Optimization of the Granular Mixture of Natural Rammed Earth Using Compressible Packing Model

Abstract

Rammed earth (RE) construction is an ancestral technique that allows for the building of durable and resistant constructions. RE buildings are sustainable and environment-friendly, and ensure energy optimization during the construction cycle. For these reasons, many of the following RE characteristics are studied: mechanical strength, seismic resistance, and thermal performance. However, the mix design of RE soils has been rarely studied. There is practically no scientific approach that allows for defining precise dosages of clay, silt, sand, and gravel used in RE materials. The broader aim of this article is to determine a scientific mix design method to find an optimal RE granular mixture. The compressible packing model (CPM) is applied to study the effect of every granular class on compactness and define the optimum mixture. Many tests have been conducted such as the Modified Proctor, compression test, and ultrasonic velocity pulse test to evaluate the relevance of this model. The results suggest many granular corrections for RE material that considerably enhance compactness and unconfined compressive strength (UCS). It was found that one granular correction provides a 137% increase in the initial UCS. Therefore, this approach enables the defining of which granular class is to be added or reduced to optimize the mechanical properties of RE.

1. Introduction

Conventional construction materials have shown serious and continuous effects on the environment for many decades, especially global warming, an increase in non-renewable waste materials, and energy consumption [1]. Engineers, constructors, and architects are expressing increasing interest in sustainable solutions, especially natural and unconventional construction materials like earth. Compared to concrete, natural earth materials have shown their relevance and low energy demand during the construction life cycle (extraction, transportation, construction, exploitation, demolition, and recycling) [2].

All over the world, earth construction is considered a cultural heritage and technical know-how that should be preserved and transmitted from generation to generation. Rammed earth (RE) is among the principal types of earth construction present in many regions such as central and western Asia, southern Europe, north Africa, Australia, and American continents. RE materials are defined as a mixture of clay, silt, sand, gravel, and water moisture with the possibility of adding stabilizing agents such as lime or cement at limited proportions. This granular mixture is compacted in formwork in successive layers to obtain at least 98% of the optimum dry density [3]. Unstabilized Rammed Earth (URE) constructions have proven to be durable, resistant, and energy efficient even without the use of added stabilizing agents. However, Stabilized Rammed Earth structures (SRE) have generally higher properties than URE. Therefore, scientists are reconsidering this material to study its compactness, density, durability [4], compressive strength [5], Young modulus, and seismic behavior [6]. These characteristics depend essentially on the used soil type [7], the shape of aggregates [8], and compaction energy [9]. To understand RE mechanical behavior, it is important to choose the granular mixture which ensures optimal compactness. Several norms and technical documents were published to assess the main recommendations and guidelines to respect RE constructions, which mainly focus on specifying the soil characteristics and procedures of construction of RE walls.

In all these norms and standards, the main requirement to justify soil suitability for RE construction is to define the granular composition through the particle size distribution (PSD) and compare it to the limits given by each normative document. The limits specify the ranges of clay, silt, sand, and gravel proportions. According to these standards, the percentages of each RE constituent should belong to specified intervals to justify soil suitability for RE constructions.

However, a notable difference in the limits of intervals of granular classes is observed among the main standards. It is necessary to predict the best granular composition with optimal compactness, a lower void ratio, and optimal mechanical strength through the Compressible Packing Model (CPM). The CPM considers important parameters to predict the compactness of every granular mixture to use in RE constructions. It considers the elementary compactness of each granular class and its dosage, the compaction index K (Which describes the compaction energy used on site), and the wall effect. In this article, the CPM is applied to find the optimal choice of the constituents’ dosages for RE materials that will provide the best compactness and mechanical strength.

The aim of this study is to reach a notable enhancement of the compressive strength and Young’s modulus as well as increase the compactness of natural RE material. Moreover, a practical method is proposed to correct an extracted RE material and optimize its granular mixture.

