Due to the popularity of civil works with basements today, the use of software for geotechnical problems: retaining walls of excavation pits is no longer strange. After one or two projects, construction designers can see that the decisive factor to give the results of the problem (displacement of retaining walls, internal forces in retaining walls) that closely reflect the actual operation of the excavation pit retaining structure (obtained from the results of displacement and stress monitoring in retaining structures) is the choice of the type of soil model and the physical and mechanical indicators included in the calculation. Therefore, the topic here does not mention issues related to software operation techniques, or the theoretical basis that these software use to model the soil and how the soil interacts with the retaining structure. Here we only present the application method for design engineers: choosing the calculated values ​​of the physical and mechanical indicators of the ground, as input for the ground model in a convincing way, based on Vietnamese codes and reflecting most accurately the working of the actual ground.

1. Soil density

Includes natural density $\gamma_{unsat}$ and saturated density $\gamma_{sat}$. Enter the calculated values ​​of these parameters according to the calculation process of TCVN 9362:2012 as presented in the topic here.

2. poisson ratio ν

In which the small number corresponds to dense sand and hard clay. From that, we can deduce the coefficient:

$$\beta=1-\frac{2\nu^2}{1-\nu}$$

3. Internal friction angle φ and unit cohesion c

Due to the limitations of the rapid shear test method in accurately reflecting the actual soil behavior as presented. In excavation problems, design engineers need to calculate with effective values ​​of c and φ drawn from the results of the drainage triaxial compression test (CD).

4. Dilatancy angle ψ

This index is not included in the Vietnamese soil mechanics system, so we do not know which Vietnamese term to choose. It is tentatively called the volumetric deformation angle, which characterizes the plastic failure of soil elements.

The Mohr-Coloumb model and the HS model both require this parameter. The definition in Plaxis gives us the following way to determine this parameter:

Definition of angle ψ in Plaxis

The triaxial compression test result chart shows the relationship between volumetric strain εv and axial strain in the compression direction ε1

From the triaxial compression test result chart above (with sandy soil that must be compressed according to the drainage consolidation diagram – CD), determine the Dilatancy angle according to the following definition formula:

$$\sin\psi=\frac{\Delta\varepsilon_v}{-2\Delta\varepsilon_1+\Delta\varepsilon_v}$$

With $\varepsilon_v=\varepsilon_1+\varepsilon_2+\varepsilon_3$ according to soil mechanics theory.

As illustrated in the diagram, take the following values:

$\Delta\varepsilon_v=0.048-0.004=0.044$ and $\Delta\varepsilon_1=-0.09-(-0.03)=-0.06$

Note when choosing the calculation value for ψ: According to the Plaxis manual, except for over-consolidated soils, clay soils usually have an angle of ψ≈0.

For sandy soils, the design consultant needs to request the testing unit to provide the ε1−εv chart from the drained triaxial compression test (CD) to determine ψ according to the above formula. For sandy soils from quartz minerals, the approximate correlation $\psi\approx{\varphi-30^o}$ can be used. For sandy soils with an internal friction angle $\varphi<30^o$, the angle ψ is almost 0.

5. Contact surface coefficient $R_{inter}$

At the soil-structure contact surface (retaining wall, foundation…), soil elements behave differently from the outside soil. Plaxis software takes this phenomenon into account by including the Rinter multiplier in the physical and mechanical parameters compared to the normal soil elements outside.

Plaxis suggests some values ​​depending on the types of soil/structure contact surfaces as follows:

6. Permeability coefficient

Permeability coefficient is important for design problems involving seepage, typically excavations within the depth of the groundwater level, which need to take into account the impact of lowering the groundwater level when excavating on the stress-deformation state of the foundation soil during excavation construction.

In Plaxis, the permeability coefficient is entered horizontally (kx) and vertically (ky). It is best for the design consultant to request the survey unit to provide these data through the results of consolidation compression tests or field seepage tests.

Some values ​​recommended by Plaxis can be found in the document “Advanced course on Computational Geotechnics Singapore” – National University of Singapore – 23-25 ​​November 2011, as follows:

Permeability coefficient according to soil void ratio

Plaxis allows the following correlation between permeability coefficient k and void ratio e:

$$\log\left[\frac{k}{k_o}\right]=\frac{\Delta e}{c_k}$$

with $c_k=10^{15}$

You can refer to the document “Basic Soil Mechanics” by R.Whitlow with the empirical formula for filter sand proposed by Hazen:

$$k=C_kD_{10}^2 (mm/s)$$

where $D_{10}$ is the effective diameter (mm)

$C_k$ is an empirical coefficient depending on the nature of the soil:

Note when entering the input data for the permeability coefficient:

• The minimum and maximum values ​​of the permeability coefficients of the soil layers in the model must not differ by more than 105.
• To simulate a layer of material that is almost impermeable (e.g. concrete), entering a permeability coefficient value of 1000 is sufficient.

