Evaluation of prediction model and machine learning methods on the shrinkage of self-compacting concrete

A_c

cross-sectional area of concrete specimen

C

number of regression trees

cm

cementitious material in the prediction formulae

F

space of regression tree

F(x)

gradient boosting (GB) model

f_cm28

28-day compressive strength

f_cm28,T

temperature-adjusted 28-day compressive strength

f_j

fraction of value xj

f_k

independent tree structure and leaf weights

$f^RFKx$

linear random forest (RF) model

h

notional size

h_T

temperature-adjusted notional size

h(x)

base learner or weak learner

K

number of classes

k

number of trees

k_h

humidity factor for shrinkage

k_h,T

temperature-adjusted humidity factor for shrinkage

L(y, F(x))

loss function iteration

l

differentiable convex loss function

p_i

fraction or probability of samples belonging to class i

q

tree structure

S

entropy

T

temperature (°C)

T_cur

curing temperature

T_l

number of leaves

t

current age of concrete

t_c

age at exposure

U

cross-section perimeter of concrete specimen

w

leaf weight

x

sample

y_i

target value

$y∼i$

pseudo-response

$y^i$

predicted value

β_as(t)

time function of autogenous shrinkage

β_s(t − t_c)

time function for drying shrinkage

γ, λ

constants to control the degree of regularisation

ε_au(t,t_c)

autogenous shrinkage strain at t days

ε_au(t_c)

autogenous shrinkage strain at t_c days

ε_au0(f_cm28)

notional autogenous shrinkage coefficient

ε_sh(t,t_c)

drying shrinkage strain

ε_shu

ultimate drying shrinkage strain

ρ

learning rate

Ω(A)

set of categorical values of attribute A

Shrinkage is the reduction in concrete volume owing to moisture loss in the absence of external stress. When concrete is exposed to ambient temperature and humidity, shrinkage strain can lead to internal cracking, which negatively affects dimensional stability and long-term durability (Abdalhmid et al., 2019; Mehta and Monteiro, 2006). This effect is particularly pronounced in mixtures with a high content of cementitious material and a low content of coarse aggregate – characteristics typical of self-compacting concrete (SCC) (Güneyisi et al., 2010; Leemann and Hoffmann, 2005; Leemann et al., 2011; Rozière et al., 2007). To address these issues, several models have been developed (Aashto, 2017; ACI, 1992, 2008; Baweja and Bažant, 1995; Bažant and Wan-Wendner, 2015; CEB, 1991; fib, 2013; Gardner and Lockman, 2001; JSCE, 2007). Additionally, several models have been proposed to predict the shrinkage of SCC, such as the modified JSCE model (Aslani and Maia, 2013), the modified CEB90 model (Poppe and Schutter, 2005) and the modified Aashto model for the shrinkage of steam-cured SCC (Khayat and Long, 2010). However, these SCC-specific models still exhibit limited accuracy in predicting shrinkage, largely owing to the constraints and variability of the experimental data on which they are based.

Estimating shrinkage behaviour using machine learning (ML) offers significant advantages for long-term predictions. As a result, several ML methods have been applied in studies of concrete structure deformation. ML involves enabling computers to modify or adapt their actions to become more accurate over time (Kristiawan and Aditya, 2015). In other words, ML allows software applications to improve predictive accuracy without being explicitly programmed, relying instead on the design of efficient and accurate prediction algorithms. These learning techniques are data-driven, combining fundamental concepts from computer science with ideas from statistics, probability and optimisation (Mohri et al., 2018). ML methods can be broadly categorised into supervised learning, unsupervised learning and reinforcement learning (Marsland, 2015; Russell, 2018). For regression tasks – a supervised learning technique – various models have been applied to predict the shrinkage of SCC, including decision trees (DTs), the random forest (RF), extra trees (ET), gradient boosting (GB) and extreme gradient boosting (XGBoost).

• DT algorithms include ID3 and C4.5, introduced by Quinlan (1986, 1993), and Cart (classification and regression tree) (Marsland, 2015). ID3 (Iterative Dichotomiser 3), developed from Hunt's concept learning system framework, was the predecessor to C4.5, which generates a classifier in the form of a DT algorithm.

• The RF, originally developed by Ho (1995) and significantly extended by Breiman (2001), is an ensemble tree-based method developed to address the limitations of Cart (Ahmad et al., 2018).

