Heave analysis of shallow foundations founded in swelling clayey soil at N’Gaous city in Algeria

A swelling soil is generally defined as a soil that has a potential to increase in volume under increasing water content.^[1,2] Clay soils consist of various minerals with a high affinity for water such as kaolinite, illite and montmorillonite. Moreover, the mechanism of hydration from a certain state induces significant swelling, especially montmorillonite mineral, which has most of the problems of swelling soils.^[3] Swelling soils are found in many parts of the world, particularly in arid and semi-arid areas, where moist conditions happen after long periods of desiccation. In the literature, several studies have been conducted on problems related to swelling soils, for example, Nelson and Miller,^[1] Nelson et al.,^[2] Chen,^[3] Fredlund et al.^[4]

The high swelling pressure, causes differential structures’ heaving, in particular light structures of low stiffness built on shallow foundations.^[5] However, this heaving induces costly damage, most of which is cracks in walls and slabs. Algeria, like other countries with a dry climate, also suffers from the problem of soil swelling. Several cases of damages have occurred in recent years in many parts of the country (e.g., Medea, Batna, Tlemcen, Oran).^[6,7,8]

Soil-structure interaction is assured by the foundations, which have the important role of transferring loads to the supporting soils. For this reason, when studying the foundation swelling soils of a construction, should interest with their mechanical behavior under the applied loads, also take into account that swelling strain of clay soils occur over time as a function of soil properties^[9] (e.g., mineralogy, structure, suction, plasticity and dry density, permeability), also the environmental conditions (e.g., moisture variations, climate, vegetation).^[10]

Many researchers have developed expressions to calculate the total heave on an unsaturated swelling soil by taking into account the soil suction, for example, Mitchell and Avalle,^[11] Mckeen,^[12] Fityus and Smith,^[13] Briaud et al.,^[14] Vanapalli et al.^[15] Moreover, expressions for saturated swelling soil based on oedometer tests have been reported in literature, for example, Fredlund,^[16] US Department of the army,^[17] Nelson et al.,^[18] Ejjaaouani and Shakhirev,^[19] Baheddi et al.,^[20] where each researcher proposes an expression based on the type of swelling tests that have been performed on the oedometer.

In literature, several researchers have numerically investigated the problem of shallow foundations on unsaturated swelling soils using 2D finite element analysis, for example, Hung and Fredlund,^[21,23] Masia et al.,^[22] Abed,^[24] Nowamooz et al.^[25] So, they conducted a hydro-mechanical study to estimate the effect of the drying-wetting path on the shrinkage and heave of the shallow foundations. Nevertheless, few studies are found providing a detailed behavior of shallow foundations on swelling soils using 3-D numerical modelling. This is due to the lack of a particular behavior model of swelling soil in recent years. Therefore, this complexity along the soil behavior, has encouraged the use of a simple method for modeling swelling soil in the present study. This method based on the simulation of swelling pressure in vertical direction of all soil mass.

In this study, isolated shallow foundations such as square, rectangular and circular footing were analyzed due to the limited study of their mechanical behavior in swelling soil by researchers. In addition, the majority of damaged structures with low stiffness are based on this type of shallow foundations. All these reasons made us choose 3-D numerical modeling.

A short description of geotechnical properties of the soil located at N’Gaous city in Batna Province, Algeria, of the analytical approach according to the soil stress state and the equations used for predicting total heave based on oedometer test is presented. A three-dimensional numerical model using the finite difference code FLAC was used to analyze the total heave. The numerical results obtained were compared with the analytical results proposed in the literature.

The studied swelling clay is located in N’Gaous city near Hospital (35° 34′ 04.8″ N, 5° 36′ 60.0″ E), which is located 77 km west of Batna Province, Algeria. Most part of this region is recognized by the abundance of active soils and the damage related to the light structure, as in Fig. 1. Table 1 summarizes the physical and mechanical properties of the undisturbed soil samples taken in the present study.

