CiteSeerX — NOTATION

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THE COLLEGE OF AERONAUTICS CRANFIELD

CERTAIN METHODS OF STRESS-ANALYSIS OF RECTANGULAR MULTI-WEB BOX BEAMS

by

K.H. GRIFFIN

TECHNISCHE H C " " " ' ^ " ' " " ' '

VLIECTUIGïO

2 • :7 REPORT No. 108

EECiaiBER 1956

T H E C O L L E G E Q g A E R O N A U T I C S C R A N F I E L D

An Assessment of C e r t a i n Methods of S t r e s s - A n a l y s i s of R e c t a n g u l a r Multi-Yfeb Box Beams *

K.H. G r i f f i n , B . S c . , A . P . R . A D . S .

SUlVlt.'L'JlY

The stress distribution in an unswept multi-vreb box under shear load applied at the centre of a rigid tip rib is examined, and compared with results obtained by a method which replaces the shear webs by a sheeir-carrying continuum. For the case considered the agreement between the results from the two methods is

satisfactory for boxes "w^ose geometry is typical of normal practice,

A method of performing root constraint calcvilations on such a box is outlined and results of an elementary analysis of this type are compared with experiment.

Paper presented to the IX International Congress of Applied Mechanics, Brussels, September, 1956

- 2 -

N O T A T I O N b half depth of b o x

0 half w i d t h of b o x 1 length of b o x t skin thickness

t thickness of internal webs t thickness of end webs

O

A area of internal booms A area of end booms

o

n n-umber of colls (n + 1 = n-umber of webs) i web or skin panel identification number x,y, z coordinates defined in Pig.I

Z applied shear force

T ,S,Tp stress-resultants defined in skin z = b P. load in boom i

q. shear flow in web i.

Q. modified web shear tlavr (eq, (2) )

u,v displacement components in plane of skin z = b.

w. z-wise deflection in web i.

P.,G. functions defining spanwise constraint stress distribution

$ . f. $p. functions defining chordwise constraint stress distribution H. constants defining web shear flow distribution

E Young's Modulus or Poissons R a t i o

r = 0-/2(1 + o-)

\ = t / t

a = nA / 2 c t

c ' = c + (n - l)A/2t + A ^ t

^ = oZ/2ribc»

e = e t /ribt 6 = ( 6 / 2 ) ^ Y = sinh 5

( 9 ( 2 H. e ) ) Q = 1 + 0 ^

C = n x / 2 c , Tl = n y / 2 c , TI- =

1. Introduction

A form of construction for thin aircraft wings v/hich is widely used at the present time is that which makes use of a number of shear webs to stabilise the compression surface. This . multiplicity of shear-carrjdjig material provides a higher order

of redundancy than that associated with a single-cell box, and various methods have been proposed for dealing with this complexity

(References 2,3).

A comiiion method of dealing with a continuous structure reinforced by a large number of discrete elements (e,g, fuselage reinforced by stringers) is to replace the discrete elements by a continuous distribution of material having the same overall properties. Hemp (Reference 1) has used this technique in dealing vrith the multi-web box, in that he replaces the discrete shear webs by a shear-carrying continuum. The solution obtained gives a choï'd-vd.se variation of shear in this continuum which is exponential, and it is natural to enquire as to the ax3curacy with which the continuum can reproduce the behaviour of the finite number of webs carrying such a rapidly varying loading. The present investigation deals with a particular case dealt with in Reference 1, namely that of the uniform rectangular raulti-web box under shear load applied centrally at a rigid tip rib. This

loading case is considered under exactly the same assumptions as those used in Reference 1, except that the continuum does not replace the discrete webs,

2, The Rectangular I.tulti-Web Box under Tip Shear

The box is shown in Figure 1, which gives the overall dimensions and shows the co-ordinate system used, and the

direction of action of the applied force Z which acts through the line x = I, y = c. The root x = 0 of the box is assiomed

-5-

to bc rigidly built in, The box has (n+l) webs, and web 'i'

is in the plane y = 2ic/n. Webs 1,2,3,4 ..., n-1 have thickness t and boom area A, while webs O, n have thickness t and boon

w

area A . All the webs are assiomed to have no bending stiffness, but to carry a iinif orm shear flov/ q., The contribution of the web

material to the bending stiffness is allowed for in the boom areas A and A . The skins z = +b have thickness t, and v/e identify panel 'i' of the skin as lying between \rebs 'i-i' and 'i'. The

stress-resultants T., S, Tp illustrated in Figure 1 have the positive directions shovm in the skin z = +b.

