- Research
- Open Access
A probabilistic bridge safety evaluation against floods
- Kuo-Wei Liao^{1}Email author,
- Yasunori Muto^{2},
- Wei-Lun Chen^{1} and
- Bang-Ho Wu^{1}
- Received: 27 October 2015
- Accepted: 18 May 2016
- Published: 18 June 2016
Abstract
To further capture the influences of uncertain factors on river bridge safety evaluation, a probabilistic approach is adopted. Because this is a systematic and nonlinear problem, MPP-based reliability analyses are not suitable. A sampling approach such as a Monte Carlo simulation (MCS) or importance sampling is often adopted. To enhance the efficiency of the sampling approach, this study utilizes Bayesian least squares support vector machines to construct a response surface followed by an MCS, providing a more precise safety index. Although there are several factors impacting the flood-resistant reliability of a bridge, previous experiences and studies show that the reliability of the bridge itself plays a key role. Thus, the goal of this study is to analyze the system reliability of a selected bridge that includes five limit states. The random variables considered here include the water surface elevation, water velocity, local scour depth, soil property and wind load. Because the first three variables are deeply affected by river hydraulics, a probabilistic HEC-RAS-based simulation is performed to capture the uncertainties in those random variables. The accuracy and variation of our solutions are confirmed by a direct MCS to ensure the applicability of the proposed approach. The results of a numerical example indicate that the proposed approach can efficiently provide an accurate bridge safety evaluation and maintain satisfactory variation.
Keywords
- Bridge safety
- Flood-resistant reliability
- MCS
- Bayesian LS-SVM
Background
In Taiwan, the bridge safety evaluation for floods is often a two-step procedure. The first step is to examine bridge safety through a preliminary inspection evaluation form (PIEF). If the overall assessment score from the PIEF does not meet a predefined standard, the evaluation should proceed to an advanced investigation such as pushover analysis to ensure the safety of the bridge. The PIEF consists of several items that are potential threats for bridge safety. Each evaluated item is allocated a weight to indicate its relative importance. The sum of all of the weights is 100. The items in the PIEF proposed by Chern et al. (2007) include the scouring depth, the foundation type, the attack angle of the river flow, the presence of protective facilities at the river bank and bed and the presence of a dam upstream. Among all of the items, the scouring depth has the highest weight and is considered as the most influential factor. Thus, the goal of this study is to investigate the safety of a scoured bridge. To fulfill this purpose, the strengths of a bridge structure such as the strengths of the pile shear stress, the pile axial stress, the horizontal displacement on the pile head, the soil bearing and the soil pulling force need to be carefully considered. In addition, the corresponding demands for the aforementioned strengths are water surface elevation, water velocity, local scour depth, wind load and soil properties that should be included in the safety evaluation as well. Because the variation of the main channel location is not considered, the worst case scenario is used. In other words, this study does not analyze every pier in a bridge; instead, only the pier that has the highest risk is analyzed. Because a one-dimensional hydraulic model is adopted here, all of the piers in the same river cross section share the same water level and velocity. The pier with the lowest river bed profile is selected for analysis because it has the largest flow depth, resulting in the largest scour depth.
Many researchers have evaluated bridge safety using a probabilistic approach. For example, Carturan et al. (2012) used a stochastic finite element method to retrieve the reliability of a bridge. Wu et al. (2011) used a most probable point-based (MPP-based) reliability method to evaluate the reliability of a levee system in the Keelung River. Adarsh and Reddy (2013) used advanced first-order and second-moment (AFOSM) and a Monte Carlo simulation (MCS) to evaluate the safety of artificial open channels. Davis-McDaniel et al. (2013) used a fault-tree model to perform bridge failure risk analysis for bridges in South Carolina. The results showed that flood, scour, overloading, corrosion of post-tensioning tendons and earthquakes are the top five critical factors. Saydam et al. (2013) evaluated the bridge superstructure safety via a probabilistic approach, in which the component and system level failures were both considered. Zhao et al. (1997) integrated the linearly constrained MCS with unit hydrograph theory and routing techniques to evaluate the reliability of hydraulic structures.
