# Damage Simulation Analysis of Canal Concrete Lining Plates Based on Temperature-Stress-Water Load Coupling

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (i)
- A considerable number of research studies focus on the numerical analysis of the canal concrete lining under the action of frost heave, and there are few studies on simulating the temperature and stress changes of the lining plate under the coupling action of different temperatures and water loads. This paper takes an engineering project as an example, and ABAQUS finite element software was used to calculate, analyze and summarize the changes and distribution rules of the temperature and stress fields of the canal lining plate under the action of multi-factor coupling.
- (ii)
- The stress concentration position of the entire canal concrete lining plate is determined by simulation, and the most likely position of damage to the canal concrete lining plate under the coupling action of temperature and load is determined. This can be used as a reference for further research on canal concrete lining construction quality control.
- (iii)
- Utilizing the in situ test of canal lining carried out by other scholars, the numerical model of this paper is verified. The results show that the simulated value is in close agreement with the field measured value, which establishes the validity and rationality of the model in this paper.

## 2. Project Overview

## 3. Numerical Simulation

#### 3.1. Basic Assumptions and Theories

#### 3.1.1. Basic Assumptions

- (1)
- Through the collection of geological data in this region, it was found that the groundwater at the bottom of the canal is located at depth and it is difficult for the groundwater to migrate into the canal foundation soil so the migration of groundwater is not considered.
- (2)
- The concrete lining plate of the canal is simplified as a thin shell structure, and the self-weight of the concrete lining plate of the canal is not considered in the analysis.
- (3)
- The heat transfer mode of canal foundation soil is heat conduction, and the heat released during water phase transformation in the canal lining is not considered during freezing in winter.

#### 3.1.2. Theory of Temperature Field Calculation

- (1)
- Boundary conditions of the first kind.

- (2)
- Boundary conditions of the second kind.

- (3)
- Boundary conditions of the third kind.

#### 3.1.3. Water Load

#### 3.1.4. Constitutive Model of Materials

- (1)
- Constitutive model of silty soil

- (2)
- Constitutive model of concrete.

#### 3.2. Establishing a Numerical Model

#### 3.2.1. Selection of the Numerical Model and Related Parameters

- (1)
- Determine the boundary and value range of the numerical model.

- (2)
- Determine the calculation parameters of silt.

- (3)
- Determine the calculation parameters of concrete.

- (4)
- Determine the surface heat release coefficient of concrete.

#### 3.2.2. Determine Geometric Dimensions

#### 3.2.3. Determine Boundary Conditions

#### 3.2.4. Determine the Calculation Conditions

- (1)
- Temperature and stress changes in the concrete lining plate of the canal under no water condition during the construction and maintenance period.

- (2)
- Temperature and stress changes in the concrete lining plate of the canal under water-filled conditions during the operation period.

#### 3.3. Results and Analysis

- (1)
- Temperature field

^{6}MPa).

- (2)
- Stress field.

_{0}. There is a linear distribution of freezing force and other forces τ

_{0}along the tangential direction of the slope plate [29]. The slope plate length is L, the thickness is B, and the bottom plate thickness is B. In this paper, the left shady slope is taken as an example. It is assumed that point A is the top of the slope (mainly supported by the normal freezing force and temperature stress acting on the top), and point B is the bottom of the slope (mainly supported by the interaction between the lining bottom plate and the slope plate). The final calculation diagram is shown in Figure 13.

- (1)
- The reaction force at the support.

- (2)
- Axial force of slope plate.

- (3)
- Slope plate shear force.

- (4)
- Bending moment of slope plate.

