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Evaluation of shear flow in composite wind turbine blades
G. Fernandes da Silva, J.C. Marín ⇑
, A. Barroso
School of Engineering, University of Seville, Camino de los Descubrimientos s/n, 41092 Seville, Spain
a r t i c l e i n f o
Article history:
Available online 23 February 2011
Keywords:
Thin-walled section
Shear flow
Wind turbine blades
Finite element analysis
a b s t r a c t
The present work addresses the evaluation of the shear flow as an extension of the Jourawski’s formula.
This idea is developed here for the case of a multi-celled composite thin-walled section. Firstly, the expli-
cit formulation of the shear flow due to shear forces and torsion is derived, noting the simplificative
hypothesis adopted. Then, the implemented model is verified by means of a benchmark problem with
a known analytical solution. Finally, this model is utilized to evaluate the shear flow on an actual blade
configuration, comparing the results obtained with those of a Finite Element model of the same blade,
with a similar discretization.
Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Due to its slender geometry wind turbine blades are usually
modelled as elastic beams. The actual development of wind energy
has caused the recent advances in structural design of blades. The
geometric complexity of blades and the extended use of composite
materials in its manufacture have led designers to opt for numer-
ical tools (finite elements (FE)) to perform structural analysis.
Although FE can be appropriate for a fixed configuration analysis,
the use of this numerical tool during the whole design process is
very laborious, because it requires a great number of modifications.
For that reason, the development of simplified analytical beam
models has been an important research subject.
In literature, there can be found many works devoted to aniso-
tropic thin-walled beam models. There are entire books [1] and full
chapters [2] dedicated to this topic. A review of several beam the-
ories has been presented by Kapania and Raciti [3]. Schardt [4] pre-
sented a formulation indicated for stability problems. El Fatmi and
Zenzri [5] applied the exact beam theory to composite beams anal-
ysis. Chandra and Chopra [6] expanded the Vlasov theory to ana-
lyse a two-cell composite rotor blade, the structural response of
the blades were tested under bending and torsion loads to provide
experimental correlation with the analytical results. Kollár and
Pluzsik [7] derived an explicit formulation for the stiffness matrix
of thin-walled open or closed section composite beams with arbi-
trary layup. Volovoi and Hodges [8,9] applied the variational-
asymptotic method to anisotropic thin-walled beams with open
section [8] and with multicelled closed section [9]. Validation re-
sults for this method have been presented by Yu and Hodges
[10]. In this work sectional stiffnesses are evaluated, including
shear correction factors. Following this research line, Yu et al.
[11] developed a generalized Vlasov theory for composite beams
with arbitrary both section geometry and layup, based on the var-
iational-asymptotic method. Jung et al. [12] used a mixed approach
that combines both the stiffness and the flexibility formulation,
including shear deformation effects, elastic couplings and warping
restraint effects. Recently, Carrera et al. [13] presented a beam ele-
ment, based on the Carrera unified formulation, with the capability
to model thin-walled airfoil sections. This element has been ap-
plied to free vibration analysis by Carrera et al. [14]. Barbero
et al. [15] employed kinematic assumptions consistent with the
Timoshenko beam theory to generate beam stiffness coefficients
for thin-walled laminated open and closed sections subjected to
bending and axial loads. Shear flow distribution in section has been
evaluated and a shear correction factor has been derived. In this
work only symmetric sections and balanced symmetric laminates
are considered. Massa and Barbero [16] developed a strength of
materials formulation as an extension of the work in Ref. [15],
including arbitrary shape of section and consideration of torsion
loads. Salim and Davalos [17] expanded Vlasov theory to the anal-
ysis of open and closed laminated sections with arbitrary both
geometry and layup. In this formulation transverse shear deforma-
tions are included.
The common object of these works is the displacements’ evalu-
ation, whose knowledge is relevant in the aeroelastic analysis of
structures like helicopter rotor blades. These works are rigorous
theories with the following basic assumptions: kinematic assump-
tions consistent with the Timoshenko beam theory, the use of Clas-
sical Lamination Theory (CLT) to account for the different stiffness
of each laminate, and the consideration that the cross section is di-
vided in segments corresponding to the different laminates.
The structural design of wind turbine blades usually takes as a
starting point the configuration of the outer surface defined by the
0263-8223/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compstruct.2011.02.002
⇑ Corresponding author. Tel.: +34 954482137; fax: +34 954461637.
E-mail address: jcmarin@esi.us.es (J.C. Marín).
Composite Structures 93 (2011) 1832–1841
Contents lists available at ScienceDirect
Composite Structures
journal homepage: www.elsevier.com/locate/compstruct

4. Author's personal copy
aerodynamic design, analysing the elastic problem uncoupled.
Therefore, the knowledge of approximate displacements (axis dis-
placement, and rotation of the section) can be sufficient. Con-
versely, the knowledge of stress and strain at all points of the
section is essential for an accurate blade design.
Following the basic assumptions mentioned above, Paluch [18]
developed an explicit formulation for the longitudinal normal
strain and stress, using the elastic centre (or mechanical centre
of gravity) concept. This model is applied to wind turbine blades
design, comparing the equivalent stiffness of the sections and the
natural frequencies calculated with the experimentally measured
for bending and torsion load cases. Following this concept, Cañas
et al. [19] used this Strength of Materials (SM) model for the design
of a 700 kW wind turbine blade. In this work, a comparison be-
tween stress results obtained using the SM model with those ob-
tained using a FE model with similar discretization is carried out,
resulting in a good agreement between both results. Likewise,
the evaluated strains are verified with experimental measure-
ments obtained through strain gages in a bending test on a full
scale (1:1) prototype, resulting in a good agreement of the results.