Although many studies have shown that a reliability analysis is necessary for evaluating bridge safety, few of the studies in Taiwan have adopted this approach. Thus, this study builds a reliability analysis framework to reflect several important bridge safety issues through a case study. For example, only reliability analysis takes the uncertainty or variation of the influencing factors into consideration in the safety evaluation process. Based on the current study, uncertainty plays an essential role in bridge safety. Many issues such as the pile shear stress, the pile axial stress, the pile head horizontal displacement, the soil bearing and the soil pulling force, are involved in a bridge failure; therefore, a system reliability analysis is needed. Thus, the five performance functions above are considered in our bridge system. Among these five performance functions, variables such as the water surface elevation, water velocity, local scour depth, wind load and soil property are pertinent and treated as probabilistic density functions. Although MPP-based approaches are often adopted in a reliability analysis, they are not suitable in the current study because of the nonlinearity and complexity of the analyzed problem. Sampling approaches such as MCS and importance sampling (IS) are potential reliability analysis tools in the current study. However, the cost of such approaches is often unaffordable for a practical problem. Recently, many studies have utilized a response surface in reliability analysis. For example, Sun et al. (2016) analyzed the reliability of a 2.5D/SiC composite turbine blade using a response surface built by support vector machines (SVM). Zhao and Qiu (2013) used the center point of the experimental points to control the construction of their response surface for reliability-based optimization. This study adopts Bayesian least squares support vector machines (LS-SVM, Suykens et al. 2002) to build the response surface, followed by an MCS. Both the accuracy and variation of the proposed approach are ensured by comparing the solution from the MCS.
The five performance functions, the random characteristics of a bridge against floods, the construction of a response surface and the proposed reliability analysis process are described below, followed by a demonstration via a numerical example.
Descriptions of the performance functions
Load types
A deterministic analysis approach (e.g., the analysis procedure provided in the design code) uses the load and reduction factors (i.e., LRFD in steel structure design) to consider the uncertainty in the variables (i.e., loads or material properties). Because this study utilizes reliability analysis to take the uncertainty into consideration, the vertical load (both dead and live loads) and wind load are calculated according to the provisions of “The Bridge Design Specifications” without using the load factors (2009). The bridge that is considered in the numerical example is a 4-lane bridge with a span of 35 ms.
Performance functions
f _{ s } and q _{ b } estimated by the N values
Driven pile | Bored pile | Implant-type pile | |
---|---|---|---|
f _{ s } | \(\hbox{min} \left[ \hbox{N}/5, 15 \right]\) | \(\hbox{min} \left[ \hbox{N}/5,15 \right]\) | \(\hbox{min} \left[ \hbox{N}/5,15 \right]\) |
q _{ b } | 30 N | 7.5 N | 25 N |
The demands of the pile strength [Eqs. (2), (3) and (4)] are calculated based on Chang’s simplified lateral pile analysis (Chang and Chou 1989). However, the boundary conditions that are defined in Chang’s method (Chang and Chou 1989) are not exactly the same as in the situation that is considered here. For example, in Chang’s method, the external force is a concentrated force and is applied at the pile head, which is not applicable when scouring occurs, as shown in Fig. 2. To use Chang’s formula, an equivalent force of the hydrodynamic pressure is calculated, for which the detailed description is as follows.
Random characteristics of a bridge failure against floods
Illustration of difference in reliability analyzed by MCS and FORM
Method | P _{ f } |
---|---|
MCS | 3.64 × 10^{−3} |
FORM | 2.68 × 10^{−3} |
- 1.
In the calculation of the soil bearing capacity and the pile head displacement demand (δ_{D}), q _{ b } (the allowable vertical pressure), introduced in Eq. (5), and the coefficient of the horizontal subgrade reaction (k _{h}) are required. Here, q _{ b } and k _{h} are functions of the value of N, as shown in Table 1 and Eq. (12). In general, the value of N is not a deterministic number from an on-site survey, as illustrated in “SPT-N values and the wind load” section. As a result, the soil bearing capacity and δ _{D} should be measured using a probabilistic approach.
$$k_{\text{h}} = \frac{{\text{502}N^{{\text{0}\text{.37}}} + \text{691}N^{{\text{0}\text{.406}}} }}{\text{2}}$$(12) - 2.
