VIETNAM NATIONAL UNIVERSITY, HANOI VIETNAM JAPAN UNIVERSITY NGUYEN CONG KIEN EXPERIMENTAL AND NUMERICAL STUDIES ON SETTLEMENT AND DISPLACEMENT OF SOFT GROUND IMPROVED BY PVD UNDER THE
INTRODUCTION
Problem statement
Vietnam has enjoyed relatively stable economic growth in recent years, fueling a surge in demand for infrastructure development and modernization across the country This momentum drives investment in transportation networks—roads and bridges—along with ports, urban facilities, and industrial zones to support a growing economy However, many of these critical projects are situated on soft ground, presenting substantial challenges related to soil deformation If soils are not properly treated or reinforced, such conditions can threaten structural stability, cause differential settlement, or even lead to failure due to excessive lateral or vertical displacement, underscoring the importance of ground improvement and robust geotechnical solutions for Vietnam’s infrastructure resilience.
Ground improvement is essential before construction on soft soils to mitigate settlement risks and ensure safe, timely projects A range of soil improvement techniques is available, each with distinct advantages, limitations, and appropriate application contexts Among them, Prefabricated Vertical Drains (PVDs) combined with vacuum preloading and surcharge embankment loading stand out as an especially effective approach This hybrid method accelerates the consolidation of soft clay layers, eliminating prolonged settlements that can last for years and otherwise delay construction timelines and inflate project costs.
The combined use of vacuum pressure and embankment loading, complemented by prefabricated vertical drains (PVDs), accelerates the rate of consolidation, minimizes residual settlement, and enhances bearing capacity, while offering environmental benefits compared with many conventional ground-improvement methods (Figure 1.1).
Since its introduction in the 1980s, the vertical drainage technique has become a widely adopted approach for soft ground improvement in geotechnical engineering projects across East Asia and worldwide, including Vietnam, Japan, and South Korea (Asaoka, 1978) Prefabricated Vertical Drains (PVDs) lie at the core of this method and consist of a plastic core wrapped with geotextile fabric, forming efficient vertical drainage paths for pore water By providing these vertical channels, PVDs accelerate the consolidation of low-permeability soils, improving settlement control and stability (Hansbo, 1981; Jamiolkowski et al., 1983).
Figure 1.1 Illustration of ground improved by PVDs combined with surcharge and vacuum pressure
Prefabricated vertical drains (PVDs) are installed in soft soil layers using mandrel-assisted equipment to create artificial drainage channels through compressible soils These channels hasten the dissipation of excess pore water pressure, increase effective stress, and thereby enhance the ground's mechanical properties The installation is often combined with vacuum preloading and surcharge embankment loading to intensify hydraulic gradients, accelerate consolidation, and ultimately raise the bearing capacity of the treated ground.
Soft-ground improvement projects using prefabricated vertical drains (PVDs) under a combination of surcharge loading and vacuum pressure identify settlement and lateral displacement as the key design parameters Lateral displacement is particularly critical in urban areas, where it must be tightly controlled to protect adjacent structures When vacuum pressure is applied through PVDs, the surrounding soil tends to move toward the vacuum zone, potentially causing surface cracks at the interface between treated and untreated zones (Chai et al., 2005; 2013) Such damage has been observed in real projects, including road crack lines at the Thu Thiem project in Ho Chi Minh City and at the Nhon Trach 3&4 Power Plant project in Dong Nai Province (Figures 1.2 and 1.3).
Figure 1.2 Crack lines due to vacuum pressure at Nhon Trach 3&4 Power Plant
Figure 1.3 Crack lines due to vacuum pressure at Four Road project
Necessity of the study
Many soft ground improvement projects using prefabricated vertical drains (PVDs) under the combination of surcharge loading and vacuum pressure have been implemented in urban or industrial areas with existing services In several cases, unexpected ground movement has caused damage to nearby structures, leading to construction delays and increased costs The ground movement and potential structural damage could be avoided if the influence zone were predictable; however, there are currently no reliable numerical or analytical methods to determine such an influence zone.
Improved Area Not Yet Improved
Although several analytical models have been proposed (Chai et al., 2013; Lu et al., 2019; Xu et al., 2025), they primarily estimate displacement at the embankment toe based on central settlement, which does not fully capture the extent of lateral deformation Field evidence shows that vacuum pressure can induce significant inward movement and cracking outside the treated area (Chu & Yan, 2005).
Ground improvement for soft ground is highly effective for large-scale projects and is expected to be widely adopted worldwide, including Vietnam, making practical and reliable estimation of settlement and lateral displacement essential This study, titled "Experimental and numerical studies on settlement and displacement of soft ground improved by PVD under the combination of surcharge loading and vacuum pressure," develops robust experimental and numerical methods to predict ground responses under PVD-based improvement The research directly addresses practical needs by improving the reliability of design and implementation of PVD-based ground improvement under combined surcharge and vacuum loading.
Objectives of the study
The study has three main objectives as follows
1) To establish advanced numerical models (using PLAXIS 2D) for evaluating settlement, pore water pressure, and lateral displacement of the grounds improved by PVDs under the surcharged loading and vacuum pressure
2) To determine the influence zone of lateral displacement adjacent to the embankment under the combined loading
3) To build analytical equations to predict the lateral displacement as a function of depth extending from the embankment toe.
Scope of the Study
Site 1: Nhon Trach 3&4 Power Plant, Dong Nai Province
Site 2: Road 3, Roud Road Project, Thu Thiem Ward, Ho Chi Minh City
Site 3: Vinh Thanh Industrial Park, VSIP Project, Can Tho Province
The scope of this study covers the following main tasks
Field data were collected from soft ground improvement projects in Vietnam that employed prefabricated vertical drains (PVDs) under a combination of vacuum pressure and surcharge loading, with settlement and lateral displacement measured to assess performance Instrumented sites captured consolidation behavior and lateral movements, enabling evaluation of the effectiveness of vacuum-assisted PVD techniques in improving soft soils The results provide practical insights for geotechnical design, settlement control, and lateral displacement management in Vietnamese soft ground projects and offer guidance for similar environments applying PVD-based methods under combined loading.
Establish a numerical analysis model in Plaxis 2D to predict settlement and lateral displacement and verify the reliability of the model using the monitored data
Develop analytical equations to predicting lateral displacement of ground adjacent to the improved zones.
Research methodology
Experimental Approach: This approach includes the implementation of field experiments to identify key model parameters and monitor the lateral displacement of soft ground under treatment conditions
Numerical models are used to simulate the lateral displacement of soil adjacent to zones improved with Prefabricated Vertical Drains (PVDs) under the combined effects of surcharge loading and vacuum pressure This numerical approach enables assessment of the interaction between drainage efficiency and soil response, providing insights into settlement behavior and the effectiveness of PVDs under dual loading conditions.