Taking into account the good applicability of the model used for
the calculation of normal longitudinal stresses, the present work
addresses the evaluation of the shear flow as an extension of the
Jourawski’s formula. This idea, suggested by Massa and Barbero
[16] is developed here for the case of a multi-celled composite
thin-walled section. Firstly, the explicit formulation of the shear
flow due to shear forces and torsion is derived, noting the simpli-
fications adopted. Then, the obtained model is verified by means
of a problem with a known analytical solution. Finally, this model
is utilized to evaluate the shear flow on an actual blade configura-
tion, comparing the results obtained with those of a FE model of
the same blade, with a similar discretization.
2. Basic model assumptions
The configuration being considered here, i.e. a wind turbine
blade, is schematically represented in Fig. 1. If we take a section
of the blade, subjected to external loads, we can obtain the resul-
tant forces and moments acting on the section by equilibrium
condition.
In order to take into consideration the complex nature of the
blade laminate, the cross-section was discretized in rectangular
elements, as it is shown in Fig. 2. This was done in such a way as
to represent accurately the geometry of the cross-section and its
laminate configuration.
For this strength of materials model we shall make the assump-
tion that plane sections remain plane after deformation. With this
assumption the longitudinal strains exx vary linearly through the
thickness and can be expressed as:
exx ¼ a þ b Á y þ c Á z ð1Þ
where x is the longitudinal axis direction. Additionally, the assump-
tion of plane sections implies that transversal normal strains eyy and
ezz are neglected. Therefore the longitudinal normal stress rxxi for
the element i can be evaluated as:
rxxi ¼ Ei Á exx ð2Þ
where Ei is the equivalent Young’s Modulus of the i element (ob-
tained from CLT).
Evaluating the longitudinal normal force N as summation of the
normal stress rxx on the section:
N ¼
Z
A
rxx Á dA ¼
Xn
i¼1
Z
Ai
Ei Á ða þ b Á y þ c Á zÞ Á dAi ð3Þ
where A is the cross-section area, Ai is the area of element i, and n is
the number of elements of the section. Similarly, for the bending
moments Mz and My:
Mz ¼
Z
A
rxx Á y Á dA ¼
Xn
i¼1
Z
Ai
Ei Á ða Á y þ b Á y2
þ c Á y Á zÞ Á dAi ð4Þ
My ¼
Z
A
rxx Á z Á dA ¼
Xn
i¼1
Z
Ai
Ei Á ða Á z þ b Á y Á z þ c Á z2
Þ Á dAi ð5Þ
The static moments of area of element i, mzi and myi, can be ex-
pressed as:
mzi ¼
Z
Ai
y Á dAi myi ¼
Z
Ai
z Á dAi ð6Þ
The moments of inertia of element i, Izzi, Iyyi, Iyzi, can be expressed
as:
Izzi ¼
Z
Ai
y2
Á dAi Iyyi ¼
Z
Ai
z2
Á dAi Iyzi ¼
Z
Ai
y Á z Á dAi ð7Þ
Therefore, the three Eqs. (3)–(5) can be put in matrix form:
N
Mz
My
0
B
@
1
C
A ¼
Pn
i¼1
Ei Á Ai
Pn
i¼1
Ei Á mzi
Pn
i¼1
Ei Á myi
Pn
i¼1
Ei Á mzi
Pn
i¼1
Ei Á Izzi
Pn
i¼1
Ei Á Iyzi
Pn
i¼1
Ei Á myi
Pn
i¼1
Ei Á Iyzi
Pn
i¼1
Ei Á Iyyi
0
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
A
Á
a
b
c
0
B
@
1
C
A ð8Þ
For an isotropic section in tension the stress resultant has necessar-
ily to pass through the gravity centre since the modulus of elasticity
is the same for the entire section. However, for a blade cross-section
made of composite materials each element has a different elastic
modulus and, therefore, the stress resultant does not necessarily
Mflap
Mlag
Vlag
N
z
x
y
Vflap
Fig. 1. Loads and moments acting on the blade.
C.G.
E.C.
Element i
Ei,Ai,mi,xi,yi
dz
dy
i
Fig. 2. Section discretization in elements, and centre of gravity and elastic centre
locations.
G. Fernandes da Silva et al. / Composite Structures 93 (2011) 1832–1841 1833

5. Author's personal copy
pass through the gravity centre. In our model the elastic centre (i.e.
mechanical centre of gravity) is calculated for each cross-section,
referring all the variables to it as opposed to the centre of gravity.
