Tabulation of some layouts and virtual displacement fields in the theory of Michell optimum structures

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CoA. Note A e r o . No. 161

THE COLLEGE OF AERONAUTICS

C R A N F I E L D

TABULATION O F SOME LAYOUTS AND VIRTUAL D I S P L A C E M E N T

F I E L D S IN T H E THEORY O F M I C H E L L OPTIMUM STRUCTURES

by

CoA Note Aero. No. l6l

The_^College of Aeronautics

Cranfield

Tabulation of some layouts and virtual displacement

fields in the theory of Michell optjjnum structures

by

-H.S.Y. Chan

The tables produced in connection with this work have not been included in this Note. Anyone requiring the full set should apply to:

The Librarian,

The College of Aeronautics, Cranfield,

CoA Note A e r o . No. l 6 l

February^_1961j-DEPARIMENr OF AIRCRAFT DESIGN

T a b u l a t i o n of some l a y o u t s and v i r t u a l d i s p l a c e m e n t

f i e l d s i n t h e t h e o r y of M i c h e l l optirnxmi s t r u c t u r e s "

by

-H.S.Y. Chan, B.A.

The author would like to thanlc the Machine Tool Industry Research Association for permission to publish this note, which reports work carried out under a contract awarded by the Association.

Contents

Page No.

Notation

P a r t I - I n t r o d u c t i o n 1

1 . Formulation of a boundary - value problem 1

2 . Method of s o l u t i o n 2

3 . Transformation formulae for the co-ordinate systems 3

k. P a r t i c u l a r s o l u t i o n s k

ka. F i e l d generated by two orthogonal c i r c u l a r arcs k

kh. F i e l d generated by two orthogonal curves 5

5 . Form of tabxilation 7

References 8

Appendix A - Proof of formula (?) 9

Appendix B

-

Analysis of results in (4b). 10

Figures

Part II - Tables

1. Tables of Bessel functions I (z), Ii(z)/z, zli(z)

2. Tables of Lommel functions Ui, U2

3. Tables of co-ordinates x n ,

x^s,

y n , yi2

k.

Tables of virtual displacements

VL^I, '^22f

U23

5. Tables of co-ordinates X21,

^22, J2i>

Yaa

Notation

CC, P

X, y

A, B

U , V ^X, 1*2

e

e

V-)

i , ( . )

V"'"'

r(z)

curvilinear co-ordinates Cartesian co-ordinates

Lamè' s parameters (defined by eq. (l))

virtual displacements along a, P directions

radii

strain

arbitrary angle between 0 and /k

k-th order Bessel function

k-th order modified Bessel function

k-th order Lommel function of two variables

Gamma function

Xix^ Xi2* yxi* yi2 numerical quantities defined by eq. (l8)"

U2x^ '^^22^ ^23 numerical quantities defined by eq. (23)"'

1

-Part 1 - Introduction

1. Formulation of a boundary - value problem

In the theory of two-dimensional Michell optimum structures, there arises the problem of calculating the lines of principeQ. stresses and the virtual displacements which are analogous to slip lines and velocities in plane plastic flow. A detailed analysis of the problem has been given in reference 1 and 2. Both analytical and numerical methods of calculation are given in reference 3'

Two special problems are considered here, in which the strain fields are generated from two given orthogonal curves with negative initial

curvatures. A curvilinear co-ordinate system (a, p) is defined such that the magnitudes of the parameters a., P are chosen to be the angles between the tangents of the curves and two fixed directions ox, oy, which are cartesian axes with the same origin as (a, p ) . (See figure l ) .

If A, B are the radii of curvature of the a, P curves then the linear element ds is defined by

ds^ = k^da.^ + B^dp^ (l)

where A, B are related by

I = B, i = A , (a)

and in the problems considered here A(O;, O ) , B ( 0 , p) are known functions.

From (2), it is easy to show that

Tlie virtual displacement field (u, v) along a-p directions is governed by (eq. ( A ? ) of reference 2 ) .

0 ^

J ö ^ " " = "2eB

1

W

where e i s the maximum s t r a i n . I t i s assumed here t h a t the d i r e c t s t r a i n

along an a-curve i s -e and t h a t along a p-curve i s e .

