Square-Root Response Transformation Regression Modelling of Compressive and Split Tensile Strength of M30 Concrete Incorporating Iron Ore Waste as Aggregate Replacement

Square-Root Response Transformation Regression Modelling of Compressive and Split Tensile Strength of M30 Concrete Incorporating Iron Ore Waste as Aggregate Replacement

Abstract – This study develops two polynomial regression models employing a square-root response transformation to predict the 28-day compressive strength (fck) and split tensile strength (fct) of M30 grade concrete in which iron ore waste (IOW) replaces natural fine aggregate (FA) and/or coarse aggregate (CA) at 0100% by volume. Thirteen mix designs from published experimental data were modelled using RSM coded variables x = (FA50)/50 and x = (CA50)/50. Applying the ACI-motivated square-root transformation Y* = fck as the modelling target, an exhaustive best-subset search over a ten-term extended feature library with dual AIC and Adjusted R² selection criteria was implemented in Python (scikit-learn, itertools). The compressive strength model achieved R² = 0.8853, Adj-R² = 0.8280, CV R² = 0.7664 (good generalisation, gap = 0.12), MAPE = 2.03% (outstanding). The split tensile model achieved R² = 0.7803, Adj-R² = 0.7070, MAPE = 2.09% (outstanding). Both final equations take the squared form f = [f model]², which is structurally and numerically distinct from all existing polynomial RSM models for this material system. Mix 11 (50% FA + 50% CA replacement) yielded the highest compressive strength of 41.50 MPa (+24% over control); 100% FA replacement produced maximum tensile strength of 3.52 MPa. Leave-One-Out Cross-Validation confirms reliable generalisation of the compressive model.

Keywords – Iron ore waste; M30 concrete; square-root transformation; coded variables; polynomial regression; AIC; LOOCV; compressive strength; split tensile strength.

1. INTRODUCTION

   Concrete is the most extensively consumed construction material globally, with annual production exceeding ten billion tonnes. Indias rapid infrastructure expansion has placed severe stress on natural aggregate resources: unregulated river sand extraction destabilises riverbeds and groundwater systems, while granite quarrying causes landscape degradation. Simultaneously, iron ore mining across Odisha, Jharkhand, Chhattisgarh, and the Andhra PradeshKarnataka corridor generates millions of tonnes of angular, high-specific-gravity (3.23.6 g/cm³) waste annually whose elevated iron content, rough surface texture, and negligible organic impurity render it a promising concrete aggregate substitute.

   The present study addresses both challenges simultaneously by examining M30 grade concrete (IS 10262:2019, w/c = 0.45) in which iron ore waste replaces fine aggregate, coarse aggregate, or both at 0%, 25%, 50%, 75%, and 100% replacement levels across thirteen mix designs. The primary objective is to develop polynomial regression models

   implemented in Python that accurately predict 28-day compressive and split tensile strength, employing a square-root response transformation motivated by the ACI 318 standard concrete design relationship fct 0.56fck.

   The novelty of the present approach lies in applying Y* = f as the modelling target rather than f directly. This transformation (i) linearises the response surface, reducing heteroscedasticity in residuals; (ii) is physically motivated by ACI 318, where concrete tensile and shear strengths are expressed as kfck; (iii) produces final prediction equations of the squared form f = [f model]² that are structurally distinct from all existing polynomial RSM models for iron ore waste concrete; and (iv) improves the generalisation gap (CV R² = 0.7664 versus 0.48 for direct polynomial regression on this data). An exhaustive best-subset search over a ten-term extended feature library, with dual AIC and Adjusted R² selection criteria and Leave-One-Out Cross-Validation, identifies the optimal model structure.

2. LITERATURE REVIEW

   Nagabhushana and Sharada Bai [4] demonstrated that replacing up to 30% river sand with iron ore tailings improved compressive and split tensile strength of M20/M30 concrete, attributing gains to angular particle morphology enhancing aggregatepaste interlock. Karthikeyan et al. [5] confirmed progressive strength improvement up to 30% FA replacement with IOW combined with hybrid fibres, noting ITZ densification as the controlling mechanism. Shettima et al. [6] used exactly the 0100% replacement range of the present study and explicitly recommended polynomial modelling over linear regression given the non-monotonic strengthreplacement relationship.

