Clay minerals of natural soils are made up of silicon, aluminium, and/or iron and magnesium. They often have flat, layered structures with different shapes. Each clay particle in an expansive soil acts like a thin plate with negative charges on its surface and positive charges on its sides. The composition of clay can be thought of as different combinations of two basic building blocks. These building blocks are presented as simplified models for understanding the minerals’ structures.

Expansive soils, characterized by their significant volume changes (heave) upon wetting and shrinkage upon drying, constitute a major source of concern for civil engineering projects, impacting infrastructure stability and durability. These soils present complex geotechnical challenges due to their diverse mineralogical composition, sensitivity to moisture variations, and unpredictable behaviour under changing environmental conditions.

The primary clay mineral behind the expansive behaviour of these soils is the presence of smectite clays, particularly montmorillonite. These clays possess a 2:1 layered structure with interlayer spaces that readily accommodate water molecules, causing the layers to expand and the soil volume to increase. Upon drying, the interlayer water evaporates, resulting in layer contraction and soil shrinkage. This cyclic wetting and drying process can lead to significant differential movements, differential settlements, and the formation of desiccation cracks in expansive soils.

Therefore, clay minerals are an important microscopic factor affecting the engineering behaviour of soils. This article explores the common clay minerals and how their properties affect the expansive behaviour of soils.

Building Blocks of Clay Minerals

The building blocks have two basic elemental units: the silicon tetrahedron and the alumino-magnesium octahedron. They are shown in a simplified way in Figure 2. The silicon tetrahedron consists of silicon and oxygen atoms. Since silicon has a 4⁺ valence, it can bond with negative ions like oxygen (O^2-) or hydroxyl (OH^–), as seen in Figure 2(a). The relative sizes of the silicon and oxygen atoms make this structural unit take the shape of a tetrahedron.

The alumino-magnesium octahedron is made up of aluminium or magnesium atoms with hydroxyls around them, as seen in Figure 2(d). These atoms are placed so that they can be imagined as forming an octahedral shape.

In the silicon tetrahedron in Figure 2(a), the oxygen atoms have an unfulfilled chemical bond each. The oxygen atoms at the bottom of a tetrahedron are shared with nearby tetrahedra, and the resulting arrangement of tetrahedra makes sheets, as seen in Figure 2(b). Sharing of oxygen atoms among the tetrahedra satisfies the oxygen atoms at the bottoms of the tetrahedra, but the oxygen atoms at the tops still have unsatisfied bonds, as seen in Figure 2(b). So, the top face of the silica sheet can form chemical bonds with positive cations.

The octahedral units share hydroxyls to make a sheet structure, as seen in Figure 2(e). The placement of the octahedral units is such that the hydroxyls in the sheet structure have fulfilled chemical bonds. The central cation in the octahedral sheet can change.

The building blocks that are used to show the crystalline structure of the different clay minerals are represented by schematic symbols in Figures 2(c) and 2(f). By changing the way these two building blocks are arranged, different clay minerals can be formed.

Types of Clay Minerals

The different minerals are grouped according to the order of the sheets. For this article, we will only consider three basic clay minerals:

1. Kaolinite
2. Illite
3. Montmorillonite

Figure 3 shows simplified diagrams of the structures of these three minerals. The bonding between the different building blocks is very important for the behaviour of the different minerals.

The term bentonite is often used for expansive soils. This term means clays that have a lot of montmorillonite. Bentonite is a very plastic, swelling clay material that mainly has montmorillonite. It is mined for business and is used for many things like drilling fluids, slurry trenches, cosmetics, paint thickeners, and more. Not all expansive soils are bentonite, but they are often called bentonite, usually by people who are not engineers.

Kaolinite Clay Mineral

Kaolinite group minerals feature a 1:1 layered structure. Each layer consists of a single tetrahedral silica sheet directly bonded to an octahedral alumina sheet. Unlike the 2:1 structure of smectites like montmorillonite, the absence of an intervening octahedral sheet results in stronger interlayer bonding and minimal swelling behaviour.

The bonds between the silica and octahedral sheets affect the size of the mineral particles in a soil. The kaolinite structure in Figure 4 shows that there is a strong bond between the octahedral sheet and the top of the silica sheet. This is because of how the atoms are arranged and the fact that some of the hydroxyls in the octahedral sheets can be replaced by the oxygen atoms at the top of the silica tetrahedra. So, there is a strong chemical bond between the octahedral sheet and the silica sheet.

