Design of Prediction Models for Estimation of the Strength of the Compressed Stabilized Earth Blocks

Abstract

Compressing a mixture of soil, water, and stabilizer forms compressed stabilized earth blocks (CSEBs), a modernized earthen construction material capable of performance similar to that of engineered masonry with added sustainability achieved by usage of raw materials on-site, reduction in transportation costs of bulk materials to the build site, and improved thermal performance of built CSEB structures. CSEBs have a wide range of potential physical properties due to variations in base soil, mix composition, stabilizer, admixtures, and initial compression achieved in CSEB creation. While CSEB construction offers several opportunities to improve the sustainability of construction practices, assuring codifiable, standardized mix design for a target strength or durability remains a challenge as the mechanical character of the primary base soil varies from site to site. Quality control may be achieved through creation and testing of CSEB samples, but this adds time to a construction schedule. Such delays may be reduced through development of predictive CSEB compressive strength estimation models. This study experimentally determined CSEB compressive strength for six different mix compositions. Compressive strength predictive models were developed for 7-day and 28-day CSEB samples through multiple numerical models (i.e., linear regression, back-propagation neural network) designed and implemented to relate design inputs to 7-day and 28-day compressive strength. Model results provide insight into the predictive performance of linear regression and back-propagation neural networks operating on designed data streams. Performance, robustness, and significance of changes to the model dataset and feature set are characterized, revealing that linear regression outperformed neural networks on 28-day data and that inclusion of downstream data (i.e., cylinder compressive strength) did not significantly impact model performance.

1. Introduction

Compressed stabilized earth blocks (CSEBs) are a version of earthen construction that can be used as a strong [1], durable [2], and energy efficient [3] alternative to contemporary masonry construction materials such as brick and concrete masonry units (CMUs). Through careful specification of mix composition, additives, initial compression, and curing conditions, CSEBs can be created to meet CMU minimum compressive strength values [4], thermal efficiency targets [3], and desired embodied carbon [5].

CSEBs are well suited for remote locations, where using site-specific, inorganic earth reduces the reliance on material delivery and transportation carbon footprint [5]. However, using site-specific materials and on-site creation processes increases material variability and, ultimately, performance relative to manufacturing in environmentally controlled production plants. ASTM C39 “Standard Test Method for Compressive strength of Cylindrical Concrete Specimens” [6] highlights the difference in predictability between on-site creation and production plant creation; concrete cylinders created on-site have a 20% greater acceptable coefficient of variation than concrete cylinders created in laboratory conditions. This relationship between on-site creation and laboratory creation is expected to hold for CSEBs, though ASTM C1314 “Standard Test Method for Compressive Strength of Masonry Prisms” [7] does not offer specific coefficient of variation values.

The uncertainty of estimated CSEB compressive strength affects design and must be considered; masonry building codes [8,9,10] require knowledge of unit compressive strength to determine masonry compressive strength through the use of equations and tables. Unit compressive strength can be predicted using either non-destructive metrics [11,12,13] or a combination of destructive and non-destructive metrics [14,15,16].

Non-destructive metrics include ultrasonic pulse velocity [11,13], electrical resistivity [11], rebound value [13], mix composition [12,14,16,17], and chemical makeup [12,18]. Destructive prediction models often use the same features with the addition of compressive strength values from a sample of the same mix composition but different sample type (i.e., cube vs. cylinder) to estimate the compressive strength. Geometric variation is common in concrete analysis where conversion factors are used to convert cylinder and cube specimen compressive strengths [14,15,16]. In concrete analysis, the conversions often comprise a single conversion factor between 1 and 1.5, depending on concrete compressive strength and aspect ratio [14,15,16]. Conversions have also been developed for unstabilized compressed earth blocks and were found to be dependent on mix composition, density, and relative strength [19].

Prediction models can use linear regressions [17,20] and neural networks of many different types, including decision tree [12], support vector [13], ANN [20,21], gene expression [22], MLP [23], and EANN [24]. Linear regressions describe relationships between linear features [25], requiring data preprocessing to linearize non-linear variables (i.e., initial compression [26], water-to-cement ratio [27], etc.), while neural networks accommodate non-linear relationships through non-linear activation functions and multiple layers [28]. However, neural networks typically require more data than linear regressions to train robustly [29], and collecting data experimentally can be time consuming. Additionally, linear regressions typically offer greater transparency than neural networks as linear regressions provide an equation relating inputs and output, whereas neural network models provide little, if any, meaningful physical insights of output dependence on physical relationships.

