# In Situ Creep Model Testing for the Tunnel Anchor Foundation of Xingkang Suspension Bridge in Luding of China

Authors: Lina Wen; Qiangong Cheng; Qiang Cheng; Xifeng Guo; Bin Zhang
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

Due to the limitations of geography and geology, cast concrete tunnel anchors were used to provide counterforces for Xingkang Suspension Bridge foundation at the left bank of Daduhe River. In this study, the in situ creep tests were conducted on two model tunnel anchors at a scale of 1:10 near the real working anchor site. Thus, the long-term deformation of the real working tunnel anchors installed at the bridge foundation could be determined from the creep test of model tunnel anchors. The creep tests were conducted under three different loads and lasted for 102.2 h, 167.5 h, and 189.4 h, respectively. The model anchor, the surrounding rock, and their interface were all monitored and measured during the creep testing. In addition, the numerical calculation, in which the Burger creep constitution was used for describing the surrounding rock and the Mohr–Coulomb criterion for describing the concrete anchor, was performed to further evaluate the long-term stability of the real working tunnel anchors. The numerical calculations are in good agreement with the laboratory testing results, and the creep deformations of the anchor and the surrounding rock have the same order of magnitude. The results show that the tunnel anchor and surrounding rock of Xingkang Bridge are in a stable creep state under the three different loads.


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## Body

## 1. Introduction

The Xingkang Bridge, located in Luding County of China, is a key engineering project of the Yakang Expressway connecting Ya'an City and Kangding City. The primary part of this bridge is a large-scale single-span steel truss suspension bridge with a length of 1100 m. Due to the limitations of geography and geology, use of several concrete tunnel anchors was adopted to provide counterforces for the bridge foundation at the left bank of Daduhe River. Compared to the cast concrete gravity anchor often employed in practice of engineering, the concrete tunnel anchor is rarely applied to the foundation of suspension bridge. The tunnel anchor seems to be in the shape of a thermos bottle plug, and it has a short side at the front and a long one at the rear in lengthwise section. Using the geometry of a bottle plug could increase the pullout capacity of an anchor. Thus, the mechanical properties of the rock mass surrounding the anchor could be efficiently used to provide the counterforce needed to guarantee the safety of the bridge foundation.It is well known that the safety of suspension bridges depends on the stability of bridge foundation to some extent [1, 2]. In other words, the long-term deformation and stability of tunnel anchor would play a key role in the Xingkang Bridge. Currently, highest attention was paid to the design technology of suspension bridge [3], the reliability analysis of bridge structure [4, 5], and the stability evaluation of bridge foundation [6]. However, studies on the tunnel anchor were rarely reported in the limited literature. Han et al. [7] conducted laboratory model tests for the tunnel-type anchorage in soft rock. Zhang et al. [8] carried out numerical calculation of the bridge foundation related to the tunnel anchor. However, the in situ creep model test has not been conducted in the field for the tunnel anchor. Moreover, the long-term deformation of tunnel anchor related to creep would influence the safety running of suspension bridge [9–11].The creep phenomenon and long-term deformation of soft rocks and weak rock mass have been of great concern in practice of engineering since the 1930s. Griggs [12] found that some soft rocks (e.g., claystone, mudstone, and argillaceous slate) would yield creep once the external load reached 12.5–80.0% of the failure load in laboratory compression tests. Maranini and Brignoli [13] conducted a series of uniaxial and triaxial compression creep tests on limestone samples, observing the propagation of fracture under a low confining pressure and the collapse of pore under a high stress, and pointed out the creep deformation that led to these phenomena. Li and Xia [14] performed uniaxial compression creep and relaxation tests for four types of rocks (i.e., siltstone, marble, red sandstone, and mudstone) and found that the rock creep characteristics with three variation stages (i.e., first decreasing, then becoming stable, and finally increasing) were similar to each other under a constant stress condition. Tomanovic [15] developed a time-dependent creep model for mudstone on the basis of substantial laboratory creep tests, and the creep deformation at any stage could be well described in this model. Chen et al. [16] adopted optimisation techniques to promote the rheological constitutive and got the long-term mechanical parameters of shale. Based on the Burger creep model, Yi et al. [17] gave a scaling criterion under both two- and three-dimensional conditions when using rheological parameters, and this criterion demonstrated the rheological behaviour of material under isotropic scaling. In addition, Yang et al. [18] performed several triaxial compression creep tests on the fractured rock masses and proposed a creep damage model, and this model differentiated the creep behaviour of fractured rocks from that of intact rocks.In this study, the in situ creep tests were performed on the tunnel anchor model at a scale of 1:10 considering the geological conditions at the installation site of the actual tunnel anchor, i.e., Xingkang Bridge foundation near the bank slope at the left bank of Daduhe River. The creep deformations of the tunnel anchor model and the surrounding rocks were collected during testing. Furthermore, we employed the commercially available software FLAC3D to calculate the creep deformations based on the Burger creep model. The creep deformation pattern of actual tunnel anchor and its influence on the long-term stability of Xingkang Bridge were analysed in terms of experimental measurements and numerical calculations. These results are helpful to the reinforcement design of the foundation section of Xingkang Bridge in the Yakang Expressway. This bridge, opened in October of 2018, is in good operation up to date.


## 2. Engineering Geology

The design and reinforcement of bridge foundation depend primarily on the engineering geology conditions. The geological investigations show that there exist strong tectonic activity, frequent seismic activities, and high-intensity earthquakes since the late Pleistocene epoch in the region of the engineering project. The natural slope with a height of 400 m was formed due to the U-shaped bending at the left bank of the Daduhe River near the bridge foundation. The steep slope was covered with significant deposits of 3 to 5 m depth at the middle and lower positions. The underlaid rocks are primary composed of strongly altered adamellite. The plan view and profile of engineering geology are plotted in Figures1 and 2, respectively. Around the bridge site, we have not found any large-scale landslide, and only a few small-scale soil landslides happened about 1 km away.Figure 1
Geography and geology near the tunnel anchor site of the Xingkang Bridge.Figure 2
Engineering geology profile at the I-I position of Figure1.The in situ model tunnel anchors are located at approximately 100 m below the real working anchors of Xingkang Bridge foundation (Figure3). The surrounding rocks of the tunnels excavated for the model anchors and the real working anchor are both composed of altered adamellite with a similar structure (Figure 4). In the model tunnel, three altered granite walls exist in the adamellite, and no groundwater activity was found. The surrounding rocks are relatively soft and prone to fracture and in medium-to-strong weathered state. Despite no fault, three groups of rock joints (33°∠72°, 57°∠36°, and 312°∠67°) exist from the exposed rocks on the tunnel surface of the model anchor. The joint widths are approximately 0.2–0.3 m, with most of them closed and a few slightly opened. When soaking the rocks in water, nail scratches could be easily made on the surface of rock. In addition, we collected the boring rock samples to perform physical and mechanical property tests in the laboratory and carried out sonic velocity measures in the boring holes. The testing results are given in Table 1. Among the parameters, ρn is the density, E is the elastic modulus, σc is the compression strength, σt is the tension strength, μ is the Poisson ratio, c is the cohesion, φ is the friction angle, and w is the water content.Figure 3
Locations of model anchor and real working anchor tunnels.Figure 4
Surrounding rocks around (a) real working anchor tunnel and (b) model anchor tunnel.Table 1
Physical and mechanical parameters of rock samples.
Parameterρn (g/cm3)E (GPa)σc (MPa)σt (MPa)μC (MPa)tanφW (%)Range2.54∼2.681.06∼5.3922.4∼46.72.5∼3.90.26∼0.310.6∼1.20.71∼0.921.07∼1.82Average2.572.1332.253.200.280.820.781.38Standard deviation0.061.848.960.580.020.230.090.31


## 3. In Situ Creep Test of the Tunnel Anchor Model

In order to determine the creep characteristics of the tunnel anchor and the surrounding rocks, the in site creep tests of two model anchors were conducted. The creep tests comply with the similarity theory in physics. The gas–liquid hydraulic loading system was used for the creep testing method, and grating displacement sensors and dislocation meters were employed to monitor the creep deformation.