In Figure 3a, the chemical bonds at the bottom of the silica sheet are satisfied, and so, the bond between the octahedral sheet and the bottom of the silica sheet is mainly formed by weaker hydrogen bonds. That is called a weak bond. So, when the mineral breaks, the sheets will split at the weak bond. However, the bonds are strong enough that the clay particles can have a number of building blocks in each clay particle. Therefore, the kaolinite particles are relatively big with a lateral size of up to 1 micron (μm) or more, and a thickness of 1/3 to 1/10 of the lateral size.

Characteristics of Kaolin Clay Mineral

• High chemical stability: Kaolinite exhibits exceptional resistance to weathering and chemical attack due to its strong covalent bonds and low reactivity.
• Low cation exchange capacity (CEC): Compared to smectites, kaolinite possesses a significantly lower CEC due to the absence of exchangeable cations in the interlayer space.
• High whiteness and brightness: The pure aluminium silicate composition and minimal iron impurities endow kaolinite with excellent whiteness and brightness, making it desirable for applications requiring high optical qualities.
• Good rheological properties: The platy morphology and surface properties of kaolinite contribute to its ability to thicken and suspend particles in aqueous suspensions, essential for its use in various industrial processes

Montmorillonite Clay Mineral

Montmorillonite is a member of the smectite family, a group of 2:1 layered silicates characterized by a tetrahedral silica sheet sandwiched between two octahedral alumina sheets. This layered structure gives rise to several unique properties, including a large surface area, high cation exchange capacity (CEC), and the ability to intercalate guest molecules between the silicate layers.

The bond between the two silica sheets at the bottom is made by weaker van der Waals forces for the montmorillonite, and it is marked as “very weak” in Figure 2c, unlike just “weak.” Because of this, montmorillonite particles may have only one or two sets of building blocks in thickness. So, the clay particles may be very thin, about 10 Ångström or less.

The unit cell of montmorillonite consists of two silica tetrahedral sheets linked to a central octahedral alumina sheet via shared oxygen atoms. This basic structure forms repeating layers stacked upon each other with weak van der Waals forces holding them together. The negative charge on the basal oxygen atoms is balanced by exchangeable cations (commonly Na⁺, C^a2+, or Mg²⁺) residing in the interlayer space. These cations contribute to the high CEC of montmorillonite, enabling its selective adsorption of various cations and organic molecules.

Characteristics of Montmorillonite

1. Swelling: The presence of exchangeable cations and hydration of interlayer space allows montmorillonite to expand significantly upon contact with water, leading to increased volume and plasticity.
2. Adsorption: The high surface area and CEC of montmorillonite facilitate the adsorption of various pollutants, toxins, and organic molecules, making it useful for environmental remediation and water purification.
3. Cation exchange: The selectivity of montmorillonite towards specific cations can be exploited for soil amendment, nutrient retention, and catalysis.
4. Intercalation: Guest molecules, such as polymers and drug molecules, can be inserted into the interlayer space of montmorillonite, leading to the formation of nanocomposites with unique properties.

Illite Clay Mineral

Illite comprises a group of mixed-layer clay minerals with structural features mirroring both smectites and micas. Its basic 2:1 layer unit resembles that of smectites, featuring a single tetrahedral silica sheet sandwiched between two octahedral alumina sheets. However, unlike smectites, illite possesses interlayer potassium ions occupying vacant octahedral sites, resulting in stronger interlayer bonding and reduced swelling behaviour.

The illite particle has a similar structure to the montmorillonite particle. But in illite, the bond between the silica sheets at the bottom is made by potassium cations that are shared by nearby sheets. The potassium ions are the right size to fit into the gaps in the bottom of the silica sheet made by the tetrahedra. Sharing the potassium ions between the silica sheets makes a strong bond. So, the kaolinite and illite minerals are much less expansive than the montmorillonite.

Characteristics of illite clay minerals

• Intermediate swelling behaviour: Compared to highly swelling smectites, illite exhibits minimal swelling upon hydration due to the presence of fixed interlayer potassium ions and stronger layer bonding.
• Moderate cation exchange capacity (CEC): While lower than smectites, illite possesses a notable CEC due to the exchange of cations on the basal plane and edges, influencing its interactions with various cations and organic compounds.
• High thermal stability: Due to its strong Si-O-Al bonds and stable interlayer potassium ions, illite exhibits exceptional thermal stability, making it suitable for high-temperature applications.
• Excellent rheological properties: The platy morphology and surface properties of illite contribute to its ability to thicken and suspend particles in aqueous suspensions, valuable for its use in drilling fluids and other industrial processes.

