Reliability Analysis of the Porto Velho Pier - Rio GrandeRS ThroughLimit Equilibrium and Finite Element Method
Seatific 2024, Vol. 4, Issue 2, pp. 5; doi.org/10.29187/2792-0771.1032
Abstract
Keywords: Porto Velho Pier in the city of Rio Grande - RS; Stability Analysis; Seawall; Reliability Analysis; Limit Equilibrium Method; Finite Element Method.
1. Introduction
In traditional deterministic analyzes of slope stability, uncertainties related to the problem are commonly neglected. A better understanding of these uncertainties, whether intrinsic or epistemic, has become an object of great interest in geotechnical research in the last two decades through the use of reliability analyzes to assess the probability of slope failure (Dyson and Tolooiyan 2019, Hostettler et al. 2019, Li et al. 2019). According to Vanmarcke (1977), there are many uncertainties involved in slope engineering, especially with regard to the spatial variability of geotechnical properties and the uncertainties associated with deterministic calculations to estimate the safety margin of slope stability. The assessment of slope stability is affected as the variability
of the soil affects the analysis systematically or randomly. Thus, geotechnical variability is complex, resulting from various sources of uncertainty, such as the mechanical and physical properties of soils and rocks, which are naturally dispersed, plus inaccuracies in transformation models and uncertainties related to human error (Phoon and Kulhawy 1999, Yang et al. 2020, Ansari et al. 2021). System failure occurs when the slope slides along a critical surface. Thus, slope reliability analysis is defined as a system reliability analysis problem in which the overall failure probability (or system failure probability, Pf) of a slope, considering several potential sliding surfaces, is of interest and is greater than the failure probability of any individual potential sliding surface as a result of system effects (Diltlevsen 1979, Cho 2013, Zeng et al. 2015, Metya et al. 2017).
*Corresponding author. *E-mail address: antiqueiraneta@gmail.com Published by Yıldız Technical University Press, İstanbul, Türkiye This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Several authors have been studying the use of reliability analysis. Haldar (2019) presents an overview of foundation design methodologies highlighting methodologies such as FORM (First Order Reliability Method) and the Monte Carlo method. The author highlights the importance of foundation designs based on reliability, then considering the variability and spatial correlation of the soil, leading to rational design decisions. Beloni et al. (2017) analyzed the geotechnical reliability of the pile foundation of a port pier located in the city of Rio Grande, RS. In their study, the probabilistic distribution of the bearing capacity was assessed using Bayesian theory concepts. The authors also emphasized the importance of reliability studies for accurately evaluating the safety of any engineering project. In 2000, the Porto Velho pier, in Rio Grande/RS (Fig. 1), began to show excessive displacements towards the channel and the formation of cracks in the sidewalk, possibly related to the rupture of the structure (Fig. 2). The objective of this work is, therefore, to analyze the stability of the Porto Velho pier through a reliability analysis, as well as to infer the evolution of the safety factor as an erosion process occurs at the base of the pier structure. The stability analysis will consist of using two approaches, the limit equilibrium method (LEM) and the finite element method (FEM). From these, a reliability analysis will be carried out using the Monte Carlo method (MMC) and the first order first moment method (FOSM). This study will enable the understanding of the development of port structures' stability, thus providing a basis for decision-making regarding future mitigation projects for the pathological manifestations observed along the structure. Furthermore, the study may also contribute to future investigations on stability analysis, using reliability methods in gravity-type port structures.
