Incremental Method for Structural Analysisof Joined-Wing Aircraft

Zahra Sotoudeh∗ and Dewey H. Hodges†

Georgia Institute of Technology, Atlanta, Georgia 30332-0150

DOI: 10.2514/1.C031302

Joined-wing aircraft are characterized by statically indeterminate structures, i.e., structures with multiple load

paths. A newway of analyzing these configurations is introduced. This new formulation is based on the fully intrinsic

equations, which introduce neither singularities nor infinite-degree nonlinearities caused by finite rotation. The

formulation makes use of an incremental form of the kinematical equations, which allows preservation of the main

advantageous features of the fully intrinsic equations. Themethod is applied and verified for a joined-wing structure.

Nomenclature

a = deformed-beam aerodynamic frame of referenceB = deformed-beam cross-sectional frame of referenceBi = unit vectors of deformed-beam cross-sectional frame of

reference (i� 1, 2, 3)b = undeformed-beam cross-sectional frame of referencebi = unit vectors of undeformed-beam cross-sectional frame

of reference (i� 1, 2, 3)Ca = short for CBa, the direction cosine matrix of frame B

with respect to frame aCBA = direction cosine matrix of frame B with respect

to frame AEA = extensional stiffness for isotropic beamEI� = bending stiffness for isotropic beam about x� (�� 2, 3)e1 = column matrix b 1 0 0 cTe2 = column matrix b 0 1 0 cTe3 = column matrix b 0 0 1 cTF = column matrix of internal force measures in Bi basisf = column matrix of distributed applied force measures in

Bi basisGJ = torsional stiffness for isotropic beamg = gravitational constantH = column matrix of cross-sectional angular momentum

measures in Bi basisI = cross-sectional inertia matrixi = inertial frame of referenceii = unit vectors for inertial frame of reference (i� 1, 2, 3)K = column matrix of deformed-beam curvature measures in

Bi basisk = column matrix of initial curvature and twist measures in

bi basisM = column matrix of internal moment measures in Bi basism = column matrix of distributed applied moment measures

in Bi basisP = column matrix of cross-sectional linear momentum

measures in Bi basisr = column matrix of position vector measures in bi basisu = column matrix of displacement vector measures in bi

basis

V = column matrix of velocity measures in Bi basisyac = offset of aerodynamic center from the beam reference

line along b2

� = column matrix of extension and transverse shearmeasures (1-D generalized force strain measures)

� = identity matrix� = column matrix of elastic twist and curvature measures

(1-D generalized moment strain measures)� = mass per unit length�� = offset of center of mass from the beam reference line

along b� = column matrix of small incremental rotations� = column matrix of cross-sectional angular velocity

measures in Bi basis0 = partial derivative with respect to x1� = partial derivative with respect to time

I. Introduction

T HE joined-wing concept, as introduced by Wolkovitch [1],features diamond shapes in the planform and front views. High-

altitude long-endurance (HALE) aircraft usually have high-aspect-ratio wings, resulting in greater flexibility than conventional aircraft.Recently, the joined-wing concept has been revisited as a lighteralternative configuration for HALE aircraft. The analysis of suchaircraft requires the development of nonlinear analysis and specialdesign tools. Because of the unusual topology of joined-wing air-plane configurations, the effects of structural deformation on thestatic aerodynamic and aeroelastic behavior are more difficult topredict. Deformation of the structure at certain locations mayproduce large changes in angle of attack at other locations of thelifting surfaces. Efforts to minimize structural weight may createaeroelastic instabilities that are not encountered inmore conventionalaircraft designs. For a joined-wing aircraft, the first sign of failuremay be in the buckling of the aft member as the structure is softened.Flutter and divergence may also become problems in these membersdue to the reduction in natural frequencies as they go into compres-sion. As the aircraft becomes more flexible, the nature of thegeometric structural nonlinearities become more important.

Several analyses have been developed to address the uniquefeatures of joined-wing aircraft. The oldest appears to be in 1991 [2],in which a parametric study of aerodynamic, structural, and geo-metric properties of joined-wing aircraft is performed. Rather thancite individual works in the 1990s, we refer here to a survey of workdone through 2001 by Livne [3] of works pertaining to joined-wingaircraft and their aeroelastic behavior.

After 2001, we note here works pertaining primarily to structuralaspects separately from those that consider aeroelastic phenomena.Primarily structures oriented studies include [4,5], which focus ondesign of a joined-wing configuration with consideration of differentstructural and geometric properties. Patil [6] performed a nonlinear

Received 8 November 2010; revision received 12 April 2011; accepted forpublication 14 April 2011. Copyright © 2011 by Zahra Sotoudeh and DeweyH. Hodges. Published by the American Institute of Aeronautics andAstronautics, Inc., with permission. Copies of this paper may be made forpersonal or internal use, on condition that the copier pay the $10.00 per-copyfee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,MA 01923; include the code 0021-8669/11 and $10.00 in correspondencewith the CCC.

∗Graduate Research Assistant, Daniel Guggenheim School of AerospaceEngineering. Student Member AIAA.

†Professor, Daniel Guggenheim School of Aerospace Engineering. FellowAIAA.

JOURNAL OF AIRCRAFT

Vol. 48, No. 5, September–October 2011

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structural analysis of a joined-wing using a mixed formulation andcompared his results with experimental data [7]. Lee and Chen [8]performed a study on buckling phenomena in joined-wing aircraft.Green et al. [9] used an equivalent static load and beam theory inoptimization of joined-wing aircraft. Finally, there are severalexperimental works on joined-wing aircraft [7,10,11] that are pri-marily structural in nature.

