Prediction of Performance of Compressed Earthen Construction Using Compressed Stabilized Earthen Cylinders (CSECs)

Abstract

Earthen construction is one of the earliest and most ubiquitous forms of building. Compressed stabilized earth blocks (CSEBs) combine compressed components including inorganic soil, water, and a stabilizer such as Portland cement, and can achieve greater strength than other earthen construction methods. Typically, site-specific soil comprises the bulk material in CSEB construction, which minimizes the quantity of construction materials that need to be provided from off-site and motivates this type of building material for remote locations. However, onsite manufacturing and innate soil variability increase the variability of CSEB mechanical properties compared to more standardized building materials. This study characterizes the effects of varying mix compositions and initial compressions on the density, compressive strength, and variability of compressed stabilized earth cylinders (CSECs) created from sandy soil. CSEC samples comprising nine mix compositions and four levels of initial compression provide data for the (i) statistical evaluation of strength, density, and variability and (ii) development of predictive equations for density and compressive strength, with R² values of 0.90 and 0.89, respectively.

1. Introduction

An estimated 8–10% of the world’s population, or 810 million people, live in earthen buildings and the estimate increases to 20–25% in developing countries [1]. Earthen construction can be strong and durable [2,3], energy efficient [4,5], and can benefit from the inclusion of industrial and agricultural waste products [6,7,8]. These factors establish CSEB construction as a sustainable building solution that has been implemented and studied successfully in several regions including the U.S. [2], Spain [3], India [4], and North Africa [7]

Compressed stabilized earth blocks (CSEBs) represent a modernization of earthen construction combining site-specific soil, stabilizer, water, and initial compression to eclipse compressive strength and durability characteristics of other traditional forms of earthen construction such as adobe [9], rammed earth [10], cob [11], and unfired clay bricks [12].

During the curing process, the stabilizer, Portland cement for this study, forms calcium silicate hydrate compounds that increase the strength and durability of the block [3,13,14]. The inclusion of Portland cement and the process of initial compression promote strength development beyond traditional earthen construction techniques [2,15]. Manip ulation of water, stabilizer, and soil amendments (e.g., sand, clay) allows for mix compositions that achieve unit compressive strengths equal to or greater than the minimum compressive strengths for concrete masonry units [2].

The initial compression process increases density and reduces porosity [15]; density and porosity relate to the strength [15,16] and durability [17,18] of cement-based materials. Density, and thus compressive strength and durability, are functions of creation and curing parameters including water content, Portland cement content, soil type and amount, initial compression, admixtures, time, temperature, and humidity. Increasing the water content of a soil mix increases density to a maximum, defined as the optimum moisture content; further increases in water content decrease the density [19]. Increasing Portland cement content increases density as Portland cement grain sizes are smaller than soil grain sizes [20]. Cement particles that are sufficiently smaller than the larger particles in a distribution can be enclosed by the large particles [21]; Portland cement fills voids between soil particles, adding mass without increasing volume until voids fill or are sufficiently coated; further increasing Portland cement content increases volume.

Water content and cement content are important factors in determining a CSEB sample’s compressive strength. The water-to-cement ratio has a well-documented exponential relationship with compressive strength for concrete and a typical water-to-cement ratio for concrete lies between 0.4 and 0.6 [22]. As CSEB requires more water for mixing and hydration, CSEBs typically possess water-to-cement ratios that double or triple that of concrete [2,7,23].

Building codes use compressive strength, density, and variability of CSEB to determine the viability of blocks. The 2021 International Building Code, The New Zealand Standards, and the Indian Standards have required minimum compressive strengths for CSEBs shown in

Table 1. CSEB code requirement summary.

	International Building Code 2021	New Zealand Standards 4298:2024	Indian Standards 1725:2020
Code Reference	IBC 2021	NZS 4298:2024	IS 1725:2020
Jurisdiction	North America	New Zealand	India
Compressive Strength	2 MPa min.	1.4 MPa min.	3.5 MPa min.
Density	N/A	$1400{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}$ min.	$1750{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}$ min.
Variability	N/A	Used to adjust characteristic strength	No sample more than 15% below average
Durability	N/A	Thickness to not decrease by more than 5% or 30 mm during building lifespan	Lose no more than 3% mass within testing period

[24,25,26]. Additionally, the New Zealand Standards and the Indian Standards have requirements or equations that use density and variability to determine whether a CSEB is viable [25,26].

