Assessment of Stiffness and Strength Parameters for the Soft Soil Model of Clays of Cameroon
This paper focuses on the advanced modelling of soft Cameroon clays for the global prediction of the behaviour of geotechnical structures. A comprehensive set of experimental data on Cameroon subsoils from the oedometer and triaxial tests are analyzed in this paper in order to determine the stiffness and strength parameters for the Soft Soil model. It is based on 71 soil samples taken from both sides of the construction sites of several major structures across the territory. At the first approach, the soil samples taken were analyzed in a geotechnical laboratory to obtain physical and mechanical identification parameters specific to each soil type. The results obtained reveal that the analyzed soils are generally compressible clays. In the second phase, the law of behaviour of the Soft Soil model was used to characterize these Cameroon soils. Its parameters were obtained after calibration of the results of the laboratory tests obtained in first approach. The results obtained in this article can be compared to the different models obtained on clays soils around the world. The parameters are of the same order of magnitude as other clays modelled around the world.
1.1. Interest and Needs of Soils Modeling
In the practice of numerical modelling, not just any law of behaviour can be considered as an acceptable approximation of any real behaviour, even after calibration of the parameters. The soil has a stress-strain relationship that is nonlinear and irreversible from a certain threshold. For numerical modelling to provide a realistic estimate of behaviour, it must use advanced behavioural laws adapted to each type of soil. In the context of this article, we have dealt with several types of compressible soils with different mechanical behaviours. In this paper, an advanced law of behaviour called the Soft Soil model, which describes the behaviour of the compressible soils in a realistic way, will be highlighted. Present, several civil engineering and geotechnical works are under construction in Cameroon. The predictive evaluation of the behaviour (settlements, lift, and stability) of these infrastructures is often carried out by empirical methods or conventional analytical methods. These do not always take into account the realistic behaviour of the material or the overall behaviour of the work and impose simplifying assumptions of behaviour. From these shortcomings in the study of structures, it is important to determine the parameters of advanced soil models that are not currently available in Cameroon in order to effectively predict the behaviour of geotechnical structures considering successive phases of construction. This global approach involves defining a specific constitutive law for each soil type, determining its parameters using classical geotechnical laboratory data and modern numerical technics. The parameters of the advanced Soft Soil model will be determined from 71 soil samples taken from construction sites across Cameroon. These parameters will constitute a national database of exploitable compressible soil parameters that can be used in numerical modelling to predict the overall behaviour of geotechnical structures.
1.2. Literature Review on Compressible Soils
The analysis of the consolidation of clay for the prediction of the foundation settlement of a structure is of particular importance in geotechnics. The first rational approach to this problem, based on the principle of effective stress, was proposed by Terzaghi . After this, an intensive research has been conducted in this area. A logical extension of Terzaghi’s one-dimensional consolidation theory to a 3D situation is due to the complete coupled porous-elastic formulation of Biot . The two theories of Terzaghi and Biot postulate that soil behaviour is linearly elastic. For global problems, most researchers are turning to digital methods. Schiffman and Arya , using Terzaghi’s one-dimensional consolidation model (1D), conducted a research using the finite difference method and the finite element method. Desai proposed a nonlinear model for the soil and implemented this model in a finite element computation program to deal with the one-dimensional consolidation problem. Melanie’s model , developed at LCPC to represent the behaviour of natural clays by finite element calculations, is an anisotropic elastoplastic model with hardening, which allows the resolution of consolidation problems. It is derived from the modified Cam-Clay model. Soft soils, which are normally consolidated, are known for their very high compressibility [5–9]. It is obvious that creep is important for problems that show a significant primary settlement: this is the case of road construction, foundations, dikes on compressible soils, or dams where strong primary settlements are followed by creep settlements years later [8, 10–12]. In other cases, dams or buildings may initially be based on over-consolidated soils; the primary settlements are then relatively small. In addition to foundation settlement problems, creep plays an important role in slope stability problems. Several natural slopes with a low safety factor often show continuous displacements due to creep under constant mechanical or hydraulic conditions [13, 14]. In recent years, several research studies have been carried out around the world to determine the advanced soil parameters for feeding advanced soils models implanted in geotechnical software for the calculation of geotechnical works in construction or in the process of being monitoring. This is the case with the stiffness and strength parameters for hardening and the Soft Soil model of soft and stiff Bangkok clays investigated by Adachi and Oka and Suched et al. [13, 15]. These authors conducted several series of isotropically consolidated drained and undrained compression tests at the Asian Institute of Technology on soil samples from Bangkok, and then exploited the results of these tests to obtain the advanced parameters of these soils for the advanced calculation of geotechnical works in Bangkok. Nallathamby and Minna  modelled and exploited the advanced parameters of soft anisotropic soils for their exploitation in the calculation of the experimental embankment of Murro in Finland. The results were compared with those measured during the test phase of the work. This Murro test embankment was constructed on a 23 m deep deposit of medium sensitive clay near the town of Seinäjoki in Western Finland. The embankment has been monitored for a long time, since it was built in 1993, and it has been subjected to several studies. The almost normally consolidated clay is overlain by a 1.6 m thick overconsolidated dry crust, and the underlying thick clay layer is almost normally consolidated and relatively homogeneous. The groundwater table is estimated to be at 0.8 m below the ground level. Murro clay is highly strain anisotropic and time dependent . Many other studies have been investigated for clays [17–19]. A large number of Cc − (Cc is the compression index, and is the natural water content) correlations have been proposed by researchers for different soft clays around the world, but comparisons of these correlations and reasons for differences between them are rarely reported. The Cc − relationships of marine soft clays from eight China’s coastal cities have been investigated by Gao and Chen . It is found that the north coast clays have larger slope of the Cc − relationships (about 0.02) than the south coast clays (about 0.008).