Those analyses and investigations after 2001 that deal withaeroelastic effects begin with Weisshaar and Lee [12], who inves-tigate the effects of joined-wing aircraft geometry, mass distributionand structural design on aeroelastic flutter mechanisms and aircraftweight. They also showhowweight, strength, and stiffness should bedistributed for an effective design. Their work relied on two differentmethods: a Rayleigh–Ritz method and the static and dynamicaeroelastic analysis capabilities in ASTROS (Automated StructuralOptimization System) [12]. Cesnik et al. [13,14] introduced anapproach to effectively model the nonlinear aeroelastic behavior ofhighly flexible aircraft. The analysis was based on a nonlinear finiteelement framework in which nonlinear strain measures are the pri-mary variables instead of displacements and rotations. The resultinglow-order formulation captures large deflections of the wings alongwith the unsteady subsonic aerodynamic forces acting on them. Anintegrated process is presented in [15] that advances the design of anaeroelastic joined-wing concept by incorporating physics-basedresults at the system level. For example, this process replacesempirical mass estimation with a high-fidelity analytical mass esti-mation. Elements of nonlinear structures, aerodynamics, and aero-elastic analyses were incorporated along with vehicle config-uration design using a traditional finite element analysis. Demasi andLivne [16] focused on the aeroelastic behavior of joined-wingaircraft with particular attention to the effect of structural non-linearity on divergence and flutter. Reference [17] used a modalreduction method and meanwhile tries to capture nonlinearityeffects. Later, using the same method, Demasi and Livne [18] per-formed an aeroelastic analysis of a joined-wing aircraft model.Reference [19] presented a parametric study on aeroelastic behaviorof two types of joined-wing aircraft. Reference [20] studied a gustresponse sensitivity analysis for a joined-wingmodel. Reference [21]used an incremental method to revisit some of the parametric studiespresented by [2]. Finally, a formulation for a symmetric and balancedmaneuvering load alleviation scheme, taking into account aircraftflexibility, is derived in [22].

In parallel work [23], the fully intrinsic equations were shown tobe well suited for analysis of HALE aircraft wings, since they arebeamlike structures with large deformations. Unlike other nonlinearbeam analyses, however, the fully intrinsic equations do not have

displacement or rotation variables. While this may make themunsuitable for some applications, their advantages are important tonote. There are no infinite-degree nonlinearities in the formulation; infact, the highest-degree nonlinearities are only second-degree.Second, there are no singularities associated with finite rotation.References [23,24] present a brief literature review of fully intrinsicequations. The concept of fully intrinsic equations for dynamicsbeams goes back over 25 years before the publication of [23], at leastback to the work of Hegemier and Nair [25]. However, the equationsof [23] appear to be unique:

1) They constitute a geometrically exact, fully intrinsic, dynamicformulation including initial curvature and twist, shear deformation,rotary inertia, and general anisotropy.

2) Their use is explicitly suggested for a dynamic formulationwithout their being augmented with some form of angular dis-placement variables [24], such as orientation angles, Rodriguesparameters, or the like used in both displacement and mixedformulations [26].

The special case of joined-wing aircraft presents a challenge for afully intrinsic formulation because of its static indeterminacy. Theabsence of displacement and rotationvariables can create amismatchin the number of quantities that must be specified at the boundariesversus the information known there. For example, a formulation withvelocity variables instead of displacement variables presents nochallenge in a dynamic formulation, but in a static problemwhere allvelocities are zero, there is insufficient information at the boundariesto solve the resulting equations. Hence, analysis of a joined-wingaircraft using the fully intrinsic equations boils down to analyzingstatic behavior of a statically indeterminate structure. In this paper thesolution of statically indeterminate structures using the fully intrinsicequations is addressed, and the method is applied to joined-wingaircraft as an example of its capability.

II. Theory

A. Fully Intrinsic Equations

Figure 1 shows a beam in its undeformed and deformed states. Ateach point along the undeformed-beam axis, a frame of referenceb�x1� is introduced; and at each point along the deformed-beam axis,a frame of reference B�x1; t� is introduced. Fully intrinsic equationscontain variables that are expressed in the bases of frames b and B[23] and can be written in compact matrix form as

F0B � ~KBFB � fB � _PB � ~�BPB (1a)

Fig. 1 Sketch of beam kinematics.

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M0B � ~KBMB � � ~e1 � ~���B �mB � _HB � ~�BHB � ~VBPB (1b)

V 0B � ~KBVB � � ~e1 � ~���B � _� �0B � ~KB�B � _� (2)

���

�� R S

ST T

� ��FBMB

�(3)

�PBHB

�� c�� �� ~�

� ~� I

� ��VB�B

�(4)

Equations (1a) and (1b) are partial differential equations for linearand angular momentum balance, respectively. Equations (2) arekinematical partial differential equations. Equations (3) and (4) areconstitutive equations and generalized velocity-momentum equa-tions. This is a complete and closed set of algebraic and first-orderpartial differential equations. The strain- and velocity-displacementequations are implicit in the intrinsic kinematical partial differentialequations [23].

As mentioned before, fully intrinsic equations include neitherdisplacement nor rotation variables. However, displacement at anypoint and direction cosines for anyvector of interest can be calculatedeither during a simulation or as a postprocessing step. For example,the direction cosines of bi and Bi may be found as

�Cbi�0 � � ~kCbi �CBi�0 � �� ~k� ~��CBi (5)

and the measure numbers of position vectors for the undeformed anddeformed beam may be found from

r0i � Cibe1 (6a)

�ri � ui�0 � CiB�e1 � �� (6b)

The following frames of reference are used in this formulation:1) For the inertial frame of reference i, the unit vector i3 is in the

opposite direction of gravity.2) For the undeformed-beam cross-sectional frame b, the unit

vectorb1 is tangent the undeformed-beam reference line, and the unitvectors b2 and b3 are parallel to the undeformed-beam cross-sectional plane, in which stiffness and inertia matrices are calculated.

3) For the deformed-beam cross-sectional frameB, unit vectorsB2

and B3 are parallel to the plane closest to the material points in thedeformed beam that make up the cross-sectional plane of theundeformed beam.

4) For the aerodynamic frame of reference a, aerodynamic lift andmoment are defined in this frame. Unit vectors a2 and a3 are definedin the airfoil frame with a2 parallel to the airfoil zero-lift line and a3

perpendicular to it.