The analysis of Appendix A leads us to the following expression for the distribution of the applied shear force Z between the webs.

Qi = ^

) , w / 2\-1+2\a-2a,-(n+2X-l)T . . r,-2.s

(1+a-T)+0 — — ) {(3 +fr )

where Q. i s defined as^^ q. ( i = 1,2,3, , . , , n - l )

=f <Ji (1 = o,„)

The various parameters are defined in the list of notation,

] ]

/3 = 1 + 6 -^j V0(2 + 0)

0 is defined by 0 = ct /nbt = (a /2b) (t V t ) where a is the web pitch, 6 is therefore essentially a "ejfiape" parameter, describing

the geometry of the internal cells, and we shall find that the distribution of V/eb shear flow given by (1) is noticeably sensitive to 0, Another important parameter in (l) is T = Ö / 2 ( 1 + o").

If the web shear distribution is evaluated by the usual "Batho"

method of equating the twist in each cell to zero, as in Refs, 2,3,

the solution is given by equation (l) with T zero. .As will be seen

later the term in T has an important effect on the magnitude of the departure of the shear distribution from the uniform. The term

in T arises because the skin shear strain, Tsdiich has to correspond with the stress obtained using equilibritim considerations, contains a

term arising from the chordvd.se Poisson contraction due to "the linearly varying bending stress. It is clear that if the box is built in to

a completely rigid root then this Poisson contraction will be res-tarain- ed and so the importance of the ein in T will be reduced. This is a matter for ejcperimental investigation,

Comparison with the 'Continuum' Theory

If v/e vinrite Y = 2 |log p\ , (2) may be throvm into the farm

Q. = ? A-B c o s h ( 2 i - n ) y / ( cosh nY(2X-1+tanh nY/tanhY)y (3) v/here A = ( I + C - T ) , B = (2X-1+2\a-2a - ( n + 2 \ - l ) T ; )

Yfe may transform the expression (in equation (28) of Reference I) for the shear s-bress in the continuum into our notation

(using i as a running co-ordinate = ny/2c) to give

q = (r^/Za) Ï A- Bcosh(2i-n)5 /(cosh n5(2\-1+tanh n6/5)J (4) where 6 = (0/2)^ = sinh Y

The close relationship in form between equations (3) and (4) may be noted, and vre may immediately compare Y afi<3. 6 for varying 0 to find the difference in the ' shape' of the shear distribution given by the "two methods.

The six-cell box v/hich provided the experimental results noted in Reference 1 had b = 4 ins,, c = 18 ins., t = 0,159 ins,, t. = t = 0,0575 ins,, a = a^ = 0,1h06. ¥e thus have 0 = 0.2712, and so Y = 0,3604, 5 = 0.3682. In this case, then, the

'continuum' method gives a two per cent variation in the

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exponential term in the v/eb shear distribution, which is certainly satisfactory.

It m i l bo seen that in both (3) and (4) the term in B indicates the departure of the web shear distribution from the imiform and that this departure is greatest when i = 0 or n, We may compare the difference in this term between the tv/o expressions for the extreme cases n = 2 and n large,

For n = 2 1/(1 + tanh 2y /tanhy) = 0,3589

1/(1 + tanh 25/6) ^ O.37OO (3.10üdifference) For n large 1/(1 + l/tanhY) = O.2568

1/(1 + 1/6) = 0,2691 (4.&;i difference)

Thus for this value of 6 the continuum theory gives a reasonably close approximation to the depar-ture from uniform shear in the vrebs, vifhich does not vary greatly in accuracy with number of cells.

