In contrast, the Hardening Soil Model offers an effective simulation of the nonlinear stress-strain response in soft soils.This research aims to establish both the conventional compressi
INTRODUCTION
Problem statement
The Red River Delta in Vietnam is predominantly composed of highly compressible soil layers, including soft and silty clays, characterized by their fine-grained texture and low permeability These soils pose substantial geotechnical engineering challenges due to their low shear strength, high compressibility, and susceptibility to significant deformation under load, complicating the design and construction of infrastructure foundations amid rapid urbanization and coastal development in the region One-dimensional consolidation analyses have limitations in addressing complex geotechnical problems involving multi-dimensional settlement behavior, while the Hardening Soil Model provides a more sophisticated framework by simulating the nonlinear stress–strain response of soft soils.
Soft and silty clays in the RRD exhibit stress- and strain-rate-dependent behavior that is also shaped by environmental factors such as fluctuating groundwater tables Traditional geotechnical models, which assume constant soil stiffness, fail to capture the nonlinear and stress-dependent response of these soils, potentially leading to inaccurate settlement predictions, misjudged bearing capacity, and compromised safety and longevity of infrastructure projects Moreover, the RRD’s dynamic hydrological conditions, with seasonal variations in water tables, demand advanced constitutive models that can accurately simulate both drained and undrained soil behavior to ensure reliable design and performance.
The Hardening Soil (HS) model has emerged as a promising tool for addressing these challenges by incorporating stress-dependent stiffness and effectively capturing the nonlinear deformation characteristics of soft soils compared with conventional models However, its successful application relies on accurately determining and calibrating its parameters, which are highly specific to local soil conditions In the region of RRD, existing studies on geotechnical properties are limited, resulting in a lack of comprehensive data on HS model parameters for these soils.
Soft clays and silty clays in the RRD demonstrate both stress-dependent behavior and strain rate dependency, meaning their stiffness and strength can vary significantly with the rate at which they are loaded This rate-dependent response, in addition to traditional stress sensitivity, has important implications for geotechnical analysis and design in the RRD, as higher loading rates can increase apparent strength and stiffness while slower rates reduce them Engineers must account for loading rate effects when evaluating settlement, stability, and overall performance of foundations and earthworks on these clays.
Research questions
1) What are the characteristics of common compressibility parameters (Cc, Cr, Cs, σp', and OCR) and hardening soil model parameters (c', ', Ψ, m, E50 ref, Eoed ref, etc) of the clays in the Red River Delta?
2) What are the possible correlations between the Hardening Soil model parameters and other soil properties (compressibility parameters, strength, and deformation characteristics)?
The Necessity of the Research
1) There was no systematic experimental investigation of the Hardening Soil model parameters and their correlation with the specific clay formations found in the RRD
2) The Hardening Soil model parameters are a critical necessity to accurately predict soil behavior under varying loading conditions
Some potential academic contributions to society are;
1) To contribute to a deeper understanding of the behavior of clays in the Red River Delta, including their stress-strain behavior, strength, and deformation characteristics
2) To improve the accuracy of numerical simulations for geotechnical engineering problems in the Red River Delta.
Objectives of the Research
1) To evaluate the typical compressibility parameters of clays (Cc, Cr, Cs,𝜎 and OCR) and some hardening soil model parameters (c', ', Ψ, m, E50 ref, Eoed re, etc) of clays at the test sites in the Red River delta
2) To determine possible correlations between Hardening soil model parameters and other basic soil parameters of the clays for practical applications
Scope of the study
Perform physical tests, oedometer tests, and consolidated drained triaxial tests
- Analyze the experimental and numerical test results
- Evaluate the key input parameters of the Hardening Soil Model for soft clays
- Make possible correlations between Hardening soil model parameters and other basic soil parameters of the clays for practical applications
- Study sites for this research are taken from the supervisor’s research projects
LITERATURE REVIEW
Geological condition of the Red River Delta
Located in northern Vietnam, the Red River Delta has a long, complex depositional history shaped by diverse geological and hydrological processes over thousands of years The delta features a flat topography with a mean elevation of about 8–10 meters above mean sea level, punctuated by occasional hills rising 20–50 meters Its geology can be understood by considering the region’s tectonic history, the properties of its sedimentary deposits, and the rock types present.
This study focuses on the soil types of the RRD and aims to evaluate the geotechnical behavior of its soft clays for heavy construction Holocene deltas exhibit varying morphologies and sedimentary facies driven by changes in coastal settings and sediment discharge rates (Tanabe et al., 2003) Contemporary coastal hydrodynamics support a three-type classification of these deltas—fluvial-dominated, wave-dominated, and tide-dominated—based on the framework of Galloway (1975) as cited in Tanabe et al (2006).
Figure 2.1 Map showing the major geological characteristics of the Red River Delta, Vietnam (Li et al 2006, Tanabe et al 2006) Modified from Tanabe et al (2006)
Within the RRD, the fluvial-dominated delta is characterized by meandering rivers, meandering levee belts, floodplains, and fluvial terraces, with stronger fluvial flux than tide-dominated and wave-dominated deltas The sediments in this region consist of gravelly sand and mottled clay.
Five tidal flats, marshes, and tidal creeks and channels—both abandoned and active—are present in the region, with sediments that include shell and wood fragments The wave-dominated delta comprises sandy spits, alternating beach ridges, inter-ridge marshes, and back marshes or swamps, and it experiences strong wave energy driven by the summer monsoon winds Sedimentation consists of tide-influenced sand and mud deposits, showing an upward increase in sand content and a growing amount of wood fragments, as noted by Phach et al (2020) and Tanabe et al (2006).
Figure 2.2 Alluvial delta plain with four paleo shorelines with ages of 3–2.5 Ka: 1.5–1
Ka, 0.7–0.5 Ka, and 0.3–0.1 Ka belonging to the highstand systems tract (amhHSTQ22
Figure 2.2 documents the geological history of the Red River Delta (RRD) as an alluvial delta plain, highlighting four paleoshorelines that formed during the middle to late Holocene in response to sea‑level regression Geological mapping and early Quaternary studies classify the RRD sediments into five formations: the Early Pleistocene Le Chi Formation; the Middle–Late Pleistocene Ha Noi Formation; the Late Pleistocene Vinh Phuc Formation; the Early–Middle Holocene Hai Hung Formation; and the Late Holocene Thai Binh Formation (Yen et al., 2021) Sediment cores from Hanoi, Hung Yen, Thai Binh, and Nam Dinh provinces further clarify the stratigraphy, revealing transgressive alluvial deposits that unconformably overlie the eroded surface of regressive deposits at the TST/arLST boundary.
Tanabe et al (2006) studied the Holocene evolution of the Song Hong delta system in northern Vietnam using seven sediment cores, each 30–70 m deep, and identified four main stratigraphic units: Unit 0 with Late Pleistocene shallow-marine sediments; Unit 1 with Latest Pleistocene fluvial sediments; Unit 2 consisting of Holocene estuarine sediments containing shell and wood fragments; and Unit 3 comprising Holocene deltaic sediments with tide-influenced sand and mud deposits, showing increasing sand content and wood fragments upward The locations of these seven cores are shown in Figure 2.3, and the soils encountered are predominantly fine-grained, soft materials such as clay, clayey soil, silt, silty soil, and sandy soil.
Figure 2.3 The location of the seven cores (Tanabe et al., 2006)
Theoretical Background of the Hardening Soil Model (HS Model)
The Hardening Soil model, developed by Schanz in 1999, is an advanced elastoplastic constitutive model used to simulate the stress–strain behavior of soils and soft soils It moves beyond elastic theory to deliver a more accurate depiction of soil mechanics by incorporating three key features: plasticity theory to account for permanent deformation, soil dilatancy to capture volume changes during shear, and a nonlinear, stress- and strain-dependent stiffness (hardening) that reflects how soil resistance evolves under changing load conditions.
(3) a yield cap, which defines the material's response under hydrostatic pressure
These fundamental characteristics enable the model to simulate complex soil behavior more realistically It's a more precise model as it can better predict displacements by
Nonlinear stress–strain behavior of soil and the evolution of plastic strains are key aspects of constitutive modeling Kondner (1963) first showed that in drained triaxial tests, axial strain and deviatoric stress can be described by a hyperbola The HS model extends this idea by adopting a hyperbolic stress–strain relationship and incorporating stress level dependency, delivering improvements over traditional bi-linear curve representations.