B. Boundary Condition Challenges

Figure 2 shows sketch of four different configurations of HALEaircraft structures. These configurations can be easily modeled as acombination of beams. Configurations 1 and 2 show a flying wingand a conventional aircraft, respectively. These configurations arestatically determinate so that in the static case, the equilibrium equa-tions [i.e., Eqs. (1)] are sufficient to solve these structures. Moreover,in a flying wing or a conventional configuration, there are sufficientboundary conditions on force,moment, velocity and angular velocitybecause each beam has at least one free end. This facilitatesnumerical solutions for solving steady-state problems [27]. On theother hand, configurations 3 and 4 are joined-wing configurationsand obviously statically indeterminate structures. In static analysiswhen velocity and angular velocity are identically zero, Eqs. (3) and(4) are trivially satisfied. Since these structures are staticallyindeterminate, equilibrium equations are insufficient for solving forthe behavior. An incremental method is introduced to overcome thisdifficulty associated with finding the static equilibrium state ofstatically indeterminate structures such as joined-wing aircraft. After

the equilibrium state is found, the fully intrinsic equations can belinearized about the static equilibrium state for dynamical analysis.The incremental method is based on repeatedly solving linear sys-tems of equations as the load is gradually increased. The governingequations for dynamics of small motions about the equilibrium statecan then be reduced to a generalized eigenvalue problem.

C. Incremental Method

The incremental method consists of sets of linear equations ofmotion, which are obtained by dropping all time derivatives from thegoverning equations and linearizing them. Thus, the fully intrinsicequations of motion become

�F0B � ~�KB �FB � ~�FB �KB � �fB � ~��B�PB � ~�PB ��B

�M0B � ~�KB �MB � ~�MB�KB � � ~e1 � ~��� �FB � ~�FB �� � �mB � ~��B

�HB

� ~�HB��B � ~�VB �PB � ~�PB �VB (7)

and the fully intrinsic kinematical equations are

�V 0B � ~�KB �VB � ~�VB �KB � � ~e1 � ~��� ��B � ~��B �� � 0

��0B � ~�KB ��B � ~��B

�KB � 0

(8)

making use of the linear constitutive equations

�����

�� cR S

ST T

� ���FB�MB

�(9)

and generalized velocity-momentum equations

��PB�HB

�� c�� �� ~�

� ~� I

� ���VB��B

�(10)

In these equations the ��� quantities are known from the previousloading step, and the �^� quantities are the unknowns at each step. Anexception to this is that �f and �m are small, specified increments ofapplied force and moment.

In the incremental method, equations that govern incrementaldisplacement and rotation must also be included. These have theform

�� � �q0B � ~�KB �qB � � ~e1 � ~��� � B ��� � 0B � ~�KB � B (11)

Although incremental displacements and rotations are introduced,the governing equations are linear, and there are neither infinite-degree nonlinearities nor singularities associated with introducingfinite rotation. Hence, the two main advantages of the fully intrinsicequations, namely, avoiding nonlinearities of orders higher thansecond and avoiding singularities, are kept.

Fig. 2 Sketch of different configurations of HALE aircraft structures.

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Equations (7–11) should be solved at each step due to anincremental loading. After each step all variables except the direction

cosine matrix, �q, and � are updated using relations of the form

�Xnew � �Xold � �X (12)

and q and do not need to be updated. The direction cosinematrixCis updated using

Cnew � �� �~� �Cold (13)

It turns out that this first-order update for direction cosine matrix Chas been sufficient in all cases run so far; however, a second-orderupdate may be used if desired. Displacement can be calculated byeither using Eq. (6b) at the end of the solution procedure as apostprocessing task or by updating a variable such as �q with �q atevery step, so that

�qnew � �qold � CiB �q (14)

1. Modeling Gravity Using the Incremental Method

As mentioned before, externally applied loads should be applied

incrementally in this method. These externally applied loads �f and �mmay include any kind of applied forces, such as gravity, thrust, oraerodynamic forces. For modeling dead forces such as gravity, thedirection cosine matrix plays an essential role. A distributedgravitational force is written as

fgi ���gi3 � Bi so that fg ���gCBie3 (15)

Thus, the incremental term may be written as

�f g �� ���g�CBie3 � �g �CBie3 (16)

Here, ���g�CBie3 is an inhomogeneous term, with ���g� as the

incremental value in each step; and�g �CBie3 is a homogeneous term.

Note that CBi � CBbCbi � CCbi and �C�� ~� C. If there is an offsetbetween the center of mass and the beam reference line, then themoment caused by gravity can be developed in the same way, viz.,

mgi� ���B� � ���gi3� � Bi so that mg ���g ~�CBie3 (17)

where � takes on values 2 and 3, and repeated indices are summedover their range. Thus, the incremental term may be written as

�mg �� ���g� ~� CBie3 � �g ~� �CBie3 (18)

2. Modeling Aerodynamic Force/Moment in the Incremental Method

A two-dimensional (2-D) aerodynamic model is used to calculatethe aerodynamic loads generated by wings and control surfaces suchas flaperons. The quasi-steady aerodynamicmodel has been changedto an unsteady model by adding the effect of induced flow from the2-D induced-flow model of Peters et al. [28], along with apparentmass/inertia terms in the force andmoment equations. Thefinal forceand moment equations, respectively, take the form [29]

fa � �b

8<:

0

��Cl0 � Cl���VTVa3 � Cl��Va3 � �0� � Cd0VTVa2�Cl0 � Cl���VTVa2 � 2 _Va3b=2 � Cl�Va2�Va3 � �0� � 2Va2�a1

b=2 � Cd0VTVa3

9=; (19)

and

ma � 2�b2

8<:�Cm0

� Cm���V2T � Cm�VTVa3 � b

Cl�8Va2�a1

� 2�b232

_�a1� b

8_Va3�

0

0

9=; (20)

and where Va2 and Va3 are the second and third elements of velocityvector in the aerodynamic frame of reference, and VT����������������������V2a2� V2

a3

q.