To address the uncertainty of predicting CSEB compressive strength, this study develops data-driven, predictive models, including linear regression and back-propagation neural networks, that estimate the compressive strength of CSEB samples. Predictive models estimate CSEB strength values through (1) prediction using mix design input variables (i.e., mix design, including water, cement, and soil proportions, and initial compression values) and (2) prediction using factored compressed stabilized earth cylinder (CSEC) strengths. Predictive model types 1 and 2 are compared to determine performance using varied combinations of input variables, varying preprocessing and inclusion of data quantity, and, lastly, usefulness of including CSEC strength when predicting CSEB compressive strength. Additionally, sensitivity to untrained mix compositions is assessed, illustrating scenarios in which prediction models struggle to accurately predict compressive strength for untrained mix compositions.

2. Materials and Methods

2.1. Material Preparation and Specimen Creation

All CSEC and CSEB samples were created from north Texas soil classified as SP (i.e., poorly graded sand) according to the Unified Soil Classification System (USCS) [30] or A-3 (i.e., fine sand) according to the American Association of State Highway and Transportation Officials (AASHTO) [31]. The same soil comprised CSEC sample creation in previous investigations of small-scale cylindrical strengths as functions of mix design and initial compression [17].

Table 1. Mix compositions used for CSEC and CSEB sample creation.

Mix Composition	Water (%)	Cement (%)	Soil (%)	CSEC	CSEB
1	6	3	91	Yes	No
2	6	6	88	Yes	No
3	6	9	85	Yes	No
4	8	3	89	Yes	Yes
5	8	6	86	Yes	Yes
6	8	9	83	Yes	Yes
7	10	3	87	Yes	Yes
8	10	6	84	Yes	Yes
9	10	9	81	Yes	Yes

lists the CSEC and CSEB mix compositions evaluated in this study. The mix compositions are defined by their percent masses of water content, cement content, and soil content. CSEB samples use mix compositions 4–9 due to the low compressive strength results reported for mix compositions 1, 2, and 3 in [17]. A mix composition prediction sensitivity analysis retains mixes 1–3 in Section 3.4.

CSEB materials (soil, water, cement) were prepared in a similar manner to [17]. The CSEB samples were created in a volume-controlled, single-compression, manual press, the Auram 3000. The press compressed to an estimated 2 MPa of initial compression, producing samples with dimensions of 298 mm × 146 mm × 114 mm. Four through-hole Transducer Techniques LWO-10 load cells provided estimates for initial compression forces in the compression chamber. Figure 1 depicts the CSEB press and loadcell configuration. After creation, the CSEB samples were placed in ice chests for curing in a temperature- and humidity-controlled environment.

An MC-500PR concrete compression testing machine from Gilson Company inc., Lewis Center, OH, USA (Figure 2) was used to test the CSEB samples in direct compression. Samples were uniaxially loaded between a spherically mounted flat platen and a flat platen driven by a hydraulic piston from below the specimen. The compression tests were conducted at a load rate of 0.2 ± 0.05 MPa/s. Samples were loaded at a constant rate until the load value dropped to 20% of the sample’s peak load. Platens were cleaned between tests to reduce uneven loading.

2.2. Linear Regression Development and Model Definition

Linear regression models relating mix design variables and initial compression (model input) to compressive strength of CSEC were developed using the process outlined in [17]. In summary, potential regression variables, selected based on relevant physics or previous literature, were systematically introduced one at a time. Regression performance and changes to regression parameters (e.g., coefficients and basis) were evaluated for stability and significance after each variable addition, and additional variables were included until performance improvements were no longer significant or all candidate variables had been exhausted. This section presents model definitions, including parameter selection and associated confidence intervals.

2.2.1. Linear Regression Model 1: CSEC and CSEB Strength Prediction

CSEC compressive strength was found to be predicted well by relating sample density $\left(\mathsf{\rho }\right)$, cement content $\left(\mathrm{C}\right)$, and linearized water-to-cement ratio $\left(\mathrm{W}{\mathrm{C}}^{\prime }\right)$ [17]. An ordinary least squares (OLS) regression was used to develop the CSEC prediction Equations (1) and (2), which are recalculated from [17] based on mix compositions 4–9, to keep parity between CSEC and CSEB prediction Equations (4) and (5). Equations (1) and (2) use the data presented in [17], while Equations (4) and (5) use the data presented in the Section 3 of this paper. Figure 3 depicts inputs, outputs, and model nomenclature for linear regression model 1.

OLS regressions assume variable linearity, and water-to-cement ratio is known to have a non-linear relationship with compressive strength for concrete samples [27]. This was converted to a linear relationship through Equation (3); A and B are experimental parameters for a given age, material, and cure conditions [27] converted to metric units [32].

