Consolidation by Vertical Drains Considering the Rheological Characteristics of Soil under Depth and Time-Dependent Loading
Abstract
:1. Introduction
2. Basic Assumptions and Mathematical Modelling
- (a)
- Compressive strain occurs only in the vertical direction, and the radial sections of the cell remain radial during the consolidation; that is, the equal vertical strain condition is valid.
- (b)
- Both vertical and radial flows obey Darcy’s law.
- (c)
- The radial flow in the vertical drain is neglected and the radial flow from the soil into the vertical drain at any depth is equal to the corresponding increase in flow up the vertical drain.
- (d)
- The constitutive relationship of the soil follows Equation (2).
- (e)
- The radial coefficient of permeability of the smear zone is smaller than that of undisturbed soil, but the other physical properties of the soil in the smear zone are the same as those in the undisturbed zone.
3. Solutions
3.1. Solutions for Instantaneous Loading
3.1.1. Average Excess Pore Water Pressure under Instantaneous Loading
3.1.2. The Overall Average Degree of Consolidation under Instantaneous Loading
3.2. Solutions for One-Step Loading
3.2.1. Average Excess Pore Water Pressure under One-Step Loading
- (i)
- Method 1
- When (loading phase),
- When (constant loading phase),
- (ii)
- Method 2
3.2.2. The Overall Degree of Consolidation under One-Step Loading
- When (loading phase),
- When (constant loading phase),
3.3. Solutions for Multi-Step Loading
3.3.1. Average Excess Pore Water Pressure under Multi-Step Loading
- For (loading phase),
- For (constant loading phase),
3.3.2. The Overall Degree of Consolidation under Multi-Step Loading
- For (loading phase),
- For (constant loading phase),
3.4. Solutions for Cyclic Loading
3.4.1. Average Excess Pore Water Pressure under Cyclic Loading
- For (loading phase),
- For (constant loading phase),
- For (unloading phase),
- For (zero-loading phase),
3.4.2. The Overall Average Degree of Consolidation under Cyclic Loading
- For (loading phase),
- For (constant loading phase),
- For (unloading phase),
- For (zero-loading phase),
4. Special Cases
4.1. Simplification of the Model
- If (i.e., ), the Merchant model is adopted. Hence, the corresponding coefficients given as Equations (21)–(28) will be simplified as follows:
- If (i.e., ) or (i.e., ), the Maxwell model is considered. Thus, the corresponding coefficients given in Equations (21)–(28) will be simplified as follows:
- If and (i.e., and ) or and (i.e., and ), the linear elastic model is adopted. Hence, the corresponding coefficients given in Equations (21)–(28) will be simplified as follows:
4.2. Solutions for Different Constitutive Models and Loading Types
5. Model Validation
6. Consolidation Behavior Analysis
6.1. Consolidation Behavior Analysis under One-Step Loading Condition
6.2. Consolidation Behavior Analysis under Cyclic Loading Condition
7. Discussion
8. Conclusions
- Based on Barron’s theory of equal strain consolidation, a four-element model was used to consider the rheological characteristics of soil, and a set of analytical solutions was developed for consolidation with vertical drains under depth and time-dependent loading. The increase in additional stress is a function depending on both time and depth. It is assumed to vary linearly with depth, and several time functions are considered to represent different loading cases, which include instantaneous loading, one-step loading, multi-step loading, and cyclic loading.
- The consolidation rate is accelerated with the decrease in loading time and the increase in (the value of the top-to-bottom additional stress ratio). With the decrease both of the modulus of the spring in the Kelvin body and the viscosity coefficient of the independent dashpot, the rheological behavior becomes more and more obvious at the later stage of consolidation. The rate of consolidation becomes faster at an early stage but slower at a later stage, with the increase in the viscosity coefficient of the dashpot in the Kelvin body.
- For cyclic loading, the consolidation degree in each cycle reaches a maximum at the end of unloading and reaches the minimum at the beginning of the loading. When the number of cycles increases to a certain value, the variation form of consolidation degree curves will tend to be stable.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Model | Average Excess Pore Water Pressure and Overall Average Degree of Consolidation |
---|---|
Merchant model | |
are given in Equations (21), (22), (66) and (67), respectively. | |
Maxwell model | |
are given in Equations (71), (73), and (75), respectively. | |
Linear elastic model | |
are given in Equations (71) and (77), respectively. |
Model | Average Excess Pore Water Pressure and Overall Average Degree of Consolidation |
---|---|
Merchant model | |
are given in Equations (21), (22), (66) and (67), respectively. | |
Maxwell model | |
are given in Equations (71), (73) and (75), respectively. | |
Linear elastic model | |
are given in Equations (71) and (77), respectively. |
Model | Average Excess Pore Water Pressure and Overall Average Degree of Consolidation |
---|---|
Merchant model | |
, , and are given in Equations (21), (22), (66) and (67), respectively; and | |
Maxwell model | |
, and are given in Equations (71), (73) and (75), respectively; and | |
Linear elastic model | |
and are given in Equations (71) and (77), respectively; and |
Model | Average Excess Pore Water Pressure and Overall Average Degree of Consolidation |
---|---|
Merchant model | |
are given in Equations (21), (22), (66) and (67), respectively; and | |
Maxwell model | |
are given in Equations (71), (73), and (75), respectively; and | |
Linear elastic model | |
are given in Equations (71) and (77), respectively; and |
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Shen, S.; Hu, Z. Consolidation by Vertical Drains Considering the Rheological Characteristics of Soil under Depth and Time-Dependent Loading. Sustainability 2023, 15, 6129. https://doi.org/10.3390/su15076129
Shen S, Hu Z. Consolidation by Vertical Drains Considering the Rheological Characteristics of Soil under Depth and Time-Dependent Loading. Sustainability. 2023; 15(7):6129. https://doi.org/10.3390/su15076129
Chicago/Turabian StyleShen, Siliang, and Zheyu Hu. 2023. "Consolidation by Vertical Drains Considering the Rheological Characteristics of Soil under Depth and Time-Dependent Loading" Sustainability 15, no. 7: 6129. https://doi.org/10.3390/su15076129