2

-2. Method of solution

The second order partial differential equation

5 ^ - H = 0 (5)

with boundary values H(a,0), H(0,p) f or — < '^M can be solved by means

of Riemann' s method, which gives

H(a,p) = H(0,0)lj2^) + ƒ IJ2V^I^DP) ^ ^ ^ d ^ +

o

ƒ ij2v/^(p::;^) ^^l^dn +ƒy^ij2/(5qW-Ti"))G(§,n)dedq

(6)

+

o o o

Generally, no e x p l i c i t s o l u t i o n w i l l be a v a i l a b l e . If, however,

G{^,r\) i s of the form

k

.2

the double i n t e g r a l i n (6) can always be reduced to a single i n t e g r a l

thanks to •.

1^ j\(2/Q:-g)0-ri)) . ( ! ƒ . Ij2/ïn)dë = 0, (7)

o

the proof of which is given in appendix A. Then it may be possible to find explicit expressions for the integrals in terms of Bessel functions.

In what follows, two transformation formulae taken from references h and 5 are of great value.

/ Jy(<l sin ^)J (p cos ^).sin ^.cos"^ ^ .d^ =

. , 1 V ~ » >x , V /V s X+2r V + 2 S T / / 2,^2

2 T(k+l+r)r(ti+l+^) ) ) i->.,^,4.i^/v...._N

o

r=o s=o 2 r'.s'.r(X+iS-l)r(v+s+l)(p2+q^)2 2 2

3

-where (a) = a(a+l)(a+2)....(a+n-l), (a) = 1 ; and

V 00 , W\ 1/ , \ - 2 r' ('2^ - 2(v+m) (z+w) . J,(Vz+w) = ) — T — . z . J„^„(Vz) v^ ' i_^ m'. v+m^ m=o

(9)

5. Transformation formulae for the co-ordinate systems

The co-ordinate system (a, p) is related to the Cartesian (x, y) by (Eq. (22) of reference 2 ) .

(a ^\ 1 ox 1 oy

la ^\ 1 2Z 1 OX

^^" (P-^^ = A SS = - i ÖP

or in integral form

ix-xj + l(y-y^) = J A.Exp(l(p^)}dc«, ( l l )

o

where

r P

x^+ly^ = x(0,p) + l y ( 0 , p ) = i j B(0,p).Exp(ip)dp ( l 2 )

o

\Jhen A is the Bessel function I ( 2 v ^ ) , the integral ( l l ) can be

represented by means of Lommel's function of two variables.

J I^{2/ö^).Bxpil{^<x))da = Exp{i(p<i:)}2 (i)^"""". f | j . i J ^ T ^ )

o k=l

Exp{i(p-o:)} Ui(aD;,2lV^) + i U2(a2;,2iV^)

(13)^

•'• See, for instance, reference 5, pp. 537/543« The Lommel function is defined by:

,-•. ^ k+2m

k

-k. Particular solutions

Two particular solutions of strain fields are considered here, the geometries of which are presented in figures 2 and 3« Detail calculations of the second case are given in Appendix B so as to demonstrate the procedures.

{ka) In the case when the strain field is generated by two orthogonal circular arcs of different radii, (figure 2 ) :

Ai(o!,0) = ri Bx(0,p) = r2 (11^) then Ax(a,p) » ri I^(2v^) + r2 ,.^| I x ( 2 ^ ) BX(Q:,P)

= ri,J|

I I ( 2 / Q P )

+ r2l^(2^)

(15)

which reduces, when rx = r2^ to equation (kk) of reference 2. If the virtual displacements on the boundary are given as

ux(a,0) = - eri(aci!+l) vx(a,0) = eri

ux(0,P) = - er2 + e(ra-ri)(cos p - sin p) vi(0,p) = er2(2p+l) - e(r2-ri)(cos p + sin p ) ,

and A, B are given by (l5), then

ui(a,p) = eri(l+aD;)l (2v'ap) 2er2/^ Ix(2i/^) -- e(r2--rx) [Ui(2p,21i/^) + U2(2p,2lV^)]

(16)

{

vx(a,p) = eri[I ( 2 v ^ ) + 2 v ^ Ii(2»^)] + 2 e r ^ I (2i/Sp) -- e(r2--ri) [Ui(2p,2iV5p) -- U2(2P,2iV^)] ,

(17)

which reduces again, when r^ = r2 and e is replaced by - e, to equation (63) of reference 2.

