SAWE Paper No. 3410 Category No. 22 Category No. 10 INITIAL SIZING OPTIMISATION OF ANISOTROPIC COMPOSITE PANELS WITH T SHAPE STIFFENERS J. Enrique Herencia, Marie Curie Research Assistant Paul M. Weaver, Reader in Lightweight Structures Michael I. Friswell, Sir George White Professor of Aerospace Engineering University of Bristol (UK) For presentation at the 66 th Annual Conference of Society of Allied Weight Engineers, Inc. Madrid, Spain, 26-30 May, 2007 Permission to publish this paper, in full or in part, with credit to the authors and the Society must be obtained, by request to: Society of Allied Weight Engineers, Inc. P.O. Box 60024, Terminal Annex Los Angeles, CA 90060 The Society is not responsible for statements or opinions in papers or discussions at the meeting. This paper meets all the regulations for public information disclosure under ITAR and EAR.
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SAWE Paper No. 3410
Category No. 22
Category No. 10
INITIAL SIZING OPTIMISATION OF ANISOTROPIC COMPOSITE PANELS WITH T SHAPE STIFFENERS
J. Enrique Herencia, Marie Curie Research Assistant
Paul M. Weaver, Reader in Lightweight Structures
Michael I. Friswell, Sir George White Professor of Aerospace Engineering
University of Bristol (UK)
For presentation at the
66th Annual Conference
of
Society of Allied Weight Engineers, Inc.
Madrid, Spain, 26-30 May, 2007
Permission to publish this paper, in full or in part, with credit to the authors and the
Society must be obtained, by request to:
Society of Allied Weight Engineers, Inc.
P.O. Box 60024, Terminal Annex
Los Angeles, CA 90060
The Society is not responsible for statements or opinions in papers or discussions at the
meeting. This paper meets all the regulations for public information disclosure under
1 Abstract This paper provides an approach to perform initial sizing optimisation of anisotropic
composite panels with T shape stiffeners. The method divides the optimisation problem
into two levels. At the first level, composite optimisation is performed using
Mathematical Programming (MP), where the skin and the stiffeners are modelled using
lamination parameters accounting for their anisotropy. Skin and stiffener laminates are
assumed to be symmetric or mid-plane symmetric laminates with 0, 90, 45, or -45 degree,
ply angles. The stiffened panel is subjected to a combined loading under strength,
buckling and practical design constraints. Buckling constraints are computed using
Closed Form (CF) solutions and energy methods (Rayleigh-Ritz). Conservatism is
partially removed in the buckling analysis considering the skin-stiffener flange
interaction and decreasing the effective width of the skin. Furthermore, the design and
manufacture of the stiffener is embedded within the design variables. At the second level,
the actual skin and stiffener lay-ups are obtained using Genetic Algorithms (GAs),
accounting for manufacturability and design practices. This two level approach permits
the separation of the analysis (strength, buckling, etc), which is performed at the first
level, from the laminate stacking sequence combinatorial problem, which is dealt
efficiently with GAs at the second level.
2 Introduction Aerospace manufactures are increasingly employing laminated composites to replace
metallic materials in primary structures in order to reduce aircraft weight. Composite
stiffened panels, especially with T shape stiffeners, are commonly used to design flight
primary structures such as wings or fuselages. In general terms, composite materials
present high specific strength and stiffness ratios [1] and offer the advantage over their
metallic counterparts of being stiffness tailored. This latter feature is closely associated
with their design and manufacture. Laminated composite materials have been restricted to
symmetric or mid-plane symmetric laminates with 0, 90, 45, and -45 degree, ply angles,
due to practical manufacturing requirements. In addition, the design of composite
stiffened panels becomes more complex when considering the manufacture of the
stiffener.
Composite optimisation is a non-linear problem. A number of optimisation techniques
have been developed over the years to design composite structures [2-29]. In the
seventies, early attempts on optimisation of laminated fibre composites were performed
by Schmit and Farshi [2,3]. They optimised symmetric laminated fibre composite
materials having homogeneous and orthotropic properties, considering ply thicknesses as
continuous variables. Stroud and Agranoff [4] followed the same trend and optimised
composite hat-stiffened and corrugated panels using non-linear mathematical techniques
with a simplified set of buckling equations as constraints. The width and thickness of the
elements of the dimensioned cross section were the design variables. Laminates were
assumed to present orthotropic properties. However, composites might exhibit certain
degree of flexural anisotropy. Ashton [5] initially showed the effect of the flexural
anisotropy on the stability of composite plates. Chamis [6] concluded that neglecting
3
flexural anisotropy of the composite in the evaluation of buckling behaviour could lead to
non-conservative results. Nemeth [7] characterised the importance of flexural anisotropy
and provided bounds within which its effect would be significant. Weaver [8,9] recently
developed CF solutions to include the effect of flexural anisotropy on compression and
shear loading. Laminate flexural anisotropy is intrinsically related to its stacking
sequence.
