
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39266595
70537
10.1038/s41598-024-70537-w
Article
Effects of degraded durability on the long-term stability of in-service slopes with reinforced concreted support structures
Ye Wenya 763425011@qq.com

1
Ma Yongzheng 107723274@qq.com

1
Qin Cuigui 2
Wang Huajun 2
Li Chunguang 3
Ding Zhouxiang 4
1 grid.412189.7 0000 0004 1763 3306 Ningbo University of Technology, Ningbo, 315211 Zhejiang China
2 Zhejiang Engineering Survey and Design Institute Group Co. Ltd., Ningbo, 315012 Zhejiang China
3 grid.9227.e 0000000119573309 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, 430071 Hubei China
4 PRI Engineering Corp., 920 28 St NE #22, Calgary, AB T2A 6K1 Canada
12 9 2024
12 9 2024
2024
14 2129715 2 2024
19 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Engineering slope stability issues typically exhibit the impact of deteriorating durability on the susceptibility of slopes to failure. A thorough investigation was essential to explore theoretical and experimental aspects of slope durability degradation and its implications on long-term stability. Hence, a durability model was developed to accommodate slope stabilization using reinforced concrete (RC) support structures. This model was grounded in classical durability principles for RC structures. Subsequently, a model test was conducted to compare the responses of a standard slope model with a weakened counterpart subjected to environmental impacts. According to the proposed methodology for slope durability and stability, a case study involving future durability and stability predictions was performed. It was found that the theoretical solutions for the carbonation or neutralization (CN) velocity, depth, and penetration time agreed well with model test results. The slope surface displacements of the weakened slope with deteriorating coefficients between 0.6 and 0.9 were 4 to 8 times those of the standard slope, demonstrating significant degradation in stability. The case study indicated a steady reduction in the safety factor, at a rate of 2.3 to 2.4‰ per year throughout the slope’s service life. Finite-element-based predictions also suggested the potential for corrosion of slope anchor bolts within 20 years and breakage within 30 years, at an average rate of 7.5‰ per year in the ultimate bearing capacity. These findings highlight the need for timely maintenance and reinforcement interventions to ensure the long-term durability of operational slopes.

Keywords

In-service slopes
RC support structures
Durability model
Model test
Finite-element simulation
Factor of safety
Subject terms

Civil engineering
Natural hazards
the Open Fund Project of the Chinese State Key Laboratory of GeoMechanics and Geotechnical EngineeringZ020020 Z020020 Z020020 Ye Wenya Ma Yongzheng Li Chunguang issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Slope stability is one of the most intricate and demanding fields in geotechnical engineering1,2. The long-term stability of slopes may progressively deteriorate due to decreased shearing strengths of outcropping strata and structural bearing capacities, leading to sudden slope failures when critical stability thresholds are exceeded. For instance, statistical data from a coastal province in East China recorded hundreds of geological disasters, including landslides, slope collapses, and debris flows from 2008 to 2020. The slope failures mentioned above are primarily driven by various environmental factors such as acid rain, corrosive groundwater, and ambient loads. In this paper, slope durability is referred to as the evolution of slopes from corrosion to failure due to reduced mechanical performance. The study of slope durability encompasses weathering of outcropping strata, vegetation loss, steel corrosion, and fracturing of slope support structures. Typical patterns of intricate slope durability and consequent slope failures are illustrated in Fig. 1.Fig. 1 Slope degraded durability or instability patterns: (a) outcropping strata weathering; (b) steel corrosion; (c) instability with fracturing and failure.

In the context of natural slopes, slope durability issues primarily refer to the degraded properties of outcropping strata under environmental weathering conditions. Numerous researchers have concentrated on the weathering mechanisms of slope geology and their consequential influence on slope durability and stability. For instance, Cano et al.3 probed into the weathering properties of outcropping with various lithologies in the Flysch of Alicante, Spain, leading to the development of new weathering indices to predict the post-excavation weathering behavior of slopes. Bryson et al.4 employed the Jar Slake test to assess the durability of shale through time-dependent changes in electrical conductivity measurements. Ahmad et al.5 studied the classification of weathering and durability for igneous and metamorphic rocks using indices such as slake durability, impact strength, and micropetrographic parameters. Momeni et al.6 carried out comprehensive laboratory tests to investigate the effects of weathering on the durability and deformability properties of granitoid rocks. Recent research has focused on blending natural geological materials with artificial cementing agents to enhance weathering resistance. For example, Sivakumar7 applied biological cementation technology to slope protection, particularly under acid rain conditions. Gowthaman et al.8 investigated the effect of wet-dry cycles on the mechanical behavior of bio-cemented soil. Aside from weathering, other factors, such as environmental traffic loads, have garnered attention in practical applications. Xu et al.9, for instance, analyzed the weakening effect of dynamic traffic loads on the strength of slope rock masses through fatigue tests, indicating the significance of dynamic traffic loads for slope durability and stability.

