Document Type : Research Article
Authors
^{1} Department of Mechanical Engineering, Nnamdi Azikiwe University, Awka, Nigeria
^{2} Department of Industrial and Production Engineering, Nnamdi Azikiwe University, Awka, Nigeria
Abstract
Keywords
The Graft of ANNFEM Technique in Macromechanics of Multioriented Natural Fiber/Polyester Laminates
C. Emeka Okafor ^{a*}, C. Chukwutoo Ihueze ^{b}
^{a}Department of Mechanical Engineering, Nnamdi Azikiwe University, Awka, Nigeria
^{b}Department of Industrial and Production Engineering, Nnamdi Azikiwe University, Awka, Nigeria
KEYWORDS 

ABSTRACT 
Laminate Ply Forces Macromechanical Analysis Plantain Fiber Composite Artificial Neural Network Finite Element Method 
Low weight and high strength requirements are prime target design objectives in strength demanding applications. Skillful design of low density, low weight and ecofriendly natural fiber composites could provide an alternative material route to the actualization of lighter structures. The present study proposed ANNFEM computational framework for the macromechanical analysis of multioriented Plantain Empty Fruit Bunch Fiber Laminate (PEFBFL) and Plantain Pseudo Stem Fiber Laminate (PPSFL). Control factors were numerically varied using Finite Element Method (FEM) and the resultant FEM models which encapsulated material properties of the laminate was streamlined into Artificial Neural Network (ANN) training scheme. A standard feedforward backpropagation network was adopted and the ANN model consists of stacking sequence, laminate aspect ratio and fiber orientation as input variables while the selected network outputs variables include average stress and displacement. The laminate constitutive equation was developed which enabled the establishment of laminate load deformation affiliation and equivalent elastic constants. The damage onset for individual lamina was detected by the maximum principal stress theory and the overall laminate strength of 40.12 N/mm^2 was obtained for PEFBFL and 32.16N/mm^2 for PPSFL. On the whole, there was steady reduction in laminates elastic modulus which points to compromised stiffness in material principal axis arising from gradual failure of the plies, this trend continued until the last ply failure occurred in ply 3 and 4 at 90 degrees in tensile mode of transverse direction. Stresses and displacements observed using CLT agree very closely with predictions of ANN. 
Natural fibers have been traditionally utilized over the years to address human needs such as in production of sacks, ropes, bags, rugs, floor covers, carpet backs, binder twines, hats, mats, baskets, stuffing material for upholstery and mattresses due to their low cost, low density, fair specific modulus, high toughness, nontoxic, recyclability and availability compared with synthetic fibers [1; 2]. Additionally, these properties have become the center of attraction for the surge in utilization of natural fibers in reinforcing polymer matrix for application in automotive [3], civil works and transportation sector [4], oil and gas sector [5], wind energy industry [6], housing sector [7], maritime sector [8], power and telecommunication industries [9]. Although, the greater part of the natural fibers such as those extracted from plantain plant are hydrophilic naturally leading to poor interfacial bond with hydrophobic polymer matrix [10], it can still be used successfully as a reinforcement after effective chemical treatment [11].
Continued exploration of postharvest plantain plant residues in the development of engineering materials such as reinforced composites has been on the increase. Natural fibers extracted from bio wastes offer significant advantage over woody biomass, since they have low microfibril angle, small lumen mean, high cellulose content, high crystallinity and relatively longer fibers which gives better control of fiber layup and orientation [12; 13; 14]. The use of biodegradable and locally available Plantain Pseudo Stem (PPS) and Empty Fruit Bunch (EFB) fibers from renewable agricultural sources in fabrication of Plantain Empty Fruit Bunch Fiber Reinforced Composite and Plantain Pseudo Stem Fiber Reinforced Composite have been reported and assessed [15; 16].
Extensive research evidence has thereafter emerged regarding various physical and mechanical properties of plantain fiber reinforced composites [15; 17; 16; 18; 11; 19; 20; 21; 22]; nonetheless, majority of these studies has concentrated on the properties of single plies which consists of unidirectional fiber reinforced lamina. Generally, reinforced composites in which fibers are aligned in one direction would be very weak in the transverse direction; consequently, any small transverse load arising from uneven lateral contraction can initiation crack. There is therefore need to combine some of the best aspects of unidirectional plantain fiber reinforced laminas in order to achieve a more robust material, this is because incorporation of variously oriented plies in one laminate will definitely lead to significant advantages when compared with conventional monolithic materials [23].
The boundless coalescence of ply orientations and stacking sequence obtainable at the design stage of laminates provides the necessary design pliancy to achieve high modulus industrial materials [24]. In this regard, laminated composites can be tailored for directional stiffness properties in longitudinal and transverse directions via combination of axis and off axis laminates. This is essential to guard against catastrophic failure in a way that peradventure any of the layers fail first, the composite can still endure more loads prior to the eventual failure of last ply [25]. Obviously, the failure stress and stiffness properties of plantain fiber reinforced laminates can be acquired through numerical techniques [26; 27]. Basically, the principles of laminated composite material mechanics can be learned at a macroscale where mechanical behaviour of the laminate and the structure are the primary focal point of investigations [28]. This provides superior initial guess for the designer in terms of average properties of the composite material [29; 30; 31]. Advanced numerical and computational tools have evolved over the years for application in solving diverse composite design problems and skilful utilization of these tools would further diminish the design and improvement time alongside the expense of experimentation expected in the design development phase of laminates.
