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Manufacturing cost-effective weight minimization of composite laminate using uniform thickness and variable thickness approaches considering different failure criteria

  • Vipin Kumar Tripathi
  • Nishant Shashikant KulkarniEmail author
Original Paper
  • 54 Downloads

Abstract

The present study aims at providing a solution to manufacturer and designer for selecting manufacturing cost-effective uniform ply thickness and finding necessary optimum ply angle stacking sequence when ply thicknesses and ply angles are available in discrete form, in two stages using combined variable thickness (VTA) and uniform thickness approach (UTA). The manufacturing cost is indirectly included in the analysis by restricting maximum number of layers and resizing ply thicknesses accordingly, while minimizing weight. Maximum stress theory, Tsai Wu theory, and Tsai Hill theory are used as constraints in the analysis independently. The capability of the conventional genetic algorithm is improved using direct value-coded representation for chromosomes to handle multiple design variables of different nature in discrete form, simultaneously. The effectiveness of combined VTA–UTA strategy in this regard is demonstrated with multiple examples.

Keywords

Composite laminate Weight minimization OptiComp Uniform thickness approach Variable thickness approach 

List of symbols

E11

Elastic modulus in longitudinal direction in GPa

G12

In-plane shear modulus in GPa

E22

Elastic modulus in transverse direction in GPa

\(\mu_{12}\)

Major Poisson’s ratio

SLt

Tensile strength in longitudinal direction in MPa

SLc

Compressive strength in longitudinal direction in MPa

STc

Compressive strength in transverse direction in MPa

STt

Tensile strength in transverse direction, MPa

SLts

Shear strength, MPa

\(\theta\)

Ply orientation angle

t

Thickness of each ply, mm

\(\rho\)

Mass density, kg/m3

a

Length of laminate, mm

b

Width of laminate, mm

T

Laminate thickness, mm

1 Introduction

Composite laminate plate is a structural element made up of multiple fiber reinforced polymer (FRP) laminas/plies connected together to provide required engineering properties for an application. The light weight associated with high stiffness and strength along the direction of the reinforcement is the main advantage of using composite laminates. The properties of composite laminate depend on various design variables such as total number of plies/laminas, stacking sequence, ply angles, thickness of each ply, etc. The composite laminate can be tailor-made using combination of these design variables to suit the application under consideration.

A composite laminate subjected to in-plane loads Nxx, Nyy and Nxy is shown in Fig. 1. In figure, X, Y and Z denote global co-ordinate system of the composite laminate, while 1 and 2 represent local co-ordinate system for individual lamina. Axis 1 of the local co-ordinate system is along the length of the fiber and axis 2 is perpendicular to local axis 1.
Fig. 1

Global and local co-ordinate systems for composite laminate

Majority automotive applications demand maximum strength to weight ratio for achieving maximum material performance in terms of durability of structure, fuel economy, cost, etc., in any structural application. To satisfy this major design requirement, many researchers (Almeida and Awruch 2009; Tripathi and Kulkarni 2014; Pelletier and Vel 2006; Lopez et al. 2009; Akbulut and Sonmez 2008; Naik et al. 2008; Omkar et al. 2008; Fakhrabadi et al. 2013; Assie et al. 2011; Seresta et al. 2007; Badalló et al. 2013; Walker and Smith 2003; Farshi and Herasati 2006; Irisarri et al. 2009; Khosravi and Sedaghati 2008; Barroso et al. 2017; Gillet et al. 2010; Adali et al. 1995; Lopez et al. 2011; Vo-Duy et al. 2017; Javidrad et al. 2018; Rocha et al. 2014) have taken weight minimization objective during design optimization of composite laminate for different loading conditions.

A composite laminate can be designed using two approaches; uniform thickness approach (UTA) and variable thickness approach (VTA). All laminas in a laminate will have same thickness in UTA, while in VTA, the laminas in the laminate may have same or different thicknesses. The use of ply thickness as user-defined discrete variable is rarely observed (Almeida and Awruch 2009; Walker and Smith 2003; Vo-Duy et al. 2017) so far in the available literature because of manufacturing difficulty and mathematical complexity. The comparison of both the approaches yields that the number of design variables in VTA becomes more than the design variables in UTA. Moreover, the nature of variables, i.e., ply angle and ply thickness is different. The ply angle is an integer number, while ply thickness is a real number. Ghiasi et al. (2009) and Nikbakt et al. (2018) concluded that genetic algorithm is the most popular and suitable method for design optimization of composite laminates.

Walker and Smith (2003) carried out minimization of weighted sum of deflection and weight of composite laminate subjected to normal loading using Tsai Wu theory as constraint considering fiber angles and layer thicknesses, both in discrete form as design variables using binary-coded genetic algorithm. Pelletier and Vel (2006) proposed integer-coded genetic algorithm for design optimization of composite laminates after exposing limitations of binary-coded genetic algorithm in catching exact incremental value of design variables commonly used in engineering applications. Lopez et al. (2009) carried out weight minimization of composite laminates using UTA subjected to in-plane loading conditions using direct value-coded genetic algorithm. The number of layers and fiber angles are used as design variables while designing conventional 0°, ±45°, 90° composite laminates subjected to Tsai Wu, Puch and maximum stress theory as constraints. The concept of direct value-coded representation of chromosomes used by Lopez et al. (2009) is extended in this article considering ply thicknesses along with ply angles as design variables. These two design variables are of different nature also. The concept of crossover and mutation operators presented by Pelletier and Vel (2006) for integer coded genetic algorithm is modified and adopted in the current simulation. Farshi and Herasati (2006) and Khosravi and Sedaghati (2008) adopted two-level strategy for weight minimization of composite laminate plate considering fiber angle and thickness of each layer as design variables.

