Cartesian forms follow from (A.7) and ( A . I O ) and with a particular choice of origin can be written,

, K (k+i)\!fi-/3 (k+l/k) ^

and . K (i-l/k)ilr2+a (k+l/k)+'jt/2k ^ + ^y ~ (ki-1) ®

S i i b s t i t u t i n g from (3.7) for ^-i and ijfa and i d e n t i f y i n g the r e s u l t i n g e x p r e s s i o n s , which are s p e c i a l cases of (A.3)» we find t h a t Ki = 0 and

i n t r o d u c i n g p o l a r coordinates p , co . The equations for the coordinate curves i n p o l a r coordinates follow from (3.14)» which g i v e s ,

21 -P = ... K ~ ka-/S/k +k' * 0) = a+/S -tan"^(l/k) and so we find, kü>./9(k+l/k)+k tan W k ) a K -ü/k+a(k+l/k)-(l/k)tan"''(l/k) /g-curve s p = . e ^ ^ i i \ i / \ r / > . . . (5.15)

We see that both these sets of curves are equi-angixLar spirals with angles tan'~'(l/k) and tan'"^(-k) respectively. The two sets are orthogonal and circulate the origin in opposite directions.

The solutions obtained so far, besides rioling out inflexions, rule out the case where a set of coordinate curves are straight lines. If one of the a-curves is a straight line r ^ = 0 or by (3.4) Pi(ot) = 0 on this line, which means of course rr^ = 0 everywhere and so all the a-curves are straight. This means by (3.5) that,

A = P^(a) , B = P^(/5) P3(a) + G(/9) where P3 and G are arbitrary functions.

Choosing a as the length along our straight lines, i.e. taking 'P^{a) = a , and since 9i!f

9/5 = ^ziP)> which does not vanish if the y9 curves have ho inflexions, choosing /? as the angle ijf2 > i.e. taking p2(/5) = 1, we can write:

A = 1 , B = a + G()9) ... (3.16)

tan cp = rdö/dr, where <p i s the angle between the radiios vector and the tangent,

ÏECHKISCHE HCGESCHCCL

VLIEGTUIG EO'JWK'JNDS _ 2 2 - Kanaabir-at 10 ~ C£LFT

The a-curves are s t r a i g h t l i n e s depending upon a s i n g l e parameter /9 and so envelope the "evolute" of the /3-curves for iirtiich they are the normals. The j9-curves are of course the " i n v o l u t e s " and b o t h s e t s of coordinate curves are i n t h i s case completely defined by the "evolute" which by (3.16) has the equation

a + G(y9) = 0 . . . (5.17) The evolute must of course be outside the region surveyed by our

coordinate system.

An interesting special case is obtained when the evolute

degenerates to a point. Our coordinate curves then become the set of rays through the point and the set of concentric circles. This is the layout used in Fig, 2,2 and our present result shows that this is a Michell optimum design,

The case where both sets of coordinate curves are straight lines is almost trivial. Here we can take a,/9 as Cartesian coordinates aj^i write,

A = B = 1 ... (5.18) In summary we can say that the lajrouts of Michell optimum

structures, for the case where the associated principal strains are equal and opposite and where inflexions are ruled out, take the forms defined by, firstly, (5.8) and (5.11)» ifliiich depend upon two arbitrary functions, and includes the special case (5.12), secondly (3.16), which depends upon one arbitrary function and finally (5.18) which has nothing arbitrary about it at all,

23

-§3.2 Conditions of Equilibrium ' • . ' - .

The considerations of §5.1 give guidance for the choice of layout for an optimum structi:ire. The determination of the required sizes of members results from a consideration of equilibrium conditions. The investigation of this matter necessitates a choice of structural form, In this section we will continue to deal with frameworks, but will specialise our studies to the case of continuous distributions with

perhaps concentrated members along isolated lines, for example along edges. We shall thus be treating plane structures consisting of double arrays of closely spaced fibres, which for the optimum case must lie along the lines of principal strain for Michell's virtual deformation of (2.6), i.e. along the a and /9 coordinate cvirves considered in §5.1.

The case where Michell's principal strains are eqioal must be

considered first. Here there is no restriction upon layout at all as long as the structure transmits the applied loads by menbers entirely in

tension or alternatively compression. The example of §2.5 shows that there may well be an infinite number of alternative structures, which by Maxwell's Theorem are of equal weight. It is quite clear that this multiplicity is a general property and so our problem is really to pick out simple yet adequate designs from the infinite possibilities. We shall therefore restrict ourselves to orthogonal layouts of fibres,

Since our structural elements are continuously distributed, their magnitiode is properly described by their equivalent thicknesses t^ and tg in the a and /3 directions respectively. This means for example that across a width Bd^ normal to the a-direction, there pass members whose total cross section area is t^Bd/9. Now in the present case the stress

-

-4.

in all the members could be f. and so, in the notation of (B.4)»

T^ = t^f , Tg = "^2^+ » S = 0 ...(5.19)

t t

Substituting in ( B . 4 ) we then obtain,

which may be written,

9tj. 1 9B ,. . s - 9t, 1 9A /. ^ s ^ f-z on\

The boiondary conditions ( B . 5 ) for the case where there is no edge member becomes,

f.t^sine = F sine + P, cose -N

* " ^ j ... (5.21) f.t„cose = P cosQ - p. sine J _{t 2 " n t}

Our problem for any given layout is to find solutions t^ > 0 and

t2 > 0 of (5.20) which satisfy (5.21) on the boundary. We shall discuss the possibility of resolving this problem in §5.5.

