Study on the Characteristics of Effective Internal Friction Angles of Silty Clay from the Yellow River Delta and the Inverse Method of CPTu Testing

Abstract

This study focuses on the silty clay of the Yellow River Delta, conducting laboratory experiments to explore the strength characteristics of typical silty clay in the Yellow River Delta. The study utilizes CPTu calibration chamber tests to systematically reveal the features of cone tip resistance (q_t), side friction resistance (f_s), and pore water pressure (u₂) of the silty clay. This research provides a theoretical basis for Delta investigation in the Yellow River region. The study compares the applicability of existing CPTu inversion methods and highlights the NTH method’s advantages in evaluating silty clay’s effective internal friction angle. Supported by indoor calibration chamber tests, the study confirms the reliability of the NTH method in estimating the effective internal friction angle under normally consolidated conditions while also identifying its limitations. These research findings offer data support for the in situ rapid and accurate estimation of design parameters, like the internal friction angle and undrained shear strength of the silty clay in the Yellow River Delta. Moreover, they provide insights for obtaining other crucial in situ data.

1. Introduction

The CPTu test can provide three independent parameters for various soil media: cone tip resistance q_c, side wall friction resistance f_s, and pore water pressure u₂. The formula can also determine additional parameters, including net cone tip resistance q_net, pore pressure ratio B_q, and normalized cone tip resistance Q_t. The CPTu test can be used to evaluate a wide range of geotechnical engineering parameters, such as the soil classification (SBT), gravity (γ), overconsolidation ratio (OCR), undrained shear strength of clay (s_u), effective internal friction angle of sand (φ′), Young’s modulus (E), static lateral pressure coefficient (K₀), consolidation coefficient (c_v), and the evaluation of soil liquefaction [1,2].

Numerous engineering projects characterize the Yellow River Delta. In this region, silty clay is distributed widely [3]. Establishing an inverse method for determining the mechanical parameters of silty clay based on in situ testing is of significant importance. In the research on inverse parameter determination through in situ testing of silty clay in the Yellow River Delta [4,5], there is currently a need for more studies on internal friction based on the CPTu method. The internal friction angle is primarily obtained through sampling tests [6,7,8]. The internal friction angle is crucial in the inverse determination of lateral earth pressure coefficients and other related factors [9,10]. It is of great value to study the inversion method of the internal friction angle of silty clay in the Yellow River Delta based on in situ testing [11].

To assess the effective internal friction angle φ′ of different soils (including sand, silt, and clay) under the drained to undrained penetration circumstances, a plastic solution of effective stress was designed for the CPTu test. Janbu and Senneset [12] at Norges Tekniske Hgskole (NTH) suggested this method, and Senneset and Janbu [13], Senneset et al. [14], Sandven [15], and Sandven and Watn et al. [16] perfected it. However, the NTH method is now only applied in Norway and a small portion of Sweden, and it is still not widely employed in other regions. Since soil’s effective internal friction angle remains relatively consistent for soils with similar mineral compositions and particle distributions, this experiment extracted representative soil samples from a silty clay area site. Through indoor calibration chamber tests, it is possible to calibrate the suggested values for the effective internal friction angle so that they are suitable for this specific region. This approach offers a more efficient method for obtaining in situ parameters such as the effective internal friction angle and undrained shear strength during future exploration activities in the same region. Consequently, the aim of this paper has four aspects: (a) checking and evaluating the procedure of effective internal friction angle φ′; (b) data interpretation by the NTH method; (c) Comparing the interpretation value of the effective internal friction angle φ′ of silty clay in the Yellow River Delta in the indoor calibration chamber test with the φ′ reference value obtained from the indoor unit test.

