Optimum design of a multicell box subjected to a given bending moment and temperature distribution

CoA Note No.82

THE COLLEGE OF AERONAUTICS

CRANFIELD

OPTIMUM DESIGN OF A MULTICELL BOX SUBJECTED

TO A GIVEN BENDING MOMENT AND TEMPERATURE

DJSTRIBUTION

^' / D. J. JOHNS

T H E C O L L E G E O F A E R O N.A U T I C 3

C R A N F I E L D

Optimum Design of a Multicell Box Subjected to a Given Bending Monrent and Temperature Distribution.

b y

-D. J. Johns, B.Sc., M.I.A.S.

SUm'[/\RY

The optimum geometry of a multicell box of given depth, under a given bending moment and temperature distribution, is obtained. The method is general enough to permit the skin thickness to be either specified, e.g. by stiffness requirements, or not.

2 -CONTEWTS Page Surnmaiy •\ L i s t of Symbols 3 I n t r o d u c t i o n é Previous analyses é 2 . 1 . Bending S t r e s s e s 6 2 . 2 . Thermal S t r e s s e s 8 Q\jalitative Assessment of the Thermal Problem 9

Assumptions 10 Analysis 11

5.1. Bending Stresses 11 5.2. Thermal Stress 11 5.3. Buckling Stress of Skin Panels 14

5.4. Weight of the Box 14

Optim.isation 15

Example 17

Discussion 18 L i s t of References 19

Appendix; Limitations on the Analysis Imposed by

Various Assumptions 21

Tables 27-30 Figures

LIST OP SB/EOLG A Area of I - s e c t i o n =. A_, -f /i,.,

S \i A^ Area of skin panels

A... Area of web

b„ Width of s k i n p a n e l s b,-r Depth of box

B Stinictur-al chord of beam, d e f i n e d i n eq, 2 , 1 . c n e a t c a p a c i t y p e r u n i t volume 0 C o n s t a n t i n eq, 5 . 2 . D E ^S \ 2 C o n s t a n t i n eq.A. Youjigs Modij-lus Secant Modulus = 2 d cr Tangent Modulus = -^-rr

f Distance from web mid-plane to the near edge of the rivet shanks in a web angle

I Moment of inertia for each cell k Thermal conductivity

K A constant used for determining the buckling stress o" , defined in eq.5.11

K„ A constant used in Ref. 3 for determining the buckling stress cr ^ defined in eq. 5.12

K.. A constant used in Ref. 4 for determining the web crushing stress, introduced in eq, A.2

m Bending moment per unit chordwise length applied on section M Bending moment on each cell = mbs

» 4 ~

L i s t of Symbols continued.

p Number of bays i n t o v/hich s t r \ j c t u r a l chord B i s s p l i t P 17eb m a t e r i a l maximum shear s t r e s s

q P e n e t r a t i o n depth. (See eq. 2.4)

r^ Ratio of depth/vri-dth of each c e l l = t^-jA»q r . R a t i o of v/eb t h i c k n e s s / s k i n thickness = t,-/tj^ R a' (3 ~ •§) a function defined a f t e r eq. 5.7

s Shear force per u n i t chordwiselength S Shear force on each c e l l = sb„

t Time

t Trajisit time, defined i n eq. 2.5 t_ Skin thickness

b

t ^ \7eb thickness

w

T Temperatijre at d i s t a n c e y from n e u t r a l axis of I beam T Minimum temperature i n t h e web

T Maximtim skin temperature

T" § E a ( T - T ) , a function introduced a f t e r eq. 5.10

w

¥ Yfeight of eo.ch c e l l , per unit length spanwise

P (2b3t3 ^ b,.^t^.p

y Co-ordinate in pLane of I - s e c t i o n measured along the web from the n e u t r a l ajcis (see Pig, 2),

L i s t of Symbols c o n t i n u e d . a C o e f f i c i e n t óf e x p a n s i o n tt S t r a i n e S t r a i n a t which £„, = ^ E H P o i s s o n s R a t i o p d e n s i t y of t h e m a t e r i a l T$ P l a s t i c i t y c o r r e c t i o n f a c t o r ( s e e eq. 5.12) cr B u c k l i n g s t r e s s of s k i n p a n e l s 0 B Bending m t r e s s i n s k i n (compressive) CL Thermal s t r e s s i n s k i n ( c o m p r e s s i v e ) c( Thermal s t r e s s i n vreb ( T e n s i l e ) S E T cr S t r e s s a t d i s t a n c e y from n e u t r a l a x i s y O" s t r e s s a t v/hich E^ = •§• E 2 l b 6 i r, r , = 7 - d e f i n e d a f t e r eq. 5.19 b t Ag 1 b t 1 + 2 iV^.

1 + Vé v ^

6

-Optimum Design of a Multicell Box Sub.iected to Bending and Thermal stresses.

1, Introduction.

Many authors Viave considered the optimisation cf multicell T/ing

structures in which no allowance was made for thermal effects. The results of various such investigations will be discussed first and the influence of thermal effects on the optiinum designs v/ill be discussed qualitatively.

A simplified analysis will then be presented from which the

quantitative effects of kinetic heating on optimum design can be assesaed. The present analysis will be concerned only \a.th multicell beams coirrposed of thin sk:ji members and full depth webs. This will probably be a practico.l method of construction for vrLngs in which thermal effects

are not too severe. No suggestions are made for ways to alleviate thermal stresses, nor v/ill the advantage of post and/or stringer stabilisation of skins conïE).ared with full depth v/ebs be shwm. It should be noted, of course, that a vast field of structural optimisation a\7aits the ingenious designer of the future who v/ill undoubtedly consider any

optimisation from the standpoint of high temperature materials versus

insulation versus heat suppression, diversion ajid removal techniques(Ref, 1), He \»a.ll also have to consider alternative methods of construction.

Such an analysis will be formidable and, unless the basic information used is exact, and the assumptions made are realistic, false conclusions may be dra-vim. Many studies and investigations have been made to determine

individual and combined effects of seme of the above considerations and it is hoped that this note v/ill also help to elucidate some of the problems of structural design,

2, Previous Analys'-'S 2.1. BendJ.ng Stresses.

The maximum strength and structural efficiency of miilticcll structures, ignoring thermal effects, has been studied previously by other investigators (Refs. 2 - 6 ) . Their findings will not be discussed in detail here but som.e comments may be opportione.