To check the failure of the laminate while minimizing its weight, researchers working in the field have proposed various criteria (theories) for predicting failure of composite laminates, which may have some minor or major weaknesses. Hinton et al. (2002) and Soden et al. (2004) compared and assessed different leading theories for predicting failure in composite laminates under complex states of stress against experimental evidence through few selected test cases. In the present study, first ply failure criterion is adopted to predict failure of laminate. The laminate is considered as failed when any single ply of the laminate fails. Among all these theories maximum stress theory, maximum strain theory, Tsai Hill theory and Tsai Wu theory are the most preferred theories for predicting failure of laminate and used by majority of researchers (Almeida and Awruch 2009; Tripathi and Kulkarni 2014; Pelletier and Vel 2006; Lopez et al. 2009; Akbulut and Sonmez 2008; Naik et al. 2008; Omkar et al. 2008; Fakhrabadi et al. 2013; Nicholas et al. 2012; Sebaey et al. 2011; Khosravi and Sedaghati 2008; Barroso et al. 2017; Gillet et al. 2010; Lopez et al. 2011) for in-plane loading conditions. The strength-based theories such as maximum stress theory (MS), Tsai Wu theory (TW) and Tsai Hill theory (TH) are considered and applied individually as constraints in the current study. A brief description of these theories follows (Mallick 2007).

1.1 Maximum stress theory

According to this failure theory, a composite lamina will fail, when any one of the principal stresses developed in the lamina reaches to its limiting value. The lamina will fail, if
$$\begin{aligned} \sigma_{11} & = S_{\text{Lc}}\; {\text{or}} \;\sigma_{11} = S_{\text{Lt}} , \\ \sigma_{22} & = S_{\text{Tc}} \;{\text{ or}} \;\sigma_{22} = S_{\text{Tt}} , \\ \tau_{12 } & = S_{\text{Lts}} , \\ \end{aligned}$$
where \(\sigma_{11}\) and \(\sigma_{22}\), are normal stresses developed in direction 1 and 2, respectively, while \(\tau_{12 }\) is the shear stress developed in plane 1–2 for individual lamina.

1.2 Tsai Wu theory

This failure criterion is based on Von Mises yield criterion. According to this theory, the lamina under consideration will fail, when following condition is satisfied:
$$F_{1} \sigma_{11} + F_{2} \sigma_{22} + F_{6} \tau_{12 } + F_{11} \sigma_{11}^{2} + F_{22} \sigma_{22}^{2} + F_{66} \tau_{12 }^{2} + 2F_{12} \sigma_{11} \sigma_{22} = 1,$$
where \(F_{1} ,F_{2} , F_{6} ,F_{11} , F_{22} , F_{66} , \;{\text{and}}\, F_{12}\) are the coefficients which can be calculated using strengths of the lamina in different directions (Mallick 2007).

1.3 Tsai Hill theory

According to this theory, the lamina will fail when following condition is satisfied:
$$\frac{{\sigma_{11}^{2} }}{{ S_{\text{Lt}}^{2} }} - \frac{{ \sigma_{11} *\sigma_{22} }}{{ S_{\text{Lt}}^{2} }} + \frac{{\sigma_{22}^{2} }}{{ S_{\text{Tt}}^{2 } }} + \frac{{\tau_{12 }^{2} }}{{ S_{\text{Lts}}^{2} }} = 1.$$

In this study, theories of failure are used as constraints to be satisfied for safe design. A procedure, based on first-order shear deformation theory, used to find the stresses for developing constraint equations is explained in the next section (Mallick 2007).

2 Analysis of composite laminate

The stresses and strains developed in each lamina are calculated using classical lamination plate theory. The additional geometric parameters required in the analysis of laminate apart from Fig. 1 are shown in Fig. 2.
Fig. 2