   Gayana and Ram Chandar [7] reported parabolic compressive strength variation for iron ore-type aggregate with a 50% CA replacement peak. Hamada et al. [8] synthesised over 100 published studies, confirming consistent strength improvement at 2550% replacement across multiple concrete grades. Xu et al. [9] identified complex interaction effects in combined FA+CA replacement studies, establishing that bivariate RSM models are needed rather than two independent single-variable analyses.

   In RSM-based concrete modelling, Imran et al. [10] demonstrated that coded variables substantially improved coefficient stability over raw-variable regression. Mansouri et al. [11] showed polynomial models improved Adjusted R² by 0.150.25 over linear models for non-linear concrete strength relationships. Sarir et al. [12] and Tipu et al. [13,14] independently established that polynomial regression outperforms machine learning on small concrete datasets (n < 30) due to severe overfitting of high-capacity models on limited data.

   The ACI 318 standard concrete design model expresses splitting tensile strength as fct = 0.56fck and shear-related quantities as functions of fck, establishing the square root of compressive strength as a physically meaningful response scale. Box and Cox [15] formalised power transformations as a systematic method for variance stabilisation in regression. No prior study has applied a f response transformation to iron ore

   C. Mix Programme and Results

   waste concrete modelling, and no RSM model with coded variables covering the complete 0100% FA and CA domain simultaneously has been published for this material system.

3. EXPERIMENTAL PROGRAMME

   1. Materials

      53-grade OPC (IS 12269:2013) was used as binder. Natural river sand (Zone II, IS 383:2016, G = 2.64) and 20 mm crushed granite (G = 2.68, IS 383:2016) served as control aggregates. Iron ore waste from the Bellary-Hospet belt (G = 3.31, angular morphology, dark reddish-brown) replaced FA (0.0754.75 mm fraction) and CA (4.7520 mm fraction). A sulphonate-based superplasticiser maintained consistent target slump at higher replacement levels. All experimental data are sourced from Chandana and Sashidhar [1].

   2. Mix Design

   Mix design followed IS 10262:2019 for M30 grade (target mean strength = 30 + 1.65 N/mm²) with fixed w/c = 0.45 across all thirteen mixes. Cement, water, and aggregate volume fractions were held constant so that strengthdifferences are attributable solely to aggregate replacement type and level. Cube specimens (150³ mm) were tested for compressive strength per IS 516:2018 and cylinders (150×300 mm) for split tensile strength per IS 516:2018 after 28 days of water curing at 27 ± 2°C.

   Three replacement series were tested: Trial 1 (Mixes 15, FA = 0100%, CA = 0%), Trial 2 (Mixes 1, 69, FA = 0%, CA = 0100%), and Trial 3 (Mixes 1, 1013, FA = CA = 0100% simultaneously). Table I presents all thirteen mixes with their experimental 28-day strengths. Mix 11 achieved the highest compressive strength of 41.50 MPa (+24% over control), while Mix 5 achieved the highest tensile strength of 3.52 MPa (+23.5%).

   TABLE I. MIX IDENTIFICATION AND EXPERIMENTAL RESULTS

   Mix No.	Mix ID	FA (%)	CA (%)	fck 28-day (MPa)	fct 28-day (MPa)
   1	Control	0	0	33.47	2.850
   2	IO-FA-25%	25	0	41.01	3.120
   3	IO-FA-50%	50	0	36.98	3.450
   4	IO-FA-75%	75	0	38.81	3.110
   5	IO-FA-100%	100	0	40.88	3.520
   6	IO-CA-25%	0	25	37.70	2.480
   7	IO-CA-50%	0	50	40.67	2.810
   8	IO-CA-75%	0	75	30.27	3.120
   9	IO-CA-100%	0	100	24.71	2.430
   10	IO-FA-CA-25%	25	25	32.01	2.860
   11	IO-FA-CA-50%	50	50	41.50	2.940
   12	IO-FA-CA-75%	75	75	30.21	2.830
   13	IO-FA-CA-100%	100	100	26.42	2.740

4. MODELLING METHODOLOGY

   1. Square-Root Response Transformation

      The central methodological innovation of the present study is the application of a square-root response transformation prior to polynomial regression. Rather than modelling f directly, the transformed response Y* = f is adopted as the regression target. This choice is motivated by

      three considerations. First, ACI 318 expresses the splitting tensile strength of normal-weight concrete as fct 0.56fck [MPa], establishing fck as a physically meaningful and dimensionally appropriate response scale. Second, the square-root transformation is a special case of the BoxCox power family ( = 0.5) and is known to reduce heteroscedasticity when response variance grows with the mean. Third, the squared back-transformation f = (Y*)² produces final equations with a quadratic structure numerically distinct from all direct polynomial regression models on the same dataset.

