Geometrically-Exact, Intrinsic Theory forDynamics of Curved and Twisted,

Anisotropic BeamsDewey H. Hodges∗

Georgia Institute of Technology, Atlanta, Georgia 30332-0150

AbstractA formulation is presented for the nonlinear dynamics of initially curved and twisted anisotropic

beams. When the applied loads at the ends of and distributed along the beam are independent ofthe deformation, neither displacement nor rotation variables appear – an “intrinsic” formulation.Like well known special cases of these equations governing nonlinear dynamics of rigid bodiesand nonlinear statics of beams, the complete set of intrinsic equations has a maximum degree ofnonlinearity equal to two. Advantages of such a formulation are demonstrated with a simple exam-ple. When the initial curvature and twist are constant along the beam, two space-time conservationlaws are shown to exist, one being a work-energy relation and the other a generalized impulse-momentum relation. These laws can be used, for example, as benchmarks to check the accuracy ofany proposed solution, including time-marching and finite element schemes. The structure of theintrinsic equations suggests parallel approaches to spatial and temporal discretization. A particu-larly simple spatial discretization scheme is presented for the special case of the nonlinear staticbehavior of end-loaded beams that, by virtue of the Kirchhoff analogy, leads to a time-marchingscheme for the dynamics of a pivoted rigid body in a gravity field. This time-marching schemeconserves both the angular momentum about a vertical line passing through the pivot and totalmechanical energy, while the analogous spatial discretization scheme for the nonlinear static be-havior of end-loaded beams satisfies analogous integrals of deformation along the beam span.Remarkably, a straightforward generalization of these discretization schemes is shown to satisfyboth space-time conservation laws for the nonlinear dynamics of beams when the applied loads areconstant within a space-time element.

∗Professor, School of Aerospace Engineering. Fellow, AIAA. E-mail: Dewey.Hodges@AE.GaTech.edu

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Nomenclaturebi unit vectors fixed in cross-sectional frame of undeformed beam

unit vectors fixed in cross-sectional frame of deformed beamdirection cosine matrix with elements Cij

Bi · bj

⌊1 0 0⌋T

distributed applied force vector per unit length along beamcolumn matrix with elements fi

f ·Bi

cross-sectional stress resultant force vectorcolumn matrix with elements Fi

F ·Bi

cross-sectional inertial angular momentum vectorcolumn matrix with elements Hi

H · Bi

3×3 cross-sectional inertia matrixcross-sectional mass moments and product of inertiaundeformed beam curvature and twist vectorcolumn matrix with elements ki

k · bi

deformed beam curvature and twist vectorcolumn matrix with elementsKi

K · Bi

distributed applied moment vector per unit length along beamcolumn matrix with elementsmi

m ·Bi

cross-sectional stress resultant moment vectorcolumn matrix with elementsMi

M · Bi

vector of cross-sectional inertial linear momentumcolumn matrix with elements Pi

P · Bi

3×3 cross-sectional flexibility coefficient matrices, Eqs. (3) and (4)kinetic energy per unit lengtharbitrary instants in timestrain energy per unit length

Bi

C

Cij

e1

f

f

fi

F

F

Fi

H

H

Hi

I

i2, i3, i23k

k

ki

K

K

Ki

m

m

mi

M

M

Mi

P

P

Pi

R, S, TT

t1, t2U

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u displacement vector of reference linecolumn matrix with elements ui

u · bi

inertial velocity vector of reference linecolumn matrix with elements Vi

V · Bi

⌊0 x2 x3⌋T

position coordinates along b2, b3 from reference line to cross-sectionalmass centroid3×3 identity matrix⌊γ11 2γ12 2γ13⌋T

extensional strain of the reference linetransverse shear strain measures of the reference linecolumn matrix with elements κi

Ki − ki

mass per unit lengthinertial angular velocity vector of deformed beam cross-sectional framecolumn matrix with elements Ωi

Ω · Bi

∂( )/∂t

∂( )/∂x1

antisymmetric 3×3 matrix associated with a column matrix; Eq. (2)

u

ui

V

V

Vi

x

x2, x3

∆

γ

γ11

2γ12, 2γ13

κ

κi

µ

Ω

Ω

Ωi

˙( )

( )′

( )

IntroductionThe intrinsic form of the geometrically-exact, nonlinear equations of equilibrium for beams has

been established for well over a century, going back to Kirchhoff and Clebsch.1 Their classical for-mulation, which includes extension, twist, and bending, was extended to include transverse sheardeformation by Reissner.2 The equations of motion for dynamics of beams appear to have firstbeen written in intrinsic form in Ref. 3. These equations were solved in displacement form. Latera mixed formulation was created based on the intrinsic motion and constitutive equations, bothspatial and temporal,4 with spatial and temporal kinematic equations adjoined by Lagrange multi-pliers. This formulation has been used, for example, in the analysis of fixed-wing aeroelasticity5, 6

as well as helicopter rotor dynamics and aeroelasticity.7, 8 The advantages include low-order non-linearities in the equations of motion and the absence of displacement and finite-rotation variablesfrom the equations of motion. Still, the adjoined kinematical equations must contain displacementand finite-rotation variables in order to be used. Although one may find that having such variables

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appear only in a subset of the equations an attractive feature, the kinematical equations must con-tain infinite-degree nonlinearities unless Euler parameters or the direction cosines themselves areused as finite-rotation variables. These latter approaches add more rotational variables and moreLagrange multipliers, thus creating more unknowns. Finally, the high-order nonlinearities and/oradditional unknowns inherent in these approaches make analytical solutions more laborious.