2

Gerard's analysis for integral beams shov'/ed that optimum design exists when r, = 0.4 r, . He also concluded that the optimum number of webs is given by the equation

p.(4p + 1) = 5(-^f 2.1.

v/here p i s t h e number of bays i n t o which t h e s t r u c t u r a l chord B i s s p l i t i . e . B = p . b s ,

•n --^_ - 2 . ^ - T) —§ 2 2

b, _{W ^T \}

Tltierefore r = 1 and r^, = 0.4^ A l i m i t i n g asstmiption of t h i s a n a l y s i s i s t h a t the web buckl.ing and skin buckling s t r e s s e s are equal,

Ref, 3 a l s o d e a l t v/ith i n t e g r a l beams and i s notev/orthy in t h a t i t d e l i n e a t e s ranges of the parameters n , r^. f o r v/hich t h e phenomenon of v/eb buckling occurs. This phenomenon resxiLts in lev/ values of the buckling s t r e s s coefficient K , values of v/hich a r e shovm i n Pig, 1 (taken from Ref, 3 ) .

Rosen'^ has mad.e a most thorough a n a l y s i s of the ultiniate s t r e n g t h of m u l t i c e l l wings and has considered t h e d e l e t e r i o u s effects of v/eb buckling and v/eb crushing on t h e strength of the s t r u c t u r e . To avoid the former, he suggested as a c r i t e r i o n t h a t r, < 3^"^, hut l a t e r shov/ed t l i i s c r i t e r i o n t o be conservative: For the l a t t e r phenomenon he showed the ranges of r, and r, for v/hich web crushing i s more

^ b t ° c r i t i c a l than web buckling.

The a n a l y s i s i s quite general in t h a t skin thickness i s an

independent pajrameter v/hich may be specified by s t i f f n e s s requirements, Althotigh Rosen has considered i n t e g r a l breams, in general, he discussed

the e f f e c t s of adding heavy attachiaent members betv/een v/eb and skin, or usi_iig formed channel v/eb membei's. He shov/ed, t h a t v/hen such a t t a c h -ment members a r c added t o an optimism design (based on his o r i g i n a l a n a l y s i s ) , i t r e s u l t s in c e r t a i n circumstances in a more e f f i c i e n t design,

5

Semonian and Anderson have considered t h e use of formed channel v/ebs i n more d e t a i l and they have shown the large e f f e c t s that f l e x i b i l i t y

of the web attachment flanges can have on the s t a b i l i t y and u l t i m a t e s t r e n g t h of m u l t i c e l l beams. Ihey have demonstrated, experimentally and t h e o r e t i c a l l y , t h a t very lov/ values of the c o e f f i c i e n t K„ r e s u l t as t h e f~distance increases. This effective r i v e t offset distance may be defined as the distance from t h e v/eb mid-plane t o the near edge of t h e r i v e t shanlcs,

I t v/as shov/n t h a t , i n order t o achieve t h e buckling s t r e s s e s p r e d i c t e d by t h e i n t e g r a l beam theory of Ref, 3, t h e quantity __f__

^ S ^ t must be l e s s than 0,1 8, otherv/ise, f a i l u r e v/ill occ\ir i n the wrinkling mode r a t h e r than i n local bxjckling,

8

-More r e c e n t l y , Houghton and Chan have reconsidered the a n a l y s i s of Ref.3 and, using the values of the buckling s t r e s s c o e f f i c i e n t K_ from Ref, 3 , they have produced a more systaM.tic, n o n - i t e r a t i v e process for determining t h e optimum geometry and weight of m u l t i c e l l beams under pure bending. Their analjrsis e f f e c t i v e l y itnplied an assumption of

i n t e g r a l beams but i s applicable to beams i n which f n AR b r~ '^ • •

S t

For convenience, t h e present analjrsis v/ill adopt the same n o t a t i o n and p r e s e n t a t i o n as Ref, 6, and have s i m i l a r l i m i t i n g assxmiptions.

2,2, Thermal S t r e s s e s .

Previous analyses of thermal s t r e s s on m u l t i c e l l s t r u c t u r e s v/ill not be discussed here since the majority of them do not present the

thermal sti'ess d i s t r i b u t i o n s in a form s u i t a b l e for the needs of t h e present a n a l y s i s . Since the b a s i s cf a.ny thermal s t r e s s a n a l y s i s i s Icnowledge of thc-tcmperature d i s t r i b u t i o n , i t i s proposed t o adopt the a n a l y s i s of Biot since t h i s offers a. r e l a t i v e l y simple expression f o r the temperatiore d i s t r i b u t i o n i n a s e c t i o n .

Biot a c t u a l l y considered the problems of a uniform slab and a t y p i c a l i n t e g r a l I-section„ The increased complexity of t h e l a t t e r s o l u t i o n

does not encourage i t s use and i t i s believed t h a t l i t t l e error v / i l l accrue i f h i s s o l u t i o n t o the slab problem i s followed,

I t must be r e a l i s e d t h a t t h e r e are tv/o d i s t i n c t pha&es t o any heating problem in a s e c t i o n such as an I-beam. I n t h e f i r s t phase the tomperatvire has not yet begun t o r i s e a+ t h e c e n t r e of the v/eb (a symmetrical beam v/ith symmetry of heating i s assumed) and everything occurs as i f the v/eb depth v/ere i n f i n i t e . During t h i s phase the temperature d i s t r i b u t i o n

i s approximated by

Ty = T^ [ l - ( • ^ ^ ) ) ' for (d-y) < q

T = 0 f o r (d-y) > q 2.3 v/here the temperatures are measured above the initial strainless level and q is called the "penetration depth" at which the temperature is just beginning to rise. Biot shov/ed that the expression for the penetration

depth is

q = 3.36^1

f-

2.4

and he defined the time at v/hich the terriperature a t the c e n t r e of the v/eb begins to r i s e as the " t r a n s i t time" t, i . e . v/hen q = d

cd^

I n t h e second phase of heating the temperature a t t h e centre of the v/eb i s denoted by T and t h e temperature d i s t r i b u t i o n i s approximated by

Q

This l a t t e r expression has been assumed by Lcmpriere and i s , of course, only applicable for times g r e a t e r than the t r a n s i t time.

