REPRINTED FROM WIND ENGINEERING - University of …dds11/6-S1152-Symons-final.pdf · Comparison of Theoretical and Numerical Buckling Loads for Wind Turbine Blade Panels by Nicholas

Comparison of Theoretical and Numerical BucklingLoads for Wind Turbine Blade Panels

by

Nicholas Gaudern, Dr. Digby D. Symons

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WIND ENGINEERINGVOLUME 34, NO. 2, 2010

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Comparison of Theoretical and Numerical BucklingLoads for Wind Turbine Blade Panels

Nicholas Gaudern, Dr. Digby D. Symons*

University of Cambridge, Department of Engineering,Trumpington Street,Cambridge, CB2 1PZ

WIND ENGINEERING VOLUME 34, NO. 2, 2010 PP 193–206 193

ABSTRACTTheoretical and finite element (FE) methods for predicting buckling of wind turbine blades

are compared. The theoretical method considers the blade skin as separate panels

(idealised as cylindrically curved, simply-supported, and under uniform axial compression);

established theory provides the critical load. This approach is compared to FE models of

individual panels and representative aerofoils. The FE calculation for an idealised panel

agrees with the theory (within 10%). Idealising the panel curvature as cylindrical makes

little difference to the critical load (<5%). Adoption of a more realistic load distribution over

the length of a blade has greater influence: the buckling strain at the root of a blade under a

cubically varying bending moment distribution (e.g. flap-wise bending of a tapered blade

under uniform incident wind pressure) is 15% higher than that for an idealised panel. The

idealised panel method is therefore a conservative method suitable for preliminary design.

*Corresponding author.

Tel: +44 1223 760502; fax: +44 1223 332662. E-mail address: [email protected]

NOMENCLATUREAA,, BB,, DD Sub-matrices of laminate stiffness matrix

E Young’s Modulus

G Shear Modulus

HH Buckling parameter matrix (see eqn. 4)

Iyy 2nd moment of area of blade cross-section

L Length of chord of a curve

NN Vector of axial loads

Nx Critical buckling load in the longitudinal direction per unit width of plate

Ny Transverse (pressure) loading

MM Vector of moments

R (Approximate) radius of curvature of panel

SS Buckling parameter matrix, SS = AA––11 (see eqn. 4)

S Distance along aerofoil section curve

b Width of panel

c Chord

t Aerofoil depth (thickness)

tw Thickness of panel

u, v, w Displacements

x, y, z Blade/panel co-ordinate system

ε Vector of direct-strains

κ Vector of curvatures

λ Number of buckled half-waves in the longitudinal (x) direction

η Number of buckled half-waves in the transverse (y) direction

ν Poisson’s ratio

1. INTRODUCTIONBuckling is a failure mode of particular interest for the designers of large composite wind

turbine blades. The cross-sections are typically composed of thin shells, and since the blade is

subjected to large flap-wise bending moments the surface panels near the blade root are

particularly vulnerable to elastic instability (see, for example, Berggreen et al [2007],

Lindenburg [2003] or Thomson [2009]). Accurate prediction of buckling loads by the finite

element method is possible (and is almost certainly cheaper than testing), but may still be a

relatively costly iterative step in the design process. It is therefore common practice for the

calculation of the critical buckling loads of blades to be simplified during the initial design

phase, where speed of calculation takes precedence over absolute accuracy.

A methodology based on the buckling of a simply-supported, cylindrically curved panel is

often applied (e.g. Burton et al [2001]): the leeward surface of the blade is considered to be

divided into three individual panels, separated by the (usually) two structural webs present

(Fig. 1). These panels are treated as being simply-supported along all edges, infinite in length,

of uniform cylindrical curvature, and loaded purely axially; a critical buckling strain for each

panel is calculated using established elastic stability theory (i.e. Timoshenko and Gere [1961]).

The bending moments in the blade that would produce these critical compressive strains may

then be obtained straightforwardly.