This paper presents an alternative approach, one in which displacement and finite-rotation vari-ables do not appear. Of course, they or a subset of them can be added to the formulation in orderto make possible their recovery; even so, depending on the problem, it may be possible to avoidnonlinearities of order greater than two and still have fewer equations and unknowns than wouldbe necessary for the usual approach applied to the most general case.

The paper is organized in the following way. First, the intrinsic formulation for beams is re-viewed, including the equations of motion, the spatial and temporal constitutive equations, andthe spatial and temporal kinematical equations needed to close the formulation. From a specialcase of the kinematical equations used in Ref. 4, the intrinsic kinematical equations are derived.Advantages of this formulation are demonstrated with a simple example. The equations are thenused to obtain two space-time conservation laws. The remainder of the paper deals with discretiza-tion schemes. A simple discretization scheme for the nonlinear statics of an end-loaded beam ispresented that, by virtue of the Kirchhoff analogy, gives rise to a time-marching scheme for thedynamics of a rigid body. The relationship of these schemes to integrals of the motion or deforma-tion is then discussed. Finally, these discretization schemes are generalized to form a space-timediscretization scheme, the relationship of which to the space-time conservation laws is presented.

Intrinsic Formulation for BeamsEquations of Motion

Consider an initially curved and twisted beam of length ℓ undergoing finite deformation asshown in Fig. 1. The displacement vector beam of the reference line is denoted by u(x1, t) wherex1 is the running length coordinate along the undeformed beam axis of cross-sectional centroidsr . The orthogonal set of basis vectors for the cross section of the undeformed beam is denotedby bi(x1) where b1 is chosen to be tangent to the reference line. The orthogonal set of basisvectors for the cross section of the deformed beam is denoted by Bi(x1, t) where B1 is not ingeneral tangent to the reference line of the deformed beam R. The direction cosines are denotedas Cij(x1, t) = Bi · bj . It is important to note that when the beam is in its undeformed state thatthese two sets of unit vectors coincide so that C reduces to the identity matrix. The warping ofthe cross section, both in and out of its original plane, is taken into account in the calculation ofthe cross-sectional constitutive law, discussed below, such that the warping displacement is notconstrainable at the boundaries. This constitutive law is thus suitable only for beams with closed

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cross sections.The equations of motion in matrix form, as derived in Ref. 4, are given by

F ′ + KF + f = P + ΩP

M ′ + KM + (e1 + γ)F + m = H + ΩH + V P(1)

where ( )′ denotes the partial derivative with respect to the axial coordinate x1, and ˙( ) denotesthe partial derivative with respect to the time t; column matrices F = ⌊F1 F2 F3⌋T , M =

⌊M1 M2 M3⌋T , γ = ⌊γ11 2γ12 2γ13⌋T , K = ⌊K1 K2 K3⌋T , P = ⌊P1 P2 P3⌋T , H =

⌊H1 H2 H3⌋T , V = ⌊V1 V2 V3⌋T , Ω = ⌊Ω1 Ω2 Ω3⌋T , and e1 = ⌊1 0 0⌋T ; f = ⌊f1 f2 f3⌋T ;m = ⌊m1 m2 m3⌋T ; γ11 is the extensional strain of the reference line; 2γ12 and 2γ13 are thetransverse shear measures; and, finally, the tilde over certain terms denotes an antisymmetricmatrixof the components of the column matrix of the same name, namely,

K =

⎡

⎢⎣0 −K3 K2

K3 0 −K1

−K2 K1 0

⎤

⎥⎦ (2)

All the unknowns are functions of x1 and t. It is helpful to relate the various column matricesto their associated vector quantities and the cross-sectional basis vectors of the undeformed anddeformed beam. The various indexed scalar variables have the following meanings: Fi = F · Bi

with F(x1, t) being the resultant force of all tractions on the cross-sectional face at a particularvalue of x1 along the reference line,Mi = M ·Bi withM(x1, t) being the resultant moment aboutthe reference line at a particular value of x1 of all tractions on the cross-sectional face,Ki = K ·Bi

with K(x1, t) being the curvature of the deformed beam reference line at a particular value ofx1 such that B′

i = K × Bi, Vi = V · Bi with V(x1, t) being the inertial velocity of a point ata particular value of x1 on the deformed beam reference line, Ωi = Ω · Bi with Ω(x1, t) beingthe inertial angular velocity of the deformed beam cross-sectional frame such that Bi = Ω × Bi,Pi = P · Bi with P(x1, t) being the inertial linear momentum of the material points that make upthe deformed beam reference cross section at a particular value of x1, Hi = H · Bi withH(x1, t)

being the inertial angular momentum of all the material points that make up a reference crosssection of the deformed beam about the reference line of that cross section at a particular value ofx1, fi = f · Bi with f(x1, t) being the applied distributed force per unit length, and mi = m · Bi

withm(x1, t) applied distributed moment per unit length. More details are given in Ref. 4.

