Application of Clay–rubber Mixtures for the Transportation Geotechnics—the Numerical Analysis

Two cohesive soils from Southern Poland were selected for the tests. One of them of a characteristic red color (hence the usable name—red clay (RC) [29] came from Triassic deposits in Patoka, near Częstochowa. While the other soil, named by us—kaolin (K), came from the Porcelain Factory in Tułowice. According to a classification consistent with the standard PN-EN ISO 14688-2:2006 [34] and the unified soil classification system (USCS) [3], these are respectively: –

Swelling (RC)—clay with silt (siCl) [34]/clay with high plasticity (CH) [3]: d₁₀ = 0.0008 mm, d₅₀ = 0.0045 mm, d₉₀ = 0.02 mm, C_c = 0.63, C_u = 100, f_Cl = 29%, f_Si = 71%, PL = 25%, LL = 75%, σ_sp = 97 kPa, FS = 31.50%,

A detailed list of all parameters and a description of the tests from which they were obtained can be found in the works of Jastrzębska [18], Jastrzębska and Tokarz [24].

One tire waste size was used in soil–rubber mixtures: granulate (G) 1–5 mm (Fig. 1). The rubber additive originated from the local shredding companies and contained negligible amounts of textile parts. The parameters characteristic for granulate (according to [24]) indicating their homogeneous nature are as follows: –

Granulate (G)—d₁₀ = 1.12 mm, d₅₀ = 2.0 mm, d₉₀ = 4.0 mm, C_c = 0.8, C_u = 2.1.

Figure 1:

The specific gravity of rubber was approximately 1.15 g/cm³, which falls within the range of values given by Akbulut et al. [2] and Kalkan [26].

For the purposes of numerical analyses, the following mixtures of pure soil (RC or K) with the addition of 0, 10, and 25% rubber waste (G) in relation to the total mass were prepared: –

RC-G-10 (90% red clay with 10% addition of granulate 1–5 mm);

The proper specimens from RC and K, and their mixtures with granulate (RC-G and K-G) were prepared by cutting the specimens directly from a block prepared in a Proctor apparatus or by pushing out from the cylindrical forms pressed into Proctor's mold (Fig. 2). A detailed description of the samples preparation was presented in the works of Jastrzębska [18], Jastrzębska and Tokarz [24]. It is worth noting that with the higher content of rubber waste, the density of soil–rubber mixtures decreased accordingly: RC − ρ = 1.85 g/cm³, RC-G-10 − ρ = 1.62 g/cm³, RC-G-25 − ρ = 1.46 g/cm³, K − ρ = 1.94 g/cm³, K-G-25 − ρ = 1.52 g/cm³. This fact is important in the case of loading a weak subsoil with embankments made of soil–rubber mixtures.

Figure 2:

The full research program included triaxial tests: cyclic and monotonic CU (consolidated, undrained) and monotonic UU (unconsolidated, undrained) was conducted according to the Standards PN EN ISO 17892-9 [36] and PKN-CEN ISO/TS 17892-8:2009 [33], respectively. Each sample (tests CU) was saturated, initially by flushing with deaerated water, and after by using back pressure procedure (at the end of the back pressure procedure, Skempton parameter was equal to B = 0.86–0.96). The cyclic triaxial tests (CU) were carried out with a low frequency (f ≈ 0.001 Hz, and it is low enough to exclude the presence of dynamic phenomena) and a constant stress amplitude equal to A = 0.35q (where q is a deviator stress corresponding a current axial strain ɛ₁ = 1%, which corresponds to the moment when the cyclic load action started), at a constant displacement rate (controlled deformation) equal to v_s = 0.9 mm/h. After cyclic loading, the test was continued under a monotonic load until an axial strain of 15% was achieved. To avoid the impact of swelling, in the case of UU triaxial tests the specimens were not saturated with deaerated water prior to shearing realized at a constant displacement rate equal to v_s = 7.2 mm/h (UU). Each series of UU tests was carried out at confining stresses equal to σ₃ = 50 kPa, 100 kPa, and 200 kPa. Whereas each series of CU tests was conducted at confining stresses equal to σ′₃ = 20 kPa, 50 kPa, and 80 kPa (for RC and RC-G) and σ′₃ = 100 kPa, 200 kPa, and 300 kPa (for K-G). The selection of confining pressure σ′₃ = 100 kPa, 200 kPa, and 300 kPa for K–rubber waste mixture tests refers to earlier Jastrzębska's tests conducted on pure K [17,19,20,21]. In turn, the choice of confining pressure σ′₃ = 20 kPa, 50 kPa, and 80 kPa for RC–rubber waste mixture tests sought to verify whether the expansive soil (weak soil) with the addition of rubber waste could be used for road or railway embankment construction, where the real transferred loads are usually less than 100 kPa.