Structure of thesis
This thesis consists of five main chapters, which are as follows:
LITERATURE REVIEW
Overview of ground improvement using Prefabricated Vertical Drains
2.1.1 History of PVDs in ground improvement
Vertical drainage for soft ground improvement began with D J Moran's 1925 sand drain method, designed to accelerate consolidation by shortening drainage paths W Kjellman advanced the technology in the late 1930s with prefabricated vertical drains (PVDs) using a cardboard core and filter material, a concept that has evolved into modern geotextile-wrapped plastic drains widely used in geotechnical engineering today To address the limitations of traditional fill preloading, Kjellman proposed vacuum preloading in 1952, which applies atmospheric pressure within a sealed system to consolidate soft soils more efficiently Depending on site conditions, preloading strategies may use fill, vacuum pressure, or a combination of both For soils with low shear strength, staged construction and surcharge loading are often employed to enhance consolidation and control post-construction settlement.
Figure 2.1 Schematic illustration of a PVD (Han 2015)
(a) Layfield Wick Drain (b) Geosupply Wick Drain
Figure 2.2 The common types of PVD used in practice (Thang 2023)
In practical applications, vertical drains are commonly arranged in square or triangular grid patterns To simplify analysis, the area influenced by each drain is represented by an equivalent cylindrical unit cell, with its diameter de estimated from the corresponding influence area For a square pattern, de = 1.05 d, while for a triangular pattern, de = 1.13 d.
Figure 2.3 Layout of vertical drain (Han 2015) where: d e = equivalent influence diameter d = spacing between two adjacent vertical drain
Figure 2.4 Axisymmetric unit-cell drainage model considering smear (Han 2015)
Installing vertical drains disturbs the surrounding soil and creates a smeared zone—an area with reduced permeability compared to undisturbed soil This decrease in permeability results from remolding during the installation process, as illustrated in Figure 2.4, and it can influence drainage efficiency and soil behavior.
Conventional radial consolidation theory assumes circular drains, so band-shaped vertical drains with a rectangular cross-section (width a and thickness b) must be converted to an equivalent circular drain with diameter dw This equivalent diameter enables the use of the theoretical formulations developed for circular drainage conditions, allowing consolidation analyses of band-shaped drains to be conducted within the circular-drainage framework The concept is illustrated in Figure 2.5, where the conversion to an equivalent circular diameter dw facilitates applying standard circular drainage formulations.
Figure 2.5 (a) Conceptual illustration of band-shaped PVD and equivalent diameter of rain well (Indraratna et al., 2005); (b) Assessment of equivalent diameter of band shaped vertical drains (Indraratna et al., 2005)
Figure 2.5 illustrates the key dimensions of a band-shaped prefabricated vertical drain (PVD) and demonstrates the method used to convert its rectangular cross-section into an equivalent circular form for analytical purposes.
The equivalent diameter of the drain is determined using the following standard equation, which is derived based on the assumption of a circular cross-section perimeter (Hansbo, 1979):
Atkinson and Eldred (1981) introduced a reduction factor to be incorporated into Equation 2.3 in order to account for the corner effect, where flow lines exhibit rapid convergence; subsequently, based on finite element analyses, Rixner et al (1986) suggested corresponding modifications.
The extent of the smeared zone around a vertical drain is governed by the mandrel geometry used during installation and by the soil’s properties, including soil type and sensitivity To predict this zone, researchers have established several empirical relationships that relate the smeared zone diameter to the drain or mandrel diameter Among these, the Hansbo correlation—introduced in 1981 and refined in 1997—has become the widely adopted standard for evaluating smear zone dimensions.
Installing prefabricated vertical drains (PVDs) inevitably disturbs the surrounding soil, primarily by remolding that reduces radial permeability This disturbed zone, commonly called the smear zone, exhibits significantly lower hydraulic conductivity than the undisturbed soil To quantify this permeability reduction within the smear zone, researchers such as Hansbo (1981), Onoue (1992), Indraratna and Redana (1998), and Hird and Mosely (2000) have formulated expressions to approximate the permeability reduction in this region.
10 where: k s = permeability of the smear zone k r = radial permeability of undisturbed surrounding soil, typically = 2 - 6
Consolidation theory for PVDs without vacuum
2.2.1 Consolidation due to vertical flow
The governing partial differential equation describing axisymmetric flow of pore water in a saturated soil medium Figure 2.6 can be expressed as follows:
(2.7) where: c r = coefficient of consolidation in a horizontal (radial) direction c v = coefficient of consolidation in a vertical direction r = radial distance u = excess pore water pressure at a distance of r u = average excess pore water pressure t = time
The degree of consolidation, also referred as the rate of consolidation, is defined as:
= − = − (2.8) where: u 0 = initial excess pore water pressure u t = remaining excess pore water pressure at time t
The overall degree of consolidation of soil due to vertical and radial flow can be calculated as follows (Carillo, 1942):
U vr = overall degree of consolidation
U v = degree of consolidation in a vertical direction
U r = degree of consolidation in a radial direction
Figure 2.6 Asymmetrical consolidation theory (Han, 2015)
The degree of soil consolidation due to vertical flow can be calculated using the Terzaghi one-dimensional consolidation theory (Terzaghi, 1943):
cv is the coefficient of vertical consolidation, a fundamental parameter in consolidation analysis that governs the rate at which a soil layer settles under loading as excess pore water drains The drainage distance, h_dr, represents the longest vertical path water must travel to escape the soil due to vertical flow When drainage occurs from both the top and bottom surfaces, the correct h_dr value is half of the soil thickness between these two drainage surfaces Properly defining cv and h_dr is essential for predicting consolidation settlement and pore-water pressure dissipation in geotechnical design and soil mechanics.
The average degree of consolidation with the time factor under a uniform distribution of initial excess pore water pressure can also be presented in Figure 2.6 (Tezaghi, 1943):
2.2.2 Consolidation due to radial flow (Barron, 1948)
Barron (1948) introduced an analytical solution to estimate the average degree of consolidation resulting from radial (horizontal) flow toward a vertically installed, freely draining sand drain:
U r = average degree of consolidation due to radial flow n = diameter ratio (n = d e/d w)
2.2.3 Consolidation due to radial flow (Hansbo, 1981)
Hansbo (1981) introduced an analytical expression for estimating the average degree of consolidation due to radial drainage The model explicitly incorporates the effects of smear zone formation and drain resistance, offering a more accurate consolidation analysis for drained soils.
