Gravity wharf failure mechanism and safety analysis considering the wave-structure-soft-soil-foundation interaction

2.1.1. Engineering case study

The case-study wharf is located on the eastern coast of China. It is a gravity structure with a height of 15 m and a width of 12 m. The water depth is 12 m, the backfill prism behind the structure is 8 m wide, the riprap bed beneath the structure is 3 m deep, and the friction coefficient between the riprap and wharf is 0.5. The environment of the project is complex, with deep water and large waves. A geological exploration report indicated that a thick, soft soil layer was distributed around the foundation. Simulation conditions reflecting a harsh wave environment with a 25-year return period, design wave height H_1% of 6 m, and average wave period of 6 s were applied. The initial simulation parameters are listed in Table 1.

2.1.2. Selection of the computational domain and boundary conditions

A cross-sectional diagram of the structure model is shown in Fig. 1. Contact points were set in the contact area between the structure and soil. Considering that the elastic modulus of the structure was significantly larger than that of the soil, the contact surface on the structure was designated as the primary contact surface, and that on the soil was the secondary contact surface. The constitutive relationship between the structure and soil was described by a Mohr-Coulomb ideal elastic-plastic model using eight-node three-dimensional reduced integral solid elements. To eliminate the influence of boundary conditions on the results, the surface of the foundation was assigned a free boundary, the bottom of the model was assigned a fixed boundary, the front and rear sides of the model were assigned limited boundaries, and the left and right sides of the model were assigned symmetrical boundaries.

Fig. 1Model wharf structural section

2.1.3. Soil constitutive model and material parameters

The riprap foundation bed material can be considered a homogeneous continuous medium that can only withstand compressive stress; therefore, the constitutive relationship of the soil was simplified to a linear elastic model. The model parameters included an axial stiffness $EA$ of 1.5×10⁷ kN/m, stiffness $E$ of 3.125×10⁵ kN/m, and Poisson’s ratio $\mu$ of 0.15. When the shear stress in any plane of the soil reached the soil shear strength, the foundation was considered to be destroyed.

To ensure fast calculation convergence, the elastic modulus of the gravity structure was set to 10 times that of the riprap backfill. Owing to the bite force between the riprap backfill, the adhesive force was set to 20 kPa. The friction between the gravity structure and riprap backfill was simulated using the built-in interface function, and the maximum shear stress was calculated as:

where $c$ is the cohesion along the interface, $A$ is the interface contact area, $\varphi$ is the interface friction angle, $p$ is the pore water pressure, and $F_n$ is the positive interface stress.

Owing to the absence of cohesion between the riprap foundation bed and the gravity structure, $c$ was set to 0. For the interface angle between the gravity structure and riprap backfill, the friction coefficient $\tan \varphi$ was set to 0.5, with $\varphi =$ 27°. For the interface function, $k_n$ and $k_s$ were considered 10 times the maximum modulus of the adjacent grid stiffness. The wavelength $L$ was determined to be 56.2 m using the following equation:

where $g$ is the gravitational acceleration, $\overline{T}$ is the average period, and $d$ is the water depth.

The geology of the soil layers was simulated as presented in Table 2 from top to bottom in the model, while the mechanical parameters of the soil body are based on the soft soil tests described in Section 2.2.

2.1.4. Stress application

The equivalent load method was used to convert the dynamic pressure generated by waves into an equivalent static load, that is, the maximum pressure on the soil surface under wave action was calculated as the static load applied to the nodes on the soil surface of the model structure. Considering the effect of peak waves, wave forces are applied to the vertical walls and bottom of gravity structures using the method specified in the JTS 145-2015 Port and Waterway Hydrological Code [26]. The periodicity of waves was simulated by changing the load over time, and the cyclic weakening law of soft-soil strength with wave cyclic loads was simulated by changing the internal friction angle. The unit grid diagram of the model is shown in Fig. 2, and the pressure distribution diagram is shown in Fig. 3.