2. Inverse Method of Effective Internal Friction Angle Based on the CPTu Test

Over pore pressure (Δu > 0) will appear in soft to hard intact clay during the CPTu test. Janbu and Senneset [12] proposed an effective stress-critical plastic solution for CPTu under undrained conditions, which can be used to evaluate φ′. The cone tip resistance coefficient N_m is defined as:

$$N_{\mathrm{m}}=\frac{N_{\mathrm{q}}-1}{1+N_{\mathrm{u}}{B}_{\mathrm{q}}}=\frac{q_{\mathrm{t}}-{\sigma }_{\mathrm{v}0}}{{{\sigma }^{\prime }}_{\mathrm{v}0}+a^{\prime }}$$

(1)

where

B_q—normalized pore pressure parameter;

N_u—cone tip resistance;

$a^{\prime }$—Effective gravity;

$c^{\prime }$—Effective cohesion.

The tip bearing capacity factor N_q and the pore pressure bearing coefficient N_u are defined as:

$$N_q=K_{\mathrm{p}}\exp \left[\left(\mathsf{\pi }-2\beta \right)\tan {\phi }^{\prime }\right]$$

(2)

$$N_{\mathrm{u}}=6\tan {\phi }^{\prime }\left(1+\tan {\phi }^{\prime }\right)$$

(3)

$$K_{\mathrm{p}}=\left(1+\sin {\phi }^{\prime }\right)/\left(1-\sin {\phi }^{\prime }\right)$$

(4)

where K_p is the passive transverse stress coefficient; β is the plasticizing angle (−40 < β < +30), which defines the size of the soil failure area [13]; the plastic solution can explain the parameters (added together) of effective stress molar Coulomb strength, and it applies to all soil types, including sand, silt, clay, and mixed soil [17,18]; Parameter N_q is also the tip-bearing capacity factor in pile foundation; when β = 0°, N_q is the same as the deep foundation classical method (Terzaghi equation); and parameter N_u is the bearing coefficient of pore pressure. Sandven [15] used six onshore and one offshore Norwegian clay samples to calibrate the NTH method analysis model, and the calibration results were in line with the results of finite element simulation.

At that time, the parameter N_m was the same as the normalized cone tip resistance coefficient Q_t, and Q_t was widely used in the analysis of CPTu test data [19,20]. Therefore, the relationship with CPTu parameters (Q_t and B_q) is shown in Formula (5), and the relationship is shown in Figure 1.

$$Q_{\mathrm{t}}=\frac{{\tan }^2({45}^{\circ }+{\phi }^{\prime }/2)\exp (\pi \tan {\phi }^{\prime })-1}{1+6\tan {\phi }^{\prime }(1+\tan {\phi }^{\prime })B_{\mathrm{q}}}$$

(5)

Equation (5) cannot be applied in practice [21], so an approximate formula is proposed, which can be used when 20 ≤ ${\phi }^{\prime }$ ≤ 45 and 0.1 ≤ B_q ≤ 1.0. The formula is shown in Formula (6):

$${\phi }^{\prime }\approx 29.5B_{\mathrm{q}}^{0.121}\left(0.256+0.336B_{\mathrm{q}}+\lg Q_{\mathrm{t}}\right)$$

(6)

The approximate solution is shown in Figure 1, which shows that the approximate value of sum B_q is in good agreement with the theoretical value.

3. Yellow River Delta Silty Clay Effective Internal Friction Angle Test

The experimental soil used in this article is excavated from the shore of the Yellow River Delta region. The calibration of its basic physical and mechanical parameters can provide a research basis for interpreting subsequent CPTu in situ test data parameters. The sampling area is shown in Figure 2.

3.1. Basic Parameters of Silty Clay

A comprehensive set of geotechnical tests were performed on silty clay in the Yellow River Delta, encompassing density, water content, specific gravity, particle size, and liquid-plastic limit tests. The findings of the combined particle size and liquid-plastic limit tests are outlined below. The test results of silty clay in the Yellow River Delta are shown in Figure 3. Figure 3 shows the uneven coefficient of soil C_u = d₆₀/d₁₀ = 7.3 < 10, and the soil particles are relatively uniform. The curvature coefficient C_c = d₃₀²/(d₁₀·d₆₀) = 1, and the curve continuity is good. The fundamental physical and mechanical parameters are presented in

Table 1. Basic physical and mechanical parameters of silty clay in the Yellow River Delta.

