Verification of Variational–Asymptotic Sectional Analysisfor Initially Curved and Twisted Beams

Ravi Kumar Kovvali∗ and Dewey H. Hodges†

Georgia Institute of Technology, Atlanta, Georgia 30332-0150

DOI: 10.2514/1.C031539

The variational–asymptotic method is a powerful mathematical method to construct asymptotically correct

models of dimensionally reducible structures. Variational–asymptotic method is used to rigorously split a general

three-dimensional nonlinear elasticity problem into a two-dimensional linear cross-sectional analysis and a one-

dimensional nonlinear beam analysis. This operation results in the replacement of a three-dimensional model with a

reduced-order model in terms of an asymptotic series of certain small parameters inherent to the structure.

Variational–Asymptotic Beam Sectional Analysis (VABS) is an engineering software package that uses the

variational–asymptoticmethod to perform a generalized cross-sectional modeling of any initially twisted and curved

anisotropic beam having arbitrary cross-sectional geometry, with transverse shear and Vlasov refinements. Several

new cases of verification are presented. The capability of VABS to recover pointwise three-dimensional stressfields is

demonstrated for an orthotropic cantilever and verified against known solutions from anisotropic elasticity theory.

Next, a form of verification is proposed for beams with either initial twist or curvature by using two different

modeling approaches: one based on curvilinear coordinates and the other on Cartesian coordinates for initially

twisted beams or piecewise-Cartesian coordinates for initially curved beams. Excellent correlation of results between

the two approaches indicates the correct modeling of effects of initial curvature and twist in both cross-sectional and

global beam analyses. Further, a numerical verification of results for initially twisted and/or curved beams is

provided by comparing values of one-dimensional variables extracted from three-dimensional finite element

analyses with results from a one-dimensional, nonlinear beam analysis (using cross-sectional constants fromVABS).

Finally, numerical examples are shown to highlight the importance of including curvature measures in the cross-

sectional modeling by comparing with finite element results.

I. Introduction

B EAMS come under the category of dimensionally reduciblestructures, with one dimension (wavelength of deformation and

radius of curvature/twist) much larger than the cross-sectionaldimensions. Typical examples in the aircraft industry include highaspect-ratio wings and helicopter rotor blades. Because of thecomplexity of the interior region of such beamlike structures, theiranalysis and designmay be thought to be best carried out using three-dimensional (3-D) finite element analyses (FEA) given that suchanalyses provide for high-fidelity modeling of complex geometriesand accurate, reliable results. However, there are some obviousdrawbacks: invariably all FEA tools are computationally expensivewhen compared with beam-modeling tools, often by 2–3 orders ofmagnitude. Another aspect often overlooked is how labor-intensiveeven the modeling process can become, especially for complexlayups and geometries. An obvious and popular alternative hasbeen conventional beam-modeling techniques. Although they arecomputationally less expensive, the results are seldom satisfactory,especially for composite structures, which are frequently the cases ofinterest. An ideal methodology would combine the relativelyinexpensive nature of beam-modeling tools with the ability toachieve high-fidelity in modeling procedures á la finite elementanalyses, resulting in an efficient, reliable analysis tool with no ad

hoc kinematic assumptions typically associated with standard beam-modeling tools.

Such an asymptotically exact methodology, called the Varia-tional–Asymptotic Beam Sectional Analysis (VABS), has beendeveloped over the last two decades with the objective to modelrealistic rotor blades, creating the best possible set of elastic constantsfor an equivalent beam analysis from a detailed representation of thecross-sectional plane. Additionally, it can also recover detailed stressand strainfields based on inputs from a one-dimensional (1-D) globalanalysis. It uses the variational–asymptotic method [1] (VAM) as itsmathematical basis. TheVAM is used to split a general 3-D nonlinearelasticity problem for a beamlike structure into a two-dimensional (2-D) linear cross-sectional analysis and a 1-D nonlinear beam analysisby taking advantage of certain small parameters inherent to thestructure (typically a=l and a=R, where a is a characteristic cross-sectional dimension, l is the wavelength of deformation and R is theradius of curvature/twist). VAM applies an asymptotic expansion interms of these small parameters of the energy functional instead ofthe system of differential equations [2–5], thereby making themodeling procedure more compact, less cumbersome andvariationally consistent. (The term “variationally consistent” is usedhere to mean that all unknown variables follow naturally from anappropriate minimization problem, based in turn on a variationalprinciple.) The development ofVABSwas first mentioned in [6]. Thecross-sectional modeling capability was later extended to includerefinements such as transverse shear and effects of initial curvatureand twist [7–10].

