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Produced by the NASA Center for Aerospace Information (CASI)
(NASA-CR-175620) SINFLE TFEGBETICAL 30DELE N85-22382

FCR COBFOSIT6 ROTOR BLADES Final Report

(Georgia Inst. of Tech,) 35 p HC A03/NF 401

CSCL 01C Dnclas

63/05 14765

"SIMPLE THEORETICAL MODELS FOR COMPOSITE ROTOR BLADES"

R. Rao Valisetty and Lawrence W. Rehfield

Center for Rotary Wing Aircraft Technology

School of Aerospace Engineering

Georgia Institute of Technology

Atlanta, Georgia 30332

Final Report 4 v7 k* , ^y

Grant No. NAG-1-398 ^^\^r^^1

November 1984
PREFACE

G

This report summarizes the development pment of theoretical rotor blade

structural models for designs based upon composite construction. Care has been

exercised to include a member of nonclassical effects that the author's previous

experir+re indicated would be potentially important to account for. A model,

representative of the size of a main rotor blade, is analyzed in order to assess the

importance of various influences. The findings of this model study sLggest that for

the slenderness and dosed cell construction considered, the refinements are of

little importance and a classical-type theory is adequate. The potential of elastic

tailoring is dramatically demonstrated, so the generality of arbitrary ply layup in

the cell wall is needed to exploit this opportunity.
INTRODUCTION

Composite material systems are now the primary materiais for helicopter

rotor system applications. Bearingless rotor designs proposed for the LHX

helicopter are an example. In addition to reduced weight and increased fatigue

lifA, these materials provide designs with fewer parts which means increased

service life and improved maintainability. Also, in terms of manufacturing, it is

possible to achieve more general aerodynamic shapes including flapwise variation in

planform, section and thickness.

The aeroelastic environment in which rotor blades operate consists of

inertial, aerodynamic and elastic loadings. Because of the directional nature of the

composite materials, it is possible to construct rotor blades with different ply

orientations and hybrid combinations of materials exhibiting coupling between

various elastic modes of deformation. For example, if the fibers are placed

asymmetrically in the upper and lower portions of the blade, there will be a twist

induced by flapwise bending. This provides a potential for improving the

performance of a lifting surface through aeroelastic tailoring of the primary loadbearing structure. Aeroelastic tailoring of a composite structure involves a design

process in which the materials and dimensions are selected to yield specific

coupling characteristics which in turn enhance the overall performance of the

structure. The design of such advanced structures requires simple and reliable

analytical tools which can take into consideration the directional nature of these

materials. In this report, a comprehensive bending theory is presented to aid in the

design of composite rotor blades.

Most of the existing analyses are formulated for isotropic metal blades. A

general tendency is to extend these analyses to composite blades with bending and

shear stiffnesses appropriately modified. This, however, does not describe the
anisotropic character of the composites. One of the earlier efforts to account for

this particular feature of composite wings is rnaee by Weisshaar 1 . He has developed

an engineering theory incorporating the bending-torsion coupling effect for the

study of divergence of swept forward wings. Later Mansfield and Sobey2 have

presented an account of the stiffness characteristics of a cylindrical tube

t

representing a helicopter blade composed of a number of plies of arbitrary layup.

This theory is derived within the context of Batho-Bredt engineering analysis of thin

walled structures and small displacements.

The above theories appear to be adequate for predicting the overall

I

` response of slender blades. These theories, however, do not have the capability to

predict the stresses or the response with adequate precision when the blade is

I not slender. A complex three-dimensional stress field develops in such blades

which is further complicated by the anisotropy of the materials and inhomogeneity

i of blade construction. Finite element analyses are reported in Refs. (3) and (k) to

obtain stiffnesses and stresses for thick walled blades. For an accurate

determination of the stresses, one needs to employ a large number of elements

which makes this type of analyses computationally unacceptable for design

purposes.

Hong and Chopra 5 have studied the effects of fiber orientation on the

flutter of rotor blades. This analysis is based on the nonlinear kinematics of 

Hodges and Dowe116.

