Calculation and Analysis of Load Transfer Characteristics of Tensile Anchors for Geotechnical Anchoring Systems
Abstract
:1. Introduction
2. Basic Equations for Anchor Load Transfer
3. Analytical Solution for Anchor Load Transfer
3.1. Elastic State Solutions
3.2. Plastic State Solution
3.3. Debonding State Solution
4. Example Verification and Engineering Application
4.1. Example Verification and Analysis
- (1)
- At a pull-out load of P = 160 kN, which is below the elastic limit load (Ptk), the interface around the anchoring section is in an elastic state. Consequently, the axial force gradually decreases to 0 towards the far end. There is no axial force distribution beyond the anchoring depth of 1.045 m, representing the limit length (Lt) of the elastic state. The axial force curve exhibits a concave form under these conditions.
- (2)
- As the pull-out load increases, reaching P = Ptk, the shear stress at the top of the anchoring section hits the ultimate shear strength τ1. At this juncture, the anchoring section begins to soften, and the portion near the top of the anchoring section enters the plastic state. When the pull-out load reaches P = 250 kN, a section of the anchor undergoes plastic softening. This specific segment, known as the plastic part, is characterized by a length (L1) equal to 0.168 m, forming a slightly convex curve on the axial force distribution curve. This curvature arises because the shear stress in the plastic softening part is less than the ultimate shear strength τ1, resulting in a reduced transmission rate of axial force to the distant part of the anchoring section. The segment with an anchorage length greater than 0.168 m remains in the elastic state, and the axial force is transmitted to an anchorage depth of 1.213 m.
- (3)
- As the pull-out load continues to rise, reaching P = Psk, the shear stress at the top of the anchoring section decreases to the residual shear strength τ2. At this critical point, the anchoring section is on the brink of cracking and sliding, and the region near the top of the anchoring section enters a debonding state. At a pull-out load of P = 300 kN, the length of the anchor undergoes debonding, with the length of the debonding part (L2) measuring 0.225 m. This debonding part is represented as a straight line on the axial force curve since the shear stress in the slipping section remains constant. Within the segment with an anchoring length between 0.225 m and 0.432 m, the material still remains in a plastic state, and the length of the plastic part (L1) is 0.207 m. The portion with an anchoring length greater than 0.432 m remains elastic, and the axial force is transmitted to an anchoring depth of 1.477 m.
4.2. Engineering Application
5. Conclusions
- (1)
- The step-wise linear-nonlinear bond-slip composite model is presented in the form of a single-peak curve. As the pull-out load increases, the anchor surface sequentially transitions through the elastic stage, the plastic stage, and the debonding stage. The model adeptly captures the elastic characteristics of the elastic stage, the softening and nonlinear attributes of the plastic stage, and the residual characteristics of the debonding stage.
- (2)
- In the pulling process, when the pull-out load is small, the anchorage interface remains in an elastic state, and the maximum shear stress is situated at the pulling end of the anchor. Consequently, the axial force experiences a rapid decline along the anchoring depth. As the pull-out load surpasses the elastic limit load, the plastic state initiates from the pulling end of the anchor, and the maximum shear stress is located at the boundary point between the elastic and plastic states. Upon exceeding the plastic limit load, the debonding state begins from the pulling end of the anchor, leading to the continued development of axial force into deeper sections. Furthermore, it is noteworthy that most measured values of axial force are less than the calculated values, indicating that the prediction method in this paper tends to be on the side of safety.
