Advancements in Shear Strength Interpretation, Testing, and Use for Landslide Analysis

Abstract

Landslides are devastating natural disasters that result in loss of life, property damage, and community disruption. They have global impacts, causing fatalities and economic losses, particularly in mountainous regions near densely populated areas. Landslides can be caused by natural factors, including water saturation from heavy rainfall, snowmelt, and changes in groundwater levels, as well as seismic activity such as earthquakes and volcanic eruptions. Human activities, such as altering drainage patterns, destabilizing slopes, and removing vegetation, also contribute to landslides. Construction and development on slopes, over-steepening, and improper land management practices can further increase the risk of landslides. A key component in understanding the stability of slopes will be knowledge of the shear strength of the soils involved. However, to do so, it will be necessary to understand the various measuring methods of shear strength, loading conditions, and other parameters. Different methods and tests are employed to determine the shear strength of soil, depending on the specific conditions and objectives. Direct shear tests are often utilized to measure peak and fully softened shear strengths. Triaxial tests, on the other hand, are suitable for assessing both peak and fully softened shear strengths under drained or undrained conditions. Generally, the ring shear device is preferred for measurements of the residual shear strengths. However, multiple reversal direct shear tests and specifically modified direct shear tests as well as triaxial tests have also been utilized for this purpose. The cyclic simple shear test is recommended as an effective technique for replicating in-situ conditions to investigate the cyclic resistance and post-cyclic shear strengths of soils. Several correlations have been developed in the literature to estimate various shear strengths, including the fully softened and residual shear strengths of soil, as summarized in this paper. These correlations utilize parameters such as the liquid limit, plasticity index, mineralogy, clay fraction, and effective normal stress. The undrained shear strength of over-consolidated soils can be captured with the use of the Stress History and Normalized Soil Engineering Properties (SHANSEP) method. Extending this approach with the use of the normalized undrained strength ratio can result in two correlations that can capture the undrained shear strength. The paper also presents correlations for the true and base friction angles to estimate the shear strength using Hvorslev’s theory. This allows for a departure from the use of the cohesion intercept and friction angle in the Mohr-Coulomb failure envelope, both of which are dependent on the over-consolidation ratio. The power function effectively represents the cyclic strength curves in soils with the curve fitting parameters a and b defining their shape and position. A correlation between the normalized undrained strength ratio and post-cyclic effective stress ratio to assess the undrained shear strength after cyclic loading was also introduced. This correlation was shown to also capture the effects of excess pore pressure dissipation and reductions in shear strength induced by a second cyclic load.

1 Background

Landslides are natural disasters that can have catastrophic consequences, causing loss of life, immense property damage, and disrupting communities. They are characterized by the rapid movement of soil, rock, and debris down slopes and have the potential for far-reaching impacts on both human and natural environments. Understanding the severity of landslides and their implications is essential for implementing proactive measures to mitigate risks and enhance resilience in vulnerable areas.

Landslides caused approximately 11,500 fatalities per year between 2004 and 2016 worldwide (United Nations Office for Disaster Risk Reduction, UNDRR 2019). It is important to note that the number of fatalities can vary greatly depending on the severity and location of the landslide, as shown in Fig. 1. This figure shows that the risk of deaths from landslides increases in highly mountainous regions in proximity to dense neighborhoods. In particular, Fig. 1 shows that the highest mortality risks appear in Asia near the Himalayas and in South America by the Andes. Some examples of disastrous landslides in the past decade are provided in Table 1. In addition, to the location of the failure, the table also provides some information about the volume of material involved and the number of fatalities caused.

Fig. 1

Mortality risk from landslides (Source: Ritchie et al. 2022)

Table 1 Selected disastrous landslide events in recent history

In addition to the fatalities, landslides can cause extensive damage to buildings, roads, and other infrastructure. The economic impact can be significant. For example, in the United States, landslides cost an average of $1–2 billion annually in direct costs and indirect economic losses (USGS 2022). Figure 2 shows the total economic damages resulting from landslides around the world in 2004, 2010, and 2016. As seen, the cost can be much higher in countries with less developed infrastructure and limited resources for disaster response and recovery.

Fig. 2

Economic damages as a percent of the gross domestic product (GDP) in (top) 2004, (middle) 2010, and (bottom) 2016 (Source: Ritchie et al. 2022)

Other impacts by landslides cannot be captured quantitatively. For example, landslides can have long-lasting effects on the environment. They can alter landscapes, destroy vegetation, and disrupt ecosystems. The sediment and debris carried by landslides can also affect rivers and streams, leading to water pollution and affecting aquatic habitats. The disturbance they cause to communities can lead to the displacement of residents, loss of livelihoods, and social and economic instability. Infrastructure damage can hamper transportation, communication, and access to essential services, affecting the overall well-being of the impacted areas.

2 Causes of Landslides

Landslides can be attributed to two main categories of causes: natural factors and those influenced by human activities. In certain instances, landslides may arise from a combination of both factors, either exacerbating their occurrence or contributing to their initiation.

Landslides are often initiated by the saturation of slopes due to the infiltration of water. This saturation process can be triggered by various elements, such as heavy rainfall, snowmelt, fluctuations in groundwater levels, and changes in surface water near coastal areas, earth dams, lakes, reservoirs, canals, and rivers. The connection between landslides and floods is tightly intertwined as both are influenced by precipitation, runoff, and the saturation of soil by water. Floods can contribute to landslides by eroding stream and river banks and saturating slopes through overland flow. On the other hand, landslides can lead to flooding when displaced rocks and debris obstruct water pathways, causing water to accumulate. Moreover, the debris from landslides can impede natural water channels, diverting the flow and resulting in localized erosion and flood-like conditions. Landslides can also have additional consequences, including the potential to generate tsunamis, cause reservoirs to overflow, and reduce the capacity of these reservoirs to store water. Steep slopes affected by wildfires are particularly susceptible to landslides due to a combination of factors, including the loss of vegetation due to burning, changes in soil chemistry, and subsequent saturation of slopes from various water sources.

The occurrence of earthquakes significantly increases the likelihood of landslides due to multiple factors. The seismic shaking itself can directly trigger landslides, while susceptible sediment layers may undergo liquefaction, and soil materials can experience dilation caused by the shaking, facilitating rapid water infiltration. Another significant hazard is the formation of landslide dams in streams and rivers downstream of steep slopes. These dams occur when the earthquake dislodges rocks and soil, obstructing the natural flow of water and causing it to accumulate behind the dam, leading to upstream flooding. However, due to their inherent instability, these landslide dams can undergo rapid or gradual erosion, eventually failing catastrophically and releasing the stored water suddenly with severe downstream damages and impacts.

Volcanic activity can trigger devastating landslides, such as volcanic debris flows (lahars), where melted snow mixes with rock, soil, ash, and water, rapidly descending steep volcano slopes and causing widespread destruction. These events can also lead to the collapse of volcanic structures, resulting in rockslides, landslides, and debris avalanches, with the potential to generate submarine landslides and tsunamis.

Human expansion onto new land and the development of residential areas, towns, and cities is another primary contributor to the occurrence of landslides. Activities such as altering drainage patterns, destabilizing slopes, and removing vegetation are common human-induced factors that can initiate landslides. Over-steepening slopes through excavation and excessive loading can also lead to slope instability. Additionally, human activities like irrigation, landscaping, reservoir management, leaking pipes, and improper excavation or grading on slopes can trigger landslides, even in previously stable areas.

3 Planning and Design of Landslide Stabilization Works

To understand the mechanics involved in the landslide process and design appropriate prevention (or control) measures, slope stability analysis is performed. The methods of slope stability analysis could be simple equations that can be solved in available spreadsheets to complex equations that require robust numerical analysis platforms. Regardless of the method used, common parameters needed for the analyses include slope geometry/topography, potential depth of the sliding surface, location of the groundwater table, soil density, external forces causing destabilization, and shear strength properties of soil.

Shown in Fig. 3 is a photograph of the Portuguese Bend Landslide, among the largest active landslides in the USA. This landslide is located near Los Angeles in California. We use very simple to complicated equations to plan and/or design stabilizing measures for such landslides. To initiate any slope stability analysis, we need to first obtain the cross-sectional details of the landslides that include the ground surface, groundwater table, and sliding surface profiles of the landslide. The ground surface profile is obtained from the topographical map/survey. Groundwater surface profiles can be obtained from groundwater surveys through boreholes. The sliding surface profile is obtained from borehole profile information.

Fig. 3

Picture of Portuguese Bend Landslide, among the largest active landslides in USA

Shown in Eq. 1 is a simplest formula to calculate stability of the slope, using the cross-sectional details explained in Fig. 4. The entire slope is divided into a number of vertical slices and stability calculations are performed for each slice and later summed up. In Eq. 1, F is factor of safety of slope, L_n is the length of the sliding surface, W_n is weight of the slice, α is angle of inclination of the sliding plane, F_h is horizontal components of all external forces, u_n is pore water pressure at the sliding surface, and c and ϕ are the shear strength parameters (cohesion intercept and internal friction angle, respectively), which will be described in more detail in the subsequent sections.

Fig. 4

A typical cross-section of a landslide area in Japan

$$ F=\frac{\sum \limits_{n=1}^{n=p}\Big({c}^{\hbox{'}}\Delta {L}_n+\left(\left({W}_n Cos\, {\alpha}_n-{F}_h\, \sin \alpha -{u}_n\Delta {L}_n\right)\tan {\phi}^{\hbox{'}}\right)}{\sum \limits_{n=1}^{n=p}{W}_n\, \sin \, {\alpha}_n+{F}_h\, \cos \, \alpha } $$

(1)

4 Brief Overview of Shear Strengths

The shear strength of soils plays a critical role in slope stability analyses. It is vital to evaluate the potential for slope failures and to determine the appropriate measures to mitigate risks. The internal resistance of a soil to a shearing force is known as the shear strength of soil. This resistance is jointly developed because of the friction between soil particles and the work that is required to cause changes in volume of the soil. Both the friction between individual grains of the soil mass, or the grain-grain friction, and the interlocking of the grains will contribute to the first component of the resistance. On the other hand, electromagnetic attraction between the soil particles and the effects of over-consolidation will result in the development of the second component of the resistance. As expected, factors that affect either or both of the components of the resistance will result in changes to the shear strength of soil. Some notable factors that will strongly influence the shear strength include the density (or void ratio), the overburden pressure (or confining stress), grain shape and roughness and stress history.

To clearly illustrate the shearing behavior of soil and its strength, it is important to understand the soil formation process. Soils are formed through the weathering of rocks. As such mineralogical composition of the soil will depend on the parent rocks. Let’s take an example of a soil layer formed through sedimentation of soil particles that were transported down by the water bodies. As these particles sediment and accumulate, they create soil layers at the bank of the water body, as presented in Fig. 5. Upward arrow in left image in Fig. 5 shows that the soil is continuously filled up as the sedimentation process continues. However, the top layer of the soil is also lost during erosion and landslide processes, as shown by the downward arrow in right image in Fig. 5.

Fig. 5

(Left) Soil sedimentation process; (right) loss of soil due to various natural causes

The mechanism how soil density changes with an increase in soil depth and the effect on its shear strength is presented in Fig. 6. Effective normal stress is the overburden pressure acting on the soil element at any location, which is sum of the unit weight of soil mass multiplied by the corresponding depth of each layer minus any pore water pressure acting the soil element (unit weight of water multiplied by the depth of water above the soil element). Void ratio is calculated as the volume of void space in the soil mass divided by the total volume of only the solid soil components. With an increase in the height of sedimentation, effective normal stress increases and that will cause a reduction in the volume of void space by squeezing the air or water filling the pore spaces out. As such, the void ratio of soil element decreases with an increase in soil sedimentation depth or the effective normal stress, as presented in Fig. 6. Let’s consider soil element x in Fig. 5. With an increase in effective normal stress on x as the sedimentation depths increase to the level y, level a and the top surface, the void ratio decreases logarithmically. Shear strength of the soil also increases linearly with the increase in the effective normal stress, as presented in Fig. 6. When a certain depth of the same soil mass is lost due to various natural phenomenon such as erosion or landslides, the effective normal stress decreases and the corresponding void ratio of the soil increases. However, the void ratio of soil element at x does not rebound back to the original void ratio observed during the sedimentation process. Slope of the rebound (or swelling) curve is around one fourth to one sixth of the slope of the original (or virgin) curve. As such, soil already at x, overburdened with the depth up to point b (after the loss of soil depth) in Fig. 5 has higher density and shear strength than it was under the condition presented at point a although the total overburden depth is the same. Soil element at depth x under the overburden pressure due to soil depth presented at point a and b are called normally consolidated and over-consolidated soils, respectively. In other words, normally consolidated soils (soil at point x during sedimentation process shown on the left side in Fig. 5) does not experience effective normal stress more than the current effective normal stress, while the over-consolidated soils (soil at point x shown on the right side in Fig. 5) has experienced higher effective normal stresses in the past compared to the current effective normal stress. For the same effective normal stress, over-consolidated soils exhibit higher shear strengths than the normally consolidated soils, as shown in Fig. 6. Shear strength of the soil is generally denoted as S_u in geotechnical engineering practice and it has a significant role in stability of the slopes.