This article is therefore organized into the following four sections. Section 2 presents the summarized review of standards, technical norms, and scientific articles related to RE, as well as the mixture design methods which include the CPM. Section 3 aims to apply the CPM in a case study of RE soil extracted from a region in the south of Morocco. It also presents the experimental protocol conducted to define the elementary compactness of each granular class required for CPM. Additionally, experimental compactness and compressive strength are then measured in a laboratory to evaluate the relevance of this model. Section 4 provides a discussion of the obtained results and summarizes the approach to follow to design an optimal RE soil. Finally, the conclusion presented in Section 5 summarizes the notable enhancement results and some limitations of the model and foresees the perspectives for the progress of RE material mix design.

2. Review of Standards, Norms, Articles, and Mix Design Methods for RE Construction

2.1. Summarized Review of Norms, Standards, and Scientific Publications Related to RE Mix Design

In the main standards and norms about rammed earth, soil suitability for construction is generally justified by the PSD and some soil identification tests (Atterberg limits, Proctor test, and Methylene blue stain test). Some technical documents, such as the CRA-Terre guide [10], also provide generic criteria that must be considered, such as a red color of the soil, having relatively low moisture, and being non-organic. “NZS 4298: Materials and Workmanship for Earth Buildings “(NZS 4298) [11] and “The Australian earth building handbook” (HB 195) as well as the Moroccan regulation “Seismic Earth Construction Regulation” (RPCT 2011) justify soil suitability for RE construction by a minimum and a maximum percentage of each granular class: clay, silt, sand, and gravel (

Table 1. Normative material requirements for rammed earth.

	Gravel (%)	Sand (%)	Silt (%)	Clay (%)
RPCT 2011	2–10%	32–58%	8–16%	8–26%
NZS 4298	50–70%		15–30%	5–15%
CRA-Terre guide	0–15%	40–50%	20–35%	15–25%
HB 195	45–75%		10–30%	0–20%

).

In the literature, Ciancio et al. [12] conducted many experimental tests to prove the insufficiency of PSD as a soil suitability criterion. Ten artificial soils were prepared in a way to respect four guidelines: Bulletin 5 [13], HB 195 [3], NZS 4298 [11], and requirements recommended by Walker et al. [14]. Suitability was analyzed via three tests: the dry unconfined compressive strength (UCS), acceleration erosion test, and drying shrinkage test. All the non-stabilized samples (five of ten samples) had a UCS less than the minimum value required in the guidelines even if the material requirements were satisfied.

Hall and Djerbib [15] conducted a series of tests to evaluate the dry density and compressive strength of several mixtures suitable for RE. Each mixture contains a mass of sand, pea gravel, and clay defined according to NZS 498. The results enable the analysis of the effects of clay, sand, and gravel dosages on mechanical performances. Mixtures with a high gravel dosage have better dry density especially when the sand/clay ratio is limited. On the other side, samples with an important sand dosage (70 to 80%) and a gravel dosage of up to 10% are characterized by lower dry density and compressive strength. However, better UCS was observed in some samples where clay dosage was higher than the limits given by HB 195, RPCT 2011, and the CRA-Terre guide.

On the other side, the shape of aggregates and the compaction energy are important parameters to evaluate soil suitability [16]. A study by Koutous et al. [8] on the effect of grain shape on the mechanical behavior of RE shows that the stiffness increases when aggregates with a rounded shape are used while angular-shaped aggregates provide strength to the material.

However, few researchers define RE granular composition via a scientific approach considering RE technique and soil composition. For example, T. Bui et al. [17] conducted an experimental study to analyze the mechanical behavior of RE walls using 50% of extracted earth soil, 30% of gravel 4/11 mm, and 20% of sand 0/4 mm. These dosages were designed according to the Andreasen & Andersen packing model [16] suitable for concrete. In another study to characterize earth concrete and yellow earth, N. Soro [18] has considered the Absolute Volumes method, which is specific to concrete material as well as the Substitution method which considers replacing cement with clay.