CSEBs that are manufactured on-site need to meet compressive strength, density, variability, and durability requirements. Compressive strength and variability testing can add at least one month to the design process as samples have to be created, cured, and tested. Potential delays (e.g., iterative mix design and testing cycles) limit the implementation of CSEBs as designers and builders have options with more standardized materials, such as brick or CMU, that require less field validation. Field validation and construction times are convincingly reduced through the development of predictive CSEB compressive strength and density characterization relationships. Such relationships are critical to bringing CSEB construction to a level of standardization of other building materials. To develop these relationships, this study examined the effects of water content, cement content, soil content, initial compression, and cure time on compressive strength, density, and variability of CSECs. These effects are then developed into predictive equations.

2. Materials and Methods

In order to develop the predictive equations, all mix variables were systematically combined to establish unique mix designs from which cylindrical samples were created. Cylindrical forms were selected for ease of initial compression control, and because cylinders are a common shape for compression testing [27].

The CSEC samples for this study were created using soil sourced from North Texas, due to ease of access, ability to acquire large quantities of soil, and similar geographic origins to soil studied in other studies [2].

Table 2. Particle gradation.

Sieve no.	Diameter (mm)	Total wt. (g)	% Passing
3/8”	9.525	0	100.00
4	4.750	0	100.00
8	2.360	75.5	91.94
16	1.180	162.5	82.65
30	0.600	182	80.57
40	0.425	308	67.11
50	0.300	505.5	46.02
80	0.180	751	19.81
100	0.150	789	15.75
200	0.075	885.5	5.45
pan	N/A	936.5	0.00

and Figure 1 contain particle gradation results as conducted per ASTM D2487-17 [28]. The soil was classified as SP (i.e., poorly graded sand) according to the Unified Soil Classification System (USCS) [28], or A-3 (i.e., fine sand) according to the American Association of State Highway and Transportation Officials (AASHTO) [29]. The soil in this study is relatively sandy compared to many other CSEB studies in

Table 3. Soil classifications used in other studies.

Sample Source	Fines%	USCS	AASHTO
Present Study	5%	SP	A3
Sitton et al., 2018 [2]	9%	SC	A-2-7
Cid-Falceto et al., 2012 [3]	11%	SP	A-1-b
R.K. Preethi and Reddy 2020 [6]	50%	CL	A6
Mansour et al., 2016 [15]	60%	CL+ML	A4
Gonzalez-Lopez et al., 2018 [14]	30%	SC	A-2-6
Ruiz et al., 2018 [30]	12%	SM	A-1-b
Narayanaswamy et al., 2020 [31]	43%	SC	A-2-6

; sandy soils are pervasive in many regions and possess different characteristics for which suitable mixes should be designed.

This study evaluated nine different mix compositions with varying quantities of a single soil type, water, and Portland cement, listed in terms of percent mass in

Table 4. Mix compositions by percent mass.

Mix	Portland Cement	Water	Soil
1	3%	6%	91%
2	6%	6%	88%
3	9%	6%	85%
4	3%	8%	89%
5	6%	8%	86%
6	9%	8%	83%
7	3%	10%	87%
8	6%	10%	84%
9	9%	10%	81%

. For each mix composition, samples were created at four different initial compressions (2.2 MPa, 4.4 MPa, 8.8 MPa, and 13.2 MPa) and allowed to cure for 7-day and 28-day durations. This range encompasses initial compressions achieved with a manual press [32] and automatic press initial compressions [2]. For each of the 36 combinations of mix composition and initial compression, 10 samples were created, with 5 tested at 7 days and 5 tested at 28 days.

Before creating samples, the soil was dried in an oven overnight, which removed latent water and allowed for more accurate moisture control in the mix composition. Dried soil was then passed through a No. 4 sieve to remove any particles larger than 4.75 mm as CSECs are typically created with a mixture of clay, silt, and sand with no gravel present [2,26,33]. The dried and sieved soil was proportioned by mass (+/− 0.5 g) according to the mix compositions listed in Table 4.