Rahimi, Hussain, Meon and Mahmud [32] reported that ANSYS software is capable of predicting the failure of composite laminates based on Maximum Stress and TsaiWu theories. Saxena and Kirtania [33] used finite element analysis and analytical methods in evaluation of symmetric crossply laminated composite plates, authors computational results compared closely with the theoretical results. Other researchers has reported comparable results between finite element and analytical methods for interlaminar stresses in laminated composite beam [34], first ply failure of laminated composite plates [35], crossply laminate of boron/epoxy composite failure under uniformly distributed load [36], failure of interply hybrid laminated composite [37], failure of carbon fiberreinforced polymer laminated composite thinwalled Ibeams [38] and failure of fiber/epoxy composite laminate interface using cohesive multiscale model [39]. It is necessary to integrate finite element analysis and artificial neural network in order to boost accuracy and speed in design process.
Various macromechanical models such as elasticity models [40], plasticity models [41] and nonlinear constitutive model [42] have also been formulated to embody the constitutive affiliations of laminates. HsuanTeh and LungShen [42] carried out constitutive modelling of composite laminates under uniaxial compression, authors detected damage onset using TsaiWu and the maximum stress criterion. Dimitrienko [43] developed universal models of laminated composites constitutive relations with discrete information on main conjugated stressstrain tensors pairs. Development of a constitutive equation for a novel plantain fiber reinforced laminate is required for establishment of the laminate load deformation affiliation and equivalent elastic constants.
There is also need to relate the externally applied force on plantain fiber reinforced laminate to the resulting deformation of strain stresses at ply level, this will provide the best standing to determine whether any ply has failed using appropriate failure theory. Different failure theories have been used in the literature to evaluate failure initiation in fiber reinforced laminates; Rajanish, Nanjundaradhya, Ramesh and Bhaskar [44] classified them into limit/noninteractive (maximum stress and maximum strain); interactive (TsaiHill and TsaiWu) and partiallyinteractive (HashinRotem and Puck) theories. The validity and criteria for selecting any of these theories is well documented [45; 46; 47]. The noninteractive maximum principal stress criterion was adopted in the present study to account for various ply failure modes inherent in plantain fiber reinforced laminate.
The fiber mass fraction is determined from eq. (1) and the ply thickness in terms of mass fraction of fibers is presented in eq. (2).
where is the density of matrix while and are the density and optimal volume fraction (50%) of fiber respectively. Properties and characteristics of the plantain fiber reinforcement–polyester matrix with detailed implementation of Archimedes principle has been reported elsewhere [15; 18] and by knowing the density of polyester resin as 1200kg/m^{3}, the ply thickness from eq. (2) and specific mass of fiberis presented in Table 1.
The individual layers of laminate composites are held together by matrix to form a plate. The layers of the laminate composites referred in the present study are of thesame constituent materials having different reinforcing pattern. The matrix and fiber part of the laminate composite are respectively polyester and plantain fiber with properties described in Tables 2 and 3. Obviously, the analysis of laminate depends on the properties of individual layers and the detailed mechanical properties of single layer of Plantain Empty Fruit Bunch Fiber Reinforced Composite (PEFBFC) and Plantain Pseudo Stem Fiber Reinforced Composite (PPSFC) have been reported by [16].
The classical Central Composite Design (CCD) method was used to determine the number of numerical experiments to be evaluated. CCD is a useful design of experiment (DOE) template which minimizes the required number of iterations for full material assessment. It involves varying the stacking sequence from (30/0/90)s to (30/45/90)s, laminate aspect ratio from 1.31 to 0.79 and fiber orientation from 30^{0} to 90^{0} as shown in Fig. 1. This control factor arrengement gave a total of 26 experimental runs.
The Finite Element Analysis was conducted in ANSYS Parametric Design Language (APDL). A 3D 4node shell 181 element with six degrees of freedom was used in creating the model following the symmetry of layer code shown in Fig. 1a. According to Abhishek [35], SHELL181 is useful in evaluation of thin to moderatelythick shell structures. The material model was assigned to structural linear elastic orthotropic behaviour and then the shell section was modified to conform to fiber angle orientations and layup.
Table 1: Plantain fiber ply parameters determined by application of Archimedes principle
FIBER SOURCE 
(g) 
(mm^{3}) 
Density kg/m^{3} 
Ply Thickness (mm) 
PEFB fiber 
20.397 
53364.8 
381.966 
0.108 
PSTEM fiber 
20.670 
58364.8 
354.151 
0.228 
Table 2: Mechanical properties of single layer plantain fibers reinforced polyester composites [16]
Composites 
Properties 