Fig. 6

Void ratio and shear strength variation in soils with effective normal stress

When we take a soil sample from location x on the right side of Fig. 5, confine it in a container, apply a certain amount of vertical stress and then apply horizontal stress incrementally to increase the horizontal displacement along a shearing plane, horizontal stress required to obtain certain displacement increases consistently up to a peak value. At that point, the soil starts rupturing and it will require less horizontal stress to deform the soil further. The required horizontal stress become constant at certain amount of soil displacement. This shear stress is called residual shear stress. This shear stress—shear displacement relationship is presented in Fig. 7. If we take the soil from location x at left side of Fig. 5 and follow the same experiment, we obtain similar behavior. However, the magnitude of peak shear stress is much less than that observed for the soil from the right side, although the residual shear stress remains the same. That is mainly due to the over-consolidation effect explained earlier and illustrated in Fig. 6. If we completely remold the soil sample from location x at the right side of Fig. 5 and repeat the experiment, the shear stress-shear displacement curve will follow the trend observed for the soil at location x collected from right side of Fig. 5. If we repeat the experiments by increasing the amount of effective normal stress (σ’), shear stress required for the over-consolidated peak, normally consolidated peak, and residual shear stress conditions also increase corresponding to the increase in normal stress, as presented in Fig. 7.

Fig. 7

Shear stress—shear displacement (left) and shear stress-normal stress (right) relationship for over-consolidated and normally consolidated soils

Materials were found to rupture because of a combination of applied normal and shear stresses (Mohr 1900). Mohr’s work resulted in a relationship that expressed the shear stress at failure (or shear strength) as a function of the applied normal stress and resulted in a curved failure envelope. Coulomb (1776) noted that it was sufficient to estimate the shear stress on a failure plane (τ) to be linearly related to the normal stress (σ). This resulted in the Mohr-Coulomb failure criteria, as defined by Eq. 2 and represented by the straight lines on the right side of Fig. 7. In Eq. 2, c represents the cohesion intercept and ϕ is the internal friction angle. The soil mass will not fail if the combination of the normal and shear stresses falls below the Mohr-Coulomb failure envelope. However, the soil mass fails if this combination plots above the envelope.

$$ \tau =c+\sigma \tan \phi $$

(2)

The Mohr-Coulomb failure criteria is applicable for both drained and undrained loading conditions. Under drained loading conditions, loads are applied on a soil mass a rate sufficiently slow to ensure that no excess pore pressures are induced. Drained loading conditions result when loads in the field are applied very slowly or when the loads applied are present on the soil mass for a sufficiently long duration to ensure that all excess pore pressures that were generated have been allowed to dissipate. On the other hand, when loads are applied rapidly such that the generated excess pore pressures cannot dissipate, undrained loading conditions will prevail. While Eq. 2 is applicable for undrained (or also called total) conditions, the Mohr-Coulomb failure criteria is expressed in terms of Eq. 3 for the drained (or also called effective) conditions. In Eq. 3, σ’ is the effective normal stress acting on the soil mass, which can be computed as the difference between the total normal stress and pore pressure (u). The shear strength parameters are the effective stress cohesion (c’) (Fig. 7) and the effective (drained) angle of internal friction (φ).

$$ \tau ={c}^{\prime }+{\sigma}^{\prime}\tan {\phi}^{\prime }={c}^{\prime }+\left(\sigma -u\right)\tan {\phi}^{\prime } $$

(3)

Soils will exhibit different behaviors during various stages of the shearing process, which will affect their shear strengths. It will be necessary to select the appropriate shear strength when performing stability analyses or designing mitigation strategies. At beginning of the shearing process, the soil mass will exhibit nearly elastic behavior. After this initial stage, the soil will reach its peak shear strength (Fig. 7) or the maximum shear resistance that can be developed because of interparticle friction and interlocking. With continued shearing, the relationship between the shear stress and shear displacement will show a point of inflection corresponding to a post-peak strength loss (Fig. 7). This was stated to correspond to the fully softened shear strength by Duncan et al. (2011) and indicates that the soil is at its critical state. Peak shear strength of soils is the maximum shear strength at its natural state. In general, if the (fine-grained) soil in the field is in a over-consolidated state, it will exhibit both a cohesion intercept and internal friction angle when it is sheared in the drained condition under varying effective normal stresses and failure envelope is generated. The cohesion component is mainly due to over-consolidation and diagenesis during the formation process. Normally consolidated soils do not exhibit a cohesion intercept.

The fully softened shear strength was first identified by Skempton (1970) when changes in the water content were observed to cause a softening of the soil mass. Skempton (1970) further noted that this softening results in a reduction of the shear strength of the soil mass to a strength corresponding to when the soil is in its critical state. In this condition, additional shear displacements do not occur in the soil mass as the water content continues to fluctuate. Furthermore, Skempton (1970) found that the peak shear strength parameters of a normally consolidated clay are numerically equal to the fully softened shear strength parameters. Peak shear strength of the soil at location x in left side of Fig. 5, presented as the lower peak in Fig. 7, is the fully softened shear strength of soil at location x, which is irrespective of the over-consolidation state.

Continued shearing of the soil mass for large amounts of displacement will cause a reorientation of the particles. In particular, these particles will become aligned with the direction of shearing. When the long axis of clay particles becomes aligned with the shearing direction, it results in the formation of polished slickenslide surfaces. Under these conditions, the shear strength will result in a constant minimum value or the residual shear strength (Fig. 7).

Until now, an implicit focus on the paper has been on the static shear strengths. However, the application of cyclic loads, such as those induced by earthquakes, blasting, wave action, machine vibrations, etc., affects the shearing resistance of the soil mass. In particular, the resistance of the soil mass during the application of the cyclic loads and the available resistance after cyclic loads will both be necessary to evaluate the stability of slopes under seismic loading as well as post-earthquake slope stability analyses.

Cyclic resistance, or the ability of a soil mass to withstand cyclic loading, is typically represented with the use of cyclic strength curves. Cyclic strength curves are obtained by plotting the number of cycles required to reach a particular failure criterion against a measure of the severity of the cyclic loading. Failure criteria are typically associated with a level of strain induced in the soil mass, but can also correspond to specific conditions like liquefaction or a particular pore pressure ratio. The severity of the cyclic loading is typically expressed in terms of a cyclic stress ratio (CSR), which is equal to ratio of the amplitude of the cyclic shear stress to the overburden pressure. Like with the Mohr-Coulomb failure envelope, the soil mass will not fail if the combination of the number of cycles of the cyclic load and the applied cyclic stress ratio plot below the cyclic strength curve. However, it will fail if this combination plots above the cyclic strength curve.

The application of cyclic loads will typically occur under undrained conditions. That is, the loads induced by the earthquake, for example, are so rapidly applied that excess pore pressures generated will not have sufficient time to dissipate. Furthermore, the undrained shear strength typically available in the soil mass after cyclic loading (or the post-cyclic undrained shear strength) will not be equal to the undrained shear strength before the application of the cyclic loads. Thus, knowledge of the post-cyclic undrained shear strength of the soil mass will also be necessary in the evaluation of the stability of slopes.

Ajmera and Tiwari (2021) summarized the soil properties required to perform slope stability analyses. In particular, they noted the different properties needed for stability calculations associated with the end of the construction, during multistage loading and over the long term. Furthermore, these properties were discussed for both free-draining and impermeable materials. For free-draining layers, regardless of whether the analyses are performed for the end of construction, multistage loading or long-term conditions, the same information is needed. Specifically, for these materials, the total unit weights should be used, external water pressures and internal pore pressures from steady state seepage analyses should be included and effective shear strength parameters should be used.

For the impermeable layers, the properties will vary depending on the condition being analyzed. In particular,

• End of construction: The total unit weight should be used. External water pressures will be included, but the internal pore pressures should not be. The total shear strength parameters should be utilized.

• Multistage loading: Like the end of construction, the total unit weights should be used, while external water pressures are incorporated and internal pore pressures are not. However, the shear strength will be represented by the undrained cohesion.

• Long term: Similar to free-draining layers, the total unit weights will be used. Steady state seepage analyses should be used to determine the external water pressures and internal pore pressures, both of which will be incorporated into the analyses. For this condition, the effective shear strength parameters should be used.

In general, peak shear strength of soil is used to evaluate the stability of an intact slope/ground (although it is rare to get a perfectly intact slope). Fully softened shear strength is used if the slope exhibits cracks/fissures but has not been slid in the past (or first-time potential slides), while residual shear strength is used for reactivated slopes/slides, stiff fissured clays (or slopes having brittle clay materials) that exhibit progressive failure mechanism. Cyclic (or post-cyclic) shear strength parameters are used to evaluate seismic (or post-cyclic) stability of the slopes. The subsequent sections will discuss about how those different shear strength parameters are measured in the laboratory or are estimated if laboratory measurements are not possible or available. Estimating these soil properties with easily measurable soil parameters are advantageous when sufficient soil samples to run the shear tests or shear test devices are not available.

5 Shear Strength Measurements

Several methods and tests are available in order to measure the shear strength of soil. The selection of the measurement method will depend on a number of factors including the type of soil, the shear strength(s) needed in the design or analyses, and the field conditions and expected mode(s) of failure. This paper will provide a short summary of several of these methods and tests.

5.1 Direct Shear Tests

Direct shear tests are commonly used to measure the shear strengths of soils under drained conditions because they are widely available, are easy to run and to interpret, and are relatively quick. However, in a direct shear test, the shear strength is always measured on the horizontal shearing plane. This plane may not be the weakest potentially resulting in an overestimate of the shear strength. Additionally, as the sample is sheared, the displacement causes a reduction in the cross-sectional area. Direct shear tests have been used to measure the peak, fully softened and residual shear strengths of a soil mass although it is not ideal to measure the residual shear strength of soil with the direct shear device. Figure 8 shows the photographs of typical direct shear devices available in practice.

Fig. 8

Pictures of direct shear devices that apply normal stress using (top) weights placed on loading arm and (bottom) pneumatic system

To conduct direct shear tests, it is necessary to either obtain an undisturbed sample from the field or to use remolded samples. Remolded samples may be natural samples that have been reconstituted or be samples prepared in the laboratory, as Tiwari and Ajmera (2011) did using dry clay minerals. Samples may be saturated, dry or prepared to some desired moisture content. Undisturbed samples may need to be trimmed to the dimensions associated with the specifications and requirements of the direct shear device available for the testing. It will be necessary to record the dimensions and weight of the sample before the testing procedures commence.

Once the sample is placed in the direct shear box, the assembly should be placed in the apparatus. If the sample is saturated, the sample will be submerged in a water bath in the assembly to ensure that it remains saturated during the entire testing process. However, when testing dry or moist samples, the water bath will not be necessary. After the direct shear box has been placed in the apparatus and the displacement and load transducers have been assembled, the test will commence with a consolidation (or vertical compression) phase. During the direct shear test, it will be necessary to measure the horizontal and vertical displacements and the shear force requiring the use of at least two displacement sensors, such as dial gauges or LVDTs, and one load sensor, such as a proving ring or load cell.

In the consolidation phase, a vertical load representing the normal stress is applied to the soil sample. This is achieved by placing weights on a loading arm or a pneumatic system. Figure 8 contains images of two direct shear systems: one applies a normal stress with the placement of weights (top image in Fig. 8) and the other uses a pneumatic system (bottom image in Fig. 8). The normal stress may need to be applied in increments allowing sufficient time between the increments for pore pressures to dissipate to ensure that the sample is capable of sustaining the next normal stress without sample extrusion.

ASTM D3080/D3080M (2011) provides details about the specifications for a direct shear test along with some specifics about conducting a direct shear test. Tiwari and Ajmera (2011) provided a detailed description of the methods they adapted to measure the fully softened shear strength. In particular, they prepared their samples by mixing dry powdered minerals with sufficient distilled water to achieve an initial moisture content equal to the liquid limit of the soil mass. This mixture was allowed to hydrate before it was transformed into a slurry using a batch mixer. The slurry was, then, evenly placed in the direct shear box using a vinyl cake decoration bag with a metal tip. To consolidate the slurry, incremental pressures were applied with real-time consolidation observations made using an automated data acquisition system integrated with the direct shear device. The shearing stage commenced after the primary consolidation at the desired normal stress was completed and was allowed to continue until the fully softened shear strength (or peak of the normally consolidated samples) was reached. It is recommended that at least four tests be conducted at different normal stresses to determine the friction angle and cohesion for the soil mass.

