Scholars at home and abroad have summarized the laws of pore water pressure, stress and deformation of surrounding rock of underground projects in different construction strata and different calculation methods, but there is little research on the stability of surrounding rock during excavation through the sudden change area of rock mass grade under the fluid solid coupling effect of shallow tunnel. In this paper, based on the change of seepage field, stress field and displacement field of tunnel surrounding rock, COMSOL Multiphysics software is used to conduct fluid structure coupling simulation analysis on the construction process of shallow tunnel.

#### **2 Fluid Structure Coupling Equation**

#### *2.1 Seepage Field Equation*

Based on Biot's consolidation theory, the governing differential equation corresponding to fluid structure coupling can be expressed by Darcy's law of fluid motion. We simplify the groundwater seepage mode into laminar flow, and Darcy's law is:

$$q\_i = -k \frac{\partial}{\partial \mathbf{x}} (p - \rho \mathbf{g} \mathbf{x}) \tag{1}$$

In the formula, *qi* refers to seepage velocity vector, *k* refers to permeability coefficient of medium, *p* refers to pore water pressure, ρ refers to liquid density, *g* refers to gravity acceleration component.

For the convenience of calculation, the rock soil layer can be regarded as a porous elastic medium. According to the principle of seepage mechanics, the seepage field control equation is [10]:

$$\nabla \cdot -\frac{k\_m}{\mu} \cdot (\nabla p - \rho \mathbf{g}) + \left(\frac{\alpha - \phi}{k\_s} + \frac{\phi}{k\_i}\right) \frac{\partial p}{\partial t} + \alpha \frac{\partial \varepsilon}{\partial t} = q\_m \tag{2}$$

In the formula, *km* is the permeability of porous media, α is the Biot coefficient, φ is the porosity of porous media, ε is the strain component, *qm* is the sink term of the fluid.

The change of volume strain will cause the change of fluid pore pressure, on the contrary, the fluctuation of pore pressure will also cause the occurrence of volume strain [11]. The constitutive equation of porous elastic medium is:

$$
\Delta \sigma\_{ij} + \alpha \Delta p \delta\_{ij} = \mathbf{H}\_{ij}(\sigma\_{ij}, \Delta \xi\_{i,j}) \tag{3}
$$

In the formula, Δσ*ij* is the stress variation, Δ*p* is the change of pore water pressure; δ*ij* is Kronecher factor, H*ij* is a given function, Δξ*i*,*j*is the total strain.

#### *2.2 Stress Field Equation*

The control equation of porous elastic material model is:

$$-\nabla \cdot \sigma = \rho \text{g} \tag{4}$$

In the formula, σ is the stress tensor, ρ is the density of liquid, *g* and is the acceleration of gravity.

The displacement boundary condition is expressed as:

$$|u| = u\_l \tag{5}$$

In the formula, *ul* is the displacement at the boundary. The stress boundary condition is expressed as:

$$|\sigma\_{ij} \cdot n\_j| = T\_i \tag{6}$$

In the formula, *n j* is the cosine of the angle between the stress and the normal of the projection plane. *Ti* is the surface force on the boundary.

The displacement and velocity of rock mass particle at the initial time (any selected time) are respectively expressed as:

$$
\mu|\_{t=0} = \mu\_i \tag{7}
$$

$$\frac{\partial u}{\partial t\_i}|\_{t=0} = v\_i \tag{8}$$

In the formula, *ui* is the displacement of rock mass particle when t = 0, v*i* is the velocity of rock mass particle when t = 0.

#### **3 Project Overview**

Taking the Er-Jia-Gou Reservoir headrace tunnel in Harbin as an example, the headrace tunnel has a total length of 2410 m, a bottom width of 4.0 m, a tunnel height of 4.0 m, and a tunnel cross section in the shape of a city gate [12]. According to the actual project plan, the buried depth of the tunnel top in some tunnel sections is 15 m, and the surface passing by is partly mountain platform. The lithology of this part of rock is strong and weak coarse-grained granite, and the rock is relatively soft. The rock mass is saturated due to long-term farming and rainwater immersion [13]. According to the construction records, when the headrace tunnel was excavated to 0+628, serious water seepage occurred in the top arch, accompanied by large area of rock collapse.

#### **4 Numerical Simulation Analysis**

#### *4.1 Model Establishment and Calculation Scheme*

In this paper, COMSOL software is used to simulate and analyze the excavation section of the headrace tunnel, study the stability of surrounding rock in the sudden change area of rock mass grade during shallow tunnel construction, and find out the influence range of the sudden change area. According to the tunnel design, the width of the tunnel bottom is 4.0 m, the tunnel height is 4.0 m, the burial depth (the distance between the tunnel top and the ground) is 15 m, the portal is in the form of a city gate, and the calculation range of the model is set as 40 m × 40 m × 60 m. The tunnel simulation model is shown in Fig. 1, in which 0–60 m, 100–160 m are ordinary areas, and 60–100 m are sudden change areas of rock mass grade. The working condition settings are shown in Table 1, of which working condition 2 is the control group. See Table 2 for physical and mechanical parameters of rock mass.

**Fig. 1** Simulation model of a 15 m deep tunnel

## *4.2 Assumptions for Numerical Calculation*

#### *4.3 Boundary Condition*

#### **5 Result Analysis**

The tunnel is excavated in full section. In order to facilitate analysis, a monitoring section is set up at 5 m behind the tunnel face to study the pore water pressure, stress and vertical displacement variation of the monitoring section during the construction of the tunnel with a buried depth of 15 m. The COMSOL steady state mode is selected for numerical calculation.

#### *5.1 Change of Seepage Field*

Pore water pressure data is measured at 5 m from the monitoring section of the tunnel to the left arch waist. This location is selected as the research object to calculate the distribution law of pore water pressure under different working conditions. It can be seen from Fig. 2 that the distribution law of pore water pressure curve under working condition 1 and 3 is basically consistent. At a distance of 20 m from the abrupt change area, the pore water pressure under working condition 1 and 3 has a sudden change, which is greatly reduced after entering the abrupt change area of rock mass grade. It can be seen from the comparison of data under working conditions 1 and 3 that the higher the rock mass grade in the abrupt change area, the greater the amplitude of pore water pressure fluctuation during construction.

Figure 3 shows the dynamic distribution of the pore water pressure of Y–Z monitoring section under different driving distances under working condition 3. The initial pore water pressure of surrounding rock before excavation is layered and increases with the depth from top to bottom. After excavation, the pore water pressure inside

**Fig. 2** Distribution of pore water pressure at the research location of the monitoring section with a buried depth of 15 m

**Fig. 3** Dynamic change of pore water pressure of monitoring section under working condition 3

the tunnel is zero, forming a certain pressure difference with the pore water pressure outside the tunnel, causing the balance of the original seepage field to be broken, and finally forming a low pressure area similar to a funnel around the tunnel.

When the tunnel excavation is advanced to the sudden change area, the maximum pore water pressure at the research location of the monitoring section is reduced from 74.7 to 56.2 kPa, a decrease of 24.8%. Under the effect of pressure difference, groundwater is easy to penetrate into the tunnel, causing softening of surrounding rock and stress reduction. Therefore, in the actual construction process, grouting and plugging shall be carried out in time at the leakage location to prevent water inrush and mud leakage. For micropores that cannot be grouted, two layers of waterproof coating can be applied for plugging.

#### *5.2 Change of Stress Field*

It can be seen from Fig. 4 that the distribution law of stress curve under working condition 1 and 3 is similar, and the stress of surrounding rock is affected and gradually concentrated at the place 20 m away from the fracture zone when the tunnel

**Fig. 4** Distribution of the first principal stress in the monitoring section of the 15 m deep tunnel

is excavated. With the sudden change of rock mass grade, the first principal stress of the tunnel surrounding rock monitoring section under working condition 1 and 3 decreases by 47% and 24% respectively. It can be seen from the data analysis under working conditions 1 and 3 that the higher the rock mass grade in the mutation area, the more frequent the stress mutation.

Figure 5 is the dynamic diagram of the first principal stress of the monitoring section under condition 3. During construction, the initial stress balance is broken and the stress is redistributed. With the advance of excavation, the stress is gradually concentrated. The maximum compressive stress is concentrated at the bottom corner of the tunnel, the stress distribution at the vault and around is small, and the maximum tensile stress is concentrated at the bottom plate of the tunnel. During the excavation of the tunnel, the compressive stress increases to a certain value and then decreases suddenly. This is because at the bottom of the soft rock strength, when the stress level exceeds the bearing range of the surrounding rock, the surrounding rock presents a yield state, resulting in a stress drop. Therefore, in the process of tunnel excavation, stress monitoring and reinforcement measures should be taken in the stress concentration area to prevent the surrounding rock from reaching the stress limit and causing instability and damage.

#### *5.3 Change of Displacement Field*

As shown in Fig. 6, during the construction of shallow tunnel, the vertical displacement at the top of the tunnel is greater than that at the bottom, and the distribution law of the displacement curve under working condition 1 and 3 is similar. At a distance of 20 m from the abrupt change area, the vertical displacement of combination mode 1 and 3 is affected and gradually increases. It can be seen from the Fig. 6a that the vertical displacement and settlement of the vault in combination mode 1 increased by

**Fig. 5** Dynamic change of the first principal stress of the monitoring section under working condition 3

1.487 mm before and after entering the sudden change area, and the vertical displacement and settlement of the vault in combination mode 3 increased by 2.31 mm. In the Fig. 6b, the vertical displacement and settlement of the arch bottom in combination mode 1 increased by 0.97 mm after entering the sudden change area, and the vertical displacement and settlement of the arch bottom in combination mode 3 increased by 0.93 mm. According to the comparative analysis of the data under working condition 1 and 2, when the rock mass grade in the mutation area is the same, the lower the rock mass grade in the ordinary area, the smaller the vertical displacement. It can be seen from the comparison of data under working conditions 2 and 3 that when the rock mass grade in the general area is the same, the higher the rock mass grade in the mutation area, the greater the vertical displacement.

#### **6 Conclusion**

In this paper, the construction process of the diversion tunnel of Er-Jia-Gou Reservoir in Harbin is simulated by fluid structure coupling simulation. Combined with the

**Fig. 6** Vertical displacement change of monitoring section of 15 m deep tunnel

change data of pore water pressure, the first principal stress and vertical displacement, the following three conclusions are drawn:

mass grade as far as possible, and focus on support and reinforcement within 20 m from the sudden change area to ensure the safety of tunnel construction.

### **References**

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## **Non-destructive Detection of Grouting Defects Behind Shield Tunnel Wall by Combining Ground Penetrating Radar with Seismic Wave Method**

#### **Qing Han, Yihui Huang, Chaochen Li, Peiquan Yang, Xiongjie Deng, Dengyi Wang, Fuan Ma, and Hufeng Shi**

**Abstract** The shield construction is a mature tunnel construction technology. Grouting behind segment wall synchronously during shield construction is an important measure to control stratum deformation, reduce ground settlement and prevent tunnel water seepage. The quality of grouting can significantly affect the quality of tunnel construction. The method of grouting quality judgment in early tunnel construction is inefficient and uncertain. To ensure the safety of tunnel structure, it is an urgent technical innovation to study the non-destructive testing technology for grouting effect of shield segment wall. Nowadays, the general Non-destructive testing methods include ground penetrating radar detection and seismic wave detection. These two methods can meet most of the needs of grouting defect detection behind the wall, but also have their own advantages and disadvantages. Therefore, this study proposes the combination of ground penetrating radar and seismic wave method of shield tunnel wall after grouting defects Non-destructive detection method, first, ground penetrating radar is used for scanning probe shield wall after grouting, and then according to the seismic wave method is used to further verify the results for some suspected injury, while ensure the detection efficiency to improve recognition accuracy. Finally, the defect detection and verification of the proposed method was carried out based on the shield segment model test platform, and this proposed method was successfully applied to the actual project of a metro line in Foshan.

**Keywords** Tunnel engineering · Shield · Back-wall grouting · Non-destructive detection · Ground penetrating radar · Seismic wave

Q. Han · Y. Huang

CCCC Foshan Investment and Development Co., Ltd, Foshan 528000, China

C. Li · P. Yang (B) · X. Deng Guangdong Construction Engineering Quality and Safety Inspection Station Co., Ltd., Guangzhou 510599, China e-mail<EMAIL_ADDRESS>
D. Wang Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China

F. Ma · H. Shi Guangxi Nonferrous Survey and Design Institute, Nanning 530031, China

S. Wang et al. (eds.), *Proceedings of the 2nd International Conference on Innovative Solutions in Hydropower Engineering and Civil Engineering*, Lecture Notes in Civil Engineering 235, https://doi.org/10.1007/978-981-99-1748-8\_8

#### **1 Introduction**

Shield method is a construction method of tunnel excavation with shield machine.  At present, it has become a main means of tunnel construction in soft soil (saturated soft clay, muddy soft soil, silty clay, saturated sandy silt, silty sand, etc.). In the process of shield, synchronous grouting is needed to reduce the stratum loss. At the same time, the grouting layer can form an effective waterproof layer around the segment peripheral to prevent groundwater from penetrating into the tunnel. However, in the process of grouting, it is difficult to directly observe the grouting effect, quantitatively control the grouting amount and ensure the grouting effect by injecting mortar through the reserved grouting holes in the segments. When the grouting quantity is insufficient, the gap of the pipe wall will not be fully filled, and the cavity will be generated behind the lining wall, resulting in uneven stress on the segments, cracking of the segments and water seepage accidents in the tunnel [1].