Figure 2.4 Hyperbolic stress-strain relationship in primary loading for a standard drained triaxial test (Schanz, 1999)
The HS model incorporates stress-dependent stiffness, with soil stiffness increasing as confining pressure rises, delivering a more realistic representation of soil behavior under varying stress It also includes plastic hardening mechanisms to capture the enhanced strength and stiffness that develop under cyclic loading due to plastic deformation Its deformation calculation accounts for both elastic and plastic strain, enabling accurate simulation of soil mechanical behavior through loading and unloading cycles.
The Hardening-Soil Standard Model captures how soils behave under loading, based on laboratory observations of soil samples Its constitutive parameters tie directly to intrinsic soil properties that can be measured with established tests, notably triaxial compression and oedometer experiments By grounding the model in these well-defined physical characteristics, the Hardening-Soil Standard Model provides a practical framework for predicting soil response under different stress states and loading histories.
Soil Model extends the classical Mohr-Coulomb model by incorporating additional parameters to comprehensively describe complex soil behavior
Table 2.1 List of the input parameters for the hardening soil model
Parameters Symbol Unit Needed tests to determine the parameter
1 Effective Cohesion c ' kPa Consolidated Drained triaxial tests
2 Effective Friction angle ° Consolidated Drained triaxial tests
3 Angle of dilation ° Consolidated Drained triaxial tests
4 Secant stiffness at reference pressure E 50 ref kPa Consolidated Drained triaxial tests
5.Unloading/ reloading stiffness E ur ref kPa Consolidated Drained triaxial tests
6 Power for the stress-level dependency of stiffness m - Typical range of value: 0.5 ≤ m ≤ 1
7.Poisson’s ratio for unloading/reloading ur - Consolidated Drained triaxial tests
8 Reference stress for stiffnesses p ref kPa Default: p ref = 100 kPa, the atmospheric pressure
10 Coefficient of lateral stress in normal consolidation
12 Swelling index or reloading index C s - Oedometer test
13 Tangent stiffness for primary oedometer loading ref
The Hardening Soil model is commonly used for analyzing settlement prediction, foundations, slopes, and excavations by accounting for soil stiffness changes depending on the pressure it's under
2.2.1 Hyperbolic Relationship of Stress-Strain in CD Triaxial Test
Under triaxial loading, σ1′ is the major compressive principal stress while the minor principal stresses are equal, with σ2′ = σ3′ In the early stage of a triaxial test, the HS model adopts a hyperbolic relationship between vertical strain ε1 and deviatoric stress qf as its fundamental basis.
Figure 2.5 The physical meaning of parameters a and b: (a) hyperbolic stress-strain curve; (b) transformed hyperbolic stress-strain
Under initial deviatoric loading, soil's stiffness reduces while it concurrently develops permanent plastic deformation The curve, Kondner (1963) proposed, can be described by a simple hyperbolic equation
By incorporating the strength parameters c' and φ', the Mohr-Coulomb failure criterion underpins the qf equation referenced earlier While the asymptotic stress qult would, in theory, require an extremely large strain to attain, soils typically fail at far smaller strains When q = qf, the failure criterion is met, and perfectly plastic yielding occurs The failure ratio Rf, defined as the ratio of qf to qa, must always be less than 1, with a common default value of Rf = 0.9.
The shear mechanism's hardening yield function f1 is defined by a hyperbolic approximation of the vertical strain ε1 and the deviatoric stress q observed in a standard drained triaxial compression test (Figure 2.5) The definitions of the yield functions f12 and f13 are given by Equations 2.6 and 2.7, respectively Frictional hardening in these functions is governed by the plastic shear strain γp, as defined in Eq 2.8, and the resulting yield condition is expressed as follows:
Although plastic volumetric strain 𝜀 is never exactly zero in reality, its magnitude is small for hard soils when compared with the axial strain Consequently, the simplification presented in Equation 2.7 usually provides a sufficiently accurate result for practical purposes When the hardening parameter (γ_p) is held constant at a certain value, the points where the yield conditions ƒ12 and ƒ13 both equal zero define a curve known as the yield locus when plotted on the p′-q plane.
In drained triaxial tests where the minor principal stress σ3 and the intermediate principal stress σ2 are held constant, the elastic Young’s modulus Eur remains constant for unloading and reloading The resulting elastic strains are then determined by the standard elastic stress–strain relationships that incorporate Eur as the constant modulus along these loading paths, enabling a consistent description of material stiffness throughout the unloading–reloading sequence under fixed σ2 and σ3.
These equations specifically address strains that arise from deviatoric loading They do not account for the strains that occur during the initial phase of the test Although the
Hardening-Soil model predicts elastic volume changes governed by Hooke's law during the initial isotropic compression phase, a behavior not captured by Equation 2.11 in its current formulation In a triaxial test, applying a deviatoric load to the sample produces axial strain that is the sum of an elastic component described by the relevant equation.
2.11) and a plastic strain (as per Equation 2.10) This leads to the following relationship:
This relationship is precisely true only when there are no plastic volume changes, meaning the plastic volumetric strain (ε1 p ) is zero
2.2.2 Plastic volume strain behavior under triaxial loading
Characteristic of all plasticity models, the Hardening-Soil model incorporates a relationship between these plastic strain rates, specifically between 𝜀 ° and 𝛾 ° , and this flow rule takes a linear form p sin m p
The specification requires further detailing of the mobilized dilatancy angle ( m ) For the present model, the expression is given by:
12 sin sin sin 1 sin sin m cv m m cv
(2.14) where 𝜑 is the friction angle at critical state, a density-independent material constant, while 𝜑 is the mobilized friction angle
The presented formulations adhere to the well-known stress-dilatancy framework proposed by Rowe (1962, 1971) and later expanded upon by Schanz & Vermeer (1996)
Within this theory, the material contracts at low stress ratios (φ_m < φ_cv) and dilates at high stress ratios (φ_m > φ_cv) When failure occurs—the mobilized friction angle φ_m reaches the failure angle φ—the system reaches a critical state described by Equation (2.16) This equation reveals a sine-based relationship among the involved angles, linking φ_m, φ_cv, and φ to show how friction mobilization governs the transition from contraction to dilation at failure The key takeaway is that the deformation mode depends on whether φ_m lies below or above φ_cv, with failure marked by φ_m = φ and captured by the trigonometric form in Equation (2.16).
1 sin sin 1 sin sin cv cv cv
At the critical state, the friction angle φcv is calculated from the friction angle φ and the dilatancy angle φ PLAXIS automates this calculation, delivering φcv without any manual input The analysis, however, requires the user to provide the ultimate friction angle and the ultimate dilatancy angle.
2.2.3 CD triaxial test-derived stiffness parameters for primary loading
Because the stress–strain response is highly nonlinear during initial loading, stiffness is commonly described by the secant modulus E50, a parameter that depends on confining stress under primary loading conditions Instead of the initial tangent modulus Ei, which is difficult to measure at small strains, the secant modulus E50 is used E50 is defined by a specific equation and provides a practical measure of stiffness that captures how confining stress influences material response in the early loading regime.
(2.17) Where E50 = Secant stiffness at 50% of failure stress,
E50 ref = Secant stiffness (reference value for primary loading), and
13 m = Power for the stress-level dependency of stiffness
The Hardening Soil model defines a reference stiffness modulus, E50, based on a reference minor principal stress p_ref, while the actual stiffness depends on the minor principal stress σ3' representing the effective confining pressure in triaxial testing Unlike fixed constants, stiffness parameters in the Hardening Soil model are reference values that adjust with stress conditions Typically, the reference pressure (p_ref) is assumed to be 100 kPa, which is the default setting in Plaxis; however, this can lead to input values that do not reflect realistic in-situ conditions Fortunately, users have the flexibility to select a suitable reference stress level to better match the specific project context.
2.2.4 Secant modulus for unloading-reloading paths in CD triaxial tests
Triaxial test analysis employs a distinct stress-dependent modulus for unloading- reloading stress paths
'cot ' ref m ur ur ref
Equation (2.18) defines Eur_ref as the experimentally-derived reference modulus for unloading/reloading cycles, associated with the defined reference pressure σ_ref Using Eur_ref in this way allows the model to treat the material response during unloading and reloading as purely non-linear elastic.