Table 2. Confidence intervals for Equations (1) and (2) parameters.

	Equation (1)		Equation (2)
Parameter	Upper CI	Lower CI	Upper CI	Lower CI
$\rho$	0.006	0.004	0.007	0.005
$C$	0.449	0.334	0.508	0.340
$WC\prime$	−0.064	−0.110	−0.051	−0.116
Constant	−11.499	−9.186	−9.598	−13.010

states the parameter confidence intervals for Equations (1) and (2) at the 95% confidence level.

Adjusted $R^2$ and mean absolute percentage error (MAPE) values are reported for each predictive model. Adjusted $R^2$ penalizes predictive models for additional variables and is utilized to ensure additional variables provide meaningful improvements. MAPE is reported for completion and comparison to neural network predictive models as adjusted $R^2$ struggles to adequately represent performance of non-linear models.

$${\mathrm{f}}_{\mathrm{c}}^{\prime }=0.005\mathsf{\rho }+0.392\mathrm{C}-0.087\mathrm{W}\mathrm{C}\prime -10.342\hspace{1em}\hspace{1em}{\mathrm{R}}^2=0.920\hspace{1em}\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}=0.189$$

(1)

$${\mathrm{f}}_{\mathrm{c}}^{\prime }=0.006\mathsf{\rho }+0.424\mathrm{C}-0.084\mathrm{W}\mathrm{C}\prime -11.304\hspace{1em}\hspace{1em}{\mathrm{R}}^2=0.876\hspace{1em}\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}=0.212$$

(2)

$$\mathrm{W}\mathrm{C}\prime ={\displaystyle \frac{\mathrm{A}}{{\mathrm{B}}^{\mathrm{W}\mathrm{C}}}}={\displaystyle \frac{97.93}{{7.2}^{\mathrm{W}\mathrm{C}}}}$$

(3)

$$\begin{array}{c}\mathsf{\rho }=\mathrm{S}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e} \mathrm{D}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}\\ \mathrm{C}=\mathrm{C}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}\\ \mathrm{W}\mathrm{C}=\mathrm{W}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{t}\mathrm{o} \mathrm{C}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{R}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\\ {\mathrm{f}}_{\mathrm{c}}^{\prime }=\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{S}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\end{array}$$

CSEB samples were created with consistent initial compression, reducing density variability relative to CSEC samples. As a result, the CSEB prediction Equations (4) and (5) have reduced sensitivity to changes in density and an increased sensitivity to changes in $\mathrm{C}$ and $\mathrm{W}\mathrm{C}\prime$ relative to the CSEC prediction Equations (1) and (2).

Table 3. Confidence intervals for Equations (4) and (5) parameters.

	Equation (4)		Equation (5)
Parameter	Upper CI	Lower CI	Upper CI	Lower CI
$\rho$	0.007	−0.001	0.006	−0.001
$C$	1.032	0.375	1.564	1.006
$WC\prime$	−0.100	−0.341	−0.288	−0.505
Constant	−0.816	−13.279	−1.010	−13.619

lists the parameter confidence intervals for Equations (4) and (5) at the 95% confidence level. The confidence intervals for density include both positive and negative values; this means that density could have an insignificant impact on CSEB predictions. This would likely not be an issue when developing CSEB predictions for samples with multiple initial compression values, as seen with the CSEC prediction parameter confidence intervals.

$${\mathrm{f}}_{\mathrm{c}}^{\prime }=0.003\mathsf{\rho }+0.704\mathrm{C}-0.220\mathrm{W}\mathrm{C}\prime -7.521\hspace{1em}\hspace{1em}{\mathrm{R}}^2=0.822\hspace{1em}\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}=0.219$$

(4)

$${\mathrm{f}}_{\mathrm{c}}^{\prime }=0.002\mathsf{\rho }+1.285\mathrm{C}-0.396\mathrm{W}\mathrm{C}\prime -7.135\hspace{1em}\hspace{1em}{\mathrm{R}}^2=0.888\hspace{1em}\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}=0.166$$

(5)

2.2.2. Linear Regression Model 2: CSEB Inclusion of CSEC Data

CSEC samples are smaller, easier to transport, and require less capacity for destructive testing than CSEB samples. Conversion factors, similar to what are used in relating concrete specimens of varying geometry (e.g., cube vs. cylinder), relating CSEC and CSEB compressive strengths allow results from smaller, easy to transport and test specimens to inform expected properties of a CSEB. The experimental CSEC to CSEB conversion factors range between 1.2 and 3.0 depending on mix composition and cure time, as seen in Figure 4 and

Table 4. CSEC to CSEB conversion factors.