The stress in the soil mass is affected by the previous loading history. When a new foundation is constructed, the new load can either increase or decrease the existing stresses in the soil. The response of the soil mass to the new load depends on the previous stress history. Therefore, it is important to understand the stress imprint of the soil mass before designing a foundation.

The term “stress imprint” refers to the state of stresses that are locked into the soil structure. These stresses are locked in because the soil particles have rearranged and formed bonds. The bonds prevent the stresses from being released, even when the load is removed. The stress imprint can be either positive or negative.

A positive stress imprint is created when the soil is loaded to a higher stress than it is currently experiencing (normally consolidated soil). A negative stress imprint is created when the soil is unloaded to a lower stress than it is currently experiencing (overconsolidated soil).

The stress imprint can affect the behaviour of the soil in a number of ways. For example, a positive stress imprint can make the soil more resistant to shearing, while a negative stress imprint can make the soil more susceptible to settlement.

The stress imprint can be difficult to measure, but there are a number of techniques that can be used. One common technique is to use a pressuremeter. A pressuremeter is a device that is inserted into the soil and used to measure the stresses in the soil.

In-Situ Stresses in Soil

In situ, the vertical stresses act on a horizontal plane at some depth z. These can be computed in any general case as the sum of contributions from n strata of unit weight γ_i and thickness z_i as;

p_o = ∑γ_iz_i —— (1)

In the context of the formation of soil deposits, the area of the land where soil accumulates is typically extensive, and the depth of the deposit keeps increasing until either the accumulation process or the internal weathering process halts. This transition leads to a gradual downward compression of the soil at any specific depth.

Likewise, the vertical stress also increases due to this compression, and in almost all instances, the unit weight of the soil is a function of depth. Given the substantial lateral extent of the deposit, there is limited justification for notable lateral compression to take place.

Consequently, it is reasonable to anticipate that the vertical locked-in effective stresses (p’_o) would surpass the effective lateral stresses (σ’_h) at the same location. This relationship between horizontal and vertical stresses can be defined as the ratio of the two.

K = σ_h/p_o ——– (2)

which is valid for any depth z at any time.

K₀ Conditions

Over geological time the stresses in a soil mass at a particular level stabilize into a steady state and strains become zero. When this occurs the vertical and lateral stresses become principal stresses acting on principal planes. This effective stress state is termed the at-rest or K₀ condition with K₀ defined as;

K₀ = σ’_h/p’_o ——– (3)

Therefore, K_o conditions refer to the state of stresses in a soil mass when there is no lateral strain. This means that the soil is not allowed to deform horizontally. K_o conditions are typically found in undisturbed soil, where the soil has not been subjected to any significant lateral stresses.

Figure 2 shows the range of K₀ and the relationship between p_o and σ_h in any homogeneous soil. The figure also shows the qualitative curves for preconsolidation in the upper zone of some soil from shrinkage/chemical effects. The figure clearly illustrates the anisotropic (σ_v ≠ σ_h) stress state in a soil mass.

In the figure above, although the linear vertical (also called geostatic) pressure profile is commonly used, the p’₀ effective pressure profile is more realistic of real soils since γ usually increases with depth. The lateral pressure profile range is for the geostatic pressure profile and would be curved similarly to the p’_o curve for real soils.

K₀ conditions are critical for:

• Retaining wall design.
• Excavation support.
• Earthquake-induced lateral pressures.
• Soil-structure interaction.

Because of the sampling limitations, it is an extremely difficult task to measure K₀ either in the laboratory or in situ. Some field methods are available, but note that they are very costly for the slight improvement—in most cases—over using one of the simple estimates following. In these equations use the effective angle of internal friction ‘ and not the total stress value.

Jaky (1948) presented a derived equation for K₀ that is applicable to both soil and agricultural grains (such as corn, wheat, oats, etc.) as;

K₀ = [(1 – sinφ’)/(1 + sinφ’)] × ( 1 + 2/3sinφ’) ——– (4)

which has been simplified—and erroneously called “Jaky’s equation”— to the following:

K₀ = 1 – sinφ’ ——— (5)

This equation is very widely used and has proved reasonably reliable in comparing initial to back-computed K₀ values in a number of cases and for normally consolidated materials. Kezdi (1972) suggests that for sloping ground Jaky’s equation can be used as follows:

K₀ = (1 – sinφ’)/(1 + sinβ) ——– (6)

where β is the angle with the horizontal (with sign) so that K₀ is either increased or reduced as site conditions dictate. This reference also gives a partial derivation of the Jaky equation for any interested user.