Numerical analyses were performed by the finite element method using the Z_Soil 2020 program [7]. In the case of the stability analysis, a simplification was used in the form of assuming a plane strain state, which is characteristic for linear objects [5,41,44]. When analyzing the VSS plate test, the assumption of axial symmetry was adopted, which is also correct for a homogeneous substrate (at least to the depth of the test range, which is defined as 2 D, where D is the diameter of the steel rigid plate used in the test. Quadrangular, four-node finite elements were used for the analyses. In each node, the searched unknowns were the components of displacements in two directions: vertical and horizontal. A simple, elastic-perfectly plastic model with a Coulomb–Mohr yield surface was used for the analyses. Since the analysis of the single-component soil medium was limited to the analysis, the state of total stresses was analyzed, without separation into stresses taken by the soil skeleton and water in the pores. Hence, the behavior of the soil medium will be described by the integer values of the strength parameters: cohesion c_u and friction angle ϕ_u. The Coulomb–Mohr model describes the shear strength of the soil very well, hence the use of this model in stability analyses gives sufficiently accurate results [13]. The behavior of the soil medium in phase of elastic work is described by two more parameters: Young's modulus E and Poisson ratio ν.

A common problem when performing numerical analyses is the selection of parameter values used in the computation. The use of advanced constitutive models makes it necessary to specify various parameters that often do not have a simple physical interpretation, which makes it impossible to determine their values on the basis of typical laboratory or field tests. Hence, the great desire to use simple models for which it is possible to precisely determine the values of model parameters [12]. Of course, then it is necessary to perform practical verifications that will confirm the correctness of the performed numerical analyses.

In the stability analysis of the slopes of the embankments, the computations were divided into two stages. The first one is the generation of the state of primary stresses, which arise from the self-weight of the soils forming the road embankment. The current state of the embankment was taken into account—the analysis of the change in the state of stress and the resulting settlements of the subsoil under the embankment during the erection of the earth structure was omitted [23]. The next step is the analysis of the stability by the strength reduction method c–ϕ. The adopted numerical procedure assumes a gradual (in predetermined calculation steps) reduction of the cohesion value and the internal friction angle according to the following relationship (3.1) [44,47,53]: (3.1) Fs=c0cf=tanϕ0tanϕf, {F_s} = {{{c_0}} \over {{c_f}}} = {{tan {\phi _0}} \over {tan {\phi _f}}}, where: F_s—safety factor, which is a measure of safety due to stability; c₀, ϕ₀—initial values of cohesion and internal friction angle; c_f, ϕ_f—values of cohesion and friction angle at the moment of completion of computation due to the lack of convergence of iterative procedures.

As a result of the decrease in the strength parameters, plastic zones, which appear in part of the model, are eliminated in subsequent iterative steps using the phenomenon of stress redistribution. The procedure is carried out until the iterative procedures do not converge, which means that the computation are completed. The results of the analyses are the value of the safety factor F_s obtained at the end of the calculations and the location of the slide surface. The abovementioned location is estimated on the basis of the analysis of the map of total displacements at the moment of loss of stability of computation [30].

The procedure of estimating the value of the F_s factor adopted above has a great advantage compared to the so-called strip methods (Fellenius, Bishop, Janbu, Morgenstern-Price, etc.), where the user must first define the location of the slide surface [40]. In the presented method, the shape of this surface is the result of computations. The obtained stability loss mechanism corresponds to the smallest value of the safety factor (each hypothetically different failure mechanism will have a greater factor).