For engineering slopes reinforced with support structures, the durability of these structures is a vital determinant of slope stability, complementing concerns regarding the durability of slope strata as previously mentioned. The predominant support structures for slopes are RC structures, encompassing anti-slide piles, anchor bolts, lattice beams, and retaining walls10. The degradation of structural durability is caused by the complex environment conditions, characterized by temperature and humidity variations attributable to climate change, deteriorating weathering conditions typified by rainstorms and acid rain, groundwater, traffic loads, external loads, and the stripping of vegetation10, especially the typical environmental conditions of slopes, marked by dry–wet cycles and high stress levels, can accelerate chemical or physical stress corrosion processes acting on support structures. As a consensus on the global scale, escalating levels of greenhouse gases precipitate global warming, leading to alterations in the atmospheric environment, including shifts in temperature, humidity, and CO2 concentration. These alterations can engender adverse weather phenomena, such as intensified rainstorms and acid rains. The corrosive properties of acid rain, coupled with CO2 concentrations, impart a deleterious impact on the durability performance of RC structures and augment the probability of corrosion and structural fracturing11–14. Consequently, the durability of in-service slopes supported using RC structures is susceptible to the influence of global warming. Huo et al.15 confirmed this point by proposing a diffusion equation that accounts for the attenuation of H+ concentration, validating the diffusion outcomes through accelerated corroded tests, and predicting the service life of slopes. Chen et al. (2009) conducted acid rain erosion simulation tests to study the impact of acid rain on anti-slide piles, thereby predicting the service lifespan of such structures. Overall, the current research has extensively analyzed structural durability. Future studies, however, require further clarification on the relationship between structural durability and slope stability.

This study examines the mechanisms governing slope durability and the consequential influence on slope instability. First, a quantitative theoretical model was developed to describe the degradation patterns of durability in slope support structures. Given that most support structures are RC variants, the developed durability model is established upon the classical durability theory applicable to RC structures. A physical model test was then executed to investigate the effects of degraded durability on slope stability. Finally, the developed durability model and the methodology for analyzing slope durability and instability were applied to a case study of an actual highway slope supported by RC structures, located in a coastal province in East China. The future instability features of the case slope were studied, demonstrating the relationship between slope-degraded durability and consequent instability.

Theoretical durability model of RC structures

Slope support structures typically consist of RC material, with their deterioration primarily characterized by concrete carbonation or neutralization (CN), rebar corrosion, corrosion-induced swelling, fracturing, and ultimate strength failure16,17. Factors influencing structural durability encompass atmospheric environmental conditions, stress states, material ratios, and construction practices. Critical parameters of structural durability include CN depth, velocity coefficients, and corrosion rate, among others. Among these, the CN depth is a crucial deterioration parameter that governs structural performance.

The existing body of literature provides various theoretical expressions for the CN depth14,18,19, supported by statistical study20 and experimental investigations21–23. In indoor and outdoor environments, the CN depth X is posited to be directly proportional to the square root of time t, as described by Fick’s first law20,24:1 X=kt

where k is the proportionality coefficient:2 k=C0·Πki

where C0 is a constant; Π represents the product of deterioration factors ki, i = 1 ~ n, and n denotes the total number of the factors.

As outlined above, the deterioration factors are linked to both environmental conditions and the inherent properties of the RC member. These factors include environmental temperature, relative humidity, CO2 concentration, concrete compressive strength, concrete stress at the CN depth, and construction quality, among other pertinent variables.

As per the Chinese standard (GB∕T 51355–2019) on durability assessment for existing concrete structures23,24, the following semi-empirical formula for the coefficient k is proposed:3 k=3kjkco2kPkskθkRHkcu

where kj is the corner correction coefficients (kj= 1.4 for corner positions of the concrete structure while kj=1.0 for non-corner positions);kco2 is the CO2 concentration influence coefficient; kP denotes the coefficient of curing and pouring;ks represents the stress influence coefficient;kθ=θ1/4, where θ is the ambient temperature in °C;kRH=1-RHRH1.5,whereRH is the relative humidity of the environment in %; and kcu=58/fcu-0.76, where fcu is the characteristic value of compressive strength for concrete cubes in MPa.

When the CN depth (X) exceeds the thickness of the protective layer, the deterioration initiates with rust corrosion, followed by swelling, cracking, and ultimately failure. This progression delineates the predicted service lifespan of concrete structures. The total service lifetime (tcr) comprises the carbonation time (ti), from the onset of carbonation to the initiation of rust corrosion, and the deterioration time (tc), from the onset of rust corrosion to the stage characterized by swelling and cracking of the protective layer. Consequently, tcr is expressed as follows:4 tcr=tc+ti

where tc=HcHfHdHTHRHHmtr, In this equation, Hc is the coefficient associated with the thickness of the concrete protective layer; Hf is the coefficient of concrete strength;Hd denotes the rebar diameter coefficient; HT is the ambient temperature coefficient; HRH is the relative humidity coefficient; Hm represents local environment influence coefficient;tr denotes the number of years in the deterioration period with all the coefficients equal to 1.0.

The carbonation time (ti) is expressed as:5 ti=ψc-x0k2

where c is the thickness of the protective layer; x0 denotes the carbonation residue; ψ is the correction factor proposed in this study to account for the slope geological environment;ψ can be assigned a value of 1.0 when the support structure is exposed to the atmospheric environment. Equation (5) is equivalent to Eq. (1) for ψ=1.0.

The carbonation residue x0 is an important parameter determining the initial time of rust corrosion. The following formula (6) for x0 is proposed25:6 x0=4.86-RH2+1.5RH-0.45·c-5lnfcu-2.30

The corrosion rate is a critical deterioration parameter in the context of RC structures, directly impacting their durability performance. Specifically, the corrosion rate can be categorized into two types. The corrosion rate, denoted as v0, corresponds to the period from the initiation of corrosion to the commencement of protective layer cracking. In contrast, the corrosion rate, v1, corresponds to the period after the occurrence of protective layer cracking. Existing cracks generally accelerate the corrosion process; however, some research reveals that corrosion products can seal the crack, thereby limiting the influence of crack opening on corrosion development26.