Generally, the perfectly linear elastic behaviour law with a Mohr–Coulomb- (MC-) type failure criterion is used in geotechnical engineering calculations. This law defines the behaviour of the soil based on 5 parameters: Young’s modulus (E), the angle of friction (φ), the cohesion (c), the dilatancy angle (ψ), and the Poisson ratio (). However, several studies [13, 20–23] have shown that this model does not represent the nonlinearity of the actual behaviour of the soil and imposes that the module in loading is the same as that in unloading. In the case of a structure exposed to cyclic loadings, for example, case of foundations of industrial buildings receiving vibrating machinery, wave breaking on a dike, and traffic load, unloading areas play a predominant role in determining its behaviour. This simplification therefore has a consequence in the models for the prediction of the real behaviour of the structure in question. The choice of the realistic behaviour model for this study meets two requirements:(i)On the one hand, that of better representing the behaviour of compressible soils in Cameroon compared to the Mohr–Coulomb model(ii)On the other hand, that of identifying the parameters of the Soft Soil model from the results of triaxial or oedometric tests on the materials selected in several construction sites throughout the national territory
Several studies have shown that the dedicated Soft Soil (SSM) model for the realistic representation of the behaviour of compressible soils is implemented in several calculation codes . It will be highlighted in 71 soil samples taken in the central, south, and Littoral regions of Cameroon. From the classical geotechnical laboratory results on soils, the parameters of this advanced model will be determined according to the relationships linking several geotechnical parameters presented in detail later in this article.
3. Formulation of the Soft Soil Model (SSM)
In this section, we start from the classical formulation of the Soft Soil Creep model (SSCM) to the determination of the parameters of the Soft Soil model. The Soft Soil model makes it possible to take into account the work hardening of the soft clays but not the secondary consolidation; this results in the evolution of the axial strain in an oedometric test as a function of time, after the end of the primary consolidation. This deformation evolves according to the logarithm of time (at least for observable time scales).
3.1. Generalization of the Differential Law for 3D Creep
The 3D model is a generalization of the 1D model. We will adopt the stress invariants for the pressure = soct and the deviatoric stress . With soct and τoct being the normal octahedral stress and the octahedral shear stress, respectively, these invariants are used to define a new equivalent mean constraint called :where .
Figure 1 shows that the measured stress is the constant on the ellipses in the plane -q. Actually, we have the modified Cam-Clay model ellipses amended by Roscoe and Burland . The soil parameter M represents the slope (of what is called critical state line as shown in Figure 1). Equation (1) is used for the deviatoric stress q:where φcv is the angle of critical friction or constant volume. Using the above definition of q, the equivalent pressure is constant over an ellipse and in the principal stress space. To extend the theory in 1D to the general case in 3D, we will now focus on the case of normal consolidation encountered in an oedometer. This value of M is a practical value calculated by default. Moreover, the finite element calculation code is used for the analysis of geotechnical structures: PLAXIS [24, 26] makes it possible to calculate an approximate value of (coefficient of resting land under normal consolidation conditions), which is the value of M calculated from equation (2). In general, the value of calculated by the program is greater than that calculated by Jacky’s formula . Otherwise, one could enter a value of to calculate the value of M by using Brinkgreve formula developed in the following PLAXIS calculation code :
The load surfaces are ellipses with associated flow (increment of normal deformation at the ellipse) while for rupture, the flow is not associated (that is, why it is necessary to enter the dilatancy angle, possibly zero, which corresponds to the plastic flow at constant volume).