The basic in situ test used in road construction is a load test of a rigid VSS steel plate with a diameter of 300 mm. In road jargon, it is called a test of the bearing capacity of the subsoil or road pavement structure. This test allows the assessment of the condition of the native soil layer or aggregate layer to a depth of approx. 0.5–0.6 m below the level of the steel plate base. The layers of the communication embankment are assessed in the same way, and a single test allows to visualize the condition of 1–2 layers of the embankment. The test loads are carried out in two stages. The first is the primary load, which in steps of 50 kPa reaches a value of 250–450 kPa depending on the type of the tested element (soil layer, embankment layer, lower or upper layers of the pavement structure). The next stage of the analysis is the secondary loading after unloading. In this stage, repeated pressures of approx. 100 kPa lower than in the first stage are applied. The results of the analyses are the values of the E₁ and E₂ stiffness modules obtained from the primary and secondary loading of the plate, respectively, calculated according to the formula (3.2): (3.2) E1,E2=ΔpΔs, {E_1},{E_2} = {{\Delta p} \over {\Delta s}}, where: Δp—load increment [MPa], Δs—settlement increment [mm].

In addition to the above modules, the result of the analyses is also the value of the strain ratio determined from the relationship (3.3): (3.3) I0=E2E1. {I_0} = {{{E_2}} \over {{E_1}}}. When making road embankments, the requirements are E₂ ≥ 40 MPa for the lower layers of the road embankment and E₂ ≥ 60 MPa for layers located up to 2.0 m below the bottom base of the pavement structure [38]. In the case of the strain ratio, these requirements come down to I₀ ≤ 2.2 for the layers located directly under the pavement structure and I₀ ≤ 2.5 for deeper layers. The above values for the I₀ index are quite conservative. Treating the soil–rubber mixture as multigrained soils, I₀ ≤ 3.0 would be a sufficient requirement. However, it should be pointed out that the requirements regarding the I₀ coefficient are not much justified for layers located deep under the pavement structure. Some engineers treat this value as a measure of compaction (i.e. a measure of the quality of contractors’ works), however, from the point of view of soil mechanics, it is a very rough approximation having little in common with the actual properties of the corpus of the road embankment.

The numerical model used for the stability analysis is shown in Fig. 3. A road embankment with a height of 6 m was analyzed. Owing to the symmetry of the problem, half of the model was taken, assuming the axis of symmetry running through the center of the embankment. It should be noted that in order to be able to take into account the above simplification, the shape of the embankment, the course of layers in the subsoil, and the corpus of the embankment, as well as the assumed boundary conditions must be characterized by symmetry. The built numerical model had 2777 finite elements. Generally, the stability of a given slope is determined by the geometric shape and the strength and deformation properties of the materials that build the embankment. In the case of a weaker subsoil, the properties of the subsoil on which the embankment was erected also have a large impact on its stability. Hence, the need to take into account the subsoil to a depth that will ensure no influence of boundary conditions on the obtained analysis results. Similarly, the area of the base of the embankment should be considered. In the built model, a distance of 25 m from the base of the embankment and 20 m into the ground under the embankment was assumed. Standard geotechnical boundary conditions were used, which assume that displacements in both directions are prevented in all nodes on the lower edge of the model and horizontal displacements in nodes on both side edges are not possible. In the stability analyses, the occurrence of exploitation load on the road surface was additionally taken into account, which was assumed as a uniform pressure of 25 kPa exerted on the part of the embankment crown occupied by the surface structure.

Figure 3:

With numerical analyses of VSS test load simulations, the dimensions of the geometrical model (Fig. 4) of the analyzed model are correspondingly smaller. They result from the dimensions of the plate used in the tests, the radius of which is: D/2 = 150 mm. The defined model has dimensions of 5 D = 1.5 m in the horizontal direction and 10 D = 3.0 m in the vertical direction. These dimensions are definitely larger than those traditionally used in the analysis of foundation–soil cooperation [13]; however, in the present case, due to these dimensions, it was possible to thoroughly analyze the extent of the impact of the plate into the subsoil and determine the influence of the deeper layers on the obtained results. The number of defined finite elements in this case was 2434.

Figure 4:

Similar to Das and Singh [9] and Tajdini et al. [46], a random change in the internal friction angle and cohesion is visible (Table 1). The results confirm the opinion that testing the soil–rubber mixture intended for engineering applications is necessary each time. Table 1 summarizes the strength parameters (internal friction angle ϕ and cohesion c) obtained from CU/UU triaxial tests. On their basis, the parameters presented in Table 2 were adopted for numerical analyses. Carefully estimated average values characterizing various cement–soil mixtures based on cyclic tests were adopted, which characterize the properties of the material under the conditions of characteristic interactions of communication structures. The parametric analysis performed in the following part will allow to assess the influence of a given parameter on the obtained results in numerical analyses. The considerate range of parameter values includes all the values obtained in tests for the analyzed materials.

