On the application of oblique coordinates to problems of plane elasticity and swept-back wing structures

CHNISCHE HOGESCHOOL VUEGTUIGBOUWKUNDE

REPORT No. 31

t2 Juli 1950

THE COLLEGE OF AERONAUTICS

CRANFIELD

ON THE APPLICATION OF OBLIQUE

CO-ORDINATES TO PROBLEMS OF PLANE

ELASTICITY AND SWEPT BACK WINGS

by

W. S. HEMP. M.A.

of the Department of*Aircraft Design

TECHNISCHE HOGESCHOOL VLIEGTUIGBOITATKUNDE REPORT No. 31 J a n u a r y > 1 9^0 T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D

On the Application of Oblique Coordinates to Problems of Plane Elasticity and Swept-Back Wing Structures

-by-¥/.S. Hemp, M.A.

of the Department of Aircraft Design — oOo—

SUMMARY

The object of this report is two-fold. On the mathematical side it seekc to illustrate the use of oblique coordinates in applications to Elasticity and Structure Theory. On the practical side it seeks to provide methods by which designers can solve problems of

stress distribution and deflection for the case of swept-back wing structures, whose ribs lie parallel to the

direction of flight.

The report is divided into three parts. In Part I the mathematical basis is developed. Formulae are derived which express the fundamental

concepts and relations of Geometry, Kinetics, Statics and plane Elasticity in terms of vector components in oblique coordinates. In Part II, the results obtained in Part I are applied to a uniform, syrmrietrical,

rectangular section, swept-back box. A complete theory of stress distribution and deflections is obtained for the case of loading by 'normal' forces and couples* applied to the ends of the box. Some consideration is also given to problems of constraint against warping. In Part III the main results of Part II are generalised to cover the case of a more representative wing struc-ture. This represents an extension of the usual Engineer's Theory of Bending and Torsion to cover the

case of swept-back wings v\fith ribs parallel to the flight direction. Practical procedures based upon this extension are laid down for stress distribution and deflection calculations. These will have the

same validity for swept-back wings, as the usual design approximations have for the unswept case.

Forces whose directions and couples where planes are normal to the plane of sv/eep-back

-2-TABLE OF CONTENTS

Page

Summary 1 Definitions of Symbols Employed 3

Part I Generalities and Applications to Problems of

Two-Dimensional Elasticity 6

§ 1,1. Geometry • 6 ë 1.2. Kinematics 6 § 1.3. Statics 8 § 1.^. Stress-Strain Relations 9

§ 1.5. Compatability Relation for the Stress

Resultants 10 ê .1.6. Application to Certain Plate Problems 11

Part II Applications to Simple Swept-Back Box Structures Ik

§ 2,1, Description of a Simplified Structure. Notation ^^.

§ 2,2, Theory of the Simplified Structure 1^. § 2.3, Simple Loading Conditions:- (1) Constant Couple 1?

§ 2.k. Simple Loading Conditions'.- (2) Bending by a

z-wire force. I8 § 2,5, Analysis of the Deflections for the Simple

Loading Conditions 20 § 2. 6, Internal Systems of Stress 22

§ 2,7, Approximate Calculation of Root Constraint for

the Case of Loading by a Constant Couple. 21+ Part III Applications to Swept Back Wing Structures 27 § 3.1. Generalisation of the Engineering Theory of

Bending and Torsion to Include the Case of

Swept Back \¥ing Structures 27 § 3.2. Procedure for Practical Stress Analysis 33

-3-Definition of the Symbols Employed Geometry. Dimensions

0(-<:,y,z) Main system of oblique Cartesian Coordinates (see Fig. 1) 0(X,Y,z) Auxiliary system of oblique coordinates (see Fig.1) a Angle between the axes Ox, Oy.

(x,y,z) Coordinates of a point referred to axes 0(x,y,z) ,- Unit vectors in the directions Ox, Oy, Oz, OX, OY

respectively, Position vector Length of vector dr Angle between dF and i

Length of the material element dr after strain Length of a plate or box measured x-wise

Half width of a plate or box measured y-vi/ise

Half depth of a box measured z-wise or, in particular, the half depth of the spar y = c

Half depth of the spar y = - c Ordinate of the skin line

Mean value of J^' over width - c ^ y ^ c . Thickness of the skins

Thickness of the spar vv^ebs or in particular, thickness of the web y = c

Thickness of the web y = - c Thickness of a diaphragm rib

A Section area of spar flange or in particular, section area of flanges at y = c.

A' Section area of flanges at y = - c A Section area of stringer

s ^ A^ Section area of rib flange

a Stringer pitch measured parallel to the ribs ap Rib pitch measured parallel to the stringers >R " '^R/aj^

T) Parameter defining the point of action of a certain shear stress distribution in a box. (see equation (\k3))

T)^ Parameter defining a torque axis for use in the calculation of twist (see equation (159))

Kinematics

u Displacement vector

(u,v,v/) Oblique components of u U = u + V cos a V = u cos a + V p Rotation vector /(p,q,r) Oblique .... i , 3 , k , i ^ , j r d s 0 dS 1 c b b '

s(y)

Ï

t *w

\r

'R

-k-C , -k-C J j c ^R'^"R ^^Ro K^ jKg Sv1 Ww

A

(p,q,r) Oblique components of p or in particular, components of rib rotation.

Constants defining a rigid body movement in a plane (see equation 17)

Components of rib displacement in the plane of the rib in y and z directions respectively.

= (^'R)y=o

Arbitrary constants occuring in expression for ^Ro (equation (65))'

Components of web displacement in directions x and ^w^^w J 2 respectively. Undashed'. - y = c , dashed,*- y = -c.

W Rigid body translation of a rib in z-direction. Defined as W = WRQ in Part II and W = (ww+w^)/2 in Part III

Wg 'Additional deflection due to shearing' (see § 3.3(5)). CO Warping displacement function (eq./ll8)) or warping

displacement itself (equation 131)

0)^,0)2 Functions of y occuring in expression for 00 (equation I32)

Section distortion function (equation II8) or

section distortion displacement itself, (equation 131) A>| s ^ Functions of y occuring in expression for A

(equation 132)

p-,P2 Constants in expression for p (equation 132) q.,q2 Constants in expression for q (equation 132) Y 'Shear Deflection' constant occuring in expression

for ?/ (equation I32)

e Strain in arbitnry direction

*lY^;ë„,r><3.^^r Strain components in oblique coordinate, system _{XX yy xy} 0(x,y).

^YY'^xY Strain components in rectangular coordinate system 0(x,Y)

(^ Rotation of an element dr. Statics

F Force vector

(X,Y,Z) Oblique components of P. Also in Parts II, III, Z is used as resultant z-wire force across a

section of a box.

(L,M,N) Oblique components of a couple - axis 0(x,y,z)

(L.,M.) Oblique components of a couple - axis 0(X,Y). Used also as resultant couple acting across a section

of a box.

T.,Tp,S(=S,=S2) Stress resultants in a plate refered to oblique axes 0(x,y) (see Fig. 3)

0 Stress function, (see equation 22) _ /T^^Tg,

-5-T^jTgsS Stress resultants in a plate refered to axes 0(x,Y) (see Fig. k)

T^ ,T" ,T^,T'^,S' ,S" Functions of y occuring in expressions for T^,T2^S in equation (37)

S„ Shear per unit length in the ribs, estimated per unit span (x-wise)

S,. .S' Shear per unit length in the v/ebs y = c, y = - c respectively.

L> Couple component (oblique) about an X-wise axis through a point y = -jC, z = o on a cross-section of a box

(equation lii.5)

L^ Ditto about axis through y = r) c, z = o (equation I59). Elasticity. Influence Coefficients

E Young''-s Modulus 0 Poisson's Ratio

a. . Matrix relating stress resultants and strains (equation 2k)

(a..) Part of a.. arising from the plate (equation 27) ^^ii^R Part of a.. arising from the reinforcing members

(equation 28)

A.. Matrix inverse to a.. (equation 31)

a,.,a... = a,-,a^. Special combinations of a. . (equation 120)

11 1 3 3T 3J; ' 13 C.. Matrix relating rates of rotation of the ribs with

the couple transmitted in a box (see equation 99» 100, 157» I58, 160 and I6I)

C.^ Constant in formula for P. (equation I57)

1 'Second Moment of Area' for a swept box (equation 142) Miscellaneous Parameters and Constants

A.(1=0,1,2,3,4) constants in expressions for linearly varying stresses in a plate. (see equation (kO) and i 2.k)

M- Constant defining the rate of die-away of a special stress system (see equations '4k» '47)

[X. Sequence of values of \ic defined by equation (1 lU-) '

^. (i=1 ,2,3,i4.) Values of A satisfying equation ^6) ! B. (i=0,1 ,2,3,ii.) Arbitrary constants in equations 43» 47» j B^.(i?j = 0,1,2,3,4) Coefficients of the linear equations for B^

(See. equations IO8, 109, 110, 111, 112, II3) p.( ) Cofactors of Bi^j in the determinant B.^ 1 .

C Sequence of arbitrary constants (equations II6, 117) P.,Q.(i=1,2,3) Constants relating rates of rib rotation to

Couple transmitted and .section warping (eq; 125,126-),. D Denominator in expressions for p., Q. (eq. (126)) R>,R2>P Constants in the warping equation 127. (see 128)

-6-Part I. Generalities and Applications to Problems of Tïi/0-Dimensional Elasticity

1.1. Geometry

The frame of reference used in this report is a system of oblique Cartesian coordinates. This system is shown in Fig. 1. The basic axes are 0(x,y,z). The angle xOy has magnitude a. The axis Oz is at right angles to the plane xOy, and is such that a rotation virhich brings Ox into the position Oy is right-handed about Oz. Use is also made of auxiliary axes 0(X,Y) lying in the plane xOy and such that Of'Xsyja) and 0(x,Yz) form systems of right-handed rectangular cartesian axes.