U_r is the average degree of consolidation achieved by radial flow toward the vertical drain The diameter ratio of the smeared zone to the vertical drain, n_s = d_s / d_w, defines how far the disturbance extends radially from the drain The radial permeability of the undisturbed surrounding soil is k_r, while the radial permeability of the smeared soil is k_s, and these properties control how easily pore water can move toward the drain The depth z indicates where in the ground the degree of consolidation is computed, and h_dr is the longest drainage distance due to vertical flow that governs the consolidation time Together, these parameters determine the radial consolidation behavior: higher k_r relative to k_s, larger h_dr, and a wider smeared zone (larger n_s) alter the rate and extent of U_r at depth z.
Q c = discharge capacity of a vertical drain
2.2.4 Consolidation due to radial flow (Indraratna, 2005)
Indraratna (2005) proposed a solution for an average degree of consolidation after modifying Hansbo’s theory (1981):
4 3 h ax h ax ax s ax s ax h ax w h ax h ax s ax w k k n n s s s s s n s k n n k n n k l q n k k n s l s k q
14 u = the average excess pore water pressure
T h = dimensionless time factor for horizontal drainage
= parameter representing the geometry of the vertical drain k h = horizontal permeability coefficient in the undisturbed zone k s = horizontal permeability coefficient in the smeared zone q w = discharge capacity
1 = initial overburden pressure due to preloading (kN/m 2 )
= − = − + − + (2.19) where U r = average degree of consolidation due to radial flow
Equivalent plane strain parameters
Finite element modeling of embankments is commonly carried out under the plane strain assumption However, the actual consolidation behavior surrounding vertical drains is inherently axisymmetric
Figure 2.7 Conversion of an axisymmetric unit cell into plane strain condition
To better capture vertical drain behavior in a two-dimensional finite-element analysis, it is necessary to establish an equivalence between axisymmetric and plane-strain conditions Indraratna and Redana (1997) achieved this by converting the axisymmetric drain layout (as depicted in Figure 2.7) into a plane-strain counterpart The conversion was accomplished by adjusting the soil permeability parameters and adopting a unit cell width of 2B in the plane-strain model In this formulation, the half-widths of the drain and the smear zone in the plane-strain condition are defined as bw = rw and bs = rs, respectively.
Indraratna & Redana (1997) represented the average degree of consolidation in plane strain condition as follows:
U hp = average degree of consolidation due to radial flow u 0 = initial excess pore pressure u = pore pressure at time t (average values)
T hp = time factor in plane strain k hp = undisturbed horizontal equivalent permeability
' k hp = smear zone equivalent permeability
The geometric parameters , and (flow term), are given by:
= − (2.24) where: q z = the equivalent plane strain discharge capacity
By assuming the magnitudes of R and B to be the same, Indraratna and Redana
(1997) presented a relationship between k hp and k hp ' , as follows:
2 ln ln 0.75 2 hp h hp hp h h h w k k lz z k k k k n s lz z s k q
Neglecting well resistance in Equation 2.25 by removing all terms involving l and z reveals that the smear zone's influence is captured entirely by the permeability ratio between the smeared soil and the undisturbed soil, i.e., the ratio that expresses how hydraulic conductivity changes due to the smear.
' ln ' ln 0.75 hp hp hp h h h k k k n k s k s k
Neglecting both smear and well-resistance effects in the preceding formulation yields a simplified expression for the ratio of plane-strain permeability to axisymmetric permeability Hird et al (1992) proposed the following relationship:
The effect of well resistance is evaluated separately and expressed as an equivalent discharge capacity under plane strain conditions This can be quantified using the following equation:
In vacuum preloading applications, the equivalent vacuum pressure is considered identical for both plane strain and axisymmetric conditions.
Mechanism of excess pore water pressure variation under fill and vacuum preloading
Figure 2.8 presents the profiles of excess pore water pressure and effective stress under fill loading and vacuum preloading It shows that at the moment of fill loading, the initial hydrostatic pore water pressure uz0 is increased by the additional vertical fill stress Δσz, and the pore water pressure decreases with time to uz(t) at time t, with the remaining positive excess pore water pressure defined as Δuz = uz(t) − uz0.
Figure 2.8 Variations of excess pore water pressure and effective stress profiles under
(a) fill preloading and (b) vacuum preloading (Chu and Yan, 2006)
Consequently, the vertical effective stress in the geomaterial increases from the initial vertical overburden stress σ′z0 to a higher value at time t, reflecting the vacuum-induced change Simultaneously, the initial hydrostatic pore water pressure uz0 decreases immediately by the applied vacuum pressure −us, and the pore water pressure rises with time to uz(t) at time t, with the remaining negative excess pore water pressure given by Δuz = uz(t) − uz0 As a result, the overall stress–pore-pressure state evolves, altering the effective stress distribution within the geomaterial.
18 effective stress in soil increases from ' z 0 ( )z to z ' 0 ( ) u z + − s u z ( ) at time t, Jie Han
Consolidation theory for PVDs with vacuum
Indraratna et al (2005) proposed the average of consolidation in plane strain condition by:
and are determined by Eq 2.22 and Eq 2.23 u 0 = initial excess pore pressure u = pore pressure at time t (average values) k 1 = vacuum reduction factor in vertical direction
T hp = time factor in plane strain p op = applied vacuum pressure
U r = average degree of consolidation due to radial flow
Numerical simulation method in Plaxis 2D
In Plaxis 2D, plane strain or axisymmetric analyses can be conducted using triangular elements with either 6 nodes or 15 nodes The 6-node element, which is the default option for 2D models, computes the stiffness matrix via numerical integration with three Gauss points.
Gaussian points In contrast, the 15-node element employs integration over twelve stress points, as illustrated in Figure 2.9 (Bentley, 2022)
Figure 2.9 The triangle element type in Plaxis 2D: (a) 15-Node and (b) 6-Node 2.6.2 Material models
The material model plays a crucial role in analyzing geotechnical problems, with different models suited to various analysis scenarios and soil profiles Selecting the appropriate soil model enhances result accuracy and calculation reliability by aligning analyses with realistic conditions In Plaxis 2D, a range of material models helps users tailor analyses to specific geological conditions This thesis applies the Soft Soil Model to soft ground (such as soft clay and silt) and uses the Hardening Soil model to simulate good ground and embankments.
The Hardening Soil model is an advanced constitutive model used to simulate the behavior of soils across a wide range from soft to stiff Unlike simpler elastic–perfectly plastic models, the Hardening Soil model captures strain-dependent stiffness and strength, producing a nonlinear stress–strain response under both monotonic and cyclic loading It incorporates dilation and hardening/softening behavior, enabling realistic predictions of foundation settlement, embankment stability, and slope stability under varying loads By using strain-dependent parameters, this model provides accurate representation of soil response in geotechnical design analyses and is widely applied in engineering practice.