Tsai et al. [10] introduced an alternative representation of the stiffness properties of a
laminated composite by the use of lamination parameters. Miki and Sugiyama [11]
proposed the use of lamination parameters to deal with the discrete laminate stacking
sequence problem. They dealt with symmetric and orthotropic laminates. Optimum
designs for constraints such as in-plane stiffness or buckling, were obtained, from
geometric relations between the lamination parameters feasible region and the objective
function. Fukunaga and Vanderplats [12] used lamination parameters and MP techniques
to carry out stiffness optimisation of orthotropic laminated composites. Cylindrical shells
under combined loading were used as a practical application. Haftka and Walsh [13] used
integer programming techniques to carry out laminate stacking sequence optimisation
under buckling constraints on symmetric and balanced laminated plates. They used zero-
one integers as design variables that were related to stiffness properties via lamination
parameters and showed that the problem was linear. Flexural anisotropy was limited to
manually modifying the optimum design and they used the branch and bound method to
solve the problem. Nagendra et al. [14] extended that work and optimised the stacking
sequence of symmetric and balanced composite laminates with stability and strain
constraints. The drawback of integer programming techniques is that they require large
computational resources especially when structure complexity increases. Fukunaga and
Sekine [15] used MP techniques and lamination parameters to maximise buckling loads
under combined loading of symmetrically laminated plates including the bending-
twisting couplings (flexural anisotropy). They verified the negative effect of the flexural
anisotropy on the buckling load of panels under normal loading and showed that under
shear and shear-normal loading flexural anisotropy could increase or decrease the critical
buckling load.
Le Riche and Haftka [16] and Nagendra et al. [17,18] adopted a different approach. They
employed GAs to solve the discrete lay-up optimisation problem. GAs are search
algorithms based on the mechanics of natural selection and natural genetics [19], which
do not require gradient information to perform the search. GAs are widely used for their
ability to tackle search spaces with many local optima [20] and therefore a non convex
design space. Furthermore, Nagendra et al. investigated the application of a GA to the
design of blade stiffened composite panels. VIPASA [21] was used as the analysis tool
and results were compared with PASCO [22], which uses VIPASA as the analysis tool
and CONMIN [23] as optimiser. It was concluded that the designs obtained by the GA
offered higher performance than the continuous designs. However, it was recognised that
great computational cost was associated with the GA. More recently, Liu et al. [24]
employed VICONOPT [25] to perform an optimisation of composite stiffened panels
under strength, buckling and practical design constraints. A bi-level approach was
adopted. VICONOPT was employed at the first level to minimise the panel weight,
4
employing equivalent orthotropic properties for the laminates with continuous thickness,
whereas at the second level laminate thickness were rounded up and associated to pre-
determined design lay-ups.
A two level optimisation strategy combining lamination parameters, MP and GAs, was
initially proposed by Yamazaki [26]. The optimisation was split into two parts. Firstly, a
gradient based optimisation was performed using the in-plane and out-of-plane
lamination parameters as design variables. Secondly, the lamination parameters from the
first level were targeted using a GA. In this paper, volume, buckling load, deflection and
natural frequencies of a composite panel were optimised without accounting for either
membrane or flexural anisotropy. Autio [27], following a similar approach to Yamazaki,
investigated actual lay-ups. Commercial uni/multiaxial plies were considered and certain
lay-up design rules were introduced as penalties in the fitness function of the GA.
However, with the exception of Fukunaga and Vanderplats [12], none of the previous
authors considered the feasible region in the lamination parameter space that relates in-
plane, coupling and out-of-plane lamination parameters.
Liu et al. [28] employed lamination parameters and defined the feasible region between
two of the four membrane and bending lamination parameters to maximise the buckling
load of unstiffened composite panels with restricted ply angles. They compared their
approach against one using a GA and concluded that the use of lamination parameters in
a continuous optimisation produced similar results to those obtained by the GA except in
cases where laminates were thin or had low aspect ratios. Diaconu and Sekine [29]
performed lay-up optimisation of laminated composite shells for maximisation of the
buckling load, using the lamination parameters as design variables and including their
feasible region. They fully defined, for the first time, the relations between the
membrane, coupling and bending lamination parameters for ply angles restricted to 0, 90,
45, and -45 degrees.
The authors’ previous work [30], based upon a two level optimisation approach, which
couples MP with GAs, has shown that composite anisotropy can be used to improve
structural performance. Design constraints such as laminate failure strength, local and
global buckling as well as practical design rules were considered. Buckling was
addressed by Finite Elements (FE) and CF solutions. It was shown that CF solutions
introduced a high degree of conservatism in the buckling analysis and hence heavily
penalised the optimum solutions. However, CF solutions significantly increased the
computational efficiency.
The purpose of this paper is to provide an approach to perform initial sizing optimisation
of anisotropic composite panels with T shape stiffeners. The method divides the
optimisation problem into two levels. At the first level, composite optimisation is
performed using MP, where the skin and the stiffeners are modelled using lamination
parameters accounting for their anisotropy. Skin and stiffener laminates are assumed to
be symmetric or mid-plane symmetric laminates with 0, 90, 45, or -45 degree, ply angles.