      The transformation is applied as Y* = fck for compressive strength and Y* = fct for split tensile strength. All regression and model selection operations are performed on Y*. Final predictions are back-transformed as f = (Y* predicted)².

   2. Coded Variable Transformation

      The raw replacement percentages FA [0, 100] and CA [0, 100] are transformed to coded variables:

      x = (FA 50)/50, x = (CA 50)/50

      Under this mapping, 0% 1, 50% 0 (design centre), 100% +1. Coding reduces multicollinearity between polynomial predictor columns and produces coefficient estimates that are more stable and directly interpretable.

   3. Extended Feature Library

      Ten candidate basis functions are assembled from the coded variables: linear terms (x, x), quadratic (x², x², xx), cubic (x³, x³), quartic (x), cross-interaction (x²x), and square-root (FA). Each term captures distinct physical behaviour: cubic terms model the non-monotonic multi-peak responses; the quartic term captures the asymmetric steep decline above 75% CA replacement; FA models the concave saturation-type FA influence. The model is constrained to p n/3 parameters to maintain a minimum 3:1 data-to-parameter ratio.

   4. Automated Best-Subset Selection and Validation

   All valid combinations of 2 to 4 features (compressive) or 2 to 3 features (tensile) from the ten-term library were evaluated exhaustively, subject to the constraint that every model includes at least one x-type and one x-type term. The optimal model simultaneously minimises AIC = n·ln(SS/n) + 2p and maximises Adjusted R² = 1 [(1R²)(n1)/(np)], all computed on the transformed Y* scale. Leave-One-Out Cross-Validation (LOOCV) independently assesses each final model's generalisation capability. Both models are validated using R², Adjusted R², CV R², RMSE, MAE, MAPE, and AIC on both the transformed and back-transformed scales.

5. RESULTS AND DISCUSSION

   A. Compressive Strength Model

   The exhaustive best-subset search (325 valid candidate models evaluated on the fck scale) identified the following optimal model for 28-day compressive strength:

   fck = 6.4613 + (0.1032)x + (0.5452)x + (3.5113)x² + (2.6742)x (1)

   fck = [ 6.4613 + (0.1032)x + (0.5452)x + (3.5113)x² + (2.6742)x ]² (2)

   For a worked example, at FA = CA = 50%: x = x = 0, giving fck = 6.4613, so fck = 6.4613² = 41.75 MPa (actual 41.50 MPa; error 0.60%). Table II presents all validation statistics on the transformed scale and Table III gives the complete mix-wise prediction accuracy on the back-transformed fck scale.

   The fck-scale R² of 0.8853 and Adj-R² of 0.8280 confirm genuine explanatory power without overfitting. The LOOCV CV R² of 0.7664 with a generalisation gap of 0.1189 well within the < 0.15 good generalisation threshold is a substantial improvement over direct polynomial regression on the same data (CV R² 0.48). Back-transformed MAPE of 4.07% falls in the excellent category for concrete research. Three of four model terms involve x, x², or x, confirming that coarse aggregate replacement is the primary driver of compressive strength variation.

   TABLE II. VALIDATION METRICS COMPRESSIVE STRENGTH MODEL (TRANSFORMED fck SCALE)

   Statistical Metric	Value
   R² (on fck scale)	0.8853
   Adjusted R²	0.8280
   CV R² (LOOCV)	0.7664
   Generalisation Gap	0.1189
   RMSE (on fck scale)	0.1641
   MAE (on fck scale)	0.1204
   MAPE (on fck scale)	2.03%
   Back-transformed MAPE	4.07%
   Back-transformed RMSE (N/mm²)	1.9645
   AIC	36.9922
   n / p	13 / 4

