Reliability-based Design Method Research Articles

A simplified reliability-based design method for multiple design variables considering different limit states

A generalized reliability model comprising the objective, constraint, and judgment functions is established for the reliability index approach (RIA), taking parameters’ properties of engineering practice and negative reliability index into consideration. Based on this, the reliability-based design (RBD) problem with multiple design variables is translated into the solution to the nonlinear equations, and a simplified method consisting of a simple variant of the Newton iteration method and the finite difference method (FDM) is proposed. Numerical examples are presented to verify the efficiency of the proposed reliability approach and to determine the incremental step size for FDM. RBD of a simply supported beam is illustrated and the variabilities of design variables are investigated considering the uncertainties in the manufacturing process and practical operations. Results reveal that the variations of the design variables should not be ignored. Moreover, analysis results show that the design value might not intuitively increase with the increase of its coefficient of variation (CoV), and it might not increase with the increasing reliability requirement for problems involving multiple variables. The reasons for this phenomenon are very complicated, and it is a systematic problem. One should be aware of this phenomenon, and specific analysis is required for specific problems.

Research on the Equivalent Criterion for Ratios of Safety Margin in Terms of the Factor of Safety and the Reliability Index Under Lognormal Distribution

The ratio of safety margin is the bridge between deterministic design and reliability design. However, the equivalence criterion between the ratios of safety margin in terms of the factor of safety (FOS) and the reliability index is ambiguous. This paper considers the traditional and reliability-based design method at the same safety level. A general form of the FOS following lognormal distribution is adopted. It was found that the variability of FOS is affected by the normalized gradient and the coefficient of variation of random variable. The ratios of safety margin in terms of the FOS and the reliability index are equivalent in the specific project if the normalized gradient does not vary with the design parameters. The feasibility of applying the ratio of safety margin may be increased in engineering practice.

Reliability-Based Design Method for Phase Mask With Periodic Array of Tapered Slits

Reliability-Based Design Method for Phase Mask With Periodic Array of Tapered Slits

A reliability-based mechanical-empirical design method for flexible pavements containing cement-treated magnesite mine tailings as subgrade

The large-scale mining of magnesite generates substantial quantities of magnesite mine tailings (MMT), which pose a significant threat to soil, water, and air quality. Utilising cement-treated MMT as a subgrade material presents a promising solution to address this environmental challenge. However, the existing mechanical-empirical design methods cannot be directly employed due to the uncertainties associated with the various design parameters particularly the behaviour of cemented MMT. This research introduces a novel reliability-based MEM design method to design flexible pavements incorporating cemented MMT as subgrade. A three-layered flexible pavement configuration, with a middle granular layer sandwiched between the top bituminous layer and the bottom stabilised subgrade, was examined. The response surface model and finite element model were developed to determine the fatigue and rutting strains of the pavement. Monte Carlo simulation was adopted to compute reliability. Further, a sensitivity analysis was performed to probe the contribution of input parameters on the reliability of pavement. The developed methodology was illustrated with a case study. Reliability analysis revealed that the cemented MMT pavement achieved reliabilities of 97.44% and 96.27% for fatigue and rutting criteria, respectively, under a design traffic load of 30 million standard axles (msa). Additionally, the sensitivity analysis identified the modulus of elasticity of the granular layer and bituminous layer as the most critical input variables. Thus, the developed design methodology for pavements incorporating MMT enables the engineers to design MMT-based flexible pavements considering the uncertainties.