However, the idealised panel geometry differs greatly from that of a real blade; one

example is that in reality the blade skin is continuous over the webs. Although this will affect

the calculation of a critical buckling load for the panel, a more critical assumption may be that

of the loading condition: a blade subjected to a realistic wind loading will have a moment

distribution that varies over the blade length, giving an axial compressive load that varies

longitudinally in the surface panel.

194 COMPARISON OF THEORETICAL AND NUMERICAL BUCKLING LOADS FOR

WIND TURBINE BLADE PANELS

z

y

Figure 1: Cross-section of a typical blade.

The aim of this study is to investigate, through a case study, the extent to which the

assumptions outlined above can affect the calculation of a critical buckling load. This

investigation will be performed by comparing the theoretical idealised panel method to

results obtained using finite element analysis of both individual panels and full blade models..

The theoretical buckling load for the idealised panel is calculated following the method

described in the ESDU standard 94007 [1994].

If the finite element models are defined carefully then the analysis results can

generally be expected to be of a high level of accuracy: exact geometry can be used, the

boundary conditions can be more appropriate, and a realistic load distribution can be

applied. The findings of this investigation provide a quantitative assessment of the validity

of the panel method assumptions, and may alert designers to the potential magnitude of

the difference between the buckling loads predicted and those that are more likely to be

encountered in reality.

2. PANEL BUCKLING THEORY2.1. Calculation of Critical Buckling LoadThe panel buckling theory used in this investigation follows the method given by the ESDU

standard 94007 [1994]. Note that Timoshenko and Gere [1961] and Burton et al [2001] also give

theory for the buckling of curved panels, but are restricted to either isotropic or symmetrical

composite laminates respectively. The ESDU theory is more general.

The calculation is suitable for finding the axial buckling loads of rectangular, curved,

simply-supported composite panels subjected to axial in-plane loading. The theory is based

upon elastic, thin plate, small deflection, classical laminate theory in which the panel is

assumed to behave as a homogeneous orthotropic material whose axes of orthotropy are

aligned with the edges of the plate. The theory takes account of direct in-plane, flexural, and

twisting deformations governed by the AA and DD stiffness matrices as defined in classical

laminate theory; see, for example, Kollár and Springer [2003]. Through-thickness shear

deformations are disregarded.

The load-deformation relationships for any laminate can be written:

(1)

where NN is the vector of axial loads, MM is the vector of moments, AA, BB, and DD describe the

laminate properties, ε is the vector of direct-strains, and κ is the vector of curvatures.

The assumption is made that the boundary conditions of the problem can be satisfied by

buckling modes with sinusoidal displacement distributions of the form:

(2)

where x and y are the in-plane co-ordinates; u and v are the corresponding in-plane

displacements and w is the out-of-plane displacement (see Fig. 2). λ and η are parameters

associated with the number of buckled half-waves along the longitudinal (x) and transverse

(y) directions respectively. For example, η = nπ/b where b is the width of the panel and n is the

number of half-waves across the width of the plate.

u x y

v x y

w

cos( ) sin( )

sin( ) cos( )

s

∝∝∝

λ ηλ η

iin( ) sin( )λ ηx y

NN

MM

AA BB

=

BB DD

εεκκ

WIND ENGINEERING VOLUME 34, NO. 2, 2010 195

The critical buckling load in the longitudinal direction per unit width of plate ( Nx ) is found

using eqn (3):

(3)

where

(4)

and R is the radius of cylindrical curvature. Ny is the transverse load per unit length of plate, in

equilibrium with any applied external pressure loading p such that Ny = −pR .

Equation (3) should be solved for a range of values of λ and η, the minimum value found

will provide the best estimate of the critical buckling load. In the case of a panel that

represents a surface panel of a wind turbine blade, the panel width b is defined, for example,

by the distance between the internal webs of the blade. Discrete values of η should therefore

be selected that correspond to integer values of the number of transverse half waves n. The

panel has no defined length, and may be considered to be effectively infinite; the variable λshould therefore be allowed to vary continuously (or at least in small steps) to a value much

larger than η.