Soil Layer	Moisture Content (%)	Liquid Limit (%)	Plastic Limit (%)	Plasticity Index	Initial Void Ratio
Silty clay	23.00	33.50	19.00	14.50	0.67

. Combining the information from Table 1 and Figure 4, it can be concluded that the soil in question is a low-liquid-limit silty clay. The consolidation test results is shown in Figure 5.

3.2. One-Dimension Consolidation Test

Through conversion by the e-lgp curve, index C_s = 0.0704 can be obtained, which is converted into the isotropic expansion index κ = C_s/ln10 = 0.00306; Compressibility index C_c = 0.119 is converted into logarithmic hardening modulus λ = C_c/ln10 = 0.0518. The maximum void ratio on the regular consolidation state line is converted to e_N = 0.89.

3.3. Triaxial Shear Test

The equipment used in the triaxial shear test in this paper is the GDS triaxial apparatus. The triaxial element is the Bishop and Wesley stress path, which is used to test cylindrical samples with a diameter of 50 mm and a height of 100 mm.

Figure 6 shows the triaxial test results of the consolidation–drainage shear test (CD) of silty clay in the Yellow River Delta under the expected consolidation-effective stress, σ_v′ = 100 kPa.

Figure 6 illustrates the stress path of the Yellow River silty clay in p′-q space in the CD test; σ₁ = σ₂ = σ₃ during consolidation, so the deviatoric stress q = 0. During the shearing process, the poor pore connectivity of silty clay causes the excess pore pressure to dissipate more slowly than the volume strain, and the pore pressure gradually reaches the maximum value. Then, with the intervention of the back pressure system, the water is forcibly discharged and the pore pressure gradually decreases, so the average stress first decreases, then gradually increases.

According to the test data, the critical state line is made, and its slope M = 1.08. According to Formula (7), the effective internal friction angle φ′ of silty clay in the Yellow River Delta can be calculated as 27.2°; analysis shows that the effective internal friction angle of silty clay in the region varies within the range of 26.7° to 40°, as seen in Figure 7.

$$M=\frac{6\sin {\phi }^{\prime }}{3-\sin {\phi }^{\prime }}$$

(7)

Based on the above consolidation test and triaxial shear test results, the basic model parameters κ, λ, e_N, and M of silty clay in the Yellow River Delta can be obtained [24]. Zhang et al. [5] proposed that according to the conventional modified Cambridge consolidation theory, the undrained shear strength of clay can be determined by the void ratio e of soil:

$$s_{\mathrm{u}}=\frac{M}{2}\exp \left(\frac{e_{\mathrm{N}}-(\lambda -\kappa )\ln 2-e}{\lambda }\right)$$

(8)

The basic model parameters of silt in the Yellow River Delta are shown in

Table 2. Basic Model Parameters.

κ	λ	eN	M
0.0306	0.0518	0.89	1.08

:

According to Formula (8), the undrained shear strength value s_u = 19.66 kPa, corresponding to the triaxial test can be obtained.

4. CPTu Calibration Test

4.1. Test Device

This paper designs and develops an indoor CPTu calibration chamber device for experimental research to provide a reference for internal friction angle inversion in the field, as shown in Figure 8.

The top of the consolidation apparatus is equipped with a cylinder that applies pressure to the soil by adjusting the air pressure, simulating the stress state of the soil at different depths. The device can provide a maximum vertical load of 200 kPa. At the same time, the probe penetration system is designed to be used to drive the probe at a penetration rate of 0.1~10 mm/s [25] to simulate different drainage conditions in the soil.

Figure 8 is a miniature CPTu device used in this experiment. The probe can measure cone tip resistance q_c and pore water pressure u₂. The diameter of CPTu used in the calibration room test is d = 16 mm, and the cone angle is 60°. The miniature cone penetration tester used in this research is a subtractive CPTu. The force sensor and pore pressure sensor directly measure the cone tip resistance and pore water pressure, respectively. The cable passing through the hollow steel shaft is connected to the cone tip and connected to the cone penetrometer actuator. Figure 9 illustrates the experimental procedure.