	Mix Composition
	4	5	6	7	8	9
7-Day	1.22	2.61	1.78	2.44	2.29	1.57
28-Day	2.37	1.79	1.55	3.05	2.08	2.19

.

To address the conversion factors changing with mix composition, a pair of linear regression equations were developed to predict CSEB compressive strength based on CSEC compressive strength. CSEB prediction Equations (6) and (7) were developed using the process outlined in [17]. Figure 5 depicts input, output, and model nomenclature for linear regression model 2.

Both Equations (6) and (7) use the average compressive strength of CSEC samples created from 2.2 MPa of initial compression, ${\mathrm{f}}_{{\mathrm{c}}_{\mathrm{C}\mathrm{y}\mathrm{l}}}^{\prime }$. Equation (6) uses ${\rho }_{Block}$ to predict 7-day compressive strength, while Equation (7) uses water content to predict 28-day compressive strength. The water content sensitivity is present in Equation (7) because cement hydration requires water and is a time-dependent process. $C$ and $WC\prime$ are present in $f_{c_{Cyl}}^{\prime }$ prediction Equations (1) and (2).

Table 5. Confidence intervals for Equations (6) and (7) parameters.

	Equation (6)			Equation (7)
Parameter	Upper CI	Lower CI	Parameter	Upper CI	Lower CI
$f_{c_{Cyl}}^{\prime }$	1.470	0.518	${\mathrm{f}}_{{\mathrm{c}}_{\mathrm{C}\mathrm{y}\mathrm{l}}}^{\prime }$	2.085	1.459
${\rho }_{Block}$	0.009	0.001	$\mathrm{W}$	0.568	0.140
Constant	−2.211	−14.603	Constant	−1.027	−4.698

lists the parameter confidence intervals for Equations (6) and (7) at the 95% confidence level.

$${\mathrm{f}}_{{\mathrm{c}}_{\mathrm{B}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}}^{\prime }=0.994{\mathrm{f}}_{{\mathrm{c}}_{\mathrm{C}\mathrm{y}\mathrm{l}}}^{\prime }+0.005{\mathsf{\rho }}_{\mathrm{B}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}-8.407 {\mathrm{R}}^2=0.802\hspace{1em}\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}=0.2506$$

(6)

$${\mathrm{f}}_{{\mathrm{c}}_{\mathrm{B}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}}^{\prime }=1.772{\mathrm{f}}_{{\mathrm{c}}_{\mathrm{C}\mathrm{y}\mathrm{l}}}^{\prime }+0.354\mathrm{W}-2.862\hspace{1em} {\mathrm{R}}^2=0.873\hspace{1em}\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}=0.2134$$

(7)

$$\begin{array}{c}{\mathrm{f}}_{{\mathrm{c}}_{\mathrm{C}\mathrm{y}\mathrm{l}}}^{\prime }=\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e} \mathrm{C}\mathrm{S}\mathrm{E}\mathrm{C} \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\\ {\mathsf{\rho }}_{\mathrm{B}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}=\mathrm{C}\mathrm{S}\mathrm{E}\mathrm{B} \mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}\end{array}$$

Figure 6 and Figure 7 show how linear regression performance changes with variables included for 7-day and 28-day samples. The selected variable performance is highlighted orange, and MAPE is represented by the blue line.

2.3. Neural Network Development

Feature set selection affects neural network model performance [33,34,35]. The neural network predictive models of this study were designed and evaluated using three different feature sets (FS) listed in

Table 6. Feature sets used for compressive strength prediction.

	Features
FS1	$\rho ,C,WC\prime$
FS2	$\mathit{log}\left(IC\right),WS,WC$
FS3	$W,C,IC$

$\rho = SampleDensity$. $C = CementContent$. $W = WaterContent$. $IC = InitialCompression$. $WS = WatertoSoilRatio$. $WC = WatertoCementRatio.$

. FS1, FS2, and FS3 use features that are known to impact density and compressive strength from the literature review. FS1 comprises the same features as the CSEC and CSEB linear regression models. These features have already been linearized for use in linear regression and offer a direct performance comparison between linear regression and neural network models.

FS2 uses the natural logarithm of initial compression, water-to-soil ratio, and water-to-cement ratio. Water-to-soil ratio and water-to-cement ratio are known to have non-linear relationships with density [36] and compressive strength [27], respectively. The natural logarithm of initial compression is known to have a linear relationship with density [26], which is linearly related to compressive strength [26]. This feature set was chosen as a combination of several basic mix components (water content, cement content, soil content, and initial compression) expressed as ratios to highlight known relationships: water-to-soil ratio with density [36] and water-to-cement ratio with compressive strength [27].