Recent theoretical research? has contributed a new appreciation for other

nonclassical effects in addition to transverse shear deformation in bending-related

behavior. A nonclassical contribution to axial stress and transverse normal strain

affect response in planar bending situations significantly for certain combinations

of geometry and stiffness for homogeneous structures. Incorporating these

influences, theories were developed for homogeneous plates 8, stiffened plates and

s
laminateslo,ll. Figure 1 depicts the response contributions incorporated in these

theories. Stresses and displacements are improved beyond linear (with respect to

thickness coordinate) distributions, yet the overall equations retain the character

of an engineering theory. This is a distinguishing feature for it allows the

integration of nonclassical influences on a simple basis.

The nonclassical Influences relevant to rotor blades are those due to

I transverse shear, bendi r

ngg-- related related warm stretch warpin g, in gg-- related related war in warping and torsion

related warping. Laminated composites are in general strong and stiff in the plane

of lamination and weak and flexible in the transverse direction. Consequently,

transverse shear deformation becomes much more pronounced. Bending -related

section warping also affects reponse in a similar way, but it is due to the fact that

bending strain does not strictly correspond to planar deformation. Torsion-related

warping arises whenever a section is restrained against out of plane deformation.

The key to improving the stress predictive capability of a theory Is to account for

these effects correctly. With this background, the present work is undertaken. A

comprehensive theory is developed within the context of small displacements for a

single cell composite blade model. The distributions for the warping displacement,

axial stress and shear flow are improved.

The starting point is the engineering theory of Mansfield and Sobey l. The

kinematic procedure to develop the equations follows that of Ref. (8). An example

is presented to illustrate the potential for aeroelastic tailoring and the significance

of nonclassical influences for rotor blades.

OVERVIEW

A typical composite rotor blade cross section appears in Figure 2. For a

preliminary design, it is common to consider the forward structural hox as the

primary stiffness producing and load bearing structure. The remaining structure is
considered only as a mass contribution, along with tuning weight which is added to

favorably alter the dynamic characteristics. Therefore, the foregoing development

j is for a single cell beam which is shown in Figure 3 together with a reference

,

coordinate axes system.

The classical development of the theory of thin wailed beams under

I

combined loading is based on two hypotheses. The first one is the Euler-Bernoulli

hypothesis which states that plane cross sections normal to the axis of the beam

remain plane during bending-related deformation and that these sections remain

undistorted in their own plane. The second one is due to Saint-Venant which states

that the applied torque is carried by a uniform shear flow and the cross sections are

free to warp in torsion-related deformation. According to the first hypothesis, the

only significant stress is the axial stress, a x. This stress is effective in carrying

:i bending and stretching-related loads. For thin walled beams, these assumptions

lead to the following results:

u(x, Y, z) = U - YV Ix - zWjx 2w m x (I)

v(x, Y, z) = V - z ^ (1a)

w(x, yr z) = W y^ (lb)

N - M x (2)

- 21?

where w = 2 f rds (2a)

Here, u, v and w denote the components of displacement of any point on the cross

section in the directions of the coordinate axes x, y and z, respectively. Similarly,

U, V and W represent the respective displacement components of the origin of the

(x, y, z) coordinate system. The twist of the cross section is denoted by ^ . In

Saint-Venant-type torsion, the variation of this variable is linear with respect to
the axial coordinate, X. U, V, W and m are functions of x only. It is convenient to

describe the variation of any given parameter on the cross section with respect to a

circumferential coordinate, s, whose origin can be arbitrarily selected. As shown in

Figure 3, the perpendicular from the origin of the (x, y, z) coordinate system on to

the tangent at any point s on the circumference is denoted by r. Nxs is the shear

flow. It is the shear stress resultant per unit length of circumference. M x

is

the applied torque and fl is the area enclosed by the box.

There are additional assumptions made regarding the stress state in the

structure. Let s and n denote circumferential and tangential coordinates so that (x,

s, n) forms an orthogonal coordinate system. With respect to this coordinate

system, o x and 

oxs 

are the dominant stresses in the structure. The axial stress, 

a x, is due to bending and stretching and the shear stress, axs, is due to the applied

torque. In the classical development, the remaining stresses -- a n, a s' ons and axn 

are assumed to be zero because their influence on the response is negligible. This

is justified for the following reasons.

In view of the nature of the construction, the structure cannot develop and

suuppPPort a ns . Therefore this stress is neglected. g The skin of the structure is

usually very thin. Therefore, the skin is assumed to be in a state of plane stress

with respect to the n-coordinate direction, which means that an and onx are zero.