Brooker and Ireland (1965) (for normally consolidated clay) suggest;
K₀ = 0.95 – sinφ’ ——– (7)

Alpan (1967) (for normally consolidated clay) suggests;
K₀ = 0.19 + 0.233 log₁₀ Ip ——– (8)

An equation similar to Eq. (8) is given by Holtz and Kovacs (1981) as;
K₀ = 0.44 + 0.0042I_p ——– (9)

where I_p is the plasticity index of the soil in percent.

We can readily derive a value for K₀ in terms of Poisson’s ratio based on the definition of K₀ being an effective stress state at zero strain. From Hooke’s law, the lateral strain in terms of the effective horizontal (x, z) and vertical (y) stresses is;

ε_x = 0 = 1/E_s(σ_x – μσ_y – μσ_z) = ε_z ——– (10)

For a cohesionless, soil μ is often assumed as 0.3 to 0.4, which gives K₀ = 0.43 to 0.67, with a value of 0.5 often used. It is extremely difficult to obtain a reliable estimate of K₀ in a normally consolidated soil, and even more so in overconsolidated soils (OCR > 1). A number of empirical equations based on various correlations have been given in the literature. Several of the more promising ones are:

Alpan (1967) and others have suggested that the overconsolidated consolidation ratio K_0,OCR is related to the normally consolidated value K_0,nc in the following form;

K_0,OCR = K_0,nc × OCRⁿ (2-23) ——– (11)

where n = f(test, soil, locale) with a value range from about 0.25 to 1.25. For overconsolidated sand, n can be estimated from Figure 3.

For cohesive soil, Wroth and Houlsby (1985) suggest n as follows:

n = 0.42 (low plasticity — I_P < 40%)
n = 0.32 (high plasticity — I_P > 40%)

However, n ≈ 0.95 to 0.98 was obtained from in situ tests on several clays in eastern Canada. Mayne and Kulhawy (1982) suggest that a mean value of n = 0.5 is applicable for both sands and clays and that n = sinφ’ is also a good representation for sand. Their suggestions are based on a semi-statistical analysis of a very large number of soils reported in the
literature.

The exponent n for clays was also given by Alpan (1967) in graph format and uses the plasticity index I_P (in percent). The author modified the equation shown on that graph to obtain;

n = 0.54 × 10^(-I_P/281) ——– (12)

And, as previously suggested (for sands), we can use;

n = sinφ’ ——— (13)

Conclusion

Geostatic conditions refer to the stress state within a soil mass due to the weight of the overlying soil. These stresses are primarily vertical and compressive, but they also induce horizontal stresses due to the frictional resistance of the soil particles. The magnitude of these stresses depends on the depth of the soil layer, the unit weight of the soil, and the geometry of the soil mass.

The K₀ condition, also known as the at-rest earth pressure, represents a specific stress state within a soil mass where there is no lateral strain. This condition occurs when the soil is horizontally constrained, such as by a retaining wall or a layer of bedrock. In K₀ conditions, the horizontal effective stress (σ’_h) is typically about 0.5 to 0.7 times the vertical effective stress (σ’_v). The factors affecting K₀ values are the overconsolidation ratio, type of soil, confining pressure, stress history, and effective stress state.

An earth slope can be either a naturally occurring incline or one that is artificially constructed through excavation or engineered fill to produce an embankment. A slope failure refers to the downward displacement of a section of the slope mass in relation to the mass below the sliding surface. Therefore, slope stability assessment is an important aspect of earthworks and geotechnical designs in civil engineering works.

The magnitude of a slope failure ranges from a few meters in height to the displacement of a substantial portion of a high land or mountain. An instance of this is the 1974 Rio Mantaro landslide in Peru, which consisted of a sliding mass of 6 kilometres in length, 2 kilometres in height, and a volume of 1.5 billion cubic meters.

Figure 1 shows a significant slope failure that took place in Oso, Washington, USA, on March 22, 2014. The failure surface displays a rotational sliding surface.

This article focuses on the stability analyses of unreinforced soil slopes under static loading.

Causes of Slope Failure

A slope failure occurs when the external shear stress (or sliding moment) exceeds the internal shear strength (or resisting moment) of the slope.

Shear stress > Shear strength or:
Rotational moment > Resisting moment

The following factors may increase the shear stress or sliding moment:

• Additional surcharge at the top of a slope.
• Application of lateral force that may be caused by seepage, earthquake, or pile driving.