(MPa) 
(MPa) 
(MPa) 
(MPa) 
(MPa) 
(MPa) 
(MPa) 
G_{12} (MPa) 

PEFBFC 
410.15 
37.3397 
33.69 
14922 
7030.962 
9990.10 
0.38 
19.3100 
3622.99 
PPSFC 
288.10 
33.1330 
29.24 
13027.5 
6817.175 
9146.305 
0.29 
15.5700 
3332.84 
Note: are tensile strengths in the longitudinal and transverse directions respectively.
Table 3: Mechanical properties of plantain fibers and polyester resin [16]
Property 
Polyester resin 
Young modulus (MPa) 
2000 – 4500 
Tensile strength (MPa) 
40  9 0 
Compressive strength (MPa) 
90 250 
Tensile elongation at break (%) 
2 
Water absorption 24h at 20 °C 
0.1  0.3 
Flexural modulus (GPa) 
11.0 
Poisson's ratio. 
0.37 – 0.38 

Plantain Pseudo Stem Fibers 
Young modulus (MPa) 
23555 
UTS (MPa) 
536.2 
Strain (%) 
2.37 

Plantain Empty fruit bunch Fibers 
Young modulus (MPa) 
27344 
UTS (MPa) 
780.3 
Strain (%) 
2.68 


Fig. 1: (a) Stacking sequence and (b) Boundary Condition for PEFBFL (Nodes = 825, Elements = 768)
Each layer thickness corresponds to the ply thickness as specified in Table 1, finite element model was created using rectangular cross section and the meshing was done for the entire laminate at a time. In the postprocessing phase, results of laminate stresses in x direction and deformation are viewed and plotted at nodes as shown in Fig. 2.
The data acquired were fitted to the empirical polynomial regression model of eq. (3).
where Y is responses (stress and displacement); β_{0},β_{i }(i = 1, 2,…..k) and β_{ij }(i = 1, 2,…k; j = 1, 2,…k) are the model coefficients; x_{i }and x_{j }are the coded independent variables. Also, the adequacy of the model was checked using the coefficient of determination (R^{2}).
The Neural Network model was trained using the raw numerical dataset obtained from the finite element simulation. The weighted sum of inputs arriving at each neuron was passed through an activation function to generate an output signal. A total of 17 (65.4%) of experimental results was used to train the network, 5 (19.2%) of the experimental result was used to validate the training while the remaining 4 (15.4%) was used for testing. In order to reduce the deviations of response predictions, a trial and error method was utilised to establish the suitable number of neurons required in the hidden layer, this was achieved by repeating the training cycle at varying number of neurons in the hidden layer. The Multilayer perceptron which was trained with backpropagation algorithm has three input neurons representing the design independent variables (stacking sequence, laminate aspect ratio and fiber orientation), a single hidden layer of ten neurons and an output layer consisting of two neurons representing the stress and displacement as shown in Fig. 4. The network performance was assessed using Mean Square Error (MSE) and coefficient of determination (R^{2}). Error minimization was achieved by Levenberge–Marquardt (LM) method. The LevenbergMarquardt algorithm uses an approximation to the Hessian Matrix in a Newtonlike update shown in Eq. (4).
where J = Jacobian matrix containing first derivatives of the network errors, e = vector of network errors. At , eq. (4) becomes the regular Newton’s method using an approximate Hessian matrix, however at , eq. (4) becomes gradient descent with a small step size. The mean sum of squares of the network errors was captured using Eq. (5).


Fig. 2: (a)Laminate Stresses and (b) Displacement in x direction for PEFBFL
Fig. 3: Flowchart showing the graft of ANNFEM technique
Fig. 4: Artificial neural network with 3 input variables, 1 hidden layer of 10 neurons and 2 output variables.
Following classical laminate theory [48; 49; 29], the variation of direct strain in x and y is given in eq. (6). Classical Laminate Theory (CLT) is a predictive means used in the analysis of laminate behaviour based on its stacking sequence and material properties. The theory assumed that each layer of the laminate is both quasihomogeneous and orthotropic; the thickness of the plate is much smaller than any of its characteristic dimension; the displacements arising from mechanical loading is small compared with thickness of the plate; the inplane strains are small compared with unity; transverse shear and normal strain are negligible; no warping exist; each ply obeys Hooke’s Law and the plate thickness is constant throughout the laminate. Eqs. 620 provides the mathematical background for application of CLT.

and shear strain in xy plane reads:
Combining eqs. 6 7 in a matrix form gives:
Integrating inplane force intensities and moment over the thickness of ply p in x,y and xydirections reads:
The knowledge of , and on the plies is necessary for complete evaluation of eqs. (9 – 11). Considering two possible loading conditions of longitudinal (direction 1) and transverse (direction 2) in the matrix, the resulting direct strains from Hooks law are respectively and where = major Poisson’s ratio and = minor Poisson’s ratio. Hence the application of both direct stresses and will yield corresponding strains as follows:
And the stressstrain relationship for an orthotropic ply can be expressed as:
where
Combining eq. (13) with eq. (8) we have:
Eq. (14) describes the association between the applied force (N) and the resulting midplane laminate deformations. While the extensional and coupling stiffnessis presented in eqs. (15) and (16) respectively.
In similar consideration for moment about specific axis for ply 1to n in Fig. 1 gives:
Eq. (17) gives the bending stiffness term ( which depends on transformed reduced stiffness term and the ply cordinate terms.
Inversion of the laminate stiffness terms in eq. (13) yields the required transformed reduced compliance terms , so considering a fully populated extensional stiffness matrices (strain component of eq. (14)
Where
Considering a fully populated bending stiffness matrices (k component of eq. 14)
Where
3.1. CCD optimum conditions
The optimization exercise for stress and displacement was conducted utilizing the flexibility of the Design Expert optimization tool function. Eq. (3) was solved for the best solutions. A usual approach which involves selecting the best solution was adopted and the chosen optimal solutions gave stacking sequence = (30/45/90)s, laminate aspect ratio = 1.31, fiber orientation = 90^{0} and desirability = 1. The laminate with optimum characteristics was further analyzed by applying the Classical Laminate Theory along the thickness of specified fiber source.
The training record was used to plot the performance progress. The Mean Square Error of the trained network for PEFBFL strength prediction was 0.46, having a regression coefficient of 0.99 while the network for PEFBFL displacement prediction has Mean Square Error of 4.55442E09 and regression coefficient of 0.93 (Figs. 5 & 6). Similarly, The Mean Square Error of the trained network for PPSFL strength prediction is 0.6557, having a regression coefficient of 0.96 while the network for PEFBFL displacement prediction has Mean Square Error of 1.25768E09 and regression coefficient of 0.94 (Figs. 7 & 8). In all, the training data indicates a good fit and the validation results also show reasonably high R^{2} values that are close to 1. The regression coefficient measures the correlation between the predicted responses (outputs) and the experimental responses (targets).
Fig. 5: (a) Actual value of PEFBFL stress versus predicted (b) Regression plot for validation
Fig. 6: (a) Actual value of PEFBFL displacement versus predicted (b) Regression plot forvalidation
Fig. 7: (a) Actual value of PPSFL stress versus predicted (b) Regression plot for validation
Fig. 8: (a) Actual PPSFL value of displacement versus predicted (b) Regression plot for validation
The optimal set of plantain fiber reinforced laminate with thickness (t) which is composed of six stacked plies in the order is represented in Fig. 9.
Minor Poisson’s Ratio is the strain resulting from stress in the axial direction. The minor Poisson’s ratio for single layer plantain fibers reinforced polyester composites using eq. (12) and relevant data in Table 1 is found as:
Similarly, the ply reduced stiffness matrix () for a single layer of PEFBFC (N/mm^{2}) and single layer PPSFC (N/mm^{2}) is also evaluated and presented in eq. (21) For PEFBFC and For PPSFC in eq. (22).
Hence the transformed reduced stiffness term for the multioriented plantain fiber/polyester laminate described in Fig. 10 is obtained from eq. (13) and presented in Table 4 for PEFBFL and Table 5 for PPSFL.
Fig. 9: Elements of the laminate showing point force and moment strengths
Fig. 10: Six layer PEFB fiber symmetric laminate code (mm)
Table 4: values (N/mm^2) for PEFBFL (30/45/90)s
Ply 
Fiber Orientation (Degree) 