The case where Michell's principal strains are equal and opposite and where our intersecting fibres carry opposite signed stresses f. and -f , will have stress resultants given by,

c

Tl = tif^ , T2 = -taf^ , S = 0 ... (5.22)

Or, of course (-f ). However (5.20) is the same in both ca^es, c

25

-Equations (B,4) aJid (B,5) give for t h i s c a s e ,

and,

f . t i s i n e = P sine + F. cose

f to COS e = -F cose + p. sin(

c '^ n t

]

... (5.24)

These equations are similar to those for the previoiis case, but it

must be remembered of course that here the coordinate curves are limited to the special forms studied in §5.1. In the general case covered by (5.6) we can write,

t c

These are of the same form as (5.6) i t s e l f and so possess i n t e g r a l s

analogous to (5.8) and ( 5 . I I ) . We then find,

a /9

Af t2 = Ki(a)+ K2(/3)- [ d?

] Jf (^-a)(/9-n) idTi

c

• - / a s /

o o

a a 13 > • • (5.25)

B f . t i = K^yS)- aKzO?)- / Ki(?)d? - / öE,f[K^{^)+ K2(n)J(?-a)

X J'l (?-a)09-ri)]cin

where Ki , K2 are a r b i t r a r y and A,E,J are given by ( 5 . 8 ) , ( 5 . 9 ) , (5.11 )•

For the s p e c i a l case of (5.12) we find from (5.25) t h a t ,

26

-It is convenient to transform this as follows

„ ,

-ka-tvS'/k „ , -ka+)9/k

f^t^ = T^ e , f^tz = T2 e ^' -)

J

... (5.26)

Comparison with (5.6) gives once more a general s o l u t i o n of the form,

a /? -.

-172 =LH(a)+L2(/9)-[dJ[Li(?)+L2(ri)lJ[(?-a)(/9wn)jd'n j

J J . . . (5.27)

a • a /9 {

V k = LK^)-aL2(^)-fLU5)dC-fd^f[Li(C)+L2(Ti)Ke-a)jM(?-a)(^-n)ldTi )

where L^ and L2 are arbitrary functions. A special solution axialogous

to (5.12) can be written,

where k^ and K^ are positive cjonstants

^ , =. a ^ , =. K^ e-^-i"^/^^ . . . (5.28)

The s p e c i a l case of (5.16) gives for ( 5 . 2 5 ) :

± [ V i U + G(/9)n + f^t, = 0 , ^ ( f ^ t , ) = 0

which yields,

where, P(a) and G^ (/S) are arbitrary functions. Finally, the particular

case of (5.18), gives the obvious result that t^ and tg must be constants.

27

-§3.3 Porm-uJ-ation of the General Problem

The problem of determining the optimum arrangement of f i b r e s , -vrfiich

w i l l e q u i l i b r a t e a given system of forces applied to the boundary of a

r e g i o n , can now be formiiLated. The two systems of f i b r e s must c a r r y

constant s t r e s s e s fi and f2 which must have the v a l u e s ,

f- = f, or - f , f2 = f, or - f . . . (5.50)

^ t c ' 2 t c \^ /

t

The layout of the f i b r e s must determine an orthogonal c u r v i l i n e a r

coordinate system (a,/?) for wliich the functions A,B s a t i s f y , by (A. 17)

with C7= 0 and (5.2)

_{d_A ^ N _a_/1 9Av}

ga^'A da' "^ 9/9^B d^' ~

9/1 9B

9a"A aa'^ " 9/?^B 9/3^

0 (fif2>0)

= 0 (fif2<0)

. . . (3.51)

The equivalent thicknesses t^ and t2 of the f i b r e s miost s a t i s f y the

d i f f e r e n t i a l equations of equilibriiom, which by ( B , 4 ) take t h e form,

da

(Bfitj) - | | f 2 t 2 = 0 , ^ ( A f 2 t 2 ) - | f . t , = 0

. . . (3.52)

The boundary, vfhich we assiome known in the intrinsic form <p = 9(0"), where 9 is the angle between the reference direction used for ijr^ in Appendix A and the positive direction along the tangent to the boundary, must be expressed as in (A.20) in terms of b^ (cr) and b2(cr). The relevant equations are (A,22), (A. 16) and(A.1l), with CJ = 0 ; we can

thus write,

This is of course not essential for the case fif2>0, but is assumed in the interest of simplicity! 1

28

-{bi(cr),b2(o-)l

„(o-) = [ ( . i % da + i P d/5) H- e

(«o /9o)

cosö = A{b^(cr), b2(cr)| . b^ (er) sine = B{bi(a-), b2(a-)l . b2(cr)

where ( « O J ^ O ) i s the p o i n t a t which ifr^ i s assumed t o be z e r o . The equilibrium conditions a t the boundary must be s a t i s f i e d . The appropriate equations follow from (B,5) w i t h T = 0 i . e .