• The ET algorithm, reported by Geurts et al. (2006), is a relatively recent ML technique designed as an extension of the RF, offering a lower risk of overfitting (Ahmad et al., 2018).

• GB, developed by Friedman (2001), is another ensemble ML technique employed for regression and classification problems based on a boosting system.

• The XGBoost algorithm, proposed by Chen and Guestrin (2016), describes a scalable, end-to-end tree boosting system.

In recent years, a few studies have applied ML algorithms to model the time-dependent deformation behaviour of concrete. Bal and Buyle-Bodin (2013) used artificial neural networks to predict shrinkage, while Karthikeyan et al. (2008) used the same algorithm for modelling the time-dependent deformation of high-performance concrete.

In this work, a new SCC prediction model – named fib-SCC – was developed by calibrating the fib Model Code for Concrete Structures (MC2010) model (fib, 2013). The objective was to create a model that accurately fitted an expanded SCC laboratory database with minimum error. The performance of the fib-SCC model was compared against established models, including the fib MC2010 model (fib, 2013), the B4 model (Bažant and Wan-Wendner, 2015), the ACI model (ACI, 1992) and several other SCC-specific models. The F-SCC model was further validated through experimental data involving mixtures with varying fly ash (FA) contents (20–55% by weight of cement). Furthermore, the efficiency of the fib-SCC model was also compared with the previously listed ML techniques.

DTs are one of the most important methods of supervised learning and are used widely for classification, which uses attribute values to partition the decision space into smaller subspaces in an iterative manner. A DT has a flowchart-like tree structure, starting with a root node and gradually growing to a final classification (or leaf) (Maimon and Rokach, 2005; Yang, 2019). The DT can be drawn in different ways, but they are almost all variants of the same principle – algorithms build the tree in a greedy manner, starting at the root and choosing the most informative feature at each step.

The split information of an attribute is measured by the entropy (S) as follows:

The Gini index (GI) can be expressed as:

The information gain is defined as:

In Equations 1–3, p_i is the fraction or probability of samples belonging to class i, K is the number of classes, Ω(A) is the set of categorical values of attribute A, f_j is the fraction of value x_j ∈ A with j = 1, 2,…, K and S_xj is entropy, which has a similar to formula to the entropy S (Yang, 2019).

The RF adds a randomness algorithm to the model while growing the trees. The ‘random’ part of the name RF refers to the fact that each tree uses a random portion of the original data as training. Instead of searching for the most important feature based on node splitting, the method creates the best feature among a random subset of features for viable split points at each node in the trees produced (Watt et al., 2020).

The RF regression can be presented as (Ahmad et al., 2018; Ao et al., 2019):

where $f^RFCx$ is the linear RF model, x is the sample and C is the number of regression trees.

The ET algorithm builds totally randomised trees whose structures are independent of the output values of the learning sample (Geurts et al., 2006) and chooses optimal cut-points, which are responsible for a large proportion of the variance of the induced tree (Berrouachedi et al., 2019). The ET method is quite similar to a RF; it only differs in constructing the DT.

GB was derived by Friedman (2001) and is an ensemble ML technique employed for regression and classification problems based on a boosting system. The principal boosting algorithm is AdaBoost (Marsland, 2015), which was first proposed by Freund and Schapire (1996), who reported that boosting can significantly reduce the error of any ‘weak learner’ algorithms that consistently generate classifiers that are only slightly better than random predictions. When a weak learner can be implemented efficiently, boosting provides a tool for aggregating weak hypotheses to approximate a gradually better prediction for larger information (Shalev-Shwartz and Ben-David, 2014). Therefore, GB can produce a competitive, highly robust and interpretable statistical process in many areas of ML. For a regression problem, the GB algorithm can be improved as follows (Friedman, 2001).

In these equations, F(x) is the GB model, m is an iteration, ρ is the learning rate, h(x) is a base learner or weak learner, L( y, F(x)) is the loss function and $y∼i$ is the pseudo-response (i = N ).

The XGBoost algorithm, proposed by Chen and Guestrin (2016), describes a scalable end-to-end tree boosting system. The method is similar to GB, as it builds an additive expansion of the objective function by minimising the loss function. However, XGBoost, which is known as one of the best-performing algorithms for supervised learning, focuses only on DTs as base classifiers, and a variation of the loss function is used to control the complexity of the trees (Bentéjac et al., 2021).