It is essential for the aerodynamic reasons to preserve the airfoil shape. The

rigidity of the cross section is maintained by an internal structure, such as a

honeycomb filling which does not add measurably to the overall stiffness of the

structure. Therefore, the circumferential normal stress, a s, cannot develop. It is

also not possible to account for this stress within the limitations of a one

dimensional theory. In the new theory, these stress assumptions are retained. 

In order to account for the transverse shear, bending-related section

warping and torsion-related section warping, a priori knowledge of a x and o

xs 
' 1 6

is necessary. Statically equilibrating stresses can be developed based on the

assumptions given by Equations ' I ) and (2). As a first approximation,it Is assumed

that these stresses are sufficient to estimate the nonclassical effects. A brief

summary of the procedure for developing these stresses follows. This procedure is

similar to that of Mansfield and Sobey l up to Equation (16). It serves to introduce

notation and provides a basis for the new theory. A flow diagram of the subsequent

development of the new theory is shown in Figure 4.

DEVELOPMENT OF ENGINEERING STRESSES

The skin of the single cell tube is considered to be a laminate built from

distinctly different layers or lamina. For atypical lamina within this laminate, the

relationship between stresses and strains referred to x, s coordinate system may be

expressed in the following matrix form

o x QII Q12 Q16 ex

as Q12 Q22 Q26 es

o x Q16 Q26 Q66 syxs

(3)

Where ex and es are the normal strains in x and s coordinate directions. yxs is

the engineering shear strain along the contour of the cross section. Qij are the

stiffnesses which depend upon the fiber orientation and elastic moduli of the

lamina. Equation (3) is derived in Ref, (12).

Now, the layers are assumed to deform together so that the skin can be

represented as a homogeneous anisotropic material represented by averaged

material properties. Let N s, Nx and N xs be the stress resultants per unit length in

s-coordinate direction. Then,
N x r All Al2 A16 ex

N s = I Alt A22 A26 es (4)

Nxs IL A16 A26 A66 Yxs

Where

N

Aij = r

J

Q^^) dn; i,j=l,2, and 6 (5)

k-1-1

and N denotes the number of layers in the skin. N s is negligible in unpressurized

slender structures, so It is set to zero. Equation (4) Is contracted to the following

result:

Nx f A11 - Al2Al2/ A22

Nxs I( A16 - Al2A261A22

A16 - Al2A261A22 l 'E x

A66 - A 26A26/A22J ^Yxs}

(6)

It is convenient to have Equation (6) expressed in the following form:

t

Nx Hll H21 

X

H H

fEx

N

xs 2 xs

where

H22 = 1/(A66 -A 26A 26JA22)

H 21 = - (A16 -A 12 A26/A22)H22

HII = All - Al2Al2/A22 + (A16 - Al2A26/A22)H21

(8a)

The resultant loads acting on a given cross section are defined by the

following integrals:

(7)
(N, M y, M z) = f NXO, z, y)ds (8b)

Where, N is the tensile load along x-axis, M y is the bending moment about y-axis

and Mz is the bending mom^,rnt about z-axis.

The location of the origin of the y,z axes and their orientaPion are at

present arbitrary. The ensuing analysis is greatly simplified If they are selected so

that

f(y, z, yz)H11 ds = (01 0, 0) (9)

The resulting locus of origins of each section, then, defines the tension axis of the

blade and the new y, z axes are the principal axes of bending. Now the stress

resultants are evaluated with the aid of equations (l) and (7).

M

N = U ,x' Hilds -

X H21ds (10)

J

2N

r

M

M y = - W ,xx Hie 2ds - X H21zds (11)

1 2R

M

M z = -V ? xx 

f

H ll y 2 ds - X H21 yds (12)

2?i

Since the cross sections undergo only a rigid body rotation and also because

the axial displacement, u, is continuous along the contour of the cross section, the

rate of twist is evaluated from the following forrnula(13)

= 1 f Yxsds (13)

' x 2St

and Equations (1) and (7) to obtain

^'x =(U,xP21ds-W,xxfH21zds

M

- V ,xx f H 21 y ds + x f H22ds )/2 R (14)

2 Sl

Equations (2), (10), (11), (12) and (14) are now recast in the following form to obtain
the overall engineering clastic law for the structire