Referring the section’s coordinates to the so-called elastic cen-
tre the coupling terms between axial force and bending moments
disappear. Then, the above expression (8) takes the following form:
N
Mz
My
0
B
@
1
C
A ¼
Pn
i¼1
Ei Á Ai 0 0
0
Pn
i¼1
Ei Á Izzi
Pn
i¼1
Ei Á Iyzi
0
Pn
i¼1
Ei Á Iyzi
Pn
i¼1
Ei Á Iyyi
0
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
A
Á
a
b
c
0
B
@
1
C
A ð9Þ
From expression (9) we can derive an explicit formulation for the
longitudinal normal stress rxxi, using Eqs. (1) and (2), and the val-
ues of a, b, c obtained from (9).
rxxi ¼ Ei
N
EA
þ
1
kyz
My
Xn
i¼1
EiIyzi À Mz
Xn
i¼1
EiIyyi
!
Á y
""
þ Mz
Xn
i¼1
EiIyzi À My
Xn
i¼1
EiIzzi
!
Á z
##
ð10Þ
where EA and kyz is defined by:
EA ¼
Xn
i¼1
Ei Á Ai kyz ¼
Xn
i¼1
Ei Á Iyzi
!2
À
Xn
i¼1
Ei Á Iyyi Á
Xn
i¼1
Ei Á Izzi ð11Þ
In Figs. 3–6 the longitudinal normal flow Nx (sum of longitudinal
normal stress rxx along the thickness) acting on an actual blade con-
figuration (obtained from Ref. [20]), is represented for four different
sections located as described in Fig. 7. The four sections are com-
prised of: root section (0.975 m from the first section), middle sec-
tion 1 (2.4 m from the first section), middle section 2 (5.8 m from
the first section), and extreme section (10.8 m from the first
section).
As it can be observed in Figs. 3–6, the results of the SM model
are compared with those of the FE model, showing a good agree-
ment. Only a certain difference can be found in one localized zone,
corresponding to the joint between the spar and the contour sec-
tion. Such a difference can be considered reasonable taking into ac-
count the three-dimensional nature of this zone, that can not be
completely described by a one-dimensional model like SM. Addi-
tionally, the transversal normal flow Ns, calculated with the FE
model, is represented in Figs. 3–6 and demonstrates that the trans-
versal normal stress can be neglected when compared to the longi-
tudinal normal stress, as it is assumed in SM model.
3. Shear flow evaluation
Consider a slice of a composite laminate thin-walled beam, of
size dx along the beam direction, thickness e(s) and distance s along
the shape of the cross-section, as it is shown in Fig. 8. Now, let us
consider a portion of this slice with cross-sectional area AÃ
. The
forces acting on this element have been represented in Fig. 9.
Therefore, we can derive an equilibrium equation:
-1200
-800
-400
0
400
800
1200
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Elements
NormalFlow(KN/m)
SM
Nx (FE)
Ns (FE)
1
2
3456
7
8
9
10
11
12 13 14 15
16
17
18
1
2
3456
7
8
9
10
11
12 13 14 15
16
17
18
Fig. 3. Normal flow comparison for root section.
-1200
-800
-400
0
400
800
1200
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Elements
NormalFlow(KN/m)
SM
Nx (FE)
Ns (FE)
1
2345678910
11
12
13
14
15 16 1718
19
20
21
22
1
2345678910
11
12
13
14
15 16 1718
19
20
21
22
Fig. 4. Normal flow comparison for middle section 1.
-2000
-1500
-1000
-500
0
500
1000
1500
2000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Elements
NormalFlow(KN/m) SM
Nx (FE)
Ns (FE)
1
2345678910
11
12
1314
1516
171819 20
21
1
2345678910
11
12
1314
1516
171819 20
21
Fig. 5. Normal flow comparison for middle section 2.
-1200
-800
-400
0
400
800
1200
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Elements
NormalFlow(KN/m)
SM
Nx (FE)
Ns (FE)
1
234567891011
12
131415 16
171819 20 21
1
234567891011
12
131415 16
171819 20 21
Fig. 6. Normal flow comparison for extreme section.
1834 G. Fernandes da Silva et al. / Composite Structures 93 (2011) 1832–1841

6. Author's personal copy
qðsÞdx À qð0Þdx þ Fr þ
@Fr
@x
dx À Fr ¼ 0 ) qðsÞ ¼ qð0Þ À
@Fr
@x
ð12Þ
where q is the shear flow (sum of shear stress rxs along the thick-
ness), and Fr is the resultant normal force in the area AÃ
. Mathemat-
ically, both variables can be expressed as:
qðsÞ ¼
Z
eðsÞ
rxsðx; sÞ Á de; Fr ¼
Z
AÃ
ðsÞ
rxxðx; y; zÞ Á dA ð13Þ
Expression (12) indicates that the total shear flow q is composed by
two terms, q(0) and the derivative of Fr. For a typical blade cross-
section the first term, q(0), represents the closed section shear flow
qc, and the last term, qo(s), corresponds to the open section shear
flow.
3.1. Open section shear flow
In this section we shall derive the expression that will be used to
calculate the open section shear flow distribution in the blade sec-
tions. The derivative of Fr can be evaluated in a discrete manner as
it is shown in expression (14), taking into account the stress discon-
tinuity between the different laminates present in area AÃ
. The
number of elements included in this area AÃ
has been denoted by nÃ
.
@Fr
@x
¼
@
@x
Z
AÃ
rxxdA ¼
@
@x
XnÃ
i¼1
Z
Ai
rxxidAi
!
ð14Þ
Substituting the expression of rxxi (10) in Eq. (14) we obtain:
@Fr
@x
¼
@
@x
XnÃ
i¼1
Z
Ai
Ei
N
EA
þ
1
kyz
My
Xn
i¼1
EiIyzi À Mz
Xn
i¼1
EiIyyi
!
