ANOVA
ANOVA results and significance of factors
The assessment of the effect of the various core architectures and key FDM printing parameters on mechanical performance of 3D-printed PLA specimen is the main aim of this study. There are five dependent variables and four independent variables; maximum load, ultimate tensile strength (UTS), youngs modulus, tensile stress at the break, and tensile strain at a break is determined against four independent variables; core type, layer height, print speed, and raster angle.
Prior to performing the analysis of variance, the assumptions of normality and homogeneity of variance were checked. The Shapiro-Wilk test was performed to check the normal distribution of response variables, and Levene’s test was used to check the homoscedasticity. The results (Table 12) indicated that Young’s modulus followed a normal distribution (p > 0.05), whereas maximum load, UTS, tensile stress at break, and tensile strain at break showed slight deviations from normality (p < 0.05). However, Levene’s test confirmed that most of the response variables met the homogeneity of variance assumption. For Tensile Stress at break, the statistical test of Levene suggested that the variance of the data is unequal (p = 0.005), with increased variance in HASV specimens with 00 raster angle. Given that group sizes are equal (n = 3 per group) and the model F-ratio is large, this isolated violation is not expected to impact on the validity of the conclusions of the ANOVA for this response variable. Since for balanced experimental designs, mainly for the comparison of means, the statistical analysis is believed to be valid because usually the classical statistical analysis is robust to deviations of the normality condition.
JMP Pro software was used to perform one- way analysis of variance (ANOVA) with the purpose to assess the importance of each of the factors in the experiments. Statistical significance was tested at a confidence level of 95%, which corresponds to a significance level of α = 0.05. The p-values of all the response variables of the models were less than 0.0001 (Table 13), which would mean that the level of statistical significance is very high and one could reject the null hypothesis in each case. R2 (adjusted) values were between 0.71 and 0.93, which show that the models explained the majority of the observed variance, with one exception (tensile stress at break, which is lower, R2 = 0.71 yet is acceptable to interpret). The F-ratios ranged between 12.18 and 62.04, which depicts a significant effect of factors especially in tensile strength and strain (Table 16).
Core structure and raster angle were found to be the most important as the responses showed consequential F-ratios and effect sizes in all (p < 0.001). Print speed also proved to be statistically reliable in the majority of responses; however, it did not have a significant effect as core type and raster angle. Layer height was not significantly different (p ≈ 1.0), with close to zero coefficients- possibly because of cross-effects with other factors or failure to make enough independent variation in the test matrix. The raster angle with 00 produced the highest constants in terms of stiffness and strength, and 900 and 450 produced negative results thereby confirming anisotropy of FDM printed structures.
The percentage contribution of each factor was calculated as (SS_factor /SS_total )×100 using Type III partial sums of squares obtained from the JMP Pro effect tests output, in order to quantify the relative influence of each process parameter on the mechanical response variables. Following ANOVA, Tukey’s Honest Significant Difference (HSD) post-hoc test (α = 0.05) testing was performed in JMP Pro of all statistically significant factors: Core structure, Raster angle and Print speed—to identify specific pairwise differences between factor levels. Layer height was not included for post-hoc analysis since it did not have a statistically significant effect for any of the five responses (p ≈ 1.0).
The Tukey HSD post-hoc test results found in Table 14 show statistically supported pairwise differences between factor levels for all five of the mechanical responses. For Core Structure, HASV was consistently in class A which turned out to be the ones with the highest values for Maximum Load, UTS, and Young’s modulus, while they ranked last for ductility (Tensile Strain for Break, class C), finding a statistically significant tradeoff between strength and ductility. ASVC and ASR came out to be the most ductile cores (group A for Tensile Strain) and this is consistent with the Taguchi S/N analysis. For Raster Angle, 00 was significantly better than both 450 and 900 for all strength related responses (p < 0.001), however no significant difference was found between raster angle levels for Tensile Stress at Break (all group A, p > 0.05) corroborating the evidence that core geometry, rather than print direction, controls local stress distribution at fracture. For Print Speed, the higher speeds (65 and 70 mm/s) showed significantly higher strength and the 60 mm/s showed significantly higher ductility, so there is a trade-off between the speed of processing and the elongation at break.
These inferences are supported by visual boxplot analyses (Table 15), which indicate that differences between the means of the various levels of factors are notable in the case of core type and raster angle. The HASV cores with 00 raster orientation always produced the best maximum load, UTS and modulus. Layer height in its turn yielded overlapping distributions and insignificant effect. According to model diagnostics, there is a significant lack-of-fit in terms of some of the cases considered but the global pure error ratio is low and there is no clear graphical trend in terms of the residual plots. There is random distribution of the studentized residuals, which validates hypotheses of normality and homoscedasticity. These conditions justify the models in terms of main-effect analysis and contingencies.
