Constant resistance large deformation anchors
The mechanical properties of NPR materials, also known as tensile expansion materials, are determined by the material’s internal structure. Figure 1 shows a typical NPR material, where the inner concave honeycomb structure expands and deforms laterally when an axial tensile force is applied, resulting in the NPR phenomenon.
The NPR anchor/rope is a composite device with a unique negative Poisson's ratio structure, shown in Fig. 1. As can be seen from Fig. 2, the NPR anchor consists of a constant resistance body (installed in a casing), a strand installed on the constant resistance body, a casing (the inner diameter of which is slightly larger than the diameter of the large end of the constant resistance body), a tray (used to transfer the deformation of the rock mass to the casing) and a fastening nut (force transmitting device).
For the overall structure of the NPR anchor cable, the constant resistance body is always fixed on the free face side of the slope. When the NPR cable is subjected to deformation of the rock mass, its steel-stranded wire is subjected to stretching, and when the cable reaches the ultimate bearing capacity, the constant resistance body begins to slide inside the resistor. The diameter of the small end of the constant resistor is a little smaller than the inner diameter of the casing, and the diameter of the large end is slightly larger than the inner diameter of the casing. As a consequence, the casing expands radially while it is squeezed by the constant resistor during axial stretching. The resulting sliding distance is the deformation of the anchor cable, and the reverse resistance acts as the constant resistance of the anchor cable.
The constant resistance of the NPR cable at work is only related to the structural parameters of the constant resistance body and the elastic parameters of the constant resistance sleeve. According to (He et al.2016), the expression for the constant resistance of NPR cable is as follows.
$${P_0}=2\pi f{I_c}{I_s}$$
1
where Ic is the geometric parameter of the constant resistance body and Is is the elastic constant of the constant resistance sleeve, given by the following equations:
$${I_c}=\frac{{a{h^2}}}{2}\cos a+\frac{{{h^3}}}{3}\sin \alpha$$
2
$${I_s}=\frac{{E({b^2} - {a^2})\tan \alpha }}{{\alpha \left[ {{\alpha ^2}+{b^2} - \mu ({b^2} - {a^2})} \right]}}$$
3
Where α is the half cone angle of the constant resistance body; h is the length of the constant resistance body; α is the diameter of the small end of the constant resistance body; b is the diameter of the large end of the constant resistance body; E and µ are the modulus of elasticity of the constant resistance sleeve and Poisson's ratio, respectively. From equations (1)-(3), it can be seen that under static loading conditions, the constant resistance of the anchor cable is related to the static friction factor between the constant resistance body and the sleeve, and the geometrical parameters of the constant resistance body and the geometrical and elastic constants of the sleeve.
When the constant resistance body undergoes relative slip inside the constant resistance sleeve, the constant resistance sleeve expands radially. This expansion releases the constant resistance of the NPR cable stably and continuously during the working process, ensuring that the anchor cables do not fail due to tension in the strands during deformation.
Model design
Finite Element Model of NPR Anchor Cable
In this paper, ABAQUS finite element software was used to numerically simulate and analyse the NPR cable, performing STATIC and GENERAL quasi-static analyses. The ideal elastic-plastic model was introduced for the material of the constant resistance body and sleeve, and the Johnson-Cook (JC) model (Chen G et al. 2005) was adopted to describe the dynamic behaviour of the metallic material (the constant resistance sleeve) of the body from low to high strain rates, which was axially stretched at a rate of 20 mm min− 1 until it was completely pulled out of the sleeve.
The model consists of a round table constant resistance body, a thick-walled constant resistance body sleeve, and a steel stranded wire. The constant resistance body is made of special hardened steel 45Cr, with a modulus of elasticity of 208 GPa and a Poisson's ratio of 0.280. The constant resistance sleeve is made of 20-gauge steel, with a modulus of elasticity of 206.9 GPa and a Poisson's ratio of 0.290. The anchor strand is made of prestressing steel, with a radius of 11.2 mm, and the standard value of tensile strength of 1860 MPa.
The constant resistance of the NPR anchor is mainly provided by the sliding friction between the constant resistance sleeve and the constant resistance body. This friction is generated during the sliding of the resistor inside the sleeve, as the diameter of the large end of the constant resistor is slightly larger than the inner diameter of the sleeve. The mechanical and geometric parameters of NPR cable are listed in Tables 1 and 2. The model is shown in Fig. 3.