In the above discussion v/e have compared web shear flov/

given by this paper with (one v/eb pitch) x (local value of continu\jm shecjr stress at v/eb line) given by Reference 1,

Strictly speaking, one should integrate the result of equation (4) over one lialf web-pitch either side of the v/cb line, adding in the case of v/ebs 0 and n the contribution of the 'special' edge vrebs. If this is done the differences noted above are increased to 13.3^ and 6,C^ respectively. Thus it may be said that in the continuum theory it is numerically favourable, though

logically incorrect, to multiply the local value of the continuum shear stress by the appropriate v/eb pitch. This artifice,

hovTCver, will lead to a distribution of vreb shear flovv v/hich does not exactly balance the applied shear force Z,

It should be noted that the percentage 'errors' given above are only differences in the term multiplied by the constant B, If v/e return to the oxaraple of Reference 1, and evaluate A and B numerically (n = 6) we find that if o- = I/3,

B = 0,1250<< A = 1.0156 and v/e shall expect the differences in the final results of equations (3)and (4) to be proportionately less. Remembering that in this case n = 6, equation (3) gives Q = 0,9827 K,- The local value of shear stress in (4) leads to q^ = 0,9811^(0,1^ difference), v/hile the 'integrated' value gives q^

=0,9800^ (0,3^ difference). Thus the overall difference between the tv/o methods is much reduced in this case. It may be remarked that this masking of the difference betv/een the two methods is most apparent when n =* 7 and that in the torsion case (equation (25), Reference 1) there is no constant term present and thus the ori^rLnal figures given are more appropriate. We may note that if T = 0

("Batho Theory") B = 1.000 and so A and B are strictly comparable in magnitude and the above reduction does not occur.

The above example provides quite close agreement between the results of the tv/o methods, A value of 0 equal to unity is not at all unreasonable structurally, and if v/e take a ten cell box for which 0 = 1, a = a = 0,10, X = 1 we have Y = 0.6585,

6 = 0,7071 ( ;. 6 = 1 . 0 7 3 8 Y ) and q^ = 1.1132^ by (10)

= 1.1303s hy (11) ( l o c a l ) (l.5?S)

= 1.1378? "by (11) ( i n t e g r a t e d ) (2.2%) For a fo^ur c e l l box of the seme geometry

q^ = 0.8368g by (10)

= O.8I9OS by (11) ( l o c a l ) ( 2 , 1 ^

• = 0.8081? by (11) ( i n t e g r a t e d ) (3.4%)

For the t o r s i o n case the above differences v/ould be i n c r e a s e d to the order of 15/S . We may thus conclude t h a t the accuracy of the continuum i n following the behavioior of the d i s c r e t e shear webs of a multi-web box i s q u i t e reasonable f o r a box having p r a c t i c a l l y s i g n i f i c a n t geometry. I t may be s a i d t h a t the

_9-

number of v/ebs is less significant than the geometry of the cells as reflected in the parameter 9 .

All the foregoing may be said to be concerned with the 'beam theory' solution of the raulti-v/eb box built in at its root imder a shear load applied at a rigid tip rib. The solution coxold have been obtained by finding the vertical deflection at the tip of each web with the tip rib absent and some simple distribution of the applied shear among the v/ebs, and then seeking the statically zero web shear distribution necessary to make all the tip deflections equal. This approach to the problem will be foxmd useful v/hen considering constraint effects a.t the root.

It is clear from the expression for u in Equation (A, 9) that the solution v/e have obtained involves an equal v/arping of all cross-sections and that the distribution of shear flov/ between the webs v/ill affect this v/arping. Since the bending stresses corres- ponding to T raust be reacted at the root of any box, it is clear

that there must be a greater or lesser amount of restraint of the warping of the root section. This restraint is incompatible v/ith

the results of equation (A,9) and so a statically zero modification to the elementary stress distribution must be added in order to make the root deformations consistent v/ith the boundary conditions. Hemp in Reference 1 suggests a modification to T v/hich varies parabolically in the chordwise direction and has an arbitrary spanv/ise variation

which may be determined by variational methods. This solution involves no shear in the v/ebs, and is correct for a box v/hero I is large.