For the steady-state solution, the applied aerodynamic force andmoment will be, respectively,

fa�

�b

8>><>>:

0

��Cl0�Cl���VTVa3�Cl�V2a3�Cd0VTVa2

�Cl0�Cl���VTVa2�Cl�Va2Va3�2Va2�a1b=2�Cd0VTVa3

9>>=>>;

(21)and

ma � 2�b2

8<:�Cm0

� Cm���V2T � Cm�VTVa3 � b

8Cl�Va2�a1

0

0

9=;(22)

So for the incremental method �fa and �ma are

�f a1 � 0 (23)

�fa2 ��b��Cd0

�V2a2

�VT��Cl0 � �Cl� � �Va3 �Va2

�VT� Cd0 �VT

��Va2

��b���Cl0 � �Cl� � �V2

a3

�VT� 2Cl�

�Va3 �Cd0

�Va2�Va3

�VT

� �Cl0 � �Cl�� �VT��Va3 (24)

�fa3 � b���Cl0 � �Cl�� �V2

a2

�VT�Cd0

�Va3�Va2

�VT

� �Cl0 � �Cl�� �VT � Cl� �Va3 � b ��a1

��Va2

� b���Cd0

�V2a3

�VT��Cl0 � �Cl�� �Va2 �Va3

�VT

� Cd0 �VT � Cl� �Va2��Va3 � b2� �Va2

��a1(25)

�ma1� 2b2��2�Cm0

� �Cm� � �Va2 � 18bCl�

��a1 �Va2

� 2b2��2�Cm0� �Cm�� �Va3 � Cm� �VT �Va3

� 14b3�Cl�

�Va2��a1

�ma2� 0 �ma3

� 0 (26)

Fig. 3 Sample discretization.

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where �VT �����������������������V2a2� �V2

a3

q. Applied loads fa and ma should be

transferred to the B frame by use of

fB � Cafa mB � Cama � Ca ~yacfa (27)

D. Discretization

One can use a simple central difference to get a discretized form ofthese equations (see Fig. 3). Consider a variable X. Let the nodal

value of X after discretization be represented by X̂nl and X̂nr for node

number n at the right and left sides of node. For the nth element onemay write

X0 � X̂n�1l � X̂nrdl

X� �Xn � X̂n�1l � X̂nr

2(28)

Variables are assumed to be different at right and left sides of eachnode, so one can easily account for discontinuity at each node. Nodalforce and moment discontinuities can be addressed by having onevectorial equation for balancing the force and another for balancingthe moment at each node [29].

For each step of the incremental method, the equations reduce to aset of linear algebraic equations (in the case of constant steady-stateor static analyses), which can be solved easily.

E. Stability Analysis

A generalized eigenvalue problem can be derived by linearizingthe discretized, fully intrinsic equations about a constant steady-statesolution, which is computed using the incremental method. Since theeigenvalue problem represents a dynamics problem, the fullyintrinsic equations work well for the vibration and forced response ofstatically indeterminate structures. One needs simply to replace dis-placement boundary conditions with boundary conditions on veloc-ity and to similarly replace boundary conditions on rotation withboundary conditions on angular velocity.With the use of velocity andangular velocity to describe geometric boundary conditions, how-ever, zero frequencies may occur that are due to lack of enoughboundary conditions on force and moment in a statically indeter-minate structure.

III. Verification of Incremental Method

In this section the incremental method is first verified by study of aclamped–clamped nonrotating beam under a distributed load and aclamped–clamped rotating beam. As a second example, the incre-mental method is verified against available experimental results [7]and against results obtained from the mixed formulation [6],

including eigenvalues. The simple aerodynamic model is verifiedagainst that found in NATASHA (Nonlinear Aeroelastic Trim andStability for HALE Aircraft). Validation studies of NATASHA maybe found in [30]. Here, the incremental method is applied to aclamped–free beam, and results obtained are compared against thoseof NATASHA.

All units are in an English system inwhichmass is in slugs, time isin seconds, force is in pounds, and length is in feet, unless otherwisespecified. However, the input data and results obtained and reportedin the paper are correct in any consistent system of units.

Table 1 Beam properties (English units)

Properties Values

Length 20Axial stiffness 1,322,000Torsional stiffness 0:0221 � 105

Out-of-plane bending stiffness 0:0172 � 105

In-plane bending stiffness 1:0989 � 105

Mass per unit length 0.0127Mass polar moment of inertiaper unit length

0.0011

b3

b1

Fig. 4 A clamped–clamped beam under distributed load.

Table 2 Mixed-formulation results for clamped–clamped beam

(English units)

Number ofelements

F1 M2 F3 u3 CPU time

10 1040.2602 16.5395 99.8917 0.3572 1.522320 1046.0439 16.3877 99.9122 0.3556 3.023230 1047.1065 16.3518 99.9168 0.3553 4.309240 1047.4775 16.3387 99.9185 0.3552 5.729050 1047.6490 16.3325 99.9194 0.3552 7.626580 1047.8347 16.3257 99.9202 0.3551 16.9257100 1047.8775 16.3242 99.9204 0.3551 22.4882120 1047.9008 16.3233 99.9205 0.3551 28.9707140 1047.9148 16.3228 99.9206 0.3551 40.9120160 1047.9239 16.3225 99.9207 0.3551 52.5070400 1047.9489 16.3215 99.9208 0.3551 506.3128

101

102

103

−102

−101

−100

−10−1

−10−2

Number of steps

Per

cent

age

diffe

renc

e in

axi

al fo

rce

(F1) Number of elements=10

Number of elements=40

Number of elements=80

Fig. 5 Convergence of axial force.

101

102

103

−10−1

−10−2

−10−3

−10−4

Number of steps

Per

cent

ag d

iffer

ence

in s

hear

forc

e (F

3) Number of elements=10

Number of elements=40

Number of elements=80

Fig. 6 Convergence of shear force.

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A. Verification of the Incremental Method for a Clamped–Clamped

Nonrotating Beam

In this examplewe illustrate the benefit of the incremental methodin obtaining a steady-state solution for a statically indeterminatestructure. For this purpose the easiest example is a clamped–clampedbeam. This problem is inherently nonlinear and serves our purposevery well. A beamwith the properties given in Table 1 is undergoinga distributed transverse force of 10 lb=ft, as shown in Fig. 4.

