Temperature Control Simulation of Rock Wall Crane Beam Based on Cooling Water Pipe

Abstract

Currently, the problem of concrete cracks is one of the common quality problems that is difficult to solve in the field of water conservancy and hydropower engineering. In order to study the distribution law of concrete temperature field during the construction period of underground cavern rock wall crane beams, different cooling water pipe arrangements were adopted to simulate and analyze the internal temperature and stress of concrete. The results show that the three-layer parallel cooling water pipe arrangement has the best cooling effect and can effectively control concrete cracking.

1 Introduction

In the process of concrete pouring and hardening, the hydration reaction of cement emits large heat. This chemical reaction leads to the change of the concentration of each component (such as the hardening of cement gel) and causes the structure to heat up. Conversely, an increase in temperature will exacerbate the rate of hydration reaction. The hydration reaction will rapidly increase the temperature of concrete to 70 ℃. Subsequently, as the hydration rate slows down and the temperature decreases, thermal shrinkage occurs, leading to the generation of thermal stress. Under the dual driving forces of chemistry and thermodynamics, the physical characteristics of temperature, strength, stiffness, creep, and shrinkage of newly poured concrete will undergo significant changes, which can easily lead to cracking [1]. A large number of engineering crack treatment and investigation results show that 80% to 90% of cracks in concrete structures are caused by the tensile stress generated during the cooling process of concrete construction exceeding the tensile strength of concrete.

The rock wall crane beam of the underground powerhouse of a hydropower station is a structure that uses grouting long anchor rods to anchor the reinforced concrete beam body on the rock wall. Its inclined surface is conducive to forming a wide upper and narrow lower factory structure, which can effectively narrow the span of the factory, reduce the excavation amount of the factory, reduce the engineering cost, and maintain the stability of the surrounding rock. During the construction period, temperature stress is prone to cracking. Currently, China mainly focuses on three aspects of crack prevention and resistance of concrete: ① improving the crack resistance performance of concrete from the material aspect, increasing its tensile strength and ultimate tensile value, reducing its hydration heat and shrinkage value; ② Propose reasonable structural types and dimensions from the perspective of structural design; ③ Take corresponding temperature control measures during construction to reduce adverse factors that cause concrete cracking [2].

The thermal conductivity of concrete is poor, making it difficult to dissipate in a short period of time and form a nonlinear temperature field inside. In the initial temperature rise stage, it is in a plastic flow state, and the uneven temperature distribution will not generate significant tensile stress in the concrete; However, during the process of temperature decrease, concrete has already reached a certain strength. The changes in temperature field and other constraints limit the free shrinkage of concrete, resulting in a certain amount of temperature stress. As the surface temperature is often lower than its internal temperature, significant tensile stress will be generated on the surface of concrete. If it exceeds the ultimate tensile strength of concrete, surface cracks will occur, which will further develop into penetrating cracks [3, 4]. Years of research have shown that the main cause of such cracks is the combined effect of concrete shrinkage (self generated volume strain) and environmental temperature changes and constraints. Therefore, the accuracy of temperature field simulation is crucial for the study of crack formation in concrete.

In recent years, research on concrete temperature and stress has been continuously deepening, and different scholars have made their research on this issue more specific and targeted. Tasri conducted research on thermal stress and temperature gradient caused by the space between the cooling pipes and cooling water temperature of post cooling components [5]. Myers and Charpin treat the heat transfer problem between cooling water pipes and concrete as plane strain and analyze the effects of cooling water flow rate and pipe material on concrete temperature [6]. Zeng has studied the effect of pouring speed on temperature stress [7]. Qiang has achieved certain results in the calculation of water pipe cooling algorithms [8].

Previous research on the effect of cooling water pipes on the temperature and stress fields of concrete has been relatively limited. This study mainly compares the layout of different cooling water pipes, Propose the best method to meet the requirements of temperature control and crack prevention.