### 3.1. Model Preparation

In order to avoid the influence of model tunnel anchor on the real working tunnel anchor in the process of the creep tests, the two model anchors were placed under the real anchors. Their height difference is approximately 100 m. The location relation between the model anchor and the real one is plotted in Figure3.The creep test of tunnel anchor model complies with the similarity theory in physics. The main parameters (strength, load, geometry, and elastic modulus) of the model anchor are determined from (1)–(4) referring to the paper by Lei et al. [19]:(1)Rm=Rp,where Rm and Rp are the strengths of model tunnel anchor and real working anchor, respectively:(2)Nm=NpC2,where Nm and Np are the loads applied on model tunnel anchor and real working anchor, respectively; C is a similarity constant, and the C-value is taken to be 10 in this study:(3)lm=lpC,where lm and lp are the geometric sizes of model tunnel anchor and real working anchor:(4)Em=Ep,where Em and Ep are the elastic moduli of model tunnel anchor and real working anchor, respectively.The tunnels for model anchor and real working anchor share exactly the same shape and axis direction (115°∠36°). The top and bottom of model anchor tunnels in cross section are both designed as arc shapes while the side walls are vertical. The two tunnels of model anchors (see Figure5), used for performing the in situ creep test, were excavated 2.70 m away from each other (axial distance). The cross sections at the front and rear side of the model tunnel anchor are plotted in Figure 6. The geometry size of front face is 1.48 m × 1.56 m, and that of rear face is 1.77 m × 1.90 m. The length of the model tunnel anchor is 4.00 m. The radii of the arc at the top and bottom of front section are Rf1 = 0.74 m and Rf2 = 0.92 m, and those of rear section are Rr1 = 0.89 m and Rr2 = 1.24 m. The lengths of front and rear anchor chambers are 6.30 m and 0.60 m, respectively.Figure 5
(a) Left tunnel and (b) right tunnel of model anchors.Figure 6
(a) Front-side surface and (b) rear-side surface of a single model anchor.
(a)(b)


### 3.2. Creep Testing Method

The gas-liquid hydraulic loading system, used for the creep testing method, was made by Jinan Saisite Equipment Co., Ltd. This loading system could offer a high loading capacity, a high-pressure servo control accuracy, and a convenient control function. The detailed configuration of the loading system is shown in Figure7. The test data were collected automatically through the pressure sensor and data acquisition systems during the loading-unloading process.Figure 7
(a) Loading system: data acquisition system, (b) site datum beam of model anchor, (c) temperature monitoring system, and (d) jack oil system of field loading system.According to the design requirement, a single main cable should bear a tension ofNp = 2.67 × 105 kN under the real working condition. Therefore, the design tension of a single cable in the model is transformed to Nm = 2.67 × 103 kN on the basis of (2). After the anchor hole was excavated, 20 jacks were subsequently installed in the rear anchor chamber. To ensure that the model anchor can reach the ultimate failure condition, 10 jacks (3 × 103 kN per jack) were used to provide the load on a single anchor. By connecting the oil circuit to multiple cylinders and integrating all the jacks in parallel, a maximum load is 11.0 times the design load in the test, e.g., 11.0 Nm. The control of load abided by the subsequent load steps of 1.0 Nm, 3.5 Nm, 7.0 Nm, 9.0 Nm, and 11.0 Nm in the long-term cyclic loading tests and corresponding creep tests for the model anchor. In fact, the surrounding rock around the rear anchor chamber has been locally damaged when the load reached 7.8 Nm. In addition, we could judge that the rear section has also reached its ultimate bearing capacity from the observed significant deformation. Thus, the maximum effective load could be taken as 7.0 Nm. The creep test lasted for 102.2 h, 167.5 h, and 189.4 h under the loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm, respectively. The creep test under each loading condition was terminated when the fluctuation of deformation was within ±3 μm.After completing the tests at the three loading stages of 1.0Nm, 3.5 Nm, and 7.0 Nm, the instrument was given a rest for 12 to 24 h before conducting the corresponding creep test. During the creep test, the creep test results of the anchor and the surrounding rock were analysed by monitoring their long-term deformations. The creep test data were collected every 10 min. The loading process lasted for at least 5 days, and the test was terminated when the deformation was stable (approximately fluctuation within ±3 μm).


### 3.3. Creep Test Results and Analysis

#### 3.3.1. Monitoring Point Position on the Tunnel Anchor

Grating displacement sensors were used to monitor the creep deformation of the model anchor during the test. Nine sensors were installed on the front anchor face, rear anchor face, and front rock mass along the tensile direction. A long steel beam with length of 12 m was used as a reference beam. The layout of the grating displacement sensors is plotted in Figure8. Four measuring points, i.e., g3 (located at the front face of the left anchor), g4 (located at the rear face of the left anchor), g6 (located at the front face of the right anchor), and g7 (located at the rear face of the right anchor), were used in the creep test on the anchor.Figure 8
Positions of grating displacement sensors.


#### 3.3.2. Creep Behaviour of the Anchor under Different Loading Stages

The creep deformations at the front and rear faces of the left and right anchors were measured by the grating displacement sensors under loads of 1.0Nm, 3.5 Nm, and 7.0 Nm. The stress of measuring point g7 was the maximum among all measuring points and the creep deformation was measured at this point every 24 h. The creep deformation and creep rate measured at point g7 under different loading conditions were analysed, as presented in Figure 9.Figure 9
(a) Deformation and (b) rate of pointg7 measured during creep testing.
(a)(b)As can be observed from Figure9, the instantaneous deformation is smaller than the creep deformation under a load of 1.0 Nm. When the load is increased to 3.5 Nm and 7.0 Nm, however, the instantaneous deformation becomes significantly larger and the fraction of creep deformation becomes smaller in the total deformation. With increasing loading level, the duration of elastic creep experienced by the anchor becomes smaller while the creep deformation and the change in creep rate both become larger.


#### 3.3.3. Influence of the Location of Model Anchor Hole on the Creep Behaviour

The two model anchor tunnels excavated for the model anchors are separated by 2.70 m in the in situ test, as indicated in Figure8. The creep deformations measured at points g3,g4,g5, and g7 are illustrated in Figure 10.Figure 10
Comparison of creep deformations between two model anchors.As shown in the figure, both the left and the right anchors show similar values of creep deformations at all loading conditions. With the same loading level, there is only a subtle difference between the creep deformations measured at different points. Meanwhile, the creep deformations measured at the front and rear faces of the model anchor are almost consistent with each other. The average values of creep deformations measured atg3 and g4 are 0.51 mm, 0.94 mm, and 1.45 mm, respectively, under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. The average values of creep deformations measured at g6 and g7 are 0.57 mm, 0.87 mm, and 1.24 mm, respectively, under the three loading conditions.


#### 3.3.4. Monitoring Point Position on the Surrounding Rock

The creep behaviour at the surface of the surrounding rock was tested using grating displacement sensors at five measuring points, namely,g1,g2,g5,g8,andg9. The layout of these measuring points is also illustrated in Figure 8. The creep behaviour of the inside of the surrounding rock was tested using multipoint vibrating wire displacement transducers. Four measuring holes were drilled inside the rock, and the drilling holes have a diameter of 76 mm. The two drilling holes of D1 and D2 were arranged along the tensile direction. The other two drilling holes of D3 and D4 were arranged along the vertical direction towards the ground surface. The layouts of these drilling holes are plotted in Figure 11.Figure 11
Multipoint displacement transducers along the (a) vertical direction and (b) tensile direction.The creep deformations on the surface and inside of the surrounding rock were tested under loads of 1.0Nm, 3.5 Nm, and 7.0 Nm. The statistics were conducted from the collected creep deformation data of points D1-3 (the locations of the maximum stress) every 24 h. Figure 12 shows the creep deformation and creep rate measured at these points under the different loading conditions. Figures 13 and 14 indicate the creep deformations on the surface and inside of the surrounding rock.Figure 12
(a) Deformation and (b) rate of points D1-3 measured during creep testing.
(a)(b)Figure 13
Final creep deformation of pointsg1,g2,g5,g8,andg9.Figure 14
Final creep deformation along the (a) tensile direction and (b) vertical direction in the surrounding rocks.
(a)(b)As shown in Figure12, the overall range of creep deformation is relatively small for the surrounding rock under the loads of 1.0 Nm and 3.5 Nm. However, the creep range becomes relatively large when the load is increased to 7.0 Nm. The creep deformation of the surrounding rock is found to increase with increasing load. In addition, a higher load leads to a larger initial creep rate of the surrounding rock as well as a more significant change in the creep rate.It could be seen from Figure13 that the maximum creep deformations measured at point g5 (located along the tensile direction in the rock mass between the anchors) are 0.49 mm, 0.85 mm, and 1.10 mm, respectively, under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. The creep deformation of the surface surrounding rock measured at other locations along the tensile direction tends to decrease with increasing distance from model anchor. In other words, the creep deformation becomes larger as it gets closer to the anchor.Figure14 shows that the creep deformation of the surrounding rock continues to increase with increasing load along both the tensile and vertical directions. In particular, the creep deformation along the vertical direction is larger than that along the tensile direction under the loads of 1.0 Nm and 3.5 Nm. When the load is increased to 7.0 Nm, however, the creep deformation along the tensile direction becomes slightly larger than that along the vertical direction.In summary, both the creep deformation and the creep rate of the surrounding rock tend to increase with increasing load level. Meanwhile, due to the sandwiching effect from the two anchors on the left and right sides, the creep deformation at the surface of the surrounding rock tends to become smaller with increasing distance from the anchor. In contrast, the creep deformation becomes larger as it gets closer to the anchor.