Similarly, f _{ s } in the calculation of the soil pulling capacity is a function of the value of N, as shown in Table 1. Furthermore, because the scour depth is a random variable, the pile length embedded in the ground is also a random variable. As indicated by Eq. (13), the number of soil stratums (n) to be considered in the analysis depends on the scour depth. As a result, F _{ s } (the friction resistance force on the pile surface) will become a non-continuous function. Thus, it is inappropriate to use the MPP-based approach to calculate the reliability.
where h _{i} is the ith soil stratum thickness (m), N _{i} is the N value for the ith soil stratum and n is the number of soil stratums embedded in the ground.$$F_{s} = A_{s} f_{s} = D\pi \sum\limits_{i = 1}^{n} {} h_{\text{i}} \times \text{min}\left[ {\frac{{N_{\text{i}} }}{\text{5}}\;\text{,}\;\text{15}} \right]$$(13) - 3.
As mentioned earlier, Eqs. (2), (3) and (4) can be applied to only one of Chang’s equations. The applicability of Chang’s equation depends on whether the pile is embedded in the ground or not. In other words, once scouring occurs, Chang’s formula should be changed to meet the required boundary conditions. As a result, the corresponding performance functions should be modified accordingly. In this case, the MPP-based approach is not suitable for reliability analysis.
This study applies response surface methodology (RSM) to replace the existing performance functions to improve the efficiency of an MCS. Before explaining the response surface method that was adopted here, the random variables in this study are introduced, which are the water surface level, water velocity, local scour depth, wind load and SPT-N values.
Water velocity and water surface level
Local scour depth
Using the 9 scouring depth formulas and 30 pairs of water levels and water velocities, 270 samples of the scour depth are obtained. The sample mean and standard deviation are used as the mean value and standard deviation of the scour depth in the following reliability analysis. The total scour depth of the pier is usually the sum of the depths of the local scouring, contraction scouring and general scouring (i.e., scouring without structures). The local scouring often dominates the overall scouring depth, and thus, other scouring is not considered at present (Alipour et al. 2013; HEC-18 2012; NRCS of USDA 2010). The correlation coefficients between the scour depth and water level and between the scour depth and water velocity are found to be 0.93 and 0.92, respectively.
SPT-N values and the wind load
- 1.
The first stratum (immediately below the riverbed): the thickness is approximately 1.3 m, and the SPT-N value has an approximate range of 0–10.
- 2.
The second stratum (below the first stratum): the thickness is approximately 20 m, and the SPT-N value has an approximate range of 2–28.
- 3.
The third stratum (below the second stratum): the thickness is approximately 19.6 m, and the SPT-N value has an approximate range of 3–16.
- 4.
The SPT-N value of the soil stratums below the third stratum is approximately 50.
Because there is no more detailed information for the SPT-N values in the investigated bridge site, the SPT-N values in the first three stratums are assumed to follow a uniform distribution that has the upper and lower bounds indicated above.
Regarding the wind load, the force on the bridge is calculated by multiplying the wind-induced pressure by its corresponding area that is affected by the water level. Once the water level is determined as described in “Water velocity and water surface level” section, the projected area of the wind load can be obtained. The wind load is a random variable because of the uncertainty in the water level.
The proposed reliability approach and a numerical example
Response surface methodology (RSM)
As shown in Fig. 8, this study first utilizes Bayesian LS-SVM to build the RSM, followed by the classical MCS. The equations needed for constructing a general RSM are described in this section. The purpose of RSM is to replace the performance functions. To fulfill this goal, the equations mentioned in “Descriptions of the performance functions” section are needed in the establishment of the particular RSM in the current study. To be specific, five Bayesian LS-SVM RSMs were built to replace the five performance functions considered here.
Case study
The Shuangyuan Bridge was selected as our case study. Several piers of the Shuangyuan Bridge were seriously damaged during Typhoon Morakot. Thus, the motivation of selecting this bridge is to determine its reliability to provide a more comprehensive explanation of its failure. Information about the bridge before restoration can be found in Liao et al. (2014).