2 Theory and Method

2.1 Theory of Temperature Field Calculation

Heat conduction is a specific heat transfer method, in which heat is transferred from the high-temperature area to the low-temperature area inside an object, satisfying the heat conduction equation:

$$ \frac{\partial T}{{\partial t}} = \frac{\uplambda }{\rho c}\left( {\frac{{\partial^{2} T}}{{\partial x^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{{\partial^{2} T}}{{\partial z^{2} }}} \right) + \frac{\partial \theta }{{\partial t}} $$

(1)

In the formula: $\lambda$ is the thermal conductivity of concrete, $c$ is the heat capacity of concrete, $\rho$ is the density of concrete, $\theta$ is the heat of concrete hydration, and $T$ is the temperature of concrete.

In the heat conduction of solids, the heat flow rate (the amount of heat per unit area per unit time) is directly proportional to the temperature gradient, but the direction of the heat flow is opposite to the direction of the temperature gradient. The heat flow rate transmitted by concrete through the boundary of water pipes can be expressed as:

$$ q = - \lambda \frac{\partial T}{{\partial x}} $$

(2)

Considering the variation of water temperature along the cooling water pipe, the heat change per unit length of cooling water per unit time can be expressed as:

$$ T_{out} = T_{in} - \frac{\lambda }{{q_{w} \rho_{w} c_{w} }}\iint\limits_{s} {\frac{\partial T}{{\partial x}}}ds $$

(3)

The heat transferred from the pipe wall to the water flow per unit time can be expressed as: due to hydration, under adiabatic conditions, the temperature rise rate of concrete is:

$$ \frac{\partial \theta }{{\partial \tau }} = \frac{Q}{c\rho } = \frac{Wq}{{c\rho }} $$

(4)

In the formula: is the adiabatic temperature rise of concrete, W is the amount of cement per unit of concrete, q is the hydration heat released per unit weight of cement per unit time.

2.2 Calculation Theory and Solution Method for Concrete Creep Stress

The theory of elastic creep stress can effectively describe the constitutive relationship of concrete. The strain increment of concrete during the time period $\Delta t$ is:

$$ \left\{ {\Delta \varepsilon_{n} } \right\} = \left\{ {\Delta \varepsilon_{n}^{e} } \right\} + \left\{ {\Delta \varepsilon_{n}^{c} } \right\} + \left\{ {\Delta \varepsilon_{n}^{T} } \right\} $$

(5)

Among them: $\left\{ {\Delta \varepsilon_{n}^{e} } \right\}$ represents the increment of elastic strain; $\left\{ {\Delta \varepsilon_{n}^{c} } \right\}$ is incremental creep strain; $\left\{ {\Delta \varepsilon_{n}^{T} } \right\}$ is increment of temperature strain.

Zhu Bofang pointed out that the sum of elastic strain increment and creep strain increment can be written as:[9]

$$ \left\{ {\Delta \varepsilon_{n}^{e} } \right\} + \left\{ {\Delta \varepsilon_{n}^{c} } \right\} = \left\{ {\eta_{n} } \right\} + \frac{{1 + C\left( {t_{n} ,\overline{{\tau_{n} }} } \right)}}{{E\left( {\overline{{\tau_{n} }} } \right)}}\left[ Q \right]\left\{ {\Delta \sigma_{n} } \right\} $$

(6)

Among them: $\left\{ {\eta_{n} } \right\} = \sum\nolimits_{s = 1}^{m} {\left[ {1 - \exp \left( { - r_{s} \Delta \tau_{n} } \right)} \right]} \left\{ {\omega_{sn} } \right\}$, $\left\{ {\omega_{sn} } \right\}$ is the state variable that changes over time.