Consolidation
Consolidation is the gradual compression of soil particles under sustained vertical pressure, with water draining from the pore spaces between particles as the soil settles The laboratory test by Terzaghi (1943) determines how saturated soils compress and settle when subjected to increasing vertical stresses in confined conditions This evaluation is typically conducted in a specialized apparatus called an oedometer, which applies controlled loads and records settlements The results help engineers understand soil behavior under load and inform geotechnical design related to consolidation effects.
Figure 2.6 The spring-piston analogy demonstrates the fundamental principles of one- dimensional consolidation in saturated soils (Das, 2010)
Figure 2.7 illustrates the three characteristic phases of time-dependent deformation observed during consolidation under each load increment
Stage I- Initial compression or Immediate settlement occurs immediately after a load is applied, primarily due to preloading effects, and takes place before any drainage occurs during laboratory tests
Stage II- Primary consolidation is the time-dependent process where pore water pressure is transformed into effective stress as excess pore water is expelled from soil voids This long-term settlement is associated with theoretical curves for most clays and addressed by Terzaghi's consolidation theory Unlike immediate settlement, primary consolidation permanently alters soil structure, often leaving residual swelling, making it the focus for many settlement estimations However, for organic soils like peats, secondary compression can be more significant over extended periods, underscoring the varying responses of soil types to loading and time
Stage III- Secondary compression initiates once all excess pore pressures have fully dissipated and occurs as soil particles slowly reorganize into more stable configurations The reliability of settlement estimates during this stage is generally lower compared to those based on primary consolidation
Figure 2.7 Time-Deformation Response During Consolidation Under Constant Load Increment (Das, 2010)
2.3.1 Consolidation theory (Vertical Drainage case)
The initial consolidation theory for saturated clay soils was introduced by Karl Terzaghi,
(1925), based on several key assumptions:
1 The clay layer is compositionally uniform
2 The clay exists in a fully saturated state, with all voids filled with water
3 The clay (solid particles) and pore water are assumed to be volumetrically incompressible
4 Darcy's law governs the hydraulic flow characteristics of the soil
5 The hydraulic conductivity, denoted as kv, remains constant throughout the process
6 Soil compression occurs only vertically with minimal displacement
7 There exists a linear relationship between the effective vertical stress (σv′) and the void ratio (e)
8 The soil behaves in a non-viscous manner, meaning it is not affected by time or strain rate
The vertical consolidation coefficient (cᵥ) for soils with vertical drainage may be calculated using the expression;
Terzaghi's consolidation theory provides the following formulation for degree of consolidation;
Based on the above two equations, the vertical coefficient of consolidation can be expressed as:
t (2.23) where, cv = vertical coefficient of consolidation; Tv = time factor for vertical consolidation; h or 𝐻𝑑𝑟 2 = the length of the longest drainage path
Fine-grained soils typically exhibit vertical consolidation coefficients (cv) between 10^-3 and 10^-1 m²/day (Briaud, 2013) Field observations show that settlement in clay tends to occur more quickly than predicted by one-dimensional consolidation theory, because that theory does not account for horizontal pore-water dissipation and the actual geometry often diverges from the model assumptions As a result, three-dimensional effects are significant, and settlement rates calculated with conventional one-dimensional methods may be inaccurate.
The compression index is defined as the slope of the linear portion of the void ratio (e) versus the logarithm of the effective consolidation pressure (log σ'v) curve This e–log σ'v relationship is a key input for settlement calculations in geotechnical engineering, as it governs how much soil will compress under load In practice, the e–log σ'v curve is often assumed to be linear in the higher pressure range, providing a simplified yet reliable basis for estimating settlements.
The e-log σ' curve can be either upward- or downward-concave, controlled by the soil’s plasticity index and its initial moisture condition Very compressible clays and soils with initial water contents below their liquid limit exhibit pronounced nonlinearity in their compression behavior Both the compression index (Cc) and the recompression index (Cr) tend to decrease with increasing plasticity index, largely due to soil fabric disturbances during sampling and laboratory preparation.
Equation (2.24) defines the void-ratio change during virgin compression of normally consolidated soil: e1c and e2c are the void ratios at vertical effective stresses σ'v1c and σ'v2c, respectively, with σ'p < σ'v1c < σ'v2c in the normally consolidated range In this range, the compression index can be calculated from the change in void ratio (Δe = e2c − e1c) and the logarithm of the vertical effective-stress ratio (log(σ'v2c/σ'v1c)), providing a means to quantify soil densification under increasing vertical stress.
Figure 2.8 Idealized void ratio (e) versus log effective pressure (log σₚ') relationship from oedometer testing (after Mayne et al., 2001)
The recompression index plays a vital role in determining the final consolidation settlement of a clay layer when it experiences an increase in vertical effective stress (σv′
) It represents a simplified e-log(σv′) curve obtained from a standard oedometer test
Understanding the recompression behavior of clay, quantified by the recompression index Cr obtained from consolidation tests, helps reliably estimate the ultimate settlement of clay layers under increased vertical effective stress (σv′) The recompression index Cr describes how clay recompresses under small load increments after preconsolidation, enabling engineers to model long-term settlement under higher stresses Accurate Cr determination from properly conducted consolidation tests improves the reliability of settlement predictions for foundations, embankments, and other structures resting on clay soils.
e1r and e2r are the void ratios measured at two distinct points on the curve, corresponding to vertical effective stresses σ'v2r and σ'v1r, respectively The stresses σ'v1r and σ'v2r lie within the recompression range and satisfy σ'v1r < σ'v2r < σ'p.
Preconsolidation stress (σp′), as defined by Casagrande (1936), represents the maximum vertical effective stress that a soil element has experienced throughout its stress history, essentially indicating the greatest past overburden pressure under which the soil was consolidated This parameter can be determined through in-situ investigations like the Cone Penetration Test with pore pressure measurement (CPTu) dissipation test, as well as laboratory procedures such as the oedometer test.
Although many techniques exist for evaluating preconsolidation stress, this study focuses on two approaches derived from the one-dimensional consolidation test: Casagrande's 1936 method and Silva's 1970 method By comparing these methods, the work aims to assess their effectiveness in estimating preconsolidation pressure, clarify the interpretation of consolidation data, and discuss the implications for predicting soil behavior under future loading.
Casagrande's method, developed in 1936, uses a graphical procedure on the e-log p′ curve to determine the pre-consolidation pressure The process begins by locating on the curve the point a where the radius of curvature is minimum From point a, a horizontal line ab is drawn, followed by a tangent line ac at the same point Next, line ad is constructed as the angle bisector of the angle formed by ab and ac The straight-line portion of the e-log p′ curve is then projected backward to intersect line ad at point f The abscissa of point f represents the pre-consolidation pressure, i.e., the maximum past effective stress the soil has experienced without undergoing plastic deformation.
One key limitation of Casagrande's method is the difficulty in accurately identifying the maximum curvature point on the e-log p' curve This challenge arises from changes in void ratio that occur under applied pressure both before and after the preconsolidation pressure, and is amplified when the soil specimen has been disturbed during drilling and sampling procedures Such disturbances alter the soil's structure and behavior, making it harder to determine the true preconsolidation stress.
Figure 2.9 Casagrande (1936)’s method (Dung and Giao, 2005)
Silva's method for determining preconsolidation pressure uses a stepwise construction on the soil's e–log(σv′) plot Start by drawing a horizontal line from the initial void ratio e0 Next, draw a tangent to the virgin compression portion (the normally consolidated range) of the consolidation curve and extend it to intersect the horizontal line from e0; this intersection is labeled point a From a, drop a vertical line downward to meet the test curve at point b Then draw a horizontal line from b until it intersects the virgin compression line at point c The effective vertical stress σv′ corresponding to the abscissa of point c is the preconsolidation pressure σp′.
Figure 2.10 The method proposed by Silva (1970) (Dung and Giao, 2005)
The overconsolidation ratio (OCR) is determined by the ratio of the preconsolidation stress (σp′) to the present vertical effective stress (σv0′) It can be expressed mathematically as:
Kulhawy and Mayne (1990), as cited in Mayne's 2007 work, established a correlation between data obtained from the Cone Penetration Test with pore pressure measurement (CPTu) and the overconsolidation ratio (OCR)
Within geotechnical analysis, the Coefficient of Consolidation Ratio (COCR) typically falls in the 0.2–0.3 range In this framework, qt denotes the cone stress adjusted for pore water pressure at the shoulder and acts as a total-stress parameter, while σv represents the total stress due to the weight of the overlying soil and σv′ the effective stress carried by the soil skeleton For undisturbed clay deposits, the overconsolidation ratio (OCR) can be estimated using a correlation that links preconsolidation pressure to the net cone tip resistance, as proposed by Mayne (1995) and Demers & Leroueil (2002, with Mayne (2007) further refining the relationship).