U ^x SII 0 0

S14 N

t

-Wry 0

S22 0

S24 MY

I -V0xxf 0 S3a S34 Mz

b f

f 0,x 514 524 S34 S44 Mx

S is a symmetric flexibility matrix whose nonzero elements are given by

S11 = l/ f H11ds

S14 = Sll/(2 of H21ds)

S22 = 1/ f H11 z

2

 ds

S24 = S22/(217 f H21 ds)

S33 = 1/fHlly2ds

S34 = S33/(2ff f H 22 y ds)

S44 = 1 2 H22ds + 1 S14JH21 ds 43F 2$

+ ?

A S

24 f 21 z ds + ?2 S24 f H 21 y ds (16)

The development dscribed so far is reported in Ref. (1). The constitutive

equations (15) are to be supplemented with the following equilibrium equations

N ,x + n = 0 (17)

M y y x = Qz

(18)
Mz,x = Qy (19)

Qz x + q

z = 0 (20)

Qylx + q = 0 (21)

M xOx + m = 0 (22)

to form the engineering theory for single cell laminated beams under combined

loading. Here, n represents the intensity of axial distributed loading, Q and Q 

represent shear stress resultants, q and q represent the distributed loads and m 

represents the distributed torque.

In order to proceed with the development of the new theory, it Is necessary

to have statically equilibrating stress resultants. The axial stress resultant can

be represented in terms of overall stress resultants with the aid of Equations 05),

(1) and (7)

Nx = fl + f2M y + f 3M z + f4Mx (23)

where

fl

= H11Sll

f2=

z HIIS22

f 3 = y HIIS33

f4 = HIISI4 + z HIIS24 + y HII S34 - 1-121/2 S1

(24)

The nonuniform shear flow due to bending and stretching can be derived from the

following equilibrium equation

Nx x + N xs, s = 0 (25)
and an additional requirement that the shear flow be equivalent to the applied

torque. This result can be given as

Nxs 

Mx f Nx,x ds + 1 fr ds f Nx,x ds (26)

A 2h

With r denoting the perpendicular from the origin to the tangent at s. The axial

stress term can be eliminated from above with the aid of Equation (25).

M

N xs = x - f5 N,x - f6M y ^ x - f7M z ^ x - fBMx,x (27)

20

where

fi Ai-4 = )

ds , fr dIf

f6-4)ds ; i=5,6,7,8 (28)

Equations (23) and (27) provide stress resultants which are in static equilibrium. A

comparison of Equations (2) and (27) reveals that the two expressions for the shear

flow are different. The new shear flow is nonuniform along the contour of the 

cross section and this is due to bending and stretching loads. The nonuniform shear

flow in turn affects the twist; its contribution can be evaluated from Equations (13),

(7), (15) and (28). The result can be expressed as

, t x = S14N +S 24M y +S 34 M 

z 

+S44Mx

+ f9N,x +f 10 M Y'x + f 11 Mz x + f 12 mx,x (29)

where

fi = - ! fH 21 f(i-4) ; i=9,10,11,12 (30)

29

f1U and fil will be zero if the shear resultants pass through the shear center. This

does not, however, indicate that twist will not develop under bending and stretching

related loads. The presence of coupling includes this possibility.
DEVELOPMENT OF THE NEW THEORY

The theoretical developmental process is described in the form of a flow

chart in Figure 4. The fundamental assumption that permits this development is

that the transverse shear strain can be estimated from the statically equivalent

stresses given by Equations (23) and (27). it is also assumed that the cross sections

undergo a rigid body rotation. if n is the tangential displacement in the sdirection, then

n = rO (31)

The linear strain-displacement relation for the transverse shear strain is given by

Y y = ups + n ix (32)

The following result emerges upon integration of u 's with the aid of Equation (31)

u = U - zWyx - yV ,x - 20 ^,x + J Yxsds (33)

where

W =2frds (34)