The dry components (Portland cement and soil) were mixed in an electric concrete mixer. Once the dry components had reached a homogeneous mixture, water was slowly added, and mixing continued for another 5 min until the moisture was evenly distributed. While mixing, water-laden soil would collect on the perimeter of the mixer and be scraped periodically to promote evenly mixed and hydrated mixes.

The hydrated mixture was placed in 50.8 mm diameter cylindrical molds with a top and bottom plunger and compressed to the desired initial compression load. The double plunger arrangement allows for compaction from both the top and bottom, reducing the impact of friction between the mixture and mold wall, thus producing more consistent samples [2,33].

Samples were labeled with mix number, initial compression, and sample number. The samples were then placed in an ice chest for temperature regulation and humidity control. The measured temperature was consistently between 23 °C and 26 °C with 100% relative humidity, shown in Figure 2. At the end of curing (e.g., 7 or 28 days), the ends of each sample were sanded planar to encourage direct compression. Planeness was checked twice in perpendicular directions on both the top and bottom of the cylinder. Once sanded, height and weight measurements were collected, and sample densities were calculated. The creation and curing process is summarized in Figure 3.

Samples were tested to failure under direct compression in an MC-500PR Gilson 500 concrete compression testing machine using a process modeled after ASTM C39, the compression protocol for cylindrical concrete specimens [27]. Samples were uniaxially loaded between a spherically mounted, flat platen above and a flat platen driven by a hydraulic piston from below. The testing apparatus is shown in Figure 4. The compression testing machine is load-controlled, and compression tests were conducted at a rate of load 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. Data collected from each test included runtime, peak load, and load rate. Platens were wiped clean between tests to help prevent uneven loading.

3. Results and Discussion

Compressive strength, density, and variability values of CSEC inform the viability and performance of a mix composition as a CSEB [24,25,26]. The compressive strength, density, and variability of nine different mix compositions are reported with statistical significance comparisons.

3.1. Compressive Strength

Compressive strength is one of the key metrics for determining the viability of masonry building materials [24,25,26]. Figure 5 and Figure 6 are box and whisker plots for 7-day and 28-day compressive strength, respectively. The box represents the inner quartile of results and the whiskers extend to the farthest sample within 1.5 inner quartiles of the upper or lower quartile; the outliers beyond the whiskers appear as open circles. Outliers for compression testing on such a heterogeneous material with small sample sizes are not unexpected, and are only removed in instances of obvious measurement error or malformed specimens, which were not observed in this study.

Trends in Figure 5 and Figure 6 illustrate, with notable exceptions, that increases in water content, cement content, initial compression, and cure time increase compressive strength. This is the expected result for stabilized soil as Portland cement requires water to hydrate and form calcium silicate hydrate compounds [13] that bind the aggregate together, thus increasing compressive strength and durability [3,14].

Average compressive strength values and standard deviations are reported in

Table 5. The 7-day compressive strength results.

7-Day Compressive Strength (Mpa)
	Initial Compression
	2.2 MPa		4.4 MPa		8.8 MPa		13.2 MPa
MIX	$\mathbf{Avg} {\mathit{f}}_{\mathit{c}}^{\mathbf{\prime }}$	$\mathit{\sigma }$	$\mathbf{Avg} {\mathit{f}}_{\mathit{c}}^{\mathbf{\prime }}$	$\mathit{\sigma }$	$\mathbf{Avg} {\mathit{f}}_{\mathit{c}}^{\mathbf{\prime }}$	$\mathit{\sigma }$	$\mathbf{Avg} {\mathit{f}}_{\mathit{c}}^{\mathbf{\prime }}$	$\mathit{\sigma }$
1	0.11	0.04	0.18	0.04	0.29	0.12	0.33	0.11
2	0.42	0.11	0.63	0.07	0.62	0.06	0.72	0.15
3	0.40	0.07	0.41	0.13	0.60	0.18	0.66	0.27
4	0.51	0.06	0.45	0.05	0.66	0.08	0.71	0.04
5	0.51	0.05	0.73	0.11	1.00	0.14	1.33	0.14
6	0.98	0.07	1.33	0.15	1.56	0.13	1.86	0.29
7	0.54	0.08	0.25	0.09	0.92	0.12	0.96	0.04
8	1.29	0.11	1.52	0.10	2.03	0.07	2.05	0.08
9	2.08	0.17	2.62	0.13	2.68	0.11	3.40	0.52

and

Table 6. The 28-day compressive strength results.