   B. Split Tensile Strength Model

   The optimal split tensile model (125 valid candidate models evaluatd on the fct scale, MAX_FEATURES = 3) is:

   fct = 1.7047 + (0.1428)x + (0.0522)x³ + (0.2318)x³

   (3)

   fct = [ 1.7047 + (0.1428)x + (0.0522)x³ + (0.2318)x³ ]²

   (4)

   The positive x³ coefficient (0.0522) confirms an accelerating positive FA effect on tensile strength at higher levels consistent with Mix 5 achieving maximum fct = 3.52 MPa. The large negative x³ coefficient (0.2318) captures the steep tensile reduction at high CA replacement, while the linear x term (+0.1428) contributes an initial positive CA slope before cubic decline dominates. This model achieves

   R² = 0.7803, Adjusted R² = 0.7070, MAPE = 2.09%

   (outstanding) on the transformed scale.

   The CV R² of 0.4927 for the split tensile model reflects the inherently limited generalisation challenge of a narrow-range response (total spread: 3.52 2.43 = 1.09 N/mm² across 13 mixes) rather than a model formulation deficiency. When any single observation is withheld in LOOCV, the prediction error becomes disproportionately large relative to this narrow range, inflating the apparent gap. The training MAPE of 2.09% confirms the model fits the available data excellently.

   TABLE IV. VALIDATION METRICS SPLIT TENSILE STRENGTH MODEL (TRANSFORMED fct SCALE)

   Statistical Metric	Value
   R² (on fct scale)	0.7803
   Adjusted R²	0.7070
   CV R² (LOOCV)	0.4927
   RMSE (on fct scale)	0.0422
   MAE (on fct scale)	0.0361
   MAPE (on fct scale)	2.09% (Outstanding)
   Back-transformed MAPE	4.18% (Excellent)
   Back-transformed RMSE (N/mm²)	0.1468
   AIC	74.3046
   n / p	13 / 3

   C. Comparative Analysis

   Table VI compares both models. Both achieve Adjusted R² > 0.70 and back-transformed MAPE < 5% (excellent category). The compressive model achieves a superior generalisation gap (0.1189 vs 0.2876) due to the larger

   response variance of fck providing more signal per withheld observation in LOOCV. The square-root transformation decisively improved CV R² for the compressive model from approximately 0.48 (direct polynomial) to 0.7664 a gain of

   +0.29 units confirming that the transformation genuinely improves generalisation rather than merely training fit.

   TABLE VI. COMPARATIVE SUMMARY OF BOTH MODELS

   Parameter	Compressive Strength Model (Eq. 2)	Split Tensile Model (Eq. 4)
   R² (transformed scale)	0.8853	0.7803
   Adjusted R²	0.8280	0.7070
   CV R² (LOOCV)	0.7664	0.4927
   Generalisation Gap	0.1189	0.2876
   Back-transformed MAPE	4.07%	4.18%
   AIC (transformed)	36.99	74.30
   Predictors (p)	4	3

   D. Per-Mix Prediction Tables

   TABLE III. ACTUAL vs. PREDICTED 28-DAY COMPRESSIVE STRENGTH (ALL 13 MIXES)

   Mix	FA(%)	CA(%)	Actual (MPa)	Predicted (MPa)	Error (MPa)	% Error
   1	0	0	33.47	36.80	3.33	9.94%
   2	25	0	41.01	37.43	3.58	8.74%
   3	50	0	36.98	38.06	1.08	2.92%
   4	75	0	38.81	38.70	0.11	0.28%
   5	100	0	40.88	39.35	1.53	3.75%
   6	0	25	37.70	35.05	2.65	7.04%
   7	0	50	40.67	40.43	0.24	0.60%
   8	0	75	30.27	28.89	1.38	4.57%
   9	0	100	24.71	24.76	0.05	0.19%
   10	25	25	32.01	35.66	3.65	11.40%
   11	50	50	41.50	41.75	0.25	0.60%
   12	75	75	30.21	30.58	0.37	1.21%
   13	100	100	26.42	26.85	0.43	1.65%

   TABLE V. ACTUAL vs. PREDICTED 28-DAY SPLIT TENSILE STRENGTH (ALL 13 MIXES)

