Application of the Hardening Soil Model in Deep Excavation Analysis

A technical guide covering theory, parameter determination, comparisons with Mohr-Coulomb, real case studies, solved examples, and best practices aligned with modern geotechnical standards.

1. Introduction

Deep excavations are among the most challenging problems in urban geotechnical engineering. Retaining walls, diaphragm walls, sheet piles, and multi-level strutting systems interact with soil in a highly non-linear, stress-path-dependent manner. Accurate prediction of wall deflection, ground settlement, and basal heave is essential to protect adjacent structures and ensure serviceability.

For decades the Mohr-Coulomb (MC) model has been the workhorse of routine design because of its simplicity. However, it treats soil as linear-elastic perfectly plastic and uses a single constant modulus. Real soil behaviour is markedly different: stiffness is stress-dependent, non-linear even at small strains, and distinctly different in primary loading versus unloading-reloading.

The Hardening Soil (HS) model, developed by Schanz, Vermeer & Bonnier and implemented in PLAXIS (and several other finite-element codes), overcomes many of these limitations. It is an isotropic hardening elasto-plastic model that incorporates:

• Hyperbolic stress-strain relationship for primary loading
• Separate unloading-reloading modulus
• Stress-dependent stiffness via a power law
• Two yield surfaces (shear hardening + volumetric/cap hardening)
• Mohr-Coulomb failure criterion

Key Insight from Teo & Wong (2011): Back-analyses of three major Singapore excavations (Rochor Complex, Lavender MRT Station, Hougang-Buangkok Tunnel) demonstrated that the HS model consistently produced more realistic wall deflections, ground-settlement profiles and basal heave than the MC model when parameters were selected using a simplified practical approach.

2. Overview of the Hardening Soil Model

2.1 Theoretical Framework

The HS model is formulated within classical plasticity theory. For any stress increment the total strain increment is decomposed into elastic and plastic parts:

\[ d\varepsilon = d\varepsilon^e + d\varepsilon^p \]

Plastic straining occurs when the stress point reaches either the shear yield surface or the cap yield surface.

flowchart TD A[Stress Increment Δσ] --> B{Primary Loading?} B -->|Yes| C[Shear Hardening Surface] B -->|Yes| D[Cap Hardening Surface] B -->|No - Unloading/Reloading| E[Purely Elastic
using E_ur] C --> F[Plastic Shear Strain γ^p] D --> G[Plastic Volumetric Strain ε_v^p] F --> H[Flow Rule → Dilatancy] G --> I[Associated Flow on Cap] H --> J[Update Yield Surfaces] I --> J E --> K[Update Stresses] J --> K

2.2 Yield Surfaces

The model employs two independent yield surfaces:

1. Shear yield surface – expands with plastic shear strain (frictional hardening)
2. Cap yield surface – expands with plastic volumetric strain (density hardening)

The shear yield function takes the form of a hyperbolic relationship that approaches the Mohr-Coulomb failure envelope asymptotically:

\[ f_s = \frac{q}{E_{50}} \frac{1}{1 - q/q_a} - \frac{2q}{E_{ur}} - \gamma^p \le 0 \]

where \( q_a = q_f / R_f \) and \( R_f \) is typically 0.9.

3. Model Parameters & Determination

The HS model requires the following primary parameters (Table 1).

Parameter	Description	Typical Test	Recommended Default / Range
\( c' \)	Effective cohesion	CD / CU	0 for sand; measured for clay
\( \phi' \)	Effective friction angle	CD / CU	28°–45° (density dependent)
\( E_{50}^{ref} \)	Secant stiffness at 50 % strength (pref = 100 kPa)	CD triaxial	See correlations
\( E_{oed}^{ref} \)	Tangent oedometer stiffness at 100 kPa	Oedometer	≈ \( E_{50}^{ref} \) if no data
\( E_{ur}^{ref} \)	Unloading-reloading stiffness at 100 kPa	CD with unload-reload loops	≈ 3 \( E_{50}^{ref} \)
\( m \)	Power for stress-level dependency	CD / Oedometer	0.5 (sand) – 1.0 (clay)
\( \nu_{ur} \)	Unloading-reloading Poisson’s ratio	–	0.2
\( \psi \)	Dilatancy angle	CD	\( \phi' - 30^\circ \) (sand); 0 (clay undrained)
\( K_0^{nc} \)	At-rest earth pressure coefficient (normally consolidated)	–	\( 1 - \sin\phi' \)