(2.28) Table 2.2 Soil terminology applied to stress history, FHWA-NHI-16-072, 2017
According to FHWA-NHI-16-072 (2017), soil can be classified based on its overconsolidation ratio (OCR) An OCR of 1 indicates normally consolidated soil Lightly overconsolidated soil has an OCR between 1 and 2.
Literature review of the studies on the soil characteristics of RRD
Nguyen and Khin (2023) investigated the compression behavior of soft clay deposits in the Red River Delta, focusing on determining the coefficient of consolidation (Cv), a key parameter for ground-improvement techniques employing vertical drains (PVDs) Their findings indicate that the non-graphical method delivers the most reliable CV measurements for assessing consolidation in these soils.
Soft clays in the Red River Delta are typically low-plasticity and normally consolidated, a characteristic that governs their compressibility behavior The study establishes quantitative correlations between key soil properties, notably the Compression Index (Cc) and the Recompression Index (Cr), with Cc = 0.3864 and Cr = 0.1208 × Cc (Cr ≈ 0.0467) These relationships enable efficient characterization of the soils’ compressibility and inform design and construction practices in similar geotechnical contexts.
2.4.1 Literature review of the study on HS Model parameter calibration
Table 2.3 Studies on previous Hardening-Soil model parameter calibration
1 The hardening soil model: Formulation and verification
The HS model incorporates stress- dependent stiffness, a multi-surface yield function, and isotropic hardening dependent on both shear and volumetric plastic strain, respectively
Oedometer tests and consolidated drained triaxial tests were used for model calibration
The HS Model stimulates complex soil behavior and can be applied the various geotechnical problems such as slope stability, excavation planning, and foundation designs
2.Stiffness and strength parameters for the hardening soil model of soft and stiff Bangkok clays(Surarak et al.,
This research analyzes experimental data on Bangkok subsoils to determine the parameters needed for the HS model in PLAXIS finite element software
The study focuses on soft and stiff clay layers at various depths
Oedometer tests and consolidated undrained triaxial tests were used for model calibration.
This study provides valuable data for geotechnical engineers working on Bangkok-based projects, with parameters that can be integrated into geotechnical practice to bolster the reliability of numerical simulations in Bangkok.
3 Parameter Analysis on the Hardening Soil
Model of Soft Soil for
Foundation Pits Based on Shear Rates in
This study identifies suitable parameters for the Hardening Soil (HS) model to analyze soft soils in Shenzhen Bay, China, and develops a robust parameter set for PLAXIS-based numerical simulations of foundation pit excavations.
Oedometer tests and triaxial tests were used for model calibration
The findings of this research can be applied to numerical analysis of foundation pits with similar geological conditions, improving the accuracy and reliability of the simulation
4 Numerical simulation with hardening soil
The study analyzes experimental data on
The calibrated parameters of the HS
Clay obtained from conventional tests
Saleh et al (2021) used marine clay to identify the parameters required for calibrating the HS model in PLAXIS, a finite element software package Drained triaxial tests were conducted to calibrate the model, and the calibrated HS model was able to accurately simulate the mechanical behavior of marine clay.
2.4.2 Conclusions from the literature review
The Hardening Soil (HS) model is a robust, accurate tool in geotechnical engineering for simulating complex soil behavior Laboratory testing is essential for determining HS model parameters, with calibration against lab data—primarily triaxial compression and one-dimensional consolidation (oedometer) tests—being central in the literature This calibration enables the HS model to more accurately capture the stress–strain response of soils under shear deformation, boosting the reliability of geotechnical simulations.
The Red River Delta is dominated by fine-grained sediments, resulting in soils with low permeability, low strength, and high compressibility that are vulnerable to slope instability A literature review reveals no systematic experimental investigation of Hardening Soil model parameters and their correlation with the specific clay formations found in the Red River Delta, Vietnam This gap underscores the need for an experimental study on the hardening soil model parameters of clays in the Red River Delta, aimed at establishing clear correlations between clay characteristics and soil behavior to improve slope stability analysis and geotechnical design in this region.
METHODOLOGY
Gathering Geological Insights of the Red River Delta (RRD)
Geotechnical engineering problems are inherently tied to the geological conditions of the proposed project area, making the collection and review of geological data essential for this study To characterize the Red River Delta (RRD), information was compiled from a range of sources, including academic papers, peer‑reviewed journals, and credible online references, to provide a comprehensive overview of its geology and geotechnical properties.
This study investigates the distribution and formation of soft clay layers to understand the characteristics of soft clays By integrating observed geological data with geotechnical data from available field tests and laboratory analyses, the data are analyzed to identify the formation processes and key characteristics of these soft clay deposits.
Figure 3.1 Flow chart showing the analyzed steps
Sampling
Undisturbed soil samples were collected for this study from Nam Dinh (BMIZ), Hai Duong (LCIZ), and Hanoi (ML) to capture regional variability The sampling followed ASTM standards, which specify procedures for obtaining samples suitable for different laboratory tests The chosen method depends on soil type and the planned tests, ensuring that the samples are representative and fit for purpose.
In cohesive soil layers containing stiff clays encountered at roughly 2-meter intervals, undisturbed soil samples are obtained by inserting a thin-walled metal tube sampler into the soil formation using a hydraulically operated piston, in accordance with ASTM D6519 For soft or sensitive soils, such as soft organic clays, thin-walled tube samplers are used in line with ASTM D1587 Standard Penetration Tests (SPT) employing a 51 mm diameter sampler may be carried out additionally to obtain supplementary soil property data.
After retrieval, the ends of the cores are trimmed to remove disturbed soil, and the samples are cut to meet the minimum dimensions required for the intended tests If needed, extrusion from the sampling tube is performed to minimize disturbance Following trimming or extrusion, the samples are cut into 100 mm lengths, sealed with wax or paraffin to prevent moisture loss, and stored in a humidity-controlled room until testing Adhering to ASTM standards throughout these steps ensures proper collection and handling of soil samples for reliable laboratory analysis, with the resulting specimens used for physical property tests, triaxial tests, and oedometer tests.
Laboratory tests
A comprehensive laboratory study on soft soils from three Red River Delta research sites was conducted to determine the Hardening Soil Model parameters using undisturbed soil samples The experimental program comprised physical properties tests, consolidated drained triaxial compression tests, and oedometer tests, all performed in a controlled-temperature environment to minimize thermal effects A standards table documents the procedures used to determine the soils' physical and mechanical properties.
Table 3.1 List of physical and mechanical properties tests
No Test Name ASTM Standard Properties
7 Secant stiffness reference value (E50 ref)
3 One-dimensional consolidation test by using the incremental loading method
4 Swelling or Rebound Index (Cs)
6 Reference Tangent stiffness (Eoed ref)
The moisture content of soil represents the weight of water divided by the weight of dry soil particles within a specified volume w s w W
Where Ww = water weight and Ws = soil sample weight
The unit weight (γ) of soil is calculated as the total weight (W) divided by the total volume (V) w
Where w = total soil weight, V =volume, wₛ = dry solids weight, γd= dry unit weight, and w = moisture content
(3) Specific gravity of soil (Gs)
The specific gravity test measures the density and specific gravity of soil samples
Begin by weighing 100 g of soil that has been pre-sieved to 0.425 mm or smaller and preparing empty pycnometers for testing Place 10 g of soil into each pycnometer and record the combined weight Add distilled water to each pycnometer until the liquid reaches one-third of the height, then shake thoroughly to mix To eliminate air bubbles, boil the pycnometers for 30 minutes, cool them to room temperature, and note the stabilized temperature After cooling, add distilled water to each pycnometer until the liquid level reaches the neck's shoulder, then determine the weight.
Equation 3.5 determines the soil particle density (ρs) from a set of pycnometer measurements Here, m1 is the mass of the empty pycnometer, m2 is the mass of the pycnometer with dry soil, m3 is the mass of the pycnometer with the soil–water mixture, and m4 is the mass of the pycnometer with water The density of distilled water at the testing temperature is ρw = 0.997 g/cm^3, and ρs denotes the particle density of the soil specimen By combining these masses with ρw in Equation 3.5, the soil particle density ρs is obtained.
