![Matrix plot of the stacking angles (θ 1-θ 4 ) and the resulting maximum deflection (w max ) and fundamental frequency (f 1 ) for Case 1.a (a/h = 20, [0] 4 ).](https://www.researchgate.net/publication/322894070/figure/fig5/AS:589593376202754@1517581423806/Matrix-plot-of-the-stacking-angles-th-1-th-4-and-the-resulting-maximum-deflection-w-max_Q320.jpg)
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Journal of
composites science
Article
Assessing Static and Dynamic Response Variability
due to Parametric Uncertainty on Fibre-Reinforced
Composites
Alda Carvalho 1, Tiago A.N. Silva 2ID and Maria A.R. Loja 3,*ID
1Grupo de Investigação em Modelação e Optimização de Sistemas Multifuncionais (GI-MOSM),
Instituto Superior de Engenharia de Lisboa, CEMAPRE, ISEG, Universidade de Lisboa, 1200-781 Lisboa,
Portugal; acarvalho@adm.isel.pt
2Grupo de Investigação em Modelação e Optimização de Sistemas Multifuncionais (GI-MOSM),
NOVA UNIDEMI, Faculdade de Ciência e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica,
Portugal; tan.silva@fct.unl.pt
3Grupo de Investigação em Modelação e Optimização de Sistemas Multifuncionais (GI-MOSM),
IDMEC-Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
*Correspondence: amelialoja@dem.isel.ipl.pt; Tel.: +351-962-564-688
Received: 31 December 2017; Accepted: 26 January 2018; Published: 1 February 2018
Abstract:
Composite structures are known for their ability to be tailored according to specific
operating requisites. Therefore, when modelling these types of structures or components, it is
important to account for their response variability, which is mainly due to significant parametric
uncertainty compared to traditional materials. The possibility of manufacturing a material according
to certain needs provides greater flexibility in design but it also introduces additional sources of
uncertainty. Regardless of the origin of the material and/or geometrical variabilities, they will
influence the structural responses. Therefore, it is important to anticipate and quantify these
uncertainties as much as possible. With the present work, we intend to assess the influence of
uncertain material and geometrical parameters on the responses of composite structures. Behind
this characterization, linear static and free vibration analyses are performed considering that several
material properties, the thickness of each layer and the fibre orientation angles are deemed to be
uncertain. In this study, multivariable linear regression models are used to model the maximum
transverse deflection and fundamental frequency for a given set of plates, aiming at characterizing
the contribution of each modelling parameter to the explanation of the response variability. A set of
simulations and numerical results are presented and discussed.
Keywords:
response variability of composites; parametric uncertainty characterization; multivariable
linear regression models; composite laminates; static and free vibration analysis
1. Introduction
In a global perspective, the growth verified in the usage of composite materials may be attributed
mainly to the transportation and construction industries, although in other areas such as medical and
health technologies they are becoming more relevant. Within the manufacturing processes, some are
witnessing a higher development; namely, resin transfer moulding (RTM) and glass-mat-reinforced
thermoplastics (GMT), as well as the long-fibre-reinforced thermoplastics (LFRT) [
1
]. According to
the composites industry report for 2017 [
2
], since 1960 the composites industry has grown 25 times,
whereas the aluminium and steel industries grew less than 5 times. These numbers denote an important
reality landscape on the increasing use of composite materials, confirming a continuous need for deeper
holistic research to enhance the understanding of these kinds of materials [3].
J. Compos. Sci. 2018,2, 6; doi:10.3390/jcs2010006 www.mdpi.com/journal/jcs
J. Compos. Sci. 2018,2, 6 2 of 17
The need for materials with better mechanical properties has already led to the development
of glass fibres—most often used as reinforcement—with a tensile strength 2–3 times higher than the
traditional ones for fulfilling specific operation requirements, such as those posed by the blades of wind
turbines, bicycle frames, and the diverse automotive and aerospace parts. Simultaneously, lightweight
materials have become very attractive as they simultaneously meet regulatory requirements for
emission reduction, fuel economy and safety. For instance, in the automotive and aerospace industries,
carbon-fibre-reinforced polymers (CFRP) have been the primary beneficiary. However, the cost of
carbon fibres still constitutes a disadvantage and these materials are not fully recyclable at the end of
their life cycle.
The use of composite materials in the most diverse areas poses different questions depending on
the nature of the specific application. Moreover, the great heterogeneity intrinsic in the constitution
of these kinds of materials in conjunction with the usual manufacturing processes is deemed to be
responsible for the significant variability in the structural responses when compared to those of
a structure made of homogeneous traditional materials, such as metals, for instance.
Attempting to consider this uncertainty and to assess its effects using different approaches, several
published works can be found. Mesogitis et al. [
4
] presented a review about the multiple sources of
uncertainty associated with material properties and boundary conditions. In this work, the authors
presented numerical and experimental results concerning the statistical characterization and influence
of uncertain inputs on the main steps of the manufacturing process of composites, including defects
induced by the process itself.
In the context of more focused work, we refer to Noor et al. [
5
] who proposed a two-phase
approach and a computational procedure for predicting variability in the nonlinear responses of
composite structures associated with variations in the geometric and material parameters of the
structure. To this aim, the authors considered a hierarchical sensitivity analysis to identify the
parameters with greater influence on the responses. After this screening stage, the selected parameters
were fuzzified and a fuzzy set analysis was performed to determine the variability of the responses.
The problem of uncertainty propagation in composite laminate structures was studied by António
and Hoffbauer [
6
]. They considered an approach based on the optimal design of composite structures
to achieve a target reliability level. In this work, the uniform design method (UDM) was used to
study the space variability using a set of design points generated over a design domain centred
on the mean values of the random variables. An artificial neural network (ANN) was developed
based on supervised evolutionary learning with the input/output patterns of each UDM design point.
This ANN was used to implement a Monte Carlo simulation (MCS) procedure to obtain the variability
of the structural responses. The use of ANN was also considered by Teimouri et al. [
7
] to investigate
the impact of manufacturing uncertainty on the robustness of commonly used ANN in the field of
structural health monitoring (SHM) of composite structures, namely concerning the thickness variation
in laminate plies. The ANN SHM system was assessed through an aerofoil case study based on the
sensitivity of location and size predictions for delamination with noisy data. Mukherjee et al. [
8
]
studied the influence of material uncertainties in failure strength and reliability analysis for single- and
cross-ply laminated composites subjected to only axial loading. These authors have categorized the
uncertainty at different scales, although in [
8
] they only considered ply level uncertainties. Note that
these uncertainties are included as random variables and the strength parameters of the composite are
derived through uncertainty propagation considering both Tsai-Wu and maximum stress criteria. MCS
was performed to quantify the effect of those uncertain parameters. In [
9
], the authors were concerned
with the prediction of the uncertainty induced by the manufacturing process on the effective elastic
properties of long fibre-reinforced composites with a thermoplastic matrix. Carvalho et al. [
10
] studied
the uncertainty propagation in functionally graded material (FGM) plates with an approach that can
be viewed as the precursor of the present work.
In the present work, the goal was to study the uncertainty propagation of laminate material
properties as well as geometric parameters related to the thickness and fibre orientation or stacking
J. Compos. Sci. 2018,2, 6 3 of 17
angle of each ply. These modelling parameters have specific contributions to the simulated linear
static response and, therefore, on the characterization of its variability. To enable the simulation of
uncertainty on the modelling or input parameters, a random multivariate normal distribution was
used to generate the set of input parameters, ensuring independence. The obtained results intend to
enable a more comprehensive understanding of the influence of uncertain modelling parameters on
the variability of structural responses.