#### 3.3.5. On the Interface between Tunnel Anchor and Surrounding Rock

To analyse the creep characteristics of the interface between model tunnel anchor and surrounding rock, six dislocation meters were installed between each model anchor and its contact surfaces with the surrounding rock. Specifically, these meters were arranged on three separate cross sections located at the top arc, bottom plate, and side wall of anchor. Figure15 displays the detailed layout of the dislocation meters.Figure 15
Dislocation meters around the two model anchors.The creep deformations of the interface between anchor and surrounding rock were tested under loads of 1.0Nm, 3.5 Nm, and 7.0 Nm. The creep deformation statistics were collected from measuring point WCJ6 (the location with the maximum stress between the anchor and the surrounding rock) every 24 h. The creep deformation and creep rate measured at point WCJ6 under the different loading conditions were then analysed (Figure 16). The creep deformations measured at other locations under the different loading conditions are plotted in Figure 17.Figure 16
(a) Deformation and (b) rate of point WCJ6 measured during creep testing.
(a)(b)Figure 17
Final creep deformations of interfaces for (a) left anchor and (b) right anchor.
(a)(b)As shown in Figure16, a similar pattern is found for the creep deformation of the anchor, the surrounding rock, and the interface. However, the change in the creep deformation of the interface with increasing load level is slower than that of the anchor and the surrounding rock. In addition, the dislocation experiences a longer elastic period. These findings demonstrate a strong clamping effect from the anchors and surrounding rocks.As indicated in Figure17, the creep deformation of the dislocation tends to increase with increasing load level at all measuring points. Under a load of 1.0 Nm, the creep deformations of the dislocation measured at different points are relatively close to each other. When the load is increased to 3.5 Nm, there is already a significant difference between the creep deformations of the dislocation measured at different points. When the load is further increased to 7.0 Nm, the deformations measured at points WCJ1 and WCJ7 are significantly smaller than those measured at other locations while the measurements taken at the remaining points are still close to each other. This feature demonstrates that the creep deformation of the dislocation is minimal at the top section of the front face of the left and right anchors. Such behaviour is consistent with the deformation rule of anchors.


## 3.1. Model Preparation

In order to avoid the influence of model tunnel anchor on the real working tunnel anchor in the process of the creep tests, the two model anchors were placed under the real anchors. Their height difference is approximately 100 m. The location relation between the model anchor and the real one is plotted in Figure3.The creep test of tunnel anchor model complies with the similarity theory in physics. The main parameters (strength, load, geometry, and elastic modulus) of the model anchor are determined from (1)–(4) referring to the paper by Lei et al. [19]:(1)Rm=Rp,where Rm and Rp are the strengths of model tunnel anchor and real working anchor, respectively:(2)Nm=NpC2,where Nm and Np are the loads applied on model tunnel anchor and real working anchor, respectively; C is a similarity constant, and the C-value is taken to be 10 in this study:(3)lm=lpC,where lm and lp are the geometric sizes of model tunnel anchor and real working anchor:(4)Em=Ep,where Em and Ep are the elastic moduli of model tunnel anchor and real working anchor, respectively.The tunnels for model anchor and real working anchor share exactly the same shape and axis direction (115°∠36°). The top and bottom of model anchor tunnels in cross section are both designed as arc shapes while the side walls are vertical. The two tunnels of model anchors (see Figure5), used for performing the in situ creep test, were excavated 2.70 m away from each other (axial distance). The cross sections at the front and rear side of the model tunnel anchor are plotted in Figure 6. The geometry size of front face is 1.48 m × 1.56 m, and that of rear face is 1.77 m × 1.90 m. The length of the model tunnel anchor is 4.00 m. The radii of the arc at the top and bottom of front section are Rf1 = 0.74 m and Rf2 = 0.92 m, and those of rear section are Rr1 = 0.89 m and Rr2 = 1.24 m. The lengths of front and rear anchor chambers are 6.30 m and 0.60 m, respectively.Figure 5
(a) Left tunnel and (b) right tunnel of model anchors.Figure 6
(a) Front-side surface and (b) rear-side surface of a single model anchor.
(a)(b)


## 3.2. Creep Testing Method

The gas-liquid hydraulic loading system, used for the creep testing method, was made by Jinan Saisite Equipment Co., Ltd. This loading system could offer a high loading capacity, a high-pressure servo control accuracy, and a convenient control function. The detailed configuration of the loading system is shown in Figure7. The test data were collected automatically through the pressure sensor and data acquisition systems during the loading-unloading process.Figure 7
(a) Loading system: data acquisition system, (b) site datum beam of model anchor, (c) temperature monitoring system, and (d) jack oil system of field loading system.According to the design requirement, a single main cable should bear a tension ofNp = 2.67 × 105 kN under the real working condition. Therefore, the design tension of a single cable in the model is transformed to Nm = 2.67 × 103 kN on the basis of (2). After the anchor hole was excavated, 20 jacks were subsequently installed in the rear anchor chamber. To ensure that the model anchor can reach the ultimate failure condition, 10 jacks (3 × 103 kN per jack) were used to provide the load on a single anchor. By connecting the oil circuit to multiple cylinders and integrating all the jacks in parallel, a maximum load is 11.0 times the design load in the test, e.g., 11.0 Nm. The control of load abided by the subsequent load steps of 1.0 Nm, 3.5 Nm, 7.0 Nm, 9.0 Nm, and 11.0 Nm in the long-term cyclic loading tests and corresponding creep tests for the model anchor. In fact, the surrounding rock around the rear anchor chamber has been locally damaged when the load reached 7.8 Nm. In addition, we could judge that the rear section has also reached its ultimate bearing capacity from the observed significant deformation. Thus, the maximum effective load could be taken as 7.0 Nm. The creep test lasted for 102.2 h, 167.5 h, and 189.4 h under the loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm, respectively. The creep test under each loading condition was terminated when the fluctuation of deformation was within ±3 μm.After completing the tests at the three loading stages of 1.0Nm, 3.5 Nm, and 7.0 Nm, the instrument was given a rest for 12 to 24 h before conducting the corresponding creep test. During the creep test, the creep test results of the anchor and the surrounding rock were analysed by monitoring their long-term deformations. The creep test data were collected every 10 min. The loading process lasted for at least 5 days, and the test was terminated when the deformation was stable (approximately fluctuation within ±3 μm).


## 3.3. Creep Test Results and Analysis

### 3.3.1. Monitoring Point Position on the Tunnel Anchor

Grating displacement sensors were used to monitor the creep deformation of the model anchor during the test. Nine sensors were installed on the front anchor face, rear anchor face, and front rock mass along the tensile direction. A long steel beam with length of 12 m was used as a reference beam. The layout of the grating displacement sensors is plotted in Figure8. Four measuring points, i.e., g3 (located at the front face of the left anchor), g4 (located at the rear face of the left anchor), g6 (located at the front face of the right anchor), and g7 (located at the rear face of the right anchor), were used in the creep test on the anchor.Figure 8
Positions of grating displacement sensors.


### 3.3.2. Creep Behaviour of the Anchor under Different Loading Stages

The creep deformations at the front and rear faces of the left and right anchors were measured by the grating displacement sensors under loads of 1.0Nm, 3.5 Nm, and 7.0 Nm. The stress of measuring point g7 was the maximum among all measuring points and the creep deformation was measured at this point every 24 h. The creep deformation and creep rate measured at point g7 under different loading conditions were analysed, as presented in Figure 9.Figure 9
(a) Deformation and (b) rate of pointg7 measured during creep testing.
(a)(b)As can be observed from Figure9, the instantaneous deformation is smaller than the creep deformation under a load of 1.0 Nm. When the load is increased to 3.5 Nm and 7.0 Nm, however, the instantaneous deformation becomes significantly larger and the fraction of creep deformation becomes smaller in the total deformation. With increasing loading level, the duration of elastic creep experienced by the anchor becomes smaller while the creep deformation and the change in creep rate both become larger.


### 3.3.3. Influence of the Location of Model Anchor Hole on the Creep Behaviour

The two model anchor tunnels excavated for the model anchors are separated by 2.70 m in the in situ test, as indicated in Figure8. The creep deformations measured at points g3,g4,g5, and g7 are illustrated in Figure 10.Figure 10
Comparison of creep deformations between two model anchors.As shown in the figure, both the left and the right anchors show similar values of creep deformations at all loading conditions. With the same loading level, there is only a subtle difference between the creep deformations measured at different points. Meanwhile, the creep deformations measured at the front and rear faces of the model anchor are almost consistent with each other. The average values of creep deformations measured atg3 and g4 are 0.51 mm, 0.94 mm, and 1.45 mm, respectively, under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. The average values of creep deformations measured at g6 and g7 are 0.57 mm, 0.87 mm, and 1.24 mm, respectively, under the three loading conditions.