5

-The c o - o r d i n a t e s ( a , P ) a r e then i-elated to the C a r t e s i a n ( x i , y x ) by

r.a

( x i - x i o ) + i(yx - yxo) = / Ax(a,p) . Exp{l(p-a)3 da o

= r i Exp{i(p-a)}.[Ui(aD:,2iv^) + i ^2{2a,2hfö^)] ( l 8 )

+ r2 4Exp{i(p-a)} [ 1 U i ( a z , 2 i V ^ ) + U^(aa,2lT/^)] - Exp(ip) I

and

P

xxo + i yio = 1J ^2 Exp{ap) dP = r2[Exp{ip} - l] (19) o

Combining (18) and (19) gives the following form for Xx, yx^

xx + i yx = (rx xii + r2 X12) + 1 (rx yxx + ^2 Jxz) (18)'""

Numerical values of Xxx» Xx2^ yxx> yx2 are tabulated in Part II.

(to) In the case -vdien the strain field is generated by two orthogonal curves of initial radii of curvature

A2(Q!,0) = rx + rxl (2/2ëa) + TZJ^ I X ( 2 / ^ )

{ r

-B2(0,P) = r2 + r2l^(2i/(rt-2e)p) + r i ^ ^ l^i^/lj:^)^),

(20) where 0 is an arbitrary angle (O < 0 < r), (Figure 3 )

-then from (B.6)

A2(a,P) = rx{lQ(2v'a(p+20)) + I^(2/p(a-ht-20))) +

^

- - { N / ^

'^

(S^^^TS^^T:^))

-. ^ ix(2y5(pï^))}

B2(a,P; ri,r2j 20,jt-20) = A2(P,o:; r2,rx; it-20,20)

(21)

6

-case considered in reference 6 ) .

U2(a,0) = - erx[(l+a3:+2jt-ij0) + 2x I (2/20a)]

e r 2 [ I (2/20a) + 2/2Bcc Ix(2/2ëa)]

-o

- e(r2-ri)[Ux(a3!,2lV^ia) - U2(ax,21/200)]

V2(a,0) = e r i [ l + 2 / 2 0 a Ix(2/20a)] + ersil+ke) I (2/20a) +

+ e ( r i - r 2 ) [ U x ( a 3 ; , 2 i V ^ ) + U2(a::i,2i7a9a)]

(22)

U2(0,p) = - er2[l+2/(rt-a9)"p Ix(2/(^-20 ) p ) ] - erx(l+at-i|0 ) l (2/(^-20 )p)

- e(r2-rx)[Ux(2p,2i7(«-20)p) + U 2 ( 2 p , 2 i / ( 7 ^ S l p ) ]

V2(0,p) = er2[(l+2p+l40) + 2p I (27(rt-20)p)] +

+ e r i [ l (2/(it-20)p) + 2/(«-20)p I x ( 2 / ( « - 2 0 ) p ) +

o

+ e(rx-r2)[Ux(^,2iV(TC-20)p) - U2(2P,2iy(rt-20)p),

then a f t e r the c a l c u l a t i o n s given in Appendix B, the r e s u l t s a r e

U2(o:,P) = eri[(l+aa+2rtl40)l (2/p(Q;ht29) + 20! I (2/o:(p+20)]

-- er2 I (2/a(p+20) + 2/a(p+20) Ix(2/a(p+20) +

+ 2/p(a-ht-20) Ix (2/p(a-Ht-20)]

- e(r2-rx)[Ux(aa,2iV'a(p+20)) - U2(aa,21-/o:(p+20)) +

+ Ux(2p,2i7p(a-ht-20)) + U2(2p,2iVp(a-Ht-20))]

V2(o;,p; r i , r 2 ; 20,jt-20) = - U2(p,aj r 2 , r i ; «-20,20) (23)

These depend upon Ux2^ U22> ^^23 defined by

U2(a,p) = - e[riU2x + r2Ua2 + (r2-rx)u23] (23)"'

numerical values of which are tabulated in Part II.