20 stiffness and irreversible plastic deformations When subjected to primary deviatoric loading, soils exhibit a reduction in stiffness while developing plastic strains
In drained triaxial tests, the relationship between axial strain and deviatoric stress is effectively described by a hyperbolic law, a concept first introduced by Kondner (1963) and later formalized in the widely used Duncan and Chang (1970) hyperbolic model The Hardening Soil model offers notable improvements over the traditional hyperbolic formulation by anchoring the framework in plasticity theory rather than elasticity, capturing soil dilatancy effects, and incorporating a yield surface cap to better represent soil behavior under varying loading conditions A key feature of the Hardening Soil model is a stress-dependent stiffness that follows a power-law relationship, enabling more accurate simulation of stiffness evolution with changing stress states.
E ref = Plastic straining due to primary deviatoric loading ref
E oed = Plastic straining due to primary compression ref
E ur = Elastic unloading/reloading c '= Cohesion
A key characteristic of the Hardening Soil Model is the dependence of soil stiffness on the stress level Under oedometer test conditions, this relationship can be expressed, for instance, as follows:
E =E p (2.32) where p ref is a reference pressure, in Plaxis p ref = 100 stress units
The Hardening Soil model is fundamentally based on a hyperbolic relationship between the vertical strain, 1 , and the deviatoric stress, q, observed during primary
21 drained triaxial loading Experimental results from standard drained triaxial tests typically produce stress-strain curves that can be represented by the following expression:
− = − for: q < q f (2.33) where: q a = Asymptotic value of the shear strength q f = ultimate deviatoric stress
The parameter E 50 represents the stiffness modulus during primary loading and depends on the confining stress It can be expressed by the following equation:
50 50 cos sin cos sin m ref ref
E ref = Reference stiffness modulus corresponding to the reference confining pressure p ref
3 = the confining pressure in a triaxial test
For unloading and reloading stress paths, another stress-dependent stiffness is used:
' cos 3sin cos sin( ) m ref ur ur ref
E ur = Reference Young’s modulus and reloading, corresponding to the reference pressure p ref
In many practical cases it is appropriate to set E ur ref equal to 3E 50 ref , these relationships are plotted in Figure 2.10:
Figure 2.10 Hyperbolic stress-strain relation in primary loading for a CD test
The soft soil model applies to near-normally consolidated clays, clayey silts, and peat, drawing on Cam-Clay and Mohr-Coulomb theory It posits a logarithmic relationship between volume strain and mean effective stress during one-dimensional compression, instead of the traditional logarithmic relation tied to void ratio This framework helps describe the complex, time-dependent behavior of soft soils under loading, combining the elasticity, plasticity, and consolidation concepts essential for accurate geotechnical analysis.
Within the Soft Soil model, the slope of the isotropic unloading–reloading curve is defined as a function of the soil’s modified swelling index κ*, which relates to volumetric strain behavior Although the swelling index κ and the modified swelling index κ* differ, the ratios λ/κ and λ*/κ* remain equivalent, ensuring consistent description of the material’s volumetric response.
To illustrate the yield function, Plaxis 2D models a triaxial stress state characterized by 2 ' = 3 ' Under these conditions, the yield function 𝑓 f is defined by Equation 2.39
Figure 2.11 Relationship between volumetric strain and mean effective stress
Figure 2.12 The Soft Soil Model: (a) Yield surface of the Soft Soil model; (b) Total yield contour of Soft Soil model (Galavi & Brinkgreve, 2014)
Soft Soil model parameters include the characteristic compression index and swelling index of soft soils, as well as the failure criteria parameters from the Mohr-Coulomb model Overall, the model requires this specific set of parameters to describe the material's compression and swelling behavior together with its Mohr-Coulomb-based shear strength criteria.
*= modified swelling index c '= effective cohesion
Within the Soft Soil Model, the modified compression parameters λ* and κ* can be obtained either from the original Cam-Clay parameters or from data derived from a one-dimensional compression test This relationship corresponds to the internationally recognized one-dimensional compression and swelling parameters, Cc and Cs, respectively.
Recent studies
To evaluate the lateral displacement and the influence zone of this method, numerous studies have been conducted to simulate and analyze soil consolidation under various loading conditions, and a table below summarizes notable research on numerical modeling and experimental analysis, illustrating how each study makes significant contributions while also revealing certain limitations.
Table 2.1 Overview of research on Lateral Displacement in PVD – improved Ground with Vacuum and Surcharge Loading
No Author Paper title Achievements Limitations
2D and 3D numerical modeling of combined surcharge and vacuum preloading with vertical drains
An 2D and 3D numerical model (ABAQUS) is used to analysis to predict the settlement, pore-water
Not yet consider the change of influence zone
No Author Paper title Achievements Limitations pressure, and the lateral displacement
Class A and C predictions for Ballina trial embankment with vertical drains using standard test data from industry and large diameter test specimens
The radial consolidation performance of the soft clay foundation beneath the Ballina
Embankment was successfully predicted using both analytical and numerical methods
Not yet consider the change of influence zone
Not yet consider to vacuum pressure
A study on lateral displacement of ground adjacent to areas consolidated by PVDs under surcharge loading and vacuum pressure
The method for calculating lateral displacement by Chai et al 2013 was examined
A numerical model was used to estimate the settlement and lateral displacement
The influence zone was investigated when presenting CDM wall
The results for calculating lateral displacement from Chai’s method is not reliable
The lateral displacement from the numerical does not match well with the measured data
Displacement of ground induced by surcharged loading and vacuum pressure
Evaluate Chai’s method to predict maximum lateral displacement on the ground surface
A numerical model was established to evaluate the settlement and lateral displacement
Numerical results did not capture well with monitoring data The relationship between influence zone and loading ratio is not clear
Lateral displacement under combined vacuum
An analytical approach has been developed to estimate the probable
The analytical equation only estimates maximum
No Author Paper title Achievements Limitations pressure and embankment loading range of maximum lateral ground displacement beneath an embankment edge net lateral displacement
Not yet consider the lateral displacement as a function of depth and outside of the improved area
Not yet consider the influence zone
6 Xu et al (2025) Prediction method for lateral displacement of PVD-improved ground under vacuum preloading
A new methodology was applied to calculate the lateral displacement, utilizing fundamental preloading conditions and soil properties
The methodology is proposed based on a unit cell so it can-not predict well the lateral displacement beyond the embankment Not yet consider the influence zone
Despite extensive studies on soft ground improvement using PVDs, several gaps remain:
• Limited research exists on lateral displacement beyond the improved area and its influence zone
• The development of the influence zone under vacuum pressure and surcharge loading remains unclear
• There is lack of analytical equations to predict the lateral displacement as a function of depth extending from the embankment toe, which can be practically applied
METHODOLOGY
General description of the methodology
This study evaluates the lateral displacement and influence zone in soft ground improved with prefabricated vertical drains (PVDs) under vacuum and surcharge loading, employing a combination of numerical modeling, parametric studies, and analytical analysis The methodology follows a structured approach illustrated in Figure 3.1, outlining data collection, model selection, validation, parametric studies, and the development of final recommendations.