In summary, the stress-strain relationship of concrete can be obtained as follows:

$$ \left\{ {\Delta \sigma_{n} } \right\} = \frac{{E\left( {\overline{{\tau_{n} }} } \right)}}{{1 + E\left( {\overline{{\tau_{n} }} } \right)C\left( {t_{n,} \overline{{\tau_{n} }} } \right)}}\left[ Q \right]^{ - 1} \left( {\left\{ {\Delta \varepsilon_{n} } \right\} - \left\{ {\eta_{n} } \right\} - \left\{ {\Delta \varepsilon_{n}^{e} } \right\} + \left\{ {\Delta \varepsilon_{n}^{c} } \right\}} \right) $$

(7)

The basic idea of using sequential coupling method to solve the stress field of concrete is to first use the HETVAL subroutine to solve the temperature field of the structure, and then import the temperature field to solve the stress field using the UMAT subroutine. Calculation model and calculation parameters.

2.3 Simulate Cooling Water Pipes

The diameter of the cooling water pipe is set to 28 mm, which is much smaller than the size of the concrete structure. The key to the accuracy of the calculation results lies in the precise establishment of a large volume concrete model containing the cooling water pipe. Therefore, the substructure method can effectively solve the complex structural problems mentioned above. The substructure method, also known as superelement technique, divides complex structures into several small modular substructures, each of which is connected by boundaries [10]. When solving, first fix the common boundary, calculate the stiffness matrix of each substructure relative to the common boundary, and then use the finite element balance equation to assemble the stiffness matrix of the substructure through the common boundary, forming the overall balance equation of the entire structure.

Finally, use the temperature of the common boundary node solved as the specified temperature to solve the internal temperature of the substructure. This study used an octagonal water pipe structure to simulate, and the grid diagram of the division is shown in Fig. 1. And take one layer of units as the cooling water pipe, as shown in Fig. 2.

Fig. 1.

Schematic diagram of the sub-structure

Fig. 2.

Simulation diagram of cooling water pipe

3 Finite Element Simulation Analysis

3.1 Calculation Model

Due to the spatial structural characteristics of the rock wall crane beam, it is advisable to use the three-dimensional nonlinear finite element method for calculation. Meanwhile, the finite element method can effectively simulate the nonlinear temperature field formed during the pouring and hardening process of rock wall crane beams. Extend horizontally from the side wall to the interior of the surrounding rock to three times the anchoring depth of the system anchor rod, that is, take the thickness of the surrounding rock as 27.0 m; Vertically, the section height of the rock wall crane beam extends 4 times upwards and downwards from the top and bottom of the rock wall crane beam, that is, 9.0 m upwards and downwards respectively; The axial direction is taken as 1.5 times the width of a crane, which means the longitudinal length is taken as 11.0 m. In order to ensure high computational accuracy, the grid size of the rock wall crane beam and its surrounding area should not exceed 1/16 of the height of the rock wall crane beam, with a grid size of 0.19 m. Among them, the Mohr Coulomb yield criterion is used for the concrete of the rock wall crane beam and the nearby rock mass. A thin layer element with a thickness of 10cm is used to characterize the contact surface between the rock wall crane beam and the surrounding rock. The elastic modulus is taken as 1/10–1/2.5 of the surrounding rock and the Mohr Coulomb yield criterion is used. In three-dimensional finite element calculations, the rock mass and rock wall crane beam both use spatial eight node hexahedral elements. The overall coordinate origin of the model is selected in the middle of the rock wall crane beam, with the X-axis transverse to the river, the Y-axis longitudinal to the river, and the Z-axis vertical upwards. The specific overall structural model and the detailed three-dimensional finite element calculation mesh of the rock wall beam are shown in Figs. 3 and 4.

Fig. 3.

Structural model

Fig. 4.

Finite element mesh of rock wall beam

3.2 Boundary Conditions and Material Parameters

The thermal parameters of rock wall beam concrete include: concrete specific heat, thermal conductivity, surface heat dissipation coefficient, and concrete adiabatic temperature rise;

The adiabatic temperature rise model of concrete adopts an exponential function model, as shown in the following equation:

$$ \theta = \theta_{0} \left( {1 - e^{{ - a\tau^{b} }} } \right) $$

(8)

Among them: $\theta_{0}$ is final adiabatic temperature rise, $\tau$ is Time, $a,b$ is time related parameters. The parameters of the concrete adiabatic temperature rise model are taken as: 46.0, 0.499, and 1.068.