These last research studies consider RE as concrete material. Nevertheless, besides the PSD criteria, many parameters such us the compaction energy and the formwork are to be evaluated to design a compact RE wall. There are currently no relevant studies that propose a scientific approach to define clay, silt, sand, and gravel dosages to optimize RE material’s compactness. Current practices explain the need to develop a scientific mix design method for RE. Hence, the choice of dosages of each component must be carefully studied to ensure better mechanical performance.

A preliminary investigation of the mix design methods applied to all granular materials is necessary. It presents some differences between concrete and RE materials. Moreover, three main mix design models used for concrete that can be applied for all granular materials are presented below.

2.2. Mix Design Theory and Models for RE Material

There are some resemblances between RE material and concrete: the two materials contain a volume of water, voids, and several granular classes. The mixture must ensure maximum compactness and sufficient compressive strength. Besides, both materials resist better to compression than tension. However, their granular composition is not similar because of the presence of an important quantity of fine constituents including clay and silt in RE soil. The process to prepare the mixture is also different. While the concrete mixture is a recipe prepared with separated granular classes, RE materials are extracted in situ and used directly with some approximate corrections. A particle size analysis and a laser granulometry test are to be conducted to have PSD which provides the size of the aggregates and the proportion of each granular class.

Additionally, concrete is usually poured while rammed earth is compacted. Therefore, the compaction index and the wall effect should be reconsidered when using the RE technique. The first step of the mix design method for RE consists of evaluating all the factors that influence its mechanical performance, especially the compaction energy. Then, theoretical models to design concrete are analogically used for RE mix design.

Some early models like the Linear Packing Density Model of Grain Mixtures [19] and the Mooney Viscosity model [20] suppose that compactness and viscosity are related to grain texture, grain roughness and compactness, and the dosages of each granular class. However, compactness is the result of an experience related to a given material, according to a certain execution process.

A synthesis of the two previous models contributed to the development of a more satisfactory model like the Solid Suspension model (SSM) [21]. Many specifications made this model more practical because of the consideration of the compaction process and the wall effect.

An improved version of this model was proposed by De Larrard [22] by reconsidering all the parameters mentioned above in addition to the compaction index quantifying the compaction intensity. This model is called the compressible packing model (CPM).

The CPM introduces virtual compactness γ which is the maximum compactness that a granular stack can achieve when each particle is placed into the mix, one by one. Virtual compactness varies from 0.64 for a stack of spherical grains to 0.74 for a compact hexagonal stack [23]. The model predicts the compactness of a mixture from the PSD and the elementary compactness of each class C_i, measured in the laboratory [24]. It defines the virtual compactness of a mixture composed of n granular classes when a given class i is dominant (Equation (1)). Then the virtual compactness γ of the mixture can be deduced (Equation (2)).

$${\mathsf{\gamma }}_{\mathrm{i}}=\frac{{\mathsf{\beta }}_{\mathrm{i}}}{1-{{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{i}-1}\left(1-{\mathsf{\beta }}_{\mathrm{i}}+{\mathrm{b}}_{\mathrm{ij}}{\mathsf{\beta }}_{\mathrm{i}}\left(1-\frac{1}{{\mathsf{\beta }}_{\mathrm{j}}}\right)\right)·{\mathrm{y}}_{\mathrm{j}}-{{\displaystyle \sum }}_{\mathrm{j}=\mathrm{i}+1}^{\mathrm{n}}\left(1-\frac{{\mathrm{a}}_{\mathrm{ij}}{\mathsf{\beta }}_{\mathrm{i}}}{{\mathsf{\beta }}_{\mathrm{j}}}\right)·{\mathrm{y}}_{\mathrm{j}}}$$