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Constitutive Equations

It is not necessary to retain all the variables in Eq. (1). For the purposes of the present dis-cussion, the generalized strains and momenta will be eliminated. For small strain, constitutiveequations implied in Ref. 4 are linear and are here written in the form

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

γ11

2γ12

2γ13

κ1

κ2

κ3

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

R11 R12 R13 S11 S12 S13

R12 R22 R23 S21 S22 S23

R13 R23 R33 S31 S32 S33

S11 S21 S31 T11 T12 T13

S12 S22 S32 T12 T22 T23

S13 S23 S33 T13 T23 T33

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

F1

F2

F3

M1

M2

M3

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(3)

Here the coefficients R11, R12, . . ., T33 are cross-sectional flexibility coefficients that can be ob-tained from a variety of means.9, 10 This equation may also be written as

γ

κ

=

[R S

ST T

] F

M

(4)

Similarly, the generalized momentum-velocity relations4 are⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

P1

P2

P3

H1

H2

H3

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

µ 0 0 0 µx3 −µx2

0 µ 0 −µx3 0 0

0 0 µ µx2 0 0

0 −µx3 µx2 i2 + i3 0 0

µx3 0 0 0 i2 i23

−µx2 0 0 0 i23 i3

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

V1

V2

V3

Ω1

Ω2

Ω3

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(5)

where µ is the mass per unit length, x2 and x3 are offsets from the reference line of the cross-sectional mass centroid, and i2, i3, and i23 are cross-sectional mass moments and product of inertia.This equation may also be written as

P

H

=

[µ∆ −µx

µx I

]V

Ω

(6)

where ∆ is the 3×3 identity matrix, and x = ⌊0 x2 x3⌋T .

Closing the Formulation

As developed in Ref. 4, the formulation was closed by using a set of kinematical relations.This was undertaken by means of the introduction of displacement variables u = ⌊u1 u2 u3⌋T

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and a suitable set of angular displacement measures. For the latter, Rodrigues parameters werepreviously used,4 but here the change in orientation is left in terms of the direction cosine matrixC. The kinematical relations are a set of generalized strain-displacement equations that relate γ

and κ to u and C and a set of generalized velocity-displacement equations that relate V and Ω to u

and C.

Generalized Strain-Displacement Equations

The generalized strain-displacement relations are of the form

γ = C(e1 + u′ + ku) − e1 (7)

andκ = −C ′CT + CkCT − k (8)

where κ = K − k and k = ⌊k1 k2 k3⌋T ; ki = k ·bi, and k(x1) is the initial curvature/twist vectorat x1, such that b′

i = k × bi. Thus, k1 is the initial twist and k2 and k3 are the initial curvaturemeasures.

Generalized Velocity-Displacement Equations

The generalized velocity-displacement relations are of a similar form, viz.,

V = Cu (9)

andΩ = −CCT (10)

It should be noted that the terms in the generalized velocity-displacement relations of Ref. 4 relatedto the frame motion have been set equal to zero. This means that the displacement and rotationmeasures are relative to an inertial frame and that the unit vectors bi are fixed in that frame.

Derivation of Intrinsic Kinematical EquationsThe derivation of intrinsic kinematical equations can be undertaken by careful elimination of all

displacement and rotation variables from the generalized strain- and velocity-displacement equa-tions. The detailed operations closely resemble the development of the transpositional relations inRef. 4 and are only outlined here. Starting with differentiation of Ω with respect to x1, one obtains

Ω′ = −C ′CT − CC ′T (11)

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Similarly, differentiation of κ with respect to t yields

˜κ = −C ′CT − C ′CT + CkCT + CkCT (12)

These equations involve C ′, which can be expressed in terms of K using Eq. (8), and C, whichcan be expressed in terms of Ω using Eq. (10). Both equations contain C ′, which can be eliminatedwith algebraic manipulation to yield a single relation between Ω′ and κ that, when simplied, isgiven by

Ω′ + KΩ = κ (13)

In a similar manner, V is differentiated with respect to x1, yielding

V ′ = C ′u + Cu′ (14)

and γ is differentiated with respect to t, leading to

γ = C(e1 + u′ + ku) + C(u′ + ku) (15)

This time u = CT V and that (e1 + u′ + ku) = CT (e1 + γ) are used, and again Eqs. (8) and (10)are used to eliminate C ′ and C, respectively. Now, one can eliminate u′ and find a single relationbetween V ′ and γ, namely,

V ′ + KV + (e1 + γ)Ω = γ (16)