### 3.3.4. Monitoring Point Position on the Surrounding Rock

The creep behaviour at the surface of the surrounding rock was tested using grating displacement sensors at five measuring points, namely,g1,g2,g5,g8,andg9. The layout of these measuring points is also illustrated in Figure 8. The creep behaviour of the inside of the surrounding rock was tested using multipoint vibrating wire displacement transducers. Four measuring holes were drilled inside the rock, and the drilling holes have a diameter of 76 mm. The two drilling holes of D1 and D2 were arranged along the tensile direction. The other two drilling holes of D3 and D4 were arranged along the vertical direction towards the ground surface. The layouts of these drilling holes are plotted in Figure 11.Figure 11
Multipoint displacement transducers along the (a) vertical direction and (b) tensile direction.The creep deformations on the surface and inside of the surrounding rock were tested under loads of 1.0Nm, 3.5 Nm, and 7.0 Nm. The statistics were conducted from the collected creep deformation data of points D1-3 (the locations of the maximum stress) every 24 h. Figure 12 shows the creep deformation and creep rate measured at these points under the different loading conditions. Figures 13 and 14 indicate the creep deformations on the surface and inside of the surrounding rock.Figure 12
(a) Deformation and (b) rate of points D1-3 measured during creep testing.
(a)(b)Figure 13
Final creep deformation of pointsg1,g2,g5,g8,andg9.Figure 14
Final creep deformation along the (a) tensile direction and (b) vertical direction in the surrounding rocks.
(a)(b)As shown in Figure12, the overall range of creep deformation is relatively small for the surrounding rock under the loads of 1.0 Nm and 3.5 Nm. However, the creep range becomes relatively large when the load is increased to 7.0 Nm. The creep deformation of the surrounding rock is found to increase with increasing load. In addition, a higher load leads to a larger initial creep rate of the surrounding rock as well as a more significant change in the creep rate.It could be seen from Figure13 that the maximum creep deformations measured at point g5 (located along the tensile direction in the rock mass between the anchors) are 0.49 mm, 0.85 mm, and 1.10 mm, respectively, under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. The creep deformation of the surface surrounding rock measured at other locations along the tensile direction tends to decrease with increasing distance from model anchor. In other words, the creep deformation becomes larger as it gets closer to the anchor.Figure14 shows that the creep deformation of the surrounding rock continues to increase with increasing load along both the tensile and vertical directions. In particular, the creep deformation along the vertical direction is larger than that along the tensile direction under the loads of 1.0 Nm and 3.5 Nm. When the load is increased to 7.0 Nm, however, the creep deformation along the tensile direction becomes slightly larger than that along the vertical direction.In summary, both the creep deformation and the creep rate of the surrounding rock tend to increase with increasing load level. Meanwhile, due to the sandwiching effect from the two anchors on the left and right sides, the creep deformation at the surface of the surrounding rock tends to become smaller with increasing distance from the anchor. In contrast, the creep deformation becomes larger as it gets closer to the anchor.


### 3.3.5. On the Interface between Tunnel Anchor and Surrounding Rock

To analyse the creep characteristics of the interface between model tunnel anchor and surrounding rock, six dislocation meters were installed between each model anchor and its contact surfaces with the surrounding rock. Specifically, these meters were arranged on three separate cross sections located at the top arc, bottom plate, and side wall of anchor. Figure15 displays the detailed layout of the dislocation meters.Figure 15
Dislocation meters around the two model anchors.The creep deformations of the interface between anchor and surrounding rock were tested under loads of 1.0Nm, 3.5 Nm, and 7.0 Nm. The creep deformation statistics were collected from measuring point WCJ6 (the location with the maximum stress between the anchor and the surrounding rock) every 24 h. The creep deformation and creep rate measured at point WCJ6 under the different loading conditions were then analysed (Figure 16). The creep deformations measured at other locations under the different loading conditions are plotted in Figure 17.Figure 16
(a) Deformation and (b) rate of point WCJ6 measured during creep testing.
(a)(b)Figure 17
Final creep deformations of interfaces for (a) left anchor and (b) right anchor.
(a)(b)As shown in Figure16, a similar pattern is found for the creep deformation of the anchor, the surrounding rock, and the interface. However, the change in the creep deformation of the interface with increasing load level is slower than that of the anchor and the surrounding rock. In addition, the dislocation experiences a longer elastic period. These findings demonstrate a strong clamping effect from the anchors and surrounding rocks.As indicated in Figure17, the creep deformation of the dislocation tends to increase with increasing load level at all measuring points. Under a load of 1.0 Nm, the creep deformations of the dislocation measured at different points are relatively close to each other. When the load is increased to 3.5 Nm, there is already a significant difference between the creep deformations of the dislocation measured at different points. When the load is further increased to 7.0 Nm, the deformations measured at points WCJ1 and WCJ7 are significantly smaller than those measured at other locations while the measurements taken at the remaining points are still close to each other. This feature demonstrates that the creep deformation of the dislocation is minimal at the top section of the front face of the left and right anchors. Such behaviour is consistent with the deformation rule of anchors.


## 3.3.1. Monitoring Point Position on the Tunnel Anchor

Grating displacement sensors were used to monitor the creep deformation of the model anchor during the test. Nine sensors were installed on the front anchor face, rear anchor face, and front rock mass along the tensile direction. A long steel beam with length of 12 m was used as a reference beam. The layout of the grating displacement sensors is plotted in Figure8. Four measuring points, i.e., g3 (located at the front face of the left anchor), g4 (located at the rear face of the left anchor), g6 (located at the front face of the right anchor), and g7 (located at the rear face of the right anchor), were used in the creep test on the anchor.Figure 8
Positions of grating displacement sensors.


## 3.3.2. Creep Behaviour of the Anchor under Different Loading Stages

The creep deformations at the front and rear faces of the left and right anchors were measured by the grating displacement sensors under loads of 1.0Nm, 3.5 Nm, and 7.0 Nm. The stress of measuring point g7 was the maximum among all measuring points and the creep deformation was measured at this point every 24 h. The creep deformation and creep rate measured at point g7 under different loading conditions were analysed, as presented in Figure 9.Figure 9
(a) Deformation and (b) rate of pointg7 measured during creep testing.
(a)(b)As can be observed from Figure9, the instantaneous deformation is smaller than the creep deformation under a load of 1.0 Nm. When the load is increased to 3.5 Nm and 7.0 Nm, however, the instantaneous deformation becomes significantly larger and the fraction of creep deformation becomes smaller in the total deformation. With increasing loading level, the duration of elastic creep experienced by the anchor becomes smaller while the creep deformation and the change in creep rate both become larger.


## 3.3.3. Influence of the Location of Model Anchor Hole on the Creep Behaviour

The two model anchor tunnels excavated for the model anchors are separated by 2.70 m in the in situ test, as indicated in Figure8. The creep deformations measured at points g3,g4,g5, and g7 are illustrated in Figure 10.Figure 10
Comparison of creep deformations between two model anchors.As shown in the figure, both the left and the right anchors show similar values of creep deformations at all loading conditions. With the same loading level, there is only a subtle difference between the creep deformations measured at different points. Meanwhile, the creep deformations measured at the front and rear faces of the model anchor are almost consistent with each other. The average values of creep deformations measured atg3 and g4 are 0.51 mm, 0.94 mm, and 1.45 mm, respectively, under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. The average values of creep deformations measured at g6 and g7 are 0.57 mm, 0.87 mm, and 1.24 mm, respectively, under the three loading conditions.


## 3.3.4. Monitoring Point Position on the Surrounding Rock

The creep behaviour at the surface of the surrounding rock was tested using grating displacement sensors at five measuring points, namely,g1,g2,g5,g8,andg9. The layout of these measuring points is also illustrated in Figure 8. The creep behaviour of the inside of the surrounding rock was tested using multipoint vibrating wire displacement transducers. Four measuring holes were drilled inside the rock, and the drilling holes have a diameter of 76 mm. The two drilling holes of D1 and D2 were arranged along the tensile direction. The other two drilling holes of D3 and D4 were arranged along the vertical direction towards the ground surface. The layouts of these drilling holes are plotted in Figure 11.Figure 11
Multipoint displacement transducers along the (a) vertical direction and (b) tensile direction.The creep deformations on the surface and inside of the surrounding rock were tested under loads of 1.0Nm, 3.5 Nm, and 7.0 Nm. The statistics were conducted from the collected creep deformation data of points D1-3 (the locations of the maximum stress) every 24 h. Figure 12 shows the creep deformation and creep rate measured at these points under the different loading conditions. Figures 13 and 14 indicate the creep deformations on the surface and inside of the surrounding rock.Figure 12
(a) Deformation and (b) rate of points D1-3 measured during creep testing.
(a)(b)Figure 13
Final creep deformation of pointsg1,g2,g5,g8,andg9.Figure 14
Final creep deformation along the (a) tensile direction and (b) vertical direction in the surrounding rocks.
(a)(b)As shown in Figure12, the overall range of creep deformation is relatively small for the surrounding rock under the loads of 1.0 Nm and 3.5 Nm. However, the creep range becomes relatively large when the load is increased to 7.0 Nm. The creep deformation of the surrounding rock is found to increase with increasing load. In addition, a higher load leads to a larger initial creep rate of the surrounding rock as well as a more significant change in the creep rate.It could be seen from Figure13 that the maximum creep deformations measured at point g5 (located along the tensile direction in the rock mass between the anchors) are 0.49 mm, 0.85 mm, and 1.10 mm, respectively, under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. The creep deformation of the surface surrounding rock measured at other locations along the tensile direction tends to decrease with increasing distance from model anchor. In other words, the creep deformation becomes larger as it gets closer to the anchor.Figure14 shows that the creep deformation of the surrounding rock continues to increase with increasing load along both the tensile and vertical directions. In particular, the creep deformation along the vertical direction is larger than that along the tensile direction under the loads of 1.0 Nm and 3.5 Nm. When the load is increased to 7.0 Nm, however, the creep deformation along the tensile direction becomes slightly larger than that along the vertical direction.In summary, both the creep deformation and the creep rate of the surrounding rock tend to increase with increasing load level. Meanwhile, due to the sandwiching effect from the two anchors on the left and right sides, the creep deformation at the surface of the surrounding rock tends to become smaller with increasing distance from the anchor. In contrast, the creep deformation becomes larger as it gets closer to the anchor.