Therefore, how to effectively detect the grouting effect behind the shield segment wall is of great significance to ensure the safety and quality of the tunnel. According to the past experience, in the construction process, whether the grouting pressure meets the design requirements and whether the grouting hole overflows or not is often used to judge whether the grouting is full. In the later stage, the grouting quality behind the shield segment wall is mainly verified by observing whether the segment seeps and adopting the drilling method. However, these methods have great limitations, which can't accurately and intuitively reflect the grouting situation behind the wall. In order to ensure the safety of the tunnel structure, it is an urgent technical innovation to study the nondestructive testing technology of the grouting effect behind the shield segment wall, and its technical achievements can be applied in guiding and monitoring the grouting construction behind the shield segment wall. So far, the ground penetrating radar (GPR) method has been widely used to detect the cavity behind the lining wall of mine tunnel. In some places, GPR method has also been applied to the quality detection of grouting behind the shield tunnel segment wall. This method can intuitively reflect the grouting situation behind the segment wall, and can adjust the corresponding parameters according to different structures. It has high detection efficiency and wide coverage, and can quickly survey the grouting defects behind the shield segment wall [2]. However, due to the complexity of segment structure (such as multi-layer steel bar structure, complex and changeable slurry) and the existing technology, there are great difficulties in detecting the density of shield segment grouting, the location of void area and the degree of void. The detection results will be disturbed, which leads to the problem that the defects cannot be identified accurately. With the research of scholars in recent years, seismic wave method has become a potential nondestructive testing method for grouting behind the wall [3]. Seismic wave method is a method to detect underground engineering geological problems by using the characteristics of medium transmitting elastic waves. A small hammer is used as an excitation source to knock on the concrete surface to generate compression waves, and then a receiving sensor placed near the excitation source receives the reflected seismic waves. By analyzing the received data to analyze the defects, this method can effectively avoid some interference. However, the detection efficiency of this method is low, and it can't be widely used. Both methods have their own advantages and disadvantages. Therefore, it is one of the main purposes of this study to combine the two to achieve complementary advantages.

This project relies on a metro line in Foshan, Guangdong Province to carry out scientific research. This project is the backbone of Foshan's east–west route, with a total length of 556 km and 27 stations. It is the east–west trunk line connecting the central group and the express rail passage connecting Foshan and Guangzhou. In the past, through the analysis of the radar signal characteristics of the foundation electrical properties, main interference and grouting defects of shield tunnel in rail transit, it is found that some defects cannot be detected accurately if only the dip radar is used to detect the grouting defects behind the wall. Therefore, it is particularly important to combine the dip radar with seismic waves to ensure the grouting quality of the tunnel.

## **2 Working Principle of Ground Penetrating Radar and Seismic Wave Method**

#### *2.1 Basic Principle of Ground Penetrating Radar Detection*

The principle of geological radar detection is that high-frequency electromagnetic wave signals with wide band and short pulse are transmitted to the ground through the transmitting antenna, when the electromagnetic wave encounters the underground interface with electrical differences (i.e. dielectric constant, electrical permeability, etc.) or other target bodies (such as hollows, cavities, metals, etc.), electromagnetic wave phenomena such as reflection and diffraction will occur. The response signal is picked up by the receiving antenna and recorded on the computer. According to the time domain and frequency domain characteristic information of electromagnetic wave, such as waveform, phase, amplitude and frequency spectrum, we can obtain the distribution of different underground electrical bodies and interpret the internal structure of the medium [4]. The schematic diagram of detecting the effect of grouting behind the segment wall by geological radar is shown in Fig. 1.

The nearsightedness expression of electromagnetic wave velocity in medium is:

$$\nu = \frac{c}{\sqrt{\varepsilon\_y}} \tag{1}$$

where, *c* is the speed of light in vacuum; ε*y* is the relative dielectric constant; ν is the wave velocity of electromagnetic wave in medium (m/ns).

The formula for calculating the depth of underground interface or target body is:

**Fig. 1** Schematic diagram of detection of grouting effect behind the wall of ground penetrating radar

$$\mathbf{Z} = \frac{\boldsymbol{\nu} \cdot \boldsymbol{t}}{2} \tag{2}$$

where *Z* is depth (m); *t* is the two-way travel time of reflected wave (ns).

According to the propagation characteristics of electromagnetic waves, electromagnetic waves will be reflected and transmitted at different electrical interfaces, and their reflection coefficient R and transmission coefficient T are expressed as follows:

$$\mathbf{R} = \frac{\sqrt{\varepsilon\_1} - \sqrt{\varepsilon\_2}}{\sqrt{\varepsilon\_1} + \sqrt{\varepsilon\_2}} \tag{3}$$

$$\mathcal{T} = \frac{2\sqrt{\varepsilon\_2}}{\sqrt{\varepsilon\_1} + \sqrt{\varepsilon\_2}} \tag{4}$$

where, R is reflection coefficient; T is transmission coefficient; ε1 is the dielectric constant of the upper layer; ε2 is the dielectric constant of the lower layer.

It can be seen from the above formula that the reflected energy of electromagnetic waves depends on the difference of dielectric constants of adjacent strata. The greater the difference, the more obvious the reflection, and the easier it is to identify the position of interface and target. Similarly, in practical engineering, the detection effect is influenced by the limited space of shield tunnel, the complex structure of tunnel segments, the radar wave scattering of field obstacles, non-radar electromagnetic fields such as wires and cables, various unfavorable factors of radar wave absorption caused by specific geological surrounding rock media, and the problem that the radar secondary reflected wave signal of shield segment overlaps with the reflected wave signal of grouting layer behind the wall, all of which will lead to problems such as the radar can't distinguish and identify grouting layer thickness and grouting defect signal. Therefore, if the ground penetrating radar is used alone to detect the grouting defects behind the wall, some grouting defects may not be identified.

#### *2.2 Principle of Seismic Wave Method*

Seismic wave method is a method to detect underground engineering geological problems by using the characteristics of medium transmitting elastic waves. A small hammer is used as an excitation source to knock on the concrete surface to generate compression waves, and then a receiving sensor placed near the excitation source receives the reflected compression waves. During acquisition, the excitation point and the reception point are always equidistant. When there are bad geological bodies (cavities, unconsolidations, etc.) under the ground, the seismic waves show anomalies from two aspects of kinematics and dynamics. The in-phase axis is staggered, the wave group changes, the amplitude increases or decreases, and even diffraction occurs, and the travel time of elastic waves will also increase or decrease [5]. When displaying the waveform, the software is used to compress (stretch) the seismic wave, and the reflected energy is expressed in different and changeable colors, so as to achieve the effect of visually displaying the shape of underground geological bodies. The schematic diagram of seismic wave method is shown in Fig. 2. In recent years, some scholars have studied the relationship between the reflection law of seismic waves and the vibration, defect size and defect depth of concrete structures [6]. Moreover, the seismic wave method has been successfully applied to the nondestructive inspection of many civil structures, which also does not provide ideas for scholars who study behind-wall grouting. In 2016, Yao et al. began to study the grouting of segment wall thickness by using seismic waves [7], and made further improvements in 2018 [8]. It was found that compared with the ground penetrating radar (GPR) method, the application of seismic waves in the quality detection of segment wall grouting is theoretically better than that of GPR. For example, seismic wave has a better detection ability for unconsolidated grout, because the liquid grout still has a better elastic coefficient difference with the surrounding soil, which can produce obvious reflection. However, in the ground penetrating radar method, many scholars suggest that the grout should be consolidated for a period of time before a better detection result can be obtained. In addition, if the defects in the grouting layer are air-filled, the seismic wave method will have better detection results for the monitoring of the solidified grouting body. The same reason is that the difference of elastic parameters between the air and the grouting layer is greater than the difference of dielectric coefficient, which can produce better reflection. Compared with electromagnetic waves, elastic waves will not be strongly interfered by steel bars, and signals will not be shielded by steel bars, so they have better detection ability for targets behind steel bars.

However, this method also has some limitations in detecting the grouting defects behind the shield wall. For example, the generation of seismic waves depends on knocking, which leads to the low detection efficiency of seismic wave method, and can't achieve scanning detection like ground penetrating radar. The coupling condition of the sensor receiving signals by seismic wave method on the concrete surface is also stricter than that of radar, which further limits the detection efficiency of seismic wave method. At the same time, at present, the impact echo method is mainly based

on time–frequency analysis, which can only be used as an auxiliary means of other detection methods, and it is difficult to conduct comprehensive detection independently. Therefore, if the ground penetrating radar and seismic wave are combined, we can complement each other's advantages to achieve better detection results.

#### **3 Example Analysis of Test Platform Detection**

In order to select the appropriate radar parameters and be familiar with the image analysis of seismic wave detection defects, we will use the method proposed in this paper to detect the wall thickness grouting defects of test platform of full-scale shield segment model in advance. This platform is a test platform of shield segment ring designed and built by Hu et al. [9] in 2016 with a scale of 1:1. The platform adopts shield segments and synchronous grouting materials of a tunnel in Nanning Rail Transit Line 1 project. The shield segment ring is composed of one uncapped block, two adjacent blocks and three standard blocks, with an outer diameter of 6.0 m, an inner diameter of 5.4 m, a wall thickness of 0.3 m, a ring width of 1.5 m, a concrete strength of C50 and an impermeability grade of P12. By embedding polyvinyl chloride (PVC) pipe in the gap of the building behind the segment wall to simulate the grouting cavity behind the wall, and by not grouting in some areas to simulate the grouting behind the wall. The test platform is shown in Fig. 3.

#### *3.1 Detection Results of Ground Penetrating Radar*

Data acquisition of ground penetrating radar (GPR) is carried out for a certain section of shield segment test platform. In order to highlight the contrast, we use the 400 MHz and 900 Hz antennas of American radar for data acquisition. After data processing steps, such as removing DC drift, removing direct wave, adjusting gain, normalizing

**Fig. 3** Test platform for post-grouting detection of full-size shield segment

distance, band-pass filtering (100–800 MHz), and moving average, the ground penetrating radar image is formed. The comparison between radar image and grouting defect is shown in Fig. 4.

(c) Image of 900 MHz antenna of platform ground penetrating radar

**Fig. 4** Schematic diagram of defects in test section and radar images

Comparing the data images collected by the two groups of radar antennas with the defects of the test platform, it is found that the ground penetrating radar can accurately identify the defects, and the set hole defects show typical hyperbolic strong reflection on the ground penetrating radar image, while the grouting layer defects show the same phase axis and staggered segments on the ground penetrating radar image. Both kinds of radars can clearly identify grouting defects.

#### *3.2 Detection Results of Seismic Wave Method*

When seismic wave data acquisition is carried out for a certain section of shield segment test platform, acceleration acquisition is adopted, and the data is played back to the computer for data format conversion. The format data of each single-channel column detector is converted into seismic wave data format, and each single-channel data is spliced into profile data. The comparison between the collected wave train diagram and the defect diagram are shown in Fig. 5.

By analyzing the seismic wave profile data of the test platform, it is found that the grouting defects of the test platform can also be accurately identified by seismic wave method. According to the image analysis, when there are holes or missing grouting layers in the grouting layer, the seismic wave profile is mainly characterized by the discontinuity of in-phase staggered segments, and the seismic wave is reflected at low frequency.

The ground penetrating radar (GPR) method and seismic wave method are used to detect the grouting defects behind the test platform wall. It is found that both methods can accurately and clearly identify the defects, and the set cavity defects show typical hyperbolic strong reflection on the GPR image, and the grouting layer defects show the same phase axis and staggered segments on the GPR image. Both kinds of radars can clearly identify grouting defects. However, when there is a cavity

**Fig. 5** Comparison between defect schematic diagram of test section and wave train diagram of seismic wave method

in the grouting layer or the grouting layer is missing, the seismic wave profile is mainly characterized by discontinuous in-phase staggered segments, and the seismic wave is reflected at low frequency. From the analysis of test results, it can be seen that these two methods can be applied to practical projects.

### **4 Case Study**

In order to further verify the engineering applicability of the method proposed in this paper, this project relies on a metro line in Foshan, Guangdong Province to carry out scientific research. The 770–870 ring on the left line of section1 and the 305–405 ring on the right line of section2 are selected as the experimental sections of this scientific research project.

Among them, the groundwater in section1 is mainly lateral supplied by river water, and has a close hydraulic connection. It is a weak-medium water-bearing (permeable) layer. The buried depth of the groundwater table in the upper muddy soil, muddy silt and cohesive soil layer is 0.6–3.7 m; The buried depth of confined water in the lower sand and gravel layer is 0.2–4.2 m, and the buried depth of confined water is close to that of phreatic water. The tunnel mainly passes through muddy soil, muddy silty sand, muddy silty soil, silty sand, medium coarse sand, round gravel, pebbles and strongly weathered argillaceous siltstone. The thickness of shield segment used in Section1 is 350 mm; The design thickness of grouting layer behind shield segment wall is 125 mm; Grouting material behind shield segment wall is made of cement, fly ash, bentonite, fine sand and water according to a certain proportion.