The specific steps for implementing general flowchart as follows:
The research begins with comprehensive data collection, gathering soil properties, field monitoring records, and numerical modeling parameters This data forms the foundation for selecting and optimizing the numerical model, enabling data-driven modeling, informed parameter estimation, and improved model performance.
• Select and optimization of numerical model:
To predict the performance of soft ground improved with prefabricated vertical drains, a numerical model was selected and optimized to simulate its response under vacuum preloading and surcharge loading Model parameters were calibrated against field data to ensure accurate predictions of performance This calibrated modeling approach supports reliable assessment and design of PVD-enhanced ground under combined loading conditions.
• Analysis and verification with field data:
The chosen numerical model undergoes rigorous analysis and validation using field data from real-world case studies This verification confirms that the model reliably captures real-world conditions, providing a robust basis for subsequent parametric studies and the development of analytical equations.
• Influence zone and analytical equation:
Different loading cases and various sections extending from the toe of the embankment are selected to capture lateral displacement and define the influence zone These measurements are then combined with existing theories to establish a predictive relationship for the influence zone and the lateral displacement as a function of depth A comparison of the lateral displacement results obtained from experimental, numerical, and analytical methods will be presented.
A comprehensive procedure will be outlined to determine the influence zone and the depth-dependent lateral displacement extending from embankment toe, based on key parameters of practical relevance to engineers.
Numerical model in PLAXIS 2D
In PLAXIS 2D, vacuum consolidation is modeled by decreasing the groundwater head since the software uses atmospheric pressure as the reference and does not directly model air pressure; this approach produces numerically induced negative pore pressures (suction), which serve as an approximation of the vacuum effect.
29 perfect vacuum conditions are unachievable in practice, effective vacuum pressures between 60 and 90 kN/m 2 are commonly realized
Vacuum consolidation in practical projects is represented in 2D or 3D numerical simulations by modeling groundwater flow or a fully coupled flow-deformation system that includes vacuum drains In these models, each drain is assigned a groundwater head up to 10 meters below the phreatic surface, corresponding to a theoretical vacuum pressure of 100 kN/m^2 Although the spacing between drains can vary in the model, it should be selected to minimize groundwater head variation within the vacuum zone As a general guideline, drain spacing should be kept to less than one-quarter of the drain length to ensure effective vacuum distribution.
Vacuum consolidation, implemented through reduced groundwater head boundary conditions or lowered heads in vacuum drains, can be incorporated within the following analysis types in Plaxis:
• Plastic analysis (with Steady-state groundwater flow selected for pore pressure computation)
• Consolidation analysis (also using Steady-state groundwater flow for pore pressure)
• Fully coupled flow-deformation analysis
To ensure proper simulation of vacuum consolidation, all requirements for groundwater flow modeling must be fulfilled:
• Material property datasets must include non-zero permeability values
• Hydraulic boundary conditions, such as groundwater heads and closed flow boundaries where applicable, must be properly defined
To accurately simulate suction effects in vacuum consolidation, the “Ignore suction” option muse be disabled within the deformation control settings of the respective calculation phase
In PLAXIS 2D, modeling vacuum pressure in Prefabricated Vertical Drains (PVDs) is achieved by using the vacuum drain option rather than the conventional normal drain Unlike normal drains, the vacuum drain allows you to assign a groundwater head that is lower than the drain's physical elevation, creating negative pore water pressure (suction) within the soil and enabling vacuum-induced consolidation By specifying a reduced groundwater head at the drain location, the vacuum effect is introduced into the model, enabling the analysis of soil behavior under vacuum-assisted preloading conditions.
In a consolidation analysis where pore pressure is set to Phreatic, vacuum drains behave as normal drains They influence only the dissipation of excess pore water pressure, while the steady-state pore pressure distribution is determined by the global water level and the local cluster settings.
Some studies simulate vacuum pressure by lowering the water table, but this approach is not accurate because reducing the water table to mimic suction results in zero pore water pressure in the affected soil, which does not realistically represent the vacuum consolidation process.
RESULTS AND DISCUSSIONS
Application of numerical simulation method in the three case studies
Numerical modeling is used to analyze case studies of projects implemented in Vietnam, with the simulated results for settlement, lateral displacement, and pore water pressure compared against field monitoring data to assess the accuracy and applicability of the numerical model.
4.2.1 Nhon Trach 3&4 Power Plant – Zone P1.5 (NT P1.5)
❖ The model and soil profile:
A numerical simulation model for the ground improvement of NT P1.5 was developed for the cross-section depicted in Figure 4.19, with the resulting results shown in Figures 4.20 and 4.21 The parameters of the model are detailed in Table 4.3.
Figure 4.19 The numerical model of NT P1.5 Table 4.3 The input soil parameters of the model (NT P1.5)
Soil parameter Unit Embankment Fill Sand Layer 1 Layer 1b
Sand Sand Soft Clay Sandy
Model type - Hardening Soil Hardening
Soil Soft Soil Soft Soil
Drainage type - Drained Drained Undrained
Tangent stiffness, E oed ref kN/m 2 3500 5000 - -
Unloading stiffness, E ur ref kN/m 2 15000 15000 - -
Coeff of horizontal permeability, k x m/day 4 1 3.5×10 -5 4.5×10 -5
Coeff of vertical permeability, k y m/day 2 0.5 1.5×10 -5 2.0×10 -5
Figure 4.20 Simulated settlement result of NT P1.5 from the numerical model
Settlement analysis reveals a maximum downward vertical settlement of about 1.3 meters beneath the embankment crest, as indicated by the settlement contour The deformation gradually diminishes toward the boundary of the treated area, demonstrating effective consolidation under the combined effects of vacuum pressure and surcharge loading.
Figure 4.21 Simulated lateral displacement result of NT P1.5 from the numerical model
The lateral displacement peaks at approximately -400 mm, concentrated near the embankment toe This substantial horizontal ground movement extends beyond the PVD-improved zone, signaling potential impacts on adjacent structures and utilities if not properly controlled The findings underscore the need for mitigation and monitoring to safeguard nearby infrastructure and ensure the embankment performs safely under lateral loading.