Additional geometric parameters of laminate

In Fig. 2, t1, tj and tn denote thicknesses of first, jth and nth lamina; h0 denotes distance from the laminate mid-plane to the top of the first lamina while \(h_{1 }\) is the distance from the laminate mid-plane to the bottom of the first lamina. \(h_{j - 1}\) is the distance from the laminate mid-plane to the top of the jth lamina and \(h_{j }\) is the distance from the laminate mid-plane to the bottom of the jth lamina. \(Z_{j}\) is the distance from the laminate mid-plane to the mid-plane of the jth lamina. The stiffness matrix for individual lamina \(\left[ { \bar{Q}} \right]\) is calculated from the material properties as,
$$\left[ { \bar{Q}} \right] = \left[ {\begin{array}{*{20}c} {\bar{Q}_{11} } & {\bar{Q}_{12} } & {\bar{Q}_{16} } \\ {\bar{Q}_{12} } & {\bar{Q}_{22} } & {\bar{Q}_{26} } \\ {\bar{Q}_{16} } & {\bar{Q}_{26} } & {\bar{Q}_{66} } \\ \end{array} } \right].$$
(1)
The elements of the stiffness matrix \(\left[ {\bar{Q}} \right]\) are calculated as,
$$\begin{aligned} \bar{Q}_{11} & = Q_{11 } \cos^{4} \theta + 2\left( {Q_{12} + 2Q_{66} } \right) \sin^{2} \theta \cos^{2} \theta + Q_{22} \sin^{4} \theta , \\ \bar{Q}_{12} & = Q_{12 } ( \sin^{4} \theta + \cos^{4} \theta ) + \left( {Q_{11} + Q_{22} - 4 Q_{66} } \right) \sin^{2} \theta \cos^{2} \theta , \\ \bar{Q}_{22} & = Q_{11 } \sin^{4} \theta + \, 2\left( {Q_{12} + 2Q_{66} } \right) \sin^{2} \theta \cos^{2} \theta + Q_{22} \cos^{4} \theta , \\ \bar{Q}_{16 } & = \left( {Q_{11} - Q_{12} - 2 Q_{66} } \right) \sin \theta \cos^{3} \theta + \left( {Q_{12} - Q_{22} + 2 Q_{66} } \right) \cos \theta \sin^{3} \theta , \\ \bar{Q}_{26} & = \left( {Q_{11} - Q_{12} - 2 Q_{66} } \right) \cos \theta \sin^{3} \theta + \left( {Q_{12} - Q_{22} + 2 Q_{66} } \right) \sin \theta \cos^{3} \theta , \\ \bar{Q}_{66} & = \left( {Q_{11} + Q_{22} - 2Q_{12} - 2 Q_{66} } \right) \sin^{2} \theta \cos^{2} \theta + Q_{66} ( \sin^{4} \theta + \cos^{4} \theta ), \\ \end{aligned}$$
where \(\begin{aligned} Q_{11} = \frac{{E_{11} }}{{1 - \mu_{12} \mu_{21} }},\;Q_{22} = \frac{{E_{22} }}{{1 - \mu_{12} \mu_{21} }}, \mu_{21} = \mu_{12 } \frac{{E_{22} }}{{E_{11} }}, \hfill \\ Q_{12} = Q_{21} = \frac{{\mu_{21} E_{11} }}{{1 - \mu_{12} \mu_{21} }} = \frac{{\mu_{12} E_{22} }}{{1 - \mu_{12} \mu_{21} }},\;Q_{66} = G_{12} . \hfill \\ \end{aligned}\)
The extensional stiffness matrix A, coupling stiffness matrix B and bending stiffness matrix D for laminate are calculated as,
$$A \, = \mathop \sum \limits_{j = 1}^{n} (\bar{Q}_{j} )\left( {h_{j } {-}h_{j - 1} } \right),$$
(2)
$$B \, = \frac{1}{2}\mathop \sum \limits_{j = 1}^{n} (\bar{Q}_{j} )\left( {h^{2}_{j } {-}h^{2}_{j - 1} } \right),$$
(3)
$$D = \frac{1}{3}\mathop \sum \limits_{j = 1}^{n} (\bar{Q}_{j} )\left( {h^{3}_{j } {-}h^{3}_{j - 1} } \right).$$
(4)
Above three matrices are used to calculate the mid-plane strains [\(\varepsilon^{0}\)] and curvatures [K] as given as follows:
$$\left[ {\varepsilon^{0} } \right] = \left[ {A_{1} } \right]\left[ N \right] + \left[ {B_{1} } \right]\left[ M \right],$$
(5)
$$\left[ K \right] \, = \left[ {C_{1} } \right]\left[ N \right] + \left[ {D_{1} } \right]\left[ M \right],$$
(6)
where \(\left[ {\varepsilon^{0} } \right] = \left[ {\begin{array}{*{20}c} {\varepsilon_{xx}^{0} } \\ {\varepsilon_{yy}^{0} } \\ {\varepsilon_{xy}^{0} } \\ \end{array} } \right],\;\left[ N \right] = \left[ {\begin{array}{*{20}c} {N_{xx} } \\ {N_{yy} } \\ {N_{xy} } \\ \end{array} } \right],\;\left[ M \right] \, = \left[ {\begin{array}{*{20}c} {M_{xx} } \\ {M_{yy} } \\ {M_{xy} } \\ \end{array} } \right],\)
$$\left[ {A_{1} } \right] = \left[ {A^{ - 1} } \right] + \left[ {A^{ - 1} } \right]\left[ B \right]\left[ {\left( {D^*} \right)^{ - 1} } \right] \, \left[ B \right]\left[ {A^{ - 1} } \right],\;\left[ {B_{1} } \right] = - \left[ {A^{ - 1} } \right]\left[ B \right] \, \left[ {\left( {D^*} \right)^{ - 1} } \right],$$
$$\left[ {C_{1} } \right] = \left[ {B_{1} } \right]^{\rm T} ,\left[ {D_{1} } \right] = \left[ {\left( {D^*} \right)^{ - 1} } \right]{\text{ and }}\left[ {D^*} \right] \, = \, \left[ D \right] \, {-} \, \left[ B \right]\left[ {A^{ - 1} } \right]\left[ B \right].$$
The mid-plane strains and curvatures can produce actual stresses \(\left[ \sigma \right]_{j}\) and strains \(\left[ \varepsilon \right]_{j}\) for any jth lamina in global co-ordinate system:
$$\left[ {\rm E} \right]_{j} = \, [\varepsilon^{0} ] + Z_{j} \left[ K \right],$$
(7)
$$\left[ \sigma \right]_{j} = \left[ {\bar{Q}} \right] _{j} \left[ \varepsilon \right]_{j} ,{\text{ where }}\left[ \sigma \right]_{j} = \left[ {\begin{array}{*{20}c} {\sigma_{xx} } \\ {\sigma_{yy} } \\ {\tau_{xy} } \\ \end{array} } \right]_{j} .$$
(8)
The first ply failure criterion is based on failure of a single lamina. Following relations are used for predicting the stresses developed in individual lamina in local co-ordinate system:
$$\sigma_{11} = \sigma_{xx} \cos^{2} \theta + \sigma_{yy} \sin^{2} \theta + 2\tau_{xy} \sin \theta \cos \theta ,$$
(9)
$$\sigma_{22} = \sigma_{xx} \sin^{2} \theta + \sigma_{yy} \cos^{2} \theta - 2\tau_{xy} \sin \theta \cos \theta ,$$
(10)
$$\tau_{12 } = \left( { - \sigma_{xx} + \sigma_{yy} } \right)\sin \theta \cos \theta + \tau_{xy} \left( {\cos^{2} \theta - \sin^{2} \theta } \right).$$
(11)