It is convenient to introduce unit vectors

l,D,k,i.^,j. lying in the directions 0x,0y,0z,0X,0Y respectively. These quantities satisfy, as is easily shown, the following

relations.-i. = i cosec a - j cot a '}

^

V)

D-1 = - i cot a, + d cosec a J 2 2 2 i = j = k = 1 , i. Ó = cos a, j.k = k. i = 0 (2) ixi = jxj = kxk = 0 7 / (3) ixj = k sin a, jxk = i^, kxi = ^^ ^

The position vector r of a point with coordinates (x,y,z) maybe

written.-r = xi 4- yj + zk (4) If the length of the differential vector dr be denoted by ds,

we find from (4) and

(2).-2 —(2).-2 (2).-2 (2).-2 (2).-2 (2).-2

ds = dr = (dx i + dyj + dz k) = dx + dy + dz + 2dxdy cos a

The vector dr is a unit vector. For the special ^^J ds

case in which this vector lies in the plane Oxy (i.e. when dz/ds = 0) and is inclined at an angle 6 to the axis Ox, vi^e find for the components dx, d^; the

formulae.-ds formulae.-ds

dx = sin('a-6) , djl = sin 9 /g\ ds sin a ds sin a ^ '

The relations (6) may be established using (2) and the formulae i.dr = cos 6 and D-dr = cos(a-9), or by a simple trigonometrical

ds ds calculation.

1.2. Kinematics

Any vector may be expressed, as ir. (4) , as_a linear combination of i,j,k. _The displacement of a point u and the rotation about an axis p may be

V .7

-7-u = -7-ui + v;j + wk p = pi + qj + rk

(7) The combinations (u,v,vi') and (p,q,r) may be termed the

'components' of the vectors in the axes 0(x,y,z), but care

must be exercised to avoid applying formulae applicable only to rectangular axes to these quantities. The lengths of vectors are given by formulae like (5). The component u is not the projection of u in the direction Ox', this last is given by u + V cos a. _ If the axis of p passes through 0, then_the displacement u induced at a point with position vector r is given by.-_ _ _

u = p X r (8) (7) into (8) and making use of (3), (1)

Substituting from (4) we find.

-U = u + V cos a = (qz - ry) sin a V = u cos a + V = (rx - pz) sin a viT = (py - qx) sin a

_J

(9) where U, V are the 'projections' of u in the directions Ox, Oy respectively.

In the remaining portions of this paragraph v/e shall restrict our attention to positions and displacements in the plane xOy. Use Virill be made of our previous notation, with the understanding that z components, such as w, z etc., are taken equal to zero,

_ If the plane xOy is subjected to a displacement u(x,y), a point at r will move to r + u. The length of an element dr will change to dS v/here,

2 -2

ds = dr , dS^ = (dr + du)^ (10) Neglecting terms of second order in the displacement we find for the strain e in the element dr the

formulae.-e = 2 2 dS - ds 2ds^ dr du • — * — ^ ds ds (11)

S u b s t i t u t i n g from (4) , (7) ( w i t h z = w = O) and u s i n g (2) vYe

f i n d . - o n ?/here a n d e = e ^xx = U =

-(ft) ^Mi) 'Mil

9U: _av 3x yy d7 u + V cos a, = 91 au ®xy 3x "^ 9y V = u cos a + V (12) /

The q u a n t i t i e s e , e and e may be termed 'components of XX Ho x y

s t r a i n ' , s i n c e the complete d e f o r m a t i o n i s d e f i n e d i n t e r m s of them. The formulae i n t h e second l i n e of (12) a r e f a m i l i a r , b u t I t must be n o t i c e d t h a t U, V a r e not t h e t r u e d i s p l a c e m e n t

components.

The d i r e c t s t r a i n e^yj i n t h e d i r e c t i o n OY may be * o b t a i n e d from ( 1 2 ) , by making use of (6) w i t h 9 = 1^/2,

We f i n d . - g 2

= e.... c o t a + e,„. cosec a - e „ „ cot a cosec a

'YY X X _{yy xy} (13)

-8-The rotation / ^ of an element dr is given (see Pig.2) by the

formula.-iB^ = Ü sin(a-9) - tl 3in 9 (14)

Using (14), (12) and (6) we can show that the shear strain e Y associated with the directions Ox, OY is given

by.-V = ^^^^9=0- ^^h=%/2 = - 2^\x=°t ^ + V °°sec a (I5) When the strain components satisfy a compatibility

relation,-i!!M ^!l!xxl ^'^V^ (16)

- 2~ + ^T"

ax9y 9y ax

the second line of (12) may be solved for the displacements U, v. The 'complementory function' for this integration is a

'rigid body motion'.-^

U = Cy + C^ , V = - Cx + C2 (17) where C, C/, C2 are arbitary constants. The results

(16), (17) are identical with those for rectangular coordinates and the usual proofs apply.

1. 3, Statics

A force F may be v/ritten,

P = Xi + Yj + Zk (18) *

If this force acts at the point r, its moment about the origin 0 is r X F. Using ( 4) (18) (3) and (1) we find,

r X P = L^i^ + M.^ j^ + Nk = Li + Mj + Nk J

L^ = yZ - zY M^ = zX - xZ N = (xY-yX) sin a [ (19) L = L. cosec a - M. cot a M = -L^ cot a + M. cosec a The conditions, for equilibrium of a system of forces are

-j^p = o, ^vx'F = o. Reference to (I8) ,. (19) shows that these may be

written.-g . x = £ Y = ' è z = o •^

/ (20) £(yZ - zY) -^(zX - xZ) = g(xY - yX) = 0 J

These equations have the same form as for rectangular axes, Turning now to tvi/o-dimensional .questions, we define the stress resultants T. , S. , T2 and Sg for a plate.

These are the oblique components of forces per unit length, acting across normal sections paralleT to axes Ox and Oy, situated in the middle surface of the plate. The sign

convention for these forces is shown in Fig. 3^ Consider an element of the plate (dx, dy). The forces acting upon it are shown in Pig. 3. The forces on the edges are determined by the stress resultants', the body force is given by (Xi + Yj)dxdy. Application of the rules of (2o) gives us the following

/differential ..,

* A translation u = cosec'^a ^(c. - Cg cos a)i + (C2- c.cosa)j5 and a rotation about 0, p = - Ck cosec a'. (srfc(9)).

where and

-9-differential equations of

equilibrium.-7

9T^ asg 9x~ "^ ay" ^ ^ 9S^ 9^2 '1 S (say) (21)

The similarity with the equations in rectangular coordinates will be noticed. If X = Y = 0 we can satisfy (21) by

introducing a stress function(2^ such that, T^ = a_| ,

ay 9x2

s = - dj.

axay

(22) It_is convenient also to introduce stress

resultants T,, Tgj S refered to axes 0(x,Y). The specification of these is shown in Fig. 4. The relations between the

un-barred and barred stress resultants may easily be shown to be,

-T

1.4,

1 T, sin 0. + Tg cos a cot a - 2S cos a Tg = ^2 cosec a

S = S - Tg cot a

Stress - Strain Relations

(23) f

J

In § 1.2 we studied a system of plane strain refered to oblique axes 0(x,y). We now interpret these results as refering to the mean strain across the thickness of a uniform plate. Such a state of strain in a plate will give rise to stresses and stress resultants and in § 1.3 we studied the properties of these forces when refered to our oblique axes. the material of our plate is elastic and obeys the Generalised Hooke's Law, then the stress resultants T. , Tr, and S will be

I f

r e l a t e d to the s t r a i n components

l i n e a r equations of the form.-

e s XX "yy ^^^

e by homogenous xy

^1

'-

^11 ^xx ^ ^ 2 V

"

*

- ^3 V

^2 = ^2^ ^xx ^ ^22 "yy + ^23 "xy •> (24) S = a-,, e + a-,o e + a_, e

31 XX 32 YY 33 xy

where as we s h a l l show l a t e r ,

= a ^ 3 31 ^ (25) For the special case in which the plate is isotropic with thickness t. Young's Modulus E and Poisson's Ratio o, known theory applied to the rectangular axes 0(x,Y)

gives.-Et / \ zr Et / ("\x ^ ° ";YY) ' ^2 = ^(^,YY ^ ^ "xx' | (1-0^) (1-0^) V(26) ^1 =

-)1

Et 2(1+ö)

'xY

/Substitution ....