However, since there is nov/ variation of bending stress across the chord there must be a corresponding differential bending of the webs (equation

( A , $ ) with q. = O ) , and this is incompatible with the assvmrption of a rigid tip rib,

4, Root Constraint in Shear

Follov/ing a remark made earlier, it would seem that a

reasonable approach to the problem of the root constraint of a multi-web box under tip shear is to solve first the root constraint problem for an arbitrary distribution of the applied shear between the webs. The web shear distribution is then chosen to make the web tip deflections including those due to the constraint stresses compatible with the conditions applied by the tip rib. The method chosen has been the familiar one of specifying a. distribution of bending stress across the cross-section with spanvd.se variation determined variationally using the Theorem of Minimum Strain Energy, It is clear that the choice of the cross-sectional variation of stress must be sufficiently general to allov/ for effects arising from the as yet unknov/n v/eb shears,

ExtendinfT, the method of Reference 1 to allov/ T. to vary _{u 1} with y as an even order polynomial brings considerable algebraic

difficulty because of the effect of the web booms in disturbing the simplicity of the skin shear stresses. It vra.s decided to allow T. to vary linearly betv/een sirbitrary values at the web lines. For the n-cell box this v/ill mean that the constraint stresses are

specified by either r/2 or (n-1 )/2 unknov/n functions of x, the equations governing v/hich must be obtained by minimising the strain energy xmder appropriate conditions. This v/ill lead to linear differential equations each involving all of the unknown functions.

It is therefore profitable to endeavour to minimise the coupling

betv/een these functions, Yfc may do this by building up our constraint stresses from distributions vAiich are themselves each statically zero and v/hich cover a minimum number of skin bays.

In Appendix B such distributions are derived and a method of solution is indicated. Calculations of this type have been performed in the case of the six-cell box already mentioned and the results are plotted in Figure 2. Section A is at 4.5 in. and section B at 27 in. from the root of the box whose length is 54 in.

-11-

The conclusions of tlxls necessarily crude approach should be treated with caution, but they indicate that the effect of v/eb shear distri-

-bution on the constraint stresses is not negligible.

5. Further Developments

The results of a simplified calculation such as that just described v/ill serve as a lead in the solution of the coniplete constraint problem involving the functions P. (g) of Appendix B as well as the G. (?). The methods of this paper arc eminently

suitable for examining the effect of flexibility of the tip rib;

and they v/ill also handle v/ithout extra difficulty the problem of the multi-v/eb box carrying concentrated loads applied to its T/ebs at stations distant from ribs, a. case vAiich has proved unsuit- -able for the continuxjin approach. The methods of the first part of the paper may also be used to investigate the acciuracy of the

continuum approach for further loading cases (e.g. the box under distributed normal pressure).

6, Conclusions

For the simple loading case discussed, namely the box under uniform shear, the method of replacing uniformly spaced webs by a 'shear continuum' is sufficiently accurate for the purposes of stress analysis. The effect of stresses due to any constraints must, hov/ever, be trJccn into account v/hen evaluating the shear distribution in the webs.

References

1. Hemp, W.S.

2. Benscoter, S.U,

3. Samson, D,R,

4. Gurudutt, M,

Stress Analysis of Multi-V/eb Boxes.

Fifth Anglo-American Aeronautical Conference. Inst.Aero.Scs. 1955.

Numerical Transformation Procedures for Shear-flov/ Calculation.

Jnl.Aero.Scs. 13, 1946, pp.438-443.

The Analysis of Shear Distribution for Multi-Cell Beams in Flexure by means of Successive Numerical Appro ximo.tions.

Jnl.Royal Aero.Soc. 58, 1954. pp.109-121.