These local stress values obtained using Eqs. (9)–(11) for individual laminas are used for constructing all the constraint equations as mentioned earlier.

3 OptiComp and genetic algorithm (GA)

‘OptiComp’, presented by Tripathi and Kulkarni (2014), is a comprehensive optimization module developed by both the authors of this article using Matlab for optimal design of composite laminates can handle variety of laminate design problems involving different design objectives, constraints and design variables. In OptiComp, following changes have been made in conventional GA to suit the problem of composite laminate design optimization.
  1. (i)

    Direct value-coded representation is used for chromosomes instead of binary-coded representation. This kind of representation facilitates handling of multiple design variables of different nature in discrete form.

     
  2. (ii)

    Direct value-coded chromosome representation can exactly catch increment value for any design variable provided by user within the limit bounds. This is difficult in binary representation.

     
  3. (iii)

    Single-point crossover function and mutation function are defined in such a manner so as to suit the chromosome representation.

     
Let \(\theta_{\text{L}}\) and \(\theta_{\text{U}}\) be the lower and upper limiting values of the ply angles, while \(t_{\text{L}}\) and \(t_{\text{U}}\) be the lower and upper limiting values of the ply thicknesses. Let the increment values of ply angles and ply thicknesses within the given limiting bounds be \(\Delta \theta\) and \(\Delta t\), respectively, as provided by user. The vectors of acceptable values of ply angles (\(\theta_{\text{s}} )\) and ply thicknesses (\(t_{\text{s}} )\) within the given limit bounds can be developed as follows:
$$\theta_{\text{s}} = \left\{ {\theta_{{ 1 {\text{s }}}} ,\theta_{{ 2 {\text{s}}}} ,\theta_{{ 3 {\text{s}}}} ,\theta_{{ 4 {\text{s}}}} , \ldots ,\theta_{\text{ns}} } \right\},$$
where \(\theta_{{ 1 {\text{s}}}} = \theta_{\text{L }} ,\theta_{{ 2 {\text{s}}}} = \theta_{L} + \Delta \theta , \theta_{{ 3 {\text{s}}}} = \theta_{\text{L}} + 2 \times \Delta \theta , \ldots , \theta_{\text{ns}} = \theta_{\text{U}} .\)
$$t_{\text{s}} = \left\{ {t_{{ 1 {\text{s}}}} ,t_{{ 2 {\text{s}}}} ,t_{{ 3 {\text{s}}}} ,t_{{ 4 {\text{s}}}} , \ldots ,t_{\text{ns}} } \right\},$$
where \(t_{{ 1 {\text{s}}}} = t_{\text{L }} ,t_{{ 2 {\text{s}}}} = t_{\text{L}} + \Delta t , t_{{ 3 {\text{s}}}} = t_{\text{L}} + 2 \times \Delta t, \ldots , t_{\text{ns}} = t_{\text{U}} .\)
If the laminate is made up of ‘n’ number of plies, then each chromosome will have 2 × ‘n’ number of elements. The first ‘n’ number of elements of the chromosome are the randomly selected ply angle values from the series (\(\theta_{\text{s}} )\) while the remaining elements are the randomly selected ply thickness values from the series (\(t_{\text{s}} ).\) The first and second half of the chromosome have given separate treatment during crossover and mutation as shown in Fig. 3. In UTA all the genes of the chromosomes of size ‘n’ are represented only by the allowable angle values selected randomly.
Fig. 3

Chromosome representation, crossover and mutation in OptiComp

4 Combined VTA–UTA strategy and results

In UTA, ply angles and number of plies are treated as design variables, while in VTA, the number of plies, ply angles and ply thicknesses are treated as design variables. The lamina thicknesses available with the manufacturer are normally in discrete form and the manufacturers are interested in making laminate with uniform thickness laminas. The present study explores usefulness of VTA to find minimum required thickness of laminate to satisfy given strength constraints which will afterwards leads to value of cost-effective uniform thickness of lamina to be used in UTA.