Gs = Specific gravity of the soil specimen
Finally, perform calibration steps to determine the water density at the recorded temperature This calibration provides an accurate measurement of the soil sample's specific gravity and density, metrics that are essential for a wide range of geotechnical engineering applications.
Particle size distribution testing identifies soil fractions by size to determine the appropriate soil classification, using both sieve analysis and the hydrometer test The main equipment includes a balance, cylinder, hydrometer, thermometer, and sieve Approximately 200 g of wet test soil is dried in an oven for at least 24 hours After drying, about 30 g of soil is ground to powder and washed through a No 200 sieve in a container, with the soil remaining on the sieve dried in an oven for 24 hours to be used in sieve analysis The soil passing through this sieve is placed into a cylinder and mixed with a small amount of sodium hexametaphosphate as a dispersant For the hydrometer reading, the cylinder containing the soil and dispersant solution must be thoroughly shaken, a stopwatch started at time zero immediately after shaking, and any air bubbles on the solution surface promptly removed.
The triaxial test measures soil shear strength under controlled drainage conditions A cylindrical soil specimen, enclosed in a membrane, is positioned within a triaxial cell and subjected to confining fluid pressure The sample is then axially loaded until it fails Drainage is controlled during the test Typically, at least three samples are tested under varying confining pressures to establish the failure envelope describing shear strength as a function of normal stress The triaxial apparatus enables true three-dimensional stress control (σ₁ ≠ σ₂ = σ₃) through simultaneous cell pressure (σ₃) and axial load (σ₁) application, with σ₂ assumed equal to σ₃ in cylindrical samples
During saturation, all three principal stresses are maintained equal to the applied back pressure to ensure full specimen saturation At the shear stage, the major principal stress (σ₁) equals the sum of applied axial stress and confining chamber pressure The deviator stress (q) is defined as the difference between major and minor principal stresses.The intermediate and minor principal stresses (σ2 and σ3, respectively) are equal and match the confining (chamber) pressure
Figure 3.2 Details of a typical triaxial cell (Head, 1986)
Figure 3.3 Principles of triaxial compression tests: (a) application of stresses, (b) representation of principal stresses, (c) usual arrangement for effective stress tests, and (d) representation of total and effective stresses (Head, 1986)
In saturated soils, voids are filled with water, while partially saturated soils contain both water and air The shear strength is resisted exclusively by the soil skeleton (interparticle forces), whereas the normal stress is carried by the combination of particle contacts (the effective stress, σ') and pore water pressure (u) In a triaxial test, shear strength is typically determined from total stress, but effective-stress analysis is possible when complete drainage is allowed (making pore pressure zero at failure) or when pore pressure is measured during the test By recording pore pressures in triaxial testing, total stresses can be converted to effective stresses, revealing the true frictional resistance between particles.
(1) Consolidated drained triaxial compression test
In a consolidated-drained (CD) triaxial test, also known as a consolidated-slow test, the procedure begins by opening the drainage valve and applying confining cell pressure to the soil specimen The specimen is allowed to fully consolidate under this pressure, with careful monitoring throughout the process After consolidation, the deviator stress is gradually increased while maintaining open drainage.
A series of consolidated drained triaxial compression tests will be performed to examine the stress-strain behavior of the soil samples The strength parameters c',ϕ', and E50,
Secant stiffness values (32) are derived from drained-condition triaxial tests Before a soil specimen is trimmed and mounted in the triaxial testing device, the base of the cell is connected to the necessary pressure supply lines and other experimental arrangements After retrieval from a controlled humidity environment, samples are meticulously trimmed to the desired dimensions after carefully removing any paraffin and wax coatings.
Cylindrical specimens for testing must adhere to defined dimensions and composition The minimum diameter is 33 millimeters (1.3 inches), and the specimen height must be between twice and 2.5 times the average diameter Individual height and diameter measurements must stay within 2% of the average to ensure dimensional consistency In addition, all particles within the specimen must be smaller than one-sixth of its diameter.
Two porous stones with high air-entry values are positioned at the top and bottom of the specimen, along with Whatman No 40 filter paper side drains for circumferential drainage as specified by the standard The prepared specimen is enclosed in a double-membraned assembly with a silicone grease layer between the membranes to provide an effective seal High-strength Gabo O-rings are used to form the top and bottom seals against the cap and pedestal, respectively, in accordance with ASTM D7181-20 specifications.
Figure 3.4 The triaxial specimen is fitted with filter paper side drains in two configurations: (a) vertical drains; (b) spiral drains (Head, 1986)
Figure 3.5 Step-by-step procedure for installing the rubber membrane onto the cylindrical soil specimen (Bardet, J P., 1997)
After assembling the triaxial chamber, two key steps ensure proper test setup: first, gently contact the axial load piston with the specimen cap using multiple light touches to align the piston with the cap, being very careful to avoid applying any significant force (less than 0.5% of the expected failure load) to the specimen and monitoring the deformation indicator during these adjustments; second, fill the chamber with the chamber liquid, using de-aired water to enable pressure transmission around the specimen The triaxial testing procedure then proceeds through three phases: saturation, consolidation, and shearing.
Figure 3.6 Key phases in effective stress triaxial testing: (a) saturation, (b) consolidation, (c) undrained compression, and (d) drained compression (Head, 1986)
Geotechnical testing often requires saturating soil samples with water to accurately measure their behavior The saturation process uses a vacuum to fill the voids with water and remove air pockets that could skew the results Even after saturation, some air remains trapped between the specimen and the rubber membranes, and potentially within the saturated porous stones and pre-saturated filter paper Saturation was successfully achieved through the application of vacuum.
Back pressure was set to 300 kPa while the confining pressure remained constant, maintaining a 25 kPa differential between confining and back pressures during the test Both confining pressure and back pressure were increased progressively in 25 kPa increments When the back pressure reached 300 kPa, the specimens were allowed to sit for at least 24 hours.
Numerical Analysis
To accurately represent soil response to ground movement, the Hardening Soil model will be used, with the study focusing on directly determining clay soil parameters through a comprehensive laboratory testing program This program comprises triaxial compression tests, with consolidated drained tests considered depending on the specific clay behavior, and oedometer tests (one-dimensional consolidation) to characterize compression and stiffness The resulting data will calibrate and validate the Hardening Soil model, ensuring reliable predictions of soil deformation and stress changes during deep excavation.
3.4.2 Parameters of Hardening-Soil Model
The Hardening Soil model uses the same failure criteria as the Mohr-Coulomb model—effective cohesion c′, effective friction angle ϕ′, and dilatancy angle Ψ—but it differs in its plastic hardening behavior While the Mohr-Coulomb model assumes a constant Young’s modulus E, real soils exhibit stiffness that increases with confining pressure σ₃′, so stress levels must be evaluated to select appropriate stiffness parameters The Hardening Soil model avoids this parameter-selection complexity by defining stiffness relative to a reference stress with σ₃′ = pref.
(a) Stiffness for Primary Loading, E50 ref
During primary loading, the stress–strain response exhibits marked nonlinearity The Hardening Soil model replaces the experimentally challenging initial tangent modulus Ei with E50, a stress-dependent secant stiffness that better captures this nonlinearity The stiffness parameter E50 corresponds to the secant modulus at half the failure stress (qf/2).
Hardening Soil model defines the reference stiffness modulus E_ref 50 at a reference confining pressure p_ref, with PLAXIS defaulting p_ref to 100 kPa The actual stiffness is not fixed; it varies as a function of the minor principal stress σ'3, equivalent to the confining pressure experienced in a triaxial test.
(b) Reference Tangent Stiffness for Primary Oedometer Loading (Eoed ref)
The oedometer modulus (Eoed) is calculated from one-dimensional compression test results using:
'cot ' ref m oed oed ref
Here, Eoed is the tangent stiffness modulus for primary oedometer loading, while the reference stiffness Eoed,ref represents its value at the reference vertical stress σ'v = p_ref In addition, section (c) discusses the secant stiffness for unloading/reloading paths, which characterizes the material response along those loading trajectories.