Characterization of model uncertainty of the p-y method for reliability-based design of offshore wind turbines under the serviceability limit state

Recent studies show that the API p-y method may underestimate the pile-soil initial stiffness of offshore wind turbine (OWT) monopiles in clay while overestimate that in sand. To facilitate the reliability-based design of OWT monopiles with the API p-y design method, a two-stage evaluation method of the model bias factor for the p-y method (εpy) of monopiles under the serviceability limit state (SLS) was presented. The model bias factor εpy is derived by comparing field test data with high-fidelity 3D finite element model (FEM) results, and the p-y method predictions, addressing the issues of limited field test data for OWT monopiles. Results show that the mean of the model bias factor εpy depends on the characteristics of piles, the soil properties, and the composition of the layered ground. The systematic dependency is expressed by a trend function and the residual model bias factor follows a lognormal distribution with a mean of 1.01 and a COV of 0.14. The significance of the model uncertainty is illustrated using a case study. The results of the field example show that the design in clay sites may be considerably conservative and may be less conservative in sand sites if ignoring the model uncertainty. The result of this work can provide a basis for promoting reliability-based design for OWT monopiles.

System reliability-based robust design of deep foundation pit considering multiple failure modes

Recently, reliability-based design is a universal method to quantify negative influence of uncertainty in geotechnical engineering. However, for deep foundation pit, evaluating the system safety of retaining structures and finding cost-effective design points are main challenges. To address this, this study proposes a novel system reliability-based robust design method for retaining system of deep foundation pit and illustrated this method via a simplified case history in Suzhou, China. The proposed method included two parts: system reliability model and robust design method. Back Propagation Neural Network (BPNN) is used to fit limit state functions and conduct efficient reliability analysis. The common source random variable (CSRV) model are used to evaluate correlation between failure modes and determine the system reliability. Furthermore, based on the system reliability model, a robust design method is developed. This method aims to find cost-effective design points. To solve this problem, the third generation non-dominated genetic algorithm (NSGA-III) is adopted. The efficiency and accuracy of whole computations are improved by involving BPNN models and NSGA-III algorithm. The proposed method has a good performance in locating the balanced design point between safety and construction cost. Moreover, the proposed method can provide design points with reasonable stiffness distribution.

Reliability-based design method for shear capacity of steel reinforced engineered cementitious composites (ECC) beams

Engineered cementitious composites (ECC) is famous for its excellent tensile deformability and crack width control ability. Steel reinforced ECC beams under shear loading exhibit brittle behaviors, which leads to a lack of structural safety. The reliability-based shear capacity model of steel reinforced ECC beams needs to be established for ensuring the structural shear safety. In this research, a test database of steel reinforced ECC beams under shear loading was constructed based on the given selecting principles. Through considering the shear resistance provided by fibers, the shear capacity model of steel reinforced ECC beams was preliminary established. The statistical characteristics of random variables were described and determined, and reliability index was calculated through Monte Carlo simulation. It is found that reliability index increases with increasing ECC strength, while decreases with increasing stirrup strength or stirrup ratio, especially at the high partial factor of ECC. For different target reliability indexes, the recommended partial factors of ECC are obtained through calibration. This research supplies a useful reference for the shear design of steel reinforced ECC beams in the actual engineering.

Reliability-based design method for marine structures combining topology, shape, and size optimization

The uncertainty and extremity of ocean conditions make the design of marine structures a challenging process. Reliability-based optimization is a powerful approach to enhance material utilization and ensure system survivability. In this paper, a novel reliability-based design optimization (RBDO) method that combines topology, shape, and size optimization is proposed to achieve the design of marine structures. Within the framework of decoupled sequence optimization and reliability assessment, the developed method employs an innovative multi-technology combined optimization Scheme and a new reliability assessment algorithm. The combined optimization Scheme is established by simultaneously incorporating topology, shape, and size optimization technologies to further improve the lightweighting effect. The advanced arc interpolation algorithm is developed, which dynamically selects the advanced mean value method or arc interpolation method based on the trend judgment criterion to accelerate the calculation for the most probable point in reliability assessments. The RBDO examples of the wind turbine jacket and platform stiffened plate are utilized to verify the proposed method's applicability and validity, and the results are compared with those of other approaches. The findings prove that this RBDO method can effectively address reliability-based design problems of marine structures and is superior in providing lightweighting structures compared with others.

