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Renewable Energy 71 (2014) 133e155

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Renewable Energy

journal homepage: www.elsevier .com/locate/renene

Structural design of spars for 100-m biplane wind turbine blades

Perry Roth-Johnson, Richard E. Wirz*, Edward LinUniversity of California at Los Angeles, Mechanical & Aerospace Engineering, Energy Innovation Laboratory, Los Angeles, CA, USA

a r t i c l e i n f o

Article history:Received 12 July 2013Accepted 13 May 2014Available online

Keywords:Wind turbine bladesSpar designBiplane inboard regionMinimum total potential energyBeam and cross-sectional analysisStructural efficiency

* Corresponding author.E-mail address: wirz@ucla.edu (R.E. Wirz).

http://dx.doi.org/10.1016/j.renene.2014.05.0300960-1481/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

Large wind turbine blades are being developed at lengths of 75e100 m, in order to improve energycapture and reduce the cost of wind energy. Bending loads in the inboard region of the blade make largeblade development challenging. The “biplane blade” design was proposed to use a biplane inboard regionto improve the design of the inboard region and improve overall performance of large blades. This paperfocuses on the design of the internal “biplane spar” structure for 100-m biplane blades. Several sparswere designed to approximate the Sandia SNL100-00 blade (“monoplane spar”) and the biplane blade(“biplane spar”). Analytical and computational models are developed to analyze these spars. Theanalytical model used the method of minimum total potential energy; the computational model usedbeam finite elements with cross-sectional analysis. Simple load cases were applied to each spar and theirdeflections, bending moments, axial forces, and stresses were compared. Similar performance trends areidentified with both the analytical and computational models. An approximate buckling analysis showsthat compressive loads in the inboard biplane region do not exceed buckling loads. A parametric analysisshows biplane spar configurations have 25e35% smaller tip deflections and 75% smaller maximum rootbending moments than monoplane spars of the same length and mass per unit span. Root bendingmoments in the biplane spar are largely relieved by axial forces in the biplane region, which are notsignificant in the monoplane spar. The benefits for the inboard region could lead to weight reductions inwind turbine blades. Innovations that create lighter blades can make large blades a reality, suggestingthat the biplane blade may be an attractive design for large (100-m) blades.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In order to improve energy capture and reduce the cost of windenergy, wind turbines have grown dramatically over time. Longerblades are being developed to enable large, multi-MW wind tur-bines capture more energy. The rated power of wind turbines hasgrown almost linearly for the past 30 years [1,2]. At the time ofwriting, a 73.5-m blade had been manufactured for a 6-MWoffshore wind turbine [3,4]. Blades are expected to grow evenlarger in the future; current investigations include the develop-ment of a 20-MW turbine [2] and the design of a 100-m blade [5].However, the development of large blades at this scale is chal-lenging. For a simple geometric upscaling of a blade and assuming aconstant tip speed ratio, the bending stresses due to aerodynamicforces are independent of blade length, and the bending stressesdue to gravitational forces increase linearly as blades get longer [5].These bending loads are highest in the inboard region, near the root

of the blade. Hence, the inboard region needs an efficient structurethat can support the bending loads, provide favorable aerodynamicperformance, but also limit overall blade mass (and cost). The bladecannot be too heavy, because excess blade mass increases inertialfatigue loads, which decrease the blade lifetime. Heavier bladesalso tend to cost more to manufacture and transport, which limitsthe economic benefits of large wind turbines. This paper in-vestigates a new biplane structural design of the inboard region forlarge wind turbine blades.

Recent research suggests that improving performance in theinboard region of the blade may improve the overall blade perfor-mance. Conventionally, the inboard region used thick airfoils tosupport large flapwise bending loads [6e8]. While the aero-dynamic performance of thick airfoils is generally poor, this is thestandard compromise between structures and aerodynamics inblade design. The inboard region is primarily designed for struc-tures: thick cross-sections (with a large second moment of area inthe flapwise direction) support the large flapwise bending loads.(The inboard region design is also influenced by manufacturingconstraints related to the root attachment, as well as transportationrequirements that limit the maximum allowable dimensions.) In

Table 1Definition of design parameters for the biplane blade.

Parameter name Symbol Description

Joint length rj Length from blade root to jointSpan R Length from blade root to tipJoint length-to-

span ratiorj/R Fraction of blade span containing

the inboard biplane regionGap g Length between the upper and lower

chord lines of the biplaneChord c Length from the leading edge to

trailing edgeGap-to-chord ratio g/c gap between biplane elements,

normalized by chord

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155134

order to improve the performance of the inboard region within thecontext of these constraints, blunt trailing edge “flatback” airfoilshave been designed to address both the structural and aerodynamicperformance of the inboard region [9,10]. Structural advantages offlatback airfoils include a larger sectional area and increasedsectional moment of inertia for a given airfoil maximum thickness;aerodynamic advantages include a larger sectional maximum liftcoefficient, increased lift curve slope, and reduced sensitivity tosurface soiling [11].

Multi-element airfoils for the inboard region have also beenproposed as an alternative concept that may improve performance.Recent aerodynamic investigations [12,13] have shown that multi-element airfoils (which are specifically designed for wind turbines,and are usually thick airfoils with thickness-to-chord ratios of15e30%) have greater lift-to-drag ratios and maximum lift co-efficients compared to standard thick airfoils for wind turbines.Another study found that multi-element airfoils fitted to theinboard region improved the annual energy production of a 10-MWturbine [14]. Hence, it may be possible to increase the maximumpower production, improve start-up performance, and improveoverall efficiency of blades for megawatt-size turbines. Thesestudies showed aerodynamic improvements, but did not explicitlyexamine the structural considerations for large blade designs.

Wirz proposed a conceptual design for wind turbine blades thathas the potential to improve the structural and aerodynamic per-formance of the inboard region: the biplane blade [15]. A similardesign was independently proposed by Grabau [16]. The biplaneblade design uses a biplane inboard region that transitions into ajoint, which is connected to a monoplane outboard region. Pre-liminary structural analysis suggests that a biplane blade could beexpected to achieve a 30% decrease in blade tip deflection [17,18].This indicates that for a monoplane blade of fixed length, it may bepossible to construct a lighter biplane blade with an equal tipdeflection. These benefits can lead directly to weight and cost re-ductions for large blades. Preliminary aerodynamic analysisshowed that biplane airfoils can also achieve significant increasesin lift [19], similar to the multi-element airfoils discussed above.

While its potential benefits are interesting, the unconventionalconfiguration of the biplane blade makes it challenging to designand analyze. Hence, a newmethod is needed to design and analyzethe biplane blade. It is not initially apparent exactly how thebiplane should be incorporated into the overall blade structure,because several design parameters need to be specified for theinboard biplane region. This paper focuses on two important designparameters (see Fig. 1 and Table 1): the joint length rj (the spanwisedistance between the blade root and the mid-blade joint), and thegap g (the flapwise distance between the upper and lower elementsof the biplane).

To design the biplane blade, the authors used a “structures-first”design approach: (1) design the internal “biplane spar” structure,and (2) fit the airfoil exterior over that spar (Fig. 2). (This approachis the reverse of the design process used for the Sandia SNL100-00reference blade [5], an example of a conventional “monoplaneblade”. The Sandia blade was designed by first choosing an airfoilfamily for the aerodynamic exterior of the blade, and then fitting a

Fig. 1. Two major design parameters for the biplane blade: joint length (the spanwisedistance between the blade root and the mid-blade joint), and the gap (the flapwisedistance between the upper and lower elements of the biplane).

“monoplane spar” structure inside these airfoils.) For the biplaneblade, the two-step structures-first approach will give an initialdesign for the biplane blade. This design will be refined later byiterating through these two steps several times, in order to arrive ata final design for the biplane blade that provides desirable aero-structural performance. In some respects, the structures-firstapproach is similar to the “Aerosolve” design process proposedfor designing conventional monoplane blades [11]. No aerodynamicanalysis of the biplane blade is presented here. Instead, this paperfocuses on the first step of the structures-first approach (structuraldesign of biplane spars), which lays the foundation for futurestudies of biplane blades.

Following on from the preliminary investigations, this workaims to show that 100-m biplane spars can provide improvedflapwise structural performance, relative to a conventional 100-mmonoplane spar. First, this work introduces a new analyticalmodel for the displacement field of simplified biplane spars(“biplane beams”), which reveals how biplane spars carry loadsdifferently from conventional cantilevered monoplane spars. Sec-ond, a computational model is developed to investigate the designof 100-m biplane spars made of composite (anisotropic) materialsand use realistic cross-sections that taper the spar thickness fromroot to tip. An approximate buckling analysis is used to verify thatcompressive loads in the inboard region do not exceed criticalbuckling loads. This extends earlier research [17,18] that consideredsimplified 50-m biplane spars made of isotropic materials and useduniform rectangular cross-sections (which did not taper the sparthickness from root to tip). Earlier research did not considerbuckling. Last, the computational model is used to perform aparametric study of different biplane spar configurations. Fifteenbiplane spar configurations were constructed by varying twodesign parameters (the joint length rj and the gap g), while theblade span was held constant. This parametric study investigatedthe effect of these design parameters on the flapwise structuralperformance of biplane spars.

2. Methods

Both analytical and computational models were used to inves-tigate the structural design of biplane spars. The analytical modelgives physical insights into the structural behavior of the biplanespar, which are not clear from the computational model alone. Theanalytical model used the method of minimum total potential en-ergy [20] to derive the displacement field of a biplane beam (asimplified structure that approximates a biplane spar, using threeEulereBernoulli beams and a rigid joint). The computational modelused beam finite elements and cross-sectional analysis to createmore complex models of monoplane and biplane spars made withcomposite materials and realistic cross-sections that taper the sparthickness from root to tip. These computational models were usedin two ways: (1) to compare the structural performance of a

Fig. 2. Example design approach for a conventional wind turbine blade (left), and the “structures-first” approach to design biplane blades (right). This paper focuses on the first stepof the structures-first approach.

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155 135

monoplane spar to biplane spars, and (2) to perform a parametricstudy of different biplane spar configurations.

2.1. Analytical model

Themethod of minimum total potential energy [20] was used toderive the displacement field of a simplified biplane spar, or“biplane beam” (Fig. 3). This method is often used for structuresthat are difficult or cumbersome to analyze with free body dia-grams, because it offers a systematic framework to obtain theequilibrium equations and boundary conditions for the structure.The resulting boundary value problem can be solved analytically forthe displacement field. The derivation of the analytical model isoutlined below.

2.1.1. GeometryConsider a “biplane beam” structure (Fig. 3), composed of an

inboard biplane region (0 � x � rj) and an outboard monoplaneregion (rj � x � R). The inboard biplane region is made up of twobeams separated by a gap g: the “upper biplane” (UB) beam and the“lower biplane” (LB) beam. The UB and LB beams are both canti-levered at x¼ 0 and both affixed to a rigid joint at x¼ rj. This joint isassumed to be very thin compared to the span R. The outboardmonoplane region is made of one beam: the “outboardmonoplane”(OM) beam. The OM beam is affixed to the rigid joint at x ¼ rj, andfree at x ¼ R.

2.1.2. LoadsThree transverse distributed loads (qUB(x), qLB(x), and qOM(x))

are applied to the UB, LB, and OM beams, respectively. To approx-imate the flapwise bending moment (the primary load on windturbine blades [21]), the transverse distributed loads were assumedto be constant along each of the spans of the OM, UB, and LB beams.