The groundwater in the Section2 is mainly lateral supplied by river water, and has a close hydraulic relationship. It is a weak-medium water-bearing (permeable) layer. The stable buried depth of water level is 0.2–6.1 m (elevation—1.24~2.53 m), and the annual variation range is about 1.5 m. The tunnel mainly passes through fine powder sand, fully weathered argillaceous sandstone, strongly weathered argillaceous sandstone, moderately weathered argillaceous sandstone and moderately weathered (gravel-bearing) sandstone. The thickness of shield segment used in Section2 is 300 mm; The design thickness of grouting layer behind shield segment wall is 125 mm; Grouting material behind shield segment wall is made of cement, fly ash, bentonite, fine sand and water according to a certain proportion.

#### *4.1 Setting of On-site Detection Parameters*

The actual detection will combine the ground penetrating radar and seismic wave method to realize the complementary advantages of the two methods. In order to achieve a better detection effect, the appropriate detection parameters will be selected according to the research of the test section and the actual geological conditions on site. Among them, the American SIR series radar with relatively stable performance and good measurement effect is selected, and at the same time, the antennas with center frequencies of 200, 400 and 900 MHz are selected for the detection comparison test research. The site drawing is shown in Fig. 6.

As for the instrument selection of seismic wave method, according to the nature and conditions of this task, RSM-EDT(A) column buried depth detector produced by Wuhan Zhongyan Technology Co., Ltd., which has relatively stable performance, good measurement effect and deep measurement depth and supports two data acquisition modes of speed and acceleration, is selected for seismic wave method data acquisition. The site drawing is shown in Fig. 7.

**Fig. 6** Field test photos of ground penetrating radar

**Fig. 7** Field test photos of seismic wave method

#### *4.2 Detection Results*

The on-site detection section is section1 and section2. The total length of the survey lines by the ground penetrating radar method and seismic wave method is 13545.5 m and 44.8 m, respectively. Several representative detection results are analyzed.

**Detection Results of Section1**. The left line of section1 is located at 7 o'clock azimuth longitudinal survey line, and the ground penetrating radar method is used to detect the scene, and the American SIR 400 and 900 MHz radars are used to collect data. After the data is collected, it is played back to the computer, and after data processing steps such as DC drift removal, direct wave removal, gain adjustment, distance normalization, band-pass filtering and moving average, the ground penetrating radar image is formed, as shown in Figs. 8 and 9. The preset radar wave speed is 0.12 m/ns. Seismic wave method is to collect data from ring 825 to ring 840 by column detector, and speed method is used for data collection. After the data is played back to the computer, the collected velocity signal is differentiated to obtain acceleration signal, and the data format is converted. The format data of each single-channel column detector is converted into seismic wave data format, and the single-channel data is spliced into profile data, as shown in Fig. 10.

It can be seen from the radar images of two frequencies that the radar reflection wave at the lower part of the segment with survey line mileage of 90–91.5 m, 94.5– 96 m, 97.5–99 m, 102–103.5 m (corresponding to ring numbers 830, 833, 835, 838) appears strong reflection at low frequency. It is speculated that the grouting layer in this range is not dense, and it may be filled with mud locally. From the wave diagram of seismic data, it can be seen that there are low-frequency reflections and in-phase staggered sections in multiple data in the 828–830 and 833 rings. It is inferred that the grouting in this area is not dense, which corresponds to the anomalies reflected

**Fig. 8** 400 MHz antenna image of 7 o'clock azimuth radar on the left line of Section1

**Fig. 9** 900 MHz antenna image of 7 o'clock azimuth radar on the left line of Section1

**Fig. 10** Normal wave train diagram of seismic wave from ring 825 to ring 840 at 7 o'clock azimuth on the left line of Section1

by the ground penetrating radar method, but fails to correspond one by one. This shows that the velocity signal is used in data acquisition by seismic wave method, and then the acceleration signal is differentiated after playing back the computer, which may result in signal distortion and can't fully reflect the abnormality. The ground penetrating radar can avoid this problem.

**Detection Results of Section2**. The instrument parameters of ground penetrating radar (GPR) and seismic wave method are consistent with those of section1 on 3 o'clock longitudinal survey line of the right line of section2. The detected GPR image and the normal wave train diagram of seismic wave are shown in Figs. 11, 12 and 13.

From the above image results, it can be seen that when the ground penetrating radar is used, the 400 MHz antenna image shows that the in-phase axis of the radar reflection wave at the lower part of the segment with 70.5–72 m survey line mileage (corresponding to the ring number 352) is staggered and discontinuous. It is speculated that the grouting layer in this range is not dense, but the 900 MHz antenna has no obvious response to this defect due to excessive interference of double-layer steel bars inside the segment. From the wave train diagram of seismic data, it can be seen

**Fig. 11** 400 MHz antenna image of 3 o'clock azimuth radar on the right line of Section2

**Fig. 12** 900 MHz antenna image of 3 o'clock azimuth radar on the right line of Section2

**Fig. 13** Normal wave train diagram of seismic waves from ring 350 to ring 354 at 3 o'clock on the right line of section2

that there are low-frequency reflections and in-phase staggered sections in many data in the 351 and 352 rings, and the seismic wave data in the 351 and 352 rings in the isoline diagram and spectrum analysis diagram are obviously low-frequency, so it can be inferred that grouting is not dense in this area. From the results of this section, it can be shown that the electromagnetic wave of the ground penetrating radar will be interfered by steel bars, and it is easy to produce misjudgment, and the seismic wave method can make up for this defect. If the signal results detected by the two radars are inconsistent, the seismic wave method can be used for verification.

#### **5 Conclusions and Suggestions**

Non-destructive testing is a very suitable method for detecting the quality of grouting behind the shield tunnel wall. In this paper, based on a metro line in Foshan, we study two methods for detecting the quality of grouting behind the shield wall: ground penetrating radar method and seismic wave method. By comparing the advantages, characteristics and limitations of each method between the test section and the actual project, we have summarized several conclusions of applying the ground penetrating radar method and seismic wave method to the grouting detection behind the wall.

The ground penetrating radar can accurately identify the grouting defects behind the shield tunnel wall, and can adjust the corresponding parameters according to different structures, with high detection efficiency and wide coverage, and can quickly make a general survey of the grouting defects behind the shield segment wall. However, the resolution of identification is greatly affected by the difference of dielectric constants of adjacent strata, and the electromagnetic wave is easily interfered by shield materials, resulting in misjudgment of identification.

Seismic wave method can also accurately identify the grouting defects behind the shield tunnel wall. At the same time, seismic waves will not be strongly interfered by steel bars, signals will not be shielded by steel bars, and there is still a good detection ability for targets behind steel bars. At the same time, in some media, seismic waves can produce better reflection than electromagnetic waves. However, this method has stricter requirements for sensors and signal processing, and its detection efficiency is low, so it can't achieve scanning detection.

Combining the advantages and disadvantages of the two methods and the research results of the test section, it is found that by using the two methods comprehensively, we firstly use the ground penetrating radar method to scan the shield tunnel, and then use the seismic wave method to further detect the suspicious defect section according to the identification results, which can accurately identify various defects of grouting behind the segment wall.

#### **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## **Service Performance Evaluation of Rubber Floating Slab Track for Metro in Operation**

**Qiuyi Li and Wei Luo**

**Abstract** Guangzhou metro line 1 is one of the first to adopt floating slab track in the Chinese mainland, it has been in operation for more than 20 years up to now. In order to obtain the vibration reduction performance of the floating slab track after long service, systematic field test and laboratory test are carried out. The results show that: (1) The rubber bearing of floating slab track keeps good appearance. (2) The measurement results of shore hardness and elongation at break show that, the rubber bearing has a certain degree of hardening after long service. Other mechanical properties of the rubber bearing still meet the design requirements. (3) The first natural frequency of the floating slab track is 33.9 Hz, its vibration reduction effect keeps good, the test result is 12.9 dB. The effective damping frequency band of the floating slab track is above 35 Hz. (4) The vertical dynamic displacement amplitude of rail and track bed is lower than the limit value given in relevant specifications. (5) The test results show that after 20 years of service, the floating slab track is in good condition, it's still qualified to maintain normal service.

**Keywords** Metro · Floating slab track · Vibration reduction · Service performance · Experimental study

### **1 Introduction**

In 1997, the floating slab track was introduced into China (mainland) and first used in Guangzhou metro line 1 [1–4]. Up to now, its service time has been more than 20 years, the operation of 6 type A trains. The floating slab track of Guangzhou metro line 1 uses rubber bearings as vibration isolation elements. Considering the extreme environment (wet, ozone, etc.) in underground line tunnels, the aging, deformation and damage that may occur in the long-term service of rubber materials are directly

Q. Li (B) · W. Luo

China Railway Siyuan Survey and Design Group Co. Ltd., Wuhan 430063, China e-mail<EMAIL_ADDRESS>
Railway Service Safety Key Laboratory of Hubei Province, Wuhan 430063, China

S. Wang et al. (eds.), *Proceedings of the 2nd International Conference on Innovative Solutions in Hydropower Engineering and Civil Engineering*, Lecture Notes in Civil Engineering 235, https://doi.org/10.1007/978-981-99-1748-8\_9

**Fig. 1** Typical cross section of floating slab track of Guangzhou metro line 1

related to the safe service and vibration reduction performance of the floating slab track [5–7]. It is necessary to evaluate the performance of the floating slab track comprehensively and systematically after long service.

The research team to which the author belongs has carried out an experimental study on the floating slab track of Guangzhou metro line 1.

#### **2 Design Overview of Floating Slab Track**

A floating slab track of 1110 m is laid on the main line of Guangzhou metro line 1, which is a single rubber bearing floating slab track. Each floating slab is 2.95 m in length, 2.8 m in width, 0.335 m in thickness at the center and 0.3 m at the edge, and weighs about 6.5 t. The bottom of the standard floating slab is prefabricated with 4 circular grooves, and 4 rubber bearings are installed. The rubber bearings are natural rubber products, with a diameter of 400 mm and a thickness of 75 mm. The design static stiffness of the rubber bearing is 12–16 kN/mm, and the lateral static stiffness of the lateral buffer rubber pad is 3.2–4.2 kN/mm [2]. The single toe spring fastener is adopted for the section of floating slab track. The typical cross section of the track is shown in Fig. 1.

#### **3 Field Test and Analysis**

#### *3.1 Appearance Detection of Rubber Bearings*

Electronic video endoscope was used to detect the appearance of rubber bearings. As shown in Fig. 2, the electronic video industrial endoscope consists of a display, a joystick, a cable and a probe, among which the probe is generally composed of two parts: a light source and a miniature camera. The endoscope works as follows:

A tiny camera takes pictures of the scene under the light source, and then converts the images into digital signals that are transmitted by cable to the display screen [8].

By using the sampling method of "separate one and pump one" (separate one floating slab and pump one floating slab for detection), 50% of the floating slab track of line 1 was extracted and its rubber bearings were tested. The appearance images of the rubber bearing obtained on site are shown in Figs. 3 and 4. The main conclusions of appearance detection are as follows:

**Fig. 3** Appearance detection image 1 of rubber bearing

**Fig. 4** Appearance detection image 2 of rubber bearing

## *3.2 Laboratory Test of Rubber Bearings*

Three rubber bearing samples were obtained from line 1 in the way of "take one for one", and then the mechanical property parameters of rubber bearings were tested in the laboratory. The test items included: appearance size, Shore hardness, tensile strength, elongation at break, permanent deformation under constant compression, static stiffness and dynamic stiffness. Figure 5 shows the rubber bearing sample, and the rubber bearing stiffness testing device is shown in Fig. 6.

The laboratory test results of the mechanical properties of the rubber bearing are shown in Table 1. The difference between the size of the bearing and the design size after permanent deformation is less than 1 mm, and the tensile strength and static stiffness still meet the design requirements. The Shaw hardness value of the rubber bearing material is about 15% higher than the designed upper limit value, and the elongation at break is about 9.5% lower than the designed lower limit value, indicating that the rubber bearing material has a certain degree of hardening after a long time of service.

**Fig. 5** Rubber bearing sample

**Fig. 6** Rubber bearing stiffness detection device

**Table 1** Laboratory test results of mechanical properties of rubber bearings

#### *3.3 Natural Vibration Characteristics of Floating Slab Track*

In this study, the field test of the vibration characteristics of the floating slab track was completed. The block length of line 1 floating slab track is 2.95 m. Three rows of measuring points are arranged, as shown in Fig. 7, with 6 measuring points in each row and 1 vertical acceleration sensor installed in each measuring point. The rigid force hammer is used as the excitation equipment, as shown in Fig. 8, and the measuring points 3, 4, 15 and 16 are used as the four hammering points.