Under these conditions, we have , and from relation (1) we derive the following relations:where and is the generalized preconsolidation pressure, this parameter being proportional to the one-dimensional case. For a known value of , can be computed from σ′, and can also be calculated from .
Instead of the parameters A, B, and C of the one-dimensional model, we will now use the parameters κ and λ, defined by the following relations:where is the same Poisson’s ratio.
3.2. Formulation of Elastic Deformations in 3D
The 1D model can be extended to obtain the 3D model, but until now, this has not been performed for elastic deformations. In order to obtain a 3D model for elastic deformations, the elastic modulus Eur is defined as dependent on the stress  by
3.3. Parameters of the Soft Soil Model
As soon as the ultimate limit criterion f (σ′, c, φ) = 0 is reached, the instantaneous plastic strain speed develops in accordance with the flow law with = (σ′, ψ); the parameters of plastic flow of the material are as follows: c′ is the effective cohesion, φ′ is the Mohr–Coulomb friction angle, and ψ is the dilatancy angle. For fine and cohesive soils, the dilatancy angle is usually small and can therefore often be taken as zero. The Soft Soil model therefore requires the following hardware constants.(i)Break parameters as in the Mohr–Coulomb model c: shear strength (kPa) φ: angle of shearing resistance (°) Ψ: angle of dilatance (°)(ii)Basic stiffness parameters λ: modified compression index used in the constitutive “Soft Soil” model К: modified swelling unloading-reloading index in the constitutive “Soft Soil” model(iii)Advanced parameters : Poisson’s ratio in unloading/loading (by default, = 0.15 for clay), : the ratio , the coefficient of lateral earth pressure in normally consolidated condition, M: soil parameter representing the slope (so-called critical state line).
3.4. Modified Swelling Index and Modified Compression Index
These parameters can be obtained from an isotropic compression test or an oedometric test. When the logarithm of the stress is plotted against the deformation, the curve can be approximated by two straight lines. The slope of the normal consolidation curve gives the modified compression index λ, and the slope of the discharge curve (or swelling) can be used to calculate the modified swelling index κ. It should be noted that there is a difference between the modified indices κ and λ and the original parameters of the Cam-Clay model, κ and λ. Relationship with the Cam-Clay model parametersand relationship with classic parameters
There is no exact relationship between the isotropic compression indices κ and κ and the one-dimensional swelling index because the ratio of horizontal to vertical stress changes during unidimensional discharge. For the approximation, it is assumed that the case of average stress during the discharge is a case of isotropic stress, that is, to say the horizontal and vertical stresses are equal.
This synthesis on the formulation of the Soft Soil model, an advanced behaviour model implemented in several computation codes, shows that it is a sufficiently simple model to be able to determine its parameters from a classical geotechnical study (triaxial and oedometric tests). In this model, there is no wedging parameter without a physical signification as often found in other models. The determination of its parameters can require optimization techniques. The user must focus on two choices: one is inherent to geotechnics in general, and the other is realistic digital simulation. The determination of the geotechnical parameters to be entered in a calculation code is not different from a choice of “manual” calculation parameters for a compaction calculation, for example. Some of the parameters are different in their expressions but always remain connected to classical geotechnical constants. The models of behaviour developed are distinguished mainly by the type of mechanical problem to be solved on the soil or on structures in soil, by the number and type of parameters that characterize them. The least “current” parameter is presumably the dilatancy. In fact, the choice of the model of behaviour depends on the problem posed, which model of behaviour is to be used for which geotechnical problem? This question is not so simple because there is no “universal” model capable of reproducing the behaviour of any type of the geotechnical work.