FS3 uses the three basic features of mix design cement content, water content, and initial compression and was chosen to examine how well a neural network can predict compressive strength with no feature preprocessing.

Another common form of prediction is to use destructive testing on one sample type to inform the prediction of another sample type [14,15,16,19].

Table 7. Feature sets used for compressive strength conversion.

	Features
FS1A	${\mathrm{f}}_{{\mathrm{C}}_{\mathrm{C}\mathrm{y}\mathrm{l}}}^{\prime },\mathsf{\rho },\mathrm{C},\mathrm{W}\mathrm{C}\prime$
FS2A	${\mathrm{f}}_{{\mathrm{C}}_{\mathrm{Cyl}}}^{\prime },\log \left(\mathrm{I}\mathrm{C}\right),\mathrm{W}\mathrm{S},\mathrm{W}\mathrm{C}$
FS3A	${\mathrm{f}}_{{\mathrm{C}}_{\mathrm{C}\mathrm{y}\mathrm{l}}}^{\prime },\mathrm{W},\mathrm{C},\mathrm{I}\mathrm{C}$

$f_{C_{Cyl}}^{\prime } = AverageCompressiveStrengthofCSEC$.

lists alternate versions of the feature sets from Table 6 that include the compressive strength of CSEC cylinders. Comparing results from FS1-3 to FS1A-3A provides context to whether inclusion of CSEC compressive strength data improves CSEB compressive strength prediction performance.

2.3.1. Neural Network Architecture

This study uses a back-propagation neural network (BPNN) design for its ability to model non-linear functions [37]. The BPNN was designed in Python [38] using tools from the sklearn [39] and keras [40] libraries. The BPNN code utilized 5-fold cross-validation and early stopping tools to develop a performance range and limit overfitting, respectively.

Neural network architecture has a large influence on network performance [29], and a 5-fold cross-validation grid search varying the number of hidden layers and neurons per hidden layer was conducted to determine the optimal network architecture. The neural networks evaluated had either 1 or 2 hidden layers, with 2, 8, 16, 24, 32, 40, 48, 56, or 64 neurons in each layer. The grid search found that there was no significant improvement when including a second hidden layer or having more than 8 neurons in the first hidden layer. Figure 8 depicts the change in mean absolute percent error relative to changes in neuron quantity in hidden layer 1. The hollow circles in Figure 9 represent samples that are 1.5 times the inner quartile range below the first quartile or above the third quartile.

Figure 9 depicts the chosen neural network architecture, which has 1 dense hidden layer with 8 neurons using ReLU activation between the input layer and hidden layer and linear activation between the hidden layer and output layer to predict a continuous value.

2.3.2. Neural Network Datasets

The dataset used in this study comprises 60 CSEB and 240 CSEC samples, a relatively small collection of experimental data points to train a neural network. To address the small dataset, two methods of generating additional synthetic data were evaluated. Normally distributed synthetic data was created by generating compressive strength data points that follow a normal distribution with the same mean and standard deviation as the experimental data for each mix composition, age, and initial compression combination.

Additionally, a conditional tabular generative adversarial network (CTGAN) was used to generate synthetic data. CTGAN is a deep learning-based synthetic data generator able to generate synthetic data with high fidelity [41]. In both normally distributed and CTGAN augmented cases, the ratio of experimental to synthetic data is 1:1. Figure 10 shows the data pool and nomenclature for each neural network model type. DS1, DS2, and DS3 all use the same network architecture shown in Figure 9. Figure 10 also shows the nomenclature for referencing a specific neural network combination of a dataset, feature set, and cure time.

3. Results

3.1. Compressive Strength

Unit compressive strength is one of the key metrics for determining viability of masonry building materials [8,9,10]. Compressive strength of CSEC samples was shown to increase as water content, cement content, and cure time increase [17]. Figure 11 and

Table 8. CSEB compressive strength results.

CSEB Compressive Strength (MPa)
	7-Day		28-Day
Mix	$\mathbf{Avg}{\mathbf{f}}_{\mathbf{c}}^{\mathbf{\prime }}$	$\mathsf{\sigma }$	$\mathbf{Avg}{\mathbf{f}}_{\mathbf{c}}^{\mathbf{\prime }}$	$\mathsf{\sigma }$
4	0.61	0.03	0.88	0.16
5	1.33	0.24	2.04	0.60
6	1.74	0.48	2.28	0.39
7	1.31	0.20	1.27	0.37
8	2.96	0.38	3.87	0.62
9	3.26	0.57	4.84	0.74

show that this trend continues for CSEB samples.