Using B i o t ' s anal3?sis i t c m be shov/n t h a t a convenient expression for T for t > t^ v/hen T i s an a r b i t r a r y function of time i s

m I s

T

m - . 4 . 5 7 t ^ T ^ = T^ - 1.075 t , T^ 2,7

T is determined neglecting the presence of the web and T is therefore obtained using eq. 2.7. These values v/ill prove sufficiently accurate for most project studies,

Pijnally, it should be emphasised, that the temperature of the slrin is a function of the skin thickness t„. No exact theory v/ill be presented here for the determination of skin temperature under arbitrary fli.ght conditions since this has adequately been covered in the literature. 3. Qualitative Assessment of the Thermal Problem,

In the past, vdng structures have been designed on the strength and stiffness criteria appropriate to their various flight histories, In general this has m.eant that a certain maximum bending moment and shear force loading has had to be satisfied and the structural designer has not been particularly concerned at what time of the flight the worst conditions arose. In designs hov/ever where thermal effects must be considered, the time element may be all important,

In such designs it v/ill be necessary to consider together the

variation v/ith time of both the normal manoeuvre load stresses and thermal stresses. Hence, the v/orst design conditions for the structure can be assessed at various times. In other v/ords it will not do to add the

most severe thermal and bending stresses unless they occijr simultaneously. Prom this one can visualise a structure, designed on manoeuvre loads

alone, negotiating satisfactorily a flight progranme v/hich produces high themal stresses, providing these stresses occur at a time v^en the manoeuvre loads are low. This might possibly occur in a long range, high speed interceptor which v/ould experience high thermal stresses early in its flight at a time of lov/ manoeuvre; and at interception -a time of high m-anoeuvre, the v/hole structure woiold h-ave re-ached its equilibrium temperature, the thermal stresses v/ould be negligible and

10

-VlJEGTUiGBCüWKiJNDE KanaahtrF.at 10 - DlILf'f

Because of such considerations, no attempt v/ill be made to consider the thermal stress patterns arising from specific flight programmes. Instead, the thermal stress terms v/ill be introduced quite generally as functions only of the maximum skin temperature, T , and the minimum v/eb temperature, T .

' m 4 . Assumnptions,

The assumptions made in t h e follov/ing a n a l y s i s are :

-4A.. The design c r i t e r i o n for t h e v/ing i s one of biKskling s t a b i l i t y of the compression skin under bending and thermal s t r e s s e s The skin thickness i s specified and l e f t independent, Tliis enables other, s t i f f n e s s c r i t e r i a t o be s a t i s f i e d . Shear s t r e n g t h ajid s t i f f n e s s i s assumed t o be covered.

iiB. The section i s i d e a l i s e d as rectangxolar with i t s depth prefixed by aerodynamic c o n s i d e r a t i o n s .

2tC. Both t h e skins and the webs are f u l l y e f f e c t ï g e i n taking bending and the s t r e s s i s d i s t r i b u t e d according t o t h e Engineers Theory, The effects of any angles, \vhich might make the skin t o web j o i n t v/ith a formed channel v/eb, on the v/eight and s t i f f n e s s of t h e

section have been neglected,

liD. The Vvddth of the box i s s u f f i c i e n t l y large i n comparison vidth t h e

depth for t h e panel buckling c h a r a c t e r i s t i c s t o be assumed t o be the same as t h a t of a box of i n f i n i t e width. This assumption enables Pig. 1 taken from Ref. 3 t o be used. Evidence given by Ref. 9 suggests t h i s assumption i s v a l i d provided the box has a t l e a s t 3 c e l l s .

2)E. The top and bottom skins are of t h e same thickness and therefore the n e u t r a l a x i s i s c e n t r a l . This l a t t e r ass^umption a l s o a p p l i e s when thermal s t r e s s e s and increases in temperature are applied t o the section.

1)F. The same m a t e r i a l i s losed for t h e skins and the webs,

The l i m i t a t i o n s placed on the a n a l y s i s by c e r t a i n of t h e assumptions are discussed in the appendices,

5. Analysis.

5,1, Bending Stresses due to Manoeuvre Loads,

Because of assumption 4D it is convenient to take as our typical structural element an I section from the beam of Pig, 2.

Let the bending manent applied on each element be M, and the moment per unit length (chordwise) toe m.

M = mbs.

Then following the asstoiriptions of Ifi and 4S, the compression stress in the skin due to bending i s given by

cj. _ M \

B " I 2 •

The moment of i n e r t i a of each elemert i s

^ = T T \ ^ / + 2 b g tg ( - ^ )

or 1 = { V^S^S ^ ^ + i

^b^t)-TT IÏ1 h „ b,r, m Hence ov, = S W = r- . o — Ö -I- — — , ^ ^ 1

Vs^^ + i ^b ^t^

5.2. Thermal Stress.

5.2.1. First Heating Phase (t<t,,)

• • • 111 • i w •! iw •! II ^(m •— • • • » • n I !• I l l

With uniform skin temperature, the temperature distribution through the I beam is given by

T^ = T, [1 -

(^)

] ' far (d-y) < q -.

y s «- q -> 2.3 = 0 for (d-y) U q J.

12

-The temperature is ass\mied constant axially so that there is only axial stress due to differential expansion. It is assumed also that the beam considered is long, and the section shovm in Pig, 2 is free from any end effects.

To determine the thermal stress distribution, Hookes Lav/ is assumed, then, if plane sections remain plane the strain across the section is given by

e = ^ + Cl T = Constant = C ,

y E ^ ^ y

5.2

For zero net thrust on the section,

0- d A = 0 ,

_5,3

where dA i s e l e m e n t a l a r e a of I beam of t o t a l a r e a A,

From eq. 5 . 2 , c = E C - E o c T , which, v/hen s u b s t i t u t e d i n t o eq. 5 . 3 , g i v e s C = E « T d A E d A Hence cr = E y E a T d A y d A a T

_3,K

Eq. 5 . ^ i s p e r f e c t l y g e n e r a l and allcv/s f o r v a r i a t i o n of t h e p r o p e r t i e s E, a Tivith t e m p e r a t u r e or v/ith y .

I f t h e m . a t e r i a l p r o p e r t i e s a r e assumed c o n s t a n t and independent of y and T eq. 5 . 4 becomes

^ y -E a • A ; T d A y - A T y

5.5

Therefore, using eq, 2.3 E a ""y A T^ i 2 bg t g ^ 3 t^^ q

-AT f l - ( ^ ) 1

_{^ J}

c r cr = s E aT r y — A —

^Vs*fv-^[^-(^)]

J

Hence, f o r t h e s k i n , the eompressive thermal s t r e s s i s

^ •| E a T . \ , R ,

A

v/here R = ^ ( 3 - f ) .

5.6

5.7

5.2,2, Second Heating Phase t > t.