Table 1
Mechanical parameters of NPR cable
|
Elastic modulus /GPa
|
Poisson's ratio
|
Tensile strength
/MPa
|
Constant resistance/kN
|
|
206
|
0.28
|
730
|
490
|
Table 2
Geometric parameters of NPR cable (mm)
|
The different components of the NPR structure.
|
|
The constant body
|
Large end diameter
|
94
|
|
Small end diameter
|
84
|
|
Small end length
|
81
|
|
Large end length
|
25
|
|
The constant resistance sleeve
|
Outer diameter
|
120
|
|
Inner diameter
|
87
|
|
length
|
100
|
|
steel stranded wire
|
diameter
|
22.4
|
Both the constant resistance body and the constant resistance sleeve are modelled by 8-node C3D8R 3D solid cells, and the mesh model is constructed as shown in Fig. 4.
Boundary conditions and loading mode
The outer side of the constant resistance sleeve of the NPR cable is axially restrained. The constant resistance body and the strand are selected for displacement coupling; the strand and the body have equal deformations. The coupling point is located at the centre point of the small end of the constant resistance body. The constant resistance body can slip along the sleeve from end A to end B under the pulling force of the strand. Coupling the constant resistance body with the strand, a force is applied on the steel strand to the centre point O at the other end of the steel strand to simulate the concentrated load of the steel strand, as shown in Fig. 6. The anchor cable is always kept with its axis on the same horizontal line. The boundary condition of the constant resistance sleeve is set to U2 = UR3 = 0, as shown in Fig. 5, where x-axis is the axial direction of the anchor cable. An axial tensile rate of 20 mm/min is applied at the free end of the anchor cable. The type of contact between the constant resistance body and the constant resistance sleeve is the face-to-face contact. The tangential friction coefficient of the contact surface between the constant resistor and the sleeve is 0.1, and the normal direction is a rigid contact.
Model validation
(1) Constant resistance evolution law
The diagrams of the axial pullout force with respect to the axial displacement, extracted from the test and the numerical simulation, are shown in Fig. 7. The test results are taken from reference. The tensile process can be divided into three stages: elastic enhancement stage, constant resistance output stage, and destructive decline stage.
Elasticity enhancement stage: In this stage, the axial displacement through which the constant resistor slides in the transverse resistor sleeve varies between 0 ~ 100 mm and the tensile force of the strand shows a rapid increase trend. The constant resistance body gradually slides into the sleeve. The pulling force of the steel strand is smaller than the initial static friction, and the sleeve undergoes approximate elastic expansion and deformation.
Constant resistance output stage: the axial displacement varies between 100–1000 mm. The pull-out force of the strand reaches 490 kN. The sleeve wall and the constant resistance body are in complete contact, producing relative sliding. The contact area between the two is constant, so that the pull-out force is roughly constant.
Destruction decline stage: when the displacement of the constant resistance body exceeds 1000 mm. The constant resistance body starts to be pulled out until it is completely detached from the sleeve. The contact area between the two parts gradually decreases, and the pull-out force is reduced to zero.
The numerical simulation results of the NPR anchor cable pulling force are compared with the experimental ones, found in reference. It can be noted in Fig. 6 that the constant resistor shows large fluctuations in both curves as it enters the sleeve and slips out of the sleeve. The numerical simulation curves corresponded well with the measured curves at each stage of the experiment. In the static pullout test, the constant resistance body is subjected to axial tension and contact with the sleeve, and friction loss occurs. This loss leads to a decrease of the constant resistant force. In the numerical simulation, the constant resistance body is assumed to be an elastic body, which can avoid the decrease of constant resistant force due to friction.
(2) Constant resistance sleeve strain law
In order to further study the evolution of the radial deformation, circumferential deformation and axial deformation of constant resistance sleeve in the static pulling numerical simulation, the axial displacement of the sleeve is segmented every 50 mm, and a total of 21 displacement segmentation points are arranged. The numerical simulation results of Fig. 8 show that there are different degrees of deformation of radial X-X and Y-Y. The deformation in the X-X direction varies between 3.22 mm-3.67 mm, and the deformation in the Y-Y direction ranges between 3.25 mm-3.68 mm. Both deformations are relatively smooth because the numerical simulation process assumes that the constant resistance body is made of an elastomeric material, which reduces the wear of the sleeve.
With the increase of axial pullout load, the deformation of the constant resistance sleeve increases rapidly and reaches the peak value. Subsequently, a weak rebound phenomenon occurs as shown in Fig. 8a and b. Finally, the plastic deformation of the constant resistance sleeve continues to stabilise with an average value of 3.530 mm. The elastic rebound of the barrel was about 0.12 mm, which was much larger than the average value of its elastic deformation of 0.072 mm. This deformation characteristic confirmed the unique negative Poisson's ratio configuration of the NPR anchor structure.