2. Materials and Methods
2.1. Fibre-Reinforced Composites
The typical configuration of laminated fibre-reinforced composite material is illustrated in Figure 1
where an exploded view of generic three-layered laminate with arbitrary ply orientation angles
is presented.
J. Compos. Sci. 2018, 2, x FOR PEER REVIEW 3 of 17
static response and, therefore, on the characterization of its variability. To enable the simulation of
uncertainty on the modelling or input parameters, a random multivariate normal distribution was
used to generate the set of input parameters, ensuring independence. The obtained results intend to
enable a more comprehensive understanding of the influence of uncertain modelling parameters on
the variability of structural responses.
2. Materials and Methods
2.1. Fibre-Reinforced Composites
The typical configuration of laminated fibre-reinforced composite material is illustrated in
Figure 1 where an exploded view of generic three-layered laminate with arbitrary ply orientation
angles is presented.
Figure 1. Exploded view of a three-layered fibre-reinforced composite material.
In Figure 1, the laminate in-plane directions are denoted by x- and y-directions; also visible is
the angle θ defined between the positive senses of the fibre longitudinal direction within each ply
and the y-direction. The possibility of considering different materials for different plies, allied to the
ability to vary the stacking angles of each ply, allows to some extent for customized materials that
result in structures with improved mechanical performance.
In the present work, the study focused on a carbon fibre-reinforced composite material that is
available in the market, the properties of which are given in Table 1.
2.2. Constitutive Relations and Equilibrium Equations
Due to the characteristics of the plate structures to be analysed, the first-order shear deformation
theory of plates and shells (FSDT) will be considered. Accordingly, the stress–strain relationships for
each ply in the laminate coordinate system can be written as:
σ
σ
σ=Q
Q
Q
Q
Q
Q
Q
Q
Q
ε
ε
γ
; σ
σ=Q
Q
Q
Q
γ
γ
(1)
with the transformed reduced elastic stiffness coefficients given in the literature [11,12]. The
coefficients σ stand for the stress tensor components and ε and γ represent the normal and total
shear strains, respectively. To overcome the through-thickness constant prediction of the transverse
shear stresses, a shear correction factor of 5/6 is considered.
To obtain the equilibrium equations required for linear static and free vibration analysis, the
Lagrangian functional is considered: L=U+V−T (2)
where U denotes the elastic strain energy, V the potential energy of the external transverse applied
loads and T the kinetic energy. Considering Hamilton’s principle [11,13,14] we have:
Figure 1. Exploded view of a three-layered fibre-reinforced composite material.
In Figure 1, the laminate in-plane directions are denoted by x- and y-directions; also visible is the
angle
θ
defined between the positive senses of the fibre longitudinal direction within each ply and the
y-direction. The possibility of considering different materials for different plies, allied to the ability
to vary the stacking angles of each ply, allows to some extent for customized materials that result in
structures with improved mechanical performance.
In the present work, the study focused on a carbon fibre-reinforced composite material that is
available in the market, the properties of which are given in Table 1.
Table 1. Carbon fibre prepreg laminate properties (IM7/8552 UD Hexcel composites).
E11 (GPa) E22, E33 (GPa) G12 , G13 (GPa) G23 (GPa) ν12,ν13 ν23 ρ(kg/m3)
161 11.38 5.17 3.98 0.32 0.44 1500
2.2. Constitutive Relations and Equilibrium Equations
Due to the characteristics of the plate structures to be analysed, the first-order shear deformation
theory of plates and shells (FSDT) will be considered. Accordingly, the stress–strain relationships for
each ply in the laminate coordinate system can be written as:
σxx
σyy
σxy
=
Q11 Q12 Q16
Q12 Q22 Q26
Q16 Q26 Q66
εxx
εyy
γxy
;"σyz
σxz #="Q44 Q45
Q45 Q55 #" γyz
γxz #(1)
J. Compos. Sci. 2018,2, 6 4 of 17
with the transformed reduced elastic stiffness coefficients given in the literature [
11
,
12
]. The coefficients
σij
stand for the stress tensor components and
εii
and
γij
represent the normal and total shear strains,
respectively. To overcome the through-thickness constant prediction of the transverse shear stresses,
a shear correction factor of 5/6 is considered.
To obtain the equilibrium equations required for linear static and free vibration analysis, the Lagrangian
functional is considered:
L=U+V−T (2)
where
U
denotes the elastic strain energy,
V
the potential energy of the external transverse applied
loads and T the kinetic energy. Considering Hamilton’s principle [11,13,14] we have:
δ
t2
w
t1
(U+V−T)dt =0 (3)
After the functional minimization and some mathematical manipulations, the free vibration and
linear static equilibrium equations for a discretized domain can be written as:
(K−ω2
iM)qi=0
Kq =F(4)
where
M
is the mass matrix,
K
represents the elastic stiffness matrix of the structure,
F
denotes the
generalized load vector and
q
represents the generalized degrees of freedom vector. The i-th natural
frequency is represented by
ωi
and
qi
is the corresponding mode shape. Regarding a set of boundary
conditions, it is possible to obtain the nodal generalized displacements.
2.3. Simulation of Modelling Parameters Uncertainty
The variable responses from a set of real specimens were simulated by considering the uncertainty
in the material and geometrical properties of a laminated composite. In the present work, we focused on
the study of the uncertainty propagation on the material properties, ply thicknesses and stacking angles.
Each modelling parameter has a specific effect on the simulated response, either static or dynamic,
and therefore on the characterization of their variability. Thus, to simulate the uncertainty in the
material and geometrical properties, a set of modelling parameters
X
was sampled from a multivariate
normal distribution. Hence, the modelling parameters were sampled considering
X∼N(µ,Σ)
; that is,
X
is distributed as a normal variable with the mean values
µ
(Table 1) and the covariance matrix
Σ
.
Additionally, the correlation matrix, equal to the identity, is given to ensure independence among the
modelling parameters. Note that a Latin hypercube sampling (LHS) with the ability to ensure the
independence between variables [
15
] was used to sample 30 observations from a multivariate normal
distribution. This sample size is not a rule but a guideline. It is a good compromise in the sense that it
was sufficient to support the significance of the results while keeping the problem at a reasonable size
for dealing with experimental test data.
2.4. Forward Propagation of the Uncertainty
The sampling procedure was carried out to obtain different samples, aiming at simulating several
plates made of different combinations of properties that are used with different aspect ratios
(a/h)
;
note that
a
stands for the length of the plate edge and
h
for its thickness. The mean values of the
material properties of the composite materials used are given in Table 1. Tables 2and 3summarize
the case studies concerning the stacking angles and individual thicknesses. After obtaining the
samples for all the defined case studies, we computed the necessary finite element analysis to evaluate
the maximum transverse deflection and a set of natural frequencies, followed by an assessment of
the correlation coefficients obtained for all case studies. It is important to note that the uncertain
parameters were simulated with a coefficient of variation (CoV) of 7.5% for all the material properties
J. Compos. Sci. 2018,2, 6 5 of 17
(see nominal values in Table 1) and ply thicknesses (Table 3). Regarding the stacking angles, we
considered a standard deviation of 2 degrees (Table 2).
Table 2. Case studies with uncertain stacking angles (θply ).