Á y
"""
þ Mz
Xn
i¼1
EiIyzi À My
Xn
i¼1
EiIzzi
!
Á z
##
Á dAi
#
ð15Þ
Considering the definition of resultant forces and moments for the
section represented in Fig. 10 then the equilibrium equations
become:
@N
@x
¼ ÀPx;
@My
@x
¼ Vz;
@Mz
@x
¼ Vy ð16Þ
where Vy and Vz are the resultant shear forces in the section. Replac-
ing (16) in Eq. (15) we obtain:
@Fr
@x
¼
XnÃ
i¼1
Ei Á
Z
Ai
À
Px
EA
þ
1
kyz
Vz
Xn
i¼1
EiIyzi À Vy
Xn
i¼1
EiIyyi
!
Á y
""
þ Vy
Xn
i¼1
EiIyzi À Vz
Xn
i¼1
EiIzzi
!
Á z
##
Á dAi ð17Þ
Extracting from the integral in the above expression the terms that
do not depend on the section coordinates (y, z), we obtain the fol-
lowing expression:
@Fr
@x
¼ À
Px
EA
XnÃ
i¼1
EiAi þ
XnÃ
i¼1
Ei
kyz
Vz
XN
i¼1
EiIyzi ÀVy
XN
i¼1
EiIyyi
!
Á
Z
Ai
yÁdAi
þ Vy
XN
i¼1
EiIyzi ÀVz
XN
i¼1
EiIzzi
!
Á
Z
Ai
zÁdAi
#
ð18Þ
Using the expressions of the static moments of area (6), the above
Eq. (18) becomes:
@Fr
@x
¼ À
Px
EA
XnÃ
i¼1
EiAi þ
1
kyz
Vz
XN
i¼1
EiIyzi À Vy
XN
i¼1
EiIyyi
!
Á
XnÃ
i¼1
Eimzi
þ Vy
XN
i¼1
EiIyzi À Vz
XN
i¼1
EiIzzi
!
Á
XnÃ
i¼1
Eimyi
#
ð19Þ
Following the generalized assumption made for isotropic beams,
that the term associated with the distribution of axial load Px is neg-
Root
section
Middle
section 1
Extreme
section
Middle
section 2
Fig. 7. Location of the four sections in the blade.
y
z
e(s)
x
x
n
s
ss
dx
Fig. 8. Cross-section scheme.
dx
q (s) dx
q (0) dx σxs(0)
σxs(s)
Fσ+dFσ
Fσ
e(s)
σ σ
Fig. 9. Force equilibrium diagram in a slice portion.
x
z
y
NVz
Vy
Mt
Mz
My
Fig. 10. Scheme of the resultant forces and moments in section.
G. Fernandes da Silva et al. / Composite Structures 93 (2011) 1832–1841 1835

7. Author's personal copy
ligible, the first term of the above equation is ignored. The Eq. (20)
for the open section shear flow becomes:
qoðsÞ ¼ À
1
kyz
Vz
XN
i¼1
EiIyzi À Vy
XN
i¼1
EiIyyi
!
Á
XnÃ
i¼1
Eimzi
þ Vy
XN
i¼1
EiIyzi À Vz
XN
i¼1
EiIzzi
!
Á
XnÃ
i¼1
Eimyi
#
ð20Þ
3.2. Closed section shear flow
In this section we shall derive the expression that will be used
to calculate the closed section shear flow distribution in the blade
sections. Considering an arbitrary multi-cell thin-walled section, as
it is shown in Fig. 11, the requirement of single-valued displace-
ment for an isotropic beam cross-section, with length S along the
s coordinate, is defined by:
Z S
0
cxs:ds ¼ 0 )
Z S
0
rxs
G
ds ¼
1
G
Z S
0
qðsÞ
eðsÞ
ds ¼ 0 ð21Þ
The evaluation of the above relation (21) for a cell ‘j’ produces the
following equation:
qj
c Á
Z
sj
ds
eðsÞ
À qi
c Á
Z
sij
ds
eðsÞ
À qk
c Á
Z
sjk
ds
eðsÞ
¼ À
Z
sj
qoðsÞ
eðsÞ
ds ð22Þ
where qj
c; qi
c and qk
c are the closed section shear flows corresponding
to ‘j’, ‘i’ and ‘k’ cells respectively, sj is the length of the medium line
of the cell ‘j’, sij and sjk are the common lengths between the cell ‘j’
and the adjacent cells ‘i’ and ‘k’.
Similarly it is possible to derive an expression for the closed sec-
tion shear flow for a beam made of a composite laminate material
i.e. an anisotropic section. The difference in this case is that the
shear modulus cannot be eliminated from the expression, as it is
shown in Ref. [21]. In fact, an equivalent shear modulus Gxs(s) is
calculated for each element (using CLT). The requirement of sin-
gle-valued displacement for an anisotropic section, assuming that
the transverse normal flow Ns and the moment flows at the thick-
ness of the laminate Mx, Ms and Mxs are negligible, is defined by:
Z S
0
co
xsds ¼
Z S
0
aÀ1
31
 Ã
Nx þ aÀ1
33
 Ã
Nxs
À Á
ds ffi
Z S
0
aÀ1
33
 Ã
Á qðsÞds
¼
Z S
0
qðsÞ
GxsðsÞeðsÞ
ds ¼ 0 ð23Þ
In the above equation we have assumed that the coupling of the
normal and tangential flows (that appear according to CLT) will
not be taken into account in order to simplify the formulation.