Concrete surface heat dissipation coefficient: ${96}0{\text{ kJ}}/\left( {{\text{m2}} \cdot {\text{d}} \cdot\,^\circ\text{C}} \right)$ for the horizontal plane, multiplied by 1.08 for the vertical plane. When there is a template on the concrete surface, take 1/3 of the value without the template. The surface heat dissipation coefficient of the surrounding rock is taken as ${5}00{\text{ kJ}}/({\text{m2}} \cdot {\text{d}} \cdot\,^\circ\text{C})$.

The growth curve of concrete elastic modulus with age adopts a composite exponential model form, namely:

$$ E(\tau ) = E_{0} (1 - e^{{ - c\tau^{d} }} ) $$

(9)

In the formula: $E_{0}$ represents the final elastic modulus, it is recommended to take 36.00 MPa; $c,d$ is two parameters related to the growth rate of the elastic modulus, with values of 0.28 and 0.52, respectively. Table 1 lists the calculation parameters of the material.

Table 1. Calculation Parameters

The following formula is used for the creep degree of concrete：

$$ \begin{gathered} C\left( {t,\tau } \right) = \frac{0.23}{{E_{0} }}\left( {1 + 9.2\tau^{ - 0.45} } \right)\left[ {1 - e^{{ - 0.3\left( {t - \tau } \right)}} } \right] \hfill \\ \quad \quad \quad + \frac{0.52}{{E_{0} }}\left( {1 + 1.7\tau^{ - 0.45} } \right)\left[ {1 - e^{{ - 0.005\left( {t - \tau } \right)}} } \right] \hfill \\ \end{gathered} $$

(10)

The stress relaxation coefficient of concrete is expressed by the following equation:

$$ K\left( {t,\tau } \right) = 1 - \left( {0.4 - 0.6e^{{ - 0.62\tau^{0.17} }} } \right)\left( {1 - e^{{ - \left( {0.2 + 0.27\tau^{ - 0.23} } \right)}} \left( {t - \tau } \right)^{0.36} } \right) $$

(11)

The underground powerhouse of the pumped storage power station is located in a deep area above 200 m underground. In the absence of a heat source, the temperature inside the cave should change according to a regular pattern around the average temperature throughout the year. The average temperature in Weifang is 12.8 ℃. Considering that there is a significant difference between the temperature inside the cave and the temperature due to the influence of construction machinery. Based on past experience, it is assumed that the temperature inside the cave is approximately between 24 and 29 ℃, and varies with the four seasons according to a regular pattern.

3.3 Calculation Results

Four types of cooling water pipe arrangements are shown in Fig. 5. Scheme 1: No cooling water pipe as control group. Scheme 2: Adopting a double-layer parallel cooling water pipe layout method. Scheme 3: Adopting a single-layer vertical cooling water pipe layout method. Scheme 4: Adopting a three-layer vertical cooling water pipe layout.

Fig. 5.

Diagram of different cooling water pipe layout schemes

Temperature Field Calculation.

Select four key points of the rock wall beam for analysis, as shown in Fig. 6.

Fig. 6.

Schematic diagram of key points on rock wall beams

A three-dimensional calculation model of the rock wall beam of a pumped storage power station was constructed based on the actual design scheme, using thermal calculation parameters of concrete and foundation. At the same time, based on the actual construction situation and corresponding boundary conditions, a three-dimensional transient temperature field simulation calculation was carried out during the construction period of the rock wall beam. Specific analysis was conducted on the temperature of the cave, the temperature calculation results after pouring the rock wall beam, and the stress results at key points.