An Investigation of the Influence of the Flexibility of the Tip Rib on the Stress-Distribution in a Multi-Span Box . College of Aeronautics Thesis (Unpublished) 1955.

APPENDIX A„

Analysis of Tfeb Shear D i s t r i b u t i o n

The bending s t r e s s e s are d i s t r i b u t e d as i n t h e Engineer's Beam Theory, t o give

T = -Z( £ - x ) / 4 b c ' ; Tg = 0 (A 1 ) where

c ' = c + (n-l)A/2t + A y t (A 2)

dealing with a loading case v/hich is symmetrical about y = o we have n .

q^_. = q. ( a l l i ) ; ) q^ = Z/Tb i=D

We may i n t e g r a t e the equation of x-v/ise equilibrium i n t h e skins (3T / 9 x + 9S/9y = O), and allow f o r t h e equilibrium conditions appljdng at the web booms t o find t h a t the shear flow i n skin panel ' i ' i s given by

Si(y) = 2/^j - (2/^°') ( y + A / t + (i-i)A/t) (A 3)

It v/ill be noted that this expression satisfies the symmetry conditions S . . (2c - y) = - S.(y). The stress resultants

n+1-i ^ •''' i^"'^

given by (l), (3) automatically satisfy the requirements of compatibility of displaoement. The strains in this skin panel are given by

- z ( ^ - x ) ^ _ 0- z(e-x)

^xx - iiEtbc' ' °yy " 4Etbc'

i-1 A (A 4)

^ V i = Et (^2_. "^y" ^ ^ V " "^ \

0=0

I n t e g r a t i n g t h e f i r s t two of (A 4) v/e obtain

u^(x,y) = - Z(^x - ^^)/4Etbc' + f^(y) v(x,y) = oZ(^-x) (y-c)/Etbc' .

where f.(y) is an as yet unknov/n function of y. We substitute from ( A 5) into the last of (A 4 ) .

=SKY=2)

^ f.(y) _ 2(l±g)/Y', ^ - i l ± ^ / y ^ ^ ^ ( i . O ^

4Etbc' + ^i^"^^ - Et l / ^ ^ j J 2Etbc' (^ + t + "^^ ^''t (1=1,2,3 ...,n)

Integrating f!(y) betv/een 2(i-l)c/n and 2ic/n, and using (A 5) we find that

(A 5)

(A 6)

1 V n y 1 \ n

^:j

_ 2(1 40-) - Et

where r= a-/2(l+o-).^

i-1

_ Z _ _ 4bc'

(2i-l)(

n

A

(i-i)ê-

t •" '- "t • n

(i = 1,2,3, ...., n) (A 7)

On examination of the above process it will be seen that

the term in r in (A 7) which leads to that in equations (3), (4) referred to in the main text, derives from the shear strain v/hich arises from the Poisson contraction varying with x in the second of (A 5 ) .

Summing equations (A 7) from 1 to i, v/e find that u^(x, 2ic/n)

/ ^s 4(1 t-o-)c 1 ? ^ ' , ^ Zic f. ^ o n(i-1 ) A / L j=0

(i=1,2,3, ...., n)

•i) rj (A 8)' We may again note that symmetry is satisfied since

u^_^ (x, 2(n-i)c/n) = u^ (x, 2ic/n).

A3

We must no'^ consider the shear v/ebs. We assume that they experience no direct strain in the z-direction, and v/e have already postulated that they cai-ry no direct stress in the x-direction though naturally they must extend in that direction with the skins. Clearly, thepi,the v/eb shear strains are given by

2(l+cr)q^At^,^ = 9w/9x + u^(x,2io/n)/b (i = 1,2,3, ..., n-l) 2(l-KT)qy'Et^ = dwjd-x +• u^(x,0)/b

where w. is the z~wise displacement at web i. Since the root is built in v/e have u r: v/ = 0 at x = 0 and since v/e assume that the load Z is applied at a rigid rib v/e must have

3w/9y = 0 at X = 1. Substituting from (5) in (6) we find that 9w./''9x - 9v7. /9 x is independent of x and so the above condition implies that