Substitution from (26) in (23) expresses T, , Tgs S in terms

of e^^, e^YJ ^xY* ^^® °-^ ("'3)' (''5) throv'o our relations into the form (24) and so determines the a.. for the isotropic plate, Denoting these results by (a. .) Y/e

fiAd.-^ fiAd.-^ ijfiAd.-^p

2 2 1, cos a+osin a,- cos a Et . 3 / 2 . 2

^^,„ — 9 — cosec-^a.j cos a + o s m a, 1 - cos a

^ P (1-0^) , 2 . 2

^ ' ' - c o s a , - c o s a , 1+cos g-osin a (27)

m the case where the plate is reinforced by closely spaced stringers of section area A at a pitch a running parallel to Ox, and by closely spaced ribs of co§tion area A^ at a pitch a^^ running parallel to Oy^E, then, if the material of the reinforcements has modulus E, loads of magnitudes respectively EA_ e and EA,., e will appear in the stringers and ribs,

s XX K yy

Distributing the stringers and ribs continuously we generate stress resultants T. = EA e^^/a^ and T^ = EA_ e^^ 7a^ and

I s XX s c- K yy K

so for a reinforced plate we must add to (27) the matrix (a. .)r.

given by. - • "^ / EA /a o o \

/ s'^ s

\ o o o /

(a..)^ = ! o E ^ a j , o J (28)

The complete matrix for a plate reinforced in the directions Ox, Oy is

thus.-a. . = (a. .) + (a. .)r, (29)

The equations (24) may be solved for e , e , e yielding.-XX yy xy

^xx = ^ 1 ^ 1 ^ -^2^2 '- ^ 3 ^

"yy = ^^21^1 + A22T2 + A23S f (30)

^xy

-

^^31^1

-"

^'32^2

^

^33^

J

^ ^^^' ^ /^22^33'^23' '^23^31"^21^33' ^21^32"^31''^22 N

^ij " j~¥7T[ ^13^.^2-^12^33'''ll''33"''?3' ^12^3l~''l l''32 j (31) l^a^2^-^23-^22^i3'a^^a2^-a^^a23, a^^a22-a^2 /

1,5. Compatibility Relation for the Stress Resultants

The strain components must satisfy (I6). It follows from (30) that the stress resultants must

satisfy,-2 satisfy,-2 satisfy,-2 \ A^^ d_^ + ^21 ^ - ^\31 -^ 1^1

ay ax axay/

A^2 i ! ^ + A22 9!^ - A32 i i - V s

ay ax axay/

' (

4

A-,3 af_ + A23 a l , - A33 a f _ \ s = o (32)

: 1^! Q^^ ' g x a y / y^^ ^^^ ^^^^

a^ and a„ are measured parallel to Oy and Ox respectively, S K

-11-In the case where a stress function 0 exists we can substitute from (22) into /32)

obtaining.-A22 L^g - 2A23 ^^^ + (2^H2 + A33)aV - 2A^3aV +A^-|9V = 0

9x^ ax^ay ax ay axay^ gy (33)

1.6. Application to Certain Plate Problems

The theories of displacement, strain and stress devel-oped in the previous sections are particularly applicable to plates whose boundaries consist of parallelograms. Let us therefore turn our attention to a plate whose edges lie along the. lines x = o, x = l, y = +c. (Pig. 5)

W e shall not seek here to solve problems with given boundary conditions, but following the 'inverse' method of St. Venant, shall impose certain restrictions on the stress distribution and examine the consequences. However, with an eye on applications to wingo, we shall restrict our discussion to solutions which

satisfy.-T2 = 0 viThen y = + c • (34)

Let us begin with the simplest of all cases in which the stress resultants are constant.^ Equation (34) then implies

that T2 = 0 everywhere. The edges x = o,l are loaded by uniform T. and S, while the edges y = ± c are loaded by

a \miform S. Writing T2 = o in (30) we find the following formulae for the constant strain

components.-"xx = ^1^1 + ^M3^

"yy = ^^21^1 + ^^23^ \ (^5)

"xy " ^31'^1 ^ ^33^

The displacements follov/ from (12). The complementory function for this integration is given by (17). We thus

find.-" = ^xx-^ ^ (find.-"xy ^ C)y + C^

(36) V = e^^^ . y - Cx + C,

yy '

As a second example l e t u s c o n s i d e r a n o t h e r case i n which X = Y = 0 and assume t h a t t h e s t r e s s r e s u l t a n t s v a r y

l i n e a r l y w i t h x. We w r i t e

T^ = X T:J + T'J] , T2 = X T^ + T^, S = X S' + S" (37)

where T', T"', TL, T'^, S' and S" are functions of y.

Substituting in f21) with X = Y = 0 and using (34) we easily show that,

T' = - ^ " T^ = T'^ = S' = 0 (38) /Substituting from (37) •••

1 2 -# ^ S u b s t i t u t i n g from (37) and u s i n g (38) we f i n d t h a t . ^ 1 ^ 2

^1 = - ™'; ^'2^ + h^ + ^4

S" = ^ Apy + A , y + A 2 " ^ ' ^ - - 0

(39)

(41) v/here A. ( i = 0 , 1 , 2 , 3 , 4 ) a r e a r b i t r a r y c o n s t a n t s . S u b s t i t u t i n g from ( 3 8 ) , (39) i n t o (37) we o b t a i n , T^ = - A2(xy + \ ^ y ) - A^x + A y + A^ — ^11 Tg = 0 I (4o) 2 S = -^Agy + A^y + AQ S u b s t i t u t i n g i n f30) and u s i n g (12) v/e f i n d t h e f o l l o w i n g e x p r e s s i o n s f o r (U,V).

^ " ^o('S3^ ^ ^'33^^ + '^4(^11^ + 'bl*^^

-^ A^\- lA^^x + A^3xy + i(A2^ + A33)y •> + A^iA^^xy + ^ A^^y ) ^

+ Ag j - iA^^xVMi3xy^+(i/^)(M2i- r ^ + M33)y^/+(Cy+c^)

2 2 V = A^.A^-^y + A^.Ag^y + A^(-A^3X - A2^xy + ^ A23y )

2 2 + iA3(-A^^x + A2^y )

+ A^iV^k^^x^-^A^^xy^ <V%¥-2r A

^^)Y^>+{-CX+C^)

(^2)

As a third and last example, let us consider a case where the stress decrease exponentially from the root x = o

(i.e. vary as e"^-^, where the real part of \i is positive,') For the sake of possible applications to the box structures

of Part II, we introduce a body

force.-X = 0, Y = - B^e"^^ (43)

where B i s a c o n s t a n t , v/hich may b e a complex number,.

A p a r t i c u l a r s o l u t i o n of e q u a t i o n s (21) and (32) i s e a s i l y shown t o b e

The d i s p l a c e m e n t s c o r r e s p o n d i n g t o (44) follov/ from (30) and (12) We f i n d .

-U = - ^ 3 ^o e - ^ ^ , V = ^11 . ^o e - ^ ^ (45)

^22|^j| i^ ^22|^j| 7

- 1 3 "

where use h a s b e e n made of t h e a l g e b r a i c t h e o r e m t h a t t h e c o f a c t o r s of (A. .! a r e g i v e n by a. . / l a . .t . To o b t a i n a complementary f u n c t i o n YW make use of (33)• Assuming t h a t 0 v a r i e s a s e^ ^•'^"^"•^•', we f i n d t h a t

A^^ X^^ + 2A^3 }s^ + (2A^2 + ^33) K^ + 2A3y\ + A22 Denoting t h e r o p t s of (46) by /\ . (i=1 , 2 , 3 , 4 ) we f i n d a s o l u t i o n of (33) i n the f o r m . - "^

= 0

(^6)

0 = e'

-|i,X

4,

> ^ ' .1^

Xj-y

_(^7)

i=1 w h e r e B . a r e a r b i t r a r y c o n s t a n t s ( c o m p l e x n u m b e r s ) , T h e s t r e s s r e s u l t a n t s f o l l o w f r o m ( 2 2 ) .

-T. = nV^^^B.Aje^ V , -T. = ^2^-'^^yB.e^>iy,

I f-— 1 1 £L I— 1 o 2 -\xx S = M- e

^BiXie^^'l'^

(itS)

T h e c o r r e s p o n d i n g d e f l e c t i o n s a r e f o u n d to b e .

-.^^ky

U = - n e - ^ ' ^ B . e ^ ' ^ i y ( K ^ A ^ ^ + /\.A^3+ A ^ g ) + C y + C^ V = [le-^'^y^ e^^'iy(/\^ A 2 ^ + ^^2-5^ ^ 2 2 ^ - C x + C 2 ""1 i— Ai

(^9)

J

Imposing the condition (34) upon our complete solution we

find.-la-^A 22

vi^hich g i v e s two e q u a t i o n s f o r t h e c o n s t a n t s B . . The i m p o s i t i o n of f u r t h e r b o u n d a r y c o n d i t i o n s a t y = + c would e n a b l e t h e s o l u t i o n t o be completed. T h i s development i s r e s e r v e d u n t i l the t h e o r y of P a r t I I i s f o r m u l a t e d .

-14-Part II Applications to Simple Swept Back Box Structures 2.1, Description of a Simplified Structure. Notation

In part II we shall apply the results developed in Part I to the study of stress distribution and deflection

problems for a uniform swept box. Such a simplified structure, Y/hile not reproducing all the characteristics of an actual

wing structure, will reveal those properties peculiar to sweep back.

The structure to be considered is a uniform rectangular section box swept back through an angle 7i;/2 - a', (see Pig, 6) Reference axes 0(x,y,z), of the kind defined in § 1.1, are so disposed that the faces of the box are given by y = +c, z = + b and the ends by x = o, x = 1. The faces z = + b are termed

'skins'. They have thickness t and are reinforced by x-wise closely spaced stringers of section area A and y-v/ise pitch a , and by y-vi'ise closely spaced rib booms of section area A^ and x-wise pitch ap. The faces y = + c are termed 'spar ;veüs'. They have thickness t and are as'sumed to carry only shear

stresses. Such direct load carrying capacity as they may possess will be assumed integrated v/ith the 'spar flanges', which run

along the four edges of the box and have a cross sectional area A, The corresponding rib booms on the skins z = + b are joined by 'rib vi^ebs' thickness tp, vi^hich are assumed to carry only shear stresses. These rib webs are of course rigidly attached to the spar webs. The materials of all the components are assumed to have Yoxing' s Modulus E and Poisson's Ratio o.