Practical Determination Protocol (Teo & Wong + Brinkgreve)

1. Perform (or obtain) consolidated-drained triaxial compression tests at three confining pressures.
2. Extract \( E_{50} \) from each stress-strain curve at 50 % of peak strength.
3. Plot \( \ln E_{50} \) versus \( \ln\left(\frac{c'\cos\phi' + \sigma_3'\sin\phi'}{c'\cos\phi' + p^{ref}\sin\phi'}\right) \). Slope = \( m \), intercept = \( \ln E_{50}^{ref} \).
4. From oedometer tests extract tangent modulus at vertical stress = 100 kPa → \( E_{oed}^{ref} \).
5. If unload-reload loops are available, take \( E_{ur}^{ref} \); otherwise set \( E_{ur}^{ref} = 3E_{50}^{ref} \).

4. Governing Equations

4.1 Stress-Dependent Stiffness

All three stiffness moduli are stress-dependent according to the power law:

\[ E_{50} = E_{50}^{ref} \left( \frac{c'\cos\phi' + \sigma_3'\sin\phi'}{c'\cos\phi' + p^{ref}\sin\phi'} \right)^m \]

\[ E_{ur} = E_{ur}^{ref} \left( \frac{c'\cos\phi' + \sigma_3'\sin\phi'}{c'\cos\phi' + p^{ref}\sin\phi'} \right)^m \]

\[ E_{oed} = E_{oed}^{ref} \left( \frac{c'\cos\phi' + \sigma_1'\sin\phi'}{c'\cos\phi' + p^{ref}\sin\phi'} \right)^m \]

where \( p^{ref} = 100 \) kPa is the reference pressure.

4.2 Shear Hardening Law

\[ f_s = \frac{1}{E_{50}} \frac{q}{1 - q/q_a} - \frac{2q}{E_{ur}} - \gamma^p \le 0 \]

4.3 Flow Rule (Non-Associated for Shear)

\[ \Delta\varepsilon_v^p = \sin\psi_m \cdot \Delta\gamma^p \]

where \( \psi_m \) is the mobilised dilatancy angle.

4.4 Cap Hardening

\[ \Delta\varepsilon_v^p = \frac{\Delta p}{H} = \left( \frac{1}{K_c} - \frac{1}{K_s} \right) \Delta p \]

\( K_c \) is related to the compression index \( C_c \) and \( K_s \) to the recompression index \( C_r \).

5. Limitations of the Mohr-Coulomb Model and Improvements by HS

flowchart TD subgraph MC["Mohr-Coulomb Limitations"] A1[Linear elastic until failure] A2[Single constant modulus] A3[No distinction loading/unloading] A4[No stress-dependency] A5[Incorrect 1-D compression] A6[Sensitive to Poisson's ratio] end subgraph HS["Hardening Soil Improvements"] B1[Non-linear before failure] B2[Three independent stiffnesses] B3[E_ur ≠ E_50] B4[Power-law stress dependency] B5[Cap yield surface] B6[ν_ur only affects elastic part] end MC --> HS

Aspect	Mohr-Coulomb	Hardening Soil	Practical Consequence in Excavations
Stress-strain curve	Bi-linear	Hyperbolic + elastic unload	HS captures progressive yielding near wall
Stiffness	Constant E	Stress-dependent	HS correctly softens soil inside excavation
Unloading	Same E	Much higher E_ur	HS reduces basal heave and wall toe movement
Ground settlement	Often symmetrical / overestimated	Realistic trough	Critical for adjacent buildings
Plastic zones	Over-predicted	Realistic	Avoids false impression of extensive failure

Classic Illustration (Teo & Wong): When a surcharge is applied behind a shallow excavation, the MC model produces a nearly symmetrical settlement trough because the soil remains elastic. The HS model correctly reduces stiffness in the higher-stress zone nearer the excavation and yields a non-symmetrical, more realistic profile.

6. Proposed Correlations for Analysis of Deep Excavations

When high-quality laboratory data are unavailable, the following practical correlations (adapted from Teo & Wong and Brinkgreve et al.) may be used.