Within the range of mix compositions tested in this study, increases to water content, cement content, and cure time increase unit compressive strength. This increase has limits, as water contents beyond the optimum moisture content will decrease sample density [36], which is directly related to compressive strength [26]. Similarly, increasing water content without increasing cement content can lead to a reduction in compressive strength as the water-to-cement ratio is known to have a negative non-linear relationship with the compressive strength of cement-stabilized materials [27].

Welch’s ANOVA and Games–Howell post hoc tests, at the 95% confidence level, were used to assess the statistical significance of changes to water content, cement content, and cure time on unit compressive strength. These statistical tests were chosen due to the small sample size and unequal variances. Increasing water content from 8% to 10% has a significant impact on compressive strength. Similarly, increasing cure time from 7-day to 28-day increases compressive strength significantly. Cement content only has a significant impact on compressive strength when increasing from 3% to 6% cement. Increasing from 6% to 9% cement does not have a significant impact on compressive strength. As expected, these results are similar to the CSEC results in [17].

3.2. Linear Regression Compressive Strength Prediction

Figure 12 shows the relationship between predicted and experimental compressive strength using Equations (1) and (2) for CSEC samples. Figure 13 and

Table 9. CSEC linear regression prediction MAPE performance for 7-day and 28-day samples separated by mix composition.

	Mix 4		Mix 5		Mix 6		Mix 7		Mix 8		Mix 9
IC	7-Day	28-Day	7-Day	28-Day	7-Day	28-Day	7-Day	28-Day	7-Day	28-Day	7-Day	28-Day
2.2	0.68	1.05	0.17	0.12	0.03	0.14	0.36	0.59	0.14	0.05	0.06	0.15
4.4	0.61	0.43	0.23	0.17	0.07	0.15	0.72	0.11	0.12	0.23	0.06	0.09
8.8	0.14	0.20	0.28	0.11	0.08	0.11	0.10	0.13	0.03	0.11	0.03	0.20
13.2	0.04	0.06	0.16	0.16	0.08	0.12	0.13	0.33	0.07	0.13	0.15	0.15

separate the model performance by mix composition and initial compression, revealing that low water mix compositions (4 and 7) tend to have greater MAPE values than mix compositions 5, 6, 8, and 9. This is most evident with low initial compression samples.

Figure 14 shows the relationship between predicted and experimental CSEB compressive strengths. Equations (4) and (5) are used to calculate the predicted values. Figure 15 and

Table 10. CSEB linear regression prediction MAPE performance separated by mix composition.

	Mix 4	Mix 5	Mix 6	Mix 7	Mix 8	Mix 9
7-Day	0.39	0.27	0.23	0.22	0.08	0.12
28-Day	0.10	0.31	0.14	0.19	0.12	0.13

capture prediction performance separated by mix composition and cure time. Notably, CSEB performance lacks the spikes in MAPE value for low water content mix compositions.

3.3. Neural Network Compressive Strength Prediction

Predictive neural networks were created using the feature sets, datasets, and network architecture shown in Table 6, Figure 9, and Figure 10, respectively.

Table 11. Neural network MAPE performance for DS1, DS2, and DS3 datasets.

	DS1				DS2				DS3
	CSEC		CSEB		CSEC		CSEB		CSEC		CSEB
	7-Day	28-Day	7-Day	28-Day	7-Day	28-Day	7-Day	28-Day	7-Day	28-Day	7-Day	28-Day
FS1	0.165	0.125	0.139	0.112	0.139	0.112	0.280	0.279	0.199	0.153	0.269	0.424
FS2	0.182	0.150	0.131	0.119	0.131	0.119	0.172	0.213	0.180	0.141	0.192	0.295
FS3	0.166	0.165	0.146	0.124	0.146	0.124	0.193	0.267	0.191	0.137	0.246	0.406

lists the mean absolute percent error for neural networks predicting CSEC and CSEB compressive strengths using datasets DS1, DS2, and DS3.

In almost all cases, 28-day sample predictions have a lower MAPE value than 7-day samples, with the notable exception of CSEB samples in DS2 and DS3. Additionally, in DS2 and DS3, CSEB samples have greater MAPE values than their CSEC counterparts. However, the DS1 CSEB predictions outperform their CSEC counterparts, and the 28-day predictions outperform the 7-day predictions. This discrepancy is likely due to the combination of synthetic data present in DS2 and DS3 and the low quantity of CSEB data.

3.4. CSEC to CSEB Conversions

Figure 16 plots the relationship between predicted and experimental CSEB compressive strength for 7D and 28D samples using LR2 Equations (6) and (7). The 7-day graph has clustered groupings for each mix composition, while the 28-day graph has horizontal strata for each mix composition because the 7-day predictive equation includes ${\rho }_{Block}$, while the 28-day predictive equation does not.