## 3.3.5. On the Interface between Tunnel Anchor and Surrounding Rock

To analyse the creep characteristics of the interface between model tunnel anchor and surrounding rock, six dislocation meters were installed between each model anchor and its contact surfaces with the surrounding rock. Specifically, these meters were arranged on three separate cross sections located at the top arc, bottom plate, and side wall of anchor. Figure15 displays the detailed layout of the dislocation meters.Figure 15
Dislocation meters around the two model anchors.The creep deformations of the interface between anchor and surrounding rock were tested under loads of 1.0Nm, 3.5 Nm, and 7.0 Nm. The creep deformation statistics were collected from measuring point WCJ6 (the location with the maximum stress between the anchor and the surrounding rock) every 24 h. The creep deformation and creep rate measured at point WCJ6 under the different loading conditions were then analysed (Figure 16). The creep deformations measured at other locations under the different loading conditions are plotted in Figure 17.Figure 16
(a) Deformation and (b) rate of point WCJ6 measured during creep testing.
(a)(b)Figure 17
Final creep deformations of interfaces for (a) left anchor and (b) right anchor.
(a)(b)As shown in Figure16, a similar pattern is found for the creep deformation of the anchor, the surrounding rock, and the interface. However, the change in the creep deformation of the interface with increasing load level is slower than that of the anchor and the surrounding rock. In addition, the dislocation experiences a longer elastic period. These findings demonstrate a strong clamping effect from the anchors and surrounding rocks.As indicated in Figure17, the creep deformation of the dislocation tends to increase with increasing load level at all measuring points. Under a load of 1.0 Nm, the creep deformations of the dislocation measured at different points are relatively close to each other. When the load is increased to 3.5 Nm, there is already a significant difference between the creep deformations of the dislocation measured at different points. When the load is further increased to 7.0 Nm, the deformations measured at points WCJ1 and WCJ7 are significantly smaller than those measured at other locations while the measurements taken at the remaining points are still close to each other. This feature demonstrates that the creep deformation of the dislocation is minimal at the top section of the front face of the left and right anchors. Such behaviour is consistent with the deformation rule of anchors.


## 4. Numerical Calculation

In order to validate the results of the in situ creep test, a 3D numerical model comprising the left anchor and the surrounding rock was solved on FLAC3D platform. In addition, the long-term deformations of the actual full-size tunnel anchors of the Xingkang suspension bridge were evaluated by adopting an empirical method.


### 4.1. Constitutive Model and Parameters

The in situ testing data show that the surrounding rock exhibits a viscoelastic characteristic when applying the loads of 1.0Nm, 3.5 Nm and 7.0 Nm to the model tunnel anchor. In other words, the creep model should comprise both viscous and elastic elements. In this study, the four-parameter Burger creep model was employed to characterise the aforementioned rheological curve to obtain the corresponding creep parameters. The four-parameter Burger creep model is formed by combining the Kelvin model and Maxwell model [20], as illustrated in Figure 18.Figure 18
Schematic diagram of four-element Burger’s model.The Kelvin model is suitable for describing the relaxation process under a constant stress condition [21], while the Maxwell model is suitable for describing the creep process under a constant stress condition [22]. In the real situation, an object would undergo both creep and relaxation processes when bearing external forces. The strain generated on a Burger material is equal to the sum of the strains generated from both the relaxation and creep processes; see the following equation:(5)ε=εk+εm,where εk and εm are the strains generated from the Kelvin and Maxwell models, respectively, and ε is the total strain generated from the Burger creep model.When the anchor of a suspension bridge is affected by an axial tensile stressσ1 along the main cable, the deformation of the anchor will induce a radial stress σ2 on the surrounding rock. The creep constitutive equation of the anchor under the combined effect of σ1 and σ2 could be written as follows [23]:(6)εt=2σ1+σ29Kv+2σ1−3σ29E21−exp−E2η2t+2σ1−3σ29E1+2σ1−3σ2η1t,where Kv is the bulk modulus (GPa); E1 and E2 are the elastic moduli (GPa) used in the Maxwell model and Kelvin model; and η1 and η2 are the coefficients of viscosity (GPa·h) used in the Maxwell and Kelvin models.To examine the relationship between the axial tensile stressσ1 on the main cable and the radial stress σ2 on the surrounding rock, the cross section of the anchor is approximately considered as a circle, while the anchor is approximately considered as a wedge body. The direction parallel to the axis of the anchor is set to be the x-axis. The radial direction along the cross section of the anchor and perpendicular to the x-axis is set as the r-axis. The structure and stress condition of the tunnel anchor used in the suspension bridge are given in Figure 19.Figure 19
3D calculation model comprised of model anchor and surrounding rock.Taking any cross section of the anchor, the axial tensile stress could be calculated by(7)σ1=FA=Fπr2,where F is the force applied on the anchor; A is the area of anchor cross section; and r is the radius of anchor cross section.The axial and radial deformations are calculated by(8)ε1=σ1E=FEπr2,(9)ε2=με1=μσ1E=μFEπr2,where E is the elastic modulus of the anchor concrete, and μ is the Poisson ratio.The radial displacement is calculated by(10)s2=rε2=μFEπr.According to the assumption of Winkler’s foundation model [24], the axial tension exerted by the surrounding rock on the anchor is expressed as(11)σ2=Ks2=μFKEπr=μFK0Eπr2,where K is the elastic resistance coefficient of the surrounding rock, and K0 is the unit elastic resistance coefficient of the surrounding rock.When substituting (7)-(11) into (6), we thus obtain the following creep constitutive equation:(12)εt=F2E−3μK09EE2πr21−exp−E2η2t+F2E−3μK09EE1πr2+F2E−3μK0η1Eπr2t+2FE+μK09EKvπr2.The 3D numerical model comprising the left anchor and the surrounding rock was solved on FLAC3D platform. The geometry size of this model is 56.8 m × 20.0 m × 47.2 m, as illustrated in Figure 19. Tetrahedral cells with a side length of 0.5 m were used to mesh the anchor and the surrounding rock in the model. The remaining region was meshed with tetrahedral cells with a side length of 2 m. There are 37,518 nodes and 216,509 elements in total.According to the measured results in Figures9, 12, and 16, the creep parameters of surrounding rocks were calculated by (12) and are listed in Table 2. When solving the numerical model shown in Figure 19, we used the Mohr–Coulomb criterion to judge the critical yield stress state of the tunnel anchor and surrounding rock. The corresponding input parameters are listed in Table 3. According to the test results and safety reservation requirement, the values of ρ0, μ0, and E0 are selected from the average values of ρ, μ, and E; σt0, c0, and φ0 are selected as 10, 50, and 90 percent of σt, c, and φ, respectively. The concrete grade of tunnel anchor is C35, and its values are referred to at China national standard (GB/T 14902-2012) [25].Table 2
Input parameters of Burger’s model used for surrounding rocks.
LoadE1 (GPa)η1 (GPa·h)E2 (GPa)η2 (GPa·h)1.0Nm0.44639005.0420.313.5Nm0.23301003.1510.117.0Nm0.15203001.544.02Table 3
Input parameters of surrounding rocks and concrete tunnel anchor.
Positionρ0 (g/cm3)σt0 (MPa)c0 (MPa)tanφ0μ0E0 (GPa)Surrounding rock2.570.320.410.700.282.13Concrete tunnel anchor2.451.571.431.280.2131.5In addition, only the gravitational stress field was considered in the calculation, and the normal displacements of the nodes at bottom and sides were constrained. The results obtained from the numerical simulation were compared with the experimental measurements to determine the creep deformation and reveal the creep deformation pattern of the anchor and surrounding rock.