Parameter	Sand	Clay (Undrained Analysis)
\( c' \) (kPa)	0	\( c_u \)
\( \phi' \) (°)	\( 28 + 0.15 D_r (\%) \)	0
\( E_{50}^{ref} \) (MPa)	\( 6 \times e^{0.025 D_r (\%)} \)	\( 0.8 E_u \) (with \( E_u / c_u = 250 \)–300)
\( E_{oed}^{ref} \)	\( E_{50}^{ref} \)	\( E_{50}^{ref} \)
\( E_{ur}^{ref} \)	\( 3 E_{50}^{ref} \)	\( 3 E_{50}^{ref} \)
\( m \)	\( 0.45 + 0.003 D_r (\%) \)	0
\( \psi \) (°)	\( \phi' - 30 \)	0
\( K_0 \)	\( (1-\sin\phi')\sqrt{OCR} \)	\( (1-\sin\phi')\sqrt{OCR} \)

Important Note on Undrained Analysis: Setting \( \phi' = 0 \) and \( c' = c_u \) is a pragmatic approach that works well for excavation problems. However, it is not suitable when consolidation, long-term strength changes, or complex effective-stress paths are important. In such cases a proper effective-stress HS analysis with careful selection of \( \phi' \) and the cap is required.

7. Real-Life Case Studies

7.1 Rochor Complex, Singapore (6.3 m deep)

Sheet-pile wall with three levels of preloaded struts. Marine clay layers dominated the response.

• HS model with \( E_u / c_u = 250 \) matched measured wall deflection well.
• MC model required \( E_u / c_u = 300 \) and still over-predicted ground settlement and basal heave.
• HS produced significantly fewer plastic points – a more realistic indication of the extent of yielding.

7.2 Lavender MRT Station (15.7 m deep)

1 m thick diaphragm wall, six levels of struts. Upper and lower marine clay over dense silty sand.

• HS with \( E_u / c_u = 250 \)–300 gave excellent agreement with inclinometer profiles.
• HS predicted larger (more realistic) surface settlement but smaller basal heave than MC.

7.3 Hougang-Buangkok Cut-and-Cover Tunnel (13.3 m deep)

Sheet-pile wall, four levels of struts in Old Alluvium and residual soils.

• Both models could be calibrated to match wall deflection, but HS with \( E_u / c_u = 250 \)–300 produced superior settlement and heave predictions.
• Strut forces were slightly better matched by the HS model.

International Confirmation

Recent comparative studies (Oued Smar metro station, Algiers; Kwai Chung, Hong Kong; Taipei National Enterprise Center) consistently show that the HS (and especially HS-small) model reproduces measured wall deflections within 5–12 %, while the Mohr-Coulomb model often overestimates displacements by 100 % or more when the same “calibrated” stiffness is used.

8. Solved Examples

Example 1 – Determination of \( E_{50}^{ref} \) and \( m \) from Triaxial Data

Given: Three CD triaxial tests on a medium-dense sand (\( \phi' = 36^\circ \), \( c' = 0 \)):

• \( \sigma_3' = 50 \) kPa → \( E_{50} = 28 \) MPa
• \( \sigma_3' = 100 \) kPa → \( E_{50} = 42 \) MPa
• \( \sigma_3' = 200 \) kPa → \( E_{50} = 62 \) MPa

Solution:

The stress ratio term is \( \frac{\sigma_3' \sin\phi'}{p^{ref} \sin\phi'} = \frac{\sigma_3'}{100} \).

Taking natural logarithms:

\[ \ln E_{50} = \ln E_{50}^{ref} + m \ln\left(\frac{\sigma_3'}{100}\right) \]

Linear regression of \( \ln E_{50} \) vs \( \ln(\sigma_3'/100) \) yields:

• Slope \( m \approx 0.55 \)
• Intercept \( \ln E_{50}^{ref} \approx 3.74 \) → \( E_{50}^{ref} \approx 42 \) MPa

These values can now be used directly in the HS model.

Example 2 – Estimating Parameters for a Dense Sand from Relative Density

Given: Dense sand with \( D_r = 75\% \).