Settlement values predicted by the numerical model align closely with the field measurements shown in Figure 4.22, demonstrating the model’s accuracy in predicting settlement behavior These results indicate that the numerical approach is a reliable method for evaluating settlement responses under the studied conditions and support its use in design and assessment.
Figure 4.22 Comparison between simulated and monitored results at NT3&4 – P1.5
The pore water pressure at both depths exhibits a trend similar to that observed in the field data However, at a depth of -4.75 m, a significant discrepancy is evident
Field Data Numerical Result Depth -4.75 m
Between days 70 and 140, the numerical prediction fails to capture the PWP rebound observed in the field data, suggesting that the numerical model may be missing a key influencing factor at this depth At -7.9 m, the agreement with field measurements improves, though minor differences still exist.
Figure 4.23 Lateral displacement comparison between simulation and monitoring at
Overall, lateral displacements recorded at the three inclinometer (IC) locations show a strong agreement between numerical simulations and field measurements, confirming the accuracy of the modeling approach A moderate discrepancy is observed at IC2, particularly in the upper 6 meters, indicating localized factors influencing the deviation.
4.2.2 Nhon Trach 3&4 Power Plant – Zone P1.2 (NT P1.2)
❖ The model and soil profile:
A numerical simulation model of the ground improvement for NT P1.2 was developed for the cross-section shown in Figure 4.24 The numerical results are presented in Figures 4.25 and 4.26, and the model parameters used in the analysis are given in Table 4.4.
Figure 4.24 The numerical model of NT P1.2 Table 4.4 The parameters of the model (NT P1.2)
Soil parameter Unit Embankment Fill Sand Layer 1 Layer 1b Layer 2
Sand Sand Soft Clay Clay Sandy
Hardening Soil Soft Soil Soft Soil Soft Soil
Drainage type - Drained Drained Undrained
Unloading stiffness, E ur ref kN/m 2 30000 24000 - - -
Soil parameter Unit Embankment Fill Sand Layer 1 Layer 1b Layer 2 permeability, k x
Coeff of vertical permeability, k y m/day 0.75 1 3.0×10 -5 2.7×10 -5 2.9×10 -5 Permeabilty change index, C k - - - 1.17 1.17 1.17
Pre-overburden pressure, POP kPa 0 0 0 0 0
Table 4.5 The parameters of the CDM Zone
Parameters Symbol Unit Column 1-3 Column 4
Model type - Linear Elastic Linear Elastic
Unsaturated unit weight unsat kN/m 3 17.5 19
Figure 4.25 Simulated settlement result of NT P1.2 from the numerical model
Figure 4.25 shows the settlement contour for NT P1.2, revealing a maximum vertical deformation of approximately -1.5 meters predominantly beneath the embankment crest The settlement distribution is relatively uniform within the improved area, indicating effective consolidation, and the presence of a CDM (Cement Deep Mixing) technique is noted.
51 zone at the embankment toe appears to have played a crucial role in limiting excessive differential settlement near the slope
Figure 4.26 Simulated lateral displacement result of NT P1.2 from numerical model
Although the lateral displacement reaches about -400 mm, it shows a more confined and stable pattern compared with NT P1.5 This improvement arises from the CDM column zone, which enhances lateral stiffness and acts as a barrier to restrict horizontal deformation near the embankment toe Consequently, the influence zone is narrower, reducing potential impact on adjacent areas.
Figure 4.27 illustrates the strong agreement between measured and simulated settlement curves for NT Zone P1.2 Both data sets show rapid settlement during the initial 40 days, followed by gradual stabilization, indicating consistent settlement behavior over time The numerical results closely follow the field data trend, with only minor deviations, demonstrating good model validation and reliability of the settlement predictions.
Figure 4.27 indicates that the numerical pore water pressure (PWP) trends at depths of -5.5 m and -9.20 m closely align with field measurements Both depths show an initial rapid PWP dissipation followed by gradual stabilization At -5.5 m, there is a notable discrepancy between about 60 and 110 days, where the numerical results underestimate the measured PWP data In contrast, the -9.20 m results exhibit better consistency throughout the observation period Overall, these findings suggest that the numerical model captures the dominant pore water pressure behavior well.
Figure 4.27 Comparison of results from simulation and monitoring data at NT3&4 –
Figure 4.28 Lateral displacement comparison between simulation and monitoring at
Figure 4.28 shows the lateral displacement results for this zone, highlighting varying levels of agreement between numerical predictions and field measurements At IC1, a substantial discrepancy is observed, particularly in the upper 6 meters, where numerical values are notably lower than the field data This mismatch is likely due to the influence of Cement Deep Mixing (CDM) ground improvement, which was not fully captured in the simulation By contrast, IC2 and IC3 exhibit better alignment, with the numerical results reasonably following the measured profiles These findings underscore the importance of accurately representing ground improvement zones in numerical models to improve the prediction of lateral displacement.
❖ The model and soil profile:
The numerical simulation model for the ground improvement of Four Road is conducted at the cross-section shown in Figure 4.29, and the numerical results are presented in Figures 4.30 and 4.31 The parameters of the model are shown in Table 4.6.
Figure 4.29 The numerical model of Four Road Project Table 4.6 The parameters of the numerical model (Four Road)
Soil parameter Unit Embankment Layer 1 Layer 2 Layer 3
Layer name - Surcharged Sand Fill Sand Soft Clay Sandy Clay
Model type - Hardening Soil Mohr-
Drainage type - Drained Drained Undrained A Undrained A
Soil parameter Unit Embankment Layer 1 Layer 2 Layer 3 angle, '
Coeff of horizontal permeability, k x m/day 2 2 2.0×10 -5 0.5
Coeff of vertical permeability, k y m/day 1 1 1.0×10 -5 0.25
Pre-overburden pressure, POP kPa 0 0 0 0
Figure 4.30 Simulated settlement result of Four Road from the numerical model
At Four Road, the settlement contour shows a maximum vertical deformation of approximately −3.0 meters beneath the embankment The settlement then diminishes with increasing distance from the center of loading, reflecting the typical behavior of soft ground subjected to surcharge and vacuum preloading without additional reinforcement.