1 
30 
13269.72 
9036.24 
4247.60 
3490.59 
2193.77 
1472.53 
2 
45 
10944.78 
10944.78 
4455.81 
3699.00 
2116.74 
2117.00 
3 
90 
7544.11 
16011.07 
3622.99 
2866.00 
0.00 
0.00 
4 
90 
7544.11 
16011.07 
3622.99 
2866.00 
0.00 
0.00 
5 
45 
10944.78 
10944.78 
4455.81 
3699.00 
2116.74 
2117.00 
6 
30 
13269.72 
9036.24 
4247.60 
3491.00 
2193.77 
1472.53 
Table 5: values (N/mm^2) for PPSFL (30/45/90)_{s}
Ply 
Fiber Orientation (Degree) 

1 
30 
11388.02 
8139.69 
3948.85 
2687.50 
1762.22 
1050.92 
2 
45 
9558.52 
9558.52 
4154.19 
2892.84 
1624.17 
1624.17 
3 
90 
7131.53 
13628.2 
3332.84 
2071.49 
0 
0 
4 
90 
7131.53 
13628.2 
3332.84 
2071.49 
0 
0 
5 
45 
9558.52 
9558.52 
4154.19 
2892.84 
1624.17 
1624.17 
6 
30 
11388.02 
8139.69 
3948.85 
2687.5 
1762.22 
1050.92 
Since the laminate in this study is symmetric, the coupling described in eq. (16) tends to zero while the extensional and bending stiffness terms are computed using the ply thickness and centroidal values described in Fig. 10 and tabulations in Table 6 for PEFBFL and in Table 7 for PPSFL.
The extensional stiffness and bending stiffness as defined in eqs. (15) and (18) are computed using plies 1, 2 and 3 in turn doubling the result in consideration of ply symmetry to get total laminate extensional and bending stiffness written in matrix form in eq. (23) to (26) for PEFBFL and PPSFL.
Evaluation of eqs. (19) and (20) gives the transformed reduced compliance and bending stiffness matrices for PEFBFL and PPSFL as shown in eq. (27) to (30).
The laminate equivalent elastic constants are therefore computed for membrane and bending modes using appropriate expressions in Table 8.
3.4. Macroscopic assessment of ply failure based on Maximum Principal Stress Theory
Ply strength analysis was performed using Maximum Principal Stress Theory, the use of this noninteractive failure criterion was justified because of the need to ascertain probable laminar mode of failure. The load required to cause the first ply failure was obtained from the strength ratio of the plies. The maximum strength ratio was subsequently used to calculate the load factor and the load intensity required to cause the first ply failure.
Table 6: Ordinate values for PEFBFL (30/45/90)_{s}
Ply 
Fiber Orientation (Degree) 
(mm) 
(mm) 

1 
30 
0.108 
0.27 
0.00798 
2 
45 
0.108 
0.162 
0.00294 
3 
90 
0.108 
0.054 
0.00042 
4 
90 
0.108 
0.054 
0.00042 
5 
45 
0.108 
0.162 
0.00294 
6 
30 
0.108 
0.27 
0.00798 
Table 7: Ordinate values for PPSFL (30/45/90)_{s}
Ply 
Fiber Orientation (Degree) 
(mm) 
(mm) 

1 
30 
0.228 
0.57 
0.07506 
2 
45 
0.228 
0.342 
0.02766 
3 
90 
0.228 
0.114 
0.00395 
4 
90 
0.228 
0.114 
0.00395 
5 
45 
0.228 
0.342 
0.02766 
6 
30 
0.228 
0.57 
0.07506 
Table 8: Equivalent elastic constants of laminates
Elastic constant 
Membrane mode 
Bending mode 