fi tn s i n e = P sine + P, cose -^

^ ^ 1 ... (5.54)

f2to cose = F cose - p. sine J

"^ '^

n t

Finally the solution for t^ and t2 must obviously satisfy the conditions:

ti > 0 , tg > 0 ... (5.35) The mathematical problem presented by the equations (5.50)

to (5.55) is a formidable one. Furthermore we have no guarantee that, for any boundary and any distribution of force, a solution exists. Special solutions given in §5.4 below show that the problem can be solved in certain cases, but a study of the boiindary conditions (5.54) shows that for some loading cjonditions, solutions cannot exist tmless the boundary has special forms. A pertinent problem would thus appear to be the determination of the restriction that must be imposed upon the forces and the boundaries to ensure the existence of a Michell optimum design,

Some guidance can be obtained in relation to the ambiguities of (5.50) and to other more general matters by a consideration of (5.54).

29

-I f P, = 0 everywhere, we have:

6 = 0 a n d f j t g = F

• .. . or e = V 2 and f ^ t i = P^ ( . . . (5.56)

or f.t^ = foto = P

and so if P has the same sign, say positive, everywhere, we must

take fJ = f5 = f . If on the other hand P is sometimes positive

^ ^ t

n

and sometimes negative, the boimdary must be built up of pieces of

coordinate curves, a-curves (9 = 0) say where P > 0 and /5-curves

(6 = V 2 ) for P < 0 . We shall then have f2 = f„ and f. = -f and

' n •^ T ^ c

furthermore the boundary must have right angled corners at all the

zeros of P ! If P = 0 everywhere we first notice that 6 = 0 and

tg = 0 or 6 = ^ 2 and t^ = 0 at the zeros of P, i.e. one of the

coordinate curves must touch the boundary at these points. At these

zeros too tanö and cot6 change sign and so f^t^ has the same sign

everywhere and likewise fgtg

,

but opposite to f^ t^. It follov/s that

one of f^, f 2 is f. and the other (-f ) , everywhere on the boiondary,

In the intermediate cases where the resultant of P , P, is neither

n t

normal nor tangential to the boundary one can by an appropriate choice

of e arrange that the signs of f^, fg are the same or opposed, but

variation of the general direction of the external farce from outwards

to inwards will give.rise to similar problems like those induced by

the zeros of F .

n

The equations (5»5l) vAiich determine the coordinate curves have,

t

at least in their form appropriate to f^:^<0 , been studied thoroughly

in §5.1.

t

30

-Reasonably general integrals have been obtained, although their form is not too convenient. However, if one adopts the point of view that the business of optimim structures is still partly an art, then the material obtained can form the basis for the construction of an enormous variety of layouts, which coji be used as judgement, intuition or even hunches may direct, as trial arrangements for the solution of any given problem. Alternatively, they can be used as in §5.4» below, to construct artificial problems,

Once appropriate values of f^ and f2 are chosen and a layout decided with determinate A and B, then oiK" remaining problems can be resolved. The problem presented by (3.33)> that of determining values of a^ /9 and 6 on the bound.ary can be resolved, if not analytically, at least graphically, by drawing out the boundary and superimposing a grid of a and ^ curves. The problem of the determination of the sizes of members t^ oxid tg, equations (5.52), (3.54)» can then be resolved, perhaps by the analysis of §5.2, but certainly by the usual numerical methods of integrating step by step the hyperbolic differential

equations along their characteristic lines. This last step however faces us with new difficulties. It is not usual to have to integrate hyperbolic differential equations subject to boundary conditions on closed curves aaid rightly so, since, as is easily seen, restrictions have to be imposed upon the possible boundary values on different parts of a closed curve. Consider the problem of integrating (5.52) in the region bounded by the curve ABCD of Pig.5.1. This curve is the transform of our real bounding curve in a plane where a, /? are rectangular

coordinates. We assume by (5.54) that the valxoes of t^ ard tg are known on ABCD. Since t^, t2 are known on AB they can be found at all points

51

-within the closed region ABE bounded by the curve AB and the

characteristics AE, BE through its end points. Similarly values of

t^, t2 can be found in the closed region BCP using the given values on

BC. We shall then have two distinct determinations of t^ and tg on the

line BP. These must agree with one another and so must imply

restrictions upon the boundary conditions on BC i.e. on P and P •

In fact a knowledge of t^, t2 on BP and tg alone on BC determines t^

and t2 in BCP and in particular on BC! It may be that a specially

chosen layout will avoid these difficulties, but this is a question

which cannot be answered with our existing knowledge,

§3.4 Special Solixtions

The general problem is clearly too difficult to solve and so,

as in other fields, we must turn to "inverse methods", which assume a

solution, or at any rate a layout, and examine what particular problems

are solved by this assumption,

Let us begin with the special case of (5.16) which arises when

G = 0. Here the evolute becomes a point and the coordinates (a, /? )

become polar coordinates, with radii and concentric circles for

coordinate curves. With the values A = 1, B = a the equations (5.52)

give, ...