Verification studies for prismatic, isotropic beams have beencarried out in sufficient detail [11,12], but the authors are not aware ofrigorous verification studies for initially twisted and curved beams.Even for orthotropic prismatic beams, there has been extensivenumerical verification but none against solutions from the theory ofelasticity. The present effort presents verification against elasticitysolutions for orthotropic prismatic beams. A recently updated energytransformation to the generalized Timoshenko form by VABS [13]was shown to affect certain cases of prismatic beams and, almostalways, all beams with initial curvature and twist. The present paper

Presented at the Structures, Structural Dynamics and MaterialsConference, Denver, 4–7 April 2011; received 8 June 2011; revisionreceived 12 November 2011; accepted for publication 14 November 2011.Copyright © 2011 by the authors. Published by the American Institute ofAeronautics andAstronautics, Inc., with permission. Copies of this papermaybe made for personal or internal use, on condition that the copier pay the$10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 RosewoodDrive, Danvers, MA 01923; include the code 0021-8669/12 and $10.00 incorrespondence with the CCC.

∗Graduate Research Assistant, Daniel Guggenheim School of AerospaceEngineering.

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

JOURNAL OF AIRCRAFT

Vol. 49, No. 3, May–June 2012

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seeks to verify these capabilities of VABS, viz., the correct modelingof how initial curvature and twist affect global behavior of beams.Several cases are presented, which are essential to demonstrate thatVABS can produce results with an accuracy comparable with that of3-D finite element codes (and theory of elasticity solutions, whenavailable).

The purpose of this paper is to present results from studies thatseek to verify VABS; hence, no details regarding the theoreticalfoundation for initially twisted/curved beams are repeated here.Details of the formulation may instead be found in [7,10,13–15]. Seework cited by Hodges [15] for a more complete history of thedevelopment and verification of VABS up to 2006 as well as thecorresponding 1-D theory of beams. However, Fig. 1 gives anoverview of the beam-modeling procedure. Note that in all theverification cases considered, the cross-sectional analysis tool VABSis used to construct a generalized Timoshenkomodel, represented bya 6 � 6 stiffness matrix that, along with the 6 � 6 cross-sectionalmassmatrix, is then input into the 1-D geometrically exact, nonlinearequations of motion for beams [14]. Results from such a 1-D globalanalysis, such as 1-D displacements, cross-sectional stress resultants,and generalized strains can then be fed back to VABS in order toperform a 3-D recovery analysis to calculate pointwise displacement,strain and stress fields from the global behavior of a 1-D global beamanalysis.

II. Bending and Torsion of an Orthotropic Cantilever

Verifications against solutions from the theory of elasticity aresuperior to verifications against numerical solutions, becausenumerical solutions such as 3-DFEAhave issues such as locking thatmight be significant depending on the type of analysis and choice ofelements. Analytical modeling of isotropic bars with elliptical andrectangular cross sections carried out in [12] showed that VABSresults correlate well with those from theory of elasticity. Here, weconsider the modeling of an orthotropic cantilevered strip beamsubjected to, in turn, a unit tip force and a unit tip torque. Analyticalsolutions exist for such orthotropic beams (or a single-layer

ply, as the case may be), as derived by St. Venant and given byLekhnitskii [16].