Fig. 3. Schematic of the biplane beam str

The constant load distribution approximates an “instantaneoussnapshot” of a gust load on a wind turbine blade. These gust loadsoften lead to the worst-case scenario of all the design load casessuggested by international certification standards of wind turbineblades [22], as was shown in Ref. [5]. The magnitude of the inboarddistributed loads was assumed to be half of the magnitude of theoutboard distributed load (Eq. (1)). In other words, the constantload distribution that would normally be present on the inboardregion of a conventional monoplane blade was equally distributedamong the UB and LB beams.

qOMðxÞ ¼ q0

qUBðxÞ ¼12q0

qLBðxÞ ¼12q0:

(1)

The transverse distributed loads only simulate flapwise bendingloads on the biplane spar. Consider a “full” biplane blade, withairfoils added over a biplane spar. The flapwise bending response ofa full biplane blade is unlikely to be much different from the flap-wise response of a biplane spar, so flapwise loads are considered inthe present study. However, edgewise loads and torsional responsewill likely be different in the full biplane blade, so they were notinvestigated here with the analytical model for the biplane spar.

2.1.3. Engineering propertiesEach beam is assumed to be made of an isotropic, linear elastic

material. The UB beam has bending stiffness EIUB(x) and extensionstiffness EAUB(x). Similarly, the LB and OM beams have stiffnessesEILB(x), EALB(x), EIOM(x), and EAOM(x).

In order to make the analytical model tractable, some assump-tions were made for the engineering properties and loads on the

ucture used for the analytical model.

vOMðxÞ ¼ q0x4

24EI�Rq0x

3

6EIþR2q0x

2

4EI

þ q0rjx

6EI�Aag2þ4Ib

���3AR2ag2þ3ARag2rj�Aag2r2j

�12IR2bþ6IR2þ12IRbrj�6IRrj�4Ibr2j þ2Ir2j�

q0r2j

�2 2 2

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155136

biplane beam, leading to simpler equilibrium equations and naturalboundary conditions. It was assumed that EA and EI were constantalong each of the spans of the OM, UB, and LB beams. Furthermore,the inboard biplane properties were scaled to the outboardmonoplane properties with constants a and b, such that

a ¼ EAUB

EAOM¼ EALB

EAOM

b ¼ EIUBEIOM

¼ EILBEIOM

:

(2)

þ48EIb

�Aag2þ4Ib

� 12AR abg �16ARabg rj

þ2ARag2rjþ6Aabg2r2j �Aag2r2j þ48IR2b2�24IR2b

�64IRb2rjþ32IRbrjþ24Ib2r2j �12Ibr2j�

(4c)

2.1.4. Kinematic assumptionsThe UB, LB, and OM beams were modeled with EulereBernoulli

beam theory. In addition to transverse bending, all beams wereallowed to stretch axially.

The rigid joint has three degrees of freedom: transversedisplacement, rotation, and axial displacement (Fig. 4). The jointeffectively links the movement of all three beams at x ¼ rj. At thejoint, the transverse displacements and rotations of all three beamsmust be equal. Additionally, at the joint, the axial displacements ofthe UB and LB beams are related to the axial displacement of theOM beam and the rotation of the OM beam.

uUB�rj� ¼ uOM

�rj�þ 1

2gv0OM

�rj�

uLB�rj� ¼ uOM

�rj�� 1

2gv0OM

�rj� (3)

2.1.5. Solution for transverse and axial displacementsAn expression for the total potential energy of the structure and

applied loads was developed and minimized [23,24]. Then, theminimum total potential energy was used to obtain the equilibriumequations and boundary conditions. Python-based software (thesymbolic math package SymPy 0.7.2. [25] and the web-basednotebook environment of IPython 0.13 [26]) was used to helpderive the solution to these equilibrium equations and boundaryconditions. The resulting analytical solution is given below.

The transverse displacements are

vUBðxÞ ¼q0x4

48EIb� Rq0x3

12EIbþq0x2

�3ARag2rj � Aag2r2j þ 12IR2b

�24EIb

�Aag2 þ 4Ib

�(4a)

vLBðxÞ ¼q0x4

48EIb� Rq0x3

12EIbþq0x2

�3ARag2rj � Aag2r2j þ 12IR2b

�24EIb

�Aag2 þ 4Ib

�(4b)

Fig. 4. Schematics for the kinematic assumptions at th

The axial displacements are

uUBðxÞ ¼gq0x

�3R2 � 3Rrj þ r2j

�6E

�Aag2 þ 4Ib

� (5a)

uLBðxÞ ¼ �gq0x

�3R2 � 3Rrj þ r2j

�6E

�Aag2 þ 4Ib

� (5b)

uOMðxÞ ¼ 0: (5c)

Compare Eqs. (4) and (5) to the analytical solution for a cantileverbeam of length R, with uniform axial stiffness EA and bendingstiffness EI, under a constant distributed load q0 [27]; here, thisreference beam is referred to as a “monoplane beam”.

vmonoplaneðxÞ ¼q0R4

24EI

��xR

�4� 4

�xR

�3þ 6

�xR

�2�(6a)

umonoplaneðxÞ ¼ 0 (6b)

2.2. Computational model

Computational models (Fig. 5) were constructed to includesome more complex design features of the biplane spar that couldnot be included in the analytical model (such as tapering of the sparfrom root to tip, a smooth biplaneemonoplane transition at the

e rigid joint (x ¼ rj) used for the analytical model.

Fig. 5. Surface geometry of spars used for the computational model. Three views are shown for each spar: edgewise (top), flapwise (middle), and isometric (bottom).

Fig. 6. The spar is defined to include the two spar caps, the two primary shear webs,and the root buildup adjacent to the “box-beam” formed by the spar caps and shearwebs. Note: figure is not to scale.

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155 137

joint, and anisotropic composite cross-sections). A computationalmodel of a monoplane spar was also constructed for benchmarkcomparisons against these biplane spars. These computationalmodels more accurately represent the real structures inside biplaneblades and conventional blades than the analytical models used inSection 2.1. The computational models were analyzed with 1Dbeam finite elements and 2D cross-sectional analysis (describedfurther in Section 2.2.4). An approximate buckling analysis wasused to verify that compressive loads in the inboard region do notexceed critical buckling loads.

The first step of the structures-first approach centers around thedesign and analysis of spars (Fig. 2). The 100-m Sandia researchblade [5] was used as a template to design these spars. The Sandiablade was chosen because many details of its design were madepublicly available, including its cross-section geometries, compos-ite lay-up, and material properties. The Sandia blade is an openplatform that can be modified to create new blade designs, makingit ideal for this investigation.

2.2.1. AssumptionsThe main assumption behind the structures-first approach is

that the structural performance of a spar is representative of thestructural performance of an entire blade. This assumption isreasonable because the spar is the primary load-bearing compo-nent in the blade [28,29]. Furthermore, to the authors' bestknowledge, the airfoil shells on the Sandia blade are not intended tobe load-bearing; they merely transfer the aerodynamic loads fromthe wind to the internal spar structure. The spar is composed ofspar caps on the top and bottom, and shear webs on the left andright. The spar caps are made of the primary load-carrying unidi-rectional fibers, and the shear webs stabilize the spar caps undershear forces; this creates a closed-box section.

In this analysis, the spar is defined to include the followingstructural components from the Sandia blade: the two principalshear webs, the two spar caps, and the portion of the root buildupthat is adjacent to the spar caps (Fig. 6). The root buildup wasincluded because the spar caps are very thin near the spar root, andthe root buildup is needed to provide more stiffness in the rootstructure. All other components in the Sandia bladewere neglected.

Three other assumptions weremade to simplify the geometry ofall spars. First, a rectangular cross-sectionwas assumed for all spars(Fig. 7). In other words, the curvature of the spar caps and root

buildup was neglected. The maximum thickness of each airfoilprofile was used to determine the height of each rectangular cross-section in the spar. Second, twist of the cross-section along the spanof the spar was not considered (i.e. the spar had zero twist fromroot to tip). Third, the cylindrical root shown in Fig. 1 was neglectedto limit the number of design parameters in this analysis; futurestudies will be necessary to quantify its effect on structuralperformance.

To evaluate the validity of these assumptions, the spanwiseproperties of the monoplane spar (Table B.1) and the Sandia blade(stations 7e30, reported in Table 2 of Ref. [30]) were compared toeach other. In the flapwise direction, these properties match welleverywhere except near the root where, as expected, the mono-plane spar is less stiff than the full Sandia blade. The monoplanespar root has a lower stiffness because it neglects triaxial laminatesin the leading and trailing edges of the root buildup, which provideadditional flapwise stiffness. In the edgewise and torsional di-rections, the properties of themonoplane spar across its entire spanhave similar distributions, but are about 1ste2nd orders ofmagnitude smaller than the Sandia blade. This result is expected,since the leading edge panel and trailing edge reinforcement

Fig. 7. All spars were assumed to have rectangular cross-sections, made of the components and materials shown above. Glass fiber reinforced plastics (GFRP) and structural foamwere used to construct all spar cross-sections. Note: figure is not to scale.

Table 2Material property data for glass fiber reinforced plastic (GFRP) composites andstructural foam [5,31,32].

Material Fabric/resin Lay-up E1(GPa)

E2(GPa)

G12

(GPa)n12 r

(kg/m3)

Uniaxial GFRP E-LT-5500/EP-3 [0]2 41.8 14.0 2.63 0.28 1920Biaxial GFRP Saertex/EP-3 [±45]4 13.6 13.3 11.8 0.49 1780Triaxial GFRP SNL Triax [±45]4[0]2 27.7 13.65 7.2 0.39 1850Foam e e 0.256 0.256 e 0.3 200

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155138

(which provide edgewise stiffness) and the airfoil shells (whichprovide torsional stiffness) were neglected from the monoplanespar.

Based on these results, the monoplane spar is an acceptablestructural approximation of the Sandia blade in the flapwise di-rection. It is clear that the monoplane spar is not a good approxi-mation in the edgewise and torsional directions, which is alimitation of the first step in the structures-first approach. Never-theless, in the absence of the detailed aerodynamic exterior, themonoplane spar is a useful 1st-order characterization of the Sandiablade's structural interior. Bearing this in mind, themonoplane sparcan be used as a 1st-order baseline for comparisons against biplanespars in this study to examine the impacts of the biplane bladedesign.

2.2.2. Design of sparsA monoplane spar was designed to approximate the Sandia

blade and 15 biplane spars were designed to approximate differentbiplane blade configurations. Material properties (Table 2) andlaminate dimensions of all spars were based on the Sandia blade[5]. The material properties of the glass fabrics and epoxy resinmaterials are also described in more detail in the DOE/MSU Com-posite Material Fatigue Database [31,32]. Because weight is one ofthe important limiting parameters that determines the length ofwind turbine blades [33], all monoplane and biplane spars hadequal cross-sectional areas at a given spanwise station (Fig. 8), andhence, equal mass per unit span. This allowed for a self-consistentcomparison between their structural performance.

2.2.2.1. Monoplane spar. Fig. 9 summarizes the procedure used todesign and analyze the monoplane spar (Fig. 5(a)). First, the ge-ometry, lay-up, and materials of the Sandia blade were used toconstruct a monoplane spar. Second, the cross-sectional di-mensions of the spar caps, shear webs, and root buildup weredefined. Third, each cross-section in the spar was meshed andanalyzed to compute its stiffness properties. Fourth, a beam finiteelement model for the monoplane spar was defined. Fifth, asimulated load case was applied to the beam model. Finally, theanalysis results were postprocessed to obtain the 1D deflections,1Dbending moments, 1D axial force resultants, and 2D cross-sectional

stresses. See Section 2.2.4 for details on the software and analysesused in this procedure.