where P^, P2 are arbitrary functions. On circiiLar boundaries (6=^2)

and radial boundaries (9aO) we have by (5.34),

circular boundaries F = fnti, P^ = 0

n ' t

radial boundaries P = foto, F.^ = 0

n ^ ' t

52

-The part of the solution which depends on P^ can be used to illustrate the points made in discussing (5.56). If we take P^ > 0 , but with

Pi < 0, v/e must write f., = f and f2 =: -f . Taking the region bounded t c

by two r a d i i and tvro c i r c l e s , wo have the case where the r a d i a l

boundaries are under normal p r e s s u r e and the c i r c u l a r boundaries under normal t e n s i o n , and, as p r e d i c t e d , we have foixr r i g h t - a n g l e d c o r n e r s .

The case where Pi = - P a ( P > 0 ) , Pg = 0 r e q u i r e s f^ = f2 = - f „ and solves the problem of a c i r c u l a r diBc under r a d i a l pressure P by f i l l i n g i n the c i r c l e with f i b r e s of constant equivalent thicknesses

t i = ta = P/f^ . . . (5.59)

The case P^ = 0 , F2 >0 consists of radial spokes

transmitting tensions of varying amounts. The point a = 0 is now a singular point and is in general a "centre of pressure" and the point of application of a "concentrated force". Applied to a wedge this solution gives a concentrated tension whose line of action lies within the angle of the wedge. The case where P2 is constant gives a

symmetrical load, whose reaction can be collected by a circular member in compression. Adding radial edge members gives us Pig, 2.2 once again.

A second example is furnished by (5.12) which gives coordinate curves in the form of eqiai-angular spirals (5.15). A special solution for t^ and tg for the case f^ = f , f2 = -f is given in (5.26) ard. (3,28). Let us adopt this with k^ = k. We then obtain by (3.14),

V ^ = ^ c ^ = (1 ? f a ) p 2 ... (3.40) The origin p = 0 must be excluded. Let us consider a region of the plane bounded internally by a circle p = constant. The loading at this boundary will be of constant magnitude by (5,40) and of constant

- 55

i n c l i n a t i o n t o the radiios by the defining p r o p e r t y of equi-angular s p i r a l s and so w i l l c o n s i s t of a uniform normal tension and a imiform t a n g e n t i a l t r a c t i o n . For a s u i t a b l e value of k (k = 1, when f.= f )

w C

the normal tension will be zero and we shall be left with th3 tractions whose resultant will be a couple, (Pig,5.2).

The origin p = 0 is thus a "centre of pressure" and a centre of

"concentrated torque". This solution can be applied to the transmission of torque from the inner boundary to the outer boundary of a circular ring (Pig,5,2).

3 4

r

{

K

E

F j

C

FIG

3-FIG 3-2

-

-4. Plates §4.1 Michell Type Theorem for a Plate

Consider the set of plates P of varying thickness t, which equilibrate given coplanar external forces with a stress

t

distribution, having principal stresses f., and fg» which satisfy the maximum shear stress criterion of yielding, namely,

(Greatest minus least of f^, fg, O) < Y ... (4.1) where Y is the tension-compression yield stress. We assiome that

there is a plate P* of the set P, with principal stresses ff , f* , wliich satisfies the conditions:

(1) (Greatest minus least of f* , f* , O) = Y

(2) There exists a -virtual strain sjrstem, with principal strains e.) and eg and principal directions coinciding with those of the principal stresses ff , f| , whose magnitude depends upon a constant nvunber e > 0 and is such that, if

e, - e.i(l-2v^)-t^? e9(l-2i^l)-yej_

^2 = -^ \ i : ; 2 ) '

then, in regions (a) where f*>f*sO and f* = Y, one has ei= e, 62= 0 (b) ( c ) (d) ( e ) ( f ) f*>0> f* " f*-f* = Y, " f*>f*>0 " f* = Y, " ei= e , £2= - e €<= 0 , 62= e t*>0> f* " f*-f* = Y " " e i = - e , €2= e -f* = Y " " ei= 0 , Ê2= - e O > f*>f* O > rp> f* -f* = Y " " e i = - e , 62= O

. . . (4.2)

t

- 3 6

Let us now apply the virtual strain system of condition (2) in (4.2) to all the plates P . Since the virtioal work of the external forces is the same in all cases, the increments of strain energy induced in the plate will also have the same value. If the stress components in a general plate P , referred to the principal directions of our virtroal strain, are f.,.,, f22 and f., 2 » then the increment of strain

energy Ï7 for this plate is given b y ,

W = 2E

E E

^^11+pr:^)(®i+ ^^2)^ + ^^22+ ^ïi;;r)(®2+ ^^i)l

"R "R /

-2^^^^11+(TIU)(®1 "^^^2)Hf22 + (^r;j)(e2+yei )l+2(l+v)f?2

-Jf2^+f|2-2i;f,,f22+2(l+u)f22ll dA

where E i s Young's modulus, v i s P o i s s o n ' s r a t i o and the i n t e g r a t i o n

w i t h r e s p e c t t o area i s taken over the region of the p l a n e , which i s

occupied by the p l a t e P. Developing the terms i n the integrand and

neglecting second order terms i n the s t r a i n s , we f i n d , using tb£

d e f i n i t i o n s of ei and ez i n (4.2) t h a t ,

W

= ƒƒ t(f,,

. . . (4.3)