Figure 17 and

Table 12. LR2 prediction MAPE performance separated by mix composition.

	Mix 4	Mix 5	Mix 6	Mix 7	Mix 8	Mix 9
7-Day	0.71	0.07	0.29	0.18	0.12	0.14
28-Day	0.15	0.31	0.13	0.23	0.13	0.14

show the conversion performance separated by mix composition and cure time. LR1 and LR2 prediction models have insignificantly different MAPE values across the range of mix compositions tested. The inclusion of CSEC compressive strength data does not significantly impact the CSEB compressive strength prediction performance. This is likely caused by the differing specimen creation that includes the effects of single or double compression and geometry profiles; understanding in situ initial compression values is a critical future need.

Neural network CSEC to CSEB conversion performance was evaluated using the feature sets listed in Table 7. Alt FS1–3 differ from FS1–3 by including the compressive strength of CSEC samples.

Table 13. CSEC to CSEB conversion performance for neural networks.

	DS1		DS2		DS3
	7-Day	28-Day	7-Day	28-Day	7-Day	28-Day
FS1A	0.398	0.242	0.339	0.227	0.260	0.171
FS2A	0.229	0.223	0.231	0.321	0.227	0.191
FS3A	0.236	0.456	0.219	0.288	0.208	0.255

lists the MAPE values for feature sets FS1A-FS3A.

3.5. Comparing CSEB Compressive Strength Prediction Models

Figure 18 shows that LR1 models outperform LR2 models on both 7-day and 28-day samples and that the inclusion of CSEC compressive strength data does not improve linear regression model performance. Additionally, Figure 19 and Figure 20 show that inclusion of CSEC compressive strength has no significant impact on neural network performance for 7-day or 28-day samples, respectively. The horizontal dashed lines represent LR1 and LR2 model performances, with linear regression models outperforming neural networks for 28-day samples. The 7-day neural network models using FS1, FS2, FS1A, and FS2A outperformed linear regression models.

Inclusion of downstream data (CSEC compressive strength) failed to significantly improve performance partially because of differences in initial compression methods. While CSEC and CSEB samples were subjected to 2.0 MPa and 2.2 MPa target initial compressions, the creation mechanism differed as CSEC samples were created in double compression and CSEB samples were created in single compression. The change in compression method [42] affected how friction developed between sample and mold, influencing density and compressive strength [24].

Changes in dataset (DS1, DS2, and DS3) have insignificant effects on performance, but DS2 has reduced variation relative to DS1 and DS3. Similarly, changes in feature set have insignificant effects on performance, but FS2 tends to slightly outperform FS1 and FS3.

Table 14. Mean and standard deviation of model performance for 7-day and 28-day samples.

7-Day						28-Day
	Avg	$\mathsf{\sigma }$		Avg	$\mathsf{\sigma }$		Avg	$\mathsf{\sigma }$		Avg	$\mathsf{\sigma }$
LR1	0.219	N/A	LR2	0.251	N/A	LR1	0.166	N/A	LR2	0.183	N/A
DS1-FS1	0.319	0.109	DS1-FS1A	0.398	0.119	DS1-FS1	0.358	0.094	DS1-FS1A	0.310	0.115
DS1-FS2	0.247	0.089	DS1-FS2A	0.229	0.052	DS1-FS2	0.259	0.097	DS1-FS2A	0.314	0.109
DS1-FS3	0.283	0.109	DS1-FS3A	0.236	0.109	DS1-FS3	0.429	0.299	DS1-FS3A	0.389	0.243
DS2-FS1	0.280	0.035	DS2-FS1A	0.339	0.054	DS2-FS1	0.279	0.057	DS2-FS1A	0.246	0.031
DS2-FS2	0.172	0.057	DS2-FS2A	0.231	0.083	DS2-FS2	0.213	0.065	DS2-FS2A	0.225	0.053
DS2-FS3	0.193	0.023	DS2-FS3A	0.219	0.070	DS2-FS3	0.267	0.050	DS2-FS3A	0.206	0.044
DS3-FS1	0.269	0.013	DS3-FS1A	0.260	0.024	DS3-FS1	0.424	0.255	DS3-FS1A	0.229	0.050
DS3-FS2	0.192	0.038	DS3-FS2A	0.227	0.033	DS3-FS2	0.295	0.144	DS3-FS2A	0.238	0.033
DS3-FS3	0.246	0.124	DS3-FS3A	0.208	0.057	DS3-FS3	0.406	0.183	DS3-FS3A	0.258	0.027

lists the mean and standard deviations for linear regression and neural network model performance for 7-day and 28-day samples.