Table A.1 lists the laminate dimensions for each of the mono-plane cross-sections (Fig. 7) that make up the monoplane spar. Themonoplane spar (Fig. 5(a)) had a span of 91.9 m, and a maximumheight of 5.3 m at the root. Although the spar was derived from the100-m Sandia blade, the spar is slightly shorter because it beginsjust past the blade root (spar station 1 corresponds to 2.4% spanalong the Sandia blade) and terminates just before the blade tip(spar station 24 corresponds to 94.3% span along the Sandia blade)[5]. Table B.1 lists the spanwise properties for the monoplane spar,which were calculated with 2D cross-sectional analysis asdescribed later in Section 2.2.4.

2.2.2.2. Biplane spar. Fig. 10 summarizes the procedure used todesign and analyze the biplane spar (one configuration is shown inFig. 5(b)), which is similar to the procedure used for the monoplanespar. First, the geometry, lay-up, and materials of the monoplanespar (which were derived from the Sandia blade) were used toconstruct a biplane spar. The outboard monoplane region of thebiplane spar was left unchanged from the monoplane spar. Theinboard monoplane region of the monoplane spar was replacedwith an inboard biplane region in order to construct a biplane spar[34]. Second, the cross-sectional design parameters were defined.The biplane cross-section included an upper and lower element,which were assumed to be identical in this study. Third, each cross-section in the spar was meshed and analyzed to compute its stiff-ness properties. Only the upper element of each biplane cross-section was meshed and analyzed, because it was identical to thelower element. Fourth, a beam finite element model was defined,

Fig. 8. All cross-sections had equal areas, so all spars had the same mass per unit span. Note: figure is not to scale.

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155 139

which included the inboard biplane region of the biplane spar. Fifth,a simulated load case was applied to the beam model. An equiva-lent load was defined for the inboard biplane region so that thetotal load on the biplane spar was the same as the monoplane spar.

Fig. 9. Design and analysis procedure for the monoplane spar. The software used in thisanalysis, and DYMORE for 1D beam finite element analysis) is typeset in italics. The surface

Finally, the analysis results for the biplane spar were postprocessedto obtain the same quantities that were obtained for the mono-plane spar. See Section 2.2.4 for details on the software and ana-lyses used in this procedure.

procedure (TrueGrid for cross-sectional mesh generation, VABS for 2D cross-sectionalgeometry of the Sandia blade was taken from Ref. [5].

Fig. 10. Design and analysis procedure for the biplane spar. The software used in this procedure (TrueGrid for cross-sectional mesh generation, VABS for 2D cross-sectional analysis,and DYMORE for 1D beam finite element analysis) is typeset in italics.

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155140

Two types of biplane cross-sections (Fig. 8) were constructed tobound the design space for biplane spars. A “half-height” biplanecross-sectionwas constructed, so that the upper and lower elementsof the biplane cross-sectionwere half as tall as themonoplane cross-section. A “full-height” biplane cross-section was also constructed,so that the upper and lower elements of the biplane cross-sectionwere the same height as the monoplane cross-section. Overall, theupper and lower elements of the full-height biplane cross-sectionare thinner-walled versions of the monoplane cross-section.

Fig. 11 illustrates the edgewise views of two sample biplanespars constructed from half-height and full-height biplane cross-sections. The half-height cross-sections form a more slenderinboard biplane region than the full-height cross-sections. Onceairfoils are fitted over each of these spars, the half-height cross-

Fig. 11. Edgewise views of two biplane spars constructed with half-height biplane cross-sejoint length-to-span ratio of rj/R ¼ 0.452 and a gap-to-chord ratio of g/c ¼ 1.25.

sections will offer more of an aerodynamic benefit, while the full-height cross-sections will offer more of a structural benefit. Noattempt was made in this study to find the optimum biplane cross-section height. Rather, these two configurations bound the designspace for the biplane spar, and the optimum configuration issomewhere in between.

Table A.2 lists the laminate dimensions for a biplane spar madewith half-height biplane cross-sections. Fig. 12(a) shows how thesecross-sections are distributed along the biplane spar in the span-wise direction (x1) and the flapwise direction (x3). Table B.2 lists thespanwise properties of this biplane spar, which were calculatedwith 2D cross-sectional analysis, as described later in Section 2.2.4. ,Fig. 12(b), Table A.3 and Table B.3 give the corresponding details fora biplane spar made with full-height biplane cross-sections.

ctions (top) and full-height biplane cross-sections (bottom). Both biplane spars have a

Fig. 12. One-dimensional representations of biplane spars with a joint length-to-span ratio rj/R ¼ 0.452 and gap-to-chord ratio g/c ¼ 1.25. One-dimensional reference lines areplotted as thick solid green lines. Spar station locations are plotted with dashed magenta lines. The solid black lines with x-symbols show the height of each spar cross-section(cross-section height ¼ hSW þ 2hRB) at each spar station. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155 141

In order to evaluate the effect of joint length and gap on thestructural performance of biplane spars, 15 configurations of theinboard biplane regionwere made (Fig. 13). Each configuration hada different value for the joint length-to-span ratio (rj/R ¼ 0.245,0.274, 0.452, 0.540, and 0.629) and gap-to-chord ratio (g/c ¼ 0.75,1.00, and 1.25). The procedure in Fig. 10 was repeated for eachdifferent pair of values for rj/R and g/c. The span (R ¼ 91.9 m) andreference chord (c ¼ 7.628 m, taken to be the maximum chord inthe Sandia blade) were held constant for each configuration. Eachconfiguration used “full-height” biplane cross-sections. The loca-tion of the biplaneemonoplane transition region varied, dependingon the configuration.

Fig. 13. Fifteen biplane spar configurations w

2.2.3. Load profiles and boundary conditionsSimple flapwise loads were used to compare the structural

performance of biplane spars and the monoplane spar, followingthe same argument used in the analytical model (Section 2.1.2). Aconstant load distribution was statically applied to each spar(Fig. 14). The maximum magnitude of the constant load distribu-tion was 1000 N/m. To load the monoplane spar and the biplanespar in an equivalent manner, it was assumed that the loads onthe biplane inboard region of the biplane spar were equallydistributed among the upper and lower elements (Fig. 14(b)). Allspars were cantilevered at the root (x1 ¼ 0 m) and free at the tip(x1 ¼ R ¼ 91.9 m). These flapwise constant load distributions were

ere designed and analyzed in this study.

Fig. 14. Load profiles applied to spars.

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155142

also used to estimate the structural efficiency of biplane spars(Section 2.2.6).

Static tip loads were also used to estimate the structural effi-ciency of biplane spars. Two load cases were considered: a flapwisetip load and a torsional tip load (Fig. 15). Again, both spars werecantilevered at the root and free at the tip. The magnitude of theflapwise tip loadwas varied between 1000N< Fflap<100,000N, andflapwise deflectionswere calculated. Themagnitude of the torsionaltip load was varied between 1000 N-m < Ttwist < 100,000 N-m, andboth twist angles and edgewise deflections were calculated.

This paper did not investigate biplane spars under edgewiseloading. While edgewise loads are important for large blades, theyare beyond the scope of this paper because the airfoil shells havebeen neglected. The airfoil shells contain reinforcements along theleading and trailing edges (Fig. 6), which help resist edgewise loads.Therefore, it would be inappropriate to load biplane spars in theedgewise direction without the additional support of the re-inforcements in the airfoil shells. Edgewise load cases will beconsidered in future studies that include both the biplane spar andthe airfoil shells to model the complete biplane blade. This paperalso did not model the dynamic response of biplane spars, but thisshould be considered in the future for fatigue analyses.

2.2.4. 1D beam models and 2D cross-sectional analysisEach spar was modeled with geometrically exact beam finite

elements [35] in DYMORE 3.0, a flexible multibody dynamics finiteelement program [36,37]. These elements were 3rd-order ele-ments, and were based on a generalized Timoshenko beam modelwith six degrees of freedom per node [38]. DYMORE performed a

Fig. 15. Tip loads applied to spars: flap

Fig. 16. Computational meshes used for finite elemen

nonlinear static analysis to solve for the deformed configuration ofeach spar under loading.

Fig. 16 compares the 1D beam models for the monoplane sparand a biplane spar. The spanwise location of the end-nodes waschosen to match the spanwise locations of the blade stationsdefined for the Sandia blade [5]. In the outboard monoplane re-gions, the monoplane and biplane spar meshes are identical.However, in the inboard region, the biplane spar mesh has an upperand lower assembly of beam elements, which merge together intoone node at the biplaneemonoplane joint. Non-Uniform RationalB-Spline (NURBS) curves were used to construct the curved beamelements in the biplaneemonoplane transition region.

The 2D cross-sectional properties of all beam elements werecalculated with the VABS 3.6 software [39,40], also known as Vari-ational Asymptotic Beam Sectional analysis. VABS uses the varia-tional asymptotic method [41,42] to calculate the mass and stiffnessmatrices for arbitrary beam cross-sections. This work used VABS forcross-sectional analysis in the samemanner as in the authors' earlierinvestigations [17,18], except some additional features of VABS wereused here that were not used earlier. The curvature and pre-twistcorrections were considered to calculate the mass and stiffnessmatrices for cross-sections that belonged to the curved beam ele-ments, and then the 1D beam analysis results from DYMORE wereused in VABS to recover the cross-sectional stresses [40].

DYMORE and VABS were selected as structural analysis toolsbecause they are fast and accurate, when compared to three-dimensional finite element analysis (3D FEA) [38,43]. This was amajor requirement in order to conduct a parametric analysis of all15 biplane spar configurations. While it is true that 3D FEA is highly

wise (top) and torsional (bottom).

t analysis. (Note: mid-element nodes not shown).

Table 3Scaling factors for the engineering properties of the biplane beam, used in theanalytical model.

Biplane cross-section a b

Half-height 1/2 1/8Full-height 1/2 1/2

1 The values of a and b depend on the geometry of the biplane cross-section. Inorder to choose these values for both biplane beams, the analytical model used thesame cross-sections illustrated in Fig. 8, assuming nominal values of bRB ¼ 1.81 m,hSW ¼ 4.84 m, bSW ¼ 0.10 m, hSC ¼ 0.15 m, hRB ¼ 0 m, and E ¼ 40.0 � 109 Pa. Thesenominal values were used to calculate EA and EI for each cross-section, whichtogether with Eq. (2) give the appropriate values for a and b.

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155 143

accurate, it was judged to be too time consuming to set up themodels and compute the results. DYMORE and VABS were alsoselected because they have successfully modeled helicopter blades[36,44,45] and wind turbine blades [35,43,46,47]. This 1Dmodelingtechnique has also been previously validated against 3D FEA forbiplane spar structures [17,18]. Furthermore, several researchershave also used 1D beam finite elements to model a joined wing[48e52]; which has a joint that resembles the joint of the biplaneblade.

2.2.5. Buckling analysisAs a result of the approach to conserve mass per unit span, the

biplane spar cross-sections have thinner walls than the monoplanespar cross-sections (Fig. 8), motivating a buckling analysis of thebiplane spar. Analytical formulas and representative structureswere used to estimate two critical buckling criteria: (1) a “global”beam buckling load, and (2) a “local” plate buckling stress.

The global beam buckling load was estimated with the Eulerbuckling formula for a simply supported beam under uniaxialcompression (Eq. (7)), where Pcritical is the buckling load, EI is theflapwise bending stiffness and l is the length of the beam [53]. Thelower biplane element in the biplane spar is subjected to the largestaxial compressive loads. Therefore, this analysis finds the criticalbuckling load for a representative beam similar to the lower biplaneelement.