In combined VTA–UTA strategy, initially VTA is used to find minimum laminate thickness for any predefined number of layers. The number of layers associated with each ply thickness necessary to sustain applied load condition can be calculated by Eq. (12) described later. Manufacturing cost-effective lamina thickness is then obtained considering constraint on maximum number of laminas put by manufacturer and design requirement. The optimal ply angle stacking sequence for that selected lamina thickness is then obtained by UTA. The effectiveness of this strategy is tested in this article against different load cases and theories of failure. The material properties of symmetric carbon/epoxy laminate of length 1000 mm and width 1000 mm, used during analysis given by Lopez et al. (2009) are provided in Table 1.
Table 1

Material properties for carbon/epoxy laminate

Property

E 11

E 22

G 12

\(\mu_{12 }\)

S Lc

S Lt

S Tc

S Tt

S Lts

\(\rho\)

Value

116.6

7.673

4.173

0.27

1701

2062

240

70

105

1605

The load cases mentioned in Table 2 are already studied by Lopez et al. (2009) for weight minimization of the conventional \(0^{\circ} , \pm 45^{\circ}\) and \(90^{\circ}\) laminate with uniform ply thickness 0.1 mm. These load cases are reconsidered in this article for finding cost-effective laminate weight using combined VTA–UTA strategy. In the current study, the ply angle increment value is reduced to 15° within the range − 75° to 90° to increase the feasible search space in discrete form. Let the available ply thicknesses with the manufacturer be 0.05, 0.075, 0.1, 0.125 and 0.15 mm. The limiting value of number of layers for a particular load case is obtained from reference results (Lopez et al. 2009) and applied in VTA analysis.
Table 2

Different load cases used in the study

Load case

Longitudinal force (N/mm)

Transverse force (N/mm)

Shear force (N/mm)

LC1

3000

3000

0

LC2

3000

3000

500

LC3

3000

3000

1000

LC4

− 3000

− 3000

0

LC5

− 3000

− 3000

500

LC6

− 3000

− 3000

1000

As described in VTA–UTA strategy, initially the problem is solved with VTA. The problem statement for VTA is
  1. 1.
    Find [\(\theta_{n}\), \(t_{n}\)],
    $${\text{to minimize weight }}W\, = \,\rho \cdot a \cdot b \cdot T ,\;\left( {T = \mathop \sum \limits_{i = 1}^{n} t_{i} } \right).$$
     

Subjected to satisfying:

Maximum stress theory/Tsai Wu theory/Tsai Hill theory,
$$\begin{aligned} - 75^{\circ} \le \theta_{n} \le 90^{\circ} \;({\text{Ply angle increment value}}\;15^{\circ} ), \hfill \\ 0.0 5\le t_{n} \le 0.15\;({\text{Ply thickness increment value}}\;0.025\;{\text{mm),}} \hfill \\ n\, = \,1, \ldots , \, N_{\text{max} } \, \left( {N_{ \text{max} } {\text{selected from reference result}}} \right). \hfill \\ \end{aligned}$$
The incremental value of ply thickness within the limiting values can be chosen as per availability or by user choice. The representative optimum ply angles and ply thicknesses stacking sequences obtained by VTA for load cases LC2 and LC5 are given in Table 3. The accuracy of finding laminate thickness in VTA can be improved by reducing ply thickness increment value when sole objective of the analysis is to find minimum required laminate thickness. For Tsai Hill theory, maximum number of layers used in the current analysis are same to that of Tsai Wu theory as reference results are not available for it.
Table 3

Optimum results for LC2 and LC5 using VTA

Load case

Failure criteria (Nmax)

Stacking sequence

Average thickness mm (T mm)

Reliability

Weight (kg) by OptiComp

LC2

MS (68)

\(\theta_{n}\) = [75/60/15/60/− 60/90/− 60/15/45/75/− 60/75/15/30/0/60/− 75/45/75/45/0/− 15/30/15/90/− 302/0/15/− 60/60/75/02]s

\(t_{n}\) = [0.12/0.125/0.1/0.0752/0.1/0.075/0.15/0.075/0.12/0.05/0.0752/0.15/0.075/0.1/0.05/0.125/0.1/0.075/0.125/0.05/0.15/0.1252/0.1/0.075/0.15/0.05/0.1/0.125/0.05]s

0.0963 (6.55)

0.65

10.512

LC2

TW (72)

\(\theta_{n}\) = [30/75/− 45/− 15/45/60/30/75/− 15/60/− 15/75/45/45/− 75/− 15/−  60/60/90/45/0/75/0/452/30/75/302/− 45/75/− 30/60/− 60/15/− 45]s

\(t_{n}\) = [0.125/0.075/0.125/0.15/0.052/0.1/0.15/0.1/0.075/0.1/0.075/0.1/0.125/0.1/0.125/0.05/0.1/0.125/0.05/0.125/0.1/0.075/0.1/0.125/0.052/0.15/0.05/0.125/0.15/0.0752/0.1/0.075/0.150]s

0.0979 (7.05)

0.6

11.315

LC2

TH (72)

\(\theta_{n}\) = [− 45/02/15/− 15/752/− 30/− 60/45/152/− 60/75/− 60/90/− 75/15/45/75/15/45/− 45/152/90/75/60/− 45/75/02/30/45/75/30]s

\(t_{n}\) = [0.075/0.125/0.075/0.125/0.075/0.1/0.15/0.12/0.125/0.075/0.125/0.075/0.15/0.1/0.075/0.125/0.075/0.125/0.0752/0.1/0.075/0.05/0.1/0.125/0.1/0.05/0.125/0.12/0.075/0.1/0.075/0.125/0.1]s