7. Module of Deformation

This is an important parameter that has the greatest impact on the results of the construction design problem: displacement and stress in the ground. Therefore, the design engineer needs to pay special attention to choosing the calculation value of this index.

In addition to the important notes used as a basis for correcting the deformation module results obtained from laboratory experiments to be closer to the actual working of the ground as presented in the topic “determining the physical and mechanical indexes of the ground”, it is necessary to pay attention to the definition of the deformation module parameters as input for the Geo5 and Plaxis software.

According to the software definition, the deformation module without lateral expansion (oedometer):

$$E_{oed}=\frac{\sigma_2-\sigma_1}{\varepsilon_2-\varepsilon_1}$$

Whre: εi are unit strains from the stress-strain relationship of the soil element. TCVN often determines the deformation modulus according to the void ratio e when conducting uniaxial compression tests, so it is necessary to find a way to convert from ε to e. According to the definition of the void ratio, the following relationship can be drawn:

$$\varepsilon_2-\varepsilon_1=\frac{e_2-e_1}{1+e_1}$$

According to the definition of deformation module of TCVN:

$E=\frac{1+e_1}{a}\beta$    $a=\frac{e_2-e_1}{\sigma_2-\sigma_1}$

hence:

$$E_{oed}=\frac{1+e_1}{a}=\frac{E}{\beta}; E_{def}=E$$

(def: deformation)

8. Parameters of the Hardening Soil model

The calculation diagram for the design of the foundation structure in the excavation problem using Plaxis software needs to be performed with the Hardening Soil model (HS model). The reason is that during the excavation process, the soil works according to the unloading – reloading diagram. Unloading when the soil in the excavation is taken out and reloaded when constructing the excavation wall support system. During this working phase, the deformation modulus of the soil is much higher than in the case of normal loading (experiments show that it is about 3 to 5 times higher than the normal deformation modulus). Therefore, if the design consultant uses the Mohr-Coulomb model, the displacement and deformation results of the foundation will be much higher than the actual monitoring because it cannot represent the unloading – reloading process of the foundation during the excavation process. Using the HS model allows to overcome this limitation and gives results closer to the actual monitoring.

Compared with the Mohr-Coulomb model, the number of soil physical parameters used as input for the HS model is greater and is interpreted as follows:

Triaxial compression test should be performed according to drained consolidation (CD) scheme.

Distinguish between Secant and Tangent modules

Determine $E_{50}^{ref}$ and $E_{ur}^{ref}$ from the triaxial compression test results with chamber pressure level $\sigma_3=p^{ref}$. Determine $E_{oed}^{ref}$ from the non-flanged consolidation compression test at pressure level $p^{ref}$

Typically, the test results show that the unloading – reloading section is linear as shown in the triaxial test graph above.

How to determine $E_{50}^{ref}$ value from triaxial compression test result chart

How to determine conventional failure strength from triaxial compression test results on soil samples

Define another way to determine $E_{oed}^{ref}$ from the uniaxial compression test results graph

Formula as defined for HS model:

$$E_{oed}=E_{oed}^{ref}\left(\frac{\sigma_y}{p^{ref}}\right)^m$$

Note the stress invariance according to soil mechanics theory as follows: $\sigma’_1-\sigma’_3=\sigma_1-\sigma_3$

When selecting construction design calculation parameters, it is necessary to note the empirical correlations as below.

For normally consolidated clay:

$$m=1$$

$$E_{oed}^{ref}\approx\frac{1}{2}E_{50}^{ref}$$

$E_{oed}^{ref}\approx\frac{50000kPa}{I_p}$ at chamber pressure $p^{ref}=100kPa$

$E_{oed}^{ref}\approx\frac{500kPa}{w_L-0,1}$ according to Vermeer

For Sand:

$$m\approx0,5$$

$$E_{oed}^{ref}\approx{E_{50}^{ref}}$$

for all Soil types:

$$E_{ur}^{ref}=(3\sim5)E_{50}^{ref}$$

$$K_o^{NC}=\frac{\Delta\sigma’_x}{\Delta\sigma’_y}=\frac{\Delta\sigma’_3}{\Delta\sigma’_1}\approx1-sin\varphi$$

Converting CU test results to CD:

In case the geological survey unit does not have the conditions to conduct a drained triaxial compression test (CD) but can only conduct an undrained consolidation test (CU), the design engineer can convert the value obtained from the undrained consolidation test results (Eu) to the value according to the drainage diagram (E) according to the soil mechanics correlation as follows:

$$E_u=E’\frac{1+\nu_u}{1+\nu’}=E\frac{1,495}{1+\nu}$$

Determining the module from the $logp-\varepsilon$ diagram:

In reality, in Vietnam, there are no conditions to conduct a 3-axis test with both unloading and reloading sections according to the CD diagram, but it is more common to use the unloading non-consolidation compression test (oedometer). For clay, the oedometer test results are usually given in the form of a semi-logarithmic $logp-\varepsilon$ diagram as follows:

Graph of the results of the unconfined consolidation compression test in $logp-\varepsilon$ form with loading and unloading sections

Slope of the graph in the loading section:

$$\varepsilon_y=A_{\varepsilon}log\left(\sigma_y\right); A_{\varepsilon}=\frac{\varepsilon_2-\varepsilon_1}{log\left(p_2\right)-log\left(p_1\right)}$$

$$\varepsilon_y=A_{\varepsilon}\frac{ln\left(\sigma_y\right)}{ln10}\longrightarrow\frac{d\varepsilon_y}{d\sigma_y}=A_{\varepsilon}\frac{1}{ln10}\frac{1}{\sigma_y}\longrightarrow{E_{oed}}=\frac{d\sigma_y}{d\varepsilon_y}=\frac{ln10}{A_{\varepsilon}}\sigma_y=\frac{ln10}{A_{\varepsilon}}p^{ref}\left(\frac{\sigma_y}{p^{ref}}\right)$$

The Hardening Soil model is defined as follows:

$$E_{oed}=E_{oed}^{ref}\left(\frac{\sigma_y}{p^{ref}}\right)^m$$

hence m=1 and $E_{oed}^{ref}=\frac{ln10}{A_{\varepsilon}}p^{ref}$

In case the survey unit provides the results in the form of a logp−e chart, the design engineer uses the conversion from ε to e as follows:

$$A_e=\frac{e_2-e_1}{log\left(p_2\right)-log\left(p_1\right)};\varepsilon_2-\varepsilon_1=\frac{e_2-e_1}{1+e_o}\longrightarrow{A_{\varepsilon}}=\frac{A_e}{1+e_o}\longrightarrow{E_{oed}^{ref}}=\frac{ln10}{A_{\varepsilon}}p^{ref}\left(1+e_o\right)$$

Do the same with the unloading section on the graph to find $E_{ur}^{ref}$

🔍Note that all E module values ​​obtained from the above calculations are based on the results of laboratory tests, and need to be multiplied by an adjustment factor to obtain values ​​close to the actual working of the ground as presented above. Can be compared with standard values ​​from TCVN 9362:2012 for deformation module values $E=E_{def}={\beta}E_{oed}$

Determination of the exponential coefficient m

Find the Eoed value at 2 pressure levels $\sigma’_y=p^{ref}=100kPa$ and $\sigma’_y=200kPa$ by drawing tangent lines to the graph at 2 abscissas (the definition of $E_{oed}$is the tangent module) we have:

$$E_{oed}^{\sigma’_y=100kPa}=\frac{320-0}{1.4\%-0.33\%}=29 900kPa$$

$$E_{oed}^{\sigma’_y=200kPa}=\frac{400-0}{1.4\%-0.47\%}=43 000kPa$$

From the definition formula: $E_{oed}=E_{oed}^{ref}\left(\frac{c\cos\varphi-\sigma’_y\sin\varphi}{c\cos\varphi+p^{ref}\sin\varphi}\right)^m$

for Sand: c=0 $\longrightarrow{E_{oed}}=E_{oed}^{ref}\left(-\frac{\sigma’_y}{p^{ref}}\right)^m$

Substitute the number and we get:

$$\frac{E_{oed}^{\sigma’_y=200kPa}}{E_{oed}^{ref}}=\left(\frac{\sigma’_y}{p^{ref}}\right)^m\longrightarrow\frac{43000}{30000}=\left(\frac{200}{100}\right)^m\longrightarrow{m}=0,5$$

9. Experimental requirements to provide data for HS model

The physical and mechanical properties that can be determined from the triaxial compression test: E-modulus, strength (c,φ), dilatancy angle (ψ)

Non-expansion consolidation compression test:

This is not a rapid non-expansion compression test as is often done in geological survey reports to determine the value of the deformation modulus E. This is essentially a slow compression test. In case the survey unit is not qualified to conduct the unloading – reloading test, the design consultant should request them to conduct the unloading – reloading for this non-expansion compression test.

Diagram of non-expansion compression test

Graph of the results of the uniaxial compression test for sandy soil showing the relationship $\sigma’_{yy}-\varepsilon_{yy}$ at loading and unloading levels

References:

• PLAXIS Introductory Course – 10-12 July 2013 – Da Nang, Vietnam
• Manual of Geo5, Plaxis construction design software
• TCVN 9363:2012 “Survey for construction – Geotechnical survey for high-rise buildings”