To this end, consider a cantilever with material properties anddimensions, as shown in Table 1. The beam ismodeled as a single plywith 0 deg in VABS and is subjected to F3 � 1 lb. (bending) andM1 � 1 lb � in: (torsion). The structural and inertial properties ascalculated by VABS for a generalized Timoshenko model [10] areprovided in Table 1. The beam’s reference axis is along x1, noting thatspecification of the length of the beam is not necessary as the solutionis independent of the length (as long as it is sufficiently slender to beclassified as a beam and the cross section considered is not near theends of the beam). To compare results from a series solution based onlinear theory of elasticity in [16] (Eqs. 49.10 and 31.16), we plot therecovered 3-D shear stress fields from VABS (�12 and �13), resultingfrom the applied sectional loading, along the depth and width of saidcross section passing through the centroid.

When the beam is subjected to a tip force, we obtain, as expected, aparabolic variation of the shear stress �13 through the thickness(along x3) as indicated in Fig. 2a. However, since this beam is loadedalong x3 instead of along x2, we also need to consider the variation of�13 across the width (where there would be negligible variationfor “in-plane” loading). As indicated by Fig. 2b, VABS correctlycaptures this variation too: it stays nearly constant across the widthbut shoots up significantly at the ends (while maintaining a parabolicvariation along x3 at every station along x2). It is also worthy to notethat the �12 distribution along the width of the cross section is quitesignificant and cannot be neglected, as shown in Fig. 2c for x3 � 0:05in. This trend is also antisymmetric about the x2 axis to satisfy a zero-net-force condition along x2.

The beam is now subjected to a tip torque, and again thedistributions of the shear stresses throughout the cross section arecompared. In particular, we look at variations of �12 and �13. Asexpected, Fig. 3a shows that �12 varies linearly through the thicknesseven for an orthotropic beam, at least at points away from the ends.However, this is not the only stress quantity that is significant. It turnsout that the �13 distribution is also very significant at the faces wherex2 ��1 in:, which is shown in Fig. 3b. Again, the correlationbetween VABS and the analytical solution is quite satisfactory.

III. Verification for Beams with Initial Curvatureand Twist: Approach 1

The methodology for verification proposed in this section isapplicable to beams with both initial twist and curvature, albeit, withminor differences; in either case, however, the method of analysis isstrongly dependent on the choice of coordinates used. In particular,we analyze the beam once using a curvilinear coordinate system andonce using a Cartesian coordinate system, making sure the twoanalyses are consistent. Care must be taken, however, to understandthe meaning and implications of selecting one system over the other,which is explained briefly below.

For initially twisted beams, analysis using curvilinear coordi-nates involves choosing a coordinate system such that the local

1-D displacements, generalized strains, & stress resultants

2-D cross-sectional analysis (linear)

2-D cross-sectional elastic and inertia

constants

1-D beam analysis (nonlinear)

2-D warping & strain recovery relations

3-D recovery analysis

3-D stress, strain, & displacement fields

cross-sectionalgeometry, 3-D elastic constants, & density

initial twist & curvature

loads & boundary conditions

Fig. 1 Beam analysis procedure.

Table 1 Material properties, dimensions, and elastic constantsa

Material properties VABS output

E11, psi 1:873 � 107 S11 (lb) 3:746 � 106

E22, psi 1:364 � 106 S22 (lb) 1:247 � 105

G12, psi 0:7479 � 106 S33 (lb) 9:093 � 104

G13, psi 0:6242 � 106 S44 (lb � in:2) 4:819 � 102

�12 0.30 S55 (lb � in:2) 3:122 � 103

�23 0.30 S66 (lb � in:2) 1:249 � 106

� (lb � sec2=in:4) 1:450 � 10�4 � (lb � s2=in:2) 2:956 � 10�5

Width along x2, in. 2.0 i2 (lb � s2) 2:464 � 10�8

Thickness along x3, in. 0.1 i3 (lb � s2) 9:856 � 10�6

aThe 6 � 6 stiffness matrix obtained from VABS has stiffness values Sij(i; j� 1; 2; :::; 6) arranged as 1: extension; 2,3: shear; 4: torsion; 5,6: bending; andinertia properties �: mass per unit length; and i2; i3: cross-sectional mass moments ofinertia.