Figure 4.31 Simulated lateral displacement result of Four Road from the numerical model
Influence zone and analytical equation
A parametric model was developed to investigate the influence zone and variations of the lateral displacement
Figure 4.39 Cross – Section of parametric model
MonitoringNumericalVSIP - IC4Inward Outward
Figure 4.40 Loading information of parametric model
t = the starting day of adding p v – the starting day of adding p s
Table 4.8 The parameters of the numerical model (Parametric model)
Soil parameter Unit Embankment Fill Sand Layer 1 Layer 2
Layer name - Surcharged Sand Fill Sand Soft Clay Hard Clay
Depth m +2.50 to 0 0 to -1 -1 to -9 -9 to -14.5
Model type - Hardening Soil Hardening
Soil Soft Soil Soft Soil
Drainage type - Drained Drained Undrained A Undrained
Soil parameter Unit Embankment Fill Sand Layer 1 Layer 2
Coeff of horizontal permeability, k x m/day 4 1 4.5×10 -5 4.5×10 -5
Coeff of vertical permeability, k y m/day 2 0.5 1.5×10 -5 2.0×10 -5
Pre-overburden pressure, POP kPa 0 0 0 0
This section analyzes the influence zone on the ground surface (B) shown in Figure 4.41 using a parametric model that varies the surcharged loading (p_s), vacuum pressure (p_v), and time increment (Δt), with the corresponding data presented in Tables 4.9 and 4.10.
Figure 4.41 The influence zone caused by vacuum pressure and surcharged loading
B = Influence zone on the ground surface (m) due to vacuum pressure (p v ) and surcharged loading (p s )
zi = the lateral displacement (mm) responding with depth zi at distance L extending from embankment toe
Figure 4.42 Lateral displacement and influence zone from the parametric numerical model
Figure 4.42 presents the lateral displacement contours and the influence zone derived from a numerical simulation The influence zone is defined as the horizontal extent from the embankment toe to the point where lateral displacement decreases to approximately 10% of the maximum value observed at the toe, with a minimum threshold of 20 mm imposed to ensure only areas with significant deformation are included This approach, consistent with Chu et al (2015) and Indraratna et al (2018), delineates the region most affected by lateral soil movement, which is essential for evaluating ground deformation and planning instrumentation.
The practical significance of the influence zone arises only when the lateral displacement at the embankment toe is sufficiently large If the displacement is relatively small—for example, less than 5 cm—the extent of the influence zone defined by the 10% criterion may be minimal or negligible To prevent underestimating the affected area, this study applies a supplementary 20 mm threshold, especially in soft ground conditions where deformation tends to localize near the embankment toe (Miura & Chai, 2002).
Table 4.9 Case studies for investigating the influence zone
No p s p v p total p v /p s kPa kPa kPa -
Table 4.10 Case studies for changing starting time of loading
Day Day Day Day Day Day
The author analyzed 10 cases by varying p s and p v (Table 4.9) to evaluate the width of influence zone and the lateral displacement at the embankment toe
Figure 4.43 shows a strong logarithmic relationship between the lateral displacement at the embankment toe and the ratio p v /p s , with a correlation coefficient R 2
A strong fit (R^2 ≈ 0.97) shows that as pv/ps increases, lateral ground displacement grows, indicating that vacuum pressure markedly governs lateral movement at the embankment toe The increase is more pronounced when pv/ps < 1.5, after which the rate of displacement growth becomes more gradual, revealing a nonlinear soil response likely due to stress-dependent stiffness and mobilization of shear strength under varying loading The fitted equation provides a practical analytical tool for estimating lateral displacement at the embankment toe based on pv/ps and loading conditions These findings underscore the importance of controlling vacuum pressure and surcharged loading in embankment design to manage pressure-induced ground deformation Overall, the results contribute to a better understanding of pressure-induced ground deformation and support safer, more effective design of embankments.
Figure 4.43 Correlation between the lateral displacement at the embankment toe and ratio p v /p s
Figure 4.44 Correlation between width of influence zone (B) and p v /p s ratio
Figure 4.44 presents a strong linear relationship between the width of the influence zone and the pv/ps ratio, with a high correlation coefficient (R^2 = 0.97) The fitted equation is y = B = 11(pv/ps), indicating that the influence zone expands proportionally as vacuum pressure increases relative to surcharged loading This suggests that higher vacuum pressure not only increases deformation but also broadens the affected area Unlike the nonlinear behavior observed in lateral displacement at the embankment toe, the width pv/ps follows a linear trend.
La te ra l Di spl ac em ent a t E m ba nkm ent T oe (m m )
W idt h of Inf lue nc e Z one (m )
Inwa rd Out wa rd
Approximately 69% of the influence zone responds linearly to the stress ratio Notably, when p_v/p_s is around 0, meaning p_v approaches 0 kPa, the influence zone transitions from an inward movement toward the improved area to an outward expansion This finding is useful for defining the extent of ground improvement or instrumentation during the design process.
Figure 4.45 demonstrates that lateral ground displacement decreases significantly as ΔL increases When vacuum pressure and surcharge loading are applied simultaneously (Δt = 0), lateral displacement is reduced by approximately 50% compared with the case where Δt > 10 days, highlighting the inefficiency of delayed loading.
Figure 4.45 Correlation between lateral displacement on ground surface and L by changing time
Vacuum consolidation achieves greater reduction in ground lateral movement when surcharge loading is applied early or concurrently with the vacuum pressure Therefore, surcharge loading should be applied at the same time as, or shortly after, the vacuum pressure to minimize ground lateral displacement.
Define K 0 as the ratio of the effective horizontal stress ( h ' 0 ) to the effective vertical stress ( v ' 0 ) at in-situ condition as:
L at era l di spl ac em ent on ground s urfa ce (m m )
Figure 4.46 demonstrates that the lateral response of a soil element is governed by the horizontal stress ratio K compared with the at-rest earth pressure coefficient K0 If K is less than K0, outward lateral displacement occurs; if K equals Ke and Ke exceeds K0, the displacement is inward, as noted by Lu et al (2019).