The output function of the XGBoost algorithm can be expressed as:

The loss function is calculated as:

Regularisation of the model to avoid overfitting is expressed as:

where $y^i$ is the predicted value; y_i is the target value; k is the number of trees; f_k corresponds to an independent tree structure and leaf weights; q is the tree structure; w is the leaf weight; T_l is the number of leaves; F is the space of regression tree; l is a differentiable convex loss function, and γ and λ are constants to control the degree of regularisation (Chen and Guestrin, 2016).

The materials used included Portland cement (PC), FA, fine aggregate (river sand), coarse aggregate (limestone) and superplasticiser (SP), all supplied by local companies in Vietnam. The PC was blended with FA to form a cementitious material. The cement (PC40) conformed to type I (ASTM, 2019a), while the FA (obtained from a coal-fired thermal plant) satisfied the requirements for type F (ASTM, 2019b). The river sand and limestone (used as fine and coarse aggregate) met the requirements of ASTM C1611 (ASTM, 2018a). The main physical properties of the aggregates are summarised in Table 1. The liquid SP was Sika Viscocrete SKY 8713, which was used as a high-range water-reducing agent with a specific gravity of 1.15 g/cm³.

In fib MC2010 (fib, 2013), the 28-day compressive strength is considered a key parameter for predicting concrete shrinkage. To investigate the influence of compressive strength on shrinkage behaviour, the mix proportions of the SCC were designed to achieve various strength grades based on the method proposed by Okamura and Ouchi (2003). The target strength grade was varied from 30 MPa to 70 MPa, encompassing both normal- and high-strength concretes (ACI, 1998). All the mixtures incorporated FA as a mineral admixture, ranging from 20% to 55% by weight of the cementitious material. Five mixes were developed, with the mix proportions presented in Table 2.

The properties of the fresh SCC mixes were evaluated based on ASTM C33 (ASTM, 2018b) using the slump flow and the flow time at 500 mm spread (T₅₀). Concrete specimens were cast without rodding or tapping the sides of the moulds (ASTM, 2015). A plastic sheet was placed over the top surface of all specimens to minimise moisture loss owing to evaporation.

The compressive strengths were evaluated by testing 150 mm × 150 mm × 150 mm cube specimens according to TCVN 3118-2022 (VSQI, 2022). The specimens were demoulded after 24 h and then cured in a moisture room at a temperature of 23 ± 2°C and relative humidity (RH) above 95% until the designated test age.

Both autogenous shrinkage and total shrinkage strains were manually measured using a length comparator, satisfying ASTM C157 (ASTM, 2017). Shrinkage tests were conducted on prism specimens measuring 75 × 75 × 285 mm. After demoulding at 24 h, the specimens for autogenous shrinkage tests were sealed with plastic sheets to prevent water loss. The initial length of the specimens was measured at this point using a comparator. All specimens were then placed in a drying chamber maintained at a temperature of 23 ± 2°C and RH of 60 ± 4%. Shrinkage measurements were recorded at predefined intervals according to the designated schedule.

The slump flow values of all the mixes varied significantly, from 650 to 760 mm, while the flow time (T₅₀) was 4–8 s, as shown in Table 2. Based on visual inspection, there was no segregation of the SCCs in the fresh state. The increased flowability can be attributed to the spherical and glassy structure of the FA particles (El-Chabib and Syed, 2013; Khatib, 2008). The SP dosage also played a critical role in reducing the water-to-cementitious materials (w/cm) ratio and ensuring the slump flow met the requirements for SCC.

The compressive strengths of the mixes are also provided in Table 2. The results indicate that FA enhanced the strength of the concrete, particularly at later ages, owing to its pozzolanic reactivity. Compared with the 28-day compressive strength, the 180-day compressive strength increased by 13% to 39%, depending on the FA content. These results demonstrate that a higher FA content contributes to greater long-term strength development in SCC. Similar findings were reported by Jawahar et al. (2013).

As shown in Figure 1(a), the autogenous shrinkage of the specimens varied significantly depending on the FA content and the w/cm ratio. M30 mix, which had the highest FA content (55% by weight of cement) and a w/cm ratio of 0.31, exhibited the lowest autogenous shrinkage. In contrast, M70 mix (with the lowest FA content (20%) and a w/cm ratio of 0.28) exhibited the highest autogenous shrinkage. These results indicate that increasing the replacement level of FA reduces autogenous shrinkage, consistent with the findings of Chan et al. (1998). The observed increase in autogenous shrinkage may be attributed to the pozzolanic reactivity of both FA and cement at later ages. Consequently, the consumption of water in the micropores of the concrete continued through the hydration reaction of the cement (Tazawa, 1999). Additionally, the autogenous shrinkage values were reduced with a decrease in cement content, which was partially replaced by FA (Lee et al., 2003). Autogenous shrinkage also may be influenced by the presence of SP, as noted by Esping and Löfgren (2006).