This problem has been solved by a mixed formulation, in whichthe geometric boundary conditions are expressed easily in terms of

displacement and rotation parameters. Table 2 shows values of axialforce F1, bending moment M2, and transverse displacement u3 atmidspan and shear forceF3 at the beam root for different numbers ofelements using a mixed formulation. This problem is also solved bythe present incremental method. Results from the incrementalmethod are compared with those of the mixed formulation with 400elements. The out-of-plane bending moment and axial force havetheir maximum values at midspan, and the shear force has itsmaximumvalue at the clamped ends. Hence, the errors are calculatedfor axial force and bendingmoment at midspan and shear force at theroot (x1 � 0).

101

102

103

10−1

100

101

102

Number of steps

Per

cent

age

diffe

renc

e in

ben

ding

mom

ent (

M2)

Number of elements=10

Number of elements=40

Number of elements=80

Fig. 7 Convergence of out-of-plane bending moment.

0 5 10 15 20 25−3

−2

−1

0

1

2

3 x 10−3

X1 [ft]

Axi

al d

ispl

acem

ent [

ft]

Mixed Formulation

Fully Intrinsic Formulation

NS=100NS=50

NS=400

Fig. 8 Axial displacement along the beam.

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

X1 [ft]

Out

of p

lane

dis

plac

emen

t [ft]

Mixed FormulationFully Intrinsic FormulationNS=100NS=50NS=400

Fig. 9 Out-of-plane displacement along the beam.

0 5 10 15 20 251000

1010

1020

1030

1040

1050

1060

X1 [ft]

Axi

al fo

rce

[lb]

Mixed Formulation

Fully Intrinsic Formulation

NS=100NS=50

NS=400

Fig. 10 Axial force along the beam.

0 5 10 15 20 25−100

−50

0

50

100

X1 [ft]

She

ar fo

rce

[lb]

Mixed FormulationFully Intrinsic FormulationNS=100NS=50NS=400

Fig. 11 Shear force along the beam.

0 5 10 15 20 25−120

−100

−80

−60

−40

−20

0

20

X1 [ft]

Ben

ding

mom

ent [

lb ft

]

Mixed FormulationFully Intrinsic FormulationNS=100NS=50NS=400

Fig. 12 Out-of-plane bending moment along the beam.

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Figures 5–7 show convergence of the axial and shear forces andbending moment using the incremental method. As one can seeclearly, the error decreases rapidly with an increase in the number ofsteps. Figures 8 and 9 show axial and out-of-plane displacement forthe problem at hand. Displacements are calculated by virtue ofEq. (14). Also, Figs. 10–12 show axial force, shear force, and out-of-plane bending moment for the problem at hand.

The norm of CBiCiB ��, which should be identically zero, waschecked for each node at the end of solution procedure. It approx-imates zero with a very good accuracy, so that orthogonality ofdirection cosine matrix is preserved in the incremental method.Better accuracy is always achievable by using a second-order updateof C at each step instead of a first-order update. Table 3 shows thetwo-norm of CBiCiB �� for the problem under consideration with20 elements and for different numbers of steps when a first-orderupdate is used for updating the direction cosinematrix. Table 4 shows

Table 3 Orthogonality error for CBi using first-order update

Node number 10 steps 100 steps 1000 steps

Node 1 0 0 0Node 2 6:66E � 007 6:65E � 008 6:65E � 009Node 3 1:89E � 006 1:89E � 007 1:89E � 008Node 4 2:90E � 006 2:90E � 007 2:90E � 008Node 5 3:35E � 006 3:35E � 007 3:35E � 008Node 6 3:20E � 006 3:19E � 007 3:19E � 008Node 7 2:57E � 006 2:56E � 007 2:56E � 008Node 8 1:69E � 006 1:69E � 007 1:69E � 008Node 9 8:37E � 007 8:36E � 008 8:36E � 009Node 10 2:23E � 007 2:22E � 008 2:22E � 009Node 11 0 0 0Node 12 2:23E � 007 2:22E � 008 2:22E � 009Node 13 8:37E � 007 8:36E � 008 8:36E � 009Node 14 1:69E � 006 1:69E � 007 1:69E � 008Node 15 2:57E � 006 2:56E � 007 2:56E � 008Node 16 3:20E � 006 3:19E � 007 3:19E � 008Node 17 3:35E � 006 3:35E � 007 3:35E � 008Node 18 2:90E � 006 2:90E � 007 2:90E � 008Node 19 1:89E � 006 1:89E � 007 1:89E � 008Node 20 6:66E � 007 6:65E � 008 6:65E � 009Node 21 0 0 0

Table 4 Orthogonality error for CBi using second-order update

Node number 10 steps 100 steps 1000 steps

Node 1 0 0 0Node 2 1:11E � 014 8:88E � 016 5:77E � 015Node 3 8:93E � 014 1:11E � 015 1:24E � 014Node 4 2:10E � 013 1:67E � 015 2:55E � 015Node 5 2:81E � 013 4:44E � 016 1:03E � 014Node 6 2:56E � 013 1:33E � 015 7:22E � 015Node 7 1:65E � 013 2:22E � 016 1:61E � 014Node 8 7:17E � 014 1:67E � 015 2:31E � 014Node 9 1:73E � 014 4:44E � 016 1:51E � 014Node 10 8:88E � 016 7:77E � 016 6:33E � 015Node 11 0 0 0Node 12 8:88E � 016 7:77E � 016 6:33E � 015Node 13 1:73E � 014 4:44E � 016 1:51E � 014Node 14 7:17E � 014 1:67E � 015 2:31E � 014Node 15 1:65E � 013 2:22E � 016 1:61E � 014Node 16 2:56E � 013 1:33E � 015 7:22E � 015Node 17 2:81E � 013 4:44E � 016 1:03E � 014Node 18 2:10E � 013 1:67E � 015 2:55E � 015Node 19 8:93E � 014 1:11E � 015 1:24E � 014Node 20 1:11E � 014 8:88E � 016 5:77E � 015Node 21 0 0 0

101 102 10310−1

100

101

102

103

Number of steps

Com

puta

tiona

l tim

e [s

]

Number of elements=10

Number of elements=40

Number of elements=80

Fig. 13 Computational time for incremental method.