Case a/h Stacking Sequence µθply σθply
1.a
20
[0]4
nominal values 2◦
1.b [0/90]s
1.c [0/90]2
2.a
100
[0]4
nominal values 2◦
2.b [0/90]s
2.c [0/90]2
Table 3. Case studies with uncertain ply thicknesses (hply ).
Case a/h Stacking Sequence µhply CoVhply
3.a
20
[0]4
0.131 mm 7.5%
3.b [0/90]s
3.c [0/90]2
4.a
100
[0]4
0.131 mm 7.5%
4.b [0/90]s
4.c [0/90]2
It is important to mention that the sample for the modelling parameters was the same for all
the case studies related to the stacking angles. For the cases related to the uncertain ply thicknesses,
another sample was used but again it was the same for all the related cases. This was done to enhance
the comparison between case studies.
2.5. Multivariable Linear Regression Model
As mentioned, the response variability of the laminated composite plates may be due to
the uncertainty associated with several materials and geometrical parameters. Thus, the use of
a multivariable linear regression model allows for the use of a probabilistic substitute model with
less computational cost. Therefore, for a specific structural response
Y
, the maximum deflection
or a natural frequency and regarding a set of predictors
X
, which can be material and geometrical
properties, the model is generally given as:
Y=β0+β1X1+. . . +βkXk+ε(5)
where subscript
k
is the number of independent variables used to explain the dependent variable
Y
.
The coefficients
βi
represent the regression coefficients and
ε
is the residual or error term. The coefficient
β0
is the intercept that corresponds to the value predicted for the structural response
Y
when the
independent variables are zero. The remaining regression coefficients represent the partial slopes,
which denote the influence of an independent variable
Xi
on the response
Y
. The residual
ε
is assumed
to follow a normal distribution with a zero mean and constant variance
σ2
denoted as
ε∼N(0, σ)
.
It is also relevant to mention that the independent variables
Xi
must be uncorrelated. Therefore, if these
model assumptions are validated, a response prediction
ˆy
can be estimated from the sampled values
xi
with a random residual. The residual
ε=y−ˆy
can thus be used to estimate the regression coefficients
and to validate the model assumptions using the method of least squares [16].
Such a probabilistic model is a multivariable linear regression model. Based on a specific sample,
it is possible to determine estimates for each regression coefficient
βi
, as well as for the coefficient
of multiple determination
R2
, which gives a measure of the response variability that is explained
J. Compos. Sci. 2018,2, 6 6 of 17
by the regression model. The
R2
coefficient and the adjusted
R2
(Adj.
R2
) are outputs of the linear
regression model.
According to inferential statistics, the sampled results can be generalized to the population.
The analysis of variance (ANOVA) provides the significance of the model based on the p-value
of the F-test. If the model is significant, it means that at least one of the slopes is nonzero; thus,
we can conclude that the predictors considered in the model are relevant. Under these conditions,
the t-test gives the significance of each individual independent variable or model parameter. Moreover,
it is possible to construct confidence intervals for the slopes. Once the model has been chosen,
the assumptions must be verified for the residuals to assess the validity of the model [16].
3. Results and Discussion
The results presented in the present Section are focused on the assessment of the influence of
the parameter uncertainty on the maximum transverse displacement
wmax
and on the fundamental
frequency
f1
of a carbon fibre-reinforced composite plate. Based on the methodology presented in
Section 2.3, the material and geometrical properties were simulated using a sample of 30 observations,
as referred. With the sampled modelling parameters, we carried out a set of finite element analysis to
build a sample of the maximum transverse displacement and natural frequencies for each of the cases
identified in Section 2.4. The finite element analysis was carried out using nine-node quadrilateral plate
finite elements based on the FSDT as described in Section 2.2. In the linear static analysis, a unitary
uniform transverse pressure loading was applied. In all the presented case studies, the plate is simply
supported. Note that the reference to a ply number is related to the stacking sequence order illustrated
in Figure 1, where the first ply is the lower one considering an ascending stacking order. Unless stated
otherwise, the aspect ratio (a/h) of the plates was set to 20.
The results for the different case studies are discussed based on the analysis of the correlation
coefficients obtained for different plates and uncertain parameter sets. In the following matrix plots,
significance codes were used to ease the results interpretation. Thus, absolute values of correlation
coefficients above 0.30 are marked with “*”, above 0.50 with “**” and above 0.75 with “***”.
3.1. Uncertainty in the Material Properties
The first case was focused on characterizing the influence that uncertain material properties may
have in the maximum transverse displacement and natural frequencies of the plate. To this purpose,
we assumed that the plates were built from a unique unidirectional composite layer with the material
properties’ mean values presented in Table 1. In this case study, the stacking angle was assumed
to be unaffected by uncertainty, whereas the material properties and the total thickness of the plate,
considered as a single layer, were deemed to be uncertain. Hence, if the referred modelling parameters
vary, it is possible to compute the scatter plots of both parameters and responses and the respective
correlation coefficients, as well as their histograms. These results are organized in the matrix plot of
Figure 2.
As a first observation, it is important to conclude on the independence among the modelling
parameters, which present a Gaussian pattern with nearly null linear correlation coefficients among
each other and consistent scatterplots. This was expected according to the uncertainty simulation
described in Section 2.3.
From the matrix plot of Figure 2, it is possible to conclude that the responses are highly correlated
(0.85), which was an expected result. It is also important to note the influence of the plate thickness,
which plays a very significant role here for both responses: the maximum deflection (0.95) and the
fundamental frequency (0.80). Although with a lower significance, the fundamental frequency is
correlated with the density (
−
0.37) and with the longitudinal elasticity modulus
(E11)
(0.36). Besides
the plate thickness, only the elasticity modulus is slightly correlated with the maximum deflection
with a correlation coefficient of 0.25.
J. Compos. Sci. 2018,2, 6 7 of 17
Considering now the static analysis of the unidirectional composite plate where all modelling
parameters are uncertain, a set of correlation coefficients between each of the material and geometrical
parameters and the maximum transverse displacement, along with the corresponding scatter plots,
are presented in Figure 3. Note that in Figure 3the different cases for different sets of uncertain
parameters are considered; all means that all of the modelling parameters are uncertain, as in Figure 2;
all hply (fix) means that all modelling parameters are uncertain except the ply thickness, which is kept
at its nominal value; the cases where a single property is identified means that only that parameter is
uncertain and all the others are kept at their nominal values.
J. Compos. Sci. 2018, 2, x FOR PEER REVIEW 7 of 17
Figure 2. Matrix plot of the modelling parameters and the resulting maximum deflection (w ) and
fundamental frequency (f) (unidirectional plate, a/h = 20, all input parameters uncertain).
Considering the first row of the matrix plot in Figure 3 where all of the modelling parameters
are uncertain, we conclude that all the parameters except the density (1.00) are responsible for
explaining, to some extent, the whole variability in the transverse displacement. This was an expected
conclusion as in a static analysis situation the self-weight of the plate is discarded; the density
parameter does not influence the maximum deflection of the plate.
It is important to note the high influence of the plate thickness, which presents a high correlation
value (0.96) to the maximum deflection. As seen in Figure 2, the longitudinal elasticity modulus (E)
is the second most significant parameter, although with a correlation coefficient much lower than the
one corresponding to the ply thickness. As the ply thickness has the highest influence on the
mechanical response of the plate, we proceeded to another study where this modelling parameter
was fixed to its nominal value and only the remaining ones could vary. This study aimed to improve
the understanding of the relative importance of the other parameters. The results are presented in
Figure 4.