Figure 1(b) shows the development of total shrinkage over the test period of up to 150 days. According to the experimental results, mixes M30 and M40 exhibited the lowest total shrinkage during the early stage (up to 28 days). M70 demonstrated the highest total shrinkage of all the mixes. These outcomes indicate that the combined effects of FA content, w/cm ratio and SP dosage significantly influence the total shrinkage of SCC. Şahmaran et al. (2007) also reported a decrease in the total shrinkage of SCC with an increase in FA.

Total shrinkage and autogenous shrinkage are recorded in unsealed and sealed specimens, respectively. For sealed specimens, moisture evaporation is prevented, allowing shrinkage to occur primarily owing to the self-desiccation process within the pores of the hydrated cement. In contrast, the shrinkage in the unsealed specimens is mainly owing to water extraction from the hardened concrete, especially from voids smaller than 50 nm in diameter (Minless et al., 2002). The complied SCC shrinkage database included 1328 datasets for total shrinkage and 282 datasets for autogenous shrinkage. This comprehensive database was compiled from experimental results reported in numerous publications spanning from 1995 to early 2024. Each database was organised into two sheets: ‘Info’ and ‘Data’. The former included information on each shrinkage dataset, structured in a format consistent with the Northwestern University database (Hubler et al., 2015a).

The parameter ranges of the expanded SCC shrinkage database were selected for normal-weight concrete without the use of expansive additives or shrinkage-reducing agents. These ranges of the parameters (28-day compressive strength (f_cm28), the cementitious material in the prediction formulae cm, w/cm ratio, aggregate-to-cementitious material ratio (a/cm), sand-to-aggregate ratio (s/a), test duration (t − t_c), volume-to-surface area ratio (V/S), RH and the age at the start of testing or exposure (t_c)) are listed in Table 3. Additionally, the table provides the median and mean values for each parameter. The distribution of these parameters is shown in Figure 2. The 28-day compressive strengths encompassed both normal- and high-strength concretes spanning a wide range (17–110 MPa) with a peak frequency around 50 MPa. The mean values of cm, w/cm, a/cm and s/a ratios were approximately 500 kg/m³, 0.36, 3.4 and 0.5, respectively.

Supplementary cementitious materials are commonly incorporated into SCC. According to the collected data, approximately 79% of the SCC mixtures included fillers and mineral admixtures; pure cement was used in only 21% of the mixes. Among the fillers, limestone powder was the most frequently used, whereas slag and FA were the predominant additives, known for enhancing the properties of SCC. Another key parameter was the aggregate type, which varied in mineralogy and could be classified into diabase, granite, limestone, quartzite and sandstone.

The shrinkage strain was primarily measured in specimens produced with normal cement (type R), which accounted for approximately 76% of the SCC shrinkage database. In contrast, slow-hardening cement (type SL) and rapid-hardening cement (type RS) were used in only about, respectively, 15% and 9% of the samples in the database. Owing to the predominance of type R cement in the database, this study focused exclusively on optimising shrinkage prediction models using type R cement. On the other hand, the limited availability of data on cement types SL and RS suggests that further experimental studies are needed before these can be included in analytical or predictive modelling.

Statistical coefficients are commonly used to evaluate the performance of shrinkage prediction models. In this study, two primary metrics were employed – the determination coefficient (R²) (Hami and Pougnet, 2020; Montgomery et al., 2012) and the normalised root mean square error (NRMSE) (Dietterich et al., 2017; Rizi and Granitzer, 2017). The optimisation process was based on minimising the prediction error according to these coefficients. However, statistical results can be significantly affected by the distribution of datasets. Typically, laboratory databases are dominated by short-term test results, small specimen sizes and early-age measurements. The percentage weights of these datasets was confirmed by the SCC database, as illustrated in Figure 2. Relying solely on unweighted error statistics may lead to biased model evaluation. To achieve unbiased data and error statistics, a weighting scheme was applied during model optimisation (Hubler et al., 2015b; Wendner et al., 2015). The weighing factors were based on key independent variables, including the test duration, environmental humidity, specimen size and age at the start of drying (Bažant and Li, 2007).