#### 3.4. Sensitivity Analysis of the Model to Changes in Input Parameters

^{16}(4

^{5}) was selected for the test design, namely: modulus (33,600, 30,800, 28,000, 25,200), Poisson’s ratio (0.24, 0.22, 0.2, 0.18), specific heat capacity (0.312, 0.286, 0.26, 0.234), coefficient of thermal conductivity (3000, 2750, 2500, 2250), and coefficient of thermal expansion (0.0036, 0.0033, 0.003, 0.0027). The orthogonal experiment design [31] is widely used in experiments covering multiple influencing factors and multiple levels of conditions. This method uses an orthogonal table to determine the test scheme and combines the basic principles of statistical analysis to calculate and analyze the test results so that the strength of the influence of each influencing factor on the test results can be obtained. In this paper, we intend to use the orthogonal test design to determine the scheme (see Appendix A), and then use the maximum stress of the concrete lining slab of the canal as the evaluation index to conduct the sensitivity of the finite element model caused by the change of the input parameters (i.e., each influencing factor). The water load was set as 1/2 canal depth (1.1 m) for the analysis.

_{1j}refers to the sum of the test indicators (maximum stress) corresponding to the j-th influencing factor at the first level, and j corresponds to the Elastic modulus (A), Poisson’s ratio (B), Specific heat capacity (C), Coefficient of thermal conductivity (D), and Coefficient of thermal expansion (E). Similarly, S

_{2j}, S

_{3j}, and S

_{4j}can be obtained in sequence.

_{1A}= 1.34 + 1.28 + 1.22 + 1.19 = 5.03

_{2A}= 1.23 + 1.21 + 1.2 + 1.23 = 4.87

_{3A}= 1.18 + 1.17 + 1.16 + 1.17 = 4.68

_{4A}= 1.14 + 1.12 + 1.11 + 1.09 = 4.46

_{1j}is the arithmetic mean value of the canal concrete lining plate corresponding to the j-th influencing factor at the first level, that is, s

_{1j}= S

_{1j}/n = S

_{1j}/4. Similarly, s

_{2j}, s

_{3j}and s

_{4J}are calculated using influencing factor A as an example:

_{1A}= S

_{1A}/4 = 5.03/4 = 1.2575

_{2A}= S

_{2A}/4 = 4.87/4 = 1.2175

_{3A}= S

_{3A}/4 = 4.68/4 = 1.17

_{4A}= S

_{4A}/4 = 4.46/4 = 1.115

_{j}is the range value of the j-th factor, ${\mathrm{R}}_{\mathrm{j}}=\mathrm{max}\left\{{\mathrm{s}}_{1\mathrm{j}}{,\mathrm{s}}_{2\mathrm{j}}{,\mathrm{s}}_{3\mathrm{j}}{,\mathrm{s}}_{4\mathrm{j}}\right\}-\mathrm{min}\left\{{\mathrm{s}}_{1\mathrm{j}}{,\mathrm{s}}_{2\mathrm{j}}{,\mathrm{s}}_{3\mathrm{j}}{,\mathrm{s}}_{4\mathrm{j}}\right\}$, and A is taken as an example to calculate:

_{A}= 1.2575 − 1.115 = 0.1425.

#### 3.5. Model Validation

_{V}is the validation uncertainty, and E is calculated as:

_{E}is the experimental uncertainty, and U

_{SPE}is the experimental uncertainty in the use of input data. In general, this value is commonly negligible [35]. U

_{S}is the simulation uncertainty.

## 4. Discussion and Conclusions

_{E}is used as the expanded uncertainty at the 95% confidence level, and the formula used is:

- (1)
- Under the condition of no water load, the temperature distribution in the middle of the canal floor is more uniform, and the overall temperature of the sunny side slope plate is higher than that of the shady side slope plate. Under the action of water load, it is found that the variation of temperature field is consistent with that under the action of no water load, and the overall trend of plate temperature on the sunny side slope is higher than that of the shady side slope.
- (2)
- In the case of no water load, the variation of the stress field of the canal lining was analyzed. According to the established numerical model of canal concrete lining, the stress of the lining plate at the foot of two sides of the slope is large, and the stress of the middle canal bottom plate is small. The shady side slope plate was selected for analysis, and it was found that from the bottom to the top, the stress value on the canal slope gradually decreased, and reached the maximum value at the foot of the lining plate slope. Simplified analysis with mechanical knowledge shows that the shear force and axial force of the lining plate reach the maximum at the slope foot, and the maximum bending moment occurs at the lining plate near the slope foot. In order to prevent the maximum tensile strain of the concrete lining plate at the slope toe from exceeding its allowable tensile strain and causing expansion cracks in the lining plate, special attention should be paid to its prevention.
- (3)
- The load conditions were set to 1/2, 2/3 of the canal water filling depth and full load (i.e., 1/2 canal depth, 2/3 canal depth, full load), and the simulation analysis was carried out. The maximum stress position is located in the middle of the canal slope toe and the canal bottom. Once these stresses exceed the allowable stress value of the concrete plate itself, damage results, and these are the weakest positions of the entire canal concrete lining plate. In the later stages of design, construction, and maintenance, relevant technical measures should be taken to consolidate the structure at these locations.
- (4)
- In order to verify the accuracy of the model, the method of comparing error values is used for verification. The results show that the temperature and stress values output by the model are in close agreement with the in situ test results, and the correlation between the measured value and the simulated value is above 0.98 (Correlation coefficient R
^{2}). The proposed model is reasonable and correct, and can be successfully used in practical cases. - (5)
- When the input parameters change, in order to determine the influence of each parameter on the sensitivity of the model, the maximum stress of the canal concrete lining plate is used as the evaluation index to analyze the sensitivity of the model. The results show that the factors affecting the sensitivity of the numerical model from strong to weak are: Elastic modulus > Poisson’s ratio > Coefficient of thermal expansion > Coefficient of thermal conductivity > Specific heat capacity.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Serial Number | Influencing Factors | ||||
---|---|---|---|---|---|

Elastic Modulus (A) | Poisson’s Ratio (B) | Specific Heat Capacity (C) | Coefficient of Thermal Conductivity (D) | Coefficient of Thermal Expansion (E) | |

1 | 33,600 | 0.24 | 0.312 | 3000 | 0.0036 |

2 | 33,600 | 0.22 | 0.286 | 2750 | 0.0033 |

3 | 33,600 | 0.2 | 0.26 | 2500 | 0.003 |

4 | 33,600 | 0.18 | 0.234 | 2250 | 0.0027 |

5 | 30,800 | 0.24 | 0.286 | 2500 | 0.0027 |

6 | 30,800 | 0.22 | 0.312 | 2250 | 0.003 |

7 | 30,800 | 0.2 | 0.234 | 3000 | 0.0033 |

8 | 30,800 | 0.18 | 0.26 | 2750 | 0.0036 |

9 | 28,000 | 0.24 | 0.26 | 2250 | 0.0033 |

10 | 28,000 | 0.22 | 0.234 | 2500 | 0.0036 |

11 | 28,000 | 0.2 | 0.312 | 2750 | 0.0027 |

12 | 28,000 | 0.18 | 0.286 | 3000 | 0.003 |

13 | 25,200 | 0.24 | 0.234 | 2750 | 0.003 |

14 | 25,200 | 0.22 | 0.26 | 3000 | 0.0027 |

15 | 25,200 | 0.2 | 0.286 | 2250 | 0.0036 |

16 | 25,200 | 0.18 | 0.312 | 2500 | 0.0033 |

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**Figure 1.**Flowchart of the simulation process of a canal concrete lining under multi-field coupling.

**Figure 4.**Finite element calculation model diagram after boundary and displacement conditions are applied.

**Figure 5.**Temperature cloud diagram of the concrete lining in the canal under no water load during the construction and maintenance period.

**Figure 6.**Stress cloud of the canal concrete lining under anhydrous load during the construction and maintenance period.

**Figure 7.**Schematic diagram of the location distribution and numbering of the slope foot of the canal lining plate.

**Figure 8.**Temperature stress of each node at the right slope foot of the canal concrete lining board.

**Figure 11.**Temperature stress of each node in the middle of the left side of the canal concrete lining slope plate.

**Figure 12.**Temperature stress of each node in the middle area on the right side of the canal concrete lining slope plate.