In silty clay, when the ratio of the diameter of the soil sample to the diameter of the CPTu probe is greater than 25, the influence of boundary effects can be ignored. The design of this consolidation apparatus takes this factor into consideration. The probe diameter is designed to be 16 mm, and the soil sample diameter is 600 mm, resulting in a diameter ratio of 37.5. This effectively eliminates the influence of rigid boundary effects.

4.2. Test Plan

In this paper, the CPTu penetration test at the same rate is carried out mainly for silty clay in the Yellow River Delta. One group of tests can conduct four penetration tests for one kind of soil. According to the normalized penetration rate formula V = vD/c_v [26,27], a high-speed CPTu penetration test (completely undrained) is designed, and the test load of each group is 100 kPa, which simulates the soil with the same depth on site. The specific test scheme is shown in

Table 3. Indoor CPTu test plan.

OCR	Undrained Shear Strength su (kPa)	Coefficient of Consolidation cv (cm2/s)	Penetration Rate (mm/s)	Drainage Condition
1	19.66	0.0053	5	Completely undrained

.

5. Test Results and Analysis

5.1. CPTu Test Results

The variation regulation of cone tip resistance with depth during penetration is shown in Figure 10. At the beginning of the test, when the penetration is about 50 mm, the cone tip resistance increases sharply, and the maximum values achieved by four repeated tests are quite different, ranging from about 480 kPa to about 350 kPa. It is mainly the over-consolidation of surface soil that causes this phenomenon. When CPTu penetrates to 50 ~ 150 mm, q_c still fluctuates, but the range is reduced. Until the penetration depth exceeds 150 mm, q_c presents a stable state and enters the stable penetration zone. The q_c test results of the same CPTu indoor test are in good agreement, although there are some deviations. The cone tip resistances q_c of the four tests are 312 kPa, 323 kPa, 320 kPa, and 322.4 kPa. The relationship curve between the excess pore pressure Δu and the depth h is shown in Figure 10. Unlike the cone tip resistance q_c, the excess pore pressure Δu gradually increases in the 0 ~ 150 mm range. In this interval, the increased speed of Δu in the four tests is different, and the gap is significant. When CPTu penetrated 150 mm, the difference in test results gradually decreased and reached stability. The final pore pressures Δu are 150.3 kPa, 148 kPa, 142 kPa, and 153.7 kPa.

5.2. Inversion Results of NHT Method

The cone tip resistance coefficient (N_m = Q_t) is the slope of the straight line in Figure 11, with the net cone tip resistance q_net in the stable penetration section as the ordinate and the effective overburden pressure as the abscissa, as shown in Figure 11a. In this example, if a straight line is forced through the origin ($c^{\prime }$ = 0), the normalized cone-tip resistance obtained is Q_t = 2.03, 2.10, 2.11, and 2.13. Similarly, the pore pressure parameter B_q is the slope of the straight line in the graph with the measured excess pore pressure Δu₂ = u₂–u₀ as the ordinate and q_net as the abscissa, and B_q = 0.55, 0.67, 0.66 and 0.69 are obtained, as shown in Figure 11b. Taking the values of B_q and Q_t into Figure 12, it can be concluded that silty clay’s effective internal friction angle in the Yellow River Delta is 26.8° according to the NTH method. The triaxial test is carried out on undisturbed soil samples of soft clay, and the reference value of the effective internal friction angle in the laboratory is obtained and compared with the value obtained by the NTH method, as shown in Figure 13. The results show that the evaluation results of the NTH method are in good agreement with the laboratory results.

5.3. Inversion of Undrained Shear Strength Based on CPTu

Undrained shear strength s_u is a critical design parameter in geotechnical engineering construction, which the existing CPTu cannot directly measure. Presently, the most commonly used means is to interpret it by adjusting the direct measurement parameter q_c of CPTu. Due to the influence of the pore pressure effect and overlying stress, the cone tip resistance must be corrected by the following formula.

$$q_{\mathrm{t}}=q_{\mathrm{c}}+u_2\left(1-\alpha \right)$$

(9)

$$q_{\mathrm{net}}=q_{\mathrm{t}}-{\sigma }_{\mathrm{v}0}$$

(10)

where q_t is the corrected cone tip resistance; q_c is the measured cone tip resistance; u₂ is the pore pressure at the cone shoulder position; α is the ratio of the cross-sectional area between the probe top column and the cone bottom, the value in this article is 0.8; q_net is the net cone tip resistance; and σ_v0 is the overlying stress.

