1 Introduction

In order to break the regional division and strengthen the regional connection, China has established a large trans-river and sea channel in the coastal areas as the connecting instrument of multi-regional economy, which greatly improves the resilience of the coastal areas’ economy. Among the large trans-river–sea passages that can be used commercially at present, immersed tube tunnel has the advantages of strong formation adaptability, short construction period and strong ability to resist natural disasters at sea, which has gradually become the preferred scheme for large trans-river–sea passages in the future (Kurbatcky & Pestriakova, 2019; Lai et al., 2017; Ren et al., 2019; Sýkora et al., 2018). But during the operation of the tunnel, fire, as a major disaster, also occurred frequently. Since the immersed tube tunnel is a long and narrow structure located in an underground or underwater environment (Zhu et al., 2019), once a fire occurs, there will often be significant casualties and huge economic losses, and the bearing capacity and service life of the tunnel will be affected as well. Therefore, it is inevitable to consider the impact of fire in the immersed tube tunnel, and fire prevention is an important issue that must be faced in the construction and operation of cross-river–sea tunnels all over the world.

The fire safety of immersed tunnels is being increasingly studied worldwide (Vuilleumier et al., 2002; Xu et al., 2018). Broadly speaking, methods in the literature of immersed tube tunnel fire can be divided into experiment, numerical simulation and combination of experiment and numerical simulation. Experimental method can observe the results intuitively, while the numerical simulation can save a lot of cost and time, and realize the setting of complex working conditions. At present, there are 1:1 full-scale model (Xu et al., 2018) and 1:5 scale model (Duan et al., 2021; Lin et al., 2021) for immersed tube tunnel of Hong Kong–Zhuhai–Macau Bridge in the literatures. The temperature field distribution (Jiang et al., 2018; Xi et al., 2020), structural damage analysis (Guo et al., 2016), smoke flow law (Lee et al., 2010; Liu et al., 2021; Xu et al., 2019), mechanical ventilation performance (Li et al., 2021), personnel evacuation (Tian & Jiang, 2018) and fire resistance of immersed tube tunnel (Choi et al., 2011) have been studied extensively.

Moreover, fire spalling of tunnel concrete is more a material-level behavior than a structural-level behavior. The moisture inside the concrete will migrate when it is exposed to fire. At the same time, the compressive strength, elastic model, thermal expansion rate and other changes of concrete have an impact on the fire resistance of the overall structure (Akca & Özyurt, 2018; Chen et al., 2020; Kamen et al., 2008; Ryu et al., 2018). Nevertheless, existing researches mainly describe and statistics the macroscopic phenomena of tunnel structure under fire, although the research focus has gradually shifted to mesoscopic component of concrete that can reflect the overall mechanical properties of concrete structures (Jin et al., 2020; Wang et al., 2018a; Wang et al., 2018b; Zhao et al., 2011). Meanwhile, few scholars have discussed the mesoscopic deterioration characteristics of immersed tube tunnels under fire.

The paper established a mesoscopic model of immersed tube tunnel roof, which takes immersed tunnel roof as the research object and considers the decisive influence of meso-structure on macroscopic mechanical behavior. Through the establishment of this model, the mesoscopic damage law of fire sinking tube tunnel roof can be clarified, and the gradual damage process of micro-cracks initiation, development, expansion and connection in fire sinking tube tunnel roof can be revealed.

2 Material Parameters and Methods

2.1 Thermodynamic Parameters of Reinforced Concrete at High Temperature

Different from the general macro-model, the analysis of reinforced concrete model under fire needs to consider the high-temperature mechanical properties of materials, in which the thermodynamic parameters mainly include specific heat capacity, thermal conductivity, convective heat transfer coefficient and thermal expansion coefficient.

In this paper, thermal conductivity and specific heat capacity of concrete at high temperature recommended by European standards are adopted. Thermal expansion coefficient is obtained by transforming the coefficient of thermal expansion studied by Yan, 2015 and Xiao et al., 2018. In addition, only convective heat transfer is considered. Convective heat transfer coefficient is calculated using the empirical formula proposed by Duan, 1985. Specific figures are shown in Table 1. Meanwhile, thermal conductivity, specific heat capacity and linear expansion coefficient of steel bars are shown are shown in Table 1.