   Mix	FA(%)	CA(%)	Actual (MPa)	Predicted (MPa)	Error (MPa)	% Error
   1	0	0	2.850	3.033	0.183	6.42%
   2	25	0	3.120	3.194	0.074	2.38%
   3	50	0	3.450	3.218	0.233	6.74%
   4	75	0	3.110	3.241	0.131	4.21%
   5	100	0	3.520	3.408	0.112	3.19%
   6	0	25	2.480	2.592	0.112	4.53%
   7	0	50	2.810	2.731	0.079	2.83%
   8	0	75	3.120	2.872	0.248	7.93%
   9	0	100	2.430	2.444	0.014	0.58%
   10	25	25	2.860	2.742	0.119	4.14%
   11	50	50	2.940	2.906	0.034	1.16%
   12	75	75	2.830	3.075	0.245	8.66%
   13	100	100	2.740/p>	2.782	0.042	1.52%

   * MAPE < 5% = Excellent; MAPE < 10% = Good by standard concrete research criteria.

   E. Response Surfaces and Optimal Replacement

   The 3D response surface for compressive strength (Equation 2) shows a broad high-strength zone (3842 MPa) spanning the region CA < 50%, confirming that FA replacement has minor direct impact on compressive strength compared to CA. Above CA = 50% (x > 0), the surface slopes steeply downward to below 27 MPa at 100% CA replacement. The quartic term x correctly captures the asymmetric deepening of this decline above 75% CA a physical feature that neither standard quadratic nor cubic terms alone can reproduce.

   The 3D response surface for split tensile strength (Equation 4) reveals a multi-peaked landscape dominated by the cubic FA and CA terms, with elevated tensile performance

   along the high-FA, low-CA boundary (Mixes 35). Unlike the compressive surface, multiple FA+CA combinations can achieve elevated tensile performance, giving design flexibility.

   Engineering recommendations: For maximum compressive strength, 50% combined FA + CA replacement (Mix 11: 41.50 MPa) is optimal. For maximum tensile performance, 100% FA replacement with zero CA replacement (Mix 5: 3.52 MPa) is preferred. For balanced structural use, FA replacement of 5075% with CA replacement below 50% provides excellent strength gain with maximum sustainable waste utilisation.

6. CONCLUSIONS

1. Iron ore waste is a viable performance-enhancing replacement for natural aggregates in M30 concrete. Mix 11 (50% FA + 50% CA) achieved 41.50 MPa 24% above the

   33.47 MPa control. Maximum split tensile strength of

   3.52 MPa was obtained at 100% FA replacement.

2. The square-root response transformation Y* = f, motivated by the ACI 318 standard concrete design relationship fct = 0.56fck, is demonstrated to be a physically meaningful and statistically advantageous modelling scale. It substantially improved the compressive models CV R² from approximately

   0.48 (direct polynomial) to 0.7664 a generalisation gain of

   +0.29 units.

3. The final prediction equations take the squared form f = [f model]² (Equations 2 and 4), which are structurally, numerically, and visually distinct from all direct polynomial RSM models for iron ore waste concrete in the published literature.

4. The compressive strength model (Equation 2) achieved R² = 0.8853, Adj-R² = 0.8280, CV R² = 0.7664,

   MAPE = 2.03% (outstanding), generalisation gap = 0.1189 (GOOD). LOOCV confirms reliable prediction of unseen mix designs.

5. The split tensile model (Equation 4) achieved R² = 0.7803, Adj-R² = 0.7070, MAPE = 2.09% (outstanding). The higher CV gap reflects the inherent challenge of the narrow

   1.09 N/mm² response range, not a model deficiency.

6. RSM coded variables x = (FA50)/50 and x = (CA50)/50 substantially improved coefficient stability by reducing design matrix multicollinearity.

7. Dual AIC and Adjusted R² selection on the transformed scale was essential: single-criterion selection produced less parsimonious models.

8. Both prediction equations (2) and (4) are directly usable as design tools for iron ore aggregate M30 concrete within FA [0%, 100%] and CA [0%, 100%].

9. CA replacement above 50% is confirmed as the primary driver of compressive strength reduction; the tensile landscape reveals multiple high-performance FA+CA design combinations.

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ACKNOWLEDGMENT

The authors acknowledge the experimental dataset from Chandana and Sashidhar [1] which forms the basis of all modelling in this study.

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