28-Day Compressive Strength (Mpa)
	Initial Compression
	2.2 MPa		4.4 MPa		8.8 MPa		13.2 MPa
MIX	$\mathbf{Avg} {\mathit{f}}_{\mathit{c}}^{\mathbf{\prime }}$	$\mathit{\sigma }$	$\mathbf{Avg} {\mathit{f}}_{\mathit{c}}^{\mathbf{\prime }}$	$\mathit{\sigma }$	$\mathbf{Avg} {\mathit{f}}_{\mathit{c}}^{\mathbf{\prime }}$	$\mathit{\sigma }$	$\mathbf{Avg} {\mathit{f}}_{\mathit{c}}^{\mathbf{\prime }}$	$\mathit{\sigma }$
1	0.22	0.07	0.18	0.04	0.29	0.06	0.43	0.07
2	0.47	0.06	0.62	0.07	0.49	0.16	0.85	0.23
3	0.53	0.16	0.55	0.07	0.73	0.10	0.94	0.16
4	0.37	0.03	0.53	0.03	0.87	0.10	0.97	0.09
5	1.14	0.08	0.97	0.08	1.59	0.31	1.54	0.14
6	1.48	0.19	1.61	0.14	1.99	0.17	2.83	0.39
7	0.42	0.03	0.67	0.02	0.95	0.07	0.98	0.08
8	1.86	0.10	1.66	0.12	2.17	0.23	2.30	0.26
9	2.21	0.34	2.55	0.22	4.05	0.56	3.87	0.12

for 7-day and 28-day samples, respectively.

CSEC compressive strength values measured in this study are less than the minimum compressive strengths for bricks [34] and concrete masonry units [35] and compare reasonably with strength ranges for various forms of earthen construction [3,4,5,6,7,8,9,13,14]. Averaged over all initial compressions, the lowest strength mix compositions include 1, 2, 3, 4, and 7 and comprise mixes with either (1) low cement content (i.e., 1, 4, 7) or low water content (i.e., 1, 2, 3). Low-cement-content mixes exhibited low compressive strength as expected; while mixes 2 and 3 had higher cement contents (i.e., 6% and 9%, respectively), their low water content did not adequately hydrate the cement. Similarly, the additional water content in mix 4 increases strength beyond mix 1; however, additional water in mix 7 does not result in additional strength. This suggests the 3% cement content of mixes 1, 4, and 7 is fully hydrated between 6% and 8% water.

Mixes 5, 6, 8, and 9 comprise the stronger specimens. Examining the higher-strength mixes with constant water content (i.e., 5 & 6 and 8 & 9), increasing cement increased strength. Similarly, holding cement content constant (i.e., 5 & 8 and 6 & 9), increasing water increased strength.

3.1.1. Variability of Compression Results

Variability that is unexpected and unaccounted for can lead to the development of structural weak points and eventual failure [36]. Variability considerations appear in the Indian and New Zealand codes as a requirement and a means of adjusting out-of-plane wall strength.

In this study, variability is quantified through the coefficient of variation (CV) of compressive strength or density and is defined as the ratio of standard deviation to the mean. It is a measure of relative spread and is a useful metric for comparing sample sets with different mean compressive strengths or densities.

Table 7. Compressive strength coefficient of variation.

Compressive Strength CV
	Initial Compression
	2.2 MPa		4.4 MPa		8.8 MPa		13.2 MPa
MIX	7-Day	28-Day	7-Day	28-Day	7-Day	28-Day	7-Day	28-Day
1	0.34	0.34	0.23	0.22	0.40	0.21	0.33	0.17
2	0.25	0.13	0.12	0.13	0.10	0.34	0.20	0.27
3	0.18	0.30	0.32	0.13	0.31	0.14	0.40	0.17
4	0.13	0.08	0.11	0.05	0.12	0.12	0.05	0.10
5	0.09	0.07	0.16	0.10	0.14	0.19	0.11	0.09
6	0.07	0.13	0.11	0.09	0.08	0.09	0.15	0.14
7	0.14	0.08	0.38	0.03	0.13	0.07	0.04	0.08
8	0.09	0.06	0.07	0.07	0.04	0.11	0.04	0.11
9	0.08	0.15	0.05	0.09	0.04	0.14	0.15	0.03

provides CV values for compressive strength results for each mix, 7- or 28-day cure times, and all initial compression values.