The stiffened panel is subjected to a combined loading under strength, buckling and
practical design constraints. Buckling constraints are computed using CF solutions and
5
energy methods (Rayleigh-Ritz). Conservatism is partially removed in the buckling
analysis considering the skin-stiffener flange interaction and decreasing the effective
width of the skin. Furthermore, the design and manufacture of the stiffener is embedded
within the design variables. At the second level, the actual skin and stiffener lay-ups are
obtained using Genetic Algorithms (GAs), accounting for manufacturability and design
practices. This two level approach permits the separation of the analysis (strength,
buckling, etc), which is performed at the first level, from the laminate stacking sequence
combinatorial problem, which is dealt efficiently with GAs at the second level.
3 Panel geometry and loading As in Ref. [30] the composite stiffened panel is assumed to be long, wide and composed
of a series of skin-stiffener repeating elements under combined loading. Each skin-
stiffener element consists of three flat plates (skin, stiffener flange and web) that are
assumed to be rigidly connected along their longitudinal edges. The skin-stiffener
element is assumed to model the panel’s behaviour. Figure 1 defines the skin-stiffener
element geometry, the material axis as well as the positive sign convention for the
loading.
Figure 1. Skin-stiffener repeating element.
b
a
Fc
Ny
Nxy
x
y
0o
45o90o
A
A'
Stiffener
Skin
bsf
hsw
tsw
t
tsf
Skin
Stiffener web
Stiffener flange
Section AA'
6
Due to the stiffener’s manufacture, four different stiffener configurations are considered.
The stiffener is manufactured as a back to back angle (Fig. 2a), adding capping plies in
the stiffener flange (Fig. 2b), or extra plies in the stiffener web (Fig. 2c), and finally the
combination of the previous configurations (Fig. 2d).
Figure 2. T-shape stiffener type.
4 Laminate constitutive equations The Classical Laminate Theory (CLT) [1] is applied to the skin, the stiffener flange and
web respectively, assuming laminates are symmetric or mid-plane symmetric. Thus,
⋅
=
κε o
D
A
M
N
0
0 (1)
where [ ]A is the membrane stiffness matrix, [ ]D is the bending stiffness matrix, { }N is
the vector of the in-plane running loads, { }M is a vector of the running moments, { }0ε is
the vector of in-plane strains and { }κ is the vector of the middle surface curvatures.
The membrane and bending stiffness matrices can be expressed in terms of material
stiffness invariants (U) and eight lamination parameters (ξ) [10]. Plies are considered
orthotropic and with fibre angles restricted to 0, 90, 45, and -45 degrees. As a result, the
lamination parameters are further reduced to six. Hence,
twa
b
c
d
ta
ta
7
⋅
−
−
−
⋅=
5
4
3
2
1
3
3
2
21
2
21
26
16
66
22
12
11
0002
0
0002
0
1000
001
0100
001
U
U
U
U
U
h
A
A
A
A
A
A
A
A
A
AA
A
AA
ξ
ξξξξξξξ
(2)
⋅
−
−
−
⋅=
5
4
3
2
1
3
3
2
21
2
21
3
26
16
66
22
12
11
0002
0
0002
0
1000
001
0100
001
12
U
U
U
U
U
h
D
D
D
D
D
D
D
D
D
DD
D
DD
ξ
ξξξξξξξ
(3)
The material stiffness invariants (U) are given as follows,
⋅
−
−−
−−
−
⋅=
66
22
12
11
5
4
3
2
1
4121
4161
4121
0404
4323
8
1
Q
Q
Q
Q
U
U
U
U
U
(4)
The ply stiffness properties (Q) are related to the ply Young’s modulus and Poisson
ratio by the following equations,
2112
11
111 νν ⋅−
=E
Q (5)
2112
2212
121 ννν
⋅−
⋅=
EQ (6)
2112
22
221 νν ⋅−
=E
Q (7)
1221 QQ = (8)
8
1266 GQ = (9)
11
22
1221E
E⋅=νν (10)
The membrane and bending lamination parameters are given by the following
integrals,
[ ] [ ] dzh
h
h
A ⋅⋅= ∫− 22
321 2sin4cos2cos1
ϕϕϕξ (11)
[ ] [ ] dzzh
h
h
D ⋅⋅⋅= ∫− 22
23321 2sin4cos2cos12
ϕϕϕξ (12)
where ϕ represents the fibre orientation angle at position z and h is the laminate thickness.
5 Optimisation strategy The optimisation strategy follows Ref. [30] and is presented in Fig. 3. This strategy
divides the optimisation into two levels. At the first level, dimensions and optimum
lamination parameters of the skin-stiffener element are found by employing gradient
based techniques (MP). At the second level, a GA code is used to target the optimum
lamination parameters to obtain the actual laminate stacking sequence for both the skin
and the stiffener.