2. 2, Theory of the Simpljfled Structure

We shall limit ourselves in what follows to cases in which the displacements occuring in the skins z = + b are equal and opposite to one another. The notation applied to plates in Part I will here be applied to the 'skin' z = b,

Corresponding values of displacements and stresses for z = -b, can then be obtained by reversal of sign,

Let us begin by considering the rib webs. These are to be treated as continuously distributed int he x direction, The 'thickness' of ribs within an element dx will thus be

'Yn^ix where,

7R = tpAR (50

The shear per unit length carried by the rib web, within dx will be written Spdx ?/here S is a function of x only. The y and z components of displacement in the plane of the rib webs will be denoted by v.^ and Y/^ respectively. These definitions

are illustrated in Pig, 7, The relation between Sp andt he displacements is

clearly,-E'/-D / 9V^ 9w \ R _{2(1+o) V az ay}

The kinematics of a 'pure shear carrying plate' are not well defined. We shall therefore in the interests of simplicity, assume that Wp is independent of z, thus attributing a limited rigidity to the ribs. Experience v/ith the theory of unsvfept

-15-boxes suggests that this restriction is not of any real significance. Differentiation of (52) with respect to z

then shows that Vp is a linear function of z. and so, remembering that i.rib displacement s must conform with those in the skins at /

z = + b, we find.- / v _R Vz/b

Equations (52) and (53) then

yield.-2 ( i + o ) s ^ 1 y w. _R

FT.

:R y - ÏÏ J ^

'^y

-^

^Ro

(53) (5^) R where w _{Ro ^ R'^y=o i s a f u n c t i o n of x.}

We t u r n now to t he spar webs c o n s i d e r i n g f i r s t of

a l l t h e siirface y = c. The x and z wise d i s p l a c e m e n t components i n t h e p l a n e of t h e s p a r w i l l be w r i t t e n u^ and w r e s p e c t i v e l y . Conformity w i t h t h e r i b d i s p l a c e m e n t s i m p l i e s . - w

w = (w_,)

w ^ R'y=c ₍₅₅₎

The component w i s t h u s i n d e p e n d e n t of z and s o , j u s t a s i n t h e case of t h e ^ r i b webs, we deduce t h a t u . i s l i n e a r i n z and t h u s i s g i v e n b y .

-u _w ( U ) _ . . z

y=c

(56)

s i n c e spar web d i s p l a c e m e n t s must a g r e e w i t h those i n t h e s k i n s a t z = + b . The s h e a r p e r u n i t l e n g t h i n t h e s p a r web w i l l be w r i t t e n S and i s r e l a t e d t o u , and w b y , -W W W \ ~ 2(" E t „ / a u aw \ w I w w] [1+a)\ az 9x / (57)

The n o t a t i o n f o r the s p a r web i s i l l u s t r a t e d i n P i g , 8. S i s a f u n c t i o n of x only and i t s v a r i a t i o n i s b r o u g h t about by t h e

shear Sp a p p l i e d by t h e r i b s .

y i e l d s t h e e q u a t i o n . - E q u i l i b r i u m of an element dz dx

dx

0 ₍₅₈₎

Substituting from (54) into (55) and from (55), (56) into (57) into (58) v/e

find,-d^s R <R ^R ETC + R dx ct 2c(1+a)

b

\9x

L

"B tdx + n y=c o' dx = 0

We shall denote corresponding quantities for the surface y ; by the same symbols as for y = c, but v/ith a dash added

(i.e. u', w' and S'). The equations corresponding to (55) (59) ar'^.- ^ (59) • c, w' W ' = (W-n) R^y=-c (60) u' = (U) w ^ y=-c (61) /s'

H

1 6 -S ' = w

Et / au' aw' \

W ƒ w w ] TTöTv az ax / E t 2 ( 1 + 0 ) ( 6 2 ) d S ' w d x + S p = O

(63)

- d S j ^ T d x c t

ET,

^ + — S R + R w 2 c ( 1 + a )

1 ,au

b^ax

1

f dh

d w. .2 y = - c i — ? ? d y + — 2 d x Ro

b J dx

o = O ( 6 4 ) T r a n s f o r m i n g ( 5 9 ) a n d ( 6 4 ) we o b t a i n t h e f o l l o w i n g e q u a t i o n s f o r Wp a n d Sp i n t e r m s o f t h e d i s p l a c e m e n t s i n t h e s k i n z = b .

-w.

RO = i^,(J ^^y + "fvdy)- ^ jf(U)^^^+(U)y_^5jdx + K^x . K2 (65)

0 0 0 ^ \ ^R , TT- S ETj, d x c t R w 4 b c ( 1 + o ) = 3^7

_r^au

9 x - c y = c

y=-cj

(66)

where K. , K2 are arbitrary constants.

The equations governing the behaviour of the skin z have already been developed in Part I. The external force

(X,Y) arises in this case from shear flows Sp applied by the ribs. We have in

fact.-= b

X = 0, Y = - S.

R

(67)

The boundary conditions at the edges y = + c can be obtained by considering the equilibrium of elements dx of the spar flanges. The balance of y - components

gives.-('^2)y=-.c = 0 y=+i (68) The x w i s e b a l a n c e of f o r c e s i s s h o w n i n P i g , 9 . We t h u s f i n d . -S + ( -S ) = E A / a e \ax - T ? i A / 9 e y = c S ' - (S)^^ = E w ^ y = - c

//i!xx\

Vax /

r69)

( 7 0 ) y = - c F o r m u l a e f o r S , S ' i n t e r m s o f U , V , S „ a n d W-, w e r e o b t a i n e d Vv' w R Ro i m p l i c i t l y d u r i n g t h e d e r i v a t i o n o f ( 5 9 ) ( 6 4 ) . T h e s e may b e e x p r e s s e d a s S + S ' = E t K, W W w 1

2l^^o)

^ ( 7 1 ) / S - S ' = ^ w w

1 7 -E t S - S ' = w w w

2TiTST

4 ( 1 + o ) c dS^ 1 - c 9V ax ö-y w h e r e u s e h a s b e e n made of ( 6 5 ) . Our b o u n d a r y c o n d i t i o n s (69) a n d ( 7 0 ) c a n t h e n b e w r i t t e n .

-(721

(S) - (S) = EA ^ ^y=c ^ ' y = - c E t K, w 1

d Re ) + (e ) \ "^LJL

^l^ x x ^ y = c x x V - c ) 2 ( 1 + 0 ) ^^^^ (S) + (S) = EA d ( ( e ) - ( e ) '\ "• •'y=c ^ ' y=-c 'Z~} x x ^ y = c ^ xx''3?-=-c..) dx ^ • - E t w

2TTT^L

(U) - ( U ) 4 ( 1 + o ) c dS„ 1 b + EYp dx uop I p 9V ; ^ - - b J 97 ^y _{b J ax} - c (74) The m a t h e m a t i c a l p r o b l e m p r e s e n t e d b y o u r swept b o x i s t h u s r e d u c e d t o a p l a t e p r o b l e m of t h e t y p e s t u d i e d i n P a r t I w h e r e t h e ' b o d y f o r c e ' Y = - Sp i s g i v e n , b y e q u a t i o n (66) a n d t h e b o u n d a r y c o n d i t i o n s a t t h e e d g e s y = + c a r e g i v e n b y ( 6 8 ) (73) a n d ( 7 4 ) . F i n a l l y l e t u s w r i t e f o r m u l a e f o r t h e s t a t i c r e s u l t a n t of t h e f o r c e s a c t i n g a c r o s s a s e c t i o n w i t h c o o r d i n a t e X, T h e s e r e d u c e t o a f o r c e Z . k a t t h e c e n t r e of t h e s e c t i o n ( x , o , o ) and a c o u p l e L . . i . A M , . j , w h e r e . - 1 1 1 1 a n d , ^ ^ ^ w ^ l @ = 2 b ( s ^ ^ + s ; ) = . - ^ ^ = 2bc.(S^ - S ; ) - 2b J S dy c M^ \ = 2bEA

_{(-xx)

y=c + (e ) i + 2b i T, dy y _{^ x x ' y = - c \ J 1 " -^} y=-c^ c

J

- c (75)

(76)

It is to be remarked tha,t v/e h"ve found it convenient to use the oblique axes OX, OY for defining the couple. If it is desired to write the couple Li + Mj using the axes Ox, Oy, then the necessary transformation is given in (19),

2.3. Simple Loading Conditions.- (1) Constant Couple. We now apply the results of the f irst example in

plate theory of § Ï.6 to a problem of swe-ot boxes. The constant stresses T. and S of this example will be assumed to be acting in the skin z = b. The corresponding strains and deflections are given in equations (35) and (36). The body force Y = - Sp

is zero in this case. Substituting Sp = 0 and the values of U, V given in (36) into (66) , v/e find This equation identically

satisfied.- Since S and e are constant equation (73) shows

that K.^ = 0 and so by (75)-^?hat Z = 0. Equation (74) shows that;

" = - 2 ^ y b ( 1 + o ) E t c w (77) E q u a t i o n s (69) a n d (7©) shov/b t h a t / Sw =