2.2. Calculation of Idealised Panel GeometryThe actual curvature κ = 1/R of an aerofoil panel must be approximated by a cylindrical curve

if eqn. (3) is to be applied. This may be achieved using the following formula:

(5)

where S is the distance along the curve, and L is the length of the chord of the curve. Eqn. (5)

is an approximation derived from the exact relationship for a circular curve. The estimated R

κ � �=−( )1 24

S

S L

S

HH SS SS SS SS

HH

114

22 12 332 2

114

12

2 2= + + +

=

( ( / ))λ λ η η

/

( )

λ

λ λ η η

2

224

11 12 332 2

2242 2

R

HH DD DD DD DD= + + +

SSSS 11= A−−

N Nx y= − ( )+ +

/ /HH HH HH2

12 11 222 2η λ



z, w

R

x, u

w = v = 0

w = v = 0

w = 0

w = u = 0

y, v

Nx

Figure 2: Geometry of idealised cylindrically curved panel.

for a circle is conservative, i.e. larger than the true R, with a fractional error of no more than

6% for a 180° panel. We note in passing that some designers (e.g. Bir [2001]) take an even more

conservative approach to preliminary design and approximate the panel as flat (zero

curvature).

3. FINITE ELEMENT MODELLINGA series of finite element (FE) analyses have been performed, which are intended to

model, by varying degrees of approximation, the buckling of a surface panel of a wind

turbine blade of particular aerofoil section under flap-wise bending. The panel chosen is the

central panel (panel 2) of the NACA 0012 symmetrical aerofoil (Jacobs et al [1933]) shown

in Fig. 3. The section has a chord of 2 m, thickness-to-chord ratio (t/c) of 0.45, wall thickness

tw = 0.016 m, and webs positioned at 0.5 m and 1.2 m from the leading edge. For simplicity,

the wall material is assumed isotropic with a Young’s modulus E of 140 GPa and Poisson’s

ratio ν of 0.3. The complexity of each test analysis was progressively increased in an

attempt to provide increasing realism: from models of single panels to models of complete

blades. A list of the finite element tests is given in Table 1 and a summary of the main

parameters is given in Table 2.

For each test, a finite element analysis was carried out using the ABAQUS/Standard

software [SIMULIA, 2009]. For both full-blade and panel tests, “S4R” shell elements were used

exclusively to define the analysis mesh (the wall thickness was assumed thin compared to the


Figure 3: Geometry of NACA 0012 test section.

Panel 1

Panel 2

Panel 3

0.016 m0.7 m

2 m

Table 1: Summary of finite element testsTest Test Type Curvature Boundary Conditions Loading

1 Single panel Cylindrical Simply supported on edges Axial compression2 Single panel Aerofoil section Simply supported on edges Axial compression3 Single panel Aerofoil section Encastré on edges Axial compression4 Uniform chord

aerofoil Aerofoil section Encastré at root Tip moment5 Uniform chord

aerofoil Aerofoil section Encastré at root Tip force6 Uniform chord

aerofoil Aerofoil section Encastré at root Uniform Load7 Tapered aerofoil Aerofoil section Encastré at root Tip force8 Tapered aerofoil Aerofoil section Encastré at root Uniform Load9 Tapered aerofoil Aerofoil section Encastré at root Linear load

overall dimensions of the structure). A constant mesh density was used, and a sensitivity test

of one of the buckling analyses to this parameter was also performed. The subspace

eigensolver in ABAQUS was used with the maximum number of iterations set at 1000; only the

first eigenvalue was requested.

3.1. Panel TestsFirstly, a direct comparison between calculation methods was made by comparing the

buckling strain calculated by ABAQUS for a simply-supported panel with perfectly

cylindrical curvature (test 1) to the value given by the ESDU theory; the approximate

radius of curvature R was found using eqn. (5). This was then followed by an ABAQUS

calculation for a panel with curvature matching that of a section of the real aerofoil (test 2).

A comparison between the idealised cylindrical curvature and the actual geometry of

panel 2 of the NACA 0012 aerofoil is shown in Fig. 4 (note that the vertical axis is

exaggerated).