For the plate P* we have f ^., = f * , f22 = ^ f and so taking account of the conditions of (4.2) we find,

'v7* = t * Y e d A = Y e V *

... (4.4)

where W* and t * apply t o P* and V* i s i t s volume,

37

-YeV* = / / t ( f i i e i + f22e2)dA

= fftfiiedA + f f t ( f n - f22)edA + jjtfzz^^

(a) (b) (c)

+ /*/'t(f22- fii)edA + fft(-f22)edA + fft(-fi^)edA . . . (4.5)

(d) (e) (f)

where the region of i n t e g r a t i o n has been s p l i t in-to the regions

designated (a) to (f) i n (4.2) and xjise has been made of the values of

e-i , £2 given i n t h i s same eq\iation. Now by the p r o p e r t i e s of

p r i n c i p a l s t r e s s e s and (4.1) we see t h a t ,

| f i l I < max. [ I f^ I, I fa I ] < Y and s i m i l a r l y [fgal < Y.

Again by (4.1) the max, shear s t r e s s i n the plane of the p l a t e

cannot exceed Y / 2 and so by a known formula,

This implies |f,^ ,f^^| < y . Hence,

± f n < Y , ± f22 < Y , ± ( f i i - f22) < Y . . . ( 4 . 6 )

Applying (4.6) to (4.5) we deduce ttiat,

YeV* < f/tYedA = YeV

where V is the volume of a general plate P. Since Ye is positive we have,

V* < V ,,. (4.7) The plate P* which satisfies (4.2) has as small a volume as any

58

-The actual volume V* can be calculated from -the virtual work of

the external farces. If the forces are specified as in §2,2 by P. and

if the virtual displacements, at the points of application of these

forces, corresponding to the strain system of (4*2) are ev. then, the

virtual work is \ P. ev. and so by (4.4) we find,

i

V* = ^

V F ^

.

V^

... (4.8)

§4.2 Development of the Theory for a Plate

The virtual strain system of (4.2) has principal strains e^ ,

6 2 given by,

e, - (l"y^)Kl-2v^)e.-^

veA ^

_ (l-y^)>e,H-(l-2i;;) e^]

... (4.9)

where e.) ,

^z

are defined in (4.2). We see that the strains are

constant and further that (0^-62) does not vanish. Substitution in

( B , 5 ) then gives an eqiiation, which combined with (A.I7) -with ex = V 2 »

gives, just as in (5.2) the equations

9

A

9 /I 9A>, _ /, .-.>,

= 9^ % 9 ^ ) = ° •••

(

^

-

^

^

^

The theory of §5.1 is thus valid for this case too and the principal

s-fcrain lines determined there can be used as principal stress lines

for optimum plate designs,

59

-f

T^ = tf^ , T2 = tf2 , S = O ... (4.11) in ( B , 4 ) , which yields,

^(Btf,) - g tf2 = O , ^(Atfg) - I tf, = O ,.. (4.12)

In addition by (4.2) the stresses must satisfy one of the follovidng, f 1 = ± Y , f2 = ± Y , f^ - f2 = ± Y ... (4.15) This gives us three equations for t, f^ , f2, which would appear to

be sufficient. However the strains correspondong to f^ and f2 must be compatible. This means by ( B , 5 ) and (4.IO) that,

a|!|ète-.f,)i.|!||(f,-.f.)i

Finally by (4.11) and ( B , 5 ) we have at an unreinforced boimdary, tfisinÖ = P sine + P.^cose

n t tfocose = F cos6 - P.sine

'^ n t

"j ... (4.15)

Our present problem has yielded a superfluity of equations. Equations (4.IO), (4,12), (4.15) and (4,14) are six equations between

the five unknowns A, B, t, f.| and fg. This is very restrictive on kinds of solutions and may well mean that only very special

distributions of P , P can be accommodated in (4.15). One way of n "C

obtaining consistency is, as we shall see in §4.5» to assume that f^ and f2 are constants. This makes (4.14) an identity, but removes fi and f2 from the list of variables,

w

-§4,5 Constant S t r e s s Soltitions

Let us now ass\ame t h a t the s t r e s s e s f^, fg are constant.

Equation (4.15) gives one r e l a t i o n between them, b u t does not fix them

completely, Hov/ever ( 4 . l ) r e s t r i c t s t h e i r values to c e r t a i n r a n g e s ,

but among a l l the p o s s i b i l i t i e s , every r a t i o f^/fg between -«> and + »

can occur. We can thus leave the values of f^ and f2 open for the

moment, u n t i l indeed we have to consider the boiondary c o n d i t i o n s ,

The equilibrium equations (4.12) i n t e g r a t e i n the forms:

^ [ f l l o g t + ( f i - f 2 ) log B | = 0 , • ^ [ f 2 l o g t -(^1-^2) log A] = 0

We see immediately t h a t i f f., = fg »

t = constant (fi=f2) . . . (4.16)

Substituting in the boundary conditions (4.15) and writing by (4.15) f^ = f2 = + Y we then find

F^ = ± t Y , P^ = 0 ,,, (4.17)

The case f^ = fg thus solves any problem of toniform normal s-tress applied to a boundary by the not unexpected solution of a uniform thickness plate!