As indicated in the current study, such optimization of core geometry and raster angle coupled with a calibrated print speed significantly affects the mechanical properties of poly lactic acid (PLA) FDM samples. The results show that in the case of the current design, the HASV core geometry with 00 raster angle and a print velocity of ~ 65 mm/s will produce the best tensile properties. Layer height did not have a discernible effect under current conditions, but could be varied in future, as an independent parameter, to help understand interactions.
Table 16 presents the ANOVA summary for each mechanical property as obtained from JMP Pro software. ‘MODEL’ refers to the variability explained by the regression model including all factors, ‘ERROR’ indicates residual variance, and ‘TOTAL’ is their sum. F and P-values are as reported in the output. All models are highly significant (p < 0.0001).
A further quantification of the relative influence of each factor on the mechanical responses is provided in the percentage contribution analysis (Table 17). Raster Angle is found to be the leading contributor for Maximum Load (53.59%), UTS (55.58%), Young’s modulus (49.35%), and Tensile Strain at Break (50.18%) which confirms the leading role filament deposition direction plays in tensile performance of FDM printed auxetic structures. Core Structure accounts for the major contribution (30.01%) to Tensile Stress at break because of the role of geometric architecture in the control of the local stress distribution at the fracture. Print Speed shows an average but steady contribution on all responses (1.94–4.93%), while Layer height shows 0% contribution due to design singularity in L18 mixed orthogonal array. The low error contributions (7.31–28.64%) prove good model fitting and experimental reproducibility.
Optimization using a least square fit method
JMP Pro software was used to identify the fit model tool to determine the optimum values. The initial stage of this process was defining the input parameter (type of core, the height of the layer, print speed, and raster angle) as an independent parameter and output parameters (maximum load, UTS, young’s modulus, tensile strain, tensile stress) as dependent parameters. The model was established as the Standard Least Squares, which means that the application did an ordinary least squares regression.
Figure 14 shows the scatter of dots plotting position in relation to actual and predicted values of each of the mechanical properties. The red lines are in form of data representing best-fit regression lines. The lines depict the anticipated values of the given actual values of the model, and the red-shaded parts around the regression lines are the 95% confidence intervals. The vertical differences between every data point and the regression line are considered the residuals between the observed and the predicted data. The scatter plots indicate that the relationship between the observations and respective regression lines is relatively close, and hence the goodness of fit for all the mechanical properties. The R-squared values show that about 91% variation in maximum load (R2 =0.91), 92% in UTS (R2 =0.92), 90% in Youngs modulus (R2 =0.90), 71% in tensile stress at break (R2 =0.71), 93% in tensile strain at break (R2 =0.93) is explained by the model. These large R-squared measures underline the excellent correlation between the expected and the observed values.
The very low P-values (< 0.0001) indicate that all regression models are statistically significant. The Root Mean Square Error (RMSE) values estimate the average prediction errors, with RMSE = 71.177 for maximum load, 0.852 for UTS, 46.371 for Young’s modulus, 1.585 for tensile stress, and 1.173 for tensile strain. Lower RMSE values indicate better model accuracy. As indicated by the P-values metric that is lower than 0.0001 in all regression models, the present study shows that this kind of the statistical analysis is appropriate. The values obtained by Root Mean Square Error (RMSE), which show average errors of the prediction, lead to RMSE = 71.177 as the maximum load, 0.852 UTS, 46.371 young’s modulus, 1.585 tensile stress, and 1.173 tensile strain. The smaller the RMSE the more accurate the model.
Figure 15 illustrates the prediction profiler plot that was essential in the identification of best combination of core type, layer height, print speed and raster angle that maximizes desirability with respect to predetermined mechanical property requirements. According to the graph, there is one parameter set that gives an optimal overall performance. The best combination of the optimized output shown in Table 15 shows that combination of HASV core structure, 0.2 mm layer height, 65 mm/s Print speed and 00 raster angle provide the best balance reaching maximum loads of 1241.264 N, ultimate tensile strength (UTS) of 16.11 MPa, Young’s modulus of 963.936 MPa, tensile stress of 8.338 MPa and tensile strain of 6.816%. These findings also provide more comprehensive advice on future work aimed at ensuring utmost-performance of 3D-printed PLA components.
Prediction profiler plot for various mechanical properties.
Prediction profiler plot.
The optimized results are illustrated in Table 18, which offers practical insights into potential future applications.
Layer height (0.2 mm) is listed for practical fabrication reference only. ANOVA indicated that it was statistically indifference for total of five response (p = 1.0), no optimal level is formally designated.
Discussion on optimized settings
Design parameters
The HASV auxetic core was confirmed to be the most optimal core as it experienced the best internal support, load distribution, and energy absorption which led to the highest load carrying capacity, UTS and modulus values in comparison to all the tested geometries. Deposition of the filaments along the tensile loading axis at the 00 raster angle allowed maximum stiffness and strength due to optimal load transfer and reduced stress concentration, which verified the anisotropic nature of FDM parts.