5.1 First level - Gradient based optimisation MATLAB [31] is employed to conduct the gradient based optimisation. The non-linear
mathematical optimisation problem can be stated as follows,
Minimise )(xM�
Subject to 0)( ≤xG j
�
Gnj ,....,1=
u
ii
l
i xxx ≤≤ ni ,....,1= (13)
where M is the objective function and represents the mass of the stiffened panel per unit
of width (or skin-stiffener repeating element), Gj are the inequality constraints such as
laminate strength, local and global buckling or practical design rules, and x�
is the vector
of the design variables.
9
Figure 3. Optimisation flow chart.
5.1.1 Objective function
The objective function is the mass of the skin-stiffener repeating element. The mass as a
function of the design variables, materials properties and geometry is given by,
( ))()()( xAxAaxM stgstgskinskin
���
⋅+⋅⋅= ρρ (14)
with
btAskin ⋅= (15)
swswsfsfswsfstg htbtAAA ⋅+⋅=+= (16)
where sfA is the area of the stiffener flange, swA is the area of the stiffener web, t is the
thickness of the skin, sft is the thickness of the stiffener flange, swt is the thickness of the
stiffener web, b is the stiffener pitch, sfb is the width of the stiffener flange and swh is the
height of the stiffener web.
1 Gradient based
optimisation
Loads GeometryMaterial
properties
2 GA
optimisation
1 level
Constraints
check
NO
YES
Final local
design
1 level Constraints
2 level Constraints
10
5.1.2 Design variables
The manufacturing requirements of the stiffener are embedded in the design variables.
The design variables for the skin-stiffener repeating element, depending on the stiffener
type, are listed in Table 1 [30], noting that ξi are lamination parameters, (e.g. Ref. [10]).
Table 1. Table of design variables.
Design variables
x�
Stiffener type Stiffener
configuration
Skin Stiffener
flange Stiffener
web h
[ ]DA,
321ξ
ta bsf
[ ]DA,
321ξ
hsw
[ ]D
321ξ
a
ht =
asf tt =
asw tt ⋅= 2
b
As stiffener type a, knowing that
asf tt =
asw tt =
h
[ ]DA,
321ξ
ta bsf
[ ]DA,
321ξ
tw
hsw
[ ]D
321ξ
c
ht =
asf tt =
wasw ttt +⋅= 2
d
As stiffener type c, knowing that
asf tt ⋅= 2
5.1.3 Design constraints
The design constraints considered are: lamination parameters feasible region, laminate
failure strength, buckling and practical design constraints.
11
5.1.3.1 Lamination parameters feasible region
The lamination parameters feasible region is extracted from Ref. [30]. Thus,
012 ,
2
,
1 ≤−−⋅ DADA ξξ (17)
012 ,
2
,
3 ≤−+⋅ DADA ξξ (18)
( ) ( ) ( ) 011414
≤−⋅−⋅−− A
i
D
i
A
i ξξξ 3,2,1=i (19)
( ) ( ) ( ) 011414
≤+⋅+⋅−+ A
i
D
i
A
i ξξξ 3,2,1=i (20)
( ) ( ) ( ) 012121612 2121
4
21 ≤−−⋅⋅−−⋅⋅−−−⋅ AADDAA ξξξξξξ (21)
( ) ( ) ( ) 012121612 2121
4
21 ≤++⋅⋅++⋅⋅−++⋅ AADDAA ξξξξξξ (22)
( ) ( ) ( ) 032321632 2121
4
21 ≤+−⋅⋅+−⋅⋅−+−⋅ AADDAA ξξξξξξ (23)
( ) ( ) ( ) 032321632 2121
4
21 ≤−+⋅⋅−+⋅⋅−−+⋅ AADDAA ξξξξξξ (24)
( ) ( ) ( ) 012121612 2323
4
23 ≤+−⋅⋅+−⋅⋅−+−⋅ AADDAA ξξξξξξ (25)
( ) ( ) ( ) 012121612 2323
4
23 ≤−+⋅⋅−+⋅⋅−−+⋅ AADDAA ξξξξξξ (26)
( ) ( ) ( ) 032321632 2323
4
23 ≤−−⋅⋅−−⋅⋅−−−⋅ AADDAA ξξξξξξ (27)
( ) ( ) ( ) 032321632 2323
4
23 ≤++⋅⋅++⋅⋅−++⋅ AADDAA ξξξξξξ (28)
( ) ( ) ( ) 01141 3131
4
31 ≤−−⋅−−⋅−−− AADDAA ξξξξξξ (29)
( ) ( ) ( ) 01141 3131
4
31 ≤++⋅++⋅−++ AADDAA ξξξξξξ (30)
( ) ( ) ( ) 01141 3131
4
31 ≤+−⋅+−⋅−+− AADDAA ξξξξξξ (31)
( ) ( ) ( ) 01141 3131
4
31 ≤−+⋅−+⋅−−+ AADDAA ξξξξξξ (32)
The above constraints are imposed on the skin, the stiffener flange and web laminates,
respectively. Further details on these constraints can be found in Ref. [11,29].