Assuming from now on that f., / f2 and writing,

we see that the eq\ailibrium conditions imply, A = P^(a)t^'' , B = F^(y9)t^2

41

-transformation like (A. 18) namely,

« = Pi(a) , ^ = F2(/?)

which is reversib.le3 since from the nature of A,B ani t, P^ and F2 are one signed namely positive, we find

A = A P U » ) » B = B P^(/9)

and so, omitting the bars, Me can write finally,

A = t , B = t ... (4.19)

The e q u a t i o n s ( 4 . I O ) i n t e g r a t e i n the form

when F 3 , F4 a r e a r b i t r a r y f u n c t i o n s . S u b s t i t u t i o n from ( 4 . 1 9 ) g i v e s on i n t e g r a t i o n , s i n c e p.) + P2 + 1 = 0 , 2pi+1 2p +1

1 2 ^ ^ =/?F3(«)+F5(a), X s i f i n ^ ' =aP4(/9)+Ps(^)

- | l o g t = /9P3(a)+ F 5 ( a ) , - | l o g t = aF^(^)+ ¥^{fi) . . . ( 4 . 2 1 )

where F^ and Fg a r e arbi-fcraiv f^.inctions. The c o n s i s t e n c y of ( 4 . 2 0 ) and ( 4 . 2 1 ) g i v e r e s p e c t i v e l y ,

•r2p;+i^f2p";';ïT = y^M+

^SU)}UM-^

P6(^)J ' ... (4.22)

if2

-The specirJL case pi = pg = - o ^ i v e s , by ( 4 , 1 8 ) , f< = - f a .

Equation (4.15) then shows t h a t we mvist have

f^ = ± V 2 , f2 = q: V 2 . . . (4.24)

which may be compared to the Michell s o l u t i o n of Section 5 with

f., = f „ , f2 = - f . Equation (4.25) has the s o l u t i o n ,

P3(a) = c^a + C3 , P4(^) = c^/5 + C2 , P5(a) = Cga + C4

'^eiis) = CgyS + C4

where c , , C2» C3, c^ are c o n s t a n t s , and so by (4.19)» (4.21) we can

w r i t e ,

A = B = t ' ' ' ^ = e°i«^-^2a+C3/5+c, ^^^ ^^^35)

A s p e c i a l case can be stiodied \asing (5.12) with k = 1 i f

we -write,

c^ = 0 , C2 = 1 » C3 = -1 , e '* = K

The l i n e s of p r i n c i p a l s t r e s s are equi-angular s p i r a l s wi-th angle

r/h- [see ( 5 . 1 5 ) ] . F i g . 5.2 applies t o t h i s case and shows t h a t the

e x t e r n a l loading for a c i r c u l a r hole i s a uniform t a n g e n t i a l -traction,

•whose resviltant i s a torque. Comparison between (4.25) and (5.14)

shows t h a t t v a r i e s a s ,

t = A - . . . (4.26)

where p is the distance from the centre of the circle. We thus see that the optimum design for -the transmission of a uniformly applied torque from a circxilar hole in a plate consists of a plate whose thickness varies inversely as the sqioare of the distance from the centre of the hole. Inspection of (5.40) for the case of a fibre mesh shows direct comparison with this case.

Another i n t e r e s t i n g example can be obtajxied by -writing,

Ci = C3 = 0

Equation (A.I6) gi-ves -z^ = 0 and so the a-curves are s-traight l i n e s .

Also T j ^ = C2 and so ijf2 = c-/? for an appropriate reference d i r e c t i o n .

Equation

g i v e s ,

C2

measuring S2 from /? = 0. This i s a c i r c l e of radius p given by,

= J_3C2a+C4

'^ 02

O-JT principal directions are thus radii and concentric circles and

the load on a circular hole is a uniform normal tension or pressure.

Equation (4.25) shows that the optdjnum variation of t is given by,

t = 2 2 ... (4.27)

°2P

Another special case is gi-ven b y p., = 0 , Pg = -1 or

fg = 0 , f.} = + Y. For this we have,

A = 1, B = t"" = a

F^(0) + Fe(l3)

... (4.28)

The a-c-urves are straight lines which envelope an e-volute given b y

a P4(/?) +

FeiP)

= 0 - ... (4.29)

and the /9-c\irves are the involutes. The degenerate case of

concentric circles and radii can be applied to a wedge under tension

due to normal "traction on the circiiLar boundaries, but not -to a hole in

a plate since in that case the displacements given by ( B . 2 ) are not

consistent; we have in fact a dislocation.

' 4 4

-The general case of a r b i t r a r y pi , pz y i e l d s by ( 4 . 2 2 ) ,

^^(«^ = (2p,!l)lc2a+C3) ^ ( « ) = ( 2 p , ! i ? ( c , a . C 3 y

^"'^"^ = (2p2!i?(ci/?-K>J ^^(^) = ( 2 p 2 ! l ) ^ c , ^ . c , )

where c^, C2, C3, C4 are arbi-trary constants.