Initial compression and cement content show no consistent trends affecting CV. Increasing water content does reduce CV. For example, Figure 7 illustrates a large reduction in CV between 6% and 8% water content and a smaller reduction in CV between 8% and 10% water content. This suggests that lower amounts of water increase heterogeneity, and thus variation, by insufficiently hydrating the cement in the mix and reducing consistent development of calcium silicate hydrate compounds. Sufficient water content promotes lower CV values.

Cure time has a smaller but statistically significant impact on CV, as 28-day samples are more consistent than 7-day samples. A similar result has been seen in concrete samples where 28-day samples have a lower CV than 7-day samples [37]. Longer curing times permit more complete hydration of cement and lead to more consistent strength values.

3.1.2. Statistical Significance of Compression Results

Statistical significance tests, at the 95% level, were conducted with Welch’s ANOVA and the Games–Howell post hoc test. These statistical tests were selected because the samples violate the assumption of homogeneity of variances. The low-cement-content mix compositions (i.e., 3%) were found to be significantly different from medium- and high-cement-content mix compositions (i.e., 6% and 9%) for all initial compressions and cure times, as seen in Figure 8. Colored highlighted cells represent statistical significance and indicate p-value <0.05. Similarly, the low-water-content mix compositions (i.e., 6%) were found to be significantly different from medium- and high-water-content mix compositions (i.e., 8% and 10%) for all initial compressions and cure times, as seen in Figure 9.

Water-to-cement ratio (WC) has a significant impact on compressive strength as well, but the relationship is not readily characterized by WC alone, as shown in Figure 10a and Figure 11a, which illustrate the relationship between WC and compressive stress for 7-day and 28-day samples, respectively. Mixes with identical water content (Figure 10b and Figure 11b) exhibit an inverse relationship between strength and WC; mixes with identical cement content (Figure 10c and Figure 11c) show an increase in strength with increased WC.

Figure 12 shows the 95% statistical significance level between mix compositions at the same initial compression for both 7-day and 28-day cure times. Figure 12 reveals that when created with the same initial compression (regardless of the magnitude), mix compositions 2, 3, 4, and 7 often have compressive strengths that are not statistically different from each other. These mixes represent four of the five low-strength mixes, and all perform statistically similarly due to low water or low cement contents; mix 1 is statistically lower and represents the extreme, low-cement, low-water mix design. Mix compositions 6 and 8 are not statistically significantly different from each other for 75% of the results. These two mix compositions comprise separate and distinct combinations of cement, water, and soil. This indicates that multiple different mix compositions can achieve similar compressive strengths.

Figure 13 illustrates the 95% statistical significance level between initial compressions within a single mix composition. Changes in initial compression for mix compositions 1, 2, and 3 had statistically significantly different results in only 25% of results. Compression magnitude does not appreciably affect the strength of these low water mixes as their cement content is not adequately hydrated. The medium- and high-water-content mix compositions, 4–9, have statistically significantly different results much more often, 75% and 67%, respectively. A change in initial compression has a greater impact on mix compositions with more water content than mix compositions with less water content.

3.2. Density Results

Density is an important characteristic of building materials and is closely tied to compressive strength [15,16], durability [17,18], and thermal characteristics [15]. Density requirements also appear in some building codes [25,26]. Density was measured prior to compression testing and is graphically shown in Figure 14 and Figure 15 and listed in

Table 8. The 7-day density results.