Therefore, the term with the normal flow was ignored as an initial
approximation such that the expression becomes:
qj
c Á
Z
sj
ds
GxsðsÞeðsÞ
À qi
c Á
Z
sij
ds
GxsðsÞeðsÞ
À qk
c Á
Z
sjk
ds
GxsðsÞeðsÞ
¼ À
Z
sj
qoðsÞ
GxsðsÞeðsÞ
ds ð24Þ
Let as consider a two-cell cross-section blade, as it is shown in Fig. 12.
Then the expressions for the closed section shear flow become:
q1
c Á
Z
s1
ds
GxsðsÞeðsÞ
À q2
c Á
Z
s12
ds
GxsðsÞeðsÞ
¼ À
Z
s1
qoðsÞ
GxsðsÞeðsÞ
ds ð25Þ
À q1
c Á
Z
s12
ds
GxsðsÞeðsÞ
þ q2
c
Z
s2
ds
GxsðsÞeðsÞ
¼ À
Z
s2
qoðsÞ
GxsðsÞeðsÞ
ds ð26Þ
We now have two equations and two unknowns q1
c ; q2
c
À Á
and can,
therefore, solve for the closed section shear flows in cells 1 and 2.
For a blade cross-section without a stiffener i.e. with only one
cell, such as the blade tip or root, the expression is much simpler:
q1
c Á
Z
s1
ds
GxsðsÞeðsÞ
¼ À
Z
s1
qoðsÞ
GxsðsÞeðsÞ
ds ð27Þ
3.3. Evaluation of shear flow due to torsion
The shear flow so far has been defined as a result of the shear
forces acting on the sections. We must now add the shear flow gen-
erated by the torsion in the blade sections. Throughout our analysis
we assumed uniform torsion along the blade. This is mainly be-
cause analytical and numerical studies developed by other authors
(see Refs. [16,11,22]) indicate that it is sensible to consider uniform
torsion for a closed section cantilever beam such that warping can
be neglected in cross-sections far-off the constrained root.
There are two types of torsion that contribute to the total shear
flow: the torsion caused by the twisting moment exerted along the
longitudinal axis of the blade and the torsion caused by the mo-
ment induced by the shear forces. This is due to the fact that in
our analysis we consider the shear force resultant at each section
acting through the elastic centre which does not coincide with
the shear centre. Therefore, there is a moment induced around
the shear centre caused by the product of the shear resultant by
its perpendicular distance from the elastic centre to the shear
centre.
The requirement of single-valued displacement due to torsion,
for an isotropic thin-walled section is defined by:
Z S
0
cxsds À 2Xh ¼ 0 )
1
G
Z S
0
qtðsÞ
eðsÞ
ds ¼ 2Xh ð28Þ
where h is the unitary angle of torsion and X is the sectorial area of
section i.e. the area enclosed by the medium line of the section. For
an arbitrary multi-cell thin-walled section, as it is shown in Fig. 11,
the application of Eq. (28) for a cell ‘j’ becomes:
qj
t Á
I
sj
ds
eðsÞ
À qi
t Á
Z
sij
ds
eðsÞ
À qk
t Á
Z
sjk
ds
eðsÞ
ds ¼ 2GhXj
ð29Þ
where qj
t; qi
t and qk
t are the shear flows induced by torsion at cells ‘j’,
‘i’ and ‘k’ respectively, and Xj
is the sectorial area of ‘j’ cell. The ap-
plied torque Mt can be expressed by equilibrium of moments as:
Mt ¼ 2
XN
i¼1
qi
tXi
ð30Þ
In the same way as for the closed section shear flow due to shear
forces, it is possible to derive an expression for the shear flow in-
1 i j k N
1 i j k N
1 i j k N
Fig. 11. Multi-cell beam cross-section.
1
2
1
2
Open section
shear flow
Closed section
shear flow
Fig. 12. Representation of the open and closed section shear flows.
1836 G. Fernandes da Silva et al. / Composite Structures 93 (2011) 1832–1841

8. Author's personal copy
duced by the torsion for a beam made of a composite laminate
material i.e. an anisotropic section. Again, an equivalent shear mod-
ulus Gxs(s) is calculated for each element. Expression (28) for a com-
posite laminate thin-walled section can be written:
Z S
0
co
xsds ¼
Z S
0
aÀ1
31
 Ã
Nx þ aÀ1
33
 Ã
Nxs
À Á
Á ds ffi
Z S
0
aÀ1
31
 Ã
Á qtðsÞds
¼
Z S
0
qtðsÞ
GxsðsÞeðsÞ
ds ¼ 2Xh ð31Þ
Similarly, the coupling of the normal and tangential flows will not
be taken into account. Therefore, the term with the normal flow
was ignored as an initial approximation such that the expression
becomes:
qj
t Á
I
sj
ds
GxsðsÞeðsÞ
À qi
t Á
Z
sij
ds
GxsðsÞeðsÞ
À qk
t Á
Z
sjk
ds
GxsðsÞeðsÞ
ds
¼ 2Xj
h ð32Þ
Considering the same two-cell cross-section blade represented in
Fig. 12, then the expressions for the two cells 1 and 2 become:
q1
t Á
Z
s1
ds
GxsðsÞeðsÞ
À q2
t Á
Z
s12
ds
GxsðsÞeðsÞ
¼ 2h Á X1
ð33Þ
À q1
t Á
Z
s12
ds
GxsðsÞeðsÞ
þ q2
t Á
Z
s2
ds
GxsðsÞeðsÞ
¼ 2h Á X2
ð34Þ
The equilibrium of moments can be used to obtain a third equation:
Mt ¼ 2 q1
t X1
þ q2
t X2
ð35Þ
We now have three Eqs. (33)–(35) and three unknowns q1
t ; q2
t ; h
À Á
and can, therefore, solve for the shear flows induced by the torsion
in cells 1 and 2.