The tenperature distribution may be approximated by

T = T (^)^ + T f - af^ . 2.6 y s^d'^ m *^ _1 F o l l o w i n g a n a n a l y s i s i d e n t i c a l t o t h a t i n t h e f i r s t h e a t i n g p h a s e , t h e t h e r m a l s t r e s s d i s t r i b u t i o n i s g i v e n b y cr = : - E a ( T - T ) ^y ^ s m' '^d^ 3A 5 . 8 Hence, f o r t h e s k i n , t h e c o m p r e s s i v e t h e r m a l s t r e s s i s ^ = I ^ '^ (^s " ^m) ^ 5 . 9 5 . 2 . 3 . Stminary. The f o l l o w i n g g e n e r a l e x p r e s s i o n d e f i n e s t h e o a n p r e s s i v e t h e r m a l s t r e s s i n t h e slcin a t a l l t i m e s cr^ = T R - ^ , 1 r r cr - T R S • ^t- 5 10 "^T - ^ "^ 1 + i r , r ' ^ - ^ ^ or b t where = I E a (3J - T ) , m' and f o r t < t , T = 0 , R = -^ ( 3 - •§) 1 m ' 2 ^-^ d'^ and f o r t > t , T^ ;^ 0 , R = 1 , m '

14

-10 A similar but more d e t a i l e d a n a l y s i s has been made by Hoff . He considered a beam i n v/hich skin and v/eb material v/ere d i s s i m i l a r , and he allowed for t h e decrease i n skin temperature a t t h e junction

with t h e v/eb. Unfortunately h i s r e s u l t s cannot be presented i n a concise form, but f o r t h e case of a uniform beam h i s r e s u l t s do not d i f f e r

g r e a t l y from those above. At time t = 0 the anal3rses correspond

e x a c t l y and a t the t r a n s i t time (beyond wMch Hoff s a n a l y s i s i s not v a l i d ) t h e discrepancy i s , f o r a l l lilcely configurations, l e s s than 10^.

Because of the s i m p l i c i t y of the present analysis t h i s discrepancy v/ill be ignored.

5 , 3 . Buckling S t r e s s of Skin P a n e l s .

The buckling s t r e s s c for t h e skin panel i s usiially given as

^c = ^ ^ s ( ^ ) ' 5.11

where E_ is the secant modulus ( = ~ ) usually used in connection v/ith this type of buckling (Ref,1l) and K is a constant depending only on the panel configuration.

The critical stress is given in Ref 3 as

12(1 - ^^) ^ \ ^

The value of 77 E i n eq, 5.12 corresponds approximately to t h e secant modulus E„. Therefore by comparing eqs. 5.11 and 5.12

b

K = ^ - — 5.13

12(1 -^^)

Pig. 1 reproduces the curves that give the values of K„ from Ref. 3 b

and Table 1 t a b i ü a t e s the calciilated values of K (from eq. 5.13) for a range of parameters r, , r , . (assuming Poissons Ratio, /i, = 0 , 3 ) .

5,4. Weight of Box.

The v/eight (per unit l e n g t h spanwise) of each c e l l i s W = P ( 2 b g t g ^ b ^ ^ t ^ p ,

Dividing t h i s by t h e width of the c e l l "b- gives

Q w = — = 2 p t (1 + 2 r r )^ 5 . i 4 bg - - ^ " s v ••' .^ -b-t>

w t^

^ ^

2 7 K ;

= ^ (^ +

* v ^ ) . 5.15

T7

é. Opt imisat i on.

The object of t h i s i n v e s t i g a t i o n i s t o obtain the skin thickness t „ , web thickness t and v/eb spacing b„, for a box of given depth b,.., t o r e s i s t applied bending s t r e s s e s and thermal s t r e s s e s so t h a t t h e weight \7 i s a minimum.

A l t e r n a t i v e l y , i f sldn thickness i s specified by s t i f f n e s s r e q u i r e -ments, t h e following eq-uations present t h e c o r r e c t approach t o s a t i s f y

strength and s t i f f n e s s requirements and give the required beam proportions. I t i s recognised t h a t f a i l u r e v/ill occur soon a f t e r t h e t o t a l

conipression s t r e s s , o:,, i n t h e skin reaches t h e bijickling s t r e s s cr

b Lf

given in eq, 5.11. Therefore cc, is used as the maxdmum permissible value cr„.

b

Using eqs. 5.1 and 5.10 t h e t o t a l conpressive s t r e s s i n the skin i s given by or o-_ = + , 5.16 O-g = C^ + CT^, 0- "^

^' ' Vs(^^K-t)

of cr i s 0

-c= '^^s ^

+ . T R ^ r^r^

(1 ^. ^

V t )

5.11 Rearranging eqs, 5. "11 sjnd. 5.16 gives

= e ^ Yir^Xrr-^ 5.17

'S ^ ^ \

/ rr, \ / \ T \ A T R -^ r, r.

m \ / W

1 . e l V V - A ^ I . .. TR.9 I 5.19

'•»]

vol ere 0 = a" r, r.

^ = ' * ^ V t

16

-The procedure t h e r e f o r e v d l l be (Given r—^ , R) t g • % (a) Ass\ime values of r— , r, r,

b... b' t ¥

(b) Hence determine K, from Table 1, and e from eq. 5.17

(c) From the compressive s t r e s s s t r a i n curve for t h e m a t e r i a l , determine

ex corresponding to e

(d) The value of t „ assumed i s iJised t o determine T, t

S —

(e) Substituting values of o" , r—• , r, , r, , T, R into eq. 5.19 / \ w

solve for ( — — \ W *S

Then keeping — and the product r, r . constant, nev/ values of r, , r . ï7

enable a nev/ value of K t o be found and t h e above procedure repeated. Therefore, since

•^\- = - § (1 + ^ r . r j , 5.15

the above procedure v/ith rr- , and the product r r constant means t h a t -^—r— i s also constant. Therefore the above procedure enables the

^ ' \ / m \

v a r i a t i o n off—r ) v/ith r (or r . ) t o be obtained for constant weight. Hence t h e "most e f f i c i e n t " s t r u c t u r e f o r given parameters! T— j , ( r r, )

i s obtained.

Similar analyses using different values of the product r, r, enable the "optimum" structure, for a given _S , to be determined.

The "optimum" structure is that "most efficient" structure v/hich has a maximum value of/ m \ equal to the applied value.

If the value of S is not specified, the above processes are repeated for various values of t . The combinations of t and the product r, r, ,

b b D X

\ \ ^

for various "optimum" structures enables the optimum _ ^ and other beam ^7

parameters to be found. To apply the above method an example will now be performed,

7. E::ample

A light alloy (DTD,687) structure is considered. The idealised stress strain curve for the material is given in Fig, 3 and it is

assumed that the teuperatures and heating times involved in this analysis are sufficiently lov/ to neglect effects due to deterioration of the

material properties,

Let it be given that the beam has a value of _ ^ _1_ specified by stiffnes_s requirements; a flight prograirme is assumed v/hich produces a value of ÏR = 20,000 Ib/in^ at the time considered. The problem is to find the optimum structure which v/ill simultaneously sustain a value

Tables 2, 3 and Fig. 4 present the results of the calculations using eqs. 5.17 and 5.19 as described in Sect. 6, Also shov/n in Table 3 are the values of/ m \ attainable in the absence of thermal stresses.