5.1.3.2 Laminate failure strength
Failure strength constraints are introduced by limiting the laminate in-plane strains
longitudinally, transversally and in shear, for both tension and compression. CLT is used
to calculate the laminate strains under the applied in-plane loads. Hence,
12
{ } [ ] { }NA ⋅= −10ε (33)
The strength load factor is given by the ratio between the allowable and applied strain.
Hence,
j
i
j
aij
i 0εε
λ = CTi ,= ; xyyxj ,,= (34)
where aε is the allowable strain, 0ε is the applied strain, x, y, and xy represent the
longitudinal, the transversal and the shear direction, respectively. Note that T and C
denote tension and compression.
For both tension and compression cases, failure strength constraints are given by,
011
≤−j
iλ CTi ,= ; xyyxj ,,= (35)
These constraints are applied to the skin, the stiffener flange and web laminates.
5.1.3.3 Buckling constraints
An energy method (Rayleigh-Ritz) and CF solutions have been employed to evaluate
buckling constraints. Both local and global buckling have been addressed. Local buckling
assesses the individual element failure (skin, stiffener web and skin-stiffener flanges)
whereas global buckling considers the failure of the stiffened panel as a whole.
5.1.3.3.1 Local buckling
Local buckling of the stiffened panel comprises buckling of the skin between stiffener
flanges, the skin-stiffener flanges, and the stiffener web. The local skin-stiffener
interaction is partially accounted for by considering the effect of the stiffener flanges over
the skin. In this particular case, the stiffener flanges will act as a reinforcement stiffening
up the skin.
5.1.3.3.1.1 Buckling of the skin
The skin between the stiffener flanges is assumed to be a long flat plate simply supported
along the edges under normal and shear load. In this case, the length of the plate is a, and
the width of the plate is the difference between the stiffener pitch (b) and the stiffener
flange width (bsf). Weaver [8,9] has recently provided a comprehensive set of CF
solutions for long flexural anisotropic plates under compression and shear loading.
Additionally, Weaver [8] details a procedure to identify exactly the critical uniaxial
compression load.
13
5.1.3.3.1.1.1 Normal buckling
Weaver [8] approximated the critical buckling load of a long anisotropic plate with
simply supported conditions along the edges and under normal loading as follows,
22112
2
DDb
KN x
cr
x
π= (36)
where Kx is a non-dimensional buckling coefficient calculated by an iteration scheme.
5.1.3.3.1.1.2 Shear buckling
The critical shear buckling load is taken from Ref. [9] and has the following expression,
4 3
22112
2
DDKb
N xy
cr
xy
π= (37)
where Kxy is the non-dimensional shear buckling coefficient. In the case of negative shear
the shear buckling coefficient is calculated assuming that the sign of each ply angle is
reversed.
The following expression [32] is used to address the interaction,
2
1
+
=
cr
xy
xy
cr
x
x
pb
N
N
N
N
RF (38)
The constraint for the local buckling of the skin, is given by
01 ≤− pbRF (39)
5.1.3.3.1.2 Buckling of the skin-stiffener flanges
The skin and the stiffener flanges are assumed to behave as a flat plate consisting of three
contiguous strips with simply supported conditions along the external edges. Figure 4
shows the loading, material axis and cross section geometry of this arrangement.
Capey [33] considered the effect of the thickness variation across the width on the
longitudinal buckling load. Analytical solutions for isotropic materials were provided and
a practical cross section as shown in Fig. 4 was approximated by a symmetric cross
section.
14
Figure 4. Skin-stiffener flanges plate.
In this paper and as in Ref. [33], it is assumed that the neutral axis passes through the
centre of each of the strips of the plate. Furthermore, since the laminates at the strip edges
will present a certain degree of unsymmetry, smeared properties are assumed and the
reduced bending stiffness approach [34] is taken. Thus,
sfsme ttt += (40)
sfsme bb = (41)
[ ] [ ] [ ] [ ] [ ]*1*** BABDD sme ⋅⋅−=−
(42)
with
[ ] [ ] [ ]sfskin AAA +=* (43)
[ ] [ ] [ ]sfskin
sfA
tA
tB ⋅−⋅=
22
* (44)
Nx
b
a
Ny
Nxy
0o
x
y
45o
90o
A
A'
tsme
t
bsme/2
b
Section AA'
15
[ ] [ ] [ ] [ ] [ ]sfskin
sf
sfskin At
At
DDD ⋅+⋅++=44
22
* (45)
where [ ]skinA is the membrane stiffness matrix of the skin, [ ]sfA is the membrane stiffness
matrix of the stiffener flange, [ ]skinD is the bending stiffness matrix of the skin, and [ ]sfD
is the bending stiffness matrix of the stiffener flange.