A Reliability-Based Robust Design Optimization Method for Rolling Bearing Fatigue under Cyclic Load Spectrum

Reliability-based robust design methods have been widely used in the field of product design; however, they are difficult to apply to the fatigue reliability design process of rolling bearings due to the problems of determining fatigue accumulated damage caused by the internal cyclic time-varying load distribution of rolling bearings and the computational cost of time-varying reliability. Therefore, a reliability-based robust design method for rolling bearing fatigue failure is proposed, which derives the formula for fatigue accumulated damage of a rolling bearing under cyclic load spectrum and significantly reduces the computational cost of rolling bearing time-varying reliability compared with existing methods. First, the state response of a rolling bearing under random design parameters is obtained by finite element simulation. Then, the adaptive kriging method is used to characterize the correlation between the random parameters and the state response. The Miner fatigue cumulative damage theory is improved and the rolling bearing fatigue time-varying equation of state under cyclic load spectrum is derived. Subsequently, a fatigue time-varying reliability model based on an improved fourth-order moment method is developed, and a reliability robust optimization design method is proposed. Finally, a rolling bearing example is presented to demonstrate that the method achieves time-varying fatigue reliability design under cyclic load spectrum and effectively improves the reliability and robustness of the product design.

A simple Monte Carlo simulation method for geotechnical reliability-based design

ABSTRACT This paper proposes a simple reliability-based design approach from a quantile value perspective. The proposed approach allows the design variable to be efficiently determined without implementing repetitive “trial and error” procedures for a given target reliability index. Only a single run of Monte Carlo simulation is needed to calculate the threshold values of the design variable after the design variable function (DVF) is derived from the limit state function. The threshold values are then sorted in ascending order, and a quantile value is chosen as the desired design variable such that MCS samples with threshold values larger, or smaller than this value are undesired samples. Three illustrative examples are presented following the development of the proposed method and its implementation procedure. Results are validated by calculating the failure probabilities of desired designs using direct MCS, and they are also favorably comparable with those reported in literature. This method can play a complementary role to existing RBD methods due to its advantages of simplicity, efficiency, and robustness.

Reliability-based serviceability limit state design of spread foundations under uplift loading in cohesionless soils

ABSTRACT The design of foundations is often governed by the serviceability limit state (SLS) requirements of the supported structure, particularly for large spread foundations. This paper aims to develop a reliability-based SLS design method for spread foundations under uplift loading in cohesionless soils. A probabilistic framework was adopted for the empirical characterisation of the compiled load-displacement curves and the quantification of the associated uncertainties. By using the obtained statistics of the curves, reliability analysis was carried out with Monte-Carlo simulations to calibrate the resistance factors within the load and resistance factor design (LRFD) framework. The calibration results showed that the embedment ratio of the foundation and the fitting errors of the empirical model, which were previously unaddressed in the literature, had notable effects on the calibrated SLS resistance factors. The relationship of the SLS with the ultimate limit state was assessed, including the governing limit state at each allowable displacement level, and the probability of ultimate failure of the foundation at the SLS condition. By considering the relationship between the limit states, the procedures for determining the design resistance factor and foundation capacity were proposed.

A level set driven concurrent reliability-based topology optimization (LS-CRBTO) strategy considering hybrid uncertainty inputs and damage defects updating

With the diversification of engineering structure performance requirements and the continuous development of structural design refinement, it puts forward higher requirements for lightweight design of structures in terms of concurrent design and reliability design. It is necessary to develop advanced design technology. This paper focuses on the influence of multi-source hybrid uncertainties and defect damage in the design process and proposes a concurrent reliability-based topology optimization design method. Based on the collocation method, the propagation analysis of hybrid uncertainties is realized. A performance reduction method considering defect damage is proposed. Furthermore, the parameter iteration strategy of the optimization criterion method is improved. Based on the sensitivity analysis of a series of parameters, the gradient optimization algorithm is used to solve the optimization model. Finally, three numerical examples are given to illustrate the effectiveness and necessity of the proposed method.