**Figure 15.**The stress cloud diagram of the canal concrete lining when the water filling depth is 2/3 of the canal depth.

**Figure 17.**Comparison of the measured and simulated values of the canal concrete lining plate temperature.

**Figure 18.**Correlation between simulated and measured values of temperature at each node of canal concrete lining.

**Figure 20.**Correlation between simulated and measured values of stress at each node of canal concrete lining.

Elastic Modulus (MPa) | Poisson’s Ratio | Shear Modulus (MPa) | $\mathbf{Thermal}\text{}\mathbf{Conductivity}\text{}\mathbf{Coefficient}\text{}\mathbf{W}/(\mathbf{m}\cdot \mathbf{k})$ | Coefficient of Thermal Expansion $1/\mathbf{K}$ | Specific Heat Capacity $\mathbf{J}/(\mathbf{kg}\cdot \mathbf{K})$ |
---|---|---|---|---|---|

20 | 0.33 | 8 | 641.5 | 0.2 × 10^{−2} | 0.1 × 10^{−3} |

Elastic Modulus (MPa) | Poisson’s Ratio | Specific Heat Capacity $\mathbf{J}/(\mathbf{kg}\cdot \mathbf{K})$ | Coefficient of Thermal Conductivity $\mathbf{W}/(\mathbf{m}\cdot \mathbf{k})$ | Coefficient of Thermal Expansion $1/\mathbf{K}$ |
---|---|---|---|---|

2.8 × 10^{4} | 0.2 | 0.26 | 2.5 × 10^{3} | 0.3 × 10^{−2} |

Serial Number | Maximum Stress (MPa) |
---|---|

1 | 1.34 |

2 | 1.28 |

3 | 1.22 |

4 | 1.19 |

5 | 1.23 |

6 | 1.21 |

7 | 1.20 |

8 | 1.23 |

9 | 1.18 |

10 | 1.17 |

11 | 1.16 |

12 | 1.17 |

13 | 1.14 |

14 | 1.12 |

15 | 1.11 |

16 | 1.09 |

Calculated Index | Result Value | ||||
---|---|---|---|---|---|

A | B | C | D | E | |

S_{1j} | 5.03 | 4.89 | 4.8 | 4.83 | 4.85 |

S_{2j} | 4.87 | 4.78 | 4.79 | 4.81 | 4.75 |

S_{3j} | 4.68 | 4.69 | 4.75 | 4.71 | 4.74 |

S_{4j} | 4.46 | 4.68 | 4.7 | 4.69 | 4.7 |

n | 4 | 4 | 4 | 4 | 4 |

s_{1j} | 1.2575 | 1.2225 | 1.2 | 1.2075 | 1.2125 |

s_{2j} | 1.2175 | 1.195 | 1.1975 | 1.2025 | 1.1875 |

s_{3j} | 1.17 | 1.1725 | 1.1875 | 1.1775 | 1.185 |

s_{4j} | 1.115 | 1.17 | 1.175 | 1.1725 | 1.175 |

R_{j} | 0.1425 | 0.0525 | 0.025 | 0.035 | 0.0375 |

Number (Location) | Simulated Value (MPa) | Measured Value [28] (MPa) | Validation Error |
---|---|---|---|