The Rayleigh-Ritz (RR) method [34] is used to perform the local buckling analysis. The
RR method is based on the principle of minimum potential energy. The potential energy
of a system has at equilibrium an extremal value [32]. For the neutral equilibrium the
potential energy due to bending ( )TV is balanced by a factor ( )λ of the work done by the
external loads ( )TW . Hence,
0=⋅− TT WV λ (46)
The potential energy due to bending is given by,
∫∫ ⋅⋅
∂∂∂
+∂∂
+∂∂
∂∂∂
+
∂∂
∂∂
+∂∂
∂∂
+∂∂
∂∂
⋅=Area
T dxdy
yx
wD
x
wD
y
wD
yx
w
y
w
y
wD
y
w
x
wD
x
w
x
wD
V2
662
2
262
2
16
2
2
2
2
2
222
2
2
2
122
2
2
2
11
4
2
2
1 (47)
where ijD are the bending stiffness terms, and w is the out of plane displacement.
The work done by the external loads is given by,
∫∫ ⋅⋅
∂∂
⋅∂∂
⋅⋅+
∂∂
+
∂∂
⋅−=Area
xyyxT dxdyy
w
x
wN
y
wN
x
wNW 2
2
122
(48)
For the solution procedure, the out-of-plane displacement shape is represented by a
double sine Fourier series, since it satisfies the simply supported boundary conditions at
the external edges. Thus,
∑∑
⋅
=n
j
mn
m
i b
yn
a
xmAw
ππsinsin (49)
16
where mnA are undeterminated coefficients.
The critical buckling load is given by the lowest value or critical factor ( )crλ , which is
obtained by minimizing Eq. (46) with respect to the mnA coefficients. Hence,
( ) 0=⋅−∂∂
TT
mn
WVA
λ (50)
This provides an eigenvalue problem inλ . The smallest non-zero solution is the critical factor ( )crλ . Therefore, the critical buckling load is given by,
{ } { }NN crcr ⋅= λ (51)
In this case, as the plate consists of three strips, the expression for the potential energy
and external work is given as a sum of the potential energy and external work of each of
the plate’s strips. Hence,
∑=
=3
1i
iT VV (52)
∑=
=3
1i
iT WW (53)
The initial and final widths of the plate’s strips to carry out the integration over the plate
width, are given by,
−
=
2
2
0
sme
smei
bb
bb and
−=
b
bb
b
b sme
sme
f 2
2
(54)
with 3,2,1, =fi .
Note that the bending stiffness of the strips at the edges and at the central strip, are given
by [ ]smeD and [ ]skinD , respectively.
Once the critical buckling factor is identified, the skin-stiffener flanges buckling
constraint is expressed as,
17
01 ≤− cr
sfλ (55)
5.1.3.3.2 Buckling of the stiffener web
The stiffener web is assumed to be a long flat plate simply supported along the short
edges and one long edge and free at the other long edge under normal loading. The
critical buckling load per unit of width [35] is given by,
−⋅=
22
2
26
662
12
D
DD
hN
sw
cr
sw (56)
The buckling load factor for the stiffener web is given by the ratio of the critical and
applied load. Hence,
sw
cr
swcr
swN
N=λ (57)
where swN is the normal load applied at the stiffener web per unit of width.
The stiffener web buckling constraint is calculated as follows,
01 ≤− cr
swλ (58)
5.1.3.3.3 Global buckling
Global buckling of the stiffened panels is assessed by an interaction equation considering
column and overall shear buckling.
5.1.3.3.3.1 Column buckling
The stiffened panel is assumed to behave as a wide column with pinned ends. The critical
buckling load accounting for the shearing force induced at the stiffener web during
buckling [4,36] is given by,
sw
xysw
e
e
cr
GA
P
PP
⋅+
=1
(59)
18
where eP is the Euler buckling load for a column and swxyG is the shear modulus of the
stiffener web.
The Euler load expression for a column is given by,
2
2
a
EIP ce
⋅=
π (60)
where cEI is the longitudinal bending stiffness of the stiffened panel. Following Ref.
[37], it can be demonstrated that,
−−⋅+⋅⋅⋅+
−⋅⋅++⋅⋅+=
2
3
2
11
2
11
2212
1
2
thzAhtE
tzEA
d
bzEA
d
bEI
swcgswswsw
sw
x
cg
sf
xsfsf
sf
cg
skin
xskinskinc
(61)
in which, skind11 and sfd11 are terms of the bending stiffness compliance matrix of the skin
and the stiffener flange, sf
x
skin
x EE , and sw
xE are the Young’s modulus of the skin, the
stiffener flange and web respectively, and cgz is the centroid of the skin-stiffener section.
5.1.3.3.3.2 Shear buckling
The stiffened panel is assumed to be infinitely long with simply supported conditions
along the long edges. The critical shear load is taken from Ref. [30]. Hence,
422
3
2
2
DDKa
N csh
cr
sh
π= (62)
where cD is the longitudinal bending stiffness ( )cEI per unit of width (b) and shK is the
non-dimensional shear buckling coefficient.