Logarithm of time versus displacement graphs are used to monitor the dissipation of excess pore pressures that were generated due to the application of the normal stress on the soil mass. An example of this graph is shown in Fig. 9, which captures the three stages of deformation. The first stage corresponds to initial compression. This is usually a result of preloading occurring during the sample preparation process and the placement of the top platen on the sample. Primary consolidation occurs during the second stage. During this stage, the applied loads are gradually transferred from the pore water to the soil skeleton as the pressurized water exits the soil mass. The third stage is secondary compression. Deformation observed in this stage is a result of the plastic readjustment of the soil fabric. Each increment of the consolidation phase of a direct shear test is usually maintained until the primary consolidation is complete.

Fig. 9

Example of logarithm of time versus vertical displacement curve obtained during consolidation

When the primary consolidation is complete under the target normal stress, a horizontal force will be applied at a constant rate. The rate should be selected to be slow enough to ensure that pore pressures generated during the shearing process dissipate when the failure of the sample occurs. The vertical displacement versus logarithm of time graph from Fig. 9 can be used to compute the shearing rate. In particular, the time to failure (t_f) is computed using Eq. 4, in which t₅₀ is the time required for the specimen to achieve 50% consolidation under the maximum normal stress. Equation 5 is then used to compute the shearing rate (R_d) with knowledge of the expected horizontal displacement at failure (d_f). This is in alignment with the recommendations in ASTM D3080/D3080M.

$$ {t}_f=50{t}_{50} $$

(4)

$$ {R}_d=\frac{d_f}{t_f} $$

(5)

During the shearing process, the horizontal displacement, vertical displacement and applied shear force should be measured and recorded. The shearing process continues until the shear strength (peak or fully softened) is obtained or for a maximum deformation of 10% of the height of the sample (ASTM D3080/D3080M). The data collected is used to develop horizontal displacement versus shear stress and horizontal displacement versus vertical stress graphs, like the examples provided in Figs. 10 and 11. The shear stress can be computed by taking the quotient of the applied shear force with the area of the sample.

Fig. 10

Shear stress versus horizontal displacement graphs for normally and over-consolidated samples

Fig. 11

Vertical displacement versus horizontal displacement grants for normally and over-consolidated samples

The differences in the behavior of normally consolidated and over-consolidated clays can be seen Figs. 10 and 11. Specifically, over-consolidated soils will tend to exhibit a clear peak in the shear stress versus horizontal displacement. After the peak strength is achieved, these soils will show a reduction in the shear stress with continued horizontal shearing. Furthermore, over-consolidated soils will tend to exhibit dilative tendencies, in that the volume of the soil mass will increase as the sample is sheared. On the other hand, normally consolidated soils will tend be contractive in nature illustrated by an increase in the vertical displacement with horizontal displacement (Fig. 11).

The peak or fully softened shear strength can be obtained from Fig. 10. When this shear strength is plotted against the normal stress for each test conducted, it will be possible to obtain the Mohr-Coulomb failure envelope as shown in Fig. 12. The y-intercept of the best-fit line through the data will yield the cohesion intercept. The cohesion intercept of normally consolidated soils should be zero. The inclination of this line from the horizontal will give the friction angle.

Fig. 12

Examples of Mohr-Coulomb failure envelopes

The direct shear device has also been used to measure the residual shear strength with the customized test process developed as outlined in Meehan et al. (2011). Some other authors have also presented the residual shear strengths of soils they tested using a multiple reversal direct shear testing. In doing so, they have sheared their samples to the displacement limit on the device and returned back the specimen to the starting position by reversing the loading. The soil specimen is once again sheared to the maximum displacement allowed by the direct shear device at the shearing rate slow enough to maintain the drained loading condition. This process is repeated until the residual shear strength is obtained. However, such multiple reversal direct shear tests are not recommended to obtain the residual shear strengths. It is recommended to obtain the residual shear strength of soils using the ring shear device. As with the peak and fully softened shear strength measurements, it is recommended that at least four direct shear tests be conducted to determine the parameters associated with the residual shear strength.

5.2 Triaxial Tests

Another commonly available soil testing system is the triaxial device. Triaxial tests represent the field situation better than the direct shear device. However, the test procedures are more complicated and more expensive compared to the direct shear tests. The peak and fully softened shear strengths have both been frequently measured with the use of consolidated drained (CD) and consolidated undrained (CU) triaxial tests. It should be noted that triaxial tests will be difficult to perform on soft samples or on samples at low confining pressures. Furthermore, CD testing of low permeability clays may be lengthy in duration due to the amount of time required to back pressure saturate the sample as well as the required strain rate to ensure dissipation of pore pressures generated during the shearing process. The time required for back pressure saturation will also be a concern when conducting CU triaxial tests on such low permeability soils.

ASTM D7181 (2020) contains the specifications for CD triaxial tests. In addition, it also summarizes the procedures followed to conduct these tests. Typically, CD triaxial tests will require cylindrical soil samples with heights that are at least double the diameter. The sample is then sandwiched between two porous stones encompassed in a thin rubber membrane, as presented in Fig. 13. The assembly is placed within a water-filled cylindrical chamber and the pressure of the water in the chamber is increased in order to apply a cell (or confining) pressure on the sample (Fig. 14).

Fig. 13

A triaxial test sample assembly

Fig. 14

Illustration of a triaxial testing mechanism

After the sample is mounted, back pressure saturation is used to saturate the sample. In this process, back pressure is applied to the sample as water is forced into the voids within the soil mass resulting in a displacement of the air (or other gasses) present. The water pressure must be carefully controlled to ensure that the sample is not damaged during the saturation process. The saturation is verified with the use of Skempton’s pore pressure coefficient or B-value. This value is a measure of the changes in pore pressure that result due an increase in the cell pressure. If the sample is saturated, the B-value will be close to one. Lower B-values will indicate that the void spaces are filled with both water and air and thus, the sample is not saturated. Typically, the sample is said to be saturated if the B-value is greater than 0.95, but in some cases, lower B-values may also be accepted.

Once the sample has been saturated, it will be consolidated by increasing the pressure of water in the chamber to the required confining pressure. The completion of the primary consolidation is monitored using real-time data collected and recorded by an attached computer system. At this point, the sample will be sheared at a rate slow enough to ensure drained loading conditions. Termination of the shearing process typically coincides with the measurement of the peak deviator stress or the application of 20% axial strain. The axial strain criterion results from the maximum allowed deformation in the soil sample before the applied deviatoric stresses become eccentric in nature. The process is repeated for at least three, preferably four, samples to allow for the determination of shear strength parameters.

The procedures for conducting CU triaxial tests are nearly identical to those described above for the CD triaxial test. They can be found described within ASTM D4767 (2020). There are some major differences between the CD and CU triaxial tests. The biggest is that the drainage valve is closed during the shearing process in a CU triaxial test. In contrast, during the CD triaxial test, this valve is left open allowing volume change to occur (or the pressurized pore water to drain out). These changes in the volume are measured by collecting the drained pore water from the sample. However, since the valve is closed in the CU triaxial test, volume change is not possible and pore pressures will be generated with the soil mass, instead. Instead of measuring the volume change, the CU triaxial test will measure the pore pressures generated. Finally, since volume change is not allowed in a CU triaxial test, the test can be conducted at a faster rate than CD triaxial tests, which must allow sufficient time during the shearing process for excess pore pressures to dissipate and the resulting volume change to occur. Thus, CD triaxial tests are typically conducted at shearing rates that are about eight to ten times slower than the shearing rates used in CU triaxial tests.

Data related to the axial deformation, cell pressure (σ₃’) and deviator force are recorded during the shearing process. In the CD triaxial test, the change in the volume of the sample is also recorded, whereas, the pore pressure is recorded in the CU triaxial test. Since volume change is permitted in the CD triaxial test, the cross-sectional area of the sample will not remain constant over the duration of the shearing process. The area (A) can computed using Eq. 6. In Eq. 6, A_c is the initial cross-sectional area of the sample, ε_a is the axial strain and ε_v is the volumetric strain. The axial and volumetric strains can be computed using Eqs. 7 and 8 as ratios of the axial deformation (ΔL) to the original length (L) and the change in volume (ΔV) to the original volume (V), respectively.

$$ A=\frac{{\mathrm{A}}_c\left(1-{\varepsilon}_v\right)}{1-{\varepsilon}_a} $$

(6)

$$ {\varepsilon}_a=\frac{\Delta L}{L} $$

(7)

$$ {\varepsilon}_v=\frac{\Delta V}{V} $$

(8)

The area from Eq. 6 and the deviator force (P_d) will make it possible to determine the deviator stress (σ_d) from Eq. 9. An example of the variation of this deviator stress with axial strain is shown in Fig. 15. Figure 16 contains an example of the corresponding volumetric strains with axial strains. The trends in Figs. 15 and 16 will be similar to the shear stress versus horizontal displacement and vertical displacement versus horizontal displacement graphs obtained from a direct shear test. Specifically, normally consolidated samples will tend to exhibit plateau in the deviator stress and compressive volume change behavior. This is similar to that shown for the sample tested under a confining pressure of 1500 kPa in Figs. 15 and 16. On the other hand, over-consolidated samples will have a clear peak followed by strain softening behavior (that is, a reduction in the deviator stress with an increase in the axial strain). The volume change behavior will be dilative indicating an increase in the volume of the sample. When the sample in Figs. 15 and 16 was tested under a confining pressure of 1000 kPa, it exhibited behavior typical of over-consolidated soils.

Fig. 15

Example of deviator stress versus axial strain graphs obtained from CD triaxial tests

Fig. 16

Example of volumetric strain versus axial strain graphs obtained from cd triaxial tests

$$ {\sigma}_d=\frac{P_d}{A} $$

(9)

The deviator stress at failure can be used to compute the effective axial stress at failure (σ₁’) as shown in Eq. 10. The axial stress at failure and the corresponding confining pressure represent the normal stresses on the vertical and horizontal planes of the soil element tested allowing for the construction of a Mohr’s circle. Such Mohr’s circles will then be constructed for each CD triaxial test conducted. An example is shown in Fig. 17. The best-fit line tangent to these Mohr’s circles will be the Mohr-Coulomb failure envelope, whose intercept will be equal to the effective cohesion and the slope can be used to compute the effective friction angle.

Fig. 17

Example of deviator stress versus axial strain graphs obtained from CD triaxial tests

$$ {\sigma}_1^{\prime }={\sigma}_3^{\prime }+{\sigma}_d $$

(10)

The procedures for reducing the data from the CU triaxial test is similar to that described above for the CU triaxial test. However, since volume change is not permitted during the shearing process, the pore pressure generated in the sample is recorded in lieu of the volume change. Thus, the CU triaxial test will allow for the determination of both the total and effective shear strength parameters. The cross-sectional area of the sample at any point in the shearing process can be computed using Eqs. 6 and 7. It is should be noted that since volume change is not permitted, the volumetric strain will be equal to zero. Equation 9 can then be used to determine the deviator stress.

Figure 18 presents an example of the deviator stress (Fig. 18 top) and pore pressure (Fig. 18 bottom) versus axial strain graphs obtained from a series of CU triaxial tests. In an over-consolidated soil, the curve for the deviator stress versus axial strain would be expected to display a peak similar to those trends shown in Figs. 10 and 15 for the direct shear and CD triaxial tests, respectively. Furthermore, negative pore pressures could be expected to develop in the sample as it tended to exhibit dilative tendencies.

Fig. 18

Example results from CU triaxial test for (top) deviator stress versus axial strain and (bottom) pore pressure versus axial strain

The total confining pressure (σ₃) will be equal to the cell pressure applied on the sample at the end of the consolidation phase (or at the instant that the shearing process commenced). The effective confining pressure (σ₃’) can be computed as the difference between the total confining pressure and pore pressure (u) recorded, as shown in Eq. 11. Equations 10 and 12 are then used to compute the effective and total axial stresses, respectively.

$$ {\sigma}_3^{\prime }={\sigma}_3-u $$

(11)

$$ {\sigma}_1={\sigma}_3+{\sigma}_d $$

(12)

The total and effective confining and axial stresses at failure will provide the information necessary to draw the respective Mohr’s circles at failure. This is illustrated with the example in Fig. 19. As with the CD triaxial test, a best-fit line drawn tangent to the effective stress Mohr’s circles will facilitate the measurement of the effective cohesion intercept and the effective friction angle. Similarly, the best-fit line tangent to the total stress Mohr’s circle will indicate the corresponding undrained shear strength parameters.

Fig. 19

Example of Mohrs’ circles and failure envelopes obtained from CU triaxial tests

Meehan et al. (2011) used triaxial tests to measure the residual shear strength of samples using polished slickenslide surfaces. To do so, they constructed a special mold to be used during the sample preparation that would prevent significant disturbance. This mold allowed the sample to be cut using a wire and polished to form slickenslide surfaces. Some figures of the process are provided in Fig. 20. In their study, Meehan et al. (2011) constructed the mold to have a failure plane inclined at 55° from the horizontal based on the expected residual friction angle for the soils they were testing. To create the polished surface, they sheared both parts of the sample for a total displacement of approximately 2 m along a frosted glass. The sample was then reassembled and placed in a triaxial cell in order to measure the residual shear strengths.