Figure 1

The soil was classified as a highly plastic silty clay (CH), in accordance with the Unified Soil Classification System (USCS). The experimental results using the conventional one-dimensional oedometer according to ASTM standard D 4546-08^[26] produced various curves giving the swelling strain versus time (see Fig. 2), which shows the free swelling strain. As well as the swelling strain as a function of vertical pressure, which shows the swelling pressure for a null swelling strain (see Fig. 3). Following this method, a single-undisturbed sample was loaded at a very low stress level σ_i=1 kPa, the soil was wetted and allowed to swell, until swelling ceases. This vertical swell was registered as the free swelling strain. After that, the vertical pressure was then applied in increments to gradually consolidate the soil sample. The pressure required to consolidate the soil sample to its initial volume (ɛ_sw=0 %) is defined as a swelling pressure. The free swelling strain and swelling pressure were 8.86% and 218 kPa, respectively, as shown the curve in Figure 2 and Figure 3.

Figure 2

Figure 3

In the calculation of the total heave of shallow foundations after soil swelling, it is necessary to consider the soil stress state under the foundation as well as the swelling pressure. The soil stress state is determined by the calculation of two stress components, mentioned as follows: the geostatic stresses σ_z,g, which are increasing linearly with depth and can be computed using the classic equation σ_z,g=γ_sw×z. The loading stresses σ_z,load are due to the construction weight, decrease with depth^[27] and can be computed using Equation (1).^[28] In addition, this simple approach is generally related to the foundation found on a horizontal surface and homogeneous soil. (1)σz,load=σ0.B.L(B+Z)(L+Z){\sigma _{z,load}} = {{{\sigma _0}.B.L} \over {(B + Z)(L + Z)}} where σ₀ is the distributed load applied at the foundation level; B is the width of the foundation; L is the length of the foundation and z is the depth considered below the foundation level.

Baheddi et al.^[20] determined that the vertical swelling strain of saturated soils occurs in the boundaries of the swelling zone II (see Fig. 4), where the total stress σ_z,t is less than the swelling pressure σ_sw as indicated by the Equation (2): (2)σz,t=σz,g+σz,load <σsw{\sigma _{z,t}} = {\sigma _{z,g}} + {\sigma _{z,load}}\, < {\sigma _{sw}}

Figure 4

3.1

Prediction of total heave

In the late 1950s, heave prediction methods were first developed as extensions of methods used to estimate volume changes due to consolidation in saturated soils using results of one-dimensional oedometer (consolidation) tests.^[2,29]

There are several methods to measure the swelling pressure by oedometer tests for swelling soils according to ASTM standard D 4546-08,^[26] among which is the constant volume (CV) method. The swell-consolidation (CS) method and loaded swell (LS) method, which need many identical samples, have also been used.^[30,31,32] The swell and swelling pressure determined from these tests are the main parameters used to compute the total heave. The total heave of the homogeneous soil profile S_sw is equal to the sum of the increments heaving Δh_i for each elementary layer h_i, as shown in Figure 5. So, it is necessary to take into account in calculation the total stress variation σ_zi,t in the middle of each elementary layer under the center of the foundation.

Figure 5

The depth at which the swelling pressure equals the total geostatic stress is defined as the depth of potential heave H, as indicated by Equation (3),^[33] this depth represents the maximum depth of the active zone. (3)γsw×H=σsw{\gamma _{sw}} \times H = {\sigma _{sw}}

In the case of shallow foundations founded in a swelling soil, the depth of potential heave H is determined in free-field; after that, loading stresses σ_z,load transmitted by the foundation are added to the total stress σ_z,t for computation of foundation heave. This was used in analytical calculations.

In this study, the clayey soil is assumed to be homogeneous. The depth of potential heave (H= σ_sw/γ_sat) equals 11 m for σ_sw = 218 kPa and γ_sat = 20 KN/m³, which was obtained experimentally, whereas the depth was divided into equal layers h_i=1m for all the analytical calculations. It would be preferable to choose a layer thickness as small as possible to increase the accuracy of the calculations.^[2] Two equations were used in the present work for predicting the total heave based on swell-consolidation (CS) test.