0.0979 (7.05)

1

11.315

LC5

MS (36)

\(\theta_{n}\) = [− 60/90/− 75/− 15/15/90/15/− 45/− 60/45/− 60/15/90/45/− 60/− 15/− 15/0]s

\(t_{n}\) = [0.125/0.1/0.125/0.05/0.075/0.1/0.0754/0.15/0.125/0.05/0.0752/0.1/0.05/0.15]s

0.0917 (3.3)

0.65

5.296

LC5

TW (32)

\(\theta_{n}\) = [45/− 15/− 60/0/− 45/− 30/30/− 75/− 60/30/602/− 45/− 60/30/− 30]s

\(t_{n}\) = [0.052/0.1252/0.05/0.1/0.125/0.15/0.125/0.12/0.075/0.125/0.1/0.05/0.075]s

0.0953 (3.05)

1

4.895

LC5

TH (32)

\(\theta_{n}\) = [30/15/75/30/− 75/− 452/− 15/− 45/15/− 60/15/− 45/− 60/− 75/45]s

\(t_{n}\) = [0.125/0.15/0.125/0.1/0.153/0.125/0.075/0.05/0.125/0.0752/0.15/0.1/0.125]s

0.1156 (3.7)

0.7

5.938

Obtained optimum laminate thicknesses ‘T’ and optimum weights using VTA are provided in Table 4 for all the load cases.
Table 4

Minimum laminate thickness and optimum weight obtained using VTA

Load case

Failure criteria

Number of plies used (Lopez et al. 2009)

Average thickness mm (T mm)

Weight (kg) by OptiComp

LC1

MS

68

0.0963 (6.55)

10.512

TW

72

0.0979 (7.05)

11.315

TH

72

0.0979 (7.05)

11.315

LC2

MS

68

0.0963 (6.55)

10.512

TW

72

0.0979 (7.05)

11.315

TH

72

0.0979 (7.05)

11.315

LC3

MS

72

0.093 (6.7)

10.757

TW

76

0.0934 (7.1)

11.395

TH

76

0.0934 (7.1)

11.395

LC4

MS

36

0.0917 (3.3)

5.296

TW

32

0.0953 (3.05)

4.895

TH

32

0.1156 (3.7)

5.938

LC5

MS

36

0.0917 (3.3)

5.296

TW

32

0.0953 (3.05)

4.895

TH

32

0.1156 (3.7)

5.938

LC6

MS

40

0.0838 (3.35)

5.376

TW

36

0.0848 (3.05)

4.895

TH

36

0.1042 (3.75)

6.018

In the next stage, analysis has to be carried out by UTA. The required lamina thickness for UTA has to be selected from available lamina thicknesses. Maximum number of layers required to sustain applied load condition for any particular lamina thickness can be calculated as,
$$N_{ \text{max} } = \frac{T}{\text{Selected lamina thickness}}.$$
(12)
The obtained value of ‘Nmax’ has to be rounded as even number on higher side so as to obtain a balanced symmetric laminate. Detailed procedure of selection of cost-effective lamina thickness for LC5 using Tsai Hill theory as constraint is explained here. The minimum laminate thickness obtained for this case from VTA is 3.7 mm. The required number of layers and possible laminate thickness for each available lamina thickness are given in Table 5.
Table 5

Possible configurations of LC5 laminate using different lamina thicknesses

Ply thickness (mm)

0.05

0.075

0.1

0.125

0.15

Required number of layers

74

50

38

30

26

Possible laminate thickness (T)

3.7

3.75

3.8

3.75

3.9

The two major considerations while designing composite laminate plate as structural member are structural consideration and manufacturing consideration. As per first, the obtained laminate thickness should be very close to required minimum thickness so as to achieve maximum strength to weight ratio. The cost of the composite laminates consists of two main parts: material cost and manufacturing cost. The major part of the material cost includes the cost of the raw material for composite laminates. On the other hand, the manufacturing cost is the function of the manufacturing time including the time lost in press from preparation, layer cutting, layer sequencing and final working (Fakhrabadi et al. 2013). Manufacturing cost can be reduced by controlling the maximum number of layers. Here an attempt has been made to satisfy both the requirements.

The possible configurations provided in Table 5 show that lamina thickness 0.05 mm gives exact minimum laminate thickness, but it does not satisfy constraint on maximum number of laminas imposed by manufacturer, i.e., 32. The lamina thicknesses 0.125 mm satisfies the requirements of manufacturer in terms of number of layers (32) and also provides laminate thickness very close to minimum laminate thickness. So, lamina thickness 0.125 mm is the cost-effective solution for this case. The average value of the ply thickness can be calculated from ply thickness stacking sequence obtained in VTA as,
$$t_{\text{avg}} = \frac{3.7}{32} = 0.115\, {\text{mm}}.$$
Table 5 shows that the available ply thicknesses above average ply thickness value results in equal or less number of plies compared with number of layers decided by manufacturer and possibly good result in terms of laminate thickness in majority cases. Now the earlier problem is solved using UTA to get necessary ply angle stacking sequence. The problem statement for UTA is
  1. 2.
    Find [\(\theta_{n}\)],
    $${\text{to minimize weight }}W\, = \,\rho \cdot a \cdot b \cdot T ,\left( {T = \mathop \sum \nolimits_{i = 1}^{n} t_{i} } \right).$$
     