The unloading-reloading stiffness modulus is determined using the following equation
'cot ' ref m ur ur ref
Eurref is the reference elastic modulus for unloading-reloading cycles, normalized at the standard stress p ref (typically 100 kPa)
ANALYSIS RESULTS
Introduction
This laboratory study analyzed soft and medium-stiff clay samples collected from the Nam Dinh (BMIZ), Hai Duong (LCIZ), and Hanoi (ML) regions It presents the basic soil properties and classification test results Laboratory tests performed at the test sites included: (1) water content of soil, (2) unit weight of soil, (3) specific gravity of soil, (4) particle size distribution of soil, (5) consolidation with vertical drainage (VD) on intact samples, and (6) consolidated drained triaxial test on intact samples It also details the oedometer and consolidated drained triaxial tests used to determine hardening-soil model parameters, and discusses possible correlations between the hardening soil model parameters and basic soil physical parameters for practical applications Figure 4.1 displays the locations of all study sites.
Figure 4.1 Map of the study sites
Physical properties profiles
Physical property profiles for each soil layer (L1, L2, L3, etc.) across the three study sites are presented in the accompanying figures Using borehole records and laboratory tests, the soil layers at these sites were classified, providing a structured understanding of subsurface materials.
(a) BMIZ site (b) LCIZ site (c) ML site
Figure 4.2 Soil layer profiles ((a) BMIZ site, (b)LCIZ site, and (c) MI site)
At the BM site, the depths 0–1.8 m (L1) and 1.8–6.9 m (L2) are clayey silt, while 6.9–23.4 m (L3) and 23.4–25 m (L4) are silty clay layers The in-situ moisture content of the soft clay specimens at BM ranged between 22.75% and 31.44%, while the total unit weight for silty clay (L3) was determined to be 17.75 to 19.39 kN/m³.
Figure 4.3 Physical property profiles (BMIZ site)
At the LC site, the soil profile comprises five depth intervals—0–1.0 m (L1), 1.0–2.4 m (L2), 2.4–5.8 m (L3), 5.8–15.5 m (L4), and 15.5–20.0 m (L5)—characterized as clayey-silty and silty-clay layers The measured water content of the undisturbed soft clay samples ranges from 26% to 43%, while the total unit weight spans 16.5 to 19.2 kN/m³.
Figure 4.4 Physical property profiles (LCIZ site)
Figure 4.5 Physical property profiles (ML site)
At the ML site, the soil profile from the surface to 25.0 m comprises five depth intervals: 0–2.8 m (L1), 2.8–6.0 m (L2), 6.0–14.7 m (L3), 14.7–20.6 m (L4), and 20.6–25.0 m (L5), characterized as clay-sandy, mud-sandy clay, and fine sand layers, respectively The soft clay samples (ML) exhibit a water content (wn) ranging from 35.1% to 58.6%, and a total unit weight between 16.0 and 18.1 kN/m³.
Table 4.1 Summary of Physical Properties for Soil Samples Across Test Sites
Unit weight of soil,w (kN/m 3 )
Evaluation of the Compressibility characteristics of soil samples
Conventional oedometer tests with incremental loading were conducted to characterize the compressive stress–deformation behavior of soft clay under constrained lateral strain conditions The accompanying table lists the number of samples used to estimate key consolidation and compressibility parameters, including pre-consolidation pressure (σp'), compression index (Cc), recompression index (Cr), swelling index (Cs), and the over-consolidation ratio (OCR).
Table 4.2 Undisturbed Sample Counts from Field Investigation Sites
No Study sites Available Undisturbed Samples
Based on the available samples from the three study sites (as outlined in Table 4.2), the parameters Cr, Cc, CS, σ'p, and OCR were determined The outcomes of the one-dimensional compression tests are graphically represented in Figure 4.6, where the void ratio (e) is plotted as a function of the effective consolidation pressure (σ'v) To evaluate the compressibility parameters from the laboratory test data, Silva's (1970) method was employed in this study.
(a) Void ratio vs Axial stress (BM U3) (b)Void ratio vs Axial stress (BM U4)
(c) Void ratio vs Axial stress (BM U5) (d)Void ratio vs Axial stress (BM U6)
(e) Void ratio vs Axial stress (BM U7) (f)Void ratio vs Axial stress (BM U8)
(g) Void ratio vs Axial stress (LC U1) (h)Void ratio vs Axial stress (LC U2)
(i) Void ratio vs Axial stress (LC U3) (j)Void ratio vs Axial stress (LC U4)
(k) Void ratio vs Axial stress (LC U5) (l)Void ratio vs Axial stress (ML U3)
(m) Void ratio vs Axial stress (ML U4)
Figure 4.6 Pressure-Void Ratio Relationships from Oedometer Tests: BMIZ, LCIZ, and
Figure 4.6 shows the representative void ratio-stress relationships (log scale) for clay soils, with plots of void ratio e versus the effective consolidation pressure σ'v derived from oedometer test results and supported by data in Table 4.3 Figure 4.7 then illustrates the distribution of Cr, Cs, and Cc values for clayey soil types across the three study sites.
Table 4.3 Oedometer Test Results Summary
Figure 4.7 Results Cc, Cr, and Cs from three study sites (BMIZ, LCIZ, and ML)
Test results show that the maximum pre-consolidation pressure for each sample was established using Silva's method and Becker’s method The over-consolidation ratio (OCR) values for the silty clay, measured at depths up to 20.0 meters, range from 0.415 to 1.103 Data from the LC site indicate that the OCR for the clay layers varies within the upper 13.6 meters from 0.654 to 0.955 The OCR value is smaller than one, as determined by the Silva method.
OCR values were determined using the Becker method For depths up to 20.0 m, the clay OCR values range from 1.214 to 1.875 LC site data indicate that the over-consolidation ratio (OCR) of the clay layers within the upper 13.6 m ranges from 1.002 to 1.649 The determined values are presented in Table 4.4.
Table 4.4 Summary of the Preconsolidation pressure and Overconsolidation ratios
Depth(m) Silva’s method Becker’s method σꞌv (kPa) σꞌp (kPa)
Figure 4.8 Results of σP' and OCR from Oedometer test (By Silva’s method)
Figure 4.9 Results of σP' and OCR from Oedometer test (By Becker’s method) σ' vo (kPa)
LCIZ BMIZ ML σ' vo (kPa)
Isotropic Consolidated-Drained Triaxial Shear Characteristics
Soft Clay Specimens under Drained Conditions was analyzed by performing the Consolidated Isotropic Drained (CID) compression triaxial tests Figure 4.10 presents the detailed information regarding sample selection for the tests
Figure 4.10 Analysis Flowchart of Consolidated Drained Triaxial Tests
4.4.1 Evaluation of the strength parameters from the CD triaxial test
To determine soil shear strength, Consolidated Drained (CD) triaxial tests are used to characterize effective cohesion (c′), effective internal friction angle (ϕ′), and the dilatancy angle (Ψ) The test program typically includes at least two or three saturated cylindrical soil specimens subjected to different effective confining pressures (σ3′) Each specimen is slowly sheared under drained conditions until failure, with axial stress, axial strain, and volumetric strain recorded continuously to capture the complete material response.
The stress-strain response is nonlinear from the start and shows no peak, with deformation extending to large strains and eventually leveling off to a near-constant stress beyond 15% axial strain Failure stress is defined as the axial stress at 15% axial strain At failure, the major principal stress σ1′ is obtained from the deviator stress qf = (σ1′ − σ3′), where qf is the deviator stress at failure For each specimen, Mohr circles are plotted from the measured σ1′ and σ3′ values to represent the stress state The Mohr-Coulomb failure envelope is τf = c′ + σn′ tan(ϕ′).
An envelope formed by a line tangent to these circles defines the strength parameters: the intercept with the shear-stress axis yields the effective cohesion c′, and the angle of inclination with the normal-stress axis yields the angle of shearing resistance under effective stresses φ′ The deviator-stress–shear-strain relationships obtained from drained tests on all specimens are presented in Figures 4.10–4.12.
The deviator stress versus axial strain response of silty clay and clayey silt was obtained under confining pressures of 100, 200, and 400 kPa (Figures 4.10 and 4.11), while mud and sandy clay were tested at 50, 100, and 200 kPa (Figure 4.12) The silty clay reached peak deviator stresses of 904.2 kPa at an 18% shear strain and 1,776.5 kPa at 20% strain For clayey silt, peak deviator stresses were 350 kPa at 16% shear strain and 900 kPa at 20% strain The mud and sandy clay specimens produced peak deviator stresses of 150 kPa at 14% shear strain and 600 kPa at 18% strain.