According to the SCC shrinkage database, data subsets were created to apply weightings, ensuring unbiased evaluation of statistical coefficients for a reliable model based on the independent variables of shrinkage type. Optimisation was first performed for dependent parameters on cement type. Then, an additional coefficient was introduced to improve the predictive accuracy of the original fib MC2010 model. In this study, the w/cm ratio – a key factor influencing shrinkage development – was selected to adjust the original model (Hubler et al., 2015a). The entire optimisation process was conducted sequentially for both autogenous shrinkage and drying shrinkage formulas.

The fib-SCC model was developed within the parameter ranges derived from the SCC database, as presented in Table 4, including curing temperature (T_cur), f_cm28, cm and w/cm ratio.

The calibration was carried out on the fib MC2010 model, originally designed to predict the shrinkage of normal- and high-strength SCC. In the modified model, the w/cm ratio was incorporated in the formulas for both autogenous and total shrinkage. Additionally, the humidity factor of shrinkage in the fib-SCC model covered a broader range than that in the original fib MC2010 model, which had a minimum applicable RH of 40%. The updated model was adjusted to fit the SCC database with a minimum RH of 27%. Furthermore, shrinkage parameters dependent on cement type were optimised to achieve the lowest NRMSE, based on the appropriately filtered datasets. The F-SCC model for predicting shrinkage in SCC is defined by the following formulas.

Total shrinkage strain is calculated as:

where t is the current age of concrete (days), t_c is the age at exposure (days), ε_sh(t,t_c) is the drying shrinkage strain, and ε_au(t,t_c) and ε_au(t_c) are the autogenous shrinkage strains at t days and t_c days, respectively.

Autogenous shrinkage is expressed as:

where ε_au0(f_cm28) is the notional autogenous shrinkage coefficient, β_as(t) is the time function of autogenous shrinkage, and α_as and α_aw are parameters dependent on the cement type.

The independent variables of the autogenous shrinkage database, including the test duration (t − t_c) and the start of the test (t_c), were divided into subsets, as summarised in Table 5. The new parameter, w/cm ratio, with an average value of 0.36 based on the SCC database, was also added to the formulas of the fib-SCC model. Based on the subsets of the database, the parameter α_aw in Equation 17 was optimised with values of 0.7 and 0.55 corresponding to, respectively, normal-strength concrete (f_cm28 ≤ 55 MPa) and high-strength concrete (f_cm28 > 55 MPa), as can be seen from the results shown in Figures 3 and 4, respectively. Consequently, the prediction capacity of the modified model based on the w/cm ratio was improved compared with the original model, as shown in Figures 5(a) and 5(b).

The optimisation of drying shrinkage was carried out in the same way as autogenous shrinkage. However, compared with autogenous shrinkage, drying shrinkage involves the consideration of a greater number of independent variables, including test duration, specimen size (V/S), RH and the start of the test. These variables were divided into subsets, as summarised in Table 6.

Drying shrinkage is expressed as:

In Equations 18–27, ε_shu is the ultimate drying shrinkage strain (10⁻⁶ m/m), k_h,T is the temperature-adjusted humidity factor for shrinkage, k_h is the humidity factor for shrinkage, RH_T is the temperature-adjusted RH (%), T is the temperature (°C), f_cm28,T is the temperature-adjusted 28-day compressive strength (MPa), f_cm28 is the 28-day compressive strength (MPa), β_s(t − t_c) is the time function for drying shrinkage, h_T is the temperature-adjusted notional size (mm), h is the notional size (mm), A_c is the cross-section area of the concrete specimen (mm²), U is the cross-section perimeter of the concrete specimen (mm) and α_ds1, α_ds2, α_dw, α_as and α_aw are parameters dependent on the cement type, as provided in Table 7.

In the case of autogenous shrinkage, as illustrated in Figure 5, the NRMSE of the fib-SCC was 62.24%, which was the lowest error of the evaluated models. The fib MC2010 and B4 models yielded significantly higher NMRSE values of 114.35% and 105.09%, respectively, based on 245 datasets of SCC with cement type R. In other words, the errors of the fib MC2010 and B4 models were approximately 84% and 69% higher than that of the fib-SCC model, respectively. Notably, the fib MC2010 model tended to underestimate autogenous shrinkage, with many data points clustering along the x-axis. In contrast, the fib-SCC model displayed a more uniform scatter of data around the standard line and most of the data points were distributed well in an acceptable error range of ±40% (indicated by the two red dashed lines), indicating a remarkable improvement for the fib-SCC model. Although the B4 model showed a reasonable level of prediction, its data points were more widely spread, reducing overall accuracy.