2.1. Deterministic method
The most common techniques for slope stability analysis are deterministic. These techniques assume the constancy of the variables used in the calculation model. This methodology usually uses the Limit Equilibrium Method (LEM) to determine the minimum safety factor for the slope failure surface (Ortigão and Sayão, 2004). According to Cho (2010), slope stability problems are commonly analyzed using the LEM. The failing soil mass is divided into a number of vertical layers (slices) to calculate the factor of safety, which is defined as the ratio of the shear strength stresses to the mobilized shear stresses, so the static equilibrium between the slices and the mass as a whole are used to solve the stability problem. However, all slice methods are statically indeterminate and, as a result, require assumptions to solve the problem. Furthermore, LEM does not consider the stressstrain behavior of the materials involved in the analysis. Another technique for slope stability analysis is the Finite Element Method (FEM). According to Zaman et al. (2000) the FEM represents a powerful alternative approach to slope
stability analysis. This method is accurate, versatile and requires fewer a priori assumptions, especially regarding the failure mechanism. In this method, the changing stress-strain condition in the soil is considered, thus focusing mainly on the failure mechanism during a slope failure (Duncan, 2014). According to Griffiths and Lane (1999), in this method the shear strength parameters of the soil c' and tan ϕ' are reduced by a Strength Reduction Factor (SRF) until failure occurs (Equation 1). In this study, PLAXIS 2D (2016) software was used for stability analysis. It is understood that the value of the safety factor is the same as the value of the SFR at the time of failure.
2.2. Probabilistic method
Probabilistic procedures for slope stability analysis vary in their assumptions, limitations, ability to deal with complex problems and mathematical complexity. Most of them, however, fall into one of two categories: approximate methods (First Order Second Moment Method, Point Estimation Method) and Monte Carlo Simulation (El-Ramly, 2002). According to Da Re et al. (2001), reliability analysis provides a systematic method for evaluating the combined influences of uncertainties in the parameters that affect the factor of safety. Thus, probabilistic analysis evaluates the stability conditions of slopes, taking into account the errors associated with the nature of the problem and the variability of the characteristics of the slope and its soil. Through this analysis, the safety of a slope is characterized by the value of the safety factor (FS) based on average values corrected for probabilistic parameters, or by the value of the reliability index (β), which implicitly involves the behavior of a function of random parameters, which defines the state of safety of a slope. Thus, the reliability index describes the stability of the slope by the number of standard deviations separating the average factor of safety from its failure value, which is defined as 1. The reliability index can also be defined as a way of normalizing the factor of safety in relation to its uncertainty. Equation 2 shows the calculation for determining the reliability index (Abbaszadeh et al. 2011). (2) Where β represents the reliability index, E[FS] the expected value of the safety factor and σ[FS] the standard deviation of the factor of safety. Table 1 illustrates the relationship between the reliability index and the probability of failure, assuming a normal distribution for the safety factor. The variability of the parameters is given as a function of the coefficient of variation (COV). Duncan et al. (2014) defines the COV as the ratio between the standard deviation (σ) and the arithmetic mean (μ) of a sample, so this parameter designates the dispersion of the data in relation to the mean, and its result is expressed as a percentage (Eq. 3).
Figure 1. (a) Continental, state and municipal identification of the studied location; (b) Location of the Porto Velho Pier in the municipality of Rio Grande; (c) Identification of the study area. In the Monte Carlo Method (MMC), the stability of the structure is calculated by generating a large number of random data for the input variables (such as friction angle and cohesion), since the probability distribution of these variables is known. As the data is generated, stability is analyzed using deterministic methods, which also makes it possible to determine the measures of central tendency corresponding to the factor of safety, as well as the corresponding probability of failure (Cho, 2010). There are two important aspects to MMC. The first refers to the search for the critical surface for each set of randomly generated input data values, which involves significant computational effort, making it impractical. The way commonly used to resolve this difficulty is to take as the critical rupture surface the one obtained by the deterministic method, which is therefore independent of the values of the probabilistic analysis input data set (El Ramly, 2001). According to Li et al. (2013) the probability of system failure (Pf,s) is frequently calculated using a large, but finite, number of potential sliding surfaces. Let S1, S2, ..., SNs denote N_s potential slip surfaces that are considered in the limit equilibrium analysis of slope stability. Then, the slope can be considered as a series system consisting of Ns
components (i.e. S1, S2, ..., SNs). System failure occurs when any component (i.e. S1, S2, ..., SNs) fails. The probability of system failure (Pf,s) can be calculated using Equation 4.