Source 

Result 
Source 
Result 

PEFBFL 
PPSFL 
PEFBFL 
PPSFL 

93450.41 
8514.19 
10611873.8 
9466.81 

107955.6 
9654.24 
8542388.10 
7876.59 

38634.96 
3640.82 
3814649.15 
3654.32 

0.25 
0.22 
0.30 
0.27 


0.29 
0.25 
0.24 
0.22 

0.28 
0.24 
0.38 
0.34 

0.19 
0.16 
0.25 
0.20 

Notably, the load factor is inversely proportional to the strength ratio only for Maximum Principal Stress Theory. However, for TsaiHill failure criterion, the load factor is inversely proportional to the square root of strength ratio. For Hoffman criterion and TsaiWu criterion, the relationship between load factor and strength ratio has combination of quadratic and linear functions. Hence for initial design development phase of laminates, Maximum Principal Stress Theory is adjudged valid.
Considering only membrane load, a partially populated compliance matrices is obtained from eq. (19) and by applying theordinate values for PEFBFL ply laminate (30/45/90)_{s} in Fig. 6:
6859.86
7774.292
2662.503
2791.012
And a membrane equivalent Young’s modulus (E_{x}) in the direction of the applied load is
where is the total thickness of the laminate (t_{p}× number of layers) and . An axial load of 476N is applied at first instance to the 25×19.05 (LW) rectangular section of Fig. 1, the force intensity is the axial force per unit width of the section ( as and ). Then the laminate midplane deformations strains at the laminate reference axes are calculated for each ply using eq. (31).
Or
3.3.1.1.Computation of PEFBFL ply strain in material axes:
Eq. (32) is the constant strain distribution through the laminate thickness in all the three symmetric plies. However, the ply strain in the material reference axes is related to the material axes 12 in eq. (33). Symmetric plies are combined because only a membrane load is considered and the strain in the material axes are constant through the thickness, so once the computation is done for a definite ply configuration, the rest of the plies with the same configuration will have identical value, hence
For plies 1 and 6 at 30^{o}
For plies 2 and 5 at 45^{o}
For plies 3 and 4 at 90^{o}
3.3.1.2. Computation of PEFBFL ply stresses in material axes:
The ply stresses in the material axis is related to the ply strains in the material axes as shown in eq. (34).
Therefore the ply strains in material axes 12 for each ply are used to establish the stressstrain relationship as follows:
For plies 1 and 6 at 30^{o}
For plies 2 and 5 at 45^{o}
For plies 3 and 4 at 90^{o}
Using the Maximum Principal Stress Theory which suggests that the ply will fail when any stress value in the material axes exceeds their respective ultimate strength. Such that
The left hand side of eqs. (35) represents the strength ratio (SR). The maximum Strength ratio (SR) for the applied stress is factored in to obtain the load factor. Table 9 showed the ply stresses associated strength ratios at for PEFBFL, it can be seen that maximum strength ratio of 0.9596 occured in ply 2 at 45 degrees in shear mode. Therefore at no ply failure has yet occurred, hence the load intensity that can cause first ply to fail according to Maximum Principal Stress Theory in the 45 degree plies in shear mode is , furthermore, since the laminate is symmetric with membrane load only, the failure of ply two and ply five are predicted simultaneously. Similar computations were carried out for PPSFL and result presented in Table 10.
Table 9: Ply stresses and associated strength ratios in PEFBFL @ and
ply 
θ 
e1 
e2 
e12 
σ1 
σ2 
σ12 
SR.1 
SR.2 
SR.12 
1 
30 
0.002719 
0.00016 
0.00443 
44.002 
9.0161 
16.048 
0.1073 
0.2411 
0.83112 
2 
45 
0.001441 
0.00144 
0.00513 
27.19 
15.00 
18.53 
0.0663 
0.4009 
0.95969 
3 
90 
0.00112 
0.00399 
6.26E19 
6.43 
26.96 
0.00 
0.01567 
0.7208 
1.2E16 
4 
90 
0.00112 
0.00399 
6.27E19 
6.43 
26.96 
0 
0.01567 
0.7208 
1.20E16 
5 
45 
0.001441 
0.00144 
0.00512 
27.19 
15 
18.53 
0.0663 
0.4009 
0.95969 
6 
30 
0.002719 
0.00016 
0.00443 
44.002 
9.0161 
16.048 
0.1073 
0.2411 
0.83112 
Table 10: Ply stresses and associated strength ratios in PPSFL @ and
ply 
θ 
e1 
e2 
e12 
σ1 
σ2 
σ12 
SR.1 
SR.2 
SR.12 
1 
30 
0.00144 
0.00014 
0.00225 
20.146 
4.0099 
7.5124 
0.04912 
0.10722 
0.38904 
2 
45 
0.00079 
0.00079 
0.00260 
14.92 
8.23 
9.43 
0.03638 
0.22003 
0.48834 
3 
90 
0.0005 
0.00209 
3.19E19 
2.19 
14.32 
0.00 
0.00533 
0.38278 
6.0E17 
4 
90 
0.0005 
0.00209 
3.19E19 
2.19 
14.32 
0 
0.00533 
0.38278 
6.0E17 
5 
45 
0.00079 
0.00079 
0.00260 
14.92 
8.23 
9.43 
0.03638 
0.2200 
0.48834 
6 
30 
0.00144 
0.00014 
0.00225 
20.146 
4.0099 
7.5124 
0.04912 
0.10721 
0.38904 
Table 11: values (N/mm^2) for PEFBFL when ply 2 and 5 has completely failed
Ply 
θ (Degree) 