If the ply thickness is not affected by uncertainty, it is possible to observe in Figure 4 that in these
conditions the longitudinal elasticity modulus (E) presents a very high correlation (0.99) with a
maximum deflection of the plate. It is also a significant parameter concerning the fundamental
frequency, although in this case the correlation coefficient between the fundamental frequency and
the material density is higher, −0.79 against 0.61. An inverse correlation (minus sign) is observed
between the density and the fundamental frequency, as expected.
Another interesting result concerns the correlation between responses. Although they present a
significant correlation, this value is not as high as when the thickness was deemed to be uncertain.
Figure 2.
Matrix plot of the modelling parameters and the resulting maximum deflection
(wmax)
and
fundamental frequency (f1)(unidirectional plate, a/h = 20, all input parameters uncertain).
Considering the first row of the matrix plot in Figure 3where all of the modelling parameters are
uncertain, we conclude that all the parameters except the density (1.00) are responsible for explaining,
to some extent, the whole variability in the transverse displacement. This was an expected conclusion
as in a static analysis situation the self-weight of the plate is discarded; the density parameter does not
influence the maximum deflection of the plate.
It is important to note the high influence of the plate thickness, which presents a high correlation
value (0.96) to the maximum deflection. As seen in Figure 2, the longitudinal elasticity modulus
(E11)
is the second most significant parameter, although with a correlation coefficient much lower than
the one corresponding to the ply thickness. As the ply thickness has the highest influence on the
mechanical response of the plate, we proceeded to another study where this modelling parameter
was fixed to its nominal value and only the remaining ones could vary. This study aimed to improve
the understanding of the relative importance of the other parameters. The results are presented in
Figure 4.
If the ply thickness is not affected by uncertainty, it is possible to observe in Figure 4that in
these conditions the longitudinal elasticity modulus
(E11)
presents a very high correlation (0.99) with
a maximum deflection of the plate. It is also a significant parameter concerning the fundamental
frequency, although in this case the correlation coefficient between the fundamental frequency and the
material density is higher,
−
0.79 against 0.61. An inverse correlation (minus sign) is observed between
the density and the fundamental frequency, as expected.
J. Compos. Sci. 2018,2, 6 8 of 17
Another interesting result concerns the correlation between responses. Although they present
a significant correlation, this value is not as high as when the thickness was deemed to be uncertain.
J. Compos. Sci. 2018, 2, x FOR PEER REVIEW 8 of 17
Figure 3. Matrix plot of the maximum deflection (w (m)) for different sets of uncertain parameters
(unidirectional plate, a/h = 20).
Figure 4. Matrix plot of the modelling parameters and the resulting maximum deflection (w ) and
fundamental frequency (f) (unidirectional plate, a/h = 20, all modelling parameters uncertain except
the ply thickness).
3.2. Uncertainty in the Layer Orientation
In this section, we considered that the plate was built from a laminate with four layers, as already
mentioned in Section 2.4. In the first stage of analysis, we assumed that the stacking angles of each
layer are affected by uncertainty. The computed results are presented in Figure 5, which presents the
sampled values for a set of laminated plates modelled according to Case 1.a (Table 2).
Figure 3.
Matrix plot of the maximum deflection
(wmax (m))
for different sets of uncertain parameters
(unidirectional plate, a/h = 20).
J. Compos. Sci. 2018, 2, x FOR PEER REVIEW 8 of 17
Figure 3. Matrix plot of the maximum deflection (w (m)) for different sets of uncertain parameters
(unidirectional plate, a/h = 20).
Figure 4. Matrix plot of the modelling parameters and the resulting maximum deflection (w ) and
fundamental frequency (f) (unidirectional plate, a/h = 20, all modelling parameters uncertain except
the ply thickness).
3.2. Uncertainty in the Layer Orientation
In this section, we considered that the plate was built from a laminate with four layers, as already
mentioned in Section 2.4. In the first stage of analysis, we assumed that the stacking angles of each
layer are affected by uncertainty. The computed results are presented in Figure 5, which presents the
sampled values for a set of laminated plates modelled according to Case 1.a (Table 2).
Figure 4.
Matrix plot of the modelling parameters and the resulting maximum deflection
(wmax)
and
fundamental frequency
(f1)
(unidirectional plate, a/h = 20, all modelling parameters uncertain except
the ply thickness).
3.2. Uncertainty in the Layer Orientation
In this section, we considered that the plate was built from a laminate with four layers, as already
mentioned in Section 2.4. In the first stage of analysis, we assumed that the stacking angles of each
J. Compos. Sci. 2018,2, 6 9 of 17
layer are affected by uncertainty. The computed results are presented in Figure 5, which presents the
sampled values for a set of laminated plates modelled according to Case 1.a (Table 2).
J. Compos. Sci. 2018, 2, x FOR PEER REVIEW 9 of 17
Figure 5. Matrix plot of the stacking angles (θ1–θ4) and the resulting maximum deflection (w )
and fundamental frequency (f) for Case 1.a (a/h = 20, [0]4).
As already mentioned in the previous case study, the individual histograms show a Gaussian
behaviour for the stacking angles, which are uncorrelated between themselves as shown by the
scatterplots and the corresponding correlation coefficients. It is again relevant that the correlation
coefficients related to the modelling parameters are close to zero (Figure 5), which means that their
independence is verified. This is consistent with the uncertainty simulation described in Section 2.3.
From Figure 5, we conclude that the stacking angles with higher correlations to the maximum
transverse deflection are the first three in the stacking, although there is not a significant
predominance from a statistical point of view. It is also visible that the angles of the inner layers
provide an inverse effect when compared to those of the outer layers.
To assess in a more detailed way the influence of each ply, we computed several combinations
and considered different sets of uncertain parameters. These sets assumed that all the stacking angles are
uncertain (All) and that only one ply at a time would have an uncertain orientation (θ1–θ4), as shown in
Figure 6. Note that the sample with the maximum transverse displacement given in Figure 5 is the one
in Figure 6 with the combination of all stacking angles being uncertain (All).
Figures 6 and 7 present the same study for moderately thin and thin unidirectional plates,
respectively. The presented matrix plots show different varying patterns for the maximum transverse
displacement. Both figures show that the fourth fibre angle has the highest correlation.
For a better understanding, Table 4 presents the correlation coefficients for Cases 1.a and 2.a. We
observe that the correlation coefficients related to the second ply angle θ are higher than those for
the first (θ) and third (θ) ply angles.
Figure 5.
Matrix plot of the stacking angles (
θ1
–
θ4
) and the resulting maximum deflection
(wmax)
and
fundamental frequency (f1)for Case 1.a (a/h = 20, [0]4).
As already mentioned in the previous case study, the individual histograms show a Gaussian
behaviour for the stacking angles, which are uncorrelated between themselves as shown by the
scatterplots and the corresponding correlation coefficients. It is again relevant that the correlation
coefficients related to the modelling parameters are close to zero (Figure 5), which means that their
independence is verified. This is consistent with the uncertainty simulation described in Section 2.3.
From Figure 5, we conclude that the stacking angles with higher correlations to the maximum
transverse deflection are the first three in the stacking, although there is not a significant predominance
from a statistical point of view. It is also visible that the angles of the inner layers provide an inverse
effect when compared to those of the outer layers.
To assess in a more detailed way the influence of each ply, we computed several combinations and
considered different sets of uncertain parameters. These sets assumed that all the stacking angles are
uncertain (All) and that only one ply at a time would have an uncertain orientation (
θ1
–
θ4
), as shown
in Figure 6. Note that the sample with the maximum transverse displacement given in Figure 5is the
one in Figure 6with the combination of all stacking angles being uncertain (All).