7

-and

(X2-X20) + i(y2-y2o) = / ^2 Exp{i(p-a)}dO!

o

= rx{Exp(i(p-Q!)}.[Ux(a3;,2iVo;(p+20) + Ux( 2(a+Tt -20 ) ,2iVp (a-Ht -20 ) +

+ 1 U2(aD;,2iVaIp+20) + 1 U2(2(a+rt-20),21V'p(a-20))]

- Exp(ip}.[Ux(2(«-20),2lVp(«-20)) + 1 U2(2rt-20 ),2l/p(it-20 ))]}

+ r2{Exp{i(p-a)}.[i Ux(a3!,2iVa(p+20)) + 1 Ux(2(a+jt-20),2iVp(a-ht-20)) +

+ u^(aD;,2lT/a(p+20) + U (2(a+rt-20),2iVp(a-ht-20)]

- E x p ( i p } . [ l Ui(2(it-20),2i/p(rt-a9)) + UQ(2(7t-20),2iVp(jt-20)) + l ] }

(2i.)

X20 + i y20 = 1 / B2(0,p).Exp{ip}dp

o

= rx(Exp(ip}.[Ux(2p,2i7p(rt-20)) + 1 Uo(2p,2iyp(n-20))] - 1} +

+ r2{Exp{ip).[l+i Ui(2p,2iVp(jt-29)) + U2(2p,2iyp(rt-20))] - 1}

(25)

Combining {2k) and (25) gives e:q)ression of the form

X2 + i y2 = (rxX2x + r2X22) + i(rxy2x + r2y22) {2k)''

5 . Form of t a b u l a t i o n

Numerical q u a n t i t i e s in equations ( i B ) ' , (23)"' eind {2k)"' have been

tabulated using Oe^_P_as parameters varying from 0 to 135 degrees and 0 = /k.

Two a u x i l i a r y t a b l e s of "Bëssël'"and Lömmêï f ü n c t i ö n s ' a r ë ' a f so "given, so

t h a t i t i s p o s s i b l e to c a l c u l a t e nijimerically expressions (15) and ( l 7 ) .

The t a b u l a t i o n s of equation (18)"' had been checked, when rx = r 2 , with

the t a b l e given in p . 350 of reference 3i and the t a b u l a t i o n s of equations

(23)" and {2k)"' had been p a r t i a l l y checked by the s p e c i a l case of reference

6.

All c a l c u l a t i o n s were c a r r i e d out on a Pegasus d i g i t a l computer, using

a l i b r a r y code for Bessel functions; and equations (18) "' and ( 2 ^ ) ' are

c a l c u l a t e d by numerical i n t e g r a t i o n using the Gauss formula. I t i s believed

t h a t the e r r o r s of c a l c u l a t i o n s nowhere exceed 0.1'^.

8

-References

1. Hemp, W.S.

2. Chan, A.S.L.

3. Hill, R.

k.

Bailey, W.N.

5. Watson, G.N.

6. Chan, H.S.Y.

Theory of structural design.

College of Aeronautics Report 115^ 1958.

The design of Michell optimum structiires.

College of Aeronautics Report ll|-2, 196O.

Mathematical theory of plasticity.

Clarendon Press, Oxford, 1950.

On integrals involving Bessel functions.

Quart. J. Math., Oxford series. Vol.

9,

1938, pp. lin/li^7.

Theory of Bessel functions.

Cambridge Univ. Press, 19^^.

Optimum Michell frameworks for three

parallel forces.

College of Aeronautics Report Aero. 167,

1963.

9

-Appendix A Proof of formula ( 7 )

I n p r o v i n g formula ( 7 ) ^ u s e h a s been made of t h e f o l l o w i n g r u l e s c o n c e r n i n g B e s s e l f u n c t i o n s : d { z " .Ij^(z)) = z " .I^^j^(z)dz k k d{z . L ( z ) ) = z . L T(z)dz k^ ' k - 1 ^ '

I'i(z) + \ Ii(z) - [1 + (f ) ' ] l j z ) = 0

( A l ) (A2) (A3) I t f o l l o w s ;

| - ƒ l^(2/(a.|)(p-Ti)).(^)2 y2yi^)d|

o i_ k = - / " " ( p ^ ) I l ( 2 - / ^ n ( p T ) ) . ( | ) ^ I k ( 2 v ^ ) d i + 1+k i t t + I ( 2 / ( a - 0 ( P - T l ) )

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

d | -a ,Q;I k

| f y Ix(2/(aU(PTi)) (|)2 y 2 / i ^ ) d |

-1+k k

è) ^ I'(2/iïï) - ! {hf I.(2/iïï)

n T 1.

d ^ ( S ^ l^Ix _p-Ti

)2/i^:Fy(:p^)}

o

1^(2/1?)