NPR cable-loess medium coupling model
Calculation model
Based on the three-dimensional model of NPR cable in Fig. 2, a coupling model is established, as shown in Fig. 9, simulating the NPR cable-anchorage and loess medium with solid elements. The constant resistance sleeve is selected to be 1000 mm long, with inner and outer diameters 87 mm and 120 mm, respectively. The length of the constant resistance body is 106 mm, the diameter of the big end is 94 mm, and the diameter of the small end is 84 mm. In order to ensure that the NPR cable system will work properly, and establish the coupling model of NPR cable - loess medium interaction, the cement mortar, the anchor and the loess medium are selected to be homogeneous, isotropic and continuous elastic materials. The Mohr-Coulomb yield criterion constitutive model is used. The contact surface normal is set as "hard contact", and the tangential friction formula is represented by the Coulomb friction model.
Solid elements were used to simulate all components of the model. In this paper, it is assumed that the radial deformations of loess and anchoring agent are the same. The influence of the anchor cable-anchorage- loess medium interaction mechanism during the radial expansion of the sleeve body, caused by the relative sliding of the constant resistance body and the constant resistance sleeve, is simplified. The NPR cable is surrounded by a homogeneous loess material of 1 m×1 m×1 m, and the outer surface of loess is constrained (UX=UY=UZ=0). The loess close to the constant resistance body is axially constrained. The sleeve of the NPR cable is anchored to the soil with the filling material of cement mortar, which has an anchorage strength of 1300 kPa and pullout strength \({P_a}=2\pi rl{\tau _f}\)according to the Technical Specification for Construction Slope Engineering (Guang-xin Li 2016). The axial velocity of 20 mm/min is applied at the distal end of the steel strand to carry out the static tensile test. The material parameters of the finite element model soil and the grouting of the anchored section are listed in Table 3. The calculation model and boundary conditions are illustrated in Fig. 9 and Fig. 10, respectively.
Table 3
Mechanical parameters of loess medium and cement mortar
|
parameters
|
Elastic modulus /MPa
|
cohesion /kPa
|
Poisson's ratio
|
internal friction angle (o)
|
density /(kg/m3)
|
|
Q2 loess
|
50
|
35
|
0.27
|
22
|
1860
|
|
cement mortar
|
7500
|
1000
|
0.17
|
30
|
2500
|
Numerical results of NPR cable-loess medium coupling model
(1) Constant resistance evolution law
Figure 11 reveals the changing law of the pulling force with respect to the axial displacement of NPR cable and NPR cable-loess medium coupling model. Under the axial constant pulling speed of 20 mm/min, it is firstly observed that the growth rate of both tensile forces is relatively large, while the load of the steel stranded wire has not yet reached the initial static friction. As the constant resistance body gradually slips inside the sleeve, the tension slope of the media-free NPR cable decreases gradually. Regarding the coupled NPR cable - loess medium model, the constraint effect of the loess medium outside the sleeve barrel causes the slope of the pulling force curve to increase. When the constant resistance body completely enters the sleeve, the structure in the negative Poisson's ratio state enters the constant resistance deformation stage. At this time, as the contact area reaches the maximum, the tension of the steel stranded wire remains constant. The loess medium has a restraining effect on the NPR cable, and the constant resistance of the NPR cable-loess medium interaction model fluctuates greatly, with an average value of about 608 kN, which is much large than that of the constant resistance of the media-free NPR cable model (490 kN).
(2) Sleeve strain evolution law
As the NPR anchor cable enters the loess medium model, the sleeve deformation rapidly increases until it reaches its peak with an increase in axial displacement. Then a small rebound is noted. Finally, a plastic deformation is formed, with an average value of 2.574 mm, which is smaller than the average value of plastic deformation of the media-free NPR cable (3.530 mm). This is due to the fact that the cement mortar outside the barrel of the sleeve is in contact with the loess medium, and the plastic deformation of the barrel is reduced. The constraints provided by the medium attenuate the circumferential expansion of the barrel. Figure 12 illustrates the evolution of the deformation of the NPR cable and the NPR Cable-Loess Medium.
In the axial direction of the sleeve barrel, three annular constant sections (i.e, S1 = 250 mm, S2 = 500 mm and S3 = 750 mm) were selected to monitor the radial displacement change of the constant resistance sleeve. The displacement of the three sections increased gradually as the barrel moved from the fixed end (proximal end) to the free end (distal end). The maximum axial displacement of the barrel reaches 5.517 mm, and the corresponding axial strain is only 0.006, which is insignificant compared to the radial strain of the barrel (0.06). The circumferential displacement cloud of NPR cable and NPR cable-loess is shown in Fig. 13.