In this study, the calculation of the corrosion rate v0 is suggested as follows22,23:7 v0=a0Kc·m·0.75+0.0125θ·RH-0.452/3·C-0.675·fcu-1.8·

where the constant parameter a0 is evaluated as 7.35 in the atmospheric environment and suggested to be approximately 11.0—15.0 in the slope geological environment with acid underground water. Kc is the position coefficient, with Kc=1.6 for the corner position and 1.0 for non-corner positions. The variable m is the local environmental coefficient, which can be assigned a value between 2.5 and 4.5 in humid or acid rain environments.

The rate of corrosion v1 after swelling and fracturing is expressed as the following formula (8), wherein ψ′ is the correction coefficient to account for the slope geological environment.8 v1=ψ′(4.5-340v0)v0

Hence, the depth of corrosion δ is calculated as follows:9 δ=v0t-titi≤t≤tcrv0tcr-ti+v1t-tcrt≥tcr

Moreover, the crack width w can be described as:10 w=(δ-0.008cd-0.00055fcu-0.015)/0.086

Furthermore, the parameter of corrosion ratio η can be defined as the ratio of the corroded part to the whole rebar section:11 η=4δ/d-4δ2/d2

where d is the diameter of the rebar, and δ is the corrosion depth.

Existing studies have shown that the strength of the RC structure member and the bearing capacity degrade steadily as the corrosion ratio (η) increases. Therefore, the corrosion ratio is regarded as one of the crucial indicators for evaluating the durability performance of RC structures. Suppose a degraded index (Qi) with an initiate value Qi0, where i = 1,…,n, and n is the total number of the indices of RC structure. The degraded index (Qi) is then calculated as:12 Qi=ζiQi0

where ζi is the deterioration coefficient of the index Qi0. The coefficient ζi is the fitting result of durability indicators such as the corrosion ratio (η). The linear fitting relationship is usually expressed as ζi=1-λiη, where λi is a fitting constant. For example, the degraded yield strength (fy) of corroded rebars with an initial value fy0 has the following fitting relationship during certain accelerated durability tests, as described by Eq. (13). Other indices, such as the elastic modulus and the elongation rate, may follow similar degradation laws 27–29.13 fy=fy01-0.029ηwithη≤5%

Since the durability degradation law of RC structures may be factually complicated and nonlinear, other durability indicators may be included besides the corrosion ratio η, such as the non-uniform distribution mode or corrosion locations30. For simplicity, this study intends to use Eq. (12) with the fitting coefficient ζi, which is chosen to calculate degraded performance indicators of the slope RC support structures.

Model test for slope durability and stability

Background of the prototype slope

To investigate the impact of degraded durability on slope stability, this study carried out a physical model test comparing the behaviors of a typical prototype slope with those of a weakened counterpart, which was projected to represent the slope’s conditions over the next 30 to 50 years. In this model test, both slopes were subjected to several environmental deteriorating factors. For modeling convenience, a prototype slope with a steep angle of 45° to 55°and a height of 55 m was simulated. The slope strata cover two types of rock or soil layers: The Moderately Weathered Tuff (MWT) layer and the upper Strongly Weathered Conglomerate (SWC) layer, which has a thickness of 40 m and a dip angle of 30°. To reinforce the slope, anchoring measures were implemented, and an artificial layer of Sprayed Concrete (SC) with a thickness of 0.2 to 0.4 m was applied to cover the slope surface.

Predicting the future deteriorative state of the slope is challenging due to the complexity and uncertainty involved. Therefore, the weakened slope over the next 30 to 50 years is assumed to have indefinite degrading coefficients. Some existing research has assessed the degree of deterioration in weathering rocks or RC structures under the impact of environmental factors such as ordinary rainwater, acid rain, cyclic loading and unloading31. These studies can serve as references for selecting degrading coefficients. In this model, the degrading coefficients for the MWT, SWC, and SC layers are assumed to be approximately 0.6 to 0.9 over the next 30 to 50 years. The related prototype slope parameters are listed in Table 1.Table 1 The prototype slope parameters in different states.

Service states	Outcropping layers	Unit Weight/kN/m3	Compression strength/MPa	Elastic Modulus/GPa	Poisson’s Ratio	Cohesion/kPa	Internal Friction Angle/°	
Current state	MWT	23	105.9	58.9	0.21	1180	40	
SWC	25.4	52.06	16.5	0.32	300	38	
SC	21	22.5	24.5	0.2	1980	51.3	
Future state	MWT	21.74	95.31	53.01	0.21	1062	37.1	
SWC	24.34	41.65	13.2	0.33	240	32.0	
SC	19.23	13.5	14.7	0.2	1188	36.8	

Test scheme and implementation

Model slope design

In the model test, the similarity ratio of the model slope to the prototype one is set at 1:50, and the dimensions of the model box are 1.5 m × 1.5 m × 1.8 m. Figure 2a shows the model box, which can contain two model slope entities of the same size. Zone A is designated for the standard model slope, while zone B is designated for the weakened slope. Figure 2b displays the specific dimensions of the model slope: The bottom of the model slope measures 0.75 m × 1.5 m; the top measures 0.75 m × 0.8 m; the height is 1.1 m; and the slope includes steps with heights 0.3 and 0.7 m, and a step width of 0.05 m. The anchors are arranged in a 3 × 3 grid with a dip angle of 35°. The simulation materials used are copper tubes with an outer diameter of 15 mm and a wall thickness of 1 mm for slope A, and aluminum tubes with an outer diameter of 16 mm and a wall thickness of 1 mm for slope B. The physical model is shown in Fig. 2c.Fig. 2 Design diagram and physical entities: (a) model box; (b) model slope; (c) physical model.