Fig. 20

Images of (a) specialized mold, (b) sample cut by a wire, (c) polishing process and (d) slickenslide surfaces created (Source: Meehan et al. 2011)

The shear strengths measured in a triaxial test can be significantly affected by the end-platen restraints (Meehan et al. 2011; Bishop and Henkel 1962; Barden and McDermott 1965; Chandler 1966; Skempton and Petley 1967). Specifications to conduct triaxial compression tests using a free top platen on specimens that will fail in a single plane shearing mechanism were provided by Bishop and Henkel (1962) and adapted by Skempton and Petley (1967)) and Chandler (1966). A free end approach was presented by Barden and McDermott (1965). In this approach, a greased membrane was used. The membrane allowed the ends of the specimen to expand radially, while permitting the specimen to experience a small amount of lateral movement during the shearing process. Meehan et al. (2011) used a both ends free platen approach in their study. This approach minimized the effect of lateral platen restraint while the sample was sheared along a single plane.

To use the both ends free approach, Meehan et al. (2011) constructed special base and top platens that permitted unrestrained lateral movement of the specimen during the shearing process. Figures 21 and 22 contain schematics and pictures of the triaxial test device set-up used. As seen from these figures, free lateral movement was permitted by placing ball bearings in a thin vacuum grease. They opted to use 3.2 mm diameter steel ball bearings placing seventy of them in each of the top and bottom platens. Their system resembles that utilized by Bishop and Henkel (1962), who used rollers. The use of ball bearings allows for unrestrained movement in any horizontal direction. In addition, the system used by Meehan et al. (2011) has a free base.

Fig. 21

Both ends free triaxial test set-up schematic from Meehan et al. (2011) (a) before and (b) after shearing

Fig. 22

Pictures of the both ends free triaxial system from Meehan et al. (2011). (a) placement of ball bearings on a platen, (b) connection of top platen with triaxial device, (c) connection of the bottom platen with triaxial device and (d) entire set-up

5.3 Direct Simple Shear Tests

Direct simple shear tests have been used to measure the peak and fully softened shear strengths. This device offers several advantages compared to the direct shear and triaxial tests. Similar to the direct shear apparatus, the test setup is relatively straightforward. However, in direct simple shear tests, the sample is allowed to fail along its weakest plane. Furthermore, the cross-sectional area remains constant throughout the shearing process. This provides results that are more accurate. Additionally, simple shear tests require smaller soil samples than triaxial testing, resulting in shorter consolidation times. It is important to note that commercial availability of this device is currently limited.

The direct simple shear tests are conducted following the guidelines outlined in ASTM D6528 (2017). Tiwari and Ajmera (2014) conducted tests on ten different soil samples using the direct simple shear apparatus. The procedures used in their study closely resemble those described by Tiwari and Ajmera (2011) for direct shear testing of similar soil samples. Fig. 23 shows the simple shear device used by Tiwari and Ajmera (2011) and samples prior and after shearing. In the direct simple shear tests performed by Tiwari and Ajmera (2014), the soil sample is carefully placed within a rubber membrane and positioned inside a stack of Teflon® rings. In other simple shear apparatuses, the sample may be laterally confined by a wire mesh. Subsequently, this assembly is securely placed into the direct simple shear box, where the consolidation process takes place. This consolidation process may be designed to occur incrementally. After the sample is subjected to the desired normal stress and the primary consolidation is complete, shearing of the sample is induced by applying a shear force at a desired strain rate. The shearing phase is conducted under undrained conditions. By maintaining undrained conditions, any changes in the pore water pressure are captured and considered in the analysis. Per ASTM D6528, the shearing process is continued until the peak shear strength is obtained or for a maximum of shear strain of 25%.

Fig. 23

(top) a typical simple shear device, (bottom left) sample preparation and (bottom right) sample after the shearing process is completed

During shearing phase of the direct simple shear test, the sensors on the device will record the horizontal displacement (Δx) along with the normal (P) and shear (T) forces. In the data reduction, this information is used to compute the shear strain (γ), the effective normal stress (σ’), pore pressure (u) and shear stress (τ) on the sample. To do so, Eqs. 13 to 16, respectively, are utilized. In these equations, σ_c’ is the consolidation pressure. As noted before, constant volume is maintained during the shearing phase of a direct simple shear test. Thus, the changes in the effective normal stress are found to be equivalent to the pore pressure developed in the sample (Airey and Wood 1987; Bjerrum and Landva 1966, and Dyvik et al. 1987).

$$ \gamma =\frac{\Delta x}{H} $$

(13)

$$ {\sigma}^{\prime }=\frac{P}{A} $$

(14)

$$ u={\sigma}_c^{\prime }-{\sigma}^{\prime } $$

(15)

$$ \tau =\frac{T}{A} $$

(16)

Using the parameters measured from the direct shear test, stress-strain curves and pore pressure versus strain curves should be prepared for each sample tested. Figure 24 contains an example of the shear strain versus shear stress curves obtained from a series of direct simple shear tests, while Fig. 25 contains the corresponding shear strain versus pore pressure curves. These figures correspond to normally consolidated samples and thus, result in the compressive tendencies in the pore pressure that is generated during the shearing process. Figure 26 provide an example of the results obtained from a series of direct simple shear test conducted on samples at different over-consolidation ratios.

Fig. 24

Example of the shear stress versus shear strain curves obtained from simple shear tests conducted on normally consolidated samples

Fig. 25

Example of the pore pressure versus shear strain curves obtained from simple shear tests conducted on normally consolidated samples

Fig. 26

Example of the shear stress-shear strain and pore pressure versus shear strain curves obtained from simple shear tests conducted on over-consolidated samples with a pre-consolidation pressure of 1200 kPa (Source: Ajmera et al. 2018, with permission from ASCE)

The maximum shear stress in Figs. 24 and 26 correspond to the shear strength of the sample. When this shear strength is plotted against the corresponding consolidation pressure, the Mohr-Coulomb envelope associated with total stress is obtained. On the other hand, when the corresponding effective normal is used instead of the consolidation pressure, the Mohr-Coulomb failure envelope obtained allows for the determination of the effective shear strength parameters. An example of the total and effective stress Mohr-Coulomb envelopes is shown in Fig. 27. As before, the y-intercept will indicate the cohesion intercept, while the slope of the line can be used to compute the friction angle for the total and effective stress conditions.

Fig. 27

Example total and effective stress Mohr-Coulomb failure envelopes from simple shear tests

5.4 Ring Shear Tests

The ring shear device has emerged as the preferred technique for quantifying residual shear strength due to its unique ability to continuously shear samples in a single direction while maintaining a consistent cross-sectional area (Watry and Lade 2000). In particular, researchers, such as Bishop et al. (1971), have emphasized that only the torsional or ring shear test can genuinely replicate residual conditions within laboratory settings. Similarly, Lupini et al. (1981) asserted that the ring shear device offers the most precise measurement of residual shear strength, minimizing any potential ambiguities. In fact, Hawkins and Privett (1985) concluded that ring shear tests exhibit superior repeatability compared to alternative methods. However, it is worth noting that the non-uniform distribution of stress across the annular samples used in these tests poses a potential challenge.

The development of the ring shear device for assessing residual strength has been extensively documented in various publications, including the works of Bromhead et al. (1999), Hvorselv (1939), and Boucek (1977). The initial development of the ring shear test is credited to Arthur Casagrande and was later reported by Hvorselv (1939) according to Harris and Watson (1997). Bishop et al. (1971) introduced a specific type of ring shear device capable of measuring residual shear strength under large displacements. To expedite drainage, LaGatta (1970) and Bromhead (1979) incorporated smaller, thinner samples into their designs. While Stark and Eid (1994) proposed additional modifications, it has been argued by Bromhead et al. (1999) that these alterations deviated from the device’s intended simplicity and user-friendly nature. Anayi et al. (1989) further suggested modifications to the Bromhead ring shear device advocating for doubled sample thickness and changes to the torque arm shape to accommodate larger samples. Steward and Cripps (1983), Hawkins and Privett (1985), Stark and Eid (1993), and Dewhurst et al. (1996) also proposed their own modifications to the ring shear device.

As a result of the parallel development of different ring shear devices (Fig. 28), each variation presents distinct advantages and disadvantages. For instance, the design presented by Bishop et al. (1971) allows for adjustable gaps between the top and bottom rings, thereby facilitating the testing of wet specimens. Shearing occurs at the mid-height of the sample in this particular device. Conversely, the device predominantly developed by Bromhead (1979) shears the soil samples in close proximity to the upper platen. Notably, this design enables the achievement of the residual condition at smaller strain levels.

Fig. 28

Different types of ring shear devices available in practice

Data collection during the shear phase of the ring shear test consists of measurements of the axial deformation, torque angle (θ) and shear force. The horizontal displacement (Δx) is then computed using Eq. 17, where r_avg is the average radius of the sample found using Eq. 18 as the average of the internal (r_i) and external (r_o) radii of the annular sample. Next, the shear stress is computed using Eq. 19 as a function of the shear force, the average radius and the polar moment of inertia (J). The polar moment of inertia of an annular sample can be computed from Eq. 20. These results are used to develop horizontal displacement versus shear stress and axial deformation versus horizontal deformation curves.

$$ \Delta x=\frac{\pi \left({r}_{avg}\right)\left(\theta \right)}{180^{{}^{\circ}}} $$

(17)

$$ {r}_{avg}=\frac{r_o+{r}_i}{2} $$

(18)

$$ \tau =\frac{T{r}_{avg}}{J} $$

(19)

$$ J=\frac{\pi \left({r}_o^4-{r}_i^4\right)}{2} $$

(20)

Figure 29 shows an example of the horizontal displacement versus shear stress obtained from ring shear tests. The residual shear strengths obtained at large displacements from these curves are used to plot a Mohr-Coulomb failure envelope. A typical Mohr-Coulomb envelope from the ring shear test is shown in Fig. 30.

Fig. 29

Shear stress versus horizontal displacement curve examples from ring shear tests (Source: Tiwari et al. 2020a, b, with permission from ASCE)

Fig. 30

Example of the Mohr-Coulomb failure envelope for the residual shear strength (Source: Tiwari et al. 2020a, b, with permission from ASCE)

ASTM D6467 (2013) provides detailed procedures for conducting ring shear tests to measure the residual shear strength of cohesive soils. Multiple methods can be employed, involving different normal stresses, to establish the failure envelope and determine the shear strength parameters. In their study, Tiwari and Marui (2004) examined three testing methods for the ring shear device: separate tests for each normal stress as well as a multistage ring shear test with increasing or decreasing loads. Although the residual friction angle remained consistent across the testing methods, variations were observed in the cohesion intercept. Tiwari and Marui (2004) underscored the advantages of employing a multistage ring shear test, including enhanced consistency by employing the same sample for all normal stresses and reduced sample volume requirements in comparison to that required to conduct separate ring shear tests. These benefits corroborated the findings of Bromhead (1992) and Harris and Watson (1997).

Tiwari and Marui (2005) provided detailed procedures for conducting ring shear tests to measure the residual shear strength, which are briefly summarized here. The study utilized a ring shear device based on the Imperial College and Norwegian Geotechnical Institute (NGI) model. This device was originally described by Bishop et al. (1971). Several modifications were made to the device to enhance its functionality including the ability to accommodate larger sample sizes, transparent outer rings for external observation of the soil sample during shearing, a wide range of shearing rates, and a pore-water measurement system at the shearing plane. For their experiments, natural soil samples passing through a 2 mm sieve or soil mixtures were mixed with distilled water to achieve a water content close to the liquid limit (Tiwari and Marui 2005). These samples were then loaded into the ring shear device and compacted. Consolidation was carried out incrementally until more than 95% consolidation was achieved at each step. After determining the residual shear strength at the highest of the desired normal stresses, Tiwari and Marui (2005) employed the reducing load multistage ring shear testing method to measure the residual shear strengths at lower normal stresses. This method is described in detail in Bromhead (1992) and Tiwari and Marui (2004).

5.5 Cyclic Simple Shear Tests

The cyclic simple shear test was found to be most suitable for replicating the in-situ cyclic stresses caused by ground shaking, by several researchers including Seed and Peacock (1971), Finn et al. (1971), Silver et al. (1976), Vucetic (1988), and Song et al. (2004). Based on these findings, the cyclic simple shear apparatus is recommended to capture the cyclic and post-cyclic response of soils. Ajmera et al. (2017, 2019) used a cyclic simple shear device that confines the soil specimen in a stack of Teflon®-rings and a latex membrane to conduct testing on a number of laboratory prepared and natural samples.