1 8

-S = - -S; -S' = -S (78) Assuming f o r s i m p l i c i t y t h a t U = V = O when x = y = o

and t h a t w„ = 0 , when x = o we f i n d from (65) t h a t , KG ^ ° 2b U s i n g ( 5 3 ) J ( 5 4 ) , ( 5 5 ) 5 ( 5 6 ) , (60) a n d ( 6 I ) we f i n d , . Vp = (^Vy'-^ " ^^^^^

-R - # • - ' - b--y- # - y ' J (0°)

\ ) % { = [e^r^± (^xy ^ C ) c | z A (81) w 1 „ 2 w | e o Cc e c / Q O \ > -2^^ x^ + — X - yy (^2) 2b i b ^ 2b v/ w NJ The m a g n i t u d e s of t h e s t r e s s r e s u l t a n t s T, a n d S f o l l o v / s from ( 7 6 ) , We f i n d . -T , = M^ + EAA^, , 2c ^ 4 b c ( 1 + ^ ^ ^ 1 ) S = -(83)

The formulae developed in this paragraph together with (35)? (36) solve the stress distribution and deflection problems for the case vfhere our simplified swept box is loaded by constant couples,

2,4. Simple Loading Conditions.- (2) Bending by a z-wire force. We noY/ apply the results of our second example of

§ 1,6 to our sT/ept box. The stress resultants for the face z = b are assumed given by equation (40). The deflections for this face are then given by (41) and (42). Since Y = 0 for this solution we have Sp = 0 as in § 2.3. Substituting from (41) and (42) into (667 we find that Sp = 0

implies.-Ap = 0, A, = - 3 ^13 A. (84) 3 2 A

^^11

-19-Substituting from (30), (4o) into (73) and recalling (75) we find, ^1 =

-4bc [1 + ^fili'

(85) S u b s t i t u t i n g f r o m ( 3 0 ) , ( 4 o ) , ( 4 1 ) , (42) i n t o ( 7 4 ) a n d r e c a l l i n g (84) we f i n d .

-c = .f^(±^ + ^ ) A -hi A

( E t c 2 / 0 2 w E q i i a t i o n s (69) (70) a n d (75) g i v e . -S -i w / V Z _ A^

S.J = 5 b - °

W -^ 4 (86) (87)

If we assume that our force Z is located along the line

X = 1 J, y = o i.e. applied oentrally at the top rib, we find by (19) that

.-1 j L^ = 0, M^ = - Z(l-x)

(88) Substituting in (76) and using (88), (85) we find that

.-A^

=

0, A^

=

A^i

(89)

S u b s t i t u t i n g f r o m ( 8 4 ) , ( 8 5 ) , ( 8 6 ) , (89) i n (40) , ( 4 l ) , (42) a n d ( 8 7 ) \Ye f i n d . -3 A . , Z ( l - x - 2 - 1 ^ y ) A '^i = - 11 M. - —i^ s 4bc ll + EAA^^ ] 4bG>1+EAA 11 2 A 11

_l.

S = -

IZ.

(90) 4b c

|1+EAA^J

; U = -4bcTT+ËAA^^ 'I

(

~c

)

A ^ ^ l x + i A 3 ^ 1 y - i A^^x '^A^^xy

+KA21 -f A - 1 hi)

11 V = -4 b c i 1 + E A A , . •)

\l

^A3^1x + A2^1y - 5A^3X - A2^

S = S ' = Z/4b w w ' + i ( A 2 3 - i A2.^ A^3 ) y ^11 / W h e r e i n ( 9 1 ) ( 9 1 ) ( 9 2 )

-20-Where in (91) we have assumed u = V = 0 when x = y = o. The equations (9o) show that the conditions at the tip x = 1 are not exactly those corresponding to 'freedom', even from direct stress. For our solution to be valid^equal and

opposing couples must be applied to the faces z = + b by loads normal to the rib x = 1, not to mention linearly varying shear loads applied parallel to this rib. However, the effects of this self-equiiibrating system will die away as one proceeds along the span and so our solution me.y be considered practically valid at (say) a distance 2c from the end,

Substituting in (65) we find^assuming Wp_^ = 0 v/hen

X = o, Z VI

^° 4bc|i+ ^ ^ i .

c 3L

^ x ^ l 4 x ) 4

2b ^ / Et ,4c(1+a)(1+ EAA^^) + c^(2A2^ + A - 3 A^h I X \ (93) 2 ^ ^ 1 The remaining deflections can be written down using (53)? (5^)>

(55)5 (56)» (60) and (61), but since the formulae are lengthy we shall not give them here.

2,s. Analysis of the Deflections for the Simple Loading Conditions

The deflections at any plane section (coordinate x) of our box may be analysed into the sum of a translation, a rotation, a vi^arping from the plane and a distortion in the plane of section. Let us consider a translation Wk and a

rotation pi + qj, where 'f/jp and q are functions of x. These will produce displacements at our section given

by,-U = qz sin a, V = - pz sin a, w = W + py sin a (94) Where use has been made of (9) and the rotation has been

located at (x,o,o). For this one equation 0,V have a 'general' "significance as in (9) and are not confined to z = b.'

Comparison of the first of (94) with (56) and (6I) suggests the identification

, . '"V=0 - (")^^ ^ ,^ (,,)

2b s i n a Comparison of t h e second of (94) w i t h (53) s u g g e s t s . -P (Terms of V independent of y) /ngs b sin a

Comparison of the third of (94) with (54)

gives.-W = gives.-WpQ (97) and (96) again. The term in (54) containing Sp does not

-21-does not occur in (53) and gives a shear strain not a rotation. We shall adopt the definitions (95), (96) and (97) for p,q and W, Other definitions are possible, but the differences are bound up with questions of 'shear deflection' and 'root conditions', with which we are not particularly concerned here.

Let us now apply our formulae to the case of loading by a couple analysed in § 2.3. Substituting from (36) with

C J = C 2 = 0 a n d (79) v/e f i n d * gvX^ \^JIUA,-o^:"Wri^-rt-tn^'jA.i^-i'. ^ Cx cosec g P = >: > JUL X cosec a W = -2b XX 2 X (98) Substituting from (77), (35) and finally (83) we find the following relations.-dx

- dSv

dx" ^11^1 + ^^^12=^1 cosec a da d x C p ^ L ^ + C22''*''-i (59) where, p _ cosec g ] (I+0) ^11 ~ 8bc ) Et c "^ 1^ w ^33 EAA. - ^ - 13 2T3 2bc(1+ EAA^.^) ) ^12 - ^21 A,,cosec g 8 b ^ c n . - ^ ^ - ^ i ) A., cosec g '22= " b ^ c d , ^ A A ^ ~ (100)

J

The relations (99) generalise the usual curvature - bending moment and twist-torque relations valid for an unsv/ept box

(beam),

The remaining terms in the deflection formulae can be analysed into firstly a 'linear

warping'.-U = (e^y + C)y, V - 0 u "s W/ 1= i (e^y . C)cz/b ^ 7 ) i^L] ( 1 0 1 )

and a 'cross sectional distortion'.-U = 0, V = ^R = ^ y y - y ^ / b ' " y y - ^ Wj^ = - e y y

.yV2b

i^AH

(102)

The warping, which c o n s i s t s of spanYifise displacement, depends upon b o t h L^ and M.. The c r o s s - s e c t i o n a l d i s t o r t i o n c o n s i s t s of an ' a n t i e l a s t i c ' bending of the r i b s .

We turn now to the a n a l y s i s of the d e f l e c t i o n s for the case of z-wise loading at the t i p , d e a l t v/ith in § 2.4.

-22-Substituting from (91) into (95), (96) recalling (93), (97) and (100) we find,

^ = - C,pZ(l-x), - ^ cosec g = ^ = - CppZ(l-x) (103) dx ^^ dx^ dx

Recalling (88) we see that the relations (99) are valid for this case as well,' The remaining displacement terras can be analysed into firstly a 'linear

warping',-A,.My U = •^' VAA ^ V = o (104)

8bc(1+f^Vl_)

c s e c o n d l y , a ' p a r a b o l i c w a r p i n g ' . -2 U = - — (A2^ ^ A l ^ ) ( c 2 . y2) 8 b c ( 1 + EAA^^) ^ ' ^^ 2 A^^ (105) V = o

and finally a cross sectional

distortion,-2(^23-1 t2±nl

Ag^M

y (^ 2 A^^ J 2

,

r.