According to the principle of similarity, the model slope parameters are determined as depicted in Table 2, where the degrading coefficients are consistent with those adopted for the prototype slope in Table 1.Table 2 The model slope parameters.

Service states	Covering Layers	Unit Weight/kN/m3	Compression Strength/MPa	Elastic Modulus/GPa	Poisson’s Ratio	Cohesion/kPa	Internal Friction Angle/°	
Current State	MWT	23	2.12	1.18	0.21	23.6	40	
SWC	25.4	1.04	0.33	0.32	6	38	
SC	21	0.45	0.49	0.22	39.6	51.3	
Future state	MWT	21.74	1.91	1.06	0.22	21.24	37.1	
SWC	24.34	0.83	0.26	0.33	4.8	32.0	
SC	19.23	0.27	0.29	0.23	23.76	36.8	

Material mix ratio

To create the model slope equivalent materials listed in Table 2, a mixture of certain cementitious materials and aggregates of various particle sizes is employed. These materials include fine sand, barite powder, gypsum, cement32. A certain amount of water is accordingly added. The specific material mix ratios are determined through orthogonal testing. The final mix ratio for the model slope equivalent materials is presented in Table 3.Table 3 The mix ratio result for equivalent materials.

Service States	Covering Layers	Cement	Gypsum	Sand	Barite Powder	Water	
Current state	MWT	12	6	58	10	14	
SWC	8	3	64	11	14	
SC	6	4	72	6	12	
Future state	MWT	10	4	63	9	14	
SWC	7	3	66	11	13	
SC	6	2	74	6	12	

Test and monitoring

The simulated durability factors included heavy or acid rains, salt spray, cyclic loading and unloading. Figure 4 illustrates a simple artificial rain system, which can adjust the simulated rainfall by varying the pump power. In the acid rain and salt spray simulation, an acidic solution with a pH of 2.0 and salinity of 1% to 3% was mixed, including Na2SO4, NaCl (concentration of 10%), and HNO3. Figure 3 also shows the weights exerted on the slope crest; cyclic loading and unloading were simulated by stacking and removing the weights. The magnitude of load per cycle is 10 kPa, and the total load was applied 4 ~ 5 times, i.e., 40 to 50 kPa. The entire test procedure was designed with 3 different stages:(a) Test stage I: Stacking and unloading over 3–4 days (d);

(b) Stage II: Heavy rainfall for 7d, followed by natural air drying for 7d, and then restacking and unloading over 3-4d;

(c) Stage III: Acid rain and salt spray for 7d, natural air drying for 7d, and then restacking and unloading over 3-4d.

Fig. 3 Durability factor simulations: artificial rain system and load weights.

As introduced above, slopes A and B were subjected to different influencing factors during the three testing stages, resulting in a steady degradation of their mechanical and durability performances. Additionally, the responses of slope A during testing may be less pronounced than those of slope B, because the latter is presumed to be in a deteriorated state. To verify this hypothesis, detection and monitoring of the effects of durability factors were carried out. The detection and monitoring efforts mainly include changes in CN depth on the SC layer of the model slopes, as well as displacement and deformation changes. The former changes reflect the model slopes’ durability performance, while the latter changes indicate slope stability capacity.

Sampling points for examining CN depth were arranged at different heights of 0.3, 0.7, and 0.9 m on the slope surface, numbered A1, A2, and A3 for slope A, as shown in Fig. 4a and B1, B2, and B3 for slope B. The CN depth was principally detected using a 1% to 2% phenolphthalein alcohol solution. Displacement monitoring points were also arranged, as shown in Fig. 4a, with points S1 to S17 on the surface of slope A The monitoring points for slope B were likewise numbered. Displacement data were acquired by a Digital Image Correlation (DIC) device, as displayed in Fig. 4b. Moreover, a strain testing instrument was used to monitor the anchor tensile deformation behavior for slope reinforcement.Fig. 4 The slope surface sampling or monitoring points: (a) point layout; (b) the DIC device.

Analysis of test data

The CN depth

The capacity of slope support structures to resist Carbonation or Neutralization (CN) is contingent on the inherent properties of structural materials and ambient conditions. This capacity decreases with the carbonation or neutralization age, rendering the CN depth an important indicator of CN resistance capacity33. Table 4 illustrates the color changes of the sampling points on different test dates. It demonstrates that the red color inside the sampling holes became lighter after approximately four weeks of testing, indicating an increased CN depth. Furthermore, the holes on slope A displayed a darker red color compared to those on slope B, as the latter was designed to possess weaker durability performance.Table 4 The sampling points color changes.