The first three terms in Equation (33) satisfy the engineering hypothesis that plane

cross sections remain plane during deformation. The fourth term denotes the 

I

effect of restrained warping and the fifth term contains the effect of transverse

i

shear deformation. With the aid of Equations (7), 00, (15), (27) and ( 29) the nonclassical part of the displacement is expressed in terms of force resultants to 

j arrive at

u = U - zWfx - yV Ix + f 13N + f

14M y + f15M z + f16Mx

+ f17N 'x + f18My'x + f19Mz ,x + f 20M x ^ x (35)

where
f13 = -2w S14 + J H21SI1 ds

f14 = - 2w S24 +,/ H21S22 z ds

f15 = -2j)S34 + J

f

H21S 22 y ds

I

f16 = - 2w S44 +,/ (H21S14 + H21 z524 + H31 y534 + H22/2S2)ds

f17 2wf9-fH22f5ds

f18 = -2w f10 -,/('H22f6ds

f19 = -2w f

ll -J H^22f7ds

f 20 = - 2w f12 -,/ H22f8ds

The new displacement is now used in conjunction with Equations (7) and (26)

to compute the axial stress resultant

N x = HIPPIx - zW t xx - YV,xx + f13N Ox + f14MY,x + fl5Mz, x + fl6Mx,x

+ f17N ,xx + f18My ,xx + f19 M z,xx + f20Mx,xx)

M

- H21( 2S2 - f 5 N I x - f6M ytx - f7M zJx - f8M xPx) (37)

The engineering kinematic variables U, W, V must be related to the generalized

force variables. This is accomplished by enforcing the definitions in Equation (8)

(N, M y, M z) = U , x./ 1,;11(1, z, y)ds

-W,xx,f Hll(l' z, y) zds

-V,xx fHll(1 , z, y)yds

+ N^xJr(H11 f13 + H21f5)(1, z, y)ds

(36)
+ Mypxpl-Illf14 + H21f6)(11 z, y)ds

+ Mz

v xf(Hilf13 + H21f7)(1, z, y)ds

+ Mx,YJ (Hll fl6 + H2lf8)(1, z, y)ds

+ Nlxx f H

ll

f 170, y, z)ds

+ MY)xxx.// H11f18(l, y, z)ds

+ Mz xxf flllf19(1, y, z)ds

+ M

x,xxf H11f20(1, y, z)ds

+ MxJ '-H21 /2 5)(1, y,z)ds

(38)

These equations can be cast in a more familiar form by solving them for the

engineering kinematic variables.

U ,x Sll 0 0 S14 N

o xx = 0 S22 0 S24 My

V^ 0

t

0

S33 S34 Mz

Mx

S 11 S12 S13 S14 NIX

S 12

, S 22 S 23 S 24 MY,x

S 13 S 23 S 33

S1

 34 Mz.x

S 14 S 24 S 34 S 44 Mx,x
S'11 S 1 12 S r 13 S'14 Nrxy

n n rr n

S 12 S 22 S + 23 S 24 My,x

n u n u

S 13 S 23 S 33 S 34 Mz,xx

it to It

S 14 S 24 S 34 S 44 Mx,xx

i (39)

Equations (39) describe the stretching and bending of the beam. They replace the

n

classical Equations 05). The additional flexibility matrices [SSand [S ] depend only

I on the elastic modulii of the material and dimensions of the cross section. The 

expressions for these parameters can easily be deduced. For example

S 11 = 04Hilds)/ f(Hllfl3 + H 21f g)ds, etc. (40)

The axial stress resultant may be expressed in terms of generalized force

variables with the aid of Equations (39) and (37)

N x = f1N + f 2 M y +f 3 M z +f4Mx

+f 21Ntx + f22Myrx + f23MzPx + f24Mx,x

+ f25N,xx + f26My,xx + f 27M z,xx + f28Mx,xx

(41)

where

f21 = H1l(s

o

 11 + zS 12 + YS 13 + f 13 ) + H21f5

f22 = H11 (S 12 + zS 22 + YS 23 + f14) + H 21 f 6

f23 = Hll(S 13 + zS 23 + YS 33 + f 15) + H21f7

f24 = H1l(S 14 + zS 24 + YS 34 + f 16) + H2lf8

f25 = Hll(S'll + z5 r 12 + YS 13 + f17)
f26 = H11 (S 112 + ZS"22 + yS 1 23 + f18)

f27 = Hll(S 113 + ZS 1 23 + YS 1 33 + f19)

f28 = HIP 114 + ZS 124 + yS 34 + f 2 d (42)

The corrections to the axial stress resultant include the effects of bending

related warping and torsion-related restrained warping. The new shear flow can

now be estimated substituting this stress in Equation (26).