7-Day Density ${\displaystyle \frac{\mathbf{kg}}{{\mathbf{m}}^3}}$
	Initial Compression
	2.2 MPa		4.4 MPa		8.8 MPa		13.2 MPa
MIX	$\mathbf{Avg} \mathit{\rho }$	$\mathit{\sigma }$	$\mathbf{Avg} \mathit{\rho }$	$\mathit{\sigma }$	$\mathbf{Avg} \mathit{\rho }$	$\mathit{\sigma }$	$\mathbf{Avg} \mathit{\rho }$	$\mathit{\sigma }$
1	1628	44	1704	18	1791	56	1799	10
2	1694	43	1770	39	1785	13	1824	28
3	1692	23	1718	41	1795	7	1845	12
4	1761	4	1766	19	1865	13	1859	3
5	1728	10	1783	17	1854	5	1903	12
6	1740	7	1794	5	1870	11	1909	13
7	1788	8	1799	32	1880	12	1927	14
8	1838	20	1879	9	1951	5	1972	7
9	1857	4	1924	15	1965	20	1995	9

and

Table 9. The 28-day density results.

28-Day Density ${\displaystyle \frac{\mathbf{kg}}{{\mathbf{m}}^3}}$
	Initial Compression
	2.2 MPa		4.4 MPa		8.8 MPa		13.2 MPa
MIX	$\mathbf{Avg} \mathit{\rho }$	$\mathit{\sigma }$	$\mathbf{Avg} \mathit{\rho }$	$\mathit{\sigma }$	$\mathbf{Avg} \mathit{\rho }$	$\mathit{\sigma }$	$\mathbf{Avg} \mathit{\rho }$	$\mathit{\sigma }$
1	1669	34	1700	28	1778	30	1805	18
2	1715	16	1775	8	1778	22	1841	14
3	1730	27	1764	12	1830	13	1865	7
4	1698	19	1751	8	1818	11	1860	7
5	1747	9	1769	11	1857	18	1879	11
6	1784	19	1812	10	1873	11	1918	16
7	1724	12	1803	13	1875	12	1914	7
8	1857	13	1870	23	1934	13	1966	14
9	1838	20	1883	13	1958	13	1973	9

for 7-day and 28-day samples, respectively.

Typically, density increases as Portland cement content, water content, and initial compression increase. This is the expected result as Portland cement particles are much smaller than soil particles and can be more tightly packed together [20,21]. Additionally, water is known to increase soil density until the soil reaches its optimum moisture content, after which additional water content decreases the soil density [19], and initial compression has a logarithmic relationship with density [15]. Cure time does not significantly affect density.

3.2.1. Density Variability

The overall trends of CV in density measurements align closely with those observed for compressive strength, with the exception that the magnitudes of CV for density results are substantially smaller. Data in

Table 10. Coefficient of variation for density.

Density CV
	Initial Compression
	2.2 MPa		4.4 MPa		8.8 MPa		13.2 MPa
MIX	7D	28D	7D	28D	7D	28D	7D	28D
1	0.03	0.02	0.01	0.02	0.03	0.02	0.01	0.01
2	0.03	0.01	0.02	0.00	0.01	0.01	0.02	0.01
3	0.01	0.02	0.02	0.01	0.00	0.01	0.01	0.00
4	0.00	0.01	0.01	0.00	0.01	0.01	0.00	0.00
5	0.01	0.01	0.01	0.01	0.00	0.01	0.01	0.01
6	0.00	0.01	0.00	0.01	0.01	0.01	0.01	0.01
7	0.00	0.01	0.02	0.01	0.01	0.01	0.01	0.00
8	0.01	0.01	0.00	0.01	0.00	0.01	0.00	0.01
9	0.00	0.01	0.01	0.01	0.01	0.01	0.00	0.00

indicate maximum CV values of 0.03 and higher CV values for low-water-content mixes. As an illustrative example, Figure 16 depicts a very similar relationship between CV and water content to that shown in Figure 7.

3.2.2. Statistical Significance of Density Results

Analysis of the density data revealed that changes in water content have a consistently significant impact on sample density, while changes in cement content are significant only for 28-day low-cement-content mix compositions. Figure 17 and Figure 18 show the 95% statistical significance level for changes in cement content and water content, respectively.

Figure 19 shows the impact of changes in mix composition on density. Changes in initial compression have a significant impact on density in 81% of results, as shown in Figure 20. The low-water-content mix compositions have the fewest statistically significantly different results, at 67% of the results.