Test date/m/d	Slope type	Sampling point number	
#1	#2	#3	
4/23	A				
B				
5/25					
				

Figure 5 presents all the test results on CN depths over the test dates. It reveals that the CN depth of the SC covering layer increases steadily with the duration of the test, reaching a maximum magnitude of approximately 24 mm within 40 days of CN penetration time for the simulated SC layer. Slopes A and B possess the average CN depth increase rates of 0.35 and 0.52 mm/d, respectively, the later is evidently 49% lager than the former. The significant discrepancy in the rates between slopes A and B can be attributed to the inferior material properties of slope B, rendering it more susceptible to rapid carbonation.Fig. 5 The CN depth variation over date.

To quantify the CN velocity indicator k in Eq. (3), the input parameters for slope A were established as follows: kco2=1.5;θ=19.2∘C;RH=0.8;fcu=2.0 MPa, with all other parameters set to 1.0. This resulted in a durability indicator k=38.05mm/a12 (or 1.99 mm/d12). Hence, the theoretical result for CN depth X of slope A can be obtained using Eq. (1), as shown by curve A0 in Fig. 5.

For slope B, the strength value fcu was degraded to 1.3 MPa, causing the indicator k to increase by 55% compared to slope A, whereas all other input parameters remained the same as those for slope A. The theoretical indicator result for slope B is displayed by curve B0. It is evident that curve A0 approximated the average value of related test curves from A1 to A3, while curve B0 was slightly higher than the average value of test curves B1 ~ B3 in the early test phase. Overall, this demonstrates that the proposed theoretical solutions are in good agreement with the test results.

DIC displacements

The displacement data results collected by the DIC device at various testing stages are discussed below:

At test stage I, the horizontal and settlement displacement curves of slope A under increasing loads are presented in Fig. 6a–d. These figures reveal that the total displacement remains relatively minimal and shows an irregular pattern of variations. Specifically, the horizontal displacement (the left Y-axis) fluctuates within a narrow range of − 0.2 to 0.2 mm, and the settlement displacement varies from − 0.25 to 0.18 mm. These results suggest that the loads exert a negligible impact on the deformation of slope A.Fig. 6 The model slope displacements at stage I: (a) ~ (b) slope A horizontal displacements; (c)-(d) slope A settlement displacements; (e)-(h) corresponding displacements of slope B.

In comparison, the displacement results of slope B are shown in Fig. 6e ~ h. These results indicate that the horizontal Y displacements at the monitoring points increase steadily with the applied loads, culminating in a maximum total Y displacement of approximately 0.8 mm, about 4 times the displacement observed in slope A. Furthermore, the settlements (Z direction) demonstrate a significant reduction, with the maximum displacement increment per load reaching approximately 0.3 mm and the overall magnitude peaking at 1.6 mm, about 8 times the displacement in slope A. This study indicates a pronounced downward sliding trend in slope B when subjected to increasing loads on the slope crest. Additionally, the rebound magnitude after unloading was maintained between 0.15 and 0.23 mm, suggesting the presence of substantial plastic displacement.

The comparative analysis of the displacement traits of slopes A and B elucidates that slope B undergoes more pronounced and consistent displacement changes. This comparison also shows that slopes with weakened durability exhibit reduced stiffness and stability.

At test stage II, the effects of heavy rain on the displacement outcomes for slopes A and B were assessed, with the results illustrated in Fig. 7. The data indicate varying reductions in displacement at all monitoring points on both slopes, with the effect being particularly significant for the deteriorated slope B.Fig. 7 The model slope settlement displacements at Stage II: (a) ~ (b) Slope A; (c)-(d) Slope B.

For Slope A, the maximum settlement experiences a decrease from − 0.43 to 0.13 mm, representing a reduction rate of − 72% compared to the previous stage. The displacements recorded at monitoring points S1 through S8 on the upper portion of the slope are more substantial than those at points S9 to S17 on the lower portion. These results demonstrate that the upper portion of slope A is more susceptible to the effects of simulated heavy rain.

Conversely, for slope B, the maximum displacements recorded at the monitoring points witness a progressive reduction with incremental loads, particularly notable during the fifth loading step. The peak displacement reached approximately − 2.4 mm, decreasing by − 50% from the − 1.6 mm recorded at the previous stage. The comparative study of the results for slopes A and B indicates that heavy rainfall significantly impacts slope stability, especially for slopes with degraded durability.

At stage III of this study, the effects of acid rain and saline spray on slope stability were analyzed. Monitoring points on the upper portion of the slope were identified to possess an enhanced sensitivity to external loads compared to their lower portion. Consequently, only data from monitoring points S1 through S9 were employed to assess the displacement variations.

As shown in Fig. 8, the maximum displacement observed at the monitoring points of slope A diminishes to values below − 0.5 mm, representing an increase rate of − 16% compared to the previous stage. For slope B, the maximum displacement does not exceed − 3.1 mm, marking an increase of − 30% from the value recorded at the previous stage.Fig. 8 The model slope settlement displacements at stage III: (a) points S1-S8 of slope A; (b) points S1-S8 of slope B.

The investigation shows that the stiffness and stability of the model slopes were marginally affected by acid rain and salt spray factors. This observation is attributed to multifaceted constraints, including but not limited to similar material properties, experimental duration, and boundary conditions. Despite these limitations, it is deduced that in the state of durability degradation, the weakened slope B exhibits a more pronounced sensitivity to environmental factors than the normal slope A.