D ev ia to r St re ss ,(σ 1 -σ 3 )( kP a)
Realationship between Deviator Stress and Strain
D ev ia to r S tr es s (σ 1 - σ 3 )( k P a)
Relationship between Deviator Stress and Strain
D ev ia to r S tr es s ( σ 1 -σ 3 ) (k P a )
Relationship between Deviator Stress and Strain
D ev ia to r S tr es s ,( σ 1 -σ 3 ) ( K P a)
Relationship between Deviator Stress and Strain
Figure 4.11 The deviator stress-strain relationships obtained from CD tests (BMIZ site)
D ev ia to r S tr es s, ( σ 1 -σ 3 ) ( k P a)
Relationship between Deviator Stress and Strain
D ev ia to r S tr es s (σ 1 -σ 3 ) (K P a)
Relationship between Deviator Stress nad Strain
D ev ia to r S tr es s, (σ 1 -σ 3 ) (K P a )
Relationship between Deviator Stress and Strain
D ev ia to r St re ss ,( σ 1 -σ 3 ) (k P a)
Relationship between Deviator Stress and Strain
D ev ia to r S tr es s, ( σ 1 -σ 3 ) ( k P a)
Relationship between Deviator Stress and Strain
Figure 4.12 The deviator stress-strain relationships obtained from the CD tests (LCIZ site)
D ev ia to r S tr es s (σ 1 - σ 3 ) (k P a )
Relationship between Deviator Stress and Strain
D ev ia to r S tr es s, ( σ 1 -σ 3 ) ( k P a )
Relationship between Deviator Stress and Strain
D ev ia to r S tr es s, ( σ 1 -σ 3 ) ( k P a)
Relationship between Deviator Stress and Strain
D ev ia to r S tr es s, ( σ 1 -σ 3 ) ( k P a)
Relationship between Deviator Stress and Strain
D ev ia to r S tr es s, ( σ 1 -σ 3 ) ( kP a)
Relationship between Deviator Stress and Strain
D ev ia to r St re ss ,( σ 1 -σ 3 ) ( k P a)
Relationship between Deviator Stress and Strain
D ev ia to r S tr es s, (σ 1 - σ 3 )( kP a)
Realationship between Deviator Stress and Strain
D ev ia to r St re ss ,( σ 1 - σ 3 )( kP a)
Realationship between Deviator Stress and Strain
D ev ia to r St re ss ,( σ 1 - σ 3 )( kP a)
Realationship between Deviator Stress and Strain
D ev ia to r S tr es s, (σ 1 - σ 3 )( k P a)
Realationship between Deviator Stress and Strain
D ev ia to r S tr es s, (σ 1 - σ 3 )( k P a)
Realationship between Deviator Stress and Strain
D ev ia to r St re ss ,( σ 1 - σ 3 )( kP a)
Realationship between Deviator Stress and Strain
Figure 4.13 The deviator stress-strain relationships obtained from CD tests (ML site)
Effective drained shear strength parameters, particularly the friction angle (ϕ') and cohesion (c'), underpin reliable geotechnical design Using the Mohr circle method to determine these drained strength parameters, Table 4.4 summarizes the drained strength characteristics for soil samples collected from three test sites The table presents the drained strength properties of three materials—soft to medium-stiff silty clay, medium-stiff clayey silt, and soft muddy sandy clay.
Figure 4.14 Typical figure for the determination of the strength parameters by using Mohr’s circle at the three test sites
D ev ia to r S tr es s, (σ 1 - σ 3 )( k P a)
Realationship between Deviator Stress and Strain
Table 4.5 Summary of the test results from the consolidated drained triaxial tests (BMIZ sites)
Table 4.4 shows that the cohesion (c') of silty clay, from soft to medium stiff, spans 19.00 to 110.00 kPa Under consolidated drained (CD) conditions, the friction angle values for the silty clay layer range from 20.21° to 36.97°.
Table 4.6 Summary of the test results from the consolidated drained triaxial tests (LCIZ site)
Table 4.5 shows that the cohesion c' values for medium-stiff clayey silt range from 16.50 to 78.50 kPa, whereas mud and sandy clay exhibit lower cohesion, spanning 5.00 to 49.00 kPa In this study, the friction angle for clayey silt is between 12.23° and 29.88°, summarizing the geotechnical properties of these soils.
Table 4.7 Summary of the test results from the consolidated drained triaxial tests (ML site)
According to Table 4.6, the cohesion values for mud and sandy clay range from 5.00 to
49.00 kPa Under consolidated drained (CD) conditions, in the case of mud and sandy clay, the friction angle ranges from 12.84° to 41.26°
4.4.2 Evaluation of the stiffness parameters
By analyzing the resulting stress-strain curves, can be determine the deformability parameters, including the constrained modulus (Eoed)
Figure 4.15 Determination of the tangent stiffness from Oedometer tests (BMIZ site)
V er ti ca l S tr es s (k P a)
Stress- strain curve from oedometer test
V er ti ca l S tr es s (k P a)
Stress- strain curve from oedometer test
V er ti ca l S tr es s (k P a)
Stress- strain curve from oedometer test
V er tic al S tr es s (k P a)
Stress- strain curve from oedometer test
V er ti ca l S tr es s (k P a)
Stress- strain curve from oedometer test
V er ti ca l S tr es s (k P a )
Stress- strain curve from oedometer test
Figure 4.16 Determination of the tangent stiffness from Oedometer tests (LCIZ site)
V er ti ca l S tr es s (k P a)
Stress- strain curve from oedometer test
V er ti ca l S tr es s (k P a )
Stress- strain curve from oedometer test
V er ti ca l S tr es s (k P a)
Stress- strain curve from oedometer test
V er ti ca l S tr es s (k P a)
Stress- strain curve from oedometer test
Figure 4.17 Determination of the tangent stiffness from Oedometer tests (ML site)
Table 4.8 Summary of the oedometer stiffness parameter with 100 kPa as the reference stress level
Depth(m) p ref (kPa) Vertical strain, Ɛ Eoed ref(kPa)
V er ti ca l S tr es s (k P a)
Stress- strain curve from oedometer test
V er ti ca l S tr es s (k P a)
Stress- strain curve from oedometer test
The Consolidated Drained (CID) triaxial test enables comprehensive soil characterization by providing not only strength parameters but also key stiffness metrics—namely the elastic modulus (E) and Poisson's ratio (ν)—which govern soil deformation under applied stresses Under CID conditions, the elastic modulus varies with confining pressure, and this relationship is illustrated for the tested soil samples in Figure 4.18.
Figure 4.18 A typical plot of the determination of the stiffness stress dependency parameter (m)
Tables 4.9, 4.10, and 4.11 present the secant modulus determined at half of the peak deviatoric stress with the parameter measured at a reference pressure (100 kPa)
Table 4.9 Summary of the secant stiffness parameter at a reference pressure of 100 kPa (BMIZ site)
Table 4.10 Summary of the secant stiffness parameter at a reference pressure of 100 kPa (LCIZ site)
Table 4.11 Summary of the secant stiffness parameter at a reference pressure of 100 kPa (ML site)
Correlations between HSM parameters and basic soil parameters
Based on samples from three study sites listed in Table 4.1, this study conducted comprehensive laboratory investigations to determine essential geotechnical parameters, including the in-situ vertical effective stress (σvo′), initial void ratio (e₀), and natural moisture content (wₙ), as well as the soil compressibility parameters, notably the compression index (Cc).
, swelling index Cs, pre consolidation stress σp′, and over-consolidation ratio OCR) The results for each parameter are presented in Tables 4.1, 4.3, and 4.4
CD triaxial test results, including the reference secant modulus at 50% failure stress (E50,ref) and the reference oedometer tangent modulus (Eoed,ref), are tabulated in Tables 4.8 and 4.9 To assess the link between hardening soil model parameters and basic soil parameters, a linear regression line is plotted and the SSD, R², and MAPE values are used to evaluate the strength of this relationship From the E50,ref results shown in Table 4.9, the reliability correlation can be determined as illustrated by Figures 4.19, 4.20, and 4.21 The correlation results for E50,ref are summarized in Table 4.12.