Where Nt is the total number of simulations generated during the MCS; FSmin is the minimum safety factor value among the safety factor values for the Ns potential slip surfaces for a given set of random samples of uncertain parameters X (i.e., cu, c and ∅) involved in the slope stability analysis; and I{.} is the function indicator. For a given random sample I(FSmin<1) is taken as a value of 1 when FSmin<1 occurs. Otherwise, it is equal to zero. The slip surface with FSmin (i.e. critical slip surface) needs to be located between potential slip surfaces for each sample generated during MCS, and its correspondent is calculated using a deterministic slope stability analysis method, such as limit equilibrium methods (Duncan et al. 2014). The First-Order, Second Moment (FOSM) method is based on truncating the Taylor Series expansion function. According to Griffith (2007), this method provides analytical
Figure 2. (a) Displacement of the pier structure blocks. (b) Height variation along the pavement and pier crown blocks. (c) Displacement of the gravity wall towards the navigation channel. (d) Area where displacements are most intense. approximations for the mean and standard deviation of a parameter of interest, as a function of the mean and standard deviation of the various input factors, and their correlations.
Carlo method, so in order to optimize the computational demand, the software allows this analysis to be carried out in two ways, the Global Minimum and the Overall Slope.
Thus, the calculation is based on the variation in the FS caused by a small oscillation in the independent variables. The number of analyses required for this procedure is equal to n+1, where n is the number of independent values. Equation 5 shows this process.
In the Global Minimum type analysis, parameter variability is applied only to the critical rupture surface, while the Overall Slope analysis considers parameter variation for all rupture surfaces. This study will therefore focus on analyzing the Overall Slope analysis only.
Where V[FS] equals the variance of the FS, δFSii corresponds to the variance of the FS when the study variables are varied by δXi and V[Xi] means the variance of each of the variables (Xi).
Stability analysis using the FOSM method was carried out using SLIDE 6.0 and Excel software. First, deterministic analyses were carried out for each of the parameter variations required by the method and then the equations were used in Excel. Based on this methodology, it is possible to identify which parameters have the greatest influence on the safety factor.
2.3. Stability analysis using SLIDE 6.0 Software
The SLIDE 6.0 software uses LEM for global stability analysis, which can be either deterministic or probabilistic. The probabilistic analysis is carried out by applying the Monte
2.4. Stability analysis using PLAXIS 2D software
The analysis was carried out using PLAXIS 2D software, where, and the Safety Factors for each of the scenarios studied were calculated using the Finite Element Method. To do this,
Table 1. Expected performance level according to reliability index values and probability of failure
High Good Above average Below average Poor Unsatisfactory Dangerous
Probability of failure (Pf) 0.0000003 0.00003 0.001 0.006 0.023 0.07 0.16
a mesh study was first carried out. The main objective of this study is to identify the most optimized analysis model, i.e. the model that requires the least possible computational effort in order to provide the most assertive results possible. To generate the finite element mesh, it is necessary to divide the domain into several parts. For two-dimensional analysis, several triangles are generated to represent the simulation domain. In the general properties of PLAXIS 2D, the number of nodes to be analyzed per element triangle is defined. Where there is the option of using an analysis with 6 or 15 nodes per element, in this study the use of 15 nodes was adopted. Subsequently, the analysis began, in which it was identified that the fine and very fine meshes are the ones with the most assertive results. Table 2 shows the results of the mesh analysis. It can be seen that although the very coarse mesh does not show convergence on the safety factor, the difference between the result obtained by the very coarse mesh and the very fine mesh is less than 1%. Based on this analysis, it was decided to carry out all the analyses with the very fine mesh. Therefore, the modeling steps consisted of importing the cross section of the pier, as well as its geotechnical profile. Subsequently, the geotechnical parameters involved in the analysis were defined, as well as the boundary conditions, such as the water table level, for example. After defining the input data and generating the mesh. The construction stages of the model were defined, where phase 1 consisted of determining the initial stress field in the soil through the “k0 procedure”. The phase consists of the insertion of the pier structure, as well as the backfilling of the back area. In phase 3, the service and mooring loads are inserted, in this step the analysis of scenarios 1 and 2 is carried out. Subsequently, in phases 4 and 5, respectively, the analysis of the structure begins as a function of the erosive process at the base of the wall. Thus, characterizing scenarios 3 and 4 respectively.