In their study, Ajmera et al. (2017, 2019) used sample preparation procedures identical to those previously described by Tiwari and Ajmera (2011, 2014) to conduct direct shear and simple shear tests, respectively. In particular, Ajmera et al. (2017, 2019) prepared their laboratory samples by mixing dry mineral powders in the appropriate proportions based on their dry weight, followed by the addition of de-aired distilled water to achieve a liquidity index of one. The mixtures were allowed to hydrate for a minimum of 24 h. Each specimen was then placed in the cyclic simple shear apparatus, where the specimens underwent incremental consolidation under at-rest conditions. In each increment, the stress was doubled after each completion of primary consolidation until a target normal stress was reached (Ajmera et al. 2017, 2019).

Following primary consolidation at the target normal stress, the specimens were subjected to stress-controlled cyclic loading while maintaining constant volume. In Ajmera et al. (2017, 2019), a sinusoidal wave form with a frequency of 0.5 Hz was applied to generate the cyclic stresses, consistent with recommendations in the literature. The amplitude of the cyclic load was determined by the cyclic stress ratio (CSR). The cyclic stress ratio represents the ratio of the amplitude of the cyclic stress to the vertical consolidation stress. This definition aligns with the one used by Bray and Sancio (2006). Cyclic loading was continued until the specimen experienced a double amplitude shear strain of 10%. If the desired strain level was not reached within 500 cycles, the cyclic phase of the test was terminated. The termination criteria Ajmera et al. (2017, 2019) adapted is similar to the recommendations in the ASTM D5311/D5311M (2013).

During the cyclic loading phase, time series data related to the horizontal (Δx) and vertical (Δy) displacements along with the vertical (P) and shear (T) forces is recorded. The frequency (f) and time (t) data is used to calculate the number of cycles (N) using Eq. 21. Furthermore, since the frequency and period (T) of a cyclic load are interrelated, the relationship between the time data and the period is also presented in Eq. 21. Equations 13 to 16 are used to compute the shear strain, effective normal stress, pore pressure and shear stress, respectively, during the cyclic loading phase of the test.

$$ N= tf=\frac{t}{T} $$

(21)

Stress and pore pressure behavior during the cyclic phase of a cyclic simple shear test is shown in Fig. 31. The example in Fig. 31 is for a sample that was subjected to a cyclic stress ratio of 0.141. The corresponding stress-strain hysteresis loops for this sample are presented in Fig. 32. In this figure, the double amplitude shear strain or the difference between the shear strain at the peak and trough of the cyclic load, was less than 10%. This resulted in the application of 500 cycles of loading as Fig. 31 illustrated. The curve labeled effective normal stress depicts the variation in the effective normal stress over the duration of the cyclic loading, which is shown with the curve labeled applied cyclic stress. The pore pressure curve was then back-calculated. These results can be used to compute the double amplitude shear strain developed in each cycle and with the use of interpolation, it will be possible to determine the number of cycles required to reach a targeted level of shear strain in the sample.

Fig. 31

Example of stress and pore pressure behavior of sample during the cyclic simple shear test

Fig. 32

Stress-strain hysteresis loops from the cyclic phase of a cyclic simple shear test

Based on the guidelines presented by Andersen et al. (1976) and Brown et al. (1977), a subsequent phase of undrained strain-controlled static loading was initiated immediately after the completion of the cyclic loading phase in the study performed by Ajmera et al. (2019). In this phase, the top platen was kept fixed while the bottom platen was horizontally displaced at a constant rate of 5% per hour. This loading rate adhered to the recommendations specified in ASTM D6528 and aimed to determine the undrained shear strength of the specimen after it has experienced cyclic loading (or the post-cyclic undrained shear strength). The data reduction procedures will be identical to those described previously for the direct simple shear tests. As such, they are not re-produced here.

Ajmera et al. (2021) recognized that soils in-situ are typical subjected to multiple cyclic loading events instead of a single event, as assumed by Ajmera et al. (2017, 2019). That is, in an earthquake, the soil mass would be subjected to a number of foreshocks, the main shock, and several aftershocks over a short duration. Ajmera et al. (2021) noted that this sequential occurrence of seismic events has the potential to intensify the degradation of soil strength, possibly leading to subsequent failures or even more severe consequences. Thus, they conducted cyclic simple shear tests following the sample and testing procedures previously described (Ajmera et al. 2017, 2019), but rather than measuring the undrained shear strength immediately after the application of a cyclic load, their samples were subjected to another cyclic load. Different degrees of excess pore pressure dissipation were permitted between the two cyclic loading events. The second cyclic load also had a sinusoidal waveform with an amplitude determined by a second cyclic stress ratio and a frequency of 0.5 Hz. Like Ajmera et al. (2017, 2019), Ajmera et al. (2021) also terminated the cyclic loads when 10% double amplitude shear strains or 500 cycles of loading was obtained. The undrained shear strength was then measured using procedures identical to those described in Ajmera et al. (2019) immediately following the termination of the second cyclic loading phase.

5.6 Cyclic Triaxial Tests

While the cyclic simple shear test is the most representative of in-situ conditions during the application of cyclic loads, the cyclic triaxial test is more commonly used to determine the dynamic properties of soils. A picture of a cyclic triaxial test system is shown in Fig. 33, while a closer image of the sample set-up is provided in Fig. 34.

Fig. 33

Cyclic triaxial system

Fig. 34

Close-up picture of sample set-up in cyclic triaxial test

The procedures for conducting a cyclic triaxial test can be found in ASTM D5311/D5311M. Procedures described earlier for the CD and CU triaxial tests are the same as those used to prepare, back-pressure saturate and consolidate the sample to the desired confining pressure in a cyclic triaxial test. However, after the completion of the primary consolidation, the sample will be subjected to a cyclic load to obtain the cyclic characteristics following which it may be sheared under static conditions to capture the post-cyclic behavior. Similar to the cyclic simple shear test, a sinusoidal wave form is typically used to apply the cyclic loads. However, while the amplitude is still expressed in terms of the cyclic stress ratio, this ratio is computed using Eq. 22, where σ_d,cyc is the amplitude of the cyclic deviator stress applied. A number of different termination criteria exist in the literature to determine when to stop the application of the cyclic load. These criteria are often a combination of one or more of the following types of constraints: a minimum pore pressure ratio, a maximum single or double amplitude axial strain, or a maximum number of cycles of loading. The single amplitude axial strain is the maximum strain from the origin in either compression or extension during a single cycle, while the double amplitude axial strain the difference between the maximum strain in compression and extension in a single cycle of load. The recommended termination criteria in ASTM D5311/D5311M are a maximum double amplitude axial strain of 20%, a maximum single amplitude axial strain of 20% in either compression or extension or a maximum of 500 cycles of loading.

$$ CSR=\frac{\sigma_{d, cyc}}{2{\sigma}_3^{\prime }} $$

(22)

Data pertinent to the axial deformation, pore pressure and deviator force is recorded during the cyclic phase of a cyclic triaxial test. Equations 6, 7 and 9 presented previously can used to compute the corresponding stresses and strains during the test. If the sample is subjected to axial compression after the application of the cyclic loads to obtain the post-cyclic shear strength and behavior, the data reduction will be identical to that described previously for the CU triaxial test.

6 Correlations Methods to Obtain Soil Shear Strengths

The best way to obtain the shear strength parameters for slope stability analyses using the appropriate soil testing procedures explained in previous section. However, it is not always possible to obtain enough soil samples or soil testing devices to perform those tests. In such situations, if we obtain soil samples from appropriate depths even in small quantities that are enough for index property testing (e.g. hydrometer analysis, liquid limit tests, plastic limit tests, and x-ray diffraction), we can indirectly estimate pertinent shear strengths through various correlations with these properties. Some of those correlation methods are explained in the following sections.

6.1 Fully Softened Shear Strength

Correlations to estimate the fully softened shear strength have appeared in the literature since 1970. In particular, the correlations in the literature have used the following parameters to develop relationships to estimate the fully softened shear strengths:

• Plasticity index (Ladd et al. 1977; Moon 1984; and Mesri and Abdel-Ghaffar 1993; Castellanos and Brandon 2013)

• Plasticity index and mineralogy (Tiwari and Ajmera 2011, 2015)

• Liquid limit and clay fraction (Saleh and Wright 2005)

• Liquid limit and effective normal stress (Wright 2005)

• Liquid limit and mineralogy (Tiwari and Ajmera 2011, 2015)

• Liquid limit, clay fraction and effective normal stress (Eid 1996; Stark et al. 2005; Stark and Hussain 2015)

• Clay fraction and mineralogy (Tiwari and Ajmera 2011)

• Mineralogy (Tiwari and Ajmera 2011)

• Combinations of various parameters (Eid and Rabie 2016; Castellanos et al. 2016; Castellanos and Brandon 2019; Stark and Fernandez 2019).

In their comprehensive study, Tiwari and Ajmera (2011) conducted an extensive analysis of thirty-six laboratory-prepared soil samples using the direct shear testing. These samples were prepared by blending different proportions of clay minerals, particularly montmorillonite and kaolinite, with ground quartz. The primary objective of their research was to establish correlations between the fully softened shear strength, clay content, and mineralogical composition of the soils.

Tiwari and Ajmera (2011) presented several correlations in their study. Figure 35 shows the relationship they developed between the clay content, mineralogy, and the fully softened shear strength. This figure illustrated the importance of considering the clay mineralogy in estimating the fully softened shear strength. Specifically, Fig. 35 shows the significant differences, at a constant clay content, between the fully softened friction angles with mixtures of kaolinite with quartz having significantly higher strengths than the mixtures of montmorillonite with quartz. They also explored the correlation between the fully softened friction angle and the liquid limit, taking into account the specific clay mineralogy. This relationship, depicted in Fig. 36, offers a useful tool for estimating the fully softened friction angle based on the liquid limit of a given soil sample. Furthermore, the relationships are significantly better than those previously presented by Wright et al. (2007) and Stark et al. (2005).

Fig. 35

Correlation to estimate the fully softened friction angle as a function of clay content and mineralogy proposed by Tiwari and Ajmera (2011)

Fig. 36

Correlation to estimate the fully softened friction angle as a function of the liquid limit and clay mineralogy proposed by Tiwari and Ajmera (2011)

In addition to the liquid limit, Tiwari and Ajmera (2011) investigated the correlation between the fully softened friction angle and the plasticity index. They found that the relationship with the plasticity index could be generalized across different mineralogical compositions. Their proposed relationship is shown in Fig. 37. A unique contribution from the work conducted by Tiwari and Ajmera (2011) was the examination of the fully softened shear strength in terms of the clay mineralogy. They introduced a triangular correlation chart, as shown in Fig. 38, which enables the estimation of the fully softened friction angle based on the percentages of clay minerals present in the soil mass. Figure 38 further demonstrated the strong control of clay mineralogy on the fully softened shear strength. Specifically, at low kaolinite contents, the fully softened friction angle was found to only vary slightly, while at higher kaolinite contents, the fully softened friction angle was sensitive to the amount of kaolinite present in the soil mass. On the other hand, when the montmorillonite content was high, the quantity of montmorillonite present did not significantly influence the fully softened friction angle, while strong dependency of the fully softened friction angle on the amount of montmorillonite was noted in soils with low montmorillonite contents.

Fig. 37

Correlation proposed by Tiwari and Ajmera (2011) to estimate the fully softened shear strength as a function of plasticity index

Fig. 38

Proposed triangular correlation chart in Tiwari and Ajmera for the fully softened friction angle in terms of clay mineralogy

To validate the proposed correlations, Tiwari and Ajmera (2011) compared their estimated values with results obtained from over eighty natural soils. They discovered that their estimates, derived from Figs. 36 and 37, had an acceptable margin of error within 25% when compared to the measured values. Furthermore, the estimates based on the triangular correlation chart demonstrated an even higher accuracy with deviations of less than 15%.

Tiwari and Ajmera (2014) delved into the impact of pore fluid chemistry on the fully softened friction angle of soils. They conducted a comprehensive analysis and developed unified correlations to estimate the fully softened friction angle for two scenarios: soils with distilled water and soils with saline water as the pore fluid. In Tiwari and Ajmera (2014), saline water was prepared to be a solution of sodium chloride with a 0.5 M concentration.

From the results obtained, Tiwari and Ajmera (2014) developed a number of correlations to estimate the fully softened shear strength for the different pore fluids they tested. Some of these are re-produced in this paper. Figure 39 illustrates the relationship between the fully softened friction angle and the liquid limit, specifically for soils with distilled water as the pore fluid as well as those with saline water as the pore fluid. Similarly, Fig. 40 depicts the correlation between the fully softened friction angle and the plasticity index for those soils with both of the pore fluids examined by Tiwari and Ajmera (2014). Both Figs. 39 and 40 illustrated that the fully softened friction angle remained nearly constant at liquid limits greater than 150 and plasticity indices greater than 100, respectively. Under these conditions, the montmorillonite present in the soil mass may be dominating the behavior and thus, the strength available.