(1)

$$\mathsf{\gamma }=\min \left({\mathsf{\gamma }}_{\mathrm{i}}\right)$$

(2)

where γ_i is the virtual compactness of the mix when class i is dominant, and y_i is the volume fraction of a class i relative to the total volume of the solid mixture. Loosening and wall effects are considered via the coefficient a_ij (Equation (3)) and b_ij (Equation (4)) which depend on the average diameter for each granular class i (d_i).

$${\mathrm{a}}_{\mathrm{ij}}=\sqrt{1-{\left(1-\frac{{\mathrm{d}}_{\mathrm{j}}}{{\mathrm{d}}_{\mathrm{i}}}\right)}^{1.02}}$$

(3)

$${\mathrm{b}}_{\mathrm{ij}}=1-{\left(1-\frac{{\mathrm{d}}_{\mathrm{i}}}{{\mathrm{d}}_{\mathrm{j}}}\right)}^{1.5}$$

(4)

where d_i is the granular class average diameter. It is the arithmetic average of two sieves diameters that limit each granular class.

The virtual elementary compactness β_i of a class i stacked all alone is deduced from the experimental elementary compactness C_i (Equation (5)).

$${\mathsf{\beta }}_{\mathrm{i}}=\frac{{\mathrm{C}}_{\mathrm{i}}}{1-\left(1-{\mathrm{k}}_{\mathrm{w}}\right)\left(1-{\left(1-\frac{{\mathrm{d}}_{\mathrm{i}}}{\mathsf{\phi }}\right)}^2·\left(1-\frac{{\mathrm{d}}_{\mathrm{i}}}{\mathrm{H}}\right)\right)}$$

(5)

where k_w depends on the shape of aggregates. It is equal to 0.87 for rounded aggregates and 0.71 for angular ones. φ and H are the mould dimensions which represent, respectively, the diameter and the height of the mould used to measure the elementary compactness in laboratory.

Moreover, the CPM links the virtual compactness γ, impossible to reach in practice, to the real compactness C by introducing the compaction index K (Equation (6)). It is an index that depends solely on the compaction means. The coefficient K is as important as the compaction energy. Pouliot et al. [25] conducted a series of tests on roller-compacted concrete (RCC) to measure the compaction index. Values vary from 7 to 14.9 depending on the mix composition and its real porosity. An average of 12 was fixed for RCC to calculate compactness and study different mixtures for this material. In the case of RE, the compaction index could be lower because of the different compacting process used.

$$\mathrm{K}={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{K}}_{\mathrm{i}}={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\frac{\frac{{\mathrm{y}}_{\mathrm{i}}}{{\mathsf{\beta }}_{\mathrm{i}}}}{\frac{1}{\mathrm{C}}-\frac{1}{{\mathsf{\gamma }}_{\mathrm{i}}}}$$

(6)

In practice, the elementary compactness C_i is measured according to the protocols described by T. Sedran et al. [23] and J. Lecomte [24]. The virtual compactness γ_i is then defined from Equation (1). For a given compaction index K which is deduced from experimental results, the value of the real compactness C is then calculated (Equation (6)).

CPM can be a potential mix design model for RE material. Different dosages can be studied to find the optimal mix for maximal compactness. In the next section, the CPM is applied to experimental soil to define the optimal granular mixture for any extracted soil. It enables the correction of any RE granular mixture by just knowing the elementary compactness and the density and the percentages of its granular classes.

3. Materials and Methods

3.1. Context of the Study

For RE construction, it is preferable to use on-site soil from the ground or available natural soils. In Morocco, RE constructions are present in the south and southeast regions [26]. Kasbahs and battlements like Ait Ben Haddou Ksar fortified village (UNESCO world heritage since 1986) are proof of the importance of earthen construction in Moroccan culture. In some regions where the climate is rude and dry, building with earth ensures thermal and hygrometric comfort in both cold and hot seasons and is adapted to the socio-economic environment [27]. Many projects have been conducted to restore ancient buildings [28], and some local initiatives are taken out to promote earth construction and encourage populations and foreigners to choose this solution.