101 10210−2

10−1

100

101

Number of elements

Rel

ativ

e er

ror

1st bending mode

2nd bending mode

3rd bending mode

Fig. 14 Relative error in natural frequency of a clamped–clamped

beam versus number of elements.

b2

b1

Fig. 15 Top view of a clamped–clamped rotating beam; angular

velocity is about b3.

102

103

104

−101

−100

Number of steps

Per

cent

age

erro

r

Number of elements=10Number of elements=20Number of elements=40Number of elements=80

Fig. 16 Axial force convergence for a clamped–clamped rotating

beam.

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the same quantity for a second-order update. The small errors for thefirst-order update do not seem to have any deleterious effect on theoverall accuracy of the results, and the errors in the second-orderupdate approach machine precision. Figure 13 shows computationaltime vs number of steps for different numbers of elements. Figure 14shows the relative error of the first three natural frequencies of aclamped–clamped beam, calculated using the fully intrinsicequations, versus the number of elements. Clearly, convergence istaking place as the number of elements grows.

B. Verification of the Incremental Method for a Clamped–ClampedRotating Beam

A clamped–clamped rotating beam (Fig. 15) can be solved withfully intrinsic equation, although this structure is statically indeter-minate. Actually there is an analytical solution [27] for a rotating,clamped–clamped beam with no external loading. Assuming thebeam has a prescribed angular velocity in theB3 � b3 � i3 directiongiven by !3, then it means that�3, V2, and F1 are the only nonzerovariables. Here, this problem is solved with incremental method andresults of incremental method is compared versus analytical results.Governing equations can be found in [27,31]. The analytical solutionfor this problem is as follows:

�F 1 �� csc��� cos�x�� � 1

�2�V2 � csc��� sin�x�� (29)

where

��3 ��3

!3

�F1 �F1

�!23R

2�V2 �

V2

R!3

x� x1R

� �� � d��dx

�2 � �!23R

2

EA(30)

0 0.2 0.4 0.6 0.8 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Normalized coordiante along beam (x1)

Nor

mal

ized

axi

al fo

rce

Fig. 18 Axial force distribution for clamped–clamped rotating beam.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized coordiante along beam (x1)

Cho

rdw

ise

velo

city

(V

2)

Fig. 19 Chordwise velocity distribution for a clamped–clamped

rotating beam.

α

Fig. 20 Joined-wing configuration under study.

Table 5 Beam properties for configuration in Fig. 20 (English units)

Properties Values

Length of front wing 20Length of aft wing 10Joint position 10� 60

Torsional stiffness 2214Out-of-plane bending stiffness 1:1017 � 105

In-plane bending stiffness 1721.4Mass per unit length 0.012675Mass moment of inertia per unit lengthfor out-of-plane bending

1:6504 � 10�5

Mass moment of inertia per unit lengthfor in-plane bending

0.0010728

Polar mass moment of inertia per unit length 0.0010728

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tip load [lb]

Join

t dis

plac

emen

t [in

]

Experiment

Mixed Formulation

Incremental Method

Front view

Top view

Fig. 21 Joint deflection.

102

103

104

100

101

102

103

104

Number of steps

Com

puta

tiona

l tim

e [s

]

Number of elements=10Number of elements=20Number of elements=40Number of elements=80

Fig. 17 Computational time for a clamped–clamped rotating beam.

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Figure 16 shows convergence of axial force to analytical solutionfor different numbers of elements versus number of steps. Figure 17shows computational time for different numbers of elements andnumber of steps. Figure 18 and 19 shows axial force and velocity (inchordwise direction) for 500 steps and 40 elements. Analyticalsolution and incremental method solution are right on top of eachother. For these results �2 � 0:00346.

C. Validation Versus Experimental Results

Figure 20 is the case considered throughout this section. Table 5shows the structural properties of this configuration. Figure 21 showsthe joint deflection versus a varying tip load. Results from the incre-mental method are in excellent agreement with those of the mixedformulation [6]. Neither formulation perfectly matches the experi-mental data [7] after a certain point because of yielding of the joint[6]. Figure 22 shows the tip deflection of the same structure undervarying tip load for the incremental method, the mixed-formulationand experimental results. Themixed formulation and the incrementalmethod are again in excellent agreement with each other and are bothclose to the experimental results. Figure 23 shows the out-of-planebending deflection of the main wing of the same structure under aconstant load distribution [6]. Again, results from the mixed formu-lation and the incremental method are in excellent agreement.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Tip load [lb]

Tip

dis

plac

emen

t [in

]

Experiment

Mixed Formulation

Incremental Method

Top view

Front view

Fig. 22 Tip deflection.

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

x1 [ft]

Out

of p

lane

ben

ding

def

lect

ion

[in]

Incremental Method

Mixed Formulation

Fig. 23 Out-of-plane bending deflection.

0 1 2 3 4 50

0.5

1

1.5

2

Tip load, [lb] Tip load, [lb]

Tip

dis

plac

emen

t, [i

n]

Mixed Formulation

Incremental Method

Top view

Front view

0 1 2 3 4 50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Join

t dis

plac

emen

t, [i

n]

Mixed Formulation

Inceremental Method

Top view

Front view

Fig. 24 Tip deflection for nonplanar joined-wing configuration.

Table 6 Eigenvalues from fully intrinsic equations

vs those from mixed formulation

Fully intrinsicequations

Mixed formulation Percentagedifference

5.03 5.02 �0:2719.97 19.91 �0:2857.58 57.38 �0:3558.32 58.27 �0:1094.07 93.34 �0:77

101

102

100

101

102

103

101

102

100

101

102

−100

−10−1

−10−2

Percentage difference at the root of front wing

101

102

100

101

102

−100

−10−1

−10−2

Number of steps10

110

210

0

F1F2 F3

M3M2

M1

Fig. 25 Convergence of force and moments values to the mixed-

formulation solution vs number of steps for front wing root.