V

The results are plotted in Pig, 5.

From Figs. 4,5 graphs are constructed (Fig.6), shewing the variation of maximum/ m \ with n, r. , and v/ith r . Prom Pig» 6 we may deduce

V V y ^

t h e optimum s t r u c t u r e v/ith _S 1 » '''° s u s t a i n t h e bending and thermal \., ~ 40

I»

s t r e s s e s , and t h e bending s t r e s s e s alone. The r e s u l t s a r e given in Table 4.

Similar c a l c u l a t i o n s have a l s o been performed with __S _1_

and the resiiLts are presented i n F i g s . 7 - 9 and Tables 3 and 4.

I t i s i n t e r e s t i n g t o compare t h e above resiiLts v/ith those obtained by t h e a n a l y s i s of Ref. 6 f o r bending s t r e s s e s only. I n Ref. 6 skin thickness was not specified and t h e optimum s t r u c t u r e was obtained with

the optimum t_/, . The resiolts of Ref, 6 f o r / m \ >,/^o n-u/- » n •^ S/h,-. ( T--2-A = 1400 l b / i n 2 are a l s o

_{^7 V}

_\"

18

-8, Discussion.

An iterative method ha.s been proposed by which the optimum beam structure to sustain bending rjid themal stresses can be found. A review of previous v/ork in the field of structural optimisation ignoring thermal effects suggests that Refs. 4 and 6 offer the best approach for preliminary design. Ref. 4 is more general than Ref. 6 but the latter is a simpler, more systematic analysis,

The present analysis is a logical extension of the v/ork of Ref. 6 and the introduction of the thermal stress terms makes the analysis less simple, but it is still systematic. The chief merit in this analysis is that skin thickness may or may not be specified initially by stiffness requirements. In the former case the solution is more easily obtained,

An examination of the results obtained for the typical example shews several interesting results.

Prom Table 4 it can be seen that,

(a) as thermal stresses are added to the structure of Rov/ 2, the new optimum structure in Row 1, has thicker v/ebs, more closely spaced. The same result follows from Rows 4 and 3. This result is a little s^lrprising as it might have been expected that thinner webs, more closely spa.ced would be required,

(b) The structures v/ith the thicker skin are marginally lighter than those v/ith the thinner skin Simultaneously, the v/ebs are thinner and moie widely spaced. The actual differences in the numbers in

Table 4, for W , in Rov/s 1 - 4 , a.re so small, and the graphical 2pb^„

method of solution suspect to error, that it is safer to deduce that the variation of structure v/eight v/ith skin thickness is a fairly flat function.

(c) The optimum structure calcula.ted in Ref. 6 shows considerable differences from the geometries for the optimum structures of the present analjrsis. Again, it is deduced that the weight is fairly insensitive to quite large changes in the optimum geometry.

To summarise, the method of analysis presented in this note has been shown to be relatively simple and systematic, and to offer a convenient means of deciding structural shapes in the project stage, when thermal stresses are present.

L i s t of R e f e r e n c e s Meyer, J . H . Gterard, G. S c h u e t t e , E. McCulloch, J . Rosen, B.W, Semonian, J,W. Anderson, R,A. H o u s t o n , D . S . Chan, A. S„L. B i o t , M.A. L e m p r i e r e , B.M. T h e r m o e l a s t i c D i s t o r t i o n and Wing S t r u c t u r a l Design,

A e r o . Eng, Review Vol, 16 No, 9 p p . 4 6 - 5 3 . Sept, 1957.

Optimijm Number of Webs r e q u i r e d f a r a M u l t i c e l l Box Under Bending.

J o u r . A e r o , S c i . V o l , 1 5 . N o . 1 , p p . 5 3 - 5 6 J a n . 1948,

C h a r t s f o r t h e Minimum 17eight Design of Multiv/eb Wings i n Bending.

N^CA T . N . I 3 2 3 , 1947.

Analysis of t h e U l t i m a t e S t r e n g t h and Optimum P r o p o r t i o n s of Multiv/eb Wing S t r u c t x a r e s ,

NACA T.N. 3633. 1955.

An Analjrsis of t h e S t a b i l i t y and Ultima.te Bending S t r e n g t h of Multiweb Beams w i t h Formed Channel Webs.

NACA T.N.3232. 1954.

The Design of a M u l t i c e l l Box i n P\jre Bending f o r Minimum Weight, C o l l e g e of A e r o n a u t i c s Note No.74. New Methods i n Heat Flov/ A n a l y s i s vri-th A p p l i c a t i o n t o P l i g h t S t r u c t u r e s , J o u r . A e r o . S c i . V o l . 2 4 No 12 p p . 857-873 Dec, 1957. Thermal S t r e s s e s i n a Box S t r u c t u r e . C o l l e g e of A e r o n a u t i c s Note (To b e p u b l i s h e d ) .

~ 20 -L i s t of R e f e r e n c e s continued.. 9 , Eggwertz, S P , 1 0 , Hoff, N . J . B u c l d i n g S t r e s s of Box Beams u n d e r P u r e Bonding. P . P . A . ( T h e A e r o n a u t i c a l R e s e a r c h I n s t i t u t e cf Sweden) Report 3 3 . T h e n i a l Bxickling of S u p e r s o n i c Vv'ing P a n e l s . J o u r . A e r o . S c i . V o l , 23 p p . 1019 - 1028 Nov. 1956. 1 1 , S t o v / e l l , E . Z . A U n i f i e d Theory of P l a s t i c B u c k l i n g of Col-umns and P l a t e s . NACA T.N.I 556.

1 2 . Monaghan, R , J . Formulae and Approximations f o r Aerodynamic H e a t i n g R a t e s i n High

Speed P l i g h t .