### 4.2. Parameters for Sensitivity Analysis of Tunnel Anchor Deformation

Sensitivity analysis is conducted to evaluate the effect of parameters used in the Maxwell and Kelvin models on the deformation of tunnel anchor. The parameters for sensitivity analysis of tunnel anchor strain include elastic moduliE1 and E2 and coefficients of viscosity η1 and η2 used in the Maxwell model and Kelvin models. The parameters used for the sensitivity analysis are summarized in Table 4. Assumed elastic moduli E1 and E2 varying from 0.1 to 0.6 GPa and 1 to 6 are included in this study. The coefficients of viscosity η1 and η2 are assumed to be 5 × 102, 1 × 103, 5 × 103, 1 × 104, 5 × 104, and 1 × 105 GPa·h and 1, 5, 10, 20, 50, and 100 GPa·h, respectively.Table 4
Summary of the parameters used for sensitivity of strain.
Elastic moduliE1 (GPa)0.10.20.30.40.50.6Elastic moduliE2 (GPa)123456Coefficient of viscosityη1 (GPa·h)5 × 1021 × 1035 × 1031 × 1045 × 1041 × 105Coefficient of viscosityη2 (GPa·h)15102050100Load1.0Nm3.5Nm7.0NmTime (h)98168189


#### 4.2.1.E1 and η1 of the Maxwell Model

Figure20 illustrates the effect of elastic moduli E1 on the strain of tunnel anchor under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. It could be found in Figure 20 that the strains of tunnel anchor increase faster as time goes by and become stable after about 40 h. In addition, the strains decrease with the increase of E1 under the three loads. When E1 is 0.1 GPa, the stable strains under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm are 65 × 10−5, 230 × 10−5, and 461 × 10−5. When E1 increase to 0.6 GPa, the stable strains under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm decrease to 19 × 10−5, 69 × 10−5, and 137 × 10−5, decreasing by 70.7, 70, and 70.3 percent, respectively. Clearly, the variation of E1 would exert significant negative influence on the strain of tunnel anchor, and load would exert positive influence.Figure 20
Effect ofE1 on the strain of tunnel anchor under loads of (a) 1.0 Nm, (b) 3.5 Nm, and (c) 7.0 Nm.
(a)(b)(c)Figure21 illustrates the effect of coefficient of viscosity η1 on the strain of tunnel anchor under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. In Figure 21, the strains of tunnel anchor under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm increase with time. When the values of η1 are 5 × 103, 1 × 104, 5 × 104, and 1 × 105 GPa·h, the strains become stable after about 50 h. Additionally, the strain of tunnel anchor decreases with the increase of η1 under the three loads. The maximum decreasing percents under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm are 35.3, 44.3, and 47.0, respectively. Obviously, η1 would also exert negative influence on tunnel anchor strain.Figure 21
Effect ofη1 on the strain of tunnel anchor under loads of (a) 1.0 Nm, (b) 3.5 Nm, and (c) 7.0 Nm.
(a)(b)(c)


#### 4.2.2.E2 and η2 of the Kelvin Model

The effect of elastic moduliE2 on the strain of tunnel anchor shows similar trend with that of E1. In Figure 22, it could be found that the larger the value of E2 is, the smaller the strain of tunnel anchor is. The maximum decreasing percents under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm are 24.5, 21.5, and 23.5, respectively.Figure 22
Effect ofE2 on the strain of tunnel anchor under loads of (a) 1.0 Nm, (b) 3.5 Nm, and (c) 7.0 Nm.
(a)(b)(c)Figure23 shows the effect of η2 on the strain of tunnel anchor under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. When the value of η2 is smaller than 20 GPa·h, the strains progressively become stable. The strains of tunnel anchor under the three loads are the first becoming stable when η2 is 1 GPa·h. Moreover, the coefficient of viscosity η2 exerts negative influence on the strain of tunnel anchor, and the maximum decreasing percents are 39.1, 41.5, and 39.3, respectively.Figure 23
Effect ofη2 on the strain of tunnel anchor under loads of (a) 1.0 Nm, (b) 3.5 Nm, and (c) 7.0 Nm.
(a)(b)(c)


### 4.3. Calculation Results and Comparative Analysis

After assigning the parameters to the computational model, the numerical simulation was then carried out for at least 120 h under each loading condition. Figure24(a) exhibits the comparison between the experimental measurements at points of g3andg4 in the anchor and the calculation results obtained from FLAC3D. As shown in Figure 24(a), the relative errors between the calculation values and the experimental measurements are less than 20% under each loading condition. In particular, there is a minimal difference between the experimental measurements and simulation results of creep deformation at point g3(on the front anchor face) and point g4 (on the rear anchor face) under the loads of 1.0 Nm and 3.5 Nm. When the load was increased to 7.0 Nm, however, the maximum difference between the experimental measurements and simulation results on the front and rear anchor faces reached 0.2779 mm.Figure 24
Comparison between experiment measurements and simulation results of (a) tunnel anchor and (b) surrounding rocks.
(a)(b)Figure24(b) provides the comparison between the experimental measurements at points of g2 and g5 in the surrounding rock and the corresponding calculation results. As shown in Figure 24(b), the relative errors between the simulation results and the experimental measurements are less than 25% under each loading condition. The difference between the experimental measurements and simulation results at measuring point g2 is very small under the loads of 1.0 Nm and 3.5 Nm. The maximum difference between the experimental measurements and simulation results at measuring point g5 is 0.168 mm. When the load was increased to 7.0 Nm, the difference between the experimental measurements and simulation results at measuring point g5 became very small. The difference between the experimental measurements and simulation results at measuring point g2 is 0.175 mm.The above comparisons show that the creep deformation at the measuring points obtained from the numerical calculation shared the same order of magnitude and a similar trend with the experimental measurements. The experimental measurements and simulation results demonstrated the same trend in the change of creep deformation for the anchor and surrounding rock.


### 4.4. Long-Term Stability of Anchor and Surrounding Rock

Here we further evaluate the long-term creep deformation pattern of the anchor in the suspension bridge and the surrounding rock. We conducted a calculation for the model shown in Figure19 considering the loading period of 100 years so as to validate if the design of the anchor satisfies the requirements of the specification. Each loading period is taken to be 10 years. We only analyse the creep behaviours at two measuring points on the front and rear faces of the anchor (g3andg4) and two other measuring points on the surrounding rock (g2andg5). The calculation results revealed the creep deformation pattern of the anchor and the surrounding rock under a normal design load over a total duration of 100 years (Figure 25).Figure 25
Long-term creep patterns of (a) surrounding rock and (b) real working anchor.
(a)(b)It could be seen from Figure25 that there is only a slight difference between the creep deformations at measuring points g2 and g5 on the surrounding rock over the duration of 100 years. In addition, these creep deformations barely changed with increasing time, and the results indicate a stable creep behaviour. Moreover, while the creep deformations at points g3andg4 on the anchor differ slightly from each other, both of these values increase at a constant rate over time. The maximum creep deformation of the anchor in the suspension bridge reached 1.377 cm over the duration of 100 years.The Honshu–Shikoku Bridge Authority in Japan reported the calculation of allowable horizontal displacement for anchors used in ultralong suspension bridges (1000 to 1500 m long) [26], and the empirical equation is given by(13)δ=0.017L,where δ is the allowable horizontal displacement (cm) and L is the main span of the bridge (m).The main span length of Xingkang Bridge is 1100 m. According to (13), the maximum allowable horizontal displacement of the real working anchor is 18.7 cm. Therefore, the maximum allowable horizontal displacement of the scaled anchor model is 1.87 cm. Such a value is greater than the maximum creep deformation of 1.377 cm for the anchor in the suspension bridge over the duration of 100 years. Therefore, the creep behaviour of the anchor satisfies the requirement of the specification and could be categorised as a stable creep phenomenon.Clearly, the creep deformation of the anchor along the tensile direction increases with increasing level of load from 1.0Nm to 7.0 Nm. In addition, with increasing load level, the creep deformations measured at the points with maximum stress in the anchor, surrounding rock, and the interface tend to be stable eventually after a decelerating growth period. This phenomenon indicates that the real working tunnel anchor would undergo a stable creep behaviour under different load levels. Since the creep deformation of the anchor does not exceed the maximum allowable value under the normal design load, the Xingkang Bridge would be in a long-term safe running state.