Solution using Teo & Wong correlations:

• \( \phi' = 28 + 0.15 \times 75 = 39.25^\circ \)
• \( E_{50}^{ref} = 6 \times e^{0.025 \times 75} = 6 \times e^{1.875} \approx 39.1 \) MPa
• \( m = 0.45 + 0.003 \times 75 = 0.675 \)
• \( E_{ur}^{ref} = 3 \times 39.1 \approx 117 \) MPa
• \( \psi = 39.25 - 30 = 9.25^\circ \)
• \( K_0^{nc} = 1 - \sin 39.25^\circ \approx 0.37 \)

9. Codal Provisions & Design Considerations

Eurocode 7 (EN 1997) does not prescribe a specific constitutive model, but it strongly encourages the use of models that can capture the relevant aspects of soil behaviour for the limit state under consideration. For Serviceability Limit State (SLS) calculations of wall and ground movements, advanced models such as HS or HS-small are preferred.

• Partial factors may be applied either to actions (Design Approach 1 & 2) or to material properties (Design Approach 3). When using advanced models, material factoring on \( c' \) and \( \phi' \) must be done carefully because the model also contains stiffness parameters.
• Many national annexes and best-practice guides (e.g., CIRIA C760, Singapore Geotechnical Engineering Handbook) now recommend HS or HS-small for detailed design of deep excavations adjacent to sensitive structures.

Practical Recommendation: Use Mohr-Coulomb for preliminary sizing and Ultimate Limit State checks. Switch to Hardening Soil (or HS-small) for final SLS verification and prediction of ground movements.

10. Shortcomings of the Hardening Soil Model

Despite its advantages, the HS model is not perfect:

1. It can over-estimate undrained shear strength of soft clay when effective-stress parameters are used.
2. It under-estimates excess pore pressures in soft clay under undrained conditions.
3. The \( c_u / p' \) ratio is over-predicted for normally consolidated clay and under-predicted for over-consolidated clay.
4. Strength under simple shear and triaxial extension is over-estimated relative to compression.
5. For stress paths inside the yield surfaces the response is purely elastic and inherits all limitations of elasticity.

These issues are largely mitigated in practice by using the total-stress approach (\( \phi' = 0 \), \( c' = c_u \)) for short-term undrained excavation analyses in clay.

11. Best Practices & Recommendations

flowchart TD A[Project Start] --> B{High-quality lab data available?} B -->|Yes| C[Derive E50ref, m, Eoedref, Eurref from tests] B -->|No| D[Use published correlations based on Dr or Eu/cu] C --> E[Calibrate against local case histories if possible] D --> E E --> F[Run HS analysis for SLS movements] F --> G[Compare with MC for ULS structural forces] G --> H[Sensitivity study on key parameters] H --> I[Final Design]

• For clayey soils in Singapore-type conditions, \( E_u / c_u = 250 \) with the HS model has proven robust.
• Always set \( \nu_{ur} = 0.2 \) unless better data exist.
• When using HS-small, the additional small-strain parameters \( G_0^{ref} \) and \( \gamma_{0.7} \) should be estimated from seismic CPT or empirical correlations.
• Document the source of every parameter – laboratory, correlation, or calibration.

12. References

1. Teo, P.L. & Wong, K.S. (2011/2012). Application of the hardening soil model in deep excavation analysis. Underground Singapore / IES Journal Part A.
2. Schanz, T., Vermeer, P.A. & Bonnier, P.G. (1999). The hardening soil model: Formulation and verification. Beyond 2000 in Computational Geotechnics.
3. Brinkgreve, R.B.J. et al. PLAXIS Material Models Manual (various editions).
4. Brinkgreve, R.B.J., Engin, E. & Engin, H.K. (2010). Validation of empirical formulas to derive model parameters for sands. NUMGE.
5. Eurocode 7: EN 1997-1 & EN 1997-2 (Geotechnical design).
6. CIRIA C760 (2017). Guidance on embedded retaining wall design.
7. Lim, A., Ou, C.Y. & Hsieh, P.G. (2010). Evaluation of soil constitutive models for analysis of deep excavation under undrained condition. Journal of GeoEngineering.
8. Recent comparative studies: Oued Smar metro (Algiers), Kwai Chung ELS works, Taipei National Enterprise Center excavation.

Yogendra Gopal Borse

Civil Engineer | Assistant Engineer Grade-I, Maharashtra PWD

B.Tech (Civil) from VJTI Mumbai. Experienced in bridge design, road works, estimation, project monitoring and digital engineering tools. Creator of YogiPWD – practical technical resources for civil engineers.

Read full About →