APPENDIX I

L i m i t a t i o n s on t h e A n a l y s i s b y V a r i o u s A s s u n p t i o n s . The a s s u m p t i o n s v/hich p r o b a b l y have t h e g r e a t e s t i n f l u e n c e on t h e v a l i d i t y of t h i s a n a l y s i s v / i l l b e l i s t e d and t h e i r e f f e c t s on t h e a n a l y s i s d i s c u s s e d . A, 1 . I d e a l i s e d S t r u c t u r e t o b e T h i n - s k i n n e d w l t h F v i l l Depth webs o n l y a s s t a b i l i s e r s : On l a v s p e e d c o n v e n t i o n a l v/ing s t r u c t ^ u r e s , v/ing d e p t h i s s o l a r g e t h a t a l l t h e s h e a r l o a d s a r e a d e q u a t e l y talcen v d t h o n l y a few f u l l d e p t h webs and wing s k i n s t a b i l i s a t i o n i s p r o v i d a d b y s t r i n g e r s a n d / o r p o s t s .

Such a s t r u c t u r e i s •unlikely t o e x p e r i e n c e s e v e r e t h e r m a l s t r e s s e s

b e c a u s e i n o r d e r t o r e d u c e d r a g a n d a c h i e v e s p e e d s where t h e r m a l e f f e c t s a r e i m p o r t a n t t h e wing d e p t h must b e reduced t o a p o i n t v\iiere i t i s more e f f i c i e n t t o a c h i e v e s t a b i l i s a t i o n b y f u l l d e p t h webs only, A s t r u c t u r e of t h i s t y p e h a s been assumed i n t h i s a n a l y s i s ,

F o r v e r y h i g h speed s t r u c t u r e s , wing d e p t h i s o f t e n so low t h a t more s h e a r c a r r y i n g m a t e r i a l must be p u t i n t o t h e v/ing t h a n i s n e c e s s a r y f o r s t a b i l i s a t i o n a l o n e . O b v i o u s l y t h i s a n a l y s i s w i l l n o t a s s i s t t h e d e s i g n of such s t r u c t u r e s u n l e s s t h e f o l l o v / i n g c o n d i t i o n i s imposed on t h e r e s u l t s , A . 2 . The E f f e c t of Shear. N e g l e c t i n g combined e f f e c t s of d i r e c t and s h e a r s t r e s s e s i n t h e web, t h e s h e a r s t r e n g t h c r i t e r i o n t o b e s a t i s f i e d i s t h a t

< P , where P i s t h e maximum s h e a r s t r e s s and w W S i s the shear force per element. If the shear force per unit length chordwise is s = r-—

S the above criterion reduces to

^^^t ^ ( ^ ) (^ê^) A.1

x t g y V P b ^ , .

Therefore in the procedure described in Section 6, where in any particular calculation b... and t_ are constant, the minimum permissible value of

\

22

-The i n t r o d u c t i o n of such a l i m i t a t i o n vri.ll only affect the optimisation i f the optimum value of r r for any given value of t„ i s l e s s than

t h a t given by eq, A . 1 .

I f t h i s should happen, the values of r, and r , , v/hich give t h e r e q m r e d product of r, r, are chosen t o give t h e l a r g e s t p o s s i b l e value

<

\ D o ,

r—^ |, Hence f o r a given S t h e l i g l i t e s t structijre s a t i s f y i n g

V y b"

eq. A.I may give a '/alue of (T—S ) l a r g e r than required. I f _S i s

\\l J b~

m not specified the optimum structure giving the required value of r-—

% may s t i l l be found.

A.3. The B r a z i e r Effect.

I n the a n a l y s i s no allov/ance has jpeen made for v/eb crushing, or the B r a z i e r effect. Rosen i n h i s anaJysis has considered v/eb crushing and he has developed a c r i t e r i o n f o r t h i s problem, v i z .

r, > D ^

f\^^

1

o-_ , e where D

M 2 C 2) 2

7 E

I n the above expression for D, K-, t h e non-dimensional web crushing s t r e s s coefficient was taken by Rosen t o equal 3 and z and z a r e t h e co-ordinates of the point on a compressive s t r e s s - s t r a i n curve a t v/hich t h e tangent modulus i s equal t o one-lTalf t h e secant mxOduliis for a m a t e r i a l , Hence, f o r any given m a t e r i a l and value of b.-^. , r . > constant.

~S

The introduction of such a limitation into the optimisation follows readily,

A,4. The J u s t i f i c a t i o n for Using The BiJckling C r i t e r i o n of Ref. 3 . The values of K assumed i n t h i s a n a l y s i s have been determined fron Ref. 3 and, f o r s t r u c t u r e s in which buckling must be precluded, Ref. 3 offers t h e b e s t a v a i l a b l e information. I t should be pointed out however t h a t for s t r u c t u r e s i n v^ich buckling, p a r t i c u l a r l y web buckling, i s permissible t h i s present a n a l y s i s i s possibly conservative. Rosen has i n v e s t i g a t e d the u l t i m a t e strength of m u l t i c e l l beams and developed an apparently s a t i s f a c t o r y c r i t e r i o n . He shov/ed t h a t

buckling i t s e l f need not c o n s t i t u t e f a i l u r e and t h a t beams can s u s t a i n as mi.ich a s tv/ice t h e applied moment needed t o i n i t i a t e web buckling. The use of Ref. 3 i s probably j u s t i f i e d t h e r e f o r e i n studies f o r non-buckling strixïtures up t o , say, t h e proof load; but for the determination

of ultimate s t r e n g t h s Rosen's a n a l y s i s i s more a p p l i c a b l e .

For problems involving mainly thermal s t r e s s e s i t may be unreasonable t o apply the r e s u l t s of Ref. 3 for the main reason t h a t the b a s i c s t r e s s d i s t r i b u t i o n s d i f f e r in the bending and thermal problems. Hoff (Ref.lO) assimied the skin b u d d i n g s t r e s s coefficient f o r a simply-supported p l a t e ( K — 5 ' 3 . 6 2 ) . Since, in g e n e r a l , the r e s u l t s of Ref, 3 are lower than t h e p l a t e r e s u l t s , the use of Ref, 3 i n combined bending thermal problems i s probably conservative.

A,5. The Design C r i t e r i o n for the Wing being t h a t of Compression S t a b i l i t y of the Skin imder Bending and Thermal S t r e s s e s ,

I t was assumed i n the a n a l y s i s t h a t the c r i t i c a l design condition for the beam occiirred i n the heating phase and tha.t the combined bending and thermal s t r e s s e s in the skin on t h e compression side c o n s t i t u t e d the major problem.