3.1.1

Department of the Army

Technical Manual by the US Department of the Army^[17] proposed Equation (4) for the prediction of total heave: (4)Ssw=∑i=1nΔhi=∑i=1n{CDAhlog[σcsσf]}i{S_{sw}} = \sum\limits_{i = 1}^n {\Delta {h_i} = \sum\limits_{i = 1}^n {{{\left\{ {{C_{DA}}h\log \left[ {{{{\sigma _{cs}}} \over {{\sigma _f}}}} \right]} \right\}}_i}} } where S_sw is the total heave; Δh_i is the heave of layer i; h is the initial thickness of layer i; C_DA is the Department of Army heave parameter determined from the slope of the swelling strain versus pressure curve in Figure 3 (C_DA = ɛ_sw/log(σ_cs/σ_i)); σ_i is the initial pressure from CS test of layer i; σ_cs is the swelling pressure from CS test of layer i and σ_f is the final vertical normal stress of layer i (σ_f=σ_z,t= σ_z,g+σ_z,load).

The obtained value is: C_DA = 0.0886/log(218/1) = 0.037.

3.1.2

Nelson and Miller

Nelson and Miller^[1] proposed Equation (5) for the prediction of total heave: (5)Ssw=∑i=1nΔhi=∑i=1n{Cs1+ehlog[σcsσf]}i{S_{sw}} = \sum\limits_{i = 1}^n {\Delta {h_i} = \sum\limits_{i = 1}^n {{{\left\{ {{{{C_s}} \over {1 + e}}h\log \left[ {{{{\sigma _{cs}}} \over {{\sigma _f}}}} \right]} \right\}}_i}} } where C_s is the swelling index of layer i and e is the initial void ratio of layer i.

The used values are: C_s = 0.054 and e = 0.478.

In this study, analytical solutions can be sufficient to determine the total heave. However, the key limitation of these solutions is that the heave is only given in the center of the footing. So, numerical modelling is necessary because it allows to determine the final state of the soil and foundations after the swelling and to study the influence of several factors such as soil properties and geometric characteristics of foundations as well.

Numerical study was performed using the finite difference method FLAC 3D.^[34] A large number of calculation steps were used in the explicit Lagrangian resolution scheme. The maximum unbalanced force is the magnitude of the vector sum of the nodal forces for all the nodes within the mesh. When the maximum unbalanced force is small compared with the total applied force associated with stress or boundary displacement changes, the model is considered to be in equilibrium. The failure and plastic flow phenomena occur within the model when the unbalanced force approaches a constant value.

A rigid square shallow foundation of width B=1 m and thickness of 0.2 m was considered. This foundation was located at the surface of the swelling clayey soil and subjected to a distributed vertical loading σ₀ varying from 0 to 500 kPa. Because of the symmetrical nature of the problem and in order to reduce computation time, only a quarter of the system was modeled, as shown in Figure 6. The model was extended in both horizontal directions L_x=15 m, L_y=15 m and a total height H=11 m. For boundary conditions, the base of the model was constrained in all directions and the horizontal displacement was zero in the x-direction for the planes x=0 and x=15. Similarly, there was no displacement in y-direction for the planes y=0 and y=15.