For tests 1 and 2, simple supports were provided on all four edges that restrained

displacements but not rotations. All four panel edges were fixed in the vertical direction and

the unloaded edges were also prevented from in-plane displacements perpendicular to the

edge, a shell edge load of unit magnitude was applied to one of the short edges (see Fig. 2). Test 3

investigated the effect of fixity of the panel edges by also preventing rotation of the panel

edges (i.e. encastré boundary conditions). In tests 1, 2 and 3 the length of the FE panel was

arbitrarily fixed at 2 m.



Table 2: Summary of parameters used for theoretical and FE calculationsParameter ValueYoung’s Modulus, E 140 GPaShear Modulus, G 53.8 GPaPoisson’s Ratio, ν 0.32nd moment of area of whole cross-section, Iyy 0.00843 m4

(Approximate) Radius of Curvature of Panel 2, R 1.848 mCentroid height of Panel 2, z 0.374 mWidth of Panel 2, b 0.712 mThickness of Panel 2, tw 0.016 mPressure Loading, Ny 0

−0.04

−0.06

−0.08

−0.1

−0.12

−0.14

−0.02

0

0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

y

z

Aerofoil panel curvature

Idealised cylindrical curvature

Figure 4: Comparison of idealised and actual panel 2 coordinates.

3.2. Blade TestsBefore the testing of full blade models could be carried out, a method to generate the required

FE models needed to be developed. A Matlab [MATHWORKS, 2008] script was written that

could write an ABAQUS input file describing a blade defined by a small set of user-input

parameters. The script allows the user to define the aerofoil section, root and tip chord, and

thickness to chord ratio (t/c) values for a linear variation along the blade length, blade twist

distribution, wall thickness and material, and mesh density.

The basic processes of this script are as follows:

• read the aerofoil geometry file to define y and z coordinates of the aerofoil section

• transform the coordinates to the desired chord c and t/c at the root

• add nodes to define internal webs

• ‘extrude’ all of the nodes in the x direction to generate a 3D blade, re-calculating the

chord and t/c and applying a twist transformation at intervals of x along the blade

• assign shell elements to groups of four nodes

We note that similar parametric methods for generating a FE mesh of a wind turbine blade

were adopted by Bechly and Clausen [1997] and Jureczko et al [2001].

All blade tests used the NACA 0012 aerofoil section geometry shown in Fig. 3. This section

was either extruded prismatically to form a uniform chord blade for tests 4 to 6 (Fig. 5) or

extruded with a linearly tapering chord for tests 7 to 9 (Fig . 6).


Figure 5: 3D FE model of a uniform blade (tests 4, 5 & 6).

3.2.1. Uniform BladeA uniform chord blade with a length of 20m was analysed in tests 4, 5, and 6 (Fig. 5). To

investigate the effect of different load distributions on the panel buckling load a different type

of applied load was used in each test. In test 4 the blade was loaded at its tip with a flapwise

moment to give a uniform moment over the blade length (and therefore uniform compression

in the surface panel on one side of the blade). For test 5 the blade was loaded at its tip with a

concentrated force in the flapwise direction to give a linearly-varying moment. In test 6 a

uniformly-distributed load was applied over the length of the blade to give a quadratic

moment distribution. Test 6 was intended to approximate the storm-loading case for a wind

turbine where the blade is parked in one position and subject to a uniform pressure.



Figure 6: 3D FE model of a tapering blade (tests 7, 8 & 9).

In each test the blade was subject to the following boundary conditions: all of the nodes at

the root of the blade were made encastré, and the nodes forming the last section of the blade

at the tip were joined into a rigid body to prevent local deformation when concentrated

loadings were applied at this point. The concentrated tip force or moment was applied to the

node at the leading edge of the tip profile.

3.2.2. Tapered BladeA tapered blade was analysed in tests 7, 8, and 9 to increase the realism of the investigation

(Fig. 6). The blade profile at the root was the same as that used in the uniform blade tests

(a NACA 0012 profile with 2 m chord) but this was linearly tapered over the 20 m length to one

third of the root value (0.667 m). However the t/c and wall thickness were kept constant at

0.45 and 0.016 m respectively. We note that a constant wall thickness is desirable for both

manufacturing and structural reasons (Jackson et al [2005]).