Fig. 2Unit grid diagram

Fig. 3Stress distribution map

2.2.1. Triaxial shear tests

The experimental process comprised five steps: sample loading, back pressure saturation, B-value detection, consolidation, and shearing. Step 1: The undisturbed soil was pushed out of the thin-walled sampling cylinder and cut into four cylindrical specimens (numbered S1-S4) with a diameter of 75 mm and height of 150 mm. Step 2: Each sample was vacuum saturated and then loaded into the GDS dynamic triaxial shear apparatus for back pressure saturation. Step 3: After 24 h of saturation, B detection was used to check the saturation degree of the soil sample. If the pore pressure coefficient B value of the test was larger than 0.98, the soil sample was considered to have reached the saturation requirement. Step 4: Initial consolidation pressures of 100, 200, 300, and 400 kPa were applied to the soil sample for isobaric consolidation. When the pore water pressure dissipated to equal the back pressure, the soil sample was considered to have completed consolidation. Step 5: Undrained shear tests were conducted using strain control. The experimental plan and sample information are shown in Table 4.

2.2.2. Dynamic triaxial tests

To study the influence of post-consolidation structure on the dynamic characteristics of soft clay, saturated triaxial specimens were used and subjected to graded isotropic consolidation loading with consolidation stresses of 100, 200, and 300 kPa. Unidirectional excitation was performed 3 times at each consolidation stress level with vibration amplitudes of 0.005, 0.01, 0.02, 0.04, 0.08, 0.16, 0.32, and 0.64 mm, with a total of 24 excitation times. To reduce disturbance errors and ensure the relative continuity of test data, the number of test vibrations was set to five weeks for each vibration amplitude of consolidation stress at each level. The experiment adopted a stress-controlled loading method for cyclic loading, and the loading waveform was generated by the servo system. The waveform was a sine wave with a frequency of 1 Hz. During the loading process, data such as loading frequency, consolidation pressure, pore pressure, and axial deformation were collected and processed by the computer, thus resulting in a total of 3000 sets of unidirectional excitation cyclic load test results.

The results were fitted using the Janbus equation, which is typically applied to determine the relationship between $G_{dmax}$ and ${\sigma }_m^{\text{'}}$ as follows:

where $G_{dmax}$ is the maximum shear modulus, $K$ is the material constant, $P_a$ is the atmospheric pressure, ${\sigma }_m^{\text{'}}$ is the consolidation stress, and $m$ is the modulus.

The shear modulus $G$ under each consolidation stress was divided by the corresponding maximum shear modulus $G_{max}$ and normalized using the Hardin-Drnevich equation as follows:

where $\gamma$ is the shear strain and ${\gamma }_{ref}$ is the reference shear strain.

The Van Genuchten soil-water characteristic curve equation for moisture and matric suction in unsaturated soil mechanics is highly flexible in fitting data and is expressed by:

where $\mathrm{\Theta }$ is the relative moisture content; $\psi$ is the matric suction; and $\alpha$, $m$, and $n$ are fitting parameters.

When the Van Genuchten equation is used to describe the relationship between shear modulus and shear strain, it can be simplified by setting $m=$ 1. Thus, the simplified relationship between $G$ and $\gamma$ is given by:

3.1.1. Triaxial shear test

The GDS moving triaxial cyclic shear system measured and recorded the changes in each data point over time. Figs. 5 and 6 present the variations in soil sample pore pressure and soil sample volume over time under cyclic loading, respectively. The pore pressure and deformation of the soft soil clearly decreased rapidly in the early period before stabilizing.