Figure 6

The rigid square footing was made of concrete, modeled by a group of brick elements with a linear elastic constitutive model. The elastic moduli used were the shear modulus G=12.5 GPa and the bulk modulus K=16.67 GPa (equivalent to Young's modulus E=30 GPa and a Poisson's ratio ν=0.2) and the unit weight γ=25 KN/m³. It is important to note that if the footing is rigid, the parameters of concrete don’t influence the solution. The swelling clay is modelled using a linear elastic–perfectly plastic constitutive model following the Mohr–Coulomb (MC) failure criterion. The Mohr-Coulomb model is widely used in numerical modelling in geotechnical engineering due to its simplicity and accuracy. This model involves five parameters: the elastic modulus E, the Poisson's ratio ν, the cohesion C, the internal friction angle φ and the dilatancy angle ψ. Table 2 provides the model parameters used in the simulation. The elastic modulus E, which is used in the present study was conventionally estimated from the oedometer modulus E_oed and the Poisson's ratio ν. According to Hook's law, the relationship is given using Equation (6)^[35]: (6)E=Eoed(1−2v)(1+v)(1−v)E = {E_{oed}}{{(1 - 2v)(1 + v)} \over {(1 - v)}}Table 2

Soils parameters used in the numerical study.

The effective stiffness parameters represented by elastic modulus and Poisson's ratio of the soil are 10 MPa and 0.35, respectively. These two parameters are mainly affecting the evolution of soil heave.^[21] However, the strength parameters represented by the cohesion and the internal friction angle are 100 kPa and 25°, respectively. These parameters were obtained from drained direct shear tests performed on soil samples after saturation.

The discretization of the model was made by primitive elements of brick form with a local refinement of the most stressed and deformed zone, that is, in the vicinity and at the base of the footing. Many of the mesh sensitivity tests have been performed to ensure that the mesh size has no impact on the numerical results and to find the optimal mesh size. The optimal mesh size allows the modeler to spend less time in calculation. In all cases, an identical mesh size between the footing and the soil is well-defined to ensure connectivity of the nodes at the interface soil-footing. The mesh size was limited as 0.05 B to the footing surface and near the footing edge. Therefore, the number of footing elements was 100 and the mesh consisted entirely of 36850 elements and 40297 nodes, as shown in Figure 7.

Figure 7

The rigid footing was in contact with the soil through the interface element using the Mohr-Coulomb failure criterion. A rough interface along the base of the footing was adopted that had a cohesion C_int=100 kPa and a friction angle φ_int=25°, and also a normal stiffness K_n=10⁸ Pa/m and shear stiffness K_s=10⁸ Pa/m. According to Itasca,^[33] a good rule-of-thumb is that K_n and K_s be set to ten times the equivalent stiffness of the stiffest neighboring zone. The apparent stiffness of a zone in the normal direction is: (8)max[K+43GΔzmin]{max}\left[ {{{K + {4 \over 3}G} \over {\Delta {z_{{min}}}}}} \right] where K and G are the bulk and shear moduli, respectively; Δz_min is the smallest width of an adjoining zone in the normal direction.

Swelling soil was modeled in the initial state by the simulation of swelling pressure. In this phase, the model was subjected to gravitational loading, then to a vertical swelling pressure that equals 218 kPa constant throughout the entire depth was applied. During this phase, the model was monitored to ensure that no plastic points occur. After that, initial horizontal stresses were generated using K₀ equal 0.57. This ensures that horizontal stresses have been generated associated with the swelling pressure. The simulation steps included generating an initial stress condition, an interface and a footing activation followed by several compressing loading steps σ₀ applied on the footing.

The load was applied to the footing with a uniform pressure surface. The choice of this type of loading gives a good comparison with the simulated swelling pressure, which is uniformly distributed. During the modelling, the vertical displacement (heave) was followed until a constant value was reached at the end of loading. In this study, the ratio of the maximum unbalanced force taken was equal to 10⁻⁵. This ratio is recommended by Itasca to achieve equilibrium.

In this paper, a series of numerical analysis by finite-difference code FLAC were performed for isolated shallow foundations, subjected to a distributed vertical loading founded on a saturated swelling clayey soil mass following the Mohr–Coulomb (MC) failure criterion. We confirmed that the numerical computation results of the footing heave were compatible with the analytical predictions based on oedometer tests proposed in the literature. Based on the results of this numerical and analytical study, important conclusions drawn from this work include: –

The proposed numerical model based on the simulation of the swelling pressure is able to predict the heave of the soil mass loaded by a shallow foundation.