The procedure of slope durability and stability analysis

In examining slope instability precipitated by durability degradation impacts, it is essential to conduct a basic finite element numerical modeling for slope stability analysis. In this model, the slope stratum domains and support structure entities are meshed with different types of finite elements, which are assigned the corresponding material property parameter values at the current time. External loads and boundary conditions are also input into the model.

For geological materials, the elastic–plastic constitutive model with the Mohr–Coulomb strength criterion is recommended, and the general elastic constitutive model is used for structural materials. The key material parameters include natural gravity, cohesion, friction angle, elastic modulus, Poisson’s ratio, and the ultimate bearing capacity of anchor bolts. By conducting a static analysis, the slope stability results are obtained, including the safety factor deriving from the Strength Reduction Method (SRM), stress and deformation fields, internal forces within the structures, and potential modes of landslide failure.

To simulate the slope stability status in future service years, all material property parameters must be corrected according to the conditions of weathering or durability degradation. There are two kinds of correction coefficients: the slope rock and soil weathering coefficients, denoted by λi, where λi <1 and i ranges from 1 to n; and the structural durability degradation coefficients describing the carbonation and corrosion behaviors, denoted by ζj, where ζj<1 and j ranges from 1 to m. Here, n and m represent the total numbers of coefficients. Subsequently, the slope rock and soil parameters, denoted by Ai, are adjusted using the formula λiAi. Likewise, the slope support structure parameters, Bj, including the anchor bolt’s ultimate bearing capacity, are adjusted using the formula ζjBj . These corrected parameters then replace the original ones, i.e.,Ai and Bj, in this numerical model.

The procedure of analyzing slope durability and stability is illustrated in Fig. 9. For validation of the predictive simulation results, it is recommended to carry out field investigation and monitoring work alongside an effective durability model test as presented in the previous section.Fig. 9 The analysis process on slope durability and stability.

Case study

Background

The case study focuses on a highway slope located in a coastal province in China, as shown in Fig. 10a. Figure 10b presents the geological profile of the slope, which naturally extends to a height of 54 m and consists of 3 differing rocky strata. The properties of these strata are listed in Table 5.Fig. 10 The case slope profile: (a) on-site scenario; (b) design profile diagram.

Table 5 The mechanical properties of slope rocky strata.

Rock strata	Natural unit weight/kN/m3	Angle of internalfriction/°	Cohesion/kPa	Elastic modulus/GPa	Poisson’s ratio	
Silty clay with gravel sand	18.1	18.2	26.0	6.2	0.27	
Fully-weathered conglomerate ③1	20.0	23.0	30.5	11.4	0.24	
Strongly-weathered conglomerate ③2	23.5	36.0	180.0	54.0	0.21	
Moderately-weathered conglomerate ③3	25.4	41.0	700.0	230.0	0.19	

To protect the cut and exposed surface of the slope, an artificially constructed rubble-facing mortar wall was utilized. Local climatological data indicate that the project site experienced an annual average temperature of 19.2 ℃, a figure subject to potential increases due to global warming. Additionally, the ambient humidity level was recorded at 80%.

To enhance the stability and durability of the current slope, the following reinforcement measures were implemented: (1) A retaining wall was constructed with mortar rubbles backfilled at the toe of the slope. The wall comprised concrete C25 and featured a 40 mm thick protective layer alongside rebars in an outer diameter of 14 mm and MU10 mortar rubbles.

(2) Pre-stressed grouted anchor bolts (#1 to #4) were installed. These bolts have lengths ranging from 15.0 to 18.0 m and an outer diameter of 100 mm. They comprised pre-stressed threaded rebars in an outer diameter of 32 mm with a minimum tensile strength of 785 MPa. The grouting material employed is concrete C30, with a tensile strength of 3.2 MPa. The estimated ultimate bearing capacity for each anchor bolt is approximately 654 kN.

(3) Lattice beams were constructed using concrete material C25, characterized by a cross-sectional dimension of 400 mm × 400 mm. The main rebars were arranged in a 2 × 3 configuration with 8 mm diameter stirrups and 35 mm thick protective layers. The installed anchor bolts were integrally bonded with the lattice beams at their nodes. Figure 11 shows the design map of the reinforcement structures. Table 6 presents the specific mechanical parameters associated with the support structures.Fig. 11 The designed reinforcement diagram.

Table 6 Mechanical parameters of the support structures.

Types of support structures	Unit weightkN/m3	Angle of internal friction /°	Cohesion /MPa	Elastic modulus/GPa	Poisson’s ratio	
1:Retaining wall	23.0	59.2	3.1	28.0	0.20	
2:Mortar rubbles	22.0	38.0	0.7	1.5	0.22	
3:Anchor bolts	28.6	/	/	47.5	/	
4:Lattice beams	25.0	59.8	3.8	30.0	0.18	

The subsequent sections examine the projected alterations in the structural durability of the slope, as per the theoretical durability model previously established. This analysis also addresses the implications for slope stability. Initially, the investigation focuses on the stability of the current slope. Thereafter, the analysis evaluates the slope’s stability following reinforcement and during typical serviceability periods.

Durability analysis

The theoretical model presented in Section. “Theoretical Durability Model of RC structures” was employed to evaluate the durability performance of slope support structures. The input parameters related to durability were primarily determined as per the Chinese standard (GB∕T 51355–2019) for durability assessment of existing concrete structures, as outlined by Luo and Niu24 and Wang et al.34. Table 7 lists the specific values for the parameters in the model based on environmental conditions of the slope, including temperature and humidity.Table 7 The input parameter values in the proposed durability model.