Scenarios Analyzed
Through a historical search of the Port of Rio Grande's technical archives, in addition to the project for the structure of the Porto Velho Pier, 12 SPT test tests were found, all of
them of the Standard Penetration Test (SPT) type. Figure 3 shows the location of each of the tests. As mooring bollards 13 is the place where the pathological manifestations are most intense, it can be seen that this is also the place where there are the greatest number of tests. An analysis of the SPT test reports shows that they are very similar. When analyzing the SP1, SP1A and SP1B tests, it can be seen that they all have a layer of approximately 3m of sand where the SPT test is interrupted, once it reaches an impenetrable layer, because at these points where the test was carried out there is the rockfill structure at the end of the pier. Already the SP1C test goes into deeper layers, in this report it is possible to identify the other soil layers of the site, Figure 4 shows the result of the SP1B sounding report, which is similar to the SP1 and SP1A reports, and SP1C. Based on these tests, the geotechnical profile of the site was defined. By applying empirical equations that correlate the NSPT number with its properties, the average geotechnical parameters of the soil that makes up the Pier structure were defined. The rockfill parameters were the same as those used in the studies by Meirelles (2008). Therefore, for this study, a high variability of the geotechnical parameters will be considered, since previous studies have already analyzed how the COV influences the stability of the structure, where it was possible to observe that higher COVs present a greater probability of failure and lower reliability rates. Therefore, the standard deviations of each of the geotechnical parameters were calculated based on the definition of the coefficient of variation (Eq. 3). Table 3 illustrates the values adopted for each of the parameters according to the analyses carried out. It was possible to define the layout of the structure by reviewing the history of the works and the quay project. By overlaying this information with the results of the SPT tests, the crosssection of the Porto Velho pier was defined. According to Lacasse and Nadim (1998), most geological processes follow a normal or lognormal probabilistic distribution. A normal distribution was adopted for the random variables, in this case, the soil resistance parameters (undrained resistance and friction angle) as well as the specific weight. The other parameters were treated as deterministic, as in Johari (2019) (Table 4). Thus, an analysis of four scenarios is proposed, in which scenario 1 represents the original undamaged structure, considering only the variability of the soil and its respective parameters, and the rockfill protection soil is characterized
Figure 3. Geotechnical investigation (a) locations of the SPT test along the pier; (b) Identification of boreholes in front of mooring bollard 13. SPT: Standard Penetration Test.
Figure 4. Results of the soil profile of the tests SP1B and SP1C, carried out in front of mooring bollards 13. as compact sand. In order to understand how the rockfill protection soil influences the stability of the structure, it was considered that this soil is clay, so this new scenario was assigned as scenario 2. As observed in a diving survey carried out by SUPRG (2016), there was erosion at the base of the rockfill protecting the wall, so this situation was defined as scenario 3 for this study. Finally, scenario 4 is represented as an evolution of scenario 3, in which erosion of part of the quay's protective rockfill is observed. Figure 5 illustrates the section of the quay for the different scenarios analyzed.
After defining the soil parameters, the structure was modeled using SLIDE 6.0 and PLAXIS 2D software (Fig. 6). In the PLAXIS 2D software, the continuum was discretized into a mesh with 1726 ground elements 1726, 14286 nodes, the average size of the elements 2,139 m, maximum element size 4.694m and minimum element size 0.0448m. The dimensions of the structure were defined based on old projects found in the SUPRG technical collection. It is important to highlight that the dimensions of the structure remained the same for both the LEM and FEM analyses.