(A 9)

0 9v/./3x - 9w. ,/9x

= X i-r 2(ii-2:).,(q_„q_ )_iL(lt£k

Et ^^ ^ - r Etbn w

k J

A

r(.2i-l)c 2° (i-l)A (2i-n-l)c

"»' L n " "^t +" "t " ^ n

(i = 2,3,4, ..., n-l) 0 = 9w /8x - 9v//9x

Et V ^ v/

t q v/^o 4(u'y)(

Etbn

Z

A

_ ^ _ . _ ^ r ( n - l ) -

If v/e define Q^ = q^(i = 1,2,3, ...,n-l)

and write 6 = ct /nbt = a t /2bt (a = v/eb pitch) w v/ v/ ^ w -^ '

and ^ = c V 2 n b c ' , a = n/i/2ct, a = n A / 2 c t , X = t ^ t . then

^ i - 1 -2(l+e)Q^ + Q^^^ + 2 6 (1+ a - r ) = 0

( i = 1 , 2 , 3 , . . . , n - l ) )' (A 10) - ( l + 2 xe)Q^ + Q^ + 6 ^ (l+2a^+.(n-l)r ) = 0

We may solve the recurrence relations of (7) to give

Qi =S (1 + a - r )+C(/3'- +^''"'•) (^.,^) (i = 0,1,2, ,.., n)

where

/3^ - 2(l +e) /3+ 1 = 0

and v/e choose the rootjg = 1 +0 - J i e ( 2 + e ) y < 1 . We find the uhkaov/n constant C by substituting in the equation involving Q and Q only, and so obtain equation (l).

APFENDDC B

Constraint Stress Distributions

In §4 it was suggested that a constraint stress distribution coiold be conveniently built \ip as a combination of "local"

statically zero distributions varying v/ith x. If v/e ass\Jme that T varies linearly v/ith y betv/een web lines we may proceed as follows.

4

I f g= n x / 2 c , V = ny/2c, H. = r? - i+1 we may v^rite

n-1

( B 1 )

T^(x,y) = \ F ^ ( Ö ^ ^ i (^) 1=1

n-1

S(x,y) = ) F | ( g ) *. (77)

( B 2 ) 1^1

n-1

T2(x,y) = \ F^(^) $^.^(n)

P^(x) = 0 ; q^ = 0

where F . ( B,) = F . ( ^ ) and * .(v) e t c . a r e defined by

•,,(n,) = i - 6j-^ V , - 6j(l-3.,) . 6^(2-3.,^,)- 6 f (1-.,^^

( i = 2 , 3 , 4 , . . . , n-2)-

* / n p = l ö ^j "^1 ^^ 2"^ ^ ''i ^ + ^j^'i "^ 077^+7^2) - 2 83( 1 -77^) 2

*i(77,) = i i+1. ii+1

« r - ' i i *«j('-''i' (^-^-^i' - « r ' ' u i ( ^ - ' ' ' i . i ' - ' j (^-^.2^

( i = 2 , 3 , 4 , . . . , n-2)

B2

* (v7 ) - — 21 ^ " y - 30 2 i ' ^ ' y - 12

;1 . 2 2.-,„5>

5'^ ^;(l8-11T?^)-8j(7+3n2-15n2+7^^)-26^(l-n^)

,2^^ 3^ .i+1 i+2/

- «r <-i • '>^M-H^'T^^^-'''U^*K^^-'T^'-'U2>

where 60 = 1 when k = 7, O v/hen k / 1

( i = 2 , 3 , 4 , . . . , n-2) ( B 3 )

The above d i s t r i b u t i o n s are i l l u s t r a t e d i n F i g , 3 .