Parameters structural types	kj	kco2	ks	kP	θ/°C	RH	fcu/MPa	c/mm	m	Hc	Hf	Hd	HT	HRH	Hm	tr/a	Kc	ψ′	λ	
Retaining wall	1.4	1.5	1.1	1.2	19.2	0.8	25	40	4.0	2.9	1.26	1.38	1.21	0.97	0.89	2.9	1.0	2.0	1.05	
Anchor bolts	1.4	1.6	1.5	1.2	19.2	0.8	30	34	4.5	1.9	2.08	1.05	1.2	1.04	0.85	2.1	1.6	5.0	1.05	
Lattice beams	1.4	1.5	1.1	1.2	19.2	0.8	25	35	4.0	2.4	1.39	1.27	1.21	1.04	0.95	1.9	1.6	2.0	1.05	

Utilizing the developed theoretical model with these parameters enabled the calculation of durability indicators for different support structures, as shown in Table 8. The durability indicators comprise the CN velocity coefficient (k), residue depth (x0), corrosion rates (v0 and v1), carbonation time (ti), deterioration time (tc) and the total service lifetime tcr, as recommended by Cho et al.35. The estimated total service lifespans for slope lattice beams, anchor bolts, and retaining walls are 30, 23, and 47 years, respectively. Given the general 30-year design service life of highways, the slope support structures were predicted to sustain 100%, 76.7%, and 156.7% of the designed service life, respectively.Table 8 The durability indicator results.

Parameters Structural types	kmm·a-1/2	x0/mm	v0/mm.a-1	v1/mm.a-1	ti/a	tc/a	tcr/a	
Retaining wall	4.15	17.19	0.007 5	0.029 3	30.27	17.02	47.30	
Anchor bolts	4.25	17.07	0.008 1	0.070 6	15.85	6.89	22.75	
Lattice beams	4.15	14.74	0.009 8	0.022 8	23.89	6.64	30.54	

To ascertain the deterioration coefficients (ζi) for slope RC structures, it is essential to evaluate the corrosion state of steel bars. The computation process includes the following steps: (a) Determining the dimensions of rebars and the thickness of their protective layers (c); (b) Calculating the corrosion rates (v0 and v1) and the corrosion depth (δ); (c) Assessing the corrosion ratio (η); (d) Determining the deterioration coefficients (ζi), as detailed in Eqs. (9)-(12).

Table 9 gives the deterioration coefficients for the slope structures over various projected service years. Given the complexities in quantifying the durability impacts of the surrounding environment on slope rock and soil, the weathering coefficients (λi) are challenging to determine. For simplicity, the coefficients (λi) for the internal friction angle, elastic modulus, and cohesion are assumed to be 1.2 ‰, 1.5 ‰, and 3.0 ‰ per year, respectively, assuming that other physical and mechanical parameters remain constant.Table 9 The predicted corrosion depth and deterioration coefficient results.

Serviceyears (/a)	Retaining wall	Anchor bolts	Lattice beam	
δ /mm	ζi	δ /mm	ζi	δ /mm	ζi	
0	0	1.0	0	1.0	0	1.0	
10	0	1.0	0	1.0	0	1.0	
20	0	1.0	0.034	0.996	0	1.0	
30	0	1.0	0.568	0.927	0.06	0.984	
40	0.073	0.978	1.275	0.839	0.281	0.928	
50	0.207	0.939	1.981	0.756	0.509	0.871	
60	0.499	0.856	2.687	0.677	0.737	0.815	
70	0.792	0.776	3.394	0.602	0.965	0.762	
80	1.085	0.700	4.100	0.531	1.193	0.710	

Stability analysis

The finite element numerical model is illustrated in Fig. 12a for the current slope scenario, and in Fig. 12b and c for the slope with different reinforcement configurations. Notably, Fig. 12b and c show anchor bolts positioned at various positions within the slope’s rock layers. Based on the Strength Reduction Method (SRM), the safety of factors for slope stability are 1.112, 1.213, and 1.678, respectively. The increased safety coefficient for slope stability achieved with the reinforcement scheme in Fig. 12c demonstrates a more effective improvement in stabilizing slopes compared to the scheme in Fig. 12b. Hence, the design scheme in Fig. 12c can be considered optimized, as it fully meets the stability requirements.Fig. 12 The numerical models of the slope case: (a) no reinforcement; (b) original reinforcement scheme; (c) optimized reinforcement scheme.

Figure 13 indicates the results of the axial internal force in the anchor bolts for the optimized reinforcement scheme. This figure reveals a more uniform distribution of axial internal forces in anchor bolts #1 and #4 compared to anchor bolts #2 or #3. The maximum axial internal forces in anchor bolts #1 ~ #4 are 61%, 85%, 64%, and 71% of the initial ultimate bearing capacity of 650 kN, respectively. Notably, anchor bolt #2, situated beneath the slope’s interception ditch, bears the highest axial internal force among all the bolts, suggesting it is theoretically the first to fail, followed by anchor bolt #4, located on the side of the retaining wall at the slope toe.Fig. 13 The axial force results of anchor bolts.