1 
30 
13269.72 
9036.24 
4247.60 
3490.59 
2193.77 
1472.53 
2 
45 
0 
0 
0 
0 
0 
0 
3 
90 
7544.11 
16011.07 
3622.99 
2865.98 
0 
0 
4 
90 
7544.11 
16011.07 
3622.99 
2865.98 
0 
0 
5 
45 
0 
0 
0 
0 
0 
0 
6 
30 
13269.72 
9036.24 
4247.6 
3490.59 
2193.77 
1472.53 
For secondplyfailure analysis, it is assumed that all the elastic values E_{1}, E_{2} and G_{12} for the two already failed plies 2 and 5 are zero; hence the reduced and transformed reduced stiffness for plies 2 and 5 are zero. Therefore the new assumed transformed reduced stiffness term for PEFBFL is presented in Table 11.
The altered laminate configuration shown in Table 11 is still symmetric, so considering only membrane load, a partially populated compliance matrices is obtained from eq. 24. Computed ply strain and stresses in material axes is shown in Table 12, the ply stresses in the material axis is related to the ply strains in the material axes as shown in eq. 34. The ply strain in material axes 12 for each ply is hence used to establish the stressstrain relationship. Table 12 showed the ply stresses and associated strength ratios in PEFBFL at , it can be seen that maximum strength ratio of 1.27715 occured in ply 1 at 6 at 30 degrees in shear mode. Therefore at ply 1 at 6 failure has occured, furthermore, since the laminate is symmetric with membrane load only, the failure of ply one and ply six are predicted simultaneously.
Table 13 showed the ply stresses and associated strength ratios in PPSFL at , it can be seen that maximum strength ratio of 0.59418 occurred in ply 1 at 6 at 30 degrees in shear mode. Therefore at no subsequent ply failure has occured in this reduced stiffness laminate, therefore the load to cause the ply failure according to Maximum Principal Stress Theory in 30 degree plies in shear mode is .
For Thirdplyfailure analysis, it is assumed that all the elastic values E_{1}, E_{2} and G_{12} for the four failed plies 1, 2, 5 and 6 are zero, hence the reduced and transformed reduced stiffness for plies 1, 2, 5 and 6 are zero. Thus the new transformed reduced stiffness terms for the composite are presented in Table 14.
Table 12: Ply stresses and associated strength ratios in PEFBFL @ and
ply 
θ 
e1 
e2 
e12 
σ1 
σ2 
σ12 
SR.1 
SR.2 
SR.12 
1 
30 
0.00430 
0.00037 
0.0068 
69.9847 
15.1605 
24.66 
0.1706 
0.4054 
1.2771 
2 
45 
0 
0 
0 
0 
0 
0 
0 
0 
0 
3 
90 
0.00159 
0.00627 
9.6E19 
7.51 
42.73 
0 
0.0183 
1.1426 
1.8E16 
4 
90 
0.00159 
0.00627 
9.6E19 
7.51 
42.73 
0 
0.0183 
1.1426 
1.8E16 
5 
45 
0 
0 
0 
0 
0 
0 
0 
0 
0 
6 
30 
0.004304 
0.00037 
0.0068 
69.9847 
15.1605 
24.66 
0.1706 
0.4054 
1.2771 
Table 13: Ply stresses and associated strength ratios in PPSFL @ and
ply 
θ 
e1 
e2 
e12 
σ1 
σ2 
σ12 
SR.1 
SR.2 
SR.12 
1 
30 
0.00227 
0.00028 
0.00344 
31.8576 
6.7477 
11.474 
0.0777 
0.1804 
0.594 
2 
45 
0 
0 
0 
0 
0 
0 
0 
0 
0 
3 
90 
0.0007 
0.00326 
4.87E19 
2.07 
22.56 
0.00 
0.0051 
0.6033 
9.1E17 
4 
90 
0.0007 
0.00326 
4.87E19 
2.07 
22.56 
0 
0.0051 
0.6033 
9.10E17 
5 
45 
0 
0 
0 
0 
0 
0 
0 
0 
0 
6 
30 
0.00227 
0.00028 
0.00344 
31.8576 
6.7477 
11.474 
0.0777 
0.1804 
0.594 
Table 14: values (N/mm^2) for PEFBFL when ply 1, 2, 5 and 6 has completely failed
Ply 
θ 