Pcritical ¼p2EIl2

(7)

The local plate buckling stress was estimated with the bucklingformula for a simply supported orthotropic plate under uniaxialcompression (Eq. (8)) [53,54]. Here, sx,critical is the buckling stress;m is the number of longitudinal half waves; h is the plate thick-ness; R ¼ a/b is the ratio of plate length a to width b; D11 and D22are the bending rigidities about the x and y axes, respectively; D66

is the twisting rigidity; and D12 is related to the Poisson's ratio n21.It is assumed that the smallest buckling load occurs when thebuckled plate has only one transverse half wave. The lower flangein the lower biplane element of the biplane spar is subjected tothe largest axial compressive loads. Therefore, this analysis findsthe critical buckling stress for a representative plate similar to thelower flange.

sx;critical ¼p2

hm2a2

hD11m

4 þ 2ðD12 þ 2D66Þm2R2 þD22R4i; (8)

where

R ¼ a=b;

D11 ¼ E1h3

12ð1� n12n21Þ;

D22 ¼ E2h3

12ð1� n12n21Þ;

D12 ¼ E2n21h3

12ð1� n12n21Þ; and

D66 ¼ G12h3

12:

2.2.6. Structural efficiencyThe monoplane spar and biplane spars considered all had the

same length and mass per unit span. Therefore, the deflections ofeach spar can be directly compared to each other, in order to esti-mate the structural efficiency of biplane spars. A factor f was

defined as the ratio of the monoplane and biplane spar tip de-flections (under a flapwise load) or as the ratio of tip twist angles(under a torsional load).

fflap ¼ vtip;monoplane

vtip;biplane; or ftwist ¼

qtip;monoplane

qtip;biplane(9)

If f > 1, then the biplane spar is more structurally efficient than themonoplane spar; conversely, if f < 1, then the biplane spar is lessefficient. This definition of a structural efficiency factor is some-what similar to the stiffness shape factor described in Ref. [55],which compares the curvature of a shaped cross-section to that of areference beam with the same area. Instead of considering curva-tures of cross-sections, this work considers tip deflections of theentire spar structure.

3. Results

3.1. Analytical model: comparison of biplane beam and monoplanebeam

The displacement fields for two variants of the biplane beam(Eqs. (4) and (5)) and the monoplane beam (Eq. (6)) were used tocompare their structural performance. Two variants of the biplanebeam were analyzed, a biplane beam with “full-height” biplanecross-sections, and a biplane beam with “half-height” biplanecross-sections. Both variants of the biplane beam had a joint length-to-span ratio of rj/R ¼ 0.45 and a gap-to-chord ratio of g/c ¼ 1.31.However, each variant used different values of the scaling factors aand b in Eqs. (4) and (5); Table 3 summarizes the values that werechosen.1 These variants were constructed to bound the designspace for biplane beams, similar to the design space used forbiplane spars (Section 2.2.2).

Along with the scaling factors in Table 3, nominal values weresubstituted into Eqs. (4) and (6), and the resulting expressions forflapwise (transverse) displacements, bending moments, and axialforce resultants were plotted (Fig. 17). These nominal values wereassumed to be R ¼ 100 m, rj ¼ 45 m, g ¼ 10 m, q0 ¼ 1000 N/m,E ¼ 40.0 � 109 Pa, A ¼ 1.5 m2, and I ¼ 1.5 m4.

The tip deflection of the biplane beam with half-height cross-sections is about 62% less than the tip deflection of the monoplanebeam (Fig. 17(a)). The tip deflection of the biplane beam with full-height cross-sections is about 77% less than the monoplane beam.In the inboard region (5 m ( x ( 35 m), the half-height biplanebeam deflects slightly more than the monoplane beam, while thefull-height biplane beam does not.

The maximum root bending moment of both biplane beams isabout 80% less than the root bending moment of the monoplanebeam (Fig. 17(b)). Past the root of both biplane beams, the bending

Fig. 17. Comparison of beam deflections, bending moments, and axial force resultants under equivalent flapwise constant load distributions, calculated with the analytical model: amonoplane beam, a biplane beam with “full-height” biplane cross-sections, and a biplane beam with “half-height” biplane cross-sections. Both biplane beams had design pa-rameters of rj/R ¼ 0.45 and g/c ¼ 1.31, and were composed of an “upper biplane” (UB) beam, a “lower biplane” (LB) beam, and an “outboard monoplane” (OM) beam.

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moment changes sign; this occurs at x z 22 m for the half-heightbiplane beam, and at x z 25 m for the full-height biplane beam.The bending moments in both biplane beams continue to decreaseuntil they reach aminimum at the rigid joint (x¼ 45m). At the rigidjoint, there is a discontinuity in the bending moment of bothbiplane beams. Outboard of the joint (45 m � x � 100 m), thebending moments of both biplane beams are identical to thebending moment of the monoplane beam.

The axial force resultants of both biplane beams are constantand nonzero in the inboard biplane region, where the upperbiplane element is under tension, and the lower biplane element isunder compression (Fig. 17(c)). Outboard of the joint, the axial forcereturns to zero in both biplane beams. The axial force is zeroeverywhere in the monoplane beam.

3.2. Computational model

3.2.1. Comparison of biplane spar and monoplane sparThe monoplane spar, a biplane spar with “full-height” biplane

cross-sections, and a biplane spar with “half-height” biplane cross-sections were analyzed to compare their structural performanceunder the constant load distribution described in Section 2.2.3.Both biplane spars had a joint length-to-span ratio of rj/R ¼ 0.452and a gap-to-chord ratio of g/c ¼ 1.25.

3.2.1.1. Cross-sectional properties. Fig. 18 shows the flapwisebending stiffness distributions for all three spars. In the inboardregion, the biplane spars have larger effective stiffnesses than themonoplane spar, which was a key motivation to investigate the

Fig. 18. Flapwise bending stiffness distributions for a monoplane spar, a biplane spar with “full-height” biplane cross-sections, and a biplane spar with “half-height” cross-sections.Both biplane spars had design parameters of rj/R ¼ 0.452 and g/c ¼ 1.25. Cross-sectional analysis (VABS) was used to calculate stiffnesses for the monoplane spar. VABS was also usedto calculate stiffnesses of “single” elements in full-height and half-height biplane cross-sections. The parallel axis theorem was used with the stiffnesses of “single” elements toapproximate the effective flapwise stiffness distributions for both biplane spars.

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155 145

biplane blade concept. In the outboard region, all spars haveidentical stiffnesses because they each use identical monoplanecross-sections.

Cross-sectional analysis (VABS) was used to calculate the flap-wise stiffness at each cross-section in the monoplane spar. For thebiplane spars, biplane cross-sections are made of an upper andlower element. VABS was used to calculate the individual stiff-nesses of each single element (i.e. the stiffness of only the upperelement of the biplane cross-sections shown in Fig. 8, not theeffective stiffness of the upper and lower elements as a collectivewhole). Then, the parallel axis theoremwas used to approximate aneffective stiffness for the entire biplane cross-section (upper andlower elements). Since the upper and lower elements of theinboard biplane cross-sections are not attached together in thebiplane spar, this analysis is a 1st-order approximation of thebending stiffness. As will be shown later, the inboard region of thebiplane spar also supports axial loads that play a major role inimproving the effective bending stiffness of the entire structure.Nevertheless, the qualitative result in Fig. 18 still holds: in theinboard region, the biplane spars have larger effective stiffnessesthan the monoplane spar.

3.2.1.2. 1D beam analysis. Fig. 19 shows the flapwise deflections,bending moments, and axial force resultants calculated by the 1Dbeam finite element analysis from DYMORE. The tip deflection ofthe biplane spar with half-height cross-sections is about 25% lessthan the tip deflection of the monoplane spar (Fig. 19(a)). The tipdeflection of the biplane spar with full-height cross-sections isabout 35% less than the monoplane spar. The maximum rootbending moment of both biplane spars is about 75% less than theroot bending moment of the monoplane spar (Fig. 19(b)). The axialforce resultants of both biplane spars are constant and nonzero inthe inboard biplane region.

3.2.1.3. Structural efficiency. Table 4 lists the structural efficiencyfactors of both biplane spars from Eq. (9), using the tip deflectionsand tip twist angles calculated by DYMORE for each of the threeload cases (Figs. 14 and 15). For the two tip loads, the structuralefficiency factors remained constant across the range of loadmagnitudes that were considered, because the deflections andtwist angles of all spars scaled linearly with the load applied. Both

flapwise load cases show that both biplane spars are more struc-turally efficient (with 1.148 < f < 1.528) at reducing flapwise tipdeflections than the monoplane spar.

The torsional tip load shows that the biplane spars have aboutthe same structural efficiency (with f z 1.06) as the monoplanespar in the torsional direction. For the largest torsional tip load(100,000 N-m), all tip twist angles were about 5e6 degrees. Whilethese twist angles were small, in both biplane spars, torsionalmotions at the joint caused edgewise deflections in the inboardregion (Fig. 20). The largest deflections occurred just inboard of thejoint, with a magnitude of about 0.012 m. Conversely, the mono-plane spar had little edgewise deflection under the torsional load.

3.2.1.4. Cross-sectional stresses. Fig. 21 compares the cross-sectional stresses of both biplane spars and the monoplane spar.The normal components of pointwise stress fields, s11(x2,x3), werecalculated for cross-sections at each spar station. Maximum valuesin each cross-section (which correspond to tensile stresses) wereplotted above, while minimum values (which correspond tocompressive stresses) were plotted below. The maximum andminimumvalues of stress change for different cross-sections; somejumps in the stress results are observed in the root and joint re-gions. Section 4.2 discusses the possible sources of these jumps inthe data.

For the inboard region (0e42 m), the full-height biplane spartended to have larger maximum normal stresses than the mono-plane spar. In the inboard region, the half-height biplane spartended to have smaller maximum normal stresses than themonoplane spar, except for the stresses in the upper biplane regionin between spans of 26e42 m.

For the minimum normal stresses in the inboard region, themagnitude of stresses in the full-height biplane spar tended to besmaller than the monoplane spar. The half-height also tended tohave smaller stress magnitudes than the monoplane spar in theinboard region, except for the stresses in the lower biplane regionat spans of 0e6m and 25m, as well as stresses in the upper biplaneregion at 4 m.

Contour plots of the normal stresses in selected cross-sections ofthe monoplane spar and both biplane spars (Figs. 22e24) wereplotted with Tecplot 10.0 [56]. In the monoplane spar, themaximum normal stresses are concentrated in the upper spar caps

Fig. 19. Comparison of spar deflections and bending moments under equivalent flapwise constant load distributions, calculated with the computational model (DYMORE): amonoplane spar, a biplane spar with “full-height” biplane cross-sections, and a biplane spar with “half-height” biplane cross-sections. Both biplane spars had design parameters ofrj/R ¼ 0.452 and g/c ¼ 1.25.

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155146

of each cross-section, while the minimum normal stresses areconcentrated in the lower spar caps; the stresses in the shear websare near zero. For the half-height biplane spar (Fig. 23), in theinboard region (cross-sections AeC), the maximum normal stressesare concentrated in the upper spar caps of the upper biplane region(near x3 ¼ 6.1 m (A), 5.5 m (B), and 5.3 m (C)), while the minimum

Table 4Structural efficiency factors for biplane spars under three different load cases,calculated with Eq. (9). Both biplane spars have a joint length-to-span ratio ofrj/R ¼ 0.452 and a gap-to-chord ratio of g/c ¼ 1.25

Load case Structural efficiency, f, of biplane spar

Half-heightcross-sections

Full-heightcross-sections

Flapwise constantload distribution

1.315 1.528

Flapwise tip load 1.148 1.234Torsional tip load 1.057 1.059

Fig. 20. Comparison of edgewise deflections of spars under a 100,000 N-m torsionaltip load, calculated with DYMORE: a monoplane spar, a biplane spar with “full-height”biplane cross-sections, and a biplane spar with “half-height” biplane cross-sections.Both biplane spars had design parameters of rj/R ¼ 0.452 and g/c ¼ 1.25.

