Study on stability and sensitivity analysis of protective riprap layer placed around bridge pier by using reliability analysis theory

Introduction - The formation of scour hole around bridge pier is recognized as the number one cause of bridge failure. This phenomenon results from a complex flow field with large-scale turbulence structures generated by flow around the pier. On the other hand, predicting the stability of scour countermeasure around bridge pier such as riprap layer is one of the main challenges in hydraulic engineering. However, the damage mechanism of riprap layer is a very complicated phenomena and many uncertainties affect the exact estimation of different effective parameters in riprap stone design. Therefore, quantifying these uncertainties by using reliability analysis to ensure the stability of the bridge is necessary. Recently, reliability analysis methods have been highly considered due to their high ability to optimize engineering designs and improve project exploitation. Reliability analysis provides a systematic and organized framework to minimize model uncertainties. In addition, it allows the designer to determine the role of each of the effective parameters in the uncertainty of model output. This is essential for identifying important parameters in order to pay more attention to them to achieve their exact value and effect, and ultimately reduce the uncertainty of the model output.
Methodology - In the present study, the stability of riprap layer was investigated by using a reliability-based framework. Monte Carlo Simulation Technique (MCST) and First Order Reliability Method (FORM) were established to determine the stability of riprap layer against shear failure. In FORM, the reliability is measured in terms of a reliability index, β, and it is related to the probability of failure or probability of limit state violation for any limit state. In addition, MCST consists of drawing samples of the basic variables according to their probabilistic characteristics and then feeding them into the limit state function and therefore the probability of failure, Pf, can be found. MCST is considered in principle an exact method, and, FORM as an approximate method. However, FORM is computationally fast and inexpensive as compared to MCST. The first stage in reliability analysis or evaluation of the probability of a system failure is to determine a limit state or performance function between the basic random variables. In the present study, the equation presented by Karimaee and Zarrati, (2013) and Karimaee et al. (2015) is used here as limit state equation. The advantage of this equation is that it can be used for unprotected and protected piers with a collar and circular as well as rectangular piers, and aligned or skewed piers corresponding to the flow direction. The application of the present method was illustrated in an applied example which is a bridge with two piers located in the main channel and flood plain. The data for this case studies was extracted from different previous studies.
Results and discussion - At first, results showed that β values and Pf obtained using FORM are having close proximity with MCST results. Therefore, it is also appropriate to use FORM for reliability assessment of riprap layer around bridge pier. In addition, it was found that due to existing uncertainties, the stability of designed riprap size which was calculated from deterministic method was low and equal to 44% for the pier placed in the main channel and 28% for the pier in the flood plain. In addition, two equations between safety factor and reliability index was determined for riprap size design placed around bridge pier in the main channel or flood plain. Results showed that by increasing the reliability index parameter, the safety factor should be increased. Using these equations one can find out an appropriate value of safety factor for desired riprap layer reliability. For example, these equations give the safety factor corresponding to target reliability index (βT) = 3 as 2.93 and 1.46 for the riprap layer size placed around bridge pier located in the main and flood channels, respectively. Next, sensitivity analysis was performed to examine the impact of each random variable on the probability of riprap layer stability in the reliability method. Results showed that the most critical parameter affecting the reliability of riprap size was the mean flow velocity, so that by decreasing the amount of this parameter for about 75% form the Mean value, the amount of parameter β increased for more than 16 times. Therefore, this parameter needs to be determined more accurately in riprap design to decrease the failure probability of riprap layer, efficiently. Finally, the effect of parameters’ uncertainties on reliability analysis of riprap layer was investigated. Results showed that for all of the effective parameters by increasing the parameters’ uncertainties, the stability of riprap layer decreases.
Conclusion – Present reliability assessment methodology showed that due to existing uncertainties, the stability of designed riprap size which was calculated from deterministic method was not reliable as desired. In addition, an overall influence of various random variables on riprap layer’s reliability was assessed through sensitivity analysis. It was shown that if through better quality control; regular maintenance and proper care uncertainties can be minimized, reliability of riprap layer could be improved.