Table 1 Thermodynamic parameter curve of reinforced concrete at high temperature.
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For the thermal parameters of mesoscopic materials, thermal expansion coefficient, thermal conductivity and specific heat capacity of aggregate and mortar are shown in Formulas (1–4). Thermal parameters of interface layer are consistent with mortar:

$$k_{T} = 1.36 - 0.136\left( \frac{T}{100} \right) + 0.0057\left( \frac{T}{100} \right)^{2} ,$$
(1)
$$k_{aT} = \frac{770}{{0.84\left( {350 + T} \right)}} + 0.7,$$
(2)
$$C_{T} = 790 + 0.9T$$
(3)
$$C_{aT} = 625 \times \left( {1 + T} \right)^{0.075} .$$
(4)

2.2 High-Temperature Damage Constitutive Model

High-temperature damage constitutive relationships of concrete, its meso-phase materials, and steel bars are shown in Table 2.

Table 2 High-temperature constitutive relation of reinforced concrete components.
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2.3 Random Aggregate Delivery Method

In engineering application, the aggregate inside concrete is scattered and irregular in spatial distribution, and the quantity is usually large. For this reason, Monte Carlo method can be used to solve the randomness problem in modeling. In this paper, APDL language in ANSYS is used to establish the coordinates and diameters of randomly distributed aggregate, and a large number of RANDOM Numbers are generated between (0,1) by RANDOM command, and then multiplied with the spatial coordinates and radii of aggregate to obtain the coordinates and diameters of randomly distributed aggregate. In order to make the aggregate independent, non-overlapping and within a certain area of placement, the following three criteria are established, where, whether the aggregate exceeds the feeding area can be determined by Formula (5):

$$\left( {\begin{array}{*{20}c} x \\ y \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {x_{i} } \\ {y_{i} } \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {r_{i} } \\ {r_{i} } \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {d_{i} } \\ {d_{i} } \\ \end{array} } \right) \in \left[ {\begin{array}{*{20}c} {\left( {x_{\min } ,x_{\max } } \right)} \\ {\left( {y_{\min } ,y_{\max } } \right)} \\ \end{array} } \right],$$
(5)

where xi, yi, ri are, respectively, the horizontal and vertical coordinates of a circle and the radius values of a circle of a random aggregate; di is the thickness of boundary layer; xmin, xmax is the minimum and maximum of the horizontal coordinate values of the analysis area of the mesoscopic model; ymin, ymax is the minimum and maximum value of vertical coordinate values in the analysis area of the mesoscopic model.

The aggregate coincides can be determined by Formula (6):

$$\sqrt {(x_{i} - x_{j} )^{2} + (y_{i} - y_{j} )^{2} } > r_{i} + r_{j} + 2d{.}$$
(6)

The aggregate overlaps with the reinforcement can be determined by Formula (7) (assuming that the reinforcement is parallel to the X-axis):

$$\left| {y_{i} - b} \right| > r_{i} ,$$
(7)

where yi, ri is the circular ordinate and radius of a random aggregate and b is the distance between the reinforcement and the X-axis.

There will be a certain gap between the aggregates in the concrete. At the same time, there is also a weak contact transition section between the aggregate and the mortar—the interlayer. In the random aggregate model, a separate analysis area must be established for the boundary layer. In the algorithm in this paper, the initial radius of aggregate delivery is the sum of the theoretical radius and the thickness of the interface layer. Finally, the concentric circle is used to generate the aggregate consistent with the theoretical radius, so as to establish the interface layer of 1 mm.

Walraven J. C established the relationship between concrete gradation curve and aggregate interface area in a single section [Formula (8)], which can be applied in microscopic analysis for two-dimensional plane analysis:

$$\begin{gathered} P_{c} (D < D_{0} ) = P_{k} (1.065D_{0}^{0.5} /D_{\max }^{0.5} \hfill \\ - 0.053D_{0}^{4} /D_{\max }^{4} - 0.012D_{0}^{6} /D_{\max }^{6} \hfill \\ - 0.0045D_{0}^{8} /D_{\max }^{8} + 0.0025D_{0}^{10} /D_{\max }^{10} ), \hfill \\ \end{gathered}$$
(8)

where Pk is the percentage of aggregate volume in the total volume of concrete; D0 is the diameter of sieve hole; and Dmax is the maximum aggregate particle size.

The concrete mesoscopic model established in this paper simplifies the aggregate to circular, which is different from the polyhedral aggregate in engineering practice. However, scholars (Ji, 2014; Li, 2012, 2016) have obtained good results in the meso-analysis of the mechanical properties of concrete structures. We found that the analysis results using spherical and irregular polygons are similar, so it is in line with the actual situation to simplify the aggregate into a circular to a certain extent.