Reliability-based robust geotechnical design of braced excavations considering multiple failure modes

Reliability design of braced excavation is still a challenge for geotechnical community. Optimization design is a normal method to control the safety and cost of braced excavations. This study presents an advanced reliability-based robust geotechnical design method, which can consider multiple failures and uncertainty of statistical information. A universal design sample was conducted to verify the necessity of considering the uncertainty of statistical information. Ultimate limit state and serviceability limit state of braced excavations were defined, and point estimating method was used to evaluate the standard deviation of failure probabilities. Two-objective and three-objective optimization models were developed to illustrate the application of proposed methods in detail. In addition, the performance of optimization algorithms and further application of multiple-objective models were discussed. The results from this study indicate that the proposed method has a good performance in determining the optimal design with reasonable robustness and cost. New algorithms have higher efficiency in solving nonlinear and multiple-objective optimization problems than the 2nd Non-dominated sorting genetic algorithm. This study can guide the design of retaining systems of braced excavations in clay.

Reliability-based load and resistance factor design model for energy piles

Energy piles have been popular globally with functions of both pile foundation and ground source heat pumps. Although several researches have been devoted to the probabilistic design and assessment of energy piles, the corresponding procedures are too complicated for engineers. As a simple variant of the reliability-based design method, the load and resistance factor design (LRFD) approach for the geotechnical design of energy piles is presented in this study. Firstly, the load-transfer model for energy piles is developed to investigate the effect of cyclic thermal loading on the pile settlement. Then, the LRFD procedures based on first-order reliability method and target reliability method are implemented into two different constrained nonlinear optimization problems, respectively. The proposed LRFD model for energy piles is demonstrated through an example pile and a series of parametric analyses.

Load and resistance factor design versus reliability-based design of shallow foundations

ABSTRACT The load and resistance factor design (LRFD) approach employed in geotechnical design codes is often calibrated using the reliability-based design (RBD) method. However, the LRFD may not achieve the target safety level exactly. This paper makes comparisons between the LRFD and RBD by considering the ultimate and serviceability limit state design of strip foundations. The RBD is carried out via the Random Finite Element Method. It is found that the failure probability and the resulting foundation width for ULS obtained using the RBD increase with the soil spatial correlation length, gradually reaching a plateau. The comparison between the LRFD and RBD results for ULS suggests that the LRFD is conservative for low to medium soil variability, particularly at smaller correlation lengths. The reliability-based SLS design is less dependent on the soil correlation length, particularly for lower coefficients of variation of the soil elastic modulus. However, a resistance factor of 1.0 for SLS is unconservative, and resistance factors of 0.65, 0.7 and 0.8 are better aligned with the RBD when the target failure probabilities are 1×10−3, 1×10−2 and 1×10−1. The current study can be used to guide the design of shallow foundations and the calibration of the LRFD approach.

A time-varying reliability-based robust design method considering overall efficiency of axial piston pump

Axial piston pumps (APPs) are the core energy conversion components in a hydraulic transmission system. Energy conversion efficiency is critically important for the performance and energy-saving of the pumps. In this paper, a time-varying reliability design method for the overall efficiency of APPs was established. The theoretical and practical instantaneous torque and flow rate of the whole APP were derived through comprehensive analysis of a single piston-slipper group. Moreover, as a case study, the developed model for the instantaneous overall efficiency was verified with a PPV103-10 pump from HYDAC. The time-variation of reliability for the pump was revealed by a fourth-order moment technique considering the randomness of working conditions and structure parameters, and the proposed reliability method was validated by Monte Carlo simulation. The effects of the mean values and variance sensitivity of random variables on the overall efficiency reliability were analyzed. Furthermore, the optimized time point and design variables were selected. The optimal structure parameters were obtained to meet the reliability requirement and the sensitivity of design variables was significantly reduced through the reliability-based robust design. The proposed method provides a theoretical basis for designers to improve the overall efficiency of APPs in the design stage.