The random subspace method is used to identify the modes and modal parameters of the floating slab track [9]. The natural vibration frequency and damping test results corresponding to modes 1–7 of line 1 floating slab track are shown in Table 2. The first order natural frequency is 33.9 Hz, and the corresponding structural damping is 6.866%. Figure 10 shows the mode shape correlation matrix of modes 1–7. The main diagonal elements of the matrix are all 1, and the values of the other

**Fig. 7** Layout diagram of measuring points for testing natural vibration characteristics of floating slab track

elements are very small, which indicates that the mode shapes of modes 1–7 obtained through identification have good orthogonality, and the identification results have high reliability.

#### *3.4 Damping Effect of Floating Slab Track*

**Layout of Test Section and Measuring Point**. Combined with the data of lines and running vehicles, this study selected two sections as shown in Table 3 to carry out field tests. The acceleration measurement point used to evaluate the vibration reduction effect of the floating slab track is arranged at the side wall of the tunnel 1.5 m away from the top surface of the rail, as shown in Fig. 9. The acceleration sensor used has been verified as qualified product by the third party testing institution, with a range of 10 g and a resolution of 0.0004 m/s2.

**The Measurement Value**. In the Standards of *Technical Guidelines for Environmental Impact Assessment for Urban Rail Transit (HJ453-2018)*, *Standard for Environmental Vibration in Urban Areas (GB10070-88)*, and *Measurement Method for Environmental Vibration in Urban Areas (GB10071-88)*, the measurement value is the maximum value of *VL*z vibration level in the process of train passing, which is shown as *VL*zmax in Fig. 11.

The frequency range of the analysis was 1–80 Hz, and the weighting factor was the whole-body Z-weighting factor stipulated by *ISO2631/1-1985*, as shown in Table 4.

**Table 3** Profiles of each test section

**Fig. 9** Vertical acceleration sensor on tunnel side wall installed on site

**Fig. 10** Mode correlation matrix

**Table 4** Z-weighting factors stipulated in *ISO2631/1-1985*

Aiming at the vibration acceleration data of a passing event, it is divided into several 1 s length data paragraphs (which can consider certain overlap coefficient). The method of calculating the *VL*z vibration level of each paragraph data is detailed as follows.

Fourier transform is performed on the data of time history of vibration acceleration to find out the vibration components in the frequency band corresponding to each central frequency, and inverse Fourier transform is performed on the vibration components identified. Then, the effective value of acceleration *a*w corresponding to each central frequency can be obtained according to Eq. (1).

$$a\_w = \left[\frac{1}{T} \int\_0^T a\_w^2(t)dt\right]^{\frac{1}{2}}\tag{1}$$

where, *a*w(*t*) is the acceleration data whose frequency is in the frequency band corresponding to a certain central frequency, and is the time function; T is the length of measurement time.

Further, the calculation formula of *VL*z vibration level is as follows:

$$VL\_z = 20\log\_{10}\left\{\frac{\left[\sum\_{j=1}^{n} (W\_j \times a\_{wj})^2\right]^{\frac{1}{2}}}{a\_0}\right\} \tag{2}$$

where: *W*j is the weighting factor corresponding to the Jth central frequency; *a*wj is the effective value of acceleration in the frequency band corresponding to the Jth central frequency. *a*0 is the base acceleration, *a*0 = 10−6m/s2

The damping effect Δ of floating slab track is calculated by the following formula:

$$
\Delta = V L\_{z \max 1} - V L\_{z \max 2} \tag{3}
$$

In the formula, *VL*zmax1 is the maximum value of *VL*z vibration level in the passing process of trains in the ordinary track section, and *VL*zmax2 is the maximum value of *VL*z vibration level in the passing process of trains in the floating slab track section.

**Test Results**. When the train passes through, the frequency domain distribution comparison of vertical acceleration level on the tunnel side walls of Sections 1 and 2 is shown in Fig. 12. Compared with the ordinary track, the vertical acceleration

level of the tunnel side walls is significantly reduced by the floating slab track. The corresponding frequency is 63 Hz.

*VL*zmax calculation values of tunnel sidewall of Sections 1 and 2 are shown in Table 5. The test result of damping effect of floating ballast bed is 12.9 dB.

#### *3.5 Dynamic Displacement of Floating Slab Track*

At the location of Section 2 in Table 3, the floating slab track is selected to carry out dynamic displacement test. The measuring points are arranged as shown in Fig. 14, where the rail displacement is the relative displacement between the rail and the track bed, and the track bed displacement is the relative displacement between the track bed and the tunnel backfill layer. The displacement sensor is verified as qualified product by the third party testing institution, the measuring range is ± 10 mm, and the accuracy can reach 0.01 mm. The typical time-history curve of the dynamic displacement of the floating plate track bed at the time when the train passes through is shown in Fig. 13. The peaks of the dynamic displacement curve reflect the impact of the wheel when it passes through the test section. The statistical values of dynamic displacement amplitude of rail and track bed are shown in Table 6. The average vertical dynamic displacement amplitude of floating plate is 1.32 mm at the end of the slab and 1.21 mm in the middle of the slab, which is less than the limit value of 3 mm given in Technical Specification for Floating slab track (CJJ/T 191-2012). The average vertical dynamic displacement amplitude of the rail is 3.5 mm at the end of

**Fig. 12** Frequency domain distribution curve of vertical acceleration level of tunnel sidewall of Sections 1 and 2

**Fig. 14** Plane layout of dynamic displacement measuring points of floating slab track

the slab and 2.5 mm in the middle of the slab, which is less than the limit value of 4 mm given in Technical Specification for Floating slab track (CJJ/T 191–2012).

#### **4 Conclusion**

Combined with the laboratory and field tests, the current performance of the floating slab track of Guangzhou metro line 1 was systematically studied. The main conclusions were drawn as follows:

**Table 6** Statistical table of dynamic displacement amplitude of rail and track bed (unit: mm)

(5) The test results of various indexes show that, after 20 years of service, the state of the floating slab track is generally good, and it has the conditions to maintain normal service.

## **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## **Study on the Influence of the Arrangement of Thermal Insulation Floor on the Thermal Insulation and Mechanical Properties of Hollow Slab**

#### **Yuchen Liu**

**Abstract** In order to study the influence of the arrangement of thermal insulation floor on the thermal insulation and mechanical properties of hollow slab, ABAQUS is used to establish the model of thermal analysis and mechanical behavior of hollow slab. By investigating distribution of temperature, distribution of heat flux, damage and deformation of floor, deformation of mid-span deflection and other characteristics of the floor section, it is concluded that although the transmission of heat can be effectively obstructed by the thermal insulation slab, the heat will be transferred to the interior of the floor through the gap between the thermal insulation slabs. The arrangement of thermal insulation slab is not the main factor which affect the thermal insulation properties of the floor with the same coverage area. Different arrangement of the thermal insulation slab has a certain impact on the mechanical performance of floor. It is recommended to arrange the thermal insulation slab in equal sections to fully improve the contact area between steel bar and concrete, which can effectively provide the bearing capacity of the floor.

**Keywords** Thermal insulation properties · Mechanical performance · The arrangement of thermal insulation floor · Hollow slab · Finite element simulation

## **1 Introduction**

The floor system is an important part in the whole system of prefabricated structure, in addition to being the main stress component, the floor also undertakes the function of space separation, insulation and so on. At present, some enterprises and colleges at home and abroad have carried out researches on the mechanical and thermal performance of floor, and achieved certain results [1].

In mechanical performance, most of the researches focus on the flexural rigidity of floor. For example, Lu et al. [2]. explored the influence of different factors on the

Y. Liu (B)

Infrastructure Development Office, Wuhan University of Technology, Wuhan 430070, Hubei, China

e-mail<EMAIL_ADDRESS>
© The Author(s) 2023

S. Wang et al. (eds.), *Proceedings of the 2nd International Conference on Innovative Solutions in Hydropower Engineering and Civil Engineering*, Lecture Notes in Civil Engineering 235, https://doi.org/10.1007/978-981-99-1748-8\_10

flexural rigidity of composite sandwich slabs. Studies have shown that increasing the thickness of the concrete pavement, decreasing shear span, the attachment of the shear screws can increase the post-rigidity (before the slip observed) of the composite sandwich slabs. Also, they proposed a modified calculation formula of flexural rigidity. Tian et al. [3]. put forward a kind of large-span composite slab, the results show that mid-span deflection and crack width of the large-span composite slab specimen under serviceability limit state meet the requirements of the relative codes, which shows the obvious bidirectional force transmission characteristics, that can be designed as two-way slab. Yang et al. [4] proposed a type of composite slab with additional steel trusses, and carried out static test and numerical analysis on its bottom slab, which includes that the new type of composite slab can improve the bearing capacity, increase the bending stiffness and enhance the crack resistance effectively. In thermal performance such as thermal insulation and fire prevention, Kovalow et al. [5] used the ANSYS software to study the thermal performance of the hollow reinforced concrete floor, and compared it with the results of experimental study. Basing on this, an approach is proposed that allows to take into account all types of heat transfer by specifying cavities as a solid body with an equivalent coefficient of thermal conductivity. Zhao et al. [6] studied the thermal performance of composite slabs comprised of closed profiled steel decking and recycled aggregate concrete (RAC). Results indicated that the temperatures within composite slabs declined with the increasing coarse recycled aggregate (CRA) content. Finally, simplified formulas were developed to predict the cross-section temperature in RAC composite slabs.

It can be seen that although there are researches on the thermal performance of floor at home and abroad, there are few on the thermal performance of fully prefabricated floor. In particular, there is barely research on the thermal insulation performance of the fully prefabricated floor, nearly there is no more systematic and comprehensive report on it so far. Therefore, this paper intends to use finite element software to model and calculate a fully prefabricated floor which are applicated in engineering, and analyze its impact on the thermal insulation and mechanical properties of hollow slab by changing the arrangement and spacing of thermal insulation slab, aiming to provide a reference for the engineering application and design.

#### **2 Establishment of Finite Element Model**

#### *2.1 Fully Prefabricated Floor Structure*

In order to study the influence of the arrangement of thermal insulation slab on its thermal insulation and mechanical performance, we change the number, size, spacing and other parameters of its thermal insulation slab on the premise that the rate of thermal insulation coverage of the fully prefabricated hollow slab is the same (48.7%). As shown in Table 1, in which board 0 is the control board. The laying mode is shown in Fig. 1.

**Table 1** Different forms of cavity floor

#### *2.2 Finite Element Mesh*

In the model of the thermal analysis model in ABAQUS, eight node linear heat transfer hexahedron element DC3D8 is used, and two node transfer connection element DCID2 is used for reinforcement analyzing. In the model of rigidity of structural, eight node linear hexahedral element C3D8R is selected for concrete analyzing, and two node linear three-dimensional truss element T3D2 is used for reinforcement analyzing. The grid element division of the fully prefabricated hollow slab is shown in Fig. 2.

#### *2.3 Properties of Materials*

For the thermal parameters of reinforcement and concrete in the temperature field in this model, such as specific heat capacity, thermal expansion coefficient, thermal conductivity and density, can be determined by referring to reference [7]. In the model of stress analysis, the model of plastic damage in ABAQUS is adopted for concrete, the curves in specification [8] are adopted in the stress–strain model of concrete which are under compression and tension, in which the strength grade of concrete is C30; The stress–strain curve of reinforcement adopts the model of linear strengthening, and the strength grade of reinforcement is HRB335. Settings of floor contact constraints are shown in Fig. 3.

The floor is composed of the upper panel, the lower panel and the insulation material filled in the hollow. The thermal insulation materials are made of extruded

**Fig. 1** The laying mode of floor (Unit: mm)

**Table 2** Temperature of the physical parameters of materials

polystyrene foam (XPS). The thermal and physical parameters of the materials are shown in Table 2.

#### *2.4 Contact and Boundary Conditions*

In the model of thermal insulation analysis of floor, the concrete floor and the thermal insulation slab in it are connected by a surface-to-surface of tie, and concrete slab and the reinforcement fabric in it are connected by a set-to-surface of tie, which can effectively transfer the temperature of the joint. The initial temperature of the floor is uniformly set as 20 °C, the temperature of the upper surface of the floor is uniformly set as 35 °C, and the temperature of the lower surface of the floor is uniformly set as 5 °C. The convective heat transfer coefficient of the outer surface of the wall is 23.3 W/(m·K), and that of the inner surface of the wall is 8.7 W/(m·K).

In the model of stress analysis of floor, the concrete slab and the thermal insulation slab inside are connected by a surface-to-surface of tie, while the concrete slab and the reinforcement fabric in it are connected by an embedded connection. The boundary conditions of the floor are simply supported at both ends. One end of the constrained edge of the floor in the model is UI = U2 = U3 = URI = UR2 = UR3 = 0, while the **Fig. 4** Boundary conditions of floors

(a)Boundary conditions of thermal insulation (b)Boundary conditions under stress

other is UI = U2 = URI = UR2 = 0. When uniform load is applied to the surface of the floor, the boundary conditions are shown in Fig. 4.