A different load distribution was used in each test in order to introduce a different

variation of bending moment (and hence different distributions of compression in the surface

panel). For test 7 the blade was loaded at its tip to give a linearly-varying moment. In test 8 a

uniformly-distributed load was applied over the length of the blade to give a quadratic

moment distribution. In test 9 the applied load was linearly varying along the length to give a

cubically varying bending moment distribution; this load case is intended to represent a

uniform pressure distribution on the linearly tapered blade (i.e. a storm loading condition

when the turbine is braked and not rotating, see also Bir [2001]). Note that a uniform moment

case is not considered for the tapered blade; under a constant moment the tapered blade

buckles at the tip under the high compressive strain due to the reduced cross-section.

Boundary conditions for the tapered blade are the same as used for the uniform blade:

encastré at the root, and with a rigid body assigned to the tip cross-section.

4. RESULTSFigs. 7 and 8 show the buckling modes calculated by ABAQUS for the two aerofoil section

panels, one with simple supports (test 2) and one with fully-restrained edges (test 3). For the

simply-supported panel the number of buckling half-wavelengths (1 transversely and 6

longitudinally) is in approximate agreement with the number predicted by the ESDU theory

(1 transversely and about 8 longitudinally – recalling that the theoretical method is for a plate

of indefinite length and takes the longitudinal wavelength that minimizes the buckling load).

The buckling mode of test 3 shows the significant effect of the fully-restrained panel edges.

Fig. 9 shows the buckling mode for the uniform blade under a uniformly distributed load (test

5, the buckling modes for tests 6 to 9 were very similar). It is interesting to observe how the

buckling in panel 2 is complemented by a matching, out-of-phase, buckling of the adjacent

panel 3, which also includes some deformation of the trailing edge. This response of a whole

blade model suggests that a panel with fully restrained edges (i.e. test 3) is likely to be a poor

approximation to the buckling of a surface panel of a real blade.

The results of the panel and blade tests are summarised in Table 3. The values of critical

strain given are those for the central panel (panel 2) at the root of the blade, and were found

using the standard formula for direct strain:

(6)ε � �= Mz

EIyy


Figure 7: Buckling mode of simply-supported panel (test 2).

Figure 8: Buckling mode of fully-restrained panel (test 3).

where M is the critical bending moment (BM) at the blade root (obtained from the ABAQUS

eigenvalue result), z is the distance from the neutral axis of the blade to the centroid of the

panel, E is the Young’s modulus of the material, and Iyy is the second moment of area of the

blade cross-section in the flapwise bending direction.

4.1. DiscussionThe direct comparison of an FE calculation for a cylindrically curved panel (test 1) with the

ESDU theory shows reasonable agreement of the predicted critical compressive strains (10%

difference). Note that this result was obtained with the length of panel of the FE model

arbitrarily set at 2 m, whereas the ESDU calculation is for an effectively infinite length. Test 1

is now treated as a benchmark and the following tests are compared to this result.

Changing the cross-section of the curved panel to the true aerofoil section (test 2) does

have a small effect on the buckling response: a slight asymmetry of the buckling mode can

be seen in Fig. 7; however, the critical buckling load differs from the cylindrical case (test 1)

by only 4%. In contrast, the change to encastré boundary conditions (test 3) provides very



Table 3: Summary of critical buckling strain resultsLoad Critical Root Crit. comp. % Diff.

Test Type Distribution BM (MNm) Strain (×10–3) to Test 1ESDU Long cyl. panel, SS Constant – 5.24 −10%

1 Cylindrical panel, SS Constant – 5.80 –2 Aerofoil panel, SS Constant – 5.54 −4%3 Aerofoil panel, encastré Constant – 13.0 +124%4 Uniform chord blade Constant BM 17.7 5.62 −3%5 Uniform chord blade Linear BM 19.2 6.10 +5%6 Uniform chord blade Quadratic BM 20.4 6.47 +12%7 Tapered chord blade Linear BM 18.4 5.86 +1%8 Tapered chord blade Quadratic BM 19.7 6.26 +8%9 Tapered chord blade Cubic BM 21.0 6.67 +15%