Mx

N xs 2- f5N,x - f6M yjx - f7M ztx - f8Mx?x

- f29N ,xx - f30M y,xx - f31 Mz,xx - f32Mx,xx

- f33N ,xxx - f34M y,xxx - f35M z,xxx - f36Mx,xxx

(43)

where

f = f f(i_8) ds -2 ^rds ff(i-8) ds 1=29, 30.... 36 (44)

The rate of twist is re-evaluated with the aid of Equations 03), (7), (35) and (41)

IP,x = S 14N +S 24M

y + S34Mz +5 44M x

+ f9N ,x + flOMy,x + 

f

11 M z,x + f12Mx,x

• f

37N ,xx + f38M y,xx + f39M z,xx + f40Mx,xx

• f41N ,xxx + f42My,xxx + f43M z,xxx + f44Mx,xxx (45)

where
i

}} f 37 2 ds^H21(S'11 + zS 1

12 + YS" 13 + f 17) - H21f29j

f38 = IfdsIH21(S"12+zS"22 +ys

 I)

"23+f18)-H21f30^

_

t

f39 

_ 

2

ds H 21 (S 13 + ZS 23 + YS " 33 + f19) - H2lf3l

111

1

' f40 sJ H + zS" + S" +f )-H f 2 ^

d ,^ ) 21(s 14 24 y 34 20 21 32}

ll

f = ds ^- H 21 1 (i-8) } ; i = 41, 42, 43, 44 (46)

f

2'0 ) 

The governing equations for the response of single cell laminated rotor

blades can now be summarized. Overall equations of equilibrium remain same as

those in (17) - (22). Similarly, the generalized kinematic variables and force

variables also remain the same. The kinematic variables are U , -W , -V and

,x ,xx ,xx

^

Fx. The force variables are N, M y, Mz and Mx. Equations that relate these two

sets of variables are given by (39) and ( 45). These overall constitutive equations

contain the effects due to transverse shear, torsional shear and restrained warping.

New distributions for the axial displacement, axial stress and shear flow are

contained in Equations (35), (41) and (44).

APPLICATION

The following section provides an example which illustrates the relative

f

significance of nonclassical influences on overall response predictions and stress

Y

! estimates. The potential for aeroelastic tailoring is also investigated to a limited

extent. For this purpose, the rotor blade of Boeing's CH -47 helicopter is used as a
guide in sizing. The blade is modeled as a single cell beam with a uniformly

distributed load, qz , with q applied at the quarter-chord point. The end where the

blade is joined to the hub is treated as a clamped end, with the other end being

free. The dimensions of the box are shown in Figure 5.

In order to study the influence of material properties, two different

material systems, graphite/epoxy and glass/epoxy, are selected. The following

material properties are assumed for each ply:

Graphite/Epoxy:

El = 25 x 106psi, E2 = E3 = 1 x 106psi

G23 = 0.2 x 106psi, G12 = G13 = 0.5 x 106psi

v12=v13=0.25, v23 = 0.25

Glass/Epoxy:

El = 7.3 x 106psi, E 2 = E3 = 2.1 x 106psi

G12 = G

13 = 0.88 x 106psi

V12 =V 13 = 0.275, v 23 = 0.25

The E's are Young's modulii, the G's are shear modulii and v's are Poisson ratios.

The subscript "I" denotes a direction along the fibers, 11 2" and 11 3" denote directions

normal to the fiber. All plies are identical with a thickness of 0.0085 inches.

Two different layups are considered for the box skin. These are (012 , 4512)

and (016 , 45 8). The first one is more effective in torsion and the second one in

bending. Furthermore, these plies are arranged with respect to z = 0 plane in such

a way that there exists three distinctly Different couplings between elastic modes of

deformation. Because of the nature of the applied loading and skin layup, there is

no coupling between stretching and twisting or between bending about z-axis and

^<— Fry . .
`I

twisting. The only coupling is between bending about the y-axis and torsion (flapwise bending). This coupling is controlled by S24. As shown in Figure 6, three

different layups are possible depending upon whether 45 0 plies are placed

symmetrically or antisymmetrically with respect to z = 0 plane.