Fig. 21. Comparison of cross-sectional stresses along the span of three spars underequivalent flapwise constant load distributions, calculated by VABS. Maximum/tensile(top) and minimum/compressive (bottom) normal stress components are comparedfor a monoplane spar, a biplane spar with “full-height” biplane cross-sections, and abiplane spar with “half-height” biplane cross-sections. The jumps in the data areconfined to the root and joint regions. Both biplane spars had design parameters ofrj/R ¼ 0.452 and g/c ¼ 1.25.

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155 147

normal stresses are concentrated in the lower spar caps of thelower biplane region (near x3 ¼ �6.1 m (A), �5.5 m (B), and �5.3 m(C)). In the outboard region (cross-sections D and E), the normalstresses in the half-height biplane spar are identical to the

Fig. 22. Normal stresses s11(x2,x3) in selected cross-sections of the monoplane spar underchord location on Sandia blade, station 10), C: 36.3% span (station 14), D: 45.2% span (stati

monoplane spar (Fig. 22). Stresses in the half-height biplane spar(Fig. 24) show similar behaviors.

3.2.2. Buckling analysisAnalytical formulas given in Section 2.2.5 were used to estimate

two critical buckling criteria: (1) a “global” beam buckling load, and(2) a “local” plate buckling stress.

3.2.2.1. “Global” beam buckling load. This analysis considered arepresentative beam similar to the lower biplane element of thebiplane spar. The representative beam had a length l ¼ 25.2 m, anda flapwise bending stiffness EI ¼ 2.876 � 109 N-m2. Substitutingthese values into Eq. (7) gives a critical buckling load ofPcritical ¼ 44,700 kN. Fig. 19(c) shows that the largest compressiveload occurs in the lower biplane element of the half-height biplanespar, and has a magnitude of 269 kN ≪ Pcritical. Therefore, thecompressive load is far below the critical value.

The representative beam's length was chosen to be the straightlength of the lower biplane element, from spar stations 1 to 13(Table A.2). Since the bending stiffness varies along the span of thebiplane spar (Fig. 18), the representative beam's stiffness was cho-sen to be uniformly equal to the smallest value that occurs in thelower biplane element; this value occurs at 25.2 m span in thesingle element of a half-height biplane cross-section. It was ex-pected that the values chosen for l and EI in Eq. (7) would give aconservative buckling load.

3.2.2.2. “Local” plate buckling stress. This analysis considered arepresentative plate similar to the lower flange in the lower biplaneelement of the biplane spar. The representative plate had a lengtha ¼ 25.2 m, width b ¼ 1.5 m, thickness h ¼ 0.0275 m, and materialproperties equal to those for uniaxial GFRP (Table 2). Substitutingthese values into Eq. (8) gives a critical buckling stress ofsx,critical ¼ 17.4 MPa, which occurs when m ¼ 13. Fig. 21 shows thatthe largest compressive stress occurs in the lower biplane elementof the half-height biplane spar, and has a magnitude of5.4 MPa < sx,critical. Therefore, the compressive stress is well belowthe critical value.

The representative plate's length was chosen to be the straightlength of the lower biplane element. The width was chosen to bethe spar cap width, bSC ¼ 1.5 m. Since the flange thickness variesalong the span of the biplane spar (Table A.2), the representativeplate's thickness was chosen to be uniformly equal to the smallestthickness that occurs in the lower flange; this thickness occurs at

a flapwise constant load distribution. A: 0% span (root, station 1), B: 18.6% span (maxon 15), E: 71.7% span (station 19).

Fig. 23. Normal stresses s11(x2,x3) in selected cross-sections of the “half-height” biplane spar (rj/R ¼ 0.452 and g/c ¼ 1.25) under a flapwise constant load distribution. A: 0% span(root, station 1), B: 18.6% span (max chord location on Sandia blade, station 10), C: 36.3% span (station 14), D: 45.2% span (joint, station 15), E: 71.7% span (station 19).

Fig. 24. Normal stresses s11(x2,x3) in selected cross-sections of the “full-height” biplane spar (rj/R ¼ 0.452 and g/c ¼ 1.25) under a flapwise constant load distribution. A: 0% span(root, station 1), B: 18.6% span (max chord location on Sandia blade, station 10), C: 36.3% span (station 14), D: 45.2% span (joint, station 15), E: 71.7% span (station 19).

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155148

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155 149

spar station 4 (h ¼ hRB þ hSC ¼ 0.0125 m þ 0.0150 m). According toTable A.2, the lower flange is made of both uniaxial GFRP (the sparcap) and triaxial GFRP (the root buildup) between spar stations 1and 6 (35.7% of the length of the representative plate). However, forthis analysis, it was assumed that the representative plate wasmade entirely of uniaxial GFRP. These material properties (alongwith the other chosen values) gave the most conservative bucklingstress.

The plate buckling stress of the monoplane spar was alsocalculated with Eq. (8) as 69.6 MPa, for comparison against theplate buckling stress of the biplane spar. The representative platefor the monoplane spar was identical to the plate for the biplanespar, except that it had a thickness of h ¼ 0.55 m (see spar station 4in Table A.1, where h ¼ hRB þ hSC ¼ 0.025 m þ 0.030 m). Thebuckling stress of the plate for the monoplane spar (69.6 MPa) isfour times larger than the plate for the biplane spar (17.4 MPa),because the plate for the monoplane spar is twice as thick.

3.2.3. Parametric analysis of different biplane spar configurationsFifteen biplane spar configurations (Fig. 13) with “full-height”

biplane cross-sections were analyzed to compare their structuralperformance under the constant load distribution described inSection 2.2.3. In particular, the effect of two non-dimensionaldesign parameters (joint length-to-span ratio rj/R, and gap-to-chord ratio g/c) on the tip deflections and root bending momentsof each spar was evaluated (Fig. 25). Overall, tip deflectionsdecrease as the joint length-to-span ratio rj/R increases (Fig. 25(a)).For a given joint length-to-span ratio rj/R, tip deflections alwaysdecrease as the gap-to-chord ratio g/c increases. The joint length-to-span ratio rj/R has a stronger effect on the tip deflection thanthe gap-to-chord g/c ratio. The biplane spar configuration with thesmallest tip deflection has a joint length-to-span ratio of rj/R ¼ 0.629 and a gap-to-chord ratio of g/c ¼ 1.25 (configuration 3 inFig. 13).

Similarly, root bending moments also tend to decrease as thejoint length-to-span ratio rj/R increases, as long as rj/R ( 0.50(Fig. 25(b)). For a given joint length-to-span ratio rj/R, root bendingmoments tend to increase as the gap-to-chord ratio g/c increases,provided that rj/R( 0.40. If the joint length-to-span ratio grows toolarge (rj/R T 0.50), the root bending moments begin to increase ifthe gap-to-chord ratio is large (g/c T 1.0). The biplane sparconfiguration with the smallest root bending moment has a jointlength-to-span ratio of rj/R ¼ 0.540 and a gap-to-chord ratio of g/c ¼ 1.25 (configuration 6 in Fig. 13).

Fig. 25. Effect of two design parameters (joint length-to-span ratio rj/R, and gap-to-chord radistribution. Lighter filled contours show smaller tip deflections and smaller root bending mothe contour plot. (For interpretation of the references to colour in this figure legend, the re

4. Discussion

4.1. Comparison of results from analytical and computationalmodels

This paper develops analytical and computational models toanalyze the structural performance of biplane spars for 100-mbiplane blades. Results from both the analytical model (Fig. 17)and the computational model (Fig. 19) in this investigation showsimilar performance trends for tip deflections and bending mo-ments of biplane beams and spars. The tip deflections of biplanebeams and spars are much smaller than the monoplane beam andspar, respectively. However, if the inboard region is too flexible (asin the half-height biplane beam and spar), inboard deflections ofthe biplane beam and spar can be slightly larger than the mono-plane beam and spar. These inboard deflections can be tuned byadjusting b (the ratio of flapwise bending stiffness in the outboardregion to the inboard region) in the analytical model (Eq. (4)). Theroot bending moment of biplane beams and spars is also muchsmaller than the monoplane beam and spar.

Inboard of the joint, bending moments are reduced by a coun-tering moment produced by axial forces in the biplane region(Figs. 17(c) and 19(c)) and reaction forces at the joint. As theanalytical model shows in Fig. 17(b), the root bending moments forthe biplane beam (z1000 kN-m for the UB beam, andz1000 kN-mfor the LB beam) do not add up to the root bending moment of themonoplane beam (z5000 kN-m). The computational model showsa similar behavior in Fig. 19(b). The countering moment also ex-plains why the bending moments decrease inboard of the joint, inthe biplaneemonoplane transition region. Inboard of the transitionregion, the bending moments increase again and reach a localmaximum near the root. The bending moments in the biplanebeam (Fig. 17(b)) also decrease at the joint, albeit abruptly, becausethe joint was assumed to be rigid and the analytical model did notinclude a biplaneemonoplane transition region. Reduced inboardbending moments are also seen in joined wings [48], where the tailis connected to the wing to produce similar axial forces and reac-tion forces inboard of the joint.

Past the root of the biplane beam, the bending moment changessign (Fig. 17(b)) because there is an inflection point in the deflectioncurve, which is 4th-order in x (Eq. (4)). This sign reversal in thebending moment is also seen in the biplane spar (Fig. 19(b)). In theinboard region of both biplane beams, the bending moment curvesalso have the same shape (Fig. 17(b)); the curve for the full-height

tio g/c) on the structural performance of a biplane spar under a flapwise constant loadments. Red points show the location (rj/R and g/c) of each biplane spar configuration onader is referred to the web version of this article.)

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155150

biplane beam is merely shifted up from the half-height biplanebeam by a constant, which depends upon b (Eq. (4)). This behavior isalso seen in the bendingmoments for both biplane spars (Fig.19(b)).

Overall, many of the features in the deflection, bendingmoment,and axial force resultant curves from the computational model alsoexist in the analytical model, indicating good qualitative agreementbetween the two models. This suggests that the analytical modelreveals some of the underlying mechanisms for the structuralperformance of biplane spars, and may be useful as a preliminarydesign tool for future work.

4.2. Cross-sectional stresses

While the biplane spar improved the tip deflections and inboardbendingmoments, inboard stresses were not always improved. Themagnitude of inboard cross-sectional stresses can sometimes belarger in the biplane spar than in the monoplane spar (Fig. 21).However, some of these large stresses can be explained by jumps inthe data, as the maximum andminimumvalues of stress change fordifferent cross-sections. The jumps near the root are seen in themonoplane spar, as well as both biplane spars. There are also jumpsnear the joint of both biplane spars, where the biplane transitionsinto a monoplane. All jumps in the data are confined to the root andjoint regions.

For both biplane spars and themonoplane spar, the jumps in thedata near the root are due to different mesh geometries and sizes inboth the 1D beammodels and the 2D cross-sectional models; thesejumps are confined to the root region (at spans of 0e15 m, high-lighted in green). For example, the length of the 1D beam elementbetween stations 1 and 2 is only 0.2 m, while the length of the 1Dbeam element between stations 2 and 3 is 2.1 m (Tables A.1eA.3).Within a 2D cross-section mesh, the sizes of the 2D quadrilateralelements sometimes varied greatly between materials. In partic-ular, the biaxial GFRP laminate was much thinner than the rest ofthe laminates. Considering spar station 1, the biaxial GFRP laminatethickness was 0.003 m in the shear web, compared to 0.013 m forthe uniaxial GFRP laminate (the next thinnest material) in the sparcap. At the sharp inside corners of the cross-section, these twomaterials are adjacent to each other (Fig. 7). Transition quadrilat-eral elements (a feature provided in TrueGrid) were used to mini-mize the size differences between elements in different materialsin the cross-section, but Fig. 21 shows that jumps in the results stillexist.