Sxjch a d i s t r i b u t i o n of T s a t i s f i e s equilibrium c o n d i t i o n s , i s s t a t i c a l l y zero, and covers a t the most four skin bays, I t thus i s s u i t a b l e for our purpose and reduce>s the coupling

betv/een the functions F_. (B,) v/hich describe i t s sparavise v a r i a t i o n , No acco\jnt has so ^ar been taken of the c o n s t r a i n t s t r e s s e s i n the web-booms. Since t h e s e derive from t h e stra.ins due t o t h e above s t r e s s - d i s t r i b u t i o n s , the r e l a t i o n between them i s not

simple. I t i s more convenient t o specify the boom loads P.(x:) independently i n terms of unknown fimctions G.(^)

(^n-i^^^ = G^(0 j as follov/s

V^) = - f ^1^^^)' ^1 W = f(-G„(0^2G^(Ö),

^iW=f-(-<>iH(^)-2^i(^^-^i+1^^0

( i = 2 , 3 , . . . , n-2)

T^^(x,y) = 0 (B 4) S^(x,y) = ^\U) , S^(x,y) = i (-G|_^(Ö+ & [ ( ? ) ) ( i = 2 , 3 , . . . , n-2)

( i = 2 , 3 , . . . , n-2) q. = 0

^ 1

B 5

The condition of minijnum energy may be used to enforce the necessary compatibility relationship between the F. and the G..

1

Our constraint system is nov/ specified by the f •unctions P.(^), G.(^) which automatically ensure that the constraint

stresses are self-balancing. Unfortunately, although much has been done to reduce the coupling betv/een the P. and G., a

typical differential equation resulting still contains seven P, and five G.. We may reduce this complexity by ignoring the P., concentrate the bending-stress carrying properties of the skins (so far as the constraint stresses are concerned) into the suitably enlarged v/eb booms, and allov/ the skins to carry only shear (S) and chord4-/ise direct stress (Tp).

A statically correct distribution of stress which reacts the tip shear Z and v/hich is expresses in terms of unknown constants H., H^, ..., H . (li . = H.) is as follov/s

V 2' ' n-1 ^ n-l i'

nP^(x)/2c = -a^g(t'-^), nP^/2c = -«^(z'-g) (i = 1,2,3,. ..,n~l)

T^iU,y) = -^(i'-Ö

Sj_(x,y) = - S ( T ? ^ 4 ) (i = 1,2,3, ..., n)

T2i(x,y) = 0 (B 5)

% = ^(hah q^ = S(l+a) (i = 1,2,3, ...,n-l)

where V = nt/2o

Combining ( B 4) and (B 5 ) , v/riting dovm the strain energy and minimising it v/ith respect to the G. v/e obtain equations of the type

GV^^^UU ¥ J V " ( Ê ) + G V ; ^ ^ ( Ö - 12(l40-)('- GV_^(g)+ 2G^"(Ö- (^l^^^U)

+ (6/a)(a^^^U)- 4G.__^(d+ 6G.(^)- 2^JG.^^(Ö+ \^2^^n = 0 (i = 3,4, ..., n-3)

(B 6)

v/ith typical boundary conditions

G.'^^(o)+ ha'iio)^ ^u^^^)- i2(n<r)^-G'_^(o)+

GV^(O) ^ G.(Z') = G!(l') = 0

It should be noted xhat in equa.tions such as ( B 6 ) , though not in ( B 5 ) , a and a should be augmented to include the effective

0

skin area. The determinant v/hich must vanish to make these equations consistent is a regular one, and a reciirrence relation may be found v/hich leads to its evaluation for general n.

The P.(Ö inay then be found in terms of the as yet unknown H., and finally the tip deflections of the v/ebs foxond in terms of the H., The H. are then chosen to make these deflections consistent

1 1

v/ith the conditions at the tip rib,

> •

FIG. 1.

ÜSL3 DISTRIBUTIONS OF CONSTRAINT STRESS

^

I2r + \

/ +

lOh

cr

1- 1-o

UJ

U

:<

2

u

UJ

© + © + © + Q"~"+"

.+ I M

•' in

QL

o EXPERIMENTAL-REF.l

— THEORY-REF.l + CURRENT THEORY

FIG.2

SKIN DIRECT STRAIN

(loooLB. TIP