U = o , V = ^ ' ' ~ y ( 1 0 6 ) 4 b c ( 1 + EAA^^) 8 b c ( 1 + E/JV^ ^) (101 ^ v/hen e x p r e s s e d i n t e r m s of M. ( w i t h L, = o) ( 1 0 4 ) . S i m i l a r l y (102) a g r e e s v a t h t h e f i r s t The formula agrees vYith

term of (I06)", The warping of'(I05) i? analogous to that occuring in unswept boxes and will give rise to a theory of 'shear lag', just as the linear vmrping will give rise to a theory of 'end constraint' similar to that arising in the case of the torsion of unsYifept boxes, ]

2.6. Internal Systems of Stress

The third example of § 1.6 may be used to construct systems of stress for which the static resultant on a cross section is zero. We take as displacements in the surface

z = b the sum of the expressions given in equations (-4-5) and (49), Yifhere the constants B., v/hich occur in these, are

limited by the relations (5O7. Equations (43) and (67) show that

^R

^

V'^"" ^^°^^

Our assrmied solution must satisfy (66), (73) (Y/ith K^ = o by (75) and (74). Making the necessary substitutions, Y/e find,

incidentally, that the constant C of (49) is zero. The three remaining equations together with (5O) form a homogeneous

set of linear equations in the five constants B.(j=o,1,2,3,4). These equations may be written • ''

^ B..B. 0 (108)

1

2 3

-where t h e e q u a t i o n s f o r i = 0 , 1 , a r e o b t a i n e d from (50) by a d d i t i o n and s u b t r a c t i o n , t h e e q u a t i o n f o r i = 2 i s from

( 6 6 ) , t h a t f o r i = 3 from (73) and t h a t f o r i = 4 from ( 7 4 ) . The c o n s t a n t s B . . a r e g i v e n by . -10 ^00 " ° ' ^oj " ^^^^ ' " ^ " ^ j ^ ('1 = ^ ' 2 , ^ , 4 . ) A B _{I 0} 23 |i.-^A ₂₂ ( 1 0 9 ) B^ ^ = c o s h a / t ^ ( j = 1 , 2 , ^ , ^ ) (110-) B B, a ^ ^ c 2 b c ( i + o ) / - r ; 2 2 j = M' do 2/^22 ^ A

_{22hijl ^R ^-^w}

V e t ( 1 1 1 )

jci ^ ~f^ - -^13^^ - A^^X2j)sinh uX ^c ( j = 1 , 2 , 3 , 4 ) )

J J / a B ₃₀

"3J

B _'40 '22

M

WM-^ \ . s i n h ,a.A.c + :i (XjA-,WM-^ + AjAWM-^3+ A.WM-^2)WM-^°sh M-XJWM-^ E A u t c 1 Et c a . , W _ Y/ 1 1 T R ' \^ 2 ( 1 + ö ) b A 2 2 | a ^ J | i ( j = 1 , 2 , 3 , 4 ) / ( 1 1 2 ) B 4 j

V

c o s h M. + E t u w 2 b ( 1 + a )

Aj 23

A. V 2^'2 2 ^ j

K^^3)

2,a A b ( 1 + o - ^\ 2 , t w

(k-A^1 + \ 5 A i 3 + A^2) s i n h i-iX -c / J

7

E q u a t i o n s (I08) a r e s a t i s f i e d by n o n z e r o B. i f .

-J

^ d

= 0 (114)

Equation (1l4) is a transcendental equation for \x. It is very complex as inspection of (109) - (113) shows. The mathematical examination of its roots is therefore out of the question, but physical intuition, base^ upon experience Y/ith unsv/ept boxes, suggests the existence bf an infinite sequence

of roots vi/ith positive real parts, which may be

Yvritten.-) (115)

-24-> ^

They can of course be calculated n\imerically in a special case. The solution of the first four, equations of (108) gives the ratio between the B.. [il^yy^h^^^'^'^^t^^'^)

We may write ^

B. =

% Pj(^)

(116)

where C is an arbitary complex constant and Pa(M') are the cofactors of Bk. in the determinant of (1l4).

\ I

_{g e n e r a l ' i n t e r n a l system may be obtained by}

s\aramation of our r e s u l t s with respect to \x over the sequence

(115). The r e s u l t i n g displacements Ü, V for the surface z = b

may be w r i t t e n .

-U

=Z °/

-M-X M-V = -M-x ^ 3 ^ 0 ^ ^ ) A. a . . ii-i, I D ' '22 a ^ ^ P ^ C u ) ^'^22 h i j - la > 3.(^x)e^'^jy. (A2^\^^+X.A^3+A^2) M-+ c

]

+ M-(3.(^x) M-Xjy 2 i 4

K

A.

J = 1 D j^^21 . . A 3 3 + A 2 2 )

(^g7)

+c,

It must be understood in (117) that the real parts of the expressions given^are to be taken.

The s 'Yifarping' and ' analysed in § 2 Hov/ever another C cannot be ob Multiplication of (117) "by e solution (117) section disto

5» at one pa

d i f f i c u l t y ' a

)tained by the

an infinite set process might b number of terms the Vv'arping at of equations egin by limit and then pro a finite numb

could be used to remove the

rtion', from the simple solutions rticular section (say) x = o. rises here, because the constants

usual harmonic analysis.

"''^'^j and operating C ( )dy yields -c

for the C .' An alternative

ing the expansions (117) to a finite ceed by choosing the C to remove er of points on the section,

The processes sketched above are very complex and hardly practicable. Recourse must doubtless be made to approximate methods of calculation to handle problems of constraint against vmrping for swept back Yvings.

2. 7. Approximate Calculation of Root Constraint for the Case of Loading by a Constant Couple

The general methods of § 2.6. are hardly feasible for design calculations. However, an approximate calculation is possible if certain restrictions are made as to the

deformation possibilities. We assume that the section of the box can only warp and distort in its plane according to the

pattern defined in equations (101) and (102), that is, in the same v/ay as occurs when a constant couple is transmitted, with no restraint at the ends. Other modes of deformation

of the section cannot occur, in particular the rib vYobs are rigid in she-'^r (tp-T'cjo ), The deformation of the skins and spar v/ebs is then given

2 5 -U = qb s i n g + co y / c

.'^V

^ " ^ V

V

pb s i n g + ^ . y

A

w f \= qz s i n g + coz/b :Yf= W + p c s i n g - ^ c / 2 b W l — "' wj ( 1 1 8 ) w h e r e , p,q,W,cü, ^ a r e f u n c t i o n s of x. Making t h e s u p p o s i t i o n t h a t T2 = o t h e s t r e s s r e s u l t a n t s f o l l o w f r o m ( I I 8 ) . We f i n d T^ = \(- a.^3d£ 4- a ^ ^ d a ) b s i n g + a^^oül+fa^^ dco + r j 3 d ^ / y J

^ ^^ ^^ ~T~J JZ dx dxf ƒ

T2 = 0

J)±

( 1 1 9 ) o f / - dp ~ dov, . ^^-^(j)( ") a ^ i doo - d A ; w h e r e a . , = a , . - — 11 11 a 12 22 ^ 1 2 ^ 2 3 - ^ ^ 1 3 " ""31 13 a 2 2 3 3 " 33 a22

(I20J

a n d w S ' w E t S dp w i }± "AT ° s m g + q s m g +

2Xüa) / - dx

dW CÜ d6 c dx i b " dx 2b ^ (121) E q u a t i o n s ( 1 2 ) , ( 2 4 ) , ( 5 7 ) , (62) and ( I I 8 ) h a v e b e e n u s e d i n t h e d e r i v a t i o n of (119) ( 1 2 o ) a n d ( 1 2 1 ) , W r i t i n g Z = 0 i n ( 7 5 ) we f i n d f r o m ( 1 2 1 ) . - n, u-tl^ cxu ^ 0^ ^Ut.^^^ L^ \

dW dA c ^ q s i n g + ^ - ^ 2 b = ° S u b s t i t u t i n g i n (76) vife f i n d . -( E t c •I Ytf

I 2rTTo)

? . ^w/2^,,^.<:(.4 4i(122)

L . = 4bc s i n g. ( E t c _ 1 dp 2 - dq

+ ^^33J dS - ^ ^^31 ^''' ^' ^

f E t c i W

'{2(T7o)

1

+ 4\.T7f-_N - ^a33 I c M ^ = 4b s i n g. (ca.^^+ EA)_ ^ - 4b c a ^ ^ s i n g. _{d x} (123) + 4ba^3a> S u b s t i t u t i n g i n ( 6 9 ) a n d (70) we f i n d . - oi/wj(^S) - do) - d/\ „ . , . . d a.,, — + a-,-,c ^ i ^ = 31 _{d x}

₃₃

dx ) dp EAb s i n g. — | d x Et' c s m g. dq dx

+

E t a ^ , — ^ + - ^ 1 0 ) = EA d 0) 2 b ( 1 + a ) c d x

^ y

(124)

-26-Solution of (123) for d^, dg_

yields,-d x yields,-d x vifhere dE d x

da

d x p ?^L^ + PgM^ + P 3 0) Q^L^ + Q2M^ + Q CO 2 _ 4b s i n g ( c a , , + EA) D 1^2 -4 b c a ^ . s m g H - n ( 1 2 5 ) „ / I 6 b ' D s m g D V, ( E t C —2 , \ w _{} f na} • b c a ^ 3 f ^ ^ j : ; ; ^ - y - - b a 3 3 J ( c a ^ ^ + E A ) y^ Qr 4 b c s i n a f 'St c

ir—Z2TTT^ ^^^33i

( 1 2 6 ) 2 2— l 6 E t b c a . -, s i n g v/ 1 3 D ( 1 + a ) 3 2 D = 1 6 b c s i n g • E t c w C 2 T T T ^ + ^ ^ 3 3 ] ( c a ^ ^ + E A ) - b c a ^ 3 S u b s t i t u t i o n f r o m ( 1 2 5 ) i n t o t h e s e c o n d o f ( 1 2 4 ) y i e l d s . . 2 p 2 - ^ - p oj = R . L , + RpM. d x 2 1 1 2 1 ( 1 2 7 ) Yi^here R. R2 = s m EA s i n g g / E t c w

iW^ - "^^33)^1 ^ ^^31^1.

EA ( E t c

(2TTTïïy - ^^33J^2 +'^^31 ^2

( 1 2 8 ) P^ = s m g EA / E t c ( w b a , ^ l ; P-. + b a ^ .Q.