The first configuration is a balanced design. Since the 45 0 plies are placed

symmetrically, S24 is zero, which means that there is no coupling between bending

about y-axis and torsion. The other two designs are unbalanced. Depending on the

angle of orientation with respect to the load, these designs produce either enhanced

g (positive) coupling or reduced (negative) coupling. As shown in the Tables 1-3, there

are now twelve different designs depending upon the material system, relative ratio

of 00 and 450 plies and relative orientation of the 45 0 plies in upper and lower parts

of the box and placement of the load.

Under the applied uniformly distributed line load (qz = q), all these

structures undergo twisting as well as bending about y-axis. Since this is a

statically determinate problem, the flapwise bending moment and torque are

estimated as

My = - Z (x - L)2

M x = -q e(x - Q (47)

e is eccentricity of applied loading which in this case is 5 inches, The applicable

boundary conditions are given by the following:

At x = 0: W = 0 1 W x = 0, ^ = 0 (48)

Under these conditions, equilibrium equations (18), (20), (22), the overall
constitutive equations for -W^ xx from (39) and the Equation (45) for twist are

solved.

The results for the transverse displacement and the twist at the free end

are presented in Tables 1 and 2 0 respectively. These parameters are given in terms

of unit applied load, q. Similarly, the maximum values for the stress resultants N 

and Nxy at the root are also presented per unit applied load in Figure 3. The

maximum value of the axial stress occurs at the center of the web-like vertical

portion of the structure.

The present theory results for W and 0 are subdivided into three groups

as shown in Tables I and 2 to illustrate the improvements due to nonclassical

effects and coupling. In the simplest engineering approach, the coupling effects

are usually neglected. The material is assumed to be orthotropic, effectively

represented by equivalent bending and torsional rigidities. These results are

denoted "classical" in Tables I and 2. It can be concluded from this study that the

nonclassical effects influence deflection more than the coupling parameter and the

improvements from both coupling and nonclassical effects are relatively small

compared to the classical result. On the other hand, the coupling accounts for a

major portion of the twist. If it is not incorporated, major discrepancies arise in

the twist predictions. This is because the classical approach cannot delineate

between the balanced and unbalanced designs. The nonclassical influences on twist

predictions are negligible. With respect to the transverse displacement prediction,

it may be seen that the nonclassical influences are more pronounced for

graphite/epoxy material system and (0 16 , 458) layup.

Table 3 shows predictions for the maximum values of the stress resultants.

Present theory results are evaluated from Equations (41) and (43). For the sake of

comparison, stresses from Mansfield's theory 2 are also presented. These are not

available in Ref. (2), but are evaluated using Equations 0), (7) and (15). Unlike the
"classical" results for twist and deflection, Mansfield stresses include the bending- 

torsion coupling effect. Therefore, the apparent differences between the present 

theory predictions and those from the Mansfield theory are entirely due to

nonclassical influences. Failure to account for these effects leads to an under

estimation of the stresses. This could be as much as 4% for tensile stresses and 9%

0 for the compressive stresses.

The structure considered here Is very slender. The theory Is also developed

under the premises of a thin walled cylinder. This explains why the nonclassical

influences are so small In this example. Experlence7-10 shows, for beams and

plates made of composite materials, the nonclassica! Influences become important

enough to be included as the structure becomes "thicker" on a relative basis. Since

the kinematic procedure used here is the same as the one used in Ref. (7)-00), the

present theory can be expected to yield valid results even for very thick-walled

rotor blades.

SUMMARY AND CONCLUSIONS

A theory has been developed for composite rotor blades that can be

modeled as dosed cell beams. The theory accounts for nonclassical influences

related to restrained warping, transverse shear stain and bending and stretchingrelated warping. A primary influence that is included is elastic coupling that

results from arbitrary composite ply layup in the walls of the cell. To illustrate the

role of elastic coupling and nonclassical influences, a rotor blade-type model has

been created. This model is analyzed as a single cell cantilever beam under

uniformly distributed load. The following conclusions are drawn from this example:

1. Nonclassical influences and elastic coupling have a negligible effect

on transverse deflection.

2. Twist is controlled by the elastic coupling. Nonclassical influences

have a negligible impact on twist.
3. The axial stress predictions are Improved as a result of accounting for

the nonclassical Influences. The shear flow is affected only slightly

by these nonclassical influences.