Eqiiation (4.20) then g i v e s ,

\^C2a+03y

and (4.19)

/ „ \ f 2/(^1+^2) / ^ \ - f i / ( f i + f 2 )

Icza+Cjy V°2a+G3y

The resiolting coordinate curves are of s p i r a l form and as i n the

case of (4.25) -with c, / 0 r e v e a l the novel fea-ture of contadning

"points of i n f l e x i o n " . These were rviled out i n ovtr pre-vious

discrussion i n § 5 . 1 .

§4.4 Alternative Approach

A d i r e c t a t t a c k upon the problem of the optimum design of p l a t e

structtnres has been suggested by the p r e s e n t w r i t e r . Let vs r e f e r our

p l a t e t o r e c t a n g u l a r Cartesian axes 0 ( x , y ) , The components of -fche

s t r e s s tensor f , f and f , v/hen m-ultiplied by the variable

XX.' yy xy'

thicloiess t , may be derived from a stx-ess function U by,

tf = 0 , tf = 1 ^ , tf = - 0 - ... (4.52)

XX 9y2 ' yy UX2 ' xy 9x9y

This ensures t h a t equilibrium i s s a t i s f i e d throughout the p l a t e .

Compatibility of s-train req-uires,

o2 n

Af "Vf )+Af -vf ) =

. , . ( 4 . 5 3 )

imposed, giving,

f2 + f2 - f . f + 3f2 ^ 3q2 . . , (4,34)

x x y y x x y y x y ' ^

where q i s the y i e l d s t r e s s for p\ire shear. F i n a l l y equilibrium

condi-tions a t the boundary g i v e ,

l ( t f ) + m(tf ) = P , i ( t f ) + m(tf ) = F , , , (4.55)

'' xx' ^ xy' X ' ^ xy' yy y

where ( I , m ) are direction cosines for the outward normal and (P ,P ) are Cartesian components of the external traction per unit

X- y

length of boundary,

Elimination of the s-tress components gives:

9^ r'^v^ v^\ ^(^\(^ 9^u>

9'3c2H''^9x2 " 9y2>' + 9y2H''*'9y2 "^9x2^

TECHN:o..-.ri 1^ -^^:-iP H O G L : : < C 1 Ï O O L V L ! E G Ï U ; G BOUWKUNDE Kanaalitraat 10 - D£LFT

^^ ^'

-^^ 9x2 • aye ^9x9^* j

<5q'=t

... (4.57)

_d ,9U

ds ^9x

m

= -^

y

... (4.58)

where s is the are length of the boundary. Assuming that the

optimum design is given by taking the equality in (4.57)» we can

eliminate t from (4.56),(4.57) and obtain a fourth order equation for

U to be solved subject -to (4.58). It may be that a numerical process

like Relaxation co\iLd be used to resolve this formidable problem, but

we cannot say, -with ovir present knowledge, that a physically acceptable

solution exists. Again, since the equations are non-linear, it may

be that several solutions are possible, but in this case

we

can presumably

pick out the lightest one of the alternatives. Finally since we ha-v-e

not used a condition of least weight, we cannot really be sure that

ovr

t

proced-ure gives us the lightest structure . It is concei-vable that

the \ise of the inequality in (4.57) might yield a structiore of less

weight, although a. s-truct\jre which is just about to yield everywhere at

its v/orking load is clearly a very good engineering design.

Our equations can be put in-to a variational form. Let us vary

the stress components and the thickness by amounts 6f , 6f , 6f

x x ' yy' xy

and 6t subject to the conditions of equilibrium and a yielding

condition l i k e (4.54)» with only an e q u a l i t y s i g n , being s a t i s f i e d .

The maintenance of the e q u i l i b r i i m conditiors i n the v a r i e d s t a t e

y i e l d s

Equation (4.41) below only gives s t a t i o n a r y weight with the Haigh

Xieldiiiig C r i t e r i o n W = constant!

- k7

-f/Te 5 ( t f ) + e 6 ( t f ) + 2e 6 ( t f )jaxdy = 0 . . . (4.59)

JJl^xx ^ xx' yy ' yy' xy x y j

where e , e , e are the components of the s t r a i n t e n s o r .

x x ' yy' xy

Introducing the S t r a i n Energy density W, which i s given by,

W = ~ r ( f -(-f )^ + 2 ( l + y ) ( f 2 - f f )

2E (_^ XX yy' ^ ' ^ xy xx yy'^

6E

"^ XX yy' 3E ^ XX yy XX yy -^ xy'

we cm, •'ATite (4.33) i n the form,

(töW + aT5t)dxdy = 0 . . . (4.41)

In-troducing the Mises-Hencky y i e l d condition and using the second

expression for YI i n ( 4 . 4 0 ) , vre can w r i t e ( 4 . 4 I ) a s ,

t(f_+f )(5f +6f ) + r ( f +f )%|ܱ4<l'

öt dxdy = 0

. . . (4.42)

The v a r i a t i o n a l equation (4.42) can be used for approximate

s o l u t i o n of optim-um design problems. One might begin by finding a

s t r e s s function U , which s a t i s f i e s ( 4 . 3 8 ) , and which depends upon

a nuniber of nrbi-trary parameters or functions. The thickness t

follovra from (4.37)» with the equal s i g n , and the s-tress components

from ( 4 . 3 2 ) . Equation (4.42) then g i v e s , by the usual processes of

the calculus of v a r i a t i o n s , equations for the parameters or

functions, v/hich -though complicated -will \Jndoubtedly yield to the

process of numerical a n a l y s i s .