Table 3. Geotechnical parameters for the LEM Monte Carlo soil Parameters Average Soil 1: Fine sand Soil 2: Sand Soil 3: Clay Soil 4: Compact sand Soil 5: Filter (sand) Rockfill
γ (kN/m³) ∅ (°) γ (kN/m³) ∅ (°) γ (kN/m³) Su (kPa) γ (kN/m³) ∅ (°) γ (kN/m³) ∅ (°) γ (kN/m³) ∅ (°)
LEM: Limit equilibrium method; FOSM: First order first moment method; COV: Coefficient of variation.
Soil 1: Fine sand Soil 2: Sand Soil 3: Clay Soil 4: Compact sand Soil 5: Filter (unidentified material) Rockfill
4.1. Probabilistic analysis
Table 5 shows the evolution of the safety factor for each of the scenarios analyzed. It can be seen that scenario 1 had a safety factor greater than 2, very low failure probabilities and high reliability indices, and the same can be seen for scenario 2. As the analyses move on to scenarios 3 and 4, it can be seen that the safety factor values are lower than 2 and higher than 1.8, with a low probability of failure and a reliability index close to 3, which according to Table 1, characterizes an above-average reliability index. Thus, it can be seen that the safety factor decreases by 20% when the base of the rockfill is lost. It can also be seen that the value of the safety factor according to the probabilistic analysis is higher than the values found in the deterministic analysis, because the probabilistic analysis takes into account the variability of the materials. In order to understand how soil variability influences the safety factors found by the FEM, a probabilistic analysis was carried out. It is important to emphasize that this analysis was based on the safety factors calculated using the MohrCoulomb rockfill resistance criterion.
4.2. Finite element method
It should be noted that throughout the analysis, it was noticed that the values of the safety factors found were lower than the values of the safety factors found by the Limit Equilibrium Method. In order to understand the reason for this difference, a series of changes were made to the input parameters in order to identify which properties would significantly affect the safety factor: •
Change in the rockfill resistance criterion from Mohr Coulomb to Linear Elastic;
In view of all these changes, the only one that had a significant impact was the change in the rupture criterion. Table 6 shows a comparison of the safety factor values found considering the rockfill failure criterion as Mohr-Coulomb and Linear Elastic for the FEM. The table also shows the results of the deterministic LEM analysis. Figure 7 shows the failure surfaces of Scenarios 1, 2,3
Figure 5. (a) Scenario 1: Sandy soil at the base of the rockfill protection. (b) Scenario 2: Clayey soil at the rockfill protection base. (c) Scenario 3: Intermediate erosion process with loss of soil protecting the rockfill. (d) Scenario 4: Critical stage where erosion of the rockfill at the base of the structure occurs.
Figure 6. Dimensions of the PLAXIS 2D software model, the continuum was discretized into 1726 triangular elements with 15 nodes. and 4 using the Limit Equilibrium Method and the Finite Element Method, respectively. Based on these results, it can be understood that LEM is based on the premise that the safety factor is defined on the basis of the balance between the stresses and the stresses developed during the analysis. The FEM, on the other hand, considers the integration between the elements of the structure, so that they respond directly to the stress-strain relationship. Therefore, by adopting a linear elastic strength criterion for the rockfill, it is possible to see that the system behaves in a very similar way to that found in the FEM
analysis, since the linear elastic strength criterion gives the rockfill a very high rigidity, similar to what is reproduced in the SLIDE software, where the FEM analysis is carried out, in which the modulus of deformation of the materials is not taken into account, as well as the interaction between them. This analysis can be seen by comparing the results of the safety factors between the LEM and the FEM (Linear Elastic). It can be seen that the values are very close, and Figure 7 shows the rupture zones of the structure. It can be seen that the critical rupture surface of the LEM is very close to the plasticization zone of the FEM using the Linear Elastic model.
Method
LEM: Limit equilibrium method; DET: Deterministic; MMC: Monte Carlo method; FOSM: First order first moment method.