## **3 Results Obtained Through Simulation**

### *3.1 Analysis of Thermal Insulation Performance*

#### (1) Temperature response of floor

With different arrangement of thermal insulation slab, heat nephogram of crosssection of floor is shown in Fig. 5. It can be seen from the figure: (1) The color of the upper surface of floor is deeper, that is, the temperature is higher but less than 35 °C. The color of the lower surface is lighter, that is, the temperature is lower but more than 5 °C. That is because of the heat is transferred from the higher side to the lower, thus there is a certain temperature gradient. (2) The temperature on the surface of the hollow slab is mainly concentrated at the place where the thermal insulation slab is arranged. There is little difference in color in the middle layer of the floor, that is, the temperatures are basically similar. From which we can indicate that although the transmission of heat can be fully obstructed by the thermal insulation slab, the heat will be transferred to the interior of the floor through the gap between the thermal insulation slabs. (3) Although the reinforcement fabric runs through the whole floor, the thermal bridge effect of it is not obvious.

From the heat nephogram of the reinforcement in floor shows in Fig. 6, the temperature distribution of reinforcement is similar to that of floor. It can be seen that, firstly, the temperature of the upper reinforcement fabric in hollow slab is higher, which of the lower reinforcement fabric is lower. Secondly, the temperature of the reinforcement at the gap of the thermal insulation slab is lower than that of which in the thermal insulation slab, which is due to the effective transfer of the temperature of the floor from the above to the interior.

(2) The heat flux of floor

Figure 7 shows the nephogram of heat flux of floor, from which we can see the heat flux of floor presents a phenomenon similar to the thenephogram of floor temperature, that is the distribution of heat flux of floor presents well blocks around thermal

**Fig. 5** Heat nephogram of cross-section of floor

**Fig. 6** Heat nephogram of the reinforcement in the floor

**Fig. 7** Cloud chart of heat flow density of floor section

insulation slab, while the difference is that the heat flux at the gap between the thermal insulation slab is deeper, which means that the heat is effectively transferred here. It also indicates that the thermal insulation slab can obviously obstruct the tranfer of heat in cross sections of the floor. However, the thermal bridge effect will be formed along the gap. Meanwhile, the vertical reinforcement in the section also plays the role of thermal bridge, but is not obvious.

#### (3) Comparison of heat transfer coefficients

Table 3 shows the results of finite element numerical simulation of fully prefabricated hollow slab. Taking slab 0 as a reference, the ratio of heat transfer coefficients from slab 0 to slab 7 is about 1:1.04:1.03:1.03:1.02:1.03:1.04:1.04, and the difference of heat transfer coefficients between each slab is uniformly less than 5%. It is obviously that the arrangement of the thermal insulation slab is not the main factor which affect the thermal insulation properties of the floor with the same coverage area.

#### *3.2 Analysis of Mechanical Performance*

(1) Damage and deformation of floor

**Table 3** Numerical results of finite element simulation

The above research shows that the arrangement of thermal insulation slabs has little effect on the thermal insulation performance, but may affect the mechanical performance of floor. Therefore, the static loading of the floor with different arrangement of thermal insulation slab is simulated and the uniform load is applied on the upper surface. The deform0ation form of the floor is shown in Fig. 8.

**Fig. 8** Cloud chart of floor damage and deformation

As can be seen from Fig. 8, the deformation forms of the floor with different arrangement of thermal insulation slabs are basically the same. Mainly showing that the deformation of mid-span deflection is too large, and the plastic damage of the floor is mainly distributed in the mid-span section, which also indicates that cracks are firstly developed here. However, the plastic damage distribution of the floor is influenced by different arrangement of thermal insulation slab. With more thermal insulation slabs, and the more uniform the size is, the distribution of plastic damage of the floor is more uniform. For example, the distribution of plastic damage in the mid span section of slab 1, slab 2, slab 3, and slab 4 is more uniform than that of slab 0, slab 5, slab 6, and slab 7. Among them, the uniformity of the plastic damage distribution of slab 4 is the best, and that of slab 7 is the worst.

#### (2) Stress of the reinforcement in floor

Figure 9 shows the stress and deformation nephograph of the reinforcement fabric in hollow slab with different arrangement of thermal insulation slab, which shows the longitudinal reinforcement of board 7 basically reaches the yield strength, while that of board 4 still has stresses. It can be indicated that the mechanical behavior of the floor with the arrangement of thermal insulation slab of 4 is the best, of which the coordination of the joint deformation between the reinforcement and the floor is better. This is due to the fact that in the working condition of slab 4, the reinforcement have full access to the concrete and the force is uniform, while in the working condition of slab 7, the reinforcement have no access to the concrete, thus they cannot bear the force together.

#### (3) Change in the deflection of the floor

Extract the curve of the mid span deflection of the floor with the change of load, as shown in Fig. 10a, it can be seen that curves in this figure. Basically present a similar pattern. With the increasing of the load imposed on the floor, the mid span deflection of the floor also increases, and the rate of increase gradually accelerates. Taking the corresponding loads of when the mid-span deflection of the floor is 1/20 of the span (175 mm), among them, the load corresponding to plate 4 is the maximum (252.7 KN), and the plate 7 is the minimum (241.3 KN), which is basically consistent with the results of the previous analysis.

In order to further explore the magnitude of the influence, when the uniform load applied to the floor is 0.1N /mm2, the curves of full deflection of the floor along the longitudinal are as Fig. 10b, in which the deflection of board 4 is the smallest and the mid-span deflection is 223.1 mm, and the deflection of board 7 is the largest and the mid-span deflection is 283.6 mm. The difference between them is about 27%. Therefore, it is recommended that the thermal insulation slabs be evenly divided as possible when with the same area, so as to fully improve the contact between reinforcement and concrete which can effectively provide the bearing capacity of the floor.

**Fig. 9** Cloud chart of floor reinforcement stress

(a) Curve of mid span from deflection (b) deflection curve along the longitudinal direction

**Fig. 10** Deflection change of floor slab

#### **4 Conclusion**

Based on the simulating carried by finite element of mechanical performance of the fully prefabricated hollow slab, the following results are obtained:

The range of temperature is larger at the surface of the floor surface is large, but smaller within the floor, which means that although the transmission of heat can be effectively obstructed by the thermal insulation slab, the heat will be transferred to the interior of the floor through the gap between the thermal insulation slabs.

The ratio of the heat transfer coefficient of hollows slabs with different arrangement of thermal insulation panels is about 1: 1.04: 1.03: 1.03: 1.02: 1.03: 1.04: 1.04, it can be seen that the difference of heat transfer coefficient between each board is uniformly less than 5%. That is, the arrangement of thermal insulation slab is not the main factor which affects the thermal insulation properties of the floor with the same coverage area.

Different arrangement of the thermal insulation slab has a certain impact on the mechanical performance of floor. It is recommended to arrange the thermal insulation slab in equal sections to fully improve the contact area between steel bar and concrete, which can effectively provide the bearing capacity of the floor.

#### **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## **Analysis of Slurry Ratio of Rotary Digging Pile in Deep Sand Layer**

**Wanjun Zang and Jiang Wen**

**Abstract** Slurry ratio is a crucial link in the construction of bored pile, which directly determines the quality of bored pile. In order to determine the key performance parameters of the slurry required to form piles in the deep sand layer, relying on Huizhou north station engineering, an orthogonal test of slurry proportioning was designed and carried out, and SPSS statistical analysis software was used to carry out bivariate correlation analysis and multivariate stepwise analysis of the test results, combined with the slurry performance index test regression equation and using MATLAB software optimization processing, slurry optimal mix ratio and verify, the research results show that: orthogonal test screening value, software calculation value, test value is not different. Conclusion: The results show that bentonite and CMC have significant influence on slurry indexes, while Na2CO3 and PHP can adjust slurry performance to meet the slurry use standard; the optimal mix ratio is 148 g bentonite, Na2CO3 5.2 g, CMC 3.5 g, PHP 0.05 g; the slurry ratio test analysis and treatment, and the optimization mix ratio is feasible and reasonable, class I pile proportion more than 98% to meet the actual engineering requirements.

**Keywords** Mud · Orthogonal test · Correlation analysis · Regression analysis · Mix ratio

## **1 Introduction**

With the continuous development of urban construction, high-rise residential buildings and large buildings are increasing, and the pile foundation is being promoted because of its large bearing capacity. The superstructure transfers the load to the surrounding soil through the pile foundation, effectively reducing the settlement

W. Zang (B) · J. Wen

The School of Civil Engineering, Fujian University of Technology, Fujian 350118, China e-mail<EMAIL_ADDRESS>
The Key Laboratory of Underground Engineering, Colleges and Universities in Fujian Province, Fujian 350118, China

of the pile body and meeting the seismic requirements of the building. Circular excavation bored pile has the advantages of strong adaptability, no noise and high bearing capacity, and is widely used in pile foundation construction, but in practical engineering, geology conditions are complex and changeable, and quality problems emerge in endlessly. The bearing capacity of pile foundation plays a decisive role in engineering safety, and the slurry quality directly determines the quality of bored pile foundation [1].

Slurry mainly plays the role of protecting the hole wall, slag discharge and hole cleaning in the construction. At present, the research on the relationship between hole forming (pile) and slurry has made some results [2–6]. However, there is not much research on the design and analysis of slurry mix ratio, especially the optimization of different soil layers. Wang X P determined the proportion of the main component of the pile wall protection slurry suitable for the ultra-thick sand layer through the ratio test [7]; Wang J et al. determined the optimal quality ratio suitable for the volcanic ash deposition areas through the slurry performance test [8]; Li et al. adopted the uniform design test method for the mix ratio test, and obtained the optimal slurry mix ratio through analysis and optimization [9]. Most of the above scholars determine the slurry mix ratio through the analysis of the test results, but there are very few studies to determine the optimal mix ratio through further optimization and comparative verification.

Slurry preparation is a crucial part of the construction of bored pile. For different geological conditions, different proportions of slurry should be prepared. Combined with the station project of Huizhou North Railway Station, the range of bentonite and additives was determined according to the actual formation conditions, and then the orthogonal test was designed, and SPSS and MATLAB were analyzed to determine the optimal slurry ratio.

#### **2 Project Overview**

Huizhou North Station Railway is a line side flat station with elevated waiting room, including station room, platform canopy, underground contact passage, subway section, elevated ramp bridge, etc., with a total construction area of 49,998 m2. The foundation form is bored cast-in-place pile, the foundation design grade is grade A, and the design grade of the building pile foundation is grade A. The revealed strata in the proposed site are mainly Quaternary strata, and the conditions of each stratum are shown in Table 1.

As can be seen in Table 1, the sand layer of the site is deep and thick, and the rotary excavation into the hole in the deep sand layer can easily cause engineering problems such as poor hole wall integrity, fast sand precipitation, excessive sediment and buried drilling. According to the distribution of on-site strata, refer to the Technical Specification for Highway Bridge and Culvert Construction (JTG/T 3650-2020) and relevant documents [10–12], The basic performance parameters of the slurry are determined as shown in Table 2.

**Table 1** Stratigraphic distribution

**Table 2** Basic performance parameters of slurry

### **3 Trial**

#### *3.1 Test Materials*

The raw material of the slurry is calcium bentonite with large mesh and strong stability, and the admixture is selected from Carboxymethyl Cellulose (CMC) which can increase the slurry viscosity and form a film on the soil surface to prevent the hole wall from peeling; Sodium Carbonate (Na2CO3), Can control slurry PH, increase the thickness of hydration film, improve slurry stability, reduce the pollution of slurry by calcium ions and groundwater; Partially Hydrolyzed Polyacrylamide (PHP), can make the slurry colloid form a chemical film in fine sand, coarse sand and gravel soil, close the hole wall, and keep the hole wall stable.

#### **Table 4** Factor levels table

### *3.2 Test Instrument*

The test and measurement content and the required instruments are shown in Table 3.

#### *3.3 Test Design*

Orthogonal tests were determined by orthogonal tables, and the results were analyzed statistically. Among them, the same number of tests is performed at the level of either or both factors, which makes them very representative according to the orthogonal table and reduces the number of trials. The orthogonal test method in the ratio design can avoid unnecessary calculation and optimize the design process. Using four levels and four factors orthogonal test, the factors and level distribution are shown in Table 4. Among these, a is bentonite, b is Na2CO3, c is CMC and d is PHP.

16 trials were designed according to the orthogonal test table as shown in Table 5.

#### *3.4 Test Method*

According to the orthogonal test table, the bentonite, Na2CO3, CMC, PHP in addition to 1000 mL water in turn, using JJ-1 electric mixer to stir and mix at a speed of 850

**Table 5** Orthogonal test table

r/min, hydrate and expand for a certain period of time at room temperature, and test the data with reference to the measurement method given in the relevant specifications.

#### **4 Analysis of the Test Results**

#### *4.1 Test Results*

According to the 16 sets of test ratio shown in Table 4, the performance test of each formula slurry was tested separately, and the calculated mean value of the measured test data is shown in Table 6. Among them, ϕ300 and ϕ600 are the readings of the rotating viscosimeter at 300 r/min and 600 r/min, respectively, and the plastic viscosity and yield point are calculated from the two sets of data.

$$
\eta = \varphi\_{600} - \varphi\_{300} \tag{1}
$$

$$
\pi\_{\rm d} = 0.511(\varphi\_{300} - \eta) \tag{2}
$$

#### *4.2 Correlation Analysis*

Correlation analysis is one of the very mature basic theories in statistics. The interdependence between various phenomena can be manifested as functional relations or correlations among variables [13].