Incorporating the durability deterioration coefficients of slope performance from the previously discussed durability analysis enables the prediction of future slope stability with degraded slope parameters. Figure 14 illustrates the variation in the ultimate bearing capacity of anchor bolts and their actual axial internal forces over future service years. Initially, the ultimate bearing capacity is expected to remain unchanged for the first two decades because no corrosion-induced reinforcement degradation is anticipated. However, it will subsequently decrease linearly an approximate loss rate of 7.5‰ per year, suggesting the impacts of degraded durability.Fig. 14 The axial internal forces or the ultimate bearing force over service years.

Regarding the actual axial internal forces in the bolts, the initial order of magnitude is #2, #4, #3, and #1. Remarkably, the axial force in bolt #1 tends to increase steadily, approximately 5‰ per year, over the first two decades, indicating a trend toward upper slope instability due to durability degradation. In the following decade, bolt #2 is predicted to fail as its actual axial force surpasses its diminishing ultimate bearing capacity. Without intervention, this will render bolt #1 overstressed and out of service due to the collapse of bolt #2. After an additional three decades, bolts #3# and #4# are also expected to fail when the anchor bolts’ ultimate bearing capacity significantly decreases due to the continuous impacts of durability degradation.

This predictive analysis assumes predetermined environmental conditions of the slope and specific durability degradation patterns, excluding the possibility of unforeseen incidents or the implementation of preventive measures.

Table 10 presents the slope safety factor results using the SRM over the forthcoming decades. Without considering the potential fracture of anchor bolts, the safety factors exhibit a steady decline at a loss rate of 2.3 to 2.4 ‰ per year, primarily attributed to the destabilizing effect of rock and soil weathering on the slope. The failure of anchor bolts is defined as the point at which their actual axial internal forces exceed their decreasing ultimate bearing capacity. Based upon this definition, the safety factor of the slope is expected to experience a significant reduction of 31% of its total value when the failure of anchor bolts occurs after approximately 30 years. Thereafter, the safety factor will continue to decrease slowly at a rate of 2.3‰ per year. In Table 10, the character ‘N’ denotes the scenario without considering the failure of anchor bolts, while the character ‘Y’ represents the scenario with the consideration of anchor bolt failure.Table 10 Results of the slope safety factors using SRM.

Service years (/a)	Slope safety factor	Service years (/a)	Slope safety factor	
N	Y	N	Y	
0	1.664	1.664	50	1.477	1.05	
10	1.628	1.628	60	1.438	1.006	
20	1.588	1.588	70	1.403	0.972	
30	1.554	1.554	80	1.360	0.944	
40	1.518	1.066				

Conclusions

This study has extensively investigated the durability issues of slope support structures, evaluating the effects of complicated environmental factors on the durability of slope support structures and how degraded durability can lead to slope instabilities. The key findings of this research are outlined below: The developed theoretical durability model for slope RC structures quantitatively delineated the structural failure processes involving carbonization or neutralization, corrosion, and fracturing. In this model, durability indicators such as CN velocity or depth and corrosion ratio were proposed to record the structural durability state. The degradation coefficient for degraded structural, parametric properties was obtained according to linear degradation laws. The proposed model test verified that the theoretical solutions for CN velocity and depth of the sprayed concrete covering layer agreed well with the test results. The case study suggested a potential failure of the RC support structures within 30 to 60 years under specific environmental conditions. Parameter sensitivity analyses of the theoretical durability model are essential for further research.

The comparative analysis evaluated a standard slope (A) against a durability-degraded slope (B) subjected to environmental factors such as heavy or acid precipitation, salt spray, and cyclic loading and unloading. The data for CN depths and DIC displacements demonstrated that the durability and stability performances of the model slopes degraded steadily over the test period when subjected to specific environmental impacts. The theoretical CN velocity of the simulated slope SC layer was 0.1 mm/d for slope A, with a CN penetration time of almost 40 days. In slope B with material degradation coefficients of 0.6 to 0.9, the CN velocity increased by 55% compared to slope A, and the CN penetration time was correspondingly shortened. These solutions were validated by the test results. The DIC displacements of slope A increased when loading after interval impacts by simulated rainfall, and acid rain among others, while the displacements of slope B were 4 to 8 times those of slope A, showcasing heightened sensitivity to environmental impacts and potential slope instability compared to its standard counterpart.

The case study involving future durability and stability concerns of slopes has yielded acceptable predictive outcomes regarding weathering and durability deteriorations in forthcoming service years. These results justified the applicability and effectiveness of the proposed theoretical durability model for RC structures in slopes. Finite element modeling for slope stability indicates a gradual decrease in the slope safety factor at a rate of 2.3 to 2.4 ‰ per year. Additionally, the slope anchor bolts were anticipated to undergo corrosion within 20 years, and fail within 30 years, with an average rate of 7.5‰ per year in the ultimate bearing capacity. Furthermore, Long-term field monitoring and detection are crucial for validating the obtained analysis results. Accordingly, timely maintenance and reinforcement interventions are proposed to ensure slope stability.

Author contributions

Ma Y.Z. carried out the research on durability theory, and mainly wrote the manuscript text; Ye W.Y. preformed the model test, and prepared figures and tables; Qing C.G. carried out the test data analysis work; Wang H.J provided the design scheme for the investigated slope case; Li C.G. preformed the slope stability analysis with numerical tools; Ding Z.X. assisted in designing the research overall framework and paper writing.

Funding

This research was jointly supported by the Open Fund Project of the Chinese State Key Laboratory of GeoMechanics and Geotechnical Engineering (Grant No. Z020020).

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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