Fig. 7.

Diagram of the variation of temperature over time in Scheme 1

Fig. 8.

Diagram of the variation of temperature over time in Scheme 4

Figure 7 and 8 show the temperature variation patterns of the four characteristic points in Scheme 1 and Scheme 4 over time, respectively. Comparing the various schemes, it can be seen that the cooling effect of the three row vertical cooling water pipe arrangement in Scheme 4 is the best, so only the variation diagrams of the control group and Scheme 4 are shown. Throughout the entire calculation cycle, the temperature at the center position N04 without cooling water pipes was higher than the temperature at the other three key points. When the template was removed on the 7th day, there was a significant change in temperature at each key point, and then the temperature gradually decreased. In the later stage of pouring, the temperature at each key point was significantly affected by the temperature and gradually tended to be consistent. And the center point NO4 in scheme 4 has the lowest temperature, and the other key points have not exceeded 30℃, indicating that the cooling effect is very good. After pouring, the temperature gradually decreases and the hole temperature gradually decreases.

Fig. 9.

Temperature distribution map of the middle section of the rock wall beam wall after one days of pouring

Fig. 10.

Temperature distribution map of the middle section of the rock wall beam wall after 3.5 days of pouring

Fig. 11.

Temperature distribution map of the middle section of the rock wall beam wall after 7 days of pouring

Fig. 12.

Temperature distribution map of the middle section of the rock wall beam wall after 28 days of pouring

Figure 9, 10, 11 and 12 shows the temperature cloud maps of the middle section of the rock wall beam on the 1st, 3.5th, 7th, and 28th days of scheme 1.The internal temperature of the rock wall beam concrete rises rapidly three days before pouring. In the condition of no cooling water pipe, the highest temperature is about 36.25 ℃ on the first day, and reaches the highest value on the 3.5th day, about 51.46 ℃. The highest temperature is located at the core of the rock wall beam concrete beam, and the concrete temperature gradually diffuses and decreases from the inside out; As time went on, the rate of temperature decrease at each location slowed down. After removing the template, the highest temperature reached 28.74 ℃ at 28 days.

In working condition two of the cooling water pipe, the temperature reached its highest value on the first day, about 33.88 ℃. The high-temperature part is located at the corner of the rock wall beam surface where the water pipe is not buried and in contact with the air. On the third day, the temperature reached its highest value, about 41.50 ℃. After removing the template, the temperature slowly decreased and tended towards the temperature inside the cave. At 28 days, the highest temperature reached 28.73 ℃.

In working condition three of the cooling water pipe, the temperature reached its highest value on the first day, about 34.13 ℃. The concrete temperature gradually diffused and increased from the inside out with the water pipe as the center. On the third day, the temperature reached its highest value, about 42.99 ℃. After removing the template, the temperature slowly decreased and tended towards the temperature inside the cave. On the 28th day, the highest temperature reached 28.73 ℃.

The cooling effect of the cooling water pipe is most obvious in the fourth working condition of the cooling water pipe. On the third day, the temperature no longer rises, and the peak appears one day earlier, with a temperature peak decrease of 15.66 ℃. On the first day, the temperature reached its highest value, about 31.92 ℃, and the concrete temperature gradually increased from the inside out with the water pipe as the center. On the second day, the temperature reached its highest value, about 35.80 ℃, and the high-temperature part was only distributed on the surface of the rock wall beam and the front end in contact with the air. On the sixth day, the temperature inside the cave had dropped to nearly 28.80 ℃. In the first 6 days, due to the formwork covering the rock wall beam, the heat dissipation of the concrete was slow. After 7 days of pouring and removing the formwork, the heat dissipation speed significantly accelerated. After 28 days of pouring, the temperature remained basically constant. Afterwards, the temperature decreased and was basically consistent with the temperature inside the cave. After 28 days, the highest temperature reached 28.73 ℃. The final temperature stabilizes around 28 ℃ in the construction environment.