3 Establishment of Local Mesoscopic Model of Immersed Tube Tunnel

Geometric dimensions of the structure are mainly referred to the construction drawing design of Hong Kong–Zhuhai–Macao Bridge. The cross-sectional dimensions and specific reinforcement are shown in Fig. 1. In this paper, the form of reinforcement is simplified. Only the influence of main reinforcement is considered. Lines marked by blue represent HRB400 steel bars with a diameter of 40 mm, while those marked by red at the flue represent HRB400 steel bars with a diameter of 28 mm. Others are HRB400 steel bars with a diameter of 32 mm. In addition, only half of the model is selected for structural calculation in order to simplify the calculation, and symmetrical boundary conditions are designed in the center of the whole structure.

Fig. 1
figure 1

Schematic diagram of immersed tube tunnel structure.

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3.1 Set Fire Scenarios

According to the requirements of ‘Code for Fire Protection of Building Design’ GB 50016-2014 (2018 edition), Hong Kong–Zhuhai–Macao immersed tunnel belongs to I class of tunnels, whose fire resistance limit should not be less than 2 h and fire resistance limit test curve should adopt the RABT standard heating curve. Therefore, RABT standard heating curve will be used as immersed tunnel fire scenarios in this paper. The specific expression is shown in Formula (9):

$$T = \left\{ \begin{gathered} T_{0} + 236 \times t\begin{array}{*{20}c} {} & {} \\ \end{array} \begin{array}{*{20}c} {\begin{array}{*{20}c} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } & {} \\ \end{array} } & {} \\ \end{array} \;\;t \le 5\;\min \hfill \\ T_{0} + 1180{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ \end{array} } & {} \\ \end{array} } & {} \\ \end{array} \;\;\;5\;\min \le t \le 95\;\min \hfill \\ 1200 - 40.333 \times (t - 95)\begin{array}{*{20}c} {} & {} \\ \end{array} 95\;\min \le t \le 125\;\min \hfill \\ \end{gathered} \right.,$$
(9)

where t is time and T0 is the initial temperature.

3.2 Boundary Condition

Due to the short fire time, temperature of concrete on the unburned side (i. e., the junction of water and soil around immersed tunnel and immersed tunnel) will not be affected. Therefore, we ignore the heat exchange between the surrounding water and soil environment and the immersed tunnel in the calculation, and assume the boundary temperature of the concrete without fire is 20 ℃. It is assumed that the main loads borne by immersed tunnel are vertical water and soil pressure, horizontal water and soil pressure, vehicle load and self-weight. Referring to the stratum conditions of Hong Kong–Zhuhai–Macao immersed tunnel and related literature (Jiang & Chen, 2015; Zhang, 2013), the load diagram of the cross section can be calculated by using the load-structure method. Fire seat is determined to be 1.8 m high from the bottom plate in the middle of the tunnel. Temperature and load arrangement are shown in Fig. 2.

Fig. 2
figure 2

Fire scenarios and load distribution of immersed tube tunnel.

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3.3 Determination of the Most Unfavorable Position of Immersed Tunnel Roof

Obviously, it is time-consuming and laborious to establish a mesoscopic model of full-scale immersed tunnel. Moreover, the failure process of tunnel roof is often restricted by local failure, which can also reflect the overall failure to some extent. Therefore, the most unfavorable position of immersed tunnel roof under thermal coupling will be explored below. Fig. 3 is the vertical displacement diagram of each monitoring point at different times of immersed tunnel roof.

Fig. 3
figure 3

Vertical displacement diagram of each monitoring point at different times.

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It can be seen from Fig. 3 that the vertical displacement trend of the roof under the action of no fire and different fire time is consistent, which shows that fire does not cause too much influence on the overall trend of the vertical deformation of the whole tunnel roof. Although the position of the maximum displacement of the immersed tube tunnel roof will change at different times, it is roughly between monitoring point 3 and monitoring point 4. Side wall at the exhaust duct causes a large reverse bending moment on the left side of the roof, which causes the peak point of vertical displacement at the bottom of the roof to deviate from the center line of the roof. The position of the maximum settlement of the immersed tunnel roof is 10.3 m away from the structural center line (measuring point 3 is 9.775 m away from the structural center line), and the maximum settlement is 5.1 mm without fire. Within 125 min of fire, the maximum settlement is 5.6 mm, which can reflect the actual deformation characteristics of immersed tunnel to a certain extent (Jiang & Chen, 2015; Zhang, 2013).