SPSS analysis software was used to carry out bivariate correlation analysis on the test results [14], analyze the correlation degree between each factor and the performance of the slurry, and obtain the correlation coefficient. In this paper, the four factors: bentonite, Na2CO3, CMC, PHP, slurry six indexes: specific gravity, viscosity, plastic viscosity, yield point, water loss, filter cake thickness were statistically analyzed, and the correlation coefficient of the factors and each performance of slurry was obtained, as shown in Table 7.

In correlation analysis, positive correlation: correlation coefficient r > 0; negative correlation: r < 0; high correlation: | r | 0.8; moderate correlation: 0.8 > | r | 0.5; low correlation: 0.5 > | r | 0.3; weak correlation: | r | <0.3.

As shown from Table 7:

**Table 7** Correlation coefficient between various factors and various slurry performance

Among them, CMC is weakly related to the specific gravity of slurry, moderately related to plastic viscosity, yield point, and water loss, and highly related to viscosity and filter cake thickness, indicating that It shows that CMC can significantly increase the viscosity of the slurry, enhance the internal friction between the suspended particles of the slurry and between the slurry particles and the liquid phase, improve the slag carrying capacity of the slurry, and reduce the water loss.

(4) Positive correlation with PHP: slurry specific gravity, viscosity, plastic viscosity, water loss, filter cake thickness, and negative correlation with PHP: yield point. Among them, PHP is weakly correlated with slurry viscosity, specific gravity, yield point and water loss, and is weakly correlated with filter cake thickness and plastic viscosity, indicating that PHP mainly plays the role of flocculation. In this test, the lifting effect was not obvious, perhaps because the slurry PH value was not increased to above 10, and the PHP was not fully effective.

#### *4.3 Regression Analysis*

As a screening method, the stepwise regression introduces the regression equations one by one according to the effect of each performance indicator, while the nonsignificant factors are always not introduced [15].

The independent variables were bentonite, Na2CO3, CMC and PHP, and the specific gravity, viscosity, plastic viscosity, yield point, water loss and slurry skin thickness of the slurry were the dependent variables. To determine the regression equation between the respective variables and the dependent variables, the multiple stepwise regression analysis was conducted using SPSS software, and the results are shown in Table 8.

Comparing Tables 7 and 8, the magnitude of the significance of the influence of the independent variable on the dependent variable in the regression equation coincides with the correlation analysis, e.g., The regression equation of the specific gravity of slurry is composed of *x*1 (bentonite) and *x*2 (Na2CO3), which is consistent with the

*Note 1 x*1 is bentonite, *x*2 is Na2CO3, *x*3 is CMC, and *x*4 is PHP *Note 2* y1 is the specific gravity, y2 is the viscosity, y3 is the plastic viscosity, y4 is the yield point, y5 is the water loss, and y6 is the thickness of the filter cake

**Table 9** Analysis of variance of the regression equations

large correlation coefficient between bentonite, soda ash and slurry proportion in the correlation analysis. Therefore, the correlation analysis and regression analysis are suitable for this test analysis and meet the actual requirements.

#### *4.4 Regression Equation Test*

To determine the fitting effect of the regression equation and the slurry performance index, the regression equation test is required. Given the significance level α = 5%, a one-way ANOVA was performed on the dependent variable versus the independent variable composing the regression equation. For example, in the specific gravity analysis, F1, P1, F2, and P2 in the table are the results of the variance analysis of specific gravity and *x*1 (bentonite) and *x*2 (Na2CO3). The cut-off value F is obtained through the F distribution table F0.05(3,12) = 3.49, the F value obtained in the Table is compared with the F value found in Table 9, it can be seen that the value of F1 and F2 is less than 3.49 but close to the critical value, indicating that the respective variable had a significant influence on the dependent variable. And the coefficient of determination R2 is close to 1, indicating that the regression equation fits well and meets the requirements of this experiment.

#### **5 Mix Ratio Optimization**

Further, slurry specific gravity, viscosity, plastic viscosity, yield point, water loss, filter cake thickness six indicators are the standard to measure the performance of slurry. To determine the optimal slurry mix ratio, the function is optimized and the following objective function is established:

$$Z = \mathbf{y}\_n \tag{3}$$

In formula: n = 1~6. The constraints on *y* and *x* in the objective function are:

$$\begin{cases} 1.08 \le \mathbf{y}\_1 \le 1.15 \\ 22 \le \mathbf{y}\_2 \le 30 \\ 8 \le \mathbf{y}\_3 \le 13 \\ 1.5 \le \mathbf{y}\_4 \le 3.0 \\ \mathbf{y}\_5 \le 15 \\ 0.5 \le \mathbf{y}\_6 \le 2.0 \end{cases} \tag{4}$$

$$\begin{cases} 128 \le x\_1 \le 188 \\ 4.4 \le x\_2 \le 5.6 \\ 2.7 \le x\_3 \le 4.5 \\ 0.05 \le x\_4 \le 0.20 \end{cases} \tag{5}$$

According to the function relationship of *y* and *x* determined by the regression equation, the objective function was optimized by MATLAB software. After calculation, after calculation, when the material dosage is 148 g of bentonite, 5.2 g of Na2CO3, 3.5 g of CMC, and 0.05 g of PHP, the function has an optimal solution, and each index is tested. The screening results of orthogonal tests (148 g of bentonite, 5.2 g of Na2CO3, CMC 3.9 g, and PHP 0.05 g), software calculation results and test results are shown in Table 10.

According to the table, the orthogonal test screening value, software calculation value and test value do not differ much. Taking the test value as a reference, the smallest difference is the specific gravity: the deviation of the screening value is 1.77%, and the deviation of the calculated value is 0.88%; the largest difference is the thickness of the slurry: the deviation of the screening value is 21.43%, and the deviation of the calculated value is 7.14%. All of the above meet the requirements, indicating that the orthogonal test is the method of finding the principal contradiction and selecting the better scheme in this test. The optimized treatment results of MATLAB software meet the requirements of various slurry indexes, so this method is applicable for the optimization analysis of slurry mix ratio.

The slurry ratio is specifically applied to Huizhou North Railway Station project and detected by low strain and acoustic wave transmission method: 1181 Class I piles, accounting for 98.504% of the tested piles; 18 class piles, accounting for 1.495% of the tested piles; There are no III and IV piles. The proportion of pile foundation integrity testing type I pile has reached more than 98%, which has ensured the project quality and project progress, achieved good results, and accumulated certain engineering experience.

#### **6 Conclusion**

(1) Orthogonal test screening value, software calculation value and test value are not much different. With the test value as the reference, the smallest difference is the proportion: The deviation of the screening value is 1.77%, and the deviation

of the calculated value is 0.88%; the biggest difference is the thickness of the slurry skin: the deviation of the screening value is 21.43%, and the deviation of the calculated value is 14.29%. All the above meet the requirements, indicating that the orthogonal test, as a method of finding the main contradiction from multiple factors and multiple levels and selecting better schemes, has achieved good results in this test.

**Acknowledgements** This project is supported by Key Technology and Application Research Project of New Construction of Huizhou North Station of Ganzhou-Tangxia Section of Ganzhou-Shenzhen Railway (GY-H-21017).

## **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## **Numerical Analysis of Bearing Capacity of Basic Stress Unit Frame of Socket-Type Wheel Buckle Formwork Support Frame Body**

**Heng Liu, Weiwen Xu, Hongtao Xu, and Xuemiao Xiang**

**Abstract** Socket-and-socket wheel-buckle steel pipe formwork support frame is a new formwork support system, with fixed component specifications and fast erection speed, which can significantly improve the construction efficiency. Its application performance is different from that of fastener-type, bowl-buckle-type and platebuckle-type, and there are not enough experimental and numerical analysis results to consult. Therefore, based on the bearing capacity test of the basic bearing units of socket-type wheel-buckle steel pipe formwork support, the finite element software ANSYS is used to conduct numerical analysis of 12 groups of basic bearing units, and the influence degree of vertical bearing capacity of different specifications of unit supports is studied by stud spacing and wedge tightness of cross bar plug. It is pointed out that the stiffness of formwork support in the vertical and horizontal directions should not be too different in the construction process.

**Keywords** Formwork support frame · Numerical analysis · Bearing capacity test

#### **1 Introduction**

In recent years, with the continuous development of civil engineering in China, various tall and complex structures have emerged. In the pursuit of efficiency today, people begin to seek new formwork support systems to solve the disadvantages of slow construction efficiency such as fastener-type, bowl-buckle-type and pan-buckletype formwork support frames. Therefore, various formwork support systems with new structural forms have emerged as the times require. Xu et al. [1, 2] introduced a new type of plug-in formwork support frame, which realized the advantages of axial force transmission of vertical bar, good connection performance, high bearing capacity and strong applicability. Qiang [3] has studied the practical application of bolt-type formwork support and the semi-rigid connection of bolt, welding lug, welding seam and node, etc. The results show that the bolt will self-lock in the

H. Liu (B) · W. Xu · H. Xu · X. Xiang

China Construction Seventh Engineering Division. Corp. Ltd., Zhengzhou 450000, China e-mail<EMAIL_ADDRESS>
© The Author(s) 2023

S. Wang et al. (eds.), *Proceedings of the 2nd International Conference on Innovative Solutions in Hydropower Engineering and Civil Engineering*, Lecture Notes in Civil Engineering 235, https://doi.org/10.1007/978-981-99-1748-8\_12

normal work of formwork support, and its connection performance is good, and it is suggested that the connection stiffness k of node can be taken as a low value. Through analysis, it is considered that it is safe and reasonable to take K = 20 kN·m/rad. Huang [4] carried out horizontal force loading tests of two types of bolt-type plane frames under four wedge tightness conditions and two groups of basic unit load-bearing performance tests, and studied the joint characteristics and load-bearing performance of unit frames. He pointed out that when bolt-type formwork supports are used, the members are mainly subjected to axial force, and the bending moment is usually less than 0.2 KN·m. Therefore, the semi-rigid connection stiffness of the joint that affects the structural performance is the initial connection stiffness of the joint, but the initial tangent stiffness of the joint. Similarly, socket-and-socket wheel-buckle steel pipe formwork support frame is gradually applied to various construction sites, and it has similarities with plug-in type, plug-in type, plug-in type and other frame bodies in bearing mechanism. However, its structural form is simpler [5], and the outer diameter of steel pipes is 48 mm. The difference between it and other forms of steel pipe formwork supports lies in the vertical and cross bar connection joints, and the connection joints are shown in Fig. 1. There is no bracing hole at the vertical pole bearing plate, so this form of formwork support needs to use the bar of fastener rack and rotating fastener to set up bracing. Its installation and disassembly are convenient and quick, and its stability is easy to ensure, which can shorten the construction period and improve the construction efficiency. However, its cross bar is disconnected at the node, so it can't be applied to the side formwork engineering of vertical members. Therefore, the load directly borne by the cross bar is mainly a small amount of materials piled up or the passage of personnel and equipment, and its load is small, which won't cause plastic deformation of the bearing plate.

The node size is shown in Fig. 2. The inner diameter of the bearing plate is 48 mm and the outer diameter is 108 mm. The plug hole is 18 mm long and 12 mm wide, and four jacks are evenly distributed on the circumference, and the distance between adjacent holes is 90; The bearing plate thickness is 20 mm. The end of the cross bar has an arc joint of 16 mm long, 3.5 mm wide and 8 mm thick. A plug of 10 mm wide

**Fig. 1** Detail structure of socket-type wheel-buckle steel pipe formwork support frame node

**Fig. 2** Detail dimensions of socket-type wheel-buckle steel pipe scaffold node

and 90 mm long is connected to the joint, and the tail of the plug is tapered with a ϕ6 mm pin hole.

In this paper, the finite element verification is carried out for the experimental study of the basic stress unit frame of the socket-and-socket type steel pipe formwork support frame in reference [6]. The difference of joint stiffness is directly caused by the unsatisfactory contact surface during the erection of the frame body, so it is expressed by the wedge tightness of the cross bar plug in different degrees. On this basis, the variable parameter analysis is carried out with the wedge tightness of the cross bar plug and the vertical and horizontal spacing of the vertical poles as variables. During the verification process, the cross section of the pipe rack is still considered as 48 × 2.75; However, the normal wedge tightness of the cross bar plug corresponds to 7~10 times during erection in the formwork support project, and the minimum wedge tightness of the cross bar plug corresponds to 0~3 times during erection in the formwork support project.

#### **2 Finite Element Analysis Assumption**

Combined with the erection and boundary conditions of formwork support system in engineering application, the finite element software ANSYS is used for numerical analysis to make the following assumptions.

transmit shear force and bending moment through the friction at the node, so it is assumed that the stud is rigidly connected, and the junction between the stud and the crossbar is semi-rigidly connected.

(3) When checking the ultimate bearing capacity, the material properties are still based on the reference [5], and the results obtained from the unidirectional material tensile test are the actual material properties.