The soil E1 is extracted from the south of Morocco, in the Tahnaout region near Marrakech. It is provided by Argilex bioconstruction, one of the local earth construction firms in Morocco.

The following (

Table 2. Norms used in this study.

Test	Standard/Technical Document Used
Water content	NF P 94-050 [29]
Sieving	NF P 94-056 [30]
Laser granulometry	ISO 13320: 2020 [31]
Atterberg limits	NF P 94-051 [32]
Absolute density	NF P 94-054 [33]
Dry bulk density	NF EN ISO 11272 [34]
Elementary compactness	LCPC protocol [23]
Modified Proctor test	NF P 94-093 [35]
Specific gravity of soil solids by Pycnometer method	ISO/TS 17892-3: 2004 [36]
Ultrasonic pulse velocity test	NF EN 12504-4 [37]
Compressive strength	NF EN 12390-4 [38]

) shows the standards and technical documents used in this study and covered by this paper.

3.2. Experimental Soil Characterization

Sieving test (according to NF P 94-050) [30] was carried out on the original soil (E1) and another sieved soil, denoted E2, obtained by eliminating particles with diameter higher than 10 mm (Figure 1). The grain size curves show that the gravel percentage in E1 (43%) does not satisfy normative requirements while E2 respects only HB 195.

Since normative documents have precise limitations on silt and clay proportions, a laser granulometry test was conducted according to ISO 13320:2020 [31]. As a result, E1 contains 1.16% of clay and 19.45% of silt, while E2 contains 1.67% of clay and 28.04% of silt (Figure 2). The sieved soil E2 fits well into the grain size interval recommended by the Australian earth building handbook HB 195. However, E1 and E2 cannot be considered suitable for RE according to NZS 498, CRA-Terre, and RPCT 2011.

The soil classification (acc.to NF EN ISO 14 688-2 [39]), as well as the other properties of E1 and E2 including Atterberg limits (acc.to NF P 94-051 [32]), water content (acc.to NF P 94-050 [29]), optimum compaction moisture content (w_opt), and maximum dry density (ρ_opt) (acc.to NF P 94-093), are summarized in

Table 3. Soil identification.

	Proportions (%)		Liquid Limit wL (%)	Plasticity Index (Ip)	Water Content w (%)	Wopt (%)	ρopt (Kg/m3)	Dmax (mm)
E1	Clay	1.16	27.6	8.2	6.4	9.9	1872	50
	silt	19.45
	sand	36.39
	gravel	43
E2	clay	1.67	27.4	7.8	6.1	9.58	1850	10
	silt	28.04
	sand	53.11
	gravel	17.18

.

3.3. Elementary Compactness

To define the elementary compactness of different granular classes, The Central Laboratory of Bridges and Pavements (LCPC) [23] proposes an experimental protocol for fine elements (clay and silt), sand, and gravel.

According to NF P 94-056 [30], four granular classes are to be considered:

• Clay: Particle size (PS) less than 0.002 mm;
• Silt: PS between 0.002 mm and 0.06 mm;
• Sand: PS between 0.06 mm and 2 mm;
• Gravel: PS between 2 mm and 60 mm.

However, it is practically difficult to separate clay and silt in site or in the laboratory to define their elementary compactness. To propose a practical method, clay and silt are considered as one granular class that we call the “Fine elements” class (FE) (Figure 3).

3.3.1. Fine Elements (FE) Elementary Compactness

As recommended in the works of Sedran et al. [23], a need of water test is conducted. It consists of progressively mixing 715 g of fines (P_p) by adding a controlled quantity of water until the soil has a texture of a smooth dough (Figure 4). The added quantity of water P_e consequently enables the calculation of the elementary compactness C_FE of FE (Equation (7)).

$${\mathrm{C}}_{\mathrm{FE}}=\frac{1000}{1000+{\mathsf{\rho }}_{\mathrm{f}}\ast \frac{{\mathrm{P}}_{\mathrm{e}}}{{\mathrm{P}}_{\mathrm{p}}}}$$

(7)

where ${\mathsf{\rho }}_{\mathrm{f}}$ is clay or silt density (NF EN ISO 11272) [2].