## 4.1. Constitutive Model and Parameters

The in situ testing data show that the surrounding rock exhibits a viscoelastic characteristic when applying the loads of 1.0Nm, 3.5 Nm and 7.0 Nm to the model tunnel anchor. In other words, the creep model should comprise both viscous and elastic elements. In this study, the four-parameter Burger creep model was employed to characterise the aforementioned rheological curve to obtain the corresponding creep parameters. The four-parameter Burger creep model is formed by combining the Kelvin model and Maxwell model [20], as illustrated in Figure 18.Figure 18
Schematic diagram of four-element Burger’s model.The Kelvin model is suitable for describing the relaxation process under a constant stress condition [21], while the Maxwell model is suitable for describing the creep process under a constant stress condition [22]. In the real situation, an object would undergo both creep and relaxation processes when bearing external forces. The strain generated on a Burger material is equal to the sum of the strains generated from both the relaxation and creep processes; see the following equation:(5)ε=εk+εm,where εk and εm are the strains generated from the Kelvin and Maxwell models, respectively, and ε is the total strain generated from the Burger creep model.When the anchor of a suspension bridge is affected by an axial tensile stressσ1 along the main cable, the deformation of the anchor will induce a radial stress σ2 on the surrounding rock. The creep constitutive equation of the anchor under the combined effect of σ1 and σ2 could be written as follows [23]:(6)εt=2σ1+σ29Kv+2σ1−3σ29E21−exp−E2η2t+2σ1−3σ29E1+2σ1−3σ2η1t,where Kv is the bulk modulus (GPa); E1 and E2 are the elastic moduli (GPa) used in the Maxwell model and Kelvin model; and η1 and η2 are the coefficients of viscosity (GPa·h) used in the Maxwell and Kelvin models.To examine the relationship between the axial tensile stressσ1 on the main cable and the radial stress σ2 on the surrounding rock, the cross section of the anchor is approximately considered as a circle, while the anchor is approximately considered as a wedge body. The direction parallel to the axis of the anchor is set to be the x-axis. The radial direction along the cross section of the anchor and perpendicular to the x-axis is set as the r-axis. The structure and stress condition of the tunnel anchor used in the suspension bridge are given in Figure 19.Figure 19
3D calculation model comprised of model anchor and surrounding rock.Taking any cross section of the anchor, the axial tensile stress could be calculated by(7)σ1=FA=Fπr2,where F is the force applied on the anchor; A is the area of anchor cross section; and r is the radius of anchor cross section.The axial and radial deformations are calculated by(8)ε1=σ1E=FEπr2,(9)ε2=με1=μσ1E=μFEπr2,where E is the elastic modulus of the anchor concrete, and μ is the Poisson ratio.The radial displacement is calculated by(10)s2=rε2=μFEπr.According to the assumption of Winkler’s foundation model [24], the axial tension exerted by the surrounding rock on the anchor is expressed as(11)σ2=Ks2=μFKEπr=μFK0Eπr2,where K is the elastic resistance coefficient of the surrounding rock, and K0 is the unit elastic resistance coefficient of the surrounding rock.When substituting (7)-(11) into (6), we thus obtain the following creep constitutive equation:(12)εt=F2E−3μK09EE2πr21−exp−E2η2t+F2E−3μK09EE1πr2+F2E−3μK0η1Eπr2t+2FE+μK09EKvπr2.The 3D numerical model comprising the left anchor and the surrounding rock was solved on FLAC3D platform. The geometry size of this model is 56.8 m × 20.0 m × 47.2 m, as illustrated in Figure 19. Tetrahedral cells with a side length of 0.5 m were used to mesh the anchor and the surrounding rock in the model. The remaining region was meshed with tetrahedral cells with a side length of 2 m. There are 37,518 nodes and 216,509 elements in total.According to the measured results in Figures9, 12, and 16, the creep parameters of surrounding rocks were calculated by (12) and are listed in Table 2. When solving the numerical model shown in Figure 19, we used the Mohr–Coulomb criterion to judge the critical yield stress state of the tunnel anchor and surrounding rock. The corresponding input parameters are listed in Table 3. According to the test results and safety reservation requirement, the values of ρ0, μ0, and E0 are selected from the average values of ρ, μ, and E; σt0, c0, and φ0 are selected as 10, 50, and 90 percent of σt, c, and φ, respectively. The concrete grade of tunnel anchor is C35, and its values are referred to at China national standard (GB/T 14902-2012) [25].Table 2
Input parameters of Burger’s model used for surrounding rocks.
LoadE1 (GPa)η1 (GPa·h)E2 (GPa)η2 (GPa·h)1.0Nm0.44639005.0420.313.5Nm0.23301003.1510.117.0Nm0.15203001.544.02Table 3
Input parameters of surrounding rocks and concrete tunnel anchor.
Positionρ0 (g/cm3)σt0 (MPa)c0 (MPa)tanφ0μ0E0 (GPa)Surrounding rock2.570.320.410.700.282.13Concrete tunnel anchor2.451.571.431.280.2131.5In addition, only the gravitational stress field was considered in the calculation, and the normal displacements of the nodes at bottom and sides were constrained. The results obtained from the numerical simulation were compared with the experimental measurements to determine the creep deformation and reveal the creep deformation pattern of the anchor and surrounding rock.


## 4.2. Parameters for Sensitivity Analysis of Tunnel Anchor Deformation

Sensitivity analysis is conducted to evaluate the effect of parameters used in the Maxwell and Kelvin models on the deformation of tunnel anchor. The parameters for sensitivity analysis of tunnel anchor strain include elastic moduliE1 and E2 and coefficients of viscosity η1 and η2 used in the Maxwell model and Kelvin models. The parameters used for the sensitivity analysis are summarized in Table 4. Assumed elastic moduli E1 and E2 varying from 0.1 to 0.6 GPa and 1 to 6 are included in this study. The coefficients of viscosity η1 and η2 are assumed to be 5 × 102, 1 × 103, 5 × 103, 1 × 104, 5 × 104, and 1 × 105 GPa·h and 1, 5, 10, 20, 50, and 100 GPa·h, respectively.Table 4
Summary of the parameters used for sensitivity of strain.
Elastic moduliE1 (GPa)0.10.20.30.40.50.6Elastic moduliE2 (GPa)123456Coefficient of viscosityη1 (GPa·h)5 × 1021 × 1035 × 1031 × 1045 × 1041 × 105Coefficient of viscosityη2 (GPa·h)15102050100Load1.0Nm3.5Nm7.0NmTime (h)98168189


### 4.2.1.E1 and η1 of the Maxwell Model

Figure20 illustrates the effect of elastic moduli E1 on the strain of tunnel anchor under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. It could be found in Figure 20 that the strains of tunnel anchor increase faster as time goes by and become stable after about 40 h. In addition, the strains decrease with the increase of E1 under the three loads. When E1 is 0.1 GPa, the stable strains under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm are 65 × 10−5, 230 × 10−5, and 461 × 10−5. When E1 increase to 0.6 GPa, the stable strains under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm decrease to 19 × 10−5, 69 × 10−5, and 137 × 10−5, decreasing by 70.7, 70, and 70.3 percent, respectively. Clearly, the variation of E1 would exert significant negative influence on the strain of tunnel anchor, and load would exert positive influence.Figure 20
Effect ofE1 on the strain of tunnel anchor under loads of (a) 1.0 Nm, (b) 3.5 Nm, and (c) 7.0 Nm.
(a)(b)(c)Figure21 illustrates the effect of coefficient of viscosity η1 on the strain of tunnel anchor under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. In Figure 21, the strains of tunnel anchor under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm increase with time. When the values of η1 are 5 × 103, 1 × 104, 5 × 104, and 1 × 105 GPa·h, the strains become stable after about 50 h. Additionally, the strain of tunnel anchor decreases with the increase of η1 under the three loads. The maximum decreasing percents under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm are 35.3, 44.3, and 47.0, respectively. Obviously, η1 would also exert negative influence on tunnel anchor strain.Figure 21
Effect ofη1 on the strain of tunnel anchor under loads of (a) 1.0 Nm, (b) 3.5 Nm, and (c) 7.0 Nm.
(a)(b)(c)


### 4.2.2.E2 and η2 of the Kelvin Model

The effect of elastic moduliE2 on the strain of tunnel anchor shows similar trend with that of E1. In Figure 22, it could be found that the larger the value of E2 is, the smaller the strain of tunnel anchor is. The maximum decreasing percents under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm are 24.5, 21.5, and 23.5, respectively.Figure 22
Effect ofE2 on the strain of tunnel anchor under loads of (a) 1.0 Nm, (b) 3.5 Nm, and (c) 7.0 Nm.
(a)(b)(c)Figure23 shows the effect of η2 on the strain of tunnel anchor under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. When the value of η2 is smaller than 20 GPa·h, the strains progressively become stable. The strains of tunnel anchor under the three loads are the first becoming stable when η2 is 1 GPa·h. Moreover, the coefficient of viscosity η2 exerts negative influence on the strain of tunnel anchor, and the maximum decreasing percents are 39.1, 41.5, and 39.3, respectively.Figure 23
Effect ofη2 on the strain of tunnel anchor under loads of (a) 1.0 Nm, (b) 3.5 Nm, and (c) 7.0 Nm.
(a)(b)(c)