ƒ (|:^Tii(2/(^:r)(P^) {\). [i';,(2/iïï) - ^

- ( 1 -H l | ^ ) V 2 v ^ ) dg

10

-A£gendlx_B Analysis of results in {kh)

In calculating the case 4b, formula (6) gives

.P

A2(a,p) = A2(o,o)i^(2./^) + Ji^{2/j^r4¥) ^ ^ ^ 1 ^ dg + ƒ i^(2y5(p::;i7)B2(o,Ti)dq

o o

(BI)

where A 2 ( | , 0 ) ,

B 2 ( 0 , T I )

can be obtained farom ( 2 0 ) .

By using formulae ( 8 ) , (9) and ( A 1 ) ( A 2 ) , i t can be shown t h a t

ƒ I^(2VQ!(P-Ti))dTi = ^ 1

I I ( 2 V ^ ) (B2)

ƒ I^(2/a(p-n)l^(2/(:t-20)Ti)dTi = , . J ^ ^ Ix(2/p(a+rt-20

) (B3)

rt-20

ƒ I^(2V^(p::^) Ii(^(rt-20)Ti) , ^ 2 t ^ dii = I^(2/p(a-ht-20)) - I ^ ( 2 V ^ )

Q!

y Ij2/(SDP) 12(2/20!) ^ d^ =

(B4)

1+k

k=o

= , j ^ Ix(2y5(pT^)) - ..J Ix(2v^) - 20 1^(2^^)

Substituting these results into (Bl) gives

A2(a,p) = rx[l (2/a(p+207) + I (2/p(a-ht-20)) +

(B5)

+ r2

a

1 ^

Ix(&/S^pT20)) +

J ^ ^

Ix(2/p(a+«-20))

11

-S i m i l a r l y ,

B2(a,p) =

A

a-Ht-20

P

Ii(2/p(a+it-20)) +

Ix(2/a(P+29))

+ ra [l^(&/p(Q;+jt-20)) + I (2/a(p+20))]

The v i r t u a l displacements are also ceLLculated from (6)

U2(a,p) = u2(o,o)iJa/^) + ƒ ?j2/(^:np) ^^^^^^°^d| +

-p

• ƒ ij&/S(p::Ty) ^^^f^-in - 2e ƒ ƒ ij2v'(S:iT(F^) B2(ê,Ti)d|dTi

(B7)

o o

From ( B 6 ) , it is easily seen, by using (7), that

.apP

- 2e ƒ j lj2/(a-|)(p-Ti)) B2(ê,ii)d|d3i

o o

= - 2eJ j I^(&/(a-U(P-^)) 32(|,p)dëdii

o o

r.a

= - 2ep J B2(ê,p)da

(B8)

which can be integrated readily by using ( A I ) ( A 2 ) .

The first two integrals in (B7) present only one type of integration which differs from those in ( B 2 ) - ( B 5 ) and ( B 8 ) , namely,

pa

J

I^(2/(ST]P) 20T

\(^^^)

k

a

\2

"^'i^) \^^^^^^)'>

(B9)

Hence, integrating term by term gives

^a

1^(2/(0-1 )P) Ui(2è,2i/^)d| = Ui(aa!,2i-/a(p+20)) (BIO)

o

The result (23) can now be obtained by combining all the integrals together.

Finally, the (x,y) co-ordinates can be calculated easily by using (13) and (Al).

FIG. I. CO-ORDINATE SYSTEMS.

( x i . S i )

FIG.2. FIELD GENERATED FROM TWO CIRCULAR ARCS. ( R A D I I n , r a )

( " , y » )

o< xa