Figure 6 shows the performance of the models in predicting total shrinkage. The fib-SCC model not only accurately captured autogenous shrinkage but also provided enhanced predictions for total shrinkage as compared with the original fib MC2010 model. The data points of the fib-SCC model were well-distributed around the reference line, while the fib MC2010 model strongly underestimated the total shrinkage. The fib-SCC model achieved R² = 0.50 and NRMSE = 51.48%, the lowest error among the evaluated models. For comparison, the NRMSE values of the fib MC2010, B4, ACI model and modified JSCE models were approximately 59%, 75%, 61% and 78%, respectively. The B4 and ACI models generally underestimated total shrinkage, while the modified JSCE model showed a tendency to overestimate it. The modified CEB90 model demonstrated the lowest accuracy, with an NMRSE approaching 322%.

Based on the experimental results obtained in this study, the fib-SCC model demonstrated reliable performance in predicting both autogenous and total shrinkage. As shown in Figure 7, although the model slightly underestimated the measured shrinkage, the majority of the data points remained within the acceptable deviation range of ±40%. Statistical evaluation showed that the R² and NRMSE of the autogenous shrinkage were 0.77 and 30%, respectively. For total shrinkage, the corresponding values were 0.84 and 24.7%. These results confirm that the fib-SCC model effectively predicted not only across the shrinkage of the SCC database with a test duration up to 1435 days but also under the experimental conditions in the short term (150 days in this study).

ML algorithms (DT (single-tree method), RF, ET, GB and XGBoost) were also used to predict the total shrinkage of SCC. These supervised learning algorithms were selected owing to their high predictive accuracy when handling multiple input parameters. The models were implemented using regression techniques based on bagging (bootstrap aggregating) and boosting algorithms. Bagging is an ensemble meta-algorithm designed to reduce model overfitting, while boosting incrementally improves model performance by training sequentially on different subsets of the database to minimise training errors (Aggarwal, 2015).

The typical stages of the learning process are illustrated in Figure 8 (Mohri et al., 2018). Each algorithm follows an iterative, step-by-step computational procedure that may involve descriptions, equations or a combination of both (Yang, 2019). The key motivation for applying ML in shrinkage prediction lies in the ability of algorithms to continuously learn and detect patterns by processing large volumes of experimental data, unconstrained by any mathematical formulas or framework. This capability allows for more accurate shrinkage predictions, making these methods promising for practical applications.

Eight input features (w/cm ratio, cm, f_cm28, RH, V/S, t_c, T and T_cur) were considered in the optimisation process of the ML methods. These features also correspond to the key parameters in the formulae of the fib-SCC model to compare the influence of features on different types of prediction models. The role of features in the SCC database for total shrinkage prediction was also dissimilar, corresponding to the types of methods carried out by their algorithm as presented. According to the results shown in Figure 9, all the ML methods considered the RH as the most important variable influencing total shrinkage behaviour, while temperature features (T and T_cur) were the least important variables.

In the SCC shrinkage database, the total shrinkage data (8250 data points) were divided into training and validation sets using an 80:20 ratio. As a result, the training and validation sets included 6600 and 1650 data points, respectively. As already mentioned, during optimisation of the ML methods, eight input features were selected, corresponding to the key influencing parameters in the fib-SCC model.

The main parameters of the ML methods that were investigated were

• the number of estimators (n_estimators)

• the maximum number of features (max_features)

• the maximum tree depth (max_depth)

• the maximum number of leaf nodes (max_leaf_nodes)

• the column subsampling rate (colsample_bytree).

For consistency, the number of boosting stages (n_estimators) was fixed at 100 as a standard default and max_features was set to 8, corresponding to the key parameters’ influence on the fib-SCC model. The other parameters were optimised through a step-by-step tuning process aimed at minimising errors based on values of statistical coefficients. The parameters were also chosen to avoid overfitting of algorithms, ensuring robust prediction not only on the training data but also on new data.