Fig. 39

Correlation proposed by Tiwari and Ajmera (2015) to estimate the fully softened shear strength as a function of liquid limit and pore fluid fluids (with permission from ASCE)

Fig. 40

Correlation proposed by Tiwari and Ajmera (2015) to estimate the fully softened shear strength as a function of plasticity index and pore fluid fluids

Tiwari and Ajmera (2015) presented correlations, as shown in Figs. 41 and 42, that establish the relationship between the ratio of the fully softened friction angle with saline water as the pore fluid to the fully softened friction angle with distilled water as the pore fluid. These ratios are determined based on the corresponding ratios of the liquid limits and plasticity indices for the respective pore fluids. Figures 41 and 42 demonstrated that the differences in the fully softened shear strength that resulted when different pore fluids were present could be attributed to the effects of the pore fluids on the plasticity characteristics of the soil mass.

Fig. 41

Correlation proposed by Tiwari and Ajmera (2015) to estimate the ratio of the fully softened friction angle with saline water as the pore fluid to the fully softened friction angle with distilled water as a function of the same ratio for the liquid limits (with permission from ASCE)

Fig. 42

Correlation proposed by Tiwari and Ajmera (2015) to estimate the ratio of the fully softened friction angle with saline water as the pore fluid to the fully softened friction angle with distilled water as a function of the same ratio for the plasticity indices (with permission from ASCE)

6.2 Residual Shear Strength

A number of correlations to estimate the residual shear strength are available in the literature. These correlations are related to a variety of different parameters including the following:

• Clay fraction (Skempton 1964; Lupini et al. 1981; and Tika and Hutchinson 1999)

• Liquid limit (Cancelli 1977; Mesri and Cepeda-Diaz 1986; and Stark and Eid 1997)

• Liquid limit and mineralogy (Tiwari and Marui 2003; and Tiwari and Marui 2005)

• Liquid limit, clay fraction and effective normal stress (Stark and Eid 1994; Eid 1996; Stark and Hussain 2015; Eid et al. 2016; and Tiwari and Marui 2005)

• Plasticity index (Voight 1973; Kanji 1974; Kanji and Wolle 1977; Gibo 1985; Borden and Putrich 1986; Müller-Vonmoss and Løken 1989; Tiwari and Marui 2005; and Toyota et al. 2009)

• Plasticity index and effective normal stress (Hawkins and Privett 1985)

• Plasticity index and mineralogy (Tiwari and Marui 2003; and Tiwari and Marui 2005)

• Plasticity index, clay fraction and effective normal stress (Eid et al. 2016)

• Mineralogy (Müller-Vonmoss and Løken 1989; Tiwari and Marui 2005; and Tiwari and Marui 2003)

• Various combinations of a numerous factors (Collotta et al. 1989)

Tiwari and Marui (2003) conducted a comprehensive study involving ring shear tests on mixtures of bentonite, kaolin, and Toyoura sand. Their research shed light on the influence of clay mineralogy on the residual friction angles of these soil materials. In addition to their experimental conclusions, Tiwari and Marui (2003) went further and provided practical correlations to estimate the residual friction angle based on the mineralogical composition of the soil mass. They devised a triangular correlation chart, depicted in Fig. 43, which serves as a useful tool for estimating the residual friction angle by considering the specific clay minerals present in the soil. Tiwari and Marui (2003) noted that the effect of kaolin on the residual friction angle was significant when less than 10% kaolin was present beyond which the effect was negligible. They further concluded that for soils with more than 50% sand and 30% bentonite, the residual friction angle was nearly constant at 3.7° regardless of the amount of kaolin present in the soil mass.

Fig. 43

Tiwari and Marui (2003) proposed triangular correlation chart for the residual friction angle

The correlations illustrated in Figs. 44 and 45 were introduced by Tiwari and Marui (2003). They enable the estimation of the residual friction angle using the liquid limit and plasticity index, respectively. These correlations take into account the mineralogical composition of the soil mass highlighting the significance of clay mineralogy in determining the residual friction angle. Figures 44 and 45 demonstrated that the residual friction angle of soils with kaolin was greater than that of soils containing bentonite, even though both samples had the same liquid limit or plasticity index. However, when the liquid limit or plasticity index was greater than 200, the residual friction angle was at a minimum value of 3.7°.

Fig. 44

Proposed correlation by Tiwari and Marui (2003) for the residual friction angle as a function of the liquid limit and clay mineralogy

Fig. 45

Proposed correlation by Tiwari and Marui (2003) for the residual friction angle as a function of the plasticity index and clay mineralogy

In their research, Tiwari and Marui (2005) critically assessed the correlation proposed by Stark and Eid (1994) for estimating the residual friction angle. They discovered that this correlation, which relied on parameters such as clay fraction, liquid limit, and effective normal stress, had a tendency to significantly overestimate the residual friction angle, sometimes by up to five times. Moreover, Tiwari and Marui (2005) emphasized that the Stark and Eid (1994) correlation was limited in its applicability, being valid only for certain mineral types and failing to provide accurate estimates for a broader range of soils.

Tiwari and Marui (2005) introduced Fig. 46, which contains a correlation between the residual friction angle and both the clay fraction and mineralogical composition of the soil mass. The figure revealed significant variations in the residual friction angle, exceeding 20° in some cases, depending on the clay mineralogy at the same clay content. This further supports their observations of the correlations presented by Stark and Eid (1994), as previously summarized.

Fig. 46

Correlation proposed by Tiwari and Marui (2005) to estimate for the residual friction angle as a function of the clay content and mineralogy (with permission from ASCE)

Figure 47 contains the correlation presented by Tiwari and Marui (2005) to estimate the residual friction angle based the liquid limit. They noted that the correlations presented by Stark and Eid (1994) tended to overestimate the residual friction angle for soils with liquid limits greater than 200. However, the relationship in Fig. 47 could reasonably estimate the residual shear strength for all of the natural samples they tested along with over 70% of the results for the natural samples available in the literature. A relationship to estimate the residual friction angle from the plasticity index, as proposed by Tiwari and Marui (2005), is depicted in Fig. 48. As with the correlation in Fig. 47, the relationship in Fig. 48 was able to estimate the residual shear strength of natural samples with good accuracy. In particular, Fig. 48 shows that the majority of the results from the natural samples were located at the lower bound of the correlation curves developed by Tiwari and Marui (2005).

Fig. 47

Correlation proposed by Tiwari and Marui (2005) to estimate for the residual friction angle as a function of the liquid limit and mineralogy (with permission from ASCE)

Fig. 48

Correlation proposed by Tiwari and Marui (2005) to estimate for the residual friction angle as a function of the plasticity index and mineralogy (with permission from ASCE)

Tiwari and Marui (2005) also prepared a triangular correlation chart, as shown in Fig. 49a, with contours providing a method to estimate the residual friction angle by considering the clay mineralogy in the soil mass. They divided the triangular correlation chart into seven zones (Fig. 49b) that could capture the variation in the residual friction angle with respect to the amount of the various clay minerals (Fig. 49c). The majority of the natural samples they tested were found to fall into zones A and B.

Fig. 49

(a) Triangular correlation chart for residual friction angle by Tiwari and Marui (2005), (b) zones used to illustrate effects of clay minerals and (c) effects of single clay mineral in each zone (with permission from ASCE)

6.3 Undrained Shear Strengths of Over-Consolidated Clays

The Stress History and Normalized Soil Engineering Properties (SHANSEP) method, introduced by Ladd and Foott (1974) and further developed by Ladd et al. (1977), provides a means to assess the impact of the over-consolidation ratio on the undrained shear strength in soils. The undrained shear strength is estimated using the undrained strength ratio, which is defined as the ratio of the undrained shear strength to the effective vertical stress. This method takes into account the influence of the over-consolidation ratio. By analyzing the relationship between the undrained strength ratio and the over-consolidation ratio, SHANSEP allows for the estimation of undrained shear strength under different stress conditions.

Ajmera et al. (2016) extended the SHANSEP method by computing the normalized undrained strength ratio. This was calculated by normalizing the undrained strength ratio at a given over-consolidation ratio to the undrained strength ratio at an over-consolidation ratio of unity (or that for a normally consolidated soil). By doing so, Ajmera et al. (2018) presented two relationships between the normalized undrained strength ratio and the over-consolidation ratio for fine-grained soils. These relationships are shown in Fig. 50. At a constant over-consolidation ratio, the normalized undrained strength ratio for the mixtures of montmorillonite with quartz is higher than that of mixtures of kaolinite with quartz.

Fig. 50

Correlation between normalized undrained strength ratio with over-consolidation ratio from Ajmera et al. (2018) (with permission from ASCE)

In Eqs. (2) and (3), the shear strength parameters of a soil mass are significantly influenced by its stress history. When dealing with normally consolidated clays, the cohesion intercept is typically assumed to be zero. In contrast, over-consolidated clays typically exhibit a non-zero cohesion intercept. In such soils, the magnitude of the cohesion intercept generally increases with an increase in the over-consolidation ratio. The friction angle also varies between normally consolidated and over-consolidated specimens of the same soil. Hvorslev (1960) emphasized that the cohesion intercept and the friction angle in Eqs. (2) and (3) are not indicative of the actual soil properties but rather act as mathematical tools for determining shear strength. According to Hvorslev (1960), the shear strength of clays is influenced by two fundamental parameters: the true friction angle (φ_e) and the base friction angle (φ_c). These parameters remain unchanged regardless of the over-consolidation ratio.

With knowledge of the equivalent pre-consolidation pressure (σ_pe), the shear strength of the soil can be determined using Eq. 23. In the case of normally consolidated soils, the equivalent pre-consolidation pressure will be equivalent to the effective vertical stress. However, for over-consolidated soils, the equivalent pre-consolidation pressure corresponds to the effective vertical stress that would result in the same void ratio if the soil were normally consolidated.

$$ \tau ={\sigma}_{pe}\tan {\phi}_c+{\sigma}^{\prime}\tan {\phi}_e $$

(23)

Tiwari et al. (2022) presented the correlations in Figs. 51 and 52 to estimate the true and base friction angles, respectively, from the plasticity index. They pointed out that the true friction angle did not heavily dependent on the mineralogical composition of the soil mass. Furthermore, Tiwari et al. (2022) found that the true friction angle was nearly constant at 10° in soils with plasticity indices greater than 80. On the other hand, the base friction angle was heavily dependent on the clay minerals present, as illustrated in Fig. 52. An increase in the plasticity index is found to cause an increase in the base friction angle in the mixtures of kaolinite with quartz, while it was constant at 2° in the mixtures of montmorillonite with quartz.

Fig. 51

Correlation from Tiwari et al. (2022) for the true friction angle with respect to the plasticity index (with permission from ASCE)

Fig. 52

Correlation from Tiwari et al. (2022) for the base friction angle with respect to the plasticity index (with permission from ASCE)

6.4 Cyclic Shear Strength

The power function was found to be a good representative of the cyclic strength curves of soils by Ishihara et al. (1980) and Ajmera et al. (2017). In particular, Eq. 24, establishes a relationship between the CSR and the number of cycles (N) required to achieve a specific failure criteria using two curve fitting parameters a and b. These parameters play a crucial role in defining the shape and position of the cyclic strength curves. Specifically, the curve fitting parameter a primarily influences the vertical position of the cyclic strength curve, while the parameter b controls the slope or inclination of the curve.

$$ CSR=a{N}^{-b} $$

(24)

Ajmera et al. (2017) presented correlations that allowed for the curve fitting parameters a and b to be estimated with knowledge of the liquid limit or plasticity index. These correlations are depicted in Figs. 53, 54, 55, and 56. Their results indicated that an increase in liquid limit generally leads to an increase in the value of a, indicating an increase in cyclic resistance for soils with liquid limits below 100 (Fig. 53). However, for soils with liquid limits above 100, the value of a decreases, implying a decrease in cyclic resistance. Similarly, Fig. 54 shows that as the plasticity index increases, the value of a generally increases for soils plasticity indices below 60 beyond which the value of a decreases with increasing plasticity index. Ajmera et al. (2017) attributed the increase in cyclic resistance to the formation of a clay matrix around quartz particles. It should be noted that these results are based on normally consolidated samples and thus, the observed trends may vary in over-consolidated soils.

Fig. 53

Correlation proposed by Ajmera et al. (2017) to estimate curve fitting parameter a as a function of the liquid limit

Fig. 54

Correlation proposed by Ajmera et al. (2017) to estimate curve fitting parameter a as a function of the plasticity index

Fig. 55

Correlation proposed by Ajmera et al. (2017) to estimate curve fitting parameter b as a function of the liquid limit

Fig. 56

Correlation proposed by Ajmera et al. (2017) to estimate curve fitting parameter b as a function of the plasticity index

From Figs. 55 and 56, Ajmera et al. (2017) clearly demonstrate the contrasting behavior of cyclic resistance between two different clay minerals used in the mixtures they tested. For soils containing kaolinite, b increases with both liquid limit and plasticity index, indicating a steeper cyclic strength curve as the plasticity index rises. In contrast, for soils with montmorillonite, b decreases with increasing liquid limit and plasticity index, suggesting flatter cyclic strength curves with higher plasticity indices.