3.6. Mix-Out Procedure to Characterize Model Robustness

Robust, generalized models provide stable results that are relatively insensitive to expected data noise or variation. Sensitivity to untrained mix compositions is important to understand for both linear regression and neural network estimation models. To assess linear regression and neural network sensitivity to untrained mix compositions, a ‘Mix-Out’ procedure was developed, whereby a model is trained on eight of the nine CSEC mix compositions and asked to predict compressive strength values for the remaining 9th mix composition. The Mix-Out procedure was conducted on CSEC compressive strength data from [17] using mix compositions 1–9. The linear regression and neural network models used the architecture shown in Figure 9 and were evaluated using FS1-3 listed in Table 6 without any synthetic data. For both linear regression and neural network models, a 5-fold cross-validation was used to develop the range of MAPE values presented in Figure 21 and Figure 22 for linear regression and neural network models, respectively.

Mix compositions 1, 2, 3, 4, and 7 typically have worse Mix-Out MAPE than mix compositions 5, 6, 8, and 9. Separating the mix compositions into region 1 (Mix 1, 2, 3, 4, and 7) and region 2 (Mix 5, 6, 8, and 9) and conducting a Welch’s t-test reveal that region 1 has significantly greater MAPE values than region 2. Figure 23 and Figure 24 show that this occurs with all three feature sets for both linear regression and neural network models. Performance of linear regression models on data from region 1 is affected by changes in feature set more than neural network performance on data from region 1. Region 1 performance (e.g., predicted compressive strength) is more sensitive to the variables selected for regression as these mixes generally experience greater variation than those in region 2. The robustness of the neural network to variations in predicted strength as a function of selected features manifests especially in region 1, where the predicted results are more variable; linear regressions benefit from linearized features [25], and neural networks can readily model non-linear relationships [28].

The difference between region 1 and region 2 performance suggests that there is a region boundary, similar to Figure 25, where untrained mix compositions are significantly more difficult to predict on one side of the boundary than the other. Region 1 mix compositions have less compressive strength than region 2 mix compositions [17]. This lack of strength is attributed to low cement content, low water content, or both [17]. Insufficient water or cement content would cause the sample to develop compressive strength differently than samples with sufficient water and cement content. The change in compressive strength development makes untrained mix compositions from region 1 more difficult to predict than region 2 mix compositions.

4. Conclusions

CSEB construction potentially improves sustainable construction worldwide; however, to confidently engineer and build with CSEBs, samples must meet compressive strength design requirements set forth in regional building codes. Development of predictive models for CSEB compressive strength meaningfully reduces mix design development time. There are many different types of predictive models with different performance characteristics. This study evaluates the differences in CSEB compressive strength prediction performance between linear regression and neural network models. Specifically, this study compared the CSEC compressive strength results from [17] with new CSEB compressive strength results and, in the process, developed linear regression and neural network models for predicting CSEB compressive strength. Linear regression and neural network model performances were evaluated relative to changes in feature set, dataset, and inclusion of downstream data.

Changes to feature set did not significantly impact model performance, but FS2 outperformed FS1 and FS3 when not including downstream data. Inclusion of synthetic data in DS2 and DS3 models did not significantly impact model performance, but DS2 tended to have less variation than DS1 and DS3. Inclusion of downstream data (CSEC compressive strength) hindered linear regression performance and did not significantly impact neural network performance because of differences in compression method: double compression vs. single compression. Future research should focus on developing CSEC creation methods that inform CSEB predictions.

Linear regression models outperformed neural network models on 28-day samples, but performance was mixed for 7-day samples, with DS2-FS2, DS3-FS2, DS2-FS3, and DS3-FS3 outperforming linear regression when not including downstream data.

Sensitivity to untrained mix compositions was evaluated through a Mix-Out procedure, finding that the compressive strength of mix compositions with low cement content and low water content are not captured well by the other mix compositions. Potential mix compositions of interest within region 1 need to be tested specifically, while potential mix compositions within region 2 can be modeled well by the other mix compositions and do not require specific testing.

Future work should focus on developing methods of creating CSEC samples that inform CSEB compressive strength predictions, likely accomplished through adjusting friction between sample and mold or applied initial compression. Additional research into untrained mix composition sensitivity could identify upper bounds of region 2 mix compositions and determine region 1 and 2 for different soil types. Lastly, additional soil types and additional inputs resulting from microscopic characterization of soil would provide generalized relationships applicable to more diverse site-specific conditions.