## 4.2.1.E1 and η1 of the Maxwell Model

Figure20 illustrates the effect of elastic moduli E1 on the strain of tunnel anchor under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. It could be found in Figure 20 that the strains of tunnel anchor increase faster as time goes by and become stable after about 40 h. In addition, the strains decrease with the increase of E1 under the three loads. When E1 is 0.1 GPa, the stable strains under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm are 65 × 10−5, 230 × 10−5, and 461 × 10−5. When E1 increase to 0.6 GPa, the stable strains under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm decrease to 19 × 10−5, 69 × 10−5, and 137 × 10−5, decreasing by 70.7, 70, and 70.3 percent, respectively. Clearly, the variation of E1 would exert significant negative influence on the strain of tunnel anchor, and load would exert positive influence.Figure 20
Effect ofE1 on the strain of tunnel anchor under loads of (a) 1.0 Nm, (b) 3.5 Nm, and (c) 7.0 Nm.
(a)(b)(c)Figure21 illustrates the effect of coefficient of viscosity η1 on the strain of tunnel anchor under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. In Figure 21, the strains of tunnel anchor under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm increase with time. When the values of η1 are 5 × 103, 1 × 104, 5 × 104, and 1 × 105 GPa·h, the strains become stable after about 50 h. Additionally, the strain of tunnel anchor decreases with the increase of η1 under the three loads. The maximum decreasing percents under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm are 35.3, 44.3, and 47.0, respectively. Obviously, η1 would also exert negative influence on tunnel anchor strain.Figure 21
Effect ofη1 on the strain of tunnel anchor under loads of (a) 1.0 Nm, (b) 3.5 Nm, and (c) 7.0 Nm.
(a)(b)(c)


## 4.2.2.E2 and η2 of the Kelvin Model

The effect of elastic moduliE2 on the strain of tunnel anchor shows similar trend with that of E1. In Figure 22, it could be found that the larger the value of E2 is, the smaller the strain of tunnel anchor is. The maximum decreasing percents under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm are 24.5, 21.5, and 23.5, respectively.Figure 22
Effect ofE2 on the strain of tunnel anchor under loads of (a) 1.0 Nm, (b) 3.5 Nm, and (c) 7.0 Nm.
(a)(b)(c)Figure23 shows the effect of η2 on the strain of tunnel anchor under loads of 1.0 Nm, 3.5 Nm, and 7.0 Nm. When the value of η2 is smaller than 20 GPa·h, the strains progressively become stable. The strains of tunnel anchor under the three loads are the first becoming stable when η2 is 1 GPa·h. Moreover, the coefficient of viscosity η2 exerts negative influence on the strain of tunnel anchor, and the maximum decreasing percents are 39.1, 41.5, and 39.3, respectively.Figure 23
Effect ofη2 on the strain of tunnel anchor under loads of (a) 1.0 Nm, (b) 3.5 Nm, and (c) 7.0 Nm.
(a)(b)(c)


## 4.3. Calculation Results and Comparative Analysis

After assigning the parameters to the computational model, the numerical simulation was then carried out for at least 120 h under each loading condition. Figure24(a) exhibits the comparison between the experimental measurements at points of g3andg4 in the anchor and the calculation results obtained from FLAC3D. As shown in Figure 24(a), the relative errors between the calculation values and the experimental measurements are less than 20% under each loading condition. In particular, there is a minimal difference between the experimental measurements and simulation results of creep deformation at point g3(on the front anchor face) and point g4 (on the rear anchor face) under the loads of 1.0 Nm and 3.5 Nm. When the load was increased to 7.0 Nm, however, the maximum difference between the experimental measurements and simulation results on the front and rear anchor faces reached 0.2779 mm.Figure 24
Comparison between experiment measurements and simulation results of (a) tunnel anchor and (b) surrounding rocks.
(a)(b)Figure24(b) provides the comparison between the experimental measurements at points of g2 and g5 in the surrounding rock and the corresponding calculation results. As shown in Figure 24(b), the relative errors between the simulation results and the experimental measurements are less than 25% under each loading condition. The difference between the experimental measurements and simulation results at measuring point g2 is very small under the loads of 1.0 Nm and 3.5 Nm. The maximum difference between the experimental measurements and simulation results at measuring point g5 is 0.168 mm. When the load was increased to 7.0 Nm, the difference between the experimental measurements and simulation results at measuring point g5 became very small. The difference between the experimental measurements and simulation results at measuring point g2 is 0.175 mm.The above comparisons show that the creep deformation at the measuring points obtained from the numerical calculation shared the same order of magnitude and a similar trend with the experimental measurements. The experimental measurements and simulation results demonstrated the same trend in the change of creep deformation for the anchor and surrounding rock.


## 4.4. Long-Term Stability of Anchor and Surrounding Rock

Here we further evaluate the long-term creep deformation pattern of the anchor in the suspension bridge and the surrounding rock. We conducted a calculation for the model shown in Figure19 considering the loading period of 100 years so as to validate if the design of the anchor satisfies the requirements of the specification. Each loading period is taken to be 10 years. We only analyse the creep behaviours at two measuring points on the front and rear faces of the anchor (g3andg4) and two other measuring points on the surrounding rock (g2andg5). The calculation results revealed the creep deformation pattern of the anchor and the surrounding rock under a normal design load over a total duration of 100 years (Figure 25).Figure 25
Long-term creep patterns of (a) surrounding rock and (b) real working anchor.
(a)(b)It could be seen from Figure25 that there is only a slight difference between the creep deformations at measuring points g2 and g5 on the surrounding rock over the duration of 100 years. In addition, these creep deformations barely changed with increasing time, and the results indicate a stable creep behaviour. Moreover, while the creep deformations at points g3andg4 on the anchor differ slightly from each other, both of these values increase at a constant rate over time. The maximum creep deformation of the anchor in the suspension bridge reached 1.377 cm over the duration of 100 years.The Honshu–Shikoku Bridge Authority in Japan reported the calculation of allowable horizontal displacement for anchors used in ultralong suspension bridges (1000 to 1500 m long) [26], and the empirical equation is given by(13)δ=0.017L,where δ is the allowable horizontal displacement (cm) and L is the main span of the bridge (m).The main span length of Xingkang Bridge is 1100 m. According to (13), the maximum allowable horizontal displacement of the real working anchor is 18.7 cm. Therefore, the maximum allowable horizontal displacement of the scaled anchor model is 1.87 cm. Such a value is greater than the maximum creep deformation of 1.377 cm for the anchor in the suspension bridge over the duration of 100 years. Therefore, the creep behaviour of the anchor satisfies the requirement of the specification and could be categorised as a stable creep phenomenon.Clearly, the creep deformation of the anchor along the tensile direction increases with increasing level of load from 1.0Nm to 7.0 Nm. In addition, with increasing load level, the creep deformations measured at the points with maximum stress in the anchor, surrounding rock, and the interface tend to be stable eventually after a decelerating growth period. This phenomenon indicates that the real working tunnel anchor would undergo a stable creep behaviour under different load levels. Since the creep deformation of the anchor does not exceed the maximum allowable value under the normal design load, the Xingkang Bridge would be in a long-term safe running state.


## 5. Conclusions

(1)
The creep deformations measured at the front and rear faces of the anchor were consistent with each other. The average creep deformations of anchors were 0.54 mm, 0.90 mm, and 1.39 mm under loads of 1.0Nm, 3.5 Nm, and 7.0 Nm, respectively. The left and right anchors shared a similar creep deformation.(2)
The region between the left and right anchors experienced the largest creep deformations of 0.49 mm, 0.85 mm, and 1.10 mm under the three loading conditions, respectively. The creep deformation became larger as it got closer to the anchor. Under the loads of 1.0Nm and 3.5 Nm, the creep deformation along the vertical direction was greater than that along the tensile direction. When the load was increased to 7.0 Nm, however, the creep deformation along the tensile direction became slightly larger than that along the vertical direction.(3)
The maximum creep deformation was measured at point WCJ6 located on the interface between the bottom plate of the left anchor and the surrounding rock. The maximum creep deformations of the interface between anchor and surrounding rock were 0.15 mm, 0.64 mm, and 1.43 mm, respectively, under the three loading conditions. The difference between the creep deformations measured at different points on the interface between anchor and surrounding rock became more significant with increasing load.(4)
Numerical analysis shows that the creep deformation of anchor and surrounding rock shared the same order of magnitude and a similar trend with the experimental measurements. The locations with maximum stress in the anchor, the surrounding rock, and the interface experienced a creep deceleration stage.(5)
Only one type of concrete model anchor with one size (i.e., front face is 1.48 m × 1.56 m, rear face is 1.77 m × 1.90 m, and length is 4.00 m) and one type of rock (i.e., altered adamellite) is taken into consideration in this work. Hence, the presented results are applicable to limited cases, but the presented results merit close attention as a valuable source of reference in engineering projects with similar conditions.



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*Source: 8898777-2020-09-07.xml*