4. The potential for aeroelastic tailoring has been illustrated by

choosing three designs with different layups. The first one is a

balanced design with zero coupling. The second and third are

unbalanced designs; one of these two produces enhanced coupling and

the other reduced coupling. For all of these three designs, the weight

is the same and transverse deflection and maximum stresses

generated within are also approximately the same, yet the induced

twists are different.

5. The structure considered In this example is very slender. This is the

reason why the nonclassical Influences appear so small in this

example. By the experience garnered by the authors with laminated

thick plates and beams, it is believed that these influences become

more pronounced for thick walled rotor blades and the present theory

is capable of predicting them.

RECOMMENDATIONS

It appears that the nonclassical influences are of little importance in the

model. As the model approximates practical dimensions of main rotor blades, it Is

likely that the slenderness, together with dosed cell construction, are responsible

for this. As a practical matter, the classical theory of Mansfield and Sobey2 should

be adequate in most cases and should be used.

If the blade under consideration is less "slender" than the example or If

moisture-temperature effects significantly alter the stiffness properties of the

composite material, then the new theory developed herein can be used. Another

situation, the determination of higher than first few vibration modes, also requires

the use of the present, more complicated theory.
REFERENCES

1. Weisshaar, T.A., "Divergence of Forward Swept Composite Wings," Journal

of Alrcraft, Vol. 17, No. 6, 1980, pp. 442-448.

2. Mansfield, E.H. and Sobey, A.J., "The Fibre Composite Helicopter Blade,

Part 1: Stiffness Properties, Part 2: Prospect for Aeroelastic Tailoring,"

Aeronautical Quarterly, May 1979, pp. 413-449.

3. Worndle, R., "Calculation of the Cross Section Properties and the Shear

Stresses of Composite Rotor Blades;' Vertica l Vol. 6, 1982, pp. lit-129.

4. Gia Votto, V., et. al., "Anisotropic Beam Theory and Applications,"

Computers and Structures, Vol. 16, No. 1-4, pp. 403-413, 1983.

5. Hong, C-H. and Chopra, 1., "Aeroelastic Stability Analysis of a Composite

Blade," presented at the 40th Annual National Forum of the American

Helicopter Society, Crystal City, Virginia, May 16-18, 1984.

6. Hodges, D.H. and Dowell, E.H. "Nonlinear Equations of Motion for the

Elastic Bending and Torsion of Twisted Nonuniform Blades;' NASA TN D7818, December 1974.

7. Rehfield, L.W. and Murthy, P.L.N., "Toward a New Engineering Theory of

Bending: Fundamentals;' AIAA Journal, Vol. 20, No. 5, 1982, pp. 693-699.

8. Rehfield, L.W. and Valisetty, R.R., "A Simple Refined Theory for Bending

and Stretching of Homogeneous Plates," AIAA Journal, Vol. 22, No. I, 1984,

pp. 90-95.

9. Yehezkely, O. and Rehfield, L.W., "A New Comprehensive Theory for

Bending and Buckling of Stiffened Plates," to appear in Israel Journal of

Technology.

10. Rehfield, L.W. and Valisetty, R.R., "A Comprehensive Theory for Planar

Bending of Laminates," Computers and Structures, Vol. 16, Nos. 1-4, 1983,

pp. 441-447.

11. Valisetty, R.R. and Rehfield, L.W., "A Theory for Stress Analysis of

Composite Laminates," presented at the 24th AIAA/ASME/ASCE/AHS

Structures, Structural Dynamics anr' Materials Conference, Lake Tahoe,

Nevada, May 2-4, 1983.

12. Jones, R.M., Mechanics of Composite Materials, Scripta Book Company,

Washington, D.C., 1975.

13. Oder, J.T., Mechanics of Elastic Structures, McGraw-Hill Book Company,

1967, New York.
List of Figures

Figure L Mechanical Responses Incorporated In the New Theories of Refs. MW)

Figure 2. Typical Rotor Blade Cross Section

Figure 3. Coordinate Systems

Figure 4. Theory Developmental Process

Figure 5. CH-47 Rotor Blade Idealized as a Single Cell Form

Figure 6. Skin Layups
;ORUNAC PAGE IS

OF POOR QLIALITY

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