An interaction formula [32] is used to evaluate the global buckling. Thus,
2
1
+
=
cr
sh
xy
cr
c
cs
N
N
P
FRF (63)
19
The global buckling of the stiffened panel in terms of constraints is given by
01 ≤− csRF (64)
5.1.3.4 Practical design constraints
Practical design rules are taken from Ref. [30]. The design constrains considered are the
limitation of the percentages of ply angles, skin-stiffener flange Poisson’s ratio mismatch
and skin gauge.
5.1.3.4.1 Percentages of ply angles
Niu [38] suggests that in composite design at least 10% of each ply angle should be
provided. Maximum and minimum percentages of the ply angles for the skin, the stiffener
flange and web are limited. The percentages of the 0, 90, 45, and -45 degree, ply angles
for each of those elements are,
1002
⋅⋅
=h
tp i
i 45,45,90,0 −=i ; swsf ttth ,,= (65)
The design constraint imposed for the maximum and minimum percentages of 0, 90, 45,
and -45 degree, is as follows,
01max
≤−i
i
p
p (66)
01min
≤−i
i
p
p (67)
5.1.3.4.2 Skin-stiffener flange Poisson’s ratio mismatch
The reduction of the Poisson’s ratio mismatch is critical in composite bonded structures
[38]. The difference between the skin and the stiffener flange Poisson’s ratio is limited by
a small number ζ to reduce the mismatch. An acceptable value of ζ is assumed to be 0.05.
The Poisson’s ratio mismatch design constraint between the skin and the stiffener flange
is given by,
0≤−− ζνν stgf
xy
skin
xy (68)
20
5.1.3.4.3 Skin gauge
Reference [38] states that the minimum skin gauge is determined by the danger of a
puncture due to lightning strike. It suggests that a minimum skin thickness of 3.81 mm
should be used. The skin gauge constraint is implemented as follows,
01 max ≤−t
t (69)
01min ≤−t
t (70)
where maxt and mint are the maximum and minimum skin thickness.
5.2 Second level - GA based optimisation A standard GA [19,20,39] is employed at this level to solve the discrete lay-up
optimisation problem. The optimum lamination parameters obtained in the first level are
targeted to identify the laminate stacking sequence for the skin, the stiffener flange and
web. The structure of a standard GA is well reported in the literature (e.g. [20]). A typical
structure of a GA consists of: generation of a population, evaluation, elitism, crossover,
reproduction and mutation. Note that at this level the GA is applied separately to the skin,
the stiffener flange and web.
5.2.1 Fitness function
Following Ref. [30] the square difference between the optimum and targeted lamination
parameters, is used as a fitness function. Extra penalty terms are added in the fitness
function to account for design practices such as the maximum number of plies with the
same orientation stacked together. Hence,
( ) ( ) k
ki
D
iopt
D
i
D
i
i
A
iopt
A
i
A
i wfwfyf Θ+−⋅+−⋅= ∑∑∑===
4
1
3
1
23
1
2)( ξξξξ�
(71)
where y�
is the design variable vector or gene representing the laminate stacking
sequence, DA
iwf , are weighting factors and kΘ are the penalty functions terms to limit the
number of plies of the same orientation stacked together. The value of kΘ is 1, when
more than 4 plies of the same orientation are stacked together [13], otherwise it is 0.
5.2.2 Design variables - Genes
The design variables are the thickness and the 0, 90, 45 and -45, degree ply angles that
constitute the laminate stacking sequences for the skin, the stiffener flange and web.
Those variables are modelled as chromosomes in genes within the GA. The
21
corresponding encoded chromosomes to ply angles are: 1, 2, 3, 4, 5, 6 and 7 for ± 45, 902,
02, 45, -45, 90 and 0 degrees, respectively. Further details are found in Ref. [30].
6 Numerical examples Reference [30] is used to compare results obtained with the two level optimisation
approach herein presented. Material properties are described in Table 2. The composite
stiffened panel is under normal and shear loading. The normal load sheared by the skin
and the stiffener and shear loads are -3502.54 N/mm and -875.63 N/mm, respectively.
Table 2. AS4/3502 material properties as in Ref. [30].
Material AS4/3502 E11[N/mm
2] 127553.8
E22[N/mm2] 11307.47
G12[N/mm2] 5998.48
ν12 0.3
ρ[kg/mm3] 1.578 10
-6
tp[mm] 0.132
Two optimum designs corresponding to stiffener type b under buckling and ply
contiguity constraints were taken from Ref. [30] to perform a comparison. Those are
detailed in Table 3.