U, Magnitude+1.078e+00+9.884e−01+8.985e−01+8.087e−01+7.188e−01+6.290e−01+5.391e−01+4.493e−01+3.594e−01+2.696e−01+1.797e−01+8.985e−02+0.000e+00

Figure 9: Buckling mode of uniform blade under a uniformly distributed load (test 5).


significant restraint and the critical strain is more than doubled. Comparison of the buckling

modes and critical strains from test 3 with the remaining full blade analyses (tests 4 to 9)

confirms that the choice of encastré boundary conditions on a single panel is a poor

approximation to the response of a panel within a complete blade. If the internal webs of the

blade are not significantly stiffer than the blade skin (and therefore do not provide

significant restraint) then the complimentary buckling response of the adjacent panels on

the blade surface means that panel 2 behaves, to a good approximation, as if simply

supported at its edges. Note the very close agreement (within 1%) of the critical buckling

strains for test 2 and test 4.

When the full blade models are loaded with non-constant bending moment distributions

of increasing order (e.g. from constant, through linear to quadratic) the critical strain

(measured at the blade root) increases somewhat. The critical bending moment distributions

for tests 4 to 9 are plotted in Fig. 10 (uniform cross-section blade) and Fig. 11 (tapered blade). It

is clear that, although the critical moments (and hence the critical strains quoted in Table 3)

differ by up to 15% at the root, the value of the critical bending moment at around 1.5 m from

the root is almost the same in each test. We note from Fig. 9 that the buckling mode

incorporates three half-wavelengths of approximately 1 m each longitudinally; therefore it

seems clear that an even better prediction of the critical condition for buckling would be the

average compressive strain over the last, say, 3 m of the blade.

Comparison of tests 5 and 6 with tests 7 and 8 respectively allows the difference between

the uniform and the, more realistic, tapered blade to be seen. The critical bending moment

measured at the root is slightly (4%) lower for the tapered blade in both cases. It seems that

two competing effects of the blade taper are almost equal: although the reducing cross-

section of blade due to taper suggests that a particular compressive strain should be achieved

at a lower BM, this effect seems to be almost balanced by an increase in the critical

compressive strain due to the tapering width b of the panel.

0 2 4 6 8 10 12 14 16 18 20

2.5E+07

2.0E+07

1.5E+07

1.0E+07

5.0E+06

0.0E+00

x (m)

M

Constant (test 4)

Linear (test 5)

Quadratic (test 6)

Figure 10: Critical bending moment distributions for uniform blade (tests 4, 5 & 6).

4.2. Mesh Density Sensitivity CheckFor speed of analysis the FE analyses of full blades (tests 4 to 9) were performed with rather

coarse mesh densities: e.g. at the blade root shell elements with an approximate side length of

0.1 m were used to model an aerofoil cross-section with chord of 2 m. To check that the

presented results are not greatly affected by the choice of FE mesh density, test 5 (uniform

blade loaded by a tip force) was also analysed with two higher mesh densities: two times and

three times the original. The results are given in Table 4. Tripling the mesh density gave a

difference in critical strain of 3%; this was judged as satisfactory evidence that the results

presented are not significantly influenced by the chosen mesh density.



0 2 4 6 8 10 12 14 16 18 20

2.5E+07

2.0E+07

1.5E+07

1.0E+07

5.0E+06

0.0E+00

x (m)

M

Linear (test 7)

Quadratic (test 8)

Cubic (test 9)

Figure 11: Critical moment distributions for tapered blade (tests 7, 8 & 9).

Table 4: Effect of increasing mesh density on critical strain (test 5)Critical Compressive % Difference From

Test Relative Mesh Density Strain (×10–3) Original5 1 6.10 –

5.1 ×2 5.94 –2.6%5.2 ×3 5.92 –3.0%

5. CONCLUSIONSA set of finite element (FE) calculations have been performed to investigate the accuracy of

theory for an infinitely long, cylindrically curved, simply supported panel as an estimate of the

critical buckling load of a wind turbine blade under flapwise bending. The calculations were

carried out using the case study of a simplified 20m blade based on a symmetric NACA

aerofoil profile.