4. Mechanical Properties of Cameroon Soils
4.1. Geology Setting of Cameroon
The geological history of Cameroon begins with the Archaean era between 3.5 and 2.5 billion years (Ga) ago. Its different phases of development are illustrated by geological masses formed during successive orogenic cycles. It is characterized by formation of craton and mountain ranges and subsequent extension phases by the splitting of the continental crust. From the south to the north, we have (Figure 2) the following:(1)The Cameroonian southern domain consists of Pan-African metasedimentary units such as the Ntui-Betamba, Yaoundé, and Ayos-Mbalmayo-Benbis units. The layers were plunged onto the Archaean Congo Craton towards the south [27, 28] from this domain, which were affected by four stages of ductile deformation, corresponding to alternating phases of E-W to NW-SE contraction (D1, D3) and N-S to NE-SW extension D2 .(2)The Cameroonian central domain, broad area that extends between the fault of the Sanaga to the south and the Tibati-Banyo fault to the north. It consists of Archaean to Palaeoproterozoic high-grade gneisses intruded by widespread Pan-African syntectonic plutonic rocks of high-K calc-alkaline affinities . Major setbacks in this area seem to have guided the implementation of plutonism having an orthogneissification with variable intensity. This area is subjected to an advanced general metamorphism where banking is composed of gneiss and amphibolite.(3)The Cameroonian northern domain is characterized by subordinate 830 Ma old metavolcanic rocks of tholeiitic and alkaline affinities associated with metasediments known as the Poli series. This domain is characterized by three stages of deformation: an early phase D1 is associated to the metamorphism with granulite facies, medium pressure . A phase D2 dated 600–580 Ma is synchronous with an intense migmatization  and a granitization associated to a metamorphism with amphibolite facies (600°, 5–7 Kb) and greenschist facies (550°, 5 Kb). And a phase D3 is liable to the dextral setbacks E-W and drive folds N-S to E-W.
4.2. Nature of Investigated Materials
To do this, it was necessary to have on the one hand, enough experience (interesting the geological formations of the country), and on the other hand, results of tests carried out in geotechnical laboratory under perfectly controlled conditions on samples which were intact, taken from the sites of structures under construction. Data collection took place over a period of almost four years. It involves some 60 geotechnical studies (on seventy-one sites) of civil engineering and geotechnical works carried out by laboratories in various regions of the territory. The analyzed samples come mainly from the central, south, and Littoral Cameroon regions. We do not claim to have obtained an exhaustive sampling or quite representative of all types of soils likely to be encountered in Cameroon. However, the results obtained on residual soils (lateritic in the central and southern regions and alluvial in the Littoral region) indicate certain trends already allowing us to determine through the advanced modelling of soils, good parameters of the Soft Soil model. It should be noted that lateritic soils cover about 70% of the Cameroonian territory .
For example, Yaoundé, Cameroon’s capital, is made up of typical ferralitic soils at three levels, namely (i)A loose upper formation dominated by topsoil.(ii)A ferruginous, nodular, armored intermediate layer consisting of iron hydroxide and kaolinite clay accumulations from hydrolysis of rock minerals.(iii)At the lower horizon, the geological formations encountered are the grenatiferes gneiss with micas or biotite only, whose structure of the mother rock (Gneiss) is conserved in depth and more altered towards the surface. Minerals such as quartz, kaolinites, goethite, and hematite are encountered.
All these formations belong to the old metamorphic base. Each time depending on the type and scale of the project to be carried out, the cored holes, and reconnaissance wells on construction sites have revealed the pedological overview of the soils of the central Cameroon region. The different terrain profiles generally encountered reveal not only the possible combinations of the sequence of soil layers in the city of Yaoundé but also the nature of each layer. So, we find in the extent of the city: clays and lateritic reddish bass. Lateritic cuirasses are observed as well.
In the Littoral region, the sedimentary rocks form low lying and gently undulating hills along the western side of the Dibamba River. The strata have experienced tropical weathering, which has resulted in the development of a lateritic residual soil profile. This type of rock mass tropical leads to weathering profiles as presented by Fookes  and shown in Figure 3. The material viewed at site comprises weathering Grade 6, as seen in the trial pit photograph from site, with possibly weathering Grade 5 material encountered towards the base of deeper rotary boreholes. Below this material, the stratum may then transition more rapidly to weathered parent rock material. This transition was not identified in the deepest exploratory boreholes that went 30 m below the ground level. In the exploratory holes observed, there was no clear evidence for overlying transported material, so material may be weathered in situ. The following section presents the nature of the materials investigated in this paper.
The samples were collected “undisturbed” using a PVC 100 mm diameter corer and 300 mm in length Figure 4. The shear box used was 60 mm in width and 25 mm in height, which is recognizable as the small shear box apparatus to the British Standards Institution [30–32]. These investigations were conducted under perfectly controlled conditions. The soils of the central, southern (Figure 5), and Littoral regions (Figure 3) of Cameroon are represented:(i)Either reddish lateritic clays (central and southern regions)(ii)Either sandy clays or clay sand from Douala (Littoral region)
4.3. Identifications Parameters and Compressibility of Cameroon Soils
Tables 2 and 3 summarize the results of the identification, compressibility, and shear strength tests carried out on soil samples taken from 71 sites of projects under construction in Cameroon. According to the results of oedometric compressibility, soils tested in the laboratory are generally compressible. The statistical significance of these laboratory data is presented in Tables 2 and 3.