1 
30 
0 
0 
0 
0 
0 
0 
2 
45 
0 
0 
0 
0 
0 
0 
3 
90 
7544.11 
16011.07 
3622.99 
2865.98 
0 
0 
4 
90 
7544.11 
16011.07 
3622.99 
2865.98 
0 
0 
5 
45 
0 
0 
0 
0 
0 
0 
6 
30 
0 
0 
0 
0 
0 
0 
The altered laminate configuration shown in Table 14 is still symmetric, so considering only membrane load, a partially populated compliance matrices is obtained from eq. (19). Following the method used in first and second ply failure analysis, the ply stresses and associated strength ratios at , is shown in Table 15. It can be seen that maximum strength ratio of 3.218395 occured in ply 3 and 4 at 90 degrees in tensile mode of transverse direction. Therefore at ply failure has occured in PEFBFL, furthermore, since the laminate is symmetric with membrane load only, the failure of ply one and six are predicted simultaneously.
Similarly for PPSFL, it can be seen in Table 16 that maximum strength ratio of 2.690382 occurred in ply 3 and 4 at 90 degrees in tensile mode of transverse direction. Therefore at ply failure has occured in PPSFL, furthermore, since the laminate is symmetric with membrane load only, the failure of ply one and six are predicted simultaneously.
The average stress which is the overall laminate strength (is therefore failure load divided by laminate thickness which is for PEFBFL. While PPSFL is . Eq. (31) gives the corresponding strain at each ply failure as because .
Based on the macromechanical analysis of (30/45/90)_{s} laminate under tensile loading, the following conclusions are drawn:
Table 15: Ply stresses and associated strength ratios in PEFBFL @ and
ply 
θ 
e1 
e2 
e12 
σ1 
σ2 
σ12 
SR.1 
SR.2 
SR.12 
1 
30 
0 
0 
0 
0 
0 
0 
0 
0 
0 
2 
45 
0 
0 
0 
0 
0 
0 
0 
0 
0 
3 
90 
0.00306 
0.01712 
2.473E18 
0.00 
120.37 
0.00 
0 
3.218395 
4.6E16 
4 
90 
0.00306 
0.01712 
2.47E18 
0 
120.37 
0 
0 
3.218395 
4.60E16 
5 
45 
0 
0 
0 
0 
0 
0 
0 
0 
0 
6 
30 
0 
0 
0 
0 
0 
0 
0 
0 
0 
Table 16: Ply stresses and associated strength ratios in PPSFL @ and
Ply 
θ 
e1 
e2 
e12 
σ1 
σ2 
σ12 
SR.1 
SR.2 
SR.12 
1 
30 
0 
0 
0 
0 
0 
0 
0 
0 
0 
2 
45 
0 
0 
0 
0 
0 
0 
0 
0 
0 
3 
90 
0.00215 
0.0141552 
1.998E18 
6.12 
100.62 
0.00 
0.01492 
2.690382 
3.7E16 
4 
90 
0.00215 
0.0141552 
2.00E18 
6.12 
100.62 
0 
0.01492 
2.690382 
3.70E16 
5 
45 
0 
0 
0 
0 
0 
0 
0 
0 
0 
6 
30 
0 
0 
0 
0 
0 
0 
0 
0 
0 
Table 17: The average values of stress (and strain ( for PEFBFL and PPSFL
Ply 
PEFBFL 
PPSFL 

(%) 
(N/mm^{2}) 
Displ. (mm) 
(%) 
(N/mm^{2}) 
Displ. (mm) 

1 
25 
0.00016 
0.0039 
38.5803 
0.001 
25 
0.00008 
0.0021 
18.27485 
0.000523 
2 
26 
0.00021 
0.0063 
40.1235 
0.00157 
26 
0.00013 
0.0034 
19.00585 
0.000815 
3 
26 
0.00039 
0.0171 
40.1235 
0.00428 
44 
0.00032 
0.0142 
32.16374 
0.00355 
Table 18: The values of stress (and deformation for PEFBFL and PPSFL
COMPOSITE 

(N/mm^{2}) 
Displacement (mm) 


CLT 
ANN 
%Error 
CLT 
ANN 
%Error 

PEFBFL 

40.1235 
40.1298 
1.23 
0.00428 
0.004151 
3.01 
PPSFL 

32.1637 
32.5589 
1.23 
0.00355 
0.00358 
0.85 
Acknowledgements
The supportive role played by Nnamdi Azikiwe University Awka in providing enabling environment for this research is highly acknowledged.
Appendix
Table 19: The ANN training data set and response for PEFBFL

Independent variables 
Response 1 
Response 2 

Exp No. 
Aspect Ratio 
Fiber Orientation 
Stacking Sequence 
Stress (N/mm^{2}) 
ANN Predicted 
Error 
Displacement (mm) 
ANN Predicted 
Error 
1 
1.05 
60 
(30/0/90)s 
37.13 
36.6834 
0.44660 
0.00361 
0.003614 
3.87E06 
2 
0.79 
30 
(30/0/90)s 
36 
36.1061 
0.10615 
0.00315 
0.003172 
2.18E05 
3 
1.05 
60 
(30/45/90)s 
40.13 
40.0192 
0.110736 
0.00423 
0.004121 
0.00010879 
4 
1.05 
60 
(30/45/90)s 
40.13 
40.0192 
0.110736 
0.00423 
0.004121 
0.00010879 
5 
1.05 
60 
(30/45/90)s 
40.13 
40.0192 
0.110736 
0.00423 
0.004121 
0.00010879 
6 
1.05 
60 
(30/0/90)s 
36.13 
36.6834 
0.5534 
0.00299 
0.003614 
0.0006239 
7 
0.68 
60 
(30/0/90)s 
34.5 
34.5005 
0.00053 
0.00259 
0.002606 
1.61E05 
8 
1.42 
60 
(30/45/90)s 
39.13 
40.0247 
0.89474 
0.00423 
0.004198 
3.15E05 
9 
1.31 
90 
(30/0/90)s 
38 
39.8442 
1.84425 
0.00423 
0.004191 
3.88E05 
10 
1.05 
90 
(30/45/90)s 
40.13 
40.1295 
0.000424 
0.00423 
0.004193 
3.68E05 
11 
1.31 
30 
(30/0/90)s 
34.8 
34.8637 
0.06373 
0.00315 
0.003485 
0.000335 
12 
1.05 
90 
(30/0/90)s 
36.6 
36.6394 
0.03941 
0.00353 
0.003546 
1.63E05 
13 
1.05 
60 
(30/0/90)s 
37.13 
36.6834 
0.446604 
0.00315 
0.003614 
0.000464 
14 
1.05 
60 
(30/0/90)s 
37.13 
36.6834 
0.446604 
0.00423 
0.003614 
0.00061613 
15 
0.79 
90 
(30/45/90)s 
40.13 
40.1281 
0.0019 
0.00423 
0.003945 
0.00028501 
16 
1.31 
90 
(30/45/90)s 
40.13 
40.1298 
0.000107 
0.00423 
0.004151 
7.92E05 
17 
1.05 
60 
(30/45/90)s 
38.13 
40.0193 
1.88926 
0.00423 
0.004121 
0.00010879 
18 
1.42 
60 
(30/0/90)s 
37.13 
37.1604 
0.03047 
0.00423 
0.004207 
2.35E05 
19 
0.79 
30 
(30/45/90)s 
38.8 
38.6298 
0.170174 
0.00315 
0.003135 
1.46E05 
20 
0.68 
60 
(30/45/90)s 
37.13 
37.1190 
0.010906 
0.00259 
0.002642 
5.22E05 
21 
1.05 
30 
(30/0/90)s 
35.8 
35.7035 
0.096432 
0.00323 
0.003332 
0.000102 
22 
0.79 
90 
(30/0/90)s 
35 
34.9949 
0.0051 
0.00315 
0.003176 
2.64E05 
23 
1.05 
60 
(30/45/90)s 
40 
40.0192 
0.01926 
0.00422 
0.004121 
9.88E05 
24 
1.05 
60 
(30/0/90)s 
37.13 
36.6834 
0.446604 
0.00323 
0.003614 
0.000384 
25 
1.31 
30 
(30/45/90)s 
38 
37.9603 
0.039629 
0.00323 
0.003255 
2.48E05 
26 
1.05 
30 
(30/45/90)s 
38 
38.4618 
0.46185 
0.00323 
0.003163 
6.69E05 
Σ 
3.45975 
Σ 
3.44E04 