(1) Correlation between E50 ref and Depth, z
(a) E50 ref vs Depth, z (b) Predicted E50 ref vs Measured E50 ref
Figure 4.19 Correlation result between E50 ref and Depth, z
This study establishes a key relationship between the reference secant modulus (E50 ref) and sampling depth (z) for clay soils across three test sites, as described by the equation
E50 ref = 2480.537 + 9572.527exp(−0.333z) The analysis reveals an exponential decay in soil stiffness (E50 ref) with increasing depth, characterized by maximum stiffness values occurring at shallow depths.
(a) E50 ref vs σ' vo (b) Predicted E50 ref vs Measured E50 ref
Figure 4.20 Correlation result between E50 ref and σ' vo
Relationship between E 50 ref and Depth,z (m)
Depth,z (m) vs E50ref (kPa) y = 2480.537+9572.527 Exp(-0.333x)
Measured E 50 ref (kPa) σ'vo (kPa)
Regression Line σ'vo vs E50 ref
This study reveals a significant linear relationship between the reference secant modulus (E50 ref) and vertical effective stress (σ'vo), described by E50 ref = 4670.057 − 18.436σ'vo This inverse relationship indicates that stiffness decreases with increasing vertical effective stress across the three study sites.
(3) Correlation between E50 ref and (σ' v0 , OCR)
(a) E50 ref vs σ' vo (b) E50 ref vs OCR
(c) Predicted E50 ref vs Measured E50 ref Figure 4.21 Correlation result between E50 ref and (σ'vo, OCR)
This equation links the reference secant modulus to vertical effective stress and OCR through an inverse power-law relationship The negative exponents (−0.560 for σ'v0,
−0.130 for OCR) indicate that stiffness decreases with increasing stress and overconsolidation This nonlinear behavior highlights the complex stress-dependent response of these clays σ'vo (kPa)
5000 σ'vo vs E50 ref Regression Data
Relationship between OCR and E50 ref
OCR vs E50ref Regression data y = -466.298 x + 3539.378
P re di ct ed E 50 re f (k P a)
Table 4.12 Summary table for correlation between E50 ref and the basic soil parameters
Correlation equation R 2 SSD MAPE No of data points
(1) Correlation between Eoed ref and (σ' v0 , Wn)
(a) Eoed ref vs σ' vo (b) Eoed ef vs Wn
(c) Predicted Eoed ref vs Measured Eoed ref Figure 4.22 Correlation result between Eoed ref and (σ'v0, Wn)
Relationship between σ'vo and Eoed ref σ'vo(kPa)
M ea su re d E oe d re f (k P a)
3000 σ'vo vs Measured Eoed ref
Relationship between Wn and Eoed ref
M ea su re d E oe d r ef ( kP a)
W n vs Measured E oed ref Regression line y = -2364.706+2508.556x R2 =0.2534 y = 0.9671x R² = 0.9671
P re d ic te d E oe d re f (k P a)
The study establishes a correlation between the reference tangent modulus and basic soil parameters, expressed as Eoed ref = 7047.178 (σ'vo) ⁻⁰ã⁴⁵⁶ (Wn)⁻⁰ã⁵⁷² for clays at the test sites
The study reveals a significant correlation between the reference tangent modulus
(Eoed ref) and key soil parameters, showing that Eoed ref decreases with increasing water content (Wn) and effective vertical stress (σ'vo) for clays at the test sites
(2) Correlation between Eoed ref and e0
(a) Eoed ref vs e0 (b) Predicted Eoed ref vs Measured Eoed ref Figure 4.23 Correlation result between Eoed ref and e0
(3) Correlation between Eoed ref and Cc
(a) Eoed ref vs Cc (b) Predicted Eoed ref vs Measured Eoed ref Figure 4.24 Correlation result between Eoed ref and Cc
Relationship between e0 and Eoed ref
M ea su re d E oe d re f (k P a)
Exopential decay curve e0 vs Measured Eoed ref y 45.728+7333.540 Exp (-2.969 x)
P re d ic te d E oe d re f (k P a)
Relationship between Cc and Eoed ref
M ea su re d E oe d re f ( kP a)
C c vs Measured E oed ref y = 984.839 +2667.731 Exp (-5.495 x)
600 900 1200 1500 1800 2100 2400 2700 3000 P re di ct ed E oe d re f (k P a)
The study shows that the reference tangent modulus (Eoed ref) is strongly controlled by the initial void ratio (e0), following the exponential trend shown in Figure 4.23 Eoed ref declines markedly as e0 increases, revealing an inverse relationship between stiffness and soil compressibility and underscoring the critical role of void ratio in governing the mechanical behavior of clays at the test sites The same modulus also demonstrates a strong dependence on the compression index (Cc), with the relationship described by Eoed ref = 984.839 + 2667.712 × exp(-5.495 × Cc) As Cc rises, the modulus declines sharply, illustrating the nonlinear response of the material to increasing compressibility.
(4) Correlation between Eoed ref and (σ' vo , e0)
(a) Eoed ref vs σ' vo (b) Eoed ref vs e0
(c) Predicted Eoed ref vs Measured Eoed ref
Figure 4.25 Correlation result between Eoed ref and (σ'vo, e0)
Relationship between σ'vo and Eoed ref σ'vo(kPa)
M ea su re d E oe d re f (k P a)
3000 σ'vo vs Measured Eoed ref
Relationship between e0 and Eoed ref
M ea su re d E oe d r ef ( k P a)
3000 e 0 vs Measured E oed ref Regression line y = -942.12x +2662.71
P re di ct ed E oe d re f (k P a)
Figure 4.25 demonstrates that soil stiffness decreases as vertical stress increases and as the initial void ratio grows, with the void ratio exerting a stronger influence (exponent -0.567 for void ratio vs -0.408 for vertical stress) This finding highlights the combined yet distinct roles of stress history and initial soil structure in governing clay stiffness, reflecting the material’s intrinsic stress-dependent nonlinear behavior.
(5) Correlation between Eoed ref and (σ' vo , CC)
(a) Eoed ref vs σ' vo (b) Eoed ref vs Cc
(c) Predicted Eoed ref vs Measured Eoed ref
Figure 4.26 Correlation between Eoed ref and (σ'vo, CC)
Figure 4.26 presents an equation that establishes a power-law link between the reference secant modulus, vertical effective stress, and the compression index (Cc) The fitted exponents, −0.233 for σvo′ and −0.461 for Cc, show that soil stiffness decreases as vertical stress increases, with an even more pronounced decline associated with greater compressibility This scaling behavior underscores the dominant role of compressibility in controlling stiffness and has direct implications for geotechnical design and soil-structure interaction analyses.
Relationship between σ'vo and Eoed ref σ'vo(kPa)
M ea su re d E oe d re f (k P a)
3000 σ'vo vs Measured Eoed ref
Relationship between C c and Eoed ref
M ea su re d E oe d re f (k P a)
Cc vs Measured Eoed ref Regression line y = -2332.923x +2355.008
P re di ct ed E oe d re f (k P a)
77 stronger influence of the compression index (Cc) compared to stress (σvo′) underscores that soil compressibility plays a dominant role in governing stiffness reduction
(6) Correlation between Eoed ref and (e0, CC)
(a) Eoed ref vs e0 (b) Eoed ref vs Cc
(c) Predicted Eoed ref vs Measured Eoed ref
Figure 4.27 Correlation result between Eoed ref and (e0, CC)
The correlation equation Eoed_ref = 4.317×e0 − 0.036×Cc − 0.488 shows that soil stiffness is controlled by compressibility (Cc), with a strong inverse relationship (exponent −0.488), while the initial void ratio (e0) exerts only a negligible influence (exponent −0.036) For soft clays, stiffness reduction is therefore driven primarily by soil compressibility rather than porosity, underscoring the critical role of Cc in predicting geotechnical behavior Based on the Eoed_ref results in Table 4.8 and Figures 4.22–4.27, the study presents reliability correlations between Eoed_ref and the fundamental geotechnical properties, confirming the link between compressibility and stiffness across tested conditions.
Relationship between e 0 and Eoed ref
M ea su re d E oe d re f (k P a)
Relationship between C c and Eoed ref
M ea su re d E oe d re f (k P a)
Cc vs Measured Eoed ref Regression line y = -2332.923x +2355.008
P re di ct ed E oe d re f (k P a)
78 soil parameters such as Wn, e0, σ' vo , Cc Table 4.13 provides a summary of these correlation findings for Eoed ref
Table 4.13 Summary table for correlation between Eoed ref and the basic soil parameters
Correlation equation R 2 SSD MAPE No of data points