Subjected to satisfying:

Maximum stress theory/Tsai Wu theory/Tsai Hill theory,
$$\begin{aligned} - 75^{\circ} \le \theta_{n} \le 90^{\circ} \;({\text{Ply angle increment value 15}}^{\circ} ), \hfill \\ t_{n} = {\text{Selected ply thickness by designer from VTA}}, \hfill \\ n\, = \,1{\text{ to }}N_{\text{max} } \left( {N_{\text{max} } {\text{calculated for the selected ply thickness}}} \right). \hfill \\ \end{aligned}$$
The representative final optimal solutions in terms of ply angle stacking sequence obtained using UTA for cost-effective lamina thickness under combined VTA–UTA strategy for different test cases mentioned under LC2 and LC5 are provided in Table 6.
Table 6

Optimum results for LC2 and LC5 by UTA using selected ply thickness

Load case

Failure criteria (Nmax)

Optimum ply angle stacking sequence

Selected thickness mm

Reliability

Weight (kg) by OptiComp

LC2

MS (44)

[0/90/0/754/03/− 15/90/753/0/902/45/− 15/02]s

0.15

0.65

10.593

TW (48)

[0/− 15/60/90/− 15/75/02/60/0/75/− 5/90/75/0/602/0/− 15/0/903/75]s

0.15

1

11.556

TH (48)

[75/90/05/75/902/02/15/902/15/30/903/75/90/02]s

0.15

1

11.556

LC5

MS (22)

[30/− 30/75/− 45/− 302/75/0/− 30/90/75]s

0.15

0.75

5.296

TW (32)

[− 60/60/− 45/30/− 75/60/75/− 15/− 452/30/− 75/− 15/15/− 45/− 30]s

0.1

1

5.136

TH (30)

[− 60/− 452/0/− 30/452/90/30/60/− 45/75/− 15/− 452]s

0.125

1

6.018

These results show that when plies with different thicknesses are available with manufacturer, then the combined VTA–UTA strategy provides value of cost-effective uniform ply thickness to be used in the laminate so as to sustain the applied load and necessary stacking sequence in two steps only. Otherwise, manufacturer has to try each and every thickness with multiple optimization process runs to reach to the conclusion. In this sense, combined VTA–UTA strategy is computationally more efficient. The population size of 200 is used for all simulation results presented in the paper, while maximum allowable number of generations is kept as 100. Crossover probability of 0.8 and mutation rate 0.01 is used during the simulation study. Table 7 provides cost effective lamina thicknesses and corresponding weights for all the load cases mentioned in Table 2.
Table 7

Cost-effective lamina thickness and laminate weight for all load cases

Load case

Failure criteria

Cost-effective lamina thickness mm

N max

Cost-effective laminate thickness mm

Minimum laminate thickness mm using VTA

Cost-effective weight (kg) by OptiComp

LC1

MS

0.15

44

6.6

6.55

10.593

TW

0.15

48

7.2

7.05

11.556

TH

0.15

48

7.2

7.05

11.556

LC2

MS

0.15

44

6.6

6.55

10.593

TW

0.15

48

7.2

7.05

11.556

TH

0.15

48

7.2

7.05

11.556

LC3

MS

0.125

54

6.75

6.7

10.833

TW

0.15

48

7.2

7.1

11.556

TH

0.15

48

7.2

7.1

11.556

LC4

MS

0.15

22

3.3

3.3

5.296

TW

0.1

32

3.2

3.05

5.136

TH

0.125

30

3.75

3.7

6.018

LC5

MS

0.15

22

3.3

3.3

5.296

TW

0.1

32

3.2

3.05

5.136

TH

0.125

30

3.75

3.7

6.018

LC6

MS

0.1

34

3.4

3.35

5.457

TW

0.1

32

3.2

3.05

5.136

TH

0.125

30

3.75

3.75

6.018

4.1 Reliability analysis

Reliability in case of design optimization process of composite laminates can be defined as fraction of successful optimization process runs reached to the global optimum solution against total number of optimization process runs (Rocha et al. 2014). In the present analysis, it is observed that the reliability of VTA is less than the reliability of UTA. It is logical as VTA deals with more design variables of different nature. The unsuccessful attempts also predict close global optimum solutions, e.g., in LC5, the unsuccessful attempts predicted optimum weight 5.315 kg for maximum stress theory in comparison with global optimum weight 5.296 kg.

4.2 Comparison of optimum results obtained at different stages

The optimum weight results obtained for conventional 0°, ± 45°, 90° laminas with ply thickness 0.1 mm (Lopez et al. 2009), results obtained using VTA alone and cost-effective results obtained using combined VTA–UTA strategy are provided and compared in Table 8 for all the load cases. It is observed that VTA alone can produce better results in terms of optimum weights for all the load cases at the cost of manufacturing complexity. VTA results are combined outcome of reduced ply angle increment value and discrete ply thicknesses.
Table 8

Comparison of optimum weights obtained at different stages

Load case

Failure criteria

Conventional weight (kg) by Lopez et al. (2009)