(b) Upon the application of vacuum pressure
Figure 4.46 Lateral deformation mode for different stress state
The K e of the specimens during vacuum preloading can be calculated as follows:
K e = effective stress ratio after applied loading
h = horizontal effective stress before loading
v = vertical effective stress before loading
h = horizontal effective stress after applied vacuum and surcharged loading
v = vertical effective stress after applied vacuum and surcharged loading
i = horizontal pressure reduction according PVD’s length ( i = 0 at the top of PVD, i = 0.8 ~ 0.9 at the bottom of PVD)
i = vertical pressure reduction according PVD’s length ( i = 0 at the top of PVD,
i = 0.8 ~ 0.9 at the bottom of PVD)
Based on the relationship between K e /K 0 and zi /S zi , obtained from the analysis 10 cases with varying p v /p s ratios, and different L sections in parametric numerical model
The author has developed an equation to estimate the lateral displacement as a function of depth at any L position:
S zi = vertical settlement at depth zi a, b can select as following relationship:
Figure 4.47 Correlation to determine a value
Figure 4.48 Correlation to determine b value where:
L = distance extending from the embankment toe p s = surcharged loading (kPa); p v = vacuum pressure (kPa)
4.3.4 Validation of Proposed Analytical Equation
In this section, author will use the proposed equation (Eq 4.5) to calculate the lateral displacement for different project, which is given in this table below:
Table 4.11 Information about different projects for verification
Tianjin Port – Case 1 40 80 China Figure 4.49
Tianjin Port – Case 2 60 80 China Figure 4.49
SBIA Airport – Case 1 45 40 Thailand Figure 4.50
SBIA Airport – Case 2 45 50 Thailand Figure 4.50
Figure 4.49 Typical Cross Section of Tianjin Port (Chai et al 2013)
Figure 4.50 Typical Cross Section of SBIA Airport (Chai et al 2013)
Figure 4.51 shows lateral displacement profiles at three instrument locations (IC1, IC2, IC3) along Nhon Trach P1.5, comparing field monitoring data, numerical results, and the proposed analytical equation At IC1, the analytical model closely matches the field monitoring data, while the numerical results exhibit a moderate discrepancy in the upper portion of the profile This comparison highlights the analytical equation’s accuracy at IC1 and underscores differences between numerical predictions and measured displacements across the instrument locations.
Compared with field data at 4 m depth, IC2 shows strong agreement among the three profiles, with only minor deviations near the surface At IC3, displacement magnitudes are smaller overall, yet the three curves remain closely aligned throughout the depth Taken together, these results indicate that the proposed equation is reliable across varying depths and locations, particularly in deeper layers where displacement is more pronounced.
Figure 4.51 Comparison of lateral displacement profiles from field data, numerical simulation, and proposed analytical model at Nhon Trach P1.5
Figure 4.52 presents the lateral displacement at Four Road – IC1, comparing field measurements, numerical simulations, and the proposed analytical model The monitored data reveal substantially greater displacement in the upper 10 m than predicted by both the numerical simulation and the analytical model, indicating a notable discrepancy in the shallow portion between observed movements and model estimates.
Figure 4.52 Comparison of lateral displacement profiles from field data, numerical simulation, and proposed analytical model in Four Road Project
Across the full depth profile, the numerical and analytical results stay in close agreement, with the strongest alignment observed below 10 meters Importantly, the analytical model more accurately captures the displacement trend in the deeper layers These findings suggest that while the model performs well at depth, its accuracy in the near-surface zone may require further calibration.
Figure 4.53 presents lateral displacement profiles at four instrumented locations in the VSIP Project, comparing monitoring data, numerical results, and the proposed analytical equation Across all locations, the displacement magnitudes are small, typically within 0–50 mm The numerical and analytical curves align closely with each other and generally match the field data well, particularly from 2 m depth downward Slight deviations occur near the surface, especially at IC1 and IC2, where the measured displacements are marginally higher Overall, the analytical model demonstrates good agreement with the observed data.
76 demonstrates good predictive capability in low-displacement conditions across the entire profile
Figure 4.53 Comparison of lateral displacement profiles from field data, numerical simulation, and proposed analytical equation in VSIP Project
Figure 4.54 presents lateral displacement profiles derived from field data at Tianjin Port alongside several analytical models In Case 1, the proposed analytical equation tracks the measured displacement essentially across the entire depth, with only minor deviations near the surface In Case 2, although all analytical equations follow the general displacement trend, the proposed equation achieves better agreement in the deeper layers (below 6 m), while Xu (2023) slightly overestimates displacement near the surface Lu (2019) shows good agreement in the mid-depth range but diverges at both shallow and deep zones Collectively, these results indicate that the proposed model offers consistent performance across the investigated depth range.
CONCLUSIONS AND RECOMMENDATIONS
Conclusions
This study successfully meets its stated objectives, with the main findings outlined below:
Plaxis 2D was used to develop advanced numerical models, assigning constitutive models to match each soil type Specifically, the Soft Soil model was applied to the soft clay layers, while the Hardening Soil model was chosen as suitable for the sand fill layer.
Vacuum pressure during preloading was modeled in Plaxis with a drain element to effectively simulate the vacuum preloading process The resulting simulations showed good agreement with field monitoring data for settlement, lateral displacement, and pore water pressure, confirming the reliability and accuracy of the adopted modeling approach.
An in-depth study of the influence zone adjacent to the embankment toe was conducted, examining how the ratio of vacuum pressure (p_v) to surcharge load (p_s) and the time gap between the starts of vacuum application and surcharge loading (Δt) govern the ground response The findings show that the p_v/p_s ratio and Δt largely determine the extent and behavior of the influence zone, with important implications for the design and stabilization of embankments These results underscore the interaction between vacuum-induced effects and surcharge loading, highlighting the need to optimize sequencing and pressure levels to accurately predict settlement, pore pressure changes, and stability near the toe.
It was found that lateral displacement at the embankment toe ( et – Unit:mm) increased non-linearly with an increasing p v /p s ratio, as described by the analytical relationship:
Additionally, the extent of the influence zone (B – Unit: m) expands linearly with the p v /p s ratio, following the correlation:
Furthermore, the time interval (t) between the application of vacuum pressure and surcharge loading significantly affects the ground response When both were
79 applied simultaneously (t = 0), lateral displacement was reduced by approximately
50%, as demonstrated in the parametric studies
A robust analytical equation was developed to predict lateral displacement as a function of depth extending from the embankment toe, providing a practical tool for geotechnical design The model was calibrated against both numerical simulations and field measurements, and it demonstrated strong agreement with observed data, confirming its practical applicability for accurate assessment of embankment behavior.
Limitations and recommendations
The lateral displacement in the CDM zone remains poorly captured by the numerical model when compared with measured data This discrepancy stems from PLAXIS 2D simplifications, wherein CDM walls are modeled as a continuous barrier, effectively ignoring the actual column spacing present in real-world conditions.
The pore water pressure distribution within the improved zone varies significantly with the spacing of adjacent PVDs, highlighting how drain configuration influences drainage performance Therefore, further investigation is required to quantify how pore water pressure changes with horizontal distance from the PVDs, enabling more accurate drainage design and reliable predictions of soil stability in geotechnical projects.
In practice, lateral displacement of the ground surface can be limited by crack formation, a constraint not yet captured by the current finite element method (FEM) model Therefore, it is necessary to develop an advanced modeling approach that can more accurately capture this behavior.
The proposed analytical equation was developed based on a parametric model and is suitable for predicting lateral displacement in projects involving thick soft clay layers (OCR ≈ 1) and a thin filling layer (< 2 m, OCR ≈ 1) For special or complex cases, it is recommended that engineers develop a detailed numerical model to obtain more accurate results.
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