- (3)
- The load–displacement curves and axial loading distribution of the anchor were determined using the load transfer method based on the step-wise linear-nonlinear bond-slip composite model. The theoretical values were extensively validated against fieldwork published in the literature. The calculation method was applied to an actual project, revealing the shear responses of the anchor during the pull-out test. The output safety factor met the requirements of relevant specifications, affirming the reliability and practicality of the method.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Guo, F.X.; Tu, M.; Dang, J.X. Analysis and Design of Protection Device for Anchor Cable Pull-Out in High-Stress Roadways. Appl. Sci. 2023, 13, 12023. [Google Scholar] [CrossRef]
- Liu, Y.G.; Xia, K.; Wang, B.T.; Le, J.; Ma, Y.Q.; Zhang, M.L. Experimental Investigation on the Anchorage Performance of a Tension–Compression-Dispersed Composite Anti-Floating Anchor. Appl. Sci. 2023, 13, 12016. [Google Scholar] [CrossRef]
- Li, J.; Chen, S.X.; Yu, F.; Jiang, L.F. Reinforcement Mechanism and Optimisation of Reinforcement Approach of a High and Steep Slope Using Prestressed Anchor Cables. Appl. Sci. 2020, 10, 266. [Google Scholar] [CrossRef]
- Zou, J.F.; Zhang, P.H. Analytical model of fully grouted bolts in pull-out tests and in situ rock masses. Int. J. Rock Mech. Min. Sci. 2019, 113, 278–294. [Google Scholar]
- Tang, Y.F.; Jiang, D.H.; Wang, T.X.; Luan, H.J.; Liu, J.W.; Zhang, S.H. Research on the Mechanism of the Passive Reinforcement of Structural Surface Shear Strength by Bolts under Structural Surface Dislocation. Appl. Sci. 2023, 13, 543. [Google Scholar] [CrossRef]
- Yang, D.; Wang, Q.C.; Jiang, Z.Q.; Yang, D.X. A Multi-Segment Expanded Anchor for Landslide Emergency Management. Appl. Sci. 2022, 12, 12985. [Google Scholar] [CrossRef]
- Chen, J.; Li, D. Numerical simulation of fully encapsulated rock bolts with a tri-linear constitutive relation. Tunn. Undergr. Space Technol. 2022, 120, 104265. [Google Scholar] [CrossRef]
- Phillips, S.H.E. Factors Affecting the Design of Anchorages in Rock; Cementation Research Ltd.: London, UK, 1970. [Google Scholar]
- Benmokrane, B.; Chennouf, A.; Mitri, H.S. Laboratory evaluation of cement-based grouts and grouted rock anchors. Int. J. Rock Mech. Min. Sci. 1995, 32, 633–642. [Google Scholar] [CrossRef]
- Cai, Y.; Esaki, T.; Jiang, Y.J. An analytical model to predict axial load in grouted rock bolt for soft rock tunnelling. Tunn. Undergr. Space Technol. 2004, 19, 607–618. [Google Scholar] [CrossRef]
- Richard, R.M.; Abbott, B.J. Versatile elastic-plastic stressstrain formula. J. Eng. Mech. Div. 1975, 101, 511–515. [Google Scholar] [CrossRef]
- Wong, K.S.; Teh, C.I. Negative skin friction on piles in layered deposits. J. Geotech. Eng. 1995, 121, 457–465. [Google Scholar] [CrossRef]
- Ren, F.F.; Yang, Z.J.; Chen, J.F.; Chen, W.W. An analytical analysis of the full-range behaviour of grouted rockbolts based on a tri-linear bond-slip model. Constr. Build. Mater. 2010, 24, 361–370. [Google Scholar] [CrossRef]
- Chen, C.F.; Zhu, S.M.; Zhang, G.B.; Morsy, A.M.; Zornberg, J.G.; Mao, F.S. A Generalized Load-Transfer Modeling Framework for Tensioned Anchors Integrating Adhesion–Friction-Based Interface Model. Int. J. Geomech. 2022, 22, 04022036. [Google Scholar] [CrossRef]
- Huang, M.H. Analysis on Pullout Load Transfer Mechanism of Geotechnical Anchor and Its Validating Monitor with Smart FRP Anchor. Ph.D. Thesis, Harbin Institute of Technology, Harbin, China, 2014. [Google Scholar]
- Zhang, L.; Wang, M.; Zhao, H.B.; Chang, X. Uncertainty quantification for the mechanical behavior of fully grouted rockbolts subjected to pull-out tests. Comput. Geotech. 2022, 145, 104665. [Google Scholar] [CrossRef]
- Zhang, W.L.; Huang, L.; Juang, C.H. An analytical model for estimating the force and displacement of fully grouted rock bolts. Comput. Geotech. 2020, 117, 103222. [Google Scholar] [CrossRef]