For both biplane spars, the jumps in the data near midspan aredue to the discontinuity in the flapwise load profile applied near thejoint (Fig. 14(b)); these jumps are confined to the joint region (atspans of 25e42 m, highlighted in purple), where the load profileabruptly changes fromamagnitude of F0/2 in the biplane region to F0in the monoplane region. In particular, the lower biplane region haslarge compressive stresses just inboardof the joint (at a spanof33m).

Large inboard stresses can be further explained by the contourplots of normal stresses in cross-sections AeC of the biplane spars(Figs. 23 and 24); thin walls in these biplane cross-sections likelyconcentrate stresses and lead to larger stress magnitudes. Note,however, that these biplane spars were not optimized designs.Instead, they were simply designed to have the same mass per unitspan as the monoplane spar (which lead to thinner-walled cross-sections) to demonstrate some preliminary structural benefits.Therefore, these stress results present an opportunity for im-provements that are needed in future designs.

4.3. Buckling analysis

The estimated buckling criteria given in Section 3.2.2 showedthat the biplane spar does not buckle under the applied loads (given

in Section 2.2.3). The simply supported boundary conditions chosenfor the buckling analysis are conservative, since it is expected thatthe boundary conditions are somewhat stiffer than simple supports.Nevertheless, the thinner walls in the biplane cross-sections greatlyreduced the critical buckling stress, compared to thicker walls in themonoplane cross-sections. This suggests that the possibility ofweight reductions from thinner walls in the biplane spar may belimited by the buckling stability of its thinner walls.

4.4. Structural efficiency

Table 4 shows that biplane spars are more efficient than themonoplane spar at supporting flapwise loads, which arise from theincoming wind. This is a desirable and significant improvement,since wind turbine blades need adequate clearance from the towerto operate safely. For the same mass as a monoplane spar, thebiplane spar reduced tip deflections, which suggests that weightreductions are possible with a realistic biplane blade design.

Under a torsional tip load, the biplane spars performed similarlyto the monoplane spar, twisting about 5e6 degrees under a100,000 N-m torque. However, the torsional and edgewise motionsare coupled in biplane spars (Fig. 20), because the upper and lowerbiplane elements are offset from the twist axis (Fig. 8), and twist atthe joint induces edgewise deflections in the biplane inboard re-gion. Conversely, in the monoplane spar, the twist axis is centeredinside each of its cross-sections, so there are no edgewise de-flections under torsional loads. The torsional-edgewise coupling inbiplane blades is a unique structural challenge that must beconsidered carefully in future studies.

4.5. Parametric analysis of different biplane spar configurations

A parametric analysis of different biplane spar configurationsshows that the joint length-to-span ratio has a strong influence onstructural performance (Fig. 25). Tip deflections of biplane spars arereduced overall as both the joint length-to-span ratio rj/R and gap-to-chord ratio g/c are increased, but tip deflections depend morestrongly on rj/R than g/c. However, this does not imply that theinboard biplane region should extend out along the entire span ofthe spar; the results for root bending moments should also beconsidered. The parametric analysis shows that root bending mo-ments of biplane spars tend to decrease as joint length-to-spanratio increases, but only up to a point. If the joint length-to-spanratio grows too large (rj/R T 0.50) when the gap-to-chord ratio isalso large (1.0 ( g/c < 1.25), the root bending moments begin toincrease. For the range of parameters examined in this study, rj/Rz 0.5 and g/cz 1.0 give an optimal biplane spar design, with bothsmall tip deflections and small root bending moments.

4.6. Outlook & future work

The structural benefits described above suggest that the biplaneblade may be an attractive design for the next generation of largewind turbine blades. For a uniform distributed load, biplane sparswere shown to deflect less and to have smaller root bending mo-ments than the monoplane spar. These results suggest the sameconclusion from past work: for a monoplane spar of fixed length, itis likely possible to construct a lighter biplane spar with an equal tipdeflection. Within the limits of buckling considerations, thesebenefits can lead directly to weight reductions for large blades thatcan reduce the cost of the blades, the gravitational loads on theblades, as well as the inertial loads on the rest of the wind turbine.Thus, it is likely that this design will have significant structuralbenefits for large (3e7 MW) and ultra-large (8e10 MW) turbinesfor both land-based and offshore applications.

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155 151

The analytical and computational models developed here werecreated to be a platform for future research of biplane blades. Manyopportunities exist for improved biplane blade designs, particularlyto mitigate large inboard stresses and improve buckling resistance.The analytical model may also be a useful tool for future pre-liminary designs. While the 1D FEA used in this paper was suffi-ciently accurate to compare biplane spars to each other with aparametric analysis, 3D FEA should be used for future studies of thetwo biplane spars that performed best (configurations 3 and 6 inFig. 13). This will increase the fidelity of the model, especially nearthe joint of the biplane spar.

While the results from this investigation suggest that bladeweight may be reduced with the biplane blade design, more loadcases are necessary to confirm this observation. Once the airfoilshave been added over the biplane spar (step 2 of the “structures-first” approach) to form a “full” biplane blade, then actual IECdesign load cases [22] will be used. Additionally, future studies willinvestigate the torsional and edgewise bending response of theinboard biplane region, since these two responses may be coupled.Although manufacturing considerations were outside the scope ofthis paper, it is important to note that the manufacture of the jointwill require some consideration and needs to be addressed in

Table A.1Laminate schedule for the monoplane spar. All spar stations use monoplane cross-sectionthe spar tip.

Spar station Spar fraction (e) Coordinate x1 (m) Root buildup

bRB (m) hRB (m

1 0.000 0.0 1.672 0.0632 0.002 0.2 1.672 0.0553 0.025 2.3 1.672 0.0404 0.048 4.4 1.672 0.0255 0.071 6.5 1.672 0.0156 0.098 9.0 1.672 0.0057 0.133 12.2 0.000 0.0008 0.151 13.9 0.000 0.0009 0.169 15.5 0.000 0.00010 0.186 17.1 0.000 0.00011 0.215 19.8 0.000 0.00012 0.245 22.5 0.000 0.00013 0.274 25.2 0.000 0.00014 0.363 33.4 0.000 0.00015 0.452 41.5 0.000 0.00016 0.540 49.6 0.000 0.00017 0.629 57.8 0.000 0.00018 0.700 64.3 0.000 0.00019 0.717 65.9 0.000 0.00020 0.770 70.8 0.000 0.00021 0.805 74.0 0.000 0.00022 0.894 82.2 0.000 0.00023 0.947 87.0 0.000 0.00024 1.000 91.9 0.000 0.000

Table A.2Laminate schedule for a biplane spar with “half-height” cross-sections, joint length-to-spheight” biplane cross-sections; stations 15e24 use monoplane cross-sections.

Spar station Spar fraction (e) Coordinates Root buildup

x1 (m) x3 (m) bRB (m) hRB (

1 0.000 0.0 ±4.768 1.672 0.0312 0.002 0.2 ±4.768 1.672 0.0273 0.025 2.3 ±4.768 1.672 0.0204 0.048 4.4 ±4.768 1.672 0.0125 0.071 6.5 ±4.768 1.672 0.0076 0.098 9.0 ±4.768 1.672 0.0027 0.133 12.2 ±4.768 0.000 0.0008 0.151 13.9 ±4.768 0.000 0.000

future efforts. The ultimate goal of this research is to design abiplane blade with airfoils, and perform a direct comparison to amonoplane blade. This will require aeroelastic simulations, a fa-tigue analysis, and a more detailed buckling analysis.

Acknowledgments

The authors gratefully acknowledge the financial support fromthe California Energy Commission through the Energy InnovationsSmall Grant (EISG grant #56536A/09-15), as well as the UCLAEugene V. Cota-Robles Fellowship. Thanks are due to ProfessorWilliam Klug from UCLA for introducing the authors to the methodof minimum total potential energy, as well as his guidance with theanalytical model. Thanks are also due to Phillip Chiu for valuablediscussions and the artistic renderings of the biplane blade. Finally,the authors are grateful to Professor Ertugrul Taciroglu and Pro-fessor Stanley Dong from UCLA for valuable discussions on thecomputational model.

Appendix A. Laminate schedules for spars

s. Note: the coordinate x1 is in the spanwise direction, pointing from the spar root to

Spar cap Shear web

) bSC (m) hSC (m) bSW,foam (m) bSW,biax (m) hSW (m)

1.500 0.013 0.080 0.003 5.2661.500 0.013 0.080 0.003 5.2651.500 0.020 0.080 0.003 5.0091.500 0.030 0.080 0.003 4.7411.500 0.051 0.080 0.003 4.4251.500 0.068 0.080 0.003 4.0911.500 0.094 0.080 0.003 3.6801.500 0.111 0.080 0.003 3.4801.500 0.119 0.080 0.003 3.2851.500 0.136 0.080 0.003 3.0891.500 0.136 0.080 0.003 2.8821.500 0.136 0.080 0.003 2.6961.500 0.128 0.080 0.003 2.4981.500 0.119 0.080 0.003 2.0771.500 0.111 0.080 0.003 1.6721.500 0.102 0.080 0.003 1.3601.500 0.085 0.080 0.003 1.1381.500 0.068 0.080 0.003 0.9541.500 0.064 0.080 0.003 0.9101.500 0.047 0.080 0.003 0.8321.500 0.034 0.080 0.003 0.7961.500 0.017 0.080 0.003 0.7071.500 0.009 0.080 0.003 0.6511.500 0.005 0.080 0.003 0.508

an ratio rj/R ¼ 0.452, and gap-to-chord ratio g/c ¼ 1.25. Spar stations 1e14 use “half-

Spar cap Shear web

m) bSC (m) hSC (m) bSW,foam (m) bSW,biax (m) hSW (m)

5 1.500 0.0065 0.080 0.003 2.63305 1.500 0.0065 0.080 0.003 2.63250 1.500 0.0100 0.080 0.003 2.50455 1.500 0.0150 0.080 0.003 2.37055 1.500 0.0255 0.080 0.003 2.21255 1.500 0.0340 0.080 0.003 2.04550 1.500 0.0470 0.080 0.003 1.84000 1.500 0.0555 0.080 0.003 1.7400

(continued on next page)

Table A.2 (continued )

Spar station Spar fraction (e) Coordinates Root buildup Spar cap Shear web

x1 (m) x3 (m) bRB (m) hRB (m) bSC (m) hSC (m) bSW,foam (m) bSW,biax (m) hSW (m)

9 0.169 15.5 ±4.768 0.000 0.0000 1.500 0.0595 0.080 0.003 1.642510 0.186 17.1 ±4.768 0.000 0.0000 1.500 0.0680 0.080 0.003 1.544511 0.215 19.8 ±4.768 0.000 0.0000 1.500 0.0680 0.080 0.003 1.441012 0.245 22.5 ±4.768 0.000 0.0000 1.500 0.0680 0.080 0.003 1.348013 0.274 25.2 ±4.768 0.000 0.0000 1.500 0.0640 0.080 0.003 1.249014 0.363 33.4 ±2.355 0.000 0.0000 1.500 0.0595 0.080 0.003 1.038515 0.452 41.5 0.000 0.000 0.0000 1.500 0.1110 0.080 0.003 1.672016 0.540 49.6 0.000 0.000 0.0000 1.500 0.1020 0.080 0.003 1.360017 0.629 57.8 0.000 0.000 0.0000 1.500 0.0850 0.080 0.003 1.138018 0.700 64.3 0.000 0.000 0.0000 1.500 0.0680 0.080 0.003 0.954019 0.717 65.9 0.000 0.000 0.0000 1.500 0.0640 0.080 0.003 0.910020 0.770 70.8 0.000 0.000 0.0000 1.500 0.0470 0.080 0.003 0.832021 0.805 74.0 0.000 0.000 0.0000 1.500 0.0340 0.080 0.003 0.796022 0.894 82.2 0.000 0.000 0.0000 1.500 0.0170 0.080 0.003 0.707023 0.947 87.0 0.000 0.000 0.0000 1.500 0.0090 0.080 0.003 0.651024 1.000 91.9 0.000 0.000 0.0000 1.500 0.0050 0.080 0.003 0.5080

Table A.3Laminate schedule for a biplane spar with “full-height” cross-sections, joint length-to-span ratio rj/R ¼ 0.452, and gap-to-chord ratio g/c ¼ 1.25. Spar stations 1e14 use “full-height” biplane cross-sections; stations 15e24 use monoplane cross-sections.