For a blade cross-section without a stiffener i.e. with only one
cell, such as the blade tip or root, the expressions become much
simpler:
q1
t Á
Z
s1
ds
GxsðsÞeðsÞ
¼ 2hX1
Mt ¼ 2q1
t X1
ð36Þ
Finally, the total shear flow will, therefore, be the sum of the open
section, closed section and torsion shear flows:
qðsÞ ¼ qoðsÞ þ qcðsÞ þ qtðsÞ ð37Þ
4. Model verification
As a verification of the developed model the results obtained
have been compared for a problem with a known analytical solu-
tion [23, chapter 7 (71), p. 349]. Consider a solid in the form of a
hollow cylinder whose material possesses cylindrical orthotropy,
subjected to a vertical shear force at one end and a vertical shear
force and bending moment at the other end, as is shown in
Fig. 13. The analytical solution of this problem takes the following
expression for the shear stress distribution:
rhz ¼ À
P
I
r2
cos h þ
Pb
2
I
k 3
r2
b
2
À
1 À clþ3
1 À c2l l
r
b
lÀ1
þ
1 À clÀ3
1 À c2l lclþ3 r
b
ÀlÀ1
cos h ð38Þ
c ¼
a
b
l ¼
ffiffiffiffiffiffiffiffi
G12
G13
s
k ¼
3 À 2 m13G12
E11
9 À l2
ð39Þ
A glass–vinylester composite material has been considered (used in
the blades that will be analysed later for comparison purposes). The
properties for this material are: E11 = 34.5 GPa, E22 = 9.34 GPa,
m12 = 0.35, G12 = 2.7 GPa. The outer radius of the cylinder is
b = 40 cm, the inner radius is a = 39 cm, and the applied force
P = 10 KN. For the application of the SM model the cross-section
of the cylinder is discretized in 20 elements. The comparison be-
tween the model results and the analytical solution are represented
in Fig. 14 and the values are listed in Table 1, showing a good agree-
ment. The maximum value of error in the shear flow evaluated by
the SM model is about 0.84% for a thickness of 1 cm. The influence
of the variation of the thickness in the amount of the error is rela-
tively small, as it is shown in Fig. 15 and in Table 2. For a very thick
wall cylinder (thickness = 10 cm) the error only achieves a maxi-
mum value of 1.4%.
5. Application of the model to the analysis of an actual blade
configuration
As object for application of the developed SM model an actual
blade configuration (taken from Ref. [20]) has been considered.
This configuration has been chosen because we have a FE model
for such a blade, which was verified by experimental measure-
ments [19]. The results of this FE model will be compared with
the results of SM model.
The blade is mainly made out of glass fibre and vinylester resin
and is approximately fifteen metres long. The different laminates
in the blade are combinations of layers used in manufacture pro-
cess: unidirectional (0° NUFF), biaxial (0°/90° or +45°/À45°), triax-
ial (90°/+45°/À45°), random (MAT) and balsa wood (sandwich
core).
The FE model was built and analysed using ANSYS [24]. The
model was assembled using an element type denominated SHELL
99 which is a laminate type element with eight nodes, 6 degrees
of freedom at each node and the capacity to model laminates made
x
y
z
P
a
b
r
θ
P
PL
Lx
y
z
P
a
b
r
θ
P
PL
L
Fig. 13. Scheme of the problem considered for verification.
-10
-8
-6
-4
-2
0
2
4
6
8
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
nodes
Shearflow(KN/m)
Analytical solution
SM model
Fig. 14. Comparison of the shear flow distributions.
G. Fernandes da Silva et al. / Composite Structures 93 (2011) 1832–1841 1837

9. Author's personal copy
up of 100 distinct laminas (distinct orthotropic axis and/or materi-
als). This element is defined through the input of its geometric
location, the connectivity of its nodes, the sequence of the laminas
that define the laminate and the mechanical properties and thick-
ness of each lamina.
The discretization of the blade was done following two main
criteria: the first is that the location of the nodes must be as similar
as possible to the blade geometry, and the second is that zones
made up of different laminates must be modelled by different ele-
ments. For this blade the FE model was defined with 5380 nodes,
dividing the blade into 64 sections (varying the number of ele-
ments per section from 18 to 32), with a total of 2027 elements
and more than 500 types of different stacking sequences. The dis-
cretization for the FE model, which is shown in Fig. 16, has been
used for the SM model, dividing the blade into the same number
of sections, which have the same number of elements.