The conclusions of the investigation are that:

• an FE calculation for a finite length panel is in reasonable agreement (within 10%)

with the theory (which assumes a panel of indefinite length),

• approximating the aerofoil panel as cylindrically curved does not significantly

influence the buckling load (4% difference),

• an isolated panel with simple supports gives a good approximation to the buckling

mode of a surface panel within a blade (1% difference), encastré boundary

conditions are not appropriate,

• tapering of a blade (varying cross-section) does not significantly change the

buckling load relative to that of the idealised panel (4% difference),

• the critical strain in a panel at the root of a blade under a varying bending moment

distribution may be up to 15% higher than the critical strain for the equivalent idealised

panel, the average strain over the buckled region is likely to be in closer agreement.

The general conclusion is therefore that the theoretical, idealised panel method is

conservative, and an appropriate method for the preliminary design of wind turbine blades.

ACKNOWLEDGMENTSThe work presented in this paper formed part of a MEng research project at the University of

Cambridge Department of Engineering.

REFERENCESBechly M.E. and Clausen P.D. (1997) Structural Design of a Composite Wind Turbine Blade

Using Finite Element Analysis, Computers and Structures, vol 63 no. 3, pp 639–616,

Elsevier Science Ltd, UK.

Berggreen C., Branner K., Jensen J.F. and Schultz J.P. (2007) Application and Analysis of

Sandwich Elements in the Primary Structure of Large Wind Turbine Blades, Journal of

Sandwich Structures and Materials, vol 9, pp 525 – 552, SAGE Publications, London, UK.

Bir G.S. (2001) Computerized Method for Preliminary Structural Design of Composite Wind

Turbine Blades, Journal of Solar Energy Engineering, vol 123, pp 372 – 381, ASME, NY,

USA.

Burton T., Sharpe D., Jenkins N. and Bossanyi E. (2001) Wind Energy Handbook, John Wiley &

Sons, UK.

ESDU standard 94007 (1994) Elastic buckling of cylindrically curved laminated fibre

reinforced composite panels with all edges simply-supported under biaxial loading,

IHS ESDU International, 27 Corsham St., London, UK.

Jackson K.J., Zuteck M.D., van Dam C.P., Standish K.J. and Berry D., (2005) Innovative Design

Approaches for Large Wind Turbine Blades, Wind Energy, vol 8, pp 141–171, John Wiley &

Sons, UK.

Jacobs E.N., Ward K.E. and Pinkerton R.M. (1933) The characteristics of 78 related airfoil

sections from tests in the variable-density wind tunnel, NACA Report No. 460.

Jureczko M., Pawlak M. and Mezyk A. (2005) Optimisation of wind turbine blades, Journal of

Materials Processing Technology, vol 167, pp 463–471, Elsevier B.V., Amsterdam,

Netherlands.

Kollár L.P. and Springer G.S. (2003) Mechanics of Composite Structures, Cambridge University

Press, UK.

Lindenburg C. (2003) Programs for buckling strength prediction of rotor blades, ECN-

RX—03-050, Presented at European Wind Energy Conference, Madrid, Spain, 16 – 19

June, 2003.


MathWorks (2008), Matlab 7.6.0.324 (r2008a), The MathWorks Inc., 3 Apple Hill Drive, Natick,

MA 01760-2098, USA.

SIMULIA (2009) ABAQUS/Standard Users Manual, Rising Sun Mills, 166 Valley Street,

Providence, RI, USA.

Timoshenko S.P. and Gere J.M. (1961) Theory of Elastic Stability: Second Edition, McGraw-Hill.

Thomsen O.T. (2009) Sandwich Materials for Wind Turbine Blades – Present and Future,

Journal of Sandwich Structures and Materials, vol 11, pp 7 – 26, SAGE Publications,

London, UK.



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REPRINTED FROM WIND ENGINEERING - University of …dds11/6-S1152-Symons-final.pdf · Comparison of Theoretical and Numerical Buckling Loads for Wind Turbine Blade Panels by Nicholas