Spar station Spar fraction (e) Coordinates Root buildup Spar cap Shear web

x1 (m) x3 (m) bRB (m) hRB (m) bSC (m) hSC (m) bSW,foam (m) bSW,biax (m) hSW (m)

1 0.000 0.0 ±4.768 1.586 0.03321 1.500 0.0065 0.040 0.0015 5.2662 0.002 0.2 ±4.768 1.586 0.02899 1.500 0.0065 0.040 0.0015 5.2653 0.025 2.3 ±4.768 1.586 0.02108 1.500 0.0100 0.040 0.0015 5.0094 0.048 4.4 ±4.768 1.586 0.01318 1.500 0.0150 0.040 0.0015 4.7415 0.071 6.5 ±4.768 1.586 0.00791 1.500 0.0255 0.040 0.0015 4.4256 0.098 9.0 ±4.768 1.586 0.00264 1.500 0.0340 0.040 0.0015 4.0917 0.133 12.2 ±4.768 0.000 0.00000 1.500 0.0470 0.040 0.0015 3.6808 0.151 13.9 ±4.768 0.000 0.00000 1.500 0.0555 0.040 0.0015 3.4809 0.169 15.5 ±4.768 0.000 0.00000 1.500 0.0595 0.040 0.0015 3.28510 0.186 17.1 ±4.768 0.000 0.00000 1.500 0.0680 0.040 0.0015 3.08911 0.215 19.8 ±4.768 0.000 0.00000 1.500 0.0680 0.040 0.0015 2.88212 0.245 22.5 ±4.768 0.000 0.00000 1.500 0.0680 0.040 0.0015 2.69613 0.274 25.2 ±4.768 0.000 0.00000 1.500 0.0640 0.040 0.0015 2.49814 0.363 33.4 ±2.355 0.000 0.00000 1.500 0.0595 0.040 0.0015 2.07715 0.452 41.5 0.000 0.000 0.00000 1.500 0.1110 0.040 0.0015 1.67216 0.540 49.6 0.000 0.000 0.00000 1.500 0.1020 0.040 0.0015 1.36017 0.629 57.8 0.000 0.000 0.00000 1.500 0.0850 0.040 0.0015 1.13818 0.700 64.3 0.000 0.000 0.00000 1.500 0.0680 0.040 0.0015 0.95419 0.717 65.9 0.000 0.000 0.00000 1.500 0.0640 0.040 0.0015 0.91020 0.770 70.8 0.000 0.000 0.00000 1.500 0.0470 0.040 0.0015 0.83221 0.805 74.0 0.000 0.000 0.00000 1.500 0.0340 0.040 0.0015 0.79622 0.894 82.2 0.000 0.000 0.00000 1.500 0.0170 0.040 0.0015 0.70723 0.947 87.0 0.000 0.000 0.00000 1.500 0.0090 0.040 0.0015 0.65124 1.000 91.9 0.000 0.000 0.00000 1.500 0.0050 0.040 0.0015 0.508

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155152

Appendix B. Spanwise properties of spars

Table B.1Spanwise properties for the monoplane spar. These properties were calculated for each ofmonoplane cross-sections.

Sparstation

Sparfraction(e)

Stiffness properties

Flapwisestiffness(N m2)

Edgewisestiffness(N m2)

Torsionalstiffness(N m2)

1 0.000 4.834Eþ10 2.530Eþ09 9.265Eþ082 0.002 4.405Eþ10 2.393Eþ09 9.206Eþ083 0.025 3.796Eþ10 2.246Eþ09 8.647Eþ084 0.048 3.437Eþ10 2.167Eþ09 8.041Eþ085 0.071 3.852Eþ10 2.421Eþ09 7.406Eþ086 0.098 3.803Eþ10 2.577Eþ09 6.686Eþ087 0.133 3.934Eþ10 3.013Eþ09 5.893Eþ088 0.151 4.073Eþ10 3.369Eþ09 5.608Eþ089 0.169 3.844Eþ10 3.515Eþ09 5.294Eþ0810 0.186 3.806Eþ10 3.872Eþ09 5.003Eþ08

the cross-sections in the spar, using the VABS 3.6 software [40]. All spar stations use

Mass properties

Axialstiffness(N)

Mass(kg/m)

Flapwise massmoment ofinertia (kg/m)

Edgewise massmoment ofinertia (kg/m)

8.073Eþ09 7.456Eþ02 3.933Eþ03 2.818Eþ027.485Eþ09 6.961Eþ02 3.573Eþ03 2.703Eþ027.175Eþ09 6.299Eþ02 2.853Eþ03 2.476Eþ027.237Eþ09 5.804Eþ02 2.311Eþ03 2.278Eþ029.028Eþ09 6.227Eþ02 2.248Eþ03 2.254Eþ021.031Eþ10 6.409Eþ02 2.019Eþ03 2.181Eþ021.306Eþ10 7.378Eþ02 1.963Eþ03 2.252Eþ021.512Eþ10 8.251Eþ02 2.002Eþ03 2.368Eþ021.606Eþ10 8.607Eþ02 1.876Eþ03 2.389Eþ021.812Eþ10 9.482Eþ02 1.840Eþ03 2.507Eþ02

Table B.1 (continued )

Sparstation

Sparfraction(e)

Stiffness properties Mass properties

Flapwisestiffness(N m2)

Edgewisestiffness(N m2)

Torsionalstiffness(N m2)

Axialstiffness(N)

Mass(kg/m)

Flapwise massmoment ofinertia (kg/m)

Edgewise massmoment ofinertia (kg/m)

11 0.215 3.287Eþ10 3.827Eþ09 4.641Eþ08 1.805Eþ10 9.371Eþ02 1.584Eþ03 2.437Eþ0212 0.245 2.853Eþ10 3.786Eþ09 4.317Eþ08 1.799Eþ10 9.272Eþ02 1.372Eþ03 2.375Eþ0213 0.274 2.301Eþ10 3.555Eþ09 3.948Eþ08 1.692Eþ10 8.706Eþ02 1.106Eþ03 2.222Eþ0214 0.363 1.458Eþ10 3.251Eþ09 3.189Eþ08 1.564Eþ10 7.963Eþ02 6.976Eþ02 1.983Eþ0215 0.452 8.629Eþ09 2.975Eþ09 2.462Eþ08 1.450Eþ10 7.286Eþ02 4.109Eþ02 1.761Eþ0216 0.540 5.144Eþ09 2.695Eþ09 1.898Eþ08 1.326Eþ10 6.601Eþ02 2.441Eþ02 1.559Eþ0217 0.629 3.004Eþ09 2.247Eþ09 1.468Eþ08 1.105Eþ10 5.503Eþ02 1.426Eþ02 1.300Eþ0218 0.700 1.702Eþ09 1.807Eþ09 1.105Eþ08 8.857Eþ09 4.426Eþ02 8.088Eþ01 1.055Eþ0219 0.717 1.461Eþ09 1.703Eþ09 1.020Eþ08 8.341Eþ09 4.172Eþ02 6.944Eþ01 9.970Eþ0120 0.770 9.257Eþ08 1.287Eþ09 8.210Eþ07 6.182Eþ09 3.151Eþ02 4.432Eþ01 7.872Eþ0121 0.805 6.339Eþ08 9.730Eþ08 6.861Eþ07 4.539Eþ09 2.383Eþ02 3.069Eþ01 6.347Eþ0122 0.894 2.640Eþ08 5.538Eþ08 4.080Eþ07 2.376Eþ09 1.356Eþ02 1.323Eþ01 4.212Eþ0123 0.947 1.242Eþ08 3.535Eþ08 2.352Eþ07 1.354Eþ09 8.658Eþ01 6.569Eþ00 3.160Eþ0124 1.000 4.343Eþ07 2.283Eþ08 9.361Eþ06 8.026Eþ08 5.591Eþ01 2.405Eþ00 2.247Eþ01

Table B.2Spanwise properties for a biplane spar with “half-height” cross-sections, joint length-to-span ratio rj/R ¼ 0.452, and gap-to-chord ratio g/c ¼ 1.25. These properties werecalculated for each of the cross-sections in the spar, using VABS [40]. For spar stations 1e14, properties of the upper and lower biplane elements are reported separately, withrespect to the 1D reference lines in Fig. 12. For spar station 14, a curvature correction (curvature about the x2-axis, k2 ¼ 0.0020 rad/m) was applied to both the upper and lowerbiplane elements.

Sparstation

Sparfraction(e)

Upper or lowerbiplane element

Stiffness properties Mass properties

Flapwisestiffness(N m2)

Edgewisestiffness(N m2)

Torsionalstiffness(N m2)

Axialstiffness(N)

Mass(kg/m)

Flapwise massmoment ofinertia (kg/m)

Edgewise massmoment ofinertia (kg/m)

1 0.000 Upper 6.043Eþ09 1.265Eþ09 4.375Eþ08 4.038Eþ09 3.728Eþ02 4.916Eþ02 1.409Eþ02Lower 6.043Eþ09 1.265Eþ09 4.375Eþ08 4.038Eþ09 3.728Eþ02 4.916Eþ02 1.409Eþ02

2 0.002 Upper 5.521Eþ09 1.199Eþ09 4.321Eþ08 3.752Eþ09 3.486Eþ02 4.477Eþ02 1.353Eþ02Lower 5.521Eþ09 1.199Eþ09 4.321Eþ08 3.752Eþ09 3.486Eþ02 4.477Eþ02 1.353Eþ02

3 0.025 Upper 4.749Eþ09 1.124Eþ09 3.981Eþ08 3.590Eþ09 3.150Eþ02 3.566Eþ02 1.238Eþ02Lower 4.749Eþ09 1.124Eþ09 3.981Eþ08 3.590Eþ09 3.150Eþ02 3.566Eþ02 1.238Eþ02

4 0.048 Upper 4.278Eþ09 1.081Eþ09 3.588Eþ08 3.605Eþ09 2.896Eþ02 2.880Eþ02 1.138Eþ02Lower 4.278Eþ09 1.081Eþ09 3.588Eþ08 3.605Eþ09 2.896Eþ02 2.880Eþ02 1.138Eþ02

5 0.071 Upper 4.823Eþ09 1.212Eþ09 3.243Eþ08 4.521Eþ09 3.120Eþ02 2.817Eþ02 1.129Eþ02Lower 4.823Eþ09 1.212Eþ09 3.243Eþ08 4.521Eþ09 3.120Eþ02 2.817Eþ02 1.129Eþ02

6 0.098 Upper 4.752Eþ09 1.288Eþ09 2.813Eþ08 5.154Eþ09 3.201Eþ02 2.520Eþ02 1.090Eþ02Lower 4.752Eþ09 1.288Eþ09 2.813Eþ08 5.154Eþ09 3.201Eþ02 2.520Eþ02 1.090Eþ02

7 0.133 Upper 4.918Eþ09 1.506Eþ09 2.427Eþ08 6.530Eþ09 3.689Eþ02 2.453Eþ02 1.126Eþ02Lower 4.918Eþ09 1.506Eþ09 2.427Eþ08 6.530Eþ09 3.689Eþ02 2.453Eþ02 1.126Eþ02