However, simultaneously v/ith t h e compressive thermal s t r e s s e s in t h e skin t h e r e a r e t e n s i l e thermal s t r e s s e s i n t h e web. A general expression f o r the maximum t e n s i l e thermal s t r e s s e s in t h e v/eb i s given from eqs. 5.6 and 5.8 a s ;

% = H i V l ^ n ) ^^S •» H7 f) '

A

or q- - f G ^S , A. 2 ¥ - ^

24

-T h e r e f o r e a t a n y g i v e n t i m e , v/e must compare t h e maximum t e n s i l e s t r e s s i n t h e web vri.th t h e u l t i m a t e t e n s i l e s t r e n g t h of t h e m a t e r i a l , and i f t h e web t e n s i l e s t r e s s i s g r e a t e r , s k i n buckl.ing i s p r e c l u d e d and t h e analj?sis i n v a l i d . I f i t s h o u l d happen t h a t v/eb t e n s i l e s t r e s s i s t h e c r i t i c a l f a c t o r y , a more s i m p l e o p t i m i s a t i o n i s p o s s i b l e ,

0, T = 0 q = 0 and t h e t h e r m a l t e n s i l e s t r e s s i n t h e ' m ^

( a ) At t i m e t

web i s c o n s t a n t t hr ou g li out , and g i v e n by A .

, from eq. A, 2,

\ r T

A ^ 2

T h e r e f o r e tlie maximum t e n s i l e s t r e s s i n t h e v/eb i s g i v e n b y

o- = T -f ^ i *

m_ t,

s / \^ .^-^

V t

eq. 5 . 1 8

or 0- = I T M^; ^

m

h

1

hi\ U

•= I ra A . 3 T h e r e f o r e , assianing t h a t T, ("r—2"" 1 and i -~ 1 a r e g i v e n , t h e r e q u i r e d beam geometry i s d e c i d e d b y t h e v a l u e of Ö which s a t i s f i e s t h e e q u a t i o n

A, 4

|.(^)^fer)©fe

= cr u l t where o" -ij. i s t h e u l t i m a t e t e n s i l e s t r e n g t h of t h e m a t e r i a l , I f _ S i s not s p e c i f i e d , t h e optimum s t r u c t u r e i s given v/hen t h e

^ 7 w weight "pp-TT™ i s a minimum, where ^S . . . „ „ ^ *S ^ ' \ W b^ (^ ^ - - ^ V t ) = b^ ( ^ - e ) ^ - • - • • 5.15 ^P\ or W ^ ' \ = (1 + 6) 1 + e m 3+0 ^ult 3 T 2

liV

u s i n g eq. A , 4 m

6 V \ '

2 aL^(1+ e) 3 -A.5

Comparing e a . A, 3 vidth eq. 5 . 1 8 i t i s seen t h a t t h e d i f f e r e n c e betv/een t h e maximum s t r e s s e s i n weh and s k i n i s given b y

% - ^ = 1 ^

^S " -^,7 A 2 ^ V 1 + 6 A. 6 T h e r e f o r e t h e c o n d i t i o n t h a t v/eb t e n s i l e f a i l u r e i s c r i t i c a l i s g i v e n b y 3 m ( .1 - a 2 V l + 6 •^ °S < ° u l t A, 7 where cr i s t h e v a l u e of t h e maximum c o m p r e s s i v e s t r e s s i n t h e s k i n , b quoted i n T a b l e 3 . (b) At a t i m e t ^ t . T > 0 q = d I n t h i s c a s e t h e t h e r m a l s t r e s s d i s t r i b u t i o n a^cross t h e web d e p t h i s p a r a b o l i c and t h e maximjjm t e n s i l e s t r e s s o c c u r s a t a d i s t a n c e y from t h e n e u t r a l a x i s where t h e combined, b e n d i n g and t h e r m a l s t r e s s e s a r e

\ 2 a;, = JL - E a ( T ^ - T ) _{y —ïr s m'}

3

3 -t-e

1 + e , . from e q , 5 , 8

Differentiation of this equation with respect to y yields the position and value of the maximum tensile stress in the web as

and Z _ d -or = Md 2 I E a (T - T ) ^ s m^ A, 8 M d T h i s e q u a t i o n c a n be reA^vritten a s , A, 9 o- = ^ T a / "h^xz

^)^i¥{-^)[i) ^y ^-'^

As in section (a) above the beam geometry can now be decided by solution of the following equation for ®. If t„ is not specified the optimisation

b f o l l o w s s i m i l a r l y 1 2 T mZ f 3 + Q

l i ^ V M

' V^nrè-; * ^) [-J T3^

- I = cr r a t A.11

26

-Obviously the condition t h a t web t e n s i l e f a i l u r e i s c r i t i c a l i s given by eq„ A.11, I n s e r t i o n i n t o eq> A.11 of the par'ameters determined by t h e analjrsis of section 6 and t a b u l a t e d in Table 3 w i l l show whether,

TABLE 1 .

Values of K. (Taken from Ref. 6 ) .

\ - t

""^

X

0,5 1.0 1.5 2.0 2.5 3.0 .25 3.62 2.35 0.80 -.40 3.71 3.62 2.67 1.45 -— .50 3.80 3.70 3.32 2.25 1.23 — .60 3.89 3.78 3.62 2,96 2,00 — ,80 4.18 4.07 3.93 3.71 2.95 2,09 1,00 4.45 4.37 4.29 4.18 3.75 3.04

20 -T/iEDB 2, EXAIvffLE T R = 20.000 I b / i n ^ BASIC P/iRAMETERS

Vt

1.00 j 0.90 0.80 6 . 0,50 j 0.45 0.40 !

Vi

2 9 / , 3

^Vl7

^b 1,00 1.50 •S..60 1.66° 1.75 1,80 2.00 1.00 1.50 i 1.60 1.66° 1.75 1.80 2,00 1.00 1.33° 1.50 1.54 1,60 1.66° 1.75 2,00 ^ t 1,00 0,66° 0.625 0.60 0.571 0.556 0.50 0.90 0.60 0.562 0.540 0.515 0.50 0.45 0,80 0.60 0.53 0.52 0,50 0,48 0.456 0.40 K 4.37 3.72 3.54 3.40 3.14 2.98 2.25