Reliability-based design of rigid pipes installed by induced trench method with tire-derived aggregate inclusions

A reliability-based design (RBD) method for rigid pipes installed using induced trench with tire-derived aggregates (TDA) inclusion is introduced in this study. The design process is to find an optimal set of the TDA inclusion dimension (i.e., width B and thickness t) that satisfies the 0.01-inch crack serviceability limit state and ultimate failure limit state requirements of the pipe, and achieves a target probability of failure or target reliability index using Monte Carlo simulation. The uncertainty and load-dependent compressibility of TDA inclusion are considered. Two design examples are provided, which show that the serviceability limit state design controls the finial design of induced trench installation (ITI) pipes. Increasing the thickness of TDA inclusion is helpful to improve reliability of the pipes, while expanding the width of TDA inclusion might increase probability of failure. The sensitivity and flexibility of using different target reliabilities, parameter uncertainties and calculation models in the proposed RBD method are examined. The results show the importance of considering the variability and load-dependent compressibility of TDA in the proposed RBD method for the design of ITI pipes with TDA inclusions.

Modified, Reliability-Based Robust Geotechnical Design Method, in Accordance with Eurocode 7

In this paper a modification of the reliability-based robust geotechnical design (RGD) method is proposed. The intention of the proposed modifications is to simplify the method, make it less computationally expensive, and harmonise of the results with Eurocode 7. The complexity of the RGD method mainly stems from the calculation of the design’s robustness measure, which is the feasibility robustness index (ββ). Due to this fact, the replacing of the existing robustness measure with a generalised reliability index (β) is considered. It was demonstrated that β fits into the robustness concept, and is traditionally used as a construction reliability measure, making it intuitive and “user friendly”. It is proposed to conduct a sensitivity analysis using Soboli indices, with the aim of freezing the variables whose contribution to the system response variance is negligible, which will further simplify the method. By changing the robustness measure, the number of the required reliability analyses is significantly decreased. Further reduction is achieved by conducting analyses only for the designs chosen in the scope of the genetic algorithm. The original RGD method is used as an extension of traditional reliability-based design. By applying the proposed modifications, the RGD method can be used as an alternative to the classic and reliability-based design method.

Reliability-based design approach for high-rise buildings subject to earthquakes and strong winds

Current design specifications often deal with seismic and wind events in isolation, which may underestimate the failure probability. In this paper, a reliability-based approach is presented to compute the load modification factor for designing high-rise buildings in regions prone to earthquake and strong wind events. The paper illustrates the proposed procedure from the construction of hazard models to load modification calculations, which is implemented in a 42-story steel frame-reinforced concrete (RC) core tube building located in the Dali region, southwest of China. The uncertainties associated with hazard models, material properties, input loads, demand models and structural capacity are considered to improve the accuracy of failure probability estimations. The effects of individual and concurrent earthquakes and strong winds are accounted for in the reliability-based design method. Then, the load modification factor is determined by comparing the calculated failure probability with the rational target value. Finally, a suggested combined factor will be used to modify the value of the design wind load in the limit state function. More work is needed before load modification can be applied in practice. However, this study offers a practical and extensible approach to improve the future calibration of load combination criteria for designing high-performance structures against various hazards.

The Reliability-Based Design Optimization of considering Rock-Support Interaction for Rock Tunnels

Uncertainty is critical to the tunnel design. In this study, a novel reliability-based design (RBD) method was developed by integrating the rock-support interaction and its uncertainties for rock tunnels. In this method, the rock-support interactions were analyzed based on a convergence-confinement method. Uncertainties were estimated by reliability analysis using Excel Solver. Chaotic particle swarm optimization (CPSO) was adopted to search the optimal tunnel design parameters based on the rock-support interaction analysis. The proposed method for estimating the reliability index and determining the tunnel support parameters was introduced in detail. To illustrate the proposed method, it was applied to two tunnels with rock-bolt and combined support systems. The results indicated that the method could obtain accurate solutions with a dramatically improved computing efficiency, guaranteeing the tunnel stability at the same time. The developed method provides an excellent way to deal with the uncertainty in tunnel design. It was proved to be an efficient and effective method for the support designs of rock tunnels.