An empirical parameter, cone coefficient N_kt, is also needed to interpret undrained shear strength. Scholars worldwide have developed many theoretical circles, including Lu et al. [28]. The following approximate expressions are derived from the analysis:

$$N_{\mathrm{kt}}\approx 3.4+1.6\ln I_{\mathrm{r}}-1.9\Delta +1.3{\alpha }_{\mathrm{c}}$$

(11)

where I_r is the rigidity index with I_r = G/s_u; Δ = (σ_v0 − σ_h0)/2s_u; and α_c is the friction ratio of the cone-soil interface, which is usually 0.2~0.6, and this paper takes 0.5. By calculation, I_r = 890.2, Δ = 1.164, and N_kt = 12.71.

$$s_{\mathrm{u}}=\frac{q_{\mathrm{net}}}{N_{\mathrm{kt}}}=\frac{q_{\mathrm{t}}-{\sigma }_{\mathrm{v}0}}{N_{\mathrm{kt}}}$$

(12)

$$N_{\mathrm{kt}}=\frac{q_{\mathrm{net}}}{s_{\mathrm{u}}}$$

(13)

The recommended N_kt value for the silty clay layer in the Yellow River Delta was obtained by calibrating the static penetration test data q_net and the undrained shear strength obtained from indoor triaxial tests using Formula (13). The comparison with the theoretical value is well-fitted and can be used as an empirical coefficient for the silty clay layer in this region. The Comparison of N_kt inversion results with standard values is shown in Figure 14.

5.4. Limitations of NTH Method

To reliably evaluate results of the clay by the NTH method, it is necessary to measure the quality of q_t and u₂ [29]. Therefore, the porous filter element and cone cavity should be saturated during the field test [30]. In some geological environments, the evaporation belt composed of the upper dry or unsaturated soil leads to the unsaturated porous filter element, which makes the u₂ readout inaccurate. If various saturated liquids (water, glycerin, and silicone oil) with low vacuum degrees are used to prepare porous filter elements, the pore pressure reading will also be affected. The total cone tip resistance q_t also depends on the u₂ readout. Therefore, during the CPTu measurement process, attention should be paid to maintaining the appropriate saturation.

For cracked or overconsolidated soil, u₂ is usually negative and B_q < 0 accordingly [30,31]. Sandven [15] found that the NTH method cannot directly use negative B_q values”, so the NTH method should not be applied to these types of soils unless this problem can be solved from an academic point of view.

6. Conclusions

This research presents a comprehensive study on normally consolidated silty clay from the Yellow River Delta through geotechnical and laboratory CPTu tests. Utilizing results from indoor calibration chamber tests, cone tip resistance (q_c) and pore water pressure (u₂) were obtained. These values were then used to interpret essential parameters such as effective internal friction angle and the coefficient N_kt for determining undrained shear strength. These interpretations were compared with indoor test results. The research findings are summarized as follows:

• A series of indoor geological tests were conducted on samples of silty clay from the Yellow River Delta to determine their basic physical and mechanical properties.
• The NTH (Normalized Tangent Hyperbolic) method was applied using data obtained during the penetration phase of the indoor calibration chamber test. Analysis indicated that while the NTH method might yield slightly lower effective internal friction angle results for the silty clay, the overall consistency is good. It suggests the reliability of the NTH method for inferring an effective internal friction angle (φ′) for normally consolidated silty clay in the Yellow River Delta. Additionally, this approach avoids the complex and extensive field testing required to establish patterns and, inversely, model detailed parameters of a specific stratum.
• The coefficient N_kt for the silty clay in the Yellow River Delta region was successfully determined through inverse analysis. The results show excellent agreement with theoretical values, providing recommended values for N_kt. This information is essential for deriving the undrained shear strength during the inverse process and offers crucial empirical coefficients for in situ geological testing in this region.