In the optimisation process, the max_depth parameter limits the number of nodes in each tree, with the goal of achieving optimum accuracy for both the training and validation datasets, as shown in Figure 10. The optimal results showed that the max_depth values for the DT, RF and ET were significantly higher than those for GB and XGBoost. A higher max_depth value generally indicates increased complexity, which can lead to a greater risk of overfitting when applied to the SCC database. In addition, the max_leaf_nodes parameter was effective in reducing overfitting for algorithms such as RF, ET and GB, while the colsample_bytree parameter improved the predictive performance of the XGBoost method. The optimum values of these parameters are shown in Table 8.

Figure 11 shows the predicted results of the ML methods, evaluated on the validation dataset of 1650 points using the optimum parameter values. The prediction capacities of the DT, RF and ET were less efficient than those of GB and XGBoost. The data points predicted by the DT were not well scattered in the diagram, yielding R² = 0.73 and an NRMSE = 37.4%. The RF and ET, which use bagging techniques, tended to overestimate total shrinkage at later ages. The boosting algorithms (GB and XGBoost) predicted total shrinkage more effectively, with NRMSE values of approximately 23.8% and 23.7%, respectively. However, the data points predicted by XGBoost were more closely distributed around the standard line than those of the other methods.

When the SCC database with the ML methods was evaluated on k-fold cross-validation with n_splits of 5, 10 and 15 (Figure 12), the XGBoost algorithm exhibited the highest accuracy. When k-fold cross-validation was divided into ten folds (n_splits = 10), the accuracy of the algorithms in descending order was 0.52, 0.50, 0.49, 0.43, and 0.21, corresponding to XGBoost, GB, ET, RF and DT, respectively. In other words, XGBoost showed the best generalised performance among the ML methods. The DT algorithm again displayed the lowest accuracy based on k-fold validation.

Figure 13 shows the prediction results of the ML methods and the fib-SCC model applied to the entire dataset of total shrinkage (8250 data points). The NRMSEs of the DT, RF, ET, GB and XGBoost models were approximately 31.2%, 33.3%, 37.9%, 20.8% and 20%, respectively – significantly lower than the NRMSE of the fib-SCC model (51.5%), as illustrated in Figure 6(b). Moreover, the data points of the fib-SCC model showed greater dispersion around the standard line (x = y) compared with the ML methods, although the majority still fell within the boundary lines defined by the ±40% error margin (represented by the red dashed lines). This indicates that the fib-SCC model was less efficient than the supervised learning methods in predicting total shrinkage. This discrepancy can be attributed to the training methods of the algorithms. Bagging methods improve predictive performance by combining the predictions of multiple models based on the average values of multiple subsets of the training data, thereby improving the final result, while boosting methods enhance the prediction performance of weak learners by iteratively correcting the errors (Unpingco, 2016; Witten et al., 2017). In comparison, the fib-SCC model calculates shrinkage strain based solely on the relationships among the influential parameters in the explicit formulas.

In aspect of residual analysis, as shown in Figure 14, differences between the test data and the predicted values of ML algorithms are clear. The results indicated that the residual points of XGBoost and GB were distributed closely around the standard axis (horizontal red line). Consequently, these algorithms exhibited better performance than the other algorithms. The DT algorithm showed the least fit for the SCC database, with the points spread widely. The RF and ET tended to overestimate the total shrinkage in the early stage, so that the predicted values were higher than the test data, and the residual points were scattered around the standard axis. However, these algorithms underestimated the shrinkage of SCC in later stages of testing.

A new model (fib-SCC model) and ML methods for predicting the shrinkage of SCC were evaluated. The following main conclusions were drawn from this work.

• The experimental results showed that the incorporation of FA improved the workability of concrete used in proportions of 20–55% by weight of cement. FA also enhanced the compressive strength at later ages. In addition to FA, the w/cm ratio and SP dosage significantly impacted both autogenous and total shrinkage.

• Numerous data were collected in an expanded laboratory database of SCC, comprising 1328 datasets for total shrinkage and 282 datasets for autogenous shrinkage.

• The fib-SCC model, developed through calibration of the fib MC2010 model, was adapted to fit over 1200 datasets using normal cement (type R). The modified model accurately predicted both the shrinkage of the SCC database and the experimental results. However, the fib-SCC model was less efficient than the ML methods.

• Of the ML methods, the DT, RF and ET were less accurate than GB and XGBoost in predicting the total shrinkage of SCC.