The following factors may decrease the shear strength or resisting moment of a slope:

• Weathering of a rock slope.
• Discontinuities such as weak seams and faults that are developed in the slope.
• Saturation of the slope.
• Removal of lateral support of the slope, for example, the cut toe of a slope.

Types of Slope Failure

Slope failures can be classified into different types based on the shape of the failure surface and the nature of the slope movement.

• Surficial (or translational) slope failure: The sliding surface is parallel to the slope surface. This type of failure can occur when there is a weak layer of soil or rock beneath the surface, or when a loose topsoil layer rests on a hard subsoil layer.
• Rotational slope failure: A large mass of the slope rotates along a curved failure surface. This type of failure is often caused by heavy rainfall or by excavation at the toe of the slope.
• Landslide: A landslide is a large-scale slope failure that can involve multiple types of slope movements. Landslides can be caused by a variety of factors, including heavy rainfall, earthquakes, and volcanic eruptions.
• Lateral spread: Lateral spread is the lateral movement of a fractured soil mass. It is typically caused by earthquakes.
• Debris flow: A debris flow, or mudslide, is a rapid movement of soil that is entrained by flowing water or wind. Debris flows are often triggered by heavy rainfall.
• Creep: Creep is a slow and almost imperceptible movement of the failure portion of a slope. It can be caused by a variety of factors, including gravity, groundwater flow, and frost heave.
• Rock falls: Rock falls occur when rocks on a slope are mobilized by wind, runoff, or gravity. Rock falls can be very dangerous, as they can travel at high speeds and cover long distances.

Slope Stability Analysis

Slope stability analysis methods typically use the limit equilibrium approach, which means that the forces or moments that cause a slope to fail (slide) are balanced by the forces or moments that resist the slope from sliding. This is known as the critical condition. A factor of safety (FS) is used to measure slope stability and is based on the force or moment equilibrium.

FS = 𝜏_f/𝜏
or:
FS = M_resist/M_slide

where;
𝜏_f = the maximum shear stress at failure, which is equal to the shear strength,
𝜏 = shear stress that causes the sliding of a failure portion, which is caused by external loads such as gravity, foundation loading, seismic force, etc.,
M_resist = total resisting moments that resist a rotational sliding,
M_slide = total sliding moments that cause a rotational sliding.

In general, a value of 1.25–1.5 is used as an acceptable factor of safety. Alternatively, within the context of limit state design, it should be mentioned that:

E_d ≤ R_d

where;
E_d = design effect of the actions (e.g., sliding forces)
R_d = design resistance (dependent on soil strength)

The shear strength is expressed by the Mohr–Coulomb failure criterion:

𝜏_f = c + 𝜎 tan 𝜙

where;
𝜎 = total normal stress,
c = soil’s cohesion based on the total stress,
𝜙 = soil’s internal friction angle based on the total stress.

The Mohr–Coulomb failure criterion can also be expressed using the effective stress:

𝜏_f = c′ + 𝜎′ tan 𝜙′

There are two main approaches for analyzing slope stability: the total stress method and the effective stress method. The total stress method uses undrained shear strength parameters (c and ϕ), while the effective stress method uses drained shear strength parameters (c′ and ϕ′)

In slope stability analyses, two “artificial” factors of safety are defined on the basis of c and 𝜙, respectively:

FS_c = c/c_m
FS_𝜙 = tan 𝜙/tan 𝜙_m

where;
FS_c = factor of safety on the basis of c,
FS_𝜙 = factor of safety on the basis of 𝜙,
c_m = mobilized cohesion that is actually developed along a slip surface, also denoted as c_d,
𝜙_m = mobilized internal friction angle that is actually developed along a slip surface, also denoted as 𝜙_d.

In slope stability analysis, partial factors of safety can be used to account for uncertainties in the soil properties and the loading conditions. These factors are similar to those used in other engineering design codes. In a stable slope, not all of the soil’s strength is needed to resist the forces that cause it to slide.

The mobilized shear strength, which is the amount of shear strength that is actually used to resist sliding, is always less than or equal to the peak shear strength. This means that there is always a factor of safety built into the design of a stable slope. The developed or mobilized shear strength that is needed for slope stability is represented by c_m and 𝜙_m. Therefore, c_m ≤ c and 𝜙_m ≤ 𝜙.

At equilibrium, the developed or mobilized shear stress can be expressed as:

𝜏_m = c_m + 𝜎 tan 𝜙_m

At the critical condition, the entire shear strength is needed for equilibrium. So, c_m = c, and 𝜙_m = 𝜙. Therefore, the minimum value for both FS_c and FS_𝜙 is 1.0.