(3) Stress distribution law of loess medium
During the working process of NPR cable, the relative sliding between the constant resistance body and the constant resistance sleeve causes the radial expansion of the latter. This "negative Poisson's ratio structural effect" has a great influence on the stress distribution of the sleeve and the surrounding loess medium. Figure 14 shows the stress cloud map of loess medium.
In the X-Y plane of loess medium, Fig. 15 shows the von Mises stress cloud map of 6 positions from the free end to the fixed end (Li = 0, 50, 100, 500, 870, 1000 mm; and i = 1 ~ 6) through which the big end of the constant resistance body slides. The stress of the soil medium increases rapidly and reaches the peak at the position of L2, and then tends to be stabilised. The distribution law of loess medium is divided into three parts (e.g., Fig. 15, Fig. 16):
The initial stage is shown in Fig. 15(a)-(b), where the contact stress between the soil medium and the sleeve increases. Due to the constraints of the boundary conditions, the peak stress occurs at L2, which is closer to the free end, with maximum value of the soil body stress 0.629 MPa.
Decline gradually - stabilisation phase as in Fig. 15(c)-(d). With the increase of the depth of the constant resistance body sliding into the sleeve, the loess medium around the free end of the sleeve presents stress concentration. In this stability stage, the maximum value of the soil stress is 0.536 MPa. The stress value of soil is stabilised at around 0.16 MPa.
The final stage is shown in Fig. 15(e)-(f). In this stage, the constant resistance body is about to be pulled out of the constant resistance sleeve. The maximum stress value of the soil medium is 0.174 MPa, which appears to rise and then decrease.
As can be seen from Fig. 16, the soil stress shows "first increase and then rapidly decrease" and gradually tends to be stabilised as the constant resistance body slips into the sleeve. When the constant resistance body is about to be pulled out of the sleeve, the soil stress exhibits a small increase.
(4) Shear stress distribution law of loess medium
In the X-Y plane of loess medium, the displacement segmentation is observed with the diagonal of X-Y plane, S1, S2. The shear stress values in the diagonal are similar in magnitude and opposite in direction, and the conjugate symmetry phenomenon is observed in Fig. 17. "Shear stresses are symmetrical along the diagonal", i.e. equal in magnitude and opposite in direction.
The loess medium tangential stress distribution can be divided into three stages:
The first stage is the initial stage, when the loess medium shear stress maintains the anchor strength τf < 1300 kPa. In this stage, the external force is mainly undertaken by the interface bond between the cement mortar and sleeve, so the anchoring performance of the NPR cable is approximately the same as the one of an ordinary cable. During this initial stage, the loess medium shear stress grows rapidly and reaches a peak value of about 0.174 MPa (L2).
The second stage is the ring expansion stage of the sleeve barrel, when the constant resistance body completely enters the sleeve, its shear stress is stabilised at 0.060 MPa (L5). The maximum shear stress is close to the constant resistance sleeve, and gradually spreads and decreases in all directions.
In the third stage, the constant resistance body is gradually detached from the sleeve, the peak shear stress slightly recovered, reaching a peak value of 0.548 MPa and then it decreases gradually.
S1 and S2 shear stress values are roughly the same, in the opposite direction. Their diagrams follow an axisymmetric parabolic distribution, while at both sides the stress value tends to be close to 0. The loess medium shear stress distribution law curve is illustrated in Fig. 18. Due to some asymmetry in the position of the segmental fetch forces, the curves are roughly symmetrical.
(5) Characteristics of displacement field change
The cloud diagram of the displacement field in the axial direction of the loess medium is shown in Fig. 19. The deformation of loess around the sleeve at the free end reaches 9.89 mm in the X-X direction, and 11.911 mm in the Y-Y direction, confirming a large deformation in both directions
(6) Microstructure damage of loess medium
The damage mechanism of NPR cable and loess medium is described further on from the microscopic point of view. The cut surfaces of constant resistance body axial displacement are selected at Li = 50 mm, 100 mm, 150 mm as the key points of the analysis. The circular path observation with the loess medium model cut surface of the observation point is also determined. The maximum and minimum principal stresses of the loess medium in the circumferential path are calculated. Loess medium first sabotage in the weaker structure and comparatively low density, and shear stress with the development of deformation and decline, taking into account that the Geotechnical body in nature undergoes damage due to stress, the stress state is complex. The Mohr-Coulomb third strength theory is applied to measure the destruction of the soil, expressed by the following formula:
$${\sigma _1} - {\sigma _3}=\left[ \sigma \right]$$
4
Where σ1 is the maximum principal stress, σ2 is the minimum principal stress, and [σ] = 0.02 MPa.