A total of 3 material zones were adopted. When analyzing the stability of the embankment slopes, the parameters of the elements simulating the behavior of the layers of the pavement structure, as well as the elements being the subsoil, were adopted in such a way that these layers did not affect the obtained analysis results. Therefore, the situation when the slope stability is affected by a weak subsoil was not analyzed. The problem defined in this way will allow to assess the possibility of using the tested material in road construction.

The most important, from the point of view of the issues discussed in the article, are the parameters of the soil–rubber mixture from which the embankment was made. These values were estimated based on mean values obtained in laboratory tests (Table 1, [18,24]). A careful estimation of the parameter values was adopted, which allows us to treat them as derived values (experts) according to the Eurocode 7 standard. The properties of the subsoil under the embankment were assumed in such a way that the subgrade did not determine the obtained results. The results of numerical analyses will make it possible to assess the possibility of using the tested material in road construction from the point of view of the strength and deformation properties and the impact of these properties on the results of acceptance tests, as well as the safety margin due to the stability of the embankment.

The analysis of the slope stability of the road embankment was carried out in three stages. The first one is the generation of the state of primary stresses in the subsoil and the corpus of the embankment, which results from the self-weight of the soils and materials. Next, the operational load of the road surface was set in calculation steps. The third and most important stage was the stability analysis carried out using the c–ϕ strength reduction method, which was described earlier. After the computations are completed, which takes place when the numerical procedures do not converge, it is possible to estimate the location of the slip surface corresponding to the lowest value of the safety factor F_s.

The first stage of the numerical simulation of VSS plate load test tests is also the generation of the primary stress state. Owing to the dimensions of the model, in this case the obtained values are definitely smaller. Next, the load transferred to the steel plate was applied. The path of the primary load was carried out, obtaining the value of 350 kPa. Next the unloading and reloading process to the value of 250 kPa was realized. The obtained values of displacements of the plate center will allow to calculate the values of modules E₁ and E₂ and the strain ratio I₀.

The results of analyses of the behavior of the road embankment erected with using the analyzed rubber–soil material are presented in Figures 5–9. The first of them (Fig. 5) shows the values of settlements due to operational loads of the road surface structure. The maximum settlement amounts to 5.8 mm and these values are within the requirements of the relevant regulations and standards [37,38,39]. Fig. 6 shows the yield surface at the moment of loss of stability, which corresponds to the value of the safety factor F_s = 2.26. This area can be estimated on the basis of the map of total displacements. The yield surface corresponds to the sliding of earth masses along the cylindrical surface running through the corpus of the embankment. Attention should be paid to the very high value of the F_s coefficient, which results from the high strength parameters of the analyzed rubber–soil material.

Figure 5:

Figure 6:

Figure 7:

Figure 8:

Figure 9.

As part of the numerical analyses performed, the impact of the cohesion and the internal friction angle values on the value of the safety factor was also carried out. The results of these analyses are presented in the figures below. It is easy to see that the cohesion is the decisive factor in the safety margin (Fig. 7a). If a sufficiently high value of cohesion is confirmed, this material will be an excellent component that will guarantee safe use of the communication structure. It is also easy to notice the large influence of the value of the internal friction angle (Fig. 7b).

The results of the second of the performed analyses—simulation of the VSS plate test, on the basis of which the properties of the subsoil and embankment layers in road construction are determined, are shown in Fig. 8. Based on the stress–settlement relationship, from the appropriate load phases, the values of the stiffness modulus and the strain ratio can be determined. The obtained modules for the results presented in the Fig. 8 are equal to: E₁ = 24.2 MPa, E₂ = 56.7 MPa and I₀ = 2.35. They have been calculated for the appropriate ranges of stresses related to the subsoil. These results meet the requirements for the subsoil or even lower layers of the road embankment on which the road surface structure will be made (30–40 MPa—depending on the road class). To be able to use the material to make layers with higher requirements, it would have to have a higher value of Young's modulus, which is a measure of material stiffness. An important result is also the settlement map (Fig. 9), which indicates the depth of the impact of the load transmitted by the VSS plate into the subgrade. Based on this observation, it can be concluded that it is necessary to test each layer of the formed embankment, while the study of the native subsoil gives information about their deformation properties only to a small depth.