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cross-sectional coordinates follow the twist, thereby continuouslyrotating along the length of the beam. For this set of test cases weassume that the beam is uniform along the span, and the twist is small(such that jak1j � 1). Such a choice of coordinates requires a singlecross-sectional analysis for the stiffness constants, and the initialtwist measure k1 enters both the 1-D global analysis and the 2-Dcross-sectional analysis, as shown in Chapters 4 and 5 of [15].

On the other hand, use of Cartesian coordinates implies that thecross-sectional coordinates are along fixed directions in space andthus do not follow the twist of the beam. Therefore, since the cross-sectional geometry varies along the length of the beam, one needs to

carry out numerous sectional analyses along the beam to account forvarying sectional properties (due to varying orientation) as seen fromthis fixed Cartesian system. The entire beam is discretized into anumber of segments, and the sectional properties are only evaluatedat the ends of each segment and interpolated linearly within it. Asevident, in both the 1-D global analysis and the 2-D cross-sectionalanalysis, k1 is set to zero.

For initially curved beams, analysis using curvilinear coordinatesrequires choosing a coordinate system wherein the local cross-sectional coordinates continuously rotate with the beam referenceline (but do not rotate when viewed from a plane normal to the beam

0.04 0.02 0.02 0.04x3

1

σ 13

a) σ13 vs x3 ; x2 = 0 in.1.0 0.5 0.0 0.5 1.0

x2

8

9

10

11

12

13σ 13

b) σ13 vs x2 ; x3 = 0 in.

1.0 0.5 0.5 1.0x2

6

4

2

2

4

6

σ 12

c)σ12 vs x2 ; x3 = 0.05 in.

2

3

4

5

6

7

Fig. 2 Comparison of 3-D stresses �12 and �13 for bending, VABS vs theory of elasticity.

0.04 0.02 0.02 0.04x3

150

100

50

50

100

150

σ 12

a) σ12 vs x3 ; x2 = 0 in.

0.04 0.02 0.02 0.04x3

20

40

60

80

100

σ 13

b) σ13 vs x3 ; x3 = 1 in.Fig. 3 Comparison of 3-D stresses �12 and �13 for torsion, VABS vs theory of elasticity.

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reference line). This approach, again, requires only a single cross-sectional analysis for the stiffness constants, and the initial curvaturemeasure(s) k2 and/or k3 enter both the 1-D global analysis and the2-D cross-sectional analysis [15]. Unlike for beamswith initial twist,an analysis using piecewise-Cartesian coordinates also requires justone cross-sectional analysis, but with k2 and k3 set to zero, since wemodel the entire curved geometry as being piecewise linear andhence locally prismatic. There is also a difference in the 1-D globalanalysis procedure: Instead of modeling it as a single beam, the beamis thought to bemade up of numerous prismatic beams joined to eachother to make up a curved beam. Therefore, again k2 � k3 � 0. Thepiecewise-Cartesian approach entails more approximations than thecurvilinear approach, but increasing the number of beam elementsproduces very similar results for a wide variety of cases irrespectiveof the choice of coordinate system. For instance, if the mode ofdeformation considered is primarily bending, then approach 1 is apractical form of verification, whereas a case with predominantlytorsional deformation will render this approach invalid forverification purposes, although it does bring out certain importantaspects pertaining to these methodologies. The exact nature of theseadditional approximations and their implications for predictivecapability are discussed in a later section.

The crux of this methodology is as follows: In cases of both initialtwist and curvature, whereas the curvilinear coordinates approachrequires that the curvature measure numbers k1, k2 and k3 enter theanalysis, a modeling procedure using Cartesian/piecewise-Cartesiancoordinates does not use them. Therefore, invariance in the 1-Dglobal analyses results, such as static deflections under load andbeam natural frequencies, can be taken as an indication that initialcurvature and twist effects have been modeled accurately in the 2-Dcross-sectional analysis, as well as in the 1-D global analysis.