If: FS_c = FS_𝜙 = a (constant)

Then: c = a⋅c_m, and tan 𝜙 = a⋅tan𝜙_m

FS =𝜏_f/𝜏_m = (c + 𝜎 tan 𝜙)/(c_m + 𝜎 tan 𝜙_m) = a
So ∶ FS = FS_c = FS_𝜙

Methods of Analysis of Slope Stability

Different methods can be used in the analysis of stability of slopes. The methods can be generalised as;

• Mass methods
• Methods of slices
• Finite element methods

In the mass methods, the sliding soil mass is analyzed as one entity. The mass methods are applicable only to homogeneous slopes and can be employed in the analysis of finite and infinite slopes. In the analysis of finite slope, Culmann’s method is normally used for planar failure surfaces, while Taylor’s chart and Michalowski’s chart are used for curved failure surfaces.

In the methods of slices, the sliding soil mass is divided into numerous slices and the stability of each slice is analyzed. Then all the slices are combined to derive the factor of safety of the slope for the assumed failure surface. To obtain the true factor of safety of the slope, numerous trial surfaces are analyzed that provide the minimum factor of safety. The methods are applicable to homogeneous or heterogeneous slopes.

The popular methods in this category are the ordinary method of slices (Fellenius method of slices), Bishop-simplified method of slices, Bishop and Morgensten method, Spencer method with consideration of pore water pressure, Morgenstern charts for rapid drawdown, etc.

The finite element method involves modelling the soil using finite elements and simulating the failure of the slope. Computer packages such as Plaxis are capable of such models.

Method of Slices

The method of slices is a numerical procedure that has been developed to handle stability analysis of slopes where conditions are nonhomogeneous within the soil mass making it impossible to deduce closed-form solutions. Some of the non-homogeneous conditions commonly encountered are irregularity of failure planes, variable soil properties, significant variation in the distribution of pore pressure along the failure plane, irregular slope geometry, etc.

The analysis requires the selection of a trial failure plane and discretization of the resulting failure wedge into a convenient number of slices as shown in Figure 3.

The analyst is required to devise the slicing in a manner that can incorporate any nonhomogeneity within the slope so that each resulting slice would be a homogeneous entity. Then, the stability of each slice can be analyzed separately using the limit equilibrium method and principles of statics. The static analysis of the slices can be obtained in most geotechnical engineering textbooks.

Assumptions of the Method of Slices

The method of slices makes several simplifying assumptions to facilitate analysis:

1. The slope is divided into vertical slices of uniform or variable thickness.
2. The failure surface is assumed to be planar or circular for simplicity.
3. The soil within each slice is assumed to be homogeneous and isotropic.
4. The normal and shear stresses at the base of each slice are assumed to be uniformly distributed.
5. Interslice normal and shear forces are neglected or assumed to be constant.

The factor of safety of the slope on any assumed circular slip surface using the method of slices is:

The ordinary method of slices assumes that the interslice forces on both sides of each slice cancel one another. Therefore, the factor of safety derived from this method is overconservative and is lower than other methods of slices.

Worked Example

Calculated the factor of safety of the earth slope shown in Figure 4 using the ordinary method of slices. Assume the failure circle shown in Figure 5.

The selected failure surface and the number of slices are shown in Figure 5.

The solution can be presented in the tabular form shown below;

The factor of safety can then be calculated thus;

FS = (c∑col[8] + ∑col[10]tan𝜙)/∑col[9]
FS = (21 × 35.848 + 2163.786 tan 25)/1488.01 = 1761.794/1488.01 = 1.18

Conclusion

Slope stability analysis is a crucial aspect of geotechnical engineering, ensuring the stability of natural slopes, embankments, and excavations. It evaluates the likelihood of a slope failure, which can have catastrophic consequences for infrastructure, property, and human safety.

The most common approach to slope stability analysis is the limit equilibrium method, which assumes that the slope is either in a state of equilibrium or on the verge of failure. It involves calculating the driving forces that cause the slope to slide (e.g., gravitational forces) and the resisting forces that prevent sliding (e.g., shear strength of the soil).

The method of slices is a widely used limit equilibrium technique for analyzing slope stability. It involves dividing the slope mass into a series of vertical slices and analyzing the forces acting on each slice. The forces acting on each slice include its weight, the normal and shear forces at its base and sides, and any external forces such as water pressure.