Figures 6and 7present the same study for moderately thin and thin unidirectional plates,
respectively. The presented matrix plots show different varying patterns for the maximum transverse
displacement. Both figures show that the fourth fibre angle has the highest correlation.
For a better understanding, Table 4presents the correlation coefficients for Cases 1.a and 2.a.
We observe that the correlation coefficients related to the second ply angle
θ2
are higher than those for
the first (θ1) and third (θ3) ply angles.
J. Compos. Sci. 2018,2, 6 10 of 17
J. Compos. Sci. 2018, 2, x FOR PEER REVIEW 10 of 17
Figure 6. Matrix plot of the maximum transverse displacement (w ) considering different sets of
uncertain stacking angles for Case 1.a (a/h = 20, [0]4).
Figure 7. Matrix plot of the maximum transverse displacement (w ) considering different sets of
uncertain stacking angles for Case 2.a (a/h = 100, [0]4).
Table 4. Correlation coefficients obtained with uncertain stacking angles for Case 1.a (left) and Case 2.a
(right).
θall 0.12 −0.23 0.01 0.33 θall 0.16 0.18 −0.07 0.35
θ1 −0.13 −0.01 −0.12 θ1 0.26 −0.09 −0.12
θ2 −0.22 −0.04 θ2 −0.18 0.04
[0]4 θ3 −0.02 [0]4 θ3 −0.05
/ = θ4 / = θ4
Figure 6.
Matrix plot of the maximum transverse displacement
(wmax)
considering different sets of
uncertain stacking angles for Case 1.a (a/h = 20, [0]4).
J. Compos. Sci. 2018, 2, x FOR PEER REVIEW 10 of 17
Figure 6. Matrix plot of the maximum transverse displacement (w ) considering different sets of
uncertain stacking angles for Case 1.a (a/h = 20, [0]4).
Figure 7. Matrix plot of the maximum transverse displacement (w ) considering different sets of
uncertain stacking angles for Case 2.a (a/h = 100, [0]4).
Table 4. Correlation coefficients obtained with uncertain stacking angles for Case 1.a (left) and Case 2.a
(right).
θall 0.12 −0.23 0.01 0.33 θall 0.16 0.18 −0.07 0.35
θ1 −0.13 −0.01 −0.12 θ1 0.26 −0.09 −0.12
θ2 −0.22 −0.04 θ2 −0.18 0.04
[0]4 θ3 −0.02 [0]4 θ3 −0.05
/ = θ4 / = θ4
Figure 7.
Matrix plot of the maximum transverse displacement
(wmax)
considering different sets of
uncertain stacking angles for Case 2.a (a/h = 100, [0]4).
Table 4.
Correlation coefficients obtained with uncertain stacking angles for Case 1.a (left) and Case 2.a (right).
θall 0.12 −0.23 0.01 0.33 θall 0.16 0.18 −0.07 0.35
θ1−0.13 −0.01 −0.12 θ10.26 −0.09 −0.12
θ2−0.22 −0.04 θ2−0.18 0.04
[0]4θ3−0.02 [0]4θ3−0.05
a/h = 20 θ4a/h = 100 θ4
J. Compos. Sci. 2018,2, 6 11 of 17
It is also worthy to note the inversion of the correlation sign between Cases 1.a and 2.a (a/h = [20; 100]).
This happens only for
θ2
and
θ3
, which correspond to the inner layers for the unidirectional stacking
sequence [0]
4
and must be further evaluated. To evaluate the results for other stacking sequences,
the case studies presented in Table 2are considered.
From Figures 7and 8, both associated with thin plates, it is concluded that the fourth ply remains
significant in the [0/90]
S
laminate, although its significance is now shared with the first layer. Note
that both are external layers.
J. Compos. Sci. 2018, 2, x FOR PEER REVIEW 11 of 17
It is also worthy to note the inversion of the correlation sign between Cases 1.a and 2.a
(a/h = [20; 100]). This happens only for θ and θ, which correspond to the inner layers for the
unidirectional stacking sequence [0]4 and must be further evaluated. To evaluate the results for other
stacking sequences, the case studies presented in Table 2 are considered.
From Figures 7 and 8, both associated with thin plates, it is concluded that the fourth ply remains
significant in the [0/90]S laminate, although its significance is now shared with the first layer. Note
that both are external layers.
However, for moderately thin plates (comparing Figures 6 and 9), we conclude that on the
non-symmetric cross-ply laminate there is a more spread significance between stacking angles.
Nevertheless, the correlation coefficient of the fourth layer maintains a higher value. The correlation
coefficients between angles θ and θ change with the stacking sequence from around zero for [0]4
(Table 4) to almost 0.30 for [0/90]S (Table 5), and to an inverse correlation in the [0/90]2 laminate (Table 6).
Table 5. Correlation coefficients obtained with uncertain stacking angles for Case 1.b (left) and Case 2.b
(right).
θall 0.31 0.17 0.33 0.46 θall 0.32 0.14 0.33 0.49
θ1 0.17 0.29 −0.12 θ1 0.16 0.29 −0.12
θ2 −0.15 −0.28 θ2 −0.15 −0.27
[0/90]s θ3 0.27 [0/90]s θ3 0.27
/ = θ4 / = θ4
Figure 8. Matrix plot of the maximum transverse displacement (w ) considering different sets of
uncertain stacking angles for Case 2.b (a/h = 100, [0/90]S).
Figure 8.
Matrix plot of the maximum transverse displacement
(wmax)
considering different sets of
uncertain stacking angles for Case 2.b (a/h = 100, [0/90]S).
However, for moderately thin plates (comparing Figures 6and 9), we conclude that on the
non-symmetric cross-ply laminate there is a more spread significance between stacking angles.
Nevertheless, the correlation coefficient of the fourth layer maintains a higher value. The correlation
coefficients between angles
θ3
and
θ4
change with the stacking sequence from around zero for [0]
4
(Table 4) to almost 0.30 for [0/90]
S
(Table 5), and to an inverse correlation in the [0/90]
2
laminate
(Table 6).
Table 5.
Correlation coefficients obtained with uncertain stacking angles for Case 1.b (left) and Case 2.b (right).
θall 0.31 0.17 0.33 0.46 θall 0.32 0.14 0.33 0.49
θ10.17 0.29 −0.12 θ10.16 0.29 −0.12
θ2−0.15 −0.28 θ2−0.15 −0.27
[0/90]s θ30.27 [0/90]s θ30.27
a/h = 20 θ4a/h = 100 θ4
J. Compos. Sci. 2018,2, 6 12 of 17
J. Compos. Sci. 2018, 2, x FOR PEER REVIEW 12 of 17
Figure 9. Matrix plot of the maximum transverse displacement (w ) considering different sets of
uncertain stacking angles for Case 1.c (a/h = 20, [0/90]2).
The results in Tables 5 and 6 are similar, despite the difference between stacking sequences. Note
that the correlation coefficient for θ is higher in these cases, reaching values similar to those for θ
(Table 6). On the other hand, Table 5 shows that the correlation for [0/90]S presents higher values for
all stacking angles, with the value for θ remaining the highest.
Table 6. Correlation coefficients obtained with uncertain stacking angles for Case 1.c (left) and Case 2.c
(right).