Eqs. (1), (4), (5), (13) and (16) constitute a closed formulation. The beauty of the formulationis striking, especially with regard to the similarity in structure of the left-hand sides of Eqs. (1a)and (13) and Eqs. (1b) and (16). Moreover, this formulation can actually be used in the solutionof a variety of problems. For example, for situations in which the applied loads f and m and theboundary conditions on F , M , V , and Ω are independent of u and C, these equations allow thesolution of nonlinear dynamics problems without finite rotation variables. As these variables arefrequently the source of the highest degree nonlinearities and a possible source of singularities orof the need for Lagrange multipliers or trigonometric functions, to be able to avoid finite rotationvariables can be quite advantageous. Finally, this formulation leads to explicit expressions fortwo conservation laws, one of which is difficult to obtain in other ways. These laws may havepractical applications in development of computational algorithms. Before exploring this aspectof the formulation, an example is presented showing advantages of the formulation for stabilityproblems involving nonconservative forces.

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Example Showing Advantages of the Intrinsic FormulationIn this section the utility of the fully intrinsic formulation will be addressed for problems in-

volving nonconservative forces. The problem of Refs. 11, 12, recently revisited,13, 14 provides aninteresting illustration of the utility of the subject methodology for follower-force problems. In thisproblem, a cantilevered beam is loaded with a transverse follower force at its tip, as shown in Fig. 2.Considering a prismatic and isotropic beam with the mass centroid coincident with the referenceline and the principal axes of the cross section along the b2 and b3 directions, the constitutive lawbecomes ⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

γ11

2γ12

2γ13

κ1

κ2

κ3

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1

EA0 0 0 0 0

0 1

GA20 0 0 0

0 0 1

GA30 0 0

0 0 0 1

GJ0 0

0 0 0 0 1

EI20

0 0 0 0 0 1

EI3

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

F1

F2

F3

M1

M2

M3

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(17)

and the cross-sectional generalized momentum-velocity relations are⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

P1

P2

P3

H1

H2

H3

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

µ 0 0 0 0 0

0 µ 0 0 0 0

0 0 µ 0 0 0

0 0 0 i2 + i3 0 0

0 0 0 0 i2 0

0 0 0 0 0 i3

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

V1

V2

V3

Ω1

Ω2

Ω3

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(18)

These relations, along with an exact linearization of Eqs. (1), (13), and (16), are used to producegoverning equations for the stability of small motions about the static equilibrium state. Ignoringrotary inertia (i2 = i3 = 0) except in the torsional equation, considering infinite axial and shearingrigidities (1/(EA) = 1/(GA2) = 1/(GA3) = 0), and letting V (x1, t) = V (x1) + V (x1, t) and

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similarly for all other variables, one obtains the following equations, linearized in the ( ) quantities:

M ′1 + M 3M2

(1

EI2

−1

EI3

)− (i2 + i3)

˙Ω1 = 0

M ′2 − M 3M1

(1

GJ−

1

EI3

)− F3 = 0

F ′3 +

F 2 M1

GJ−

F 1 M2

EI2

− µ ˙V3 = 0

Ω′1 −

M 3Ω2

EI3

−˙M1

GJ= 0

Ω′2 +

M3Ω1

EI3

−˙M2

EI2

= 0

V ′3 + Ω2 = 0

(19)

where the equilibrium state is governed by three first-order equations given by

F′

1 −M 3F 2

EI3

= 0

F′

2 +M 3F 1

EI3

= 0

M′

3 + F 2 = 0

(20)

The compactness and ease of derivation are noteworthy, as are the beauty and symmetry of the finalequations. To appreciate the simplicity of the above formulation, one should compare it with theequations of Ref. 14. It is clear that the present formulation is considerably simpler. The simplicityof the present formulation for this problem stems from the boundary conditions’ being independentof displacement and orientation variables, a property typical in follower force problems.

It should be noted that the present formulation will provide a simple alternative for some aeroe-lastic flutter analyses, although in such cases at least one measure of orientation may be needed.This normally can be done without the introduction of a full set of orientation angles or parameterssince only one direction needs to be specified to calculate the angle of attack. The direction cosinesfor that direction, a 3×1 column matrix φ, are all one needs to introduce; their spatial derivativescan be related to the curvature measures as φ′ + Kφ = 0.

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Conservation Laws for the Intrinsic FormulationIn this section the conservation laws will be developed. To do so, an inverse form of Eq. (4) is

helpful, which can be written as

F

M

=

[R S

ST T

]−1 γ

κ

=

⎧⎨

⎩

(∂U∂γ

)T

(∂U∂κ

)T

⎫⎬

⎭(21)

where the associated strain energy per unit length can be written as

U =1

2

γ

κ

T [R S

ST T

]−1 γ

κ

(22)

Similarly, the momenta can be written as

P

H

=

[µ∆ −µx

µx I

] V

Ω

=

(∂T∂V

)T

(∂T∂Ω

)T

(23)

with the kinetic energy per unit length given by

T =1

2

V

Ω

T [µ∆ −µx

µx I

] V

Ω

(24)

The approach for development of the conservation laws involves three steps:

1. Pre-multiply the equations of motion, Eqs. (1), by the transpose of either column matrix⌊V T ΩT ⌋T or ⌊(e1 + γ)T KT ⌋T , and integrate the result over space and time.