5. Results and Discussion
In the previous sections, the parameters of identification, compression, and shear resistance were presented for the soils of Cameroon, including the transformation relations of the laboratory parameters to the parameters of the Soft Soil model for their use in a numerical calculation code. Table 3 and Figures 6–8 present the performed parameters of the Soft Soil advanced model determined for the 71 soil samples analyzed from the central, south, and Littoral regions of Cameroon. They were obtained from the raw soil mechanics laboratory results that were presented in Tables 2 and 3 and calibrated according to the relations (1–8) of the advanced model formulation, the Soft Soil model , presented in parts of this article. The statistical significance of these parameters of the Soft Soil model is presented in Table 4. The results obtained in this paper (clays of Littoral region, central region, and southern region) can be compared to the different models obtained on the clays soils around the world. The parameters are of the same order of magnitude as other clays modelled around the world (Table 5): Haney clay ; Osaka clay ; the Cubzac-les-Ponts clayey embankment [35, 36]; from the site of the dam of Flumet in the Isère in France ; and the site of Saint-Laurent des eaux , the Bangkok clays [12, 15], and the Murro test embankment clay in Western Finland  to name only those. The soft soils found in the African continent (tropical zone soils) [37–39] have been considered for these comparison in this paper: the settlement of a railway embankment on PVD-improved Karakore soft soil in Ethiopia ; the rheology of mechanical properties of soft soil in Nigeria ; and various problem on soft clayed soils in South Africa . The results of monitoring of the full scale experimental embankment on soft Douala clays (Littoral region) of Cameroon have been published recently in this domain in the Journal of Civil Engineering .
In the context of projects of great importance whose works are subjected to complex loading (vertical and horizontal components and moments applied to the foundation, dynamics loads, and cyclic loads), the analytical calculation is no sufficient to predict the behaviour of such structures. It is therefore necessary to first performed advanced soil modelling according to the results of laboratory tests and then to carry out a numerical modelling of the structure taking into account the realistic behaviour of the materials, the soil-structure interaction, and the different staged construction. For compressible soils, the Soft Soil model used in numerical modelling offers safety and comfortable results, hence the importance of advanced soil modelling used in numerical modelling for the prediction of behaviour of geotechnical structures.
In this paper, we determined the parameters of the advanced compressible soil model on 71 soil samples from various locations in the country. The parameters of the Soft Soil model determined now serve as a national database for compressible soils. These parameters can be used in projects to numerically predict the behaviour of geotechnical structures under construction or during operation. Present, the most efficient tool for predicting complex phenomena is modelling. It is therefore important to first determine the parameters of each model likely to describe the actual behaviour of the structure before using it in a computer code. The study meets national or even international needs, which is to have a database for exploitable soils for supplying calculation codes when carrying out major projects. More precise methods, such as numerical calculations, should be used when the soil-structure interaction has a dominant influence. Numerical methods have the capacity to take into account macroscopic heterogeneities of the soil (layers of different characteristics or heterogeneity); the same is true of the heterogeneity caused by different loading levels according to the points of the Massif, in the case of a soil with nonlinear behavior (variable rigidity). Numerical methods make it possible to take into account any loading geometry and the phasing construction or the progressive application of loading; they are also well suited to situations where it is necessary to study the interaction between neighboring structures, that is, to say where one is dealing with one or more structure-soil-structure interaction problems. The use of the Soft Soil model for compressible soils involves longer computation times, since the stiffness matrix of the material is decomposed in each step of the calculation. For the analyzed soils, which are compressible, the Soft Soil model correctly describes their behaviour and makes it possible to optimize the designing of the geotechnical structures, which are supported or support these compressible soils, thus demonstrating the importance of the advanced modelling of soils for predicting the behaviour of geotechnical structures. This study is a contribution to the advanced soil modelling of Cameroon; it will be extended to the case of rigid soils with the hardening soil model.
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors confirm that there are no conflicts of interest associated with this publication.
The authors acknowledge particularly the GéoMécanique team of the Laboratoire 3SR at Grenoble, France, and Geotechnical’s Laboratories and Engineering’s Offices of Cameroon for the facilities and support to perform the experiments. They also thank the people who were involved in obtaining the results from this paper.
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