## **3 Numerical Analysis of Bearing Capacity of Different Specifications of Frame Body**

#### *3.1 Definition of Material Parameters*

According to the setting of different structural analysis units in the finite element software ANSYS, Beam188 element with shear deformation is used to simulate the vertical bar, and Combin39 element is used to simulate the stiffness of the vertical bar node in the numerical analysis of formwork support. The material parameters are obtained by uniaxial tensile test: E = 2.21 × 105 MPa, fy = 359 MPa; The definition of bending moment-turning angle relationship of spring element under two working conditions of normal wedging degree and minimum wedging degree is shown in Table 1 [7].

#### *3.2 Establishment of Finite Element Model*

Taking the vertical and horizontal spacing of vertical poles and the wedge tightness of nodes as variables, the influence degree of vertical bearing capacity of socket formwork support frame is studied. As the vertical pole bearing plate spacing and cross bar size are fixed, finite element models of formwork support frames with different specifications are established, and the frame body specifications are shown in Table 2. The finite element model of each unit frame is established, as shown in Fig. 3. In the analysis, it is assumed that each vertical pole carries load synchronously, so the vertical translational degrees of freedom of the top of four vertical poles in the unit frame are coupled.

#### *3.3 Analysis of Bearing Capacity Results*

Non-linear buckling analysis is carried out on each basic stress unit frame model, and the ultimate bearing capacity of different specifications of frame bodies is obtained, which is summarized in Table 3. Comparative analysis is made according to the

**Table 2** Summary of unit rack specifications

*Note* L1-longitudinal distance; L2-horizontal distance; H-step distance; R-wedge tightness of nodes; R1-normal; R2-minimum; A-height of sweeping pole; B-height of free end

**Fig. 3** Schematic diagram of unit frame model

difference of bearing capacity caused by the change of vertical pole spacing and wedge tightness of cross bar plug in each unit frame.

As can be seen from Table 3, the unit frame 1 with small vertical pole spacing and large wedge tightness of cross bar plug has the largest bearing capacity, whereas the unit frame 12 with large vertical pole spacing and small wedge tightness of

**Table 3** Nonlinear ultimate bearing capacity of each unit frame

cross bar plug has the smallest bearing capacity, and the bearing capacity changes with the change of vertical pole spacing. According to the difference of vertical and horizontal spacing of vertical poles, the bearing capacity of unit frame is better when the stiffness difference in two directions is small; However, the bearing capacity is low when the stiffness difference between two directions is large. For example, the ultimate bearing capacity of unit frame 3 and unit frame 4 with 600 mm horizontal distance and 1200 mm vertical distance is slightly lower than that of unit frame 9 and unit frame 10 with 900 mm horizontal distance and 1200 mm vertical distance, which indicates that the deformation of unit frame develops earlier when the stiffness difference between two directions is large. In practical application, attention should be paid to the vertical and horizontal stiffness of frame body. When the stiffness difference between the two directions is large, additional pulling and connecting measures should be added.

In the numerical analysis, the ultimate bearing capacity of cross-bar plugs with different specifications under normal wedging degree is 11.93~16.16% higher than that under minimum wedging degree, and the difference is fluctuating and irregular. The nonlinear load–displacement curve of each unit frame is shown in Fig. 4. From Fig. 4, it can be seen that the smaller the vertical pole spacing of the unit frame is, the faster the bearing capacity decreases when it reaches the ultimate bearing state. On the contrary, when the vertical pole spacing is large, although the bearing capacity is relatively small, the bearing capacity can continue when the ultimate bearing capacity is reached. In order to visually compare the ultimate bearing capacity of different specifications of unit frames, the summary is shown in Fig. 5. It can be seen from Fig. 5 that, regardless of the joint stiffness, the bearing capacity will be greatly reduced when the vertical and horizontal spacing of the unit frame exceeds 900 mm. Under the normal joint wedge tightness, the bearing capacity of the unit frame decreases obviously when the vertical pole spacing increases; Under the minimum joint wedge tightness, the bearing capacity of the unit frame decreases slightly less than that of the normal wedge tightness when the vertical pole spacing is increased; When the vertical pole spacing is small, the ultimate bearing capacity of the unit frame with normal wedge tightness is much higher than that of the unit frame with minimum wedge tightness, but the difference between them is small when the vertical pole spacing of the unit frame is large.

**Fig. 4** Nonlinear load–displacement curve of each unit frame

#### **4 Conclusion**

In this paper, the general finite element software ANSYS is used to verify the vertical ultimate bearing capacity of the basic stress unit frame of the socket-type wheelbuckle steel tube formwork support frame, and the following conclusions are drawn from the analysis results.

## **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

## **Investigation of Web Hole Effects on Capacities of Cold-Formed Steel Channel Members**

**Ngoc Hieu Pham**

**Abstract** Cold-formed steel structures have been widely applied in structural buildings with advantages in manufacturing, transportation and assembly. Holes can be pre-punched in the sectional members to allow technical pipes to go throughout such as electricity, water or ventilation. This affects the capacities of these such members which have been considered in the design standards in America or Australia/New Zealand. The paper, therefore, investigates the effects of web holes on the capacities of cold-formed steel channel members under compression or bending. Their capacities can be determined according to the American Specification AISI S100-16. The investigated results are the base for analysing the effects of web hole dimensions on the behaviors and capacities of cold-formed steel channel members. It was found that the capacity reductions were obtained for compressive members with the increase in hole sizes, but the flexural capacities were noticeable increase with the increase in the hole heights.

**Keywords** Effects · Capacities · Web holes · Cold-formed steel channel members

## **1 Introduction**

Cold-formed steel members have been progressively applied in structural buildings due to their advantages compared to traditional steel structures [1]. Channel sections are the common products worldwide for many decades [2]. Holes are pre-punched in the webs of such members to allow the technical pipes such as water and electricity to go throughout with the variation of hole shapes. The presence of holes has been demonstrated to reduce the capacities of these cold-formed steel members and has been considered in the American and Australian/New Zealand design standards ([3, 4]). According to these standards, a new method has been introduced in the design of cold-formed steel members namely the Direct Strength Method (DSM) which

N. H. Pham (B)

Faculty of Civil Engineering, Hanoi Architectural University, Hanoi, Vietnam e-mail<EMAIL_ADDRESS>
© The Author(s) 2023

S. Wang et al. (eds.), *Proceedings of the 2nd International Conference on Innovative Solutions in Hydropower Engineering and Civil Engineering*, Lecture Notes in Civil Engineering 235, https://doi.org/10.1007/978-981-99-1748-8\_13

has been illustrated to be innovative compared to the traditional design method—the Effective Width Method [5]. The DSM will be used for the investigation in this paper.

The DSM allows the designer to directly predict the capacities of cold-formed steel members based on the determination of elastic buckling loads. These buckling loads can be provided by utilising several software programs for buckling analysis of coldformed steel sections such as CUFSM [6] or THIN-WALL-2 [7]. The application of these software programs in the design is reported in the works of Pham [8] or Pham and Vu [9]. For cold-formed steel sections with perforations, the determination of elastic buckling loads was studied and subsequently proposed by Moen and Schafer ([10–16]). Their research results were the base for the development of a module software program CUFSM funded by the American Iron and Steel Institute ([17, 18]). This module software program will be introduced and applied in the buckling analysis of cold-formed steel sections with perforations in this paper.

The paper presents the application of the DSM in the determination of capacities of cold-formed steel channel members with perforations according to the American Specification AISI S100-16 [3]. The rectangular hole shapes are considered and are arranged evenly with the variation of the hole sizes. The hole heights vary from 0.2 to 0.8 times of the investigated sectional depths whereas their lengths are from 0.5 to 2.0 times of the depths. The material properties are regulated in the American Specification [3]. The boundary conditions are varied to obtain different buckling modes of the investigated cold-formed steel members under compression or bending. The investigated results will be used to analyse the influence of hole sizes on the capacities of cold-formed steel channel members with perforations under compression or bending.

## **2 Determination of Capacities of Cold-Formed Steel Members Under Compression or Bending According to the American Specification AISI S100-16**

The provisions for the design of cold-formed steel members are presented in Chapter E for compression and Chapter F for bending according to the American Specification AISI S100-16 [3]. The DSM for the design of cold-formed steel members is presented in this paper.

#### *2.1 Member in Compression*

The nominal axial strength (*Pn*) is the least of three following strength values including global buckling strength (*Pne*), local buckling strength (*Pnl*), and distortional buckling strength (*Pnd* ).

#### **Global Buckling Strength (Pne)**

$$P\_{n\epsilon} = \left(0.658^{\lambda\_c^2}\right) P\_\text{y} \quad \text{if } \lambda\_C \le 1.5 \tag{l}$$

$$P\_{n\epsilon} = \left(\frac{0.877}{\lambda\_c^2}\right) P\_y \quad \text{if } \lambda\_C > 1.5 \tag{2}$$

where λ*c* = √*Py*/*Pcre*;

*Py* Is the yield strength of the gross section;

*Pcre* Is the elastic global buckling strength, is taken as the smaller of the following values.

$$P\_{\rm ey} = \frac{\pi^2 EI\_{\rm y}}{(K\_{\rm y}L)^2} \tag{3}$$

$$\begin{aligned} P\_{\text{ext}} &= \frac{1}{2\beta} \Big[ (P\_{\text{ex}} + P\_t) - \sqrt{(P\_{\text{ex}} + P\_t)^2 - 4\beta P\_{\text{ex}} P\_t} \Big] \\\ P\_{\text{ex}} &= \frac{\pi^2 E I\_x}{(K\_x L)^2}; \ P\_t = \frac{1}{r\_o^2} \Big( GJ + \frac{\pi^2 E C\_{\text{w}}}{(\mathbf{K}\_t \mathbf{L})^2} \Big); r\_o = \sqrt{x\_o^2 + y\_o^2 + \frac{I\_x + I\_y}{A\_g}} \end{aligned} \tag{4}$$

The sectional properties (*I x, I y, J, Ag, xo, yo, ro*) are determined on the basis of the gross-section. These properties for perforated sections can be determined using the "weighted average" approach as presented in table 2.3.2-1 of the specification [3] based on the ratio between the segment length of the gross-section and the net section, as follows:

$$\begin{split} I\_{\text{avg}} &= \frac{I\_{\text{g}}L\_{\text{g}} + I\_{\text{net}}L\_{\text{net}}}{L}; J\_{\text{avg}} = \frac{J\_{\text{g}}L\_{\text{g}} + J\_{\text{net}}L\_{\text{net}}}{L} \\ r\_{o,\text{avg}} &= \sqrt{\mathbf{x}\_{o,\text{avg}}^{2} + \mathbf{y}\_{o,\text{avg}}^{2} + \frac{I\_{x,\text{avg}} + I\_{\text{y,avg}}}{A\_{\text{avg}}}}; A\_{\text{avg}} = \frac{A\_{\text{g}}L\_{\text{g}} + A\_{\text{net}}L\_{\text{net}}}{L} \\ \mathbf{x}\_{o,\text{avg}} &= \frac{\mathbf{x}\_{o,\text{g}}L\_{\text{g}} + \mathbf{x}\_{o,\text{net}}L\_{\text{net}}}{L}; \mathbf{y}\_{o,\text{avg}} = \frac{\mathbf{y}\_{o,\text{g}}L\_{\text{g}} + \mathbf{y}\_{o,\text{net}}L\_{\text{net}}}{L} \end{split}$$

*Cw,net* is the net warping constant assuming the hole height *hhole\** as determined in Eq. (5), where *hhole* is the actual hole height and *D* is the sectional depth

$$h\_{hole\*} = h\_{hole} + \frac{1}{2}(H - h\_{hole})\left(\frac{h\_{hole}}{H}\right)^{0.2} \tag{5}$$

**Local Buckling Strength (Pnl)**

$$P\_{nl} = \begin{cases} \begin{array}{c} P\_{ne} \\ \left[1 - 0.15 \left(\frac{P\_{rl}}{P\_{ne}}\right)^{0.4}\right] \left(\frac{P\_{rl}}{P\_{ne}}\right)^{0.4} P\_{y} \text{ for } \lambda\_{l} > 0.776 \end{array}} \tag{6}$$

where λ*l* is the slenderness factor for local buckling, λ*l* = √*Pne*/*Pcrl* ;

*Pcrl* is the elastic local buckling load of the gross section or perforated section that can be determined using elastic buckling analyses.