IV. Verification for Beams with Initial Curvatureand Twist: Approach 2

The second and a more rigorous form of verification would be tocompare results from VABS (in tandem with a geometrically exact1-D beam analysis code such as NATASHA [17] (NonlinearAeroelastic Trim and Stability of HALE Aircraft) with 1-Dinformation extracted from a 3-D finite element analysis (such asABAQUS). For example, to compare the static tip deflection under adead load, the 1-D displacement variable from NATASHA iscompared with the average cross-sectional tip-deflection fromABAQUS. Comparing the beam natural frequencies againstABAQUS can be viewed as a verification of both the static anddynamic behavior modeling capabilities of a beam approach usingcross-sectional properties from VABS. It should be emphasized thatthe 1-D global analysis is geometrically exact; hence, the accuracy ofthe results would then solely depend on the accuracy of the stiffnessconstants provided to the beam analysis tool by VABS. Hence, goodcorrelation between 1-D global results and 3-D finite element resultswould automatically imply accurate 2-D cross-sectional modeling.

V. Results and Discussion

Several verification cases are now taken up following both theapproaches described, and comparisons between the 1-D and 3-Danalyses are conducted systematically. In this section, results areobtained and discussed for the following cases: initially twistedisotropic beams (case 1), initially curved isotropic beams (case 2),initially twisted anisotropic beams (case 3), initially curvedanisotropic beams (case 4), and a helical spring (case 5).

A. Case 1: Initially Twisted Isotropic Beams

Results from [13] showed the prediction of natural frequencies ofan initially twisted isotropic rectangular section and verified byapproach 1. However, no static deflection results were shown; hence,supplementing results from [13] with results from the current sectioncompletes the discussion. An example of a statically loaded,isotropic, initially twisted beam is now shown, verified byapproach 1. Here, we consider an isotropic square cross section of

side 0.5 in. The length of the beam is 10 in. and has an initial twist ofk1 � 0:1 rad=in: Young’s modulus is E� 2:6 � 107 lb=in:2,Poisson’s ratio is �� 0:3, and the mass density �� 7:3�10�4 lb � s2=in:4 The beam is cantilevered and subjected toexaggerated gravitational loading of 500 times its normal value.Figure 4 shows vertical and lateral deflections of the beam byanalyses using curvilinear andCartesian coordinates. As can be seen,the curves show excellent agreement.

B. Case 2: Initially Curved Isotropic Beams

An initially curved, cantilevered, isotropic beam is chosenwith thesame material properties as in case 1. However, the cross section isrectangular measuring 2 in: � 1 in: The length of the beam is now20 in. and has an initial curvature of k2 � 0:05 rad=in: Again,verification is carried out by approach 1, where a set of curvilinearcoordinates and piecewise-Cartesian coordinates are chosen to carryout a static and a dynamic analysis on the beam and the resultscompared. First, the beam is subjected to a uniform exaggeratedgravitational loading 500 times its normal value. The beamundergoes axial and transverse displacements, as shown in Fig. 5.Again, clearly, both analyses show excellent agreement with oneanother.

Natural frequencies for the same beamwere then calculated with asmall modification: the clamp at the end is removed and the beam ismade free–free. Results are plotted as a function of k2 varying from�0:05 rad=in: to �0:05 rad=in: As expected, the curves aresymmetric about k2 � 0 due to material isotropy and overallsymmetry. With the understanding that correct prediction of naturalfrequencies can be interpreted as an accurate representation of theelastic and dynamic behavior of the beam as influenced by initialcurvature, Fig. 6 clearly indicates the accuracy of modeling initialcurvature effects in VABS.