MCS reliability and its COV
100 year failure probability of the selected bridge by MCS
Sample size | 100 | 300 | 3000 |
---|---|---|---|
P _{ f } | 2.34 × 10^{−1} | 2.30 × 10^{−1} | 2.32 × 10^{−1} |
COV | 0.181 | 0.106 | 0.033 |
Reliability analyzed by Bayesian LS-SVM
The inputs and outputs of the response surfaces
Inputs | Outputs |
---|---|
Water level | Safety or failure for pile pulling force |
Water velocity | Safety or failure for soil bearing force |
Local scour depth | Safety or failure for pile shear stress |
Wind load | Safety or failure for pile axial stress |
SPT-N value | Safety or failure for pile head displacement |
Although the RSM has been utilized in many fields, the locations of the sample points significantly influence its accuracy (Zhao and Qiu 2013). Many methods, such as the factorial design, central composite design (CCD), and Latin hypercube design (LHD), are available for engineers to use. Because this study used a computer simulation to analyze bridge reliability, such a process was considered as a computer experiment. The minimum bias design (MBD) was often considered as a good principle for selecting the sample location in this case. The LHD was considered as a space-filling design, focusing inside the design region rather than in the perimeter or the extremes of the design region, and satisfies the criteria of the MBD (Myers and Montgomery 2002). Thus, the LHD was adopted here to construct the RSM for the reliability analysis.
The RMSEs of the 5 response surfaces with a sample size of 50
No. of response surface | RSM | RMSE (%) |
---|---|---|
1 | Pile pulling force | 4.80 |
2 | Soil bearing force | 4.72 |
3 | Pile shear stress | 3.30 |
4 | Pile axial stress | 3.48 |
5 | Pile head displacement | 3.45 |
System 100 year failure probability (P _{ f }) and accuracy measurements (RMSE) using the response surface with different sample sizes
Sample size | Sample range | P _{ f } (LS-SVM) | P _{ f } (Bayesian LS-SVM) | RMSE (%) |
---|---|---|---|---|
#5 | ||||
50 | μ ± 2σ | 2.57 × 10^{−1} | 2.51 × 10^{−1} | 3.45 |
80 | μ ± 2σ | 2.42 × 10^{−1} | 2.41 × 10^{−1} | 1.10 |
100 | μ ± 2σ | 2.38 × 10^{−1} | 2.37 × 10^{−1} | 0.65 |
120 | μ ± 2σ | 2.34 × 10^{−1} | 2.34 × 10^{−1} | 0.45 |
150 | μ ± 3σ | 2.33 × 10^{−1} | 2.32 × 10^{−1} | 0.32 |
Comparison of 100 year failure probability (P _{ f }) using response surface and MCS
Approach | Bounds^{a} | P _{ f } | COV |
---|---|---|---|
MCS | NA | 2.32 × 10^{−1} | 0.033 |
RSM(150)^{b} | μ ± 3σ | 2.32 × 10^{−1} | 0.01 |
RSM(80)^{b} | μ ± 2σ | 2.41 × 10^{−1} | 0.02 |
Please note that the COVs of the MCS (the values in Tables 3 and 7) were computed according to Eq. (21). The COVs of the proposed approaches, however, did not follow the binominal distribution. Thus, the COVs of the proposed approaches (the values in Table 7) were calculated from 50 simulations (i.e., performing the MCS 50 times using the established RSM).
Based on the aforementioned computation and discussion, using a response surface with a sample size of 150 can deliver accurate and efficient bridge reliability against floods. To be specific, the sample size was reduced from 3000 to 150. The computational cost can be further reduced to a sample size of 80 if a 5 % error tolerance is allowed in both the accuracy and variation.
Based on the MCS solution, the failure probability of the selected bridge was 2.3 × 10^{−1}, which was greater than the threshold value (1.00 × 10^{−3}) suggested by the International Organization for Standardization (ISO), indicating that this bridge did not have sufficient reliability, which was consistent with the failure event observed in the Marokot floods. When the proposed algorithm was designed, the sample size and range were two factors that could impact the reliability estimation. Several trial and error tests were conducted to find the appropriate sample size. To determine the sample range, it is intuitive to select a range that basically covered all possible values of the considered random variables. For example, using μ ± 3σ can cover 99.73 % of the possible values of the corresponding variables. However, because the 100-year failure probability of the considered bridge was not a small number, a sample range of μ ± 3σ could be too broad for the current probability level. Thus, a narrower sample range (e.g., μ ± 2σ) was adopted to perform the RSM-based reliability calculation, as shown in Tables 6 and 7. If a 5 % error tolerance was applicable for the accuracy and variance, this approach could effectively reduce the required sample size from 150 to 80.