θall 0.35 0.00 −0.10 0.33 θall 0.36 −0.01 −0.09 0.33
θ1 0.00 0.26 −0.13 θ1 0.00 0.26 −0.13
θ2 −0.19 0.19 θ2 −0.20 0.19
[0/90]2 θ3 −0.19 [0/90]2 θ3 −0.20
/ = θ4 / = θ4
3.3. Uncertainty in the Layer Thickness
In the present work, the variability on the maximum deflection due to uncertain ply thicknesses
was also analysed. Figure 10 shows the same type of matrix plot but for Case 3.a.
Matrix plots were constructed and analysed for all of the studied cases. However, for the sake
of simplicity, Tables 7–9 summarise the results obtained.
The correlation coefficients between samples for maximum transverse displacement for almost all
case studies are dominated by the uncertain properties of the fourth ply (Tables 7–9). A correspondence
can be observed with the results presented in the previous sections, although for the ply thickness
higher values are obtained for the correlation coefficients.
Comparing the cases with uncertain stacking angles (Cases 1 and 2) and those with uncertain
ply thicknesses (Cases 3 and 4) for different aspect ratios, there is greater consistency in the
distributions of the maximum transverse displacement for Cases 3 and 4 (Figures 10–12), which are
almost symmetric. On the other hand, for Cases 1 and 2, there are significant changes in the aspect
ratios and stacking sequences (Figures 6–9).
Figure 9.
Matrix plot of the maximum transverse displacement
(wmax)
considering different sets of
uncertain stacking angles for Case 1.c (a/h = 20, [0/90]2).
Table 6.
Correlation coefficients obtained with uncertain stacking angles for Case 1.c (left) and Case 2.c (right).
θall 0.35 0.00 −0.10 0.33 θall 0.36 −0.01 −0.09 0.33
θ10.00 0.26 −0.13 θ10.00 0.26 −0.13
θ2−0.19 0.19 θ2−0.20 0.19
[0/90]2θ3−0.19 [0/90]2θ3−0.20
a/h = 20 θ4a/h = 100 θ4
The results in Tables 5and 6are similar, despite the difference between stacking sequences. Note
that the correlation coefficient for
θ1
is higher in these cases, reaching values similar to those for
θ4
(Table 6). On the other hand, Table 5shows that the correlation for [0/90]Spresents higher values for
all stacking angles, with the value for θ4remaining the highest.
3.3. Uncertainty in the Layer Thickness
In the present work, the variability on the maximum deflection due to uncertain ply thicknesses
was also analysed. Figure 10 shows the same type of matrix plot but for Case 3.a.
Matrix plots were constructed and analysed for all of the studied cases. However, for the sake of
simplicity, Tables 7–9summarise the results obtained.
The correlation coefficients between samples for maximum transverse displacement for almost all
case studies are dominated by the uncertain properties of the fourth ply (Tables 7–9). A correspondence
can be observed with the results presented in the previous sections, although for the ply thickness
higher values are obtained for the correlation coefficients.
Comparing the cases with uncertain stacking angles (Cases 1 and 2) and those with uncertain ply
thicknesses (Cases 3 and 4) for different aspect ratios, there is greater consistency in the distributions of
the maximum transverse displacement for Cases 3 and 4 (Figures 10–12), which are almost symmetric.
On the other hand, for Cases 1 and 2, there are significant changes in the aspect ratios and stacking
sequences (Figures 6–9).
J. Compos. Sci. 2018,2, 6 13 of 17
In the cases with uncertain ply thicknesses, the correlation coefficients for the thickness of the
fourth ply (h4) overcome all the others with values near 1.0 (Figures 11 and 12), with the exception of
Cases 3.b and 4.b.
From Tables 7–9, it is possible to conclude that the fourth ply is by far the most significant parameter.
Table 7.
Correlation coefficients obtained with uncertain ply thicknesses for Case 3.a (left) and Case 4.a (right).
hall 0.10 0.17 0.24 0.97 hall 0.10 0.17 0.24 0.97
h1−0.01 −0.01 0.02 h1−0.01 −0.01 0.02
h20.04 0.09 h20.04 0.10
[0]4h30.02 [0]4h30.02
a/h = 20 h4a/h = 100 h4
Table 8.
Correlation coefficients obtained with uncertain ply thicknesses for Case 3.b (left) and Case 4.b (right).
hall 0.06 0.25 0.45 0.88 hall 0.06 0.26 0.45 0.88
h1−0.02 0.00 0.02 h1−0.02 0.00 0.02
h20.05 0.10 h20.05 0.10
[0/90]Sh30.01 [0/90]Sh30.01
a/h = 20 h4a/h = 100 h4
Table 9.
Correlation coefficients obtained with uncertain ply thicknesses for Case 3.c (left) and Case 4.c (right).
hall 0.10 0.18 0.24 0.97 hall 0.09 0.18 0.22 0.97
h1−0.01 0.00 0.02 h1−0.01 0.00 0.02
h20.05 0.10 h20.05 0.10
[0/90]2h30.02 [0/90]2h30.02
a/h = 20 h4a/h = 100 h4
J. Compos. Sci. 2018, 2, x FOR PEER REVIEW 13 of 17
In the cases with uncertain ply thicknesses, the correlation coefficients for the thickness of the
fourth ply (h4) overcome all the others with values near 1.0 (Figures 11 and 12), with the exception
of Cases 3.b and 4.b.
From Tables 7–9, it is possible to conclude that the fourth ply is by far the most significant
parameter.
Table 7. Correlation coefficients obtained with uncertain ply thicknesses for Case 3.a (left) and Case 4.a
(right).
hall 0.10 0.17 0.24 0.97 hall 0.10 0.17 0.24 0.97
h1 −0.01 −0.01 0.02 h1 −0.01 −0.01 0.02
h2 0.04 0.09 h20.04 0.10
[0]4 h3 0.02 [0]4 h3 0.02
/ = h4 / = h4
Table 8. Correlation coefficients obtained with uncertain ply thicknesses for Case 3.b (left) and Case 4.b
(right).
hall 0.06 0.25 0.45 0.88 hall 0.06 0.26 0.45 0.88
h1 −0.02 0.00 0.02 h1 −0.02 0.00 0.02
h2 0.05 0.10 h20.05 0.10
[0/90]S h3 0.01 [0/90]S h3 0.01
/ = h4 / = h4
Table 9. Correlation coefficients obtained with uncertain ply thicknesses for Case 3.c (left) and Case 4.c
(right).
hall 0.10 0.18 0.24 0.97 hall 0.09 0.18 0.22 0.97
h1 −0.01 0.00 0.02 h1 −0.01 0.00 0.02
h2 0.05 0.10 h2 0.05 0.10
[0/90]2 h3 0.02 [0/90]2 h3 0.02
/ = h4 / = h4
Figure 10. Matrix plot of the ply thicknesses (h1–h4) and the resulting maximum deflection (w )
and fundamental frequency (f) for Case 3.a (a/h = 20, [0]4).
Figure 10.
Matrix plot of the ply thicknesses (h
1
–h
4
) and the resulting maximum deflection
(wmax)
and fundamental frequency (f1) for Case 3.a (a/h = 20, [0]4).
J. Compos. Sci. 2018,2, 6 14 of 17
J. Compos. Sci. 2018, 2, x FOR PEER REVIEW 14 of 17
Figure 11. Matrix plot of the maximum transverse displacement (w ) considering different sets of
uncertain ply thicknesses for Case 3.c (a/h = 20, [0/90]2).
Figure 12. Matrix plot of the maximum transverse displacement (w ) considering different sets of
uncertain ply thicknesses for Case 4.b (a/h = 100, [0/90]S).