- Martin, L.B.; Tijani, M.; Hadj-Hassen, F. A new analytical solution to the mechanical behaviour of fully grouted rockbolts subjected to pull-out tests. Constr. Build. Mater. 2011, 25, 749–755. [Google Scholar] [CrossRef]
- Xiao, S.J.; Chen, C.F. Mechanical mechanism analysis of tension type anchor based on shear displacement method. J. Cent. South Univ. Technol. 2008, 15, 106–111. [Google Scholar] [CrossRef]
- Ma, S.Q.; Zhao, Z.Y.; Nie, W.; Gui, Y.L. A numerical model of fully grouted bolts considering the tri-linear shear bond–slip model. Tunn. Undergr. Space Technol. 2016, 54, 73–80. [Google Scholar] [CrossRef]
- Nemcik, J.; Ma, S.Q.; Aziz, N.; Ren, T.; Geng, X.Y. Numerical modelling of failure propagation in fully grouted rock bolts subjected to tensile load. Int. J. Rock Mech. Min. Sci. 2014, 71, 293–300. [Google Scholar] [CrossRef]
- Xu, C.; Li, Z.H.; Wang, S.Y.; Wang, S.R.; Fu, L.; Tang, C.N. Pullout Performances of Grouted Rockbolt Systems with Bond Defects. Rock Mech. Rock Eng. 2018, 51, 861–871. [Google Scholar] [CrossRef]
- Zhao, M.H.; Huang, Y.J.; Huang, M.H. Study on nonlinear calculation method of load transferring along tensile anchor rod base on finite difference method. J. Railw. Sci. Eng. 2018, 15, 1963–1970. [Google Scholar]
- Zhou, S.C.; Zhu, W.C.; Yu, S.S. Analysis of load transfer mechanism for fully grouted rockbolts based on the bi-exponential shear-slip model. Chin. J. Rock Mech. Eng. 2018, 37, 3817–3825. [Google Scholar]
- Zhou, B.S.; Wang, B.T.; Liang, C.Y.; Wang, Y.H. Study on load transfer characteristics of wholly grouted bolt. Chin. J. Rock Mech. Eng. 2017, 36, 3774–3780. [Google Scholar]
- Liu, B.; Huang, L.; Li, D.Y. Analytical Formulation on the Mechanical Behavior of Anchorage Interface for Full-Length Bonded Bolt. Appl. Mech. Mater. 2012, 166–169, 3254–3257. [Google Scholar] [CrossRef]
- CECS22:2005; Technical Specification for Ground Anchors. China Planning Press: Beijing, China, 2005.
Experimental Load | 0.3 Nmax a | 0.5 Nmax a | 0.7 Nmax a | 0.8 Nmax a | 0.9 Nmax a | 1.0 Nmax a | 0.3 Nmax a |
---|---|---|---|---|---|---|---|
Duration (min) | 10 | 10 | 10 | 10 | 10 | >10 | 5 |
Anchor Cable A | Anchor Cable B | Anchor Cable C | |||
---|---|---|---|---|---|
P (kN) | S (mm) | P (kN) | S (mm) | P (kN) | S (mm) |
234 | 6.79 | 198 | 5.78 | 216 | 6.52 |
390 | 14.22 | 330 | 12.58 | 360 | 14.65 |
546 | 20.07 | 462 | 19.03 | 504 | 22.74 |
624 | 27.83 | 528 | 27.12 | 576 | 30.55 |
702 | 36.08 | 594 | 34.90 | 648 | 37.44 |
780 | 42.71 | 660 | 40.25 | 720 | 44.91 |
Ptk (kN) | S1 (mm) | Psk (kN) | S2 (mm) | τ1 (kpa) | τ2 (kpa) | Design Load (kN) | Pumax (kN) | Safety Factor | |
---|---|---|---|---|---|---|---|---|---|
Anchor cable A | 234 | 6.79 | 546 | 20.07 | 191.0 | 44.5 | 650 | 1065.8 | 1.64 |
Anchor cable B | 198 | 5.78 | 462 | 19.03 | 160.3 | 39.4 | 550 | 892.4 | 1.62 |
Anchor cable C | 216 | 6.52 | 504 | 22.74 | 169.1 | 42.7 | 600 | 993.8 | 1.66 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Cheng, Z.; Wang, Y.; Zhang, K.; Wei, D. Calculation and Analysis of Load Transfer Characteristics of Tensile Anchors for Geotechnical Anchoring Systems. Appl. Sci. 2024, 14, 472. https://doi.org/10.3390/app14020472
Cheng Z, Wang Y, Zhang K, Wei D. Calculation and Analysis of Load Transfer Characteristics of Tensile Anchors for Geotechnical Anchoring Systems. Applied Sciences. 2024; 14(2):472. https://doi.org/10.3390/app14020472
Chicago/Turabian StyleCheng, Zhiyuan, Yimin Wang, Kunbiao Zhang, and Daidong Wei. 2024. "Calculation and Analysis of Load Transfer Characteristics of Tensile Anchors for Geotechnical Anchoring Systems" Applied Sciences 14, no. 2: 472. https://doi.org/10.3390/app14020472
APA StyleCheng, Z., Wang, Y., Zhang, K., & Wei, D. (2024). Calculation and Analysis of Load Transfer Characteristics of Tensile Anchors for Geotechnical Anchoring Systems. Applied Sciences, 14(2), 472. https://doi.org/10.3390/app14020472