Conclusions
A deterministic bridge design or evaluation process is often adopted in Taiwan. After the Morakot Typhoon, engineers realized that a probabilistic approach is needed to consider the uncertainty in the parameters. Therefore, this study builds an accurate and efficient reliability methodology to fulfill such a need. Bridge failure is a complicated system problem, and many different types of events should be considered. Based on the literature and PIEF suggested by an earlier study, the safety of a bridge substructure is one of the most pertinent factors in bridge reliability and is the scope of this study. The random variables considered include the water surface elevation, water velocity, local scour depth, wind load and soil property. A probabilistic hydraulic analysis and on-site survey data are used to capture the variation in these variables. The Bayesian LS-SVM is adopted to establish the response surface, in which the LHS is used to generate the samples. Compared with results from a direct MCS, the accuracy and variation of the proposed method is confirmed. In addition, the reliability obtained from the proposed algorithm for the case study indicated that the selected bridge does not have sufficient reliability, which is consistent with the failure event observed in the Marokot floods. From the reliability derivation for a simplified case and the classifier outcomes, it is observed that the limit state functions are likely to be highly nonlinear and that the sampling method is a suitable choice for reliability analysis. The MCS analysis, however, is time consuming. In the presented case, 3000 samples are needed. The proposed response surface-based reliability analysis can improve the computation efficiency with the same accuracy and variation of the traditional approach (MCS). For example, the sample size is reduced from 3000 to 150. If a 5 % error tolerance is applicable, the proposed approach can further reduce the sample size to 80.
Declarations
Authors’ contributions
KWL formulated the Bayesian LS-SVM-based MCS to perform a system reliability analysis for bridge safety evaluation. YM examined the limit state functions and ensured the accuracy of the manuscript. WLC and BHW carried out the literature search, data acquisition and numerical analysis for the demonstrated example. All authors read and approved the final manuscript.
Acknowledgements
This study was supported by the TU-NTUST Joint Research Program and the National Science Council of Taiwan under grant number NSC 102-2221-E-011 -078 -MY2. The support is gratefully acknowledged.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Adarsh S, Reddy MJ (2013) Reliability analysis of composite channels using first order approximation and Monte Carlo simulations. Stoch Environ Res Risk Assess 27:477–487View ArticleGoogle Scholar
- Alipour A, Shafei B, Shinozuka M (2013) Reliability-based calibration of load and resistance factors for design of RC bridges under multiple extreme events: scour and earthquake. J Bridge Eng 18:362–371View ArticleGoogle Scholar
- Carturan F, Islami K, Pellegrino C (2012) Reliability analysis and in-field investigation of a r.c. bridge over river Adige in Verona, Italy. Bridge Maintenance, Safety, Management, Resilience and Sustainability 2850–2855Google Scholar
- Chang YL, Chou NS (1989) Chang’s simple side pile analysis approach. Sino-Geotech 25:64–82Google Scholar
- Chern JC, Tsai IC, Chang KC (2007) Bridge monitoring and early warning systems subjected to scouring. Directorate General of highwaysGoogle Scholar
- Davis-McDaniel C, Chowdhury M, Pang WC, Dey K (2013) Fault-tree model for risk assessment of bridge failure: case study for segmental box girder bridges. J Infrastruct Syst 19(3):326–334View ArticleGoogle Scholar
- Fischenich C, Landers M (1999) Computing scour. U.S. Army Engineer Research and Development Center, VicksburgGoogle Scholar