Weight (kg) using VTA only

Weight (kg) by combined VTA–UTA strategy

% reduction in weight using combined VTA–UTA over conventional weight

LC1

MS

10.914

10.512

10.593

2.94

TW

11.556

11.315

11.556

0

TH

11.315

11.556

LC2

MS

10.914

10.512

10.593

2.94

TW

11.556

11.315

11.556

0

TH

11.315

11.556

LC3

MS

11.556

10.757

10.833

6.25

TW

12.198

11.395

11.556

5.26

TH

11.326

11.556

LC4

MS

5.778

5.296

5.296

8.34

TW

5.136

4.895

5.136

0

TH

5.938

6.018

LC5

MS

5.778

5.296

5.296

8.34

TW

5.136

4.895

5.136

0

TH

5.938

6.018

LC6

MS

6.42

5.376

5.457

15

TW

5.778

4.895

5.136

11.11

TH

6.018

6.018

The optimum weight results obtained using maximum stress theory as constraint are less than the reference weights for all the load cases and maximum 15% weight reduction is observed for LC6 over reference weight. Hinton et al. (2002) stated that Tsai Wu theory predicts enhanced strength under compression–compression biaxial loading. In line with this statement, the optimum weight results obtained by this theory in compression–compression region are less than results predicted by other theories. As rightly mentioned by Pelletier and Vel (2006), Tsai hill theory over predicts optimum weights for all the load cases.

5 Combined VTA–UTA strategy with contiguity constraint

The contiguity constraints can also be applied along with the strength-based constraints in combined VTA–UTA strategy discussed earlier. Maximum four consecutive plies can be identical in terms of ply angles and thicknesses to improve strength of the laminate through reducing inter-laminar stress and matrix cracking (Javidrad et al. 2018). So, the contiguity constraint in the form: minimum two or maximum four plies can be identical; is added in the weight minimization problems discussed earlier. The optimum results obtained with this added constraint using VTA for a representative load case 5 are provided in Table 9.
Table 9

Optimum results for LC5 using VTA considering contiguity constraint

Load case

Failure theory (Nmax)

Optimum stacking sequence

Weight (kg) and laminate thickness T (mm)

LC5

MS (36)

\(\theta\) = [(− 75/60/− 45/− 15/0/− 30/60/15/− 75)2]s

t = [(0.15/0.075/0.075/0.1/0.1/0.1/0.075/0.075/0.1)2]s

5.457 (3.4)

TW (32)

\(\theta\) = [(0/60/− 60/0/− 60/− 15/− 45/75)2]s

t = [(0.075/0.1/0.125/0.1/0.075/0.1/0.075/0.125)2]s

4.975 (3.1)

TH (32)

\(\theta\) = [(− 30/− 60/0/− 75/− 45/− 15/60/45)2]s

t = [(0.1/0.075/0.075/0.15/0.15/0.125/0.125/0.15)2]s

6.099 (3.8)

Current study deals with the design of balanced symmetric laminate. To apply the contiguity constraint in the defined form, it is necessary that, half the number of laminas in the laminate must be even in number. This aspect has to be considered, while selecting the cost-effective uniform lamina thickness along with manufacturing and design requirements. The selected cost-effective uniform lamina thickness and optimum results obtained using UTA for LC5 are provided in Table 10.
Table 10

Cost-effective optimum results for LC5 using UTA considering contiguity constraint

Load case

Failure theory (Nmax)

Optimum stacking sequence

Weight (kg) and laminate thickness T (mm)

LC5

MS (28)

\(\theta\) = [(0/90/− 15/− 75/45/− 75/− 15)2]s

5.617 (0.125)

TW (32)

\(\theta\) = [(− 60/60/− 30/45/− 30/− 60/− 45/30)2]s

5.136 (0.1)

TH (32)

\(\theta\) = [(15/75/− 30/75/− 30/− 60/15/− 60)2]s

6.420 (0.125)

It is observed that inclusion of additional contiguity constraint along with strength-based constraints, increases the minimum required laminate thickness as well as manufacturing cost-effective laminate thickness.

6 Conclusion

In the present study, a new insight on utilization of VTA for design optimization of composite laminate has been given through combined VTA–UTA strategy. This strategy can become a helpful tool to designers and manufacturers for selecting uniform ply thickness and for finding necessary optimum stacking sequence when both these variables are available in discrete form. Laminate is confined to be made within predefined number of layers to control the manufacturing cost. The weight and manufacturing cost reduction of laminate are achieved using single objective function by controlling maximum number of layers with this strategy.

The obtained results show that the laminate weight not only depends on design variables but also on choice of the failure criteria. The maximum stress theory is more sensitive towards the design variables as it does not consider stress interaction. VTA alone can produce less optimum weights for all the theories of failure and can be effectively used for finding the minimum laminate thickness required to bear the applied load. Tsai Hill theory is more conservative in weight prediction of composite laminate. Inclusion of contiguity constraint along with strength-based constraints results in increased cost-effective laminate thickness. This combined optimization methodology has ability to become useful for designing manufacturing cost-effective laminated composite structures for majority applications. The combined VTA–UTA strategy can be used with any optimization algorithm which can handle multiple design variables of different nature simultaneously.

Notes

Acknowledgements

The authors are thankful to College of Engineering, Pune and Vishwakarma Institute of Technology, Pune for providing necessary infrastructure and computational facility for this work.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Vipin Kumar Tripathi
    • 1
  • Nishant Shashikant Kulkarni
    • 1
    Email author
  1. 1.Mechanical Engineering DepartmentCollege of EngineeringPuneIndia

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