C. Case 3: Initially Twisted Anisotropic Beams

The case considered is a structurally coupled composite beamwithrectangular cross section manufactured from AS4/3501-6 graphiteepoxy. The material properties and outer dimensions of the beam areprovided in Table 2. A prismatic cantilevered beam of thisconfiguration has been experimentally tested by Minguet andDugundji [18,19]. Note that a prismatic beam with the givenconfiguration will result in bending-twist coupling. In the presentcase, we consider an initial twist k1 � 0:05 rad=in:, which furtherintroduces extension-torsion and shear-bending couplings (alongwith weak shear-torsion and extension-bending couplings). Thebeam is free on either end, and a free–vibration analysis is carried out.The decision to model the beam with free–free boundary conditionsstems from the fact that results may vary considerably depending onhow a boundary condition is applied in a finite element analysis, asdiscussed by Yu [20]. For example, there is an infinitely large varietyofways tomodel a clamped end in a 3-Dfinite element analysis, noneof them necessarily being “correct.” Theway a boundary is modeleddoes not affect long beams asmuch it does short ones, with short stripbeams affected more than short regular beams. Therefore, choosingfree–free boundary conditions eliminates the possibility thatboundary conditions can contribute to differences, especially forhigher frequencies.

Unlike the previous cases examined, we now verify VABS usingapproach 2. The beam is modeled using curvilinear coordinates inVABS and NATASHA and compared with finite element resultsgenerated using ABAQUS. Figure 7 compares the natural frequen-cies calculated for various values of k1 ranging from0 to 0:05 rad=in:The solid lines are results produced byVABS and the symbols denoteresults from ABAQUS. Note that due to a symmetric layup and thenature of the boundary conditions, results would be identical forpositive or negative values of initial twist, and varying k1 starting atzero will suffice. The first nine modes shown are in the order of first,second and third flapwise bending (F1, F2, F3), first torsion (T1),fourth flapwise bending (F4), second torsion (T2), fifth and sixthflapwise bending (F5, F6), and third torsion (T3). As should be thecase, initial twist most strongly affects the torsional frequencies.

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Fig. 4 Static deflections of an isotropic initially twisted beam; k1 � 0:1, L� 10 in:, and a� 0:5 in:

Fig. 5 Static deflections of an initially curved isotropic beam; k2 � 0:05, L� 20 in:, and a� 2 in:

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While one may not be interested in so many modes of vibration, it isworth noting that even the sixth flapwise bending mode differs fromFEA results only by about 1.5%. This is in spite of the parameter a=l(whichwe assumed to be small) not being negligibly small comparedwith unity because lwould approximately be one-sixth of the beamlength. The rest of the frequencies are all within 0.5% for all values ofk1. With the correlation being this excellent, it is also relevant toconsider the relative costs of these methods of analysis. The

ABAQUS model was meshed with 4,500 20-noded brick elements,and the code took roughly 180 s to run. On the other hand, VABStakes roughly 0.1 s to perform the cross-sectional analysis, whileNATASHA takes roughly 3 s to obtain converged results even whenusing a very large number of elements (roughly 200 along thelength). Further, taking into account the relative ease of both pre- andpostprocessing of results in VABS/NATASHA, the gains of using a2-D/1-D methodology become apparent. It should be noted thatNATASHA is aMATLAB-based code and is therefore not optimizedfor speed, so actual gains in efficiency when using a compiledprogram for the beam analysis would be much greater than thosereported here.

D. Case 4: Initially Curved Anisotropic Beams

The geometry and the material properties for this case remainexactly the same as in the previous casewith the difference being thatthe beam is now initially curved rather than initially twisted. Thismodifies the stiffness matrix by introducing shear-twist andextension-bending couplings (along with weak extension-twist andshear-bending couplings). The beam has a striplike geometry with

Fig. 6 Natural frequencies of an initially curved free–free isotropic beam; L� 20 in: and a� 2 in:

Table 2 Material properties and dimensions

Material properties Outer dimensions

E11 2:059 � 107 psi Width 1.1820 in.E22, E33 1:42 � 106 psi Thickness 0.0579 in.G12, G13 8:70 � 105 psi Length 22.047 in.G23 6:96 � 105 psi Layup �45=03s