(JiM^- ^^^33) "^3 ^ "^31^3

E t EA ^ 2 b ( 1 + o ) +

!i2(

-/ A s o l u t i o n o f ( 1 2 7 ) w h i c h v a n i s h e s a t x = o a n d r e m a i n s f i n i t e a s X "> «>0 i s . -(R L + RpM ) p - J - J . ^ 2 _ 1 (1 „ e P"") ( 1 2 9 ) GO =

-3

T h e f i r s t o f ( 1 2 4 ) g i v e s a s s u m i n g A = o f o r X = o . -^ E A b Q , s i n g - a -^ . \ ^ _ \ ? 2 l 3 ^33" 0) ( 1 3 0 ) T h e r e m a i n i n g u n k n o Y / n s a r e e a s i l y f o u n d , p . q. f o l l o Y / f r o m ( 1 2 5 ) , W f r o m ( 1 2 2 ) a n d t h e s t r e s s r e s u l t a n t s f r o m ( 1 1 9 ) a n d ( 1 2 1 ) T h e s o l u t i o n f o u n d s o l v e s t h e p r o b l e m o f ' r o o t c o n s t r a i n t ' f o r a ' l o n g ' s w e p t b o x l o a d e d b y a n y c o u p l e a t t h e t i p . I t may b e a p p l i e d w i t h t h e u s u a l a p p r o x i m a t i o n t o o t h e r c a s e s o f l o a d i n g . T h e m e t h o d u s e d h e r e may b e e x t e n d e d t o d e a l Yi/ith t h e p a r a b o l i c Y / a r p i n g o f ( 1 0 R ) a n d s o y i e l d a n a p p r o x i m a t e s o l u t i o n o f t h e s h e a r l a g p r o b l e m * f o r t h e sYifept b o x . / P a r t I I I .

-27-Part III Applications to SYv^ept Back Wing Structures

3.11 Generalisation of the Engineering Theory of Bending and Torsion to Include the Case of SY/ept Back Wing Structures

The intention of the present section is to generalise the solutions obtained in Part 11 for the simple cases of

loading (§ 2.3, § 2.4, § 2.5), to cover the case of a uniform sv;ept box, whose section bears a closer resemblance to an actual v;ing structure, than that considered previously. The box

to Y/hich Y/e shall now devote attention is shown in Pig. 10. The section has unequal spars, that at y = c has thickness t^.^ and depth 2b, v/hile that at y = -c has corresponding dimensions t' and 2b'. The skins are identical in both geometry and

elasticity and so the section is symmetrical about the y-axis, The skins may be curved, but the development below is restricted

to the case v/here dj^/dy_ is small,* where >^(y) is the ordinate. This v/ill ensure that the angle g betYi^een the stringers and

the rib-skin intersections may be treated as constant over

the skin surfaces. The., flanges of the spars y = + c v/ill have section areas A and'A' respectively.

The notation for displacements, strains, stress resultants etc. Y/ill be the same as in Part II. However, in the case of the curved skins, displacements etc. will be

treated as occuring 'in the surface'. For example V Yvill represent a displacement parallel to the tangent of the curve of cross section.

We make the folloYi/'ing assumptions Y/ith regard to di spla c ement

s.-1. Each section x = x moves as a rigid body with

displacement ViTs and rotation pi + qj. W,p,q are functions of x, the last two being quadratic and the first cubic.

2. The section is Y/arped from the plane by a displacement which is linear in x. In the skins we have U = w^(y).x + cop(y) and the v;arping in the spar v;ebs is linear in z. By a suitable •definition of q we may assume that the rotation of linear

elements of the two spar webs to be equal and opposite.

3. The section is distorted in the plane in such a way that Sp = 0 and that Y = /^^ ^y) .x + ^r,{y).

Reference to § 2,5, in particular to equations (104), (I05) and (106) shows that our assumptions are sufficiently general to deal with the loading cases and the simple box treated there. Putting our assumptions into mathematical form, we can

write.-/U = q ^ sin g + . . ..

* This implies that b - b'j /2c is small *

2 8 -U = q ^ s i n g + oo V = - p ( ^ - y g ) Sin g + W ^ + A u = qz s i n g + (co) . z/b ^ V ' y = c YY V/.. W = W + pc s i n g u.' = w w. w qz s i n g + (to) . z / b ' f _ = ?ƒ - pc s i n g (131) w h e r e , P q. W Cü

A

= P^x + ^PgX 1 2 = q ^ x + i q 2 X = YX - |-x (q^ + -rq2x)sin g (132) ü ) . x + CO2

= A,

X + A ,

The q u a n t i t i e s p . , P 2 , Q.^ , 0.2» Y a r e c o n s t a n t s , v/hile o), , ÜO^,

i^, A 2 ~ 1 i j j . i j j . e 0 P J , P p , y.^ , y . p , I a i c c;uiiiD o t i i i i / B , w j i j . x c uu, , u u p ,

A^ , A o are functions of y. The terms in (131 )• involving ii:,q are d^

dy obtained'by an applicatlcin of (9). Those inviolving

in the formula for V represent the tangential component of those parts of Wp Y/hich express rigid body motions (see Fig. 11).

The remaining component of the portions of w n are included in Cn .

The definition of W in (131) is {w + w')/2, which will differ from that used in § 2. 5 equation (97) , by a term which

depends upon the cross sectional distortion and so v/ill be

l i n _{e a r m x.} T h i s d i f f e r e n c e w i l l t h e r e f o r e not a f f e c t t h e r e l a t i o n betY/een W and q g i v e n i n (99) and (103) • T h i s r e l a t i o n h a s b e e n adopted hero and used t o d e r i v e t h e f o r m u l a f o r W i n ( I 3 2 ) , Prom e q u a t i o n s (12) and (131) we f i n d f o r t h e s t r a i n s i n t h e s k i n s , -XX (q.^ + q2x) S s i n g + co^ •e xy - ( P , . . P 2 X ) ( - , - y t § ) s i n g . r f . A , . ^ x . ^ (^33)

It follows that the stress resultants T. and S are linear in x and so assuming in accordance wltYi the findings of Part II that T2 = o and Sp = o we find from (21) Y^riting

X = Y = T2 = 0 that.-dS

^1 = - ^ 3 1 - (^l)x=o

T2 = 0

s(y)

/Equation (30) then

29Equation (30) then g i v e s .

-^xx = ^ l ( - ^ W^ ^^l)x=.o)+ ^ 3 ^ i (135)

'xy-h^(-^§^ (T,)^^,).A33S

\ C o m p a r i n g ( 1 3 3 ) a n d ( 1 3 5 ) ^ ^ d e d u c e u s i n g ( I 3 4 ) . -( q . + q o x ) S s i n g -( w J A. P 2 S i n g \ y M1 ""11 ^11 y ~ ~ " " 1 1 A , . B i n g r + 2 ( p + j ^ q ) -^^ ^ d y ^ ^11 '^ ^11 v' _{- c} y y.p Bxxj. a r - c y 1 = ( ' ^ i ) y = - c ^ ^2P2 + ^ ^ 2 ) S i n g J J^ d y - p g s i n g ( y ^ + c b ' )| - c / dcü2 ^K ^K A l + — = A 3 ^ ( T ^ ) ^ ^ ^ + A33S -f p ^ ( ^ - y ^ ) s i n g - Y g j S u b s t i t u t i n g from ( 1 3 3 ) a n d ( I 3 6 ) i n (69) and (70) ( v / i t h A' w r i t t e n f o r A) we f j n d . -^0) = - ( S ) y _ , + ^2 ^ i ^ ^ (l^ ^ EAb) S'o) = ^ S ) y _ ^ + E A ' b ' q g S i n g (137) w h e r e , . c

^ = i I ^ ^

(138)

- c

Substituting from (131), (132) in (57) and (62) (with t' written for t ) v/e find expressions for S , S' Y/hich may be

J • x i W / j - , - , \ • -I J • Y / W c o m p a r e d Y/ith (137) y i e l d i n g . -( ^ l ) y = c -( ^ = - c -r^— = - - ^ = - P2C s m g ( I + 0 ) , Y = -( l + o ) ^ ^ -( 1 + o ) q 2 S i n g -( EAb E A ' b ' 2 c ^

T " ^ t " F")^^^y=-c'^ Ê ( t + t ' + ATT

W W " ^ W W ( ^ 2 ) y = c ^ ^ 2 ) y = - c __ . ^ . J (1+o)^__ ^ ( = - p . c s i n g - - ^ ( l ^ + i ^ ) ( s ) ' I + 0 ) . (^EAb E A ' b ' 2 0 ^ )

E ^2 ^^^

^>r~

- TT— + i r t <

( Y'f W _ 11 W/ / w h e r e i n t h e

-30-Y/here in the last equation Y/e have made use of the ant j-symmetric nature of the warping in the spar webs (cf. assumption (2)

given above). Substituting now from the third of (136) into the first of (139) we f

ind.-A P2 _{2 A —}31 , _{^ 2}

^^11 "^

(140) The formulae for T. P S, S^^, S' ((136), (137))

obtained above contain the unknown constants q^ , q2, P2 (-^-])y__„ and (S) . Equations (139), (140) shov/ that two of them

are expressible in terms of the remaining tni-ee. Th^oe las-': three and hence the stresses can be determined by use of

equations of overall eq^uilibrium like (75) and (76). HoY/ever, these reg^uire modification for the present structure. We

find easily

that.-Z = 2bS,„ + 2b'S' + 2 r S f^ dy ")

w Y/ J dy •^- 1

c c /

Li = 2bcS^^ - 2b'cs; + 2 j ysfl - 2 J ^^Sdy

-c -c

\

(l4l)

M . = 2bEA(e ) + 2b'EA'(e ) + 2 f^T.dy_{1 ^ xx'y=c ^ xx^y=-c J ^ 1 "^} c ^

-c

where allov/ance has been made for the z-wise components of skin shear S d^ .