3.3.2. Sand and Gravel Elementary Compactness

To measure the compactness of sand and gravel, a precise quantity (7.5 kg of gravel and 3 kg of sand) is vibrated in a sieve shaker for 1 min under a pressure of 10 kPa applied by a steel piston (Figure 5). Then, the final volume is measured in both cases [23]. Equations (8) and (9) express the elementary compactness of sand C_s and gravel C_g.

$${\mathrm{C}}_{\mathrm{s}}=\frac{3}{{\mathrm{V}}_{\mathrm{f}}·{\mathsf{\rho }}_{\mathrm{s}}}$$

(8)

$${\mathrm{C}}_{\mathrm{g}}=\frac{7.5}{{\mathrm{V}}_{\mathrm{f}}·{\mathsf{\rho }}_{\mathrm{g}}}$$

(9)

where ${\mathrm{V}}_{\mathrm{f}}$ is the final volume after vibration and ${\mathsf{\rho }}_{\mathrm{s}}$ and ${\mathsf{\rho }}_{\mathrm{g}}$ are, respectively, sand and gravel densities measured in the laboratory.

3.3.3. Experimental Compactness and Compaction Index

The theoretical compactness C of the granular mixture predicted by CPM is compared to experimental compactness C_exp. In order to find C_exp, a pycnometer test is realized to define the specific gravity of different solid soils γ_s (acc.to ISO/TS: 17892-3: 2004). The specific gravity of dry soils γ_d of all specimens is measured after 28 days post confection and the void ratio is deduced (Equation (10)). The experimental compactness is expressed in Equation (11).

$$\mathrm{e}=\frac{{\mathsf{\gamma }}_{\mathrm{s}}}{{\mathsf{\gamma }}_{\mathrm{d}}}-1$$

(10)

$${\mathrm{C}}_{\exp }=\frac{1}{1+\mathrm{e}}$$

(11)

Once the CPM parameters are determined (Dosages, Virtual compactness, and the experimental compactness), the compaction index K_exp is determined from Equation (6). This parameter describes the compaction energy used in the laboratory and can be considered to evaluate the compactness of any recipe prepared by the experimental soil described above.

3.4. Preparation of Specimens

Three RE samples were prepared from the original soil: E1, the soil E2, as well as the optimized granular mixture E3. These samples were made in cylindric molds whose diameter is equal to 15 cm while their height is equal to 20 cm. The compaction specifications are the same as the Proctor test: five layers for a total height of 20 cm compacted with 56 blows for each layer. The water content used for each soil is the optimum water content defined by the Proctor test.

After demolding, specimens are stored and dried in the laboratory of Hassania School of Public Works (EHTP) at room temperature and humidity for 6 weeks (Figure 6).

3.5. Mechanical Tests

An ultrasonic pulse velocity (UV) test is conducted on each specimen to measure the ultrasonic pulse velocity (acc.to NF EN 12504-4 [37]). The test is carried out by direct transmission to the transducer faces and the specimen surfaces (Figure 7). Materials with a lower UV have lower density and strength. The objective of this test is to compare the UV of E2 and E3 samples and predict their stiffness and compressive strength.

The compression test was carried out on all samples with a compression testing machine with a loading speed of 1.27 mm/min (Acc.to NF EN 12390-4 [37]) after flattening their upper surface. The compression test is performed with a vertical displacement transducer to measure vertical displacements of the specimen and provide the load-displacement curve (Figure 7).

4. Results and Discussions

4.1. Experimental Compactness and RE Compaction Index

According to the LCPC protocol, CPM parameters are measured for each granular class.