4C

-5. Supgpsticns for Lines of Research

The theories discussed in previous sections exhibit many gaps both in their scope and in their foionda-tions. Application of the

theories is held -up in many cases by mathematical difficulties and in others by the need to carry out possible, but lengthy, special investigations. Future research can therefore take -two forms.

In the first place, one can fill in the details in -those parts of the subject, where the next steps are reasonably clear and where the necessary mathematical techniques are known. Secondly one can tackle the fundamental problems, attempting to clear up some of the mysteries and -to broaden the coverage of the theories. The

pedes-trian first form may well be suggestive for solution of the more profound problems of the second.

Reasonably straightforward problems inclxxie:

(1) Systematic stiody of the coordinate curves corresponding to the general integrals for A,B obtained in equations (5.8), (5.II). (2) Systematic study of the possible forms that can be assumed by

involute curves and their normals. Inspection of known solutions with a -view to application -to design problems.

(See equations (5^16), (5,29)).

(5) Use of the analog^'- •'.'d.th slip lines in plastic flow to make use of known res\ilts in tiii.s field. In particular one might study equation (5.12) with k a complex number.

(4) Detail study of the constant s-tress solutions for plates, [see eqmtions (4.25), (4.23) and (4.50), (4.51)]. This might throw some light on the use of coordinate curves with inflexions, which are ruled out in the general stijdy based on (5.8),(5.1l). (5) Development of practical methods either analytical or

graphical-niomerical for the determination of t^ and t2 vising equations of the type (3^3--) and (5.54) vrhen the coordinate curves are kno'iA-n, Study of the res-trictions imposed on -the external forces.

- 4 9

-(6) Use of the experience gained from research projects like (l)»(2), (5)»(4) above to develop the art of drawing in a set of

coordinate c\jrves to meet a given loaded boundary at

qualitatively appropriate angles so that consistent signs can * be given to f., and f2.

(7) Solution of a number of simple problems using the variational equation (4.42).

Fundamental investigations into the existing theory inciLude:

(8) Proof of an existence theorem for a Michell optimum framework to equilibrate a finite number of given fcjrces.

(9) Investigation of the equations of section §5.5» Development of techniques for their solution. Proof of an existence theorem for a Michell optimum layout of fibres.

(10) Study of the eq-uations of section §4.2 -with a view -to developing solutions for plates -with variable stresses. Existence theorem for the optimum plate,

(11) Investigation of the general problem of integrating hyperbolic partial differential equa-tions with boundary conditions on closed curves,

Investigations directed to-wards broadening the scope of the existing

theory include: , (12) Development of Michell type theorems for other types of

constr'Jiction, e,g, reinforced plates and shells,

(15) Development of less restrictive conditions than the Michell type for plates. The restrictions imposed by the theorem of §4,1 seem to rule out most practical problems.

(14) Development of theories to deal with several alternative loading conditions and stiffness requirements.

50

-6 . Co::'.'?] u.qio-p.q

(1) The c l a s s i c a l theorems, such as t h a t of Maxwell, can be used

with considerab?.e success to produce economical layouts

for frame s-truct^oï'os,

(2) The theorem of Michell can be \jsed to c r e a t e a complete theory

for the economical design of a p l a t e - l i k e s-truotiire

consistijig of a double a r r a y (weave) of c l o s e l y spaced f i b r e s ,

I t may vrell b e t h a t -the optimvim layouts can only be achieved

by r e s - t r i c t i o n s on the d i s t r i b u t i o n of the e x t e r n a l forces

or i n some cases on the shape of the p l a t e , b u t the

development of techniques for the s o l u t i o n of -the mathematical

problems involved, shotild l e a d -to many s o l u t i o n s of p r a c t i c a l

i n t e r e s t ,

(3) The extension of Michell's methods to continuous p l a t e s leads to

i n t e r e s t j j i g , b u t r a t h e r r e s t r i c t e d r e s u l t s . I t woiiLd seem t h a t

h e r e , a l e s s s p e c i a l i s e d approach i s c a l l e d for and i t may w e l l

be t h a t the v a r i a t i o n a l theorem of s e c t i o n §4.4 and i t s

associa-ted approximate methods of a n a l y s i s could be of g r e a t e r

use i n the problem of p l a t e d e s i g n .

(4) The stibject of t h i s r e p o r t offers g r e a t scope for r e s e a r c h , both

i n the d e t a i l development of the c l a s s i c a l methods and t h e i r

exterxsicn -to t h e o r i e s of g r e a t e r b r e a d t h ,