3.4 Establishment of Local Mesoscopic Model of Immersed Tunnel Roof

This article only conducts random aggregate delivery in the dangerous area of the roof (10.3 m from the centerline of the tunnel) to establish a local Mesoscopic model. In view of the weak position of the submerged tunnel roof, local meso-models are established in the middle of the roof fire zone and the upper compression zone, whose area is 300 × 300 mm. In addition, the roof of immersed tube tunnel is made of C50 concrete of two-grading. According to the existing grading regulations, the particle size ranges of 5–20 mm and 20–40 mm are selected, while the aggregate below 5 mm is included in the mortar. Seventy percent aggregate filling rate is adopted. By calculation, 192 aggregates with random size particle sizes in the range of 5–20 mm and 23 aggregates with random size particle sizes in the range of 20–40 mm can be obtained. Total number of aggregates is 215.

The model was established by ANSYS (ANSYS WORKBENCH 19.2). Homogeneous concrete, aggregate, mortar and interlayer adopt Plane188 element, while steel rebars adopt Link180 element. Foundation springs adopt Combine39 element which is set on the sides and bottom of the tunnel. In ANSYS, there is no spring element that is only subjected to compression. The author customizes its elastic resistance coefficient based on the characteristics of Combine39 element. We set the elastic resistance coefficient of the spring element to 1 N/m. When it is tensioned, the elastic resistance coefficient is consistent with the actual formation resistance coefficient, so as to approximate the simulation of only compression spring element. The horizontal trapezoidal load and the vertical load are superimposed on the boundary line of the two-dimensional plane element using Surf156 elements.

In addition, in order to facilitate classification and analysis, the meso-models on the top and lower sides are marked as U and N, respectively, as shown in Fig. 4.

Fig. 4
figure 4

Local mesoscopic model of reinforced concrete.

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This paper mainly considers the variation characteristics of the main strain in the mesoscopic model. In order to facilitate the subsequent analysis, meso-model N is divided into three parts, I, II and III. Each part is 10 cm thick, as shown in Fig. 5.

Fig. 5
figure 5

Meso-model N partition diagram.

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4 Results and Analysis

4.1 Temperature Field Distribution of Meso-Model N

In order to more clearly characterize the temperature field distribution of concrete at the bottom of the roof, the vertical measurement line is established along the thickness direction in the center of the meso-model N. It can be seen from Fig. 6 that temperature within 15 cm of the roof is below 100 °C when the RABT heating curve continues for 125 min. From Fig. 6, we can see that the heating rate of the concrete far from the fire side is lower when the temperature of the fire curve increases, and similarly, the temperature decreasing rate is also lower when the temperature is lowered. This clearly reflects the thermal inertia of concrete.

Fig. 6
figure 6

Internal temperature of model N varies with the fire time.

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Combined with Fig. 7, we found that the surface concrete temperature of about 2 cm reaches 800 °C, whose strength is completely lost. Furthermore, the concrete temperature of 2–6.6 cm is between 380 and 800 °C, and its strength is basically lost.

Fig. 7
figure 7

Temperature field distribution diagram of the roof meso-model N when exposed to fire for 125 min.

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4.2 Distribution of Maximum Principal Strain of Meso-Model N

Fig. 8 shows the maximum principal strain distribution of meso-model N under different fire time.

Fig. 8
figure 8

Distribution of the maximum principal strain of the meso-model under fire for 5/35/65/95/105/125 min: a 5 min, b 35 min, c 65 min, d 95 min, e 105 min, f 125 min.

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After 5 min of fire, the temperature increased rapidly. Large tensile strain also occurs in the lower temperature region, but the damage mainly occurs in the region III. At about 35 min, the damage was further expanded, and a large strain was generated on both sides of the aggregate, indicating that there was a large stress concentration between the two sides of the aggregate and the mortar. Large tensile strain can be observed around the aggregate in region I after 65 min of fire. When the duration of fire reaches 95 min, the constant temperature section of RABT heating curve ends, and the temperature in region I is basically equal to or close to normal temperature (20 °C). There will be no strain increase due to material deterioration caused by high temperature. Thus, this region is mainly caused by tensile damage caused by positive bending moment and thermal expansion of concrete near the bottom. On the contrary, the temperature of concrete in zone II is about 50–150 °C, which produces a certain additional compressive stress under the leading role of thermal expansion, so it remains in a relatively safe stress state. After cooling for 15 min, the concrete temperature at the bottom of the roof began to decrease and the strain began to recover.

4.3 Distribution of Minimum Principal Strain Of Meso-Model N

Fig. 9 is the minimum principal strain distribution of the meso-model N under different fire time. It can be seen from the figure that under the influence of thermal expansion, region III has been in a state of large compressive stress, and its damage development is basically consistent with that formed by the maximum principal strain distribution. Notably, there was no damage in the roof after 125 min of fire, indicating that the damage in regions II and I is mainly controlled by tensile stress.