MSE= 
0.46038 
MSE= 
4.5544E09 
Code value: (30/0/90)s =>1, (30/45/90)s=>2
Table 20: The ANN training data set and response for PPSFL
Independent variables 
Response 1 
Response 2 

Exp No. 
Aspect Ratio 
Fiber Orientation 
Stacking Sequence 
Stress (N/mm^{2}) 
ANN Predicted 
Error 
Displacement (mm) 
ANN Predicted 
Error 
1 
1.05 
60 
(30/0/90)s 
27.13 
28.44332 
1.31332 
0.00261 
0.002682 
7.23E05 
2 
0.79 
30 
(30/0/90)s 
26 
26.0954 
0.0954 
0.00215 
0.002161 
1.07E05 
3 
1.05 
60 
(30/45/90)s 
32.61 
31.83736 
0.772644 
0.00358 
0.003508 
7.15E05 
4 
1.05 
60 
(30/45/90)s 
32.61 
31.83736 
0.772644 
0.00358 
0.003508 
7.15E05 
5 
1.05 
60 
(30/45/90)s 
32.61 
31.83736 
0.772644 
0.00358 
0.003508 
7.15E05 
6 
1.05 
60 
(30/0/90)s 
28.13 
28.44332 
0.31332 
0.00299 
0.002682 
0.00030771 
7 
0.68 
60 
(30/0/90)s 
26.5 
26.18098 
0.319022 
0.00259 
0.002401 
0.00018924 
8 
1.42 
60 
(30/45/90)s 
29.13 
28.94753 
0.182469 
0.00358 
0.003572 
7.83E06 
9 
1.31 
90 
(30/0/90)s 
30 
30.00286 
0.00286 
0.00358 
0.003142 
0.00043795 
10 
1.05 
90 
(30/45/90)s 
32.61 
32.21414 
0.395862 
0.00358 
0.003573 
7.08E06 
11 
1.31 
30 
(30/0/90)s 
26.8 
26.78153 
0.018465 
0.00215 
0.002192 
4.22E05 
12 
1.05 
90 
(30/0/90)s 
28.6 
28.5736 
0.026401 
0.00253 
0.002878 
0.000348 
13 
1.05 
60 
(30/0/90)s 
29.13 
28.44332 
0.686676 
0.00215 
0.002682 
0.0005323 
14 
1.05 
60 
(30/0/90)s 
29.13 
28.44332 
0.686676 
0.00358 
0.002682 
0.00089771 
15 
0.79 
90 
(30/45/90)s 
29 
29.04575 
0.04575 
0.00258 
0.002687 
0.00011 
16 
1.31 
90 
(30/45/90)s 
32.61 
32.55896 
0.051038 
0.00358 
0.003578 
1.93E06 
17 
1.05 
60 
(30/45/90)s 
30.13 
31.83736 
1.70736 
0.00358 
0.003508 
7.15E05 
18 
1.42 
60 
(30/0/90)s 
29.13 
28.69715 
0.432848 
0.00358 
0.003407 
0.00017295 
19 
0.79 
30 
(30/45/90)s 
30.8 
30.7556 
0.044397 
0.00255 
0.002387 
0.00016264 
20 
0.68 
60 
(30/45/90)s 
29.13 
29.62188 
0.49188 
0.00259 
0.002551 
3.94E05 
21 
1.05 
30 
(30/0/90)s 
27.8 
27.62689 
0.173107 
0.00223 
0.002223 
6.65E06 
22 
0.79 
90 
(30/0/90)s 
26 
26.18772 
0.18772 
0.00215 
0.002291 
0.0001414 
23 
1.05 
60 
(30/45/90)s 
32 
31.83736 
0.162644 
0.00322 
0.003508 
0.0002885 
24 
1.05 
60 
(30/0/90)s 
29.13 
28.44332 
0.686676 
0.00223 
0.002682 
0.000452 
25 
1.31 
30 
(30/45/90)s 
30 
27.88602 
2.113975 
0.00223 
0.002957 
0.0007266 
26 
1.05 
30 
(30/45/90)s 
30 
30.01162 
0.01162 
0.00223 
0.002206 
2.38E05 
Σ 
4.128949 
Σ 
1.81E04 

MSE= 
0.655701 
MSE= 
1.2577E09 
Code value: (30/0/90)s =>1, (30/45/90)s=>2
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