When two sets of cyclic loads are applied on a sample, the cyclic resistance available during the second cyclic event is dependent on the degree to which excess pore pressures generated from the first cyclic load are allowed to dissipate before the second cyclic load is applied. Ajmera et al. (2021) presented relationships containing the cyclic strength curves for a second cyclic load for kaolinite samples. Figure 57 illustrates that an increase in the degree of pore pressure dissipation results in an increase in the cyclic resistance, as depicted by an upward shift in the cyclic strength curves.

Fig. 57

Cyclic strength curves for kaolinite samples during a second cyclic load

6.5 Post-Cyclic Undrained Shear Strength

Ajmera et al. (2019) provided a number of correlations to estimate the undrained shear strength after cyclic loading, defined as the post-cyclic undrained shear strength. However, in their correlations, this shear strength was typically normalized by the effective vertical stress prior to cyclic loading to obtain the post-cyclic undrained strength ratio. One of the correlations Ajmera et al. (2019) developed between the post-cyclic undrained strength ratio and the plasticity index is shown in Fig. 58. This figure shows that for soils with plasticity indices less than 50, an increase in the post-cyclic undrained strength ratio occurred with an increase in the plasticity index. On the other hand, when the soils had a plasticity index greater than 50, the post-cyclic undrained strength ratio decreased with an increase in the plasticity index.

Fig. 58

Correlation to estimate post-cyclic undrained shear strength with the use of the plasticity index developed by Ajmera et al. (2019) (with permission from ASCE)

Figure 59 contains the correlation that Ajmera et al. (2019) proposed between the post-cyclic undrained strength ratio and the pore pressure ratio. The pore pressure ratio is computed as the ratio of the excess pore pressure generated during cyclic loading to the effective normal stress immediately prior to the application of the cyclic load. Like Fig. 58, there is a substantial scatter around the correlation proposed. However, Ajmera et al. (2019) highlighted that this relationship indicates that an increase in the pore pressure ratio would correspond to a reduction in the post-cyclic undrained strength ratio.

Fig. 59

Correlation between post-cyclic undrained strength ratio and the pore pressure ratio from Ajmera et al. (2019) (with permission from ASCE)

A correlation between the degradation ratio and the plasticity index was also provided by Ajmera et al. (2019). The degradation ratio is ratio of the post-cyclic undrained shear strength to the static undrained shear strength. It serves as a measure of the amount of reduction in the undrained shear strength that resulted from the cyclic loading. The correlation developed by Ajmera et al. (2019) is provided in Fig. 60. In soils containing kaolinite as the clay mineral, Fig. 60 indicates that the degradation ratio rises as the plasticity index (PI) increases. However, the impact of the cyclic stress ratio on the degradation ratio was found to be negligible. Conversely, in soils where montmorillonite is the clay mineral, the degradation ratio is influenced by the cyclic stress ratio. As the cyclic stress ratio increases, there is a corresponding decrease in the degradation ratio.

Fig. 60

Correlation between degradation ratio and plasticity index developed by Ajmera et al. (2019) (with permission from ASCE)

Ajmera et al. (2019) calculated the normalized undrained strength ratio and the post-cyclic effective stress ratio for the samples they tested. The normalized undrained strength ratio is an extension of the parameter first proposed by Ajmera et al. (2016). In Ajmera et al. (2019), the normalized undrained strength ratio (Norm. USR) was obtained when the post-cyclic undrained strength ratio was normalized by the undrained strength ratio. In other words, the normalized undrained strength ratio is calculated using Eq. 25. In this equation, s_u is the undrained shear strength, σ’ is the effective normal stress, and the addition of the subscript “pc” indicates the values correspond to the post-cyclic conditions (that is, the conditions immediately following the application of a cyclic load). The post-cyclic effective stress ratio (PC-ESR) is ratio of the consolidation pressure (or effective stress prior to the application of a cyclic load) to the post-cyclic effective stress, as shown in Eq. 26. It can also be computed with knowledge of the pore pressure ratio (r_u) or the ratio of the excess pore pressure generated during cyclic loading to the consolidation pressure. The formula to do so is also provided in Eq. 26.

$$ Normalized\ USR=\frac{s_{u, pc}/{\sigma}_{pc}^{\prime }}{s_u/{\sigma}_c^{\prime }} $$

(25)

$$ PC- ESR=\frac{\sigma_c^{\prime }}{\sigma_{pc}^{\prime }}=\frac{1}{1-{r}_u} $$

(26)

Ajmera et al. (2019) proposed the correlation, as presented in Fig. 61, between the normalized undrained strength ratio and the post-cyclic effective stress ratio. Figure 61 illustrates an intriguing trend: as the post-cyclic effective stress ratio increases, the normalized undrained strength ratio also exhibits a corresponding increase. At a post-cyclic effective stress ratio value of one, the normalized undrained strength remains at one, indicating no reduction in shear strength. However, as the post-cyclic effective stress ratio rises, reaching around 20, the normalized undrained strength climbs to approximately 10.

Fig. 61

Correlation proposed by Ajmera et al. (2019) to estimate the post-cyclic undrained shear strength in terms of the normalized undrained strength ratio as a function of the post-cyclic effective stress ratio (with permission from ASCE)

The correlation in Fig. 61 was shown to be capable of capturing the post-cyclic undrained shear strength of soils subjected to two sets of cyclic loads in Ajmera et al. (2021). This is illustrated in Fig. 62, in which the post-cyclic effective stress ratio was calculated immediately after the end of the second cyclic load. Thus, the relationship, originally developed by Ajmera et al. (2019), is able to capture the effects of excess pore pressure dissipation between cyclic loads and any additional reductions in undrained shear strength induced by the second cyclic load.

Fig. 62

Use of the correlation from Ajmera et al. (2019) to estimate the normalized undrained strength ratio from the post-cyclic effective stress ratio

6.6 Various Other Correlations

Only Stark et al. (2005) and Tiwari and Ajmera (2011) explored the relationships between the fully softened and residual friction angles. However, the correlations presented by Stark et al. (2005) contain significant scatter and hence, will not be discussed. Tiwari and Ajmera (2011) primarily focused on investigating the fully softened shear strength of soils but also provided insights into the relationship between the residual and fully softened friction angles. The residual friction angles used in their analysis were derived from the work of Tiwari and Marui (2005). In their findings, presented in Figs. 63 and 64, the authors proposed new relationships that describe the difference between fully softened and residual friction angles based on the liquid limit and plasticity index, respectively. Tiwari and Ajmera (2011) further introduced normalized relationships for the difference between fully softened and residual friction angles, individually normalized by the respective angles. Figures 65, 66, 67, and 68 depict these relationships with respect to the liquid limit and plasticity indices of the soil samples.

Fig. 63

Correlation from Tiwari and Ajmera (2011) between the differences in the fully softened and residual friction angles as a function of the liquid limit

Fig. 64

Correlation from Tiwari and Ajmera (2011) between the differences in the fully softened and residual friction angles as a function of the plasticity index

Fig. 65

Correlation from Tiwari and Ajmera (2011) between the differences in the fully softened and residual friction angles normalized by the fully softened friction angle as a function of the liquid limit

Fig. 66

Correlation from Tiwari and Ajmera (2011) between the differences in the fully softened and residual friction angles normalized by the fully softened friction angle as a function of the plasticity index

Fig. 67

Correlation from Tiwari and Ajmera (2011) between the differences in the fully softened and residual friction angles normalized by the residual friction angle as a function of the liquid limit

Fig. 68

Correlation from Tiwari and Ajmera (2011) between the differences in the fully softened and residual friction angles normalized by the residual friction angle as a function of the plasticity index

7 Example of Shear Strength Estimation with Correlation Methods

In this section, we will demonstrate a few examples on how to estimate shear strength parameters of two soil samples collected from different locations and having different soil mineralogical dominations as illustrated by their activity values.

• Soil A

• Liquid limit = 60, Plasticity index = 20, Clay Fraction = 22%, Effective vertical stress at sampling location = 100 kPa, Initial void ratio (e_o) = 0.88, Over-consolidation ratio = 8, Compression index (Cc) = 0.28, and Recompression index (Cr) = 0.05.

• Soil B

• Liquid limit = 80, Plasticity index = 40, Clay Fraction = 13.0%, Effective vertical stress at the sampling location = 100 kPa, Initial void ratio = 1.30, Over-consolidation ratio = 2, Compression index = 0.43, and Recompression index = 0.08.

After an earthquake, excess pore pressures are generated in both of those soils. The pore pressure ratios immediately after the earthquake event are estimated to be equal to 0.59 for Soil A and 0.18 for Soil B. If this information is not available, the pore excess pore pressure generated from an earthquake (Δu) can be calculated using Eq. 27 from Davis and Berrill (1982). In this equation, R is the distance from the earthquake epicenter to the location of the soil element expressed in meters, N₁ is the blow counts from a standard penetration test corrected to an overburden pressure of 1 tsf (96 kPa), and M is the magnitude of the earthquake. Both Δu and σ_v’ should be expressed in kPa in Eq. 27.

During a standard penetration test, a split-spoon sampler is driven into the ground using a standard number of blows with a hammer. The resistance offered by the soil to the penetration of the sampler is recorded as the N-value (N). However, the N-value obtained is influenced by the overburden pressure or stress applied to the soil at the specific test depth. To account for this influence and allow for better comparisons between different depths and locations, the N-value is commonly corrected to a standard reference pressure of 1 tsf (96 kPa) using Eq. 28, in which σ_v’ should be in kPa.

$$ \Delta u=\frac{450}{R^2{N}_1^2\sqrt{\sigma_v^{\prime }}}{10}^{1.5M} $$

(27)

$$ {N}_1=N\sqrt{\frac{100}{\sigma_v^{\prime }}} $$

(28)

If the soil samples have activity (Eq. 29) values higher than 2–3, they are expansive soils and are montmorillonitic in nature. If the soil samples have activities less than 0.8–1.0, they are considered as kaolinitic or illitic soils.

$$ \mathrm{Activity}\ \mathrm{of}\ \mathrm{soil}=\left(\mathrm{Plasticity}\ \mathrm{Index}/\mathrm{Clay}\ \mathrm{Fraction}\right) $$

(29)

• For Soil A, Activity = (20/22) = 0.91 [Kaolinitic]

• For soil B, Activity = (40/13) = 3.1 [Montmorillinitic]

Fully Softened Shear Strength

From Fig. 36, the fully softened friction angle is calculated as follows:

• Soil A: 24°

• Soil B: 16°

From Fig. 37, the fully softened friction angle is calculated as follows:

• Soil A: 23°

• Soil B: 16°

Residual Shear Strength

From the central line in Fig. 47, the residual friction angle is calculated as follows:

• Soil A: 20°

• Soil B: 14°

From the central line in Fig. 48, the fully softened friction angle can be calculated as follows:

• Soil A: 20°

• Soil B: 14°

Let’s consider the average friction angles obtained from two methods.

• Soil A: Fully Softened Friction Angle: 23.5°

  • Residual Friction Angle: 20°

• Soil B: Fully Softened Friction Angle: 16°

  • Residual Friction Angle: 14°

Soil A:

Fully Softened shear strength: 100 tan 23.5° = 43.5 kPa.

Residual shear strength: 100 tan 20° = 36.4 kPa.

Soil B:

Fully Softened shear strength: 100 tan 16° = 28.7 kPa.

Residual shear strength: 100 tan 14° = 24.9 kPa.

Using the true and base friction angle, the fully softened shear strengths for these two soils could be estimated from the correlations presented by Tiwari et al. (2022) in Figs. 51 and 52. To obtain the fully softened shear strength, the equivalent pre-consolidation pressure will be equal to the effective vertical stress. Thus,

Soil A: ϕ_e = 19° and ϕ_c = 7°

$$ \tau ={\sigma}_{pe}\tan {\phi}_c+{\sigma}^{\prime}\tan {\phi}_e $$

$$ \tau =100\tan 7+400\tan 19=44.7\ kPa $$

Soil B: ϕ_e = 15° and ϕ_c = 2°

$$ \tau =100\tan 2+100\tan 15=30.2\ kPa $$

Values obtained from both methods are very close.