1 (left side slope toe) | 1.527 | 1.534 | 0.007 |

2 (left side slope toe) | 1.406 | 1.424 | 0.018 |

3 (left side slope toe) | 1.391 | 1.428 | 0.037 |

4 (left side slope toe) | 1.39 | 1.416 | 0.026 |

5 (left side slope toe) | 1.393 | 1.412 | 0.019 |

6 (left side slope toe) | 1.39 | 1.405 | 0.015 |

7 (left side slope toe) | 1.391 | 1.389 | −0.002 |

8 (left side slope toe) | 1.406 | 1.413 | 0.007 |

9 (left side slope toe) | 1.427 | 1.533 | 0.106 |

10 (right side slope toe) | 1.321 | 1.432 | 0.111 |

11 (right side slope toe) | 1.319 | 1.411 | 0.092 |

12 (right side slope toe) | 1.317 | 1.332 | 0.015 |

13 (right side slope toe) | 1.306 | 1.332 | 0.026 |

14 (right side slope toe) | 1.307 | 1.313 | 0.006 |

15 (right side slope toe) | 1.306 | 1.324 | 0.018 |

16 (right side slope toe) | 1.317 | 1.335 | 0.018 |

17 (right side slope toe) | 1.319 | 1.364 | 0.045 |

18 (right side slope toe) | 1.321 | 1.436 | 0.115 |

19 (left side slope plate) | 1.305 | 1.315 | 0.01 |

20 (left side slope plate) | 1.121 | 1.218 | 0.097 |

21 (left side slope plate) | 0.988 | 1.112 | 0.124 |

22 (left side slope plate) | 0.822 | 0.845 | 0.023 |

23 (left side slope plate) | 0.679 | 0.732 | 0.053 |

24 (left side slope plate) | 0.544 | 0.603 | 0.059 |

25 (left side slope plate) | 0.446 | 0.478 | 0.032 |

26 (right side slope plate) | 1.374 | 1.381 | 0.007 |

27 (right side slope plate) | 1.105 | 1.123 | 0.018 |

28 (right side slope plate) | 0.895 | 0.942 | 0.047 |

29 (right side slope plate) | 0.774 | 0.751 | −0.023 |

30 (right side slope plate) | 0.64 | 0.684 | 0.044 |

31 (right side slope plate) | 0.507 | 0.536 | 0.029 |

32 (right side slope plate) | 0.409 | 0.503 | 0.094 |

Number (Location) | Parameters of Error (%) | Value of Error (%) | |
---|---|---|---|

${\mathit{\delta}}_{\mathit{E}}$ | ${\mathit{\delta}}_{\mathit{S}}$ | $\left|\mathit{E}\right|$ | |

1 (left side slope toe) | 1.54 | 0.46 | 1.08 |

2 (left side slope toe) | 1.96 | 1.26 | 0.70 |

3 (left side slope toe) | 1.75 | 2.59 | 0.84 |

4 (left side slope toe) | 1.29 | 1.84 | 0.54 |

5 (left side slope toe) | 1.70 | 1.35 | 0.36 |

6 (left side slope toe) | 2.08 | 1.07 | 1.02 |

7 (left side slope toe) | 1.29 | 0.14 | 1.14 |

8 (left side slope toe) | 1.28 | 0.50 | 0.79 |

9 (left side slope toe) | 1.54 | 6.91 | 5.37 |

10 (right side slope toe) | 1.54 | 7.75 | 6.21 |

11 (right side slope toe) | 1.54 | 6.52 | 4.98 |

12 (right side slope toe) | 1.28 | 1.13 | 0.16 |

13 (right side slope toe) | 1.28 | 1.95 | 0.67 |

14 (right side slope toe) | 1.45 | 0.46 | 1.00 |

15 (right side slope toe) | 1.29 | 1.36 | 0.07 |

16 (right side slope toe) | 1.31 | 1.35 | 0.04 |

17 (right side slope toe) | 1.28 | 3.30 | 2.01 |

18 (right side slope toe) | 2.07 | 8.01 | 5.93 |

19 (left side slope plate) | 1.32 | 0.76 | 0.56 |

20 (left side slope plate) | 1.81 | 7.96 | 6.15 |

21 (left side slope plate) | 1.73 | 11.15 | 9.42 |

22 (left side slope plate) | 1.97 | 2.72 | 0.76 |

23 (left side slope plate) | 1.01 | 7.24 | 6.23 |

24 (left side slope plate) | 2.59 | 9.78 | 7.19 |

25 (left side slope plate) | 1.21 | 6.69 | 5.48 |

26 (right side slope plate) | 1.31 | 0.51 | 0.80 |

27 (right side slope plate) | 1.68 | 1.60 | 0.08 |

28 (right side slope plate) | 1.96 | 4.99 | 3.03 |

29 (right side slope plate) | 2.62 | 3.06 | 0.44 |

30 (right side slope plate) | 2.59 | 6.43 | 3.84 |

31 (right side slope plate) | 2.60 | 5.41 | 2.81 |

32 (right side slope plate) | 2.63 | 18.69 | 16.06 |

Number (Location) | Uncertainty of Measurement and Simulation (%) | Uncertainty (%) | |
---|---|---|---|