Fig. 13.

Diagram of the variation of the first principal stress over time in Scheme 1

Fig. 14.

Diagram of the variation of the first principal stress over time in Scheme 4

Figure 13 and 14 show the variation of the first principal stress over time for the four feature points in Scheme 1 and Scheme 4, respectively. The tensile stress of rock wall beam concrete is mainly caused by the self weight of the concrete and the temperature tensile stress. In the early stage of concrete pouring, the surface of the rock wall beam bears tensile stress and the interior bears compressive stress. And the surface tensile stress and internal compressive stress almost reach their maximum values when the concrete temperature reaches its peak, after three days of pouring. At this time, the maximum tensile stress on the outer surface is 1.38 MPa, and the maximum tensile stress is located at the contact position between the rock wall beam concrete and the crane beam rail. Due to the constraint of the lateral surrounding rock on the rock wall beam, the maximum compressive stress is located at the corner point of the transition zone between the rock wall beam and the surrounding rock, which is 2.32 MPa. The maximum compressive stress inside the concrete of the rock wall beam is 1.63 MPa, located at the core of the concrete beam. As time increases, the tensile stress on the outer surface transforms into compressive stress. After the removal of the template on the 7th day, the internal compressive stress gradually decreases and transforms into tensile stress.

The compressive stress inside the concrete of the rock wall beam in working condition five of the cooling water pipe is mostly less than 0.90 MPa, and the maximum compressive stress is located at the corner point of the transition zone between the rock wall beam and the surrounding rock, which is 1.77 MPa. The central part of the beam is located near the cooling water pipe, with a maximum compressive stress of 0.33 MPa on the first day. During the cooling water period, the compressive stress gradually decreases in the first three days. After stopping the water supply and removing the mold, the internal temperature of the concrete rises, and the outer surface temperature is basically consistent with the hole temperature. The internal compressive stress gradually increases, reaching a maximum of 0.78 MPa on the tenth day after the mold is removed. Only at the contact between the cooling water pipe and the concrete is subjected to significant tensile stress, with a maximum tensile stress of 1.17 MPa. The maximum tensile stress on the outer surface reaches 0.62 MPa at the outlet of the cooling water pipe, while the maximum tensile stress in other areas is below 0.35 MPa. In the later stage of pouring, the temperature at each location is significantly affected by air temperature, with the internal temperature slightly lower than the external surface temperature, and the maximum tensile stress on the external surface reaching 1.35 MPa.

4 Conclusion

The maximum temperature of the unconnected cooling water pipe is 51.46 ℃. The peak temperature in Scheme 2 decreases by 9.96 ℃, the peak temperature in Scheme 3 decreases by 8.47 ℃, and the peak temperature in Scheme 3 decreases by 15.66 ℃. It can be seen that the arrangement of three rows of vertical cooling water pipes has the best cooling effect. The temperature of each key point in Plan 4 did not exceed 30 ℃ throughout the entire time cycle, and the temperature at the center point of the beam decreased most significantly.

After the removal of the non cooling water pipe mold, as the heat dissipation on the surface of the rock wall beam concrete increases, the hydration heat of the concrete continues to release heat, and the temperature difference between the inside and outside of the concrete increases, resulting in compressive stress on the surface of the concrete and tensile stress on the inside. The key point where the cooling water pipe is not connected in Plan 1 is subjected to compressive stress before demoulding and tensile stress after pouring. The key points of Plan 4 experience a decrease in the peak compressive stress before demolding, while NO1 experiences a smaller tensile stress. The tensile stress decreases in the later stage of pouring. The maximum tensile stress at the core of the concrete beam in the rock wall beam without cooling water pipes is 2.50 MPa. It can be seen that the tensile stress of scheme four also decreased the most significantly, with a peak tensile stress decrease of 0.21 MPa before demolding and a maximum tensile stress decrease of 1.15 MPa in the later stage of pouring.