Table 4. CPM parameters (Soil E2).

	FE	Sand	Gravel
Density (Kg/m3)	775.5	1312	1600
Average diameter di (mm)	0.03	1.03	6
Elementary compactness Ci	0.733	0.481	0.778
Corrected elementary compactness βi	0.842	0.550	0.912

summarizes the results to implement in the model.

In the literature, compaction index values considered for concrete vary from 4.1 to 9 [22] depending on the concrete placement process (

Table 5. Different values of the compaction index in literature.

Placement Process	Pouring [40]	Tamping with a Rod [41]	Vibration [22]	Vibration and 10 kPa Pressure [22]	Roller Compacted Concrete [25]
K	4.1	4.5	4.75	9	14

). Experimental compactness C_exp of studied samples is implemented in the CPM which allows for defining the experimental compaction index. In this case, the compaction energy is characterized by an average compaction index K_exp equal to 7.48 (

Table 6. Experimental and theoretical compactness.

Recipe	FE Dosage (%)	γd (KN/m3)	Void Ratio e	Cexp	Kexp	C
E1	20.61	18.87	0.339	0.747	7.39	0.746
E2	29.71	17.14	0.374	0.710	7.58	0.706

). The obtained K_exp is considered a logical value and can be adapted for other theoretical simulations with the CPM.

By fixing K_exp, theoretical compactness is determined (Table 6). Since the compaction energy is well known (Modified Proctor), different granular mixtures can be studied to find the optimum mixture with higher compactness. Therefore, granular corrections of the original soil can be defined.

4.2. Granularity Effect on the Mixture’s Compactness

The objective of this section is to evaluate the effect of FE and define the optimum dosage that ensures higher compactness.

The first observation is that E1 is not suitable for RE according to norms and documents considered in this article, while E2 satisfies HB 195 [3] material requirements. According to the CPM, E2 compactness is around 0.706 and it is possible to simulate several mixtures to enhance the initial compactness.

The first simulation consists of fixing sand dosage and varying FE and gravel dosage. As is shown in Figure 8, the optimum FE (FE_opt) dosage that ensures the maximum compactness depends on the sand dosage. Compactness is lower when the sand dosage is higher and FE_opt takes higher values. The granular structure contains less gravel, and its stiffness and compactness are, therefore, lower.

In the second simulation, the FE dosage effect is analyzed when gravel dosage changes. As can be perceived in Figure 9, compactness increases when gravel dosage is higher. FE_opt varies from 20% (When gravel dosage is around 40%) to around 50% for a minimum of gravel. It means that it is possible to fix the gravel dosage for any extracted soil and deduce FE_opt and sand dosage using the CPM equations. If gravel dosage should be limited to ease compaction and ensure soil cohesion (Table 1), these simulations prove that material requirements and limits given in some norms can be revised to cover higher gravel dosage and achieve better compactness.

In practice, these simulations can be used to:

• Choose the right dosage and the right granular class to add or reduce to optimize compactness;
• Evaluate the effect of any granular class on compactness;
• Define the granular corrections considering available raw materials in site.

In the case of the mixture E2, with 17% of gravel, the CPM allows for prediction compactness for many mixtures with different FE dosages (from 0 to 63%). For example, Figure 10 gives different values of compactness when gravel dosage is fixed at 17%. The optimum compactness characterizes a mixture of 38% of FE and 45% of sand. This mixture E3 has a predicted compactness of around 0.782 which implies an enhancement of 10.76%. E3 samples are characterized by a dry density equal to 18.53 kg/m³, a void ratio of 0.283, and an experimental compactness of 0.779 (Comparable to the theoretical value). To prove the relevance of this result, mechanical tests are conducted to compare the mixture samples of E2 and E3. Nevertheless, E1 was not mechanically tested because it is unsuitable still for RE construction and it has been used just to determine the compaction index.