�12, �13 0.30 k1 0 � 0:05 rad=in:�23 0.34 k2 0 � 0:05 rad=in:� 1:4784 � 10�4 lb � s2=in:4 —— ——

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

100

200

300

400

500

600

700

800

Fig. 7 Natural frequencies of an initially twisted free–free anisotropic beam; VABS vs ABAQUS; L� 22:047 in: and a� 1:182 in:

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curvature k2 out of its plane, i.e., along the soft-bending direction.One of the reasons this casewas chosen is because for striplike beamsk2 affects the results more than k3. A more important reason is thatthere exist elasticity solutions for isotropic strip beams with in-planecurvature [21], against which VABS has be verified [22]. However,no analytical solutions exist for anisotropic strip beams curved out-of-plane. Thus, one has to resort to 3-D finite element procedures,against which the present case is intended to be verified. Unlike theprevious case, however, both approaches mentioned previouslywill be applied to this example. Figure 8 compares naturalfrequencies calculated by VABS/NATASHA (solid lines) againstfinite element results generated using ABAQUS (symbols). Most ofthe characteristics described for case 3 hold true including the relativecosts of the two analyses and the ordering of modes shown.

Figure 9 compares natural frequencies obtained via analyzingwithcurvilinear and piecewise-Cartesian coordinates using VABS andNATASHA (represented by solid lines and symbols, respectively).As one increases the number of elements one can observe that bothanalyses yield nearly identical results, thus verifying the capabilitiesof VABS. Just as in case 3, of all the modes of vibration, the torsional

frequencies are most strongly affected because of the initialcurvature. While the presence of initial curvature does modify theequations governing the 1-D global beam analysis (and hence thenatural frequencies), it can also be shown that it is equally importantto take into account how this curvaturemeasure affects the 2-D cross-sectional analysis (and hence modifies the sectional stiffnessproperties that appear in the 1-D global analysis) in order to predictthis variation with curvature completely and correctly. For example,Table 3 shows the predicted torsional frequencies by ABAQUS andVABS/NATASHA for an initial curvature measure k2 � 0:05. In the

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

100

200

300

400

500

600

700

Fig. 8 Natural frequencies of an initially curved free–free anisotropic beam; VABS vs ABAQUS; L� 22:047 in: and a� 1:182 in:.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

100

200

300

400

500

600

700

Fig. 9 Natural frequencies of an initially curved free–free anisotropic beam: curvilinear vs piecewise-Cartesian coordinates approaches; L� 22:047and a� 1:182.

Table 3 Effect of including curvature measure inVABS on torsional natural frequencies (in hertz)

Mode ABAQUS VABS VABSprismatic VABSpc

Torsion 1 177.60 178.31 172.14 171.78Torsion 2 373.00 373.26 360.34 360.01Torsion 3 609.46 608.10 585.72 585.17

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third column are results from a 1-D global analysis with stiffnessconstants based on a prismatic beam cross-sectional analysis.In other words, the curvature measure(s) were neglected in VABSand the natural frequencies calculated. As can be seen, there is aconsiderable difference in results; and the prediction based onprismatic stiffness constants may lead to gross inaccuracies incertain cases.

One can observe that predictions of the torsional modes betweenthe curvilinear and piecewise-Cartesian-coordinate methodologiesslowly diverge as k2 increases. This would be a manifestation of thefact that the piecewise-Cartesian coordinates approach entails moreapproximations than the curvilinear approach, viz., that initialcurvatures (and twist) affect the cross-sectional modeling. For a verylarge number of elements, the Cartesian approach would tend to aprocedure equivalent to just neglecting the curvature measures inVABS. This is evident by looking at the fourth column of Table 3.It is important to understand this limitation of the methodology. Forsimple static-load cases, this methodology works very well as theeffect of k2 on flapwise deformation is not as significant. A detailedstudy of the importance of including curvature measures in the cross-sectional modeling can be found in [7], where it is shown thatcorrections to the stiffnessmodel due to initial twist and curvature arevital for proper representation of anisotropic beams.