Fig. 9
figure 9

Distribution of the minimum principal strain of the meso-model under fire 5/35/65/95/105/125 min: a 5 min, b 35 min, c 65 min, d 95 min, e 105 min, f 125 min.

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4.4 Analysis of Strain Characteristics of Roof Meso-Model U

Fig. 10 is the minimum strain distribution diagram of the meso-model U. It can be seen from the diagram that the compressive strain inside the roof decreases continuously within 125 min of fire, and the maximum compressive strain also decreases from 0.000246 to 0.000095. This is because in the process of fire, the concrete in the bottom of the roof under the thermal expansion area undertakes most of the pressure load, so that the compressive stress of the concrete in the non-fire area has been reduced. Although the compressive strain in meso-model U is less than the strain corresponding to the peak stress during the whole fire process, there is a large concentrated strain around the aggregate. It also can be seen from the cloud image that the strain between aggregate and aggregate has a trend of connection in mortar, indicating that the degradation characteristics of meso-model U are basically consistent with that of meso-model N at room temperature.

Fig. 10
figure 10

Distribution of minimum principal strain of meso-model U when exposed to fire for 5/35/65/95/105/125 min: a 5 min, b 35 min, c 65 min, d 95 min, e 105 min, f 125 min.

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4.5 Progressive Failure Process of Meso-Model N

In order to clarify the internal deterioration of immersed tunnel roof, referring to previous research results (Li, 2016; Wang et al., 2018a; Wang et al., 2018b), it is defined that the tensile strain of each meso-material unit of concrete is greater than 0.00015, the compressive strain is greater than 0.002, and the failure occurs when the temperature is higher than 380 °C. Therefore, the progressive failure diagram of the meso-model degradation of the roof of the fire sinking tunnel is obtained (Fig. 11), in which the white part represents the failure area.

Fig. 11
figure 11

Progressive destruction of the meso-model model when exposed to fire for 5/35/65/95/105/125 min: a 5 min, b 35 min, c 65 min, d 95 min, e 105 min, f 125 min.

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After 5 min of fire, due to the rapid increase of temperature, the surface layer of the fire surface of the roof has been damaged in a large area. Combined with the analysis results of the internal temperature field of the micro-model, it is found that the main failure area is basically uniform development with 150 °C isothermal line.

After 65 min of fire, the internal micro-cracks in area I develop along the periphery of aggregate, and some micro-cracks have penetrated into the mortar area in the process of further degradation, forming large cracks. But unlike region III, aggregate in region I was not damaged.

After 95 min of fire, longitudinal penetrating cracks along the thickness direction of roof have been formed in region I, while the failure development of region III has gradually slowed down. From 95 to 125 min, the temperature of the external environment has gradually decreased to normal temperature, but the damage range of region I and region III is still expanding, and more cracks begin to erupt in region II. The maximum stress damage depth obtained in this paper is about 28 cm, and the maximum damage depth calculated by Guo et al. (2016) is 25 cm, which can prove the reliability of the simulation. At the same time, the research results (Li et al., 2011; Ma et al., 2004) on meso-damage of concrete are similar to those obtained by this simulation.

Fig. 12 shows the variation curve of damage ratio of each meso-material with fire time. The progressive deterioration process of concrete at meso-level can also be seen from the diagram.

Fig. 12
figure 12

The change curve of meso-material damage ratio with time.

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5 Conclusions

In the present study, a local mesoscopic model is established for the dangerous area of immersed tube tunnel roof. Numerical simulation results are present in this paper. Based on the phenomena and data of the simulation, following conclusions can be drawn:

  1. (1)

    In the mesoscopic model, the section loss caused by high temperature is about 6.6 cm, while the maximum stress damage along the thickness direction is about 28 cm, which is basically consistent with the relevant research results. Compared with mortar and boundary layer, aggregate damage has a certain hysteresis. In the normal temperature area, aggregates will not be damaged even if the crack penetrates.

  2. (2)

    Deterioration of concrete is a gradual process in meso-scale. In the high-temperature area on the fire side of the roof, the interlayer with lower strength is destroyed first and produces micro-cracks. However, the meso-damage of concrete in this area uniformly develops inward due to the rapid transmission of high-temperature damage. In the normal temperature area inside the roof, the concrete will produce large tensile stress driven by thermal expansion, and the micro-cracks around the aggregates will further deteriorate, forming longitudinal through cracks in the mortar.