- HEC-18 (2012) Hydraulic engineering circular no. 18. US Department of Transportation, WashingtonGoogle Scholar
- Jain SC (1981) Maximum clear-water scour around cylin-drical piers. J Hydraul Eng ASCE 107(5):611–625Google Scholar
- Jain SC, Ficher EE (1980) Sour around bridge piers at high flow velocities. J Hydraul Eng-ASCE 106:1827–1842Google Scholar
- Li TC, Liu JB, Liao YJ (2011) A discussion on bridge-closure according to the water stage. Sino-Geotech 127(1):79–86Google Scholar
- Liao KW, Chen WL, Wu BH (2014) Reliability analysis of bridge failure due to floods, life-cycle of structural systems. In: Life-cycle of structural systems: design, assessment, maintenance and management, pp 1636–1640Google Scholar
- Liao KW, Lu HJ, Wang CY (2015) A probabilistic evaluation of pier-scour potential in the Gaoping River Basin of Taiwan. J Civ Eng Manag 21(5):637–653View ArticleGoogle Scholar
- Ministry of Transportation and Communications R. O. C (2009) The bridge design specifications. A government reportGoogle Scholar
- Myers RH, Montgomery DC (2002) Response surface methodology: process and product optimization using designed experiments. Wiley, New YorkGoogle Scholar
- Natural Resources Conservation Service of United States Department of Agriculture (2010) The national engineering handbook. Natural Resources Conservation Service of United States Department of Agriculture, WashingtonGoogle Scholar
- NCREE (2010) A pilot project for building a bridge detection system via advanced technology. Directorate General of Highways, TaipeiGoogle Scholar
- Neill CR (1965) Measurements of bridge scour and bed changes in a flooding sand-bed river. In: Proceedings of the Institution of Civil Engineers, London, EnglandGoogle Scholar
- Saydam D, Frangopol DM, Dong Y (2013) Assessment of risk using bridge element condition ratings. J Infrastruct Syst 19(3):252–265View ArticleGoogle Scholar
- Sheen SW (2012) Flood frequency analysis of watersheds in Taiwan. J Soil Water Conserv Technol 7(1):11–21Google Scholar
- Shen HW, Shneider VR, Karaki SS (1966) Mechanism of local scour. Colorado State University, ColoradoGoogle Scholar
- Shen HW, Shneider VR, Karaki SS (1969) Local scour around bridge piers. J Hydraul Eng ASCE 95:1919–1940Google Scholar
- Sun Z, Wang C, Niu X, Song Y (2016) A response surface approach for reliability analysis of 2.5 DC/SiC composites turbine blade. Compos. Part B Eng. 85:277–285View ArticleGoogle Scholar
- Sung YC, Wang CY, Chen C, Tsai YC, Tsai IC, Chang KC (2011) Collapse analysis of Shuanyang bridge caused by Morakot Typhoon. Sino-Geotech 127:41–50Google Scholar
- Suykens JAK, Gestel TV, Brabanter JD, Moor BD, Vandewalle J (2002) Least squares support vector machines. World Scientific, SingaporeView ArticleGoogle Scholar
- Wang H, Wang CY, Chen CH, Lin C, Wu TR (2011) An integrated evaluation on flood resistant capacity of bridge foundations. Sino-Geotech 127:51–60Google Scholar
- Water Resources Agency (2009) Analyses of rainfall and flow discharge for Typhoon Morakot. A government reportGoogle Scholar
- Wu SJ, Yang JC, Tung YK (2011) Risk analysis for flood-control structure under consideration of uncertainties in design flood. Nat Hazards 58(1):117–140View ArticleGoogle Scholar
- Wu TR, Wang H, Ko YY, Chiou JS, Hsieh SC, Chen CH, Lin C, Wang CY, Chuang MH (2014) Forensic diagnosis on flood-induced bridge failure. II: framework of quantitative assessment. J Perform Constr Facil 28(1):85–95View ArticleGoogle Scholar
- Zhao W, Qiu Z (2013) An efficient response surface method and its application to structural reliability and reliability-based optimization. Finite Elem Anal Des 67:34–42View ArticleGoogle Scholar
- Zhao B, Tung YK, Yeh KC, Yang JC (1997) Reliability analysis of hydraulic structures considering unit hydrograph uncertainty. Stoch Hydro Hydraul 11:33–50View ArticleGoogle Scholar
- Zhao YG, Zhong WQ, Ang AH-S (2007) Estimating joint failure probability of series structural systems. J Eng Mech ASCE 133(5):588–596View ArticleGoogle Scholar