The displacement field curves at the three analysed critical points L1 = 50 mm, L2 = 100 mm, L3 = 150 mm, are shown in Fig. 20. As the displacement of the constant resistance body in the constant resistance sleeve increases, the radial expansion of the sleeve barrel increases. The barrel damage first occurs in the radial direction with tensile failure and then it develops into circumferential expansion.
Firstly, the loess medium is damaged around the sleeve near the free end and expands rapidly in all directions. The shear stress of the loess medium near the sleeve reaches the peak, and then it decreases in all directions. In the axial displacement of L1 = 50 mm, as the constant resistance body enters the sleeve, the soil inside the circumference, which is defined by a radius of r = 500 mm, and the soil within its range is damaged; the axial displacement at L2 = 100 mm is smaller and the damage is limited to an area defined by r = 400 mm. Around the circumference, the shear strength with respect to L1 is relatively increased. An axial displacement of 150 mm is observed. The soil around the sleeve is extruded to the fixed end of the extension. The loess medium is seriously damaged mostly at the free end.
study area
The landslide of Hechang Xiangyan Waterfront Community is located next to the North Street in Liulin County, in China. In addition, the geomorphology of the site belongs to loess gully area. The highest elevation of the slope is 890 m and, the lowest elevation is 790 m. The relative height difference is 100 m. The present landslide is in the form of a two-stage ladder with a width of 201 m and a length of 50 m, which is a medium-sized landslide. The surface layer of slope body is Quaternary Middle Pleistocene Q2 + 3 silty clay and silt, and the sandstone of Carboniferous Shanxi formation beneath has a sandstone dip of 40° and a dip angle of 6°. The foot of the slope was excavated in the repairing process of repairing houses, making the slope steeper, losing its support and reducing its stability, seriously threatening the lives and safety of the residents in the neighbourhood below, see Fig. 21.
Layout of monitoring system of NPR cable
In order to study the efficiency of the NPR cable application in a loess slope of an actual landslide body, monitoring, early warning and control of integrated technology research, according to the scale of the landslide body and monitoring objectives of Hechang Xiangyan Waterfront Community, a total of 6 NPR Newtonian force monitoring points, 6 GNSS surface displacement monitoring points, 1 GNSS reference station are installed at the Waterfront Community to obtain monitoring of the landslide movement. The monitoring points setting and the section distribution characteristics are shown in Fig. 22. The anchor cables are located at the foot of the platform, in the middle of the slope. The diameter of the drill hole is 150 mm, the anchoring angle is 25°, the length of the anchoring section is 14 m, the free section is 44 m, and the total length of the anchor cable is 58 m.
(1) Analysis of monitoring results
The landslide in Waterfront Community was debugged in August 2018. Since then a Newton force remote monitoring system is used, the data of which are depicted in Fig. 23. Since the monitoring started, the overall fluctuation of the curve is relatively smooth, except for the individual monitoring curve Newtonian force values, which appear similar to the "narrower pulse" fluctuation phenomenon. This occurs due to the high-precision stress sensors signal acquisition interference of the NPR cable monitoring the landslide, which is influenced by several conditions, such as rainfall, earthquakes and river scour. Large overall fluctuations are observed at the monitoring Fig. 23(c). The Newtonian force has a significant change, and after the curve is enlarged, it is noted that the fluctuation of the Newtonian force curve, which is larger for a short time, and the monitoring curve also shows a steep jump, which indicates that the cracks of the original landslide are in the developmental stage, according to the five monitoring and warning modes of the slope hazard monitoring curve change (Sun G L 2016). However, the slope of this monitoring curve in the long time scale, the slope of the curve is small, and the late stage of the Newtonian force curve tends to a steady state with time, as there is no special abrupt change point. The Newtonian force curve shows that at this monitoring location, although the slope body has cracked and expanded, it exhibits no other disturbing effects under natural conditions except for rainfall. Hence, the landslide body, as a whole, is in a stable state.
Application of Newton force monitoring system in landslide of Xiangyan Waterfront Community; NPR cable and landslide have become a whole, working together under a common force, and the change of the Newton force of the monitoring system reflects the movement of the landslide body over the sliding bed, which shows that this system can effectively monitor the loess landslide remotely. The system has been running well since its installation, and can stably detect small stress changes inside the slope body, which ensures the advanced early warning and fine monitoring ability, as well as the practical effectiveness of slope disaster prediction. Newton force early warning systems can significantly improve the prediction of a potential disaster and the mitigation of loess landslide disasters, thus ensuring the safety of neighbourhood residents at the foot of the slope.