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101

102

100

101

101

102

100

101

Percentage difference at the root of aft wing

101

102

101

102

101

102

101

102

−10−1

−10−2

−10−3

10−0.7

100.1

−100

−10−1

−10−2

Number of steps

100

F1 F2

F3

M1 M2M3

Fig. 26 Convergence of force and moment values to the mixed-

formulation solution vs number of steps for aft wing root.

101

102

100

101

101

102

100

101

Percentage difference in force and moment in joint position

101

102

10−3

10−2

10−1

101

102

−100

−10−1

101

102

10−2

10−1

100

Number of steps10

110

2

101

F1 F2F3

M1 M2 M3

Fig. 27 Convergence of force and moment values to the mixed-formulation solution vs number of steps for joint.

0 10 20 30 40 50−0.2

0

0.2

0.4

0.6

0.8

x1 [ft] x1 [ft]

x1 [ft] x1 [ft]

x1 [ft] x1 [ft]

F1 [l

b]

0 5 10 15 20 250.8

1

1.2

1.4

1.6

1.8

2

F1

[lb]

0 10 20 30 40 500

1

2

3

4

5

6

7

F3

[lb]

0 10 20 30 40 500

5

10

15

20

M1

[lb]

0 10 20 30 40 50−50

−40

−30

−20

−10

0

M2

[lb]

0 10 20 30 40 50−35

−30

−25

−20

−15

−10

−5

0

M3

[lb]

a) b)

c) d)

e) f)

Fig. 28 Plots of a–c) force and d–f) moment distributions in front wing.

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D. Verification of the Incremental Method for a Nonplanar

Joined-Wing Configuration

The incremental method is also verified for a nonplanar joined-wing configuration versus results obtained from the mixed formu-lation [6]. Figure 24 shows transverse tip and joint deflection of ajoined wing versus magnitude of tip load, respectively. This test caseis exactly the same as one in [6]. 100 steps are used to achieve theseresults.

E. Verification of Eigenvalue Analysis Versus Mixed Formulation

For validation of the eigenvalue solver, a structure the same as inFig. 20 with properties the same as in Table 5 is used. The structure isunder a constant distributed follower force of 0:5 lb=ft. 100 stepswere used to solve the steady-state equations. 80 elements were usedin the front wing and 40 in the aft. Table 6 shows the first fiveeigenvalues calculated with the incremental method based on fullyintrinsic equations and with the mixed formulation.

F. Convergence Study

Because the incremental method works by solving a sequence oflinear problems to find the steady-state solution for a joined-wingstructure under a specific loading, the number of steps plays a specificrole. The clamped–clamped example shows a very good conver-gence rate. Here, convergence of incremental method for the samestructure as Fig. 20 is studied. Table 5 shows structural properties forthe problem at hand. Both wings are loaded with a follower force inthe B3 direction, having a constant magnitude of 0.5 lb. Results arecompared with those using the mixed formulation for the samenumber of finite elements. The front and aft wing roots and the joint(i.e., the junction) are critical points in this configuration. Figures 25–27 show percentage difference with respect to mixed formulation’sresults for these three points as number of steps increases.

There are three critical points in this configuration, i.e., the twoclamped ends and the joint position (see Fig. 20). Figures 25–27show the convergence of force and moment measure numbers inthe Bi basis at these three critical points. For this study the front

0 10 20 30 40 50−3

−2.5

−2

−1.5

−1

−0.5

0

x1 [ft]

F2

[lb]

0 5 10 15 20 252

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

x1 [ft]

F2

[lb]

0 5 10 15 20 25−9

−8

−7

−6

−5

−4

−3

x1 [ft]

F3

[lb]

0 5 10 15 20 25−23

−22.5

−22

−21.5

−21

−20.5

−20

−19.5

x1 [ft]

x1 [ft] x1 [ft]

M1

[lb]

0 5 10 15 20 25−60

−50

−40

−30

−20

−10

0

10

M2

[lb]

0 5 10 15 20 25−30

−25

−20

−15

−10

−5

M3

[lb]

a) b)

c) d)

e) f)

Fig. 29 Plots of a–c) force and d–f) moment distributions in back wing.

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wing has 40 elements and the aft wing 20. The mixed-formulationresults for the same number of finite elements is taken as the refer-ence solution. Determination of the axial force F1 is an inherentlynonlinear process for a joined-wing configuration, and it thus takesmore steps to converge to the exact solution. Figures 28 and 29

show the distributions of internal force and moment in front andaft wing for the same problem. The number of steps for theseresults is 100.

G. Verification of Aerodynamics Model ImplementationVersus NATASHA

Implementation of the aerodynamic formulation in the incre-mental method is verified by a comparison of the results for aclamped–free beam under an aerodynamic load with results fromNATASHA. Table 7 shows aerodynamic properties of the beam.Figure 30 shows the force and moment distributions for this beam.The good agreement attests to the correctness of the aerodynamicmodeling in the incremental method.

IV. Example: Instability Under Follower Force

In this section the effect of loading the front wing with a followerforce in the chordwise direction is studied (resembling the thrust

Table 7 Aerodynamic properties

(English units)

Properties Values

cl0 0cl� 2cd0 0.01cm0

0.025cm� �0:25

Velocity 10 ft=sNumber of steps 500

0 2 4 6 8 10−5

−4

−3

−2

−1

0x 10−8

F1

[lb]

NATASHA

Incremental method

0 2 4 6 8 10−0.1

−0.08

−0.06

−0.04

−0.02

0

F2

[lb]

NATASHA

Incremental method

0 2 4 6 8 100

0.02

0.04

0.06

0.08

0.1

F3

[lb]

NATASHAIncremental method

0 2 4 6 8 100

0.5

1

1.5

2

M1

[lb ft

]

NATASHAIncremental method

0 2 4 6 8 10−0.5

−0.4

−0.3

−0.2

−0.1

0

X1 [ft] X1 [ft]

X1 [ft] X1 [ft]

X1 [ft] X1 [ft]

M2

[lb ft

]

NATASHA

Incremental method

0 2 4 6 8 10−0.5

−0.4

−0.3

−0.2

−0.1

0

M3

[lb ft

]

NATASHA

Incremental method

a) b)

c) d)

e) f)

Fig. 30 Plots of a–c) force and d–f) moment distributions in clamped–free wing.