Table 3. Optimum Skin-stiffener type b designs with buckling and ply contiguity
Finally, the effect of the stiffener type on the optimum design under strength, buckling
and practical design rules was evaluated. Optimum design using CF solutions were taken
form Ref. [30], and shown in Table 7. For this case, at the first level, the stiffener flange
width was freed and considered as a design variable. As previously, 225 terms (m = n =
15) were used in the double sine series for the RR method. The lower bounds for the
stiffener flange width and web height were set as 60 and 70 mm, respectively. Common
aerospace design strain levels of 3600µε in both tension and compression and 7200µε in shear were imposed. Stacking sequence constraints such as ply contiguity and at least one
set of ±45 degree plies at the outer surface of the laminates were added at the second
level. Table 8 shows the optimum designs obtained using this two level optimisation
approach.
24
Table 7. Optimum skin-stiffener designs for different stiffener types under buckling,
strength and practical design constraints from Ref. [30].
Stiff.
type
Wc/Wd
[kg] cr
bλ cr
sλ bsf hsw Lay-up
a 2.90
/3.05 1.01 0.98 60 70
Skin (65 plies)
[(±45)2/0/(±45)2/04/45/02/90/02/45/
(04/90)2/90/0/0]MS
Stiffener (30 plies)
[(±45)3/-45/03/90/02/90/0/0]S
b 2.99
/3.10 1.04 1.00 60 70
Skin (66 plies)
[(±45)2/0/45/±45/02/
-45/02/90/04/45/90/
(90/04)2/45/02]S
Stiffener (47 plies)
[(±45)5/02/±45/902/02/90/04/90]MS
c 2.79
/2.87 1.01 0.99 60 70
Skin (65 plies)
[(±45)3/-45/45/03/90/04/45/04/
902/04/45/04/90]MS
Stiffener flange (9 plies)
[±45/45/90/0]MS
Stiffener web (44 plies)
[(±45/02)2/02/-45/03/90/04/90/02]S
d 2.79
/3.02 0.99 1.02 60 70
Skin (65 plies)
[(±45)2/90/(02/45)2/04/±45/0/
903/04/-45/04/45/0/0]MS
Stiffener flange (8 plies)
[±45/0/90]S
Stiffener web (53 plies)
[±45/0/(±45)2/(04/-45)2/02/-
45/902/04/90]MS
Under these circumstances the optimum designs obtained do not differ significantly from
those found in Ref. [30]. Modest weight savings are found in both the continuous and
discrete level (max. approx. 2.6%). Nevertheless, a redistribution of the material between
the skin and the stiffener is observed. This is thought to be due to the stiffening effect of
the stiffener flanges over skin is included in the optimisation. As stated in Ref. [30], the
driving design constraint is strength. It is also seen that the stiffener type has an impact on
the design. Designs with stiffener type c are the lightest whereas designs with stiffener
type b are the heaviest. The difference between these two optimum designs is
approximately 6%. Note that in the cases of stiffener types c and d, the stiffener flange
minimum thickness was considered to be at least 4 plies. It is observed that for these two
stiffener types the thickness of flanges tended to a minimum. This suggests that, in this
case, the stiffener flanges might not be needed. However, if T shape stiffeners are used
the stiffener flanges have to provide a certain degree of integrity to the joint with the skin.
25
Table 8. Optimum skin-stiffener designs for different stiffener types under buckling,
strength and practical design constraints.
Stiff.
type
Wc/Wd
[kg] cr
bλ cr
sλ bsf hsw Lay-up
a 2.90
/3.01
1.27
(1.20) 0.98 60.01 70
Skin (52 plies)
[±45/453/±45/90/(45/02)2/02/
902/04/±45/02]S
Stiffener (42 plies)
[(±45)2/90/04/±45/(04/90)2]S
b 2.94
/3.02
1.06
(1.35) 0.99 60 70
Skin (52 plies)
[(±45)2/45/90/45/±45/452/02/
902/(04/45)2/-45/02]S
Stiffener (65 plies)
[(±45)3/04/±45/45/04/90/04/
-45/03/90/04/90/90]MS
c 2.74
/2.86
1.02
(1.03) 0.99 60 70
Skin (59 plies)
[±45/45/±45/90/02/45/±45/04/
45/02/-45/04/45/02/902/0/0]MS
Stiffener flange (7 plies)
[±45/0/90]MS
Stiffener web (65 plies)
[±45/04/(-45/02)2/02/(90/04)2/
902/04/45/02]S
d 2.75
/2.97
1.12
(1.13) 0.99 60 70
Skin (57 plies)
[±45/90/453/02/(±45)2/03/45/
04/902/04/45/0/0]MS
Stiffener flange (9 plies)
[±45/90/0/0]MS
Stiffener web (68 plies)
[(04/±45)2/02/90/0/-45/02/45/
(-45/04)2/902/02]S
As previously a FE model was set up in MSC/NASTRAN [40] following Ref. [30] to
evaluate the buckling performance of the designs. The critical load factors are in brackets
in Table 8. It is clearly seen that no buckling failure occurs. Table 9 collects thicknesses
and lamination parameters for the first and second optimisation levels. Adequate to good
agreement is found in all cases.
26
Table 9. Thicknesses and lamination parameters for optimum skin-stiffener designs
under buckling, strength and practical design constraints.