3.4. Regression Models
In the previous case studies, we assessed the correlation of the material and geometrical
parameters, assuming different uncertain sets. From those studies, it is already possible to conclude
that some parameters are more significant for the plate responses.
The present study intended to build probabilistic models to represent the unidirectional
composite plate response, both in the case of its maximum transverse deflection (w) and in the
case of its fundamental frequency (f). To this purpose, a multivariable linear regression approach
(Section 2.5) has been considered. According to this methodology, the models predicting those two
responses were initially written as:
Figure 11.
Matrix plot of the maximum transverse displacement
(wmax)
considering different sets of
uncertain ply thicknesses for Case 3.c (a/h = 20, [0/90]2).
J. Compos. Sci. 2018, 2, x FOR PEER REVIEW 14 of 17
Figure 11. Matrix plot of the maximum transverse displacement (w ) considering different sets of
uncertain ply thicknesses for Case 3.c (a/h = 20, [0/90]2).
Figure 12. Matrix plot of the maximum transverse displacement (w ) considering different sets of
uncertain ply thicknesses for Case 4.b (a/h = 100, [0/90]S).
3.4. Regression Models
In the previous case studies, we assessed the correlation of the material and geometrical
parameters, assuming different uncertain sets. From those studies, it is already possible to conclude
that some parameters are more significant for the plate responses.
The present study intended to build probabilistic models to represent the unidirectional
composite plate response, both in the case of its maximum transverse deflection (w) and in the
case of its fundamental frequency (f). To this purpose, a multivariable linear regression approach
(Section 2.5) has been considered. According to this methodology, the models predicting those two
responses were initially written as:
Figure 12.
Matrix plot of the maximum transverse displacement
(wmax)
considering different sets of
uncertain ply thicknesses for Case 4.b (a/h = 100, [0/90]S).
3.4. Regression Models
In the previous case studies, we assessed the correlation of the material and geometrical
parameters, assuming different uncertain sets. From those studies, it is already possible to conclude
that some parameters are more significant for the plate responses.
The present study intended to build probabilistic models to represent the unidirectional composite
plate response, both in the case of its maximum transverse deflection (
wmax
) and in the case of its
fundamental frequency (
f1
). To this purpose, a multivariable linear regression approach (Section 2.5)
has been considered. According to this methodology, the models predicting those two responses were
initially written as:
wmax =β0+β1E11 +β2E22 +β3ν12 +β4G12 +β5G13 +β6G23 +β7h+ε
f1=β0+β1E11 +β2E22 +β3ν12 +β4G12 +β5G13 +β6G23 +β7h+β8ρ+ε(6)
J. Compos. Sci. 2018,2, 6 15 of 17
The results obtained for the different regression coefficients
βi
are summarized in Table 10. It is
important to mention that a set of significance codes were used to classify the significance of each
regression coefficient based on the p-value of the t-test.
From Table 10, it is possible to conclude on the very high values of the adjusted
R2
. However,
concerning the maximum transverse deflection regression model, the hypothesis of independence and
normality of the residuals has been rejected, which does not happen in the case of the model for the
fundamental frequency where all of the model assumptions have been verified.
Concerning the regression model for the maximum deflection, we conclude that the most
significant parameters are the longitudinal elasticity modulus
(E11)
and the plate thickness
h
. Poisson’s
ratio (ν12)and the shear modulus G23 are the next two in terms of significance.
For the fundamental frequency, all of the parameters are significant except the shear moduli
G12
and
G23
. However, we consider the regression model for the fundamental frequency validated, even
with some nonsignificant variables.
Table 10. Multivariable linear regression models—initial case summaries.
wmax f1
Adj. R297.44% 99.77%
Model F-test p-value F-test p-value
158.4 <2.2 ×10−16 1543 <2.2 ×10−16
Estimate p-value Estimate p-value
β0−5.856 ×10−43.67 ×10−15 *** −6.162 ×10−10.07120 .
β15.492 ×10−16 2.96 ×10−8*** 2.626 ×10−11 <2 ×10−16 ***
β29.294 ×10−16 0.3557 5.999 ×10−11 4.47 ×10−6***
β39.081 ×10−50.0165 * 6.774 ×10−10.06491 .
β4−1.723 ×10−15 0.4396 1.974 ×10−11 0.37622
β5−5.241 ×10−16 0.8048 7.866 ×10−11 0.00111 **
β6−4.662 ×10−10.0822 . 3.407 ×10−11 0.19487
β71.829 ×10−1<2 ×10−16 *** 5.360 ×103<2 ×10−16 ***
β8 - - −3.563 ×10−3<2 ×10−16 ***
Residuals Independence/normality rejected OK
Significance codes: 0 “***” 0.001 “**” 0.01 “*” 0.05 “.” 0.1 “ ” 1.
Therefore, in a second stage of this study we considered alternative models considering only the
previous most significant parameters in both cases. After a forward selection process, the following
simplified models were obtained:
wmax =β0+β7h+ε
f1=β0+β1E11 +β7h+β8ρ+ε(7)
The results for these final models are presented in Table 11.
It is worth mentioning that there was no need for an alternative model in the case of the
fundamental frequency, although this was considered.
From the results in Table 11, it is possible to say that the simplified models (Equation (7)) present
high values of adjusted R2, and in both cases the residuals assumptions are verified. Therefore, these
simplified models are validated. Moreover, it can be observed that in the case of the maximum
deflection model, by considering only the thickness, we attain a model that explains 90.46% of the
plate deflection variability. For the simplified fundamental frequency model, a very high explanation
(99.317%) is obtained, continuing to observe the residuals assumptions.
It is relevant to note that an intermediate simplified model for the maximum deflection can be
given by:
wmax =β0+β1E11 +β7h+ε(8)
J. Compos. Sci. 2018,2, 6 16 of 17
where
E11
is included. However, in this case, the residuals problems persisted, although the value
of the adjusted
R2
is 97.58 %. The normality of the residuals was improved when compared to the
model in Equation (7), but the residuals independency was not guaranteed as observed in Figure 13.
Considering this, it is not possible to accept the corresponding multivariable linear regression model.
Table 11. Multivariable linear regression models—simplified case summaries.
wmax f1
Adj. R290.46% 99.317%
Model F-test p-value F-test p-value
276.1 4.917 ×10−16 1401 <2.2 ×10−16
Estimate p-value Estimate p-value
β0−4.829 ×10−4<2 ×10−16 *** 1.028 0.00254 **
β1 - - 2.606 ×10−11 <2 ×10−16 ***
β71.806 ×10−14.92 ×10−16 *** 5.343 ×103<2 ×10−16 ***
β8 - - −3.586 ×10−3<2 ×10−16 ***
Residuals OK OK
Significance codes: 0 “***” 0.001 “**” 0.01 “*” 0.05 “.” 0.1 “ ” 1.
J. Compos. Sci. 2018, 2, x FOR PEER REVIEW 16 of 17
Table 11. Multivariable linear regression models—simplified case summaries.
Adj. 90.46% 99.317%
Model F-tes
t
p-value F-tes
t
p-value
276.1 4.917 × 10
−16
1401 <2.2 × 10
−16
Estimate p-value Estimate p-value
β0 −4.829 × 10
−4
<2 × 10
−16
*** 1.028 0.00254 **
β1 - - 2.606 × 10
−11
<2 × 10
−16
***
β7 1.806 × 10
−1
4.92 × 10
−16
*** 5.343 × 10
3
<2 × 10
−16
***
β8 - - −3.586 × 10
−3
<2 × 10
−16
***
Residuals OK OK
Significance codes: 0 “***” 0.001 “**” 0.01 “*” 0.05 “.” 0.1 “ ” 1.