2. Integrate the resulting expression by parts to bring U and T into evidence.

3. Simplify the expressions using the space-time compatibility equations, Eqs. (13) and (16).

The first conservation law is derived by pre-multiplying the equations of motion by ⌊V T ΩT ⌋T ,which leads to

∫ t2

t1

∫ ℓ

0

V T

(F ′ + KF + f − P − ΩP

)

+ ΩT[M ′ + KM + (e1 + γ) F + m − H − ΩH − V P

]dx1dt = 0

(25)

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Integration by parts yields

∫ t2

t1

∫ ℓ

0

−V ′T F + V T

(KF + f

)

− Ω′T M + ΩT[KM + (e1 + γ)F + m

]dx1dt

+

∫ t2

t1

(V T F + ΩT M)dt

∣∣∣∣ℓ

0

−

∫ ℓ

0

T dx1

∣∣∣∣

t2

t1

= 0

(26)

After use of Eqs. (13) and (16) to eliminate V ′ and Ω′ and a remarkable series of cancellations oneobtains ∫ t2

t1

∫ ℓ

0

(V T f + ΩT m)dx1dt

+

∫ t2

t1

(V T F + ΩT M)dt

∣∣∣∣ℓ

0

=

∫ ℓ

0

(T + U) dx1

∣∣∣∣

t2

t1

(27)

The first term is clearly the work done by all applied forces and moments along the beam betweentimes t1 and t2, and the second term is the work done by the forces and moments at the ends of thebeam between times t1 and t2. The sum of these terms must equal the change in total mechanicalenergy, the term on the righthand side. This conservation law is of the form that one should expect.It is noted that, both in this conservation law and the other one below, ℓ can also be considered asthe length of a finite element.

Another conservation law can be found by again following the procedure spelled out above,except this time pre-multiplying by ⌊(e1 + γ)T KT ⌋T . The result from step 1 is

∫ t2

t1

∫ ℓ

0

(e1 + γ)T

(F ′ + KF + f − P − ΩP

)

+ KT[M ′ + KM + (e1 + γ) F + m − H − ΩH − V P

]dx1dt = 0

(28)

Integration by parts and simplification subject to the restriction that k is constant yields

∫ t2

t1

∫ ℓ

0

[γT P + κT H + (e1 + γ)T

(f − ΩP

)+ KT

(m − ΩH − V P

)]dx1dt

+

∫ t2

t1

(eT1 F + kT M + U

)dt

∣∣∣∣ℓ

0

−

∫ ℓ

0

[(e1 + γ)T P + KT H

]dx1

∣∣∣∣

t2

t1

= 0

(29)

Eliminating γ and κ using Eqs. (13) and (16), after cancellations similar to those in the derivation

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of Eq. (27), one finds

∫ t2

t1

∫ ℓ

0

[(e1 + γ)T f + KT m

]dx1dt

+

∫ t2

t1

(eT1 F + kT M + U + T

)dt

∣∣∣∣ℓ

0

=

∫ ℓ

0

[(e1 + γ)T P + KT H

]dx1

∣∣∣∣

t2

t1

(30)

The first term is a sort of generalized impulse of all forces and moments along the beam be-tween times t1 and t2, while the second at least appears to relate to the generalized impulse appliedat the ends of the beam between times t1 and t2. The presence of the mechanical energy per unitlength in this term, however, makes its identification problematic. The law states that total of thesemust equal to the change in the generalized momentum expression on the righthand side from t1

to t2.Eqs. (27) and (30) are the conservation laws that were to be derived in this section. They can

be used as benchmarks to evaluate the accuracy and self-consistency of any approximate solution,as well as criteria in the development of energy-preserving or -decaying discretization schemes fornonlinear structural dynamics; see, for example, Refs. 15–18 and related work cited therein.

Applications of the Conservation LawsIn this section the utility of these conservation laws will be illustrated as applied to the dis-

cretized analysis of the statics of beams, the dynamics of rigid bodies, and the nonlinear dynamicsof beams.

Kirchhoff Analogy

The Kirchhoff analogy is described by Love1 in terms of a problem posed for rigid-body dy-namics, the governing equations of which are exactly those of an inextensible beam undergoingstatic deformation. While the former is normally posed as an initial-value problem, the latter is atwo-point boundary-value problem. However, mathematically both problems can be posed eitherway.