4. Density and Compressive Strength Prediction Equations

4.1. Model Selection

Equations were developed to predict density, 7-day compressive strength, and 28-day compressive strength based on mix composition and initial compression. An ordinary least squares (OLS) regression was used to develop the regression equations and was evaluated for violations of linearity, normality of errors, and multivariate collinearity assumptions. Linearity was confirmed through well-known results from the literature (e.g., compressive strength vs. density) or by visually using data from the current study. Multivariate collinearity was confirmed through variable inflation factor (VIF) analysis. Normality of errors was evaluated via Q-Q plots and Shapiro–Wilk normality tests, chosen due to the small sample size; the normal distribution of errors was confirmed for density, but was narrowly violated for compressive strength.

Alternative regression techniques were for meaningful changes to coefficients, and improvements to $R^2$ and mean absolute percent error (MAPE). These alternatives included the generalized linear model (GLM), generalized least squares (GLS), Huber regression (Huber), and random sample consensus (RANSAC). Regression coefficients, $R^2$, and MAPE were unaffected by the model choice; therefore, OLS was utilized for this study.

First, a group of candidate independent or composite variables that are pertinent to the theory or appear in the literature were identified. Variables were systematically evaluated using measured data via the adjusted $R^2$, statistical significance, and the stability regression constants. This process continued until the improvements were no longer statistically significant. Figure 21 illustrates the details of this process for developing the linear regressions.

4.2. Density Prediction

Five variables were evaluated for inclusion in the density prediction equation: Portland cement content (C), water content (W), initial compression (IC), water-to-Portland cement ratio (WC), and water-to-soil ratio (WS). The independent variables C, W, and IC comprise the base constituents that define the sample. Composite variables WC and WS represent interactions between water and cement and water and soil and are commonly used as metrics for concrete strength analysis [22] and density prediction [19].

Table 11. Prospective variables for density prediction equation.

Variable	Type	Symbol	Range
Portland Cement Content	Independent	C	3–9% of mass
Water Content	Independent	W	6% –10% of mass
Initial Compression	Independent	IC	2.2 MPa–13.2 MPa
Water to Portland Cement Ratio	Composite	WC	0.67–3.3%
Water to Soil Ratio	Composite	WS	6.5–12.3%

lists the prospective variables that were evaluated for the density prediction equation.

C, W, WC, and WS were all found to have approximately linear relationships with density, especially at high IC levels. IC is known to have a natural logarithmic relationship with density [15] and was linearized for the purposes of conducting a linear regression by taking the natural logarithm of IC.

Figure 22 shows the adjusted $R^2$ and mean absolute percent error (MAPE) values for the tested models, and the orange highlighted bar represents the selected model. The inclusion of C and W variables offered minor improvements in adjusted $R^2$ that proved statistically insignificant, and no improvement in MAPE. Equation (1) shows that IC and WS had a positive correlation with density, while WC had a negative correlation with density. This equation performs well on the experimental data, with an adjusted $R^2$ of $0.906$, and the performance against experimental data is shown in Figure 23.

$$\rho =81.1\ln \left(IC\right)+2881.39WS-22.68WC+1448.31\hspace{1em}{R}^2=0.906$$

(1)

4.3. Compressive Strength Prediction

The five variables evaluated for density Equation (1) were evaluated for the compressive strength equation; predicted density ($\rho$) was also included. Variables are listed in

Table 12. Prospective variables for compressive strength prediction equation.

Variable	Type	Symbol	Range
Portland Cement Content	Independent	C	3–9% of mass
Water Content	Independent	W	6–10% of mass
Initial Compression	Independent	IC	2.2 MPa–13.2 MPa
Water-To-Portland Cement Ratio	Composite	WC	0.67–3.3%
Water-To-Soil Ratio	Composite	WS	6.5–12.3%
Predicted Density	Composite	$\rho$	1650 ${\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}$–2000 ${\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}$

.

C, W, and WS were all found to have a linear relationship with compressive strength. Density is known to have a linear relationship with compressive strength for both CEB and concrete samples [15,16]. WC is known to have a nonlinear relationship with compressive strength for concrete samples [22,35] and was converted to a linear relationship through Equation (2); A and B are experimental parameters for a given age, material, and cure conditions [38]. This paper uses the values for A and B proposed in [38] converted to metric units [39].