It is relevant to note that an intermediate simplified model for the maximum deflection can be
given by: w =β
+βE+βh+ε (8)
where E is included. However, in this case, the residuals problems persisted, although the value
of the adjusted R is 97.58 %. The normality of the residuals was improved when compared to the
model in Equation (7), but the residuals independency was not guaranteed as observed in Figure 13.
Considering this, it is not possible to accept the corresponding multivariable linear regression model.
Figure 13. Residuals of the regression model for w
(Equation (8)).
4. Conclusions
This work presents a study on the uncertainty propagation of the geometrical and material
parameters on the mechanical response of carbon fibre-reinforced composite laminate. The
simulation of the uncertain modelling parameters was carried out by considering a random
multivariate normal distribution.
The significance of each material and geometrical parameters on the simulated linear static and free
vibration response of a certain composite structure was assessed and, therefore, the characterization of
the response variability was analysed and conclusions were drawn.
From the obtained results, it is possible to conclude that the variability of the maximum
transverse deflection and fundamental frequency is more sensitive to laminate thickness than to other
parameters. The longitudinal elasticity modulus (E) appears as the second most significant
parameter and the density is the next, when considering the laminate fundamental frequency.
It is also important to summarize the greater sensitivity of the simulated static response to
changes on the geometrical parameters of external layers, namely the upper one. Additional
simulations were carried out for a larger sample size, confirming the presented conclusions, although
this topic should be addressed in more detail in future studies.
The multivariable linear regression analysis confirms the conclusions of the presented
correlation analysis in what concerns the influence of the material properties and the global thickness
of the laminate. Valid multivariable linear regression models were obtained for the response
Figure 13. Residuals of the regression model for wmax (Equation (8)).
4. Conclusions
This work presents a study on the uncertainty propagation of the geometrical and material
parameters on the mechanical response of carbon fibre-reinforced composite laminate. The simulation
of the uncertain modelling parameters was carried out by considering a random multivariate
normal distribution.
The significance of each material and geometrical parameters on the simulated linear static and
free vibration response of a certain composite structure was assessed and, therefore, the characterization
of the response variability was analysed and conclusions were drawn.
From the obtained results, it is possible to conclude that the variability of the maximum transverse
deflection and fundamental frequency is more sensitive to laminate thickness than to other parameters.
The longitudinal elasticity modulus
(E11)
appears as the second most significant parameter and the
density is the next, when considering the laminate fundamental frequency.
It is also important to summarize the greater sensitivity of the simulated static response to changes
on the geometrical parameters of external layers, namely the upper one. Additional simulations were
carried out for a larger sample size, confirming the presented conclusions, although this topic should
be addressed in more detail in future studies.
The multivariable linear regression analysis confirms the conclusions of the presented correlation
analysis in what concerns the influence of the material properties and the global thickness of the
laminate. Valid multivariable linear regression models were obtained for the response variables,
allowing for the identification of the most important parameters regarding the description of the
response variability.
J. Compos. Sci. 2018,2, 6 17 of 17
As a final global conclusion, it is considered that under the present assumptions, this methodological
study provides an effective tool to characterize the relative influence of each modelling parameter on the
explanation of the variability of the mechanical response predictions.
Acknowledgments:
The authors wish to acknowledge the financial support of Project IPL/2016/CompDrill/ISEL
and the support of Fundação para a Ciência e a Tecnologia through Project LAETA-UID/EMS/50022/2013, Project
UNIDEMI-Pest-OE/EME/UI0667/2014 and Project CEMAPRE-UID/Multi/00491/2013.
Author Contributions:
All authors contributed to the design and implementation of the research, to the analysis
of the results and to the writing of the manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
References
1. Schemme, M. LFT—Development status and perspectives. Reinf. Plast. 2008,52, 32–34. [CrossRef]
2.
State of the Composites Industry Report for 2017. Available online: http://compositesmanufacturingmagazine.
com/2017/01/composites-industry-report-2017 (accessed on 1 November 2017).
3.
Prashanth, S.; Subbaya, K.M.; Nithin, K.; Sachhidananda, S. Fiber Reinforced Composites—A Review.
J. Mater. Sci. Eng. 2017,6, 1–6.
4.
Mesogitis, T.S.; Skordos, A.A.; Long, A.C. Uncertainty in the manufacturing of fibrous thermosetting
composites: A review. Compos. A Appl. Sci. Manuf. 2014,57, 67–75. [CrossRef]
5.
Noor, A.K.; Starnes, J.H., Jr.; Peters, J.M. Uncertainty analysis of composite structures. Comput. Methods Appl.
Mech. Eng. 2000,185, 413–432. [CrossRef]
6.
António, C.A.C.; Hoffbauer, L.N. Uncertainty assessment approach for composite structures based on global
sensitivity indices. Compos. Struct. 2013,99, 202–212. [CrossRef]
7.
Teimouri, H.; Milani, A.S.; Seethaler, R.; Heidarzadeh, A. On the Impact of Manufacturing Uncertainty in
Structural Health Monitoring of Composite Structures: A Signal to Noise Weighted Neural Network Process.
Open J. Compos. Mater. 2016,6, 28–39. [CrossRef]
8.
Mukherjee, S.; Ganguli, R.; Gopalakrishnan, S.; Cot, L.D.; Bes, C. Ply Level Uncertainty Effects on Failure
of Composite Structures. Le Cam, Vincent and Mevel, Laurent and Schoefs, Franck. In Proceedings of the
EWSHM—7th European Workshop on Structural Health Monitoring, Nantes, France, 8–11 July 2014.
9.
Hohe, J.; Beckmann, C.; Paul, H. Modeling of uncertainties in long fiber reinforced thermoplastics. Mater. Des.
2015,66, 390–399. [CrossRef]
10.
Carvalho, A.; Silva, T.; Loja, M.A.R.; Damásio, F.R. Assessing the influence of material and geometrical
uncertainty on the mechanical behavior of functionally graded material plates. Mech. Adv. Mater. Struct.
2017,24, 417–426. [CrossRef]
11. Reddy, J.N. Mechanics of Laminated Composite Plates; CRC Press: Boca Raton, FL, USA, 1997.
12.
Loja, M.A.R.; Barbosa, J.I.; Mota Soares, C.M. Analysis of Sandwich Beam Structures Using Kriging Based
Higher Order Models. Compos. Struct. 2015,119, 99–106. [CrossRef]
13.
Loja, M.A.R.; Barbosa, J.I.; Mota Soares, C.M. Dynamic Behaviour of Soft Core Sandwich Structures using
Kriging-Based Layerwise Models. Compos. Struct. 2015,134, 883–894. [CrossRef]
14.
Loja, M.A.R.; Barbosa, J.I.; Mota Soares, C.M. Dynamic Instability of Variable Stiffness Composite Plates.
Compos. Struct. 2017,182, 402–411. [CrossRef]
15.
Iman, R.L.; Conover, W.J. A Distribution-Free Approach to Inducing Rank Correlation Among Input Variables.
Commun. Stat. B 1982,11, 311–334. [CrossRef]
16. Montgomery, D.C. Design and Analysis of Experiments; John Wiley & Sons, Inc.: Hoboken, NY, USA, 1997.
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