E. Case 5: A Helical Spring

Wenow consider the case of a beam that is simultaneously initiallytwisted and curved. Imagining a really long beam with these twocurvature measures results in what looks like a helical spring, shownin Fig. 10a. Such a “beam” is modeled in ABAQUS and VABS withan isotropic circular cross section with k1 ≠ 0 and either k2 ≠ 0 ork3 ≠ 0. The geometric andmaterial properties of the helix are shownin Table 4. Using these specifications and the Frenet–Serret formulasfor continuous, differentiable space curves, one can computethe initial curvature measures as k1 � 0:749 rad=in: and k2�4:967 rad=in: (Note that k3 could have been used instead of k2: thechoices are equivalent. Also, no initial strains are assumed.)

Figure 10b compares the natural frequencies of a hanginghelical spring (clamped–free boundary conditions) obtained from the

1-D/2-D methodology to a 3-D finite element analysis in ABAQUS.The lowest modes are predicted the most accurate, the accuracydeclining with increasing mode number. However, all thefrequencies shown in the bar graph predicted by VABS are within1% of the results fromABAQUS, yet at a far less computational cost.For example, the ABAQUS model was meshed with 11,87220-noded hexahedral elements, and the analysis took just under 180 sto complete (on a Core 2 Duo processor). On the other hand, VABSand NATASHA obtained results within 1% of the ABAQUS resultusing far fewer degrees of freedom andmuch less computing time: inthis case, 13 s. Hence, this serves as another verified case for VABSwith regard to initial twist and curvature, further demonstrating thatVABS provides a far less costly alternative to 3-D FEA tools withoutsignificant loss of accuracy.

VI. Conclusions

The purpose of this paper was to provide a more complete body ofcomputational results that serve to verify results from VABS whenused along with suitable beam analysis. There are three types ofverification herein. First, it was demonstrated the combination ofVABS and a suitable beam analysis can correctly reproduce the 3-Dstress fields of beams from anisotropic elasticity theory. Second, anovel form of verification methodology is introduced based onanalysis of the same beam on the basis of two different coordinatesystems: curvilinear and Cartesian/piecewise-Cartesian coordinatesystems. Note that curvature measures are inherent to the formerapproach, and drawbacks and limitations of such a methodology arealso pointed out. Third, results for beams featuring initial curvatureand/or twist from a 1-D beam code (NATASHA was used herein)using sectional constants provided by VABS were compared withresults fromABAQUS 3-D finite elements. Global results from a 1-Danalysis, such as natural frequencies and static deflections underload, were compared for both isotropic and anisotropic beams asfunctions of the curvature measures. VABS and beam theory(geometrically exact for the static results) were shown to provide alevel of accuracy comparablewithABAQUS at a small fraction of thecomputational cost.

Acknowledgments

This work was supported in part by the U.S. Army through theVertical Lift ResearchCenter of Excellence at theGeorgia Institute ofTechnology, the Technical Monitor of which is Michael J.Rutkowski. Technical discussions with Robert A. Ormiston aregratefully acknowledged. The authors also gratefully acknowledgecomments from a most professional, thorough, and helpful reviewer.

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a) ABAQUS model for the helical spring

1 2 3 4 5 6 7 8 9 100

1000

2000

3000

4000

5000

6000

ABAQUS

VABS/NATASHA

b) Natural frequencies: VABS/NATASHA vs. ABAQUS

Fig. 10 Modeling and frequencies of a helical spring.

Table 4 Material and geometric properties of helix

Parameters Values

Material properties

E 2:9877 � 106 psi� 0.30� 7:3921 � 10�4 lb � s2=in:4

Geometric properties

Helical radius 1:9685 � 10�1 in:Wire radius 1:9865 � 10�2 in:Helical angle 8.5744 deg

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