Table 6. Comparison of the results of safety factors for LEM and FEM FEM Scenarios 1 2 3 4
By adopting the Mohr-Coulomb strength criterion for the rockfill, there is a significant reduction in the safety factor. This is because in this analysis, the rockfill has a more realistic resistance criterion, so that the load related to the mooring effort causes a stress-strain relationship that has a direct impact on the value of the safety factor, which does not occur in the LEM analysis. It is therefore understood that the PLAXIS 2D software satisfactorily reproduces the stability analysis of the structure so that when similar analysis criteria are inferred,
FEM: Finite element method; FOSM: First order first moment method.
the results found are very close to those calculated using the SLIDE software. However, it should be noted that the analysis methodologies are very different, so that when a more realistic failure criterion is applied to the structure, the analysis results in safety factors that are lower than those calculated by LEM. This is because the FEM considers a wider range of parameters in its analysis. In addition, this method also takes into account the horizontal forces acting on the structure, which LEM does not. In order to understand how the variability of the soil influences the safety factors found by the FEM, a probabilistic analysis was carried out. It is important to emphasize that this analysis was based on the safety factors calculated using the Mohr-Coulomb rockfill resistance criterion.
Figure 7. Comparison of the rupture surfaces of the LEM and FEM. LEM: Limit equilibrium method; FEM: Finite element method.
method
Table 7 shows the results of the FOSM analysis. It shows that the factor of safety is lower than the factor of safety found in the FOSM method for the LEM. However, for both analyses it is possible to see that as the variability of the geotechnical parameters is applied, there is a reduction in the reliability index. When relating the results of Figure 7 to the results of the FOSM analysis for the LEM, it can be seen that the reliability index is very similar to that found by the LEM. It is important to highlight that even though the FEM results in lower safety factors than those found by the LEM, the structure's reliability index continues to present an aboveaverage level of performance. Therefore, it is understood that as erosion of the base of the structure occurs, its reliability reduces.
6. Conclusion
From this study it is possible to understand how the stability of the Porto Velho pier is influenced by the variability of the geotechnical parameters, as well as the probability of failure of each of the scenarios analyzed. In addition, it was possible to see the evolution of the safety factor as the loss of the quay base structure occurs. By applying the probability analyses for the Limit Equilibrium Method, it was possible to notice a convergence between the results found for the Monte Carlo and FOSM methods. The Finite Element Method analysis showed that adopting the rockfill strength criterion as Linear Elastic resulted in a plasticization area very similar to the areas of the critical rupture surfaces found in the Limit Equilibrium Method, as well as having very close safety factor values. However, by adopting the rockfill resistance criterion as being Mohr Coulomb, the safety factors showed much lower values than those calculated by the Limit Equilibrium Method. With regard to Scenario 4, the FEM identified the rupture of the structure at this stage. In view of the above, it can be seen that the variability of the parameters directly influences the probability of failure of the structure as well as its reliability index. Furthermore, it can be understood that the stability of the Porto Velho pier is influenced both by the loss of the protective soil at the base of the rockfill and by the erosion of the rockfill base itself. This research showed that for the configuration of the Porto Velho pier section, the Finite Element Method presented more assertive results when compared to the results inferred by the Limit Equilibrium Method. This is because the FEM takes into account the interaction between the elements as well as their stiffness. Through the analyses carried out and the results presented, it was identified that the rockfill strength criterion has a major influence on the safety factor. In addition, the FEM identified the rupture of the quay structure from the moment the base of the rockfill was lost. This conformation is also observed in the existing structure of the Porto Velho pier. Therefore, it is assumed that the deformations found at the site stem from the loss of the rockfill base.
Acknowledgements
The Federal University of Rio Grande and the Port of Rio Grande for sharing their technical assets to enable this research to be carried out.
Data Availability Statement
The published publication includes all graphics and data collected or developed during the study.
Conflict Of Interest
The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Ethics
There are no ethical issues with the publication of this manuscript.
Financial Disclosure
The authors declared that this study has received no financial support.
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Neta, L.A.D.S.; Fagundes, D.D.F.; Alves, A.A.M.D.L. Reliability Analysis of the Porto Velho Pier - Rio GrandeRS ThroughLimit Equilibrium and Finite Element Method. Seatific 2024, Vol. 4, pp. 5. https://doi.org/10.29187/2792-0771.1032