${\mathit{U}}_{\mathit{E}}$ | ${\mathit{U}}_{\mathit{S}}$ | ${\mathit{U}}_{\mathit{V}}$ | |

1 (left side slope toe) | 1.54 | 0.46 | 3.05 |

2 (left side slope toe) | 1.96 | 1.26 | 4.05 |

3 (left side slope toe) | 1.75 | 2.59 | 4.30 |

4 (left side slope toe) | 1.29 | 1.84 | 3.13 |

5 (left side slope toe) | 1.70 | 1.35 | 3.60 |

6 (left side slope toe) | 2.08 | 1.07 | 4.22 |

7 (left side slope toe) | 1.29 | 0.14 | 2.52 |

8 (left side slope toe) | 1.28 | 0.50 | 2.57 |

9 (left side slope toe) | 1.54 | 6.91 | 7.55 |

10 (right side slope toe) | 1.54 | 7.75 | 8.32 |

11 (right side slope toe) | 1.54 | 6.52 | 7.18 |

12 (right side slope toe) | 1.28 | 1.13 | 2.76 |

13 (right side slope toe) | 1.28 | 1.95 | 3.19 |

14 (right side slope toe) | 1.45 | 0.46 | 2.89 |

15 (right side slope toe) | 1.29 | 1.36 | 2.87 |

16 (right side slope toe) | 1.31 | 1.35 | 2.90 |

17 (right side slope toe) | 1.28 | 3.30 | 4.15 |

18 (right side slope toe) | 2.07 | 8.01 | 8.98 |

19 (left side slope plate) | 1.32 | 0.76 | 2.70 |

20 (left side slope plate) | 1.81 | 7.96 | 8.72 |

21 (left side slope plate) | 1.73 | 11.15 | 11.66 |

22 (left side slope plate) | 1.97 | 2.72 | 4.72 |

23 (left side slope plate) | 1.01 | 7.24 | 7.51 |

24 (left side slope plate) | 2.59 | 9.78 | 11.02 |

25 (left side slope plate) | 1.21 | 6.69 | 7.10 |

26 (right side slope plate) | 1.31 | 0.51 | 2.61 |

27 (right side slope plate) | 1.68 | 1.60 | 3.67 |

28 (right side slope plate) | 1.96 | 4.99 | 6.30 |

29 (right side slope plate) | 2.62 | 3.06 | 5.98 |

30 (right side slope plate) | 2.59 | 6.43 | 8.20 |

31 (right side slope plate) | 2.60 | 5.41 | 7.43 |

32 (right side slope plate) | 2.63 | 18.69 | 19.38 |

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## Share and Cite

**MDPI and ACS Style**

Li, Q.; Wu, B.; Zhou, H.
Damage Simulation Analysis of Canal Concrete Lining Plates Based on Temperature-Stress-Water Load Coupling. *Sustainability* **2022**, *14*, 9202.
https://doi.org/10.3390/su14159202

**AMA Style**

Li Q, Wu B, Zhou H.
Damage Simulation Analysis of Canal Concrete Lining Plates Based on Temperature-Stress-Water Load Coupling. *Sustainability*. 2022; 14(15):9202.
https://doi.org/10.3390/su14159202

**Chicago/Turabian Style**

Li, Qingfu, Binghui Wu, and Huade Zhou.
2022. "Damage Simulation Analysis of Canal Concrete Lining Plates Based on Temperature-Stress-Water Load Coupling" *Sustainability* 14, no. 15: 9202.
https://doi.org/10.3390/su14159202