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force of an engine). Figure 31 shows the configuration and Table 8provides the structural properties for the problem at hand. Four forcesare located at x1 � 2:5, 7.5, 12.5, and 17.5 ft, and each has a value ofF lb. Figure 32 shows the eigenvalues analysis of a clamped–free

beam (i.e., only the front wing) under this loading. The first insta-bility happens atF� 40 lb. Figure 33 shows eigenvalue analysis of ajoined-wing configuration (Fig. 31). For this case the sweep angle is50 and the joint position is at x1 � 10 ft. There is a fundamentaldifference between a single-load-path configuration (one beam)and a multiple-load-path configuration (joined-wing). The firstinstability for one beam, a static buckling type instability, occurs atF� 40 lb. However, for a joined-wing configuration the first insta-bility, which happens to also be atF� 40 lb, is a dynamic instability.For this configuration a static instability occurs at F� 74 lb,which is well beyond the first instability and, therefore, not ofsignificance.

V. Conclusions

A new way of analyzing statically indeterminate structures, i.e.,with multiple load paths such as used in joined-wing aircraft, isintroduced. The formulation is based on the fully intrinsic equationsof motion and kinematics and introduces neither singularities norinfinite-degree nonlinearities caused by finite rotation. Instead itmakes use of an incremental form of the governing equations ofmotion and kinematics, augmented by an incremental equation forchange in displacement and orientation. This formulation leads tosolution of a linear system of equations at each incremental loadingstep, thus avoiding the numerical difficulties associated with solvingnonlinear systems of equations such as finding suitable initial guessand convergence. There is also no need to parameterize finite rotationwith orientation angles, Rodrigues parameters, etc. Consequently,there are neither singularities nor infinite-degree nonlinearitiesassociated with finite rotation in the present formulation. The mainadvantageous features of the fully intrinsic equations are thuspreserved. The method is verified and applied to a joined-wingstructure. Results obtained indicate that the method is 1) capable byitself of obtaining the nonlinear static or steady motion solution forthe structural, structural dynamic or aeroelastic behavior of staticallyindeterminate structures and 2) capable of providing an accurate setof initial guesses as needed or desired for a Newton–Raphsonsolution of both statically determinate and indeterminate structures.More structural and aeroelastic studies using this capability will bereported in future work.

Acknowledgments

This work was supported in part by the NASA Dryden FlightResearch Center, with Kevin Walsh as the Technical Monitor.Technical discussions withMayuresh J. Patil of Virginia PolytechnicInstitute and State University, along with use of one of his computercodes, are gratefully acknowledged.

References

[1] Wolkovitch, J., “The Joined Wing: An Overview,” Journal of Aircraft,Vol. 23, No. 3, 1986, pp. 161–178.doi:10.2514/3.45285

[2] Kroo, I., and Smith, S., “Aerodynamic and Structural Studies of Joined-Wing Aircraft,” Journal of Aircraft, Vol. 28, No. 1, 1991, pp. 74–81.doi:10.2514/3.45994

[3] Livne, E., “Aeroelasticity of Joined-Wing Airplane Configurations:Past Work and Future Challenges—A Survey,” 42nd Structures,Structural Dynamics, and Materials Conference, Seattle, WA, AIAAPaper 2001-1370, April 16–19 2001.

[4] Blair, M., and Canfield, R. A., “A Joined-Wing Structural WeightModeling Study,” 43rd Structures, Structural Dynamics, andMaterials Conference, Denver, CO, AIAA Paper 2002-1337,April 22–25 2002.

[5] Rasmussen, C., Canfield, R., and Blair, M., “Joined-Wing Sensor-CraftConfiguration Design,” Journal of Aircraft, Vol. 43, No. 5, 2006,pp. 1470–1478.doi:10.2514/1.21951

[6] Patil, M. J., “Nonlinear Aeroelastic Analysis of Joined-Wing Aircraft,”44th Structures, Structural Dynamics and Materials Conference,Norfolk, VA, AIAA Paper 2003-1487, April 7–10 2003.

[7] Dreibelbis, B., and Barth, J., “Structural Analysis of Joint Wings,”Regional Student Conference, April 2003.

20 25 30 35 40 45 50−100

−50

0

50

100

Force [lb]

Rea

l par

t [ra

d/s]

20 25 30 35 40 45 50−100

−50

0

50

100

Imag

inar

y pa

rt [r

ad/s

]

Fig. 32 Eigenvalue analysis for one-beam configuration.

20 30 40 50 60 70 80 90 100−30

−20

−10

0

10

20

30

Force [lb]

Rea

l par

t [ra

d/s]

20 30 40 50 60 70 80 90 100−20

−10

0

10

20

Imag

inar

y pa

rt [r

ad/s

]

Fig. 33 Eigenvalue analysis for joined-wing configuration.

Table 8 Beam properties for configuration

in Fig. 31

Properties Values

Length of front wing 20Extensional stiffness 1:322 � 106

Torsional stiffness 2:2138 � 103

Out-of-plane bending stiffness 1:72146 � 103

In-plane bending stiffness 1:09890 � 103

Mass per unit length 0.012675

Topview

Front view

Fig. 31 Sketch of configuration under thrustlike loading.

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[9] Green, N. S., Canfield, R. A., Swenson, E., Yu, W., and Blair, M.,“Structural Optimization of Joined-Wing Beam Model with Bend-Twist Coupling Using Equivalent Static Loads,” 50th Structures,Structural Dynamics, and Materials Conference, Palm Springs, CA,AIAA Paper 2009-2644, May 4–7 2009.

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