Process parameters
The empirical evidence showed moderate print speed (65 mm/s) provided the best trade between the quality of the deposition and manufacturing rate. Speeds outside this range have caused a negative impact on interlayer bonding and the tensile strength and print speeds lower than the limit created minimal enhancements in performance. Since layer height was found to be statistically insignificant (p = 1.0), no optimal level is formally designated. The value of 0.2 mm is noted for practical fabrication consistency only.
Anisotropy and multi-objective performance
This overwhelming performance advantage of the 00 orientation compared to both 450 and 900 showed deep directionality in performance of printed parts. Optimized parameter set (HASV, 00, 65 mm/s, 0.20 mm) reached an estimated ultimate load of ~ 1241 N, UTS of 16.1 MPa, and elastic modulus of approximately 964 MPa, tensile stress at break of 8.3 MPa and strain at break of 6.8%, a reasonably good compromise between toughness and strength. Cohesion with Taguchi methodology and a desirability of 0.722 shows that this is a robust combination to improve the mechanical performance without losing manufacturability and as such a combination is desirable in high load functional requirements.
The results demonstrate a clear directional dependence in the mechanical performance of the printed specimens. The 0° raster orientation produced higher strength and stiffness compared with the 45° and 90° orientations, indicating the influence of filament alignment relative to the loading direction. The optimized parameter combination (HASV core structure, 0° raster angle, 65 mm/s print speed, and 0.20 mm layer height) yielded a predicted maximum load of approximately 1241 N, UTS of 16.1 MPa, and Young’s modulus of about 964 MPa, with tensile stress at break of 8.3 MPa and strain at break of 6.8%. These results indicate a balanced mechanical response between strength and ductility. The Taguchi multi-response optimization produced a desirability value of 0.722, indicating that this parameter combination improves tensile performance while maintaining practical manufacturability for load-bearing PLA components.
The results show an obvious direction dependence in mechanical performance of the printed specimens. The 0° raster orientation resulted in higher strength and stiffness than the 45° and 90° orientations with the influence of the alignment of the filaments to the loading direction.The best combination of parameters (HASV core structure, 0 degrees raster angle, 65 mm/s print speed, and 0.20 mm layer height) obtained the following results: The predicted values of maximum load, UTS value and Young’s modulus value were about 1241 N, 16.1 MPa and 964 MPa, respectively; the tensile stress at break value and the strain at break value were 8.3 MPa and 6.8%, respectively.These results indicate a balanced mechanical response between the strength and the ductility. The Taguchi multi response optimization resulted in a desirability value of 0.722, which means that this combination of parameters enhances the tensile performance while ensuring practical manufacturability in load bearing PLA components.
Relation to Taguchi Methodology and Research Significance
These optimized values align with the principles of the Taguchi design of experiments, wherein the highest signal-to-noise (S/N) ratios are used to identify robust and influential factors. The pronounced effects of the HASV core structure and 0° raster angle underscore the critical role of structural and directional parameters in additive manufacturing. The overall desirability score of 0.722 indicates a strong effectiveness and balance of the optimized parameter combination across multiple mechanical properties. This parameter combination optimizes strength while maintaining adequate ductility, making it suitable for structural applications requiring high load-bearing capacity. Presenting these findings demonstrates how a combination of well-planned experimental design and statistical optimization can effectively guide engineers toward process conditions that maximize functional performance while ensuring feasible manufacturing.
Mechanical properties of 3D-printed PLA specimens with different core structures and raster angles. (a) Maximum Load (N); (b) Ultimate Tensile Strength (MPa); (c) Young’s Modulus (MPa); (d) Tensile stress at Break (MPa); (e) Tensile strain at Break (%).
The mechanical performance of 3D-printed PLA specimens varies significantly across different core structures and raster angles, as illustrated in Fig. 16. The HASV core structure at 0° raster angle consistently demonstrates superior mechanical properties, exhibiting the highest maximum load (approximately 1100 N), ultimate tensile strength (approximately 15 MPa), and Young’s modulus (approximately 950 MPa). This finding aligns with our Taguchi analysis, which identified HASV core and 0° raster angle as optimal parameters. In contrast to some previous studies3 that found 90° raster angles optimal for certain PLA applications, our results show that the 0° raster angle consistently outperforms 45° and 90° orientations across most core structures, especially for load-bearing capacity and tensile strength. This discrepancy may be attributed to the unique infill patterns of our core structures and the specific grade of PLA used. The effect of raster angle on mechanical properties can be explained by load alignment with the printed filaments; when the tensile load direction coincides with the filament orientation (0°), the specimen exhibits higher strength and stiffness, as noted by Hossain et al.4 and Masood et al.5. However, the tensile strain at break shows a different pattern, with certain configurations like AS at 0° demonstrating higher ductility despite not having the highest strength. This inverse relationship between strength and ductility is consistent with findings from Chacon et al.5, suggesting that optimizing for one mechanical property may require compromises in others. These empirical findings validate both our ANOVA results identifying core structure and raster angle as the most significant factors (p < 0.0001) and provide the foundation for the subsequent ANN model development.