Substituting from (I36) and (137) into (1+1) and

making use of (139) and (140) we find after some

transformation>-where Z EI sin g I = 2(Ab2 + A'b'2 + EA. ^ -11 -c

5 s <v)

1

J (142) (S)v=-n = -_y=-c v/here '1 8c>: 8/ EIA ₁₁ ^ dy -c 2E(j*2_;^,^,2^^ _2 'A 3^ = 11 r 2 4

J y>? dy + ^

c y -c 1 1 'i ^^ -c o U(y).>^(y^)dy^dy( (143) EI \

I

A-

J

EI sin g

(144)

It is to be remarked that Z and L. are constant in our solution, whereas M. is linear in x. It is assumed in (144) that Z is applied at x = 1 and hence that M. is given by the expression in' (88). ^

Formulae for the stress resultants can now be /obtained.

-31-obtained. Substituting from (142), (143), (lM-4), (139) and (140) in the formulae of (136) we

find.-w h e r e a n d w h e r e S =

h

^1 ^xx -= = • -L^ . Z 8 c ^ " EIA^^ L^ - rjcZ

^xx tn

^ 1 " -^^11 ^(^1 ^ ^ , , (2y • C T y

S

0 3 -K dy

L,)]^

(145) ^ (146) Substituting in (137) we f ind.-K d y + S.„ = — = + L^ Z r AbZ ^ 8c^ EIA^^ o L^ Z ?. A'b'Z (1^7) w 8c^ EIA,

- j S dy .

I - c

The point y = rjc, z = o on a section x = x may be termed the 'shear centre' at the section. It may be remarked that -| = o Y/hen there is symmetry about the z axis. The torque L.

about an axis through the shear centre may be termed the 'Batho Torque' and is seen to be reacted by a uniform shear flov/ given by the usual Batho formula.

W^e turn now to the calculation of the deflections. Combination of (137) with the second of (139)

gives.-Y = (^) ( ^ + ^ ) ' (148) w v/

The quantity Y can then be obtained using (147). It is equal to the mean^.shear strain in the two spar webs and so the term in W(eq,(13^))- 'Yx' is the 'shear deflection', The

calculation of the rotations requires a knowledge of p., which we have not found as yet. To determine p. Y/e must consider the deformations of the ribs. The rib displacements are

calculated upon the supposition that Sp = o and that Wp is a function of y only (cf.'§ 2.2). We find by (52) that"

dT/r)

^E = - - d F , "R = "-Rt^) ^"-"^^

The displacement V a t t h e skin i s g i v e n b y .

-^ = ( - E ) z < - - R i = - s ' i ( f ) 050)

R e c a l l i n g (60) Y/e f i n d . -W^ co'... y V _R w _{+ f — dy = 0 (151)} - o ^

/Substituting

-32-Substituting from (131) in (I5I we f

ind.-I A w^ - py sin g + W - ^ j -2 dy -c ^ and

f Ai

^^^

J 72

-c ^ -c 2

7

dn = M52) (153)

NoYY the strain e in the curved skin can be calculated in two Y/ays, Firstly from the displacements V andV-n'*'

by ,a Y/ell knov/n formula and secondly from equation (30). We thus

find.-yy 9y - w = A2^T^ + A23S (15^)

S u b s t i t u t i n g from ( 1 3 1 ) , ( I 5 2 ) , (145) and (I46) and e q u a t i n g c o e f f i c i e n t s of x i n t h e r e s u l t i n g formulae we f i n d . -9A

_{1 , ^'^ fAi ^^ ^'^21 '^^}

+ ^ — 2 1 ~ïï d y • — 9y dy^ _J, K, (155) ^11 ^^ The v/hi s a t i s f i e s (I53) i s ,

A,

3 remaining terms of our identity give an equation forci2, ich we do not write here. The solution of (I55) v/hich

A, =

1

Ag^Z ( diK ) ₍₁₅₆₎

; 7 Ê ï F - ^ (y - ° ) dF

The quantity p. can now be calculated, Opeaating on (I36) with 0

J ( ) dy and using (I56), (145) and (146) we find an expression

•c r '~i

for ) (<^2^v= " ^'^2^v- +'T'(T3-'b') j . This quantity can also b( 0

J

-c

obtained from (139) using (142) and (145). Equating the two results we find for

p^.-p^ = C^^L^ - 0^2^-^ "^ "^13^ ' where cosec gi(1+a)(b/t + b'/ C. 11 '12 12""^ - -^3'

cosec g|(1+o)(b/t^ + ^'A'w) ^t^±ZJh3^1ll^ ^f^y^li

8-?c (^ 2E^c

^ '~

_2K

^ "

^31 cosec g 2A^^EI

• ^osec a j (14:01/b_^ bis ^ - A^ /A^ ^) {

^3

,se [

EC U^^ t;^ 33 13 11 .

_J

/ cosec g /+ 2^A^^EI

In all strictness w^ - v^^ - ^^ -—ti -r^, but the inclusion of the second"

R R dy

-33-+ _2a^^EI cosec g l+ö))b f' , b' f ^ . ^,, /Ab A'b' \ ) 1

(^ w o w-C ^ w w '^J /(I57) (A,^ - AJ-,/A,J C 0 0

^ ^^ 2c ^ - (^^ J ^^ ^y - ° J ^ dy - J y ^ .-,/•.

(A2^ - A3/4A.,^)

j y ^. dy

- c J

Using (140), (142) and (I57) we then find.

-ë = C.,^

J}]

4- C.,2M^

s Lj - 1-) G Z where L. =

and Tl' = - 0.^3/00^.^

Using (144), (142) and (132) we f

ind.-- ^ ind.-- cosec g = ^ = C2^L. + C22M. dx dx (156) (159) where, and '21 '22 - C^2 = cosec g

_{/EI J}

(160) (161) The formular(158) and (I60) have the same form as (99) and it

can be shov/n that the constants C. . of (I57) and (I6I) reduce given in (100) v/hen the proper specialisation is

The difference in the nev/f ormulae lies in the IS

to the forms introduced.

introduction of L^ in (I58). L^ is the y = -q^c. The intersection of tl\is line section (coordinate x) may be termed the that sectiono

moment about a line v/ith a rib wise

'centre for twist' at The aim set at the beginning of this section has now been accomplished. Formulae for stresses and deflections

have been obtained for the case of a uniform swept v/ing

structure loaded by 'normal' forces and couples at the ends, This represents a generalisation of the usual Bending-cum-Batho formulae which are used by aircraft engineers to obtain a first approxim-ation to the behaviour of unswer)t v/ings.

3.2. Procedure for Practical Stress Analysis

Consider now an actual swept back v/ing structure having two straight spars, skins reinforced by stringers and ribs parallel to the ''direction of flight'. (See Pig. 12) The v/ing possesses a small amount of taper and the dimensions of the structure vary in a gradual manner along the span. The existance of a plane of symmetry intersecting the spar v/ebs

v/ill be assumed. If no such plane exists in reality, then the actual top and bottom surfaces should be replaced by fictitious

-34-surfaces having ordinates and geometry v/hich are mean values of the real quantities for the two surfaces. This plane of

symmetry will be taken as the x,y plane of a coordinate system. The y-axis will be taken parallel to the ribs the x-axis will

intersect the traces of the ribs on the x,y plane at their mid points and the z-axis 'vill be normal- to the" x,y plane. Attention will be directed in what follows to a single rib-\/ise section with coordinate x. The geometry of this section

and of the various structural elements at this section, v/ill be described by the symbols used in § 3.1. and illustrated in Pig. 10. It v/ill be assumed that the wing is loaded by forces acting in a z-wise direction and by couples v/hose axes lie

in the x,y plane.

The procedure for estimation of the stresses at section x may be outlined as

follov/s!-1. Tabulation of the values of .the following quantities at this

section!-g,c,b,b ' ,^(y) ,t,t^^^.,t^;!^,tj^,A,A',A.g.,Ap,ag,aj^,E,ö

If any of t h e s e , apart from >^, vary a c r o s s the s e c t i o n , t h e n

mean values should be taken, Allov/ance for the bending

s t i f f n e s s of the spar and r i b v/ebs should be made by augmenting

the areas A,A' and Ap.

2. Calculation of sundry c o n s t a n t s for the s e c t i o n !

-A. s \

— , — , (a..) (equation (27)), (a.^)^^ (equation (28)), •s -R -^ P

a., (equation (29)), the determinant / a. .1 , A., (equation (€31)),

C C [ • ' - J ' - ' - d '

Z (equation (I38)), ( ^^dy, J y ^ % , ? f r,(y). / (y. )dy,dy, -'c -C -c +)

I (equation (142)), r)(equation (143)).

3. Calculation of the resultant static action across the section!

-Z sum of z-wise forces acting at points outboard of section. This acts at the centre of the section (y=o).

L.,M- Oblique components, refered to axes 0(X,Y) (see Fig,1) of the sum of the moments, about the centre of the section, of all forces and couples acting at places out board of the section. These may be calculated using the formulae of equation (19). If the external foroes are denoted by Z. and act at (x., y.) we may

(> write!- ^ ^ ^

1 i

L. = r y..z M = - > (x - x)Z

where the summation S- is v/ith respect to i over all the points x. such that x .C x. 4 1 (where x = 1 is the tip). Any'couples' must be replaced by forces before inclusion in these formulae.