Nonlinear Statics of Beams

The nonlinear equations of equilibrium for an initially straight and untwisted beam (k = 0)that is rigid in extension and shear (γ = 0) and is subjected only to end loads (f = m = 0) are

F ′ + κF = 0

M ′ + κM + e1F = 0(31)

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The constitutive law can be reduced to

M = T−1κ = Dκ (32)

where D = T−1 is taken as constant along the beam, and the orientation of each cross-sectionalframe can be expressed in terms of C where

C ′ = −κC (33)

The boundary conditions are for the end loads F (0) = F and M(0) = M where F and M areconstant 3×1 column matrices. Here F contains the measures in the Bi basis of a known appliedforce for which the measures in the bi basis remain constant as the beam deforms (i.e., a deadforce). The solution to the force equation is thus easily shown to satisfy the relation

F = CC(0)T F (34)

which indicates, as expected, that the cross-sectional force resultant has a constant direction in theinertial frame. A clamped boundary at the other end yields a two-point boundary-value problemfor which C(ℓ) = ∆.

The first conservation law, Eq. (27), is identically satisfied, while the second conservation law,Eq. (30), when specialized for this static example and for f = m = k = 0, requires that along thebeam

eT1 F + U = const. (35)

This law and its extension to initially curved and twisted beams was discussed by Love1 (art. 260,261) and referred to as an energy integral of the deformation.

In addition to the energy integral of the deformation, another integral can also be shown to exist.One first pre-multiplies the force equation byMT and adds to the result the moment equation pre-multiplied by F T . Since all other terms cancel, the result is that (MT F )′ = 0, or that, along thebeam

MT F = const. (36)

This integral reflects the fact that the section force vector F, which is constant along the beam, iseverywhere perpendicular to changes in the section moment vector M along the beam. In otherwords, the component of section moment that is parallel to the section force remains constant alongthe beam. This integral cannot be derived from either of the two conservation laws derived above.

It is interesting that some discretization schemes will satisfy these in each element while otherswill only approach the satisfaction of them as the mesh is refined. The way in which a particular

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discretization scheme will satisfy these relations may have an effect on the quality of the scheme.To the best of the author’s knowledge, however, this subject has not been investigated.

Dynamics of Rigid Bodies

Kirchhoff discovered that, regarding ( )′ as the time derivative, Eq. (31) is exactly Euler’sdynamical equation for a rigid body that

1. is free to rotate about an inertially fixed point O;

2. is subjected only to a dead force W = Wibi (for example, a gravitational force) passingthrough a point offset from O by zB1 where z andWi are constants;

3. has an inertia matrix for O expressed in the body-fixed reference frame given byD;

4. has inertial angular velocity measures κ when expressed in the body-fixed frame; and

5. has measures of inertial angular momentum about O, expressed in the body frame basis,given byM .

To further elaborate on the loading, it means that in matrix notation F = −zCW , so that

F ′ = −zC ′W = zκCW = −κF (37)

as required by the first of Eqs. (31). The boundary conditions on force and moment for the beamare initial conditions on the applied load and angular momentum for the body. One may specifythe direction cosines C(0) and time-march to an arbitrary time.

Concerning the integrals of deformation in Eqs. (35) and (36), for the rigid-body analog the“const.” in both cases now means constant in time. Eq. (35) now has the significance of conser-vation of total mechanical energy, which is the sum of the rotational kinetic energy of the body Uabout O and potential energy of the applied load eT

1 F . Eq. (36) now stipulates conservation of theangular momentum of the body about a line passing through O and parallel toW.

Discretization Schemes that Satisfy Specialized Conservation Laws

Given Kirchhoff’s analogy, it really does not matter whether one considers discretization inspace of the equations for the statics of beams or discretization in time of the equations for thedynamics of a rigid body. Considering the static behavior of a beam, the most straightforwarddiscretization is a simple finite-difference scheme such that on the interior of a beam segment, Fand M are approximated as averages of the nodal values on the left and right, denoted here byFl and Fr for the force and Ml and Mr for the moment. Thus, Eqs. (31) can be written for one

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segment asFr − Fl

∆x+ κF = 0

Mr − Ml

∆x+ κM + e1F = 0

(38)

where ∆x is the segment length along x1, and

F =Fr + Fl

2

M =Mr + Ml

2

κ =TM

(39)

It can be shown that this discretization scheme is identical to that described for the weakest possiblemixed finite element formulation, which uses piecewise constant shape functions for all unknownson the interior, discrete values of the unknowns at the ends, and piecewise linear test functions.This formulation is described in Ref. 4 and was used subsequently in several contexts.5, 7, 8

This scheme is second-order accurate along the beam. Moreover, for the rigid body this schemerepresents an implicit time-marching algorithm that is second-order accurate. In each case it sat-isfies the appropriate conservation laws. That is, the time-marching algorithm for the rigid bodyis both energy and momentum conserving. This is important in the time domain to guarantee nu-merical stability.15 Exactly what the satisfaction of these integrals implies in the space domain forthe beam statics problem is not as clear. However, failure of a proposed solution to satisfy theseintegrals would cast doubt on its accuracy. Specifically for finite element schemes, while its impli-cations across a single element are less clear, these integrals should be satisfied at least in the limitof a fine mesh.