Peak Shear Strength

Soil A: The soil with an over-consolidation ratio of 8 and an effective vertical stress of 100 kPa will have a pre-consolidation pressure (σ_p’) of:

$$ {\sigma}_p^{\prime }= OCR\times {\sigma}^{\prime }=8(100)=800\ kPa. $$

Using the compression and recompression indices, the equivalent pre-consolidation pressure is found as follows:

$$ {C}_r=\frac{e_o-{e}_p}{\log \left(\frac{\sigma_p^{\prime }}{\sigma^{\prime }}\right)} $$

So, ${e}_p={e}_o-{C}_r\log \left(\frac{\sigma_p^{\prime }}{\sigma^{\prime }}\right)$

$$ {e}_p=0.88-0.05\log \left(\frac{800}{100}\right)=0.835 $$

Where, e_p is the void ratio of the sample at the pre-consolidation pressure.

$$ \mathrm{Now},{C}_c=\frac{\left({e}_o-{e}_p\right)}{\log \left(\frac{\sigma_p^{\prime }}{\sigma_{pe}^{\prime }}\right)} $$

$$ 0.28=\frac{0.88-0.835}{\log \left(\frac{800}{\sigma_{pe}^{\prime }}\right)} $$

$$ \mathrm{Thus},\qquad {\sigma}_{pe}^{\prime }=552.6\ kPa $$

As estimated before, ϕ_e = 19° and ϕ_c = 7°. Thus, the peak shear strength is:

$$ \tau =552.6\tan 7+100\tan 19=102.3\ kPa. $$

$$ \mathrm{Soil}\ \mathrm{B}:{\sigma}_p^{\prime }=2(100)=200\ kPa $$

$$ {e}_p=1.30-0.08\log \left(\frac{200}{100}\right)=1.28 $$

$$ 0.43=\frac{1.3-1.28}{\log \left(\frac{200}{\sigma_{pe}^{\prime }}\right)} $$

$$ {\sigma}_{pe}^{\prime }=63.3\ kPa. $$

As calculated above, ϕ_e = 15° and ϕ_c = 2°. Thus, the peak shear strength will be equal to:

$$ \tau =63.3\tan 2+100\tan 15=29.0\ kPa $$

From Fig. 50, the normalized undrained strength ratio from a central line can be found to be:

• Soil A: 2.7

• Soil B: 1.1

Using Fig. 58, the static undrained strength ratios for the normally consolidated consolidations will be:

• Soil A: 0.37

• Soil B: 0.33

Thus, the peak shear strength can be calculated as:

$$ \mathrm{Soil}\ \mathrm{A}: US{R}_{OCR=8}= Norm. USR\times \frac{s_u}{\sigma_c^{\prime }} $$

$$ US{R}_{OCR=8}=2.7(0.37)=1.00 $$

$$ \tau =1.00(100)=100\ kPa $$

$$ \mathrm{Soil}\ \mathrm{B}: US{R}_{OCR=2}=1.1(0.33)=0.36 $$

$$ \tau =0.33(100)=33\ kPa $$

Both methods yield similar peak shear strengths.

Cyclic Strength Curves and Post-Cyclic Undrained Shear Strength

The relationship in Fig. 54 would indicate that Soil A has a power function parameter a of 0.28, while that for Soil B is 0.38. Using the correlation in Fig. 56, the power function parameter b can be found to be 0.087 and 0.085 for Soils A and B, respectively. Thus, the cyclic strength curves corresponding to 5% double amplitude shear strain can be drawn using Eq. 24. Specifically, the power functions are going to be:

• Soil A: $CSR=0.28{N}_5^{-0.087}$

• Soil B: $CSR=0.38{N}_5^{-0.085}$

The corresponding cyclic strength curves obtained when these equations are plotted are shown in Fig. 69.

Fig. 69

Cyclic strength curves for hypothetical Soils A and B

The following process can be used to determine the post-cyclic effective stress ratio.

Using Eq. 26, the post-cyclic effective stress ratios would be:

• Soil A: $PC- ESR=\frac{1}{1-{r}_u}=\frac{1}{1-0.59}=2.42$

• Soil B: $PC- ESR=\frac{1}{1-0.18}=1.23$

Thus, from the correlation presented in Fig. 61,

• Soil A: Norm. USR = (PC − ESR)^0.753

  • Norm. USR = 2.42^0.753 = 1.94

• Soil B:Norm. USR = 1.17^0.753 = 1.17

If the undrained strength ratio of both of these two soils prior to the cyclic loads were 0.36, then the post-cyclic undrained strength ratio would be equal to:

• Soil A: $\frac{s_{u, pc}}{\sigma_{pc}^{\prime }}= Norm. USR\times USR$

  $\frac{s_{u, pc}}{\sigma_{pc}^{\prime }}=1.94\times 0.36=0.70$

• Soil B: $\frac{s_{u, pc}}{\sigma_{pc}^{\prime }}=1.17\times 0.36=0.42$

Now, ${\sigma}_{pc}^{\prime }={\sigma}^{\prime }-u={\sigma}^{\prime}\left(1-{r}_u\right)$. Thus, the effective vertical stress immediately following the application of the cyclic loading is:

• Soil A: ${\sigma}_{pc}^{\prime }=400\left(1-0.59\right)=164\ kPa$

• Soil B: ${\sigma}_{pc}^{\prime }=50\left(1-0.18\right)=41\ kPa$

Using this, the post-cyclic undrained shear strength can be calculated as:

• Soil A: s_{u, pc} = 0.70(164) = 114.8 kPa

• Soil B: s_{u, pc} = 0.42(41) = 17.22 kPa.

These results indicate that there was a degradation in the undrained shear strength since the static undrained shear strengths were:

• Soil A: s_u = 0.36(400) = 144 kPa

• Soil B: s_u = 0.36(50) = 18 kPa.

The degradations ratios associated with these two soils are:

• Soil A: $\delta =\frac{s_{u, pc}}{s_u}=\frac{114.8}{144}=0.80$

• Soil B: $\delta =\frac{17.22}{18}=0.96$

Degradation ratios indicate that Soil A experienced a 20% reduction in shear strength as a result of the cyclic loading. On the other hand, the cyclic loading only caused a 4% reduction in the shear strength in Soil B.

Using the relationship in Fig. 60, the degradation ratios would be:

• Soil A: 0.84

• Soil B: 0.94

These values are similar to those obtained above using the other method.

8 Summary and Recommendations

Landslides pose catastrophic consequences, resulting in loss of life, property damage, and disruption of communities. Natural factors like heavy rainfall, snowmelt, and groundwater fluctuations induce water saturation, triggering landslides. Earthquakes heighten landslide risks through shaking, liquefaction, and the formation of landslide dams that can lead to flooding. Volcanic activity also contributes to devastating landslides, such as volcanic debris flows and structural collapses. Furthermore, human activities like altering drainage patterns, destabilizing slopes, and removing vegetation exacerbate landslide occurrences. The effects of landslides are also exacerbated in various regions. For example, mountainous regions face higher mortality risks, while countries with limited resources experience greater economic losses. Furthermore, these natural disasters have far-reaching environmental and societal impacts, including landscape alteration, vegetation destruction, ecosystem disruption, and water pollution from sediment and debris. Landslides displace residents, jeopardize livelihoods, and instigate social and economic instability. Infrastructure damage impedes transportation, communication, and access to vital services. Understanding landslides is pivotal for implementing proactive measures that mitigate risks and enhance resilience, safeguarding the well-being of communities.

The shear strength of soils is a crucial factor in analyzing slope stability and implementing measures to mitigate risks. Shear strength refers to the internal resistance of soil to shearing forces, which is influenced by factors such as friction between soil particles, interlocking of grains, electromagnetic attraction, over-consolidation effects, density, overburden pressure, grain shape, roughness, and stress history. During shearing, soils exhibit different behaviors, including elastic behavior, peak shear strength, post-peak strength loss, and fully softened shear strength. The fully softened shear strength occurs when changes in water content cause a reduction in shear strength as the soil approaches its critical state. Continued shearing can lead to particle reorientation and the formation of polished slickenslide surfaces, resulting in the residual shear strength. The shear strength of soil is typically represented using the Mohr-Coulomb failure criteria.

In addition to shear strength parameters under static conditions, the resistance of soil to cyclic loads is essential for slope stability analysis is seismic regions. Cyclic strength curves, representing the number of cycles to failure against cyclic stress ratio, provide information on the ability of a soil mass to withstand cyclic loading. Furthermore, knowledge of the post-cyclic undrained shear strength is also necessary to evaluate slope stability after cyclic loading. Different soil properties are required for stability analysis depending on conditions, including end-of-construction, multistage loading, and long-term analysis, as well as for free-draining and impermeable materials. The recommended parameters of each of these conditions and types of soils were summarized in this paper.

Direct shear tests are commonly used to measure the shear strengths of soils under drained conditions. These tests are widely available, easy to run and interpret, and relatively quick. However, the shear strength is always measured on the horizontal shearing plane, which may not be the weakest plane and can result in an overestimate of the shear strength. Direct shear tests can measure the peak, fully softened, and residual shear strengths of soil. ASTM D3080/D3080M provides details and specifications for conducting direct shear tests.

Triaxial tests are another commonly used method for measuring shear strength. These tests can measure both peak and fully softened shear strengths using consolidated drained (CD) and consolidated undrained (CU) triaxial tests. CD testing of low permeability clays may be time-consuming due to back pressure saturation requirements. ASTM D7181 provides specifications for CD triaxial tests, while ASTM D4767 outlines procedures for CU triaxial tests.

Ring shear tests have emerged as the preferred method for quantifying residual shear strength. These tests allow continuous shearing of samples in a single direction while maintaining a consistent cross-sectional area, closely replicating residual conditions. The non-uniform stress distribution across annular samples used in ring shear tests can pose challenges. Different variations of the ring shear device have been developed, each with its own advantages and disadvantages. ASTM D6467 provides detailed procedures for conducting ring shear tests to measure the residual shear strength of cohesive soils.

Direct simple shear tests offer advantages compared to direct shear and triaxial tests. They allow the sample to fail along its weakest plane while the cross-sectional area remains constant throughout the shearing process. These tests require smaller soil samples and shorter consolidation times. However, the commercial availability of direct simple shear devices is currently limited. ASTM D6528 provides guidelines for conducting direct simple shear tests.

A number of correlations for the fully softened shear strength are available in the literature. The key ones were summarized in this paper. These correlations provided insights into the impact of clay mineralogy. In particular, it was seen that significant differences in the fully softened friction angles between different clay mineral mixtures existed, even at a constant clay content. Furthermore, the correlations between the fully softened friction angle and the plasticity index or the liquid limit should be utilized in practice. If mineralogical compositions are not available, soil activity information should be used to identify whether the soil is montmorillinitic or kaolinitic (or illitic) from a mineralogical point of view.

The impact of pore fluid chemistry on fully softened shear strength was also presented in this paper. Correlations developed for soils with different pore fluids make it possible to estimate the fully softened friction angle in terms of the plasticity index or the liquid limit. Additionally, variations in the fully softened friction angle observed in soils with different pore fluid chemistries could be attributed to the impacts of these pore fluids on the plasticity characteristics of the soil mass.

Tiwari and Marui (2003) conducted a comprehensive study on the influence of clay mineralogy on residual friction angles. They found that the presence of kaolin had a significant effect on the residual friction angle, particularly when kaolin content was less than 10%. However, for soils with high sand and bentonite content, the residual friction angle remained nearly constant regardless of kaolin content.

The correlations based on liquid limit and plasticity index developed by Tiwari and Marui (2005) take into account the mineralogical composition of the soil mass. They showed that the residual friction angle of soils with kaolin was higher than that of soils containing bentonite, even with similar liquid limits or plasticity indices. They also noted that for soils with liquid limits or plasticity indices exceeding 200, the residual friction angle reached a minimum value of 3.7°.

The SHANSEP method allows for the estimation of undrained shear strength considering the over-consolidation ratio. This method was extended by Ajmera et al. (2018) with the introduction of the normalized undrained strength ratio. They revealed that mixtures of montmorillonite with quartz have higher ratios than mixtures of kaolinite with quartz.

The shear strength parameters in the Mohr-Coulomb failure criteria are influenced by stress history, with over-consolidated clays showing a non-zero cohesion intercept and varying friction angles. Tiwari et al. (2022) provided correlations to estimate the true and base friction angles from the plasticity index, finding that the true friction angle remained constant at 10° for soils with high plasticity indices, while the base friction angle varied with clay mineral composition. In the mixtures of montmorillonite-quartz, the base friction angle could be taken to be constant at a value of 2°.

Correlations presented by Ajmera et al. (2017) allow for the estimation of the curve fitting parameters a and b based on liquid limit or plasticity index to calculate the cyclic resistance of a soil mass. Increasing liquid limit generally increases the value of a for soils with liquid limits below 100 indicating higher cyclic resistance. However, for liquid limits above 100, a decreases. The behavior of parameter b varies with clay mineral type. For kaolinite, b increases with liquid limit and plasticity index, while for montmorillonite, b decreases. Dissipation of excess pore pressure between cyclic loads influences cyclic resistance, with increased dissipation leading to higher cyclic resistance.

Ajmera et al. (2019) provided correlations to estimate the post-cyclic undrained shear strength and introduced the normalized undrained strength ratio. The normalized undrained strength ratio is calculated by normalizing the post-cyclic undrained strength ratio by the undrained strength ratio. As the post-cyclic effective stress ratio increases, the normalized undrained strength ratio also increases. This correlation was found to capture the post-cyclic undrained shear strength after two sets of cyclic loads and thus, could consider the dissipation of excess pore pressure and the effects of the second cyclic load.