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

(2)

Following the same procedure as described for developing the density prediction, a linear regression was developed for both 7-day and 28-day samples. In both cases, the best-performing equation excluded W, WS, and IC. The excluded variables impact the compressive strength equations through their effect on density.

Figure 24 shows the compressive strength prediction performance for both 7-day and 28-day samples, with the selected models highlighted in orange. Equations (3) and (4) are the best-performing models for 7-day and 28-day samples, respectively. Some of the more complicated models had adjusted $R^2$ and MAPE values that were slightly improved over the listed models, but those proved to be statistically insignificant improvements.

$$f_c^{\prime }=0.004\rho +0.385C-0.086W{C}^{\prime }-7.521\hspace{1em}{R}^2=0.900$$

(3)

$$f_c^{\prime }=0.004\rho +0.482C-0.104W{C}^{\prime }-8.826\hspace{1em}{R}^2=0.901$$

(4)

Figure 25 shows the predicted vs. experimental compressive strength for 7-day and 28-day samples. Both predictive equations have adjusted $R^2$ values, 0.900 and 0.901, respectively. Extreme values of experimental results occur in mix 9 (Figure 25a) and in mixes 6 and 9 (Figure 25b) and represent unusually strong specimens, which can also be observed in Figure 5 and Figure 6. In general, the models’ prediction accuracy decreases when considering the edge case mixes (1 and 9); this is typical in regression modeling and the inclusion of data from additional testing of mixes in the neighborhood of these cases should improve model accuracy. Specifically, more mixes with high cement content should be incorporated.

4.4. Mix Design Guideline

Figure 26 plots density and compressive strength together and illustrates that there are many mix compositions of water and cement that will provide the same predicted density or compressive strength. Each quadrant is data for a specific initial compression. The red and yellow lines represent predicted density values. The black and gray lines represent predicted 28-day compressive strength values. The black dots represent the nine mix compositions tested in this paper; curves for density and compressive strength represent estimations from Equations (1) and (4).

According to IBC 2021, CSEBs are required to have minimum compressive strengths of 2 MPa [24]. When creating CSEBs with a manual press capable of initial compressions ~2.2 MPa, the mix composition must have at least 9% Portland cement and 8.7% water, or 6.8% Portland cement and 10% water, as seen in Figure 26.

If density is a primary consideration and CSEBs created in a manual press need to meet a minimum density of $1750{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}$ [26], then the mix composition must have at least 9% Portland cement and 7.6% water, or 3% Portland cement and 9.5% water. Figure 26 provides a useful tool for determining optimal mix compositions based on code requirements and production capabilities.

5. Conclusions

This paper studied the results of CSEC samples created from nine different mix compositions with four different initial compressions. Changes to mix composition have a statistically significant impact on compressive strength for 77% of results and on density for 68% of results. Changes in initial compression have a statistically significant impact on compressive strength for 56% of results and on density for 81% of results. Mix composition has a more significant impact on compressive strength than initial compression, and initial compression has a more significant impact on density than mix composition. Cure time does not have a statistically significant impact on the density of CSEC samples cured in a sealed container, and sample density is determined by mix composition and initial compression. Water content and cure time were found to have a statistically significant impact on the coefficient of variation for CSEC samples, with the coefficient of variation decreasing as water content and cure time increased. Predictive equations for density, 7-day compressive strength, and 28-day compressive strength were developed with R² values of 0.906, 0.881, and 0.890, respectively. Density was found to be a function of initial compression, water-to-soil ratio, and water-to-cement ratio; 7-day compressive strength was found to be a function of density, cement content, and water-to-cement ratio; and 28-day compressive strength uses the same variables as 7-day compressive strength with the inclusion of initial compression.

This work provides specific mix design guidelines for sandy soil. Other soil types are expected to have similar relationships with regard to water and cement contents, but coefficients will change as soil type changes. For example, the presence of fines or clay in a soil will affect the amount of binder required to achieve the desired strength. Density predictions would also be affected; for example, clay particles are smaller than those in a sandy soil and their volume is more susceptible to variation as a function of water content. Additional studies are needed to collect representative data from many more soil types to achieve a broad set of design aids toward standardization. Current and future efforts include characterizing the effect of the mechanism of block production on CSEB performance and variability.