Discretization Scheme that Satisfies the Space-Time Conservation Laws

A far more interesting situation arises for the nonlinear, space-time treatment of beams. Togeneralize the above to a scheme that satisfies both conservation laws, the same scheme is appliedto a rectagular element of the space-time domain such as that shown in Fig. 3. Here the subscriptsl, a, and r are retained for the “left,” “average” and “right” values of any variable in its spatial (x1)variation; while s, m, and f refer to “starting,” “mean” and “final” values for any variable in itstemporal (t) variation. Using this scheme for a rectangular space-time element, a variable such asF has four nodal values: Fls, Frs, Flf , and Flf . The values along the edges are taken as the average

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of the nodal values on that edge such that

Flm =Fls + Flf

2

Frm =Frs + Frf

2

Fas =Fls + Frs

2

Faf =Flf + Frf

2

(40)

Furthermore, the value on the interior of the element is

Fam =Fas + Faf

2=

Flm + Frm

2

=Fls + Flf + Frs + Frf

4

(41)

Such relations hold for variables M , V , and Ω as well. The intrinsic equations of motion alongwith the intrinsic kinematical equations are written in difference form as

Frm − Flm

∆x+ KamFam + fam −

Paf − Pas

∆t− ΩamPam = 0

Mrm − Mlm

∆x+ KamMam + (e1 + γam)Fam + mam −

Haf − Has

∆t− ΩamHam − VamPam = 0

Vrm − Vlm

∆x+ KamVam + (e1 + γam)Ωam −

γaf − γas

∆t= 0

Ωrm − Ωlm

∆x+ KamΩam −

κaf − κas

∆t= 0

(42)where f ,m, k, and the section constants are taken as constant over the space-time element, and∆t

is the time step.For the special cases of longitudinal and torsional dynamics of rods, one can prove that this

scheme satisfies both space-time conservation laws. The involves simply substituting the dis-cretized equations of motion and kinematical equations directly into the conservation laws, whichare satisfied identically. Unfortunately, for more general cases, because of nonlinear terms andcomplicated algebra, it does not appear to be possible to prove analytically that this discretizationsatisfies the conservation laws. However, the author investigated this by undertaking a numericalsolution of the element equations. A wide variety of loading conditions and values for initial cur-vature and twist were substituted into numerical solutions for the general case of Eqs. (42) andwere verified to satisfy both conservation laws to machine precision in every case tried. Of course,a complete study of the implications of this observation is beyond the scope of this paper.

As with the special cases, the failure of a converged result from any discretization scheme to

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satisfy these laws would certainly cast doubt on its accuracy. It is also possible that these lawscould be used to indicate regions of high error in the space-time domain. While the satisfaction ofthese laws, at least appears to be necessary in some sense, there is no claim here that these lawsare sufficient for accurate time simulation of the nonlinear dynamics of beams. High-frequencydissipation must be built in to any general-purpose scheme for time-marching of such equations.17

AcknowledgementsThe author gratefully acknowledges hours of technical discussion with his esteemed colleague

Prof. Olivier A. Bauchau, who suggested the form of the space-time discretization scheme pre-sented herein. Critical remarks by Drs. Mark V. Fulton, Mayuresh J. Patil, and Gene C. Ruzickawere quite helpful in refining the presentation.

Concluding RemarksA formulation for the nonlinear dynamics of beams, based only on intrinsic equations, has been

presented. The main advantage of this formulation appears to be that the nonlinearities are of lowerdegree. This is because one need not introduce finite rotation variables unless the loads or boundaryconditions depend on orientation. Even when they do, however, it is possible in some cases tocircumvent the introduction of a full set of finite rotation variables. Advantages of this formulationare demonstrated through setting up a nonconservative stability problem. Also, the formulationnaturally leads to two space-time conservation laws, one of which is the usual work-energy relationand the other of which is an apparently new generalized impulse-momentum relation. The work-energy relation is already known to facilitate the construction of energy-preserving and -decayingtime-marching schemes in flexible multi-body dynamics.

The similarity of the spatial and temporal operations in all governing equations to each other,and of those in the equations of motion to those in the kinematical equations, suggests a parallelapproach to space-time discretization. Using the Kirchhoff analogy, it was shown that a time-discretization scheme for the nonlinear dynamics of rigid bodies can be used in space for thenonlinear static behavior of beams. Both of these schemes satisfy integrals of motion or deforma-tion appropriate to the specialized problem, which for the dynamics problem exhibit conservationof energy and of one component of angular momentum. Of these two, only the conservation ofenergy can be derived from one of the space-time conservation laws, but it actually comes from aspecialization of the new conservation law, not the work-energy relation. Remarkably, a straight-forward generalization of these discretization schemes leads to a space-time discretization schemethat satisfies both space-time conservation laws. Evidently, both space-time conservation laws arepotentially useful in checking numerical algorithms and indicating error. Additional uses for themwill be considered in future work.