Fig. 5Variation of the soil sample pore pressure over time under confining pressures

a) 100 kPa

b) 200 kPa

c) 300 kPa

d) 400 kPa

Fig. 6Variation of the soil sample volume over time under confining pressures

a) 100 kPa

b) 200 kPa

c) 300 kPa

d) 400 kPa

The principal stress difference and principal stress ratio in each sample are depicted according to the axial strain in Figs. 7 and 8, respectively. The stress-strain relationships exhibit similar characteristics and patterns of change regardless of the applied confining pressure. Thus, the confining pressure has little influence on the stress-strain relationship. The soil compression curve shows strain reinforcement, exhibiting no softening phenomenon; therefore, the peak occurring when the axial strain reaches 15 % was considered the damage point. Under small axial strains, the principal stress difference and principal stress ratio of the soil samples increased rapidly with increasing strain, particularly when at a confining pressure of 400 kPa. This tendency may be due to the relatively large pores of soft soil, which facilitates extensive compression of the pore volume but does not fundamentally change the microstructure of the soil; thus, the shear strength remains low. High axial strain not only compresses the pore volume but also changes the microstructure of the soil by arranging more closely the soil particles, hardening the soil skeleton. Thus, the ability of the soil to resist external deformation is significantly improved, and the shear strength is enhanced.

Fig. 9 shows the effective stress paths of the soft soil specimens. The higher the confining pressure, the higher the slope of the stress path and the more evident its change. The four test soil specimens reached a critical state during the shear process, thus exhibiting unique critical state lines. Fig. 10 depicts the consolidated undrained strength envelopes of the soil specimens. As the accumulation of pore water pressure with increasing confining pressure reduced the effective stress, the Moore circle moved to the right along the transverse axis. Soil shear damage was considered to occur when the strength envelope was tangential to the Moore circle. Thus, the effective cohesion $c\text{'}$ was 6 kPa, and the effective internal friction angle was 26.5°.

Fig. 7Principal stress difference vs. axial strain curves

Fig. 8Principal stress ratio vs. axial strain curves

Fig. 9Effective stress paths

Fig. 10Consolidated undrained strength envelopes of the soft soil specimens

3.1.2. Vibration test

During each stage of vibration, the changes in the axial displacement, axial force, and pore pressure were measured to analyze the damping ratio and modulus of the soil. The trends in the vibration curves for each evaluated excitation were similar. Therefore, the results obtained under an isotropic consolidation stress of 300 kPa and vibration amplitude of 0.005 mm (as shown in Fig. 11) were considered examples for this discussion. The changes in the axial displacement, axial force, and pore pressure vibration curves were similar, following a trend of increasing to a peak before decreasing to a trough with the progression in the applied vibration cycle. All variables exhibited ideal sine wave changes that met the conditions of elastic vibration. However, a slight pore pressure oscillation and growth phenomenon occurred during the excitation process.

The variation in the shear modulus with shear strain under different consolidation stresses is presented in Fig. 12. For each level of consolidation pressure, the shear modulus decreased with increasing shear; the shear modulus under a high consolidation stress was consistently higher than that under a low consolidation stress. By using the Janbus equation to fit the relationship between the maximum shear modulus $G_{dmax}$ and consolidation stress ${\sigma }_m^{\text{'}}$, the values $m=$ 7.414 and $K=$ 74.764 were obtained, as shown in Fig. 13. Fig. 14 depicts the vibration damping ratios according to shear strain under different consolidation stresses. Regardless of the consolidation stress, the damping coefficients were relatively close and increased with increasing shear strain.

Fig. 11Variations in a) axial displacement, b) axial force, and c) pore pressure during the 0.005 mm amplitude vibration test at an isotropic consolidation stress of 300 kPa

a)

b)

c)

Using the Hardin-Drnevich equation for normalization, the fitting coefficient $R^2$ was determined as 0.9765 for ${\gamma }_{ref}=$ 0.091, indicating an acceptable degree of fit, as shown in Fig. 6(d). Finally, using the improved Van Genuchten equation for nonlinear fitting, $R^2$ was determined as 0.9800 for ${\gamma }_{ref}=$ 0.087 and $n=$ 1.12, indicating a better fit, as illustrated in Fig. 15.

Fig. 12Variations in the shear modulus with shear strain at different consolidation stresses

Fig. 13Relationship between maximum shear modulus and consolidation stress

Fig. 14Vibration damping ratios according to shear strain at different consolidation stresses

Fig. 15Shear modulus varies with damping ratio at different consolidation stresses