4.3. Mechanical Performances: Stiffness and Compressive Strength

The first test conducted is the ultrasonic pulse velocity test. It enables the evaluation of the longitudinal ultrasonic wave velocity (UV) in E2 and E3 samples to compare their stiffness.

Table 7. UV, UCS, and Young’s modulus average of E2 and E3.

Soil	UV (m/s)	UCS (MPa)	Young’s Modulus (MPa)
E2	500	1.16	31.43
E3	1180	2.75	133.51

shows that the E3 sample has a maximum UV, while the original E2 is characterized by a lower UV. Even if the UV norms are specifically used for concrete, they can be considered to find the RE stiffer sample.

The found UCS of the E2 samples is 1.16 MPa, while the found UCS of the optimized E3 samples is 2.75 MPa. The stress–strain curves of both samples are presented in Figure 11. This result means that the granular correction enables the enhancement of the UCS by 137% without adding stabilizers such as cement or lime.

A review realized by Avila et al. [42] synthesizing the major studies on mechanical properties of unstabilized rammed earth (URE) shows that RE UCS is in the range of 1 MPa to 2.5 MPa, while the Young’s modulus of URE varies from 60 MPa to 1000 MPa. It means that E3 has a high UCS compared to the results in the literature. Moreover, the Young’s modulus of E3 reaches 133 MPa and its stiffness is four times higher than E2. Even though there is no clear relationship between compactness and UCS, these results show that the higher the compactness is the better the UCS will be.

5. Conclusions

Rammed earth is an eco-friendly material to reconsider as a sustainable solution and an alternative to conventional materials. Constructors, engineers, and architects needs scientific results to make RE design and construction easy and justified. The objective of this paper was to apply the CPM on RE material to define a precise and optimal granular mixture that provides higher compactness and notably better mechanical performance. The relevance of this theoretical model is evaluated through an experimental protocol for two different recipes: The original soil (E2) and the optimized granular mixture (E3). The results show that any extracted grounded soil can be corrected by adjusting the dosages of its granular classes in order to achieve optimum compactness.

The following conclusions can be made:

• Existing norms and technical documents give suitable grading for rammed earth without considering aggregate types and their granular arrangement after compaction. Some normative dosages do not ensure optimal compactness and compressive strength and can be revised;
• Compaction index K has to be defined before starting simulations. Ramming operations in the laboratory should resemble real conditions. Further investigations and more tests could be carried out to estimate an average compaction index for the rammed earth technique;
• A significant dispersion of compaction index values could be observed especially when manual compaction is used on site. Therefore, determining its value is always necessary;
• The optimum dosage of fine elements, sand, and gravel depends on the original soil composition, the elementary compactness of each granular class, and raw granular materials to use for correction;
• In order to limit the number of simulations, the CPM can be combined with some mixed design methods such as the Bolomey method;
• In this article, the correction of the original soils E1 (D_max = 50 mm) and E2 (D_max = 10 mm) allows for achieving a high UCS of 2.75 MPa. A direct correlation between UCS and compactness is still not clear and difficult to assess. As a perspective, a large number of tests could be performed to help to link these two parameters;
• All the studied soils are natural. The found UCS can be even higher using some natural stabilizing elements like lime;
• A mix design workflow can be utilized in any RE construction project using any soil extracted from any region;
• A general workflow, proposed in Figure 12, can be applied to design an optimized granular mixture according to the CPM.

In conclusion, the CPM seems to be a potential model to design rammed earth granular structures. It considers many parameters such as compaction energy and type of used granular class which help to obtain resistant and compact RE walls.

In the future, RE material compactness and mechanical strength could be predicted by a simple granular correction with any raw granular material. Using stabilizing agents, especially lime will allow for the improvement of these characteristics. The improvement of RE mechanical performance can also be achieved by reconsidering the compaction process and energy. Further works and the extended application of this model in real projects should enhance this methodology and set up a specific RE database.