8 0.151 Upper 5.091Eþ09 1.685Eþ09 2.332Eþ08 7.562Eþ09 4.125Eþ02 2.503Eþ02 1.184Eþ02Lower 5.091Eþ09 1.685Eþ09 2.332Eþ08 7.562Eþ09 4.125Eþ02 2.503Eþ02 1.184Eþ02

9 0.169 Upper 4.804Eþ09 1.757Eþ09 2.197Eþ08 8.030Eþ09 4.304Eþ02 2.345Eþ02 1.195Eþ02Lower 4.804Eþ09 1.757Eþ09 2.197Eþ08 8.030Eþ09 4.304Eþ02 2.345Eþ02 1.195Eþ02

10 0.186 Upper 4.852Eþ09 1.942Eþ09 2.183Eþ08 9.218Eþ09 4.741Eþ02 2.300Eþ02 1.253Eþ02Lower 4.852Eþ09 1.942Eþ09 2.183Eþ08 9.218Eþ09 4.741Eþ02 2.300Eþ02 1.253Eþ02

11 0.215 Upper 4.108Eþ09 1.913Eþ09 1.907Eþ08 9.026Eþ09 4.686Eþ02 1.980Eþ02 1.219Eþ02Lower 4.108Eþ09 1.913Eþ09 1.907Eþ08 9.026Eþ09 4.686Eþ02 1.980Eþ02 1.219Eþ02

12 0.245 Upper 3.567Eþ09 1.893Eþ09 1.752Eþ08 8.994Eþ09 4.636Eþ02 1.715Eþ02 1.187Eþ02Lower 3.567Eþ09 1.893Eþ09 1.752Eþ08 8.994Eþ09 4.636Eþ02 1.715Eþ02 1.187Eþ02

13 0.274 Upper 2.876Eþ09 1.777Eþ09 1.568Eþ08 8.458Eþ09 4.353Eþ02 1.382Eþ02 1.111Eþ02Lower 2.876Eþ09 1.777Eþ09 1.568Eþ08 8.458Eþ09 4.353Eþ02 1.382Eþ02 1.111Eþ02

14 0.363 Upper 1.820Eþ09 1.621Eþ09 1.204Eþ08 7.689Eþ09 3.981Eþ02 8.720Eþ01 9.916Eþ01Lower 1.820Eþ09 1.621Eþ09 1.204Eþ08 7.689Eþ09 3.981Eþ02 8.720Eþ01 9.916Eþ01

15 0.452 e 8.629Eþ09 2.975Eþ09 2.462Eþ08 1.450Eþ10 7.286Eþ02 4.109Eþ02 1.761Eþ0216 0.540 e 5.144Eþ09 2.695Eþ09 1.898Eþ08 1.326Eþ10 6.601Eþ02 2.441Eþ02 1.559Eþ0217 0.629 e 3.004Eþ09 2.247Eþ09 1.468Eþ08 1.105Eþ10 5.503Eþ02 1.426Eþ02 1.300Eþ0218 0.700 e 1.702Eþ09 1.807Eþ09 1.105Eþ08 8.857Eþ09 4.426Eþ02 8.088Eþ01 1.055Eþ0219 0.717 e 1.461Eþ09 1.703Eþ09 1.020Eþ08 8.341Eþ09 4.172Eþ02 6.944Eþ01 9.970Eþ0120 0.770 e 9.257Eþ08 1.287Eþ09 8.210Eþ07 6.182Eþ09 3.151Eþ02 4.432Eþ01 7.872Eþ0121 0.805 e 6.339Eþ08 9.730Eþ08 6.861Eþ07 4.539Eþ09 2.383Eþ02 3.069Eþ01 6.347Eþ0122 0.894 e 2.640Eþ08 5.538Eþ08 4.080Eþ07 2.376Eþ09 1.356Eþ02 1.323Eþ01 4.212Eþ0123 0.947 e 1.242Eþ08 3.535Eþ08 2.352Eþ07 1.354Eþ09 8.658Eþ01 6.569Eþ00 3.160Eþ0124 1.000 e 4.343Eþ07 2.283Eþ08 9.361Eþ06 8.026Eþ08 5.591Eþ01 2.405Eþ00 2.247Eþ01

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155 153

Table B.3Spanwise properties for a biplane spar with “full-height” cross-sections, joint length-to-span ratio rj/R ¼ 0.452, and gap-to-chord ratio g/c ¼ 1.25. These properties werecalculated for each of the cross-sections in the spar, using VABS [40]. For spar stations 1e14, properties of the upper and lower biplane elements are reported separately, withrespect to the 1D reference lines in Fig. 12. For spar station 14, a curvature correction (curvature about the x2-axis, k2 ¼ 0.0020 rad/m) was applied to both the upper and lowerbiplane elements.

Sparstation

Sparfraction(e)

Upper or lowerbiplane element

Stiffness properties Mass properties

Flapwisestiffness(N m2)

Edgewisestiffness(N m2)

Torsionalstiffness(N m2)

Axialstiffness(N)

Mass(kg/m)

Flapwise massmoment ofinertia (kg/m)

Edgewise massmoment ofinertia (kg/m)

1 0.000 Upper 2.399Eþ10 1.180Eþ09 4.330Eþ08 4.037Eþ09 3.728Eþ02 1.951Eþ03 1.315Eþ02Lower 2.399Eþ10 1.180Eþ09 4.330Eþ08 4.037Eþ09 3.728Eþ02 1.951Eþ03 1.315Eþ02

2 0.002 Upper 2.190Eþ10 1.118Eþ09 4.311Eþ08 3.745Eþ09 3.481Eþ02 1.776Eþ03 1.263Eþ02Lower 2.190Eþ10 1.118Eþ09 4.311Eþ08 3.745Eþ09 3.481Eþ02 1.776Eþ03 1.263Eþ02

3 0.025 Upper 1.894Eþ10 1.059Eþ09 4.063Eþ08 3.588Eþ09 3.151Eþ02 1.422Eþ03 1.163Eþ02Lower 1.894Eþ10 1.059Eþ09 4.063Eþ08 3.588Eþ09 3.151Eþ02 1.422Eþ03 1.163Eþ02

4 0.048 Upper 1.724Eþ10 1.034Eþ09 3.791Eþ08 3.620Eþ09 2.904Eþ02 1.157Eþ03 1.078Eþ02Lower 1.724Eþ10 1.034Eþ09 3.791Eþ08 3.620Eþ09 2.904Eþ02 1.157Eþ03 1.078Eþ02

5 0.071 Upper 1.942Eþ10 1.171Eþ09 3.502Eþ08 4.513Eþ09 3.113Eþ02 1.131Eþ03 1.076Eþ02Lower 1.942Eþ10 1.171Eþ09 3.502Eþ08 4.513Eþ09 3.113Eþ02 1.131Eþ03 1.076Eþ02

6 0.098 Upper 1.928Eþ10 1.259Eþ09 3.179Eþ08 5.151Eþ09 3.202Eþ02 1.022Eþ03 1.049Eþ02Lower 1.928Eþ10 1.259Eþ09 3.179Eþ08 5.151Eþ09 3.202Eþ02 1.022Eþ03 1.049Eþ02

7 0.133 Upper 2.017Eþ10 1.484Eþ09 2.832Eþ08 6.531Eþ09 3.689Eþ02 1.004Eþ03 1.092Eþ02Lower 2.017Eþ10 1.484Eþ09 2.832Eþ08 6.531Eþ09 3.689Eþ02 1.004Eþ03 1.092Eþ02

8 0.151 Upper 2.101Eþ10 1.664Eþ09 2.693Eþ08 7.562Eþ09 4.125Eþ02 1.031Eþ03 1.152Eþ02Lower 2.101Eþ10 1.664Eþ09 2.693Eþ08 7.562Eþ09 4.125Eþ02 1.031Eþ03 1.152Eþ02

9 0.169 Upper 1.992Eþ10 1.738Eþ09 2.542Eþ08 8.030Eþ09 4.304Eþ02 9.703Eþ02 1.164Eþ02Lower 1.992Eþ10 1.738Eþ09 2.542Eþ08 8.030Eþ09 4.304Eþ02 9.703Eþ02 1.164Eþ02

10 0.186 Upper 1.988Eþ10 1.917Eþ09 2.399Eþ08 9.062Eþ09 4.741Eþ02 9.593Eþ02 1.225Eþ02Lower 1.988Eþ10 1.917Eþ09 2.399Eþ08 9.062Eþ09 4.741Eþ02 9.593Eþ02 1.225Eþ02

11 0.215 Upper 1.723Eþ10 1.896Eþ09 2.228Eþ08 9.026Eþ09 4.686Eþ02 8.288Eþ02 1.192Eþ02Lower 1.723Eþ10 1.896Eþ09 2.228Eþ08 9.026Eþ09 4.686Eþ02 8.288Eþ02 1.192Eþ02

12 0.245 Upper 1.501Eþ10 1.877Eþ09 2.074Eþ08 8.994Eþ09 4.636Eþ02 7.200Eþ02 1.163Eþ02Lower 1.501Eþ10 1.877Eþ09 2.074Eþ08 8.994Eþ09 4.636Eþ02 7.200Eþ02 1.163Eþ02

13 0.274 Upper 1.211Eþ10 1.762Eþ09 1.903Eþ08 8.458Eþ09 4.353Eþ02 5.808Eþ02 1.088Eþ02Lower 1.211Eþ10 1.762Eþ09 1.903Eþ08 8.458Eþ09 4.353Eþ02 5.808Eþ02 1.088Eþ02

14 0.363 Upper 7.710Eþ09 1.602Eþ09 1.548Eþ08 7.422Eþ09 3.981Eþ02 3.688Eþ02 9.726Eþ01Lower 7.710Eþ09 1.602Eþ09 1.548Eþ08 7.422Eþ09 3.981Eþ02 3.688Eþ02 9.726Eþ01

15 0.452 e 8.629Eþ09 2.975Eþ09 2.462Eþ08 1.450Eþ10 7.286Eþ02 4.109Eþ02 1.761Eþ0216 0.540 e 5.144Eþ09 2.695Eþ09 1.898Eþ08 1.326Eþ10 6.601Eþ02 2.441Eþ02 1.559Eþ0217 0.629 e 3.004Eþ09 2.247Eþ09 1.468Eþ08 1.105Eþ10 5.503Eþ02 1.426Eþ02 1.300Eþ0218 0.700 e 1.702Eþ09 1.807Eþ09 1.105Eþ08 8.857Eþ09 4.426Eþ02 8.088Eþ01 1.055Eþ0219 0.717 e 1.461Eþ09 1.703Eþ09 1.020Eþ08 8.341Eþ09 4.172Eþ02 6.944Eþ01 9.970Eþ0120 0.770 e 9.257Eþ08 1.287Eþ09 8.210Eþ07 6.182Eþ09 3.151Eþ02 4.432Eþ01 7.872Eþ0121 0.805 e 6.339Eþ08 9.730Eþ08 6.861Eþ07 4.539Eþ09 2.383Eþ02 3.069Eþ01 6.347Eþ0122 0.894 e 2.640Eþ08 5.538Eþ08 4.080Eþ07 2.376Eþ09 1.356Eþ02 1.323Eþ01 4.212Eþ0123 0.947 e 1.242Eþ08 3.535Eþ08 2.352Eþ07 1.354Eþ09 8.658Eþ01 6.569Eþ00 3.160Eþ0124 1.000 e 4.343Eþ07 2.283Eþ08 9.361Eþ06 8.026Eþ08 5.591Eþ01 2.405Eþ00 2.247Eþ01

P. Roth-Johnson et al. / Renewable Energy 71 (2014) 133e155154

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