4.22 1

3.62 3.35 3.14 2.86 2,62 1.85 4.07 3.67 3.42 3.31 3.11 2,82 2,46 1.45

m 't C a l c u l a t i o n of —r f o r d i f f e r e n t v a l u e s of r, r , S

^y>h

1.00 0.90 0.80 1 .. ^b 1.00 1.50 1.60 1.66° 1.75 1,80 2,00 1,00 1.50 1,60 1.66° 1.75 1.80 2.00 1.00 1.33° 1.50 1.54 1,60 1,66° 1.75 2.00 ... _ ^ m m H 27.3 27,3 600 796 52.2 52.0 1320 1520 56,6 55.7 1430 1630 59.0 57.5 1480 168O 60.1 58,3 1505 1700 60,1 58.3 1505 1700 56.3 55.5 1425 1620 26.4 26,4 580 758 51,0 50.9 1280 1463 53.6 53.0 1340 1523 54.5 53.9 1370 1545 54.7 54.0 1372 1552 53.0 52.6 1330 1512 46.3 ^^^.3 1150 1330 25.5 25,5 558 722 40.6 40,6 988 1150 ZjS.I 48,1 1200 1362 49.1 49.1 1230 1390 49.8 49.8 1250 1410 kQ.3 48,9 1222 1385 47.1 47.1 1170 1335 36.3 36.3 866 1028 m R H cr b^.a b^^z __ 27.0 27.0 580 769 49.7 49.7 1225 1415 53.9 53.2 1325 1515 56.2 55.4 1382 1575 57.3 56,3 1412 1600 57.3 56,3 1412 1600 53.6 53.0 1320 1508 25.2 25.2 530 704 48,6 48,6 1188 1360 51.1 50,9 1250 1423 51.8 51.5 1268 ly^l 52,1 51.9 1275 1451 50.5 50,4 1235 1410 41.1 41.1 974 1150 24.3 24.3 512 670 38.7 38.7 910 1068 45.8 45.8 1105 1265 46.8 46,8 1135 1290 47.5 47.5 1151 1310 46.6 46,6 1130 1286 Zt4.8 4^,8 1075 1235 34.6 34.6 796 955 U n i t s i n l b , i n s e X 1 O* , cr X 1 0~^ I b / i n ^ •:^ I b / i n * _ W X Mien T R = O

- 30 ^

TABLE 4 . EKi^A'PI^

GEaiETRr OP 0PTB5UM STRUCTURE TO SUSTAIN r ^ = 1400

Row Source 2 3 4 5 P i g . 6 P i g . 6 P i g . 9 P i g , 9 T a b l e 7 Ref. 6 T R 20,000 2 0 , 0 0 0 0 0 ^S ,025 ,025 .02Vf .02it4 ,0237 b t .913 .793 . 9 9 . 8 6 2 .962 ^b 1.732 1.587 1 . / / 2 1c 689 1.76 r J ^ t . 5 2 7 . 5 .575 «511 — T

w

2 \ . 0 3 6 4 .0349 .0365 .0350 ,55 ,0350

. i

Remarks

^1

*s

1 T-— s p e c i f i e d

V \

) OPTD.IJM *S/b,-. 1

Ks 5 A 3 2 I

1

•n-*^

7i

\ ^

K

> i 4 0 > \ ^ i 6 0 \ " > > O 5 l O l-S-ljj 2 O 2-5 3 0 FIG.I VALUES OF Ks. (TAKEN FROM R E F 3 ) 5 3 2 1 / / / A / / O I 2 £ 3 4 5 6 X I O

FIG. 3 STRESS-STRAIN CURVE D.T.D. 687 - 3 N.A.

V

f

^ - 1

w d iy

T I

I 6 0 0 14 OO ÜQ. I 2 0 0 lOOO 8 0 0 LIN ^ ** £ OF ^ - ^ t ^ " ^ ^ = -T O - O MAXI = I O 0 - 8 J_ O 0 , 0 0 MUM ^ . — ^ - ^ 0 LB 1 1 EFFICIENCY / • ^ ' ^ ^ ./IN.2. / > / ^

N

- ~ ~ 1 ^ • t ^ -- ^ ^

s,

\ IS 1-6 1-7 r^ 1-8 1-9

FIG.4 VARIATION OF ^ 5 WITH'T^ A N D - r r - f x

bc^2 b b t I 8 0 0 1600 m I 4 0 0 1 0 0 0 1 1 LINE OF MAXI 1 1 1 y . ts bta T R

X

" > • 9 ^ 0 8 \ _ _!_ 4 0 = 0 S/IUM ^ EFFIC /

y

:iENC / r

-f-/ ^ * ^ • " - - ^

X^

^ - — " ^

k

- . ^ \ , \ 1 2 0 0 - r - = 7 ; 1-5 1-6 1-7 X 1-8

FIG.5 VARIATION OF ^ ^ ^ I T H \ AND'^b t

1-9

I 7 0 0 I 6 0 0 P Q. O I 5 0 0 14 OO I 3 0 0 I 2 0 0 1 1 y 0

y

' ,-«• ^ ' =ro = O " « 4 0 X y ^ ^ ^ . ^ / ^ - ^ _ - . < • y ^ . - < ^ X X ' 0 y ^ W ^ 1 . i ' ' : > . / -> / ^^ 1

y

k

A

f

A

A

n>t

• 9 m

FIG. 6 VARIATION OF O P T I M U M - ^ j W T H OPTI VALUES O F ' ^ b A N D ' ^ b ' ^

l O

MUM

I 6 0 0 I 4 0 0 m bw2 I 2 0 0 lOOO 8 0 0 LINE ^ t s T R OF -^b-T ^ 1 41 = 2 0 . MAXI! k = I O 0 ^ O - 6 O O O vlUM ^ • - — *^ LB./IN EFFIC ^ ^ y 9 2 lENC^

^V

/ • ^ ' ^ ^ ^

s

^ \ \

v

N

^ . . ^

V

s

N

1-5 1-6 17 Tb 1 8 1-9 2 O

FIG.7 VARIATION OF -—g WITH T^ AND T,, T ^

I 6 0 0 I 4 0 0 m bw2 I 2 0 0 lOOO 8 0 0 •• LIN _y ^ .* bw T R 1 1 E OF MAXIMUM 1 1 1 T b T t ^ l O

y^

_ _!_ ' " 4 1 = o 0 - 9 0 - 8 ^ ^ EFFK , y y : i E N C < / f -Y , ^ / ^ v,^^ \

\j

• * « s \ ^ ^

N

\ V \ V \ 1-5 1-6 _'•7 T K r-8 1-9 2-0

m

I 6 0 0 I 5 0 0 1 4 0 0 oi 2 p I 3 0 0 a O I 2 0 0 IIOO — — -.• y ^ y^,^^ TR = ^^ y 0 ^ ^ ^ ^ / ' y ^ ^ ' < - ^ ^ ^

f

^w / L ^ y

l

^ \ • ^ 41 ^' y^'

y

y ^

y

y y /

y

y

^.^ y / / •

y)

/ /

y

f

Jy / •8 _{^ b - ^ t} I-O m

FIG.9 VARIATION OF OPTIMUM b " 2 WITH OPTIMUM VALUES OF Tj^ AND T,, nr^