For comparison purposes we applied two simple 10 KN shear
load (flap and lag directions) at tip of the blade, as it is shown in
Fig. 17 for the FE model. The idea was to use a load case easy to
model for both FE and SM models in order to achieve similarity
of the results for a simple case.
5.1. Shear flow results
The results obtained using both models (SM and FE) have been
compared for the majority of sections. Four zones in particular,
with different geometric characteristics, have been selected to
illustrate the total shear flow distribution along the blade. The
location of these four zones and its representative sections is
shown in Fig. 18.
5.1.1. Tip zone
Section 61 is located near the blade tip and it has only one cell
i.e. no stiffener. As it can be seen in Fig. 19 (values in Table 3), both
models present similar variation and magnitude, although the
maximum values of the SM model are higher than those of the
FE model.
5.1.2. Middle zone next to the tip
Sections at this zone (56, 46) have two cells i.e. they have a stiff-
ener. As it can be seen in Figs. 20 and 21 (values of Fig. 21 in Table
Table 1
Comparison of the shear flow distributions.
Node Analytical solution (KN/m) SM model (KN/m) Difference (%)
1 À7.970 À8.035 0.822
2 À7.579 À7.642 0.825
3 À6.447 À6.500 0.820
4 À4.684 À4.723 0.816
5 À2.463 À2.483 0.836
6 0.000 0.000 0.000
7 2.463 2.483 0.836
8 4.684 4.723 0.816
9 6.447 6.500 0.820
10 7.579 7.642 0.825
11 7.970 8.035 0.822
12 7.579 7.642 0.825
13 6.447 6.500 0.820
14 4.684 4.723 0.816
15 2.463 2.483 0.836
16 0.000 0.000 0.000
17 À2.463 À2.483 0.836
18 À4.684 À4.723 0.816
19 À6.447 À6.500 0.820
20 À7.579 À7.642 0.825
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 2 4 6 8 10
thickness (cm)
error(%)
error (%)
Fig. 15. Variation of the error in the shear flow evaluation with thickness.
Table 2
Variation of the error in the shear flow evaluation with thickness.
Thickness (cm) Error (%)
0.5 0.8248
1.0 0.8310
2.0 0.8356
4.0 0.8944
6.0 0.9987
8.0 1.1733
10.0 1.4036
Fig. 16. Finite element model of the blade.
Fig. 17. Flap and lag loads applied at tip of the blade.
1838 G. Fernandes da Silva et al. / Composite Structures 93 (2011) 1832–1841

10. Author's personal copy
4), both models present similar variation and magnitude and the
values of the SM model are slightly higher than those of the FE
model. In this case, the maximum value of the total shear flow is
about 50 KN/m, which is considerably smaller than the maximum
value of the total shear flow for section 61 (about 200 KN/m). The
main difference from section 61 lies in the two peaks/troughs lo-
cated approximately near elements 9 and 19. These peaks corre-
spond, respectively, to the start and end nodes of the stiffener. In
this region it is likely that 3-D effects affect the finite element mod-
el results in a way that is not contemplated in the strength of mate-
rials model.
5.1.3. Middle zone next to the root
Sections at this zone (27, 21), like in the zone above, have two
cells, i.e. they have a stiffener. The main difference between this
zone and the zone above is the thickness, that it is larger at this
zone. The effect of this larger thickness is a low level of the shear
flow in all the length of the sections, as can be appreciated for both
models in Figs. 22 and 23. The maximum value of the total shear
flow achieved at this zone is about 30 KN/m. No significant differ-
ences can be reported between the results for both models applied.
5.1.4. Influence zone of the concentrator
Sections at this zone (15, 6) present a sudden geometrical
change. In a length of 1 m, the cross-section is transformed from
2115 646 5627
middle zone
next to the root
middle zone
next to the tip
tip zoneinfluence zone of
the concentrator
14.4613.0210.275.554.202.660.97distance (m)
6156462721156section
61
Fig. 18. Location of sections considered.
section 61
-100
-50
0
50
100
150
200
250
0 5 10 15 20
Element number
Shearflow(KN/m)
FE
SM
Fig. 19. Shear flow distribution for section 61.
Table 3
Shear flow distribution for section 61.
Element no. FE (KN/m) SM (KN/m) Difference (%)
1 À64.313 À74.534 13.71
2 À58.082 À72.747 20.16
3 À48.226 À65.544 26.42
4 À38.784 À53.400 27.37
5 À27.537 À10.930 151.93
6 À5.338 56.517 109.44
7 134.180 108.064 24.17
8 168.360 157.153 7.13
9 166.420 219.782 24.28
10 144.810 226.320 36.02
11 152.230 219.060 30.51
12 107.730 162.745 33.80
13 87.844 124.969 29.71
14 66.495 85.824 22.52
15 32.010 24.406 31.15
16 À51.278 À18.897 171.36
17 À52.023 À32.073 62.20
18 À58.835 À43.677 34.70
19 À62.070 À54.936 12.99
20 À63.517 À66.501 4.49
Section 56
-100
-50
0
50
100
150
200
250
0 5 10 15 20
Element number
Shearflow(KN/m)
FE
SM
Fig. 20. Shear flow distribution for section 56.
Section 46
-100
-50
0
50
100
150
200
250
0 5 10 15 20 25
Element number
Shearflow(KN/m)
FE
SM
Fig. 21. Shear flow distribution for section 46.