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References1Love, A. E. H.,Mathematical Theory of Elasticity, Dover Publications, New York, NewYork,

4th ed., 1944.2Reissner, E., “On One-Dimensional Large-Displacement Finite-Strain Beam Theory,” Stud-

ies in Applied Mathematics, Vol. LII, No. 2, June 1973, pp. 87 – 95.3Borri, M. and Mantegazza, P., “Some Contributions on Structural and Dynamic Modeling of

Helicopter Rotor Blades,” l’Aerotecnica Missili e Spazio, Vol. 64, No. 9, 1985, pp. 143 – 154.4Hodges, D. H., “A Mixed Variational Formulation Based on Exact Intrinsic Equations for

Dynamics of Moving Beams,” International Journal of Solids and Structures, Vol. 26, No. 11,1990, pp. 1253 – 1273.

5Patil, M. J., Hodges, D. H., and Cesnik, C. E. S., “Nonlinear Aeroelastic Analysis of Com-plete Aircraft in Subsonic Flow,” Journal of Aircraft, Vol. 37, No. 5, Sept.-Oct. 2000, pp. 753 –760.

6Hodges, D. H., Patil, M. J., and Chae, S., “Effect of Thrust on Bending-Torsion Flutter ofWings,” Journal of Aircraft, Vol. 39, No. 2, Mar.-Apr. 2002, pp. 371 – 376.

7Hodges, D. H., Shang, X., and Cesnik, C. E. S., “Finite Element Solution of NonlinearIntrinsic Equations for Curved Composite Beams,” Journal of the American Helicopter Society,Vol. 41, No. 4, Oct. 1996, pp. 313 – 321.

8Shang, X., Hodges, D. H., and Peters, D. A., “Aeroelastic Stability of Composite HingelessRotors in Hover with Finite-State Unsteady Aerodynamics,” Journal of the American HelicopterSociety, Vol. 44, No. 3, July 1999, pp. 206 – 221.

9Borri, M., Ghiringhelli, G. L., and Merlini, T., “Linear Analysis of Naturally Curved andTwisted Anisotropic Beams,” Composites Engineering, Vol. 2, No. 5-7, 1992, pp. 433 – 456.

10Yu, W., Hodges, D. H., Volovoi, V. V., and Cesnik, C. E. S., “On Timoshenko-Like Modelingof Initially Curved and Twisted Composite Beams,” International Journal of Solids and Structures,Vol. 39, No. 19, 2002, pp. 5101 – 5121.

11Barsoum, R. S., “Finite Element Method Applied to the Problem of Stability of a Nonconser-vative System,” International Journal for Numerical Methods in Engineering, Vol. 3, 1971, pp. 63– 87.

12Wohlhart, K., “Dynamische Kippstabilitat eines Plattenstreifens unter Folgelast,” Z. Flug-wiss., Vol. 19, No. 7, 1971, pp. 291 – 298.

13Hodges, D. H., “Lateral-Torsional Flutter of a Deep Cantilever Loaded by a Lateral FollowerForce at the Tip,” Journal of Sound and Vibration, Vol. 247, No. 1, Oct. 11, 2001, pp. 175 – 183.

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14Detinko, F. M., “Some Phenomena for Lateral Flutter of Beams under Follower Load,” Inter-national Journal of Solids and Structures, Vol. 39, No. 2, January 2002, pp. 341 – 350.

15Simo, J. C. and Wong, K. K., “Unconditionally Stable Algorithms for Rigid Body Dynamicsthat exactly Preserve Energy and Momentum,” International Journal for Numerical Methods inEngineering, Vol. 31, 1991, pp. 19 – 52.

16Simo, J. C. and Tarnow, N., “The discrete energy-momentummethod. Conserving algorithmsfor non-linear dynamics,” ZAMP, Vol. 43, 1992, pp. 757 – 792.

17Bauchau, O. A. and Theron, N. J., “Energy Decaying Schemes for Nonlinear Beam Models,”Computer Methods in Applied Mechanics and Engineering, Vol. 134, 1996, pp. 37 – 56.

18Bauchau, O. A. and Bottasso, C. L., “On the design of energy preserving and decayingschemes for flexible, non-linear multi-body systems,” Computer Methods in Applied Mechanicsand Engineering, Vol. 169, 1999, pp. 61 – 79.

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List of Figures1 Schematic of initially curved and twisted beam undergoing finite deformation, in-

cluding cross-sectional warping . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Schematic of beam under transverse follower force . . . . . . . . . . . . . . . . . 233 Schematic of space-time finite element . . . . . . . . . . . . . . . . . . . . . . . . 24

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B 1

B 2

B 3

Deformed State

Undeformed State

r

R

R

s

r

u

x1

b 1

b 2

b 3

Figure 1: Schematic of initially curved and twisted beam undergoing finite deformation, includingcross-sectional warping

22 OF 24

x u, , a

u , a

u , a

P B

Figure 2: Schematic of beam under transverse follower force

23 OF 24

x

t

ls

lf rf

rs

lm rm

af

as

am

Figure 3: Schematic of space-time finite element

24 OF 24