#### **Distortional Buckling Strength (Pnd)**

$$P\_{nd} = \begin{cases} P\_\text{y} & \text{for } \lambda\_\text{d} \le 0.561\\ \left[1 - 0.25 \left(\frac{P\_{rd}}{P\_\text{y}}\right)^{0.6}\right] \left(\frac{P\_{rd}}{P\_\text{y}}\right)^{0.6} P\_\text{y} & \text{for } \lambda\_\text{d} > 0.561 \end{cases} \tag{7}$$

where λ*d* is the slenderness factor for distortional buckling, λ*d* = √*Py*/*Pcr d* ;

*Pcrd* is the elastic distortional buckling load of the gross section or perforated section that can be determined using elastic buckling analyses. For the perforated section, if λ*d* ≤ λ*d2*, where λ*d2* is determined as in Eq. (10) then:

$$P\_{nd} = \begin{cases} P\_{\text{ynet}} & \text{for } \lambda\_{\text{d}} \le \lambda\_{\text{d1}} \\\ P\_{\text{ynet}} - \left(\frac{P\_{\text{ynet}} - P\_{\text{d2}}}{\lambda\_{\text{d2}} - \lambda\_{\text{d1}}}\right) (\lambda\_{\text{d}} - \lambda\_{\text{d1}}) & \text{for } \lambda\_{\text{d1}} < \lambda\_{\text{d2}} \le \lambda\_{\text{d2}} \end{cases} \tag{8}$$

*Pynet* Is the axial yield strengths of the net section; λ*d1,* λ*d2* Are the slenderness factors of distortional buckling; *Pd2* Is the nominal axial strength of distortional buckling at λd2.

$$
\lambda\_{d1} = 0.561 \left( \frac{P\_{\rm ynet}}{P\_{\rm y}} \right) \tag{9}
$$

$$
\lambda\_{d2} = 0.561 \left[ 14 \left( \frac{P\_{\rm y}}{P\_{\rm ynet}} \right)^{0.4} - 13 \right] \tag{10}
$$

$$P\_{d2} = \left[1 - 0.25 \left(\frac{1}{\lambda\_{d2}}\right)^{1.2}\right] \left(\frac{1}{\lambda\_{d2}}\right)^{1.2} P\_y \tag{11}$$

#### *2.2 Member in Flexure*

The nominal moment of a beam (*Mn*) is the least of three values including global buckling moment (*Mne*), local buckling moment (*Mnl*), and distortional buckling moment (*Mnd* ).

Investigation of Web Hole Effects on Capacities of Cold-Formed Steel … 165

#### **Global Buckling Moment (Mne)**

$$M\_{ne} = M\_{\text{y}} \quad \text{if } M\_{cre} \ge 2.78M\_{\text{y}} \tag{12}$$

$$M\_{n\varepsilon} = \frac{10}{9} \left( 1 - \frac{10M\_{\text{y}}}{36M\_{c\varepsilon\varepsilon}} \right) M\_{\text{y}} \quad \text{if } 0.56F\_{\text{y}} < M\_{c\varepsilon\varepsilon} < 2.78 \, M\_{\text{y}} \tag{13}$$

$$M\_{ne} = M\_{cre} \quad \text{if } M\_{cre} \le 0.56 M\_{\text{y}} \tag{14}$$

where *My* is the yield moment of the gross sections; *Mcre* is the elastic global buckling moment that can be determined as follows:

$$M\_{cre} = \frac{\pi}{K\_\text{y}L} \sqrt{EI\_\text{y}\left(GJ + \frac{\pi^2 EC\_\text{w}}{(\text{K}\_\text{l}\text{L})^2}\right)}$$

The sectional properties are defined and determined as presented in Sect. 2.1 for the gross section and the perforated section.

#### **Local Buckling Moment (Mnl)**

$$M\_{nl} = \begin{cases} M\_{n\varepsilon} & \text{for } \lambda\_l \le 0.776\\ \left[1 - 0.15 \left(\frac{M\_{cl}}{M\_{ns}}\right)^{0.4}\right] \left(\frac{M\_{cl}}{M\_{ns}}\right)^{0.4} M\_\text{y} & \text{for } \lambda\_l > 0.776 \end{cases} \tag{15}$$

where λ*l* is the slenderness factor for local buckling, λ*l* = √*Mne*/*Mcrl* ;

*Mcrl* is the elastic local buckling moment of the gross section or perforated section that can be determined using elastic buckling analyses.

#### **Distortional Buckling Moment (Mnd)**

$$M\_{nd} = \begin{cases} M\_{\text{y}} & \text{for } \lambda\_{\text{d}} \le 0.673\\ \left[1 - 0.22 \left(\frac{M\_{\text{crd}}}{M\_{\text{y}}}\right)^{0.5}\right] \left(\frac{M\_{\text{crd}}}{M\_{\text{y}}}\right)^{0.5} M\_{\text{y}} & \text{for } \lambda\_{\text{d}} > 0.673 \end{cases} \tag{16}$$

where λ*d* is the slenderness factor for distortional buckling, λ*d* = √*My*/*Mcr d* ;

*Mcrd* is the elastic distortional buckling moment of the gross section or perforated section that can be determined using elastic buckling analyses. For the perforated section, if λ*d* ≤ λ*d2*, where λ*d2* is determined as in Eq. (19) then:

$$M\_{nd} = \begin{cases} M\_{\text{ynet}} & \text{for } \lambda\_{\text{d}} \le \lambda\_{\text{d1}} \\ M\_{\text{ynet}} - \left(\frac{M\_{\text{not}} - M\_{\text{d2}}}{\lambda\_{\text{d2}} - \lambda\_{\text{d1}}}\right) (\lambda\_{\text{d}} - \lambda\_{\text{d1}}) & \text{for } \lambda\_{\text{d1}} < \lambda\_{\text{d}} \le \lambda\_{\text{d2}} \end{cases} \tag{17}$$

*Mynet* Is the yield moment of the net section;

λ*d1* and λ*d2* Are the slenderness factors of distortional buckling;

*Md2* Is the nominal moment of distortional buckling at λ*d2*.

$$
\lambda\_{d1} = 0.673 \left( \frac{M\_{\rm ynet}}{M\_{\rm y}} \right)^3 \tag{18}
$$

$$
\lambda\_{d2} = 0.673 \left[ 1.7 \left( \frac{M\_{\rm y}}{M\_{\rm ynet}} \right)^{2.7} - 0.7 \right] \tag{19}
$$

$$M\_{d2} = \left[1 - 0.22\left(\frac{1}{\lambda\_{d2}}\right)\right] \left(\frac{1}{\lambda\_{d2}}\right) M\_{\text{y}} \tag{20}$$

## **3 Elastic Buckling Analyses for Cold-Formed Steel Channel Members with Perforations**

Elastic buckling analysis is a compulsory step to apply the DSM in the determination of capacities of cold-formed steel members. Section C20015 taken from the commercial sections is selected for the investigation, as illustrated in Fig. 1, where "C" indicates the channel section; the nominal dimensions include the depth D = 203 mm, the width B = 76 mm, the lip length L = 19.5 mm, and the thickness t = 1.5 mm. Rectangular hole shape is considered in this investigation with the hole heights varying from 0.2 to 0.8 times of the sectional depth (D), and the hole lengths varying from 0.5 to 2.0 times of the depth (D).

#### *3.1 Elastic Sectional Buckling Analyses*

The elastic sectional buckling analyses are carried out with the support of the module software program CUFSM [18]. This program requires simple input and directly provides output results including the local buckling and distortional buckling loads of both gross-section and perforated sections, as illustrated in Fig. 2. It was found that elastic local buckling loads only depend on the hole heights whereas they are hole lengths for distortional buckling loads [10]. The material properties regulated in the American Specification [3] include the strength Fy = 345 MPa, and Young's modulus E = 203,400 MPa. The elastic buckling loads, therefore, are determined and reported in Table 1.

Table 1 shows that the elastic loads are seen as an increasing trend for local buckling modes, and the opposite trend is obtained for distortional buckling modes when the hole sizes increase. The distortional buckling loads of perforated sections are always less than those of the gross section. Meanwhile, the local buckling loads of

#### **Fig. 1** Nomenclature

the net section are even higher than those of this gross section (see the hole heights of 0.5D and 0.8D for compression, or 0.8D for bending). This means that local buckling modes will occur at the net section for small hole heights and at the gross section areas between holes for large hole heights. The reason for this has been explained in previous studies [19]. Therefore, the elastic local buckling loads (*Pcrl, Mcrl*) of the perforated sections can be taken as the smaller of these load values of the gross section and the net section for the design.

#### *3.2 Global Buckling Analyses*

The C20015 section is chosen for the investigation with the length of 2500 mm. There are 05 symmetrical and even holes in the web of the investigated section as shown in Fig. 3. The variations of hole sizes have been presented above.

**Fig. 2** The CUFSM software program

**Table 1** Elastic buckling loads of C20015 section

**Fig. 3** The web hole locations of the specimen C20015

The specimen will be investigated under compression and bending. For compression, two different boundary conditions are applied to obtain two different global buckling modes. The first configuration allows the specimen to freely rotate about the strong axis (x-x) to obtain the flexural–torsional buckling mode (see Fig. 4) whereas the free rotation is for the weak axis (y-y) in the second configuration to get the flexural buckling mode (see Fig. 5). Warping displacements are restrained at two ends for both two configurations. The effective lengths, therefore, can be taken as follows: Lx = L, Ly = Lz = 0.5L for the first configuration; Ly = 0.5 L, Lx = Lz = 0.5L for the second configuration, where L is the specimen length. For bending, the specimen can freely rotate about both the strong axis (x-x) and the weak axis (y-y), and free warping displacements are applied at two ends as illustrated in Fig. 6. The effective lengths in all axes are equal to the specimen length (L). The elastic global buckling loads are determined as presented in Sect. 2 and are given in Table 2.

Table 2 shows that the buckling loads of flexural–torsional buckling modes under compression are the most significant impacts due to the appearance of web holes with the 50% capacity reduction compared to those of the gross section specimen, whereas they are about 30% reduction for the other buckling modes.

Both elastic sectional and global buckling loads calculated in this section will be used for the determination of the capacities of the investigated specimen with perforations in Sect. 4.

**Fig. 4** The first configuration for compression and flexural–torsional buckling mode

**Fig. 5** The second configuration for compression and flexural buckling mode

**Fig. 6** The configuration for bending and flexural–torsional buckling mode

**Table 2** Elastic global buckling loads of the C20015 specimen under compression or bending

F, F-T stand for flexural and flexural–torsional buckling modes; Δ% stands for the deviations between the elastic global buckling loads of the perforated section members and the gross section members

## **4 Investigation of Capacities of Cold-Formed Steel Channel Members with Perforations Under Compression or Bending**

The obtained elastic buckling loads in Sect. 3 are used to directly determine the capacities of C20015 specimen with its length of 2500 mm and the variation of end boundary conditions by applying the Direct Strength Method design as presented in Sect. 2. There are three component strength values including global buckling strengths (*Pne, Mne*), local buckling strengths (*Pnl, Mnl*), and distortional buckling strengths (*Pnd , Mnd* ) as listed in Tables 3, 4 and 5, respectively. The member capacities of the investigated specimens are the least of the above strength values as given in Table 6 and illustrated in Fig. 7.

**Table 3** Global buckling strengths of the C20015 specimen under compression or bending

**Table 4** Local buckling strengths of the C20015 specimen under compression or bending

In terms of global buckling strengths (see Table 3), the effects of web holes are insignificant with less than 5% reductions for small hole heights of 0.2D in comparison with those of the gross section members, but they become noticeable with more than 20% reductions for large hole heights of 0.8D.

In terms of distortional buckling strengths (see Table 5), the hole impacts are unchanged for the hole heights of 0.2D and 0.5D, and have minor changes for the hole heights of 0.8D, whereas the effects of hole lengths become noticeable. The

**Table 5** Distortional buckling strengths of the C20015 specimen under compression or bending

**Table 6** Member buckling strengths of the C20015 specimen under compression or bending

reductions of distortional buckling strengths are about 20% for compression and about 10% for bending in comparison with those of gross section members.

In terms of member capacities, it is found that the member failure modes are governed by local buckling modes for both cases due to the small thickness of the investigated specimen (see Tables 3 and 6). For compression, they are seen as downward trends for both two configurations if the hole dimensions increase. The capacity reductions are insignificant for small and intermediate hole heights (hhole = 0.2D and

a) Configuration 1 for compression with flexural- torsional buckling modes

b) Configuration 2 for compression with flexural buckling modes.

c) Bending configuration with flexural-torsional buckling modes.

**Fig. 7** The capacities of C20015 section members

0.5D), but they are significant for large hole heights (*hhole* = 0.8D), especially in the second configuration. For bending, it is found that the impacts of hole lengths are negligible, as seen in the minor deviations of member capacities with the variation of hole lengths. The novel point herein is the member capacities become higher for larger hole heights as seen in Fig. 7c. As seen in Table 1 for sectional buckling analyses, the local buckling moments of net sections are less than those of the gross section, which means that local buckling modes governed the member failures occurring at the net sections. Therefore, the member capacities are seen as an upward trend with the increase in the hole heights due to the increasing trend of elastic local buckling moments as discussed in Sect. 3.

#### **5 Conclusions**

The paper investigated the effects of web hole dimensions on the capacities of coldformed steel channel members under compression or bending. Variations of boundary conditions were used for the investigation to obtain different global buckling modes. The Direct Strength Method applied for the investigation was regulated in the American specification AISI S100-16. The CUFSM software program was used to support the elastic buckling analyses of the investigated section. The obtained strengths were the base for the analysis of the member behaviors. The several remarks are given as follows:

#### **References**

Twentieth international specialty conference on cold-formed steel structures. Saint Louis, Missouri, USA

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