G. Fernandes da Silva et al. / Composite Structures 93 (2011) 1832–1841 1839

11. Author's personal copy
circular to an aerofoil shape section, describing a corner in the
transition zone. As can be seen in Figs. 24 and 25 (values of
Fig. 25 in Table 5), the models present different variation and mag-
nitude of the shear flow. The reason behind this discrepancy is that
this geometry cannot be modelled as a beam; therefore the SM
model was unable to take into consideration the effect of the
concentrator.
6. Conclusions
The purpose of this work was to enhance a simplified SM model
for the analysis of wing turbine blades, in order to endow it with
the ability to evaluate the shear flows in the blade sections.
An explicit formulation of the shear flow distribution in a thin-
walled laminated composite beam of multi-celled cross-section
Table 4
Shear flow distribution for section 46.
Element no. FE (KN/m) SM (KN/m) Difference (%)
1 À6.912 À2.342 195.18
2 À6.467 À2.955 118.87
3 À6.043 À3.625 66.71
4 À3.254 À3.476 6.38
5 À3.621 À2.648 36.76
6 À3.365 À2.208 52.38
7 3.490 1.926 81.21
8 À4.877 10.705 145.56
9 À26.481 À10.360 155.60
10 22.863 23.618 3.19
11 36.700 38.088 3.64
12 32.163 39.453 18.48
13 31.083 40.232 22.74
14 31.000 40.380 23.23
15 31.829 40.265 20.95
16 24.407 35.498 31.24
17 1.818 15.579 88.33
18 À20.153 À13.686 47.25
19 9.941 10.868 8.54
20 À8.623 8.672 199.43
21 À15.122 7.979 289.51
22 À13.810 5.183 366.47
23 À10.645 3.665 390.45
24 À8.779 1.600 648.66
25 À7.185 0.659 1189.95
26 À7.313 À0.918 696.42
Section 27
-100
-50
0
50
100
150
200
250
0 5 10 15 20 25 30
Element number
Shearflow(KN/m)
FE
SM
Fig. 22. Shear flow distribution for section 27.
Section 21
-100
-50
0
50
100
150
200
250
0 5 10 15 20 25 30
Element number
Shearflow(KN/m)
FE
SM
Fig. 23. Shear flow distribution for section 21.
Section 15
-100
-50
0
50
100
150
200
250
0 5 10 15 20 25 30
Element number
Shearflow(KN/m)
FE
SM
Fig. 24. Shear flow distribution for section 15.
Section 6
-100
-50
0
50
100
150
200
250
0 2 4 6 8 10 12 14 16 18
Element number
Shearflow(KN/m)
FE
SM
Fig. 25. Shear flow distribution for section 6.
Table 5
Shear flow distribution for Section 6.
Element no. FE (KN/m) SM (KN/m) Difference (%)
1 21.102 À10.371 303.47
2 52.022 À10.036 618.37
3 52.905 À8.153 748.90
4 17.893 À6.852 361.12
5 7.810 À2.185 457.49
6 À7.112 À0.337 2007.57
7 À29.630 3.721 896.31
8 À37.987 7.327 618.45
9 À10.452 9.901 205.56
10 29.778 10.992 170.90
11 57.646 10.568 445.46
12 29.421 8.543 244.41
13 À19.965 7.269 374.65
14 À44.065 2.704 1729.84
15 À39.736 0.981 4150.56
16 À32.573 À2.874 1033.24
17 À37.484 À6.415 484.34
18 À19.804 À9.121 117.12
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12. Author's personal copy
was developed and successfully implemented into a design fortran
code previously developed. This formulation was verified by means
of a problem with a known analytical solution. A perfect agreement
was obtained for the model results respect to the analytical
solution.
As a demonstration, the SM model was used for the analysis of
an actual blade configuration. For comparison purposes the same
blade configuration was analysed with a FE model with similar dis-
cretization. The results show that both models present similar var-
iation and magnitude for the majority of the blade sections.
As it was explained in Section 3, some simplifications have been
made throughout the development of the SM model to simplify its
implementation in the fortran code. In particular by neglecting the
effects of the axial load distribution Px on the open section shear
flow, and also by neglecting the coupling of the normal and tan-
gential flows on the closed section shear flow and torsion shear
flow calculations. A further analysis could be carried out in order
to investigate the effects that adding these terms would have on
the results.
Another simplification that must be pointed out is the applica-
tion of the SM model to a blade of variable section, since the for-
mulation developed considers a beam with constant cross-
section. This fact constitutes another potential source of inaccuracy
in the model.
As it was explained in Section 5.1 the discrepancies in the re-
sults between the SM and FE models increased for sections near
the blade ‘‘corner’’ i.e. the region of the blade where the cross-sec-
tion changes from circular to an aerofoil shape section, and for sec-
tions near the blade root. As it can be seen in Figs. 16 and 17 the
corner of the blade modelled using finite elements is a 3-D model
that resembles the real geometry of blade. The same cannot be
done using the strength of materials model which is a one-dimen-
sional model. Thus, there are significant three-dimensional effects,
which cannot be modelled by the strength of materials model, that
affect the sections near the corner. Nevertheless the actual design
of wing turbine blades presents a smooth transition in root zone.
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